Real Reductive Groups I

This is Volume 132 in
PURE AND APPLIED MATHEMATICS
H. Bass, A. Borel, J. Moser, S.-T. Yau, editors
Paul A. Smith and Samuel Eilenberg, founding editors
A complete list of titles in this series appears at the end of this volume.

Real Reductive Groups I
Nolan R. Wallach
Department of Mathematics
Rutgers University
New Brunswick, New Jersey
ACADEMIC PRESS, INC.
Har court Brace Jovanovich, Publishers
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Copyright © 1988 by Academic Press, Inc.
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Library of Congress Cataloging-in-Publication Data
Wallach, Nolan R.
Real reductive groups.
(Pure and applied mathematics; v. 132- )
Includes index.
1. Lie groups. 2. Representations of groups.
I. Title. II. Title: Reductive groups. III. Series:
Pure and applied mathematics (Academic Press); 132, etc.
QA3.P8 vol. 132, etc. 510 s [512'.55] 86-32199
[QA387]
ISBN 0-12-732960-9 (v. 1: alk. paper)
88 89 90 91 9 8 7 6 5 4 3 2 1
Printed in the United States of America

To my mother
Pauline Wallach
"For as the sun is daily new and old,
So is my love still telling what is told."

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Contents
Preface xi
Introduction xiii
Chapter 0. Background Material 1
Introduction 1
0.1. Invariant measures on homogeneous spaces 1
0.2. The structure of reductive Lie algebras 4
0.3. The structure of compact Lie groups 6
0.4. The universal enveloping algebra of a Lie algebra 8
0.5. Some basic representation theory 10
0.6. Modules over the universal enveloping algebra 13
Chapter 1. Elementary Representation Theory 17
Introduction 17
1.1. General properties of representations 18
1.2. Schur's lemma 20
1.3. Square integrable representations 22
1.4. Basic representation theory of compact B groups 24
1.5. A class of induced representations 29
1.6. C™ vectors and analytic vectors 31
1.7. Representations of compact Lie groups 35
1.8. Further results and comments 39
vii

viii
Contents
Introduction
2.1.
2.2.
2.3.
2.4.
2.5.
2.A.
2.A.I.
2.A.2.
The definition of a real reductive group
Parabolic pairs
Cartan subgroups
Integration formulas
The Weyl character formula
Appendices to Chapter 2
Some linear algebra
Norms on real reductive groups
Chapter 2. Real Reductive Groups 41
41
42
48
56
60
65
68
68
70
Chaptpr 3. The Basic Theory of (g, K)-Modules 73
Introduction 73
3.1. The Chevalley restriction theorem 74
3.2. The Harish-Chandra isomorphism of the center of the
universal enveloping algebra 77
3.3. (g,K)-modules 80
3.4. A basic theorem of Harish-Chandra 82
3.5. The subquotient theorem 86
3.6. The spherical principal series 92
3.7. A Lemma of Osborne 95
3.8. The subrepresentation theorem 97
3.9. Notes and further results 100
3.A. Appendices to Chapter 3 103
3.A.I. Some associative algebra 103
3.A.2. A Lemma of Harish-Chandra 104
Chapter 4. The Asymptotic Behavior of Matrix Coefficients 107
Introduction 107
4.1. The Jacquet module of an admissible (g,K)-module 108
4.2. Three applications of the Jacquet module 112
4.3. Asymptotic behavior of matrix coefficients 114
4.4. Asymptotic expansions of matrix coefficients 118
4.5. Harish-Chandra's H-function 125
4.6. Notes and further results 130
4.A. Appendices to Chapter 4 131
4.A.I. Asymptotic expansions 131
4.A.2. Some inequalities 133

Contents
ix
Chapter 5. The Langlands Classification 137
Introduction
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
5.8.
5.A.
5.A.I.
5.A.2.
5.A.3.
Tempered (g, K)-modules
The principal series
The intertwining integrals
The Langlands classification
Some applications of the classification
SL(2,R)
SL(2,C)
Notes and further results
Appendices to Chapter 5
A Lemma of Langlands
An a priori estimate
Square integrability and the polar decomposition
137
138
140
144
149
152
156
159
163
164
164
166
168
Chapter 6. A Construction of the Fundamental Series 173
Introduction 173
6.1. Relative Lie algebra cohomology 174
6.2. A construction of (t,/C)-modules 176
6.3. The Zuckerman functors 179
6.4. Some vanishing theorems 184
6.5. Blattner type formulas 188
6.6. Irreducibility 193
6.7. Unitarizability 196
6.8. Temperedness and square integrability 201
6.9. The case of disconnected G 203
6.10. Notes and further results 206
6.A. Appendices to Chapter 6 207
6.A.I. Some homological algebra 207
6.A.2. Partition functions 211
6.A.3. Tensor products with finite dimensional representations 212
6.A.4. Infinitesimally unitary modules 220
Chapter 7. Cusp Forms on G 225
Introduction 225
7.1. Some Frechet spaces of functions on G 226
7.2. The Harish-Chandra transform 230
7.3. Orbital integrals on a reductive Lie algebra 234

X
Contents
7.4. Orbital integral on a reductive Lie group 243
7.5. The orbital integrals of cusp forms 250
7.6. Harmonic analysis on the space of cusp forms 254
7.7. Square integrable representations revisited 259
7.8. Notes and further results 264
7.A. Appendices to Chapter 7 265
7.A.I. Some linear algebra 265
7.A.2. Radial components on the Lie algebra 268
7.A.3. Radial components on the Lie group 273
7.A.4. Some harmonic analysis on Tori 277
7.A.5. Fundamental solutions of certain differential operators 282
Chapter 8. Character Theory 289
Introduction 289
8.1. The character of an admissible representation 290
8.2. The K-character of a (g, K)-module 294
8.3. Harish-Chandra's regularity theorem on the Lie algebra 296
8.4. Harish-Chandra's regularity theorem on the Lie group 311
8.5. Tempered invariant Z(g)-finite distributions on G 313
8.6. Harish-Chandra's basic inequality 320
8.7. The completeness of the nt 323
8.A. Appendices to Chapter 8 326
8.A.I. Trace class operators 326
8.A.2. Some operations on distributions 331
8.A.3. The radial component revisited 337
8.A.4. The orbit structure on a real reductive Lie algebra 342
8.A.5. Some technical results for Harish-Chandra's regularity
theorem 349
Chapter 9. Unitary Representations and (g, K)-Cohomology 353
Introduction
9.1. Tensor products of finite dimensional representations
9.2. Spinors
9.3. The Dirac operator
9.4. (g, K)-cohomology
9.5. Some results of Kumaresan, Parthasarathy, Vogan,
Zuckerman
9.6. u-cohomology
9.7. A theorem of Vogan-Zuckerman
353
354
359
365
368
373
381
388

Contents
9.8.
9.A.
9.A.I.
9.A.2.
Bibliogi
Further results
Appendices to Chapter 9
Weyl groups
Spectral sequences
aphy
XI
394
396
396
398
403
Index
411

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Preface
This book is intended as an introduction to the representation theory of real
reductive groups. It is based on courses that the author has given at Rutgers
for the past 15 years. It also had its genesis in an attempt of the author to
complete a manuscript of the lectures that he gave at the CBMS regional
conference at The University of North Carolina at Chapel Hill in June of 1981.
When the manuscript for those lectures reached over 300 pages the author
realized that the scope of the project involved much more than was expected
for a CBMS volume. We apologize to the conference board for not having
completed the volume that was expected. We, however, hope that this book
will in part fulfill the obligation.
Initially, it was our intention to present the subject of representations of real
reductive groups from the beginning to recent research, all in one volume. This
has also been beyond the ability of the author. We have opted to present the
material in two volumes in order to expand upon the original extremely terse
exposition and to include recent developments in even the more "classical"
aspects of the theory.
There are many people that have been helpful in the production of this
volume. We thank our students (both former and present) for their patience
over the years with the lectures on which this book is based. We especially
thank Roberto Miatello for all of the errors that he has found in the various
earlier versions of this material and for his many helpful comments. Hans
xiii

xiv
Preface
Duistermaat pointed out a major blunder in our original exposition of
Harish-Chandra's regularity theorem. His explanation of the method of proof
of this theorem that will appear in his forthcoming book with Kolk was very
helpful. We also thank Kenneth Gross for having organized the above-
mentioned CBMS regional conference so well. Finally, we take this
opportunity to thank Armand Borel for his editorial help, encouragement and patience
throughout the preparation of this opus.
We also take this opportunity to thank the National Science Foundation
for the summer support during the preparation of this volume.

"You do not understand my philosophy.
But that is the way science progresses
each generation misunderstands
the previous one."
— Harish-Chandra
Introduction
The representation theory of real reductive groups is one of the most
beautiful, demanding, useful and active parts of mathematics. Although there
have been many important contributors to the field. Harish-Chandra, through
his power and vision, almost single-handedly changed the field from a
backwater of physics to what it is today. For better or for worse Harish-
Chandra, in developing his awesome theory, also established the style of the
field. Few disciplines in mathematics put as much emphasis on their technical
details. This aspect of the subject makes it an extremely easy part of
mathematics to read "line by line" and a very difficult part for those who
would just like an "over-all" picture of the subject.
Although this book is a product of the Harish-Chandra legacy, we have
attempted to allow the reader to get a "feel" of the subject without necessarily
having understood every line. It is hoped that upon a first reading, the material
will be studied by "jumping" from one part, that may seem interesting, to
another. We have endeavored to do enough cross-referencing so that a reader
could open the book in the middle and understand the material there by
following the details backward. A careful reader will find mathematical gems
in unlikely places. Kostant's theorem on rt-cohomology is in Chapter 9,
Zuckerman's translation principal is in an appendix to Chapter 6, radial
component theory is in the appendices to Chapter 7, Kostant's theorem on
nilpotent orbits is in an appendix to Chapter 8.
XV

xvi
Introduction
As the title indicates, there is a forthcoming second volume which will
contain, in particular, a proof of Harish-Chandra's Plancherel theorem.
Although both volumes emphasize the analytic aspects of the theory, the
material in the volume at hand is more algebraic than the second volume. The
reader who is predominantly interested in the algebraic aspects of the theory
can read this volume without being too "contaminated" by analysis.
Let us now give a "thumbnail tour" of the present volume. Chapter 0 is a
compendium of some of the basic results that usually appear in a first course in
Lie groups and Lie algebras. It is included to establish notation and references.
The purpose of Chapter 1 is to introduce the theory of infinite dimensional
representations of Lie groups. The material presupposes no prior knowledge
of the reader. Our account is tailored to the needs of the later chapters and
since most of representation theory of general Lie groups is unnecessary to the
case of real reductive groups, the reader should be aware that this chapter is
just the tip of the iceberg. The chapter emphasizes representations on Hilbert
spaces. Basic material on smooth, analytic and "K-finite" vectors is included.
A novel aspect of this chapter is the development of the Peter-Weyl theory for
compact Lie groups as a corollary to the theory of square integrable
representations.
In Chapter 2, we introduce the class of Lie groups that will be studied
throughout the remainder of the book. In particular we make the term "real
reductive group" precise. The only prerequisites for this chapter are included
in Chapter 0. We develop the theory of parabolic subgroups and Cartan
subgroups. We take the more primitive notion to be that of parabolic
subgroup and then show how the theory of Cartan subgroups is an outgrowth.
Most of the classical groups are introduced in this chapter. We give the
Iwasawa, Bruhat and Cartan decompositions for the groups. Integration
formulas are given for these decompositions as are various versions of the
Weyl integration formula. We also include a proof of the Weyl character
formula (the standard one) since a similar proof will be used for the discrete
series in Chapter 8.
The material of Chapter 3 is the "heart" of the "algebraic" approach to
representation theory. It contains various forms of the Chevalley restriction
theorem and the Harish-Chandra homomorphism. The formalism of (g, K)-
modules is introduced. The critical notion of admissibility is developed. A
proof is given of Harish-Chandra's theorem that irreducible unitary
representations are admissible. The chapter also includes the celebrated "sub-quotient
theorem" of Harish-Chandra, Lepowsky, Rader and its corollary (in our
development), the subrepresentation theorem of Casselman. The latter result
is perhaps the most important single theorem to our development. It makes

Introduction
xvii
the theory of the real Jacquet module a viable approach to the representation
theory of real reductive groups. Also our proof of this theorem contains ideas
that will be critical to later developments in the book. The chapter also
includes the basic theory of spherical functions. Most of the material in this
chapter is algebraic or at least has algebraic statements. We have, however,
given some analytic proofs of theorems that now have completely algebraic
proofs. We indicate where the-more algebraic approach can be found in the
literature.
Chapter 4 is the core of our approach to the subject. It contains the theory
of the real Jacquet module and its consequence (in our exposition) the
asymptotic behavior of matrix coefficients. This chapter is strongly influenced
by our joint work with Casselman (which was motivated by the p-adic theory
of Jacquet [1]) and by Harish-Chandra's theory of the constant term. Indeed,
as we shall see in Volume 2, this latter theory is a consequence of the material
in this chapter. Our approach to the asymptotic expansions is module
theoretic. Special cases of the results can also be found in Warner [2]. Also a
modern account of Harish-Chandra's original approach can be found in
Casselman, Milicic [1]. The critical difference between our results and that of
Harish-Chandra is that we give asymptotic expansions of smooth matrix
coefficients rather than just "K-finite" ones.
The point of Chapter 5 is to give a proof of the Langlands quotient theorem
("Langlands classification"). This theorem reduces the classification of
irreducible (g, K)-modules to the classification of "tempered" (g, K)-modules.
The elementary aspects of tempered representations and their relationship
with square integrable representations is also given. At this point in our
development, the critical importance of the irreducible square integrable
representations has become manifest. However, in this chapter these
representations are described only in the case of SL(2,R).
Chapter 6 is devoted to a homologico-algebraic approach to constructing
"admissible" (g, K)-modules that is equivalent to that of Zuckerman using
derived functors of the "K-finite functor". Our approach follows the broad
lines of our joint work with Enright. An approach that is closer to
Zuckerman's original ideas can be found in Vogan [2]. Using, what we call
Zuckerman's functors, we construct irreducible unitary representations. These
representations had been conjectured to be unitary by Vogan (a generalization
of a conjecture of Zuckerman). Vogan gave the first proof of this result, using
Harish-Chandra's theory of tempered representations. Our proof is
elementary, and we use it as a basis for the theory of tempered representations. We
single out the families constructed from so-called "0-stable Borel subalgebras"
and call them the "discrete series". Using the theory of Jacquet module we

xviii
Introduction
prove that they are square integrable. In Chapter 8 it is shown that these
representations exhaust the irreducible square integrable representations. The
reader can go directly from this chapter to Chapter 9 which studies the
"twisted" (g, K)-cohomology with respect to unitary modules. A complete
proof (mainly due to Vogan, Zuckerman and Kumaresan) of a conjecture of
Zuckerman (that completely calculates this cohomology) is given there using
the modules constructed in this chapter.
The next step is to prove that the "discrete series" exhausts the irreducible
square integrable representations. In our approach, this is where the
analysis begins in earnest. The next two chapters are very close to the spirit of
Harish-Chandra's original approach. In Chapter 7, the basics of Harish-
Chandra's theory of orbital integrals is given. Our approach differs in one
important detail. We do not use the theory of the discrete series to prove that
the orbital integrals define tempered distributions. Instead, we use a special
case of Kostant's convexity theorem (essentially due to Thompson [1]). The
critical idea in this chapter is Harish-Chandra's characterization of the
matrix coefficients of the discrete series in terms of the vanishing of certain
integral transforms. That is, these matrix coefficients span the space of "cusp
forms". We give Harish-Chandra's formula for recovering a cusp form from its
orbital integrals. This result implies Harish-Chandra's basic theorem that says
that irreducible square integrable representations can exist if and only if there
is a compact Cartan subgroup. However, the completeness theorem must wait
for the results in the next chapter.
At this point the reader should have noted a glaring omission in the contents
of this book. The only mention of character theory has been in connection
with the Weyl character formula. Chapter 8 is devoted to Harish-Chandra's
theory of characters of admissible representations. These characters are
initially defined as distributions on the group (as traces of generalized
convolution operators). The main theorem on characters is that they are given
as integration against a locally integrable function (Harish-Chandra's
regularity theorem). Furthermore, on each Cartan subgroup this function has a form
reminiscent of the Weyl character formula. With the "local L1-theorem" in
hand we prove that the Fourier coefficients of orbital integrals of cusp forms
are multiples of characters of what we called the discrete series in Chapter 6.
The completeness theorem is now immediate.
As we observed above, Chapter 9 could be read immediately after Chapter 6.
This chapter contains a concise introduction to (g, K)-cohomology, vanishing
theorems due to Kumaresan, Enright, Vogan-Zuckerman and the complete
calculation of (g, K)-cohomology with respect to a tensor product of a finite
dimensional and an irreducible unitary representation (due to Vogan and

Introduction
xix
Zuckerman). The reader should consult Borel, Wallach [1] for an account of
the general theory and its applications to discrete groups. We include tables of
the vanishing theorems.
There are several books whose contents have significant overlaps with this
one. Knapp's recent book (Knapp [1]) approaches the subject through
examples. Since this book contains very few worked examples, we recommend
that the reader approaching the subject for the first time, study Knapp's book
in conjunction with this one. Since there are important differences in the
approaches to the material in these two books, even a more sophisticated (in
representation theory) reader would benefit from having read both. Another
important reference for the theory is Vogan [1] which covers a good deal of
the more algebraic material in this volume. Again, there is a significant
difference in emphasis and the student should benefit from a study of both this
volume and that of Vogan. There is also a third (very stylish) approach to the
subject involving sheaves of differential operators on algebraic varieties. This
theory, mainly due to Beilinson, Bernstein and Brylinski, Kashiwara is the
subject of a forthcoming book of Milicic. Other notable books on the subject
are Warner [1], [2] and Varadarajan [1]. Both of these works follow Harish-
Chandra's original methods quite closely. Warner's treatise in addition
contains a very thorough introduction to representation theory (i.e., C™-
vectors, analytic vectors, induced representations). These books (and
Helgason [1]) were valuable aids in the preparation of this work.
The literature in the field of reductive groups is vast. We have done our best
to give adequate references. However, as is the case in any growing field, there
are cases when a result has been proved (partially) by many authors. It would
be a project beyond the scope of this book to give the precise history of the
genesis of the theorems included in this book. However, in most cases the
interested scholar should be able to determine a precise chronology by
consulting the citations that we have included.
A reader who has mastered the basic graduate curriculum in mathematics
should have all the mathematical background necessary to master the material
in this volume. However, the serious student should approach this work with
an ample supply of paper and pencils. Be patient and it will be yours.

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0 Background Material
Introduction
The purpose of this chapter is to compile some of the background results,
terminology and notation that will be used in this book. We recommend that
the reader use this chapter basically for reference purposes. However, it might
be worthwhile for the reader to skim through it on his first reading to become
familiar with some of the notation and definitions. There are almost no proofs
in this chapter. Everything covered can be found with adequate explanations
in the references that we give, except for the material in Section 6. In Section 6
we give a noncommutative variant of the Artin-Rees Lemma of commutative
algebra. There is a general Artin-Rees Lemma for nilpotent Lie algebras (see
McConnell [1], Nouaze, Gabriel [1]). Lemma 0.6.4 appears for the first time in
Stafford, Wallach [1].
0.1. Invariant measures on homogeneous spaces
0.1.1. Let G be a locally compact topological group. Then a left invariant
measure on G is a positive measure, dg, on G such that
I f(xg)dg = J f(g)dg
G G
1

2
0. Background Material
for all x e G and all / in (say) CC{G). If G is separable then it is well known
(Haar's theorem) that such a measure exists and that it is unique up to a
multiplicative constant.
If G is a Lie group with a finite number of components then a left
invariant measure on G can be identified with a left invariant n-form on G (here
dim G = n). If /x is a non-zero left invariant n-form on G then the identification
is implemented by integrating with respect to /i using the standard method of
differential geometry.
If G is compact then we will (unless otherwise specified) use normalized left
invariant measure. That is, the total measure is one.
If dg is a left invariant measure and if x e G then we can define a new left
invariant measure on G, jUx, as follows:
Hx(f) = j f(gx)dg.
G
The uniqueness of left invariant measure implies that
Hx(f)=d(x)jf(g)dg.
G
with 3 a function of x which is usually called the modular function of G. If S is
identically equal to 1 then we say that G is unimodular. If G is unimodular then
we will call a left invariant measure (which is then automatically right
invariant) invariant. It is not hard to see that 3 is a continuous homomorphism
of G into the multiplicative group of positive real numbers. This implies that if
G is compact then G is unimodular.
If G is a Lie group than the modular function of G is given by the following
formula:
S(x) = |det Ad(x)|
where Ad is the usual adjoint action of G on its Lie algebra.
0.1.2. Let M be a smooth manifold and let \i be a volume form on M. Let G
be a Lie group acting on M. Then (g*/.i)x = c(g,x)fix for each g e G, x e M.
One checks that c satisfies the cocycle relation
(1) c(gh,x) = c(g,hx)c(h,x) for h,g e G, x e M.
We will write jM f(x)dx for jM f\i. The usual change of variables formula
implies that
(2) lf(gx)\c(g,x)\dx=lf(x)dx
M G
for / (say) in CC{G) and g e G.

0.1. Invariant Measures on Homogeneous Spaces
3
Let ff be a closed subgroup of G. We take M to be G/H. We assume that G
has a finite number of connected components. A G-invariant measure, dx, on
M is a measure such that
(3) j f(gx) dx = j f{x) dx, f e Q(G), 0 e G.
If dx comes from a volume form on M then (3) is the same as saying that
\c(g,x)\ = 1 for all g e G, x e M.
If M is a smooth manifold then it is well known that either M has a volume
form or M has a double covering that admits a volume form. By lifting
functions to the double covering (if necessary) one can integrate relative to a
volume form on any manifold. Returning to the situation M = G/H, it is not
hard to show that M admits a G-invariant measure if and only if the modular
function of G restricted to H is equal to the modular function of H. Under
this condition, a G-invariant measure on M is constructed as follows: let g be
the Lie algebra of G and let h be the sub-algebra of g corresponding to H.
Then we can identify the tangent space at Iff to M with g/h. The adjoint
action of ff on g induces an action Ad~ of ff on g/h. The above condition
says that |det Ad~(/i)| = 1 for all he H. Thus if ff° is the identity component
of ff (as usual) and if ju is a non-zero element of Am(g/h)* (m = dim G/H)
one can translate \i to a G invariant volume form on G/H°.
Thus by lifting functions from M to G/H° one has a left invariant measure
on M. Now Fubini's theorem says that we can normalize dg, dh and dx so that
(4) \f{g)dg= j (\ f{gh)dh]d{gH) for / e Q(G).
G G/H \H /
0.1.3. Let G be a Lie group with a finite number of connected components.
Let ff be a closed subgroup of G and let dh be a choice of left invariant measure
on ff. The following result is useful in the calculation of measures on
homogeneous spaces.
Lemma. // / is a continuous compactly supported function on H\G {note the
change to right cosetsl) then there exists, g, a continuous compactly supported
function on G such that
f(Hx)=Sg(hx)dh.
G
This result is usually proved using a "partition of unity" argument. For
details see, for example, Wallach [1, Chapter 2].

4
0. Background Material
0.1.4. Let G be a Lie group and let A and B be subgroups of G such that
An Bis compact and that G = AB. The following result is useful for studying
induced representations.
Lemma. Assume that G is unimodular. If da is a left invariant measure on A and
if db is a right invariant measure on B then we can choose an invariant measure,
dg, on G such that
Sf(g)dg= j f(ab)dadb for f e CC(G).
For a proof of this result see for example Bourbaki [1].
0.2. The structure of reductive Lie algebras
0.2.1. Let g be a Lie algebra over C. We use the notation 3(g) for the center
of g. Then g is said to be reductive if g = 3(g) ® [g, g] with [g, g] semisimple.
We recall the basic properties of g that will be used in this book with
appropriate references.
Recall that a subalgebra, h, of g is called a Cartan subalgebra if h is
maximal subject to the conditions that h is abelian and if X e h then ad X is semi-
simple as an endomorphism of g. Here, if X, Y e q then ad X(Y) = [X, Y~\
(as usual). Cartan subalgebras always exist and they are conjugate to one
another under inner automorphisms (c.f. Jacobson [1. p.273]).
If X e g then define the polynomials Dj on g by
det(t/ - ad X) = £ tsD,{X\
here n = dim g. Let r be the smallest index such that Dr is not identically zero.
Set D = Dr. X e g is said to be regular if D(X) is nonzero.
Lemma. // X is regular then ad X is semi-simple. Futhermore, the centralizer
in Qof a regular element is a Cartan subalgebra of g (Jacobson [1, p.59]).
Fix, h, a Cartan subalgebra of g. If a e h* then we set
ga = {X e g! [H, X] = a(H)X for all H e h}.
If a and ga are non-zero then we call a a root of g with respect to h, and ga is
called the root space corresponding to a. The set of all roots of g with respect to

0.2. The Structure of Reductive Lie Algebras
5
h will be denoted <t>(g, h) and called the root system of g (with respect to h). We
have
(1) g = f)0 0 9.-
<*e1>(8.W
(2) If ae<D(g,h)thendim(ga) = 1 (Jacobson [1, p.lll]).
(3) Ifa,^ecD(g,h)then[ga,g/i] = ga + /i
(Jacobson [1, p. 116]).
(4) If a e <t>(g, h) then the only multiples of a in <t>(g, h)
are a and —a (Jacobson [1, p.l 16]).
0.2.2. Let g be as above. If B is a symmetric bilinear form on g then B is said
to be invariant if
B([X, Y~\,Z) =-B(Y, \_X,Z]) for all X,Y,Ze g.
A non-degenerate invariant form on g always exists. On [g, g] one takes the
Killing form Jacobson [1, p.69] and on j(g) one takes any non-degenerate
symmetric form. The direct sum of the two forms is then a non-degenerate
invariant form on g. Fix such a form, B. Fix a Cartan subalgebra, h, in g. It is
clear that h is orthogonal, relative to B, to all of the root spaces. We therefore
see that
(1) B restricted to h is non-degenerate.
Thus, if \x e h* then we can define H^ e h by
B{H,HJ = n{H) for//eh.
We can then define a non-degenerate symmetric bilinear form ( , ) on h * by
(H,x) = B(//„, //t) for n, t e h*. One has
(2) (a,a) is a positive real number for a e <t>(g,h). (Jacobson [1, p.l 10])
Let hR denote the real subspace of h spanned by the Hx for a e <t>(g, h). Then
one has
(3) B restricted to hR is real valued and positive definite (Jacobson [ 1, p. 118]).
0.2.3. We retain the notation of the previous number. If a€%I)) we
denote by sx the reflection about the hyperplane a = 0 in h. That is,
sxH = H- (2a(ff)/(«, a))ff, for //eh.

6
0. Background Material
sx is called a Weyl reflection. The Weyl reflections have the following
properties:
(1) VD(g,h) = (D(g,h) (Jacobson[l,p.ll9]).
(2) sJ)R = hR.
We denote by W(g,h) the group generated by the Weyl reflections. W-^g,!))
is called the Weyl group of g with respect to h.
Let hR denote the subset of all H e hR such that a(H) is nonzero for all
a e<t>(g,h). Let C denote a connected component of h'R. Then C is called a
Weyl chamber.
(3) W(q, h) acts simply transitively on the Weyl chambers (Bourbaki [2,
p. 163]).
0.2.4. A subset P of <t>(g, h) is called a system of positive roots if <t>(g, h) is the
disjoint union of P and -P( = {-a|ae P}) and if whenever aJeP and
a + j8 e 0(g,h) then a + fi e P. If C is a Weyl chamber then the set of all
a e ^(g,!)) that are positive on C is a system of positive roots. Conversely, if
P is a system of positive roots then the subset of hR consisting of those H such
that a{H) > 0 for all a e P is a Weyl chamber. Thus specifying a Weyl
chamber is the same as specifying a system of positive roots.
Fix a system of positive roots, P. Then a e P is said to be simple if a cannot be
written as a sum of two elements of P. The set of all simple roots of P is called a
simple system for P or a fcasi's for the root system <t>(g,h). Let it denote the
simple system for P. Then it has the following properties (Jacobson [1, p. 120]):
(1) it is a basis for (hR)*.
(2) If p e P then 0 = £ naa with na e N.
(3) W(g, h) is generated by the sx for a e rc (Bourbaki [2, p. 155]).
0.3. The structure of compact Lie groups
0.3.1. Let G be a compact Lie group with Lie algebra g. Let gc denote the
complexification of g. Then gc is a reductive Lie algebra over C. In fact, if
( , ) is any positive non-degenerate symmetric bilinear form on g then we
define a new form on g, < , >, as follows:
(X, y> = j (Ad(g)X, Ad(g)Y)dg for X, Y, e g.
G

0.3. The Structure of Compact Lie Groups
7
Here (as usual) dg denotes normalized invariant measure on G. The
invariance of dg immediately implies that
<.Ad(g)X, Ad{g)Y) = {X, Y) for g e G and X, Y e g,
By differentiating this formula one sees that < , > is an invariant form on g.
Thus, if u is an ideal of g then the orthogonal complement to u is also an ideal
of g. Hence, dimension considerations imply that g is a direct sum of 1-
dimensional and simple ideals. This clearly implies that g is reductive.
Recall that the Killing form of g, B, is defined by the following formula:
B{X, Y) = tr ad X ad Y for X,Ysq.
Since ad X is skew adjoint relative to < , > for X e g it is clear that
B(X, X) < 0 for X e g. Also, B(X, X) = 0 if and only if ad X = 0. Thus, g is
semisimple if and only if B is negative definite. The converse is also true.
Theorem. // g is a Lie algebra over R with negative definite Killing form then
any connected Lie group with Lie algebra g is compact.
This theorem is known as Weyl's theorem. For a proof see, for example,
Helgason [1, Theorem 6.9, p.133].
0.3.2. In this book a commutative compact, connected Lie group will be
called a torus. Let T be a torus with Lie algebra t. If we look upon t as a Lie
group under addition then exp is a covering homomorphism of t onto T. The
kernel of exp is a lattice, L, in t. That is, L is a free Z module of rank equal to
dim t.
Let TA denote the set of all continuous homomorphisms of T into the circle.
If /i e TA then the differential of n (which we will also denote by n) is a linear
map of t into i'R such that fi{L) <= 2mZ. If fi is a linear map of t into i'R such
that fi(L) <= 27n'Z then fi is called integral. If \i is an integral linear form on t
then we define for t = exp(A'), t" = exp(n(X)). This sets up an identification
of integral linear forms on t and characters of T.
0.3.3. Let G be a compact, connected Lie group. Then a maximal torus of G is
(as the name implies) a torus contained in G but not properly contained in any
sub-torus of G. Fix a maximal torus, T, of G. Then tc is a Cartan subalgebra of
gc. The elements of <t>(gc,tc)are integral on t and thus define elements of TA.
Thus, we will look upon roots as characters of T. We now list some properties
of maximal tori that will be used in this book.

8
0. Background Material
(1) A maximal torus of G is a maximal abelian subgroup of G (Helgason
[1, P-287]).
(2) If T and S are maximal tori of G then there exists an element g e G such
that S = gTg1 (Helgason [1, p.248]).
(3) Every element of G is contained in a maximal torus of G. That is, the
exponential map of G is surjective. (Helgason [1, p.135].)
(4) If T is a maximal torus of G then G/T is simply connected. (This follows
from say Helgason [1, Cor.2.8, p.287].)
Let T be a maximal torus of G. Let N(T) denote the normalizer of T in G
(the elements g of G such that gTg~l = T). Let W(G, T) denote the group
N(T)/T. Then W(G, T) is called the Weyl group of G with respect to T. If
g e s e W(G, T) then we set sH = Ad(g)ff for H e t. This defines an action
of W(G, T) on t.
(5) Under this action W(G, T) = W(gc,tc) (Helgason [1, Cor.2.13, p.289]).
0.3.4. Let g be a semisimple Lie algebra over C. Then a real form of g, u, will
be called a compact form if u has a negative definite Killing form. The following
result is due to Weyl. Combined with Theorem 0.3.1 it is the basis of what he
called the "unitarian trick".
Theorem. // h is a Cartan subalgebra of g then there exists a compact form, u,
of g such that u n h is maximal abelian in u. (Jacobson, [1, p.147].)
0.4. The universal enveloping algebra of a Lie algebra
0.4.1. Let g be a Lie algebra over a field F which we will think of as R or C.
Then a universal enveloping algebra for g is a pair (A,j) of an associative
algebra with unit, 1, over F, A, and a Lie algebra homomorphism,;, of g into A
(here an associative algebra is looked upon as a Lie algebra using the usual
commutator bracket, [X,Y~\ = XY- YX) with the following universal
mapping property: If B is an associative algebra with unit and if a is a Lie
algebra homomorphism of g into B then there exists a unique associative
algebra homomorphism a~ of A into B such that a{X) = o~{j{X)).
It is easy to see that if (A,j) and (B, i) are universal enveloping algebras of g
then there exists an isomorphism, T, of A onto B such that Tj = i. Thus, if a
universal enveloping algebra exists then it is unique up to isomorphism.
The usual construction of a universal enveloping algebra of g is given as
follows: Let T(g) denote the free associative algebra over F generated by the

0.4. The Universal Enveloping Algebra of a Lie Algebra
9
vector space g. That is, T(g) is the tensor algebra over the vector space g. Let
/(g) denote the two sided ideal of T(g) generated by the elements XY-
YX - IX, y] for X,Ysq. Set l/(g) = T(g)//(g). Let i denote the natural map
of g into T(g). Let p denote the natural projection of T(g) into 17(g). Set
] = pi. Then it is easy to see that (U(q),j) is a universal enveloping algebra
forg.
The basic result on universal enveloping algebras is the Poincare-Birkoff-
Witt Theorem (P-B-W for short):
Theorem. Let Xl,...,X„be a basis of g. Then the monomials
KXX'-Kxr"
form a basis of l/(g) (Jacobson [1, p.159]).
0.4.2. In light of the uniqueness of universal enveloping algebras and P-B-W
we will use the notation U(q) for the universal enveloping algebra of g and
think of g as a Lie subalgebra of l/(g). Thus,;' will be looked upon as the
canonical inclusion.
Let l/m(g) denote the subspace of l/(g) spanned by the products of m or less
elements of g. Then l/m(g) <= l/m + 1(9) defines a filtration of l/(g). This
filtration is called the canonical filtration of l/(g). With this filtration l/(g) is a
filtered algebra (that is, Up(q)U"(q)<= l/p + ,(g)). Let Gr l/(g) denote the
corresponding graded algebra, g generates l/(g) and the elements XY — YX
are in U '(g) for X, Y e g. Hence Gr l/(g) is a commutative algebra over F. Let
S(g) denote the symmetric algebra generated by the vector space g. Then there
is a natural homomorphism, fi, of S(g) onto Gr l/(g). P-B-W implies that this
homomorphism is an isomorphism. If Xx,..., Xk are in g then set
symm(xl~-xk) = (i/k\)YJxal---xak
a
the sum over all permutations a of k letters. Then symm extends to a linear
map of S(g) to l/(g). Let q be the projection of l/m(g) into Gr l/(g). If x e S(g) is
homogeneous of degree k, then it is easily checked that q(symm(x)) = x.
Hence symm defines a linear isomorphism of S(g) onto V(q). In particular, if
X sq then symm(Arm) = Xm (the multiplication on the left hand side is in S(g)
on the right hand side it is in l/(g)). symm defines a linear isomorphism of S(g)
onto l/(g) which is called the symmetrization mapping.
We note that if a the Lie algebra (0) then U(a) = F. Let e be the Lie algebra
homomorphism of g onto a given by e(X) = 0. Then e extends to a
homomorphism of l/(g)ontoF which we also denote bye (rather thane~). eis
called the augmentation homomorphism.

10
0. Background Material
We denote by gopp the Lie algebra whose underlying vector space is g with
bracket operation {X, Y} = [Y,X~\. Then l/(gopp) = l/(g)opp (the opposite
algebra). The correspondence X h^ —X defines a homomorphism of g onto
gopp whose extension to l/(g) will be denoted xT. We note that the linear map
x i-» xT is defined by the following three properties:
(1) lr=l.
(2) XT = -X for X eg.
(3) {xy)T = yTxT for x, y e l/(g).
0.4.3. Let b be a subalgebra of g. P-B-W implies that the canonical map of
U(b) into l/(g) is injective. We can thus identify U(b) with the associative
subalgebra of l/(g) generated by 1 and b. Let V be a subspace of g such that
g = b ® V. Then P-B-W implies that the linear map
l/(b)0S(K)-»l/(8)
Given by b ® v i-» b symm(i>) for b e U(b), v e S(V), is a surjective linear
isomorphism. Hence l/(g) is the free module on the generators symm(S(K)) as
a U{b) module under left multiplication. Similarly, l/(g) is the free right U(b)
module generated by symm(S(K)) under right multiplication by U(b).
0.5. Some basic representation theory
0.5.1. One of the most useful elementary results in representation theory is
Schur's Lemma. There is a Schur's Lemma for most representation theoretic
contexts (algebraic, unitary, Banach, etc.) In this book there will be several
such Lemmas. We begin this section with a particularly useful one (usually
called Dixmier's Lemma). It is based on the following result:
Lemma. Let V be a countable dimensional vector space over C. If T is an
endomorphism of V then there exists a scalar c such that T — cl is not invertible
on V.
Suppose that T — cl is invertible for all scalars, c. Then P(T) is invertible on
V for all non-zero polynomials P in one variable. Thus if R = P/Q is a rational
function with P and Q polynomials then we can define R(T) to by the formula
P(T)(Q(T)"'). This rule defines a linear map of the rational functions in one
variable, C(x), into End(K). If v e V is non-zero and if R e C(x) is non-zero
with R = P/Q as above then R{T)v = 0 only if P(T)v = 0. Thus the map of

0.5. Some Basic Representation Theory
11
C(x) into V given by R i—► R(T)v is injective. Since C(x) is of uncountable
dimension over C this is a contradiction.
0.5.2. We now come to Dixmier's Lemma. Let V be a vector space over C.
Let S be a subset of End(K). Then S is said to act irreducibly if whenever W is a
subspace of V such that SW W then W = V or W = (0).
Lemma. Suppose that V is countable dimensional and that S a End(K) acts
irreducibly. If T e End(K) commutes with every element of S then T is a scalar
multiple of the identity operator.
By 0.5.1 there exists c e C such that T — cl is not invertible on V. Since the
elements of S preserve Ker(T — cl) and Im(T — cl) and since at least one of
the two spaces must be proper, we see that T = cl.
0.5.3. Let g be a Lie algebra over F = R or C. Then a representation of g is a
pair (a, V) with V a vector space over C and a a homomorphism of g into
End(K). The universal mapping property of l/(g) implies that it extends to a
representation of l/(g). We will write a rather than a ~ for this extension. If a is
understood we will usually use module notation for representations of Lie
algebras (and their extensions to enveloping algebras). That is, we will write xv
for a(x)v. We will then call V a Q-module or a l/(g)-module (which, of course,
it is in the usual associative algebra sense).
If V and W are g-modules we denote by Homg(K, W) the space of all g-
module homomorphisms (or intertwining operators) from V to W. That is, the
space of all linear maps, T, of V to W such that TXv = XTv for X e g and
v e V. We say that V and W are equivalent if there exists an invertible element
inHomg(K,W).
Let V be a g-module. Then a subspace, W, of V is said to be invariant if X W
is contained in W for all X e g. V is said to be irreducible if the only invariant
subspaces of V are V are (0). In this context Schur's Lemma says:
Lemma. // V is an irreducible Q-module then Homg(K, V) = C7.
Let v be a non-zero element of V. Then U(q)v is an invariant non-zero
subspace of V. Hence U(q)v = V. P-B-W (0.4.1) implies that 17(g) is countable
dimensional. Thus V is a countable dimensional. The result now follows from
Lemma 0.5.2.
0.5.4. We now concentrate on a particularly important class of Lie algebras.
A Lie algebra s over C is called a three dimensional simple Lie algebra (TDS for

12
0. Background Material
short) if it has a basis H, X, Y with commutation relations [X, Y~\ = H,
[//,X] = 2X, \_H, 7] = - 2Y. A concrete example of a TDS is sl(2,C) the Lie
algebra of 2 by 2 trace zero matrices. Here one takes
"o r
o o_
y =
"0 0"
1 0_
H =
"1
0
0"
-1_
We therefore see that if s is a TDS and if u is the real subalgebra of s with
basis X — y, i(X + Y), iH then u is isomorphic with the Lie algebra of St/(2)
(the group of 2 by 2 unitary matrices of determinant 1).
Let (a, V) be a finite dimensional representation of s (that is, dim V is finite).
Since SU(2) is simply connected, there is a Lie homomorphism a" of SU(2)
into GL(V) (the group of invertible elements of End(K)) whose differential is a
restricted to u. Let du be normalized invariant measure on SU(2). Fix ( , )
a positive non-degenerate Hermitian form (inner product for short) on V. Then
we define a new inner product < , > on V as follows:
<X vv> = | (o~(u)v,(j~(u)wydu fori;,weK.
St/(2)
Then (o~(u)v,a~(u)w> = <i>,vv> for u e SU(2) and v, weV.
Differentiating this relation gives (Xv, vv> = —(v,Xw} for X e u and b,weK Thus
if W is a s-invariant subspace of V then so is the orthogonal complement of W.
We have proved:
Lemma. // V is a finite dimensional s-module then V splits into a direct sum of
irreducible s-submodules.
The proof we have just used is a special instance of the celebrated "unitarian
trick". This trick was also used in 0.3.1.
0.5.5. Thus to describe finite dimensional s-modules it is enough to describe
irreducible ones. To do this we will use the following commutation relation in
l/(s):
(1) \_X, y] = nY"~l(H -n+ 1) for n = 1, 2,....
Let V be a finite dimensional irreducible s-module. Then H has an
eigenvalue on V of maximal real part, c. Let v be a non-zero eigenvector for H
with eigenvalue c. By the commutation relations denning a TDS we see that
HXv = (c + 2)Xv. Thus Xv = 0. On the other hand,
(2) HYnv = (c - 2n)Ynv and XY"v = n(c - n + l)Y"'lv
by (1). We therefore see that there must be a non-negative integer, m, such that

0.6. Modules Over the Universal Enveloping Algebra
13
Ymv is non-zero but Ym+lv = O.Seti;0 = i>andi>„ = yni; for n= 1,2,.... Then
(2) implies that v0,..., vm is a basis for a non-zero invariant subspace of V.
Since V is irreducible, this implies that v0,...,vmisa basis of V. (2) now implies
that tr H = (m + l)(c - m) on V. Since \_X, F] = H we must have tr H = 0 on
K. Thus c = m.
If VF is an m + 1 dimensional vector space over C with basis vv0,..., wm. We
define the endomorphisms x, y and h of W by the following formulas:
(3) xw0 = 0, xw„ = n(m — n + l)w„_ ! for n = 1,..., m;
ywn = w„ +! for n = 0,..., m — 1 and ywm = 0;
W„ = (m - 2n)w„ for n = 0,..., m.
Then it is not hard to show that x, y, h satisfy the commutation relations of a
TDS. Putting all of this together we have proved:
Lemma. Let s be a TDS with standard basis X, Y, H. Then for every strictly
positive integer m + 1 there exists up to equivalence exactly one irreducible
m + 1 dimensional irreducible s-module, W. Furthermore, W has a basis
w0,...,wm such that X, Y, H correspond to the elements x, y, h in (3)
respectively.
0.6. Modules over the universal enveloping algebra
0.6.1. Let A be an associative algebra over C. Then A is said to be (left)
Noetherian if whenever ^c- c Ik c • • • is a chain of left ideals in A then
there exists, m, such that lm = lk for all k > m.
Let g be a Lie algebra over C.
Lemma. U(#) is Noetherian.
If / is a subspace of U(g) set
Gr/ = @(/n(/i#n(/J-'(9)).
Here the notation is as in 0.4.2. If / is a left ideal of U(q) then Gr(/) is easily seen
to be an ideal in Gr l/(g). Gr l/(g) is isomorphic with S(g). The Hilbert basis
theorem implies that S(g) is Noetherian (Atiyah, Macdonald [1, p.81]). Hence
we conclude that there is m such that Gr lm = Gr lk for all k> m. But then
lm = Ik for all k > m.

14
0. Background Material
0.6.2. If A is an algebra with unit over C then an ,4-module, M, is said to be
finitely generated if there exist elements mx,..., mn of M such that M = S Anij.
Lemma. Let A be Noetherian and let M be a finitely generated A-module. If
Mi a • • • a Mn a • • • is a chain of submodules of M then there exists m such that
Mm = Mk for all k > m.
This is proved by induction on the number of generators and is left to the
reader (cf Atiyah, Macdonald [1, p.75]).
0.6.3. Let A be as in the previous Lemma. Let / be a two-sided ideal of A. We
set /* equal to the ideal in A generated by the products of k elements of /. Then
/ is said to have the Artin-Rees property (AR property for short) if whenever M
is a finitely generated /1-module and N is a submodule of M there is a non-
negative integer k such that
(1) (Ik + iM)^N = IJ(IkMi^N) for allj > 0.
If t is an indeterminate set Alt] = A ® C[t]. That is, A\t] is the algebra of all
polynomials in t with coefficients in A. If / is a two sided ideal in A then we set
/* = A + tl + t2I2 + ■■■ + tkIk + ■■■in Alt'].
Lemma. / has the AR property if I* is a Noetherian algebra.
Let M be a finitely generated /1-module. Set
M* = M + tlM + t2l2M + ■■■.
Then M* is a finitely generated /*-module. Let N be a submodule of M. Put
Nl= N + t(IMnN) + t2I(IMnN) + --- + trIr(IM niV) + -
Nk = N + t(IM nN) + - + tk(lkM n N) + tk+lI(IkM niV) + -
ThenA?! c N2 <= -is a chain of/*-submodulesof M*. There is thus a k such
that Nk+j = Nk for all / > 0. This is the AR property.
0.6.4. If n is a Lie algebra over a field then we set nt = [n, n] and
nm+i = [nm>n] for m = 1, 2, n is said to be nilpotent if there exists k such
that nk = 0.
Let g be a Lie algebra over C. Let n be a nilpotent Lie subalgebra of l/(g)
such that if X is in g then IX, n] <= it. Let / = n(7(g). Then / is a two sided ideal
in l/(8).

0.6. Modules Over the Universal Enveloping Algebra
15
Proposition. / has the AR property in (/(g).
SetgA = g + m + t2rt! + t3n2 H in l/(g)[f]. Since itj = 0for/»0,gA is
a finite dimensional Lie algebra over C. Thus if i is the natural inclusion of gA
into l/(g)[t] then we have the extension i'~ to (/(gA). It is easy to check that
i~(U(gA) = I*. Thus since (/(gA) is Noetherian, /* is also. Thus Lemma 0.6.3
implies the result.
0.6.5. We conclude this section with a particularly important construction
of (/(g)-modules. Let b be a Lie subalgebra of g. Let M be a (/(b)-module. Let
(/(g) act on U(g) ® M by left translation in the first factor. Let VM be the (/(g)-
submodule of U(g) ® M generated by the elements b ® m — 1 ® bm for
me M and b e (/(b). Then we set
(/(g)®M = ((/(g)®M)/KM.
L'(b)
We now collect some properties of this construction. Let N be a (/(g)-
module and let T be a (/(b)-module homomorphism of M into N, then
(1) Then there exists a unique (/(.q)-module homomorphism of (/(g) ®m) M
into N, TA such that TA(1 ® m) = Tm.
Indeed, put T~(g® m) = yT(m). Then Ker T~ contains VM. Hence T~
induces a (/(g)-module homomorphism 7A of (/(g)®U(h) M into N. The rest
is equally clear.
(2) Let 0->/l^>B-^C->0bea (/(b)-module exact sequence. Then
0^U(Q)(g)AXu^)(g)BA l'u(g)(g)C^0
urn (/((>) t/(b>
is a (/(g)-module exact sequence.
Let V be a subspace of g such that g = b© V. (/(g) = S(K)(g> (/(b) as a right
(/(b)-module under right multiplication (0.4.3). Thus we can look upon the
modules (/(g) (g)U(b) D as S{V) ® D for D = A,B, C. Under this identification,
aA = / ® a and j?A = / ® ;8,
the result is now clear.

This Page Intentionally Left Blank

I Elementary
Representation Theory
Introduction
In this chapter we develop most of the general representation theory that will
be needed in this book. We have attempted to make the material as elementary
as possible.
The infinite dimensional representation theory of Lie groups is a vast
subject that has been studied by many authors in that last 40 years. Thus, a
short chapter such as this one can only "scrape the surface" of the material. A
much more encyclopedic account can be found in Chapters 4 and 5 of Warner
[1]. The more general theory is not really necessary to our book, since we will
be studying mainly reductive groups.
We now give a description of this chapter. The first section is canonical
except for the introduction of the conjugate dual to a Hilbert representation.
This notion is of great importance to the representation theory of reductive
groups. In the second section we give a variant of Schur's Lemma. As we
indicated in Section 0.5 there are many variants of this Lemma. The one that
we give for irreducible unitary representations is sufficient for our purposes.
Section 3 is devoted to the most elementary properties of square integrable
representations. As we will see in the later chapters, these representations are
17

18
1. Elementary Representation Theory
the basic ingredients in the harmonic analysis of real reductive groups.
Section 4 contains the Peter-Weyl theory of representations of compact groups. It
also contains the critical (for our purposes) notion of isotypic component.
In Section 5 we study a very special class of induced representations. A good
exposition of the general theory of induced representations can be found in
Warner [1, Chap. 5]. Included in this section is Frobenius reciprocity for
compact groups. In Section 6 we introduce just enough of the theory of
smooth and analytic vectors to do the representation theory of the later
chapters. Again, the serious reader can consult Warner [1, Chap. 4] for a much
more comprehensive account. Section 7 is devoted to giving the Cartan-Weyl
classification of irreducible representations of connected compact Lie groups.
We give some details of these well-known results, since the proof we use
involves concepts that will be needed in later chapters.
1.1. General properties of representations
1.1.1. Let G be a separable, locally compact group with left invariant
measure, dg (0.1.1). Let V be a topological vector space over C. We denote by,
End(K), the space of continuous endomorphisms of V and by GL(V) the
group of all invertible elements of End(K). Then a representation of G on V is a
homomorphism, it, of G into GL(V) such that the map G x V -> V given by
g, v i—► it(g)v is continuous. That is, the homomorphism, it, is strongly
continuous. We will say that (it, V) is a representation of G.
Let (it, V) be a representation of G. Then a closed subspace, W, of V will be
said to be invariant if it(g)W is a subspace of W for all ge G. (it, V) will be said
to be irreducible if the only invariant subspaces of V are (0) and V.
If (it, V) and (a, W) are representations of G then a continuous linear map,
T, of V to W such that Tit(g) = o(g)T for all g e G is called an intertwining
operator or G-homomorphism. We use the notation HomG(K, W) for the space
of intertwining operators. We say that (it, V) and (a, W) are equivalent if there
exists a bijective element, T, in HomG(K, W) such that T~' is in HomG(W, V).
If G is a Lie group and if V is a Frechet space (c.f. Reed, Simon [1, p.132])
then a representation (it, V) of G is said to be smooth if the maps of G to V
given by g i—► it(g)v are Cx for all ve V.
1.1.2. In this book the most important class of representations that we will
study will be representations (it, H) where H is a (separable) Hilbert space.
Such a representation will be called a Hilbert representation. If (it,H) is a
Hilbert representation and if it(g) is a unitary operator for all g e G then we
call (it, H) a unitary representation.

1.1. General Properties of Representations
19
Let (it, H) be a Hilbert representation of G. Let |-| denote the operator
norm on End(ff). The principle of uniform boundedness (c.f. Reed, Simon
[1,111.9, p.81]) implies:
(1) If fi is a compact subset of G then there is a constant, Cn, such that
\it(g)\<Cn for all gen.
The definition of a representation also implies:
(2) If v, w e H then the map g i—► (it(g)v, vv> is continuous on G.
1.1.3. Lemma. Let H be a Hilbert space and let it be a homomorphism of G
into GL(H). If (it,H) satisfies (1) and (2) above then (it,H) is a representation
ofG.
If / e CC(G) then we define for v,weH the sesquilinear form nf(v,w) by
fif(v, w) = | f(g){it(g)v, w> dg.
G
Let supp / be contained in a compact subset, fi, of G. Then
l%MI ^ Ca\v\ ■ |w| 11/11,(11/11, is the L1 norm of /).
Hence there is an operator it(f) in End(H) such that
W/)l ^ Qill/lli and nf(v,w) = (it(f)v,w} fori;, weH.
If / is a function on G we set L(g)f(x) = f(g'lx) for g,xe G. Then
(1) n(L(x)f) = it(x)it(f) for feCc(G),geG.
If U is an open subset of G such that C1(U) is compact then we will use the
notation Ll(U) for the space of all / e Ll(G) such that supp/ is a subset of U.
The above considerations imply that it extends to a bounded linear map of
Ll(U) into End(H) and that (1) is satisfied.
Assume that 1 e U. If V is an open subset of U containing 1 and having the
properties that VV is contained in U and that if v e V then v'1 e V then the
map V x Ll(V) to Ll(U) given by x, f i—► L(x)f is continuous. Thus the map
of V to H given by x i—► it(x)it(f)v is continuous for / e Ll(V) and i; e H.
Let l/j- be a decreasing sequence of open relatively compact subsets of G
such that f] Uj = (1). Let {u,} be a sequence of non-negative, continuous,
functions on G such that supp Uj is contained in Uj and
\uj(g)dg= 1.

20
1. Elementary Representation Theory
Then one shows easily (using uniform continuity) that
(2) lim (n(itj)v, vv> = <y,w> for v,weH.
Let H0 denote the subspace of all v in H such that the map g i—► n(g)v is
continuous on G. Then 1.1.2(1) implies that H0 is closed in H. Now (2) implies
that H0 is weakly dense in H. Thus H0 = H.
If v, w e H, and if x, y are in a compactum fi contained in G then
||7r(x)i; - n(y)w\\ < \\n(x)v - n(y)v\\ + Cn\\v - w\\.
This completes the proof of strong continuity.
Note. The part of the proof using H0 is taken from Warner [1, p.238]. In that
reference it is shown (using the Theorem of Krein and Smulian) that only
condition (2) is needed.
1.1.4. If (n,H) is a Hilbert representation of G then we set n*(g) = (n(g)'1)*.
Then the conditions (1) and (2) of 1.1.2 are clearly satisfied by n*. Hence,
(n*,H) is a representation of G which is called the conjugate dual
representation of (rc, H). Clearly, one has
(Tt(g)v,Tt*(g)w} = <i;,vv> for v,weH,geG.
1.2. Schur's lemma
1.2.1. Let G be a topological group. In this section we study variants of
Schur's lemma that apply to unitary representations of G. The first and
simplest form is:
Lemma. Let (n,H) be an irreducible unitary representation of G. Then
HomG(H,H) = CI.
This result is easily proven using the spectral theorem. If T e HomG(H, H)
then T* is also. Since
T = (T+ T*)/2 + i(T - T*)/2i,
it is clearly enough to prove that a self-adjoint intertwining operator is a scalar.
We thus assume that T is self-adjoint. Let {Pn} be the family of spectral
projections corresponding to T (Reed, Simon [1, p.234]). Since n(g)Tn(g)'l =
T for all g e G, the uniqueness of the spectral family for T implies that each

1.2. Schur's Lemma
21
Pn e UomG(H,H). This implies that PaH = H or {0} for each Borel set in R.
It follows that there is a closed interval J = [ — a,a~\ such that P, = /. If we
bisect J then one of the two halves, say, Jx will have spectral measure /.
Continuing to bisect in this way we find a nested sequence Jx => J2 => • • • of
intervals each having spectral measure /. Since f] Jk is a point, {p}, we see that
P is supported on {p}. Hence T = p/.
1.2.2. We now give a useful refinement of the above result. For this we need
some notation. Let H be a Hilbert space. If B is a subset of End(ff) then set
B' = {XeEnd(H)\TX = XT for all TeB}.
Let B be a subalgebra of End(H) such that / e B and if Te B then T* e B.
Then Von Neumann's observation is:
(1) If v e H then (B')'p <= Cl(Bv).
Indeed, since T* e B if TeB, the orthogonal complement to Cl(Bi;) is
B-invariant. Thus, if P is the orthogonal projection of H onto Cl(Bi;) then
P e B'. Hence, if T e (B')' then TP = PT. Thus, TCl(Bi;) is a subspace of
Cl(Bi;). (1) now follows since v e Cl(Bi;).
We can now give a refinement of Schur's lemma.
Proposition. Let (n, H) be an irreducible unitary representation of G. Let D be
a dense subspace of H that is G invariant. Let T be a linear map of D into H
(there is no topology on D) such that Tn(g)v = n(g)Tv for all g e G, v e D.
Assume also that there exists a dense subspace D' of H and S a linear map of D'
into H such that
< Tv, vv> = <u, Sw} for v e D, w e D'.
Then T is a scalar multiple of I restricted to D.
Let A denote the subalgebra of End(H) spanned by the operators n(g) for
g e G. If X e A then, clearly, X* e A. Since tt(1) = I, I e A. We also note
(2) If x, y e H, X e End(H) and if S > 0 is given then there exists U e A such
that
\\Ux-Xx\\<5 and \\Uy-Xy\\<5.
Indeed, set V = H © H with the direct sum inner product. Let B = {U © U j
U e A}. Then B' is the space of operators of the form
U(x, y) = (Xx + Yy, Zx + Wy) with X, Y,Z,We A'.

22 1. Elementary Representation Theory
Now Lemma 1.2.1 implies that (A')' = End(ff). Thus it is easy to see that (B')'
is the space of all operators of the form
U(x, y) = (Zx, Zy) with Z e End(H).
(1) now implies that if Z e End(ff) then (Zx,Zy) e C\(B(x,y)). This clearly
implies (2).
Let T be as in the statement of the result we are proving. Assume that ve H
and that v and Tv are linearly independent. (2) implies that there exists a
sequence {Uj} in A such that
lim Up = v and lim UjTv = v.
Now, if w e D' then
<i>, vv> = lim <UjTv, vv> = lim <TUp, vv> = lim <Up, Sw} = <i>, Sw} = <Tv, vv>.
Since D' is dense in H this implies that Tv = v. Since this is ridiculous, we
conclude that if v e D then v and Tv are linearly dependent. This easily implies
that T is a scalar multiple of / on D.
1.3. Square integrable representations
1.3.1. Let G be a locally compact, separable group. Fix, dg, a right invariant
measure on G. Let L2(G) denote the space of all square integrable functions
with respect to dg. If / e L2(G) and if x e G define R(x)f by
R(x)f(g) = f(gx) for g e G.
Since dg is right invariant K(x) is a unitary operator for all x e G. Furthermore,
<R(x)u,i>> = | u(gx)v(g) dg,
o
which is easily seen to be a continuous function of x. Lemma 1.1.3 implies that
(R,L2(G)) is a unitary representation of G, called the right regular
representation of G.
1.3.2. If (rc, H) is a Hilbert representation of G and if v and w are in H then we
use the notation cvw for the function
g h-> (n(g)v,wy.
The functions c0jW are called coefficients or matrix coefficients of rc.
Let (;r, //) be an irreducible unitary representation of G. Then we say that
(it, H) is square integrable if it has a non-zero, square integrable matrix
coefficient.

1.3. Square Integrable Representations
23
Lemma. // (n, H) is a square integrable representation of G then every matrix
coefficient ofn is square integrable. Furthermore, there exists a unitary operator,
TeHomG(H,L2(Gj),
with closed range such that
T(H) is a subspace of C(G) n L2(G).
Fix w' and v', unit vectors in H such that cw. „- is square integrable. Set
D' = {ve H\cVtC. e L2(G)}. We note that D' contains span{n(g)w'\ge G}.
Thus D' is dense in H. Also, if v e D' then n(g)v e D',
We define a map T from D' to L2(G) by Tv = cvy. Then
Tn(x)v = R(x)Ti> for veD'.
Define on D' the inner product ( , ) given by
(v,w)= <u,w> + (Tv,Twy.
(1) D' is complete relative to ( , ).
Indeed, let {vj} be a Cauchy sequence in D'. Then {v}} is Cauchy in // and Tv}
isCauchy in L2(G). Thus, u,-converges to ve Hand Tfjconverges to u e L2(G).
In particular, a subsequence of Tvj converges pointwise, almost everywhere to
u. But
c„ >r. converges uniformly to c„,,. .
Thus, u = c„ „- almost everywhere. This implies that v e D'.
Let S denote the canonical inclusion of D' into H. Then S is clearly a
bounded linear mapping of D' into //. Let S* denote the adjoint map from H
to D'. Then S* satisfies the hypotheses of Proposition 1.2.2 with D = H. Thus
S* = al with a e R. So £)' = //. It also implies that there exists b > 0 such that
(2) <Tt;,Tvv> = b<u, vv> for v,we H.
The lemma now follows using the map bll2T in light of (2) and the already
observed fact that D' = H.
1.3.3. The above proof has as an immediate consequence the Schur
orthogonality relations:
Proposition. Let (it, H) and (a, V) be square integrable representations of G.
(1) If it and a are not equivalent then
| {n(g)x,y)> conj(O(0)z, w» dgi = 0.
a
for all x, y e H, z,w e V.

24
1. Elementary Representation Theory
(2) There exists a constant d(n) > 0 such that if x, y, z,w e H then
| (Tt(g)x,y} con)((Tt(g)z,wy)dg = d(n)~\x,zs)iw,ys).
o
Define the operators T and S by T(u)(g) = (n(g)u, y} and S(v)(g) =
(a(g)v,w}. Then the proof of the preceding result implies that there exist
t > 0 and s > 0 such that (\/t)T and (l/s)S are unitary intertwining operators
from H and V, respectively, to L2(G). It follows that there exists a positive
constant C such that
\(T(u),S(v)}\ <C||u|||M| for all u e H, v e V.
Hence, for each v e V there exists a unique A(v) e H such that
<T(u),S(i;)> = (u,A(v)} for all u e H.
It is easy to see that there exists a positive constant, a, such that (1 /a)A is a
bijective unitary intertwining operator. This proves (1).
We now assume that it = a. Then the above argument implies that
| (n(g)x,y)> cori)((n(g)z,w)>)dg = a{w,y)(x,z}
o
= b(x,z)<w,_y>.
Thus a(w,y) = (l/d)<w,_y> with d > 0. This completes the proof of (2).
1.3.4. If (rc, H) is a square integrable representation of G then the number
d(n) in 1.3.3(2) is called the formal degree of it. d(n) has an interpretation as a
generalized dimension in the theory of Von-Neumann algebras (Dixmier
[1, p.281]). If G is compact then we will see (in the next section) that d(n) =
dim H < oo.
1.4. Basic representation theory of compact groups.
1.4.1. Let HjJ < N, N < oo be Hilbert spaces then the symbol @-Hj will
mean the Hilbert space completion of the algebraic direct sum of the H} with
the inner product
<£ vv E wi) = E <»,-. WA »,-. wi e ty, j < N.
Let G be a topological group. Let (itj, Hj) be unitary representations of G for
j < N. Let H = @;ffj. Then the representation of G, rc, on H given by the
extension to H of 7r(g)(E u,) = S Ttj(g)vj is called the direct sum of (nj,Hj).
Let G be a separable, locally compact, unimodular, group with invariant
measure, dg. A sequence {u,} of non-negative continuous functions on G is

1.4. Basic Representation Theory of Compact Groups
25
called a delta sequence if the following three conditions are satisfied:
(1) supp uj+1 is contained in supp Uj and f] supp us = {1},
(2) Uj(x) = Ujix'1),
(3) Juj(flf)dflf= 1 for all;.
a
The following result is due to Gelfand, Graev, Piatetski-Shapiro (the proof
we give is due to Langlands):
Proposition. Let (n, H) be a unitary representation of G.If there exists a delta
sequence Uj on G such that each n(uj) (1.1.3) is a compact operator (c.f. 8.A.1.1) on
H then there exist unitary irreducible representations (jtj,Hj), j < N, N < oo,
such that (n, H) is equivalent with the direct sum of the (tCj, Hj). Furthermore, for
each i there are only a finite number of (itj, Hj) equivalent with (nh ff;).
Let S be the set of all collections of closed, invariant, mutually orthogonal,
irreducible subspaces of H. We order S by inclusion. Zorn's lemma implies
that there is a maximal element, T, of S. Let V be the Hilbert space direct sum
of the elements of T. Let X be the orthogonal complement to V. Then X is a
closed, invariant subspace of H. Suppose that X is nonzero. Let ubea unit
vector in X. Since lim n(uj)v = v we see that there exists i such that if u = u;
then n(u)v is nonzero. Now, (2) implies that if Q = n(u) restricted to X then Q is
non-zero and self-adjoint on X. Also, by assumption Q is compact. Let Z be an
eigenspace for a non-zero eigenvalue for Q on X (such exist by the spectral
theorem for compact self-adjoint operators c.f. Lemma 8.A.1.2). Then Z is
finite dimensional. Let R be a non-zero subspace of Z of minimal dimension
subject to the condition that R = W n Z for some closed invariant subspace,
W, of X. Let Y be the intersection of all invariant subspaces of X containing R.
If Y were reducible then Y could be written as an orthogonal direct sum A + B
with A and B closed invariant subspaces of Y. Since Q leaves invariant any
invariant subspace of X, we see that R must be completely contained in A or in
B. But this contradicts the definition of Y. Hence Y is irreducible. We have now
contradicted the definition of T. Hence X = 0 so V = H.
The last assertion follows from the fact that the non-zero eigenvalues of each
n(uj) have finite multiplicities.
1.4.2. For the rest of this section we will assume that G is compact. If
/ e C(G) and if ueL2(G) then
R(f)u(x) = | u(xg)f(g)dg = | u(g)f(X-1g)dg.

26
1. Elementary Representation Theory
Hence R(f) is the integral operator on L2(G) with kernel K(x, y) = f(x~ ly).
Since G is compact we see that R(f) is a Hilbert-Schmidt operator. Hence
R(f) is compact.
The previous proposition therefore applies to (R, L2(G)). We now derive
some consequences of that result.
Proposition. Let (it, H) be an irreducible unitary representation of G. Then
dim H < oo.
Since G is compact and the matrix coefficients of it are continuous, (it, H) is
square integrable. Thus Lemma 1.3.2 implies that it is equivalent to an
irreducible closed subspace of L2(G) which is also contained in C(G). The
result now follows from:
Scholium. Let (X,n) be a measure space with total measure I. If V is a closed
subspace of L2(X) contained in U°(X) then dim V < oo.
Let ||- • -|| denote the L2-norm and let ||- • -\\x denote the L^-norm. Then it is
clear that
(*) ll/ll <: ll/IL for/6L»(X).
Let Q be the inclusion of V into L2(X). Let W be the closure of V in U°(X).
Then (*) implies that Q extends to a bounded operator from W to V. Hence
W= V. The closed graph theorem now implies that there exists a positive
constant such that
(**) \\f\L<c\\f\\ iovfeV.
Let fi,..., fd be an.orthonormal set in V. If ^ e C for i = \,...,d then
IE IMft(x)\ < ||E iMftll < c||E ^-|| = c(I l^;|2)1/2.
Choose Hi = conj(/(x)). Then we have
Il/iWI^cdl/iWI2)1'2 fora.e.xeX
This implies that
El/WI2<c2 fora.e. xeX.
Integrating this inequality over X yields d < c2. This proves the result.
1.4.3. As we have observed, if (n, H) is an irreducible unitary representation
of G then it is square integrable.

1.4. Basic Representation Theory of Compact Groups
27
Lemma. Let (it, H) be an irreducible unitary representation of G. Then the
formal degree of it is equal to dim H.
Let d be the formal degree of it. Let vl,..., v„ be an orthonormal basis of H.
Set fij = cVUVj. Then the matrix [,/ijM] is unitary. Hence
Xl^(x)|2 = n forallxeG.
If we integrate both sides of this equation over G then 1.3.3(2) implies that
(\/d)n2 = n. Hence d = n as asserted.
1.4.4. Let GA denote the set of all equivalence classes of irreducible unitary
representations of G. If y e GA we denote by L2(G)(y) the sum of all invariant,
irreducible subspaces of L2(G) that are in the class y. The material in 1.4.2
implies that
(1) dim L2(G)(y) < oo and L2(G) = ®L2(G)(y).
1.4.5. Let y e GA and let (it, H) e y. We set d(y) = d(it) = dim H( < oo). We
put for g e G
Xy(y) = XM = tr Jt(0).
Then %y is called the character of y.
Lemma. // y, \i are in GA then
Jxy(0)conj(x„(0))d0= <5y>/J.
G
This is an immediate consequence of 1.3.3(2) and 1.4.3.
1.4.6. Let for yeGA, ay = d(y)conj(y,/). Let P denote the orthogonal
projection of L2(G) onto L2(G)(y).
Lemma. Py = R(txy).
This result is also a direct consequence of 1.3.3(2) and 1.4.3.
Corollary. If y e GA then dim L2(G)(y) = d(y)2.
By the above lemma dim L2(G)(y) = tr R(a.y). This is easily seen (using the
material in 1.4.2) to be equal to
d(y)\ conj(ay(x"'x))dx = d(y) conj(^(l)) = d(y)2.

28
1. Elementary Representation Theory
1.4.7. Let (n, H) be a unitary representation of G. If y e GA then we set H(y)
equal to the closure of the sum of all the closed, invariant subspaces of H that
are in the class y. H(y) is called the y-isotypic component of H.
Lemma
(1) H(y) = n(«y)H.
(2) H is the Hilbert space direct sum of the H(y).
Hv,weH then R(a.y)cViW = cuw with u = n(a.y)v. live H(y) then cvw is a sum
of matrix coefficients of y. Hence (n(ay)v,w) = <u,vv> for all ve H(y), we H.
This implies that H(y) is contained in it(<xy)H.
We now prove the reverse inclusion. If v e it(ay)H then R(a1,)c„ w = c0>w for
all we H. Hence c„>we L2(G)(y) for all w e H. Let Z = span{7r(g)i;|ge G}.
Then
dim span{c.,w I z e Z, w e H} < d(y)2.
This implies that dim Z < oo. Hence, Z splits into a finite direct sum of
irreducible invariant subspaces each in the class of y. This completes the proof
of(l).
We now prove (2). We note that if v e H(y), w e H(n) with y and n distinct,
then c„,w e L2(G)(y) n L2(G)(n) = {0}. This implies that <H(y), H(/z)> = 0. We
must therefore only show that the sum of the H(y) is dense in H. We label GA as
yx, y2,--- If y = fj then we set ay = a^. If u, we H then
lim £ R(Xj)cv.w = c„,w
iV->oo j<iV
in L2(G). Thus if w e H is orthogonal to the algebraic sum of the H(y) then
cv,w — 0 f°r a^ »eW. Hence w = 0. (2) now follows.
1.4.8. We conclude this section with a useful variant of the "unitarian trick".
Lemma. Let (n,H) be a Hilbert representation of G (still assumed to be
compact). Then there exists an inner product ( , )onH that gives the original
topology on H and is such that relative to( , ),n is unitary.
Define ( , ) as follows:
(v, w) = | (n(g)v, Tt(g)w} dg for v, w e H.
G
There is a positive constant C such that |n(g)\ < Cforallge G (1.1.2(1)). Since

1.5. A Class of Induced Representations
29
Tt(g)n(g ') = / we also see that ||7t(0)i>|| > C 1\\v\\ for all ge G. Hence
C~\v,v) < <u,u> <C(v,v) forallyeH.
so ( , ) defines the same topology as < , >. The rest of the argument goes
as usual (0.3.1, 0.5.4).
1.5. A class of induced representations
1.5.1. Let G be a unimodular, locally compact group. Let K and P be closed
subgroups of G such that K is compact and such that G = PK. Let 3 denote
the modular function of P (0.1.1). Let dp denote left invariant measure on P
and let dk denote normalized invariant measure on K. Then we can choose
invariant measure on G so that
\f(9)dg= I f(pk)dpdk for feCc(G) (Lemma 0.1.4).
a pxk
We extend 3 to G by setting 3(pk) = 3(p) for p e P, k e K. This makes sense
since 3(p) = 1 for p e K n P. If / is a function on K such that f(pk) = f(k) for
pe K r\P then we extend / to G by setting f(pk) = f(k) for p e P and k e K.
(1) If / is integrable on K and if f(pk) = f(k) for all p e P n X then
J/(*0W*0)d* = J/(*)ifc.
Indeed, there exists, g e CC(G) such that (see 0.1.3)
$g(pk)dp = f(k) forall/ceX.
p
For this g we also have
$g(x)dx= | g(pk)dpdk = \f{k)dk.
G PXK K
If x e G we set x = p(x)/c(x) with p(x) e P, /c(x) e K. This decomposition is not
necessarily unique, but the ambiguity will be irrelevant to our argument. We
have for x e G
| f(k) dk = | g(u) du = | g(ux) du = | g(pkx) dp dk
K G G P*K
= I g(pp(kx)k(kx))dpdk= | 3(p(kx))g(pk(xk))dpdk
pxK Px K
= \3(kx)f(kx)dk.
K
since 3(p(x)) = 3(x). This proves (1).

30
1. Elementary Representation Theory
1.5.2. Let (a, W) be a Hilbert representation of P. In light of Lemma 1.4.8, we
assume that the restriction of a to K n P is unitary. Let {H")0 be the space of
continuous functions, u, from G to W such that
(1) u(pg) = S(py'2a(p)u(g) for PeP,geG.
If u, v e {Ha)0 then we set
(2) <U,l>> = J <«(*), l>(*)>d*.
Let H" denote the Hilbert space completion of (H")0 relative to < , >.
If u e (H")0 and if g e G we set
(3) na(g)u(x) = u{xg) for all x e G.
Clearly n„(0)iie(ff'V
1.5.3. Lemma. (1) If g e G t/ien Jtff(gt) extends to a bounded operator on H".
(2) (na,Ha) is a Hilbert representation of G which is unitary if a is a unitary
representation of P.
As above we write g = p{g)k(g) with p(g) e P, k(g) e X. Since the ambiguity
in the definition of p(g) is in the compact set P n X, it follows that if fi is a
compact subset of G then there exists a compact subset, fi', of P such that p(fi)
is contained in fi'.
Let ueH and let fi be a compact subset of G. If g e fi then
(i) IW0HI2 = J IMMII2d/c = J «S(Mlk(P(/c0)M/c(M)ll2^.
K K
By the above p(kg) e (KQ)'. Hence 1.1.2(1) implies that there is a constant En
such that \a(p(kg))\ < En for g e Q. Put Z)n equal to the supremum of S112
on Xn. Then (i) implies that
(ii) \\n.(9)u\\ £ DaEa\\u\\ forged.
This proves (1).
SetCn = £)nCn. Using (ii) it is easy to see that if u, v, ze Hand if g e fi, then
l<X(#)z,u> - <na(g)z,V)\ < Cn||z|| • ||u - u||,
|<7r„(0)u,z> - <7rff(0)i;,z>| < Cq||m - v\\ ■ \\z\\.
Since it is clear that the functions cu „ for u,ce (//")0 are continuous, the above

1.6. C°° Vectors and Analytic Vectors
31
inequalities imply (see Lemma 1.1.3) that (na,Ha) is a Hilbert representation
of G.
(i) combined with 1.5.1(1) implies that if a is unitary then na is unitary.
1.5.4. The representation (ita, H") constructed above is a special case of an
induced representation. We will not have any use for a more general definition
of induced representation. Thus, in this book, induced representation will
mean the above construction. We will also use the notation
Ind» forfo.ff").
1.5.5. We now look at the special case when G is compact. Let P be a closed
subgroup of G. We may take G = K in the above construction. Let (a, W) be a
finite dimensional unitary representation of P. We study (ita, H").
Let y e GA then (with notation as in 1.4.7)
(1) H"(y) is contained in (H")0.
Indeed, if u e C(G), and if v e H then ita(u)v e (H")0. Thus (1) follows from
Lemma 1.4.7(1).
Fix (n,V\ a finite dimensional unitary representation of G. Let Te
HomG( V, H"). Then (1) implies that T(V) e (H")0. Thus we can define TA (v) =
T(v)(\) for v e H. It is clear that TA is in HomP(K, W). We have
(2) The map 7V-► TA defines a linear isomorphism of HomG(K, H") onto
HomP(V,W).
Indeed, if S e HomP(K, W) define S~ (u)(g) = S(n(g)v). Then it is easy to see
that S~ e HomG(K, H"). It is also clear that (TA)~ = T and (S~)A = S.
(2) is usually called Frobenius reciprocity. It immediately implies
(3) dimH"(y) = d(y)d\mUomP(V, W) for y e GA,(n,V)e y.
1.6. Cx vectors and analytic vectors
1.6.1. For the rest of this chapter we will be studying representations of Lie
groups. Let G be a Lie group with a finite number of connected components.
We fix a left invariant measure, dg, on G. Let (it, H) be a Hilbert representation
of G. If v e H is such that the function <f>(g) = ic(g)v is of class Cx from GtoH
then v is called a C™-vector or smooth vector for (it, H). The following result was
first observed by Garding in order to prove Theorem 1.6.2.

32
1. Elementary Representation Theory
Lemma. If f € C™(G) and if v e H then n(f)v is a smooth vector for (n, H).
Let U be a relatively compact open subset of G containing 1. Let />'([/) be
the space of all L1 functions on G with support in U. Let V be an open subset of
U such that if x e V then x~' e V and VV c= I/. Then
(1) Let / e C™(V) then the map of V to L'(G) given by F(x) = L(x)/ is of
class C™.
Indeed, if X e g (the Lie algebra of G, as usual) then we set L(X)f(g) =
d/dtt = 0(f(exp(-tX)g). Taylor's theorem implies that there is e > 0 and £ a
bounded function of t, g for |t|<e, such that f(exp(-tX)g) = f(g) +
tL(X)f(g) + t2E(t,g) for |t| < e and g e V. This implies that
\\L(x)L(X)f - (\/t)(L(x exp(tX))/ - L(x)/)||, =
||L(X)/ - (l/t)(L(exp(tX))/ - /)||, < \t\C
with C an appropriate constant for |t| < e. This implies that F(x) is of class C1.
This argument can be iterated to prove (1).
We have seen in the proof of Lemma 1.1.3 that the correspondence / to
n(f)v is a bounded linear map of Ll(U) into H. Thus the map of V to H given
by x i—► 7r(L(x)/)i; isaC" map. Since n(L(x)f) = n(x)n(f). We see that the
map of V into H given by xi—► 7r(x)7r(/)i; is of class C™. The lemma now
follows since n(x) is a bounded linear operator on H hence it is of class C™.
1.6.2. Theorem. The space of C™ vectors of H is dense in H.
As is well known, there exists a delta sequence Uj (1.4.1) consisting of C33
functions on G. Since
lim n(uj)v = v for v e //.
The result follows from the previous Lemma.
1.6.3. Let H ™ denote the space of all C °° vectors for tt. If v e H °°, and if X e g
then we set
n(X)v = d/dtt = 0 n(exp(tX))v.
Then n(X) maps Hx into H1" and it is not hard to show (using Taylor's
Theorem) that
(1) n([X, Y]) = Tt(X)n(Y) - n(Y)n(X) on H for all X, Y e g.

1.6. C" Vectors and Analytic Vectors 33
Hence (n,Hx) defines a representation of g. The universal mapping property
of l/(g) implies that n extends to 1/(3).
If D e U(q) then we set pD(v) = \\n(D)v\\ for v e Hx.
We give H™ the topology induced by the semi-norms pD for D e U(q).
1.6.4. Lemma. (1) ff00 is a Frechet space.
(2) (rc, H00) is a smooth representation of G(l.l.l).
Since U(q) is countable dimensional it is enough to show that H™ is
sequentially complete to prove (1). Let (vj) be a Cauchy sequence in Hx. If
X e g then {vj} and {Xvj} are Cauchy sequences in H. Thus there exist v,
ue H such that
lim Vj = v and lim Xi^ = u.
We note that
lim Xn(exp(tX))Vj = 7r(exp(tA'))u
and that
d/dt(e\p(tX)v = 7c(X)n(e\p(tX))v.
Hence
Vj + J 7r(exp(sA'))7r(A')fJds = 7r(exp(tA')f;J..
0
If we take the limit of this expression in/ we have
r
v + J n(exp(sX))uds = 7r(exp(tA'))f;.
0
This implies that the map 11—► n(exp(tX)v is of class C1 with derivative equal
to 7r(exp(tA'))u. Hence g 1—► n(g)v is of class C1. This argument can be iterated
to show that v is a smooth vector. Hence Hx is complete.
We now prove (2). We first observe that
(i) The map Uj(q) ® Hx -> Hx given by g, v 1—► Jt(gf)u is continuous.
We also have
(ii) n(g)n(X)v = Tt(Ad(g)X)n(g)v for g e G, Xsq and veH^.
Hence if D e l/(g), pG and if t' e H30 then
pD(n(g)v -v)= MgMAdig-'Wv - n(D)i;||.

34
1. Elementary Representation Theory
In light of (i), we have shown that (n, H™) is a representation of G. Now the
argument that we used to prove (1) completes the proof of (2).
1.6.5. Let ZG(gc) denote the subalgebra of l/(gc) consisting of those
g e U(qc) such that Ad(x)g = g for all x e G. If G is connected then
ZG(gc) = Z(gc) the center of l/(gc).
Lemma. Let (n, H) be an irreducible unitary representation of G. Then each
z e ZG(gc) acts by a scalar multiple of I on ffx.
If X e Qc we will use the notation, conj(A'), for complex conjugation of X
relative to g. That is, if X = Xx + iX2 with Xx, X2 e g then conj(X) = Xy —
iX2. We define a conjugate linear anti-homomorphism of U(qc) onto U(qc),
x*—>x*
as follows:
(1) 1* = 1,
(2) X* = -conj(X) forXegc,
(3) (xy)* = }^*x* for x, y e l/(gc).
It is clear that (Zc(gc))* = Zc(gc). If we take D = D' = Hai and T= n(z),
S = 7r(z*) then the lemma follows from Proposition 1.2.2.
If (rc, H) is a representation of G and if % is a homomorphism of ZG(g) to C
such that n(z)v = ^(z)u for z e ZG(g) and v e Hx then ^ is called the infinitesimal
character of n.
1.6.6. Let (;r, H) be a Hilbert representation of G. Then we say that v e H is
an analytic vector for (rc, //) if the function
is real analytic for all w e H. This agrees with the standard terminology
(Warner [1, p.278]) since weak analyticity implies strong analyticity.
However, we will only need this notion of analyticity in this book. We use the
notation Hm for the space of analytic vectors of H. It is clear that if v e H then
n(g)v e H and n(X)v e H for g e G and X in g. Hence, Hm is a representation
of g.
The main reason for the introduction of analytic vectors is the following
result:
Proposition. Let G be connected. If V is a Q-invariant subspace of Ha, then
C \(V) (in H) is a G-invariant subspace of H.

1.7. Representations of Compact Lie Groups
35
If W is a subspace of H we denote by W1 the orthogonal complement of W
in H. Then it is easy to see that Cl(VF) is equal to (W1)1. Let X e g, let v e V
and let w be in V1. Then there exists e > 0 such that if |t| < e then
(n(exp(tX))v,w) = £ (tnln\)(n(Xn)v, w>
and the series converges absolutely. Since w e V1 it follows that
<;r(exp(r.X>,w> = 0 for |t| < e.
The real analyticity of ti—► (n(exp(tX))v, vv> now implies that
<7r(exp X)v, vv> = 0 for all v e V, w e V1 and X e g.
This implies that V1 is invariant under the operators 7r*(exp X) for X e g (see
1.1.4 for re*). Since exp(g) generates G as a group, we see that V1 is an invariant
space for n*. Hence (K1)1 is an invariant subspace for it.
1.7. Representations of compact Lie groups
1.7.1. Let g be a reductive Lie algebra over C. We will use the notation of
section 0.2. Fix h, a Cartan subalgebra of g. Fix B, an invariant non-
degenerate form as in 0.2.2. Set <t>(g, h) = <t>. Fix P, a system of positive roots
for <t>. Let A = {a!,..., a,} be the simple roots in P.
(1) If X e ga and if Y e g_a then \_X, 7] = B(X, Y)HX.
Indeed, [ga,g-a] is a subspace of h. If H e h then B([X, Y~\,H) =
-B(X, [H, 7]) = a(H)B(X, Y). So (1) follows from the definition of Hx.
1.7.2. Lemma. Let (a, V) be an irreducible finite dimensional representation
of g. Then the elements of h act semi-simply on V.
Let for a e P, X e ga, Y e g^,, be non-zero. If H = (2/(a, a))Ha then X, Y, H
span a TDS (0.5.4), s„. Hence Lemmas 0.5.4 and 0.5.5 imply that each Ha, a e P,
acts semi-simply on V. Schur's Lemma implies that the elements of j(g) act by
scalars on V. Since the span of the Hx, a e P and j(g) is h the lemma follows.
1.7.3. We note that the argument in the proof of the above Lemma actually
proves
(1) Let (a, V) be a finite dimensional representation of g. h acts semi-simply
on V if s(g) does.

36
1. Elementary Representation Theory
Let (a, V) be a (not necessarily finite dimensional) representation of g such
that h acts semi-simply on V. If fi e h* then we set K„ = {p e V: hv = fi(h)v for
all h e h}. Then K„ is called the fi-weight space of V and if fi e h* and V^ is
non-zero then fi is called a weight of K.
We now assume that V is finite dimensional. We partially order the weights
of V by saying that ^>yif^ — yisa sum of elements of P. Let A be a weight
of V that is maximal relative to the partial order.
(2) '" 2(A, a)/(a, a) is a non-negative integer for a e P.
Indeed, let s„, X, Y be as in the proof of Lemma 1.7.2. Then I^c Vli + at.
Thus XV„ = (0). The result now follows from 0.5.4 and 0.5.5.
If A is an element of b* satisfying (2) then we say that A is dominant integral.
(3) If fi is a weight of V then 2(fi, a)/(a, a) is an integer for all a e <J>.
This also follows from TDS theory.
(4) If fi is a weight of V then so is sxfi for all a e <J>.
Indeed, let sa be as above. Let v be a non-zero element of V^. Then there
exists r>0 such that Xrv is non-zero but Xr+1v = 0. By TDS theory
(fi + 2m)(H) = m, a non-negative integer. Also, TDS theory implies that if
w = Xrv then YJw is non-zero for j = 0,..., m. Thus, the forms fi + 2(r - ;')a
are weights of V for; = 0,..., m. Since safi is on this list of weights (4) follows.
1.7.4. We are now ready to give the Cartan-Weyl classification of irreducible
finite dimensional representations of g.
Theorem. (1) If V is an irreducible, finite dimensional Q-module then V has a
unique highest weight (i.e., maximal weight), which we write as Av. Furthermore,
the Av weight space is one dimensional.
(2) // V and W are irreducible finite dimensional Q-modules then V and W are
equivalent if and only if Av = A^.
(3) // A is a dominant integral linear form on b then there exists an irreducible
finite dimensional Q-module, V, such that Av = A.
We set n+ = EaeJ, ga and n" = IaeP g_a. Then g = n" 0 h 0 n + . P-B-W
implies that
(0 l/(g)=l/(rT)U(b)l/(n + ).
We now prove (1). Let fi be a maximal weight of V. Fix a non-zero element
v € Vp. Then U(q)v = U(n~)v by (i). Since V is irreducible, this implies that

1.7. Representations of Compact Lie Groups
37
K= U(n~)v. Since the weights of I) on l/(rt~) are of the form — S n,Oy with rij
non-negative integers, and the 0 weight space consists of the scalar multiples
of 1.(1) now follows.
Before we begin the proof of (2) we will introduce a concept that will be
useful in the later chapters. Let b = h® n + . b is usually called a Borel
subalgebra of g. If fi e h* we denote by CM the 1-dimensional b-module C with
h acting by fi and n+ acting by 0. We set (0.6.5)
(ii) Af(/z)= 1/(9) ®C„ (0.6.5).
Vib)
M{fi) is usually called a Verma module. By the first part of this proof h acts
semi-simply on M(n) and the weights of M(n) are the linear forms n — S n,-ay
with rij non-negative integers. Furthermore, the ji-weight space is spanned by
1 (x) 1. Let N be the sum of the submodules of M(n) that do not intersect
C1 ® 1. Then it is easy to see that N is the unique proper maximal submodule
of M(n). Hence, M(n) has a unique, non-zero irreducible quotient which we
denote L(n).
Let V be an irreducible, finite dimensional g-module with highest weight A.
Then we have seen above that n + KA = 0. Hence there is a surjective g-module
homomorphism of M(A) onto V (0.6.5(1)). But then V is equivalent to L(A).
This implies (2).
To prove (3) we need only show that if fi is dominant integral then L(n) is
finite dimensional. So, assume that /* is dominant integral. Let a be simple root
inPandletsa = s be the corresponding TDS. Set m = n(H) + l(X, Y, //areas
above). Then
(iii) Ym(\ (g> 1) e N (the maximal proper submodule of M(n)).
Indeed, set v = ym(l ® 1). If P e A is not equal to a then \_qp, V] = 0 by the
definition of simple root. Also 0.5.5(1) implies that Xv = 0. Since the simple
root vectors generate n+ as a Lie algebra (0.2.1(3)), see that n + t; = 0. Now (i)
implies that U(q)v e N. This proves (iii).
(iv) If a is a simple root in P and if v e L(p) then U(%)v is finite dimensional.
Indeed, this is true if v is the image of 1 ® 1 in L(/4 Let us call that
element w. Set s = sa. Let Z — l/(s)w. Clearly, the union of the spaces
l/J'(g)Z is L(n). Since each of these spaces is finite dimensional and s
invariant (iv) follows.
(v) If a is a weight of L(fi) and if s e VK(g, h) then so is a weight of L(n).
This follows from (iv) using the argument proving 1.7.3(4) and 0.2.4(3).
(vi) If a and y are weights of L(/i) agreeing on hR then a = y.

38
1. Elementary Representation Theory
This is clear since 3(g) acts on L(n) by scalars.
(vii) L(n) has only a finite number of weights.
We set W = W(g,h). If a is a weight of L(n) then a is integral. 0.23(3)
implies that there is s e W such that so is dominant integral. Thus in light of
(iii) we need only show that there are only a finite number of dominant integral
weights. We may (in light of (vi)) assume that 5(g) = 0. But then the integral
forms are in a lattice in h*. If a is a dominant weight then a = \i — Q with Q a
sum of elements of P. Thus
<<7,<T> = ill - Q,a) < ill,a} = ill, {I - g> < ill, 11).
Thus the dominant weights are contained in the intersecti on of a discrete set
and a compact set. This proves our assertion.
It is not hard to show that the weight spaces of M(n) are all finite
dimensional. (One must show that the weight spaces of h on l/(n~) are finite
dimensional.) Hence (vii) completes the proof of (3).
1.7.5. Let G be a compact Lie group with maximal torus T. For the rest of
this section we will use the notation g for the complexification of the Lie
algebra of G. We will also write h for tc. Then g is a reductive Lie algebra over
C and h is a Cartan subalgebra of g. We may thus continue with the notation
of the previous paragraphs.
Let (rc, H) be an irreducible (unitary) representation of G. Then an isotypic
component for T (1.4.7) is a weight space for h. We will thus use the notation
H(ii) for the n weight space and also think of n as a character of T (0.3.2). In
particular we will look upon the highest weight of H as a character of T.
We now assume that G is connected. Let G~ be the simply connected
covering group of G. Let p. be a dominant integral functional on h that is also
T-integral (0.3.2). Then there is a representation n of G~ on L(fi) whose
differential gives the action of g. Let Z denote the kernel of the covering
homomorphism of G~ onto G. We assert that Z is contained in Ker n.
Assuming this for the moment, we have
Theorem. Let fibea dominant integral, T-integral form on h. Then there exists
an irreducible unitary representation (ft^F") of G whose differential is
equivalent to the %-module L(fi). Let y^ denote the equivalence class of n^. Then
GA = {}>„.■ n dominant integral and T-integral}.
We must show that Z is contained in Ker n^. Let p be the covering
projection of G~ onto G. Set T~ = p~'(T). Then G~/T~ is a covering space

1.8. Further Results and Comments
39
of G/T. Since G/T is simply connected (0.3.3(4)), this implies that T~ is
connected. Since Z is a subgroup of T~ we see that \i(Z) = 1. Z is easily seen to
be central, so Schur's Lemma completes the proof.
1.8. Further results and comments
1.8.1. This section contains some results that are related to the material of
this chapter. Some of them will be referenced to the literature and others will
be left as exercises to the reader. They will not be used in the body of this book.
1.8.2. The material in Section 1.3 is strongly influenced by the material in
Borel [1] on irreducible square integrable representations.
We note that there is a slightly more general notion of square integrability
which we will now discuss (we use the notation of Section 1.3). Let Z be the
center of G. Let d(Zg) be a right invariant measure on Z\G. If # e ZA then we
write L2(G;x) for tne space of all measurable complex valued functions on G
such that
f(zg) = X(z)f(9) for z e Z, g e G and
11/112= | \f(Zg)\2d(Zy)<K.
z\o
We set (itx(g)f)(x) = f(xg) for x,geG and / e L2(G; X). Then (icx, L2(G; X)
is a unitary representation of G.
Let (rc,//) be an irreducible unitary representation of G. Then Schur's
lemma implies that there exists X e ZA such that n(z) = #(z)/ for z e Z. We say
that (it, H) is square integrable modulo, the center with central character x, if
there exist v, w e H - {0} such that cvwe L2(G;x)- The analogue of
Lemma 1.3.3 is true in this context.
The orthogonality relations (Proposition 1.3.3) also have an analogue. Here
(jt, //) and (a, V) should be taken to have the same central character and the
integration should be over Z\G. The proofs are essentially the same as those
of Section 1.3.
1.8.3. As we indicated in Section 1.5, the notion of induced representation
that we introduced is a special case of a more general theory. The interested
reader should consult Warner [1], Chapter 5 for a comprehensive account of
induced representations of Lie groups and for a complete set of references to
the vast literature.

40
1. Elementary Representation Theory
1.8.4. We now use the notation of Section 1.6. Dixmier-Malliavan [1] have
proved that if (it, H) is (say) a Hilbert representation of G then H™ is the span
of the spaces it(f)H with / a smooth compactly supported function on G.
This result allows one to give a simple proof of the following result.
Theorem. Let P and K be closed subgroups of G with K compact. Let (a, W) be
a Hilbert representation of P and let (H^Y' denote the space of all smooth
elements of (Ha)0 with the topology of uniform convergence on compacta with
all derivatives. Then (ita,(Ha)'x') is a smooth Frechet representation of G that
is equivalent to (it, H™).
1.8.5. The Verma modules (1.7.4(ii)) will be studied in more detail in
Chapters 4,6 and 9. There is a vast literature on this subject. The best reference
is Dixmier [1], Chapter 7.

2 Real Reductive Groups
Introduction
In this chapter we introduce the class of real Lie groups that we will be
studying throughout this book. The definition of a real reductive group that
we give in Section 1 can be shown to be the same as that in Borel, Wallach [1,
0.3.1] if we add the condition of inner type. We have opted to give the more
cumbersome definition since it allows an extremely elementary entry into the
fine structure of these groups. We hope that the experts will not become too
impatient with our presentation of the material. To the less expert reader we
wish to issue a warning about some of the proofs in this chapter. Although, at
first sight, they seem to be complete (indeed, possibly over-detailed) there are
many points that have been left to the reader. Also the examples in this chapter
should really be looked upon as exercises.
The first section of this chapter gives the definition and basic structure of
real reductive groups. It contains the Cartan and Iwasawa decompositions of
these groups. The second section is, perhaps, the most important section of
this chapter. It introduces the notion of parabolic subgroup and of parabolic
pair. The theory of parabolic subgroups makes the harmonic analysis on real
reductive groups tractable, since it reduces many problems on a real reductive
group to corresponding problems on the Levi factors of these subgroups. In
41

42
2. Real Reductive Groups
Section 3 we show how the theory of parabolic subgroups can be used to study
Cartan subgroups of real reductive groups. The relationship between Cartan
subgroups and cuspidal parabolic subgroups is one of the basic ingredients in
Harish-Chandra's Plancherel formula. Section 4 contains integration
formulas associated to various decompositions of real reductive groups that are
consequences of the results in earlier sections. In the final section of this
chapter we show how to use the Weyl integral formula to derive the Weyl
character formula. We include this material since it contains many of the ideas
that will be used in our exposition of the theory of square integrable
representations of real reductive groups.
2.1. The definition of a real reductive group
2.1.1. Let F = R or C. Let (as usual) M„(F) denote the space of all n x n
matrices over F. Let GL(n, F) denote (as usual) the group of all invertible
elements of M„(F). Let ft,... ,fm be complex polynomials on M„(C) such that
each fj is real valued on M„(R) and such that the set of simultaneous zeros of
the fj in GL(n, C) is a subgroup, Gc, of GL(n, C). Then Gc is called an affine
algebraic group defined over R. The subgroup, GR = Gcn GL(n, R) is called the
group of real points. If in addition, g* e Gc for g e Gc then Gc is called a
symmetric subgroup of GL(n, C). We define an automorphism 6 of GR by
6(g) = (g~1)*.
Let Gc be a symmetric subgroup of GL(n, C) with real points GR. By a real
reductive group we will mean a finite covering, G, of an open subgroup G0
of GR.
Thus the statement "G is a real reductive group" carries with it all of the
above data. We will also write p for the covering homomorphism from G onto
G0. We will identify the Lie algebra of G with that of GR. Thus we can define on
g, the Lie algebra of G, an involutive automorphism, 6, given by 6{X) = — X*.
This automorphism is usually called a Cartan involution.
2.1.2. Examples
1. GL(n, R). GL(n, R) is clearly a real reductive group.
2. SL(n, R). Let SL(n, F) be the subgroup of GL(n, F) consisting of all g with
det(g) = 1. Then all of the hypotheses are satisfied by SL(n, R).
3. GL(n, C). Here we look upon C" as R2" and multiplication by i denoted by J.
Then GL(n, C) is the subgroup of GL(2n, R) given by the equations gj —
Jg = 0. We can choose the identification of C" with R2" so that J* = —J.
Thus the conditions in the definition are satisfied.

2.1. The Definition of a Real Reductive Group
43
4. SL(n,C). SL(n,C) = {g e M„(C)|det g = 1}. We leave it to the reader to
show that SL(n, C) is a real reductive group.
5. 0(p, q). Let p and q be non-negative integers with p + q = n > 0. We look
upon R" as the direct sum of Rp and R". Let Ipq be the operator on R" given by
/ on Rp and —/on R". Then 0(p,q) is given by the equations glp,qg* = ip,r
Clearly, O(n,0) = O(0, n) is compact. We write 0(n) for O(n,0).
6. SO(p,q). SO(p,q) = 0(p,q) n SL(n, R). We write SO(n) for SO(n,0).
1. U(p, q). We look upon C + q as the direct sum of C and C. These in turn
we identify with real vector spaces of twice the dimension as in Example 3.
Then U(p,q) = GL(n,C) n 0(2p,2q)(n = p + q). We write U(n) for U(n,0).
8. SU(p, q). SU(p, q) = U(p, q) n SL(n, C). We write SU(n) for SU(n, 0).
9. Sp(n, R). We take J on R2" as in Example 3. Then Sp(n, R) is given by the
equations gJg* - J = 0.
2.1.3. The above list just gives some of the so-called classical groups over R.
We now give a general "example". In the proof of the next Lemma we will use
several standard concepts that we have not yet denned. The point of this
lemma is to reassure the experts that our concept of real reductive group is the
"usual one".
Lemma. A connected semi-simple Lie group with finite center is a real
reductive group.
Let G be as in the statement. Let g be the Lie algebra of G. Let 8 be a Cartan
involution of g. Then if B is the Killing form of g, the form <X, Y> =
- B(X, BY) is an inner product. Let AT,,..., X„ be an orthonormal basis of g
relative to < , >. We use this basis to look upon g as R". Let Gc be the
automorphism group of gc. If g e Gc then
g* = 0 conj g~' conj 8
(here conj is complex conjugation in gc relative to g). Also GR = Aut(g). Set
G0 = (GR)°. Then Ad is a covering homomorphism of G onto G0- Thus all
the conditions are satisfied.
2.1.4. Let G be a real reductive group with Lie algebra g. We assume all of
the data in 2.1.1. Let B(X, Y) = tr XY for X, Y in g. If X e g then 6(X) =
— X* e g. {X, y> = -B(X,9Y) defines an inner product on g. Hence B is
non-degenerate. Set f equal to the + 1 eigenspace for 6 in g and set p equal to
the -1 eigenspace of 6 in g. Then the decomposition g = f © p is called a
Cartan decomposition of g. One has:
(1) f is the Lie algebra of a compact subgroup of G.

44
2. Real Reductive Groups
Indeed, I is the Lie algebra of GR intersected with the orthogonal group
of< , >.
(2) [I, p] <= p and [p, p] <= I.
This is clear from the definitions.
We set g„ = I © ip. Then, if we denote by B the complex bilinear extension
of B to gc (still given by the formula tr XY), B is negative non-degenerate on
g„. Clearly, (g„)c = gc. g„ is called a compact form of g. The argument in 0.3.1
implies that gc is reductive. We have shown
(3) The Lie algebra of a real reductive group is reductive.
2.1.5. We now study the global structure of a real reductive group, G. We
first look at G0. Set
K0={ge G0 \0(g) = g}.
Then K0= G0n 0(n). Hence it is compact.
Lemma. The map K0 x p -> G0 given by k, X i—► k exp X is a surjective
diffeomorphism.
Lett/ e G0.Then(/ = u exp X with u orthogonal and X self adjoint (2.A, 1.1)
Clearly, gg* = exp(2X). Let f} be as in 2.1.1. Then /j(exp 2mX) = 0 for
m= 1,2,.... Hence f}(e\ptX) = 0 for all t e R(2.A.1.2). Thus X e g. Since
X = X* = -6X, X e p. But then ue G0. Hence ueX0. We therefore see
that the map in the statement is surjective. That it is a diffeomorphism now
follows from 2.A.4.
We note that as immediate consequences of the above lemma we have:
Corollary 1. 6(G0) = G0.
Corollary 2. G0/K0 is connected and simply connected.
2.1.6. Let a be a subspace of p that is maximal subject to the condition that it
is an abelian subalgebra of g. If H e a then since H is self-adjoint H is
diagonalizable. Thus ad H is diagonalizable. That is, a acts semi-simply on g ■
under ad. If fie a* we set g* = {X e g! [ff.X] = n(H)X for He a}. Set
<D(g,a)= {lzea*i/z*0andg''*0}.
We note that 6 is —I on a. Hence, g° is ^-invariant. Thus g° =
t n g° ® p n g°. Now, p n g° = a by the choice of a. We set °m = I n g°.

2.1. The Definition of a Real Reductive Group
45
Then
(1) fl = °m@a@0g".
Let a'= {Hea |/z(H)^0,|ze«(fl, a). Fix H0eo'. Put P={/ze«(8, a) i/z(H0)>
0}. Set
n = 0 fl" and n = 0rt.
lie P
Then n and n are subalgebras of g and we have:
(2) 3 = n © °m © a ® rt.
The following decomposition of g is called the Iwasawa decomposition:
(3) g = f © a © rt.
To prove this we set q = (/ + 0)12. Then q(%) = f. Since q(n) = q(n) and
q{a) = 0 it follows that q(n ® °m) = f. Thus it is clear that g is the sum of the
spaces in (3). Counting dimensions using (2) proves that the sum is direct.
2.1.7. We now give the Iwasawa decomposition of G0- Let Nt and let Al be
respectively the connected subgroups of G0 corresponding to rt and a.
Lemma. The map -4, xN, xK0-tG0 given by a, n, k i—► nak is a surjective
diffeomorphism.
Since each H e a is self-adjoint and a is abelian, exp is a surjective
diffeomorphism of a onto Al{2.AAA). Let H0 be as above. Let v,,..., vn be an
orthonormal basis of R" consisting of eigenvectors for H0 with eigenvalues fij
in decreasing order. Then the definition of rt implies that the elements of n are
upper triangular with 0's along the main diagonal relative to the basis {u,-}.
(1) ^i^i is closed in Go ■
Indeed, let a^n, converge to g e G0. Since the diagonal entries of a^ are
those of ap aj converges to a e GL(n, R). Hence n, converges tone GL(n, R).
Since GR is closed in GL(n,R), a, n e GR. a is clearly self-adjoint hence a =
exp X with X e p. Since Ad(a) restricted to a is / and this is a polynomial
condition 2.A.2 implies that X e a. n is upper triangular with l's on the main
diagonal son = exp Y with Y nilpotent (2.A. 1.5). Now J^(exp t Y) is a
polynomial in t which vanishes at all integral t. Thus J^(exp tY) = 0 for all real t. So
Ysq. 2.1.6(2) and the shape of Y imply (1).

46
2. Real Reductive Groups
Let q denote the map in the proposition. If H, X, Y are in a, n, and I
respectively then
dqa^k(H,X, Y) = aHnk + anXk + ankY.
Thus if dqa^k{H,X, Y) = 0 then
0 = n~1Hn + X +kYk~\
Since n~lHn + X is in a + n and kYk1 is in I, Y = 0. But then H +
nXn~l = 0. Hence H = X = 0. It follows that q is everywhere regular. Hence
the image of q is open in G0. Since Kx is compact, (1) implies that the image
of q is closed. Hence q is surjective. To complete the proof we need only show
that q is injective. If ank = a'n'k' then
(aVr'an=/c'/r'.
The matrix expressions relative to the above basis imply that k = k' and
a = a'. Thus n = n'. The proof is now complete.
2.1.8. We now transfer everything to G. Let p be the covering homomor-
phism of G onto G0. Set K = p_1(X0). Let A and iV be respectively the
connected subgroups of G corresponding to a and n. We have
Theorem. (1) The map p x K -> G (//yen by X, /c i—► exp X/c is a surjective
diffeomorphism.
(2) The map A x N x K -> G (/f'yen by a, n, k*—> ank is a surjective
diffeomorphism.
That the above maps are surjective and everywhere regular follows
immediately from the above results. Thus we need only show that the maps are
injective. We first prove (1).
If e\p(X)k = exp(Xl)kl then applying p to both sides we see that exp X =
exp^) in light of Lemma 2.1.5. Hence X = Xx and thus k = kl.
We now prove (2). For this we first observe that
(a) p is a Lie group isomorphism of A onto Ax and of N onto A^.
Indeed, p is obviously a covering map. Since Ax is simply connected (a) is
obvious for A. Let H0 be as above and set a, = exp tH0. Then
lim Ad(ar)A: = 0
»-> -00
for X e n. Since A^ is generated by exp n, this implies that A^ is contractible
(use a, exp(A')ar~1 = exp(Ad(ar)Ar). Thus (a) is also true for N.

2.1. The Definition of a Real Reductive Group
47
We now prove (2). If a,n,/c, = ank then p(a,) = p{a), p(n,) = p(n) and
p(/ci) = p(/c). Thus (a) implies that a = a,,n = n1so/c = /c1.
The decomposition in (1) of G is called a Cartan decomposition of G. The
decomposition in (2) is called an Iwasawa decomposition of G.
2.1.9. The above results depend on a choice of a and a choice of P. We now
study the extent of this dependence.
Lemma. Let a t and a2 be maximal abelian subalgebras of p. Then there exists
ue K0 such that Ad^X^) = a2.
Let Hl and H2 be to a! and a2 as H0 is to a as above. Then
{Xep\lX,Hj] = 0} = aj
for ; = 1, 2. Set f(k) = B(Ad(k)HuH2) for k e K0. Since K0 is compact /
attains a minimum at u. If V e f then
0 = d/A|, = oB(Ad((expty)M)H1,H2)
= B([y,Ad(M)H,],H2) = BiyEAd^)//,,//,).
Since 7 is arbitrary and [p, p] is contained in I this implies that
\_Ad(u)Hl,H2] = 0. Hence Ad(u)a, is contained in a2. Thus Ad(u)a! == a2
since at is maximal abelian in p.
This argument is due to G. Hunt [1].
2.1.10. We return to the notation in the paragraphs preceding 2.1.9. Let
H e P. Let X e g" be such that <A\ X) = 1. Then
(1) [X,0X] = -ff,,.
Here H„ e a is denned by B(H, HM) = ^(H) for all H e a. Thus if x =
(2//z(ffM))X, y = -SAT, /i = (2/|z(fy)ff„. Then x, y, /i spans a TDS (0.5.4)
over R. There is thus a Lie homomorphism, a, of SL(2, R) into G0 such that
a(y*) — (0o(y))"'■ Let /c be the image of
0 T
_-l 0
under a. Then it is easily checked that if sM is denned by s„H = H — B(h, H)H
for H e a then Ad(/c)H = s„H.
Let N(a)= {ue X°! Ad(u)a = a}. Set W(Q,a) = {Ad{u)\a\ue N{a)}. Then
(2)
sM e VF(g, a) for all jjeP.

48
2. Real Reductive Groups
This follows from the above observations.
Let a' be the set of all H e a such that n(H) is non-zero for all p. e <t>. A
connected component of a' is called a Weyl chamber of a. If C is a Weyl
chamber then the set of all fi e <t> such that ^ is positive on C, denoted Pc, is
called a system of positive roots. If pe P and if fi cannot be written as a sum of
two elements of P then \i is called simple in P. The argument in Jacobson [1,
Thm 1, p.241] proves:
Proposition. (1) W(q, a) is generated by the s^ for p simple in P.
(2) W(q, a) acts simply transitively on the Weyl chambers of a.
2.2. Parabolic pairs
2.2.1. Let G be a real reductive group with Lie algebra g. Let 8 be a Cartan
involution of g and let g = I © p be the corresponding Cartan decomposition
of g. Let 3(g) = 3 be the center of g. Then 6(3) = 3. Hence 3 = fn3©pn3. We
set s = p n 3. Then s is called a standard split component of g. Here standard is
relative to a choice of 0. Set G+ = {g e G I Ad(t/)|3 = /}. Then it is easily seen
that G+ is a real reductive group in our sense.
2.2.2. We set X(G) equal to the set of all continuous homomorphisms of G
into the multiplicative group, R*, of non-zero real numbers. We set
°G = {geG\x(g)2 = \ for all / e X(G)}.
Put S(G) = S = exp(s). Then S is called a standard split component of G. We
have
Lemma. The map S x °G+ -> G+ given by s, g 1—► sg is a surjective Lie group
isomorphism.
Let us assume that G = G+. Let G = NAK be an Iwasawa decomposition
of G. Set °a = an [g, g]. Put °A = exp °a. We assert that
(1) °G = °ANK.
Indeed, rt and °a are contained in [g, g] hence N and °A are contained in °G.
Since K is compact it is also clear that K is a subgroup of °G. Thus the right
hand side of (1) is contained in the left hand side. Let G° be the identity
component of G. Let °g denote the orthogonal complement to s in g relative
to B. Let G' be the connected subgroup of G with Lie algebra °g. It is easily

2.2. Parabolic Pairs
49
seen that the center of G1 is contained in K. It is therefore not hard to see
that
(*) The map S x G1 ^G° given by s, g i—► sg is a surjective Lie group
isomorphism.
We also note that the Cartan decomposition of G implies if g is in G then
there is k e K such that gk is in G°.
If n e s* we set x„(exp Xt/) = exp(^(X)) for X e s, # e G1. Then (*) implies
that x„ e X(G°). We extend *„ to G by setting x„(gk) = x» for pG° and
k e K. Then it is easy to see that each x^ e X(G). Clearly
f] Kerx» = °ANK.
This implies that the left hand side of (1) is contained in the right hand side.
Since G = G+ (by our assumption) and A = S x °A as Lie groups, the
proposition follows from (1) and the Iwasawa decomposition.
2.2.3. Examples
(1) Let G = GL(n,R). Then G = G+, s = R/, S = {all a > 0} and °G =
{geG\det(g)2=l}.
(2) LetG = GL(n,C). G = G+, S is as in (1), °G = {g e G \det(g)\ = 1}.
For all of the other examples in 2.1.2, G = G+ = °G. We give one more
example.
(3) G = GSp(n, R). Let J e GL(2n, R) be as in 2.1.2 Example 3. Then GSp(n, R)
is the subgroup of all g e Gl(2n, R) such that gjg* is a scalar multiple of J. We
leave it to the reader to check that GSp(n, R) is a real reductive group. Then S is
as in the above examples. G = G+.°G = {g e G \gJg* = ±J}-
2.2.4. Leta,<D = <D(g,a), etc. be as in 2.1.6. Let m = {X e g \\_X,a] = 0}. Set
M equal to the set of all g e G such that Ad(g) is / on a. This is clearly an
algebraic condition that is invariant under taking adjoints. Thus M is a real
reductive group. The standard split component of M is A since m = a ® °m
(2.1.61 It is also clear that °M = M nK.
2.2.5. Let t be a maximal abelian subalgebra of °m. Set h0 = t © a. Set h
equal to the complexification of h0.
(1) h is a Cartan subalgebra (0.2.1) of #c.
If X e g and if X commutes the elements of h then so does 8X. Hence
X = U + V with U e I, V e p and both U and V commute with the elements

50
2. Real Reductive Groups
of h. But then V must be in a and U must be in °m. Hence V must be in t. We
have therefore shown that h is maximal abelian in g. If X is in t or a then X acts
semi-simply on gc. Thus the elements of h act semi-simply on gc. So h is a
Cartan subalgebra of gc.
Let <t>(gc, h) be (as usual) the root system of gc relative to h. It is obvious that
(2) cD(g,a) = (D(gc,h)|a-{0}.
Since the elements of <I>(gc, h) are real valued on a and take pure imaginary
values on t it follows that (see 0.2.2)
(3) bR = (it + a)n[gc,gc].
Let //, be an element of a' n hR. Let ffl5..., Hr be a basis of hR. We order
<t>(gc, h) lexicographically relative to this basis. Let R denote the
corresponding positive root system (0.2.4). Let R0 be the set of all p e <t>(g, a) such that
fi(Hi) > 0. Then R0 is a system of positive roots for <t>(g, a) (2.1.10). Then it is
clear that R\a — {0} = R0. Let A (resp. A0) be the corresponding system of
simple roots for R(resp. R0) (0.2.4, 2.1.10). Set F0 = {aeA|a|a = 0}. Then
(4) (A-F0)|a = A0.
Indeed, if p e A0 and if a e R restricts to n then a = ft + ■ ■ ■ + ft with ft
simple in R. Only one of the ft can have a non-zero restriction to a since p. is
simple in R0. This implies (4).
(5) A0 is a linearly independent subset of a*.
Indeed,if \i e h*setconj(^)(H) = conj(^(conj(//)))for// e h(hereconj(A')is
conjugation in gc relative to g). If a e R then its restriction to a is given by
(a + conj(a))/2. Let A - F0 = {a,,..., ar}. Then there is a permutation ;' i—► /
of 1,..., r such that conj(ay) = af + ZaeFo n^ix. (5) follows easily from this.
2.2.6. Let F be a subset of A0. We set aF = {// e a \ n(H) = 0 for p. e F}. Set
mF = {X e g! IX, aF] = {0}}. Put MF = {g e G | Ad(g)H = HforH e aF} and
AF = exp aF. Then MF is a real reductive group with (MF)+ = MF = 8(MF).
Also relative to 8 the split component of MF is AF.
Let RF be the subset of those roots in R0 whose restriction to aF is non-zero.
Set
nF = © 9"-
pieR
Let NF denote the connected subgroup of G with Lie algebra nF.

2.2. Parabolic Pairs
51
Lemma. (1) nFis anilpotent Lie subalgebraof g.
(2) If X enF then ad X is nilpotent on g.
Let H eaF be such that fi{H) > 1 for all p. e A0 - F. Set n = nF. Put
n, = [n,n] and nJ+! = [n,-,n,-]. Then, recall that n is nilpotent if nk = {0}
for k large. Since ad H has all of its eigenvalues greater than or equal to j on
rtj there must be an index such that n^ = {0}.
Let c be the lowest eigenvalue of ad H on g. Then (ad X)mQ is contained
in the sum of the eigenspaces of ad H with eigenvalue at least c + m. This
implies (2).
2.2.7. Set PF = MFNF. Then PF is called a standard parabolic subgroup of G.
The word standard has to do with the choices of a and R0. The pair (PF, AF)
will be called a parabolic pair (p-pair for short). Lemma 2.2.2 implies that under
the multiplication mapping MF is isomorphic with AF x °MF. We have
Lemma. (1) The map MF x NF->PF given by m, n^mn is a surjective
diffeomorphism.
(2) The map °MF x AF x NF -> PF given by m, a, nt—> man is a surjective
diffeomorphism.
It is enough to prove (1) since (1) combined with Lemma 2.2.2 implies (2).
It is an easy calculation to see that the differential of the map in (1) is
everywhere regular. Thus (1) will follow if we show that the map is injective.
So suppose that m, mx e MF and n, nx e NF and that mn = mlnl. Then
mlm~[ = n(niyl. Hence we must show that MFnNF = {1}. Let H be as
in the proof of Lemma 2.2.6 and set a, = exp tH. Then lim atna_, = 1
r->-oo
for all neNF. Since the a, are central in MF, this clearly implies that
MFnNF = {1}. So the lemma follows.
The decomposition in (2) is called a Langlands decomposition of PF. P0 is
called a minimal parabolic subgroup of G. The PF are standard relative to P0.
2.2.8. We say that a real reductive group is of inner type if Ad(G) is a
subgroup of Int(gc).
Lemma. Let Gbea real reductive group of inner type. Let (PF, AF) be as above.
(1) Mf is a real reductive group of inner type.
(2) K°PF = G.

52
2. Real Reductive Groups
Set KF = Kn MF. Then Theorem 2.1.8 implies that MF = KF(MFf. Thus it
is enough to show that Ad(XF) is contained in Int((mF)c). Set *aF = an °mF.
Let k e KF. Then there exists u e (KF)° such that Ad(u)*aF = Ad(/c)*aF. Thus
we may assume that Ad(/c) stabilizes *aF. But then Proposition 2.1.10 implies
that we may assume that Ad(/c) restricted to *aF is the identity. That is, we may
assume that k e °M. Let t be as in 2.2.5. Then if we argue as above we may
assume that Ad(/c) is the identity on t. Let g„ = I © ip. Then the connected
subgroup, G„, of Int(gc) corresponding to g„ is compact by Theorem 0.3.1.
t © ia is the Lie algebra of a maximal torus, T„ of G„. Hence, 0.3.3(2) implies
that Ad(/c) e T„ which is a subgroup of Int(gc). This proves (1).
In the first part of this proof we have shown that K = °M0K°. Thus
K°P0 = G which proves (2).
2.2.9. GL(n, R). g = M„(R). We take a to be the diagonal matrices. Let Eu
be the matrix with 1 in the i,; position and 0's everywhere else. If H is the
diagonal matrix with hh..., hn on the main diagonal we set Sj(H) = h}. Then
<D(g,a) = (e,-£J- i^j}. We take R0 = {r,t-Sj\i<j}. A0 = k,-k2, fi2-e3,...,
e„_! — e„. If m!,..., mp are positive integers adding up to n then we set
P(m,,..., mp) equal to the subgroup of all matrices in the following block
form: First we write every matrix in the form \_At ^] with ALJ an m; by m,
matrix. Then the form of the elements of P(ml,..., mp) is A-Ui = 0 for i > j.
This describes all standard parabolic subgroups of GL(n, R). We leave it to the
reader to find which subset of A0 corresponds to mu..., mp.
2.2.10. We now give a proof of an important Theorem that is usually known
as the Bruhat Lemma. This result was first proved by Bruhat for the classical
groups. The general result is due to Harish-Chandra [4]. We will follow
Harish-Chandra's original argument.
Fix (P0,A0) = (P,A), a minimal p-pair. Let NG(A) = {g e G \Ad(g)a = a}.
Set W(G,A)= NG(A)/M. We look upon W(G,A) as a group of linear
automorphisms of a. We leave it to the reader to prove that
(1) If G is of inner type then W(G, A) = W(q, a).
Use W(G, A) then we can choose s* e K such that k e s. We fix such a
choice for each s e W(G, A).
Theorem. Assume that G is of inner type. Then G is the disjoint union of the
setsPs*P,se W(G,A).

2.2. Parabolic Pairs
53
We have seen that G = K P. Thus to prove that G is the union of the asserted
subsets of G, it is enough to prove that if k e K then k e Ps*P for an
appropriate s e W = W(G,A). Fix k e K.
(2) p (the Lie algebra of P) is the sum of Ad(/c)p n p and n.
Recall that (X, y> = -B{X,6Y) defines an Ad(X)-invariant inner
product on g. Relative to this inner product p1 = On. Thus (p + Ad(/c)p)x =
0(n n Ad(/c)n) = 0(n n (ad(/c)p n p)). It is also clear that
dim(p + Ad(/c)p) + dim(p + Ad(/c)p)1 = dim g.
We therefore have
dim((p n Ad(/c)p) + n)
= dim(p n Ad(/c)p) + dim rt - dim((Ad(/c)p n p) n n)
= dim(p n Ad(/c)p) + dim rt - dim(p + Ad(Zc)p)1
= dim(p n Ad(/c)p) + dim rt - dim g + dim(p + Ad(/c)p)
= dim p + dim Ad(/c)p + dim n - dim g = 2 dim p + dim n - dim g.
Since dim n = dim On and dim p + dim On = dim g, the above equations
imply that dim((Ad(/c)p n p) + rt) = dim p. Ad(/c)p n p is a subspace of p thus
(2) follows.
(3) If X e p and if ad X has real eigenvalues (as an endomorphism of g) then
X e 3(g) + a + n.
Let H and a, be as in the proof of Lemma 2.2.6 for F = 0. Then
Ad(a,)y = Y for rem and limr_ r Ad(a,)X = 0 for X e rt. If x e g then
ad(Ad(a,)x) = Ad(a()ad x Ad(a,)_1. So ad x and ad(Ad(a()x) have the same
eigenvalues. Assume that X e p and that ad X has real eigenvalues. Then
X = Y + Z with Y em and Z e rt. If we take the limit to - oo of Ad{at)X
then we see that Y has real eigenvalues. Now Y = U + h with U e °rrt and
he a. Since ad h has real eigenvalues, this implies that ad U has real
eigenvalues. The elements of ad(°rrt) have purely imaginary eigenvalues. Hence
ad U = 0. This proves (3).
(4) If h e a' then Ad(/V)/i = h + n.
If AT en then eadXh = h + Z.j>0 (ad X)Jh/j\ e h + n. If X e n then set
5{X) = Ad(exp X)h - h. Then dd0(X) = IX,h] for X en. This implies that

54
2. Real Reductive Groups
there is an open neighborhood of 0 in n such that d(U) is an open
neighborhood of 0 in n. We now prove (4). Let X en then there exists t > 0 such that
Ad(a_()X e 5(U) (see the proof of (3)). Thus X e Ad(a,)d(U) = d(Ad(a,)U).
Hence 5 is surjective, which is the content of (4).
Let h e a'. Then (2) implies that there exists X en such that h + X e
Ad(/c)p n p. (4) says that there exists ne N such that h + X = Ad(n)h. This
implies that there exists yep such that Ad(n)h = Ad(k)y. In particular this
equation implies that ad y has real eigenvalues. Thus, (3) implies that y =
z + hi + u with z e 3(g), h^ea and u e n. As above ad(y) has the same
eigenvalues as ad^). Thus hx e a'. So (3) implies that y = Ad^Xz + h^) for
some «! e N.
Recall that g is identified with the Lie algebra of GR. Thus if g e G and if
x e g then x and Ad(g)x have the same eigenvalues, set g = n~lknx. Then
Ad(g)(z + h^ = h. Thus (if we compare eigenvalues) z e a. Thus we may use
the notation /i, for z + li,. m = ker(ad h) = Ker(ad hy). Thus Ad(g)m = m.
Since Ad g preserves eigenvalues, Ad(g)a = a. Thus there exists seW such
that g e s*M. But then k e Ns*P.
To complete the proof we must show that if t, s e W and if p, pl e P and
if ps* = t*Pi then s = t. Let h e a then Ad(pi)h = h + X, Ad(p)sh = sh + Xx
with X, Xi e n. Thus th + Ad(t*)X1 = sh + X. Ad(t*)X1 = U + V with
U en and V e 8n. Thus sh + X^ — U = th + V. Thus sh = th. Since h is
arbitrary in a, this implies that s = t.
2.2.11. We will now apply this result to prove the so-called Gelfand-Naimark
decomposition (which first was proved for general groups and minimal
parabolics by Moore [1]). Let F be a subset of A0 and let (PF,AF) be the
corresponding p-pair. Let PF = MFNF as usual.
Corollary. Assume that G = G + . The map of 6(NF) x PF to G given by x,
pi—>xp defines a diffeomorphism onto an open subset of G whose complement has
measure 0 relative to dg.
We first prove the result in the case when F = 0. We use the notation
of the last number. We observe that W is a finite set. Indeed, the elements
of W permute the roots and are completely determined by the
corresponding permutation. If s e W then Ad(s*)n = £a>0 gs*. Thus Ad(s*)n =
(Ad(s*)n) n n + (Ad(s*)n) n On. Let Us (resp. Vs) be the connected subgroup
of G with Lie algebra (Ad(s*)n) n n (resp. (Ad(s*)n) n On).
(1) s*N(s*)~1 = VSUS.

2.2. Parabolic Pairs
55
Let y(v,u) = vu for veVs, ue Us. Then dyul(X, Y) = X + Y for X e os,
yeus. Thus the image of y contains an open neighborhood of 1 in
s*N(s*)~[. Fix H e a such that sa(ff) < 0 for all positive roots, a. Set
a, = exp tH. If x€s*JV(s*)_l then lim,^ atxa_t = 1. Since, a,I^a_( = 1^
and a(l/sa_, = l/s our usual argument now implies (1).
(2) If x e ON, p e P set /?(x, p) = xp. Then /? is a diffeomorphism onto an open
subset of G.
We first assume that G = G0- Then d/?^*, 7) = xXp + xpY for AT e 0n,
7 e p. If this expression is 0 then X = —pYp'1. The right hand side of this
equation is in p and the left hand side is in On. Since these two spaces have
0 intersection, this implies that /? is everywhere regular. If /?(x,p) = j8(x,,p,)
then (x1)~1x = pxp~l. Let H, a, be as above for s = 1. If yeON and
lim(^_00 atya^t exists then it is easy to see that y = 1. On the other hand,
it is easy to see that lim(^_ x a,qa_, exists if qeP. So l=(x,)"'x =
PiP1. Thus x = x, and p = p,. So /? defines a diffeomorphism onto an
open subset of G0.
Let G now be arbitrary (subject to our hypotheses). Let ji also denote the
corresponding mapping for G. Let q be the covering homomorphism of G
onto G0- Then qji is everywhere regular by the above. Since the center, Z, of
G is contained in P and ZnON = 0, it is not hard to see that ji is a
diffeomorphism onto an open subset of G.
Fix t e W such that ta is negative for all positive roots a. Then G is the
disjoint union of the sets
(t*)-1P(fs)*P, seW.
Now P(ts)*P = N(ts)*P = (ts)*((ts)*y[)N(ts)*P = (ts)*VtsP (by (1)). (2)
implies that Vts(ts)*P is a submanifold of G of dimension dim Vts + dim P. Thus,
if Vts is not equal to ON then (f*)~'P(ts)*P is a submanifold of G of lower
dimension. Hence up to a set of measure 0, G is the union of the sets
(t*y1(ts)*(0N)P, V,s = 0N.
If Vs = ON then t~1s preserves the Weyl chamber. Thus t~ls = 1. So s = t.
Thus if Vts = ON then s = 1. The corollary now follows in this case.
Now let F be arbitrary. It is clear that 0(NF)PF => O(N0)P0. Hence,
0(NF)PF has total measure in G. Let j8f(x,p) = xp for x e 0(NF), p e PF. Since
g = 0(n) © p, the argument used to prove (2) shows, in this case, that jiF
is everywhere regular. If we use H e aF such that a(H) > 0 for a e <b(PF,AF)
and argue as we did for /? we find that fiF is injective.

56
2. Real Reductive Groups
2.3. Cartan subgroups
2.3.1. Let G be a real reductive group. Then a Cartan subalgebra g is a Lie
subalgebra, h such that hc is a Cartan subalgebra of gc. We define the
polynomials D} on g by
det(t/ - ad X) = £ tjDj(X).
Let / be the dimension of a Cartan subalgebra of gc. Then using the
theory of complex reductive Lie algebras (0.2.1) one sees that Dk = 0 for
k < I. We set D = £),. Then D is a non-zero polynomial function on g. Set
g' = {X e g D(X) + 0}. Then g' is open and dense in g. Let Int(g) denote the
group of automorphisms of g generated by the automorphisms of the form
exp(ad X) for X e g. As is well known Int(g) = Aut(g)0. If X e g is such that
ad X is a semi-simple endomorphism of gc then we say that X is semi-simple.
Lemma. (1) // X eg' then X is semi-simple and Cg(X) = {Y e g | [A', Y~\ = 0}
is a Cartan subalgebra of g.
(2) If X is a semi-simple element of g then Ca(X) is a reductive subalgebra of g
that contains a Cartan subalgebra.
(1) is an immediate consequence of 0.2.1. We now prove (2). Let u = CS(X).
Let Vc be the sum of the eigenspaces with non-zero eigenvalue for ad X in gc.
Set V = Vc n g. Then g = u © V. Let q be the map of V x u to g given by
q(y, x) = exp(ad y)x. Then
<ko,z(y>x) = adyZ + x.
Thus dqoz is surjective. The inverse function theorem implies that there are
open neighborhoods U of X in u and W of 0 in V such that q(W, V) is open
in g. Hence, if D is identically 0 on u then D is zero on g. Since this is contrary
to our assumptions we see that g' n u is non-empty. Hence (1) implies that u
contains a Cartan subalgebra h of g. Let h be the complexification of h0. Set
(D = <D(gc, b). Let <D0 = {a e <D | a(X) = 0}. Then it is clear that
Uc=D©©(9c)a-
ae<D
Let a be an abelian ideal in uc. Then, in particular, a is invariant under ad h.
Let Q be the set of all roots, a, such that (gc)a is contained in a. One has
a = bna©©(gc)a.
xeQ
If a e Q then, in particular, a(A') = 0. Thus, since a is an ideal in uc, it is a

2.3. Cartan Subgroups
57
simple matter to see that — a e Q. Since a is abelian, this implies that 2 = 0.
Thus a is central in uc. This implies that u is reductive.
2.3.2. Fix a Cartan involution 0, of g. Let B be a non-degenerate 8 and g
invariant form on g such that (X, Y> = — B(X, 6Y) defines an inner product
on g. We say that 6 is associated with B.
Lemma. // 8X is another Cartan involution of g that is associated with B then
there exists x e Int(g) so that x6x~' = 8X.
Set N = 99i. Then our assumptions imply that (NX, Y} = (X,NY}
for all X, Y e g. Thus N2 = exp W with W a self-adjoint endomorphism
of g. The condition that exp W is an automorphism of g is a polynomial
condition. Thus, since exp(mVF) is an automorphism for all integral m,
exp tW is an automorphism for all te R (2.A. 1.2). But then W is a
derivation of g. Hence W = ad X for some X e g. Since 0,^0, = N~[, 2.A. 1.2
implies that 0, exp tXOi = exp( — tX) for all t e R. We therefore see that if x =
exp(-(l/4)ad X) then x8x~[ commutes with 0,. Since both 6 and 0! are
Cartan involutions associated with B this implies that 8X = xOx'1.
The above argument is due to Mostow [1].
2.3.3. Lemma. Let I) be a Cartan subalgebra of g. Then there exists x e
Int(q) such that xh is 6-invariant.
We may assume that g is semi-simple since every Cartan subalgebra
contains the center. Let u be a compact form of gc such that u n hc is a
maximal abelian subalgebra of u (0.3.4). Let y denote conjugation on gc
relative to u and let a denote conjugation on gc relative to g. Let B denote the
Killing form of gc.Set(.Y, Y) = -B(X,yY)for X, yegc.Then( , ) is an
inner product on gc. Set N = ay. Then (X,NY) = (NX, Y) for all X, Ye gc.
Thus if we argue as in the proof of Lemma 2.3.2 we see that N2 = exp(ad X)
with X e iu. We note that aNo = AT1 and yNy = N~l. Hence the usual
argument shows that y exp t ad Xy = exp( — t ad X) and a exp t ad Xa =
exp(-1 ad X). From this it is easy to deduce that if y = exp((l/4)ad X) then
^ = yyy~' commutes with a. The restriction to g of n is a Cartan involution of
g, Ql, associated with B. Lemma 2.3.2 implies that there exists z e Int(g) such
that 8{ = z8z~'. Then x = zy is the desired element of Int(g).
2.3.4. Let h be a 0-stable Cartan subalgebra of g. Then we say that h is a
maximally split Cartan subalgebra of g if h n p is maximal abelian in p. We say
that h is fundamental if h n I is maximal abelian in I.

58
2. Real Reductive Groups
Lemma. Fundamental and maximally split Cartan subalgebras exist.
Furthermore, any two fundamental (resp. maximally split) Cartan subalgebras
are conjugate under Int(g).
The Cartan subalgebra in 2.2.5 is clearly maximally split. Let t be a maximal
abelian subalgebra of I. Let c^ be maximal abelian in p subject to the condition
that [t,at] = 0. Set b^ = t + a,. We may argue as in 2.2.5 we see that f)! is a
Cartan subalgebra of g which is clearly fundamental.
Let Ki denote Ad(X°). Let bj, ;'= 1, 2, be maximally split Cartan
subalgebras of g. Let aj = fy n p,;' = 1,2. Then Lemma 2.1.9 implies that there
exists ke Ki so that /cOj = a2. Thus we may assume c^ = a2 = a. Let t- =
I n bJ5 j = 1, 2. Then tj is maximal abelian in °m (2.1.6),;' = 1, 2. Thus there
exists m e Ad(°M°) such that mix = t2 (0.3.3 (1)). This completes the proof in
the case of maximally split Cartan subalgebras.
Let i)j, j = 1, 2 be fundamental Cartan subalgebras for g. Set tj• = I n fy
for j = I, 2. Then there exists ke Ki such that /ct, = t2. We may thus
assume that ^ = t2 = t. Let u = Cg(t). Then u is reductive (Lemma 2.3.1) and
^-invariant. Let <Xj = p n bj for j = 1,2. Then each a,- is maximal abelian in
pnu. Hence Lemma 2.3.1 implies that there exists u e Int(u) n Ki with
uai = a2 ■ The result now follows.
2.3.5. Let b be a Cartan subalgebra of g. If a e <t>(gc,bc) then a is called a
real root of h if a is real valued on h.
Lemma, h is fundamental if and only if it has no real roots.
Let h be a Cartan subalgebra of g which we assume (as we may) is 8-
invariant. Let a be a real root for h and let s be the corresponding TDS in gc.
Then s° = s n g is a 0-stable subalgebra of g isomorphic with s/(2, R). We can
clearly choose a standard basis X, Y, H of s0 such that H e h n p and
BX = - Y. Since a is real, it follows that a is 0 on t = h n f. Thus R(X - Y) + t
is an abelian subalgebra of I. This shows that if h is fundamental then h has
no real roots.
We prove the converse by induction on the dimension of g. If dim g = 0 the
result is obvious. Assume the result for all reductive Lie algebras of smaller
dimension. Suppose that b is 0-stable and that h has no real roots. Set t = h n I.
Let u = Cg(t). If t is non-zero then u is reductive, 0-stable and of lower
dimension. Thus h n u is fundamental in u. But u clearly contains a
fundamental Cartan subalgebra of g. Hence h is fundamental in g. If t = {0} then

2.3. Cartan Subgroups
59
all of the roots are real. But then g is abelian and so the result is also true in this
case.
Let a be maximal abelian in p. Fix G = ANK an Iwasawa decomposition
of G. Then a standard p-pair, (PF, AF), is said to be cuspidal if °mf has a Cartan
subalgebra, tF, completely contained in f. Set hF = tF + aF.
Proposition. Let I) be a Cartan subalgebra of g. Then there exists a standard,
cuspidal, p-pair, (PF, AF), and x e Int(g) such that xh = hF.
We may assume that h is 0-stable and that h n p is contained in a. Let <t>0
denote the set of roots of a that are non-zero onf)np = a,. Let H e c^ be such
that <x(H) is non-zero for all a e <t>0. There is s e W(q, a) so that a(sH) > 0 for all
a e P (the positive system corresponding to the choice of n, Lemma 2.1.10(2)).
Let k e s. We replace h by Ad(/c)h. Let F be the set of all a e A0(2.2.5) that
vanish on H. Then, h n p is contained in aF. The result now follows.
2.3.6. Let h be a Cartan subalgebra of g. Then a subgroup of the
form CG(h) = {g e G!Ad((/)|,, = /} will be called a Cartan subgroup of G.
A standard p-pair, (PF,AF), is cuspidal if and only if °MF has a compact
Cartan subgroup, TF. In this case HF = TFAF is a Cartan subgroup.
Proposition 2.3.5 immediately implies:
Proposition. // H is a Cartan subgroup of G then there exists a standard
cuspidal p-pair, (PF,AF), and g e G° such that gHg~x = HF.
If H is a Cartan subgroup of G then we call H fundamental (resp. maximally
split) if h is fundamental (resp. maximally split).
2.3.7. A parabolic subgroup of G is said to be maximal if it is proper and is
not properly contained in any parabolic subgroup of G. The maximal
parabolic subgroups of G are conjugate to the subgroups, PF, with F of the
form A0 — {a} with a a simple root. Proposition 2.3.6 implies that if H is a non-
compact Cartan subgroup of G and if G = °G then there is a maximal
parabolic subgroup, PF, of G such that H is Int(g) conjugate to a Cartan
subgroup of MF. This gives an inductive technique for finding all Cartan
subgroups up to conjugacy. Let us give some examples.
1. SL(n,R). Let us denote by P(mu..., mk) the intersection with
SL(n, R) = G of the groups so designated in 2.2.9. Then if k = 2, P(mum2) is

60
2. Real Reductive Groups
maximal. If n > 2 then G has no compact Cartan subgroups. The cuspidal
parabolics correspond to the cases when m};= 1 or 2 for j = 1,..., k.
2. SU(p,q). We assume that p > q> 1. We choose a to be the space of all
matrices h(tu..., tq), tjeR,j = \,...,q, given by
0
0
h
_ U
0
0
0
u
u
0
0
Set sj(h(tl,...,tq)) = tjfor ;' = 1,...,q. <t>(g,a) consists of e, + e^ for i =£ j,
±2ej for j = l,...,q and if p > q, ±es for j = 1,..., q. Choose the Weyl
chamber corresponding to
tl>t2>--->tq.
If p > q (resp. p = q) then the simple roots are e! — e2,...,e,_, — eq, eq
(resp. 2eq). Set Hj = h(tu...) with tt = 1 for i < j and t, = 0 for i > j. Then
the m's for maximal standard parabolics are of the form CB(Hj), j = l,...,q.
We leave it to the reader to describe the Cartan subalgebras of g.
2.4. Integration formulas
2.4.1. Let G be a real reductive group. Fix 6, a Cartan involution, and
G = NAK, an Iwasawa decomposition of G. Let (PF,AF) be a standard
p-pair. If [l e (aF)* and if H e aF we write a" = exp fi(H) if a = exp H. We
define pF e (aF)* by pF(H) = (±) tr(ad H\VF).
Lemma. Let dn, da, dm be respectively invariant measures onNF,AF, °MF. Let
dk be the normalized invariant measure on K. Then we can choose an invariant
measure dg on G such that
S f(y)dg = | f(namk)a~2pFdndadmdk,
O NFxAFx°MFxKF
for f e CC(G). Also if us C(K) then
|u(/c)d/c= | u(kFk(kij))a(kg)2pFdkFdk
K K*KF

2.4. Intergration Formulas
61
here if g e G and if g = nak, ne N, ae A, ke K then a(g) = a and k(g) = k.
Let dp denote a left invariant measure on PF. Then we can choose an
invariant measure, dg, on G such that
$f(g)dg= | f(pk)dpdk
a pr*K
by Lemma 0.1.4. Thus we must show that up to scalar multiple dp =
a'2"' dndadm. Lemma 2.2.7 implies that dp = h(n,a,m)dndadm with h a
smooth function on NF x AF x °M. By left invariance h is independent of n.
By definition of °MF the modular function, S, of PF is 1 on °MF. Thus dp
is right invariant under °MF. Hence h is a function of a alone. The
Jacobian of the action n>->ana_1 is det(Ad(a)|n) = a2pF for ae/lf. Thus
a 2"Fdndadm is left /lF-invariant.
We now prove the second assertion of the Lemma. According to Lemma
0.1.3 there exists a continuous compactly supported function/ on G such that
| f(pk)dp= | u(kFk)dkF, keK.
Pf Kf
Thus we have
\f(x)dx = \u(k)dk.
G K
Now,
| f{x) dx = | f(xg) dx= | f(pkg) dp dk.
a a pt x k
We write kg = na(kg)k(kg) as above, dp transforms by S under right
multiplication by elements of PF. Since d(na(kg)) = a(kg)2pF, we have
Ju(/c)d7c= | a(kg)2pFf(pk(kg))dpdk= | u(kFk(kg))dkFdk.
K PFxK KFxK
As was to be shown.
2.4.2. For our next integration formula we assume that G is of inner type.
Let R be the system of positive roots for <t>(g, a) corresponding to the choice
of n. Set a+ equal to the Weyl chamber corresponding to R (2.1.10). Set
A+ = exp(a+). If a e A, a = exp H, we set y(a) = T\aeR sinh(a(//)).
Lemma, dg can be normalized so that
\f(g)dg= | y(a)f(klak2)dkldadk2.
G K* A* * K

62
2. Real Reductive Groups
For simplicity of notation we will write M for °M. Let /?: a+ x K/M -* p
be denned by P(H,kM) = Ad(k)H. Let p' denote the range of p. Since
Ad(K)a = p, Ad(X)a+ = Ad(X)o' and Ad(X)(o - a') is a finite union
of submanifolds of lower dimension, p' is open, dense and has a
complement of measure 0 in p. It is easy to check that ft is a diffeomorphism onto p'
(Proposition 2.1.10). Let fi:K x A+ x K/M -► G be denned by n(k,a,xM) =
/ocax-1. The above remarks and Theorem 2.1.8 imply that fi is a
diffeomorphism onto an open subset of G that has a complement of measure 0. This
implies that there is a smooth function h such that
| f(g)dg = | h(k,a,x)f(kxax~l)dkdad(xM).
G K x A x K/M
Since dg is left and right invariant it is easy to see that h is a function of
only a. Let X} be a basis of n such that ad HX} = aj(H)Xj for all ;' and
H e a. Set Y, = A} + 0A}. Let Zm be a basis of m and let Hj be a basis of a.
We may look upon the Y} as a basis of the tangent space at 1 to K/M. A
direct calculation yields
^i,o.i(V,-,0,0) = (Ad(a-1y,.)o,
dfiUaA(Zm,0,0) = (ZJa,
d^,1(0,HJ,0) = (Hj)a,
^i,o,i(0,0,Y/) = ((/-Ada-1)Y/)o.
It is now easily seen that the Jacobian of \i at 1, a, 1 is
T\j(a"J - a~xj).
Hence h(a) is a constant multiple of y(a). Since we are using normalized
measure on K, M and K/M, we may replace the integration over K/M by
integration over K. Since dk is invariant d(kx) = d/c and d(k~l) = dk. The
result now follows.
2.4.3. We continue our assumptions of 2.4.2. Proposition 2.3.6 implies
that there exist 0-stable Cartan subalgebras hi,...,^ that are mutually
non-conjugate and such that every Cartan subalgebra of g is conjugate to
one of them (here conjugation is relative to Ad(G)). Let #,,..., Hr be the
corresponding Cartan subgroups. Set A/,-= {g e G\ Ad((/)bj = fy}. Then A/,-
contains Hj = and it is easily seen that Wj = Nj/Hj is a finite group.
Let /^- be a system of positive roots for <D(gc,(f)j)c)- Set ^-(H) = naeP a(H)
for H e by. Let D be is as in 2.3.1. Then \D(H)\ = I^H)!2. Since G and each

2.4. Intergration Formulas
63
Hj are unimodular, each coset space G/ff, has a G-invariant measure, dxj
(0.1.2).
Proposition. There exist positive constants Cj,j = 1,..., r and normalizations
of Lebesgue measure on g and the by such that
\f(X)dX = £>, J \D(H)\( | /(Ad xH)dx)dH, forfe CM
8 l)j \0/H| /
For the moment, fix ;', and set 1), = I), etc. Let fi:G/H x h' ->g be defined by
n(gH, h) = Ad(g)h (here h' = g' n h). We may identify the complex tangent
space at Iff to G/H with n+ + n . Translating by the elements of G allows us
to identify the tangent space at gH with this space. A direct calculation yields
dfigHM(X,Z)=Ad(g)(adXh + Z)
for X e n+ + rT, Z e h. This implies
(1) The Jacobian of [i at gH, h is D(h), up to sign.
This implies that fi is everywhere regular. The remarks preceding the
statement we are proving now imply that \i is a [W]-fold covering of its range.
Lemma 2.3.1 implies that g' is the disjoint union of the open subsets
Ad(G)(h;)c. The result now follows from (1).
The above result is sometimes called the Weyl integral formula for g.
2.4.4. We now derive the Weyl integral formula for G. We define real
analytic functions d} on G by
det(t/ - (Ad g - I)) = X tJ dj(g).
Here n = dim G. Set d = dj for j = rank(gc). We set G' = {g e G | d(g) ^ 0}.
Then G' is open, dense with complement of measure 0 in G. We retain the
notation of 2.4.3.
Proposition. There exist positive constants nij so that if dg and dhj are
respectively invariant measure on G and Hj then
I f(9)4/ = !>/ J | d(hj){ | /(ghjg-^digHj^dhj, for f € Q(G).
G Hj \G/Hj /
We fix j and for the moment drop the index ;'. Let a:G/H x ff' -> G be
defined by a(gH, h) = ghg~' (here ff' = ff n G'). We have
dagH<h(X,Z) = (Ad(0)((Ad(/r') - l)X + Z))a{glfM),

64 2. Real Reductive Groups
for X e n+ ©n", Ze h. The rest of the proof is now almost identical to
that of Proposition 2.4.3 and we leave it to the reader.
2.4.5. We now derive some integration formulas that are related to the
Gelfand-Naimark decomposition. We will use the notation of 2.4.1. We set
VF = 6NF. Fix invariant measures dn, dm, da, dv respectively on NF, °MF,
AF and VF.
Lemma. The invariant measure, dg, can be normalized so that
§f(g)dg = | a~2PFf(nmav)dndmdadv
G Nr x oMf. x Af. x Vf.
for f e Q(G). If ue C(K) then
\u(k)dk = | a(v)2pFu(kFk(v))dkFdv.
K KFXV
Let fi: NF x °MF x AF x VF -> G be denned by /x(n, m, a, v) = nmav. We have
seen (2.2.11) that fi is a diffeomorphism onto an open subset of G whose
complement has measure zero in G. Thus there exists a smooth function, h, on
NF x °MF x AF x VF such that
\f(g)dg= | f(nmax)h(n,m,a,v)dndmdadv.
G NFx°MFxAFxVF
As usual, the bi-invariance of dg implies that h is a function only of a. We may
now argue as in the proof of Lemma 2.4.1 to complete the proof of the first
integration formula.
We now prove the second one. We may replace u by the function
u(k) = | u(kFk)dkF
K
and therefore assume that u(kFk) = u(k) for kF e KF.Let a e CC(PF/KF) be such
that
\a(p)dp= 1.
p
Put h(pk) = a(p)u(/c) for pe PF,keK. Then
| f(k)dk = | h(g)dg = | a~2pFh(nman)dndmdadv
K O NF x "MF x AF x VF
= | h(pv)dpdv = | h(pa{v)k(v)) dp dv
PF x V> PF x Kf
= | a(i;)2pf7i(p/c(i;))dpdi;= | a(f;)2"f'u(/c(f;))df;.
As was to be proved.

2.5. The Weyl Character Formula
65
2.5. The Weyl character formula
2.5.1. The purpose of this section is to show how to use the Weyl integral
formula to prove the Weyl character formula. Let G be a compact Lie group.
Lemma. G is a real reductive group.
Since GA is countable (Theorem 1.7.5) we may write GA as {yi,y2,---}-
Let (itj, Vj) e jj. We set Hj = @kij Vk with the direct sum inner product. Let
Hj be the direct sum representation. Let Gj be the kernel of fij. Then G,
contains G}+1 and f] Gj = {1} (1.4.4(1)). This implies that for some index, k, the
Lie algebra of Gk is 0. Hence Gk is finite. Hence there is an index k' such that
Gk. = {1}. Let fik. = fi, Hk, = H. We look upon H as C" with the usual inner
product, ( , ). We then look upon C" as R2" in the usual way and take
< , > = Re( , ). We identify G with its image in GL(2n, R). Let / be the
set of all real valued polynomials on M2fl(R) that vanish on G. Let P be the
algebra of all real valued polynomials on M2n(R).
Let M be the set of zeros in GL(2n, R) of /. If / e / then f(X*) = g(X)
defines g e P which is clearly in /. Since M is the Zariski closure (c.f. Mumford
[1, p.l]) of G, M is an algebraic group. Thus M is a real reductive group.
Let Pc be the algebra of complex valued polynomials on M2n(R). The
Stone-Weierstrass theorem implies that the restriction of Pc to M is
uniformly dense in C(M). Let M act on Pc by mf(X) = f(Xm) for f e Pc,
X e M2fl(R) and me M. Since the space of homogeneous polynomials of a
fixed degree is invariant under the action of M and is finite dimensional we
see by 1.4.4(1) and 1.3.2 that the restriction of Pc to M is precisely the
algebraic sum of the isotypic components of L2(M). But then the space
of G-invariants in Pc restricted to M is uniformly dense in the space of G
invariants in C(M) under the right regular action. Now the condition that a
polynomial be G-invariant is itself a polynomial condition. Thus every G-
invariant polynomial is M-invariant. But then the G-invariant continuous
functions are M-invariant. This implies that C(M/G) consists of the
constants. Hence G = M.
Note. This lemma is an important part of the Tannaka duality theorem.
2.5.2. We now assume that G is connected. Let T be a maximal torus of G.
Let h = tc. Fix R a system of positive roots for <t>(gc,h). Let S e h* be half
the sum of the elements of R. Fix < , > an Ad(G)-invariant inner product

66
2. Real Reductive Groups
on g (0.3.1). Denote by ( , ) the induced symmetric non-degenerate form
on h*. Let A be the simple root system of R.
(1) 2(M/(a,a)=l for a e A.
If a e A let sa be the corresponding Weyl reflection (0.2.3). Let ji e R — {a}
then sJeR by 0.2.4(2) and 0.2.1(4). Thus saR = (R - {a})u{-a}. This
implies that sa3 = 3 — a. (1) now follows.
In particular, (1) implies that 3 is dominant integral. Hence there is a finite
covering G~ of G so that if T~ is the corresponding maximal torus then 3
is T~ integral (see 1.7.5). Define on T~ by A(t) = tan„6jl(l - (""). Then
|A(t)l2 = \d(t)\ (2.4.4). 0.3.3(1) says that, up to conjugacy, T is the only Cartan
subgroup of G. If we carefully follow the argument in 2.4.4 one finds that if
all the measures are normalized measures then (mi)~l = [VF(G, T)] = w.
We therefore have
Proposition. Let dg and dt be normalized invariant measure on G and T
respectively. Then
| f(g)dg = (1/w) | |A(T)|2 | j{gtg~')dgdt.
G TO
2.5.3. We assume for the remainder of this section that G = G~. If \i e TA
we set A(n)(t) = Z.seW det(s)ts"(W = W(G,T)). We say that fi is regular if
sn / n for s e W — {1}. It is easy to see that A(n) = 0 if \i is not regular. If n
is regular than there exists se W such that sfi is dominant integral. Let fi and
ji be integral, dominant integral and regular then
(1) | A(n)(t) conj(A(p))(t)dt = w^.
T
This is an immediate consequence of Lemma 1.4.5.
Lemma. A = A(S).
Let a be a simple root then using the material in the proof of 2.5.2(1) we
see that A(sj) = — A(t) for t e T. Now A is a sum of characters of T with
coefficients +1. The coefficient of ts is 1. The other characters that come
into the expansion are of the form 3 — q with q a sum of distinct elements
of R. Thus A = S cqA(3 — q) the sum over all q that are sums of distinct
elements of R and the coefficients cq are integers. We assert that if ^4(<5 — q) is
non-zero then A(S — q) = ±A(3) (here q is a sum of distinct elements of R).
Indeed, if se W then s(d — q) = 3 — q' with q' a sum of distinct elements

2.5. The Weyl Character Formula
67
of R. Thus we may assume that 5 — q is dominant and regular. This implies
that 2(6 — q, a)/(a, a) is a positive integer for all simple a. Hence 2.5.2(1)
implies that 2(<j,a)/(a, a) < 0 for all simple roots a. Thus (q, a) < 0 for all
a.e R. But then (q, q) < 0. So q = 0 as asserted.
We therefore conclude that A = cA(S). Proposition 2.5.2 implies that
\\A\2dt = w.
T
So (1) above implies that c = 1.
2.5.4. We now come to the Weyl character formula.
Theorem. Let y e GA and let A be the highest weight of y relative to R
(Theorem 1.7.5). Let xy be the character of y. Then
A(3)Xy = A(A + S).
We order the weights of L(A) (1.7.4) by saying that n> oif n — a is a sum
of elements of R. If /i is a weight of L(A) and if sn > A for some s e W then
sn = A. This implies that A(5)xy = A(A + 5) + f with / = I cqA(A + d - q)
where q is a sum of elements of R and A + 3 — q is dominant integral and
regular. Applying 2.5.2(1) we have
| A(d)(t)Xy(t)(conj(A(5)(t)xy(t))dt = w + | \f(t)\2dt.
T T
Lemma 1.4.5 combined with Proposition 2.5.2 now imply that
||/(t)|2^ = 0. So/ = 0.
T
2.5.5. We now show how one uses the Weyl character formula to derive the
Weyl dimension formula.
Theorem. Let y e GA have highest weight A relative to R. Then
d(y)= [] (A + S,a)/(a,a).
Clearly, ^(1) = d(y). Hence
d(y) = lim^(exp(it//,)) = lim A(A + 3)(e\p(itHs))/A(3)(exp(itHs))
r->0 r^O
= lim/l(5)(exp(it//A + ,))//l(^)(exp(it//,)).

68
2. Real Reductive Groups
But Lemma 2.5.3 implies that
A(d)(exp(itH)) = [] (exp(t(i)i(a(H))) - exp(-t(i)i(a(ff)))).
Hence d(y) = limr^0 T\teR sin(ttx(HA+s)/2)/sm(tot.(Hd)/2). The result now
follows.
2.A. Appendices to Chapter 2
2.A.I. Some linear algebra
2.A. 1.1. We put the usual inner product, ( , ), on C". If X e M„(C) then
we denote by X* the conjugate transpose of X. Then X* is the adjoint
operator to X relative to ( , ). If X = X* then we say that X is self-adjoint. If
X is self-adjoint and if (Xv, v) > 0 for all non-zero v eC" then X is called
positive non-degenerate (or positive definite). If Xe M„(C) then we write
exp X for the usual power series
Then exp defines a complex analytic mapping of M„(C) into GL(n, C).
As is well known, if X is self-adjoint then there is a unitary operator, u,
on C" such that uXu'1 is diagonal with real entries. Hence it is clear that
(1) If X is self-adjoint then exp X is positive non-degenerate.
(2) If A is positive non-degenerate then there is a self-adjoint matrix, X,
such that A = exp X.
We may assume that A is diagonal with positive diagonal entries, ax,...,an.
Take X to be the diagonal matrix with diagonal entries log^),..., log(a„).
2.A.I.2. The following lemma is due to Chevalley. It will be used several
times in this chapter.
Lemma. Let f be a real or complex valued polynomial function on M„(C).
Suppose that Y is self-adjoint and that /(exp mF) = 0 for all m = 1,2,—
Then /(exp tY) = 0 for all real t.
Let u be a unitary matrix such that uYuT' is diagonal. If we replace / by
the polynomial g(Z) = /(u'Zu) we may assume that Y is diagonal with
real diagonal entries au...,a„. We restrict / to the diagonal matrices. Our
assumption now says that /(exp^aj,..., exp(ma„)) = 0 for m= 1,2,—

2.A.I. Some Linear Algebra
69
Set p(t) = /(exp(fa!),..., exp(fa„)). If p is not identically zero then
P(t) = I bm exp(MJ with Ax >->A,
with bi non-zero. Thus if s is real and sufficiently large then
\bx exp sAx\ >
X bmexpsAn
Thus p(m) is non-zero for sufficiently large integers, m. Since this is
contrary to our hypothesis, we must have p(t) = 0 for all t.
2.A. 1.3. If X e M„(C) we define (/ - e\p(-X))/X to be the sum of the
power series
E(-l)"(l/(m+l)!)X".
Lemma. Let X,Y bee M„(C) then
d expx(y) = exp X((I - exp(-ad AT))/ad X) Y.
This result can be proved directly by manipulating power series. See (e.g.,
Wallach [1]).
2.A. 1.4. Let p„ denote the space of self-adjoint elements of M„(C).
Lemma. The map U(n) x p„ -> GL(n, C) given by u, Y h-> u exp X defines a
surjective diffeomorphism.
Let geGL(n,C). Set A = g*g. Then A is positive non-degenerate. So
A=expY with Y e p„. Set X = (\)Y and p = exp AT. Then p2 = A It
is easy to check that gp ' e U(n). Thus the map is surjective. Suppose
that g = u exp X = u' exp A". Then exp 2X = exp 2A". This implies that
exp 2mA" commutes with X for all m = 1,2, Lemma 2.A.2 implies that
exp tX' commutes with X for all real t. Hence A" commutes with A'. Thus
X and A" can be simultaneously diagonalized using a unitary matrix. Since
exp 2X = exp 2A" this implies that X = A". Thus u = u'. So the map is in-
jective. Let / denote the map we are studying. The Lie algebra of U(n) can
be identified with the Lie algebra of all skew-adjoint matrices (Y* = — Y).
If X* = -X, if Y, Z e p„ and if u e U(n) then
4fu.z(x> y) = "(* exP z + d/d't=o exp(Z + tY)).
Thus if dfuZ(X, Y) = 0 then X exp Z must be self-adjoint. But then X exp Z =
— AT exp Z. So (exp Z)A'(exp( —Z)) = —X. After an orthonormal change of

70
2. Real Reductive Groups
basis we may assume that Z is diagonal with real diagonal entries al,...,a„.
Thus the eigenvalues of T i-> (exp Z)T(exp( —Z)) are of the form exp(a,- — ak)
which are all positive so X = 0. Lemma 2.A.3 implies that
((7-exp(-adZ))/adZ)y = 0.
But (ad Z)2k + l Y is skew-adjoint and (ad Z)2kY is self-adjoint. Thus we see
that VY= 0 with
V = X (ad Z)27(2/c + 1)!.
The eigenvalues of V are of the form
X(ai-aJ-)2V(2/c+l)!,
which are positive. Thus Y= 0. So / is everywhere regular and bijective. Hence
/ is a diffeomorphism.
2. A. 1.5. Let F = R or C.UX e Mn(F) then X is said to be nilpotent if Xk = 0
for some k. If g e GL(F) then g is said to be unipotent if g — I is nilpotent.
Lemma. // Y is nilpotent then exp Y is unipotent. If g is unipotent then
g = exp Y with Y nilpotent.
It is clear that exp Y = / + YZ with [T,Z] = 0. Thus ((exp Y) - I)k =
YkZk. Thus if Y is nilpotent exp Y is unipotent. Let g be unipotent. Set Z =
^ — /. Put log(g) = Em>! Zm/m. Since Z is nilpotent this series is actually
finite. The obvious formal manipulation of power series gives exp(log(g)) = g
(it is rigorous since all series are finite). Since log(g) = ZW, log(g) is nilpotent
so the lemma follows.
2.A.2. Norms on real reductive groups
2.A.2.I. Let G be a real reductive group. Then as in 2.1.1 there exists GR a
symmetric algebraic subgroup of GL(n,R) (for appropriate n) and, p, a finite
covering homomorphism of G onto an open subgroup of GR. Furthermore, we
can choose a Cartan involution 8 of G such that p(8(g)) = p(g1)*.
On R2", which we look upon as R" + R", we put the standard inner product.
If ge GL(n,R) then we set \\g\\ = ||#® (g^1)*!! where || || is the operator
norm. If g e G then we set \\g\\ = ||p(g)||. Let K be the maximal compact
subgroup of G corresponding to 8. Let g = I ® p be the corresponding Cartan
decomposition of g. Then ||.. .|| has the following properties:

2.A.2. Norms on Real Reductive Groups 71
(1) 11*11 = llrt forge G.
(2) IMI<IMIII>'II, foTx,yeG.
(3) {ge G\\\g\\ < r} is compact for all r < co.
(4) ||fc, exp(tX)/c2|| = ||exp XH' for all ku k2 e K, X e p
and all t e R, t > 0.
These properties are easy to prove and are left to the reader.
2.A.2.2. Lemma. Let (n, H) be a Hilbert representation of G. Then there
exist constants C > 0, r > 0 such that ||7r(g)|| < C||gf||r for allg e G. (Here \\A\\
denotes the operator norm of A.)
We note that \\g\\ > 1 for all g e G. We set a(g) = log(||gi||). Then a(x) > 0,
<j{xy) < o(x) + a(y) and a(x_1) = a{x). Set ^((7) = log \\n(g)\\ for pG. Then
H(xy) < n(x) + n(y) for x, y e G.
PutBr={0€G a(0)<r}.
(1) There exists a positive constant, C, such that n(x) < C for x e Bx.
This follows from (3) above and 1.1.1(1).
(2) e~c\n(x)\ < \n(kx)\ < ec\n{x)\ for x e G, k e K.
This follows from (1) since K is contained in Bx.
Let X e p. Then a(exp tX) = ta(exp X) for t > 0. Let j be a non-negative
integer such that j < a(expA') < j + 1. Then a(exp(A'/(; + 1)) < 1. Hence
n(exp(X/(j + 1)) < C. This implies that
A*(exp X)<(j+ 1)C < C(\ + a(exp X)).
Thus, if k e K then
H(k exp X) < C + C(l + <r(exp AT)) = C(2 + <r(exp AT).
Theorem 2.1.8(1) now implies that if g eG then \\n(g)\\ < e2C\\g\\c. This
completes the proof.
The above result will play an important role in our study of matrix
coefficients of representations. The method in the above proof was suggested
by the proof of Warner [1,4.4.5.9].
2.A.2.3. We will call any function, ||- -II, on G with values in [1, 00) satisfying
(1), (2), (3), (4) of 2.A.2.1 a norm on G. We note that the proof of 2.A.2.2 implies

72
2. Real Reductive Groups
that if || || j and || ||2 are norms on G then there exist constants C > 0 and
q > 0 such that
(1) \\g\\2 * QlglU, for all ^eG.
We fix an Iwasawa decomposition, G = NAK, with a contained in p. We
assume (for the sake of simplicity) that G has compact center. Let <t>+ be the set
of positive roots of <t>(g,a) corresponding to N. Let {a.l,...,ixr} be the simple
roots in <t>+. By our assumption, the simple roots span a*. We define
Hu..., Hr e a by Xj(Hk) = djk. Let A + be as in 2.4.2.
Lemma. Let ||- • -|| be a norm on G. Then there exist n, fi e a*, with n(Hj) > 0
for all j, and positive constants Cl, C2 such that
C1fl"<||fl|| < C2ali,
forallaeC\(A + ).
In light of (1) we may assume that ||- • -|| is given as in 2.A.2.I. Let S denote
the weights of a on R2" corresponding to the representation p(g) ® (p(g)"1)*
for g e G. We partially order I by p. > /} if n{H3) > P(H3) for ; = 1,..., r. Let
Hl,...,Hd be the maximal elements of S. Then ||a|| is the maximum of the
a"J,j= l,...,d, for aeC\(A+). Set y = nx +••■ + nd.2A.2.l (3) implies that
y(Hj) > 0 for all j = 1,..., d. Hence it is clear that
a"d < \\a\\ < a" for a e C\(A + ).
2.A.2.4. Lemma. Let \ \ • • • \ \ be a norm on G. Then there exists d > 0 such that
\\\g\Vddg<K.
o
Let y be as in 2.4.2. Then y(a) < a2p for aeC\(A + ). Let n be as in
Lemma 2.A.2.3. Let d be so large that dn(Hj) > 2p(Hj) for ;' = 1,..., r. The
result is now a direct consequence of 2.4.2.

3 The Basic Theory of
(g, X)-Modules
Introduction
In this chapter we begin the representation theory of real reductive groups.
The theory of (g, K)-modules (first introduced by Harish-Chandra for
connected K and later defined in general by Lepowsky) is the connecting link
between the algebraic results of Chevalley and Harish-Chandra and group
representation theory. The main results of this chapter are Harish-Chandra's
theorem that implies that irreducible unitary representations are admissible
(Section 3.4), the subquotient theorem of Harish-Chandra, Lepowsky, Rader
(Section 3.5) and its important refinement due to Casselman (Section 3.8).
Section 1 contains the theorem of Chevalley that relates the polynomial K-
invariants to the invariants of the Weyl group. This theorem is one of the main
ingredients in Harish-Chandra's proof of the isomorphism between the center
of the universal enveloping algebra and the Weyl group invariants on a Cartan
subalgebra. This result and Harish-Chandra's determination of all
"infinitesimal characters" is the content of Section 2. In Section 3, Lepowsky's
definition of (g, K)-modules is introduced. The most important example is the
space of K-finite, smooth vectors of a Hilbert representation. The main result
73

74
3. The Basic Theory of (g, K)-:VIodules
in Section 4 is Theorem 3.4.1 which asserts that the isotypic components of a
finitely generated (g, K)-module are finitely generated as modules for the
center of the universal enveloping algebra. This theorem combined with
Schur's Lemma implies the above mentioned theorem of Harish-Chandra on
irreducible unitary representations. In Section 5 we give Lepowsky's proof of
the subquotient theorem. It also contains preliminary results on the algebraic
structure of (g, K>modules. Section 6 is devoted to an exposition of some of
Harish-Chandra's theory of the spherical principal series. The main result of
this section is the exact sequence in 3.6.6. However, the estimate in 3.6.7 will be
fundamental in later developments. The material in Section 7 will be useful in
the theory of the Jacquet module. Section 8 is devoted to a new proof of the
subrepresentation theorem of Casselman. Although this theorem appears to
be only slightly stronger then the subquotient theorem, we will see in the next
chapter that the difference between the two theorems is significant.
3.1. The Chevalley restriction theorem
3.1.1. Let G be a real reductive group. Let 6 be a Cartan involution for G
and let g = f ® p be the corresponding Cartan decomposition. Let K be as in
2.1.8. If V is a real vector space then we denote by P(V) the space of complex
valued polynomial functions on V. Let K act on P(p) by kf(X) = /(Ad(/T' )X)
for k e K, X e p and / e P(p). We denote by P(p)K the space of all / e P(p)
such that kf = f for all k e K.
Let a be as in 2.1.6. Let W = W(q, a) be as in 2.1.10. Let W act on P(a) by
sf(H) = /(s-1 H) for s e W, H e a, f e P(a). Let P(a)w denote the set of all
/ e P(a) such that sf = / for all s e W. If V is a real vector space and if W is a
real subspace of V then we define for / e P(V), Resv/W(f) to be the restriction
of / to W.
3.1.2. Theorem. Assume that G is of inner type (2.2.8). Then Resp/a is an
algebra isomorphism of P(p)K onto P(a)w-
As we have seen in 2.1.10, W = {Ad(/c)|a!/c e K, Ad(/c)a = a}. Thus
(1) Resp/a(P(p)*) is contained in P(a)w.
(2) Resp/a is injective on P(p)K.
This follows from Lemma 2.1.9.
(3) Let Hj e a, j = 1, 2. If there exists k e K such that Ad(k)Hi = H2 then
there exists s e W such that sHl = H2.

3.1. The Chevalley Restriction Theorem
75
Clearly, a and Ad(/c)a are maximal abelian in Ca(H2) n p. Since Ca(H2) is
real reductive and 0-stable (2.3.1(2)), there exists k2e(K n CG(H2))° such that
Ad(/c2)(Ad(/c)a) = a. Take s = fc2fc|0.
(4) If Hj e a, j = 1,2 and if WHl n WH2 = 0 then there exists a continuous
function / on p such that f(Ad(k)X) = f(X) for all k e K and X e p and
/(H1) = 0,/(H2)=1.
By (3), Ad(X)//! n Ad(X)//2 = 0. Thus there is a continuous function, h,
on p such that his identically 0 in Ad(K)H1 and identically 1 on Ad(K)H2. Set
f(X) = \h(Ad(k)X)dk.
K
(5) Let Hj, j = 1, 2 be as in (4). Then there exists peP(p)K such that
p(ff,) * p(ff2).
Set C = Ad(X)//! u Ad(X)//2. ThenC is a compact subset of p. Let/be as
in (4). The Stone-Weierstrass theorem implies that there is a polynomial q on p
such that
\q(X) - f(X)\ < \ for X e C.
Then p(X) = \K q{Ad{\i)X)dk defines the desired polynomial.
Let F denote the quotient field of P(a). Let L be the quotient field of
J = Resp/a(P(p)*). Let D,- be as in 2.3.1. Set f(z) = I zJ Resg/a/),-. Then the
roots of / are the elements of <f>(q, a). If \i e a* and if n vanishes on °a (2.2.2)
then ne J. Thus we see that F is a normal extension of L(see any book on
Galois theory). So L = {/ e F! of = f for all a e Gal(F/L)}, here Gal(F/L)
is the group of all automorphisms of F that are equal to / on L. By the
above, if a e Gal(F/L) then a(a*) = a*. Hence, aP(a) is contained in P(a)
for all a e Gal (F/L).
Denote by U the group of all automorphisms of P(a) that are equal to
/ on J. Then we have shown that J = {f e P(a)\af = /for all a e U}. If a e U
and if H e a then 3(f) = of(H) defines a homorphism of P(a) into C. Hence
the nullstellensatz (c.f. Mumford [1, p.3]) implies that there exists Hl such
that 5(f) = /(H,) for all / e P(a). Now, of = f for / e J, so (5) implies that
there exists s e W such that H^ = sH. We have therefore shown that if / e
P(a)w then of = / for all a e U. Hence P(a)w' is contained in J. Now (1)
implies the result.
Note. The above Theorem is the celebrated Chevalley restriction Theorem.
We note that if G is not necessarily of inner type and if we define NK(A) =
{k e K ! Ad(/c)a = a} and W = NK(A)/°M then the conclusion of the above
theorem is still true (with the same proof).

76
3. The Basic Theory of (g, X>Modules
3.1.3. We now derive a corollary to Theorem 3.1.2 which is also called the
Chevalley restriction theorem. Let g be a reductive Lie algebra over C.
Let P(g) denote the space of all complex polynomials on g. We define an
action of g on P(g) by Xf(Y) = d/dtt=0f(e\p(-t ad X)Y). Set /(g) =
{f e P(g)\Xf = 0 for all X e g}. Let h be a Cartan subalgebra of g.
Let W = W(q, h). We let W act on P(h) by sf(H) = /(s_1 H). Let /(b) denote
the W-mvariants in P(b).
Theorem. Resg/1, is an isomorphism of /(g) onto 1(1)).
Since the center of g is contained in b, we may assume that g is semi-simple.
Let gu be a compact form of g such that g„ n b is maximal abelian in g„ (0.3.4).
Set G = Int(g) which we look upon as a real reductive group. Let 6 denote
conjugation on g relative to gu. Then 8 is a Cartan involution of g (looked
upon as a real Lie algebra). If we set I = gu and p = igu then g = f ® p is the
corresponding Cartan decomposition. Since gu is a real form of g, ResB/p is an
isomorphism of /(g) onto P(p)K. Set a = b n p. Then Resb/a is an
isomorphism of /(b) onto P(a)w. The result is now an immediate consequence of
Theorem 3.1.2.
3.1.4. Example. We look at the case when g = M„(C) (the Lie algebra of
GL(n, C)). We take for I) the space of diagonal matrices. If // e b and if H has
diagonal entries hl,...,h„, then define £,-(//) = h-r Then <t>(g,b) is the set of all
Ej - ek for distinct j, k. We take <t>+ to be the set of all e; - ek for j < k. Then
if a = Ej - ek, then saH has diagonal entries, hal,...,han, with a the
permutation (j,k). We therefore see that W is the set of all permutations of
the diagonal entries. Thus, the fundamental theorem of invariant theory for
the symmetric group (Weyl [1, pp.37, 38]) implies that P(bJ^ is equal to
C[a1;..., a„], where a; is the ;'-th elementary symmetric function in the
diagonal entries of H. Recall that these functions are denned by
11 (t + hs) = £ f-^j(H).
1 < j < n
Define for X e M„(C) the polynomials pj by
det(tI + X) = YJf-JpJ(X).
Then it is clear that Res^p, = o-r Theorem 3.1.3 now implies that P(g)8 is the
polynomial algebra in pu..., pn.

3.2. Harish-Chandra Isomorphism of Center of the Universal Enveloping Algebra 77
3.2. The Harish-Chandra isomorphism of the center of the
universal enveloping algebra
3.2.1. Let g be a reductive Lie algebra over C. Let Z(g) be the center of
l/(g)(0.4.1). In this section we will give Harish-Chandra's determination of the
homomorphisms of Z(g) into C. In order to carry this out we will use the
Harish-Chandra isomorphism. In Section 6 we will give a related (but
different) mapping that is called the Harish-Chandra homomorphism.
Let h be a Cartan subalgebra of g (0.2.1). Fix R a system of positive roots for
<t>(g,h). Let n+ (resp. n ) be the sum of the ga (resp. g_J for a e R. Then
g = n+ ©l)©n".
P-B-W (Theorem 0.4.1) implies that
l/(g) = l/(l))©(rTl/(g) + l/(g)n+).
Let q denote the projection of l/(g) onto l/(h) corresponding to this direct sum
decomposition. Let 17(g)" be the set of all x e 17(g) that commute with every
element of h.
Lemma, q is an algebra homomorphism of 17(g)* into 17(h).
We enumerate <t>+ as {a,,...,ad}. Let X}, j = \,...,d, be a basis of n+
with A'yeg,.. Let Yj be a basis of rT with YJe0_a. Let Hk be a basis of
b, /c= 1,...//. If neW then set
X" = (*,)"'•••(*„)'",
y" = (y1)"'---(ydr.
If ke Nl then set
H* = (//,)*• ■■■(H,)*t.
Then P-B-W implies that the elements YmHkX" form a basis of 17(g).
(1) 17(g)" n (n-l/(9) + l/(g)n+) = 17(g)" n n" 17(g) = 17(g)" n I7(g)n+.
If x e 17(g)" then x = S am t „ YmHkX" with the sum over all m, /c, n such that
S myay = S Mj-ay. Which clearly implies (1).
Let uy e 17(g)" for ;' = 1,2. Then ulu2 = ulq(u2)(mod l/(g)n + ).

78
3. The Basic Theory of (g, K)-Modules
Hence (1) implies that
uxu2 s q(M1)q(M2)(mod(n"l/(g) + l/(g)n + ).
This is the content of the Lemma.
3.2.2. Fix an invariant from, B, on g as in 0.2.2. We define a mapping
X^X* of gontog*byB(y,A:) = X#(y)for Y e g. Then X h-> X* induces
an algebra isomorphism of S(g) onto P(g). ad induces an action of g on S(g)
as derivations. Under X i-> X* this corresponds to the action of g on P(g)
in 3.1.3. We may thus identify S(g) and P(g) as g-modules.
Let p e h* be half the sum of the elements of R. We define an isomorphism,
p, of S(h) given by p(H) = H — p(H) on h and extended to S(h) by the
universal mapping property. Since h is abelian, l/(h) is isomorphic with S(h).
Thus we will use S(h) and l/(h) interchangeably. We define a homomorphism,
y, of Z(g) into (7(h) by y = p. ° q. (Note that Z(g) is contained in (7(g)1'.)
Under our identification, the g-invariants in S(g), S(g)8, correspond to P(g)fl.
We also have an action of W = W(g, h) on S(h). The VF-invariants in S(h)
correspond to P(b)^. (Here we have replaced g by h in the above discussion.)
We write U(l))w for the W-invariants in 1/(1)) (= S(b)).
We can now give the Harish-Chandra isomorphism for the center of the
enveloping algebra.
3.2.3. Theorem. y(Z(g)) is contained in U(\))w. The map y defines an algebra
isomorphism of Z(g) onto V(\))w.
We first note that the result follows from
(*) y(Z(g)) is contained in V(\))w.
We use the standard nitration of l/(g) (0.4.2). Then Gr l/(g) = S(g). We can
therefore consider Gr q: S(g) -> S(h). The direct sum decomposition
g=l)0(n+ 0n')
is B orthogonal. Thus, under our identifications, it is an easy matter to see
that Gr q = Res8/lr Thus, if we compare the nitration to the grade and apply
Theorem 3.1.3 combined with (*) the result follows. We are thus left with
proving (*).
Let a be a simple root in R. Let mx = h + ga + g_a. Set n" equal to the sum
of the root spaces corresponding to the elements of R - {a}. Set n* equal to the
sum of the root spaces corresponding to the roots -jHorjieR — {a}. Then
g = m" ® n" ® n".

3.2. Harish-Chandra Isomorphism of Center of the Universal Enveloping Algebra 79
P-B-W implies that
1/(9) = l/(m") 0 (n"U(g) + l/(g)n").
Let q" be the linear projection of l/(g) onto l/(m") corresponding to this direct
sum decomposition. Define p" in (m')* by p"(X) = (|)tr(ad X |n„) for X e m".
Define a homomorphism, a of U(m") to itself by a(X) = X — p"{X) for
X e m*. If we argue as in the proof of Lemma 3.2.1 we see that a ° q restricted
to Z(g) is a homomorphism into Z(m"). Let y" be the Harish-Chandra
homomorphism associated with m'. Then y" ° o ° q = y. Thus if we show
that v"(Z(ma)) is contained in the sa invariants of l/(h) then (*) will follow
from 0.2.4(3). We therefore can assume that g = m', That is, R = {a}.
We are reduced to the case when [g, g] = g! is a TDS (0.5.4). Let X, Y, H be
a standard basis for g^ Set C = H2 + 2(XY + YX). Then a simple
computation shows that C is in Z(g). Let c e S(g) be the element given by the same
formula in S(g) (which we have identified with P(g)). Then Resg/1,(c) = H2.
Now W = {/,s„} and sx restricted to 3(g) is /, saH = -H. Thus Theorem 3.1.3
implies that S(g)8 = S(3(g))C[c]. If we compare the standard nitration
of Z(g) with Gr Z(g) we see that Z(g) = l/(3(9))C[C]. But it is clear that
y(C) = H2 - 1. Thus (*) is true in this case. This completes the proof.
3.2.4. We now show how one uses the Harish-Chandra isomorphism to
derive Harish-Chandra's formula for infinitesimal characters. If p. e h* then
set 1 = p o y (p extends to a homomorphism of l/(h) to C by the universal
mapping property of l/(h)).
Theorem. Let %be a non-zero homomorphism of Z(g) to C. Then there exists
pel)* such that 1 = 1^. Furthermore, if p, p' e h* then x^ = X„- if and only
if there exists s e W such that sp = p'.
Let Dj be as in 2.3.1. Set p, = Res^D,-). Set f(t) = I tjp-. If a e <D(g, h) then
/(a) = 0. This implies that l/(h) (which is identified with S(h) which is in turn
identified with P(b)) is integral over U{\))w (cf. Zariski, Samuel [1]). Hence
every non-zero homomorphism of P(l))w into C is given by point evaluation
{[pp. ci't.]). In light of our identifications, this implies the first assertion.
The second assertion follows from the observation that if h, h' e h and if
f(h) = f(h') for all f eP{\))w then there exists s e W such that h' = sh. (cf. the
proof of 3.1.2).
3.2.5. We now look at what these results say for g = M„(C). If A is an
associative algebra over C and if [o,,t] is an n by n matrix over A we set

80
3. The Basic Theory of (g, K)-Modules
det([aJit]) = S sgn(a) n"=1 aaJj, the sum over all permutations of n-letters.
We take Ejk to be the standard basis of M„(C) and look upon these
elements as being in l/(g). Let t be an indeterminate and set ajk(t) = Ejyk +
(j - I + t)5]<k. Write det([aJt(t)]) = I t"~JUj. Then the content of the
classical Cappelli identities (Weyl [1, p.42]) is that UjS Z(g). One computes
thaty(«J-) = o,y(3.1.5).
3.3. (g, K)-modules
3.3.1. Let G be a real Lie group with Lie algebra, g. Let K be a compact
subgroup of G. We recall Lepowsky's definition of a (g, K)-module. Let V be a
g-module that is also a module for K (for the moment we ignore the topology
of K). Then V is called a (g, K)-module if the following three conditions are
satisfied:
(1) k-X-v = Ad(k)X-k-v for v e V, k e K, X e g.
(2) If v e V then Kv spans a finite dimensional vector subspace of V, Wv, such
that the action of K on Wv is continuous.
(3) If Y e I and if v e V then d/dt, = 0 exp(tY)v = Yv.
If V and W are (g, K)-modules then we denote by Homg K(V, W) the space of
all g-homomorphisms that are also K homomorphisms of V to W. V and W
are said to be equivalent if there is an invertible element in Homg K(V, W).
We denote by C(g, K) the category of all (g, X)-modules with Horn in this
category given by Homfl K(V, W).
3.3.2. A (g, X)-module, V, is said to be finitely generated if it is finitely
generated as a l/(g)-module. V is said to be irreducible if the only g and K-
invariant subspaces of V are V and (0). In this context we have the following
variant of Schur's Lemma.
Lemma. Let V be an irreducible (g,K)-module. Then Homg K(V, W) = CI.
Let v be a nonzero element of V. Let Wv be as in 3.3.1(2). Then U(q)Wv is
a g and a K-invariant subspace of V. Hence, V = U(q)Wv. In particular,
this implies that V is countable dimensional. The result now follows from
Lemma 0.5.2.
3.3.3. Let V be a (g, X)-module. Let y e K". Then we set V(y) equal to the
sum of all the X-invariant, finite dimensional, subspaces of V that are in the
class of y. Lemmas 1.4.7 and 1.4.8 immediately imply

3.3. (g,K)-Modules
81
Lemma. Asa K -module, V = @fEr V(y). Here the direct sum is the algebraic
direct sum.
If y e KA then we call V(y) the y-isotypic component of V. We say that V is
admissible if dim V(y) < co for all y e KA.
3.3.4. Lemma. Let V be a (g, K )-module. Then V is admissible if and only if
dim HomK(W, V) < co for all finite dimensional K-modules, W.
Let W be a finite dimensional K-module. Let T be a X-homomorphism of
VF into V. Then T(VF) is a direct sum of irreducible X-submodules of V
(Lemma 3.3.3). Since W has only a finite number of inequivalent irreducible
quotients, there exists, F, a finite subset of KA depending only on W, such that
T{W) is contained in @ F V(y). The lemma now follows.
3.3.5. Let (rc, H) be a Hilbert representation of G. Then according to Lemma
1.4.7, H is the Hilbert space direct sum of the H(y) for y e KA. Here we are
assuming, as we may, that it \K is unitary (Lemma 1.4.8). Lemma 1.4.7(1) implies
that H(y) n Hx (1.6.3) is dense in H(y) for all y e KA. We set HK equal to the
algebraic direct sum of the H(y) n H33 for y e KA. By the above, it is clear that
HK is a dense subspace of H (resp. Hx).
Lemma. HK is a %-invariant subspace of Hv. With this structure of g and K-
modules, HK is a (Q,K)-module.
We note that HK is the space of all C^-vectors, v, of H such that n(K)v spans
a finite dimensional subspace of H. 1.6.4(H) says that if X e q, ge G and
ter then n(g)n(X)v = n(Ad(g)X)n(g)v. Thus, if v e HK, if X e g and if W„
is the span of n(K)v then Wv is contained in HK and n(X)v e ^(g)^ a finite
dimensional K-invariant subspace of H°°. The result now follows.
HK is called the space of C™, K-finite vectors of H or the underlying (g, X)-
module of //. We say that // is admissible if //K is admissible. H is said to be
infinitesimally irreducible if //K is irreducible as a (g, X)-module. If (it, H) and
(a, V) are Hilbert representations of G then rc is infinitesimally equivalent
with a if the (g, K)-modules HK and KK are equivalent.
3.3.6. Let Ve C(g,K). U fieV* then we write X • [i (resp. k • fi) for the
functional X • n(v) = -fi(Xv) (resp. k • /.i(v) = n(k~lv). Then relative to these
actions V* is a g and a X-module that satisfies the compatibility condition
3.3.1(1). We set V~ = {^e V* \ Kp spans a finite dimensional subspace}. We
may argue as we did above to see that V~ is a g and a X-submodule of K*.
Hence V~ is a (g, X)-module. V~ is called the (g, K)-dual module of K

82
3. The Basic Theory of (g, K)-Modules
We set V* equal to the space of all conjugate-linear functionals on V with g
and K acting on V* as above. We set V= {fie V*\Ky. spans a finite
dimensional subspace}. Then as above V is a (g, K)-module that is called the
conjugate dual (g,K)-module of V.
3.4. A basic theorem of Harish-Chandra
3.4.1. Let G be a real reductive group. We return to the notation of 3.1.1. Let
ZG(g) denote the subalgebra of l/(g) consisting of those elements u e U(q)
such that Ad(g)u = ufor all g e G. Notice that if G is of inner type (3.1.1) then
ZG(g) = Z(g). The purpose of this section is to prove several important
theorems of Harish-Chandra [1] the first is:
Theorem. Let V be a finitely generated (3.3.2) (g, K)-module. If y e KA then
V(y) is finitely generated as a Za(Q)-module.
The proof of this result involves several steps which we now begin. We fix V
a finitely generated (g, K)-module. Let W be a finite dimensional K-invariant
subspace of V such that V= U(q)W. In light of the material in 0.4.3, we
see that V= symm(S(p))W. We define V0 = W and Vj+l = pVs + Vs for
j = 0, 1, Then each V-} is K-invariant, pVj is contained in Vj + 1 the union of
the Vj is V. Set Gr(K) equal to the direct sum of the spaces (Vj/Vj-x), here
V^! = (0). Then Gr(K) is equivalent with V as a K-module.
Let pj be the natural projection of Vs into Vj/Vj- !. If X e p, v, w e Vj
and if pj(v) = p,-(w) then pJ+l(Xv — Xw) = 0. We may thus define an action of
each Jepon Gr(K) by Xpj(v) = pJ+1(Xv) for v e Vj.
3.4.2. We define a new Lie algebra structure on f © p as follows:
(1) If X, Y e f or if X e f, Y e p then [X, Y~\ has the same meaning as it did
ing.
(2) If X, Y e p then \_X, 7] = 0.
We denote by gc the Lie algebra f © p with commutation relations given
as in (1), (2). We form a Lie group Gc with total space K x p and multiplication
given by:
(3) (k,X)(u,Y) = (ku,Ad(u-1)X+Y), k,ueK,X,Yep.
Then Gc is a Lie group with Lie algebra gc.

3.4. A Basic Theorem of Harish-Chandra
83
Lemma. Gr( V) is a finitely generated (QC,K )-module.
Let veVj and X, Yep. Then XYpj(v) = pj + 2(XYv) = pj + 2(YXv + [X, Y]v) =
pj+2(YXv) since [AT, y]ef. Thus XYv = YXv for all ueGr(K) and X, Yep.
It is therefore clear that Gr(K) is a gc-module. Conditions 3.3.1(1), (2), (3)
are all assertions for K and they follow from the fact that V is a (g,K)-
module.
Let Gr(K)J = Vj/Vj^. Then p Gr(K), = Gr(V)j+l. Thus Gr(K) is finitely
generated.
3.4.3. We may look upon p as an abelian normal subalgebra of gc. Then
S(p) is the universal enveloping algebra of p. Clearly, S(p)K is contained in the
center of U($c).
Lemma. If ye KA then Gr(K)(y) is finitely generated as a S(p)K-module.
Let y e KA and let (p., X)e y. We look upon Uomc(X,Gr(V)) as an S(p)
and a A:-module with the actions (uT)(v) = u(Tv) and (kT)(v) = /c(T((/T»)
for ueS(p), keK and ceGr(F). As a S(p)-module Hom^X, Gr(K)) =
A'*®Gr(K) with S(p) acting on the right factor. Thus under this
action Homc(A\ Gr(K)) is finitely generated as a S(p)-module. Also
UomK(X, Gr(K)) is the space of K-invariants in Uomc(X, Gr(K)). Set L =
S(p) HomK(A', Gr(K)). Since S(p) is Noetherian (0.6.1) there exist
elements Ti,..., Td in UomK(Xy Gr(K)) such that L = I S(p)Ty(0.6.2). If Te
HomK(A:,Gr(K)) then T = I.pjTj with p,eS(p). Hence T=/cT=I(Ad(/c)pJ)T;.
for all k e K. Hence if we set for p e S(p),
p° = |Ad(/c)pd/c
K
then T= ^ (p,)0^- Since p° e S(p)K for all p e S(p), we have proved:
(*) UomK(X, Gr(K)) is finitely generated as a S(p)K-module.
Let a: HomK(A', Gr(K))® X->Gr(V)(y) be defined by a(T®x)=Tx.
Then a is surjective. Thus we see that if Tj,..., Td are as above then
Gr(K)(y) = I S(p)KTj(X). This completes the proof of the lemma.
3.4.4. Let P(g)G be the algebra of all polynomials on g, /, such that
/ o Ad(^) = / for all ge G.
Lemma. P(p)K is finitely generated as a Res^v(P(Q)G)-module.

84 3. The Basic Theory of (g,K)-Modules
Let a be maximal abelian in p. We use the conventions in 2.1.1. In particular
we identify the Lie algebra of G with that of GR. We define polynomials, qj,
on g by
det(t/ + X) = £ tjqj(X) for X e g.
Let S denote the set of all weights of a on R". Then clearly, S spans a*. If
j8 e S then S j8J'Resg/a(q;) = 0. Since g„ = 1 this implies that
(*) P(a) is finitely generated as a Resg/a(P(g)G)-module.
2.1.9 implies that Resp/a is injective on P(p)K, so the result follows
from (*).
3.4.5. We define a linear map, 3, of (7(g) to S(p) by d(symm(p)k) = e(k)p, for
p e S(p) and /c e 1/(1), here we are using 0.4.3 and £ is defined as in 0.4.4. If
u e Uj(q) (0.4.2) we set dj(u) equal to the;'-th homogeneous component of d(u).
We note that
(1) If h e U\q), v e Vk then Pj+k(uv) = 5j(u)pk(v).
Fix, B, an invariant non-degenerate form on g. As in 3.2.2, we identify P(g)
(resp. P(p)) with S(g) (resp. S(p)). Set /(p) = Resg/p(P(g)G). We look upon /(p)
as a subalgebra of S(p). Then S(p)K is finitely generated as an /(p)-module
(Lemma 3.4.4). We also note that
(2) If u e S(g)G then symm(u) e ZG(g) and (5(symm(«)) = Resg/P(u).
3.4.6. We are finally ready to complete the proof of Theorem 3.4.1.
Lemma 3.4.3 now implies that if y e KA then Gr(K)(y) is finitely generated
as an /(p)-module. Let v1,...,vd be homogeneous generators with Uj
homogeneous of degree kj. Let Vj€ Vk project onto v}. 3.4.5(1) and (2) now
imply that
©P*((Z Zc(9)fj) nK) = Z Hp)i>j = Gr(K)(y).
Hence S ZG(g)fj = K(y), which was to be proved.
3.4.7. We now derive some consequences of Theorem 3.4.1. The first is
immediate.
Corollary. Let V be a finitely generated (g,K)-module such that if ve
V then dim Zg(q)v < oo. Then V is admissible.
3.4.8. Corollary. Let V be an irreducible (g, K )-module then V is admissible.

3.4. A Basic Theorem of Harish-Chandra
85
Lemma 3.3.2 implies that the elements of ZG(g) act on V by scalars. The
result now follows from 3.4.7.
3.4.9. Before we can give the next application we must introduce some
notation and results. Let C e ZG(g) be the Casimir operator of G corresponding
to B. That is, if Xu..., Xm is a basis of g and if X1,..., Xm are denned by
B(Xj,Xk) = 5jJk then C = l.XjXj.
Theorem. Let (it,H) be a Hilbert representation of G. Suppose that if
v e HK (3.3.5) then dim C[C]i; < oo. Then HK is a subspace of the space of
analytic vectors for it (1.6.6).
Let CK be denned for (I,K) in the same way as C was denned for (g, G).
We note that
(1) If Deffjf then dim C[C,Q]i> < oo.
Set A = C — 2CK then (1) implies that
(2) If ve HK then dim C[A]i; < oo.
Fix veHK and let w e H, Set / = c„,w (1,3.2). We look upon l/(g) as
the space of all left invariant differential operators on G (as usual). Then
(2) implies that there is a monic polynomial, p, such that p(A)/=0. Let
Xu..,,Xm be an orthonormal basis of g relative to the inner product,
< , >, given by (X, Y} = -B(X,0Y). Then A = I(X;)2. Thus, in local
analytic coordinates, p(A) is an analytic elliptic operator. Analytic elliptic
regularity (Nirenberg [1, p.158]) implies that / is real analytic.
The following result is the basic theorem in the title of this section.
3.4.10. Theorem. Let (it,H) be an irreducible unitary representation of G.
Then (n, H) is admissible.
In light of Lemma 1.6.5, and the previous theorem HK consists of analytic
vectors. Let v e HK be non-zero and set V = U(q) span(7r(K)i;). Then 1.6.5
combined with Corollary 3.4.7 implies that V is an admissible (g,K)-
submodule of HK. Now G = KG0, so Proposition 1.6.6 implies that C1(K)
is a G-invariant subspace of H. Hence C1(K) = H. Since Cl(K)(y) = Cl(K(y))
for all y e KA, this implies that HK = V.
3.4.11. Theorem. Let (it, H) be a unitary representation of G. Then (n, H) is
irreducible if and only if it is infinitesimally irreducible (3.3.5). // (it, H) and

86
3. The Basic Theory of (g, K)-Modules
(a, V) are irreducible unitary representations of G then n and a are unitarily
equivalent if and only if they are infinitesimally equivalent.
Suppose that (it, H) is irreducible. Then, as we have seen in the preceding
proof, if W is a non-zero (g, X)-submodule of HK then W = HK. Suppose
that HK is irreducible. If H is reducible then H = Hi © H2 unitary direct
sum of closed, non-zero, G-invariant subspaces. Thus HK = (Hl)K® (H2)K-
This contradiction implies the first part of the result.
We now prove the second assertion. Let A be an invertible element of
HomaJ((HK, VK). Then A maps (HK)(y) to (VK)(y) for all yeKA. We may
thus define A* e Homg,K(VK,HK) by (A*v, w) = (v, Aw) for v e V(y) and we
H(y) (here we have used the admissibility of VK and HK). Then A*As
UomgK(VK, VK). Thus the first part of this theorem and Lemma 3.3.2
imply that A*A = cl with c > 0. Set T = <T1/2A Then T extends to a
unitary operator from H onto V which is clearly a K -intertwining operator.
It is easy to see that if X e g then T7r(exp(A')) = 7r(exp(A'))T on HK. Since
G = KG0, this implies that T defines a unitary equivalence.
3.4.12. Theorem. Let (n,H) be an admissible Hilbert representation of G.
Then (n, H) is irreducible if and only if it is infinitesimally irreducible.
If (n, H) is reducible then there exists a closed, proper, non-zero, G-invariant
subspace V of H, Since V is admissible it is clear that VK is proper. If HK is
reducible then H is reducible by the argument in the first part of the proof of
3.4.10.
3.5. The subquotient theorem
3.5.1. The purpose of this section is to give a proof of the celebrated
subquotient theorem of Harish-Chandra [3], Lepowsky [1] and Rader. We
first must establish some generalities about (g, X)-modules. We return to the
notation in 3.3.1. In this section l/(g) will denote the universal enveloping
algebra of gc,
Lemma. U(q)k is a Noetherian algebra over C.
Let / be a left ideal in U(q)k. Then (7(g)/ is a left ideal in (7(g), Since (7(g) is
Noetherian, there exist XjSl, j=\,,..,d such that U(g)I = I U(g)Xj.
Hence, if y e I then y = Z UjXj with us in l/(g). If k e K then Ad(k)y = y and

3.5. The Subquotient Theorem
87
Ad(k)xj = Xj. Thus we may replace us by its projection in U(q)k. This implies
that / = I U(q)kXj.
3.5.2. For simplicity, we now assume that K is connected. If y e KA
then we fix Vy e y. Set Iy = {x e 1/(1) x acts by 0 on Vy}. U y,ae KA then set
Uy-a = {x e 17(g) /x <= U(q)I„}. We note that Schur's Lemma implies that
U(l)/Iy is isomorphic with End(K,,). We look upon End(K,,) as a K-module
under left multiplication. If we apply the material in 0.4.3 and 0.6 it follows
that
(1) U(Q)/U(Q)Iy is l/(g)-module isomorphic with U(Q)(g)m)End(V).
The latter module can be considered to be a (g, K)-module if we use the
K-action, k(g ® T) = Ad(k)g ® kT. Thus, in light of (1), we may look upon
l/(8)/l/(g)/„ as a (g, X)-module.
Lemma. (l/(g)/l/(g)/)(a) = l/for'/Wfa)"'* n l/(9)/y) /or a// y, a e XA.
If V is a (g, X)-module then (since K is assumed to be connected) V(y) =
{ve V\Iyv = 0}, Let K = l/(g)/l/(g)/), and set q equal to the natural
projection of l/(g) onto K. If g e l/(g)"-' then Iaq(g) = 0. Also, if g e l/(g) and if
4^(0) — 0 then ^ e [/(g)"''. The result now follows,
3.5.3. Lemma. Let W be an admissible (g, K)-module, Let y e KA and let X
be a U(q)k and U(i)-invariant subspace of W(y). Then (U(Q)X)(y) = X.
We first observe that
(1) U(Q)™\WM=U(Q)KU(t)\WM.
Indeed, let A denote the left hand side of (1). Let B denote 1/(1) | Wly). Then B
is isomorphic to End(Kj,). Thus in particular, B is a finite dimensional simple
algebra over C. This implies that A = B'B where B' is the commutant of
B in A (for this case this result is implicitly proved in 1.2.2), It is easy to see
that B'= U(Q)K\W{y),
We now prove the Lemma. U(q)X = (U(Q)/U(Q)Iy)X which is the direct
sum of the spaces U(Q)"'yX. So (U($)X)(y) = 17(g)"* = X by (1).
3.5.4. The following result is true for general real reductive groups of inner
type. We will give the necessary modifications of the proof below in
Section 3,9,

88
3. The Basic Theory of (g, X>Modules
Proposition. Let V be an irreducible (g, K)-module. Let yeKA. Then
UomK(Vy, V(y)) is an irreducible U(Q)K-module under left multiplication.
Furthermore, if W is an irreducible (q,K)-module with W(y) non-zero and if
HomK(K),, W(y)) and UomK(Vy, V(y)) are equivalent as U(q)k-modules then V
is equivalent to W as a (g, K)-module.
The first assertion is an immediate consequence of Lemma 3.5.3. To
prove the second assertion we observe that
(1) Y = U(q) (X)U(g)KU(i) V(y) has a unique irreducible quotient.
Indeed, if X is a (g, K)-submodule of Y then 3.5.3 and 3.5.2 imply that X(y) is
either 0 or equal to 1 ® V(y). Let M be the sum of all (g, K)-submodules of Y,
X, such that X(y) = 0. Then it is clear that M is a proper (g, K)-submodule of Y
and that M contains all proper submodules. This implies (1).
Clearly, (1) implies the second assertion.
The material in 3,5.2-3.5.4 is based on Lepowsky, McCollum [1] (c.f.
Dixmier [2,9.1]).
3.5.5. We now assume that G is a real reductive group. We return to the
notation of 3.4. Let P = °MAN be a minimal parabolic subgroup of G with
given standard Langlands decomposition. Let (a,Ha) be an irreducible
unitary representation of °M. If \i e (ac)* then we denote by o^ the
representation of P given by ojjnari) = a" + pa(m) for m e °M, as A, and n e N.
We set Indp(aJ = (jtffi/J, //"•") (see 1.5,4), The representations nail are
called the principal series.
Lemma. (Ha'")K is an admissible (g, K)-module.
As a K-module, (H"-'')K is the space of K-finite vectors in the representation
induced from a on °M to K. The result now follows from 1.4.5(3).
3.5.6. Theorem. Assume that G is connected. Let V be an irreducible (q,K)-
module. Then there exist a e °MA and \i e (ac)* such that V is equivalent to a
submodule of a quotient module of (Ha-")K.
We will be devoting the rest of this section to a proof of this result. We
will be following the argument in Lepowsky [1] (cf. Dixmier [2, 9.2, 9.4]).
We note that P-B-W implies that the map l/(n)® 17(a)® 17(1)-» 17(g)
given by n, a, k i—► nak is a linear bijection. We identify U(a) ® 1/(1) with its

3.5. The Subquotient Theorem
89
image under this mapping and have
(1) l/(fl)=l/(o)®l/(f) + nl/(8).
We give U(a) ® l/(f) the tensor product algebra structure. Let p denote
the linear projection of l/(g) onto 17(a)® 1/(1) corresponding to the direct
sum decomposition in (1), (The reader should be warned that l/(a)® 1/(1),
as a subspace of U(q) has no a priori algebra structure.)
(2) If x e U(q) and if y e U(q)K then p(xy) = p(y)p(x).
We note that since [a,n~\ = n, U(a)nU(s) is a subspace of nl/(g). Modulo
nl/(g), xy = p(x)y. Now p(x) = S afc, a,-el/(a), k}e 1/(1). Thus xy =
S Oy/cyy = S a}ykj = S a}p(y)kj = p(y)p(x) (remember the warning about the
multiplication).
3.5.7. Let y e KA and let /?,, be the natural projection of 1/(1) onto 1/(1)//,
(which we identify with End(Ky)). If a e °MA we fix ffff e a. Let Pa be the
projection of Vy onto K». Let pyJk) = PJy(k)Pa. We set py = (I®Py)p
and p,,jff = (/ ® jffyj<r)p- We also set for n e (oc)*, py^ = ((n + p) ® /)p),jff.
Let T be a maximal torus in °M°, Then h = (t + a)c is a Cartan subalgebra
of gc. Fix a positive root system, <t>+, in <t>(mc,h), let pm be half the sum
of the elements of <t>+. If a e (ac)* (resp. j8e(tc)*) then we extend a
(resp. j8) to b by setting a|t = 0 (resp. j8|a = 0). If a e °MA let fi„ denote the
lowest weight of a with respect to <t>+. If [i e (ac)* and if a e °MA then we
setfi(<r,^) = A„ + n- pm.
(1) If z e Z(8) then py,„.M(z) = Zn^zJP, for /ze (oc)* and a € °M\
Here xA is as in 3.2.4.
Indeed, as in 3.2.1(1), we find that p(Z(g)) is contained in l/(m)M. Thus,
Schur's lemma implies that py_„tll(z) is a scalar multiple of Pa. We can
compute the scalar by evaluating py.a^(z) on a lowest weight vector of Vy(a).
The result now follows from the definition of Xa- (The reader should be
wary about the interchange of positive and negative roots.)
3.5.8. We now use p to compute the action of l/(g)K on H"-"(y). Frobenius
reciprocity says that the map Ti—► TA defines an isomorphism of
HomK(Kv,(H"")K) onto UomM(Vy,Ha). Here T» = T(v) = 7»(1).
(1) (n(H--")K)(l) = 0.
Indeed, if fe(H'-")K and len then Xf(\) = d/dt\t=0 /(exp tX) = 0
since/(n) = /(l) for all neN.

90
3. The Basic Theory of (g, K)-Modules
Thus if T e UomK(Vy,(H"")K) and u e 17(g)* then (1) implies that (uT)A =
(p(u)T(v))(\) = ((fi + p)®I)pyJu))T(v))(l). This equals T(py.„»i>)(l). We
have therefore shown that
(2) If Te UomK(Vy,(H^)K) and if u e l/(g)* then (uT)* = T*py.aJu). In
particular, (H"-tl)K has infinitesimal character Xcua.n)-
3.5.9. Lemma, //ue 17(g)* and if py(u) = 0 then u e U(q)k n l/(g)/r
Let T: S(p) ® 1/(1) -» l/(g) be denned by p ® /c i—► symm(p)/c. We saw in
0.4.3 that T is a linear bijection. Set q = (Resp/Q ® I)T l. Let Sj(p) denote
the space of elements of S(p) that are homogeneous of degree ;'. Let S,(p)
denote the sum of the Sk(p) for k < j.
(1) If g e symm(S,<p))l/(f) then p(g) - q(g) e I/'"" »(o) <g> 1/(1).
Indeed, let A^- be a basis for n. Then X} — 6Xj is a basis of the orthogonal
complement of a in p. Set Uj = symm(S/(p))l/(/c). If g e 17,- then
0 = 1(9) + £ (*,- - 0Xt)gt mod l/y_ „
with gj e Uj: _ x. Thus
g ee q(g) + 2 £ Xl9l - £ (X, + 0X,)fc mod l/,._,.
This implies that
g = q(g) + 2 £ *,#, mod [/,-_,.
(1) now follows, since p(q(g)) = g(#) and p(l/,-) <= l/J(a)l/(fe).
Let q,, be denned in the same way as py.
(2) If m e 17(g)* and if qy(u) = 0 then u e 17(g)* n l/(g)/r
Let /c,- in 1/(1) be such that their projections into U(l)/Iy, kj, define a basis.
Now u = Zsymm(pJ)/c;mod U(g)Iy. Thus if u is K-invariant and if qy(u) = 0
then (kp = Ad(fc)p, ke K,pe S(p))
Resp/Q(/cPj) Ad(/c)/cJ = 0 for all keK.
Hence Resp/a(kpj) = 0 for all ke K.So 2.1.9 implies that pj = 0 for all/. This
proves (2).
We now prove the Lemma. Let u e 17(g)* and suppose that py(u) = 0
but qy(u) is non-zero. We write u = ~Luj with Uj e symm(SJ'(p))l/(f). Suppose
that qy(u}) = 0 for j > r but that qy(ur) is non-zero. Then (2) implies that
p(u}) = 0 for ;" > r. Hence p (u) = p^u,.) mod l/r_ ,(a) ® 1/(1). But qy(u) =

3.5. The Subquotient Theorem
91
qy(ur) mod l/r_!(a)® U(i). Now (1) implies a contradiction. So the Lemma
follows from (2).
3.5.10. Before we can complete the proof of Theorem 3.5.6 we must
introduce a bit more notation. Let W = W(qc, h). Let 3 = p + pm. Let <I>+ be
the positive system of roots for (gc, h) compatible with n and containing <I>m.
Then S is the half sum of the elements of <I>+. Let p. be as in 3.2.1. Let q be the
map as denned in 3.2.1 using — <I>+ instead of <I>+. Then Theorem 3.2.3 says
that
(1) q(Z(ci)) = ii-iU(l))w.
If m e 17(g)* and if y e KA then set
fyJT) = det(T - py(u)).
Then fyM e U(a)\_T~\ <= l7(h)[T]. W acts on l7(h)[T] by acting on the
coefficients. Set
weW
(2) Let m = d(y)\_W~\. Then there exist elements z}, j = l,...,min
Z(g) with zm = 1 and g1M = £ q(zj)TJ.
This is clear from (1).
Set v(u) = S ZjUJ. Then
(3) o(m) e l/(9)K n l/(g)/r
Indeed, py(v(u)) = I py(zj)py(u)}. Let a e °MA. Then
Py.a(»(«)) = (/ ® K)q(Zj)Py.M'-
Here we look upon l/(h) as l/(o) ® l/(tc). Hence
So Pj,(y(y)) = 0. (3) now follows from Lemma 3.5.9.
3.5.11. We now complete the proof of Theorem 3.5.6. Let fi be an irreducible
finite dimensional representation of U(q)k such that Ker fi => U(Q)Iy n U(q)k.
Schur's lemma implies that fi(z) = x(z)l for z e Z(g). Now x = Xa f°r
some A e b* (3.2.4). Let S = {{a,p)\a e °M\ p. e (oc)* with [y:a] # 0 and
p e WA\a}. Suppose that u e U(q)k and that pJ,iff>/J(«) = 0 for all (a,p) in the
finite set S (3.2.4). Let v(u) be as above. Then il(v(u)) = 0 by (3) above.
Our assumption on u implies that x(z-) = 0 for j < m. Thus Q(u)m = 0. Hence

92
3. The Basic Theory of (g, K)-Modules
Ker fi :=> [ f](a s Ker p a ]m. The Theorem now follows from
Proposition 3.5.4, Lemma 3.5.3, 3.5.8(2) and 3.A.I.I.
3.6. The spherical principal series
3.6.1. We continue with the notation in 3.5. If a is the trivial 1-dimensional
representation of °M and if n e (ac)* then we set ita„ = n^ and Ha" = //". If
/ e L2(°M\K) then we set f^nak) = a" + pf(k) for ne N,a e A, Ice K.
If ge G and g = nak with ne N, as A, ke K then we write n(g) = n,
a(g) = a, k(g) = k. Theorem 2.1.8(2) implies that as functions on G, n, a and
k are smooth. Let 1 denote the function on K that is identically equal to 1.
Let y0 denote the class of the trivial representation of K. Then it is clear that
(1) (H")K(y0) = CI.
If \i e (ac)* then we define E^ by
(2) E„(0) = <7r„(0)l„,l„>.
We have
(3) Ztl(g) = $a(kgr + »dk for g e G.
K
Indeed, l^g) = a(g)" + p and l„(/c) = 1 for g e G, k e K.
3.6.2. Proposition. If s e W(q, a) then Hs/I = H„ for all n e (ac)*.
The proof of this result of Harish-Chandra [8] will take some preparation.
Let CC(K\G/K) denote the space of all smooth, compactly supported, K-
bi-invariant functions on G. To prove the proposition it is enough to show
that if feCc(K\G/K) then
$f(g)Zll(g)dg=$f(g)Zsll(g)dg.
o a
We compute
| f(g)Etl(g)dg = | f(g)a(kgy+»dkdg = | Rg^^dg,
G GxK G
since / is left K invariant. Let S = AN and let ds be a choice of left invariant
measure on S so that Lemma 0.1.4 applies. Then
I f(g)Kig)dg = | f{sk)a(sy+?dsdk = \jXs)a{sy+?ds
a sxk s

3.6. The Spherical Principal Series
93
by the right X-invariance of /. Now ds can be normalized such that ds =
a~2pdnda. Hence the integral we are calculating is equal to
J f(na)a>l-f'dnda.
N x a'
We set FAa) = a" \Nf(na) dn. Then we have shown
(1) \f{g)~.t,{g)dg = \Ff(a)a»da.
G A
Thus to prove the proposition it is enough to show
(2) F/exp H) = F/(exp sH) for all H e a, f e C?(K\G/K)
and s e W(q, a).
3.6.3. Let <t>+ be the positive root system in <t>(g, a) corresponding to n. Let
A0 be the corresponding set of simple roots. Let F = {a} with a e A0 and let
{Pf,Af) be the corresponding p-pair (2.2.7). We have the standard Langlands
decomposition PF = °MFAFNF (2.2.7). Let pF be denned by pF(H) =
(i) tr(ad H\„F) for He a. We set for /e CC(K\G/K),
fp(am) = a'pr | f(nam)dnF
Nf
for a e AF,me °MF. Here dnF is some fixed choice of invariant measure on NF.
We set *PF = PnMF. Here P = P0. Set KF = K n MF. Then *PF is a minimal
parabolic subgroup of MF, with Langlands decomposition *PF = °MA*NF
with * NF = N n MF. We normalize the invariant measure, d * nF on *NF so that
dn = dnFd*nF(4.A.2A).
We note that fp e C?(KF\MF/KF). Let *F9 denote "F/' for MF. Then we
have
(i) Ff = % wither.
Now s^,// = Ad(k)H with /c e XF. Thus to prove Proposition 3.6.2, it is
enough to prove it in the case when A0 = {a}.
3.6.4. Set p0 = py (3.5.7) for y the class of the trivial representation. Let ji be
the automorphism of U{a) denned by fi(H) = H + p(H)\ for H e a. Set
7o = P ' Po- Then y0 is called the Harish-Chandra homomorphism.
Lemma. The following two statements are equivalent.
(1) y0(^(fl)K) is contained in U(a)w (W = W(q,q)).
(2) 2 = S for allseW,pe (oc)*.

94
3. The Basic Theory of (g, K)-Modules
3.5.8(2) implies that u • E„ = n{p0{u)) Ek for a11 " e ^(S)*' V e <ac)*- If
ue 17(g) then u £„(1) = Ad(/c)u • SM(1). Thus, since EM is real analytic the
result follows.
3.6.5. Lemma. // A0 = (a) then y0{U(o)K) = {he 17(a) \ sah = h}.
Set gi = [g,g]. Then dim ong, = 1. Let Xx,..., Xp be an orthonormal
basis of Pi = pngj relative to B. Set CV = 1.{XJ)2. Since Ad(X) acts
transitively on the unit sphere of Pi it is clear that S(p)K is the algebra
generated by l,p n 3(g) and Cp.Now, Kery0 = 17(g)* n 17(g)!(Lemma 3.5.9).
Hence y0{U(Q)K) = y0(symm(S(pc)*:). It is an easy calculation to see that there
are constants cx # 0 and c2 such that y0(Cv) = cx(H2 + c2). The lemma now
follows.
3.6.6. In light of the reduction in 3.6.3, Proposition 3.6.1 follows from
Lemmas 3.6.5 and 3.6.4. We note that we have also proved the following basic
theorem of Harish-Chandra [8]:
Theorem. The following sequence of algebra homomorphisms is exact:
0 ^ 17(g)* n 17(g)! ^ 17(g)* ^ 17« ^ 0.
Furthermore, y0 ° symm:S(pc)K -» 17(0)^ is a linear bijection.
3.6.7. We conclude this section with an estimate on the H„ which will
be used in the next section. We set a+ = {H e a |a(H) > 0 for a e <t>+}. Let
A+ = exp a+. 2.1.8 combined with 2.1.9 imply that
(1) G = KC\(A + )K.
Thus if feC(K\G/K) then / is completely determined by its values
onC\(A + ).
Let(a*)+ = {Juea*|(Ju,a)>Oforae<D+}. Let W = W(q,q).
(2) U fie a* then there is a unique element in WfinCl((a*)+). We use the
notation \n\ for this uniquely defined element.
Proposition 2.1.10 implies that the above intersection is non-empty. We
must therefore show that if B, a e Cl((a*)+) and if sB = a for some se W
then a = B. Lemma 3.A.2.2 implies that sB = B - Q with Q = 1, c„a the
sum over a e <D+ and the c„ > 0. Thus (B, B) = (a, a) = (B - Q, a) = (B, a) -
{o,Q) < (B,o) = (B,B) - (Q,B) < (B,B). So in particular, (Q,B) = 0. But then
(B, B) = (B, B) + (Q,Q). Hence Q = 0.

3.7. A Lemma of Osborne
95
Lemma. Let fie(ac)* then |E„(a)| < a1"6"1 E0(a) for all aeC\(A + ).
3.6.1(3) implies that |SM(gi)| < HRe/I(g). Thus we can assume that ^e a*.
Proposition 3.6.1 implies that we may assume that n = \n\. Let a e A. Then
E„(a) = | a(ka)» + l>dk = | a(n)2pa(k(n)ay + l>dn
K N
by Lemma 2.4.5. Now k(n) e Na(n)~ln. Hence
E„(a) = | a(n)~" + pa(na)" + ',dn.
JV
We also note that a(na) = aa{a~lna). We have therefore shown that
(*) E„(a) = a" + p | a(ny>, + >'a(a-lna)>, + >'dn.
If a e Cl(/1 + ) then Lemma 3.A.2.3 implies that
(a(n)~la(a'lna))" < 1.
Thus
H„(a) < a"( a" | af^^fa-'na)"^ ) = a"H0(a)
by(*).
3.6.8. Corollary. If fie (ac)* and of as C\(A + ) then SM(a) < a|Re"L
7r0 is a unitary representation of G (Lemma 1.5.3). Hence S0 < 1. The
result now follows from Lemma 3.6.7.
As we shall see in the next chapter, the above estimate is very crude.
3.7. A Lemma of Osborne
3.7.1. Let G be a real reductive group. We will, as usual, denote ZG(gc)
the elements of l/(gc) fixed under the automorphisms Ad(g) for g e G. Fix
g = f©a©nan Iwasawa decomposition of g. The main result of this
section is
Proposition. There is a finite dimensional subspace, E, of U(qc) such that
l/(nc)£ZG(gc)l/(Ic) = l/(gc).
The proof of this result will take some preparation. We first give an
application of this result (which is due to Osborne) that will be useful in later

96
3. The Basic Theory of (g, K)-Modules
developments in this book. Let 6 be the Cartan involution of g
corresponding to 6 and let K be the corresponding maximal compact subgroup of G.
3.7.2. Corollary. Let V be a finitely generated, admissible (g, K)-module.
Then V is finitely generated as a U(nc)-module.
Let Vj, j = 1,..., d be a set of generators for V as a l/(gc)-module. Let F
be a finite subset of KA such that all of the Vj are contained in the sum,
W, of the isotypic components of V corresponding to the elements of F.
Then U(ic) and ZG(gc) stabilize W. Hence Proposition 3.7.1 implies that
U(nc)EW = V. Since dim EW < oo the result follows.
3.7.3. We now begin the proof of 3.7.1. Let B be an Ad(G)-invariant non-
degenerate form on g. Then B allows us to identify S(gc) with P(g). We have
seen in 3.4.4 that
(1) S(ac) is finitely generated as a Resg/Q(S(gc)G)-module.
Let a:S(gc) -» S((a ® f)c) be the homomorphism extending the linear map
of gc to (a 0 n)c given by a(X) = 0 if X e n and a(X) = X if X e (I © o)c.
Let fi be the homomorphism of S((o©f)c) to S(ac) given by n(l) = 0 and
H(H) = H if H e a. Then (keeping in mind our identification) we have
(2) The restriction of Resg/0 to S(gc)G is equal to the restriction of fi o a.
Let F be a finite dimensional subspace of S(ac) such that S(ac) =
Resg/0(S(gc)G)F. We may assume that if / e F the homogeneous
components of / are in F.
Lemma. S(nc)S(9c)GFS(fc) = S(gc).
We prove that SJ(gc) is contained in the left hand side by induction on j.
If / = 0 this is clear. Assume this for k < j. If g e SJ(gc) then
with np, aq, kr homogeneous elements of S(nc), S(ac) and S(fc) respectively.
If deg np or deg kr is positive then npaqkr is contained in the left hand side by
the inductive hypothesis. Thus we may assume that g e S(ac). By the above
there exist fk e F and zk e S(gc)G such that g = T.k Resg/Q(zt)/t. 3.1.2(2) now
implies that g — I zkfk e S(nc)Sj^ i(ac)S(fc) (see 3.5.9 for S,). The inductive
hypothesis now implies the result for /.

3.8. The Subrepresentation Theorem
97
3.7.4. We note that the above argument actually proves that
(1) £ S'(nc)S>(Qc)GF'SVc) = S;(gc).
r+s+t+u<j
We can now prove 3.7.1. Set E = symm(F). Set V = l/(nc)Zc(gc)£l/(Ic)
and Vj = Vn Uj(qc). We prove by induction on / that Vs = Uj(q). If ;" = 0
this is clear. Assume the result for j — 1. Then V'jV'^ ' = SJ(gc) by (1). Here
we have identified Gr l/(gc) with S(gc) as in 0.4.2. This completes the proof.
3.8. The subrepresentation theorem
3.8.1. We retain the notation of the preceding section. Set P = P with
Langlands decomposition P = °MAN. If (a, Ha) is an irreducible unitary
representation of °M and if n e (a)£ then (na„,//"•") will denote the
corresponding principal series representation (3.5.5). We note that as a
representation of K, Ha" is equivalent with IK(a) = H". If f e H" then we
set fll(nak) = ap + tlf(k). We look upon napi as acting on H" with action
rtajQ)f(k) = Ukg).
Lemma. Let u, ve H", then the Junction p., g -» (naitl(g)u, v~) is a smooth
function on (ac)* x G that is holomorphic in fi. Furthermore, if His a compact
subset of (ac)* then there exists a K-invariant semi-norm q on (H")00 and
k e a* depending only on fi such that
l<tff.M(^i^2)"»f>l ^ Q(u)q(v)ax
for all kl,k2eK,ae C\(A + ) (3.6.7) and peQ.
(TtaJg)u,v} = | <MM(*(*0)),i>(*)>d* = J a(kgy + "(u(k(kg)),v(k)}dk.
K K
Here we are using the notation in 3.6.1. The first assertion follows from this
formula. If we majorize the terms in this formula by their absolute values
then we have
\<naJg)u,Vy\ < $ a(kgf"' + »dk\\u\U\v\\ao
K
where H"--^ denotes the sup norm. The second assertion now follows from
3.6.1(3) and 3.6.8.
3.8.2. Let Ha%ll denote the (p, °M)-module Ha with a acting by (n + p)I
and n acting by 0. Let V be a (g, X)-module. If T e HomgJf(K, //"■") then set

98
3. The Basic Theory of (g, K)-Modules
TA(v) = T(v)(\) for ve V. It is easily seen that TA eHomvoM(V/nV,HaJ
(use 3.5.8(1)).
Lemma. The map Ti—► TA defines a bijection between HomgA;(K, #"■") and
UompoM(V/nV,HaJ.
Let S e Homp,oM(K/nK,HaJ. We set S~(v)(k) = S(kv) for k e K and ceK
It is clear that S is a X-module homomorphism. We now show that S~ is a
(g,K)-module homomorphism. If leg then S~(Xv)(k) = kS~(Xv)(l) =
S~(Ad(k)Xkv)(l). Thus we need only show that S~(Xv)(l) = (XS~(v))(l) for
all X e g and ve V. If X e p then this assertion is clear. If X e I then it is true
by the definition of S~. The Lemma now follows from the obvious formulas
(S~)A = Sand(TA)~ = T.
3.8.3. We now come to the main result of this section. This result
combined with 3.8.2 and 3.7.2 implies that every irreducible (g, K)-module is
equivalent with a subrepresentation of some //"". This is the celebrated
subrepresentation theorem of Casselman.
Theorem. Let V be a finitely generated (g, K)-module. If V = nV then V = 0.
We prove the theorem by induction on r/cR[g, g] = dim a n [g, g]. If
r/cR[g, g] = 0 then n = 0 so the result is clear in this case. We assume the
result for all G with r/cR[g, g] < r. Suppose that r/cR[g, g] = r. We assume
that V is a finitely generated (g, K)-module with V = nV. If W is a g-module
quotient of V then W = x\W, We may thus assume that G = G° and that V
is irreducible as a g-module. Schur's lemma implies that z(g) acts by scalars
on V. Hence we may assume that G is connected and semi-simple.
We note that there exists k e K such that On = Ad(/c)n (2.1.10). Thus if we
set it = On thenrTK = V. Let A0 be as in 3.6.3. Let a e A0 and let F = A0 — {a}.
Let (PF,AF) be the corresponding p-pair with PF = MFNF, as usual. Set
*nF = nnmF and nF=0nF. 3.7.2 implies that V/nFV is finitely
generated as a l/(*nF)-module. Thus if V/nFV is non-zero then V/nV =
(K/nFK)/*nF(K/nFK) is non-zero by the inductive hypothesis. We therefore
conclude that
(1) V = riFV for all F = A0 - {a}, aeA0.
3.5.6 implies that there exists an irreducible unitary representation a of
°M and fie(ac)* such that V is equivalent with a subquotient of H"K. Let
X <= Y be (g, K)-submodules of H"K such that V is isomorphic with Y/X.
Theorem 3.4.9 combined with Proposition 1.6.6 imply that C\(X) and Cl(Y)

3.8. The Subrepresentation Theorem
99
are G-invariant subspaces of H". We set H = C\{Y)/C\(X) and set n equal
to the induced action of G on H. Then (it, H) is an admissible representation
of G with HK equivalent with V as a (g, K)-module. We therefore assume
that V = HK.
(2) There exists a K-invariant semi-norm, q, on HK and lea* such that
|c„>w(a)| < q{v)q(w)ak for all v, w e HK and a e C\(A + ).
This follows from 3.8.1.
Let A0 = {a1;..., ar}. Let Hj e a be denned by ctj(Hk) = 5jtk. Then Cl(a+) =
S R+Hy(R+ = [0, oo), as usual). We look upon the a, as coordinates
on A. Then C\(A + ) = {a e A ! a" > 1 for all ;}. If F = A0 - {o^} then we
set Uj = nF. Fix ;'. The weights of a on it,- are of the form S ntat with nke N
and Mj > 1. We therefore note that
(3) If a is a weight of a on ny then a" < a~aj for a e C\(A + ).
Let Yl,..., Yd be a basis for it,- consisting of weight vectors with respective
weight pk. Since x\jV = V, if v e V then there exist vke V such that i> = S ykuk.
Hence
<K(a)u,w> = X<t(a)>kfk,w>
t
= X <Ad(a) yt0t, w> = X a^<7i(a)i;t, 7i*(yt)w>.
Set D equal to the maximum of the q(vk)q(n*(Yk)w). Then (2) and (3) imply
(4) \(Tt(a)v,wy\<dim(n)Dax-"J ioraeC\(A + ).
Set £ = Sj a,-. If we apply (4) to all j and iterate on (4) then we conclude
(5) For each p = 1, 2,... there exists a constant Dp depending on p, v, w
such that
|<7r(a)i;,w>|<Z)paA-p?
foraeC\(A + ).
We are now ready to derive the contradiction that completes the
inductive step. Fix fl7 f2 e HK. Set f{p,g) = <Jtff>/J(gi)/1,/2> for g e G and n e (oc)*.
Let u, w be non-zero elements of V. Let F be a finite subset of KA such
that i>, wel,ef K(y) = W. Let Uj be an orthonormal basis of W. Let Cp
be (dim W)2|i>||w| times the maximum of the constants in (5) corresponding
to the Uj, uk. Then (5) implies that
(6) |<7r(/c1a/c2)y, w>| < CpaXpi for all a e C\(A + ) and kt, k2 e K.

100
3. The Basic Theory of (g, X>Modules
Lemma 2.4.2, Lemma 3.8.1 and (6) now imply that the function
Hn) = | f(p,y) conj(cv,w(g))dg
a
is holomorphic on (ac)*. But na-ll is unitary for fie ia*. If S(fi) is non-zero
with fie ia* then V would be equivalent to a subrepresentation of na„ (see
the argument in 1.3.3). Lemma 3.8.2 implies that this is impossible since we
are assuming that V = nV. Hence 3 is identically 0. Now c„-w = fin,-) for
appropriate fx, f2 and p.. Hence c„ M. = 0. This is the desired contradiction.
3.9. Notes and further results
3.9.1. Let the notation be as in 3.1. In Harish-Chandra [8] it is shown
that if dim a = r then U(a)w is isomorphic with a polynomial algebra in r
generators. Thus if we apply Theorem 3.6.6 we see that (in the notation of
3.6), U{q)k/U{q)k n l/(g)f is isomorphic with a polynomial algebra in r-
generators.
If we now move to the notation of Section 3.2, then the above result for
a implies that U(l))w is isomorphic with a polynomial algebra in dim h-
generators. Thus the same is true for Z(g) (3.2.3).
3.9.2. We now move to the notation of Section 3.3. Harish-Chandra [1]
contains the following generalization of the second part of Theorem 3.411.
Theorem. Let (n, H) and (a, V) be admissible unitary representations of G.
If HK is (g, K)-isomorphic with VK then it and a are unitarily equivalent.
3.9.3. The results of Section 3.3 imply that a connected semi-simple Lie
group with finite center is type 1 in the sense of Murray-Von Neumann (see
Dixmier [1]). This implies that the abstract Plancherel Theorem for such G
has a particularly nice form.
3.9.4. In Section 3.5 the subquotient theorem was only proven for connected
groups. One can, with a bit more effort, extend the proof to the case of
disconnected groups of inner type. However, in 3.8 we have given a proof of
Casselman's theorem (which is stronger than the subquotient theorem) for
general real reductive groups.
3.9.5. In Section 3.6 we have given some of the theory of zonal spherical
functions on real reductive groups. A more complete account of this theory

3.9. Notes and Further Results
101
can be found in Helgason [2]. In 4.5 we will prove some very sharp
asymptotic results for the SM.
The transform Ff of Section 3.6 is a special case of the "Harish-Chandra
transform" which will play a basic role in the study of orbital integrals
(7.2, 7.4).
Our proof of Theorem 3.6.6 is not the standard one. Harish-Chandra's
original proof was quite algebraic. Completely algebraic proofs of 3.6.6 and
3.1.2 can be found in Lepowsky [2]. Also Theorem 3.8.3 has a proof using
the theory of differential operators on algebraic varieties due to Beilinson
and Bernstein [1]. Earlier algebraic proofs in special cases were given
Stafford, Wallach [1] for sl(n, R) and in Wallach [2] for G linear.
3.9.6. In the course of the proof of Theorem 3.4.1 we introduced the Lie
group Gc with Lie algebra gc. The group Gc is usually called the Carlan
motion group associated with G. If V is a finitely generated (g, K)-module
and if V0 is a finite dimensional, K-invariant, subspace of V such that
U(QcWo = V then we introduced a K-invariant nitration V0 a K, <= V2 a •••
of V with U,K = V and such that Gr(K) is naturally a (gc,K)-module,
finitely generated and graded as a S(pr)-module. One can thus use the usual
theory of Hilbert polynomials to find invariants of V.
We use the usual identification of S(pc) with P(pc) (the holomorphic
polynomials on pc). Let / be the ideal of all p e S(pc) such that p Gr (V) = 0.
Then it can be shown that the radical of / (p e S(pc) such that p' e / for
some r) is independent of the choice of V0. Let X(V) = {x e pc p(x) = 0
for pel} (recall our identification). It is clear that Ad(k)X{V) = X(V)
for keK.
Assume that G is semi-simple. Let Kc be the subgroup of Int(gc)
generated by exp(ad fc). If xe pc then we say that x is nilpotent if ad x is
nilpotent as an endomorphism of gc. Let .A'\pc) be the set of all nilpotent
elements in pc. Assume that V is irreducible. Then results of Kostant,
Rallis [1] imply that X(V) is contained in , V(pc). Also in the above
mentioned paper it is proved that Kc has only a finite number of orbits on
.i'(pc). One can show that the degree of the Hilbert polynomial is equal
to max{dim Kc • x\x e X(V)} which we write as Dim V. One can show
that Dim V is equal to the Gelfand-Kirillov dimension of V (Gelfand,
Kirillov [1]). The above constructs deserve further study.
3.9.7. In 3.5.2-4 we gave we developed some results of Lepowsky,
McCollum [1] that culminated in the proof of Theorem 3.5.5. We now

102
3. The Basic Theory of (g, K)-Modules
show how one can extend these results to the case of (possibly) disconnected
real reductive groups. So let G be a real reductive group of inner type and
let K be as usual. Let H(K) denote the space of all K-finite functions on K,
under the left (hence also the right) regular action of K. Then H(K) is a
representation of K under both L and R (L(k)f(x) = f(k~lx), R(k)f(x) =
f(xk)). Set H = H(q, K) = l/(gc) (g)mH(K) with the tensor product taken
with respect to the action, L, on H(K). On H we define a multiplication as
follows
K ® /,) • (02 0 h) = | 0,(Ad(*)02) ® Mk)f2(k~l -)dk.
K
A direct calculation shows that this multiplication is well defined on H and
makes H into an associative algebra over C. If V is a (g, K)-module then we
let//(X) act on V by
/ • v = | f(k)k • vdk.
K
We write (g ® f) • v = g • f • v. We leave it to the reader to show that this
defines an //-module structure on V. We have thus canonically assigned
to each (g, K)-module, V, an H-module such that H • V = V. Such an //-
module is called faithful. One can show that the above correspondence
defines an equivalence of categories between C(g, K) and the category of all
faithful //-modules. In particular, an irreducible (g, K)-module defines an
irreducible //-module and vice-versa.
We note that if we identify H(K) with 1 ® H(K) then H(K) as an algebra
under convolution (ft * f2 = L(fl)f2) is a subalgebra of //. Let j be the
natural mapping of l/(gc) ® H(K) onto U(Qc)(g)mH(K). We put on
U(Qc)K ® H(K) the tensor product algebra structure. Then j defines an
algebra homomorphism of U(qc)k ® H(K) into //. We make H into a (g, K)-
module by letting g act by left multiplication and by setting k • (g ® /) =
Ad(k)g®L(k)f.
If y e KA then set H(K)y = {/ e H(K)\f- Vy = 0}. Then the material in
1.5.4 implies that H(K)/H(K)y is isomorphic with End(K)1) as an algebra.
If a, y e KA then in analogy with the material in Section 3.5 we set
W° = {x e H!(1 ® H(K)y)x c H(l ® //(£)„)}.
The following result is prove in exactly the same way as Lemma 3.5.2.
Lemma. (ff/ff(l ® H(K)y)(a) = Ha-y/(Ha>y nH(\® H(K)y).

3.A.I. Some Associative Algebra
103
3.9.8. The key step in the next Lemma is
(1) Let W be an admissible (g, K)-module.
Then H™\WM = j(U(qc)k® H(K))\Wly).
This is proved in exactly the same way as 3.5.3(1).
Lemma. Let W be an admissible (g, K)-module. Let y e KA and let X be a
U{QC)K and K {hence H{K))-invariant subspace of W(y). Then (H • X){y) = X.
This is proved by exactly the same argument as that in 3.5.3.
3.9.9. At this point it is a simple matter to prove Proposition 3.5.4 in the
generality that we have been studying. The argument is (as usual) the same
using (1) above to prove
(1) Y = H 0mec)K^H(K))V{y) has a unique irreducible quotient.
3.A. Appendices to Chapter 3
3.A.I. Some associative algebra
3.A. 1.1. Let A be an associative (left) Noetherian algebra over C with unit.
If V is an /1-module then we set Ann(K) equal to the two sided ideal of A
consisting of those elements of A that act by 0 on V.
Lemma. Let V be an irreducible finite dimensional A module. Let W and X be
finite dimensional A-modules. If Ann(K) a Ann(W) Ann(X) then V is
equivalent with a subquotient of either W or X.
If / is a left ideal in A then we look upon A/I as an /1-module under
left multiplication. Schur's lemma implies that A/Ann(V) is isomorphic
to a direct sum of dim V copies of V. Furthermore, every irreducible sub-
quotient of A/Ann(W) (resp. A/Ann(X)) is equivalent to a subquotient
of W (resp. X). By assumption we have a natural /1-module surjection of
A/Ann(W) Ann(X) onto A/Ann(V). We also have the /1-module exact
sequence
0 -»Ann(X)/Ann(W) Ann(X) -» A/Ann(W) Ann(X) -» A/Ann(X) -»0.

104
3. The Basic Theory of (g, K)-Modules
Now Ann(X) = S Axj for appropriate elements Xj, j = \,...,d in Ann(A').
Thus Ann^) Ann(X) = S Ann(H/)xJ-. Thus every irreducible subquotient
of Ann(A')/Ann(H/) Ann(X) is a subquotient of 4/Ann(W). The Lemma
now follows.
3.A.I.2. Let V be an /1-module such that 1 e A acts by /. If v e K and if
v* e V* we define the linear functional, cBiV*, on A by c„ „*(a) = v*(av). Let
R(K) be the linear span of the cv_„. for teF, u* e K*. If a* e A* we set
a • a*(b) = a*(ba) for a, be A. This action makes A* into an /1-module.
Clearly, R(V) is an A-invariant subspace of A*.
Lemma. Let Vj, j = l,...,d be irreducible inequivalent A-modules. Then
the sum S R(Vj) is direct.
Let V be an irreducible /1-module. Let / e R(V). Then there exists vk e V,
Hk e V* with k = l,...,p such that/ = S cUk w. We note that a • c,,„, = cavv*.
Thus each of the c„t>/Jlc is either 0 or generates a submodule of A* equivalent
to V. This implies that Af is a direct sum of a finite number of irreducible
/1-modules each equivalent to V.
We now prove the result by induction on d. If d = 1 there is nothing to
prove. Assume the result for d — 1. If the sum is not direct then there is a
non-zero / e R(Vd) such that / e Et<d R(Vk). The latter sum is direct and
Af has a non-zero projection into at least one of the summands. The above
remarks now would imply that Vd is equivalent with one of the Vk for k < d.
This contradiction completes the induction and hence the proof.
3.A.I.3. If V is a finite dimensional ,4-module then define iv(a) to be the
trace of the action of a on V.
Corollary. Let Vk be irreducible inequivalent finite dimensional A-modules
for k = l,...,d. Then the functionals xVk are linearly independent.
3.A.2. A Lemma of Harish-Chandra
3.A.2.I. The purpose of this appendix is to prove an important technical
Lemma due to Harish-Chandra [1]. As usual it will be necessary to
introduce some notation and preliminary results. We, at first, assume that G
is connected and semi-simple. We also assume that G <= GR and Gc is simply
connected. Let G„ = Gcn U(n). Then G„ is a compact form of Gc which is

3.A.2. A Lemma of Harish-Chandra
105
therefore simply connected. Fix the Cartan involution 6 for G such that
K = GunG. We also fix a a maximal abelian subalgebra of p and a
corresponding Iwasawa decomposition G = NAK of G. Fix h as in 2.2.5.
Let Rbea positive root system for <t>(gc, h) compatible with the choice of
N (see 2.2.5). Let R0 be as in 2.2.5. Let A be the set of simple roots of R. We
write A = {a1,...,a(}. Let A^e h* be denned by 2(AJ-,at)/(at,at) = djJc for
all j, k. Let V' be the irreducible, unitary, finite dimensional representation
of G„ with highest weight A, relative to R (Theorem 1.7.4(3)). Set Xj = Aj\a.
We note that a a igu. We therefore have
(1) The weights of a on Vj are of the form Xj — Q with Q a sum of elements
of R0. Also the weight spaces for distinct weights are orthogonal.
Since Gc has been assumed to be simply connected, the action of G„ on
Vj extends to a representation of Gc where the action is holomorphic. We
will use module notation for all actions.
3.A.2.2. Let(a*)+ = {fie o*!(At,a) > 0 for a e R0}.
(1) If neC\((a*)+)thenn = YJCjXj with c} > 0.
This is an easy consequence of the material in 2.2.5.
The following Lemma is used in Section 3.6.
Lemma. Let fie Cl((a*)+). Let se W($,a). Then sfi = fi — Q with Q = ~Lcaa
the sum over a e R0 and ca > 0.
By (1) we need only check this for Xj. But sXj is also a weight of a on Vj. The
Lemma therefore follows from 3.A.2.1(1).
3.A.2.3. Let +a = {He a|/z(ff) > 0 for pe Cl((a*)+) - {0}}.
(1) H eCl(+a) if and only if ////)> 0 for all;.
This is a direct consequence of 3.A.2.2 (1).
We note that exp: a -» A is an isomorphism of a (as an additive Lie group)
onto A. Let log denote the inverse mapping. We are now ready to prove the
main result of this appendix.
Lemma. Let Gbea real reductive group with Cartan involution 0 and Iwasawa
decomposition G = NAK. Set N = 9{N). Let ae Abe such that log a e Cl(a+)
and let neN. Then log(a(ana-1)) - log(a(n)) e Cl( + a).

106
3. The Basic Theory of (g, K)-Modules
Since N is contained in \_G°,G°~\ we may assume that G is connected and
semi-simple. N is also simply connected so we may also assume that G is as in
the rest of this section.
Let X be the Xj weight space for a in Vj. Then MX = X and nX = 0. Hence
Vj = l/(n)X Let X = X} and set V = Vj. If fi is a weight for a on V then we write
V^ for the /^-weight space. Then
(2) v= ^®@K orthogonal direct sum.
Now an~xa~x = k(ana~x)~xa(ana~xyin(ana~1)~x. Let veX be a unit
vector. It follows that
(3) \\an-1a-1v\\ = a(ana~l)~».
On the other hand, nv = S v (n) the sum over the weights of a and
vjin) € V^. Since iieiV, vx(n) = v. We thus see that anaT xv = a~k S a"u/J(n).
Hence
Hence (3) implies that (after replacing n by n~l) a{ana~x)~/- < a(n)~*. The
result now follows from (1).

4 The Asymptotic
Behavior of Matrix
Coefficients
Introduction
In this chapter we study the asymptotic behavior of matrix coefficients of
admissible representations of real reductive groups. Although the matrix
coefficients are complicated functions on the group, they are asymptotic to
elementary functions (exponentials and polynomials). This simple form of the
asymptotic behavior has to do with the structure of the Jacquet module. The
first two sections of this chapter give the most elementary aspects of the theory
of these modules (due independently to Casselman and the author). The
Jacquet module is a replacement for the highest weight theory for finite
dimensional representations. It is "built" out of highest weight modules and its
theory involves a slight extension of that of "Verma modules". This type of
module will also play an important (but different role) in Chapter 6.
Sections 3 and 4 contain the main analytic theorems about matrix entries.
These results sharpen and extend Theorems of Harish-Chandra. We point out
that our expansions apply to more general matrix entries than those of
Harish-Chandra. We will see that this extension leads to significant
simplifications of Harish-Chandra's theory of tempered representations. The
material in Section 4 will be used in Volume II to develop Harish-Chandra's
theory of the constant term.
Section 5 contains some basic results on the Harish-Chandra H-function
and Harish-Chandra's derivation of convergence theorems for two important
107

108
4. The Asymptotic Behavior of Matrix Coefficients
types of integrals. These theorems will be used in the next chapter to prove the
convergence of the so called "intertwining operators for the generalized
principal series". The H-function is a very specific zonal spherical function
which Harish-Chandra used in his definition of the Schwartz space (7.1). As we
shall see in the next chapter, its asymptotic behavior controls that of matrix
coefficients of "tempered" representations.
Section 4 is by far the most technical in this chapter. Since the material in
this section will not be used in this volume it can be skipped without any
serious loss to the understanding of later results.
4.1. The Jacquet module of an admissible (g, K)-module
4.1.1. Let G be a real reductive group. Fix 6 a Cartan involution and K the
corresponding maximal compact subgroup of G. Let G = NAK be an
Iwasawa decomposition of G with a contained in the — 1 eigenspace of 8. Let
P = °MAN be the corresponding minimal parabolic subgroup. Let Y denote
the category of finitely generated (g, °M)-modules, V, such that if v e V then
dim U(p)v < oo. We will now derive some properties of the objects in Y. First
we need some notation.
If V is an a-module and if [i e (ac)*, then we set
K„={d£ V\(H - n(H))kv = 0 for some k and all H e a}.
V^ is called a generalized weight space for V. Set L+ be the set of all non-
negative integral combinations of elements of <t>(P, A).
Lemma. Let V e Y. Then
(1) V= ® K„ with dim K„ < oo for n e (ac)*.
(2) There exist A,,..., A„ e (ac)* such that VA is non-zero and if V^ is non-zero
then n = Aj — Q for some j < q and Q e L + .
Let >jj, j < d be a set of generators for V as a l/(g)-module. Set W =
£ U(p)vj. Then W is finite dimensional and a-invariant. Thus W = © W^.
Put n = On. Then V = U{n) W by P-B-W. As an a-module under ad, l/(n) is the
direct sum of U(n)-Q for Q e L+. It is easily seen that each U(n)-Q is finite
dimensional. Furthermore, U^^qW^ is a subspace of V^Q. Take Ai,...,A,
to be the distinct weights of a on W.

4.1. The Jacquet Module of an Admissible (g, K)-module
109
4.1.2. Let Z(g) be (as usual) the center of U(q). If V is a Z(g)-module and if i
a homomorphism of Z(g) into C then set V = {v e V\(z — x(z))kv = 0 for
some k and all z e Z(g)}. It is clear that if V is a g-module then Vx is a
g-submodule.
Lemma. Let V be a Z(Q)-module such that if v e V then dim Z(g)i; < oo.
Then V = ®VX the sum over all homomorphisms of Z(g) into C.
If v e V then the elements of Z(g) restricted to the finite dimensional space
Z(q)v form a commutative algebra of endomorphisms and hence they can be
put in simultaneous triangular form. This is the content of the Lemma.
4.1.3. Lemma. (1) If V e f then V has finite length as a U(q)-module.
(2) Let V be a (g, °M)-module such that
i. As an a-module V = ®Vtl and each V^ is finite dimensional,
ii. dim U(p)v < <x> for v e V;
iii. There exist li,--,lp, homomorphisms of Z(g) to C such that V is the
direct sum of the VXi.
Then VsT.
We have seen that if V e f" then V satisfies (2)(i). (2)(ii) is part of the
definition of V. If V is finitely generated as a l/(g)-module and satisfies 2(i)
then let F be a finite subset of (ac)* be such that W = @ eF W generates V as
a U(q) module. Then W is Z(g)-invariant and finite dimensional. Hence W is a
finite direct sum of spaces W, i= l,...,p. Thus V = 'Ll<qU(Q)Wx and
therefore satisfies 2(iii). Thus we will have proven both parts of the Lemma if
we prove that if a g-module V satisfies the conditions of (2) then V has finite
length as a l/(g)-module.
We may assume that, in (2)(iii), p = 1. Let h be a Cartan subalgebra of gc
such that h contains a. Let <t>+ be a system of positive roots for <t>(gc,h)
compatible with <D(P, A). If p. e b* then set K[>] = {veV\(H - n(H))kv = 0
for some k and all H e h}. Since he pc, (2)(ii) implies that V = ® V\_n~].
Let n+ (resp. n~) be the sum of the positive (resp. negative) root spaces. Then
b = h + n+ is a subalgebra of pr. Let S = {A e I)*! %A = %}. Then S is a
finite set (Theorem 3.2.4).
Let M be a non-zero subquotient of V. If m e M then dim U(b)m < oo.
Thus there exists a non-zero element m in M such that n + m = 0 and such
that m e M[//| for some n e h*. Thus, the definition of the Harish-Chandra

no
4. The Asymptotic Behavior of Matrix Coefficients
isomorphism (3.2.1,2) implies that p + p e S. We have therefore shown
(*) If M is a non-zero subquotient of V then there exists pe S such that
M\_p — p] is non-zero.
Set W = 0 V[p - p\ Then dim W < oo. Let K, <= K2 <= ■ ■ ■ be an
increasing chain of [/(g)-submodules of V. Then VlnW <= K2 n H-7 <= ■ • ■ is an
increasing chain of subspaces of W. Thus there exists k such that if j > k then
VjnW = VknW. But then (F,/Kt)[> - p] = 0 for ; > /c. Hence (*) implies
that Vj = Vk for j > k. The ascending chain condition is proven in the same
way.
4.1.4. We now introduce another category of g-modules. Let 3/f denote the
category of all finitely generated, admissible, (g, K)-modules. If V is a g-
module then we set K*[n] = {p. e V*! nkp = 0 for some k}.
Lemma. If V is a Q-module then K*[n] is a Q-submodule of V*. If Ve J4?
then V*[n']er.
Lemma 2.2.6(1) implies that there is a positive integer p such that
(ad(n))pg = 0. If p e K*[n] then {n)kp = 0 for some positive integer k. Hence
{n)k + pQp = 0. This implies the first assertion of the Lemma.
We now begin the proof of the second assertion. We first observe that Z(g)
is finitely generated as a ZG(g)-module under left multiplication. This is
proved using the argument in the proof of Lemma 3.4.4 and the Harish-
Chandra isomorphism (3.2.3).
Let V e #e. Then V = ®yeK, V(y). Since Kis admissible, dim V(y) < oo for
all y e K\ Clearly, ZG(g)K(y) = V(y) for all y e K\ Thus, we see that if v e V
then dim Zc(g)i> < oo. The above observation implies that dim Z(g)i> < oo
for all v e V. Since V is finitely generated, we may argue as in 4.1.3 to prove that
there exist a finite number Xi, ■ ■ ■, Xd of homomorphisms of Z(g) to C such that
V = ® V'. Each V' is finitely generated. Hence there exists a positive integer
q such that if v e V' then (z — Xi {zj)"v = 0. If % is a homomorphism of Z(g) to
C then we set xT(z) = x(zT). Then V* = ®j£d(v*s)* = ®j<d(v*)^- Hence
the same is true for K*[n]. So K*[n] satisfies (2)(iii) of 4.1.3.
K*[n] = Uk>0(V/(n)kV)*. Since Ve Jt, 3.7.2 implies that (V/(n)kV)* is a
finite dimensional (p,°M)-submodule of V*. Hence K*[n] satisfies condition
2(ii) of 4.1.3, and it is a (g, °M)-module. We are left with proving condition
4.1.3(i).
Set ^-[n] = [p e V* | (n)V = 0}. Then K*[n] is the union of the ^.[n] = Vj
and nVj <= Vj^.l. Clearly, V0 = 0, thus n^ = 0. If p e Vj and if p is not in V^ ,

4.1. The Jacquet Module of an Admissible (q, /f)-module
111
then there exist Xke n, k = 1,..., / - 1, such that 0 # A", ■■■XJ-lne K,. This
implies that if (Vj)^ ¥=(Vj_l))l then n = A - /?, -■■■-/?,•_, with A a weight of
Ki and j8k e <D(P, /I), /c = 1,...,./' - 1. Let H e a be such that a{H) > 1 for
a e <D(P, /I). Set m = max{Re A(H) \ A a weight of a on K,}. It follows that
if (% ^ (^- i)„ then Re M#) < m - ; + 1. We conclude that if n e (oc)*
then there exists ;' such that K*[n]M = (Vj)^. This shows that K*[n] satisfies
all of the conditions of Lemma 4.1.3(2).
4.1.5. If V e Jt then we set;n(K) = K*[n]. If n is understood we will use the
notation j{V). We call j(V) the Jacquet module of V. If V, W^arefg^j-modules
and if a e Homg K(V, W) then we denote by a* the g and K homomorphism of
W* to V* given by <x.*(p) = ^°afor^e W*. If V, WeJt then we set ;'(a) equal
to the restriction of a* to j(V). With these definitions V-~>j(V) is a functor
from the category 3fC to the category Y. The following result says that j is an
exact and faithful functor. It clearly implies that j is a powerful link between
our categories Y and Jf.
Theorem. (1) If V e Jt and if V is non-zero then j(V) is non-zero.
(2) //
0-/l^>B-^C-0
is an exact sequence in 3fC then
0^j{C)mmM.j(A)^0
is an exact sequence in Y.
Theorem 3.8.3 implies that if V # 0 then K/nK# 0. Since (V/nV)* is a
subspace of j(V), (1) follows. We now prove (2). It is easy to see that
0-./(C)-./(B)-,(.4)
is exact. Thus the content of (2) is that;'(/i) is surjective. We may assume that A
is a submodule of B. We must therefore show that if \i ej(A) that \i extends to
an element of j(B).
If fie j(A) then n e /l*[n] for some k (here we are using the notation in
4.1.4). Proposition 0.6.4 applies to the two sided ideal nU(n) of U(n). Since
B e Jt, Corollary 3.7.2 implies that B is finitely generated as a [/(tt)-module.
Thus there is a positive integer p such that ((n)p+kB) n A = (rt)t((n)pB n A) <=
(n)kA. Now if neAt[n] then ne(A/(n)kA)* <= (A/((n)p+kBnA)))* <=
(B/(n)p+kB)* = B*+k[n\. The result now follows.

112
4. The Asymptotic Behavior of Matrix Coefficients
4.2. Three applications of the Jacquet module
4.2.1. We retain the notation of the previous section.
Theorem. // V e J4? then V has finite length as a Q-module.
Let Vy Z3 V2 => ■ ■ ■ be a decreasing chain of submodules of V. Set Mj =
{nej(V) n(Vj) = 0}. Then M, a M2 <= ■■■ is an increasing chain of sub-
modules of j(V). Lemma 4.1.4 implies that;'(K) has finite length as a g-module.
Thus there exists k such that Mj = Mk for; > k.
The exact sequences (in Jt)
0 - Vj - K- V/Vj - 0
induce the exact sequences (in Y)
0^j(V/Vj)^j(V)^j(Vj)^0
by Theorem 4.1.5. Clearly, the image of j(V/Vj) in;'(K) is M,. We therefore see
that j(Vk) is equal to j(Vj) for all j > k. We now consider the exact sequences
(in Jf)
These sequences induce the exact sequences (in Y~)
0-7(Kt/^-)-7(K)-7(K;-)-0.
This and the above imply that if / > k then j(Vk/Vj) = 0. Thus Vk/Vj = 0 by
Theorem 4.1.5.
This result is usually proven using Harish-Chandra's regularity theorem for
characters (see Chapter 8).
4.2.2. In order to prove the next result we will need some notation. Let (a, Ha)
be a finite dimensional representation of P which we assume (without any
loss of generality) is unitary when restricted to PnK = °M. Let (na,Ha)
be the corresponding induced representation (1.5.4) of G. Let X" denote the
(g,K)-module, (H")K. We note that as a K-module, X", is isomorphic with
the space of K-finite vectors in IndM(a|oM). Thus 1.5.5(2) implies that X"
is an admissible (g, K)-module. In particular this implies that X" <= (H")<a
(1.6.6, 3.4.9).
If V is a (g, K)-module then V/nkV is naturally a (p, °M)-module for each
k = 1,2, It is also easy to see that if a is as above then there exists r such
that nrHa = 0 (cf. the argument in the proof of 2.2.6). Let T e Horn, K(V, X").

4.2. Three Applications of the Jacquet Module
113
If v e V then we set TA (v) = T(v)(l). Ker TA contains nrV. Hence TA induces
an element of Homp oM(V/nrV,Ha).
Lemma. The correspondence T\—► TA is a linear isomorphism of
HomBiJf(K,X") onto Homp,„M(K/nrK,HJ.
The proof of this result is identical to the proof of Lemma 3.8.2.
4.2.3. Proposition. Let V e ,W. Then there exists a finite dimensional
representation, a, of P and an injective (g, K)-homomorphism of V into X".
Since V is finitely generated as an [/(n)-module (3.7.2). V/nkV is finite
dimensional for k = 1, 2,.... AN is simply connected, thus V/nkV integrates
to a representation of P. Let ak denote this representation. Let Ik denote the
identity map of V/nkV onto itself and let Tk = /A . Then Ker Tk contains nkV.
We assert that f]k Ker Tk = (0). Indeed, let V^ denote f] nkV. Then V^ is a
g-submodule of V that is easily seen to be a (g, K°)-submodule that is
finitely generated as a l/(g)-module. 0.6.4 implies that there exists k > 0 such
that nk + rVn Vx = n'fn'Kn KJ for r > 0. Thus Vx = nVx. Hence 4.1.5(1)
implies that Vx = (0). This implies that f) Ker Tk = (0). Now, Ker Tt =.
Ker Tk+ ,. Hence, Ker Tt = (0) for k sufficiently large, since V has finite length
(4.2.2). This completes the proof.
The following result is due to Casselman, however his original proof was
much more complicated.
4.2.4. Corollary. Let V e J4? then there exists a Hilbert representation of G,
(n, H), such that V is equivalent to HK.
Let a be as in 4.2.3 and let T be an injective element of Homg K(V, X"). X"
is contained in (Ha)<° (3.4.9) since X" is admissible, hence C1(T(K)) is a
G-invariant subspace of H". Since X" is admissible, it is also clear that
C\(T(V))K = T(V). Take H = C1(T(K)) and n the induced action of G.
4.2.5. If V e JC and if (n,Jf) is an admissible Hilbert representation of G
such that HK is equivalent to V as a (g, K)-module then we call (it, H) a
realization of V. The content of 4.2.4 is that every V e Jt has a realization.
4.2.6. Our next application of the Jacquet module is a technically useful
criterion for admissibility due to Stafford and the author.

114
4. The Asymptotic Behavior of Matrix Coefficients
Theorem. Let V be a (g, K)-module that is finitely generated as a U(n)-module.
Then V is admissible.
Let VJ[ri] = VJ be as in the proof of Lemma 4.1.4. Then VJ = (V/nJV)*.
Hence dim VJ < oo. The argument at the end of the proof of Lemma 4.1.4
implies that K*[n] is a direct sum of generalized weight spaces and each
generalized weight space is finite dimensional. Let x be a homomorphism of
Z(g) to C. Let n e (K*[n])*. Suppose that [i e VJ but that fi is not an element
of K*_,. Then there exist elements Xken, k = l,...,j—l such that
Xi---XJ_in is a non-zero element of V*{. This implies that if (K*[n])z is
non-zero then so is (K*[n])* n V\. Now 4.1.3(2) implies that K*[n] e 'f.
Let Vab = {ve V\n(v) = 0 for all n e K*[n]}. Then V„ = f] nJV Thus
Proposition 0.6.4 implies (see the proof of 4.2.3) that nVx = Vx. Now Vx is a
g-submodule of V (Lemma 4.1.4), hence Vx is a finitely generated (g,X0)-
module. Theorem 3.8.3 now implies that Vx = 0. The proof of Theorem 4.2.1
only uses the following properties of V: it is finitely generated as a U(n)-
module and Vx = 0. That argument therefore proves that V has finite length.
Since an irreducible (g, K)-module is admissible (3.4.8), V is admissible.
4.2.7. Corollary. If V is a finitely generated, admissible (g, K)-module, then
V is finitely generated and admissible as a (g, K°)-module.
3.7.2 implies that V is finitely generated as a l/(rt)-module. The result now
follows from the previous theorem.
4.3. Asymptotic behavior of matrix coefficients
4.3.1. Let G be a real reductive group. We will assume throughout this
section that G° = °(G°). We retain the notation of the previous sections. Let
A0 be the set of simple roots of <t>(P, A). Let F be a subset of A0 and let (PF,AF)
be the corresponding standard p-pair.
Lemma. Let V e H. Then V/nF V is an admissible finitely generated
(mF, K n PF)-module.
Let *nF = n n mF. Then *nF is the "n" for a minimal parabolic subgroup
of MF. 3.7.2 implies that V is finitely generated as a l/(n)-module. Hence
K/nFK is finitely generated as a l/(*nF)-module (n = *nF©nF). The result
now follows from Theorem 4.2.6.

4.3. Asymptotic Behavior of Matrix Coefficients
115
4.3.2. Let V be an admissible (g, K)-module. Then we denote by V~ the space
of all elements, fi, of V* such that Kfi spans a finite dimensional subspace of
V* (here kn(v) = n(k~lv), as usual).
Lemma. // V e Jf then V~ e Jf.
Let VJ a VJ <= '■■ be an increasing chain of submodules of V~. Set
Vj = {v e V \VJ(v) = 0}. Then K, => V2 => ■■■ is a decreasing chain of sub-
modules of K Now K is of finite length (4.2.1). Hence there exists k such that
Vj = Vk for all j > k. Since V is admissible, VJ ={ne V~\n(Vj) = 0}. We
therefore see that VJ = VJ for j > k. Thus V~ is finitely generated. Since
V~ is clearly admissible (V~ = ® V(y)*), V~ e H.
4.3.3. Let (rc, H) be a Hilbert representation of G. Let (H00)' be the space of
all continuous linear functionals on H x. If g e G (resp. X e g) and if \i e (H x)'
then we define gp. (resp. Xp) by g^(t;) = n{n(g~' )i>) (resp. AT^(t;) = - n(n(X)v))
for u e (H™)'. Then 1.6.4(ii) implies that
(1) gXn = (Ad(g)X)gn for 0 e G, X e g and n e (ff»)'.
Set (H33)* equal to the space of all ji e (H00)' such that Kp spans a finite
dimensional space. Then (1) implies that (HX)'K is a (g, K)-module.
Let for v e H, a(v) e H' be defined by o(v)(w) = <w, i>> for we//. Then a is
a conjugate linear continuous isomorphism of H onto //'.
Lemma. // (n, H) is admissible then (H™)^ = a(HK). Furthermore,
(H*>)'k = (HK)~.
This is clear, since dim H(y) < oo for each y e KA.
4.3.4. Let V be an admissible finitely generated (g, K)-module. Let F be a
subset of A0 and let (PF,AF) be the corresponding standard p-pair. Then
V/nFV is an admissible finitely generated (mF,KF) module (here KF =
MF n K). Since aF is contained in ZM(mF), this implies that
(1) V/nrV= ® (K/rtFK)„ the sum over n e ((oF)c)*.
Furthermore, there exists d such that (//- ^(//)d(K/nFK)/I = 0 for all
\i e ((aF)c)* and H e aF.
Set £(PF, V) = {n e((aF)c)*| (K/uFK)„ * 0}.
(2) £(PF,K) = {^|Q ^e£(P,K)}.

116
4. The Asymptotic Behavior of Matrix Coefficients
Indeed, V/nV = (V/nFV)/*nF(V/nFV). If deE(PF,V) then 3.8.3 implies
that *nF(V/nFV)g + (V/nFV)g. (2) now follows.
4.3.5. LetA0 = {a,,..., ar}. Define //,,..., Hr e a by a,-(fl,-) = c5(J. If Ke Jf
then we define Av e a* by
Av(Hj) = max{ -Re M^) !ji e E(P, V~)}.
Fix a norm ||-|| on G (2.A.2.3). The following Theorem generalizes an
unpublished result of Harish-Chandra.
Theorem. Let (n, H) be a finitely generated, admissible, Hilbert representation
of G. Set V = HK and A = Av. There is a positive constant d such that if
ne^^Yx then there exists a continuous semi-norm, a^, on Hx with the
property that
\(li(it(a))v)\ < (1 + log ||fl||)"flAflr»
forveH*'andaeC\(A+).
Let \i e (HX)'K. 4.3.3 implies that fi = a(w) with w e HK. Lemmas 2.A.2.2
and 2.A.2.3 imply that there exists Sea* and C > 0 such that if x, y e H then
|<7r(a)x,);>|<Ca*||x||||);|| for a eCl(.4+).
This clearly implies that if fi e (H06)^ then there exists, a'^, a continuous
semi-norm on Hx such that
(1) |(^(7t(a))u)| < aV;(c) for veH* andae C\(A + ).
The idea of the proof is to show that if d(Hj) > A(Hj) then we can replace
5 in (1) by 5 - mXj with m = min{l/2,^(//j) - A(H;)} at the cost of possibly
changing the semi-norm a'^ and putting in a term (1 + log ||a||)''.
Let a e A0. Set F = A0 - {a}. If a = a^ then set H = Hj. Then aF = RH.
Set a, = exp(tH). If a e C\(A + ) then a can be written uniquely in the form
a = a'a, with a = exp(Z xkHk), xk > 0, Xj = 0 and t > 0.
Let q be the canonical projection of V~ onto V~/nFV~.
(2) If q(n) = 0 then there exists a continuous semi-norm t^onW" such that
\n(n(a)v)\ < a^^y'^v), for a e C\(A + ) and v e H™.
Let Xy,..., Xp be a basis of nF consisting of root vectors for a corresponding
to the roots /?!,...,/?„ respectively. Out assumption implies that \i = S Xkfik

4.3. Asymptotic Behavior of Matrix Coefficients
117
with nk e V~. Hence
\(fi(n(a)v)\ = \lXkfik(n(a)v)\ = 1-1^(71(^)^)0)1
< I \fik(n(aMAd(a'1)Xk)v)\ = I a-'\»k(it(a)it(Xk)v)\
ZXa'-'oMXJv))-
(2) now follows from 3.8.6(1).
Let for z e C, (V~/nF)V~)2 denote the generalized eigenspace for H with
eigenvalue z. Let Pz be the projection of (V~/nFV~) onto (V~/nFV~)z
corresponding to the //-weight space decomposition. Let n e V~. Then q(fi) =
S Pzq(fi). Let fiz e V~ be such that q(fiz) = Pzq{n)- Then p. - E p.z e nFV~.
We now estimate fiz(n(a)v) for each z. Set \iz = v. Let vt,..., vp be a basis for
l/(aF)g(v). We assume that v, = q(v). Let vt e K~ be such that q(vk) = vk for
/c > 2. Now
Hvk = £ ^„v„
and B = [bkn~\ has the property that
(3) (B - zl)' = 0.
We also note that
(4) ak = Hvk-YJbknvnenFV~.
Let a' e C\(A+) be such that (a'f = 1. We set
Vi(it(ata')v
F(t,a';v
and
G(t,a',v)
vp(n(ata')v)
(Ji(n(ata')v)
a„(n(ata')v)
it
F{t,a'\v) = -BF(t,a';v) - G(t,a';v)
Then
(5)
This implies that
r
(6) F(t,a'; v) = exp(-tB)F(0,a';v) - exp(-tB) J exp(sB)G(s,a';v)ds.

118
4. The Asymptotic Behavior of Matrix Coefficients
We now estimate the terms in (6). (1) implies that
(7) ||F(0,a';i>)|| < (a')sP(v) with P a continuous semi-norm on H™.
(2) implies that
(8) ||G(t,fl';o)|| < exp((5(ff) - \)t)(a')sp'(v) with p' a
continuous seminorm on H00.
(9) ||exp(sB)|| < C(l + |s|)VRez for s e R. Here p < d
(see the beginning of the proof).
This follows immediately from (3). These estimates imply that if t > 0 then
||F(t,fl';o)||< C(l + t)"e''Rez(a')dp(v)-(l + } (1 + Syes{Rez + SiH)'l)ds\
for some continuous semi-norm P on H™ and some positive constant C.
We observe that (1 + s)"e~cs is bounded by a constant C for e > 0 and
s > 0. We therefore have
(10) ||F(t,fl';o)|| < C(l + O'e"'R"(fl')*0(») + C(l + t)'e,l«U)-2'3)(a')ap(v)
for t > 0.
Here C is a positive constant and fi is a continuous semi-norm on H™.
There are now two cases.
Case I: 5(H) — | < A(H). Then there is a continuous semi-norm, fi, on H™,
such that
||F(t,fl';o)|| < (1 + t)"e,MH)(a')*p(v), for t > 0.
Case II: 3(H) — | > A(H). Then in (1) we may replace 3 by 3 — (\)a (after
having argued as above for all a). We may clearly iterate the argument
leading to (10). After a finite number of steps we will be in Case I.
If we apply this argument to all simple roots, the desired estimate follows.
4.4. Asymptotic expansions of matrix coefficients
4.4.1. In this section we show how the technique of the last section can be
refined to prove asymptotic expansions of certain matrix coefficients of an
admissible finitely generated Hilbert representation. We retain the notation
and assumptions of the previous section. Let F be a subset of A0, then we
have the corresponding standard p-pair (F>, AF).

4.4. Asymptotic Expansions of Matrix Coefficients
119
Let (rc, H) be an admissible finitely generated Hilbert representation of G.
Set V = HK. As in 4.3.3 we identify (H™)'K with V~. Set KF = KnMF.
Lemma 4.3.1 implies that V~/(nF)kV~ is an admissible finitely generated
(mF, KF)-module. Since aF is a subspace of ZM(mF), we have
(1) (nF)kV~/(nF)kV~ splits into the direct sum of finitely many generalized
weight spaces for aF. Let Ek denote the corresponding weights.
Here we write (nF)°K~ = V~.
4.4.2. Lemma. Ek + l <= {fi + x\xe<f>(PF,AF), fieEk}.
Let Sk:nF®((nF)kV~/(nF)k+[V~)^(nF)k + lV~/(nF)k + 2V~ be defined by
Sk(x <g> (u + (nF)k+' V) = xu + (nF)t + 2K~.
Then Sk is a surjective aF-module homomorphism. Since the weights of aF
on nF are precisely the elements of <t>(PF, AF), the Lemma follows.
4.4.3. Set£= (j£t.Then
(1) £ <= {n + <x\ne E(PF, V~\ a a sum of elements of <t>(PF,AF)}.
Let S = {j\oCj e F} (here we are using the notation of 4.3.5). Let L+ =
{ZjeS rijXj! rij e N}. In this notation (1) implies
(2) -E <={n-x\ne -E(PF,V~), aeL+}. Furthermore, if 3 e -Ek
then 3 = \i — a with \i e - E(PF, V~) and a = 2.JeS rijXj with Z rij > k.
If n, 3 e (aF)* then we say that n>dif p — de L+. Let £° be the set
of all maximal elements of -£(they are clearly contained in -£0 =
Set
*AF = An°MF.
Then
A = *AFAF and C\(A+) = (*AF n C\(A+)) C\((AF))+.
Let d and Av be as in 4.3.5.
Theorem. Let ae V~. If ue £°, Q e L+ and v e H™ then there exists a
polynomial of degree at most d on aF, pf^Q(H; a, v) such that
(i) The map aF ® ff°° -> C, h, v i—> p^ Q(h; a, v) is continuous and linear in v.

120
4. The Asymptotic Behavior of Matrix Coefficients
(ii) If H e (aF)+ then a(7i(exp tH)v) is asymptotic to
I exp(t/z(ff)) £ exp(-tG(ff))pM,Q(tff;ff,i>)
as t-> +00. (4.A.I.I.)
(iii) If ae*AFn C\(A + ) then
|pM.Q(H;ff,n(fl)i;)| < (1 + ||ff||)'(l + log llfllD'fl^o)
with A = Av, and B is a continuous semi-norm on H™ (depending on p. and Q).
Fix H as above. Set a, = exp tH. Put (p(H)\p e — £} = {z,-} with Re z, >
Re z2 > •••. Let for each j, kj be denned to be such that if geL+ and
Q = E«6f "«a« with s ", = ^ then Re zj > (A - 6)(tf).
Let qk denote the natural projection of V~ onto V~/(nF)kV~. Let N be a gap
in the sequence {z,-} (4.A.1.1). Set k = kN.
(I) If qk(p) = 0 then there exists e > 0 and a continuous semi-norm BN on
H1" such that
\(p(n(a,a)v)\ < exp(t(Re zN - e))(l + log \\a\\)daABN(v)
fort> l,aeCl(/l+)andf;e//(:0.
Let X,- and Bj be as in the proof of (2) in 4.3.5. Since qk(p) = 0,
p = 1, Xj. • • • Xjkpj.iiiJk for some pj; Jk e V~. Thus if a e C\(A+) then
|^(K(a)i7)| = \Llijl,...jMXji-XjR)ii(a)v)\
< £ a-fij.--fiim | H jk(n(a)7t(Xjr ■ X]R)v)\
<(1 +log||fl||)VSfl-^--^ffwii Jrt(Xjr-XjJv).
Here we have used Theorem 4.3.5. The last inequality clearly implies (I),
Suppose that qk(p) is non-zero. Let pl = qk(p),...,pp be a basis of
U(&F)<Ik(li)- If xeaF then xpj = I bJr(x)pr with bjr e ((oF)c)*. Let B(x) =
\_bjr(x)~] for x e aF. Let p. = py,..., pp e V~ be such that qk(pj) = ps. Then
XVj = £ bJr(X)Hr + lj
with jj = yp) e (nF)kV~.
Let a be as in the statement of the Theorem, then we set
F(t,a';v) =

4.4. Asymptotic Expansions of Matrix Coefficients
121
and
G(t,a';v)
Then as in 4.3.5 we have
(II)
j\F(t,a;V)
yi(n(a,a')v)
yp(n(ata')v)
■B(h)F(t,a;v)-G(t,a;v).
This implies that
(III) F(t,a;v)
t
= exp(-tB(h))F(0,a\v) - e\p(-tB(h)) | e\p(sB(h))G(s,a;v)ds.
o
Let Q be the projection of C" onto the direct sum of the generalized
eigen-spaces for —B(h) with eigen-value whose real part is less than Re zN.
Then if we argue as in the proof of 4.3.5 we find that if t > l(B = B(h))
(IV)
\\Q(e-'BF(0,a\v) - e~'B ] esBG(s,a;v)ds)\\
< exp(t(Re zN - e))(l + log \\a\\)"aAp(v
with fi a continuous semi-norm on H°°.
As in the proof of 4.3.5 we find that if R = I — Q and if t > 1 then
(V)
\Re'B G(t,a;v)\\ < e'"(\ + log ||a||)daAj8(i;)
with fi a continuous semi-norm on //"
(V) implies that
| R(esBG(s,a;v))ds
converges absolutely.
Set
F°(t,a;v) = e'BRF(0,a;v) - e"> J R(esBG(s,a;v))ds.
Then
RF(t,a;v) - F°(t,a;v) = -e~'B | R(esBG(s,a;v))ds.

122
4. The Asymptotic Behavior of Matrix Coefficients
A straightforward estimation shows that there exists e' > 0 such that
00
\\e',B | R(esBG(s,a;v))ds\\ < (1 + log \\a\\)daA exp(t(Re zN - e))p(v)
t
for t > 1 with ji a continuous semi-norm on H™.
Set fN(t,a;v) equal to the first component of F°(t,a;v). Then fN(t,a;v) =
2,j<Nexp(tZj)ujN(t,a;v), with uJN(-,a;v) a polynomial in t of degree at
most d. If t > 1 then the above inequalities imply that
(VI) |(|z(fl,fl)o) " /nM;«OI < exp(t(Re zN - e)(l + log \\a\\)daAp(v)
with j8 a continuous semi-norm on H.
If M is a gap of the sequence {z,} and M > N then the above estimates imply
that Uj N = Uj M if ;' < N. We set pz.(t; v) = plN(t, 1; v) for Af > ;'. We have at
this point shown that
(VII) fi(n(a,)v) is asymptotic to the exponential polynomial series
I exp(z;t)pZj(t; v) as t -> + oo.
We now refine the above argument to prove the Theorem. Let for e > 0,
SFc = {heaF\\\h\\ = 1 and oc(h) > e for all ae $(Pf,^f)}. If h is a non-zero
element of aF then set a{h) = h/\\h\\.
(VIII) If HuH2e SF.£ and if t, s > 0 then a{tHl + sH2) e SFit.
This is an easy consequence of the triangle inequality.
Setr(e) = ma\{A(h)\h e SFJ. (A = Av.)
(IX) If h e SFit, At e - £t then Re At(/i) < r(e) - /ce.
This follows from 4.4.3(2).
Set —k£ equal to the set of weights of aF on V~/(nF)kV~. Set Fk =
Uj>kjE. Then (IX) implies
(X) If ^ e Fk then Re n(h) < r(e) - /ce for h e SF,£.
Put Ek'e = {n e kE! Re At(/i) > r(e) - /ce for h e SF<e}. Since SF e is compact it
is easy to see that
(XI) There exists 3 > 0 such that if h e SF e and if ^ e Ek-e then Re n(h) >
r(e) — ks — 5.
Let At e V~. Fix /c > 0. Let At!,..., AtP> Ati = At.---. Atpand ri,--, yp, and B(/i)
be as above. Then the eigenvalues of B(h) are the 0(/i) with 8 e kE. Let P = Pk be
the projection of Cp onto the sum of the generalized eigenspaces for - #(•) that

4.4. Asymptotic Expansions of Matrix Coefficients
are elements of Ek,t. We set
~Hi(»)
123
F(v)
and
G(v)
yP(v)
for v e Hx.
Then (as usual),
dt
F(n(exp t%) = - B(h)F(n(exp th)v) - G(n(exp th)v).
This implies that
(XII) F(jr(exp th)v) = e'mh,F(v) - } e'{''s,B{k,G(Tt(exp sh)v)ds.
o
Set Q = Qk = I - P. The standard estimates yield
\\QF(n(exp th)v)\\ < exp(r(e) - ke)t)(l + t)2dP(v) for t > 1, h e SF,£
and P a continuous semi-norm on H™.
We also note that (in light of (XI)) if s > 0 then
\\esmG(n(exp sh)v)\\ <(1 + s)2V**",(*)"*)V<r(8)"**)0(p).
Here we have used the obvious estimates in order, and /? is a continuous semi-
norm on H33. This implies that the integral
00
I emh)PG(n(expsh))ds
o
converges absolutely and uniformly for h e SFe. We set
00
Fkh(v) = PF(v) - | esB(h)PG(7r(exp s/i)i;)ds
o
for i> e ff33 and /i e SF e. The above estimates imply that
(XIII) ||F(7r(exp tfi)p) - e',mFkh(v)\\ < (1 + t)2d exp(t(r(e) - ke))p(v),
for /i e SF e, tefl" and /? is a continuous semi-norm on ff33.

124
4. The Asymptotic Behavior of Matrix Coefficients
This implies that
(XIV) lim e'mPF(n(exp th)v) = Fkh(v) for h e SF,£.
t-> + oo
It follows that
(XV) FM(7i(exp th)) = e''mFkh(v) for h e SF,£ and v e H1".
We now assume that k has been taken so large that k — r(e) > 0. Let d
be as in (XI). Let 0 < c < 1 be such that k - r(e) - (£)<5 = (/ce - r(e))c. Let H,
and H2 e SF,£ be such that (Hl,H2} > c. (We note that <//,//'> > 0 for
H, H' e aF.) It is easily checked that if t,s > 0 then HtH, + sff2ll > cs + t.
We leave it to the reader to show (using the above) that
||exp(tB(ff2))(FtiIfl(rt(exp(tff2)i>) - PF(7r(exp tff2)i>|| < (1 + t)2de~»'2p(v)
for t > 1 and ft is a continuous semi-norm on H33.
This implies that limr^ + 00 e'B("2)Fk Hl(7r(exp tH2)t;) = Ft Hl(v). Hence
lim lim eB,,H'+sH2)F(7r(exp(tH1 + sH2))v) = FkHl(v).
S~* + 00 t~* + 00
We therefore see that
Ft.„2(7r(exp tHy)v) = exp(-tB(H1))Ft,H2(i;).
If we interchange the roles of Hy and H2, this implies that
lim exp(tB(ff2M,Ifl(jt(exp(tff2))i>) = FkiHl(v).
t-> + oo
We have (finally) shown that Fk Hl = FkIi2. Since
UfceS^'eS^ </!,/!'> >C}=SF,£
this implies that Fkk is independent of h e SFe. Set FKe equal to the common
value of the Fkh.
If we combine all of the above we have
(XVI) ||F(7r(exp th)v) - e"'B(A)FM(i;)|| < (1 + t)2de'(r(e)-ke)p(v)
for t > 1, h e SFj, and /? is (you guessed it!) a continuous semi-norm on H™.
We now note that if we choose a smaller e then Fke will not be changed. We
may therefore denote Fke by Fk. If we now combine (XVI) with (VII) the
theorem now follows.

4.5. Harish-Chandra's E-function
125
4.5. Harish-Chandra's H-function
4.5.1. We retain the notation of the previous sections. Let V be an admissible
(g, K)-module. Let V~ be as in 4.3.2. Then V~ is also an admissible (g, K)-
module. The next result gives a characterization of V~.
Lemma. Let W be a (g, K)-module. Suppose that there exists a complex
bilinear mapping b: V x W -> C such that
(1) b(Xv,w) = -b(v,Xw),b(kv,kw) = b(v,w)
for v e V, w e W, X e g and k e K.
(2) If b(v, W) = 0 then v = 0 and if b(V, w) = 0 then w = 0
(i.e., b is non-degenerate).
Then W is (g, K)-isomorphic with V~.
If w e W then set T(w)(v) = b(v,w). Then T defines a g and K-module
homomorphism of W into K*. Thus T(W) is contained in K~. The non-
degeneracy of b implies that T is injective. If y e KA let y* denote the class of
the dual representation of any representative of y. Then (2) combined with the
X-invariance of b implies that dim W(y*) = dim V(y). Hence T is surjective.
4.5.2. Let (najJ,//"•") be as in 3.5.5. Let a" denote the dual representation
to a.
Lemma. (H"-")^ is isomorphic with(H"~~l')K.
Let / e (Ha-»)K and g e(H"~-'")K. We set
<f,gy = $<f(k),g(k)}dk.
K
Then Lemma 2.4.1 implies that < , > satisfies 4.5.1(1). We leave it to the
reader to prove that < , > satisfies 4.5.1(2).
4.5.3. We are now ready to study the H-function. Let H^ be defined as in
3.6.1. We set E = 30. We have followed Harish-Chandra in giving this zonal
spherical function a special name. Lemma 3.6.7 indicates its special role. Also
the function E will be used in the definition of Harish-Chandra's Schwartz
space (7.1).

126
4. The Asymptotic Behavior of Matrix Coefficients
Theorem. There exist positive constants C and d such that
a-" < 3(a) < Cfl-'(1 + log ||a||)d
foraeC\(A + ).
Let (n,H) denote (n0,H°) (see 3.6.1). Under the pairing < , >, HK —
(HK)~. Let 10 be as in 3.6.1. Then E(g) = (n(g)l0,10>. Set V equal to the
(g,X)-submodule of HK generated by 10. Then under < , >, V~ — V.
Suppose that p + p is a weight of a on V/nV. Let a be an °M-type of the p + p
weight space of V/nV. Then Lemma 3.8.2 implies that there is a non-zero
element of HomgK(V,(H")K). Frobenius reciprocity implies that a must be the
trivial °M-type. Now this implies that E/1 = E0. Theorem 3.6.6 now implies
that p. = sO for some element in W(G, A). Hence p = 0. We therefore conclude
that (in the notation of 4.3.5) Av = —p. The upper inequality now follows
from Theorem 4.3.5.
We now prove the lower inequality. Formula (*) in 3.6.7 says that
E(a) = a" [a(nYa(a'lnaf dn.
We make the change of variables n i—► ana'1. Then we have
3(a) = a'" | (a(ana'l)a(n)'lya(n)2l}dn.
N
Lemmas 2.4.5 and 3.A.2.3 now imply the first inequality.
4.5.4. We now show how Harish-Chandra used the above result to prove the
convergence of two important integrals. These results will be used in the next
chapter to prove the conversion of the intertwining integrals of Kunze-Stein,
Knapp-Stein and Harish-Chandra. Our exposition follows that of Harish-
Chandra [8].
Theorem. Let d be as in Theorem 4.5.3. If e> 0 and if F is a subset of A0 then
| a(n)"(l - p(\oga(n))yd-edn < oo.
NF
Let h e Cl(a+). Set a, = exp th. Then 4.5.3 implies that there is a positive
constant C such that
(1) (a,)pS(ar) < C(l + t)d for t > 0.
We have seen in 4.5.3 that
(2) (a,yE(at) = [ a(n)»a(atna;l)» dn.
N

4.5. Harish-Chandra's E-function
127
We now choose h to be the element such that a(h) = 0 for a e F and a{h) = 1
for a e A0 - F. Then mF = Cg(h). Set *nF = mF n n. Then n = *nF © nF. 2.4.5
implies that we can normalize the invariant measure on *NF such that
(3) J a(*nF)2d*nF = 1.
»nf
4.A.2.1 and 4.A.2.2 imply that we can normalize the invariant measure on NF
such that if fe CC(N) then
(4) \f{n)dn= | f(*nFnF)d*nFdnF.
N »NF x flF
We assert that
(5) (fl,)"S(fl,)= | a(nra(a,nat-1rdn.
Nf
Let I(t) denote the right hand side of (5). Since a,xa^, = x for x e *NF and
x e Na(x)k(x) the obvious manipulation of (2) using (4) yields
/(t)= | a(*nF)2pa(k(*nF)nFYa(k(*nF)atna-tyd*nFdnF.
*NF*NF
Now k(*NF) is contained in KF which commutes pointwise with the a, for
t e R. Also, a(ka,na_t) = a(ar/cn/c~1a_r) for k e KF, t e R and n e NF. Since
KF is compact, dknFk~l = dnFox\ iVFfor k e KF. The obvious calculation now
yields
I(t)= { a(*nf)2pa(nF)',a(arnFa_r)pd*nFdnF.
*NF x JVF
(3) now implies (5).
In particular (5) implies
(6) | a(nFya(a,nFa-,)pdnF < C(l + t)d for t > 0.
Nf
We now use the notation in 4. A.2.4 (with the "F" there equal to 0). Then we
have for t > 0
a(a,nFa_,r2 = ||a(a,na_r)_1f;0||.
Now
\\o(a,nFa_,)v0\\2
= \\v0 + E e-j<(G(nFylv0)j\\2 < 1 + e^'lk^)"1^!!2
j>o
= 1 + e'2'a(nFy4p < (1 + e-'a(nF)-2p)2.

128
4. The Asymptotic Behavior of Matrix Coefficients
We have proved the following inequality
(7) a(a,nFa.,)>(\ + e''a(nF)'2py112 for t > 0 and nF e NF.
If r > 0 then we set (NF)r = {neNF\a{n) > r}. Then 4.A.2.3(2) implies
(8) (NF)r is compact for all r > 0.
In (7) we take t = — 2 log r for 0 < r < 1. Then (6) implies that if n e (NF)r
then a(atna_t) > 2~1/2. We therefore find that
C(l+t)d> | a(nYa(a,na^)pdn>2-1'2 | a(n)p dn
(NF)r (NF)r
which implies
(9) | a(n)»dn<C'(\ - 2 log r)d.
(NF)r
We now take rp = exp( - 2") for p = 0, 1,.... With this notation (9) implies
that
| a(n)pdn<C'(\ +2p+l)d< C"2".
If n e (NF)rp + 1 - (NF)r we have rp > a(n)p > rp+,. Hence on this same set
we have 1 + 2" < 1 - p(log a(n)) < 1 +2P+1. This implies that if e > 0 then
| a(n)"(l - p(\oga(n)yd-*dn
< C"(l + 2pyd-e2pd < C"'2~ep.
If we sum over p > 0 we find that
I a(n)»(\ - (log a(n))Yd-edn < C" ^ 2"£p < oo.
NF-(NF)ro
This implies the theorem since (iVF)r is compact.
4.5.5. We retain the notation of the previous paragraph. If g e G then
we can write g = nFmF(g)aF(g)kF(g) with nF e NF, mF(g) e °MF, aF(g) e AF
and kF(g) e K. We leave it to the reader to check that aF(g) is determined
uniquely by g but that mF(g) and kF(g) can be replaced by mF(g)k and
k~lkF(g) for k e KF. Fix a norm ||-|| on G (2.A.2).
Let SF be the "S" function for °MF. We extend SF to G by setting
EF(namk) = EF(m) for neNF, aeAF, me°MF and keK. The above
considerations imply that this extension is well denned.

4.5. Harish-Chandra's S-function
129
4.5.6. Theorem. If r > 0 and if q > d + r then
| a(n)"3F(n)(l + log ||mf(n)||)d(l - pflog a(n))'"dn < oo.
NF
We first prove that there exists a positive constant C such that
(1) 1 + log ||mF(n)|| < C(l + log ||n|| - p log a(n)) forneiVF.
Assume that \\g\\ = \\a(g)\\ with (a, F) a finite dimensional representation of
G and \\a(g)\\ is the Hilbert-Schmidt norm of a(g) relative to an inner product
on F such that a(g)* = o(6(g))~l. We choose an orthonormal basis {i>,-} of F
such that the elements of MF have block diagonal form
Ay
0
and the elements of NF have block form
7,
Then since
\W(n)o(n)*\
This implies that
\a(nF(n))a(mF(n))a(mF(n))*a(aF(n))2a(nF(n))*\\
\\a(mF(n))a(mF(n))*o(aF(n))2\\ < ||a(n)a(n)*||.
\a(mF(n))a(mF(n))*\\ < ||a(aF(n))-2||||n||2.
If we apply Lemma 2.A.2.3 and then take the logarithms, (1) follows.
(1) in light of 4.A.2.3, implies that it is enough to prove that
/ = | a(n)PFEF(n)(\ - p(\oga{n))r~qdn < co for q > d + r.
nf
By the definition of SF
/= | a(n)PF | a(kmF(n)Ydk(\ - p(log a(n)))r'qdh.

130 4. The Asymptotic Behavior of Matrix Coefficients
Now, if k e KF and if n e NF then
kn — knmF(n)aF(n)kF(n) = knk~lkmF{fi)k~laF(n)kkF(n)
with n e NF. Thus kmF(n) = mF(knk~l) and aF(kn) = aF(n). This implies that
/= | a(km-lfFa(mF(km-l)){\-p(\oga(n)y-qdndk
NFx KF
= | a(nYF(\ - p(\oga(knk-l))r-qdndk
NFx KF
< | a(n)"(l - p(\oga(n))r-qdndk < oo
Nr
by Theorem 4.5.4.
4.6. Notes and further results
4.6.1. The theory of the real Jacquet module is an outgrowth of work of
Casselman and of Casselman and the author to introduce a functor on the
category 3/f with the same exactness properties as the Jacquet module (Jacquet
[1]) in the case of p-adic groups. With this notion in hand many arguments for
the "real case" are proved in a manner quite analogous to the way they are
proved in the "p-adic case". Indeed, the material in this chapter is more
strongly influenced by Harish-Chandra's work on p-adic groups than it is by
that on real groups.
A more complete exposition of the theory of the Jacquet module can be
found in Wallach [2] and Wallach [3]. The category y, introduced in Section
1, is essentially the same as what some authors call &. This category is an
extension of the category G, which was introduced by Bernstein, Gelfand,
Gelfand [1] to study the structure of Verma modules. Further results on
Verma modules will be proved in Chapter 6. The best reference for the theory
of Verma modules is Dixmier [2, Chapter 7].
4.6.2. We use the notation of Section 4.3. If in Theorem 4.4.3 the space ff°° is
replaced by HK then the expansions in Theorem 4.4.3 can be found in
Casselman, Milicic [1] (these results sharpen earlier work of Harish-
Chandra). Their proof uses the theory of regular singularities as generalized in
Deligne [1]. If v e HK their results imply that the expansions actually converge
to the matrix entry. However, one must still prove that their expansions are
asymptotic in our sense (c.f. Borel, Wallach [1, Chapter 3]).
We will see in Volume II that Harish-Chandra's theory of the constant term
is a fairly direct consequence of Theorem 4.4.3 (in light of Lemma 7.7.5).

4. A. 1. Asymptotic Expansions
131
4.6.3. If we combine Theorem 4.5.3 with Lemma 3.6.7 then, in the notation
of 3.6.7 we have
Proposition. If v e (ac)* then
|3„(fl)| < a|Rev|-"(l + log ||a||)d for a e C\(A+).
The number d that appears in 4.5.3 can be taken to be | W(G, A)\ - 1 since
one can show that
dim(H°)K/n(H°)K = \W(G,A)\.
4.A. Appendices to Chapter 4
4.4.1. Asymptotic expansions
4.A.I.I. By a formal exponential polynomial series we will mean a formal sum
of the form
(1) £ exp(v) £ pLn(t)e-"\
\<p n>0
where pjn is a polynomial in t for each j, n.
The point here is that we do not care if the series converges. Fix such a
formal series. Then we may rearrange it in the following way:
(2) £ e\p(Ujt)pUJ(t),
with Uj e {zk - n \ 1 < k < p, n > 0, n e N}, Re u, > Re u2 > - - ■, and pu. is the
sum of the pkn with zk — n = uy
We will call N a gap of the series if uN > uN +,.
If / is a function on R then we say that / is asymptotic as t -> + oo to the
formal polynomial series given as in (1) if for each gap, N, there exist positive
constants (depending on N) C and e such that
(3) 1/(0 - £ exp(u/)pu(f)| < C exp((Re uN - e)t) for t > 1.
j<N
Notice that if N is a gap then
(4) lim exp(-tReuN)|/(t)- £ exp(u/)pu.(t)l = 0.
r -► + oo j < N
4.A.I.2. Lemma. Let
E exp(v) £ Pj,n(t)e'n'
1<j<p n>0

132
4. The Asymptotic Behavior of Matrix Coefficients
and
E exPK0 E 1j.n(t)e~
<j<«
be formal polynomial series such that z-t — zk (resp. Wj — wk) is not an integer for
j ¥= k and p]0 ^ 0, qj0 i= 0. // both formal series are asymptotic to the same
function, f(t), then p = q and after relabeling w} = z-}, pjn = qjn.
We will use the following simple fact:
(1) Let Uj be purely imaginary and let pj be polynomials for j = 1,..., n. If
Uj i= uk for j i= k and if
lim E exp(u/)p,-(t) = 0
then pj = 0 for all;'.
Indeed, set deg pj = dy Let d denote the maximum of the dj. Let aj be the
coefficient of td in p}. Set <I>(t) = I exp(Mit)ai. Then limr^ + 00 <I>(t) = 0. This
easily implies that
T
I
o
But
T
I
0
lim (1/T) j|«(t)|2A = 0.
lim (l/T)J|*(t)|2dt = 5>;|
Thus all of the as = 0. This implies that all of the ps = 0.
We now sketch the proof of the lemma. (The idea is very simple but the
notation would get out of hand if we gave all of the details.) Let Re z, = - =
Re zr > Rezr+1 > > Re zp and Re w, = = Re wu > Re wu + 1 > >
Re wq. Then
|/(t) - E exp(z,t)p;>0(OI < C exp((Re z, - e)t)
and
|/(t) - E exp(w,0«,,oWI < C exp((Re w, - e)t)
1 <j<u
for some positive constants C and e. Suppose that Re z, > Re w,. Then the
two inequalities above imply that
lim E exp((zj-z1)t)pJ-,o(0 = 0-
r -■> + oo 1 < j < r
So (1) implies that pj0 = 0 for j = 1,..., r. This is contrary to our assumptions.

4.A.2. Some Inequalities
133
Hence Re z, < Re w,. We therefore see (by symmetry) that Re z, = Re wi.
Now this implies that
lim (£ exp((Zj- - Zi)t)pJm0(t) - £ exp((w,- - w,)t)^,o(0) = 0-
r -■> + oo
(1) now implies that r = u and (after relabeling) z} = w},
Pj,o = 1j.o for; = l,...,r.
Now replace f by f(t) - S, Sj-<r exp(z/)p,0(t). We observe that if c e C
then c can be written in at most one way in the form zk — n (resp. wk — n).
Thus we can "bootstrap" the above argument to prove the Lemma.
4.A.2. Some inequalities
4.A.2.I. Before we get to the main material of this appendix, we first prove a
few results on groups with dilations. Let N be a Lie group. Then a smooth 1-
parameter group of automorphisms, a,, of N is called a family of dilations of
Nif limr^ + 00 at(n) = 1 for aline N. We collect some properties of groups with
dilations. Let A, be the corresponding 1-parameter group of automorphisms
of rt.
(1) exp is a diffeomorphism of n onto N.
Let U0 be a neighborhood of 0 in rt and let I/, be a neighborhood of 1 in N
such that exp is a diffeomorphism of U0 onto Ul. If n e N then there exists
t > 0 such that at(n) e l/,. Thus a,(n) = exp X with X e U0. Hence, n =
a_r(exp X) = exp A_,(X). So exp is surjective. If exp X = exp Y let t > 0 be
so large that A, X, A,Y e U0. Then exp A, X = a, exp X = a, exp Y = exp A, Y.
So X = Y. Since exp = a, exp A_t it is clear that exp is everywhere regular.
(2) Let rt, and n2 be A, invariant subspaces of rt whose direct sum is rt. Then
the map n, x n2 to N given by X, Y i—>exp X exp Y is a diffeomorphism
onto N.
This is proved by first showing that it is true for small neighborhoods and
then dilating as in the proof of (1).
Let D denote the derivation d/dt\, = 0A,. Then the eigenvalues of D have
strictly negative real parts. From this it is easy to see that
(3) rt is nilpotent. In particular N is unimodular.
(4) Assume that the subspace in (2) are Lie subalgebras. Let Ni and N2 be
the corresponding connected Lie subgroups. Then we can normalize the

134
4. The Asymptotic Behavior of Matrix Coefficients
invariant measures dn, dri\ and dn2 on N, N{ and N2 respectively such that if
/ e CC(N) then
\f(n)dn= | f(nin2)dnidn2.
Let h(n{, n2) = nyn2 for n, e N{, n2eN2. Then (2) implies that h is a
diffeomorphism of N, x N2 onto N. There is thus a smooth function u(n,, n2)
such that dn = udn1dn2. The left invariance of dn implies that u is
independent of n, and the right invariance of dn implies that u is independent
of n2. Thus u is constant and the assertion follows.
4.A.2.2. Let G be a real reductive group with compact center. We will use the
notation of 4.3. Let F be a subset of A0 and let (PF,AF) be the corresponding
standard p-pair. Let pF e a* be defined by pF(H) = (\) tr(ad H\„F). Let
*aF = °mf n a. Then a = *aF ® aF. We note that pF(*aF) = 0. Set NF = 8(NF).
Let H e a be such that a(H) = 1 for a e A0 set a, = exp t//. Then n i—► atna_t
defines a group of dilations on N = 8N that leaves NF invariant. In particular
exp is a diffeomorphism of nF onto NF. Let log denote the inverse map to exp
onnF.
Let B be as usual and set (X, Y} = -B(X,0Y) for X, Y e q. We set
||X|| = <A:,A:>1/2forA:eg.
If g e G let k(g), a(g) and n(g) be as in 3.6.1. We note that exp is a Lie
isomorphism of a onto A. We denote by log the inverse map to exp on A.
4.A.2.3. Lemma. There exists a positive constant C such that if ne NF
then
(1) 1 - pF(log a(n)) > C(\ +||logfl(n)||),
(2) 1 - pf(log <i(n)) > C(l + ||log n||).
The rest of this appendix is devoted to a proof of this Lemma. Let
d = dim nF. Let < , > denote the inner product on Adg corresponding to
< , > on g. Let i;0 be a unit vector in AdtiF. Let a denote Ad Ad. We note
that ff(°mf)i)0 = 0, a(nF)i;0 = 0 and o(H)v0 = 2F(H)v0 for H e a. This
implies that
(I) \\<r(g->o\\ = a(g)-2'"' for g e G.
We set <bF equal to the set of all elements of <t>(P, A) that are non-negative
integral combinations of elements of F. Put ZF = <t>(P, A) - <bF.
(II) (pF,oc)>0 foraeIF and(pF,a)>0 for aeO(P,/l)-

4.A.2. Some Inequalities
135
Let *pF be the half sum with multiplicities of the elements of <t>F .Then
p = *pF + pF. If a e F then (pF, a) = 0. If a e A0 - F then (pF, a) = (p, a) -
(*pF,a) and (*pF, a) < 0. Thus (pF,a) > (p,oc) > 0. This proves (II).
(Ill) If X e nF and if a(X)v0 = 0 then X = 0.
Write X = 1. Xa with X„ e g~". Then o{Xx)v0 is in the 2pF - a weight
space. Thus a(Xa)v0 = 0. If Xa is non-zero then this implies that (pF,a) = 0.
Since this contradicts (II), (III) follows.
4.A.2.4. Let H be as in 4.A.2.2. Let nF j be the -;' eigenspace for ad H in
nF. Set m = 2pF(H) > 0. Put V = a(U(nF))v0. Let V} denote the m — j
eigenspace for a(H) on V. Then V is the direct sum of the Vr We also observe that
V0 = Rv0 and that <J^, Vk) = 0 for j ^ k. H v e V then we can write v as
u = S i>j with Uj- e V}. If le nF we write X = 2, Xj with X,- e nFJ. Let p
denote the largest eigenvalue of — ad H on nF.
Let Xenf. Then
(1) a(exp X)v0 = y0 + E (CT(exP x)!;o)j + E (CT(exP *K)r
j<p ;>p
This implies that
(2) ||a(exp X)v0\\2 > 1 + £ ||(a(exp ^T)u0)J-||2-
j<p
Now (a(exp X)u0)j-= (7(^)110 + m^X,,..., Aj_,) with u^ a polynomial
map. This combined with the Seidenberg-Tarski theorem easily implies that
(3) There exist positive constants C and r such that if X enF then
|k(exp(X)i>0|| > C(l + ||X||r.
We will give an elementary proof of this result at the end of this appendix.
(3) combined with (I) above implies
(4) a(exp Xy2»F > C(\ + \\X\\)r for X e nF.
(2) in 4.A.2.3 now follows (after taking logarithms) from (4). We now
derive 4.A.2.3(1). We now assume that F = 0. Then ||a(exp(-A'))i;0||2
is a polynomial in X for XenF. Thus there exist positive constants
C, and s such that ||<j(exp(-AT))t;0||2 < C,(l + ||A:||)S. This implies that
-p(log(a(exp X))) < C'(l +log(l + ||X||)) for some positive constant C.
We write
log a(exp X) = -£ aa(X)Ha with a„(X) > 0 (3.A.2.3)
the sum over A0. Since (p, a) > 0 for a e A0 this implies that there exists

136
4. The Asymptotic Behavior of Matrix Coefficients
a positive constant C" such that max aa(X) < C"( —p(log a(exp X))). Thus
there exists a positive constant C2 such that
||log a(exp X)|| < C2(l + log(l + ||X||)).
Now 4.A.2.3(2) implies 4.A.2.3(1).
We are left with the proof of (3) above. That result follows from the
following Scholium.
4.A.2.5. Scholium. Let V and W be finite dimensional Hilbert spaces over
R. We assume that V is the orthogonal direct sum of subspaces V}, j = \,...,p
and that we are given injective linear maps 7} of Vj into W and polynomial maps
u. of @k<j Vk into W. Then there exist positive constants C and r such that
if Vj e Vj and v
S Vj then
1+1
TjVj + uj
I",
>C(i + |M|)'.
If p = 1 then the result is obvious. We assume the result for p — 1
and prove it for p. Let C and r' be the constants for p — 1. Fix Vj e Vj
for j<p. Set u„ = up(vl +--+0j>_1). Set S = {v e Vp\\\v\\ > 2\\up\\}. Set
E=\+-Ej<p\\TjVj + Uj(vl+-- + vj^)\\2 + \\Tpvp + up\\2. If v e S then
E>C'(1 +||i>, +"- + ^-ill)r' + (i)l|Tf;p||2. Otherwise, N| < 2\\up\\, Since
up is a polynomial map there exist positive constants H and s such that
\\u(vx +■■■ + vp
,)ll < //(l -(- llu, +
(1 +||o, +'■' +vp.
+ Vp-t
II) ^6(1+Ho
Thus
y /s
with Q = H'1. Thus, in this case £>C'(1 +\\vt + --d J)*" + g(l +|
\r/s
We therefore see that if we take r
mate is true for p with some C > 0.
min(r', r'/s, 2) then the desired esti-

5 The Langlands
Classification
Introduction
In this chapter we continue the study of the "fine structure" of admissible
(g, K)-modules. The main result is Theorem 5.4.4 which (combined with
Theorem 5.4.1) reduces the classification of irreducible (g, K)-modules to the
classification of tempered, irreducible (g, K)-modules. This is the celebrated
theorem of Langlands. The proof of this theorem rests on the asymptotic
results of the last chapter, the elementary theory of tempered representations
and the simplest aspects of the theory of the principal series. These topics are
developed in the first three sections of this chapter.
Section 4 contains the main theorem. The proof we give follows the broad
lines laid out in Borel, Wallach [1, Chapter IV]. In Section 5 we give some
direct consequences of the classification and its proof. In 5.5.3 we introduce the
notion of Langlands parameter and prove a useful order relation on the
parameters. Corollary 5.5.3 has been used to prove theorems about (g, K)-
modules by induction on the partial order on the Langlands parameters (c.f.
Borel, Wallach [1, Chapter IV]). Theorem 5.5.6 contains the basic finiteness
theorem for irreducible (g, K)-modules. This result is usually proved using
character theory. As we shall see, the a priori knowledge of this result will lead
to some simplifications in the Harish-Chandra character theory (Chapter 8).
The next two sections study, respectively, the special cases SL(2,F) with
F = R and F = C. We use elementary methods combined with the Langlands
classification to give the full classification of irreducible (g, K)-modules in
these two cases.
137

138
5. The Langlands Classification
5.1. Tempered (g, K (-modules
5.1.1. Throughout this chapter we will assume that G is a real reductive
group such that G° = °(G°). We fix an Iwasawa decomposition G = NAK.
Let (P,A) be the corresponding minimal p-pair of G. In particular, our
assumption implies that <b(P, A) spans a*. We fix a norm, ||- ■ -||, on G (2.A.2.3).
Let A0 be the set of simple roots in <t>(P,A). We write A0 = {a,,..., ar}.
Let /?,,...,/?r in a* be defined by (Pj,ock) = SJk. As usual, we set (a*)+ =
{fi \fi = I Xjfij with Xj > 0}. We set +a* = {/^ /^ = £ x^ with Xj > 0}.
Let (n,H) be an admissible Hilbert representation of G. Then we say that
(n,H) satisfies the weak inequality if there exists a non-negative constant,
d, such that if w e HK, and tefl1" (1.6.1) then \(n(g)v, w}\ <a(v)(\ +
log ||g||)''S(^) for all g e G and a is a continuous semi-norm on H™
depending only on w. Here E is as in 4.5.3. We say that (n,H) satisfies the strong
inequality if for each d > 0 and w e HK, v e Hx then \(n(g)v, w}\ < <Jd(v)(\ +
log HglD^E^) for all g e G. Here, ad is a continuous semi-norm on H™
depending only on d and w. These definitions are provisional, unitary
representations satisfying the weak inequality will be called tempered later in this
chapter. We will also see that if (n, H) is irreducible and unitary then (n, H)
satisfies the strong inequality if and only if it is square integrable.
Let V be an admissible finitely generated (g, K)-module. Let Av be as in
4.3.5. Then we say that V is tempered if Av + p e -Cl(+a*). We say that
V is rapidly decreasing if Av + p e — +(a*).
5.1.2. Proposition. Let (n,H) be a Hilbert representation of G. If HK is
tempered then (it, H) satisfies the weak inequality. If HK is rapidly decreasing
then (n, H) satisfies the strong inequality.
Let V = HK. Theorem 4.3.5 implies that there exists d>0 such that
if w e V then there exists a continuous semi-norm, aw, depending on w
such that \<Tt(a)v,w}\<aw(v)(l + log \\a\\)daA (A = Av) for all aeC\(A+).
Let Wi,...,wp be a basis of the span of Kw. Let a(v) = supkeKl,aw(kv).
n(k)w = Hgj(k)Wj with each gj& continuous function on K. (n(klak2)v,wy =
(ji(ak2)v,ii(kly~lw> = Z cox\}(gj((kl)~l))(it(ak2)v, w>. It follows that
(1) If we V and v e Hx then\(Tt(k1ak2)v,w}\ < a(v)(l + log \\a\\)daA
for a e C\(A+) and ku k2 e K.
We now prove the result. Suppose that V is tempered. Then aA < a p
for all aeC\(A+). Now, a'" < 2(a) for all aeC\(A + ) (Theorem 4.5.3).

5.1. Tempered (g, K)-Modules
139
Since E(klgk2) = E(g) for all ku k2eK, the first assertion now follows
immediately from (1).
If n e +a* then for each r > 0 there exists a positive constant Cr such that,
a" < Cr(l + log ||a||)_r for a e Cl(.4+). Hence, the second assertion is also
a direct consequence of (1).
5.1.3. Proposition. Let V be an admissible finitely generated (g, K)-module.
If V is rapidly decreasing then V splits into a direct sum V = © Vj with Vj
irreducible. Furthermore, there exist (itj,Hj) irreducible (unitary) square inte-
grable representations of G such that Vj is equivalent to (Hj)K.
Before we prove this result we must prove a lemma which will be useful in
the later chapters.
Lemma. There exists a positive constant r such that
j~(0)2(i + iogii0iir^<<x).
G
Let y(a) be as in 2.4.2. Then y(a) < Ca2p for a e C\(A+). We now apply
Lemma 2.4.2 (using the left and right X-invariance of S and ||- ■ ||)
| E(0)2(1 + log \\g\\yrdg = | H(a)2(l + log \\a\\rry(a)da
G A*
<C I a2pS(a)2(l +log||a||)-rda
<C I (1 + log ||a||)d"da.
Here we have used Theorem 4.5.3. Since the last integral is finite for r
sufficiently large the result follows.
5.1.4. We now prove the above proposition. Let (n, H) be a realization of
V (4.2.5). Let (n*,H) be the conjugate dual representation. Let V~ be the
underlying (g, X)-module of it*. Then V~ is admissible and finitely
generated (see 4.3.2). Let i>l5..., i>pbea set of generators for V~ asa(g,X)-module.
Then the set {it*(g)Vj\g e G, j = l,...,p} spans a dense subspace of H.
If v, w e V then we put
(v,w) = X | (it{g)v,Vj} con)((it(g)w,vJ})dg.
j o
The above integral converges absolutely by Lemma 4.5.3. The choice of
the Vj implies that (v, v) > 0 for non-zero v. Since dg is right invariant it

140
5. The Langlands Classification
follows that
(1) (Xv,w) =—(v,Xw) and (kv,kw)
= (v, w) for X e g, k e K, and v,w e V.
This implies that if W is a (g, X)-submodule of V then W1, its orthogonal
complement relative to ( , ), is also a (g, X)-submodule. Since V has finite
length as a (g, X)-module (4.2.1), it is clear that V splits into a direct sum of
irreducible (g,X)-modules. Since V™>j(V) and V-^>V~ are exact functors
(Theorem 4.1.5) we see that each summand of V is rapidly decreasing. Thus
to complete the proof of the proposition we may assume that V is irreducible.
Fix w a non-zero element of V. Let T(v)(g) — {n(g)v,w~y « , > is the
original inner product on H) for v e V. We have shown that T(v) e L2(G)
for all v e V. If x e l/(g) then xT(v\ = T(xv). Thus T(V) consists of smooth
vectors (1.6.1) for L2(G). The argument in the proof of Theorem 3.4.9 implies
that T(V) is contained in the space of analytic vectors for L2(G). Thus if we
set Hi =C1(T(K)) then Hx is an R(G)-invariant subspace of L2(G) (here
R(g) is right translation by g and we have used Proposition 1.6.6). Set nx(g)
equal to the restriction of R(g) to Hx. Then it is clear that (Hi)K = T(V)
and that T is a (g, X)-module isomorphism of V onto T(V). The result now
follows from Proposition 1.3.3(2).
5.2. The principal series
5.2.1. We retain the assumptions and the notation of the previous section.
Let F be a subset of A0 and let (PF,AF) be the corresponding p-pair (2.2.7).
Let (a, Ha) be a Hilbert representation of °MF which is unitary when
restricted to KF = K n Pr. Let ft e (o*F)c. We define «-HPf-°->x to be the space
of all smooth functions f:G->(Ha)°° such that f(namg) = atl + pa(m)f(g) for
neNF,aeAF,me °MF and g e G. We define for f,ge <*>HPl'-a-''
<f,gy = i<f(k),g(k)}dk.
K
Let hPf-"->1 denote the Hilbert space completion of xhPf-"-'1. Then
1.5.3 implies that if we define itPF,a^(g)f(x) = f(xg) for g, xeG then
(TtpF.a^,HPF-"'>') is a Hilbert representation of G.
We denote by /PFiff>/J the underlying (g, X)-module of (itPFmatll,HPF'a'1').
5.2.2. Lemma. // (a, Ha) is admissible and finitely generated then IpFm„mll is an
admissible (%,K)-module. Furthermore, /pFiffi/J is the space of all f e <xHPF'a-'i

5.2. The Principal Series
141
such that
(1) f(K) <= W c (//ff)K wi't/i VF a finite dimensional subspace depending only
onf.
(2) / is right K-finite.
Let / e IpF,a,v Then in particular, / is a smooth vector for itP<,J\K. Also
/ is K-finite, which means that there exist fu...,fn e IPt.„„ such that
itpF,<,itl{k)f = £ O/C0.// for /c e X. Here a; is a smooth function on X. Thus
f(k) = I aj-(Jt)j5(l). Now, if k e Xf then f(k) = a(k)f(\). Thus / satisfies (1).
(2) is an immediate consequence of the definition of K -finite vector. The
converse is equally easy and left to the reader.
We now assume that (a,Ha) is admissible and finitely generated. Let
*PF = P n °MF. Then 4.2.2 implies that (in the notion of 4.2) (Ha)K is
equivalent to a submodule of oMf.Xy with y a finite dimensional representation of
*PF (here the sub-°MF indicates that we have replaced G by °MF). Hence, as a
KF-module (Ha)K is equivalent to a subrepresentation of Ind(y|oM). This
implies that IPf.atl is equivalent as a K-module to a subrepresentation of
Ind£F(IndoM(}>|oM) = \x\d%M(y\,M). Frobenius reciprocity now implies that
/Pf „„ is admissible.
5.2.3. We now give another variant of Frobenius reciprocity which seems
to have been first observed by Casselman. We retain the notation of the last
paragraph. Let V be a (g,X)-module. If Te HomgK(V,IPf.a„) then we set
TA(v) = 7»(1). Since T(v)(ri) = 7»(1) for n e NF we see that TA(nFi;) = 0
for veV. If XemF then T(Xv)(l) = (AT(i>))(l) = d/dt\t = 0T(v)(e\p tX) =
X(T(v)(\)). Here the action is on the module (Hail)K which is (Ha)K with
aF acting by n + pF. We therefore see that TA defines an element of
HommF.K(K/nFK, (HaJK). We have
Lemma. The map T i—* TA defines a bijection between Homg k(V,IPf„jM) and
HommF.K(K/nfK, (//„,».
The proof is exactly the same as that of 3.8.2.
5.2.4. Let W be an admissible, finitely generated (°mF,XF)-module. Let
(a, Ha) be a realization of W. Then the (g, K )-module, IPFt„tll depends only on W
for each \i e (af)c. We write IPf,w.^ for this (g, X)-module.
Lemma. (IPf.w^)~ is equivalent with 1PfW~-11.
If / e lpr.w.v and if g e hF,w~, „ then we set
{f,g)=\{)\k\g(k))dk.
K

142
5. The I anglands Classification
Here, ( , ) denotes the natural pairing of W and W~. The result now follows
from 2.4.1 and 4.5.1.
5.2.5. Proposition Let V be an irreducible, tempered, (g, K)-module. Then
there exists a standard p-pair, (PF, AF), and an irreducible unitary representation,
(a, Ha), of °MF such that (Ha)K is rapidly decreasing and n e (aF)* such that V is
isomorphic to a summand of //>,,,„,,„.
Set E(V)=-E(P0, V) (4.3.4). If fie E(V) then set F(n) = {j\Re(n +p,fa)
< 0}. Let A0 e E(V) be such that F(A0) has the minimal number of
elements. Set F = {aj j e F(A0)}. Let p. denote the restriction of A0 to aF.
Then X = (K~/nFK~)_/I is nonzero, admissible, finitely generated (mF,KF)-
module. By the definition of F, Re(A0 + p, fa) = 0 for ;' £ F(A0). Thus
fi + pF = iv with v e (aF)*. Let W be an irreducible (non-zero) quotient of X.
Lemma 5.2.3 now implies that V~ is isomorphic with a submodule of IPFtWtiv.
Suppose that (X/*nFX\ is non-zero for some (e(*aF)£. Then — ( +
H e E(V). By the above Re(-£ + n + p, fa) = Re(-C + *pF, fa) for all ;'.
Thus, the definition of F implies that Re( — ( + *pF, fa) < 0 for all j e F(A0).
We conclude that W~ is a rapidly decreasing (0mF, XF)-module.
5.2.4 implies that V is equivalent with a quotient of IPf W4v . 5.1.3 implies that
W~ is the underlying (°mF,XF)-module of an irreducible, square integrable
representation, (a, Ha), since (7rPFffIV,//p-'T'v) is unitary (1.5.3). The result now
follows.
5.2.6. Corollary. Let V be an irreducible, tempered (Q,K)-module then V
is equivalent to the underlying (%K)-module of an irreducible unitary
representation.
This follows directly from the last part of the proof of the preceding
theorem.
5.2.7. Fix a subset, F, of A0. Let (a, Ha) be an admissible, finitely generated,
Hilbert representation of °MF. Let fi e (aF)£. Let (jtPFjffj/J, HPF-"-tl) be as above.
Since PF will be fixed in this number, we will drop the PF in our notation. Let
(Ha)°° be endowed with the usual topology (1.6.3). We set (//"•")„ equal to the
space of a smooth functions from G to (Ha)°° that are in °°//"■". If x e l/(g)
and if 3 is one of the semi-norms denning the topology on (//"•")„ we set
dx(f) = supkeKd(naJx)f(k)) for / e (//"•")«,. Then it is easy to see that
(//"•")„ defines a smooth Frechet representation of G, that IPf. „„ is a dense
subspace and that (//"•")„ is contained in (//"•")".
It can be shown (c.f. Borel, Wallach [1, III, 7.9] that (//"■")«, is equivalent to
(//"•")°° as a smooth Frechet module.

5.2. The Principal Series
143
5.2.8. The following result will be used in Section 4.
Lemma. Let (PF, AF) be a standard p-pair. Let (a, Ha) be a representation of
°MF satisfying the weak inequality. Let \i e (aF)* be such that Re(ji, a) > 0 for all
a e <t>(PF, AF). Then there exists a constant r > 0 such that if fu f2 e IPaii then
there is a constant C such that
\<*.Jg)fi,f2>\ < csRe„(0)(i + log \\g\\y.
In particular, if a e C\(A +) then
l<n„»/i./2>l < CaRe"H(a)(l + log \\a\\y + d.
\<rtaJg)fuf2>\= 5<fdkg],fi(k)>dk
K
SaF(kgr+»'<<r(mF(kg))f1(kF(kg),f2(k)>dk
K
< | aF(MRe" + "n<a(mF(M)/i(M%),/2Cc)>M/c
K
< C I aF(kg)R<» + <"-ZF(mF(kg))(\ + log \\mF(kg)\\Ydk
K
< C'(l + log \\g\\r' | aF(kgf^ + "-EF(mF(kg))dk.
K
Here we have used ||mF(g)|| < C'||gi||)'''. To see this, we choose (a, W) a finite
dimensional irreducible representation of G that is unitary for K and is such
that if W0 = {w e W\ a(n)w = w, ne NF} then the representation of °MF on
W0 has compact kernel. If aeAF then o(a) = axl on W0. Thus if g =
nF(9)aF(9)mF(9)kF(g) and if weWo is a unit vector then ||<T(sf_1)w|| =
a^gr'Waim^gr'Ml Hence \\<r(mF(g))w\\=aF(g)x\\<r(g-1)w\\£ C\\g\\< for
some q. We observe that sup{||a(m)w|| ] w e W0, \\w\\ = 1} is a norm on MF.
We now continue the argument. The last expression above is equal to
C'(l+log||0||)r' J aF(kg)Re"+"Fa(kFmF(k(kg)ydkdkF.
KxKf
Now a(mF(g))aF(g) = a(g) and aF(kg) = aF(g) for k e KF. Thus
\<K.Jg)fiJ2>\ < C(l + log \\g\\y' | a(kFkgyr + «e»dkdkF
K*KF
= C(\+log \\g\\Y' ^(kg^^^dk
K
= C(l+log||g||)r'3ReM(g).
The last inequality in the statement follows from Lemma 3.6.7.

144
5. The Langlands Classification
5.3. The intertwining integrals
5.3.1. We retain the notation of the previous sections. Let F be a subset of A0
and let (PF, AF) be the corresponding standard p-pair. Fix (a, Ha) a
representation of °MF that satisfies the weak inequality. We set (as usual), NF = 6(NF)
and KF = K n MF. Let <b(PF,AF) denote the set of roots of aF on nF.
Lemma. Let fi e (aF)£ be such that Re(/^, a) > 0 for all a e <P(PF,AF).
(1) lffe(H°-»)xandifWe(Ha)Kthen
I |</(n),w>|dn< oo.
NF
Furthermore the map
/.-> | (f(n),wydn
Nf
is continuous on (//"•")00.
(2) If w e (Ha)K is non-zero then there exists f e /pFiffi/J such that
| </(«), w>dn
Nf
is non-zero
We will use the notation of 4.5.5. We first prove (1). If n e NF then f(n) =
f(nmF(n)aF(n)kF(n)) with n e NF. Thus /(n) = aF(n)" + PFa(mF(n))/(/cF(n)). This
implies that
| Kf(n),w}\dn= | aF(n)Re" + "|<a(mF(n)/(/cF(n)),w>Mn
NF Nf
^ I ^(/(M«))aF(n)" + P(l+log||mF(n)||rHF(mF(n))dn
with j8 a continuous semi-norm on (Ha)x. Here we have used the weak
inequality. Set y(f) = supkeK P(f(k)).
Now aF(n)Re" < C,( 1-p(log aF(n))" for all q > 0 (4.A.2.3). Thus the
integrand is dominated by
C,r(/)af(n)SF(mF(n))(l - p(log aF(n))"'.
(1) now follows from 4.5.6.
We now prove the second assertion. Let n e CC(NF) be such that
| h(n)dn = 1.
Nf
Set f(nman) = a" + pa(m)h(n)wfor n e iVF,m e °MF,a e AFand n e NF. Extend

5.3. The Intertwining Integrals
145
/ to G by 0. Then / e (//"•% and
| (f(n),w}dn= (w,w) > 0.
Nf
Since //>,,,„.„ is dense in (//"•/i)oc, the continuity assertion in (1) now implies (2).
5.3.2. We retain the above assumptions on a and p..
Lemma. Let f e //>,,,„,„ then there exists a finite dimensional subspace, V(f),
of (Ha)K such that
| (f(h),w}dn = 0
NF
for all w e (Ha)K orthogonal to V(f).
Let w € (H„)K. If k € KF then
| (f(nk),Wydn= | (a(k)f(k-1nk),w}dn= | <a(/c)/(n),w>dn
NF Nf Nf
= | </(n),ff(* »dn.
Here we have used the invariance of dn on iVF under conjugation by KF. Let S
be the set of all elements, y, of KA such the projection of / into the y-isotypic
component of //>,,,„,„ is non-zero. Let V(f) be the sum of the (5-isotypic
components of Ha with 5 a constituent of some y e S restricted to KF. Then
since S is finite, V(f) is finite dimensional. The above formulas now imply the
Lemma.
5.3.3. The preceding Lemma implies that there exists a linear map, A>fVT,„, of
1Pf^ to (H„)K, such that
| <f(n),wydn = (pPttaJf),wy
Nf
for all / e IPf..„tll, w e (Ha)K. The calculations in the proof of 5.3.2 imply that
/W,<t.« 's a ^f-module homomorphism.
Lemma. j8Pj..,ffj/J(nF/pjffj/J) = 0. Let aPf.„„ be the corresponding linear map
of Ipf,oJ"fIpf.o,„ into (Ha)K. Then
(see 5.2.3 for notation).
Let / e /pFiffi/J and let X e nF. If w e (//„),<■ then set
7w(/)= I </(«), w>dii.
N,,

146
5. The Langlands Classification
Lemma 5.3.1 implies that yw is a continuous functional on (//"")„. Thus
yJXf) = d/dt\,=0yw(na „(exp tX)f) = 0 by the right invariance of dn on NF,
If X s °mF then yw(Xf) = d/dt\t=.0yw(7coJexp tX)f) =
| <f(nexptX),w}dn
r = 0 NF
| <a(exp tX)/(exp( - tX)n exp tX), w> dn
t=-0 NF
j <a(exp tAT)/(n), w> dn.
The last equation follows from the invariance of dn on NF under conjugation
by elements of °MF. We leave it to the reader to see that the estimates in 5,3.1
justify the interchange of differentiation and integration. We have
thus shown that pPr,oJnFIPr<aJ=0 and that pPr,aJXf) = XpP„Jf) for
X e °mf. If h e aF then we may argue in exactly the same way (taking
into account d(ana~l) = a2pFdn on NF) to find that PPFt<,.ll{hf) =
(A4 — PF)(h)PpF,„,n(f)- This completes the proof.
5.3.4. The above lemma combined with 5,2.3 implies that there exists a
(g, K)-module homomorphism jPFt„tll of //>,,,„,„ into /pF.ffi/J such that
JpF,aJf)W = Xpr.ajf)- A>so 5.3.1 implies that jPf.ail is non-zero, (This is a
critical point for later applications). We now give an important interpretation
of the above integrals that is due to Langlands (in this generality).
Theorem. We maintain the above assumptions. Let f e (//"•") ^ and let
g e /pF,ff.„. Let he aF be such that a(h) > 0 for all a e <b(PF, AF), Then
lim e^-^h\n(cxpth)f,gy= | </(n),0(l)>dn.
Here n = tiPfa)l.
Since g is K-finite, the span of g(K) is finite dimensional. Also, our
hypothesis on /implies that f(g) e(//ff)°° for each g e G. Seta, = exp th. Then
<7c(a,)f,g>=S<f(ka,),g(k)>dk = J aF(n)2"</(/c(n)a,),g(/c(n))>dn,
by Lemma 2.4.5.

5.3. The Intertwining Integrals 147
Now n = nmF(n)aF(n)kF{n) with n e NF. Hence kF(n) e NF(mF(ii)aF(n)y 1n.
This implies that
(Tt(a,)f,gy= I aF(n)^aF(n)-p-\a(mF(n)ylf{mXg{k(n))ydn
Nf
= a? + x | aF(ny-»<<j(mF(n))-lf(a_triat),g(k(n))ydn
= arp I aF(a,na_,y->'<(j(mF(a,na-t))-1f(h),g(k(atha-t))ydn.
Nf
If we could interchange the limit and integration then the result would now
follow. We are thus left with the justification of this interchange. To this end we
set, for £ a measurable subset of N,
/,(£) = { aF(arna_r)p"<a(mF(«rna_r)r'/(«), g(k(a,na_,)} dn,
E
We will show that there is an integrable function u on NF such that
(1) /,(£)< | u(n)dn forallf>0.
E
The justification for the interchange of limit and integration is then a
consequence of Vitali's convergence theorem (c.f. Dunford, Schwartz [1]). We
are left with proving (1). The transformation rule of / implies that /,(£) =
| aF(arna_r)"""aF(n)p + "<a(mf(arna _tylmF{nj)f(n), g(k(atna.t))> dn,
E
The integrand is dominated by a constant times
aF(a,na_,y-«<»EF(mF(a,na^-lmF(n))aF(riy + «<><(l + log ||mF(n)||)d.
We now analyze this expression. We first observe that
(2) EF(x-ly)= | a(kxya{kyy dk, for x, ye°MF.
Kf
Indeed,
| a(kx~ly)dk= | a(k(kx)x1y)a{kx)2pdk = | a(kyya(kxydk
KF Kf Kf
since kx = n(kx)a(kx)k(kx).
Now
| a(kx)"a(kxy dk = J a(k(n)xya(k(n)yya(n)2''dn.
KF *Nf
If we now use the fact that k(n) e *NFa(n)~lfi we have
(3) 3F(x-ly)= J a(nxya(ny)» dn,
*NF

148
5. The Langlands Classification
(3) implies that
/,(£) <
{ aF(a,na.,)p~R^a(*nmF(a,na_,))pa(*nmF(n))pv(n)aF(nY+R^dnd*n
witho(n) = (l + log Hnll)-.
Now a(mF(g))aF(g) = a(g) and *nmF(atna_t) = mF(afnna.t), hence, if we
set v(h) = (1 + log ||n||)'', then
/,(£)< | a(*natna_t)p^tla(*nn)Re>l + ''v(n)d*ndn
*NFXE
= | a(k(*n)atna^-R^a(k{*n)n)Re^pv(n)a(*n)2pd*ndn
*Nr x E
= | a(aM*n)nk(*n)la.yR^a(k(*n)nk(*ny1Y + R^v(ri)a(*ri)2pd*ridn
*NFx£
= | a(*n)2pa(a,na_r)"-Re"a(n)', + Re''(l +\og\\n\\)dd*ndn
»NFx£
= |a(a,na_r)',-Re''a(n)', + Re''(l + log ||n||)"dn.
£
Let 0 < e < 1 be such that <Re fi — epF, a> > 0 for a e <t>(PF, AF). Then
a(arna_r)p-Re"a(n)p+Re''
= a(atna^-epFa(alm.y(R^~tpF)a{nfe^tp''a{nY + tpF < a(n)p + c>'F
by Lemma 3.A.2.3.
Now 4.A.2.3 implies that for each q > 0 there exists Cq > 0 such that
a(n)E"F + " < C,a(n)"(l - log ||a(n)||)-«.
Set u(n) = a(n)p + EPF(l + log ||n||)''. Then u is integrable on N and we have
just shown that
/,(£)< | u(n)dn.
£
This completes the proof of the theorem,
5.3.5. We will see in the next section that this result is one of the main
ingredients in Langlands' classification of irreducible (g, X)-modules. The
above proof is due to Harish-Chandra [15], Special cases of this Theorem had
been proved earlier in Helgason [3] and Knapp, Stein [1]. We should point
out that in the literature just cited / is also taken to be X-finite, Since we do not
need this condition, some of our later arguments will be simpler than the
originals.

5.4. The Langlands Classification
149
5.4. The Langlands classification
5.4.1. We retain the notation and assumptions of 5.1. Let F be a subset of
A0 and let (PF,AF) be the corresponding p-pair. Let (a,Ha) be an
irreducible unitary representation of °MF such that (Ha)K is tempered (5.1.1). Let
H e (aF)£ be such that Re(ji,a) > 0 for a e <b(PF,AF). We call such a triple,
(PF,a,n), Langlands data. (We allow PF = G, that is to say F = A0.) Set
PF = MFNF.
The following theorem is a combination of a basic result of Langlands
[1] and a refinement of the result by Milicic [1].
Theorem. Assume that (PF, a, n) is Langlands data
(1) If / e/pFi<MJ and if jPf.M^(f) is non-zero then f generates 7pFiffi/J as a
(g, K)-module.
(2) JpF,a,v(IpF.a,v) is the unique non-zero irreducible (g, K)-submodule of 7Pf „ ^
which is also the unique irreducible quotient of 7pFjff . We denote this module by
Jpf.cvl-
(3) If (PF,a,n) and (PF,a',n') are Langlands data and if JpF,ffi/J is equivalent
to JpFma-mll' then F = F', ft = ft' and a is unitarily equivalent to a'.
We first show that (1) implies (2). Let Z be a proper (g, K)-submodule
of IpF,„mll. Then (1) implies that jPf.ajM(Z) = 0. Since /pFiffi/J is a non-zero
module homomorphism (5.3.1), this implies that Ker/F is the unique
maximal, proper (g, X)-submodule of /Pf.„.h. We therefore see that JPFiffi/J is
irreducible. Now, (/pF-ff~,_/J)~ = /pF.„,„ (5.2.4) and (PF,a~, —n) is Langlands
data if we replace A0 with —A0. The above now implies that 7pFiff~__ has a
unique non-zero irreducible quotient (g, K)-module, hence /pFiffi/J has a unique
non-zero irreducible (g, X)-submodule. This completes the proof of (2)
assuming (1).
5.4.2. We now prove 5.4.1(1). Let / be as in the statement of (1) above.
Let Z= 17(g) span {Kf}. Then C1(Z) is a G-invariant subspace of H = Ha".
If Z is a proper subspace of HK then C1(Z) is also proper in H. Hence
there exists a non-zero element g e HK such that (g, C1(Z)> = 0. Let
W= span{Xg}. Then <W,C1(Z)> = 0. Since kg{\) = g(k), we may therefore
assume that g(l) is non-zero. Now, jPFia.>i(f)(k) is non-zero for some ke K.
If we replace / by kf we may assume that jpF,aill(f)(l) = Ppr.a.ll(f) is
nonzero. With all of this in place we are ready to derive a contradiction.

150
5. The Langlands Classification
Let h e aF be such that a(/i) > 0 for all a e <t>(PF, AF). Set a, = exp th. Let
m e °MF. Then Theorem 5.3.4 implies that
0= lim e,("-",(*,<n(fl,m)/,ff> = <ff(m)/J(/),ff(l)>,
here it = na and /? = PPF_ail. Since a is irreducible this implies that g(\) = 0,
which is the desired contradiction.
5.4.3. We now prove 5.4.1(3) (we use the notation therein). Let VxJPFt„tllx
Jpf.o'.p'- We choose a realization (it, H) of V. Lemma 5.2.8 implies that there
is a constant d > 0 such that if v e V, v~ e V~ and if a e C\(A+) then
K„~(a)|<CaRe"-'(l+log||a||)d and \cv^(a)\<C'aR^'-"(\+\og\\a\\)d.
Let h e aF, be such that a(/i) > 0 for all a e <t>(PF., AF,). If we set a, = exp th
then we have
lim a^-Re"-ei,cvv~(a,) = 0 for all e > 0.
!-> +00
This implies (here we use 5.3.4 and 5.3.1) that (Re n + ep)(h) > Re n'(h) for
all e > 0. If we take the limit to e = 0 then we have Re n(h) > Re n'(h) for
all such h. This in turn implies that Re(ji, j8y) > Re(n', j3s) for all ;" not in F'.
Hence F is contained in F'. If we interchange the rolls of F' and F we find
that F = F' and Re \i = Re \i'. Let h be as above. Then
lim ar,lcVtV~(a,)=oi(v,v~),
r-> + oo
lim ar*'cv.Aa,) = P(v,v~),
r-> + oo
both exist and the bilinear forms a and /? are both non-zero. We may
thus choose v, v~ so that a(v,v~) is non-zero. Then limr^ + 00 af~"' —
P(v, v~)/a(v, v~). But Re(ji — n')(h) = 0. Hence fi = \i'.
We are left with proving that a % a'. Let S(resp. U) be a (g, X)-module
homomorphism of /pFiffi/J (resp. //>,,.„,„) onto K. Let teK and let / e /pr,ff„
(resp. g e /pF.ff.„) be such that Sf = v (resp. l/(g) = i;). Let ke KF and let
m e l/(°mF). Set jx = mkf and ^! = mkg. Then S^) = mkv = U(gi). Let /i
be as above. Then {itaJat)fuf)> = (na,Jat)gl,g)> for all t. If we replace /
by k'f (if necessary) we may assume that f(\) is non-zero. Theorem 5.3.4
implies that {mkj3PF^(f), f(l)~> = <mkpPF.a,Jg),g(l)y. Since k e KF and
m € U(°mF) are arbitrary, this implies that a « a'. The proof of the
theorem is now complete.

5.4. The Langlands Classification
151
5.4.4. We are now ready to state the celebrated Langlands classification of
irreducible (g, K)-modules.
Theorem. Let V be an irreducible (g, K)-module. Then there exists Langlands
data (PF, a, p) such that V is (g, K)-isomorphic with JPFiaill.
In light of the uniqueness statement in 5.4.1 the above Theorem reduces the
classification of irreducible (g, K )-modules to the classification of irreducible,
tempered (g, K)-modules. 5.2.5 reduces this question to the classification of
irreducible "rapidly decreasing" unitary representations of the °MF and the
determination of the constituents of the unitarily induced representations in
5.2.5. We will see that the "rapidly decreasing" representations are the
"discrete series" which we will parameterize in the next three chapters. The full
determination of the tempered, irreducible (g, K)-modules has been carried
out in Knapp, Zuckerman [1].
5.4.5. We now begin the proof of 5.4.4. Let V be an irreducible (g, K)-
module. Set E(V) = {— p + p\p e £(P0, V)}. We use the partial order and
notation in 5.A.I. Let A e E(V) be such that (Re A)0 is a maximal element
among the (finite set of) (ReA)0, A'eE(K). Put F = F(Re A). We will
identify F with the corresponding subset of A0. Set p = A|Q. Then Re p =
(Re A)0. Set W = (K~/nFK~)_/1 + p. Then W is a non-zero finitely generated,
admissible (mF,KF)-module (3.7.2, 4.2.6). Let 3e(*aF)^ be such that
(W/nFW)d_ll + p is non-zero. Then 3 - p + p e £(P0, V). We relabel {1,..., r)
so that F = {l,...,t}. Then *aF is the linear span of {//„,,..., H„t}. Let
Pi,...,P, be the corresponding "/?/' for *aF and (a.i,...,a.,}. Set k =
Re( — 3 + *pF). Then 5.A. 1.3 implies that there is, a subset, F', of {1,..., t}
such that
1 = - E w + E xjPj
J'ef itf
with y/j <0,je F' and Xj > 0, j £ F'. Hence
k + Re p = - X yj«j + E xjPj + E x,h
iff JtF' j>t
with tj > 0 for ;" > t. Now jS, > 0 for ; = l,...,t (5.A.1.1(1)). We assert that
F' = {1,..., t}. If not then X + Re p > — ZjeF- y}a.} + "Lj>tXjf}j. Hence 5.A.1.3
implies that (X + Re p)0 > S tjfij■ = Re p = (Re p)0. Since A was chosen
such that (Re A)0 is maximal we have a contradiction. If we "unwind" the
minus signs we have shown
(1) W~ is a tempered (°mF,XF)-module.

152
5. The Langlands Classification
The exactness of the Jacquet module (4.1.5) now implies that if Z is an
irreducible, non-zero quotient of W then Z~ is tempered. Let (o,Ha) be an
irreducible unitary representation of °MF such that (Ha)K = Z~ (5.2.6).
Lemma 5.2.3 implies that V~ is equivalent with a submodule of IPf,„-.-„•
Hence V is equivalent with a quotient of IPF.„tll. Since (PF,a,n) is Langlands
data, Theorem 5.4.1 implies that V is (g, K)-isomorphic with JPFt„ttl. This
completes the proof.
5.5. Some applications of the classification
5.5.1. In this section we will use the results of the last section to derive some
results that refine the growth conditions of Section 4.3. We will also drop the
provisional definitions of Section 5.1. We begin with the following direct
application of 5.4.4 and 5.4.1.
Theorem. Let (n, H) be an admissible Hilbert representation of G that satisfies
the weak inequality. Then HK is tempered. (See 5.1.1 for the definitions.)
If (rc, H) satisfies the weak inequality then every subquotient of (n, H) does
also. The exactness of the Jacquet module implies that HK is tempered if and
only if every irreducible subquotient of HK is tempered. Thus to prove the
Theorem we may assume that (71, H) is irreducible. According to 5.4.4 there
exist Langlands data (PF,o,n) such that HK is equivalent to JPF,„ttl. If PF is
proper then 5.4.1 combined with 5.3.4 implies that (n, H) cannot satisfy the
weak inequality. If PF = G then HK is tempered by the definition of Langlands
data.
5.5.2. In light of the above result, we will use the term tempered to describe
the weak inequality as well as the definition in 5.1.1.
The next result uses an idea due to Milicic [1].
Theorem. Let V be an admissible finitely generated (g, K)-module. Let (n, H)
be a realization of V. If fie a* is such that if ae Cl(.4 + ) then
|<7T(a)t;,w>| < Ca"(l + log ||a||)d
for v, we HK(=V) for some constants C and d (possibly depending on v, w).
Then \i > Av (see 4.3.5).
Let £ be a finite dimensional irreducible (g, X)-module with highest
weight X relative to <t>(P, A). Then a acts on £~/n£~ by —X. Clearly,

5.5. Some Applications of the Classification
153
(V~/nV~)®(E~/nE~) is a quotient of (V~ ® £~)/n(K~ ® £~). Let (e
£(P, V~). Then a acts on (V~/nV~\® (£~/n£~) by the generalized
eigenvalue C — A. This implies that V~ ® £~/n(K~ ® £~) has a nonzero (m, °M)-
module quotient of the form Ha C„A with a an irreducible finite dimensional
representation of °M. Thus there exists a non-zero (g, K)-homomorphism,
T, of K~®£~ into 7Pjffj?_A_p. By duality, there exists a non-zero (g, X)-
homomorphism of //>,„-,a + p-c into K® £.
Let A be so large that (P, a, X + p — () is Langlands data. Let a+ =
{h ea\ct(h) > 0 for a e <S>(P,A)}. 5.3.4 combined with 5.4.1 imply that there
exists a matrix coefficient, u, of K® £ such that
lim e',c-/l),',)u(expt/i)= 1, /iea+.
Now every matrix coefficient of V ® £ is a linear combination of products
of a matrix coefficient of V and a matrix coefficient of £. If / is a matrix
coefficient of £ then it is easy to see that \f(a)\ < CaA for a e C\(A + ). Our
hypothesis now implies that
\u(a)\ < Ca" + A(l + log \\a\\)d for a e C\(A + ).
Hence, for each e > 0 there exist positive constants Cl and C2(e) such that
if h € a+ then
Cig.(M-"«0(*) < |u(exp t/,)| < c2(e)e,ll, + x + '"'w.
This clearly implies that if h e a+ and if £ > 0 then
(X - Re C)(h) <(n + k + ep)(h).
If we take the limit as e -»0 and use continuity we see that n(h) > — Re (,(h)
for all h e Cl(a+). The result now follows from the definition of Av.
5.5.3. If F is a subset of A0 then we identify (aF)* with the subspace of
elements of a* that vanish on *aF.
Corollary. Let V be an irreducible (q,K)-module. Let (PF,o,n) be Langlands
data such that V is equivalent to J = .//>,,,„,„. Then Re n = (Av + p)0. We will
call Re fi the Langlands parameter of V (if V is tempered p = 0). // W is an
irreducible subquotient of /,>,,,„,„ with Langlands parameter X then X < Re p with
equality if and only if W = J.
Let (n,H) be a realization of V. Then 5.2.8 implies that
|<7i(a)t;,w>| < CaRe""(1 + log ||a||)d

154
5. The Langlands Classification
forv,weVand a e C\(A +). Hence, Rep — p > Av by the previous result. Let
A e £(K) be such that Re A is maximal among the elements of E(V). Then
Re(A + p)0 < Av + p, by the definition of Av. Hence Re p. > (Av + p)0 >
Re p. This proves the first assertion.
5.2.8 combined with 5.5.2 now implies that if W is an irreducible
subquotient of IPF<„mll with Langlands parameter X then X < Re p. If X — Re p
then W is equivalent to //>,,,„,„• with Re p' = X = Re p. Now this implies that
there exists a matrix coefficient, u, of /pF,ff!/J such that
lim e'{"~p)ih)u(e\pth) exists and
r -■» + oo
lim e'("'~p)('1)u(exp t/i) exists and is non-zero.
t ++ 00
This can only happen if p' = p. Now Theorem 5.3.4 (combined with 5.4.1)
implies that W = J.
5.5.4. The above result allows one to prove theorems about irreducible
(g, X)-modules by induction on the size of the Langlands parameter. We next
show that the notions of strong inequality, rapidly decreasing and square
integrable all coincide.
Theorem. Let (it, H) be an irreducible square integrable representation of G.
Then HK is rapidly decreasing.
Let v,weHK. Set f(g) = {n(g)v, w>. Then / is X-finite (5.A.3.1) and Xf is
square integrable for all X e U(q). Let h e Cl(a+) be non-zero. Then 5.A.3.4
implies that
lim e'p{h)f(exp th) = 0.
!-> + 00
Set V = HK. Let (e E(P,V~). Let £ be a finite dimensional irreducible
(g, X)-module with highest weight X such that (P, a, X — ( + p) is Langlands
dataforallae°MA.
Let (P, a,p) be Langlands data. Let h e Cl(a+) and let F = {a e A01 a(/i) =
0}. We leave it to the reader to show that if fu f2e lPa „ then
lim e«("-'"w<n(exptA)/1,/2>= | </1(nfc),/2(fc)>diidfc.
r-> + oo Nf*Kf
(Use the argument in 5.3.4.)
This combined with the argument in the proof of 5.5.2 implies that V ® £
has a matrix coefficient, u, such that
lim e'a~"){h)u(expth)= 1.
!-> + 00

5.5. Some Applications of the Classification
155
By the material at the beginning of this proof
lim en"-xmu(expth) = 0.
!-> +00
Hence lim,^ + x e'f-W = 0. This in turn implies that Re(p - Q(h) < 0. Since
h is an arbitrary non-zero element of Cl(a + ), we have shown that — Re ( + p
e —+a*. We conclude that Av + pe —+a*. Hence V is rapidly decreasing.
5.5.5. We will therefore drop the terms "rapidly decreasing" and "strong
inequality" for irreducible (g, X)-modules. We conclude this section with two
results that are consequences of 5.4.1(1).
Proposition. Let a be an irreducible representation of °M and let ps a*. Then
h.o.n ,s a finitely generated (g, K)-module.
If (P, a, p) is Langlands data then the result follows from 5.4.1(1). Let £ be an
irreducible, finite dimensional (g, K)-module with highest weight X such that
Re(ji + X) e (a*)+. We may assume that °M acts trivially on £*/n£*. We
define a (g, K)-module homomorphism, T, of lPm„mtl + k ® £ onto 7P-ffj/J as follows:
Let y be a non-zero element of £* such that y(n£) = 0. Then y(namgv) =
a~xy(gv) for v e £ and n e N, a e A, m e °M and g e G. Set T(f ® v)(g) =
f(g)y(gv). Then it is easy to see that T is a (g, K)-homomorphism of
Ip,o,n + x® ^ to /,>.„.„. We leave it to the reader to check that it is surjective.
Since a quotient of a finitely generated (g, X)-module is finitely generated,
and the tensor product of a finitely generated (g, X)-module with a finite
dimensional (g, X)-module is finitely generated, the result now follows.
5.5.6. Theorem. Let x be a homomorphism of ZG(g) to C. Then up to
equivalence there are only a finite number of irreducible (g,K)-modules with
infinitesimal character %.
Let V be an irreducible (g, X°)-module. Set I(V) equal to the space of all
functions / from K to V such that f(K) is contained in a finite dimensional
subspace, W, of V, f is a smooth function from K to W and f(uk) = uf(k)
for u e K°. If / e I(V) then set kf(x) = f(xk) for x, k e K. If X e g then set
Xf(k) = Ad(k)X)f(k). Then it is clear that I(V) is a (g,X)-module. Suppose
that Z is a (g, X)-module containing V as a (g, X°)-subquotient. Let U be
a (g, K°) submodule such that Z/U contains V as a submodule. Let P be a
K° invariant projection of Z/U onto V. Let q be the canonical projection of
Z onto Z/U.U ze Z we define T{z)(k) = P(q(kz)\ Then it is easy to check that
T(z) e I(V) and that T is a (g, X)-module homomorphism. We observe that

156
5. The Langlands Classification
I(V) is a finitely generated (g, K)-module. Indeed, let v e V be a non-zero
element. Let up j = 1,..., d be such that X = [JjUjK0, a disjoint union.
Set fk(ujK°) = bLkv. Then it is a simple matter to check that I(V) is generated
by the fk as a l/(g)-module. Thus we have shown that if V is an irreducible
(g, K°)-module then there are only a finite number of irreducible (g, K)-
modules that can contain V as a subquotient. Thus if we prove the result in
the case when G is connected, the Theorem will follow.
We therefore assume that G is connected. Now 3.5.8(2) implies that there are
only a finite number of pairs (a, fi) with a e (°M)A and fi e a£ such that IPa ^
has a fixed infinitesimal character. Since each of these modules is of finite
length, by the previous result, the theorem now follows from 3.8.3 and 3.8.2.
5.6. 51.(2, R)
5.6.1. In this section we show how one can use the Langlands classification
to classify the representations of SL(2, R). In the next section we will use
similar arguments on SL(2, C). Let G = SL(2, R). We take K = SO(2) and we
take P to be the subgroup of upper triangular matrices in G. We set
H
and a = RH. We put a, = exp tH and A = {a,\ t e R}.Then°M = {/,-/}. So
°MA = {1, e} with e( —/) = — 1. We identify a% with C by identifying \i with
H(H) (note that p = 1 with this normalization). If a e °MA and if fi e C then we
write /„.„ for IPa^.
If k e Z then we define yk e XA by
Ik
cos 8 sin 6
— sin 6 cos 6
,ifcS
Then XA = {yk\ke Z}. If V is a (g,X)-module we will write V(k) for V(yk).
As a X-module, Ia<ll = IndfM(a). So Frobenius reciprocity implies that
(1)
(2)
'i.„=©'i.„(2*)>
',„ = © /..„(2* + 1).
Furthermore all of the isotypic components in (1) and (2) are one dimensional.

5.6. 5X(2, R)
157
As a K-module (under Ad), gc = CI © Cy2 © Cy„2- Set
Then X = {exp i6h\6e R}. We fix x, ye gc with [/i,x] = 2x, [/i,y] = — 2y
and [x,.y] = h. Then 0(x) = -x, 0(y) = -y. We choose B(X, 7) = tr AT 7.
Thus B(h,h) = 2, B(x, y) = 1. The corresponding Casimir operator is
(3) C = (i)/i2 + xy + yx = £)h2 + h + 2yx = £)h2 -h + 2xy.
We set
"o r
0 0
Y =
"0 0"
1 0
Then
(4) C = (\)H2 + XY+ YX = [\)H2 + H + 2YX = ({)H2 -H + 2XY.
It is easy to see that
(5) Cactson/^byiO^-l).
We also note that since h acts by kl on IaJJi), (4) and (5) imply that
2xyLM(t) = ^2-(/c-l)2),
(6) 2},x|/<„M(t) = 2V-(/c + l)2).
(6) implies that if \i is not an integer or if n is an integer and if a i= e"+'
then every isotypic component of /„„ is cyclic. Indeed, xla^(k) is contained in
/„„(/( + 2) and yla.^(k) is contained in /„„(/c — 2). We therefore see that
(7) 7ffi/J is reducible only if ji e Z and a = e"+'.
5.6.2. We now introduce an auxiliary family of (g, X)-modules that will
be used to analyze the possible reduction points in 5.6.1(7). Set q = C/i© Cx
and q = C/i © Cy. If k e Z then we denote by Ck (resp. Ckj) the q (resp. q)-
module C with h • 1 = k and x • 1 = 0 (resp. y • 1 = 0). We set Vk =
U(Qc)(S)ui<,)Ck and Vk = V(9c)<g)vmCk. Then both Vk and F* are
admissible finitely generated (g, X)-modules (use the basis ymx"h" of U(qc) to see
that {/"(g> 1} is a basis of Kk). Here K acts by u • g® 1 = Ad(u)g ® yk(u) 1 for
ueK and g e l/(gc).
We observe that gc is a TDS, thus there is, up to isomorphism, exactly one
irreducible k + 1 -dimensional representation, Fk for each k e N. We will now

158
5. The Langlands Classification
prove
(1) Vk (resp. Vk) is reducible if and only if k > 0 (resp. k < 0). If Vk (resp. Vk)
is reducible then it has a unique maximal, proper submodule which is
isomorphic with V~k ~2 (resp. V~k+2\ The corresponding irreducible quotient
is Fk (resp. F|t|).
Let vk e Vk be the element that corresponds to 1 ® 1. Then Cvk =
j((k + l)2 — \)vk. Since 1 ® 1 generates Vk as a g-module this implies that
C acts on Vk by the scalar i((/c + l)2 - 1). 5.6.1(3) implies that
xy'vk = i((/c + l)2 - (k - 2r + 1)2)7'-' vk.
Thus if k < 0 then x_yri;t is non-zero for all r > 0. Now, Kk = Q)Cyrvk.
Hence, if k < 0 then Kk is irreducible. (The argument for the irreducibility of
Vk goes in the same way, with the signs reversed.) If k > 0 then (6) implies that
xyk+lvk = 0. Thus we have a non-zero (g,K)-homomorphism of V'k~2
into Vk. V~k~2 is irreducible by the observations we have already made. It is
easily seen that Vk/V~k~2 is irreducible and isomorphic with Fk. The
result for the modules Vk is proved in the same way.
5.6.3. H acts on Fk/nFk by — k and °M acts by ek. Thus
(1) Fk imbeds in /E]_t_ i and nowhere else.
Since, the admissible dual of /„„ is /„,-„, and each Fk is self-dual we see that
(2) Fk is a quotient of lak + ^.
Setfor/c>0,/ceZ,Dt = V~k-\D.k= Vk + 1. We also set/)+ 0 = r'and
0-,o='?1-
Lemma. // 7ffj|1 is reducible (that is, fie Z and a = et+') t/ien
(a) If n> 0 then Dli®D_tl imbeds in /„ „ and the corresponding quotient is
isomorphic with F)l_l.
(b) If n<0 then F_/I+1 is the unique irreducible submodule of / and
lo.n/F-n+i is isomorphic with £)„ ® D.^.
(c) If n = 0 then a = e and /E 0 is isomorphic with D+ 0 © £)-,0-
Suppose that /c > 0 and that Dk imbeds in Ia ll. Then a = ek+' and 5.6.1(5)
and the calculation of the eigenvalue of C on V~k~' imply that /^ = ±/c. If
H= —k then (1) implies that Ft_! imbeds in 7ffj/J and since (P,a,k) is Lang-
lands data, we would have the contradiction Dk x Fk_!. We have shown
(3) If k > 0 then Dt imbeds in 7ffj/J with a = et+1 and /^ = /c and in no
other I

5.7. 5X(2,C)
159
Similarly, we have
(4) If k < 0 then Dk imbeds in 7ff-/J with a = ek+' and n= —k and in no
other /„„.
Fix for the moment k > 0, a = ek+'. Then Dk and D_t are both isomorphic
with submodules of Iak. Both of these (g, K )-modules are irreducible and since
they are inequivalent (even as K -modules). It follows that the direct sum
Dk@D k is isomorphic with a submodule of Iak. As a /<-module,
Dk ® D k ® Fk_! is isomorphic with Iak. (2) now implies (a).
(b) follows from (a) and I~k x la t, Dk % D k and Fk_ i k Fk_i.
If fi = 0 then as above, the only place that D ± 0 can imbed is in 7E 0. Also as a
/<-module 7E 0 is isomorphic with D+ 0 © Z)_j0. So (c) follows as above.
5.6.4. In light of the above results and the Langlands classification, we have
(1) The non-tempered representations consist of the /„„ with Re n > 0 and
/j^Zor/ieZ and a ^ e"+1 and the Ft, /c > 0 ( = JP,Ek t+1).
Lemma 5.6.3 implies that if V = Dk or D„fe with /c > 0 then Av = — k — 1.
Thus Dk and D k are the underlying (g, X)-modules of irreducible square
integrable representations. We can now give the list of irreducible
representations of SL(2,R).
(I) The square integrable representations Dk, keZ, \k\ > 0. These are
usually called the discrete series.
(II) The unitary spherical principal series, ^ ifl,fi e R. The irreducible unitary
non-spherical principal series, 7£ i)t with jjeR- {0}.
(III) D+ 0 and D 0, the constituents of the reducible unitary principal series.
These are sometimes called limits of discrete series.
(IV) The finite dimensional representations Fk, k e N.
(V) The Ia)l with Re n > 0 and [i not an integer or/ieN and a # e"+'.
5.7. SL(2,C)
5.7.1. In this section we use elementary methods to give the classification of
irreducible (g, X)-modules for G = SL(2,C). We look upon G as a real
reductive group. Thus the Lie algebra of G, g, is looked upon as a real Lie
algebra. On the other hand, g = s/(2, C), which also has the structure of a Lie
algebra over C. We will write J for multiplication by i" on g and look upon J as
a real endomorphism.

160
5. The Langlands Classification
We choose K = SU(2) and P to be the group of upper triangular elements
of G. Let H be as in 5.6.1. We set a = RH. If t e R then we set a, = exp tH
and we take A = {a,! t e R}. We set
m(B)
e'B 0
0 «-"
Then °M = (m(0)!0eR}. We note that in this case, °M = T, a maximal
torus of K. If /c e Z then we define at e TA by <xk(m(0) = eike. Then TA =
{at! k e Z}. We look upon ac as C by identifying p with ji(//). With this
identification, p = 2.
From the representation theory of SU(2), we know that KA = {yk k e N}
with dim yt = /c + 1. We will use the following tensor product formula
repeatedly.
(1) 7k® 7]= © yk+j-2r-
0<r<min(fc.j)
The easiest way to prove (1) is to use characters. We leave this as an exercise
to the reader who has not seen this formula before.
5.7.2. If V is a (g, X)-module then we write V(k) for V(yk). We note that
g = f ® Jt and that (Ad, fc) e y2. This combined with (1) above implies
(1) If V is a (g, X)-module then gK(/c) c V(k + 2) ® V(k) ® V(k - 2). Here,
we set V(j) = 0 if ; < 0.
We write lkmll for lPm„kmll for k e Z and jjeC. We note that the multiplicity of <jj
in yk is 1 if |;| < k and k + j is even and it is 0 otherwise. Thus Frobenius
reciprocity implies that
(2) h., = © /M(2; + |*|) with dim IkJ2j + \k\) = 2) + \k\ + 1.
The subrepresentation theorem now implies that
(3) If V is an irreducible (g, X)-module then dim V(j) < j + 1.
If V is an irreducible (g, X)-module then we set k(V) = min{/c V(k) is
nonzero}. Then k(V) is called the minimal K-type of V.
(4) If V is an irreducible (g, X)-module with minimal K-type, k = k(V), then
there are two possibilities:
(i) V is finite dimensional.
(ii) V(k + 2j) is non-zero for all j > 0.

5.7. 5X(2,C)
161
Indeed, if V is infinite dimensional and if V(k + 2/) = 0 for some ; > 0
then Zr<J V(k + 2r) is g-invariant by (1). This is a contradiction.
(4) implies
(5) If Ik is reducible then it must have either a finite dimensional submodule
or a finite dimensional quotient module.
Indeed, assume that / = lKll is reducible. Then 5.5.5 implies that / has an
irreducible, non-zero submodule, V. If V is finite dimensional then we are
done. If V is infinite dimensional then V(k{V) + 2j) = I(k(V) + 2j) for; » 0.
Hence, I/V is finite dimensional.
(5) reduces the study of the reducibility of the lKtl to the determination of the
imbeddings of the finite dimensional representations in the principal series.
5.7.3. We note that [Jx,_y] = J[x, y] = [x,J_y] for x, ye%. This implies
that u = {x e gc| Jx = ix} and u = {xe gc| Jx = — ix} are commuting
ideals in gc such that gc = u © u.
Let X be as in 5.6.1. Then X, JX is a basis of nc. Also, h = CH © CJH
is a Cartan subalgebra of gc. Clearly, ad(H)X = 2X and ad(JH)X = 21X.
Define ox, o2e h* by o^H) = o2{H) = 2 and o^JH) = 2i, o2(JH) = —2i.
Then {al5a2} is a system of positive roots for <t>(gc,h). Let H}, j = 1, 2 be
defined by Oj(Hk) = 2djk. Then C^ and CH2 are respectively Cartan sub-
algebras for u and u. We have
(1) H = Hl+H2 and JH = i(Hx - H2).
We can now apply the theorem of the highest weight to see that the finite
dimensional irreducible (g, X)-modules are parameterized by pairs of non-
negative integers. We write FJ,k for a representative. We leave it to the reader
to check that
(2) As a K-module FJ,k = y} ® yk.
Now h acts on FJ,k/ncFJ-k by the lowest weight of FJ,k. Thus Hl acts by
~j and H2 acts by —k. We recall that p = 2. We have therefore proved
(3) F'-k imbeds in /k_jj_J_k_2 and it imbeds in no other principal series
representation.
The conjugate dual representation of F''k is FkJ. Thus we have
(4) FkJ is a quotient of Ik-jj+k + 2 an^ it is a quotient of no other principal
series representation.

162
5. The Langlands Classification
5.7.2(5) now implies
(5) The only reducible principal series representations are Ik-j,-j-k-2 and
Ik-j.j+k + 2 f°r j\ k non-negative integers. The first type has FJ,k as a submodule,
the second type has Fk,i as a quotient module.
Since k + j + 2 > 0 for ;', k non-negative integers, FkJ = JPik-j,k+J + 2- Let
Zk-jj+k + 2 be the maximal proper submodule of Ik~jj+k + 2.
(6) Zk_yj + k + i is irreducible.
Indeed, if it were reducible then it would contain a finite dimensional sub-
quotient module, F. Now F would have the same infinitesimal character as
FkJ. This implies that F is isomorphic with FkJ. This contradicts 5.5.3.
To complete the classification we need only identify the modules
* - J, J + * + 2 •
5.7.4. Let y e u be such that ad Hxy = ~2y. Then ad H2y = 0. Let y =
yx + iy2 with y} e g for ;' = 1, 2. If / e C^iG) then we set
L(y)f(g) = d/dt\, = 0(f(exp(tyi)g) + if(exp(ty2)g)).
(1) Suppose that k e Z, fi e C and that \(k + n) = — p with p > 0 and p e Z.
Then L(y)"(IKll) is a submodule of 4+2p,w+2p = '-„.-*• Furthermore, L(y)p
is a non-zero (g,K)-homomorphism of 7kj/J into /_„,-*.
We note that [u, y] = 0. If x e u n nc and if [x, _y] = H^ then
[L(x),L(y)'] = -pL(y)"- '(£(//,) + p - 1).
The asserted intertwining properties now easily follow. We leave the details
to the reader. Since, H^"\f, contains Cf(N) the last assertion is also clear.
We are now ready to identify the Zk_]J+k + 2.
(2) Let ;', k be non-negative integers. Then I_j_k_2 j_k is irreducible and
isomorphic with Zk-JJ+k + 2 as a (g, X)-module.
Indeed, i((-; - k - 2) + (j - k)) = -(k + 1). 5.7.3(5) implies that
I-j-k-2J-k is irreducible. Hence, (1) implies (2).
For the classification we will need one more observation which follows
immediately from 5.7.3(5).
(3) If k e Z, \i e R then Ik!pi is irreducible.
Here is the classification:
I. The tempered representations consist of the Ikill with IteZ^eR and
each is irreducible.

5.8. Notes and Further Results
163
II. The finite dimensional irreducible (g, K)-modules.
III. The /t „ with Re n > 0 and at least one of ?(fi + k) or j(n - k) is not
a strictly positive integer.
5.8. Notes and further results
5.8.1. The results in Section 5.6 are originally due to Bargmann [1]. In a very
real sense, this work of Bargmann is the first to use the "infinitesimal method"
to study representations of semi-simple Lie groups. It contains the pivotal
ideas of expanding in terms of isotypic components and the use of the Casimir
operator. It seems that Bargmann did this work on the suggestion of Pauli.
5.8.2. The results of Section 5.7 are originally due to Gelfand, Naimark [1].
In this paper the methods are of a more global nature. The point being that
every irreducible unitary representation is either the trivial one-dimensional
representation or is infinitesimally equivalent to an irreducible principal series
representation (i.e., either unitary principal series or complementary series).
5.8.3. Proposition 5.2.5 is essentially (that is after the material in 5.5 is taken
into account) a result of Harish-Chandra, Langlands [1] and Trombi [1].
5.8.4. The intertwining operators as studied in Section 5.3 are due to Harish-
Chandra. The motivation for these operators comes from the earlier work of
Kunze, Stein [1], [2], who studied these operators in the case of minimal
parabolic subgroups. See also Knapp, Stein [1]. The main point in the earlier
papers was to give a meromorphic continuation of the operators jPm„mV of 5.3.4
to allow v to be purely imaginary. This analytic continuation will be
implemented in Volume 2 of this book.
5.8.5. As was indicated in the body of this chapter the Langlands
classification is due to Langlands [1]. The formulation given involves some ideas of
Milicic, and it follows the broad lines given in Chapter 4 of Borel, Wallach [ 1 ].
To complete the classification of irreducible admissible (g, X)-modules, it is
necessary to classify the irreducible tempered representations. In light of 5.2.5
and 5.5.4 it is enough to determine the irreducible square integrable
representations (this will be completed in 8.7) and to find the equivalences
between the irreducible components of the representations IPaiv for a
irreducible and square integrable, v real. The latter part has been done by
Knapp, Zuckerman [1]. In that paper, an unambiguous parametrization of
the irreducible tempered representations is also given.

164
5. The Langlands Classification
5.8.6. Theorem 5.5.6 is usually proven using the theory of characters, in
particular Harish-Chandra's regularity theorem (8.4).
5.A. Appendices to Chapter 5
5.A.I. A Lemma of Langlands
5.A.I.I. Let V be a real vector space with inner product < , >. Let
{at!,..., ocr} be a basis of V such that <ay,at> < Ofor; =£ k. Let fik,k = \,...,r
be denned by </J>;,at> = 3Jk for j, k = 1,..., r. We define a partial order on V,
x > y if x — y = S u-s a.s with Uj > 0.
(1) Pj>0, j=l,...,r.
Let yj be the Gram-Schmidt orthonormalization of the ak. Then our
hypothesis on the ctk implies that jj > 0 for all j = 1,..., r. The definition of
the Pj now implies that </?,, yk} > 0 for all ;', k. (1) now follows.
Let C = {x e V\ <x, a;> > 0 for all j = 1,..., r}. Then C is a closed convex
cone containing no line through the origin. If x e V let Cx = {y e C \y > x).
Then it is clear that Cx is a closed, non-empty, convex and C0 = C.
(2) Let x e V. There exists a unique element x0 e Cx such that ||x0|| < \\y\\
for all yeCx.
Let zeCj. Then T={ye C^11|>"||<||^||} is compact. Hence ||---|| achieves
a minimum on T at (say) x0. If u e Cx is such that ||u|| = ||x0|| then tu +
(l-t)x0eCx for all 0<t<l. ||tu + (l-t)x0||2 = t2||«||2+2t(l-t)<«,x0> +
(1 - t)2ll^oll2 < (t|l«ll + (1 - Oll^oll)2 = IKII2 with equality if and only if
u = cx0. Thus u — x0.
5.A.I.2. We note that
(1) If G c= {1 r} then the set {zy!z; = a7,/e G; zj = fy, j $ G}
is a basis of V.
If x e K then x0 = S U;j87. Clearly, uy > 0 for all;'. We set F(x) = {j.'Uj = 0}.
(2) If;£F(x)then<x0,ft.> = <x,ft.>.
Since x0 e Cx, it is clear that if j$ F(x) and if <x0,j8y> + (x,fij) then
<x0, /?;> > <x, jSj). We thus assume this inequality. Let e > 0. If k e F(x) then
<x0 — eolj, at> = — e<a7,at> > 0.

5.A.I. A Lemma of I anglands
165
If k $ F(x) then
<x0 - etXj, at> = <x0,at> - £<a7,at>.
Hence, if we take e > 0 sufficiently small then x0 — ectj e Cx. Set u = x0 — ea;.
Then
||u||2 = <u,x0> - e<ay,«> < <u,x0> < ||u||||x0||.
Hence ||u|| < ||x0||. This contradicts 5.A. 1.1(2).
(3) Let x0 = X ujPj- Then x = E zjaj + E ujPj
JtF{x) jeF{x) itF(x)
with Zj < 0 for ;' e F(x).
(1) implies that x = £,-eFW ZjtXj + Zj$Fixj Wjfy. If ;' is not in F(x) then
<x, j8j> = <x0,/?,■>. If we now observe that
det([<ft,/Jt>M<FW])/0,
it follows that w,- = Uj for ;' £ F(x). Since x0 e Cx, x — x0 < 0. Thus Zj < 0
for j e F(x).
(4) If x, _y e K and if x > _y then x0 > y0.
We first show that if y e Cx then y > x0. Let X be the linear span of the a.}
for/' e F(x) and let 7 be the linear span of the /?, for/ ^ F(x). Then V = X + Y
an orthogonal direct sum. Let P be the corresponding orthogonal projection
onto Y. If ; $ F(x) then
Paj = aj+ E xtJat-
fceF(x)
If p e F(x) then <Pa;,ap> = (a^a,,) + IteFW xtJ<at,ap>. Hence
E xt.;<afc'ap> ^°-
keF{x)
This easily implies that xkj > 0. We have thus shown that if z > 0 then Pz > 0.
We also note that if z e Cx then Pz e Cx. Indeed, Pz > Px by the above and
if ;' e F(x), <z, Pa,) > <z, 00. Finally, if z e C, one sees easily that z > Pz.
Thus, if z e Cx then z > Pz > Px = x0.
We now prove (4). Let x > y. Let z e Cx. Then z > x so z > y. Hence z e Cy.
But then, x0 e Cr Thus, the observations above imply that x0 > y0.
(5) If G is a subset of {1,..., r) and if
x= -E-w-+ E oft
with Sj > 0, / e G and t, > 0, / £ G then G = F(x).

166
5. The Langlands Classification
Set y = T.JiGtjPj. Then yeCx. If j is not an element of G then
<x,/J;> = (yjj) > <x0,ft> by (4). But x0 6 Cx so <x0,ft-> > <x,/J,>. Hence,
<x0, j8j> = (y,Pj} for ;" $ G. It is now clear that x0 > y (use the argument in
(4)). Hence (4) implies that y = x0. This completes the proof.
5.A.I.3. We now apply the above results to root systems. We use the
notation in 5.1.1. If we replace V by a* and < , > by ( , ) then we have
proved.
Lemma. Let p. e a*. Then there exists a unique subset F(n) of {1,..., r] such
that
v- = - E ypj + E xjpj
jeF(n) JtF(n\
with yj>0 and Xj > 0. Set n0 = ^jiF{n) *fiy If <?, ne a* and if n> a then
Ho > a0.
A similar proof of this lemma has been given by Carmona [1]. An alternate
constructive proof can be found in Borel, Wallach [1, Ch. 4, Appendix].
5.A.2. An a priori estimate
5.A.2.I. If x e R" then we denote, as usual, the coordinates of x by x!,..., xn.
Set (R+)" equal to the set of all x with Xj > 0 for j = 1,..., n. If S is a subset
of {1,..., n] then we set xs equal to the element of R" with (xs)j = 0 if j'• £ S
and (xs); = Xj if ; e S. Thus x0 = 0 and x11 "» = x. If x e (R+)" we set R(x)
equal to the convex hull of the xs. Then R(x) is a rectangle whose interior is
contained in (R+)".
We will use standard multi-index notation. Thus, if I = (i1,..., i„) with
ijeN then x' = x'fx'j■ ■ ■ x'„", d' = d\ld'£■■■&',? (with 6; equal to partial
differentiation in the;'-th coordinate) and |/| = ^ + • •• + i„. (We realize that
there is an overlap in notation, so multi-indices will be denoted by /, J, K and
subsets of {1,..., n) will be denoted by S, T.) We say that / < J if ik < jk for
k = 1,..., n. We fix K = (1,..., 1). The "fundamental theorem of calculus"
implies
(1) | dKf(x)dx = (-\)"Yj(-\ff(xs) for/eCco(R")andxe(R+)".
R(x)
\S\ denotes the cardinality of S.

5.A.2. An a priori Estimate
167
5.A.2.2. If H is a subset of R" and if yeR" then we write y + H =
{y + x x e H). We also write Hs = {xs\ x e H}.
Lemma. Let S be a non-empty subset of {1,..., n}. Let e > 0 be given. Then
there exists a positive constant CeS such that if x0eCl((R + )") and if
x e ((R + )"))s with Xj - (x0)} > e for all jeS then if f e Cco(R") is such that
d'f e L'((R+)") for all I < K then
1/MlsQsI I Wf(y)\dy.
I<K (R+)" + io
It is enough to prove the result for x0 = 0, since we can translate / by x0.
So assume that x0 = 0. If heC^(R") and if h(xs) = 0 for S ¥= Tand/i(xr)= 1
then (1) above implies
(1) | d«(hf)(y)dy = (- l)'r' +J(xT).
The Leibniz formula applied to (1) yields
(2) \f(x)T)\^llCI\\Q'h\\Rlx),ai | \e)K~lf(y)\dy.
I<K (R+)"
Here C, is a constant depending only on / and n and ||- • •||R(;c),00 is the sup norm
on R(x).
Let S be fixed as in the statement of the Lemma, If x e (R+)", then we set
u(x) equal to the element given by u(x)j = x} for j e S and u(x)j = 2||x|| for
j $ S. Then u(x) e (R + )" and ||u(x)r||2 > 4||x||2 for T + S, 0.
Let a e CC(R) be such that a(t) = 0 for |t| > 2, a(t) = 1 for |t| < f and
0 < a(t) < 1 for all t. Let /} e C* (R) be such that 0 < P(t) < 1 for t e R and
p(t) = 0 for |t| < i p(t) = 1 for |t| > i Set
h(z) = (j\ll(zj/e)y(\\z\\2/\\x\
2\
If Xj > e for j e S then h(u(x)T) = 0 for T ^ S. Since u(x)s = x for
x e ((R+)")s, (2) gives an estimate for |/(x)| (use u(x) in place of x and T = S
in (2)). We must therefore show that IIB'/iIIr,,,^,)i0O is bounded by a constant
depending only on e and S for each I < K. Leibniz's rule implies that it is
enough to estimate
9JI Y\p(Zj/s)\c)'-Ja(\\z\\2/\\x\\2)
for J < I < K, This expression is 0 if {k\jk > 0} = <J> is not contained in S.

168 5. The Langlands Classification
Otherwise it is equal to
(i/e|j|)( n p'(Zj/*i)( n /f(V8A(2nzii/iixii2)|/"'"a"/"'")(iizii2/i|x|12)-
If zeR(u(x)) then ||z||2 < (1 + 4(n - |S|))||x||2. This implies the desired
estimate,
5.A.2.3. This result has as an immediate consequence the following fact,
which will be used in Section 5.5.
Corollary. Let the notation be as in the previous result. Let x e ((R+)")s then
lim f(tx) = 0.
5.A.3. Square integrability and the polar decomposition
5.A.3.I. We maintain the notation of Section 5.1. Let y(a) be denned as in
2,4.2. If / e C™(G) then we say that / is K-finite if R(K)L(K)/spans a finite
dimensional space.
Lemma. Let f e C'x'(G)be K-finite, Then f is square integrable if and only if
(1) | 7(fl)|/(fc,flJlt2)|2dfl<QO fora\\kuk2eK.
x +
We first prove that (1) for all kx, k2e K implies that / is square
integrable. Let «!,..., ud be a basis for span{R(K)L(K)f}, Our hypothesis implies
that each «,- is square integrable on A + . Now f(klak2) = R(k2)L(kl)f(a) =
S hj(k1,k2)uj(a), with h} e CX(K x K). Thus, there exists a positive constant,
C, such that \f(k1ak2)\ < CL \Uj(a)\ for kuk2 e K and a e A +. Thus / is
square integrable by 2.4.2,
Suppose that / is square integrable. Then Lemma 2.4.2 combined with
Fubini's theorem implies that (1) is true for almost every kl, k2 e K. Let S be
the set of all (kuk2) such that (1) is true. Then K x K — S has measure 0 and
if (kuk2)eS theny1/2L(/c1)R(/c2)/eL2(/l + ). Since S is dense in K x K it is
easy to see, using K-finiteness, that y1/2span(L(K)R(K)f) is contained in
L\A+). HenceS = X x K.

5.A.3. Square Integrability and the Polar Decomposition
169
5.A.3.2. Lemma. Let f e C(A) then
| y(a)\Hf(a)\2da < oo, He U(a),
A*
if and only if
| a2p\Hf(a)\2da((X>, H e U(a).
A +
Since y(a) < Ca2p for a e C\(A + ) the sufficiency of the above condition is
clear. We will use the following result to prove the necessity, (Notation as in
the previous appendix.)
Scholium. Let eu..., en be the .standard basis of R". Let a1;..., ap e (R")* —
{0} be such that ct^ej) > 0 for all i, j and a,(e,) = c); j for 1 < i, ;' < p. Set
p = a.i + ■ ■ ■ + ap. There exists a constant C such that
(1) | e"M\f(x)\2 dx < C Y | (TrsinhaiMPM2^
<R+>" |/T<p(R+>"
for all f eCco(R").
We first prove that there exists a C > 0 so that (1) is true for / e CC(R"). If
p = n then the result follows from (sin hO = 0)
| sinh x — \f(x)\2 dx = - J cosh x\f(x)\2dx.
o dx o
Since \d/dx\f(x)\2\ < \f(x)\2 + \d/dxf(x)\2 and cosh x > ex/2 for x > 0. So
the result is obvious for p = n. Assume that the result is true for p — 1 > n.
We prove it for p. If we reorder the coordinates on R" we may assume
a.p(ei) > 0. Then
| (l\ sinh a^d/dxj]f(x)\2dx
(R+r \ i )
-\—Yj a^Ci) | cosh ocj(x) Y\ sinh at(x)\ f(x)\2dx.
j (R*)" i*j
So,
ap^J | cosh ap(x) Y\ sinh a,(x)|/(x)|2dx
(R+)" i<P~l
< I (Usmha^mx^ + ld/dxjix^dx.
(R+)"

170
5. The Langlands Classification
Thus
| cosh ap(x) Y\ sinh a,(x)|/(x)|2dx
(R+)" i<p-l
<Cp I fnsinha,.(x))(|/(x)|2+|8/6xp/(x)|2)dx
("*r \ t )
withCp = 2/ap(e1).
If we replace / by exp(ap/2)/. The inductive hypothesis for p — 1 implies
the result for p. We have thus proved the existence of C such that (1) is true
for / e Ccco(R"). We now prove the result using this C. Suppose that
/ e C^R"). If the right hand side of (1) is infinite there is nothing to do. So
assume that it is finite.
Let u e C°°(R) be such that u(x) = 1 for |x| < 1 and u(x) = 0 for |x| > 4. Set
for t > 0, h,(x) = u(||x||2/t2). Then h, is smooth, ht(x) = 0 for ||x|| > It and
ut(x) = 1 for ||x|| < t. If t > 1 then \5'h,(x)\ < C, for all x e R" (the important
point is that C, is independent of r.) Indeed,
(8/6x,.)/ir(x) = u'(||x||2/t2)2xj/t2
and |x,-| < It when u'(||x||2/t2) is non-zero. We now leave it to the reader to
prove the inequality for all /.
Now
lim d'h,f(x) = d'f(x)
r-* + oo
and the preceding remarks imply that
|6^r/(x)|2 < D7 X |8-y(x)|2 forxeR"
\J\s\i\
with D, depending only on /.
Thus
J e'w|/(x)|2dx = lim | e>lx)\h,f(x)\2 dx
<limCY J (nsinha^ie'^x)!2^
I -> oo I Ijs r (R + )"
= C Y | (nsinha,(x))|8'/(x)|2dx
\I\<r (R*)"
by dominated convergence. The Scholium now follows.
We now prove the Lemma. It is clear that we may assume that °(G°) = G°.
Take a1;..., a„ to be the simple roots in <t>(P, A). Let <b(P,A) = {a1;..., ap}
(here each root is counted dim g" times). Take e; e a to be the elements denned

5.A.3. Square Integrability and the Polar Deeompositon
171
by <*,(£;) = Stj. Use this basis to identify R" with a. The Lemma is now an
easy consequence of the Scholium.
5.A.3.3. Lemma. Let f e C*(G) be K-finite then xf e L2(G) for all
x e U(q) if and only if
| a2p\xf(kxak2)\2da < oo
A*
for all k1,k2€ K and all x e t/(g).
This is an easy consequence of 5.A.3.1 and 5.A.3.2.
5.A.3.4. Proposition. Let f e C"(G) be K-finite and such that xf e L2(G)
for all x e U(c\). If he a-{0} is such that <x(h) > 0 for all a e <t>(P,A) then
lim e'mf(exp th) = 0
Set g(a) = a2p\f(a)\2. 5.A.3.3 implies that hg e Ll(A + ) for all h e U(o). The
result now follows from 5.A.2.3.

This Page Intentionally Left Blank

6 A Construction of the
Fundamental Series
Introduction
As we have seen in the last chapter, the tempered representations (in
particular the square integrable representations) are the basic "building blocks"
to construct all irreducible admissible representations (up to infinitesimal
equivalence) of real reductive groups. Except for the simple case of SL(2, R) we
gave no indication of how one might construct irreducible square integrable
representations. In this chapter we use a method that is equivalent (see 6.10) to
Zuckerman's derived functors to construct (g, K)-modules (our method is
based on the results in Enright, Wallach [2]). An exhaustive account of
Zuckerman's functors can be found in Vogan [2]. The key new ingredient in
our presentation is the a priori proof of the unitarity of the fundamental series.
This combined with our theory of the real Jacquet module leads to a proof
that the fundamental series is tempered and square integrable when the
parameters are regular and there is a compact Cartan subgroup. In Chapter 8
we will show that the square integrable representations constructed in this
chapter (which we call the discrete series) give all of the irreducible square
integrable representations of real reductive groups.
We also derive many of the algebraic properties of the derived functor
construction. In particular, we prove generalizations of Blattner's formula for
the /C-multiplicities. In our development Blattner's conjecture (a theorem of
173

174
6. A Construction of the Fundamental Series
Schmid [2] and Hecht-Schmid [1]) is proven before the characters of the
discrete series have even been defined.
In Section 10 we discuss the relationship between the material in this
chapter and the corresponding results in the literature.
There are four appendices at the end of this chapter. Two of them (3 and 4)
contain basic results of the theory. Appendix 3 is an exposition (based on the
Jacquet module) of some of the results in Zuckerman [1] on "coherent
continuation". The technique is based on unpublished joint work with
Casselman. Appendix 4 contains the theorem of Harish-Chandra [1] which
asserts that an admissible finitely generated infinitesimally unitary (g, K)-
module is the underlying (g, /C)-module of a unitary representation of the
group.
Chapter 9 is independent of the material in the next two chapters. Thus a
reader interested in the applications of the results of this chapter to (g, K)-
cohomology can go directly to Chapter 9.
6.1. Relative Lie algebra cohomology
6.1.1. Let G be a real reductive group and let K be a maximal compact
subgroup of G. Let M be a closed subgroup of K such that det Ad(m) =
det Ad(m)|m. Let C(g, M) be the category of all (g, M)-modules (see 3.3.1). If
V is a (g, M)-module then we define CJ'(g, M; V) to be HomM(AJ'(g/m), V). We
define for 0 e CJ'(g, M; V),
dp(x0,...,xJ) = YJ(-i)kxk-p(x0,...,xk,...,xj)
+ £ (-\y+sP(Lxr,xsix0,...,xr,...,xs,...,Xj).
r<s
Here, Xj e g/m and Xj is a representative in g. It is standard that (C*(g, M),d)
is a complex. The cohomology of this complex is denoted by H\q, M; V). A
complete discussion of this cohomology can be found in Borel, Wallach
[1, Ch 1]. In this section we will only discuss a variant of Poincare duality
for this theory and a few specific results that will be used in this chapter.
6.1.2. Let u)0 be a fixed element non-zero of A"(g/m), where n = dim(G/M).
We define a sesquilinear pairing of (A;(g/m)*)c with (A"~J(g/m)*)c as follows.
Let conj(jS) denote the complex conjugation of /? e (AJ(g/m)*)c relative to the
real form AJ'(g/m)*. If a e (AJ'(g/m)*)c and if p e (A"~;(g/m)*)c then we define
(a,j8) by aA conj(jS) = (a,/i)a»0.
Assume that det(Ad(m)) = det(Ad(m)|m) for m e M. If V e C(g, M) then we
define V* to be the space of all conjugate linear functionals, \i, on V such that

6.1. Relative Lie Algebra Cohomology
175
Mfi spans a finite dimensional space. There is a natural pairing < , > of V*
with V given by in, v} = n(v). We look upon Cj(q,M; V) as a subspace of
(AJ'(g/m)*)c ® V. The restriction of the tensor product of the above pairings
induces a sesquilinear pairing of C"~;(g,M;V*) and C;(g,M;V). We will
denote this non-degenerate pairing by < , >.
(1) If«eC"_-''-1(fl,Af;K*) and p e Cj(q,M;V)
then<da,j8> = (-l)j<a,48>.
This is proved by direct calculation (cf. Borel, Wallach [1, p. 15]). Let
BJ(g,M;K) = dC;~'(g,M;K) and let Zj(q,M;V) be the kernel of d on
CJ'(g,M;K). (1) implies that
(2) {Zn'\%M; V*))1 = B%M; V) and (B"-J'(g,M; V*)L = Z\q,M; V)
relative to < , >.
(2) clearly implies that
(3) < , > induces a non-degenerate pairing of H"~j(q,M; V*)
with Hj(q, M; V).
6.1.3. Let W be an (m, M)-module. We form a (g, M)-module l/(gc) (X)U(m) W
endowed with the g-module structure given by left multiplication and the
M-module structure given by
m(g ® w) = Ad(m)g ® mw.
Lemma. H>(6,M;U(ac)®vlm)W) = 0 forj<n.
This result is a special case of Lemma 6.A. 1.5.
6.1.4. We now recall another result that will be useful in the next few
sections. Let U, VeC(q,M). Suppose that Te HomgM(U, V). Then T induces
a linear map of C'(q,M;U) into Cj(g,M;K) given by Tfi{Xu...,X}) =
T(P(Xl,...,XJ)). The formula for d implies that Td = dT. So T induces a
linear map of HJ(g, M; I/) into H'(g, M; K). If
0->l/->K-> W^O
is an exact sequence in C(g, M) then the corresponding maps on the O also
induce exact sequences. The standard method of cohomology theory now
yields a long exact sequence
-> H\q, M; U) -> Hj(q, M; V) -> H\q, M; W) -> HJ+ '(g, M; U) ->

176
6. A Construction of the Fundamental Series
6.2. A construction of (I, K)-modules
6.2.1. Let X be a compact Lie group. Set H(K) equal to the space of left
(hence right) X-finite smooth functions on X. We look upon H(K) as a (I, X)-
module in two different ways. We set L(k)f(x) = /(/c"'x) and R(k)f(x) =
f(xk) for f e H(K) and x, ye X. If V is a complex vector space then we
define C^X; V) to be the space of all functions, /, from X to V such that
f(K) is contained in a finite dimensional subspace W of V and / is smooth
as a function from X to W. On Cco(X; V) we also have two actions L and
R of X given by the formulas above. We set H(K; V) equal to the subspace
of those functions in Cco(X; V) that are X-finite under both actions.
Let V be a (I, X)-module with action given by it. If u e V and if / e H(X)
then we set Lv(v®f)(k) = /(/c);r(/r>. Then LK maps K®H(X) into
ff(X; K). An obvious calculation yields
(1) Lv(it ® L)(/c) = L(/c)LK for /c e K or 1/(1).
If / e H(K; V) then /(X) spans a finite dimensional subspace 1} of V. Let
«!,..., ud be a basis of ^. Then f(k) = I. fj(k)vj. We set Qv(f) = I.fj®vj.
It is clear that Qv(f) is independent of all choices used in its definition and
that it defines a linear map of H(K; V) into H{K)® V. Set SV = QVLV.
(2) Sv o (tt ® L)(/c) = (L(/c) ® /) o Sv and SK ° (/ ® R(/c)) = (R ® n)(k) ° SK
for k e X or 1/(1).
This observation is proved by the obvious direct calculation.
(3) Sv is bijective.
It is obvious that Lv and Qv are injective. Thus Sv is injective. We prove the
surjectivity. Let / e H(K), let v e V and let v1,..., vd be a basis for the linear
span of Kv. Let nx,..., nd be the dual basis set crr(/c) = nr(n(k~l)vt). Then
SV^j® /) = £ crjf® vr. Since X is compact, we may assume that
Iy cr>J- conj(crJ) = 5rJ. So
Sk( E u; ® conj(cr,;)/j = u, ® /.
6.2.2. Let M be a closed subgroup of X. Let V e C(f, M) with action rc. Then
we look upon V ® H(K) as a (f, M)-module under it® L and also as a (I, X)-
module under I ® R. We define
rj(V)=HJ(lM;V®H(K)).
Here the cohomology is relative to the first action above. We look upon T;(K)
as a (I, X)-module under the action induced by the (I, X)-module structure

6.2. A Construction of (t, K (-Modules
177
/ ® R. Then P is a functor from the category C(f, M) to the category C(f, X).
These functors are special cases of Zuckerman's derived functors. We will
show, in the next section, that one can construct the general ones from these.
Let F e C(I, K). We define, for each;, two functors from C(f, M) to C(f, K).
The first is K-> T\V® F) = AF(V) and the second is V->T](V)®F =
BF(V).
If C and D are categories and if A and B are a functors from C to D then a
natural transformation of /I to B is an assignment X \—> T(A') for each object
leCofa morphism T(X) e HomD(A(X), B(X)) such that if S e Homc(A:, Y)
then the following diagram is commutative
A(X) T{X) > B(X)
■4(S)
B(S)
/1(7) r(V) > B(Y)
If T(A') is an isomorphism for every X e C then we say that T is a natural
equivalence.
Lemma. Let F e C(x,K) then there is a natural equivalence TF of AF with BF.
Furthermore, if W is a (i,K)-module and if S eHomtK(W,F) then, if we set
Us = rJ(S ® /), the following diagram is commutative
Aw
VS(V)
(V)^±BW(V)
AF(V)^hBF(V)
Furthermore, if X, Y e C(f, K) then
TX®Y(V) = (TX(V)®I)TY(V®X).
We note that if X is a vector space over C, which we look upon as a
(I, X)-module with the trivial action, and if
V e C(f, M) then Hj(l M;V®X) = Hj(l, M; V) ® X.
This is immediate from our definition of relative Lie algebra cohomology.
Let SF be as in the previous paragraph. We put TF(V) = H\SF). Then
6.2.1(2), (3) imply the all but the last assertion of the Lemma. We now prove the
last assertion.
A direct calculation shows that
Sx®y = (Sx®I)(I®Sy).
To complete the proof apply the cohomology functor, HJ, to both sides of this
equation and use the fact that H' takes products to products.

178
6. A Construction of the Fundamental Series
6.2.3. We now come to a critical result in this theory. We look upon 1/(1) as a
(I, X)-module under the adjoint action. If V is a (I, M)-module and if F is a
(I, X)-submodule of 1/(1) then we have a (I,M)-module homomorphism
m: V ® F ->V given by v ® y t—» yv.
Lemma. Assume that M acts trivially on A"(f/m) (n = dim I/m). Let
V e C(f, M) then the following diagram is commutative
P(K®f)-^»rj(K)
7J(K)
P(K)®f
Identity
r\v)
We first prove the result for; = n. The formula for d and combined with our
hypothesis implies that ff"(f, M; V) = (K/f K)M for V e C(f, M). Thus
r"(m): (K® I® H(K))/l(V® I ® ff(K)) -(K® H(K))/f(K® ff(K))
is given by
£ ^®xt®./;.t^ £m® ./;.*•
It is easy to see that v ® Sk(X ® /) maps under / ® m to - u ® L(X)f. Hence
mSt(£ t;, ® Xt ® /M) = - £ «,- ® L(Xt)/JJk.
Since, X}vk ® J5.t + vk ® L(Xj)fjk e f(K® ff(K)) the result follows for ; = n.
We now prove the result by downward induction on/. Assume that the result
is true for; + 1. Let Z be the kernel of the natural mapping of l/(fc) (X)^, V
onto V. Since (l/(Ic) (X) (,,„,, V) ® H(X) is isomorphic with
(A.6.1), Lemma 6.1.3 implies that rj((U(ic)(g)U{mjV)® H(K))) = 0 for
y < n. The long exact sequence of cohomology now implies that we have the
exact sequence
■■■-P(l/(fc)(g)K)-
l/(m)
This yields the following prism
0 - P'(f ® K) -
rV)/
o-^rj'(K)®i-
rj'(K)->rj'+1(Z)-
n"(m)
'(f®Z)
7J(Z)
->rj+1(Z)
\
»P'+1(Z)®I

6.3. The Zuckerman Functors
179
where the edges starting with a 0 are exact. The inductive hypothesis implies
that the square and the right-hand triangle is commutative. Thus the left-hand
triangle is also commutative. This proves the Lemma,
6.3. The Zuckerman functors
6.3.1. Let G be a real reductive group and let K be a maximal compact
subgroup of G. Let M be a closed subgroup of K such that M acts trivially
on A'opf/m, We look upon U{c\c) as a (I, X)-module under the adjoint
action. If KeC(g, M) then we have the (g, K)-module homomorphism
^(9c) ® V ^ V given by g ® v i—> gv, which we denote by m. We will also
look upon V as a (I, M)-module, We can therefore apply the functors of the
previous section to V,
Lemma. Let V e C(g, M) then there is a unique structure of a (g, K)-module on
rj(V) such that the action of (f, K) is as in the last section and the following
diagram is commutative (U = C/(g)),
F(K®li)^irj'(K)®(/
P(m)
r\v)
Identity
HK),
Let m~ be the linear map such that if m is replaced by m~ on the right arrow
in the above diagram then the diagram is commutative. We must therefore
show that m~ is a C/(g)-module structure. To do this, we analyze the following
cube
r\v® u®u) r'{m®'\ rj(V® u)
rJ(m®/)
P(K)® U® U
l®m rj'(K® U)
/TV(V)
r\v)®u
x 1
TLAV)/
r\v)®v
n'(m)
m~ ® /
P'(m)
V\V)
P(K),
All of the faces are commutative except possibly the top and front faces. The
content of the Lemma is that the front face is commutative. Since all of the
"T" mappings are isomorphisms, it is, enough to show that the top face is

180
6. A Construction of the Fundamental Series
commutative. To prove this, we factor the top face as follows:
v\v® u®u) ' '> rj(v® u)
TV{V®U)
TV(V)
r\v® u)®u v'{m) > r\v)® u
TV(V)®1
rj(V) ®u®u m ®'> rj(V) ® u.
Now apply Lemma 6.2.3.
6.3.2. The above result implies that the Tj define functors from C(g, M)
to C(g, K). They are usually called Zuckermans functors. We now give some
of their basic properties.
Lemma. Let V e C(g, M) and if Vy e y e K then
UomUK(Vy,rj(V)) = Hj(i, M, V®(Vy)*).
The Peter-Weyl theorem implies that
H(K)= ®(Vy)*®Vy
as a (I, X)-bimodule. To complete the proof we will use the following result.
Scholium. Let X be a (I, M)-module and let L be a compact Lie group such
that X also has the structure of a (l,L)-module with the two structures
commuting. If y e LA then set X[y~\ equal to the y-isotypic component of X. Then
Hi(i,M;X)= © Hl(lM;X[y-]).
It is clear that the spaces X\_y] are (I, M)-submodules of X. Also, each
space C\l,M;X) is an (I,L)-module under the action (ufi)(xl,...,xi) =
u(j8(x!,..., x,) for ue L. Clearly, d(uji) = udfi. Thus each (I, L)-isotypic
component of C*(f,M;X) is a subcomplex. Let Ey be (as usual) the projection
onto the y-isotypic component. The by the above, dEy = Eyd. It therefore
follows that //*(!,M;X) is the direct sum of the cohomology spaces of
the complexes C*(i,M;X)[y~\. Since it is also clear that C'(I, M;X)[y~\ =
C\l,M;Xiy~\), the result follows.
We now complete the proof of the Lemma. As we have observed before,
the (I, X)-structure on V ® H(K) given by / ® R commutes with the (I, M)-
structure that we are using to calculate cohomology. Thus the Lemma follows
from the above Scholium and the observation preceding it.

6.3. The Zuckerman Functors
181
6.3.3. Lemma. Let V be (%M)-module. If V is admissible then VJ(V)
is an admissible (q,K)-module. Let I = {g e U(Q)\g acts by 0 on V) then
i. rj(V) = o.
We note that CJ(l,M; V®(Vy)*) is finite dimensional if V is admissible.
Thus the first assertion follows from Lemma 6.3.2. We note that / is a (g, K)-
submodule of C/(g). Hence the second assertion follows from 6.2.2 and the
definition of the (g, X)-module structure on r\V).
6.3.4 Proposition. // FeC(g,K) and if KeC(g,M) then TF(V) is a
(g, K)-isomorphism from T\V ® F) onto VJ(V) ® F.
We must show that
VHV®F)®U
Tr(V)®l
T\V®F)
TF(V)
(F(K) ® F) ® U —» rJ(V) ® F
is a commutative diagram.
To prove this we examine the following prism
rj(v® f® c/)J3^>n(K® f)
(**)
P(K)® F® U
TV{F ® V)
P'(K® F)®U
TF(V)®l/
m
V\V®F).
TF(V)
rj(V)®F
The triangles in the diagram are both commutative with invertible
maps. The rear face is commutative by the definition of m. The bottom face
is (*). Thus, if we can show that the top face is commutative then the result
will follow.
Let A: U(q) -> U(q) ® U(q) be as in 6.A.I.I. Let for, Y, a vector space over
C, T:Y®U®U ->U®Y®U be denned by T(y® ux ® u2) = t^ ®y® u2.
We consider the following diagram
-*B,
Mi
_»n ^i^B,
/'2
^3
-»B
6
D,
D,
°-^D,
-»D.
If this is commutative and if y, nt, d are all invertible then it is easy to see

182
6. A Construction of the Fundamental Series
that if a = aAa3ix2(x.x and fi = ftftftft then
-* B
C
P
D
is commutative.
We apply this observation to the case when A = P(K® F® U), Bl =
P(K® F® U® U), B2 = P'(K® U® F® U), B3 = TJ(V® U ® F), B =
rj(V®F), c = r\v)®F®u, D, = rj(V)®F®u®u, D2 = rJ(V)®
U®F®U,D3 = rj(V) ®U®F,D= rj(V) ® F, a, = P(7 ® / ® A), a2 =
F(7 ® T), a3 = F(7 ® m), a4 = P(m ® /), y = TF(g>u, ^ = TFe>ue>u(V),
H2 = TV9F9V(V), ii3=Tv9F(V), 3 = TF(V), ft =7® 7® A, ft = 7 ® T,
ft = 7 ® m, ft = m ® 7. (The reader should write out this diagram
sideways on a piece of paper.) All of the squares except for the last one are
obviously commutative. Since the diagram (***) is the top face of (**),
we will have (finally!) proved the result if we show that the last square is
commutative. Let us write it out.
r\v® u®F) r,(m@/|) r\v®F)
(****) T;.0f(K)
r\V)®U®F-
m® /
W)
rj(V)®F
To prove that this is commutative we examine the following prism
r«HrJ(K0F)
vW®t/) | \rt\v)
H
V® U ®F)-
* rj(m)®/ *
V'{V® U) ® F > P(V)®F
rj(V)®u®F
m®I
T„(V) J I
rj(V)®F.
All of the faces are commutative except for the back one. Since this face is
(****), we are done.
6.3.5. The next result is basic to the later developments of this theory. The
idea is due to Zuckerman the result was first proved in Enright, Wallach [2],
Let dim(f/m) = p.
Theorem. Let V be a (g, M)-module. Then there is a non-degenerate sesqui-
linear pairing between rJ(V) and rp~\V#), Furthermore, if p = In with n a

6.3. The Zuckerman Functors
183
natural number and if V admits a non-degenerate (g, M)-invariant Hermitian
form then T"(K) admits a non-degenerate (g, K)-invariant Hermitian form.
We should warn the reader that the proof of this result (involving
the material in 6.1.2) will be as important to us as the statement. Let B
denote the sesquilinear perfect pairing between HJ(l, M; K® H(Kj) and
H"~j(l M;(V®H(K)f) (6.1.2). In light of the Scholium above, B
induces a perfect pairing between Hj{i,M, K®H(K))[>] and Hp~i{i,M;
(V ® H(K)f)[y~\. Now, as in the last number,
H'-^f, M; (K® H(K))#)[y] = H"-\l M; (K® ff(K))*)[y]).
Let 5 be the (I,M)-module homomorphism of V* ®H(X) into (V®H(K)f
corresponding to the tensor product of the canonical pairing of V with
V* and the L2-inner product on H(K). Then 3 is an isomorphism of
K#®H(X)[y] onto (K® H(K)f[yl This, in light of the definition of
P (6.2.2) implies the result in the special case when g = I. We will abuse
notation and denote by B the (I, X)-invariant, non-degenerate, sesquilinear
pairing of P(K) with rp~J(V#). We now prove that B is g-invariant. We
have the following commutative diagrams
P'(K® g) ® rp~i{V#) T*iV)® l> (P'(K) ® g) ® V~\V*)
V'(m)®l
p(K)® rp~j(K#)
p
m®l
-+ri{V)®rp-J{v*)
-*c.
and
rj(V) ® rp-j(v* ® g) /®7""/ }> r\v) ® (rp~j(v*) ® g)
/® rp-'(m)
P'(K)®rp"J'(K#)
I ®m
->r\v)® rp-j{v#)
c >c.
The definition of B now easily implies that
B o (P'(m)-® /) = -/J o (/ ® r>-J(m)) o (/ ® Tg) o (Tg® 7)"1.
This is the content of the first part of this result. If V admits a non-degenerate

184
6. A Construction of the Fundamental Series
(g, M)-invariant Hermitian form and if p = 2n then we can look upon P as a
sesquilinear pairing of T"(K) with itself. One checks that
/J(D,w) = (-l)"conj(/J(w,i>)).
Thus, if n is even ji is Hermitian. If n is odd multiply /? by i. This completes
the proof of the theorem.
6.4. Some vanishing theorems
6.4.1. In this section we will prove some vanishing theorems for the
Zuckerman functors.
Let G be a real reductive Lie group of inner type and let 6 be a Cartan
involution of G. Fix, h, a 0-stable Cartan subalgebra of g such that h is
fundamental. Let f be, as usual, the Lie algebra of the maximal compact
subgroup of G corresponding to 6. Let I = I n h. Let H e if. ad H is semi-simple
with real eigenvalues. We set I = {X e g', [H, X] = 0}. Let u denote the direct
sum of the eigenspaces of ad H corresponding to strictly positive
eigenvalues. We will call q = Ic + u a 6-stable parabolic subalgebra. Notice that
6 restricted to I is a Cartan involution of I and that 6u = u.
If q is a 0-stable parabolic subalgebra then qt = q n fc is a parabolic
subalgebra of fc. We set rrt = fnq = fnl and set ut = unfc. Then
qt = mc + uk.
Let L = {g eG\Ad(g)H = H). Set M = K n L. We leave it to the reader
to prove that M acts trivially on A,op(f/m). If W is an (m, M)-module then we
look upon W as a (qt, M)-module by letting uk act by 0. We set M(qt, W) =
u(ic)(S)u«lk)W- Then MK>W) is a (f,M)-module with f acting by left
multiplication and M acting by m(k ® w) = Ad(m)/c ® mw for me M, ke
l/(fc)and we W.
We note that if dim u, = n then dim f/m = 2n.
Lemma. rJ(M(qk, W)) = 0 for j < n.
As a (f, M)-module M(qk, W) ® H(K) is isomorphic with M(qt, W® H(K))
by Lemma 6.A. 1.1. Since dim fc/qt = n the result now follows from Lemma
6.A.I.5.
6.4.2. Lemma. Let V be a (f,M)-module such that V has a (t,M)-module
filtration 0 = V0 <= Vx <= V2 <= ■ ■■ with V^V^^ isomorphic with M(qk,Wj)
for some (m, M)-module Wj and \J V- = V. Then fJ( V) = 0 for j < n.

6.4. Some Vanishing Theorems
185
We first prove that P(^) = 0 for all i and all j < n. If i = 0 this is obvious.
Assume this for i then the (f, M)-module exact sequence
0-^-^+,-^+,/^-0
induces the (f, K)-module exact sequence
P'W)-P'W+1)-F"W- + 1/»fl-
Thus Lemma 6.4.1 implies the assertion for i + 1.
Now let ji e Cj(f, M, K® H(K)) with ;' < n. Then there exists i such that
P e C\t, M;V;® H(K)). The preceding results now imply that /? = da with
a e Cj~'(!, M; K® H(K)). This completes the proof.
6.4.3. Corollary. Assume that V is as in the previous Lemma and in addition
that V is admissible and admits a non-degenerate (I, M)-invariant Hermitian
form. Then P(K) = 0 for j + n.
6.4.2 implies that Y\V) = 0 for j < n and P(K) = 0 for ; > n by 6.3.5.
6.4.4. Let W be an (I, M)-module. We extend W to be a (q, M)-module by
letting u act by 0. We write M(q, W) for the (g, M)-module, U(Qc)(g)UM W
with g acting by left multiplication and M acting by m(g ® w) = Ad(m)g ® mw
for m e M, g e U(qc) and w e W.
Lemma. M(q, W) has a (l,M)-module filtration as in Lemma 6.4.2. In
particular, P'(M(q, W)) = 0 for j < n.
Let X denote the complex conjugate of X in gc relative to g. Then gc =
uf[cfu. Thus V = M(q, W) = l/(il) 0 W as an (I, M)-module. Set u„ =
{X e u 6X = - X}. Then U(u) = U(uk) symm(S(un)). Set Z0 = l/(Ic)(l ® W).
Put
Z,-+i = t/(Ic)(symm(SJ +'(«„)) ® W) + Z,-.
Notice that Z0 is isomorphic with M(qk,W). We also note that
ut(symm(SJ'+1(u"„))® W) is contained in symm(SJ+1(u„))® W + Zr Thus
modulo Zp symm(SJ+1(u„))® W is the ut-module, SJ+1(gc/ut 0 q))® W.
These observations now easily imply that Zj+1/Zj has a filtration of the
desired form. The Lemma now follows.
6.4.5. We continue our discussion with q a 0-stable parabolic subalgebra
of gc. Let h be a Cartan subalgebra of lc. Let <t> be the root system of gc rel-

186
6. A Construction of the Fundamental Series
ative to h. Fix <t>+ a system of positive roots in <t> such that if we set n+ equal
to the sum of the positive root spaces of gc relative to <t>+ then n+ contains
u. Set (<t>,)+ equal to the set of roots of Ic relative to h in $+. Put <t>(b,u)
equal to the set of weights of b on u. Let p be half the sum of the elements
of <t>+ (as usual). The following Lemma is a special case of a more general
result that allows W (below) to be infinite dimensional.
Lemma. Let W be a an irreducible (I, M)-module. Then
w=w1®---®wr
with Wj an irreducible (I, M°)-module. Assume in addition that W is finite
dimensional. Let \j be the highest weight of Wj relative to (<t>()+- // (Re A; +
p, a) < 0 for all a e <t>(h, u) and all j then M(q, W) is irreducible.
Let Wi be an irreducible, non-zero, (I,M°)-submodule of W (4.2.1). Let
Mx = {meMlmWi = Wj}. Then Mx contains M°. Hence M/Mx is finite.
Let {oi,..., or) be a set of representatives for M/Mx. We assume that ax = 1.
Then OjWx is an irreducible (I, M°)-submodule of W Let j be the smallest
index such that ojWj intersects Wj in 0. If j doesn't exist then W = Wj and
we are done. Otherwise, set W2 = OjWj. Then the sum W2 = Wl®W2 is
direct. Let i be the smallest index such that ct; Wj intersects W2 in 0. If i doesn't
exist then W2 = W Otherwise, set W3 = oi Wj. The sum W3 = Wj ®
W2 ® W3 is direct. It is now obvious how one completes the proof of the
first assertion.
For the proof of the second assertion we use
Scholium. Let Fl and F2 be irreducible finite dimensional (l,M°)-modules.
Let n(Fi) denote the set of weights of Fl relative to h. Let A be the highest
weight of F2 relative to (<I>1)+. Then Fl ® F2 splits into a direct sum of
irreducible finite dimensional (I,M°) modules with highest weights of the form
A + p with p € 71(7^).
If p, S e ^(Fi) then we write p> d if p — S is a sum of elements of (<t>|)+-
Let fi,..., fd be a basis of Fx with fj an element of the pj weight space of Fl
and such that if i" > j then pt> pj. Set n, equal to the intersection of \c
with n + . Then njj is contained in Y.(>j C/j. Let v be a non-zero element in
the weight space of F2. Set Vi = C/(lc)(EJSj- Cfk ® v). We leave it to the
reader to check that Vx = Fx ® F2. Now n^jjj® v) is contained in Vj+l.
Hence Vj/Vj+l is either zero or is irreducible with highest weight A + pj.
This proves the Scholium.

6.4. Some Vanishing Theorems
187
We now prove the second assertion of the Lemma. We will use the
notation of the first part of this proof. Since Mx contains M°, we can choose
each <jj such that Ad(<Tj)h = h and Ad(0j)*(O,)+ = (%Y• Thus ajWx is an
irreducible (1,M°)-module with highest weight OjAx- This implies that
(Re Ay + p, Re Aj + p) is independent of j.
Let il be the sum of the root spaces corresponding to the elements of
— 0(l),u). Then as an (1, M)-module M(W) = M(q, W) is isomorphic with
t/(il)® W. This implies that the highest weights of the M-isotypic
components of M(W) are of the form Aj — Q where Q is a sum of (not necessarily
distinct) elements of <t>(h, u). Let V be a non-zero (g, M)-submodule of M(W).
Then V" is non-zero. Let p be a highest weight in this space. Then, since the
infinitesimal characters of M(W) are of the form Xa we must have p + p =
s(Aj + p) for some element of the Weyl group of gc relative to h and some ;'.
This implies that (Re p + p, Re p. + p) = (Re Aj + p, Re Aj + p). But then
(Re Aj - Q + p, Re Aj-Q + p) = (Re A, + p, Re A; + p)- 2(Re A,- + p, Q) +
(Q, Q) > (Re Aj + p, Re Aj + p) + (Q, Q) by our hypothesis. Thus Q = 0.
But then V contains 1 ® W. Hence V = M(W). This completes the proof.
6.4.6. If g e U(qc) then we write conj(g) for complex conjugation of g
relative to U(q). If g e U(qc) then we set g* = {conj(g))T. We note that P-B-W
implies that
l/(gc)=l/(lf)e(ul/(9c)+ t/(9c)")-
Let p denote the corresponding projection onto U(lc).
Let W be a (1, M)-module. We now define a (g, M)-invariant, sesquilinear
pairing of M(W) with M(W*), If x, y e U(qc) and if w e W, w* e W* then
set
(x (x) w, y® w#) = (p(y*x)w, w*).
It is easily checked that if q e U(q) then
(xq ® w - x ® qw, C/(gc) ® W*) = 0
and
(l/(8c) ® W, yq ® w* - y ® qw*) = 0.
Thus, ( , ) "pushes down" to a sesquilinear pairing of M(W) with M(W*\
We will also leave it to the reader to show that this pairing is (g, M)-invariant.
Set R~(q, W) equal to the set of all (I, M) submodules, N, of M(W) such
that N n(l ® W) = 0. Then it is easily seen that if Nx and N2 e R~ then
Nx + N2 is also. Set R(q, W) equal to the sum of the elements of R~. Then

188
6. A Construction of the Fundamental Series
R e R~ and it is easily seen that
R(q,W) = {meM(W)!(m, M(W*)) = 0}
and
R(q, W*) = {m € M{W*) (M(W), m) = 0}.
Proposition. Assume that W and M(W) are irreducible. Then the form ( , )
is non-degenerate. In particular, Yi(M(W)) = 0 for j i= n.
The first assertion is an immediate consequence of the above observations.
Lemma 6.4.4 implies that Y\M(W)) = 0 and T\M(W*)) = 0 for/ < n. Thus,
the second assertion follows from Theorem 6.3.5.
6.5. Blattner type formulas
6.5.1. We retain the notation of the previous section. We also assume unless
otherwise specified that G is connected.
(1) M is connected.
Let H e it be as in the definition of q. Then M = {k e K ! Ad(k)H = H}.
Let T be the maximal torus of M with Lie algebra it. Then T is also a maximal
torus of K. If m e M then there exists m0 e M° such that Ad(m0m)t = t. Thus
Ad(m0m) induces an element s e W(K, T). Fix, <b£ , a system of positive roots
for <t>(fc,tc) such that a(H) > 0 for a e <D^. We set (D^, = d>+ n 0>(M, T).
Then s(<D^ - <D+) = (0>^ - d>+). There exists s^WfM'j) such that
SiS®^ = <I>m- Thus s^^ = <t>t . We may thus assume that Ad(m0m) acts as
the identity on t. This implies that m0m e T, since K is connected. Hence
m € M°. Thus m e M° so M = M°.
Let <t>t be a system of positive roots for <t>(fc,tc) that is compatible
with qt. If fi € it* is <b£ dominant integral and T integral then we denote
by K„ an irreducible (f, X)-module with highest weight \i. If F is a finite
dimensional (f, X)-module then we write ch(F) for the character of F restricted
to T. We will also write e" for the character 11-> t".
Let <I>^ denote <b£ n <t>(M, T) and set pm equal to the half sum of the
elements of <I>^. If y e MA fix Ey e y. If V e C(m, M) is admissible then we set
cMK)= £ dim UomM(Ey, V)y.
yeMA
This expression has meaning as a formal sum.

6.5. Blattner Type Formulas
189
Let AM = ePmnae<t>m (1 - e "). Let ky be the highest weight of y relative
to <f>n . Then the Weyl character formula says that
AMchy= £ det(s)e'(^+p").
seW(M.T)
Notice that there is exactly one term for each Weyl chamber. Thus AM chM(K)
makes formal sense on T. Furthermore, we can read dim HomM(£f, V) as the
coefficient of eXv+Pm.
Lemma. With the above notation and conventions
AKchM(M(qt,Ey) = £ det(s)e^ + '*>.
seW(M.T)
As an M-module M(qk, Ey) is isomorphic with S(uk)® Ey= ® SJ(uk) ® Ey.
chM(S\uk)® Ey) = chM(S\uk))chM(y). Now, chM(SV*)). = £«"c» the sum
over Q that are sums of j (not necessarily distinct) elements of <t>(ut,tc).
Thus
XchM(SJ-(nt))=l/ [] (1-0 (6.A.2.2).
Since AMePk~Pmnae<t>{Uk lc) (1 — e~x) = AK, the Lemma now follows.
6.5.2. Let pk denote the partition function of <t>(ut,tc) with multiplicities
equal to 1 (see 6.A.2.1).
Lemma. // there exists seW(K,T) such that s(ky + pk) — pk is <f>£-
dominant and T-integral then
X(-l);chKnM(qt,£y) = det(S)chKKsav + Pk)^k.
otherwise I (- 1);chK VjM(c\k,Ey) = 0.
We note that M(qk,Ey) has infinitesimal character Xxy + Pk- Hence the same
is true for P'(M(qt,Ey)) (6.3.3). Thus W(l, M;M(qk, Ey)® (V„)*) = 0ifX„ + pk
is not in W(K, T)(ky + pk). This implies the last assertion of the Lemma.
Fix fi such that k^ + pk e W(K, T)(ky + pk). We must compute
X (- 1)''dim H\l, M; M(qt,Ey)(x) (K„)*).
Since the cohomology we are studying is the cohomology of a finite
dimensional complex, we may apply the Euler-Poincare principle, which says
that the alternating sum of the dimensions of the graded components of the
cohomology is equal to the alternating sum of the dimensions of the graded

190
6. A Construction of the Fundamental Series
components of the complex. We are thus left with the calculation of
£ (- 1)J dim(A''(f/m)* ® M(qt, Ey) ® (K„)*))M
As an M-module M(qk, £) is isomorphic with S(ut) ® £. Thus we are
computing
X I (- iy dim(AJ'(f/m)* ® Sr(uk)® Ey® (VU)*)M.
r
Let w be the order of W(M, T) then the Weyl integral formula says that w times
the number that we are computing is
X (- iy | |AM(t)|2 ch(A'(f/m)*) ch(S'(ut)) ch(£) conj(ch KJ A.
r
Now I (-1)'ch(A'(f/m)*) = n,E^Ui0 (1 -e«)(l -«"")• Also,conj(ch K„) =
Z det(s)e~sa"+'"')/conj(AA;) by the Weyl character formula. After the obvious
algebra is done w times the number we are computing is
X det(su) | n,E„UiI)(l - t~')ch(Sr(uk))t'^+'>-H-^-+^dt.
r.s.u T
This in turn is equal to
X det(sM)pt(0 J n,6„n.0(l -r'^t^^H-^-^dt.
C.s.u T
If we apply A.6.2.2 then we have
X det(su) | tsav+'k)r,,u>' + '"')dt.
The individual integrals in the above expression are non-zero if and only if
s(J.y + pk) = u(Xn + pk). Since we are assuming that Xy + pk = t(AM + pk). The
non-zero terms are those with to st = u. Thus we have w terms each equal to
det(t). This completes the proof of the Lemma.
6.5.3. Let p„ be the partition function of <t>(u„,tc) with m(ot) equal to the
dimension of the a weight space of T in u„. Let £ be an irreducible (LM)-
module. Let my be the multiplicity of y in £ for each y e MA.
Let Wl = {s e W(K, T) \ sO£ contains 0>+},
Theorem. (Generalized Blattner Formula)
X(- D'chK(P(M(q,E)) = £ my det(s)p„((^ + pt) - s(A„ + pk))y».
The sum on the right hand side is over /leK'je MA, and s e W(K, T),

6.5. Blattner Type Formulas
191
Let aeKA then we must compute
N. = Z (- 1)' dim UomK(Va, r;(M(q, £))
= X (- 1)' dim H'(f, M; M(q, £) ® (K„)*)
= X (- 1)' dim(Ai(Ic/mc)* ® M(q, £) ® (KJ*)M.
The last number depends only on the M-module structure of M(q, £). As an
M-module, M(q, £) is isomorphic with U(u) ® E which is isomorphic with
l/(ut)S(u„)® £. This in turn is isomorphic with M(qt,(S(u„)® £)ss). Here,
if V is a (qt,M)-module then Vss is the (qt,M) module, V, with M acting as
usual but uk acting by 0. If we now reverse the argument using the above
Euler characteristics we find that Na =
5>y(- \)J dim HomK(Kff, P(M(qt, (S'(u„) ® £y)ss).
Now
ch(S'(u„) ® £,) = X myPn,r(p) det(s)e^ + s^ + ^/AM.
The sum over p e it* and s e W(M, T). Here p„ r(p) denotes the number
of ways that p can be written as a sum of r elements (with multiplicity) of
<t>(u„,tc). Since pn.r(sp) = p„_r(fA for s e W(M, T) we find that
ch(S'(u„) ® £y) = I pn,Xp) det(S)^^-"+^/AM.
Thus if n is an irreducible representation of M with highest weight Xn.
Then the multiplicity of En in Sr(iT„) ® £y is equal to the sum over all
seW(M, T), peit* with s(Xy - p + pm) = XK + pm of det(s)p„,r(/4 Since
s(pk - pm) = (pk - pm) for s e W(M, T), we see that the multiplicity of n is
Z det(s)p„,r(Ay + pk - s(XK + pk)).
seW(K.T)
Hence,
N* = Z ™y Z det(s)p„,r(/ly + pt - s(4 + pt)) •
Z(-l)'dimHomK(Kff,P(M(qt,£J).
Lemma 6.3.4 implies that I (- 1)' dim UomK(Va, n'(M(qt, £J) is 0 if there is
no t e Wl such that t(XK + pk) = k„ + pk and that it is det(t) if such a t exists.
Thus we have
N° = Z mv Z det(st)p„(Ay + pt - st(ACT + pt).
IeICseWIMJ)

192
6. A Construction of the Fundamental Series
Now, the map W(M, T) x Wl -> W(K, T),s, 11-> st is a bijection. So the result
follows from the above formula.
6.5.4. Let sK be the (unique) element of W(K, T) such that sK<S>£ = -0>+.
Proposition. Assume that E is an irreducible (I, M) module such that M(q, £)
is an irreducible (g, M)-module. Let my be as above for E. Let n = dim uk (as
usual) then
(1) IfaeKA then
dimHom(K„,rn(M(q,£))
= (- 1)" E m, I det(s)pn(ly + pk- s(Xa + pk)).
seWOC.T)
(2) Assume that q is a 6-stable Borel subalgebra and that h acts on E by p. Also,
assume that sK(p\{ + pk) — pk is <t>k dominant and T integral. Then (Va is an
irreducible (i,K)-module with highest weight a).
dim HomK(Ks(/l|l + Pk)_pt,r"(M(q,£)) = 1.
(3) With the notation and assumptions as in (2), if a e it* is Q>k -dominant
and T-integral then HomK(Va,rn(M(c\, £)) is nonzero only if a + pk =
sk(h\i + Pk) + Q \t with Q a sum of elements of — sK<t>+ (see 6.7.6).
(1) follows from Theorems 6.5.3 and 6.4.6. We now prove (2).
(1) implies that
dimHomK(Ks(/l|l + Pk)_Pk,r"(M(q,£))
= (-1)" X det(s)p„(4 + pt - ssK(/4 + pt)).
seICKJI
If P„(/4 + Pk~ ssK(p\t + pk)) is nonzero then ssK(^|t + pk) = p\t + pk -Q
with Q a sum of elements of <I>(u„, tc). On the other hand, (sK(p\t + pk), a) >
0 for ae$i+. Hence (p\t-\- pk,a) <§ for aeO^. This implies that
•^(Ht + Pk) = P-\i + Pk + ^ "aa the sum over aefDj with nx > 0. Hence
E nax = — Q. Let H e it be such that Ic( = h) is the centralizer in qc of H and
oc(H) > 0 for a e 0>(u,b) = ®+■ Then we have 0 < Inaa(H) = -Q(H) < 0.
Thus 2 = 0. Since det(sK) = (- 1)", (2) follows.
We now prove (3). If dim HomK(K„,rn(M(q,£))) > 0 then (1) implies that
there exists w e W(K, T) such that w(a + pk) = (p\t + pk) - Q with pn(Q) > 0.
Hence sKw(p + pk) = sK(p\t + pk) - sKQ. Now, sKw(a + pk) = a + pk- R
with R a sum of elements of <f>£. Thus a + pk = sK(p + pk) — sKQ + R. If
we write R = —sK( — sKR) then (3) follows.

6.6. lrreducibility
193
6.6. lrreducibility
6.6.1. We retain the notation and assumptions of the previous section. Let
b = h + u be a 0-stable Borel subalgebra and let <t>+ be the corresponding
system of positive roots. The purpose of this section is to prove
Theorem. Let A e h* be pure imaginary on h n g and satisfy the following two
conditions
(1) sK(A\{ + pk) — pk is <t>t -dominant and T-integral.
(2) Re(A + p, a) < 0 for a e <D+.
Let CA be the one-dimensional (b,T)-module with h acting by A. If n =
(\) dim K/Tthen T"(M(b,CA)) is non-zero and irreducible.
The fact that rn(M(b,CA)) is non-zero follows from Lemma 6.4.5 and
Proposition 6.5.4(2). It is the irreducibility assertion that will take up the rest
of this section.
The basic idea that we use to prove this result is due to Zuckerman (we also
use some techniques from Enright, Wallach [1]). We begin by proving the
result in a special case.
6.6.2. Lemma. Assume that in addition to the assumptions of Theorem 6.6.1,
A also has the property that if Q is a weight of t on Aj(u„) then Re(A|, + pk +
Q,oc)<0 for all a e 0>+. Then rn(M(b,CA)) is irreducible.
Set a = h n p. Put p. = A |0 and a = A |t. Notice that our assumptions imply
that Re(A, a) = (a, a) for a e 0> + . Set b„ = {H - p(H) + X H e h, X e u}
considered as a subspace of b © C. We look upon b„ as a (bk, T)-module under
(ad, Ad). Lemma 6. A. 1.3 implies that we have a (b, T)-module exact sequence
(d = dim u„ = dim b/bt))
0 -» l/(b) (X) Ad(bJbk)®Ca^U(b) (X) A'-^/bJOC, -»
U(bk) U(bk)
- U(b) (g) (bM/bk) ® C„ - l/(b) g) C„ -»CA -»0.
l/(bk) l/(bk)
We observe that if £ is a (bt, T)-module then
l/(gc) ® l/(b) (X) £ * l/(flc) ® E * l/(gc) g) l/(fc) (X) £.
t/(b) t/(bk) t/(bt) U(t) U(bk)

194
6. A Construction of the Fundamental Series
Here the "«" indicate (g, T)-isomorphisms.
SetEj=U(tc)®\J(blJbk)®Ca.
U(bk)
The above observations now imply that we have the (g, T)-module exact
sequence
0 - l/(flc) ® Ed X U(qc) (X) Ed_, -^> ■ ■ ■ A Uioc) (X) E0X M(b,CA) -» 0.
Wc) t/(fc) l/(tc)
Now £,- has a composition series by (I, T)-modules of the form M(bk, Q +/r)
with j8 a weight of t on AJ(b/bt)( = AJu„ as a T-module). The hypothesis of
the Lemma combined with Lemma 6.4.5 implies that E3 splits into a direct sum
of irreducible (I, T)-modules of the form M(bk,Ca + /j) with p as above.
As a (I, T)-module,
U(ac) (g) M(bt,Cy) * S(pc)® Af(bt,C,).
U{tc)
We therefore conclude that
(1) ni/(9c) (X) Ej) = 0 for r * n.
Uitc)
Set Z. = ^.((/(gJ^^E.) for ; = 0, 1,..., <f. Then we have the (g, 7>
module exact sequences
0^Z1^l/(gc)g)M(bt,C(J)^M(b,CA)^0
Vile)
0-»Z,+ 1-»l/(gc)(g)£,-»Z,-»0.
In light of the above observations and the results of Section 6.4 we have the
following (g, X)-module exact sequences
o-»nz,)-»rn(i/(gc) (x) M(bk,ca))-»rn(M(b,cA))-»rn+»z, ^o
o -»rn+J(Zj) ->rB+j+1(Z;+1) -» o
for ; > 1. Thus rn + 1Z, is isomorphic with rn + JZj for ; > 1. Since Z, = 0
for j > d this implies that we have the (g, T)-module exact sequence
(2) 0 -» r"Z, - r-(l/(gc) (X) M(bt,CJ) -» r-Af(b,CA) -» 0.
t/(fc)
Now the map m ° (symm ® /) of S(pc) ® (1 ® M(bk, CJ) to
U(9c)<g)mc)M(bk,Ca)
is a (I, T)-module isomorphism. Thus the map rn(m ° (symm ® /)) from

6.6. lrreducibility
195
r"(S(pc)®(1 ® M(bk,Ca) to r"(U(qc)(g)V(tc)M(bk,Ca) is a (f,K)-module
isomorphism.
This implies that
(3) l/(9c)r"(l ® Af(bt,C„)) = r-(U(Qc) (X) M(bk,C.))
U{tc)
The exact sequence (2) implies that
(4) U(Qc) (Image of T"(l ® Af(bt,C„) in T"M(b,CA)) = T"M(b,CA).
Now 6.5.2 combined with 6.4.3 imply that T"(l ® M(bk,Ca)) is isomorphic
with VSk(a + Pk)_Pk as a (f> K)-module. Thus (4) implies that
(5) l/(9c)™(b,CA)(Wpk)_pJ = T"M(b,CA).
Proposition 6.5.4 implies that y = ySk{a + Pk)-Pk occurs with multiplicity one
in T"M(b, CA). Also Lemma 6.4.6 implies that M(b, CA) has a non-degenerate
(g, T)-invariant Hermitian form. Hence Theorem 6.3.5 implies that
T"M(b, CA) has a non-degenerate (g,K)-invariant Hermitian form. In light of
(5) if N isaproper(g,X)-invariantsubspaceof T"M(b, CA) thenN(y) = 0. But
then Wis orthogonal to rnM(b, CA). SoN = 0. This completes the proof of the
Lemma.
6.6.3. In order to complete the proof of the theorem we need some
observations about Verma modules that will also be used later. Let A e h* be
such that Re(A + p, a) < 0 for a e <t>+. Let F be a finite dimensional (g, K)-
module and let p. be the lowest weight of F. We use the notation of 6.A.3 and b
to parameterize infinitesimal characters.
(1) «>F.-„M(b,CA + „) = M(b,CA).
Indeed, M(b, CA + /J)® F* is isomorphic with M(b, CA + /J ® F*). This
module has a composition series with constituents of the form M(b,CA + ll + s)
with S a weight of F*. The argument that proves (2) in 6.A.3.7 implies that
M(b,CA) is the only constituent with infinitesimal character xA+p. So (1)
follows.
Now, rn(M(b,CA + „)®F*) is (g,K)-isomorphic with rn(M(b,CA + „))®
F*. Thus (1) implies
(2) (DF,,-„r"M(b,CA+„) is isomorphic with rnM(b,CA).
Now we assume that A satisfies the hypothesis of the theorem and that
A + p satisfies the hypothesis of Lemma 6.6.2 (for example take p = —2k
with k » 0). Then (2) combined with 6.A.3.10 implies the theorem.

196
6. A Construction of the Fundamental Series
6.7. Unitarizability
6.7.1. We now drop the assumption that G is connected (until 6.7.4). Let M
be a closed subgroup of K. Let Ke C(g, M), Suppose that < , >isa(g, M)-
invariant Hermitian form on V. If y, y' e MA then (V(y), V(y')} = 0 if y is not
equal to y'. Thus < , > restricted to V(y) is a non-degenerate form. Assume
that V is admissible. Then V(y) admits a positive definite M-invariant
Hermitian form, ( , ). Thus < , > restricted to V(y) is given by (v, w> =
(Ayv, w). Clearly, Ay is self adjoint and commutes with the action of M on
V(y). Set py equal to the number of positive eigenvalues of Ay and set qy equal
to the number of negative eigenvalues. Then both py and qy are divisible
by d(y). We set
chs(K,< , »= £ d(y)-\py-qy)y.
ye M A
We note that chMK= S),EAf* ^(y)_1(Py + <?,)}'. The following result is obvious.
Lemma. < , > is positive definite if and only if chs(K, < , » = chM(K).
6.7.2. Let V e C(g, M) be admissible and assume that V has a (g, M)-
invariant non-degenerate form < , >. We assume that dim K/M = 2n with
n an integer. Then T" V has a non-degenerate (g,X)-invariant non-degenerate
form as does ®TJ V (6.3.5) we denote both by < , >.
Lemma.
chs(P"K,< , » = chs(©PK,< , »
= £sgn(ff"(f,Af;K®K*))y
y
= YJSgn(®]Ci(i,M;V®V*))y.
7
Here, if (£, < , » is a pair of a finite dimensional vector space and a non-
degenerate Hermitian form the sgn(£, < , » is the usual signature. On the
corresponding spaces we use the forms as given in 6.1.2.
The pairing on © P'V is between the pairs T1 V and r2n~JV. If ;' is not equal
to n then it is clear that
chs(P'K©r2"-J'K, < , » = 0.

6.7. Unitarizabilitv
197
This proves the first equation. For the second, we recall that
PV = © Hj(l M;V® V*) ® Vy
as a (f, K)-module (6.3.2). We recall that the sesquilinear pairing of T'V and
r2"~JV is given as follows. On Vy we put any /C-invariant inner product. On
V* we put the dual inner product. On V® V* we put the tensor product
Hermitian form. We pair Cj(i, M; V® V*) and C2""J'(I, M; V® V*) as in
6.1.2. These forms push down to cohomology. If / = n then we multiply by i (if
necessary). We then take the tensor product form on //"(I, M; V® V*) ® Vy.
It is therefore clear that the coefficient of y in chs(rn V) is sgn(H"(f, M;
V ® V*)). This proves the second equation.
Let
C= ® C\l, M; V® V*) and H = © H\l, M;V® V*).
Then ker d/im d = H. We have seen in 6.1.2 that ker d is the orthogonal
complement to im d. Thus if we set B = dC and Z = ker d then the radical
of < , > restricted to Z is B which pairs nondegenerately with C/Z. Thus
sgn(C, < , » = sgn(H, < , >). This completes the proof of the Lemma.
6.7.3. Fix q a ^-stable parabolic and let the notation be as in 6.4.1.
Lemma. We look upon AJ(Ic/nic)* as an (in,M)-module with Hermitian form
given as in 6.1.2. Then as a T-module
due^tfc/mc)*) = ch,(A"(tc/mr)*)
= (-!)» f[(l-e")(1 +0(E = *(ut,t<.)).
We note that as a T-module (f(./m()* = uk © uk. Let S = {a,,..., a„}. Let
Xj be chosen in the a, weight space of uk such that
X{ A • • ■ AXn AX, A • • • AXn = £co0
with e = 1 if n is even and e = i if n is odd.
If Q = {a;,,...,ajr}, ji<ji+u then set hq = XhA---AXjr, \Q\ = r and
<g> = ZaeQ a. AJ(ut © ut) has as a basis the elements nQAjip with g, P subsets
of I and |6| + |P|=;. If 161 + 1^1 = ;' and if \R\ + \S\ = 2n-j then
HQAjIPA[iRAjIs = 0 if Q u S is not equal to S or P u R is not equal to S.
If |Q| + \p\ = n but Q u P is not equal to I then < , > restricted to C

198
6. A Construction of the Fundamental Series
fiQAfiP + Qie_qAjie_p has signature (1,1). This, in particular, implies the
first equation in the Lemma.
Furthermore, (^QA/ZE_Q)A(^QA/IE_Q) = (—l)"~IQIea)0. The weight of
^QA/ZE_Q is — <£> + 2<(2>. We therefore see that
chs(A"(fc/mc) = (- 1)V-<E> X (- \)me2<Q> = (- 1)V-<E> L] (1 - e2*)
= (-l)V-<E> r] (1 -e")(l +e«)
=(-irn (i-ou + o-
ocel
This completes the proof of the lemma.
6.7.4. We return to the assumption that G is connected. We observed, in
6.5.1, that if V e C(m, M) is admissible and has M-invariant Hermitian form
< , > then AM chM(K) makes sense as a formal sum of characters of T.
Furthermore, if the coefficient of y e MA is m then the coefficient of
e" (ft - ly + pm) in AM chM(K) is also m.
Lemma. Let W be an irreducible (I, M)-module that admits a positive definite
Hermitian form < , >. We also assume that there exists fieil* such that
n\_l,I] = 0, ^|(np = 0, (n,a) > 0 for ael and if C, is the one dimensional
(I, [M, M~\)-module corresponding to \i then M(q, W® C_r/I) is irreducible for
all t > 0. Let < , > be the Hermitian form on M(q, W) given as in 6.4.6. Then
< , > is non-degenerate and
AM chs(M(q, W)) = AM ch(W)/n (1 - e~°)(l + e'\
/ iel
We look upon the form on M(q, W ® C_r/J as a form < , >, on
U(u) ® W. If t e R then on each weight space for t, < , >r depends
polynomially on t. lit > OthenM(q, W ® C_r/I) is irreducible and the t weight
spaces are mutually orthogonal. Let H e it be as in the definition of 0-stable
parabolic. Set l/(u)_r equal to the — r weight space of U(u) relative to the
action of H. If t > 0 then the signature of < , >r restricted to
(l/(u)_r® W)(y) is constant for each y e [M, M]A (notice that this space is
independent of t).
Let Yt,..., Yq be a basis of uk and let Xy,..., Xp be a basis of u„. We assume
that >}(resp. Xj) is in the j8,-(resp. aj) weight space relative to t. We also assume
that B(Xj,Xf) = -5lk and that B(Yj, Yf) = 5jtk. We note that B(Xj, Yk) = 0.

6.7. Unitarizability
199
Thus IXj, X*~\ = - buH„ moduloft1) and [J}, y*] = SlrHp modulo (t1) (here
orthogonal complements are taken using B).
We write AT7 = X\lX'i--Xip',YJ = Y{---Y{',\I\ =1 ijt\J\ = Z A,</> =
S ij-aj- and <J> = S ;'t/?t. It is a simple matter to see that
<x,YJ®w,x"YJ'®w'yl = t^+^sIJ,dJ,J{-\)^nj(fi,ocJynk(fi,pk)\w,w'y
+ ^/,j,/v(f,w,W)
with P7jjj(;w,w') a polynomial in t of degree at most min(|/| + \J\,
K'l + l^'l) — 1- Thus if V is a finite dimensional M-invariant subspace of W
and if t is sufficiently large then if ch^ is the signature character as an [M, M]-
module and if a' is the restriction of a e t£ to [m,m] n t we have
AMch;(l/(u)_r®K,< , >,)
£ (-l)l^-«'> + <^AMch[M,M]K
«/>+<J»(H)=r
As we have seen in the beginning of this proof, ch^ on an H-eigenspace is
independent of t > 0. This implies that if f > 0 then
AMch.(l/(u)_r®(W®C_„,))
= «-'" X (-l)|J|e-,<,>+W)AMchM^.
Hence if t > 0 then AM chs(M(q, W® C_,„)) =
e~'"I(-l)|J|e~(</> + <J>)AMchM W.
If we take t = 0 in this formula the result follows from 6.A.2.2.
6.7.5. Theorem. Let the notation and hypotheses be as in Lemma 6.1.4. Then
T"M(q, W) admits a positive definite (g, K)-invariant Hermitian form.
Let V = M(g, W). On VV we put the non-degenerate (g, X)-invariant
form guaranteed by 6.3.5, 6.4.5 and 6.4.6. We prove that this form is positive
definite. Lemmas 6.4.6, 6.4.5 and 6.7.1 imply that it is enough to show that
(1) (-l)"chsr"K = X(-l)J'chrj'K
We prove (1) by calculating the coefficient of y e KA on both sides of the
equation. For the left hand side we must calculate sgn(H"(f, M; K® V*))
which equals sgn(HomM(A"(fc/inc), M ® V*) by Lemma 6.7.2. This number
is the coefficient of ePm in
AMchs(A"(fc/mc)*®M®K*).

200
6. A Construction of the Fundamental Series
Here we use the tensor product Hermitian form for the term inside the
parenthesis. Since signature characters of tensor products obviously multiply,
we are calculating the coefficient of ePm in (P = <t(u„, t), Q = ®{uk, t))
]1 (1 - e')(l + e-°))[AM ch W \\ (1 - e~') \[ (1 + o) ch V*
aeQ J \ j xeP xeQ J
(here we have used Lemmas 6.7.3 and 6.7.4)
= (J! 0 - e'))Uu ch wjj\ (1 - e~')\ ch V*.
On the other side of the equation the coefficient of y is
X (- \)j dim W(l, M;V® V*)
= X (- 1); dim HomM(A^(fc/tc), V ® V*)
by the Euler-Poincare principle. This in turn is the coefficient of ePm in
AMX(-l)J'ch(A^(fc/tc)*)chFchF*
= (l\(\-e*)(\-e-*)\(hMchwll\(\-e-*) \\ (1 - e'')) ch V*
by Lemma 6.A.3.3. Thus the right hand side of (1) is equal to the coefficient
of ePm in
n(l- e')Uu ch win (I- e-')) ch V*
which is clearly equal to the left hand side. This completes the proof of the
theorem.
6.7.6. In order to be consistent with Harish-Chandra's parameterization
of the discrete series we must now do a bit of manipulation of Weyl
chambers. We first note that if s e W(K, T) then there exists ke K such that
Ad(k)X = sX for X e t. Since the centralizer of t in gc is h, this implies that
there is a unique element, s', of ^(gc,h) such that s' restricted to t is s. We
will identify s' with s. Let n e h* be such that \i is purely imaginary on h n g
and such that {\i + p)\t is T-integral. Let P be a system of positive roots
such that Re(/i,a) > 0 for a e P. Let Pk be the system of positive roots for
<t(fc,tc) compatible with P. Let sK be as usual. Put Q = — sKP. Set
b = be = h + u(Q) equal to the Borel subalgebra containing h and
corresponding to Q. We note that b is 0-stable. Set DP„ = f"M(b, CSk(/J+pk)).

6.8. Temperedness and Square Integrability
201
We also set (as usual pk equal to the half sum of the elements of Pk and
Pn = P\t~ Pk- We have
Theorem.
(1) Dp ^ has infinitesimal character x„-
(2) If n\t + pn — pk is T-integral and Pk-dominant integral then DP„ is not equal
to 0. Let p„ be the partition function of <t(u(P)„, t) with the multiplicity of a equal
to the dimension of the a weight space in u(P)„. If y € KA then
HomK(Vy> DP J = ZS det(s)p„(s(Av + pk) -(n\t + Pn)\
In particular, under the above condition, if y has highest weight /x\t + pn — pk
then dim HomK{Vy,DPJ = 1.
(3) Dp^ admits a positive definite (g, K)-invariant Hermitian form.
(4) // Re(/i,a) > 0 for alias P then DP lt is irreducible.
(1) is clear. (2) is a restatement of Proposition 6.5.4.(3) is a special case of
Theorem 6.7.5 (see Lemma 6.4.5). (4) is a restatement of Theorem 6.6.1.
6.7.7. Let \x be as in the statement of the previous Theorem. Assume that \i, P
satisfy the condition of 6.7.5(2). Then Theorem 6.A.4.2 and 6.7.5(3) imply that
there exists a unitary representation (7rP/J,//p") of G such that (HP")K is
(g, K)-isomorphic with DPll. We call the series of representations thus
"constructed" the fundamental series of G. If h = tc and if [i satisfies in
addition condition 6.7.5(4) then we use the notation 7i„ = nP„ (P is determined
by fi) a discrete series representation.
In the next section we will see that the nP„ are tempered and that the discrete
series consists of square integrable representations. We note that the discrete
series is usually defined to be the set of all equivalence classes of irreducible
square integrable representations of G. In Chapter 8 we will also prove that
every square integrable irreducible representation is equivalent to some n^.
Thus our terminology is consistent with usual usage.
6.8. Temperedness and square integrability
6.8.1. We retain the notation of the previous section. Let A e h* be such
that A is pure imaginary on h n g. Let P be a system of positive roots for
<t(g,h) such that Re(A,a) > 0 for a e P. Let Pk be the corresponding system
of positive roots for <t(fc,tc). We assume that A|t + p„ — pk is Pk dominant
and T-integral.
We also assume that G = °G (that is, the center of G is compact).

202
6. A Construction of the Fundamental Series
Theorem. Let (nP A, HP,A) be as in 6.7.6 then 7rPA is tempered (5.1,5.5).
By definition (HP A)K = DP A. Let F be an irreducible, finite dimensional
(g,K)-module with highest weight \i relative to P such that \id = \i. As in 6.6.3
(with the change in normalization of systems of positive roots of 6.7.6.)
one sees that if Re(A,a)>0 for all aeP then ^Fill(DpA) = DPA+ll and if
Re(A,a) > 0 and Re(A + n,a) > 0 for all a e P then <bF.,-tl(DPA+tl) = DPA.
Let P0 = M0N0 be a minimal parabolic subgroup of G. We will write A
for A0. Let h'0 be a Cartan subalgebra of m0 that is 0-stable. Set h' equal
to the complexification of ho. Fix g e Int(gc) such that g\) = h'. If a e h*
then set a' = a°g~l. 6.A.3.4 implies that ch(j(£)PiA))=i; cs(A)e_sA'/AG.
Now 6.A.3.8 and 6.A.3.7(1), (2) imply that if Re(A,a)>0 for aeP then
cs(A + n) = cs(A). Now, if Re(A,a) > 0 for a e P then nP A is unitary (6.7.5,
6.7.6). Thus, in particular, the matrix entries are bounded. Hence 5.5.2
implies that if cs(A) is non-zero then Res(A' + n') + p e Cl(+(a)*). Thus, since
there exists an irreducible finite dimensional (g, X)-module, Fk with highest
weight k/i for k = 1, 2,..., we have shown that if Re(A, a) > 0 for a e P and
if cs(A) is non-zero then Re(s(A' + k/i')\a) + p e Cl(+a*). Thus
^'|a + (l/fc)(Re(SA'|a + p)ECl(+a*).
If we take the limit as k -> oo then we have (after using the above
observations about ^f.,-^)
(i) If Re(A,a) > 0 for all as P and if n is P-dominant integral and
H° 6 = n then sjx' |„ e Cl(+a*) if cs(A) is non-zero.
Let a!,..., a, be the simple roots of P. Then 0a, = ar with j -> j' a
permutation of {1,...,!}. Let fij, j = 1,..., I be such that 2(nj,ak)/(ak,ak) = 8JJc.
Then Re A = X xjHj with x; > 0. Since Re 8A = Re A, Xj = xy. Hence
Re A = Z yji/ij + jiy) with j/, > 0. Now (i) implies that if cs(A) is non-zero
then Res((/ij- + ^j)')|„ e Cl(+a*). Hence 5.1.1 implies that nP A is tempered.
6.8.2. Theorem. Assume that tc = h. Then if (A, a) > 0 for all ae P then
nP A = n is square integrable.
Let 8 be the half sum of the elements of P. We assume by going to a
finite covering of G (if necessary) that there exists an irreducible finite
dimensional (g,X)-module with highest weight 8 relative to P. Let a be the
action of G on F. We define a representation /? of G on End F by fi(g)T =
a{g)Ta(0gYl. Let k > 0 be an integer such that if ( is a weight of a on

6.9. The Case of Disconnected G
203
End F then (fcA + (, a) > 0 for all aeP. Thus DPikA ® (End F) = © £>P.kA + c
the sum over the weights of End F taken with multiplicity.
Let f(g) = tr $(g)l. Then f(kxak2) = tr(a(a)2) for kuk2eK,aeA. Hence,
if a e C\(A + ) then f(k1ak2) > a2p. Thus, since our hypothesis and the
previous theorem imply that each DPkA+!i is tempered, we see that if c is a
infinite matrix coefficient of nPkA then \c(k1ak2)\a2p < C < oo for a e Cl(/1+).
Hence 2.4.2 implies that nPkA is square integrable. Theorem 5.5.4 implies
that if cs(fcA)is non-zero then ksA'\a e +a*. There exists p, a positive integer,
such that p is a highest weight of a finite dimensional irreducible (g, K)-
module. Hence cs(A + kpA) = c5(A) for all seW. Hence, if k is sufficiently
large, then (1 + kp)sA'\a e +a* if cs(A) is non-zero. If we divide both sides
by (1 + kp) we find that sA'|„ e +a* if es(A) is non-zero. Now Theorem 5.5.4
implies that nP A is square integrable.
6.8.3. The reader should be warned that the above statement is false if the
condition tc = h is removed. It is a good exercise to see how this
assumption was used in the above proof. Our use of tensor products with finite
dimensional representations to obtain estimates on matrix entries is based
on ideas taken from Hecht-Schmid [1]. An immediate consequence of the
preceding result is the following fundamental theorem of Harish-Chandra
[13]. (We will prove the converse in Chapter 7.)
Theorem. If G contains a compact Cartan subgroup then G has irreducible
square integrable representations.
6.9. The case of disconnected G
6.9.1. In this section we drop the assumption that G is connected. We
assume that G is of inner type and that G = °G. We also assume, throughout
this section, that there exists, T, a Cartan subgroup of G contained in K
and that for some (hence every) choice of positive roots relative to tc = h,
is T° integral (this can be achieved by going to a finite covering of G). Let
Z={geG\Ad(g) = I}.
Lemma. T = ZT°.
By definition T= {g e G\ Ad(g)\t = I}. Let t e T. If a e <t = <t(gc,h) then
Ad(t)(gc)a, = (c)c)a- Hence each ae<5extends to a one dimensional character

204
6. A Construction of the Fundamental Series
of T. Let P be a system of positive roots of <t and let A be the
corresponding system of simple roots. If x, y e T and xx = yx for all a e A then
xa = y" for all a e <t. It is clear that there exists t0 e T° such that (t0)x = t"
for all a e A. Thus t(t0)~' e 2. This implies the result.
6.9.2. Set G, = ZG°, K, = ZK°. Let TJ denote the Zuckerman functors
from C(g, T) to C(g, K), r{ those from C(g, T) to C(g, K,) and P0 those from
C(g,T°)toC(g,K°).
If V e C(q, Kt) we define IndJj^K) to be the space of all functions, /,
from K to V such that f{kxk) = kj{k) for fc, e K! and keK. If hK,
leg and /elndf.fK) we set kf{x) = f(xk) and X/(x) = (Ad(x)X)/(x)
for x e K. We leave it to the reader to show that with these actions Ind£,(I/)
isa(g, K)-module.
Lemma. If V e C(g, T) then P> = Ind*t T[ V.
We leave this as an exercise (which is an easy consequence of the definitions).
6.9.3. Let 7 e TA and fix Ey e 7. Since T° is central in T there exists
Hy e(T0)A such that t e T° acts on Ey by /iv/. Fix P, a system of positive
roots for <t, such that A = Ay = ny + p„ - pk is P-dominant. Let b = h + u
be the 0-stable Borel subalgebra of gc corresponding to —sKP. Let ke K°
be a representative for sK and let sKy be the element of TA with
representative Ey with action y(k~ltk). We set Ey equal to this T-module tensored
withCw.
Lemma. As a (g,K)-module rnM(b, Ey) = DPy is isomorphic with the
module Hom9j(DPA, rnM(b,£v))® DPA, having (g, K°) acting on the second
factor and Z acting on the first factor. Furthermore, DP y is irreducible if DP A
is irreducible.
As a (g, T°)-module M(b,Ey) is just dim E y copies of M(b, CSk(A+p)).
ThusM(b,£}A) =
Hom9,r(M(b,CSK(A + p)),M(b,£;))®M(b,CSK(A + p))
with Z acting on the first factor. Let Xy denote this Z module. It is a simple
matter to see that DP y is isomorphic with X ® DP A with Z acting on the
first factor and (g,K°) acting on the second. The lemma easily follows
from this.

6.9. The Case of Disconnected G
205
6.9.4. We let (°7rPiA,0HPA) be the unitary representation of G° associated
with DPA in 6.7.6. We form a unitary representation (1nPy,1Hp,y) of Gl as
follows: 1HPy = Xy®°Hp, with G° acting on the second factor and Z
acting on the first (the compatibility is guaranteed by the construction of
Xy). Let 1nPty denote this action.
Lemma. Set (nPy,Hp,y) = Ind^'ftp y) (unitary induction, here G/Gl =
K/Kl which is finite). Then {HP,7)K is isomorphic with VM(b,Ey) as a
(g, K)-module.
Indeed, (HPy)K = (Indgo lHp^)K = lndK(CHPy)K) = IndK T\M(b,E;).
6.9.5. Theorem. // (Ay,a) > 0 for all as P then (nPy,HPy) is a non-zero,
irreducible, square integrable representation of G with infinitesimal character #A.
Since GjGx = KjKx, for each element of GjGx we may choose a
representative in K. If k e K then Ad(fc)t is a maximal abelian subalgebra of f.
Hence there exists an element k0e K° such that Ad(fc0)t = Ad(fc)t. Thus the
representatives of the cosets can be chosen so that they normalize t. For
such k, Ad(f)|tE ^(gc,h) (we are assuming that G is of inner type). We
may thus choose a set of representatives 1 = yu-.., yd for GjGx such that
(1) yj e K and Ad( v,-)t = t.
(2) If Sj is the element of W(qc, h) corresponding to y;
then SjP contains <&t ■
(3) If Sj = sr then j = r.
We identify xDPy with the elements of DPy = VM(b,Ey) supported on
Kl. Then it is easily seen that y]DPy is isomorphic with 1DsPsy. Thus, if
y)DPy is isomorphic with y}DPy then we must have
(*) DsjP.sj7 is isomorphic with DSrPtSrr
In the left hand side of (*) the K-type with highest weight s,-(A + p) — 2pk
occurs. The highest weights of the K -types that occur on the right hand
side are of the form sr(A + p) — 2pk + srQ with Q a sum of elements
of P. Thus if (*) holds then there must exist Q as above such that
Sj(A + p) - 2pk = sr(A + p + Q) - 2pk. Now this implies that ||A + p\\2 =
||A + p + Q||2. Since (A + p, Q) > 0 this implies that Q = 0 and hence
Sj(A + p) = sr(A + p). But then Sj = sr since A + p is regular.

206
6. A Construction of the Fundamental Series
Thus the y) DPy are mutually inequivalent. This easily implies the irre-
ducibility assertion. The square integrability is clear since it can be tested
onG°.
As a (g,K°)-module (Hy)K = ® DSjPsjA and each of the summands has
infinitesimal character #A, The proof of the theorem is now complete.
6.9.6. The above discussion is a modification of the arguments in Harish-
Chandra [14, pp.176-177].
6.10. Notes and further results
6.10.1. Let M be a closed subgroup of K. Put Kx = MK°. If V e C(g, M)
then set VKi equal to the space of all v e V such that span([/(f )Mv) is the
underlying (f,M)-module of a finite dimensional representation of Ku Set
VK = Ind^F^) (6.9.2). Then V -> VK is a left exact functor from C(g, M) to
C{q,K). The Zuckerman functors are usually defined to be the right derived
functors of V -> VK (c.f. Cartan, Eilenberg [1]).
Let us recall what this means. / e C(g, M) is said to be injective if whenever
we have
0 >A-^B
P
I
with a, j8 morphisms in C(g, M) and a injective then there exists a, a morphism
of B into / in C(g, M) such that aa = /?.
In C(g, M) one has injectives given as follows. Let WeC(m,M). Put
l(W) = (Hom[/(m)([/(gc), W))M. Here g acts by right translation and M acts
by (mf){g) = m/(Ad(m) 'g). We leave it to the reader to show that I(W) is
injective. If KeC(g,M) then V injects into l(V) (we forget the g-module
structure) under the map i(v)(g) = gv. In the jargon of homological algebra
this implies that C(g, M) has enough injectives.
If V e C(g, M) then an injective resolution of V is an exact sequence
with each Ij injective. One can find such a resolution by taking 70 = I(V),
Ix = l(l{V)li(V)),eic.
We note that the cohomology of the complex
(/ok-^Ci)*-^-

6.A.I. Some Homological Algebra
207
is, up to a natural isomorphism, independent of the choice of the resolution.
The j-th cohomology space of this complex is the j-th right derived functor.
One of the key results in Enright, Wallach [2] implies that our functors TJ
are naturally equivalent with the right derived functors of VK. It is this
formulation of the Zuckerman functors that is studied in Vogan [2].
Zuckerman introduced these functors to give an algebraic analogue of the
sheaf theoretic constructions in Schmid's thesis (Schmid [1]) which proved a
substantial part of Langlands' conjecture on the discrete series.
6.10.2. Our calculations of K-multiplicities are based on Lemma 6.5.2. This
result can be sharpened as follows. If s e W(K, T) then denote by l(s) the
number of a e <bk such that sa. is negative. Then det(s) = (- l)'(s). In the
notation of 6.5.2 one has
Theorem. // there exists seW(K,T) such that s(A + pk) — pk is <£>£-
dominant and T-integral then rJM(Qk,E) = 0 if j is not equal to l(s) and
rlts)M(Qk,E) is isomorphic with VS(A+Pk)-Pk. Otherwise PM(Qk,E) = 0 for all j.
This theorem is substantially, the Borel, Weil, Bott theorem (see Enright,
Wallach [2] for a proof using the formalism in this chapter, there are also
related results in Chapter 9).
6.10.3. We now move to the situation in Section 6.7. Let g be a 0-stable
parabolic in gc. Let A e i'(I/[I,I])* be T-integral. Let CA be the
corresponding one dimensional (I, M)-moduIe. Suppose that ^ e i(l/[l,l])*
vanishes on I n p and is such that M(q, CA._t)l) is irreducible for t > 0. Then
we set T"M(g, CA) = B„(A). Theorem 6.7.4 implies that B„(A) is either 0 or it is
a (g, K)-module that admits a positive definite (g, X)-invariant Hermitan form.
This result implies a conjecture of Parthasarathy [2] and of Zuckerman
which was first proved by Vogan [3]. Our discussion follows the proof in
Wallach [4]. We will study the modules fi„(A) in more detail in Chapter 9.
6.A. Appendices to Chapter 6
6.A.I. Some homological algebra
6.A.I.I. In this appendix we will compile several results on algebraicly
induced modules that are used in this chapter. The first theorem is taken from
Garland, Lepowsky [1]. Let g be a Lie algebra over (say) C and let m be a
subalgebra of g. Let W be an m-module and let V be a g-module.

208
6. A Construction of the Fundamental Series
Lemma. The Q-modules ([/(g) (g)U(m) W)®V and [/(g) ®V{m){W ® V) are
isomorphic.
Let A: [/(g) -> [/(g) ® [/(g) be defined by A(l) = 1 ® 1 and A(X) =
X ® 1 + 1 ® X for X e g. Let S(x) = xT (see 0.4.2). Let m: [/(g) ® [/(g) ->
[/(g) be given by multiplication. We leave it to the reader to check the
following identities. (Hint: Test them on elements of the form X")
(1) (/ ® m)(A ® /)(/ ® S)(A(g)) = g ® 1 for g e [/(g).
(2) (/ ® m)(I ® S)(A ® 7)(A(0)) = 0 ® 1 for 0 e [/(g).
Recall that if X and 7 are g-modules then X ® 7 is a ^-module with action
g(i; ® w) = A(g)(v ® w). We define a mapping, a from [/(g) (X)[,(m)(W'' ® F) to
([/(g) (X)^ W) ® V by a(g ® (w ® «)) = A(g)((l ® w) ® i>). Then it is easily
seen that a is well defined and is a g-module homomorphism. We define a map
P in the opposite direction by fi((g ® w) ® i;) = p((I ® S)A(gf)((l ® w) ® i;)).
Here p is the projection of [/(g)® W® K onto [/(g) (X)^^® V). (1)
and (2) imply that a and /? are mutual inverses. The Lemma now follows.
6.A. 1.2. The next results have to do with the Koszul complex. Let V be a
finite dimensional vector space over (say) C. Let S(V) denote the symmetric
algebra over V. Let S](V) denote the elements of S(V) that are homogeneous
of degree j. We define
3: SJ(V) ® Ak(V) -> SJ+l(V) ® Ak~l(V)
by 3(u® vlAv2A---Avk) = E(- l)put;p® i^A--- Aj7pA--- Atv Here the carat
means delete.
Lemma. The following sequences are exact.
0^SJ(V)®AnV^SJ+l(V)®A"^lV^----
->S1+"-l(V)® AlV-> SJ+n(V)->0.
We look upon S](V) as the space of all polynomial functions on V* that
are homogeneous of degree j. If jieV* and if ueS](V) then we set
3/Ju(a) = d/dtt=Qu(a + t/x). Let Vj be a basis of K and let Hj be the dual basis.
We define
d: S](V) ® AkF ->• SJ'^ '(H ® Ak +' K
d(w ®v) = Y, 9^r» ® t^rAt;.

6.A.I. Some Homological Algebra
209
We leave it to the reader to check that do + od is j + n times the identity on
S](V) ® AkV. The Lemma follows from this observation.
6.A.I.3. Let g be a Lie algebra over C and let b be a subalgebra. Let W
be a g-module. We define a g-module homomorphism, 3, from (7(g) ®uib)
(AJ'(g/b) ® W) into U(Q)(g)utb)(AJ-l(Q/b)® W) by
3(0 ® X,A• • • AXj® w) = X(- 1)*"^ ® X,A-• • AikA■■■AXJ®w
+ X (- l)k+'« ® *i A • • • AlkA • • • AX; ®Xkw
£ (_ i)'+«M® [XTXJAXiA• • • AlrA• • • AlsA■■■AXj®w
r<s
here X denotes the projection of X e g into g/b.
It is an exercise (which we leave to the reader) to show that 32 = 0.
Lemma. The following is an exact sequence of ^-module homomorphisms
(n = dim(g/b)
0 - (/(g) (X)((A"(g/b)) ® W) 4 I/(g) ®((A"- '(g/b)) ® W)
[/(b) [/(b)
-►■■■• -4 (/(g) (X)((g/b) ® W) -4 (/(g) (X) W -► W -► 0.
[/(b) [/(b)
Here t/ie /ast map is the obvious natural %-module homomorphism g ® w h-> gw.
Set (/;(g) = (/y(g)(/(b). We set
Ehk=Uj(Q)(g)((Ak(Q/b)®W)
[/(b)
then 3 maps E-hk to £,-+, _ k _ ,. Let K be a subspace of g such that g = b ® V.
Then [/,(g) = symm(S,(K))I/(b). Here
S,(K) = X S*(K).
9 induces a map 3: EjJi/Ej_uk^ Ej+ lj)k_ i/£,-,*_ ,. It is easily seen that 3 is
given by the map 3 of 6.A. 1.2. Thus if u e Ejk and if u = 0 then there exists
Vi e £,_1>)k + 1 such that u — ovl e Ej_Uk, there is thus v2 e Ej_2,k+x such that
u — dvl — 3d2 g Ej-2.k> etc This implies the lemma.
6.A.I.4. Now let g => b => m with b and m subalgebras of g. Let W
be a b-module, then l/(g)(X)[/(b) W is easily seen to be isomorphic with
(/(g) (g) ((/(b) (§)„,„,, W). Thus Lemma 6.A.1.3 implies (r = dim(b/m))

210
6. A Construction of the Fundamental Series
Lemma. Let W be a b-module. There exists a Q-module exact sequence
0 -► U(q) (X) (Ar(b/m) ® W) -»• • • -» [/(g) (X) (b/m) ® W)
l/(m) l/(m)
- 1/(9) (g) W - 1/(9) (g)W-»0.
l/(m) t/(b)
6.A.I.5. Let G be a real reductive group. Let K be a maximal compact
subgroup of G and let M be a closed subgroup of K. Let g, b, m be as in
6.A.1.4 with Ad(M)b = b. Here g and m denote the complexified Lie algebras
of G and M respectively. If W is a (b, M)-module then the exact sequence in
6.A.1.4 is a (g, M)-module exact sequence.
Lemma. Let W be a (b, M)-module. Then Hj{q, M; [/(g) (g)U(b) W) = 0 for
j < dim(g/b).
We first prove the result in the special case when b = m. Let Uj and ELk be
as above. Set E} = £; 0 and C) = HomM(A'c(g/m), £,). Then dC) is a subspace
of C)X\. Let V be an M-invariant complement to m in g. Then the
corresponding graded complex is given by
d: UomM(\kV, S\V) ®W)^ HomM(A' + 1 V, SJ+l(V) ® W)
with
da(v0,.,.,vk) = Y,{- \)r(vr® l)a(v0,...,€„..., vk).
If we choose a non-zero element of A" V (n = dim V) then we can identify
Ak(V*) with A"~kV. The corresponding map is thus given as in the Koszul
complex with indices n — k rather than k. The special case of the Lemma now
follows from Lemma 6.A. 1.2.
We now prove the general case of the Lemma. Consider the exact sequence
in Lemma 6.A. 1.4. Set
Xj = a(I/(g) (X) (Mb/m) ® W)).
Then the exact sequence yields the short exact sequences
0 - Xj - t/(9) (X) ((AJ- '(b/m)) ® W) - X,_, - 0.
t/(m)
The above special case of the Lemma combined with the cohomology long
exact sequence implies that if p + 1 < n then Hp(o),M;X].l) is linearly
isomorphic with Hp+1(q, M; Xj). Thus if p + r < n then H"(g, M; X0) is linearly

6.A.2 Partition Functions
211
isomorphic with Hp+r(g,M;Xr). But Xr = U(Q)®V(m)((Ar(b/m))® W) and
*o = U(q) (g)utb) W. The result now follows.
6.A.2. Partition functions
6.A.2.I. Let V be a finite dimensional vector space over C. Let S be a finite
subset of V. To each element, s, of S we assign a positive integer, m(s), which
we call the multiplicity of s. We will think of S as containing m(s) copies of s.
We also assume that there exists /ieK* such that n(s) is real and greater than
1 for all s e S. If v e V then we define ps{v) to be the number of ways that v
can be written in the form
(1) S,+"" + Sr
with Sj e S (allowing multiplicity). Clearly ps(v) = 0 if v is not in ZS6S Ns. For
example if S = {s} and m{s) = 2 then ps{ns) = n + 1 if n e N.
Lemma. If v e V then 0 < ps(v) < oo.
If v e K were to have an infinite number of expressions in the form (1) then
there would be an infinite sequence of positive integers rx < r2 < ••• with v
having an expression of the form (1) having r, terms. But then n(v) > r; for all
j. Since this is ridiculous the lemma follows.
6.A.2.2. We retain the notation of the previous number. We look upon V as
the space of linear functionals onF*.
Lemma.
[] (1 - O-"'" = I Ps(v)e-".
seS veV
Here the expression can either by interpreted formally or a convergent series
on V+ = {v* e V* \ v*{s) > 0 for s e S}.
This is proven by doing the obvious (formal expansion) using (1 — e~s) =
Y p"s
6.A.2.3. We now assume that S is the union of two sets S, and S2 and that
each s e Sj is assigned multiplicity nij(s). We define a multiplicity on S by
setting m(s) = m^s) + m2(s). (Here my(f) = 0 if t is not an element of Sj.)

212
6. A Construction of the Fundamental Series
Lemma. ps(v) = SW6K pSl(v - w)pSl(w).
We note that the previous Lemma implies
n (1 - e-Tmis) = EI 0 - e-T""is) EI (J - e~Tm2(s)-
seS seSi seS2
If we expand the right hand side and collect terms the Lemma follows.
6.A.3. Tensor products with finite dimensional representations
6.A.3.I. In this appendix we prove several results that involve tensor
products of finite dimensional and infinite dimensional modules. The basic ideas
are due to Zuckerman [1] and Jantzen [1]. Once we have developed the
theory of characters we will prove somewhat better results. For the purpose
of this chapter the relationship with the Jacquet modules will be of great
technical importance.
Let g be a reductive Lie algebra over R with Cartan involution 6. Let h be
a Cartan subalgebra of gc. Fix, <t+, a system of positive roots for <t(gc,h)
and let b denote the corresponding Borel subalgebra. If %: Z(gc) -> C is
a non-zero homomorphism then there exists [i e I)* such that x = Xb,»
(Theorem 3.2.4, there #b„ was denoted x„)- Furthermore, #b„ = #b„. if and
only if [i' = s[i for some 56^= ^(gc,h). If bt is another Borel subalgebra
of gc then there exists g e Int(gc) such that ,9b = bx. Set \)x = gt). If \i e h*
then set an = n ° g'1. Then ;;„_„ = xbl.w Also if g' e Int(gc), if g'b = gb and
if a' depends on g' as above then xbl,^ = Zb,.ff>-
We may thus parameterize all infinitesimal characters by fixing one Borel
subalgebra, b, and one Cartan subalgebra, h, in b. In this appendix it will be
convenient to use Borel subalgebras contained in complexifications of
minimal parabolic subalgebras of g. For applications we will use other Borel
subalgebras (say 0-stable, 6.4.1).
Let p be a minimal parabolic subalgebra of g. Set n equal to the nil-radical
of p and let m = 0(p) n p. Let h0 be a Cartan subalgebra of m and set h equal
to the complexification of h0. Let b denote a Borel subalgebra of gc
contained in pc. We will use b and h for our parameterization and we will write
X^OTXb,^
6.A.3.2. Let G be a real reductive group of inner type with Lie algebra g and
Cartan involution 8. Fix (P, A), a minimal p-pair and let b and h be as in the
end of the last number. Let V be the category of (g, °M)-modules of 4.1.1. If
£ is a finite dimensional (p, °M)-module then we set M(E) = U{qc) (X)^^, E

6.A.3. Tensor Products with Finite Dimensional Representations
213
with g acting by left multiplication and °M acting by m(g ® e) = Ad(m)g ® me
for g e U(qc), e e E and m e °M. Clearly M(£) is an object in Y". Assume that
E is irreducible. Let [i be the highest weight of E (we use <t+, as above). We
denote by, 8, the half sum of the elements of <t+.
(1) M(E) has infinitesimal character x? + a-
Indeed, P-B-W implies that
U(qc) = U(mc)®(e(n)U(gc) + I/(9c)n).
Let p denote the corresponding projection onto U(mc). If z e Z(gc) and if
g ® e e M(E) then z(g ® e) = g ® p(z)e. Now p(z) is central in U(mc) hence
p(z)e = y\z)e. Let <jf and y be as in 3.2.2 and let e be a highest weight vector for
E. Then p(z)e = <?(p(z))e = q(z)e = n(q{z))e = {n + 8)(y{z))e. Thus x = X»+s-
(2) M(E) has a unique non-zero irreducible quotient, L(E). If Kef is
irreducible then there exists an irreducible, finite dimensional (p, °M)-module
such that V is isomorphic with L(E). If E and E' are irreducible finite
dimensional (p, °M)-modules then L(E) is isomorphic with L(E') if and only
if E is isomorphic with £'.
Let H e a be such that a(H) = 1 for all simple roots in <S>(P,A). M(E) =
U(6(n)c) ® £ as an a-module. Thus H acts semi-simply on M(E) with
eigenvalues of the form n(H) — n with n a non-negative integer. The n{H)-
eigenspace for H is 1 ® E. Set R~ = {N\ N a submodule of M(E) such that
N n(l ® £) = 0}. Clearly, a submodule, JV, of M(£) is in i?~ if and only if
the ^(H)-eigenspace for H on N is zero. This implies that R~ is closed under
addition. Set R(E) equal to the sum of all elements of R~. It is obvious that
R(E)eR~. It is also clear that R(£) is proper and that if N is a proper
submodule of M(£) then N e R~. This proves the first assertion. If V is a
g-module then set V" = {v e K |nt; = 0}. If Kef" is non-zero then F" is a
non-zero, finite dimensional (p,°M)-module. Let £ be an irreducible,
nonzero submodule of V". Then we have a non-zero (g, °M)-module homomor-
phism of M(£) into V. Thus if V is irreducible then V is isomorphic with
L(£). Finally, it is easy to see that £(£)" is isomorphic with £ as a (p, °M)-
module.
6.A.3.3. Let V e V. If \i e h* then we set Vfl= {veV (h- n(h))kv = 0 for
some k and all hei)}. Then dim F^ < go (4.1.3). We denote by ch V the
formal expression
L(dimF>*.
Set AG = A = e^nO - e") the product over a e <t+.

214
6. A Construction of the Fundamental Series
Lemma. Let E, n, M(E) be as in the last number. Then
chM(£) = Xdet(s)es("+,,)/A
the sum over s e W(m, h).
As an h-module, M(E) = U(6(nc)) ® E. Let p denote the partition
function of <t(nc,h) = Z (the weights of h on nc) with multiplicities equal to 1
(see 6.A.2). Let <t(nc,h) = {a,,..., a,,}. Let Y, be a non-zero element of the
-ct; weight space. Then P-B-W implies that the monomials Y"> ■■■ Y"d"
form a basis of U{6nc). Thus ch U{6nc) = Z p{n)e". So ch M(E) =
ch(£) noreI(l — e~")~l. The Weyl character formula implies that
ch(£)= X det(s)est»+S)/AM
seW(m,l))
with 8m the half sum of the elements of <5(mc, h) n <t+. Hence
ch(E) = es"-s( X det(s)es("+,,,/AM\
VeW(m,t)) /
Since (eSmS Uxel (1 - e"a))AM = AG, the Lemma follows.
6.A.3.4. Lemma. (1) Let E be an irreducible finite dimensional (p,°M)-
module with highest weight [i. Then there exist integers c5()i) such that (W =
ch L(£) = X cs(rie-s{» + d)/AG.
seW
(2) // El,...,Ei are mutually inequivalent then ch L(El),..., ch L(Ed) are
linearly independent.
(3) If V e V has generalized infinitesimal character x„ then there exist
integers cs(V) for s e W such that
ch V = X cs(V)e-st»+S)/AG.
self
If Hi,n2e h* then we write nl > \i2 if nl - n2 is a sum of (not necessarily
distinct) elements of <t+. We observe that if L(F) occurs as a subquotient of
M(£) and if £ and F are irreducible with highest weights jx and a respectively
then [i > a with equality if and only if F is isomorphic with £. Also a + 5 e
W(/x + 8) under the above condition.
Fix nef)*. If a + 8 e W(jx + 8) is a minimal element relative to the above
partial order and if £ is an irreducible finite dimensional (p, °M)-module
with highest weight a then L(£) = M(£). Indeed, any highest weight of R(E)

6.A.3. Tensor Products with Finite Dimensional Representations
215
is strictly less than a. Thus assertion (1) is true in this case. Assume that (1)
has been proved for all y + 8 e W(n + 8) with y < a. Then ch(L(£)) =
ch(M(£)) - ch(R(£)). Since the elements of V have finite length (4.1.3).
ch(R(E)) = I. ch(L(Ej) with £y a finite dimensional irreducible (p,°M)-
module with highest weight )»,• such that ys < a. Thus (1) is true for a.
We now prove (2). It is clear from the preceding lemma that ch M(E1),...,
ch M(Ed) are linearly independent. It is also clear from (1) that it is enough
to prove (2) in the case when E1,...,Ed are a set of mutually inequivalent
irreducible (p,°M)-modules such that if E is an irreducible finite dimensional
(p, °M)-module with highest weight of the form s(n + 8) — 8 then E is
isomorphic with Ej for some j. If the £,- are numbered compatibly with the
partial order on their highest weights then the matrix relating the ch M(Et) and
the ch L(£,) is triangular with ones on the main diagonal. This implies (2).
If F e V then F has finite length. Let Vx,..., Vd be the irreducible
constituents of V. Then ch F = X ch Vj. (3) now follows from (1) and 6.A.3.2(2).
6.A.3.5. Corollary. If V1,V2ei" are irreducible and if ch Fj = ch F2 then
Vx is equivalent with V2.
6.A.3.6. Let t^, denote the subcategory of all objects in -V that have
generalized infinitesimal character x„ ■ Clearly, V^ = V^ for s e W.
Lemma. Let F be a finite dimensional (Q,°M)-module and let V e V^. Then
V ® F = ®Z„ with Zae Vfl + a and the sum is over a subset, S, of the weights
of F such that {;(„ + „!<x eS} = {#„ + „ a a weight of F} and such that if
a, )8 e S and *„+„ = x„+f then a = 0.
We note that ch(F® F) = ch(F) ch(F). If a is a weight of F let m(a)
denote the dimension of the a weight space. Then m(sa) = m(a) for all s e W.
Now ch(F) = Z c5{V)es"/A. Thus ch(F® F) = I m(a) £ cs(F)es(" + <T)/A. In
light of 6.A.3.4 and 6.A.3.5 this implies that the irreducible constituents of
V ® F have precisely the infinitesimal characters described in the statement.
This implies the Lemma.
6.A.3.7. Let Fet^, and let F be an irreducible finite dimensional object
in V. Let a be a weight of F. We set <tf „ equal to the projection of F ® F
onto the summand with generalized infinitesimal character %„+„.
Lemma. Let /jef)* be such that Re(/i, a) is non-zero for all ae<S(gc,f)).
Let <t+ = {a E<t(gc,h)!Re(^,a) > 0}. Let a be the highest weight of F

216
6. A Construction of the Fundamental Series
relative to <t+ and set O" = <tF „. Then O" is an equivalence of the categories
*; and TTM + <r.
Let X, Y e *; and let A e Hom9iM(X, 7). Then
A®Ie Hom9-OM(X ® F, Y ® F).
Set <&F_a(A) = A® I restricted to <tF,(T(Ar) for a a weight of F. Then this
construction shows that each <tF „ is a functor that is easily seen to be exact.
Set, for Kef,,<»_, = %.,-,,. We put
T(X) = *_„(*•(*))
for K e V,,. We now come to the two main observations.
(1) If V e tT and if A ch K = £ cses" then A ch(<D"(K)) = £ cses("+<T).
Let h^ be the real span of the Ha for a a root. If /? e h* then we write Re /?
for the real part of the restriction of /? to hR. We have
A ch(F ® F) = X m(P) £ c,e'"'+w
the sum over the weights of F (see the proof of 6.A.3.6). If /? is a weight
of F and if there exists se W such that s{n + f$) = n + a then sRe/j-
Re ^ = a — s/i. This implies that s Re ^ — Re \i is a sum of positive roots.
Our assumption on implies that s Re \i = Re ^ — Z »va with «„, > 0 (the
sum is over <t+. Hence s Re ^ = Re \i. But then s = 1 (again by our
assumption). This implies (1).
(2) If FE*;+(Tandif A ch V = £ cses("+<T) then A ch «_„(K) = ^.e".
Indeed assume that /? is a weight of F* and that \i = s(/i + fi + a) for some
se W. We note that ft = — a + Q with Q a sum of positive roots. Thus Re \i =
s(Re n + Q). But then (Re \i, Re n) = (Re n + Q, Re \i + Q) = (Re n, Re jj) +
2(Re n, Q) + (Q, Q) > (Re n, Re n) + (Q, Q). Hence Q = 0. Thus s = 1, ^ =
— a. (2) now follows.
The upshot is that if Fef„ then ch T(V) = ch V. We assert that T is an
equivalence of tT with itself. If X, Kef^ let y: Hom9,oM(X, F®F*)->
Hom9,oM(X ® F, V) be defined by y(A)(x ®f) = (I® f)(A(x)). Here
(I®f)(v®f*) = f*(f)v, feF, f*eF*. Then 7 is a bijection. Now,
Hom9,0M(K T(F)) = Hom9,OM(K «_<r(*"(K))) = HomBoM(K, «*(K) ® F*). Set
SK = 7-1(<t'T(/)). Then Sv: K->T(K) is a (g,°M)-module homomorphism
which is 0 if and only if V = 0. If V is irreducible then T(V) is irreducible
(ch V = ch T(V)). Thus SV:V-> T(V) is an isomorphism. We note that

6.A.3. Tensor Products with Finite Dimensional Representations
217
V*-*SV is a natural transformation (6.2.2) from the identity functor to T.
Assume that we have shown that if V has length r then Sv is an isomorphism
and assume that V has length r + \. Let L be an irreducible and non-zero
submodule of V. Then we have the following commutative diagram with
exact rows:
0 * L * V * VIL * 0
0 > T(L) »T(V) * T(VIL) > 0
Since our hypothesis implies that SL and SVjL are isomorphisms this implies
that Sv is also. The Lemma now follows.
6.A.3.8. We now transfer the above material to the category jf (4.1.4). Let
j-.jf ~^,V be the Jacquet module functor (4.1.5).
If V e JC then V = © Vx (see the proof of Lemma 4.1.4) here the sum is
over homomorphisms of Z(gc) to C and V* = {ve V\(z — x{z))dv = 0 for
some d and all z e Z(gc)}. Let Px be the corresponding projection of V onto
Vx. Let Jf* denote the full subcategory of Jf consisting of objects with
generalized infinitesimal character %. Then PX:JV -^Jt* is an exact functor.
If X = X„ then we set Px = P„ and Jf* = jf".
If F e Jf is finite dimensional, if V e Jf* and if a is a weight of F then
we set Q>F,a(V) = P„+ff(F® F). Then <tF(7 is an exact functor from the
category jf" to the category Jtf + °. If i is a homomorphism from Z(gc) to
C then set %*(z) = x(zT). Note thatj-.J?*™*^. Also (*„)* = x_„.
Lemma. If V e Jt" and if(V ® F)x is non-zero then x = X„ + a for some, a, a
weight of F Also, j(<bFt<t(V)) = <tF.,_„(j(F)).
j(V ® F) = j(K) ® F* under the identification of (V ® F)* with F* ® F*.
Indeed, there exists r such that (n)rF* = 0. Thus, if \i e j(V) and if (n)*^ = 0
then (n)k + r(n ® F*) = 0. So j(F) ® F* is contained in j(K ® F). Let /,,..., fd
be a basis of F* such that nf is contained in Fi + 1 = £,■>;+! Cf. If \i e
j(V® F) then it can be written uniquely in the form X ^ ® f. Let j be the
smallest index such that Hj is non-zero then modulo V*®FJ+1, (n)k)i =
(n)Vj® fj- Thus, HjS j(V). The assertion now follows. We now prove the
lemma.
If (V ® FY is non-zero then j((V ® FY) is also non-zero and it is equal
to (j(V) ® F*Y*. The Lemma follows from this and the observations
preceding it.

218
6. A Construction of the Fundamental Series
6.A.3.9. We now assume that F is irreducible. Let /jef)* satisfy the
condition of Lemma 6.A.3.7 and let <t+ be as in the statement of that lemma.
Lemma. Let a be the highest weight of F. Then <tF „ is an equivalence of
categories between JF" and JF" + °.
Let for V e Jf", T(V) and Sv be defined as in the proof of 6.A.3.7. We note
that j{V)eir-ll so the pertinent functors on j(V) are <bF*t_<t = <b~" and
<tF(T = <&„. In the proof of Lemma 6.A.3.7 we showed that SJ(V):j(V) ->
j(T(V)) is an isomorphism. Lemma 6.A.3.8 implies that j(T(V)) = T(j(V)).
We now show that Sv is an isomorphism.
We first look at the case when V is irreducible. It is clear from the definition
of Sv that it is injective. We therefore have the exact sequence in H"
O^V-^ T(V)-> T(V)/SV(V) -► 0
induces the exact sequence in ^(Theorem 4.1.5)
0 -y(T(K)/S„(K)) -j(T(K)) ^H j(V) - 0.
As we have observed above, j(T(V)) is isomorphic with j(V). Hence, j(Sv) is
bijective. The exactness of the above sequence implies that j(T(V)/Sv(V)) = 0.
Hence T(V) = SV(V) (4.1.5(1)).
This proves that Sv is an isomorphism if Fis irreducible. The rest of the
proof is now identical to the last part of the proof of 6.A.3.7.
6.A.3.10. Corollary. We retain the assumptions and notation of the previous
number. If V e jf""1"" is irreducible then <!>*•*, _ff(K) is irreducible.
The proof of the previous result shows that <tF(T is an equivalence of
categories between H" and H" + CT with inverse functor <tF. _„.
6.A.3.11 The rest of this section is devoted to a proof of a theorem of
Kostant [3]. Although this result is not seriously used in the text, the proof
that we give is an application of the theory that we have just developed.
Let g be a reductive Lie algebra over C. We put the direct product Lie
algebra structure on g x g. Set 8{X, Y) = {Y,X). Then there is a real form
of g x g for which 8 is a Cartan involution. Indeed, let gu be a compact form
of g. We look upon g as the complexification of gu. If we consider g as a Lie
algebra over R then gc is isomorphic with g x g. 6 is the Cartan involution
associated with (g,gu). Set f = {(X,X) |Ieg}.

6.A.3. Tensor Products with Finite Dimensional Representations
219
Lemma. Under the identification, (7(g x g) with (7(g) ® (7(g), Z(g x g) is
isomorphic with Z(g) ® Z(g).
This is an easy exercise and is leu 10 me reauer.
6.A.3.12. If V is a g-module then we set Ann(F) = {ge [/(g) | gV = 0}. Then
Ann(F) is a two side ideal in (7(g). We define a g x g-module structure on
[/(g) by (X, Y)g = Xg - gY. Then U = [/(g) is a (g x g,f)-module and
Ann(V ) is a submodule under this action. Hence, (7/Ann(F) is also a
(g x g, f )-module.
If Xu Xi are homomorphisms of Z(g) into C then set (X\,Xi)(z\ ® zi) =
Xi (z 1)^2(22)-
Lemma. // K has generalized infinitesimal character x then U/Ann(V) has
generalized infinitesimal character (x,X*)-
This is also left to the reader.
6.A.3.13. We now prove the advertised result of Kostant.
Proposition. // V is a Q-module with generalized infinitesimal character x„ and
if F is a finite dimensional g module then V ® F splits into a direct sum of g
modules with generalized infinitesimal characters. Furthermore, the generalized
infinitesimal characters that can appear are of the form x^+a with a a weight
off.
We may assume that V = Uv with v e V. Fix a non-zero element f e F. Let
T{g ® A) = gv ® Af for g e U and A e End(F) then T defines a g-module
homomorphism of (U/Ann(V)) ® End(F) onto F® F. Here, g acts by left
multiplication on both factors. We make End(F) into a g x g-module by
setting (X, Y)A = XA- AY. Then the action of g on (C//Ann(K)) ® End(F)
is the same as that which is gotten under the identification of g with g x 0.
U/Ann{V) is a finitely generated (g x g,f)-module with generalized
infinitesimal character (#„,(#„)*). Hence, it is admissible (3.4.7. Corollary). Thus,
Lemma 6.A.3.8 implies that (U/ann(V)) ® End(F) splits into a direct sum of
invariant subspaces, (7,-, with generalized infinitesimal character of the form
Ofo + ff.X-d-a) with o-, <5 weights of F. Hence, V ® F = X T((7,). Since, each
T(Uj) has a generalized infinitesimal character of the correct form, the result
follows.

220
6. A Construction of the Fundamental Series
6.A.3.14 Let Jt denote the category of all g-modules. Let JC denote the
full subcategory of modules with infinitesimal character x„- The above result
implies that if F is a finite dimensional g-module and if a is a weight of F then
we can define the functor <tF „ from the category JC1 to the category JC1+° as
in 6.A.3.7.
6.A.4. Inhnitesimally unitary modules
6.A.4.I. Let G be a connected real reductive group with maximal compact
subgroup K and associated Cartan involution 8. Let g = f + p be the
corresponding Cartan decomposition of g. Let B be a (g, K)-invariant symmetric
bilinear form on g such that if we set <X, Y) = - B(X,6Y) for X, Y e g then
< , > is positive definite. We set \\X\\ = (X, X)l'2 for X e g.
Let V be a (g, K)-module endowed with a pre-Hilbert space structure,
< , > that is (g,K) invariant. Let C denote the Casimir operator of
g corresponding to B. Let H denote the Hilbert space completion of
(V,( , »•
Lemma. There exists a positive constant, e, such that if C acts on V by [il then
£||X"t;||/n!<oo
for leg with \\X\\ < e and v e V.
Let CK be the Casimir operator of f corresponding to B. If X e g then we
write X = Xx + X2 with X, e p and X2 e f. Then ||X||2 = ||Jf,||2 + ||X2||2.
Let X e g and let v e V(y) for some y e KA. Then
||X"t;||2 = IIX,*"-1^!2 + 2 Re(XlX"-lv,X2Xn-lVy + H^X"-1^2
<2||X1X"-1t;||2 + 2||X2X"-1t;||2
= -2<(X1)2X"-'t;,X"-1t;>-2<(X2)2X"-1t;,X"-1t;>
Let t be a maximal abelian subalgebra of f and let <t be the set of non-zero
weights of t on gc. Fix <t^, a system of positive roots for <t(fc, tc). The weights
of tc on U"~ '(9c) are of the form ft + •••/?„_, with ft e 0. Hence the highest
weights of [/"" '(9c) ® V(y) are of the form Xy + ft +••• + #,_, with ft e <t
(see Scholium 6.4.5).
Let Zu...,Zr be an orthonormal basis of p such that X{ = HXJIZ, and
let Wj,..., Wt be an orthonormal basis of f such that X2 = \\X2\\Wi. Then

6.A.4. lnfinitesimally Unitary Modules 221
CK=-TL (Wj)2 and C - CK = Z (Zj)2. We have
-{(Xl)2X"-1v,X"-1v)< -\\Xl\\2Y,((Zj)2Xn-lv,X"-lv) and
We therefore conclude that
||X"t;||2<-2^||X1||2||X"-1t;||2+4||X||2<QX"-1t;,X"-1t;>.
Now, (CKX"-1v,X"-1v) < n„\\X"-1v\\2 with
/i, = max{||A, + ft + - + A,_, + <y2 - ll^ll2! A,..., ft- ,eO).
Letc = max{||j8||!J8E<I)}. Then
^<2(||/v + <y2+(n-l)2C2).
Set c(y) = 2\\n\\2 + S\\Xy + 8k\\2. We have shown that
(1) ||X"t;||2 < ||X||2(c(y) + 8(n - l)2c2)\\X"-lv\\2.
(1) implies that if n > 1 then
(2) ||X"t;||2 < (n - l)2(c(y)/(n - l)2 + 8c2)||X||2||X"- 't;||2.
We may iterate (2) and conclude that
(3) ||X"t;||2 < ((n - l)!)2c(7) J] ((c(y)/(i-l)2 + 4C2)||X||2)||t;||2.
2<j<n
Let N be such that if n > N then c(y)/(n - l)2 < 1. Set C(y)2 = c{y) + 8c2
and C2 = 1 + 8c2. Then (3) implies that
(4) HA-rll < C(7)N«!C"||X|n|t;||.
Thus if we take e = 1/2C the Lemma follows.
6.A.4.2. The purpose of this appendix is to prove the following result of
Harish-Chandra[l,p.231].
Theorem. Let V be as in Lemma 6.A.4.1 and assume in addition that V is
admissible andfinitely generated. Then there exists a unitary representation
(n,H) of G such that HK is isomorphic with V as a (%K)-modu\e.
Since V has finite length as a (g, K)-module and if Vx is a (g, K)-submodule
of V then V is the direct sum of Vx and the orthogonal complement of Vx, we
may assume that V is irreducible. Hence, C acts on V by a scalar. Let (7r1? H^

222
6. A Construction of the Fundamental Series
be a realization of V as a Hilbert representation (4.2.5). Let ( , ) denote the
Hilbert space structure on Hl. We assume (as we may) that nx is unitary when
restricted to K and that (H^ = V. Let H be as in the previous number. Since
the action of K on V is unitary we may extend it to a unitary representation of
K on H. Then HK = V.
If F is a finite subset of KA then we set VF = @7eF V(y). Let EF denote the
orthogonal projection of Hx onto VF and let PF denote the orthogonal
projection of H onto VF. We note that if v e V and if w e VF then (v,w) =
<r, AFw> with AF a positive definite self-adjoint (relative to both < , > and
( , )) endomorphism of VF.
Let e > 0 be as in the previous Lemma. Set U0 = {X e g j \\X\\ < e}. If
X e U0 and if v e K then we set
Then X h-» A(X)t; is real analytic on U0.
(1) Let F be a finite subset of KA, v e V and X e U0 then PF/1(A> =
EfTr^exp X)v.
Let 0 < ^ < e be such that Z |(X"t;,w)|/n! < oo for ||X|| <8 and weVF
(3.4.9). Then (jr,(exp X)t>,w) = Z(X"t>,w)/n! for \\X\\ < 8 and t»eFf.
Thus, if ||X|| < 8 then (Jt,(exp X)v,w) = Z,(X"v,w)/n\ = Z,(EFX"v,w)/n\ =
I (EFX"v, AFw)/n\ = I (X"v, AFw)/n\ = <A(X)v, AFw) = (PFA{X)v, w)
Thus (1) is true for \\X\\ < 8. Since both sides of (1) are real analytic for X e
U0, the assertion follows.
(2) If v, w e V and if X e U0 then (A(X)v, A(X)w) = (v, w>.
We note that x, y h-» (^^(xJti^^w) is real analytic on G x G. Thus if F
is a fixed finite subset of KA such that v,w e VF then there exists 0 < 8 < e
such that if u, w e FF and if ||X|| < ^ then
X («, X" + mw)(- l)m/n!m! = («,w).
Set for X e g, expm(X) = 1 + X + • • • + Xm/m! e [/(g). If v, w e KF and if
||X|| < 5 then (A(X)v,A(X)w) =
lim <expm(X)t;, expm(A» =
m~* go
lim <t;,expm(-X)expm(X)w> =
m-* go
lim (AF1t;,expm(-X)expm(X)t;) =
m-» co

6.A.4. Infinitesimally Unitary Modules 223
(A~Fxv,w) = <i\n>>. This proves (2) for ||X|| < 8, so (2) follows from analyticity.
In particular (2) implies
(3) If ve V and if ||X|| < e then \\A(X)v\\ = \\v\\. Here ||---|| is the norm
associated with < , >.
This implies that A(X) has a continuous extension to H which we also
denote A(X).
(4) A(Xf = A(-X) and A(X)*A(X) = I for X e U0.
This follows from (2) and (3).
We assume by choosing a (possibly) smaller e that exp is a diffeomorphism
on U0. Set U = exp([/0) and let log denote the inverse function on U to exp
restricted to U0. If g e U then we set n(g) = A(\og(g)) on H. Then if k e K n [/,
n(k)v = kv for v e V. Also if g e U then n(g)* = n(g)~x = n(g~x) on H.
(5) If g!,..., g„ are either in U or K and if F is a finite subset of KA then
PF7i(g1)---7i(g„)v = Epn^g^-■■ n^g^v for all v e V.
If n = 1 this follows from (1). Assume the result for n — 1 > 1. Let f! <=
F2 <= • • • be finite subsets of KA whose union is KA. Set E} = £F and ^- = PF
with F = Fj. Let weFF then
<7r(sr1)---7r(gf„)i;,w> = lim (Pjn(g2)-■ ■ n(g„)v,n(gJ'1™).
j->co
lim <£J-7r1(g(2)--7r1(g(„)t;,7r(g(1)Mw>= lim (n(gx)Ejnx(g2)- ■ ■ nx(g„)v,wy
j->co j->oo
= lim {PF7i{gl)Ej7il{g2)---7il(gn)v,wy
= lim (£F7t1(g(1)£J-7r1(g(2)---7r1(g(„)t;,/lFw)
= (EF7l1(g1) • • • 7T,(sf„)y, /1FW) = <£i,7T1(gf1) • • • Tliigjv, W>.
This proves (5).
(5) implies that if g,,..., gf„, x!,..., xm are elements of [/ or K and if
0i"-0n = xr--xm then ^(grj)- • 7r(.9„) = 7t(x1)---7t(xJ. Thus if g e G and
9 = 9\'"9« tnen ^(0) = rc(0i)"'rc(0n) depends only on g. We clearly have:
(6) If x,y e Gthen7r(x)7r(y) = n(xy),n{x)* = 7r(x_1)and7r(fc)t; = kvfork e K
and ve V.
(7) (n, H) is a unitary representation of G
In light of (6), we need only show that if gs -> g then lim n{g^)v = n(g)v for

224
6. A Construction of the Fundamental Series
v e V. Now there exists N such that if j > N then g} = gxj with Xj e U and
lim Xj = 1. Since n(xj) = A(\og(xj)) (7) follows.
We finally note that if X e g then 7r(exp(tX))t; = A(tX)v for tX e C/0 and
v e V. Thus d/dtr = 07r(exp(tX))t; = Xt; for ve V. The proof is now complete.
6.A.4.3. We note that one can use the Cambell-Hausdorff formula to prove
(slightly more directly, but using some topology) that the conclusion of the
above theorem is valid under the hypothesis of Lemma 6.A.4.I.

7 Cusp Forms on G
Introduction
In this chapter we study a variety of integral transforms that were first
introduced by Harish-Chandra. The goal of this chapter is to lay the
groundwork for the proof that the representations that we called the discrete
series in the last chapter exhaust the irreducible square integrable
representations of a real reductive group. The first step is to introduce the space of cusp
forms on G. We show that matrix coefficients of irreducible square integrable
representations are cusp forms (eventually we will show that the space of cusp
forms is the span of these matrix coefficients). Thus the analysis of cusp forms
gives information about irreducible square integrable representations. The
key theorems in this direction are 7.6.1 and 7.5.2. These results are based on
Harish-Chandra's theory of orbital integrals which is also critical to his other
monumental achievement, the "local L1-theorem" for characters. Our main
contribution, in this chapter, to Harish-Chandra's original method is
Lemma 7.4.3, which, in particular, allows us to defer the character theory to
the next chapter. The key Lemma that allows this simplification is the result in
7.A. 1.1 which is a special case of Kostant's convexity theorem. Another
simplification in our exposition is the observation that Theorem 7.6.1 is a
consequence of the material in Appendix 5. In Harish-Chandra's original
225

226
7. Cusp Forms on G
development, an analogue of Theorem 7.6.1 is proved for all / e ^(G). The
material in Appendix 5 is then used to calculate the constant CG. The more
general theorem will be proved in Volume II.
In the first section, we introduce a general method of constructing Frechet
convolution algebras of functions on a real reductive group. We are mainly
interested in two examples, the space £f{G) (whose importance will be more
apparent in the next chapter and in Volume II) and the Harish-Chandra space
^(G) which is critical to the theory of cusp forms on G (and plays the leading
role in this chapter).
The exposition in Sections 7.3 and 7.4 is strongly influenced by the notes of
Varadarajan [1] and (of course) by the original papers of Harish-Chandra.
The key results on cusp forms are contained in Sections 5 through 7.
As usual, in this book, there are several important, but (even more technical)
results that are deferred to the appendices. The deepest are in 7.A.2 and 7.A.3.
7.A.5 contains an exposition of a technique of Gelfand-Shilov [1] for finding
fundamental solutions of certain constant coefficient differential operators.
The main result in that appendix is due originally to de Rham [1].
The Theorems in this chapter are all due to Harish-Chandra. His motivation
can only be surmised. However, the earlier work of the Russian school must
have had an important influence on this work. But it was Harish-Chandra
(and only he) who realized that the key to the representation theory and
harmonic analysis on real reductive groups is the discrete series and hence the
harmonic analysis on the space of cusp forms.
7.1. Some Frechet spaces of functions on G
7.1.1. Let G be a real reductive group with maximal compact subgroup K
and corresponding Cartan involution 6. We denote by L and R respectively
the left and right regular representations of G and (7(g) on C^iG). Let a and
b be smooth positive X-bi-invariant functions on G such that
(1) If r > 0 then the set {g e G \ a(g) < r) is compact.
(2) b(x) = b(x~l) forxeG.
(3) There exists a constant d0 > 0 such that J b(x)2a(x)~dodx < oo.
G
(4) a(xy) < a(x)a(y) for x, y e G.
(5) If x, y e [/(g) then \L(x)R(y)a(g)\ < Cx<ya(g)d^K
(6) J b(xky) dk < b(x)b(y) for all x, y e G.
K

7.1. Some Frechet Spaces of Functions on 0
111
If fe C*{G) then we set for r>0,x,ye [/(g)
P„.b.*.yAf) = SUP9.« a(gyb(gyl\L(x)R(y)f(g)\.
Let .^,b(G) be the space of all / e C'X(G) such that pa.b.x.yAf) < °° for
all x, y, r, endowed with the topology given by the above semi-norms. If
y e KA and if / e (^(G) then we set
Eyf(g) = d(y)$^(k)f(k-1g)dk.
K
Theorem. !^a,b{G) is a Frechet space.
(1) The inclusion of C^(G) into ^,(,(G) is continuous with dense image. (Here
C*(G) is given the usual topology, which will be described in the course of the
proof.)
(2) // / e <fBjk(G) and if x,ye U(q), r > 0 then h(g) = a(g)'L(x)R(y)f(g) is
in L2(G). Furthermore, the map
#.fc(G)x.<£6(G)-.S£.fc(G)
given by fx, f2 •—► f\ * f2 is continuous.
(3) If ysKA and if f e SfaJ,(G) then EJ e ,%,b(G) and the series Z EJ
converges to f in ifab(G).
Since we may use the pa.btX,y,r with x, y running through a basis of (7(g)
and r rational, to prove that ,</ = .^b(G) is a Frechet space it is enough to
show that it is sequentially complete. Let f be a Cauchy sequence in if.
The definition of the topology of // easily implies that there exists fe CX(G)
such that f converges uniformly with all derivatives to / on compacta (c.f.
the argument in 1.6.4). Let Q be an open subset of G with compact closure.
Wesetfor/isC^G),
VmW = sup9e£1 a(g)db(gyl\L(x)R(y)h(g)\.
Then pa,h,x,yJh) = supn npx,y,d(h).
Fix x, y, d. Let N be so large that (px,y<d = pa,btXty,d) if j,k>N then
Px.yAfj ~ fk) < 1- Then px,yJfj) < 1 + px,y,d(fN) for j > N. Let Q be open
in G with compact closure and let Na > N be such that npx.y<d{f — fj) < 1
for j > Na. Then npx,y,d(f) < 2 + "Px,m(/n)- Hence,
PxjAf) ^ 2 + Px.yAfs) < °°-
SofefS.
Let e > 0 be given. Let N be such that if j, k> N then px,yAfj ~ fk) < e-
Let Q be open in G with compact closure. Let Nn> N be such that

228 7. Cusp Forms on 6
nP*,yAf -/;)<£ for j > Na. If k > N and if j > Nn then
nPx,yAf ~ fk) < nPx,yAf - fj) + "Px.yAfj " /*) < 2fi.
Thus px,y,d{f- fk) < 2e for k > N. We have thus shown that if is complete.
We now sketch the (standard) argument to prove (1). Recall that the
topology on C?(G) is given as follows. A net fa -> / if there exists a compact
subset Q of G and /? such that if a > /? then supp fa u supp / <= Q and ftt -> /
uniformly on Q with all derivatives. Clearly the px y r are continuous semi-
norms on C™(G).
Let h e C°°(R) be such that h(x) = 1 if |x| < 1 and h(x) = 0 if |x| > 2. Set
for r > 0, ur(g() = h(a(g)/r). We leave it to the reader to check
(i) Let x, y e (7(g) then there exists a constants Bxy
and D(x, y) such that \L(x)R{y)ur(g)\ ^ BXtya(g)D(x-y)
for r > 1 and all g e G.
Let x, ye t/k(g) and let x,,..., xq be a basis of t/k(g). Then
Px,yAUrf - /) < Cr_1 X Px,,x,,d+D(x„x,)(/)
for r > 1. Thus ur/ -> / in if. We have thus proved (1)
The first part of (2) follows from (3) above. Let /,, f2 e if. Then
b(gyla(g)dfx * f2(g) = b(gyla(g)d J L(x)fl(z)R(y)f2(z~lg)dz.
G
The absolute value of the right hand side is at most
b(g)~' a(g)dpx,y,dl(fi )px,y,d2(f2)
times
lb{z)b{z-1g)a(zydla(z~1g)~d2dz.
G
Now (4) implies that a(z~lg) > a(z)~la(g). Hence
\b(z)b(z~xg)a(z)-d>a(z~lgyd>dz
G
= J b(k-1z)b(z-1kg)a(k~lzyd'a(z-1kgyd2dkdz
G*K
<a(gy"2 J b(z)b(z-lkg)a(zyd'+d2dkdz
< %)a(g()^21 b(z)b(z-1)a(zydl+d2dz
G
= b{g)a(gyd2\b(zfa(zyd^d2dz.
G

7.1. Some Frechet Spaces of Functions on C
229
Thus if we take d2 = d and dx > d + d0 then
Px.yjfi * k) ^ CpxAJl(fi)pUy<i2(f2).
We now prove the last assertion. Let j be fixed and let xx,..., x„ be a basis
of l/j(g). If x e U}(q) then Ad(/c)x = X Uj(k)Xj with u;- a matrix coefficient
of (AdIk, U'(%)). Fix T a maximal torus of X and <t+ a system of positive
roots for <t(fc,tc). Let X denote the set of weights of T on Uj(q). Let pk be
(as usual) the half sum of the elements of <I>+, If 7 e KA then set ^y equal to
the highest weight of y. Let CK be the Casimir operator of f relative to B\v
Then the eigenvalue if CK on any representative of y is \\ny + pk\\2 — ||pj|2.
(ii) There exists d > 0 such that £ ||^ + pfc||~d < 00,
Indeed, let nu..,,nr be a basis if 13(f)* and let ^p +,,...,^, be such that if
\x is a <t+-dominant integral, T-integral form on tc then n + pk = I, m^
with mt e Z and m; > 1 for i > r + 1. For such /*, \\n\\ > C(I(m,)2)1/2, with C
a positive constant. Thus the series that we are estimating is dominated by
Z(£(mj)2)~d/2 the sum over all m, e Z with m; > 1 for i > r + 1. This series is
easily seen to converge for d > 2q.
Set D = \\pk\\2 + Q. Then D'E.f = EyDrf. So
(*) Eyf =ll/i+ftll -2%£>X
Let Sj(v) denote the set elements of KA that occur in Vy ® Uj(q). If a e Sj(y)
then /*„ = ny + 8 with 8 a weight of the action of T on l/j(gc). This we have
(iii) If a e Sj(y) then |K + pk\\ < C,||/iy + pk|| + ^
with Cj and Dj positive constants depending only on j.
Let fetf Then a(g)'b(g)-' |L(.x)R(y) V(»)l =
a{gyb(gy' d(?) J ^(k) £ «i(fc)L(xj)i?(};)/(r' 0) dfc.
K i
Now, |zT(k)| < d(y) and |t<j(fc)| < Cx for keK. We therefore conclude that
(iv) Px,y,(Eyf)<Cxd(y)2^pXj^r(f).
In the integral above we may replace L(xj)R(y)f(k~1g) by
£ £aL(xj)i?(};)/(k-10).
<reS(y)
If we apply (iv) and (*) above we have
(v) Px,yAEJ)<Cxd(y)d(a)2 £ H^ + pk\\-" ^pDxuyAf\
<JeS(y) i

230
7. Cusp Forms on G
The Weyl dimension formula implies that d(y) < C\\ny + pk\\m with m =
|<t+|. Thus if \\ny + Pk\\ > Dj/2Cj then
(Vi) px,,AErf) ^ C*IK + ftir" + m I PD*t,,Af)-
i
This combined with (ii) easily implies that Z,yEyf converges in £f. The
argument in 1.4.7 easily implies that the above series converges pointwise to /.
This completes the proof of the theorem.
7.1.2. We now give two examples which will be most important to later
developments. We take (a, F) to be a finite dimensional representation of G
with compact kernel. We put an inner product on F, < , >, that is K-
invariant. Set \\g\\ = tr a{g)a(g)* + tr a(g'x)a{g-x)*. Then \\g\\ > 1. We
take a(g) = \\g\\ and b(g) = 1. Then the material in 4.A.1 implies that a, b
satisfy (l)-(6) in the previous number. With this choice, we denote the space
X.b(G) by £f(G). We call Sf(G) the space of rapidly decreasing functions on G.
Notice that we may use any norm (4.A.1) on G to define £f(G).
The next example is due to Harish-Chandra. Following his usage we will
call it the Schwartz space of G and denote it by ^(G). Let 3 be as in 4.5.3.
We set b(g) = E(g) and a(g) = 1 + log ||.9||. Then (1), (2), (4) of the previous
number are clear. We leave (5) to the reader. (3) follows from Theorem 4.5.3
and 5.A.3.I. To prove (6) we note that if x e G is fixed and if
K(9) = |3„(xfc0)dfc
K
(here we are using the notation of 3.6) then h^ is X-bi-invariant and if
xe[/(g)K then xhfl = n(y0(x))hfl. Thus the material in 3.6 implies that
hp = c 3^. Clearly
c = hAl) = mi(xk)dk = Zll(x).
K
If we recall that 3 = 30 then (6) now follows with equality.
The observation about arbitrary norms on G applies in this case also. The
rest of this chapter will be devoted to analysis on this space,
7.2. The Harish-Chandra transform
7.2.1. Let G be a real reductive group such that G = °G. Fix a maximal
compact subgroup K of G and let 6 be the corresponding Cartan involution.
Let 3 and ||---|| be as in 7.1.2. Let for a = 1 + log ||-|| and b = 3, pfljbjXjy,r =
ax,y,r (we use tne notation of the previous section). Let ^(G) be the
corresponding Frechet algebra of functions on G (7.1.2).

7.2. The Harish-Chandra Transform
231
Fix (P0,A0) a minimal parabolic pair for G. Let (P, A) be a corresponding
standard p-pair with P = °MAN a standard Langlands decomposition (see
2.2). Let p = pP. If / e ^(G) and if m e M, m = a°m, ae/1 and me°M then
we set
Theorem.
(1) If f e ^(G) then the integral defining f converges absolutely.
(2) If fe <${G) then f e #(M) and t/ie map f -> f is continuous from <%(G)
to <£(M).
By the definition of <#(G),
l/(»)l<ffi.i.r(/)S(»)(l+logllalir
for all r > 0. 5.3.4(2) implies that
3(xy)= $ap(kx~1)a»(ky)dk.
K
If Q is compact in G then a(fcy) < CSJ for y e Q and ke K. Thus S(xy) <
CnS(x) for y e Q, x e G, Also (1 + log \\xy\\) < C'a{\ + log ||x||) under the
same circumstances. With this in mind, we see that Theorem 4.5.6 implies (1).
We also note that if we use the above argument with the seminorms alxr,
x e U(m) then it is easy to see that f e C"°(M). We note that
(i) R(x)f = (R(x)f)p forxe[/(m),
(ii) L(x)f = (L(x)f)p for x e I/(°m).
(iii) If h e U}(a) and if hu..., hd is a basis of UJ(a)
then
L(h)fp = ^ak(h)(L(hk)f)p
with ak linear in h.
Thus to prove (2) it is enough to prove that there exists k > 0 and for each
d > 0 there exists Cd such that if / e C(G) then
(*) |/»| < CdEF(m)(\ + log IMirV ,.„+*(/).
Here P = PF and HF is as in 4.5,5.
Let h be a Cartan subalgebra of m. Let <t+ be a system of positive roots
for <t(gc,hc) such that n is contained in the sum of the root spaces for <t+.
Let 8 be the half sum of the elements of <t+. Let (ji, F) be the irreducible finite
dimensional g-module with highest weight 8. Let Gl be a finite covering group

232
7. Cusp Forms on G
of G such that y. is the differential of a representation of Gx. Fix an inner
product on F such that the compact form of Gx acts unitarily on F. If T is
an endomorphism of F then denote by ||T|| the Hilbert-Schmidt norm of T.
Let \\g\\ = \\n{g)\\. Then ||---|| is defined on G, 4.5.3 implies,
(iv) There exist positive constants CuC2,d1, d2 such that
cjlair'O + log Hgii)-'' < S(0) < cjfliro + log ||g||)^.
Let Mi be the subgroup of Gx corresponding to M. Then as a representation
of Mu F = F, + ••• + Fr, a direct sum of irreducible M, modules and F,
has highest weight 8. Relative to this decomposition of F we may also assume
that n(n) for n e N has the block form
0
/_
= / + y(n) with y(ri) in the above upper triangular block form with zeros on
the main diagonal.
Thus jiinm) = (I + y(n))n(m). So
||wn||2 = \\ii{m)\\2 + ||y(fi)/i(»n)ll2.
This implies that ||nm||2 > ||m||2. On the other hand, ||n|| = ||nmm-1!| <
||nm||||m-1||. Since ||m|| = ||0(m)|| = ||m-1|l, this implies that \\nm\\ >
Unllllmir'.Wehave
(v) ||nm||2>||n|| and \\nm\\ > \\m\\.
LetneN,ae A and m e °M, Then \\nam\\ = ||0(n)a_10(m)||. Let °A+ be the
"/1 + " for °M relative to P0n°M, A0n°M. Then 0(m) = kxaxk2 with kh
k2e Kn°M and ax e C\(°A + ). Let v0 be a unit highest weight vector for (n, F).
Then \\nam\\* = \\e(n)a~iklalk2\\1 = \\B(n)klar1al\\1. Set « = fl^'nfc,).
Then ||nam||2 > HixT'aJI2 > ||/iO>)Ma-1ai)«>oll2 = <r2'a21l/i(«>)«>oll2- Now
iT1 = n(v~l)a{V-l)k{v~l). Thus HM^oH2 = aCt;-1)-2". If we put all of these
inequalities together with (it;) we have (P = PF)
Z(nam) < apSF(m)(l + log ||m||)d'a(t;"1)''.
Hence
a~"E(nam){l + log \\nm\\y2d'
< 3F(m)(l + log IMir^'adT'Xl + log |M|)-*.

7.2. The Harish-Chandra Transform
233
Let d3 be so large that (Theorem 4.5.4, Lemma 4.A.2.3)
J fl(x)"(l +log||x||rd3dx< oo.
»(N)
We also note that there exists r > 0, C > 0 such that
l|ma|r>C||a||.
Since m is fixed, we have
a~" J H(nam)(l + log ||man||r2d+''',dn
< SF(m)(l + log ||m||rd(l + log Hall)-*.
This implies (*) above. The theorem now follows.
7.2.2. We say that / e <£(G) is a cusp form if (L(x)R(y)f)p = 0 for all proper
parabolic subgroups P (2.2,7) of G and all x,yeG. One of the key points in
the theory of Harish-Chandra is that the space of cusp forms on G is the
closure in <^(G) of the X-finite matrix coefficients of the discrete series. The
following result is a key step in this direction.
Theorem. Let f e ^(G) and assume that dim ZG(Q)f < oo. Then f is a cusp
form on G.
It is enough to prove that f{\) = 0 for all proper parabolic subgroups of
G if dim ZG(g)/ < oo. If P is a parabolic subgroup of G and N is the
nil-radical of P then N is contained in G°. Thus, since Z(g) is a finitely
generated ZG(g) module, we may assume that G is connected.
If y e XA then define
Erf(g) = d{y)lxr(k)f(k-1g)dk
K
for g e G. Then Eyf e ^(G) is left K-finite and Z £y/ converges to / in ^(G).
Thus the previous result implies that we may assume that / is left X-finite.
Thus L(U(Qc))f = V is a finitely generated admissible (g, X)-module (3,4,7).
Let P be a proper parabolic subgroup of G. Let P = °MAN be a standard
Langlands decomposition (2.2.7) of P. Set T(h) = hp for /i e K. Then it is easy
to see that T(nV) = 0 and that T is an (m, K n M)-module homomorphism
from (K/nK)®C_p into <#(M). Thus, since V/nV is admissible and finitely
generated as a (m,X n M)-module, dim U(a)T(h) < oo for all deK In
particular, this implies that T(f)(a) = Z a"p„(log a) a finite sum with fi e a J
and p„ a polynomial on a (8.A.2.10), The previous theorem implies that if we

234
7. Cusp Forms on G
set T(/)(exp H) = fi(H) for H e a then /? is rapidly decreasing on a. Now
Lemma 4.A.1.2 applied to P(tH), teR (lim,^P(tH) = 0) implies that
T(f)(a) = 0 for a e /I. Thus, in particular, /p(l) = 0.
7.3. Orbital integrals on a reductive Lie algebra
7.3.1. We retain the notation and assumptions of the previous section. Let
B denote an Ad(G)-invariant, symmetric, nondegenerate bilinear form on g
such that if <X, Y} = — B(8X, Y) then < , > is symmetric and positive
definite and B restricted to [g,g] is the Killing form of [g,g]. If X e g then
set \\X\\ = <X, X)l'2. If x e G then denote by ||x|| the Hilbert-Schmidt norm
of Ad(x). Then ||- • -|| is a norm in our sense.
We assume that f contains a maximal abelian subalgebra t that is a Cartan
subalgebra of g. The results in this section are due to Harish-Chandra (some
of the proofs differ from the originals).
Lemma. There exist non-negative integers p, q and a positive constant C such
that
||x|||(det(ad Y\p)\" < C\\Ad(x)Y + 6(Ad(x)Y\\-
for all x e G, Ye f. Here p is (as usual) the — 1 eigenspace for 6.
Let P0 be a minimal parabolic subgroup of G with 8 standard Langlands
decomposition °MAN. If a e <b(P0,A) then let na denote the corresponding
a-rootspace. Let a+ = {H e a | a(H) > 0 for a e <b(P0,A)} (as usual). Then
G = K Cl(exp a+)K. If g = klak2 with a = exp H, H e Cl(a+), kl,k2eK then
(1) Hgfll2 = dim m + 2 X dim na cosh 2a(H).
Let <*!,...,a,, be the simple roots in <&(P0,A). Let W,,...,Wdeabe defined
by cx.j(Hk) = 8jk. If a e <t(P0>-4) then a = Z m/a)^ with m/a) a non-negative
integer. Set r = max{m;(a) 11 < j < d, a e ^(/o* ^)}- Then (1) implies
(2) There exists a positive constant C such that
||0||<cfncosha/H)Y
for g = /q exp Hfc2, kl, k2€ K, H e Cl(a+).
(3) If X e I and if n is an eigenvalue of ad X on p then ||X|| > |^|.
This is clear.

7.3. Orbital Integrals on a Reductive Lie Algebra 235
(4) If X e I, x e G then
||Ad(x)X + 0(Ad(x)X||2 = 2||Ad(x)X||2 + 2||X||2.
Indeed,
||Ad(x)X + 6 Ad(x)X||2 = <Ad(x)X, Ad(x)X>
+ 2<Ad(x)X,0 Ad(x)X> + <0 Ad(x)X,0 Ad(x)X> = 2||Ad(x)X||2
- 2B(Ad(x)X, Ad(x)X> = 2||Ad(x)X||2 + 2||X||2.
As asserted.
(5) If a = exp H with H e Cl(a+) and if X e f then
||Ad(a)X + 6 Ad(a)X||2 > r-2||[Hj,X]||2(cosh a/H))2
for all j = 1,..., d.
Indeed, X = Z + Za (X, + 0XJ with ZemJ.en,. Thus (4) implies that
||Ad(a)X + 6 Ad(a)X||2 > 4(cosh a,.(H))2||XJ|2.
Clearly, ||Xaj||2>r-2||[//,,X]||2.
(5) implies
(6) ||Ad(a)X + 6 Ad(a)X||2d > C f\ (|| [//,-, X]||2(cosh a/H))2)
for H, X as above.
Let y(X) be the smallest absolute value of an eigenvalue of ad X |p for X e f.
Then ||ad(X)H;|| > y(X)\\Hj\\. Let s = dim p. (6) implies that
||Ad(a)X + 0Ad(a)X||2d> C n I|[H,,X]||2 cosh a/H)
;= l
>C||a||1"-ni|[//,,X]||2
i = i
> Cl7(X)2d||a||
i = i
\2d\\n\\llr
with C, > 0 (here we have used (2)).
Let jx1,...,jxs be the eigenvalues of adX|p, counting multiplicity,
with |^|=y(X). (3) and (4) imply that |^,| < ||Ad(a)X + d Ad(a)X||.
|det(adX|p)| = |/v"/Uso
||Ad(a)X + 6 Ad^XH2^2*-" > C1||det(ad(X)|p)||2d||a||1/r.
This implies that if p = 2 dr and q = 2rds then
||a||||det(ad(X)|p)||p < C||Ad(«)X + 6 Ad(a)X||".

236
7. Cusp Forms on G
If x e G then x = k1ak2 with ku k2eK and aeC\(A + ). \\x\\ = \\a\\,
|det(ad(Ad(fc2)AT)|p)| = |det(ad(X)|„)| and HAdf*,)^ = \\Y\\ for YeQ. The
Lemma now follows.
7.3.2. We set f" = {X e f det(ad(X)|p) # 0}. Then our assumption on t
implies that f" is non-empty. Let Yj,..., Yk be a basis for f. Let yu...,ykbe
the corresponding coordinates on f. Let Xx,..., Xn be a basis of g relative to
< , >. Let x!,...,x„ be the corresponding coordinates on g. We will use
standard multi-index notation for higher derivatives in the y and x coordinates
(see 5.A.2.1). Set for r e R, m e N and / e Cco(g)
q,m(f)= I suP;fe9 ||x + 0xiriai"/axy(X)|.
|/|<m
Proposition. Let f e C00 (g) be such that qr<m(f) < oo for all r, m then the
integral
f, f(Ad(g)Y) dg
G
converges absolutely for Y e f" and defines a smooth function, g(Y), of Y e f".
Furthermore, there exist constants a and ft such that (p, q are as above)
\\Y\\r\@'W)9(Y)\ < C/>r|det(ad Y\p)\-'qVmW(f)
with u = p\I\ + ft, v = r + q\I\ + a and w = \I\.
\(d^/dy,)f(Ad(g)Y)\ = \(Ad(g)d^/dy,)f(Ad(g)Y)\.
Ad(gp'W= I UjM$WI?>xJ)
\j\ = \i\
with
K/(0)l < Q\gfK
Thus
\(^l^y1)f(Ad(g)Y)\ < C\\g\\M X \(^IZxJ)f(Ad(g)Y)\.
The previous result says that
(*) \\g\\ < C|det(ad Y \p)\-"\\Ad(g)Y + 6 Ad(g)Y\\".
Hence
0'W)f(M(g)Y)\
<C|det(ad r|p)r"l/l||Ad(0r)y+0Ad(0r)^ll"|/| I \(d^/dxJ)f(Ad(g)Y)\.

7.3. Orbital Integrals on a Reductive Lie Algebra
237
If we use (*) again and (4) from the previous number we have
(**) II1TK31 W)/(Ad(0)r)| < C|det(ad Y|p)rp|/|^MrN,,|/|+,,d,|/|(A
with d0 such that
$\\g\\-d°dg< co.
G
If we integrate both sides of (**) the result follows.
7.3.3. We set t" = t n I". We fix a system of positive roots <t+ in <t(gc, tc).
Set n = n.,g(I)+ a. Let <S„ = {a e <t(gc,tc) |(gc)a is contained in pc}. Set n„
equal to the product of the aeO+nOn. If Het then |det(ad(H)|p)| =
\n„(H)\2. Let T be the Cartan subgroup of G corresponding to t.
Let ,y(g) denote the (usual) Schwartz space of g. That is, the space of all
/ e C°°(g) such that
UrAf) = sup*e9 ||AT I \QW/Qx'f(X)\ < oo,
|/|<s
endowed with the topology given by the seminorms n,tS.
Notice that the seminorms qrs are continuous on £^{q).
If / e y(g) and if H e t" then we set
*;(ff) = jr(ff)J/(Ad(0)ff)d0.
G
Then <Dj e C°°(t").
Let Hu...,Hr be an orthonormal basis of t and let tt,..., tr be the
corresponding coordinates on t.
Lemma. There exists a constant, u, such that if I is an r-multi-index then there
is a continuous seminorm n, on £f(Q) such that
\dW/dt'9}{H)\ < \n„(H)\-"n,(f)
for f e //(g), H e t".
We use the notation of 7.A.2.6 except that the h in the appendix is now t.
If pe%) let p be as in 7.A.2.9, Let / = {p',pe S(qc)g}. Then S(tc) is
finitely generated as an /-module (see the proof of 3.2.4). Let p} e S(tc),
j = 1,..., d, be such that S(tc) = X Ipj.
Let t' = {// e 11 n(H) # 0}. Then t' is contained in t". Let Het' and let W be
an open neighborhood of H in t'. Set U = \d(G)W. If fe ^(g)then we define
g(X) = lf(Ad(g)X)dg
G

238
7. Cusp Forms on G
for X e U, Then the preceding results imply that g e C°°(l/). Clearly,
g(Ad(x)Y) = g(Y) for Y e U, x e G. Hence Theorem 7.A.2.9 implies that
pg\w = n~lpng\w for p e S(qc)g. This implies that
(1) P<b}(H) = <&Tpf(H) for/e,r(g)and//Et',
Let H\'---H'rr (=3|/|/3t/ as a differential operator) equal Z UjPj with
u,.ES(gc)G.Then
|Pj.*J(f/)| < C\nn(H)rnj(f)
with /i- a continuous semi-norm on £f{Q) and ^ depends only on the degree
of Pj. If we make a suitable choice of n} we can replace r-} with r, the
maximum of the ry Thus
|p/I>J(H)|<C|7r„(H)r ;>>,■(/)
for H e t". Hence
ia""/at'*j(H)l < c|7r„(//)|-x^(";/)
for // e t'. Since both sides of the above inequality are continuous on t", the
result follows.
7.3.4. If U is an open subset of t then we define if {I]) to be the space of all
/eC°°(l/) such that
nv,Af) = s»Pxsv \\x\\r I |ai'i/at'/(X)| < «.
|/|<s
It is easy to see that £f(U) endowed with the topology given by the semi-
norms nVt,tS is a Frechet space. We are now ready to state (and prove) the
following basic result of Harish-Chandra.
Theorem. If f € 9'(q) then <tj e Sf(t"). Furthermore the map f i—*-<I>J from
£f(Q) to Sf(t") is continuous.
Let C be a connected component of t". We note that if H e it then
a(H)sR for aeO = 0(gc,tc). Thus if a e <t„ then ia is either strictly
positive or strictly negative on C. We define |a|c to be ia in the first case
and otherwise to be —ia. Thus \n„(H)\ = Y\\a\c(H) (the product over
a e <t+ n <t„ for He C. Fix x0 e C. Let x e C1(C). Then |a|c(x + tx0) =
|a|c(x) + t|a|c(x0) > t|a|c(x0) if t > 0. Let / e £^(q). The preceding result

7.3. Orbital Integrals on a Reductive Lie Algebra
239
now implies that if F = <&f and if q = \<&+ n <t„| then
\PF(x + tx0)\ < r""fi„(f)
for t > 0 and p e if{i). Here \ip is a continuous seminorm on ^(g).
Now dk/dtkpF{x + tx0) = {x0)kpF(x + tx0). (Don't forget that we are
identifying S(t) with the constant coefficient differential operators on t,) This
implies that if we set u(t) = pF(x + tx0) then u(k)(t) = (x0)kpF(x + tx0).
Hence
(1) u*\t) < r"V*p(/)Wxo)r.
Scholium. Let u e (^((0,1]) and suppose that
|««*»(t)| < rmak
for 0 < t < 1 and k = 0, 1, 2,.,,. TTien
|u(t)l< C(a0 + ---am+1)
/or 0 < t < 1. Here C depends only on m.
We may assume that m > 1, If m > 1 then since,
utk)(t) = utk)(\)-\utk+1)(s)ds,
we see that \u{k){t)\ < ak + lrm + 1/(m- 1) + a^k + l + ak for 0 < t < 1. Hence
we have
(2) \utk)(t)\<2rm+l(ak + l + ak) for0<£<l.
If we argue as above using (2) we find that
(3) \u(k)(t)\<2m~lrl(ak + --ak + m) for0<t<l.
If we apply (3) to the case k = 1 we find that
|«,1)(t)|<2mlog0Va1+--- + am + 1)
for 0 < t < 1. If we integrate this inequality we get the estimate asserted in
the Lemma for u(0) = u.
We now complete the proof of the theorem. The Scholium combined

240
7. Cusp Forms on G
with (1) above implies that if X e C then
uq+ 1
|p<»J(X)| < £|7r„(x0)P I /Mp(A
with E a constant independent of f. We also note that if p(X) = — B(X, X)
then p(X) = \\X\\2 for let. This implies that <b]^f(X) = \\X\\2k®Tf(X). The
Theorem now follows from 7.3.2.
7.3.5. We now study the analogous integrals for other Cartan subalgebras.
We will be constantly referring to material in Chapter 2. Let (P0,A0)
be a minimal p-pair for G. Let h be a Cartan subalgebra of g. Then
Proposition 2.3.6 implies that there exists a standard, cuspidal p-pair,
(PF,AF) and x e G° such that h = Ad(x)hF (see 2.3.6 for the terminology).
Let H (resp. HF) be the Cartan subgroup of G corresponding to h (resp. hF).
By definition, hF = tF + aF where tF is maximal abelian in °m n f. Let TF
be the Cartan subgroup of °MF corresponding to tF. Then it is easy to
see that
(1) HF= TFAF = xHx~l.
7.3.6. On HF we take the invariant measure dtF daF where dtF is normalized
invariant measure on TF and daF is Lebesgue measure corresponding to an
orthonormal basis of aF, On H we take the pull-back measure corresponding
to h *—► xhx~'. It is easily seen that this measure is independent of the choices
made in its definition. We fix an invariant measure on G and take the quotient
measure, dgH, on G/H.
Let <t+ be a system of positive roots for ^(qc^c)- Let n denote the
product of the elements of <t+. Let <t^ be the set of all real roots in <t+ (2.3.5).
Set h' = {h e h \ a(/i) # 0 for a e <»}. If h e h' then set e(h) = sgn(naa(/i)) the
product taken over <!>£. If / e ^(g) we will be using the following notation:
(1) <b?(h) = e(h)n(h) | f(Ad(g)h)d(gH)
G/H
with the domain of <tH equal to the set of all h e h' for which the integral
converges absolutely.
We note that <t" depends on the choice of <t+ but only up to a sign. If
in <t(gc,(hF)c) we choose the positive roots to be {a o Ad(x)"11 ae<S+).
Then
(2) <tH(/i) = <t? F(Ad(x)h).
Thus we loose no real generality in studying these integrals if we assume

7.3. Orbital Integrals on a Reductive Lie Algebra
241
(as we do) that H = HF. On G/AF we take the quotient measure corresponding
to our choice of invariant measure on AF. Then it is clear that
(3) <b»(h) = e(h)n(h) | f(Ad(g)h)d(gAF).
Now Lemma 2.4.1 implies that the invariant measures on K, °MF, NF can be
normalized so that
(4) <bHf(h) = e(h)n(h) J f(Ad(kmn)h)dkdmdn.
We now begin our analysis of this formula.
7.3.7. Let h e h. If n e NF then we set Th(n) = Ad(n)h - h. If n e NF then
n = exp X with X enF, If we expand the exponential series for Ad(n) = eadX
it is easy to see that Th(n) e nF.
The obvious calculation gives
{dTh)n{X) = Ad{n)[_X,W\
for n e NF, X e nF. This easily implies
(1) If det(ad(/j)|n) is non-zero then Th is everywhere regular.
Lemma. // det(ad(/i) |n) is non-zero then Th is a diffeomorphism of NF onto nF.
Furthermore there is a choice of Lebesgue measure on nF such that
det(ad(J!)|n) J f(Ad(n)h -h)dn=\ f(X)dX
Nf nr
for (say) f a rapidly decreasing function on nF.
If we show that Th is a diffeomorphism of NF onto nF then the integration
formula will follow from the above formula for the differential of Th.
Let li0eaf be such that a(/i0) > 0 for ae <&(PF,AF). Set a, = exp(th0).
The obvious calculation shows that
Th(atna-,) = Ad(at)Th(n).
Since Th is, in particular, regular at 1, there is an open neighborhood of 1
in NF and a neighborhood U0 of 0 in nF such that Th is a diffeomorphism
of C/, onto U0. Now (Jr>0 Ad(ar)[/0 = nF. Thus the above equivariance
implies that Th is surjective. Suppose that Th(nx) = Th(n2). Let t be such that
a^ja-tSUi for j = 1,2. Then Th(atnla_t) = Ad(at)Th(nl) = Ad(at)Th(n2) =
Th(atn2a^t). Thus atnla_t = atn2a_t. Hence, nx=n2. This completes the
proof of the Lemma.

242 7. Cusp Forms on G
7.3.8. We now choose <t+ such that if ae$+ and if a|0 is non-zero then
a|a e <b(PF,AF). Let X be the set of all ae$+ whose restriction to aF is
non-zero. If h e h then |det(ad h\„)\ = \YlxeE a.(h)\. It is an easy matter to
see that
(1) |det(ad h\„)\ = e(h) ]J a(h).
This combined with the previous Lemma implies that
(2) <bHf{h) = CFe{h) n «C«) I f{Ad{k)Ad(m)(h + X))dkdmdn.
If / is a smooth function on g then we set
f{X)=\f{Ad(k)X)dk for X sg.
K
Since Ad(°M) preserves dX on nF we have (in the above notation)
(3) <t«(/,) = CF [] a(h) J f(Ad(m)h + X)dmdX.
«£«*-!; 0Mxn
If / e Sf(Q) and Q = PF then we set for Z emF
(4) fQ)(Z)= $f(Z + X)dX.
If / e 5^(g) and if h e h then write h = /i_ + h+, h_ e aF and h+ e tF then
set «(Z) = u(f,h-)(Z) = f(Q)(h_tZ) for Z e °mF. We have proved:
(5) <b«{h) = CF<S>l(K)
Let <tF„ = {a e <I>((mF)c,hc)!((mF)c),j e pc}. The above calculations imply
the following result of Harish-Chandra
Theorem.
(1) The integral defining Q>" for f e £^(q) converges absolutely for h e h'.
(2) Set h" = {h e h a(h) # 0, a e <DFi„}. // / e y(g) t/ien <t? extends to a
smooth function on h". Furthermore, if f e ,y(g) then 0" e ^(h") and the map
of £f{Q) to £f{r>") given by f t—► Q>f is continuous.
This follows from the above material and Theorem 7.3.4.
7.3.9. If X e g then set det(ad X - tl) = Z t jZ)/X). Here n = dim g. Let
D = Dr(X) with r = dim h. The preceding theorem has the following
corollary which will be important in the next chapter.
Corollary. \D\ 1/2 is locally integrable on g.

7.4. Orbital Integrals on a Reductive Lie Group
243
We use the notation of 2.4.3. Then 2.4.3 says that
\f(X)dX = X cj I \nj(h)\2 I f(Ad(g)h)d(gH,)dhj
a b, ain,
in the sense that the right side converges if the left side does. Now \D(h)\ =
\nj(h)\2 for he\)}.
Let / e C™{q) be non-negative. Then the preceding theorem implies that
<bf e S((h;)) for j=\,...,r. Thus
oo > I c, I |*,(*)|<tf! = I c,- J \nj(h)\ J f(Ad(g)h)dgHjdhj
bj b' G/Hj
= l0ll^)l2 I |/>(Ad(g)fc)|-1'2/(Ad(g)fc)dgHydfc>
b; g/h,
= ||D(X)|-1'2/(X)dX.
a
Since / is an arbitrary smooth, compactly supported, non-negative function
on g the corollary now follows.
7.4. Orbital integrals on a reductive Lie group
7.4.1. We retain the notation of the previous section. We also assume that G
is of inner type. Let K" = {k e K |det((7 - Ad(fc)|p) # 0}. If h is a Cartan
subalgebra of g let H denote the corresponding Cartan subgroup. Let H' be
(as usual) the set of all h e H such that det((7 — Ad(/i))|bi) is non-zero. Put
GlH'-\ = {yhg-l\heH',geG}.
Then G[H'] is an open subset of G (see the proof of 2.4.4). The proof of 2.4.4
yields
I /(0)d0 = (-)f |det((/-Ad(JO|„i)| J f(ghg~l)dgHdh.
G[H] \WJ H G/H
Here dg is a fixed choice of invariant measure on G, we fix an invariant
measure on H and we take dgH to be the quotient (G-invariant) measure
on G/H. Also w is the order of the finite group N(H)/H where N(H) =
{geG\gHgl=H}.
7.4.2. We now assume the f contains a Cartan subalgebra t of g (under this
assumption K" is not empty). The displayed formula in the previous number
easily implies (apply it to both K and G)

244 7. Cusp Forms on 6
Lemma. There exists a positive constant, c, such that
J f(g)dg = c J |det((7 - Ad(fc))|p)| J f{gkg~')dgdk.
G[K"] K G
7.4.3. If e > 0 then we set
Ge,£={^_1!|det((Ad(t)-/)|t-)l>e},
Kt = {k e K j |det((Ad(fc) - 7)|p)| > e} and
(G")„ = G[KJ.
Fix a norm (2.A.2.1), ||---|| on G, which we assume is given as the operator
norm corresponding to a representation (n, F) of G on a finite dimensional
Hilbert space F. We also assume that n(g)* = n(6g)~l and that det n(g) = 1
for g e G.
Lemma. Let 0 < e < 1. Then
I (1 + log \\g\\)-iS(g)dg < QT*2 J y(a)(l + log \\a\\rdE(a)2 da.
(G")„.c A*
Here y is as in 2.4.2.
2.4.2 says that, up to constants of normalization, if / is integrable on G then
(1) \f(g)dg= J f(klak2)y(a)dkldadk2.
G S<A* x K
This implies that
J {\+\og\\g\\ydZ(g)dg
(G")..«
= J|det((Ad(fc)-/)|p)| J (1 + \og\\au-lkua~l\\ydE(au-lkual)dudadk
Kc K*A +
= J|det((/-Ad(fc))|p)| J (\+\og\\aka-l\\)-dE(aka-l)dadk.
Kc Ay
If X e p then n(X) is self-adjoint. If ae A then a = exp H with H e a. So
||flkfl"'|| = Hflkfl-'k-'ll = ||exp Hexp(-Ad(k)H)|| = ||e*<»>e-*A<««H||
-> el|n(//-Ad(k)H)ll/(dimF- 1) H\\ \\
Thus we see that
log Hflkfl-'ll ^ \\n(H - Ad(fc)H)||/(dim(F - 1).
Let for k e K, n(k) denote the minimum of the absolute values of the

7.4. Orbital Integrals on a Reductive Lie Group 245
eigenvalues of (Ad(fc) — 7)|p. Then we have shown
(2) There exists a positive constant C such that
logllafca-'II^QiWlogllall.
Let Hi,...,n2q be the eigenvalues of (7 — Ad(fc))|p counting multiplicity. If
we assume that k e K" then we may relabel so that |/^-| = ji(k) for j = 1, 2. It is
clear that |^;| < 2 for all i. Hence if k e Kt then
e<|/i,--/i2,l^/i(fc)222'-2.
Set C = 2~q +'. Then we have proved
(3) If fcEX£then/i(fc)>Ce1/2.
This combined with (2) and the calculations already done implies that
J (I + log \\g\\rdZ(g)dg
(G").,e
< Ce-"12 J |det((Ad(fc) - /)|p)| J y(a)(l + log \\a\\ydE(aka~l)dadk
K A
< CC'e-"12 J y(fl)(l + log \\a\\rd J E(aka~l)dkda.
A* K
Now
J E(akal)dk = E(a)E(a-1) = E(a)2
K
(see the discussion in 7.1.2).
If we put this together with the preceding inequalities the lemma follows.
7.4.4. If / e C(G) then we write
(1) f{g)=\f(kgkl)dk.
K
If / e C?(G) then we set
(2) Qf(k) = $f(gkg-1)dg
G
with domain the set of ke K for which the integral converges absolutely.
Lemma. If f € C™(G) then the domain of Qf contains K". Furthermore,
Qf e C-(X").
Set h{g) = \f(g)\. Then jh(gtg1)dg = J y(a)h(autu~xa-x)dadu.

246
7. Cusp Forms on G
The argument used to prove 7.4.3(2) proves
(3) If k e Ke and if as A then log \aka'l\ > e1/2C log ||a||.
This implies that if u is a compactly supported function on G and if
supp u is contained in {g e G\ log \\g\\ < r) then there is a constant C > 0
independent of u such that u(auku~la~l) = 0 for keKt, ueK and
log ||«|| > Ce~ll2r. Thus the integral converges for k e K".
LetXl5..., X„ be a basis of g. If Y e f then Ad(g)Y= X c/gOXywitheachcja
matrix coefficient of (Ad, g) hence
dk/dtk\t = 0f(gk exp tttT1) = (Ad(0)7)y(^^)
= {lc1(g)XJ)kf(gkg-1).
There exist constants £> > 0 and u > 0 such that |cj(g)| < £>||gf||". Hence
\dk/dtk\t=0f(gk exp tYj?-1)! < CIMI*" £ |«/(^-')|,
where u^ is a basis of Uk(qc).
This and the argument we used to prove the first assertion of the present
Lemma implies the second.
7.4.5. Fix, <t+, a system of positive roots for <t(gc, tc). We assume that p is
T-integral (this is always possible by going to a covering group of G). Set W =
W(9c,tc)then
A(t) = t" n (1 - r") = X det(s)£sp
for t e T. Set T" = T n K". If / e q°(G) then we write
(1) F}(t) = A(t)lf(gtg-1)dg for t e T".
G
The previous Lemma implies that FJeCco(T"). Set Tt = Ge<tnT =
{t e T|| A(t)|2 > e}. Clearly, T£ is a subset of T".
Lemma. Let d he suc/i t/iat
J (l+log||a||)-dH(a)27(a)da<a)
(see 4.5.3). T/ien t/iere exist positive constants C and u such that if f e C™(G)
then
j\FTf(t)\dt<Ce^aUUd(f)
(see 7.2.1 for a d).

7.4. Orbital Integrals on a Reductive Lie Group 247
Settr = aUUd.
\\F}{t)\dt< lm)\l\f(gtg-l)\dgdt.
Tc Tc G
Now, |A(t)| > Ce1/2 for t e Tt. Thus
| \FTf(t)\dt<Ce-1'2 J \A(t)\2l\f(gtg-l)\dgdt
Te Tc G
<C,~l'2a(f) I |A(t)|2f (1 +log||^-1||r,,S(^-1)Adg
TE G
= CE~l'2a(f) J (\+log \\g\\yZ(g)dg
<Cda(f)£-(d + l)l2 J 7(<i)(l+ log \\a\\ydZ(a)2 da
A +
by Lemma 7.4,3. If we take u = (d+ l)/2 the Lemma now follows.
7.4.6. If one argues as we did to prove 7.3.3(1) (using 7.A.3.7) one proves
(1) Flf(t) = y(z)F}(t) for / e Cf(G), t e T and z e Z(g).
We will now use the notation of 7.A.4.2. We label the elements of <t+ as
a,,..., a„. Then the set T in 7.A.4.2 is our T. Let B(T') be as in that number.
We can now prove the following basic theorem of Harish-Chandra.
Theorem.
(1) If feC?{G)thenFTfeB(T).
(2) The map f \-> Fj of C™(G) into B(T') extends to a continuous map of 11(G)
into B(T'\
Let V = <€(G) and W = C?(G), S(w) = Fw, A = Z(gc), 7 = 7 and a = Ca.
Lemma 7.A.4.2 implies that S(f)e B(T) for all / e C?(G) and S extends
to a continuous map of ^(G) into B(T). The result now follows.
Note: The original proof of Harish-Chandra used the theory of the
characters of the discrete series (see Varadarajan [1, Part II] for a nice
treatment of Harish-Chandra's original proof). The key new ingredient in our
proof is Lemma 7.4.3. We note that the above theorem is stated in G. Warner
[II, 8.5.6] but the proof therein is based on a result on discrete series characters
that is deferred to "Volume 3".
7.4.7. The next result will be used in Section 7.7.

248
7. Cusp Forms on G
Theorem. Let [ibe a continuous seminorm on B(T'). Then there exists d>0
and a continuous seminorm a on ^(G) such that
I H(FIwrWk < (1 + log \\g\\)d~.(g)a(f)
K
for f e <£(G) and g e G.
If we argue as in the proof of the previous theorem it is enough to show that
there exists a continuous seminorm a on ^(G) and q, d> 0 such that if
/ e <€{G) then
(*) I I \FTmg)f(t)\dtdk < c-«ff(/)(l + log \\g\\)dE(g) for each e > 0.
K TL
Now, (*) <
J J \A(t)\$\f(xtx-lkg)\dxdtdk
K Tc G
<<j(f)z~ll2l J |A(f)|2(l +\og\\xtxlkg\\)-dS{xtx-lkg)dxdtdK
with a = alA d.
We note that
iMi>iwii)'-,ir, = ii)'ir,iwi.
Thus log ||xy|| + log \\y\\ > log ||x||. Hence
(l+log||x);||)(l+log||);||)>l+log||x||.
We therefore conclude that the expression in (*) is less than or equal to
e-1/2(r(/)(l+log|M|)d J |A(f)|2f(l + log||xfx-1||)"dH(xfx-,fc9)^^^
Kr, g
<e-,/2(l+log||0|t)'3(0)ff(/) J (l + logHxID-^x)^.
Ge.c
The assertion now follows from Lemma 7.4.3.
7.4.8. As in the last section we now study the analogous results for general
Cartan subgroups. We assume (for simplicity) that for each Cartan subgroup
of G the corresponding p is integral. Let I) be a Cartan subalgebra of q and
let H be the corresponding Cartan subgroup. Set a(h) = a(/j) for a e <&(qc, hc),
h e h (here X is the complex conjugate of X e gr relative to g). Let <t+ be a
system of positive roots for <t(gr, hc) such that if a e <t+ and if a # —a then

7.4. Orbital Integrals on a Reductive Lie Group
aE<t+.Set Z= {aE<t+ia # -a}. We set
249
no- *~")
ae I
Clearly, AH(h) = ±A(h) for he H (see 7.A.3.6 for A).
If / e #(G) then set
(1) FHf(h) = AH(h) J f(ghgl)dgH.
G/H
The measure on G/H is chosen as in 7.3 and the domain is the set of all h e H
for which the integral converges absolutely. As in 7.3.6 we assume that H =
HF = TFAF with (PF,AF) as standard cuspidal p-pair. We also assume that
<D(PF, AF) = {a|„! a e Z}. As in 7.3.7 we have
(2)
F?(h) = A„(h) J f(kmnhnlm-lkl)dkdmdn.
K x°M x N
Define for he H, Th(n) = h~xnhn~x for n e NF.
7.4.9. Lemma.
(1) rh(NF) is contained in NF.
(2) // det((Ad(/j"') — /) |nF) is non-zero then Vh is a diffeomorphism of NF onto
NF. Furthermore, if f is integrable on NF then
|det((Ad(/T
/|nr)| | f{h~lnhn~x)dn= J f(n)dn.
The proof of this result is essentially the same as that of Lemma 7.3.7. We
leave the details to the reader.
7.4.10. If / e C(G) let / be as in 7.4.4. We note that ifheH then
|det((Ad(/Tl)-/)L )| =
Thus, as in 7.3.8 (3) we find that
no-^")
(i)
F»(h) = CFh" AM(h) J f{mhm'1n)dmdn
= CFh-"AM(h) J f(nmhm~l)dmdn
°M x NF
where AM is the "A" for <t+ n <I>(mc, hc)-
This implies that if we set for a e AF, m e °MF, ua(m) = fp(ma) (7.2.1) then
(2)
FHf(ta) = FTUa(t).

250
7. Cusp Forms on G
If U is an open subset of H and if g e C °°(l/) then we set for each p e l/(hc),
r >0
<lp,r,v(9) = suphe[; (1 + log P||)1P0(fc)|.
Let ^(U) be the space of all g e C°°(l/) such that qp<r<v(g) < oo for all p, r
endowed with the topology given by these seminorms.
Set H" = {heH\h"^l for all a e <D(mF,hc)}. As in 7.3.8 we now have
Theorem.
(i) The integral defining FH, for f e C™(G) converges absolutely for h e H"
and defines an element of ^(H").
(ii) Furthermore, f t—> F" extends to a continuous mapping from ^(G) to
<€{H").
7.4.11. If x e G then det(Ad(x) - (1 + t)l) = Z trdr(x). Let / be the rank of
gc and set d(x) = d,(x). The following result is a basic ingredient in Harish-
Chandra's proof of the "local L1-theorem" for characters.
Corollary. |d|-1/2 is locally integrable on G.
If f e C™(G) is a non-negative function then 2.4.4 implies that
I \d(g)\-ll2f(g)dg = X c; f \d(h)\1'2 f f(ghg-l)dgH
G Hj G/Hj
^lcji\h-"\\FH/(h)\dh.
Now /j p is bounded on the support of F"J for all j. Thus
$\d(gTmf(g)dg<CL$\F"/(h)\dh
G Hj
which is finite by the preceding theorem.
7.5. The orbital integrals of cusp forms
7.5.1. We begin this section with some calculations on SL(2,R). As in most
of this chapter, the results are due to Harish-Chandra.
Let L be a connected Lie group locally isomorphic with SL(2, R). We
identify the Lie algebra of L with s/(2,R). Set
0 1"
0 0 '
h =
0 1
1 0
H =
1
0
0
-1
X =

7.5. The Orbital Integrals of Cusp Forms
251
Let T (resp. A0, resp. N) be the connected subgroup of G with Lie algebra Rh
(resp. RH, resp. RX). Let A be the Cartan subgroup of G corresponding
to a. Set for 6 e R, t{6) = e\p(nOh). If / e (€(G) then we set FTf(t(6)) = Ff{6).
Notice that T" = T = {t(6)\e e R - Z}. A direct calculation using the
integral formulas in 7.4.3 (see also 7.4.4) yields
(1) Ff(d) = 2i sin nd J sinh(2t)/( exp( 0tt
Set u = |7r0|cosh It. Then we have for non-zero 6
0
,2<"
0 e
r2t 0
dt.
(2) Ff(6) = 2i(sin 7r0/|7r0|) J /(exp( sgn 0
z(6,u)
z(6,-u) 0
du
with z(0,«) = u + (u2 - (ne)2)1'2.
The two values (mod 2Z) for which we can have (jump) singularities are
d = 0 or 1. The above formula shows that there is no jump singularity for
6 = 1. We concentrate on the case 0 = 0. (2) implies that
(3)
lim Ff{0) = 2i f /(exp 2uX)du
e-»o+ o
lim F/0) = 2i J /(exp - 2uX)du
e-»o- o
This implies
(4)
lim Fr(0)- lim ^(0) = ! J f(expuX)du.
9-0+ 9-0- -co
If we differentiate formula (2) then we have
d
dd
Ff(6) = {6(7id cos nd - sin nd)lnQ2 sin ^0)^(0)
with lim^o E{6) = 0.
We therefore have
(5)
- 2jti(sin n6/n6)f(exp n6h) + E{d)
liml^,F/(0) = 2,i/(l).
We now interpret (4) and (5) in terms of orbital integrals on A. We set
Hf{t) = Ff(exp tH). Then 7.4.10(1) says (in this case) that
(6)
Hf(t) = e' J /(exp tH expxX)dx.

252
7. Cusp Forms on G
We therefore conclude that
(7) lim Ff(6) - lim Ff{6) = i lim Hf(t).
e->o+ e-o- <->o
The definition of Ff implies that Ff(a) = Ff(a~l). Thus
lim (d/dt)Hf(t) = 0.
(-•0
This implies
<8) ,Lt (a) w> - .!™ (ss)f'm " °" S (£)"'w
Now let CeZ(Ic) be such that yT(C)= -d2/d62 and yA(C) = d2/dt2.
Here y, is the Harish-Chandra isomorphism associated with the Cartan
subgroup J. C is (up to scalar multiple and subtraction of a scalar) the Casimir
operator of I. 7.4.6(1) and its analogue for FA combined with (7) and (8) imply
that
« .!!?. (isfF'm ~ ,VT {i>JF'm -vr' !S (*)''«*
It is this formula that we will use in the rest of this section.
7.5.2. Up to now we have been assuming that G = °G. We have made this
assumption in order to simplify the statements of the main results. We now
assume (only) that G is of inner type. Let A be a split component of G. If H is
a Cartan subgroup of G then A is contained in H. The formula for Ff(h)
is meaningful for he H.lf fe<g(G) and if a e A then R(a)f \oG=ue<£(°G).
Furthermore, F"(ha)=F""°G(h) for hsHn°G. This device allows us
to transfer our results in the case of G = °G to the more general situation.
We can now state the main result of this section. The rest of the section
will be devoted to its proof. In the course of the proof several results will
be proved that are theorems in their own right (for example the formula
generalizing (9) above).
Theorem. Let f e ^(G) and assume that F" = 0 for every Cartan subgroup
of G that is not fundamental. If H is fundamental then F" extends to a smooth
function on H.
We will in the course of this proof use orbital integrals for several different
real reductive groups. If L is a reductive group and if J is a Cartan subgroup
of L then we set LFJf for the corresponding "F/\ This will keep track of the
group over which the integration has taken place.

7.5. The Orbital Integrals of Cusp Forms
253
We prove this result by induction on the dimension of G. If dim G = 0 or
1 then G = H is the only Cartan subgroup and F" = f. So this case is trivial.
We now assume that the result has been proved for all reductive groups, L,
of inner type with 0 < dim L < dim G. If G is not equal to °G then dim °G <
dim G. Let A be a split component of G. If J is a Cartan subalgebra of °G
then J A is a Cartan subalgebra of G and every Cartan subalgebra of G is of
this form. Thus the discussion at the beginning of this number combined
with the inductive hypothesis implies the theorem in this case. We may
therefore assume that G = °G.
7.5.3. Now suppose that H is a non-compact fundamental Cartan
subgroup of G. Then we may assume that H = HF and PF is proper. Set Q = PF,
L = °MF and T = TF. If J is a Cartan subgroup of L then J A is a Cartan
subgroup of G and (7.4.10(2))
(1) FJ/(ja) = LFJR(a)fQ(j), jeJ,aeAF.
Thus the inductive hypothesis prevails. We are thus left with the case when
G contains a compact Cartan subgroup, T.
We return to the notation of the parts of 7.4 preceding 7.4.8. Let
<S„ = {a e 0+ ! (gc)a c pc}. Let aX denote the complex conjugate of X e qc
with respect to g. Then (r(gc)a = (9c)-a- Let a e <!>„, let Z e (gc)a and W — aZ.
If Z is non-zero then Z + W is a nonzero element of p (not just pc). We may
normalize Z so that a([Z, W~\) = 2. Set H = Z + W, fc = -i[_Z, W~[ and
* = (iHCZ, W] + i(Z - W)). Then one checks that H, h, X have the same
commutation relations as the elements with the same designation in 7.5.1.
Let V = RH + RX + t. Then [Ia,la] is isomorphic with s/(2,R). We can
thus use the calculations of the previous number.
Let T.= {teT\tx= 1}. Set Ta = (t e Tx\tp # lfor^ed^ -{a}}. Then T'„
exp(Rh) is open in T. Let Lx be the connected subgroup of G with Lie algebra \x.
Then Ta is in the center of Lx. Set kx(8) = exp Onh. If t = ukx(8) e T" and if
/ e CC(G) then
(2) \f(xtx~x)dg= J \f(guxK{6)x-lg-l)dxdg.
G GIL' L*
We set A«(t) = t'~a/2 n,^.,,, (1 - r"). Then
(3) Huka{S)) = 2iAx(ukx(6)) sin(7r0).
Set
Rf(g,u,d) = Ax(uk*(Q)) sin(rc0) J/(0«xM0)*~ W*-

254
7. Cusp Forms on G
Then
Ff(uK(6))= J Rf(g,u,e)dgL°.
G/L"
Let / e C?(G). Let u e T'a. Fix p e C/(tJc). We note that if |0| is sufficiently
small and positive then ukx(6) e T'. We calculate
(p/l*)F,(«fc«(0))
Let J be the centralizer in G of Ta exp RH. Then 7.5.1(9) implies that (up to a
multiplicative constant)
(4) lim phkFTf(uK(e)) - lim phkFTf(ukx(6))
e->o+ e^o-
= P I ( f)(i ^Y ' (A«(«k«(0))A.(«)-1 lim (0 F}(« exp tH).
Since both sides of (3) are continuous on ^(G), (3) holds for / e ^(G).
This is the jump condition we mentioned at the beginning of the proof of the
theorem. The above formula implies that if Ff = 0 for all non-fundamental
Cartan subgroups, H, of G then Ff is smooth in a neighborhood of each t e T'a,
a e <tj. Suppose that a e <t^. Let Ia = g n (tc + (gc)a + (gc)-J. Let La be the
connected subgroup of G corresponding to T. Then Lx is compact. We may
now argue as above and see that Lemma 7.4.4 implies that there are no
"jumps" in this case.
We have therefore shown that if F" = 0 for all non-fundamental Cartan
subgroups of G then Fj is smooth in a neighborhood of each t e T such that
tx = 1 for at most one aeO+. The theorem now follows from 7.A.4.3.
7.5.4. Corollary. Let f €^{G) be a cusp form. If H is a Cartan subgroup of
G that is not compact modulo the center of G then F" = 0. If H is compact
modulo the center of G then F" extends to a smooth function on H.
The first assertion follows from the definition of cusp form (7.2.2) and
7.4.10(2). The second is a consequence of the preceding theorem.
In the next section we will derive some consequences of this result.
7.6. Harmonic analysis on the space of cusp forms
7.6.1. Let G be a real reductive group of inner type such that G = °G. We will
use the notation of 7.A.2. Thus, we look upon S(gc) as the algebra of
differential operators with constant coefficients on g. Fix 8, K, etc. as in the
previous sections. Let q = dim f and p = dim p.

7.6. Harmonic Analysis on the Space of Cusp Forms
255
Let 0(i(G) denote the space of all cusp forms on G (7.2.2). If G has a
compact Cartan subgroup, T and if <t+ is a system of positive roots for <t(gc, tc)
then we set
m= ft HxeU(tc).
Theorem. // G has no compact Cartan subgroups then 0(£(G) = {0}. // T is a
compact Cartan subgroup of G then there is a non-zero constant CG such that if
f e °^(G) then
wF}{l)=CGf{l).
(Notice that Ff e C'iT) by 7.5.2).
This result is a special case of a much more general theorem of Harish-
Chandra which asserts a similar limit formula for any / e ^(G) with T
replaced by a fundamental Cartan subgroup (c.f. Varadarajan [1; II, p.220] for
an exposition of Harish-Chandra's original proof). We will only need the
above statement which is much easier to prove. As usual, the proof takes some
preparation. There is however, one case where the result has already been
proved. Assume that all of the Cartan subgroups of G are one dimensional.
Then it is easily checked that either G is one dimensional, g = s/(2, R) or
g = su(2). In the first case the result is obvious. In the second case it is a
restatement of 7.5.1(5). In the last case G is compact and the result is a
consequence of the Peter-Weyl theorem and the Schur orthogonality relations
(we leave this as an exercise to the reader). We will thus assume that the Cartan
subgroups of G are at least two dimensional. We also note that we can replace
GbyT'xG and extend / e <$(G) to T1 x G by f(t,g) = f(g). This will not
change the statement of the theorem but the Cartan subgroups will all have
dimension at least 2.
7.6.2. We use a pseudo-orthonormal basis of g relative to B to identify
g with R". We set P(X) = B(X,X). Then the L of 7.A.5.1 is the co of 7.A.2.8.
We set F= Fs = Fpq (7.A.5.8). Then F(Ad(g)X) = F(X) for g e G and leg.
We will also use the notation in 2.4.3. For each j let iij be as in 7.A.2.9 for h;.
Then \D(h)\ = \7ij(h)\2 for h e fy. Thus if we apply 7.6.1 and 2.4.3 we have for
(1) f{0) = Y^cj\\nj{h)\2F(h)o^ J f(Ad(g)h)dgHdh
i b} g/Hj
= I cj I |jr/fc)|e,(fc)F(fc)*5 (fc)dfc.

256
7. Cusp Forms on G
Here we are using the notation of 7.3.6. Let coj e S(\)j) be as in 7.A.2.9. Then
Theorem 7.A.2.9 implies that
i Vj
7.6.3. Let Dj be as in 7.3.9. Let r be the rank of gc (recall that we
are assuming that r > 2) and let n = dim g. For each t > 0 we set Qr =
{Xe g |£>;(X)| < t,r <j < n}.n in this number will denote 3.14....
Lemma. Suppose that G is semi-simple. IfO<t<n — l then exp restricted to
Q, is a diffeomorphism.
Lemma 7.A. 1.4 implies that Xhc'" defines a diffeomorphism of fir onto
an open neighborhood of / in Int(g). Now, Ad(exp X) = eadX. Since, Ad is a
covering homomorphism, the Lemma follows.
7.6.4. Let W be an open neighborhood of 0 in 3(g) such that exp restricted to
W is a diffeomorphism. Let Qt be as above in [g,g]. We set W, = W ©fir.
Then
(1) If 0 < t < n — 1 then exp is a diffeomorphism from W, onto an open
neighborhood, V, of 1 in G.
Let u e CC°°(R), 0 < u(s) < 1 be such that u(s) = 1 for s < {n - l)/2 and
u{s) = 0 for s > 2(n - l)/3. Let 0 e Y c C\(Y) <= W with Y open and Cl(y)
compact. Let h e C?(W) with h(X) = 1 for X e C\(Y). We define a function, ft
on gas follows: If Xe W; _ t and if X = Z + T with Ze W and Te Q^, then
P(X) = h(Z)Tlr<j<n_! «(D;(T)) otherwise ]8 is 0. Then
(2) ^eC-(9), supple »;_,.
(3) j8(Ad(g)X) = P(X) for X e g and geG.
(4) If h is a Cartan subalgebra of g then supp ft n h is compact.
The last assertion follows from 7. A. 1.3. We now introduce a function a on G
that will be used later. If X e W„_, then set a(exp X) = /?(X) otherwise a = 0.
Then a is a smooth function on G and a(gxg~l) = a(x) for x,geG.
7.6.5. If /eC°°(G) then set f~(X) = ]8(X)/(exp X) for leg. Clearly,
/~(0) = /(l). Let // be a Cartan subgroup of G. Then
(1) *?-(*) = ^^~ /J(fc)Fj?(exp /J), for / e Cf (G) and li e h".

7.6. Harmonic Analysis on the Space of Cusp Forms
257
We note that AH(exp h)/'n(h) is non-zero for /i e W„ _ , n I). Thus
7i(h)/A,j(e\p h) defines a smooth function on W,^,nl).
Since the map / k-» F" extends to a continuous map of <^(G) into %\H")
(7.4.10) we have
(2) The map / h-> <t^ extends to a continuous map of ^(G) into C^(h").
7.6.6. We now begin the proof of Theorem 7.6.1. The material in the
previous number combined with the results in 7.6.2 imply that if / e ^(G) then
(1) /(l) = X cj J Hj(h)F(h)(o^ ";'*' ^fc)F^(exp Ji)dJi.
We note that the above integrals are over compact sets.
Now (1) implies that if / e °^(G) and if G contains no compact Cartan
subgroup (recall that we are assuming that G = °G) then f(\) = 0 (7.5.4). Now,
if fs0(€{G) then R(g)f e0(€{G). Thus, if G contains no compact Cartan
subgroups then 0rS(G) = {0}. This proves the first part of the theorem.
We now begin the proof of the second part of the theorem. Recall that r > 1.
7.6.7. We assume that H, = T is compact. If / e 0<i(G) then 7.6.6(1) implies
that
(1) f(\) = cl$n(h)F(h)ol[n<V—^—x(h)FTf(exph)dh.
t A7 (exp h)
Recall that Fj e CX(T) (Theorem 7.5.2). The following result is one of the
keys to our proof.
Lemma. There exists a non-zero constant Mg such that if g e £f(t) then
J F(h)n(h)a>ln/2]g{h) dh = Mag{0).
t
(Mg will be, essentially, computed in the course of the proof.)
If p e P(tc) we look upon p as a differential operator of order 0 on t. We will
use the following commutation identities are easily proved by induction.
(2) Let X, Y be endomorphisms of a vector space then
[X\ y]= X .\(-l)k~j+lXi((adXf~iY)= X .\((adX)iY)Xk~i.

258 7. Cusp Forms on G
We now prove the lemma.
J F(h)n{h)a>ln/2]g{h)dh = J F(h)coln/217i{h)g(h)dh
t t
-lF(h)[_d}["i2\n~]g{h)dh = 1-11.
t
Now our assumption that T is a Cartan subgroup of G implies that
p (= dim p) is even and q — dim t is even (q = dim f). Thus Theorem 7.A.5.8
implies that there is a non-zero constant Bg such that
(3) I = Bg(c5["/2]-lr/2])(ng)(0).
We now compute II. We first note that since n — r is even (n — r)/2 =
[n/2] - [r/2]. We apply (2). II =
[n/2]-i /rn/2~|\
I I )(- 1)["/2W+' I F(h)oy((ad ol)ln/2]-}n)g(h)dh
;=o \ J J t
Now deg?r = [n/2] - [r/2]. Thus (ad co)["/21~''n = 0 if j < [r/2]. Hence we
may again apply 7.A.5.8 and find that II =
[n/2]-i /rn/2~|\
I ( )(- l)["/21"j+ 'B9((^-[r/21(ad d3)["/2'-^))0(O).
We now apply the second formula in (2) to the "ad" terms. We observe that the
coefficients of ad["/21_J7r vanish at 0 forj > [r/2] (see Scholium 7.A.2.9). Thus
(as the reader should check) if j > [r/2] then
co^[r/21(ad 6}[n/2]-in) • g(0) = (ad 6}ln/2]-lr/2]n) • 0(0).
We combine this with the above formulas for I and II and we have
J F(h)n(h)(5l"/2]g(h)dh = C((ad (5l"/2]-lr/2]n)g)(0)
t
with
["i-2l /[n/2]
\j=lr/2)\ J J
Now Scholium 7.A.2.9 implies that
ad d)["/21-[r/217r = 2["/21~[r/21([n/2] - [r/2])! \\ Ha.
ae<I> +
Since
.tp(-i)fc^C)=(-i)fc""(p-!)#° uk^p>o
the lemma follows.

7.7. Square Integrable Representations Revisited 259
7.6.8. We will also use
Lemma. Let W = W(qc, tc). Let ube a W'-invariant smooth function defined
on a W-invariant neighborhood of 0. Then
I ((Un}ju\o)Y\Hx = u(0) n Ha-
Let a be a simple root in $+. If F cz <t+ then we set F~ = sxF if a is not
in F and F~ = (sx(F — {a})) u {a} otherwise. Then F h-» F~ is a bijection of the
set of subsets of <t+. Let p denote the left hand side of the formula that we are
proving. Then sxp =
-1 ((n «/.)«)(0) n «/,=-?■
Fc»+ \\PtF~ J ) PtF~
(Here we have used xu(0) = (sx)u(0) for x e S(tc).) Thus sp = det(s)p for s e W.
This implies that p = q nae<1)+ Ha with q e S(tc). A comparison of degrees
shows that q is constant. If we compare homogeneous terms we see that
q = "(0).
7.6.9. If we apply 7.6.7(1) and Lemma 7.6.7 we have
(1) /(1) = c,m/ n H«).((«/A(exp.))/JFj(exp.)|*=0.
Now, p is identically equal to 1 in a neighborhood of 0. Set u(h) =
n(h)/A(exp h). Then u e Cco(Wn. x)w and «(0) = 1 (see the proof of the Weyl
dimension theorem). In light of the preceding Lemma we have completed the
proof of the theorem.
7.7. Square integrable representations revisited
7.7.1. We continue to assume that G = °G. Let <f2(G) denote the set of
equivalence classes of irreducible square integrable representations of G (1.3).
If a e E2(G) then fix (na,Ha) e a. If v, w e (H„)K then the matrix coefficient
cv,wi9) = (.no(9)v>w) is an element of r^(G) (Theorem 5.5.4) which is also Z(g)-
finite. Thus c„>M, e °C(G) (7.2.2). Theorem 7.6.1 combined with Theorem 6.8.3
implies the following deep theorem of Harish-Chandra [13].
Theorem. S2{G) is non-empty if and only if G has a compact Cartan subgroup.

260
7. Cusp Forms on G
7.7.2. In light of the above result we assume that T is a compact Cartan
subgroup of G. As in 6.9.1, we write T = ZT°. If n e TA let £„ denote the
character of n and d(^) the dimension of [i. n restricted to T° is d(/i) times a
character, A(^) of T°. If / e 0<${G) then Fr e ^(T) (7.6.3). Thus the Peter-
Weyl theorem implies (1.4.5, 1.4.7) that
(1) Ff= I (FfVinK.
Here if /i e Cco(T) is T-central then
h*{n)=\h{t)con]&ll(t))dt.
T
The second part of Theorem 7.6.1 implies that there is a non-zero constant
CG such that
(2) /(l) = CG(d>iy)(l) = CG X [] (A^aKF^^).
neTA ae<I> +
If z e Z(gc) then we have seen that
(3) F2f = y(z)Ff on T', hence on T.
(4) So(Fz/)A(^) = A(^)(y(z))(F/)-(^).
Now if a e <f2(G) and if / = c„>w, f,we (//„)K then z/ = ^ff(z)/ with %a the
infinitesimal character of a. Thus if we put all of this material together we have
proved another Theorem of Harish-Chandra.
Theorem. Let a e &i(G) then there exists [ieTA such that (A(/i), a) is non-zero
for all a e ^(Qc, tc) and such that the infinitesimal character of a is XAi)l).
7.7.3. The above theorem has an important corollary (as usual, due to
Harish-Chandra).
Corollary. Let y e KA then the number of a e S^{G) such that (HJ^y) is
non-zero is finite.
Let C be the Casimir operator corresponding to B. Let CK be the Casimir
operator for K corresponding to B restricted to f. Let Xu..., X„ be an
orthonormal basis of p relative to B. Set Cp = ~L(Xj)2. Then C = CK + Cp.
Fix y e KA and let ny be the eigenvalue of CK on any representative of y. We
note that
(1) If (7t, //) is a unitary representation of G with C acting by cl and if HK(y) is
non-zero then c < n7.

7.7. Square Integrable Representations Revisited
261
Indeed, if v e HK(y) is a unit vector then
c= c(v,v) = (Cv,v) = (CKv,v) + (Cpv,v) = ny -£ (XjV,Xjv) < ny.
If aeS2(G) then let A„ denote an element of (T°)A that gives the
infinitesimal character as in the preceding theorem. Let p be the half sum of a
choice of positive roots. Then xJC) = l|AJ|2 — ||p||2. Hence
||AJ|2<||p||2 + ^7.
We have fixed y. The A„ "wander over" the lattice (T°)A, thus the above
inequality implies that there are only a finite number of possibilities for
infinitesimal characters of square integrable representations whose y-isotypic
component is not zero. Since there are only a finite number of isomorphism
classes of irreducible (3, X)-modules with a fixed infinitesimal character (5.5.6),
the result follows from 3.4.11.
7.7.4. We also record the following implication of the main theorem of this
chapter.
Proposition. Let G = °G. If (n, H) is an irreducible tempered representation
with infinitesimal character X\ with A e (tc)* such that (A, a) e R — {0} for
a e $(flc tr) then n is square integrable.
5.2.5 implies that there exists a standard p-pair, (PF, AF), with PF = P =
°MAN a standard Langlands decomposition such that (n,H) is equivalent
with //>,„,,„ with ae<$2(°M) and /.tea*. We may assume that TF= T which is
contained in °M is a Cartan subgroup of °M. Set h = tF + a. Then relative to
bo h.„,in has infinitesimal character given by A„ + ijx with A„ an element of
(TF)A that gives the infinitesimal character of a. Our hypothesis implies that
A„ + ifi is real valued on a. Hence n = 0. If P is proper G then h must have a
real root, a (2.3.5). If aed>(gc,tc) corresponds to a then (A, a) = 0. This
contradicts our hypothesis on A. Hence P = G and the result follows.
7.7.5. We now introduce a construction that will be useful in later chapters.
Let (as usual), R and L denote respectively the right and left regular
representation of G and (7(c)) on CrJ\G). Let A(G) denote the space of all
smooth right and left K-finite functions, /, on G such that dim Z(gc)/ < 00.
Lemma. Let f e A(G) then there exists an admissible Hilbert representation
(n, H) of G and v, w e HK such that f = crw (recall that cvw(g) = (n(g)v, w».

262
7. Cusp Forms on G
Let V = (7(gc) span{R(K)f}. Then V is an admissible finitely generated
(g,K)-module under the obvious actions (3.4.7). Let
W = L(I/(9c))span{L(K)/}.
If /, e V and if f2 e W then
/i = I <V.*(* »)*(*.)/
and f2 = Z brsL(ur)L(ys)f with xm, ys e (7(g) and fc„, us e X. We assert that
Xa„l)r,s(i?(^)R(yL(Um)L(ys)/(l)
is independent of the expressions for /, and /2. Indeed, the formula in question
is
(X brMK)L(ys)h)(\)
which clearly only depends on /,. Also it is
(X am,nR(xm)R(kn)f2)(l)
which only depends on f2. We set the value equal to (f\,f2). This defines a
sesquilinear pairing of V with W.
If keK then (R(k)fuf2) =
X am,„(i?(fc)i?(xm)L(fc„)/1)(l) = X am,„(L(^')i?(xm)i?(fc„)/2)(l)
= (/„L(k-1)/2).
Similarly, if X e g then (R(X)fuf2) = -(fuL(X)f2). Thus, ( , ) is a
(g, X)-invariant pairing of V and W^.
Suppose that he V and that (/i, W) = 0. Then, (L(k)R(x)h)(\) = 0 for all
k e K and x e C/(g). Now, h is real analytic on G (see the material in 3.4.9) and
G = KG0. Hence, h = 0. Similarly, if g e W and if (V,g) = 0 then g = 0. We
have thus proved that the pairing of V and W is non-degenerate.
Let (7i, H) be a realization of V as a Hilbert representation of G. Let (n*, H)
be the conjugate dual representation of G (1.1.4). Let V be the space of
infinite vectors of {n*,H). Then the above results imply that there is a (g, K)-
module isomorphism, T, of V onto W such that if v e K and if u e K then
<i;, u> = (i;, Tu). Let w e K be such that Tw = f. Then we assert that
<n(g)v, w> = f(g) for g e G. Indeed, the left hand side is a real analytic
function, u, on G with R(x)R(k)u(l) = (n(x)n(k)v, w> = (R(x)R(k)v, w) =
R(x)R(k)f(l) for x e (7(g) and ke K. Thus « = /. This completes the proof.

7.7. Square Integrable Representations Revisited
263
7.7.6. We are now ready to prove a result of Harish-Chandra that is one of
the essential ingredients of his proof of the Plancherel theorem.
Theorem. Let f e 0(£(G). If f is right K-finite then dim Z(g)/ < oo.
If h e %'(G) is right K-finite and if \i e TA then set T(n)h(g) = (F£(g)/)A(/4
Then
(1) T(n)(zh) = \(n)(y(z))T(n)h for z e Z(gc).
(2) T(n)(R(g)h) = R{g)T{fi)h for g e G.
(3) T0i)fc e C»(G).
(4) There exists a continuous seminorm, g, on ^(G) and d such that
I T(n)h(g)\ < q(h)(\ + log \\g\\)dE(g) for 9eG.
(2) is obvious. (1) and (3) have already been observed in 7.7.2. (4) follows from
Theorem 7.4.7.
For the moment, fix y e K\ Set u = L(a7)T(/i)f (1.4.6). Then u is left and
right K-finite. We set V = l/(gc) span{i?(K)u}. Then (1), (4) and the previous
Lemma imply that V is an admissible, finitely generated, tempered (g, K)-
module. We now assume that (A(/i), a) is non-zero for all a e <&(QC, tc)- Then
Proposition 7.7.4 implies that every irreducible constituent of V is square
integrable. 5.1.3 implies that V splits into a direct sum of irreducible square
integrable (g, K)-modules. Let S(y) be the set of equivalence classes of the
constituents of V. Let F be the (finite) set of K-types of span{i?(K)/}. If
to e S(y) then there exists a e F such that Hom/i:(Fff, HJ is non-zero. Set Q =
{(a e S2(G)\HomK(V„,Hlo) is non-zero for some a e F}. Then Q is a finite
set by Theorem 7.7.3. Clearly, S(y) is a subset of Q. Let X = {n e tj | x„ = X
for some n e Q}. Put T = {/^ e TA | A(/i) e Z}. Then T is a finite set. We have
shown that if T(/i)f is non-zero then /leT.
7.7.2(2) implies that
/(0) = CG I n (A(/i),«)T(/i)/(a)
for g e G. The theorem now follows from (1).
7.7.7. It will be shown, in the next chapter, that the span of the functions u in
the course of the above proof is the span of the matrix coefficients of the
discrete series representation of G corresponding to Q (see 6.9.5).

264
7. Cusp Forms on G
7.8. Notes and further results
7.8.1. We first expand a bit on the material in Section 7.1. Let a and b be as in
that 7.1. If we drop conditions 7.1.1(3) and (6) then the space ^,b(G) is still a
Frechet space (the seminorms and the topology defined in exactly the same
way as in 7.1.1). Furthermore, it can be shown (without difficulty) that
Theorem. £fa,b(G) is a smooth representation of G x G under the left and right
regular representation.
In addition to the two examples of 7.1 one now has the spaces (€P{G) which
are given by a(g) = 1 + log \\g\\ and b(g) = E(g)2/p for 0 < p < oo. Clearly,
<£2{G) = <^(G). The spaces (€P(G) are usually called the Lp-Schwartz spaces.
7.8.2. We now look at the material in Section 7.2. The transform f was
originally introduced by Harish-Chandra in his work on spherical functions
(Harish-Chandra [9, p.595]). In this chapter, we have used this transform
basically to reduce calculations of orbital integrals to the case of a compact
Cartan subgroup. In the next chapter, we will see that we can calculate the
character of a representation induced from P in terms of fp. This will also give
a better understanding to our (unmotivated) definition of cusp form.
7.8.3. As we have seen, Theorem 7.6.1 is a powerful tool in the analysis of
cusp forms. We have also pointed out that this result is a special case of a more
general theorem of Harish-Chandra, which we now state.
Theorem (Harish-Chandra [13, Lemma 38, p.47]). There exists a non-zero
constant CG such that if H is a fundamental Cartan subgroup of G and if
fe^(G)then
lim mFf{h) = CCGf(\)
/i-i
with C a non-zero constant depending only on the choice of invariant measure
inH.
This theorem plays a basic role in the proof of Harish-Chandra's Plancherel
theorem. A full discussion will appear in Volume II of this opus.
7.8.4. Although Lemma 7.7.4 is not difficult, the result will play an important
role in our discussion of Harish-Chandra's "philosophy of the constant term"

7.A.I. Some Linear Algebra
265
(also to appear in Volume II), since it allows us to transfer the results of
Chapter 4 from matrix entries to elements of A(G).
7.A. Appendices to Chapter 7
7.A.I. Some linear algebra
7.A.I.I. We put the usual inner product < , > on C" with corresponding
norm ||---||. On End(C) we put the operator norm. The following result is
based on an ingenious trick of Thompson [1]. The use of the spectral radius in
the proof was suggested by Roger Nussbaum.
Lemma. Let X, Y e End(C) be such that X* = X and Y* = Y. Then
(i) \\exeY\\>\\ex+Y\\.
(2) If furthermore tr X = tr Y = 0 and n > 2 then
\og\\exe*\\>\\X+ y||/(n-l).
If XEEnd(C) then set r(X) = lim sup \\Xk\\l/k (Here the limit is as
k -> + oo. Also it is well known that we may replace lim sup by lim.). Then
r(X) < \\X\\ and if g e GL(n,C) then r(gXg~l) = r(X). We will also use the
fact that IKXX*)*1!! = \\X\\2k. In particular if p e GL(n,C) is self-adjoint and
positive definite then r(p) = \\p\\.
Let a, b be self-adjoint and positive definite matrices.
(i) ||at||2'c<||(a2t2)'c|| for k= 1,2,....
Indeed,
\\ab\\2k = \\((ab)(ba))k\\ = r(((ab)(ba))k) = r(ab2a2---b2a)
= r((a2b2)k)<\\(a2b2)k\\.
Since \\Xk\\ < \\X\\k, (i) implies
(ii) \\(ab)2k\\ < \\(a2b2)k\\.
This in turn implies that
(iii) \\(ab)2k\\ < \\a2kb2k\\.
If we apply (iii) to a = exp(X/2k), b = exp(Y/2k) then we have
\\((exp(X/2k) exp(y/2k))2k|| < \\exeY\\. If we now take the limit as fc ^ oo, (1)
follows.

266
7. Cusp Forms on G
To prove (2), it is enough to show that if X is self-adjoint, tr X = 0 and n > 1
then
Hex|l>eimi/n-i_
Let n!,..., n„ be the eigenvalues of X, counting multiplicity, labeled such
that l/Xi| > \fij\ for all j. Let \i be the largest eigenvalue of X. ||X|| = |^J
and ||e*|| = e". Thus we must show that jx > |/^|/(n - 1). If /^ > 0 then
Hx = H and the assertion is clear. If ^ < 0 then \ni\ = —Hi = n2 + '" +
H„ < (n — \)n, as asserted.
7.A. 1.2. The other results from linear algebra that we will need in this
chapter are of a different nature. Let U = {X e M„(C) | if n is an eigenvalue of
X then |Im /z| < n\. It is clear that U is an open subset of M„(C).
Lemma. The exp is a diffeomorphism of U onto an open subset of GL(n, C).
As is well known dexpx Y = ex((I — e~ad*)/ad X)Y. The eigenvalues of
(/ — e~ad*)/ad X are the numbers (1 — e")/n with n of the form a — y and
a, y are eigenvalues of X (here (1 - ez)/z = — 1 if z = 0). Since, (1 - ez)jz = 0
if and only if z = 2nik with k a non-zero integer, it follows that exp is
everywhere regular on U. Thus to prove the lemma, we need only show
that exp is injective on U.
If X e End(C) then X can be written uniquely in the form X = Xs + X„
with [XS,X„] = 0 and Xs diagonalizable, X„ nilpotent. If g e GL(n,C) then g
can be written uniquely in the form gsgu where gs commutes with gu and gs is
diagonalizable and gu — I is nilpotent. Suppose that X, Y e U and that
ex = eY. Then we must have exp(Xs) = exp(7s) and exp(X„) = exp(7„). But
then X„= Y„. We may thus assume that X and Y are diagonalizable. If
a, y are eigenvalues for X and if e" = ey then a — y = 2nik with k an integer.
Thus since X e U this implies that a = y. This implies that the e" eigenspace
for ex is the a eigenspace for X. If we apply this observation to Y we see that
since ex = eY, X = Y.
7.A.I.3. Let Oj denote the j-th symmetric function on C". We set ak = 0 if
k > n or k < 0. Recall that
| | iv, + ,M = X/" V.iv...... v„).
Lemma. If C > 0 and if |t7j(xi,..., x„) | < C for j = 1,..., n then \xj\ <
C + 1 for j = !,...,«.

7.A.I. Some Linear Algebra
267
The definition of the as implies that
aj(x) = oj(xu...,xn-l) + xnoJ-1(x1,...,xn-1).
After relabeling (if necessary) we may assume that |x„| > \xj\. We may also
(clearly) assume that x„ is not equal to 0.
(1) \ok-.l(xu...,xn-l)\ <(C + \ok(x1,...,xn-l)\)/\xn\.
Indeed, C > \ak(x)\ =
\Ok(Xi,..., X„_ i) + X„Ok- i(Xi,.. ., X„- i)\
> \xn\\ok-l(xu...,xn-l)\ - \ok(xu...,xn-1)\.
<T„(X) = Xi'--Xn, SO
(2) \a„^l(xl,...,x„^i)\<C/\x„\.
An easy argument using induction (1) and (2) shows that
(3) |ff,_;(x1,...,x,_1)|<cf X \x„\k)l\xn\} foTJ=l,...,n-l.
In particular, (3) implies that
\al(xl + ■■■ + x,_,)| < C(l + ••• + Ixr^VlxJ"-1.
Thus C > |Xi + ■■• + x„| > |x„| - C(l + ■■■ + |x„r2)/|x„r'. Hence
ix„r < qi + ••• + ix„r') < c(ix„r-')/(ix„i -1) if ix„i > 1.
This implies that if |x„| > 1 then |x„|" + 1 - |x„|" < C|x„|". So |x„|" + 1 <
(C + l)|x„|". Hence |x„| < C + 1 if |x„| > 1. If |x„| < 1 then it is clear that
|x„| < C + 1. This completes the proof.
7.A.I.4. If X e End(C) then define the polynomials Dj(X) by
det(tI-X) = Zti(-\riDn„j(X).
If X has eigenvalues nx,...,n„ counting multiplicity then it is easy to
see that Dj(X) = a}(nu..., n„). The preceding lemma now implies that if
\Dj(X)\ < 7i- 1 for j= \,...,n then X eU (7.A.1.2). Thus Lemma 7.A.1.2
implies
Lemma. Set Vr = {X e End(C) | \Dj(X)\ <r}. If r < n - 1 then exp is a
diffeomorphism of Vr onto an open subset of GL(n, C).

268
7. Cusp Forms on G
7.A.2. Radial components on the Lie algebra
7.A.2.I. The discussion of radial components in this appendix is based upon
the results in Harish-Chandra [6]. Let G be a Lie group with Lie algebra g. Set
I_ = G x g which we look upon as a Lie group with multiplication given as
follows
(1) (x, X)(y, Y) = (xy, Ad(jT l)X + Y) for x,yeG,X,YeQ.
The Lie algebra, I, of L is g x g with bracket given by
(2) [(X, Y), (X1, Y')-] = &X,X'l |T, A"] + [X, 7']).
L acts on g by (g, X) • Y = Ad(g)(Y + X). This makes g into an L-space. Let
DO(q) be the algebra of all differential operators on g with smooth coefficients.
IfXElthensetT(X)/(7) = d/dt/'(exp(-tX)y)|r = ofor/ECco(g).ThenTis
a Lie algebra homomorphism of I into DO(q). Hence T extends to an algebra
homorphism of U(\c) into DO(q).
If (X, Y) e I then T(X, Y) is a smooth vector field on g which we can look
upon as a smooth function from g to g. We leave it to the reader to check that
(3) T{X,Y)y = lV,X-\-Y.
In I, g x 0 is a Lie subalgebra isomorphic with g and 0 x g is a Lie subalgebra
with 0 bracket operation. Thus
(4) U(\c) = U(Qc) ® S(gc) with a complicated multiplication.
7.A.2.2. Let X!,..., X„ be a basis of g and let be the corresponding
coordinates on g. If D e DO(q) then
D = ZP,3'.
Here we use the standard multi-index notation. If I = (iu..., i„) with ij a
non-negative integer then |/| = I. ij and 3' = 3|,|/x'11 ■ ■ ■ xj,". If X e g then we set
Then Dx is a constant coefficient differential operator on g. Clearly,
T(l ® S(flc)) is the algebra of all constant coefficient differential operators on
g. We will thus identify S(gc) with the algebra of constant coefficient
differential operators on g.
It is convenient to introduce a slight twist on T. We define R(x ® y) =
T(l ® y)T{x ® 1) for x e l/(gc) and y e S(gc). If 7 e g and if u e C/(IC) then we
define i?y(u) = i?(u)y e S(gc).

7.A.2. Radial Components on the Lie Algebra
269
7.A.2.3. Now let h be a Lie subalgebra of g such that there is an ad(h)-
invariant complementary subspace, V, of h in g. We also assume that h' =
{He I) det(ad H \v) # 0) is nonempty. Set V~ = symm(S(Fc)) in l/(gc).
We filter U(lc) as usual. This filtration induces a filtration of f ®S(hc) with
(^®S(t)c)Y = I symm(S"(Fc))®S"(hc).
p + q^j
We filter S(qc) using the filtration associated to the gradation by
homogeneous degree. We denote this filtration by Sj(qc).
Lemma. If He I) then RH((^ ® S(hc))j) is contained in Sj(qc). The map
H h-> RH restricted to (f ®S(hc))^ is a polynomial mapping from h to
L((V ® S(\)C)Y, Sj(Qc). If H e h' then RH is a bisection from (V ® S(t)c))j to
s,(gc)-
We prove this by induction on j. If j = 0 then the result is obvious. Assume
the result for j — 1 > 0. Let H1,...,Hr be a basis for h and assume that
Xu..., Xs is a basis for V. If p + q = j then
KH(symm(X1,---XJr)®//jl---//jr)
= (- 1)" + "[H, Xfl] • • • [H,XJ/^, • ■ • Hjr mod S,_ .(9c).
The proof of the inductive step is now clear.
7.A.2.4. Set TH = i?H restricted to f ® S(hc) and rHJ equal to rH
restricted to the j-th homogeneous component. Then H k-» (rHj)~' is a rational
map with singularities contained in I) — h'.
Let e be (as usual) the homomorphism of l/(gc) to C given by e(l) = 1 and
E(g) = 0. We identify 1 ® S(gc) with S(gc). If p e S(gc) and if H e h' then we set
(5H(p) = (e ® /)((rH)_' (p)). Then if // e h' then (5H defines a linear map of S(gc)
into S(hc). Let (5H ;- be the restriction of 5H to Sj(qc). Then // k-» (5h j is a rational
map from h into L{Sj(qc),Sj{t)c))- If D e DO(q) then let 5H(D) = 5H(DH).
Lemma. If D e £>0(g) then there exists a differential operator 5(D) on h' such
that 5(D)H = 5H(D) for H e h'.
This is clear from the above discussion.
7.A.2.5. Let U and l)x be open subsets of g and let W be a neighborhood
of 1 in G. We assume that \d(W)U1 is contained in U.

270
7. Cusp Forms on G
Lemma. Let f e C°°(l/) be such that f(Ad(x)Y) = f(Y) for x e W and
Y e Uu If D e DO(U) and if ft = \)' nUx then
{Df)\a = b{D){f\n\
Df(H) = (DHf)(H) = (rH((rHyl(DH))f)(H). Now, K(((Ker e) n F) ® 1)/
restricted to C/, is zero by the assumed invariance of /. The lemma now follows
from the definition of 5(D).
7.A.2.6. We now assume that G is a real reductive group such that Ad(G)
acts trivially on the center of g. Let h be a Cartan subalgebra of g. Let B be as
in the definition. Set V = h1 relative to B. Let H = {g e G | Ad(g)h = h for all
h e I)} be the corresponding Cartan subgroup.
Lemma. Let h0 e h' then there exist neighborhoods U and l/, of h0 and W a
neighborhood of 1 in G such that Ad(W)Ul is a subset of U and such that if
B{U,Ul,W)= {f e CCD(U)\f(Ad(x)Y) = f(Y) for x e W and Ye I/,} then
B(U,UuW)\Vnl), = C™(VnUl).
Let p be the natural projection of G onto G/H. Let <&(gH, h) = Ad(g)h
for g e G and h el). Then it is easy to see that <t is everywhere regular on
G/H x h'. Hence there exist an open neighborhood Wx of \H in G/H and U2
an open neighborhood of h0 in h' such that <t restricted to Wj x U2 is a diffeo-
morphism onto an open neighborhood, U, of h0 in g. Let W2 be an open
neighborhood of 1 in G such that p(W2) = Wx. Let W be an open neighborhood
of 1 in G contained in W2 and such that W • W is contained in W2. Set
I/, = <b{p{W) x I/2).
We assert that if we choose a possibly smaller W then [/,n()' =
C/2.Indeed, if Ad(x)/i, = h2, hx e U2 and h2 e h'. Then Ad(x)h = h. If N =
{g e G ! Ad(g()h = h} then N/H is a finite group. Thus we may choose W
such that N r\W = H r\W. This implies the assertion. If u e C^iy then
define / on C°°(C/) by /(<I>(x,/i)) = «(/i) for x e W and /i e U2. Then / is
clearly in B(U, Uu W) and / = u on C/2.
7.A.2.7. If fe C°°(g) and if g e G then set y(gf)/(X) = /(Ad(g-1)X) for
X e g. Let T(g) denote the algebra of differential operators, D, on g such
that y(g)D = Dy(g) for all g e G.
Lemma. 5: T(g) -> £>0(h') is an algebra homomorphism.
Let n0 e n' and let U, Uu W etc. be as in the preceding Lemma. Let u e
C'iUi n h') and let / e B(U, Ux , W) be such that / = uon{/1nf)' = ft. Let

7.A.2. Radial Components on the Lie Algebra 271
Du D2eT(Q). Then (D,D2/)|n = 3(D1)((D2/)|n) since D2f e B{U, Uu W).
This in turn equals S(Di)5(D2)f\n. Now (DiD2)f(h) = 5(DiD2)u{h) for
h e ft. Thus we have shown that 8(DxD2)u = 5{Dl)5{D2)u for u e C°°(n). The
Lemma now follows.
7.A.2.8. Our next task is to derive a formula for 5{D) for £) e S(g)G =
S(g) n T(g). We first look at the element to e S(g)G with o) = 'LXjYi where
{Xk} is a basis of g and B(Xj, Yk) = 5jk.
Let $ = ®{qc, bc) and let <t+ be a system of positive roots for <t. If a e 0 we
choose £aE(gc)a such that B(£„,£_J = 1. Then [£„,£_J = Hx(B(h,Hx) =
a(h) for he be)- Let {//,} be a basis of be sucn that B{H}-,Hk) = 5jtli. Then
co = I(H,.)2 + 2Iae<I)+£a£_a.
Now
R((EXE„X + £_.£.) ® 1) = [£„, .][£-,, •] + [£_., •][£„, •]•
If X, 7 are vector fields on g then (X • Y\ = Xh Y + X(h) Y(h). Thus
r,((£a£_a + £_«£«) ® 1) = a(/i)([£«,£-«] - [£_«,£«])
- «(fc)2(£a£_« + £_«£«)
= 2(a(h)Hx - a(/i)2£.£_J.
Hence
(1) rfc(symm(£«£_„) ® 1) = *{h)Ha - «(fc)2£«£_«.
Thus
(2)r»(-2 X a(/i)^2symm(£a£_a) ® 1 + 2 £ a(fc)"11 ® H. + £ 1 ® H2) = co.
We therefore see that
(3) 5(o>) = 2Y,«(hrlHx + Y,H].
7.A.2.9. We define an isomorphism of S(gc) onto P(QC), P^~* P* by
X#(y) = B(X, y)forX, y e g. Let p \-*p* be the inverse map. If/ sS(gc) then
we define /e S(hc) by /# =/#|„. We set n(h) = Uae<b+ ct(h) for he h- We can
now state Harish-Chandra's formula for <3 (Harish-Chandra [6, Thm 1, plOO]).
Theorem. If D e S(qc)g then 5(D) = 7r~'Dn. Here a function is looked upon as
a differential operator of order 0.
We first check the formula for ox We note that
(it X hj^ = o.

272
7. Cusp Forms on G
Indeed, if s e W(qc, hc) then sn = det(s)7r. If / e P(h) and if saf = —f then
/ vanishes on the hyperplane a = 0. Thus if sf = det(s)/ for se W(qc, hc) then
f = ng with g a polynomial on h. Now E Hjn is also skew symmetric relative
to W(§c, hc). Since it has strictly lower degree than n we are forced to conclude
that it is 0.
£ (Hj)2nf = X ((Hj)2n)f + 2 X W(ty/) + * I (ty)2/-
Now H,-7r = E„e<1,+ a{Hj)a.~ln. So
ZHjnf-JlHjf+l I oT'HjY
This proves the formula for to.
To prove the full formula we use an ingenious trick of Harish-Chandra
which is based on the following Scholium which will be used in another
context.
Scholium. If f e Pj(g) then 2'p.p = (ad co)'f- Here we look upon P(g) as
multiplication operators contained in DO(q) and if x, ye DO(q) then
ad(x)y = xy — yx.
Let X e g. We must show that (ad coY(X*y = 2'(j\)X}. We compute
(ad coY(X*Y = (ad co)'"1 (ad co{X*)')
= (adcoy1( X (X*)\<id(oX*)(X*y-k-1
\0<j<k-1
Now ad coX* =21 B(X}, X) Y} = 2X. So
(adcoY(X*y = 2(ad co)j-l £ {X*fX{X*y-k-1.
0<j<k-1
It is clear that if / is a polynomial of degree strictly less than j — 1 then
(ad co)'- \f = 0. Thus if we put our calculations together we have
(I) (ad coy(X*y = 2jX(adcoy-l(X*y-1.
If we use the obvious argument by induction the scholium follows.
We now complete the proof of the Theorem. Let D e Sj(g)G. Then
£> = (l/2j/!)(adcoy/)#.Thus
5(D) = (l/23j\)(ad5(co)Y5(D#).
Now 5(D*) = £>*!(,. Thus
5{D) = (\lVj\)(ad5(co)y(D*\k)
= (\/Vj\)(ad5(co)Y(n-l(D* l^n).

7.A.3. Radial Components on the Lie Group
273
Since the Theorem has been proven for co we have
(ad 5(co)yD# |„ = n-l((ad(ai)y(D# \$n = (Vj^n^Dn
by the Scholium applied to h. This completes the proof.
7.A.3. Radial components on the Lie group
7.A.3.I. Let G be a Lie group with Lie algebra g. We put a Lie group
structure on G x G by (g, h) • (u, v) = (gu,u~lhuv). We leave it to the reader to show
that with this multiplication G x G is a Lie group that is Lie isomorphic
with the usual product group. Let L denote this Lie group. Let I be the Lie
algebra of L. We look upon G as an L-space with action (x,y)-g = x(yg)x~1.
Let T: I -> DO(G) (differential operators with smooth coefficients) be defined
by
T{X)f{g) = jt{f{cxp{-tX)g)\t = 0.
Then T extends to an algebra homomorphism of U(\c) into DO(G). A direct
calculation shows that if X, Yea, then
(1) T(X,Y)g=L(X-Ad(g)X)g + L(Y)g.
Here I is looked upon as g x g with a twisted bracket operation. Thus
U(\c) = U(qc) ® l/(gc) with the corresponding multiplication. The first
factor is U(Qc x 0) and the second is (7(0 x gc).
In this appendix we use this formalism to prove analogues (also due to
Harish-Chandra) of the results of the last appendix. The only results that are
essentially different are the last two. Thus for the most part we will leave it to
the reader to fill in the analogous arguments.
7.A.3.2. Let H be a closed subgroup of G with Lie algebra h. We assume
that g=h®F as in 7.A.2.3 and in addition that Ad(//)K = K Set
f" = Symm S(VC) in l/(gc). We filter V ® [/(hc) using the standard filtration
of I/(IC).
As before we set R(x ® y) = T{\ ® y)T(x ® 1). We look upon l/(gc) as the
algebra of all right invariant differential operators on G. That is, we identify
it with T(l ® U(qc)). If De DO(G) and if ge G then there exists a unique
DgeU{Qc) such that Df(g) = Dgf(g). Define for ueU(\c), g e G, Rg(u) =
R(u)g. Let //'={/ie//!det((/-ad(/i))|K#0}. Clearly, if h' is non-empty
then so is H'. We assume this. If he H then we write Yh for Rh restricted
to r ® I/(hc).

274
7. Cusp Forms on G
Lemma. If he H' then Vh is a linear bijection ofi^® l/(hc) onto t/(gc) which
respects the /titrations (we use the standard filtration on U(qc)).
7.A.3.3. We assume that there is a complex Lie group Hc contained in
GL(gc) such that Ad(H) = Hc n GL(g).
Lemma. The map hh->Th |r®S(i,)) e L((^ ® l/(hc))J, Uj(gc)) is real analytic in
h, factors through the homomorphism h i-» kd{h) and extends meromorphically
toHc.
This is clear from the definitions and 7.A.3.1(1).
7.A.3.4. If heH' and if xe U'(§c) set 5h(x) = (e® I){rh{x)). As in the
preceding appendix, if x e U{qc) then h\-^5h{x) is meromorphic from Hc
into U\\)C).
If D e £>0(G) then set 5h(D) = 5h(Dh).
Lemma. If D e DO(G) then there exists a differential operator 5(D) e DO(H')
such that 5h(D) = 5(D)h.
7.A.3.5. Let U and l/, be open in G and let Q be a neighborhood of 1
in G such that x(/,xM is contained in U for x e Q. As in the previous
appendix, we set B(U, UUQ) equal to the space of all C00 functions on U such
that f(xyx~l) = f(y) ioxxeQ,yeVx.
The following result is proved in exactly the same way as Lemma 7.A.2.5.
Lemma. Let f e B{U, UUQ) and let D e DO{G). Then
Df\VnH- = <5(Wk„).
7.A.3.6. We now assume that G is a real reductive group of inner type. Let h
be a Cartan subalgebra of g and let H be the corresponding Cartan subgroup.
We take V = hx relative to B. Let U be open in G such that xU x~' = U for
all xe G. Let D(U) denote the algebra of differential operators on U, D, such
thaty(g)D = Dy(g) for g e G. Here y(g)f(x) = f(xgx~l). The following result is
proved in exactly the same way as Lemma 7.A.2.7.
Lemma. 5 is an algebra homomorphism from D(U) into DO(H').
Our next task is to find a formula for 5{z) for z e Z(gc). Fix <t+ a system of
positive roots for ^(gohc). We assume that the corresponding p is the

7.A.3. Radial Components on the Lie Group
275
differential of a homomorphism of H into Cx. This can always be guaranteed
by going to a covering of G. Set
A(h) = hp n (\-h")= X det(s)/isp.
Let y be the Harish-Chandra homomorphism from Z(gc) to l/(hc). Here is
the formula of Harish-Chandra in this case.
7.A.3.7. Theorem. If ze Z(gr) then
<5(z) = A_1y(z)A.
Let p. e (hc)* be <t+ dominant integral. Again, by going to a finite covering
of G we may assume that p defines a character of H. Let o^ be the character of
the corresponding finite dimensional representation of G. If z e Z(gc) then
(1) za)l = (p+p)(y(z))ali and
(2) zo^v = <5(z)(o-JH.).
We note that A(h) is non-zero for h e H'. Set
(A-S(z).A_1)* = P* fox he H'.
We set qh = ph- y(z) e U(\)c). (1) and (2) combined with the Weyl character
formula imply that
(3) X det{s)s{p +p){qh)hsU'+l,) = 0 for he H'.
seW(g.l))
We note that the coefficients (in h) of qh extend to meromorphic
functions on Ad(//C). Let h+ denote the set of all h e h such that a{h) e R and
tx(h) > 0 for all a e <t+. Then s{k/i + p)(h) - (kp. + p)(h) -> - co as k -> + co.
Write qhj for the homogeneous component of qh of degree j. If /? e h* then
np(ph)= I.j n']P{ph j). Let <jf be the maximum of the j such that qhj is
non-zero. If h e exp(h+) then
0 = lim fc-«/!-<*+'>X det(s)s(k/i + p)(^)^s(fc"+p> = /i(q*.,).
fc -♦ GO
The /ief)* that are highest weights of irreducible finite dimensional
representations of G are Zariski dense in h*. We have shown that if
h e exp(h+) then qh = 0. Since Hc is connected and qh is meromorphic in h
this proves that qh = 0 for all he Hc.
7.A.3.8. In the next chapter we will need a generalization (also due to
Harish-Chandra) of the above theorem. Let 6 be a Cartan involution for G.

276
7. Cusp Forms on G
Let h0 e g be such that 8h0 = — h0. Then ad(n0) has real eigenvalues. Let m
be the centralizer in g of n0 and let n be the direct sum of the eigen spaces
for ad n0 corresponding to strictly positive eigenvalues. Let V = n ® On. Let
M = {geG\ Ad(a)h0 = h0}. Let M' ={meM\ det((/ - Ad(m))\v) # 0}. Let
h be a Cartan subalgebra of m and let H be the corresponding Cartan
subgroup of M (also of G). We denote by 5G M the "<5" from DO(G) to
DO(M'), 5M H the one corresponding to DO{M) to DO{H') and by 5 the one
going from DO(G) to DO(H').
Set for me M,
AGM(m) = |det(Ad(m)|n)|1/2 det(/ - (Ad(m)|n).
We define a homomorphism yfl m from Z(,qc) to Z(mc) as follows. P-B-W
implies that l/(gc) = t/(mc) ® (6ncU{Qc) + [/(gc)nc). Let q denote the
corresponding projection of l/(gc) onto U(mc). Let y\ be the homomorphism
of U{mc) to C/(mc) given by r\{X) = X - [\) tr(ad X|n) for X e m. Then
y9.m is given by r\ <> q restricted to Z(gc).
7.A.3.9. Let U be an open subset of G such that xUx~l = U for
xeG.
Proposition. Let z e Z(gc) and f eCx(M r\U) be such that f(xyx~x) =
f{y) for x e M and yeMnU. Then on M' nU we have
<5G.m(z)/ = ^GM7GM(z)AGMf.
Let x e Af n [/ n C and choose H such that x e H. Let F0 be an open
neighborhood of 0 in V such that if <I>(X) = exp XM, X e V0 then <t is a
diffeomorphism of V0 onto an open neighborhood of \M in G/M. Let
W be an open neighborhood of x in M'nU such that if u(X,y) =
exp Xy exp( —X) for X e V0 and yeW then u is a diffeomorphism onto an
neighborhood of x in G.
Let W, be an open neighborhood of x in W and let P be a neighborhood
of 1 in M such that yWy'1 is a subset of W for y e P. Let [/, = u(F0 x W^.
Set Q = exp V0P. Then if we argue as in the proof of 7.A.2.6, it is possible
to choose V0, W, Wx, P so small that
(1) B(U,Ul,Q)\w=B(W,Wl,P).
Let feB{W, WUP) and let h e B{U, UUP) be such that h = f on W.U
z e Z(gc) then <5M.H(<5G M(z)) = (5(z). Harish-Chandra's formula implies that
)(AG.1M)) = Ay(z)A1.Thus
<5M.;;(AG.MyG.M(2)(AG.M)_1 ~ SGM{z))(T = 0

7.A.4. Some Harmonic Analysis on Tori
277
for a eCr{WinH). Thus
&GMyGM{=){\iMr\f = <5g.mU)/
for f e B(W,WU P)- Thus the result has been proved on G' n U n M. Since this
set is dense in M' n U and the desired formulas are real analytic on the larger
set, the result follows.
7.A.4. Some harmonic analysis on Tori
7.A.4.I. The purpose of this appendix is to collect some technical results
that will be used in Section 7.4. Let T be a compact torus with Lie algebra t.
Then exp is a covering homomorphism (if we look upon t as an abelian Lie
group under addition). Let T = Ker(exp). Then T is a lattice in t that contains a
basis. We identify TA with {/is t*\n(D <= 2nZ}. That is, if n is such a
functional then t" = eim) if t = exp H.
Let < , > be an inner product on t. We also use the notation < , >
for the dual inner product on t*. Let Xi,...,Xn be an orthonormal basis
of t. We set A = X (Xj)2. Clearly, depends only on < , >.
It is clear that
(1) At"= -<Ju,Ju>t" for neT\
If / e C™(T) and if \i e TA then we set
/A(/i) = J/(t)r*A.
T
Let kr be the distribution on T with Fourier series
X (i + <n,nyyrt".
That is, if f e CX'(T) then
It is obvious from (1) that
(2) (/-A)fcr + 1 = fcr for all r.
The Plancherel theorem for T says that
(3) K = »•
Here,3(/) = /(l).
Lemma. Let k e N. // r > [n + k)/2 then kr e Ck(T).

278
7. Cusp Forms on G
If r > n/2 then
X (i + <^>r<a).
Thus the Fourier series defining /cr converges absolutely. Hence kr e C°(T) if
r > n/2. Xjkr has Fourier series
X (i + <^>rw-
Since
|(1 + </i,/i»-'i/i(^)| < (1 + <^>P+1/2
it follows that if r > (n + l)/2 we can differentiate the Fourier series
defining fcr term by term. So kr e C^T) if r > (n + l)/2. The lemma follows
from the obvious iteration of this procedure.
7.A.4.2. If x, y e T then set d(x, y) equal to the Riemannian distance between
x, y corresponding to < , >. That is, d{x,y) = inf{\\X — Y\\\ exp X = x,
exp Y = y}. Here ||X|| = <X, X>1/2.
We now come to the first main result of this appendix. Let V be a
Frechet space. Let au..., ad e TA - {0} be distinct. Set T' = {t e T\tXi # 1,
i= l,...,r},if e>0thenset T't = {t e T \\\ - tXi\ > efori= l,...,r}. Let A
be an algebra of continuous linear operators on V containing the identity.
Let C be a subalgebra of U(tc) such that D = I — A e C and such that
there exist pu..., pqe U(tc) such that l/(tc) = X Cp^.
Let y be a surjective algebra homomorphism of A to C. Let W be a dense
subspace of K Suppose that we have a linear map, S, of W into C^iT')
such that S(Tt;) = y(T)S(v) for v e W, T e A. Finally, we assume that there
is a continuous seminorm a on V and u > 0 such that
J |S(i;)(t)|dt< e~"a(t;)
for i; e ^ and all 0 < e < 1.
If p e C/(tc) and if / e C°°(T') then we set
Op(/) = sup,er |p/(t)|.
Let B(T') be the space of all /sC°°(r) such that ap(f)< oo for all p,
endowed with the topology given by these seminorms.
Lemma. S extends to a continuous linear map of V into B(T').
Let 37' = T - T' (as usual). If x e T set u(x) = (±) inf{d(x,y) !y e T'}. If

7.A.4. Some Harmonic Analysis on Tori
279
X e t then we denote by Br(X) the r > 0 ball in t centered at X relative to
< , >. Let r0 be such that exp is a diffeomorphism on Br(X) for all let.
If x e V then set i;(x) = min{u(x), 1, r0/2}.
Let h e C°°(R) be such that 0 < h(x) < 1 and h(x) = 1 for |x| < 2~1/2 and
h(x) = 0 for x > 1. Fix x e T and set i; = i;(x). Let x = exp X. Define
g( e (^(T) as follows: #(exp Z) = 0 if Z is not an element of Br(X) and
0(exp Z) = h(\\Z - X\\2/v2) if Ze Br(X). Let / e C»(T). If p e t/(tc) then
p/M = P0/M-
Fix peU'itc). Let d be the maximum of the orders of the p,-. Set
s = n + j + d + 1. Then ((py)r is the formal adjoint of p looked upon as a
differential operator in y see 8.A.2.7)
p/(x)=|fcs(x)'-1)£>sPa/-()')^
T
= I (p^Mx^'MjOD'/Wy + J (Py)rfcs(xy-')[Z)s,0]/(y)^.
r r
Now [£>s,gf] is a differential operator on T of the form E p7(j/)9|,|/j/'
(here we are using coordinates on T corresponding to our basis of t)
with Pi(y) = 0 if d(x,y) < v/21'2 or if d(x,y) > v. Also from the choice of g
it is clear that \^^l^yJpt(y) I < CjV~q with Q, <jfj constants independent
of y and x (see 5.A.2). We write 9|,|/3j/' = X a,,p, with au e C. We
therefore have
pf(x) = I (p,)7'M*j'~1)0(j')D7Wj'
r
+11 ((P/Wr(p,()0(p,)rMxr >i.,/(jO<**
Z.i T
Now, a^V/Mxy"1) defines an element of C°{T) for |J\ < j' + d. We
therefore find that there exist c, e C,q — qp and a constant Cp such that
|p/(x)|<Cptr"X I |c,-/O0|dj>.
If 0 < e < 1 and if x e TJ. then v{x) > Ce, with C a fixed positive constant. We
therefore have
|p/(x)|<Cpe-"XJ|Cj/(t)|^.
7"
We apply this to / = S(w) for some w e W. We have shown:
(1) If p e U(tc) then there exists a constants q(p) and a continuous seminorm
ap on K such that if 0 < e < 1 and if t e T£ then
|pS(w)(t)| < e-«"'>oi,(w).

280
7. Cusp Forms on G
If p e U(tc) then p = X y(Uj)pj with t^ e A. Thus if w e W and if t e T),
then (<jf(j) = q(p.))
\pS(w)(t)\ < X |p,.y(«;)S(W)(t)| < £ Cpe-"^»ap(Uj.W)
by (1). If we set q = max{g(./)} and ^p(i;) = Z op(u}w) then we have shown:
(2) There exists a constant q such that if p e l/(tc) then there exists a
continuous seminorm ^p on V with the property that
\pS(w)(t)\ < e-"n„(w)
for all 0 < e < 1 and all t e Te.
Let c e T and let x = exp X. Let y e t be such that exp(X + tY)eTE for
0 < t < 1. In light of (2) we can apply Scholium 7.3.4 to pS(w)(exp(X + tY))
to find that if p e U(tc) then there exists a continuous seminorm \ip on V
such that |pS(w)(t)| < np(w) for all we W. This completes the proof.
7.A.4.3. We now assume that if a,- is a non-zero multiple of txk then j = k.
Let T, be the set of all t e T such that taj = 1 for exactly one j. Then it is
clear that T u Tt = T" is open in T.
Lemma. Let f e B(T') he such that /or each p e C/(t), p/ extends to a
continuous function on T". Then f extends to a smooth function on T.
We may clearly assume that n = dim T > 2. If a:[0,1] -> T is a smooth
curve then we set
L(<x) = J||<x'(s)||<fa.
(1) If x, y e T" and if e > 0 then there exists a smooth curve joining x, y
with values in T" such that L(a) < d(x, y) + e.
To prove this we will use the following simple result.
Scholium. Let P1,...,pd be real valued linear functionals on R" that are
pairwise linearly independent. Set for i < j,
h = det
"AW Pi(y)~
Jj(x) p}(y)]
Then U = {(x,y)e R" x R"in(<y ]8f- # 0} is open and dense in R" x R".

7. A.4. Some Harmonic Analysis on Tori
281
We first show that y = II, <d fiid is not identically 0 by induction on d. If
d = 2 this is clear, so assume this for 2 < d < r — 1. We now prove it for
d = r. The inductive hypothesis implies that K = {(x,.y) !!!,<,,_, /?u # 0}
is open and dense in Rd x Rd. Since Pd-Ud is not identically zero y is not
identically 0.
We now prove the Scholium by induction on d. If d = 2 the result is clear.
Assume the result for d = r — 1 > 2. If d = r then the inductive hypothesis
implies that 0 = I11<J<„_1 /?y is not identically zero. Thus the set, V, of all
(x, y) such that 0(x, y) is non-zero is open and dense in Rd x Rd. Since
y = 6 n,-<„ pin the result now follows by the first observation in the proof
of this Scholium.
We now prove the Lemma. It is enough to find a broken C00 curve, a,
taking values in T" and joining x, y such that L(a) < d(x, y) + e/2, since
we can smooth the curve changing the length by no more than e/2. We also
observe that it is enough to prove this for x, y in T since T is open and
dense in T. Let Y} be a basis of t such that r = ZZyj. Set F =
{X = 1, XjYj',0 < Xj < 1}. Then exp defines a bijection of F onto T.
Let x, y e T and let X, 7 e F be such that exp X = x and exp 7 = y. Let
/4s) = (1 — s)X + sY. If we interchange some of the strict inequalities and
less than or equal signs used to define F we may assume that if y(s) = exp n(s)
then L(y) = d(x, y). The Scholium above implies that we may assume that
af and tXj are linearly independent on RX + R7 = P for i < j. Thus the set
of all Z e F n P such that there exist i < j such that <x,(Z) and a;(Z) are in
2niZ is discrete in P. We can thus replace \i by a a piecewise linear curve, //,
whose length differs from that of n by any arbitrarily small 3 and if 0 < s < 1
then there is at most one j such that a.j(n'(s)) e 2niZ. a(s) = exp(^'(s)) is the
desired curve.
We now prove the Lemma. Let p e U(t) and let x, y e T". Let for each
j = 1,2,..., a,, be smooth curve joining x toy in T"such that lim^^, L(oy) =
d(x, y). Now
}£(p/(*/O))<fr = p/O0-p/(x).
o at
Write a,-(£) = Z ^(t)^ then
IP/W - P/(J0I < j
A(/W)
dt <Y.\\W\XxPfW\dt
^ I ^kP(/) j II/UOIA < nL^) X a^l/).

282
7. Cusp Forms on G
Let fip(f) = nZ.k oXkP(f). Then if we take the limit as j -> oo in the above
inequality we have
(1) \pf(x) - pf(y)\ < ii„(f)d(x, y) for x, y e T".
Let x e T. Let {x,} be a sequence in T" such that lim Xj = x. Then (1)
implies that pf(xj) is a Cauchy sequence in C. Thus in light of (1) we can
define pf(x) to be the limit of this sequence. Furthermore, this extended
function clearly satisfies the estimate in (1) for all x,yeT and all p. The
Lemma now follows.
7.A.5. Fundamental solutions of certain differential operators
7.A.5.I. In this appendix we will prove a result due to De Rham [1] and
Gelfand, Shilov [1]. Our proof follows that of Gelfand, Shilov fairly closely.
We record it here for the sake of completeness and because we will use the
technique of the proof in Volume II of this opus.
Let p, q e Z, p, q > 1. Set n = p + q. We look upon R" as R" © R". Set
P(x, y) = |x|2 - \y\2, x e R", y e R". Also set L = Z 32/x? - Z d2/yf. We
will also use coordinates x,,..., x„ with xp+i= yt. Set C = {x e R"! P(x) > 0}.
The obvious calculation yields
(1) LPZ + 1 = 2(z + l)(2z + n)Pz on C for all z e C.
If / e y(R") and if Re z > 0 then we set
Pl(/) = |Pz(x)/(x)dx.
c
Lemma. P\+l(Lf) = 2(z + l)(2z + ri)P\(f) for all f e £f(Rn) and all
Re z > 0.
Set S = {x e R"!P(x) = 0}. Then S - {0} is a smooth hypersurface of
R". Let for each e > 0, SB = {x e S| ||x|| > e}. Put RB = S£[/{x e R"! P(x) > 0
and ||x|| = e}. Set C£ = {x e R" |P(x) > 0 and ||x|| > e}. Then Rc is the
boundary of C£. RE is piecewise smooth so Stokes' theorem is applicable
onQ.
Let u e C^iC) be such that u and (3/3x,)w have continuous extensions
to C1(C) and
(a) |u(x)| + Xl(9/9^,-)»WI ^ C||x||d forsomeC>0 and
d > 0 and all x e R".
(b) «(S) = {0}.

7.A.5. Fundamental Solutions of Certain Differential Operators 283
If we apply Stokes' theorem, we find that if / e £f{R") then
f u(x)(d/dxi)f{x)dx = lim J u(x)(3/3x,)/(x)dx
c ' c-or
= - J (3/3xi)u(x)/(x)dx.
c
(We leave the details to the reader.)
If Rez>l then u(x)=P=++1M and u(x) = (3/3x,-)Pz+1M satisfy (a)
and (b) above. Hence
J Pz+1(x)L/(x) = J LP* + l(x)f(x)dx.
c c
This combined with (1) above implies the Lemma for Re z > 1. Since both
sides of the equation that we are proving are holomorphic in z for Re z > 0,
the Lemma follows.
7.A.5.2. The above result implements a meromorphic continuation of
P+ for z e C. More precisely we have
Lemma. // /e^fR") then z i-» P+(/) has a meromorphic continuation
to C. The poles are contained in the union of the sets {—1,-2,...} and
{ — n/2, — n/2 — 1,...}. // n is odd then the poles are all simple. Furthermore,
P\ and Resz = wP+ define tempered distributions on R".
The first part of the Lemma follows from
(1) P\+i{Uf) = 2^(z + l)-(z + j)(2z + n)-(2z + 2(j - 1) + n)Pl(f).
Since this tells how to define Pz{f) for Re z > —j. The last assertion follows
from
\Pl(f)\< I ||x||2Rez|/(x)|dx
for Re z > 0 and (1).
7.A.5.3. We now do a different analysis of P\. Let S, (resp S2) be the unit
sphere of Rp (resp. R'). Let dax (resp. da2) be respectively the rotationally
invariant measures on S, and S2. Then (up to a constant depending only
on p, q)
go r
(1) Pz(/)= J |rp-1s"-l(r2-y2)z J f(ral,sa2)dalda2dsdr.
0 0 S, xs2
Let lc denote the characteristic function of C. (1) implies

284
7. Cusp Forms on G
Lemma. // Re z > — 1 then (lc^)z is locally integrable on R".
Let / be a non-negative, smooth, compactly supported function on R".
Assume that supp / is contained in {(x,_y)|||x|| < N, ||y|| < N}. Then
J |lcP(x)|Rez/(x)dx < C(f) J j rp~ lsq-aX(r2 - s2)Rezdsdr.
R" 0 0
If we use the coordinates r and t with s = tr for 0 < t < 1 then the second
integral becomes
] r2Rez + n- l dr\ f- l{\ - t2fez dt,
o o
which is finite for Re z > — 1.
7.A.5.4. We continue with the analysis of the previous number. Set for
r, s e R,
u(r, s) = | f(rol,so2)doldo2.
Then u e ^(R2) and u is even in both variables (1) If g e 5^(R2) is even in
both variables than h(x,y) = g(xl/2,yl/2) defines an element of £f(U) where
U = {(x,y) x, y > 0}. (See 7.3.4 for the definition of ^(U) for U open in a
Euclidean space.)
Taylor's theorem implies that if x > 0 and if y e R then there exists
0 < 6 < x such that
\(dm/dym)f(x,y)- X (dk+m/dxkdym)f(0,y)(xk/k\)\
k<N
= {xN+ 1/{N + l)\)\(dN+ l/dxN+l)f{6, y)\.
Thus if 0 < x < 1 then
(i + y2Y\(dm/ym)f(x,y)-l ^k+m/^kdym)f(0,y)(xk/k!)\<xN+lPr^N(f)
k<N
with prMiN a continuous seminorm on ^(R2).
Now
(3'1+7ax'ia};m)/(0,y) = 0
if k is odd. Thus, if we substitute x1/2 for x in the above inequality we find
that if 0 < x < 1 and if y e R then
(1 + y2Y\(dk+m/xkym)f(xll2,y)\ < Cr,k,m.

7.A.5. Fundamental Solutions of Certain Differential Operators 285
Thus, v(x,y) — f(xll2,y) defines a Schwartz function on {(x, y) i x > 0,
yeR} which is even in y. We may thus repeat the above argument in the y
variable to finish the proof of (1).
(1) implies that u(r,s) = v{r2,s2) with v e <¥(C). Hence
p\(f) = | (f r"' lsq- V - s2fv(r2,s2)dsdr.
0 0
We make the change of variables (x, y) ~ (r2,s2) and then (x, y) = (r, tr)
with r > 0 and 0 < t < 1 and obtain
P^(/)= |rz+"/2_l |t,/2"l(l -t)MMr)d£dr.
o o
Set
<t(z,r) = }t"/2-l(l - t)zt;(r,tr)dt.
o
Taylor's Theorem implies that
<|l-t|m + 1Pm(")
with pm a continuous semi-norm on Sf(C). Thus if Re z > 0 we have
v(r,tr)- I ((f-l)V./!)(9V9tJ>'(r,tr)|r =
1
<t(z,r)= X (ry//!)(3V3^M',.',)f t,/2"'(l - t)I + ^dt + EJr.z).
0<;<m 0
Furthermore, E(r,z) is holomorphic for Re z > — m and |£(r, z)| < Cm(z)qm(u)
with <jfm a continuous semi-norm on ,C/(R2) and Cm a continuous function of
z for Re z > — m. We can argue in the same way to get similar estimate on
(1 + r2)'I(a7ar,')<I)(z,r).
We therefore see that <t(z, r) has a meromorphic continuation to C with at
worst simple poles at —1, -2,.... Furthermore, $(z,-)ey(R) where it is
holomorphic and the residues at the poles are Schwartz functions.
We observe that
(*) P\(f)= ] rz + n/2-l<S>(z,r)dr.
o
We note that
GO
J rzl2+nl<S>(z,r)dr

286
7. Cusp Forms on G
is holomorphic wherever <t(z, r) is. Thus to implement the analytic
continuation of (*) away from — 1, — 2,... we may look at
i
(**) \rzt2+nlQ>(z,r)dr.
o
7.A.5.5. We now look at the case when n is odd. Then (**) above
implies that the poles other than — 1, — 2,... are at most simple poles at the
points — n/2, — n/2 + 1, — n/2 — 2, Furthermore
Resz=_„/2Pz+(/) = <t(-n/2,0).
The calculations in 7.A.5.4 imply that
<t(-n/2,0) = «(0,0) | f'2- '(1 - tydt\z=.n/2.
o
Let B(z, w) denote the classical beta function (B(z, w) = Y(z)Y(w)/Y(z + w)
and T(z) is the classical gamma function). Then the usual integral formula
for B(x, y) yields
J I"12' l(l - t)zdt = B(q/2, z + 1) = T{ql2)T{z + \)/T(z + q/2 + 1).
o
If q is even the value of this function at —n/2 is non-zero. We have therefore
shown
(1) If n and p are odd then Resz = -n/2P+(f) = Cpqf(0) with Cpq non-zero.
Lemma. Assume that n is odd. Set
F(x)=lc((-l)«x)|P(x)rl/2.
There exists a non-zero constant Cpq such that if f e £f(R2) then
\F(x)L^f{x)dx = CpJ(0).
If q is even then this follows directly from (1) and 7.A.5.2(1). If q is odd
then replace P by — P and L by — L.
7.A.5.6. We now look at the case when n is even. We first assume that p and
q are odd. As before we begin with the material in 7.A.5.4. In this case it is
clear that <t(z,r) has a pole at z = —n/2. Thus Pz+{f) has a double pole at

7.A.5. Fundamental Solutions of Certain Differential Operators
287
z = — n/2. If we argue as in the previous number, we find that
(1) (z + n/2)2P*(f)\z = n/2 = CpJ(0).
We have
Lemma. // p and q are odd then there exists a non-zero constant Cpq such
that
J lc(x)L"'2f(x)dx = CM/(0)
for all f e y(R").
7.A.5.7. We now analyse the case p and q even. In this case one checks that
<t(z,0) is holomorphic at —n/2. Thus we see that Pz(f) has a simple pole at
z = —n/2 whose residue is a non-zero multiple of f(0). On the other hand
7.A.5.2(1) implies that
Resz=_„/2PZ(/) = C(d/dz\^0)P*(L"'2f)
with C a non-zero constant. This implies
Lemma. // p and q are even then there exists a non-zero constant Cpq such
that
J lc(x) log \P(x)\L"'2f(x)dx = CM/(0)
for f e .S^(R").
7.A.5.8. We now put all of this material together, and we drop the
assumption that p, q > 1.
Theorem. Let n > 2. Let p, q be non-negative integers such that p + q = n.
Define Fpq as follows
FP,q(x) = lc(x)|P(x)|"1/2 if n and p are odd,
Fpjx) = lc(-x)| P(x)r1/2 if n and q are odd,
Fp.q(x) = IcM ^ P ar>d <5f are odd,
FP,q(x) = l°g |P(*)I if P and <? are even.
Then Fpq is locally integrable and there exists a non-zero constant Cpq such

288
7. Cusp Forms on G
that
for f e £f(R").
If p, q > 1 then the result follows from the above discussion (note the
change in the case p, q even). The only case we have not checked is n even
and q = n. We leave this to the reader (Hint: Argue as in the previous
number using C = {x \P(x) < 0}. Only the "r-integral" plays a role.)

o Character Theory
Introduction
The purpose of this chapter is to develop Harish-Chandra's theory of
characters of real reductive groups. In his early papers, Harish-Chandra,
realized that the correct infinite dimensional generalization of the usual
character of a finite dimensional representation was as a distribution given as the
trace of an operator on the representation space (see Section 8.1). Although
the definition of the character of a (g, K)-module is quite natural, it is not at all
clear how to apply it as a computational tool. The power of the character
theory of real reductive groups rests on Harish-Chandra's regularity theorem
(8.4.1). As a consequence of this theorem it can be shown that the character
of an irreducible (g, K)-module is given by a formula that is (formally) quite
similar to the character of a finite dimensional representation. Harish-
Chandra gave two important (intimately related) consequences of his
regularity theorem. The first was a characterization of tempered
representations in terms of the growth of their characters. The second was his
determination of the irreducible square integrable representations of a real
reductive group. We conclude this chapter with these applications. Our
exposition of these results does not stray very far from Harish-Chandra's
original papers.
289

290
8. Character Theory
We have benefited from Varadarajan's treatment of the regularity theorem
(Varadarajan [1]), Our exposition is a bit simpler than that of the original
since we have avoided the use of the notoriously difficult Theorem of Harish-
Chandra on analytic G-invariant differential operators that annihilate the G-
invariant functions (c.f. Varadarajan [1, Thm23, p.143, part 1]). In order to
achieve this simplification, we prove a stronger theorem on the Lie algebra
(8,3.3) than the original of Harish-Chandra. The key to our approach is
Lemma 8.A.3.7, which was suggested to us by Duistermaat.
As is usual in this book, we have included several appendices to this chapter
that either contain standard results that will be applied in the body of the work
(e.g., trace-class operators, elementary Fourier theory and basic distribution
theory). There are also several technical results (that could very well have been
included in the pertinent proofs) that we have opted to include as appendices in
order to help clarify the flow of the arguments.
8.1. The Character of an admissible representation
8.1.1. Let G be a real reductive group and let K be a maximal compact
subgroup of G. Let £f(G) be as in 7.1.2. Fix a norm, ||---|| (2.A.2.3), on G. Let
px^r denote the seminormpxya(,r of 7.1.2. with a = || - - -1| and i> = 1. Fix d such
that
\\\g\\-ddg<K.
G
Let (n,H) be a Hilbert representation of G. Lemma 2.A.2.2 implies that
there exist positive constants r and C such that
(1) \\n(g)\\ < C||0||'.
This implies that we can argue as in 1.1.3 to define, for each / e £f(G), an
operator n(f) with
(2) ||7r(/-)|| < ClPl.ljd+r(/)
with C, depending only on n.
Lemma. Assume that (n, H) is admissible and finitely generated. Let {vj} be
an orthonormal basis of H such that each Vj is contained in a K°-isotypic
component of H. Then there exists a continuous semi-norm, p, on if{G) such that
X>(/HII<;p(/) forfeST(G).

8.1. The Character of an Admissible Representative
291
Since G = KG0 and G/G° is finite, there exist kl = \, k2,...,kme K such
that G = (J, i/im fc,G° and each subset fc,G° is a connected component of G.
If / e y(G) therTwe define for each /, ft(g) = /(/c^) for g e G°. We extend /,
to G by 0. Then /; e £f(G), f = Z L(jy/,. It is also clear that the maps f >-* ft
are continuous on if{G). We note that if g e G then n(L(g)f) = n(g)n{f). We
assume (as we may) that rc restricted to K is unitary. Suppose that we have
found {Vj} and p such that the assertion of the lemma is true for / e tf{G)
such that supp / c G°. If / e ,9"(G) then £ ll^/t^ll = £ ||Z w(k,)w(/,)t|,|| <
^j lk(/i)«/ll ^ ^1£,<m P(/i)- Thus if we set q(f) = Z p(/;) then the result
follows from the special case. Now n restricted to G° is admissible and finitely
generated (4.2.7). The result will therefore follow if we prove it in the special
case when G = G°. So assume that G is connected.
Let CK be the Casimir operator of K corresponding to B|tx(. If y e Ka then
let Xy denote the eigenvalue of CK on any representative of y. Let T be a
maximal torus of K° and let P be a system of positive roots for K with respect to
T. Let p be (as usual) the half sum of the elements of P. If Ay is the highest
weight of y then
Ay = ||Ay + p||2-||p||2.
Also
d(y)= n(Av + P.«)/(P.«)-
This implies that there is a constant, C > 0, such that
<*(y)<C(Ay + ||p||2)'
withp= \P\/2.
There is a positive integer JV such that the number of y e KA with highest
weight Ay is at most N. As in 7.A.4.1
I (^ + iipii2 + ir<oo
for r > dim T/2. This implies that
(i) I d(y)2(Xy + \\p\\2 + !)-'<<» for r > 2p + dim T/2.
jeJE"
Proposition 4.2.3 says that there exists a finite dimensional representation,
a, of P0, a minimal parabolic subgroup of G, such that HK is (g, K)-isomorphic
with a submodule of X" (see 4.2 for the pertinent notation). Frobenius
reciprocity implies that
(")
dimH{y)<d{a)d{y)2 for yeKA.

292 8. Character Theory
Set D = CK + (||p||2 + 1)7. Let y e KA and let x e H(y). If / e 5"(G) then
||7r(i?(D")/)x|| = (||p||2 + /v+l)"||7r(/)x||.
Hence (2) implies that ||jr(/)x|| < ClPl,D,,r+d(/)(||p||2 + Ay + \y\\x\\.
Let {vj} be an orthonormal basis of H such that each Vj is contained is an
isotypic component of H. Let q > 2p + dim T/2. Then
LlW/RII <C,f I d(ff)d(7)2(llpll2 + A,)-«)p1>fl,,r+(l(/).
Thus (ii) implies the result.
8.1.2. Let (n,H) be as above. The preceding Lemma implies that if / e
Sf(G) then 71(f) is of trace class (8.A.1.5). We set ©„(/) = tr 71(f). Let {Vj}, p
be as above. Then |tr n(f)\ < Z ||tt(/)i|,-|| < p(f), f e .^(G). Thus 0„ defines
a continuous linear functional on if(G) which we call the distribution character
of 7%.
We may also assume that each Vj is contained in some isotypic component
of H relative to K, If y e KA then set F(y) = {j\ v} e H(y)}. Then if £v is the
orthogonal projection of H onto H(y) then
Z <tt(/)^,^> = trEy7i(f)Ey.
Set 0^(gf) = tr Eyn(g)Ey. Then 0J is a real analytic function on G and
(1) ©„(/)= I lf{g)K(g)dg for/e^(G).
Lemma. If (n,H) and (a, V) are admissible finitely generated Hilbert
representations of G such that HK and VK are (g, K)-isomorphic then &K = 0„.
In light of (1) above, in order to prove this result it is enough to show that
<t>i = 4>l for all 7 eK\
Since HK and VK are isomorphic it is clear that x</>£ = x(j)y„ for all x e
[/(g), fc e K and y e K A.
Since $1 and <f>l are real analytic and G = KG0 this implies that they are
equal.
8.1.3. The above Lemma implies that if V is an admissible finitely generated
(g, K)-module and if (n, H) is a realization of V then 0^ depends only on V. We
may therefore write &v for 0^. We will also call &v the distribution character
ofV.

8.1. The Character of an Admissible Representative
293
Lemma. Let
0-K-W-Z-0
be an exact sequence in H (4.1.4). Then &w = &v + &z.
Let (n, H) be a realization of W. We assume (as we may) that V is a sub-
module of W. Let //, = C1(K). Then Hl gives a realization of V and /////,
gives a realization of Z. As a Hilbert space /////, = (Z/,)1. The lemma
follows if we split the sum giving the trace into the part corresponding to //,
and the part corresponding to (H,)1.
8.1.4. Lemma. If K,,..., Vd are nonzero mutually nonisomorphic objects in
H then &Vl,..., &Vd are linearly independent.
Let for each j, Kj be a realization of Vj. Set for y e KA, </>£. = </>]. In light
of the material in 8.1.2, it is enough to show that for each y e KA, the nonzero
0}' are linearly independent. Fix y e KA. After relabeling we may assume that
Vj(y) is nonzero for j <r and is zero otherwise. 3.5.4 (3.9.7-9) implies that each
of the [/(g^-modules HomK(Vy,Vj) is irreducible. If x e U(qc)k then set
Hj(x) equal to the trace of the action of x on Hom/i:(K),, Vj).
A direct calculation (which we leave to the reader) yields
d(y)Hj(x) = ,x0J(l) for x e U($c)K.
Thus Corollary 3.A.1.3 implies that (f>\,..., 4>l are linearly independent.
8.1.5. If V e <ff then K is of finite length as a (g, K)-module (4.2.1). Let V =
K, => K2 => • • • => Vd => Vd+, = {0} be a Jordan-Holder series for V. If W is an
irreducible object in H then we say that the multiplicity of W in V is the number
of indices, j, such that Vj/ Vj+, is isomorphic with W. Notice that the previous
Lemma implies that the multiplicity is independent of the choice of Jordan-
Holder series. If W has positive multiplicity in V then we say that W is a
constituent of V.
Theorem. If V, W e H and if &v = &w then V and W have the same
multiplicities for their irreducible constituents.
This is an immediate consequence of Lemmas 8.1.3 and 8.1.4.
8.1.6. Let V be an irreducible (g, K)-module with distribution character &v.
If / e ,C/{G) then set r(g)f(x) = f(g lxg) for x, g e G.

294
8. Character Theory
Lemma.
(1) &y"z(g) = Qy for geG.
(2) If V has infinitesimal character x then z@v = x(z)®v f°r aU z e Z(gc)-
Let (tz, H) be a realization of V.lffe V{G) then tt(t(0)/) = n(g)n(f)n{g)~'.
Thus 0„(T(g)/) = tr k(0M/>(0)- ' = tr n(f) (Corollary 8.A.1.10).
If /ey(G) and if zeZ(gc) then n{zrf) = ji{z)ji{f) = x(z)n(f). Thus
©^(z7/) = #(z)0K(/). Hence (2) follows from the definition of the action of
[/(g) on D'(G) (8.A.2.7).
8.1.7. A continuous functional © on Sf(G) is said to be central if
0 „ T(g) = 0 for all geG.
It is said to be an eigendistribution with infinitesimal character # if z0 = x(z)®
for all z e Z(gc). Thus if K is an irreducible (g, K (-module then &v is a
central eigendistribution. In the next section we give the relation between the
K-character and the G-character. In Sections 3 and 4 we will prove several
theorems of Harish-Chandra that give the local structure of invariant ei-
gendistributions. We will then apply these results to distribution characters
to (in particular) complete the theory of the discrete series.
8.2. The K-character of a (g, K )-module
8.2.1. We retain the notation of the previous section. If y e KA then we use
the notation q for the character of y and d(y) for the dimension of any element
of y. We endow C™^) with the topology defined by the semi-norms
vD,K(/) = sup{|D/(fc)||fceK},DeI/(f)
Lemma. Let V be an admissible finitely generated (g,K)-module. If y e KA
then set mv(y) = dim Hom/i:(K),, V). If f e C™^) then the series
X my(y) I r,y(k)f(k)dk
yelC- K
converges absolutely and defines a continuous linear functional &K v on CX(K).
We will use the notation and results in the proof of 8.1.1. As in 8.1.1 we may
assume that G = G°. Set
Ty{f) = \ny{k)f{k)dk for yeK\

8.2. The K-Character of a (g, K)-Module 295
Then Ty((l + CK)rf) = (1 + A/Ty(/) for feC^iK). It is obvious that
\rjy(k)\ < d(y) for k e K. Thus
|T,(/)|<d(y)v,,K(/).
We therefore find that if we set D(r) = (/ + Q)r then
|Tv(/)|<(l+;.y)-rd(y)vD(r),K(/).
We have also seen that mv(y) < Cd{y) for all y e KA (here of course C is
independent of y). Thus the series that we are estimating is dominated by
c( I (1 + ^rrd(y)2)vmrhK(f)-
8.1.1(1) implies that the above series converges if r is sufficiently large. If r is
that large then
\®K,vif)\ ^ CrVmrhK(f)-
8.2.2. We will call &KV the K-character of K We now relate the K-character
to the distribution character. We will assume that G has a compact Cartan
subgroup. Let K" be as in 7.4.1. If / e ,(f{G) then we set
iPf(k) = |det((7 - Ad{k))\p)\lf(gkg-l)dg
G
for k e K" and equal to 0 if k e K — K".
Theorem. Let e > 0 and /et / e c/(G) be such that supp f cz Gce (7.4.3). Then
(1) trzC*'(K).
(2) er{f) = ®K.v(<Pf)-
Here we are using the same normalization of dg to define &v and [j/f.
We have seen in 7.4.4 that Qf e CX(K"). Our assumption on the support of
/ implies that Qf has compact support in K'. This implies the first assertion.
We now prove (2). Recall (7.4.4)
(i) If k e K" n G'ee, and if g e G then log \\gkg~11| > Cel/2 log ||</||
with C a positive constant independent of k and g.
If / e ,5^(G) then set F9(fc) = |det((7 - Ad(k))\v)\f(gkg-1). The argument at
the end of 7.4.4 proves
(ii) If us [/(!),£ >0 then for each r > 0 there exists a continuous semi-norm

296
8. Character Theory
vurBon £f(G) such that
\uFg(k)\<\\g\rvu,M)
for / e ff{G) with supp(/) c G£e and all keK.
Let (7r, //) be a realization of V. Let {t^v} be an orthonormal basis of H(y) for
y s KA. (2) combined with the argument in the previous number implies that if
/ e Sf(G) has support in GBe then
(iii) I
J F,(fc)<K(fc)«,.„ !*-,/><** < Vr,£(/)||0||-r
for all r > 0.
Here vr £ is a continuous seminorm on .^(G).
This implies that if / is as above and if Tg = (n\K)(Fg) then Tg is summable on
//(8.A.1.4) and HTJj < vcr(f)\\g\\-r for all r > 0. A direct application of the
integral formula in 7.4.2 implies that if v, w e H then
<7i(/>,w> = J (n(g)Tg7t(g-l)v,wydg.
G
The above inequalities allow us to interchange summation and integration
to find that
©„(/) = J ir(n(g)Tg7i(g)l)dg = J tr Tqdg.
G G
This in turn implies that
®K(/)=ffZldet((/-Ad(k))|p)|/(gkff-,)<w(k)i;j,,,i;^>dkdg.
G K j.y
In light of (iii), we may interchange the G and K integration and the
summation. The theorem now follows.
8.3. Harish-Chandra's regularity theorem on the Lie algebra
8.3.1. We retain the notation of the previous sections. If X e g and g e G
then we write gX for Ad(g)X. If ft is an open G-invariant subset of g and
if /e (^(ft) then we write r(g)f(X) = f(g~lX) for g e G and XeQ. Set, as
usual, £>'(ft)G = {Te £>'(ft)| Tz{g) = T for 9 e G}. Let D be as in 7.3.9 and
set g' = {X e g| D(X) # 0}. Put fi' = fin g'.
Let h,,..., hs be a complete set of non-conjugate Cartan subalgebras of g.
We set ft} = G(ft' n fy). Then ft' = (J ft}. If g e G and if H e ft' n fy then
set 4*/0, H) = .9//. 8.A.3.3 implies
(1) 4*j is a submersion of G x (ft' n bj) onto ft}.

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra
297
Fix dX, a Lebesgue measure, on g. We will also write dH for a Lebesgue
measure on each h;-. As in 7.A.2, we look upon S(g) as the algebra of constant
coefficient differential operators on g. Put 7(g) = S(g)G (7.A.2.8).
8.3.2. Fix a Cartan subalgebra, b, of g. Let <t = <t(gc, hc). If a e <t and if
a(h) c R (resp. a(h) c i'R) then we say that a is real (resp. imaginary). Let <&R
and <t7 denote respectively the sets of real and imaginary roots. Set TR = <bR,
T, = i<D, and r = TR u r7. Put h" = {// e h |«(//) # 0 for all a e T). Clearly,
b" ^ h'.
Lemma. Let Cbea connected component of h". Then there exist yl,...,yqeT
such that
(1) y1,..., yq are linearly independent,
(2) C = {Hel)\yj(H)>0,j=\,...,q}.
Furthermore, C n h' is connected.
If a e T then a(j(g)) = 0. We may thus assume, without loss of
generality, that g is semi-simple. Set \)R = {H e hc|a(H) e R for asO}. Then
f) = (f)u n W © O'bu n W- ^ a e Fr (resp. a e T7) then a(ihR n h) = 0 (resp.
a(hRnh) = 0). Set Xbs^b) = {// e hR nh|a(H) # 0 for a e TR} and
X('I)k) n h) = {// e ((ihR) n h) | a(H) # 0 for a e r,}. Then a connected
component of h" is of the form C, x C2 with C, (resp. C2) a connected component of
(b« n b) (resp. XCbu)n b))- Since TR and T, are both root systems the first
assertion follows from 0.2.4.
Set E = <t - (<tR u <t7). If a e X then it is clear that the real and imaginary
parts of a are linearly independent. Thus (b/)a = {H e b/| a(H) = 0} is of co-
dimension 2 in \)j. Thus
Cn(h,)' = C- (JO,,).
which is connected.
8.3.3. For the rest of this section we will assume that Ad(G) acts trivially on
the center of g. If </>,,..., </>,, are homogeneous Ad(G)-invariant polynomials
on [fl> 9] then we set for r > 0
Q(01,...,0d,r)={XE[g,g]||0i(X)|<r,i=l,...,d}.
Let U be an open connected subset of 3(g) = 5. Put Q = {X + 7|Xe [/,
^W *,,r)}.

298
8. Character Theory
Lemma. ft is connected. Furthermore, if h is a Cartan subalgebra of g and if C
is a connected component of h' then C n ft is connected.
If X eU and Y e% 0d,r) then X + t7Eft for 0 < t < 1. This
clearly implies that ft is connected. We now prove the second assertion. It
is enough to prove it in the case when g = [g, g]. Let B be a convex
neighborhood of 0 contained in ft n b. Let C be a connected component of b".
Lemma 8.3.2 implies that C is convex. Hence C n B is convex. If X e ft n C
then there exists t > 0 such that tX e B nC. Thus C n ft is connected. The
second part of Lemma 8.3.2 now implies the result.
8.3.4. Theorem. Let ft be as in the previous section. Let T e D'(ft') be
such that dim I{qc)T < oo on ft'. Then there exists an analytic function FT =
F on ft' such that
(1) T=TFon ft' (see 8.A.2.2 for TF),
(2) If b is a Cartan subalgebra of g then there exists an analytic function fi
on b" which is an exponential polynomial on each connected component of b"
(8.A.2.10) such that F|Qnb. = \D\-l/2j8.
Furthermore, if we extend F toQby setting F = 0 on ft — ft' then F is locally
integrable on ft.
We may assume that b = by 8.A.3.5 implies that
T°(pT) = |£>r,/2p|£>r/2¥,0(T)
for p e 7(g). Thus dim 7(gj(|£)| >/2vF?(T)) < oo. We have seen that S(bc) is
finitely generated as an /(g)-module. Thus dim S^c)^?^) < oo. Lemma
8.A.2.10 implies that there exists a function fy on ft' n by whose restriction
to every connected component is an exponential polynomial and is such that
y°(T) = \D\~ll2T0J. If X e ft} with X = gH, H e b-, then set j8(X) = %H).
If F= |£>r1/2j8thenT= TF on ft'.
We note that if we extend /? to ft by 0 then /? is locally bounded. We have
seen (7.3.9) that |£>r1/2 is locally integrable. The last assertion now follows.
Lemma 8.3.3 implies the asserted extension properties of each fy.
8.3.5. We now come to the main result of this section which is an extension
of a fundamental theorem of Harish-Chandra. Let Xl,..., Xn be a basis of g
and define X1 by B(Xh Xj) = 5U. Put □ = Z X;X'. Then □ e 7(g).

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra
299
Theorem. Let ft be as in 8.3.3. Let Te £)'(ft)G be such that dim /(g)T < oo
on ft' and dim C[D]T < oo on ft. Let F = Fr (8.3.4). Then T = 7>.
The proof of this result will take up the rest of this section. Before we enter
the details of the proof, we first develop some results on distributions on ft
that are supported in U ® Jf{S.A..4.2). We note that if / is a G-invariant
polynomial on g then f(X) = f(Xs) for leg (see 8.A.4.1 for Xs). Thus
ft n (3 ©.#") = U©.4\
8.3.6. Until we specify otherwise we assume that g = [g, g]. Let Jf =
0, u 02 u • • • u 0r with Oj = GXj and 0, open in Jf, 02 open in Jf — Ox,
etc. (Corollary 8.A.4.7(2)). Set Jp = (J/>P0r Then Jf is closed in 9. We may
assume that X = Xj and that X is non-zero.
Let H, X, Y be a standard basis for a TDS, u, in g (Lemma 8.A.4.1). As a
u-module under ad, g is a direct sum of irreducible submodules, Vm, with
dim Vm = nm + 1 and ^m is a natural number. The eigenvalues of ad h on Vm
are simple and are given by \xm — 2fc, for k = 0 to /^m. The — ^m eigenspace
is gy n Km and XKm is the sum of the eigenspaces for ad h with eigenvalues
strictly greater than — jim. This implies that
(1) 9 = 91'©[A',9].
Set V = QY. If a e G and if Z e K then set <D(g,Z) = g(X + Z). Then
d%,o(9, V) = g(V + [X, g]) = g. This implies that there exists an open
neighborhood, V~, of 0 in K such that X + V~ eft and 0 restricted to
G x K~ is a submersion onto its image.
We note that <t(G x V~)riA'j is open in ,/lj. Let W be an open G-invariant
subset of g such that WnJ] = Oj. Let V;= {Z e V~ \<b(g,Z)e W for
geG}. Then K;isan open neighborhood of 0 in V~ and <t(G x Vj)r,Jfj = Oj.
If X = 0 then we take Vj = ft. The main result of this number is
(2) Let Oj c ft and Xs e Oj. Let Vj be as above for X = Xj. There exists
a neighborhood, C/j, of 0 in Vj such that if we put <&j{g,Z) = g(X + Z) for
</eG and X e Lf- then
(i) <tj is a submersion onto an open neighborhood, ft;, of X in ft.
(ii) Q}nJ]=Oj.
(iii) (Xj+Uj)nOj={Xj}.
It is clear that any open neighborhood of 0 in Vj satisfies (i), (ii). We must
therefore show that we shrink Vj to satisfy (iii). If Xj = 0 take Vj = U}. We

300
8. Character Theory
therefore assume that X = X} is non-zero. Let {X, Y,H) be as above for X.
Let Wm denote the sum of the eigenspaces for ad H on Vm with eigenvalues
strictly less than nm. Set W equal to the sum of the Wm. Then ad X is a linear
isomorphism of W onto [X, g]. Ths implies that there exists a
neighborhood, W0, of 0 in W and a neighborhood U' of 0 in Uj such that
x, Z -> <fy(exp x, Z)
is a diffeomorphism of If0 x U' onto an open neighborhood of X in
g(. Let Wl be an open neighborhood of 0 in W0 such that e"diWl)X is a
neighborhood of X in ^. If we shrink W0 and I/' we may assume that
<D/exp W0, U') nJ^c exp(ad W{)X. Suppose that Z e C/' and X + Z e Os.
Then X + ZeOjn <D/exp W0, [/'). Thus X + Z = eadvX with ceW,. Hence,
<fy(l,Z) = <fy(exp i;,0). This implies that v = Z = 0. Thus we may take
C/j = l/'in order to satisfy (iii).
We now assume that g = j © [g, g]. Let U} be as above. We will now use
the notation U} for U © Uj. We will also write <tj for the map by the formula
in (2) above.
8.3.7. Let E be the vector field on g defined by
d
Ef(x + y) = jt(f(x + ty))t = l, x e j, y e [g, g].
If x,,..., x„ are linear coordinates on g such that {xj}i<, are linear
coordinates on [g, g] and {xjo, are coordinates on j then
E = £ x,-3/3x(.
*£«
Lemma. Let F be the space of all distributions supported on (3©/)nQ.
If T e F then dim C[£]T < oo and the eigenvalues of E on F are all real
and strictly less than —q/2.
Let j be fixed and let OjcQ. Let X e j © 0, and let <fy, Lf, K, Q; be as
in 8.3.6(2). Assume that 0, # {0}. Let y,,..., yd be linear coordinates on
K n [g, jj] such that yk(V n Fm) = 0 if m # k. If Z e K, write Z = IZm with
Z,eFn Fm. We note that ad HZm = ~nmZm. It therefore follows that
(i) (d<b,)a,z&n, I (i/i- + i)zj = a(A- + Z)
= <ty(g(, Z) for g e G and Z e Uj.

8.3. Ilarish-Chandra's Regularity Theorem on the Lie Algebra
301
Since <t>, is a submersion, we may define <t° as in 8.A.3.2(2). (1) implies
(2) <t°(£T) = (I (inm + \)ym d/dym)^(T).
The choices in 8.3.5(2) imply that if supp(T) c (3 © A/,) n ft then
supp <t°(T) cz [/ x {0}. Let Fj denote the subspace of those elements of F
with support contained in (3©. I^nft We prove by downward induction
that if T e Fj then dim C[£]T< 00 and the eigenvalues of £ on Fy are strictly
less than — q/2. We assume (as we may) that Or = {0}. Then 8.A.5.4 implies
that £ acts semi-simply on Fr with eigenvalues strictly less that — q < —q/2.
Assume the result for fj+1 we prove it for Fj. Let T e Fj. Then <t°(T) has
support in U x {0}. (2) combined with 8.A.5.4 implies that there exist
a,,...,aseR such that -a,> d + { Z /im such that <t°(n (£ - a,)T) = 0.
Now E (/^m + 1) = q. So X /^m = q — d. Thus —a,- > \{d + q). By the above
suppling (£ — a^T) c fj.+ ,. The result now follows.
8.3.8 Set (o(Z + X) = B(A",X) for Z e 3, X e [9,9]. Let X, be a basis of
9 such that Xj e 3 for i > <jf and B(.X;,.xy) = ef<5y with K; = ± 1. Set D, =
Zi<qr.id2/dxf and D0 = □ — □,. We look upon to as a differential
operator under multiplication. Set h = E + (q/2)I, x = —jco and y = □,. Then
a direct calculation yields
(1) [/i,x] = 2x, [/!,>'] = -2j/, [x,y] = /i.
In other words, x, y, h is a standard basis for a TDS, u. F is a u-module
that satisfies the hypothesis of Corollary 8.A.5.1 (Lemma 8.3.7). Hence
Corollary 8.A.5.1 implies
Lemma. // Te F and if p is a non-zero polynomial in one variable then
p(D,)T = 0 implies that T = 0.
8.3.9. We now record a result that will be used at the end of the proof of
the regularity theorem.
Lemma. IfSeF and if p is a non-zero polynomial in one variable such
that p{\J)S = 0 then S = 0.
We first show that if S e F C e C and if (□ - ()S = 0 then S = 0.
Assume not. Let S = Z S„ with (h~n)% = 0 for some d. Then 0 = (D-().S =
X (D0 - ()S„ + £ D,^. Let / be minimal among the n such that S^ is
non-zero. Since (h - (n - 2))d D,S^ = 0. We find that D,SA = 0. Thus

302
8. Character Theory
8.A. 5.1 implies that Sx = 0. This contradicts our definition of 1 This proves
our assertion. We now prove the lemma by induction on the degree of p. If
deg p = 0 then the result is clear. Assume the result for all non-zero
polynomials of degree d - 1 > 0. If deg p = d then p(t) = (t - Qq(t) with ( e C
and q is a polynomial of degree d — I. Thus 0 = p(D)S = (□ — ()(<jf(D)S).
Our assertion above implies that <jf(D)S = 0. The inductive hypothesis now
implies that S = 0.
8.3.10. We now prove Theorem 8.3.5 in two special cases, g = su(2) © 3
and g = s/(2, R) © 3. We do this for two reasons. First of all the proof we
give in these two cases contains most of the ideas in the proof of the full
theorem. Secondly, these two cases are needed to initialize the induction
that will be used to prove 8.3.5. In both of these cases we prove 8.3.5 under
the assumption that Ad(G) = Int(g). So we take G = St/(2) if [g, g] = su(2)
and G = SL(2, R) if [g, g] = s/(2, R).
The rest of this number is devoted to the proof of the result in the case
g = su(2) © 3. We note that g - g' = 3 © {0}. Thus T — 7> is supported on
[/©{0}. Let 5 be the element of D'{su{2)) given by 5{f) = f{0). Then
Theorem 8.A.5.2 implies that if linear coordinates on su(2)
and if we use multi-index notation then
(1) T - TF = X 9' <5 ® T, a finite sum with T, e D'(U).
Since T and TF are invariant under the adjoint action of SU(2), we see
easily that there exist T0,..., Tpe D'(U) such that
(2) T-7> = X(n,)'5<g>7}.
We now compute DTf- TUF. We shall see in the general case that this is
one of the key steps in the proof.
(3) □ TF - TaF = 3 ® S with S e D'{U).
Let 0 e Cf (Q). Put n((f>) = DTF(0) - TLJf#). Then
H(4>) = J F(X)U4>(X) - UF(X)4>(X)dX.
9
Set
Then h = Rh © 3 is a Cartan subalgebra of g and all Cartan subalgebras of g
are conjugate under Ad(SU(2)) to h. We can apply the Weyl integral formula

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra 303
(2.4.3) to find that (up to a scalar multiple)
GO
/i((/>) = | J t2F(th + Z) J n<t>{t Ad(g)h + Z)dgdtdZ
-J J f2DF(f/7 + Z)|0(f Ad(g)h + Z)dgdtdZ.
3 x G
Set *,(t,Z) = tJG#Ad(0)/i + Z)d0. Then %eC*(R), <^(0,Z) = 0,
(3/3t)<IV(0,Z) = 0(0, Z). We also note (7.3.3(1), 7.A.2.9)
<t^(t,Z) = -(82/3t2)^(t,Z) + □o^t.Z) and
tDFftii + Z) = -(d2/dt2)tF(th + Z) + tD0F(t/i + Z).
Set Q(t,Z) = tF(th + Z). Then
H(4>)=~1 ) {Q(th + Z)(d2/dt2)%(th+Z)-((t>2/(t>t2)Q(th + Z)%(th + Z)}dtdZ
GO
+| J {e(tfc + z)n0^(tfc + z)-noe(^ + z)«^ + z)}dtdz.
3 -«i
The properties of F in 8.3.4 imply that the second integral above is 0. We
calculate the first by integrating by parts twice. If / is a function on h such
that / restricted to (0, oo) x 3 extends continuously to [0,00) x 3 and f
restricted to ( — 00,0) x 3 extends continuously to (— 00,0] x 3. Then set
/±(Z) = lim f( + th + Z). The obvious integration yields
f->0 +
M4>) = -jffl.(Z) - e-(Z))(^Wo,Z)<lz
+ !(((s)e).'z»-((s)e)_<Z))','''0-z»^
The above calculations of % now imply that if we set for feCf(U),
S(/) = -f(Q + (Z)-e_(Z))/(Z)dZ
3
then \i = 3 ® S. This proves (3).
We now prove that T = TF on fi. The hypothesis of Theorem 8.3.5 implies
that there exists a polynomial p(X) = Xr + lower order such that p( □ )T = 0.
Thus p(D)F = 0 on Q'. We note that (2) implies that there exist
distributions S0,..., Sr_ , on [/ such that
p(D)TF = Tp(1)F+ X (D,)^®S,
ISr- 1

304
8. Character Theory
Hence
p(n)(T-TF) = P(n)T-p(n)TF = -Tp(L)F- £ (□,)ia®s,
!<r- 1
= - I (D,)'«®S,
i < r - 1
On the other hand (2) says that
p(D)(T - TF) = p(D)f X (n,)'«® 7]Y
If we compare the two formulas (for the same distribution) we find that the
coefficient of (□1)d+r<5 in the second formula is Td whereas the highest
derivative of 5 that occurs in the first formula is (□,)'" l5. This implies that
Td = 0. The argument can now be iterated to show that 7} = 0 for all j. This
completes the proof in this special case.
8.3.11. We now look at the case g = s/(2, R) © j. This case will be done in
essentially the same way as the previous one. However, there is the additional
complication that „¥' is not just {0}. We now begin the analysis in this case.
We set G = SL(2, R).
We note that if X e s/(2, R) then the characteristic polynomial of X
is f2 + det X. Thus if det X is non-zero then X is regular. This implies
(1) ft-ft' = [/©./K
We therefore find
(2) supp(T-TF)cI/©^K
Let F denote (as above) the space of S e Z)'(fi) such that supp S is
contained in U © Jf. Let E be as in 8.3.7. In that number we proved that if
V e F then dim C[£] V < oo and the eigen values of E on F are all real and
< — 3/2. This implies
(3) There exist Xt, i = \,...,q such that -3/2 > Xx > X2 > ■■ ■ > kq, 7} e F
such that (£ - x,)m7} = 0 for some m with T- TF = ^ 7}.
As in the previous case the key to the argument is the calculation of
DTf - TUF = [i. Set
*-G i>

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra 305
If / e Cc°°(s/(2, R)) then set (K = SO{2))
5±(f) = j j f(±s Ad(k)X)dsdk.
K 0
We also denote by 3 the evaluation at 0. We prove
(4) There exist distributions S+, S_ and S0 on U such that
H = 3+®S+ + 5_ ®S__ + 5®S0.
Set
Then Rh © 5 and RH © 5 is a complete set of non-conjugate Cartan
subalgebras of g. If we apply the Weyl integration formula then we find that
/x = cxnx + c2jx2 with Ci, c2 constants and
GO
^(0)={ j t2F{th + Z)$n<l>(tAd(g)h + Z)dgdtdZ
3 -co G
GO
-{ { t2\JF{th + Z)|0(f Ad(g)h + Z)dgdtdZ,
3 - co G
H2(4>) = \ J f2F(f// + Z)j^D<Mf \d{g)H + Z)dgdtdZ
3 - go G"
GO
-j j f2DF(f// + Z) |0(f Ad{g)H + Z)dgdtdZ.
3 - co G
We put for 0e Q0 (ft)
,*,(*, Z) = f j 4>(t Ad(g)h + Z) rf0 and
G
2%(t,Z) = \t\ $ <t>(t Ad(g)H + Z)dg.
a
We note that K = S0(2) = exp Rh. We set for 0 a continuous function on g
4>°(Y) = $ d>{Ad(k)Y)dk.
K
Then as in 7.5.1 we find that
(*) 1%(t,Z) = t]ct>0(t(0e_2x e*) + z)sinh2sds.

306 8. Character Theory
If we make the change of variables u = \t\ sinh 2s we have (as in 7.5.1)
/, l ?V ( ° u + (u2-t2yi2\ \
(*) ^sgnrHo^^^^^^ Q j + zj,,
This easily implies that if (j) = u® v then
(**) 1<t(0+,Z) = 5+(u)v(Z), ,*(()-Z) = -5_(«)«(Z)
and up to a scalar multiple I — J <t(0, Z) = u{0)v{Z).
We apply these results to the calculation of jxx . As in the previous case if we
setQ(f,Z) = tF(th + Z) then
GO
vM) = l j {e(f,z)(-a2/af2)1^(f,z)-((-a2/af2)e(f,z)1<i)(f,z)}dfdz
3 -co
GO
+ { j (e(f,z)D0(1<i)4f,z))-(n0e(f,z))1^(f,z))dfdz.
3 -co
The second term is 0 since there are nojumps in the Z-variable. If we integrate
by parts in the first integral (twice) then (**) implies that jix has the desired
form. We now analyse \i2.
7.3.8(2) implies that
GO
2<t(f,Z)= j <f,°(tH + sX + Z)ds.
- OC
Hence 2<b(t,Z) is smooth and (by the original formula) even in f. This implies
that if (j) = u ® v then
(***) 2%(0,Z) = (5++5_)(u)v(Z) and (d/dt)2%(0,Z) = 0.
Repeating the argument above in this case shows that n2 has the desired
form.
(5) E5± = -25+ and Ed = -35.
The last equation follows from 8. A.5.3. We leave the first as an exercise to the
reader (hint: use linear coordinates x, such that Xi(X)= 1).
As in the previous case, (5) implies
(6) nkTF-TlkF= X (□i)j«5+®^+ I (ai)}5_®Bj
j<k-\ j<k-\
j<k- 1
for appropriate Ajt Bj, C, e D'{U).

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra
307
Let p(x) = xr + lower order be a polynomial in one variable, x such that
p(D)T = 0. Then (7) implies
(7) p(D)7> - TP(D)F = X (□i)'S+®S,+ X (□1)'<5_®[/j
j<r-l j<r-l
with S;, Lf, F, e £>'([/).
Since p(D)T = 0, p(D)F = 0 (recall that we are extending F by 0). Hence
p(D)(T - 7» = -p(D)TF = -p(D)7> + Tp(n)F.
So (7) implies that p(D)(T- 7» = £_2>,> _2r+1 W, with EWj = jW,. On the
other hand (4) implies that
p(n)(T-TF) = p(D)(X7}).
If we expand this in terms of the generalized eigenvalues of E we find that the
term that corresponds to the lowest eigenvalue is (□ t )rTq. The corresponding
eigenvalue is -2r + /.,. Since /, < - 3/2 the above equations are consistent
only if (D,)']1, = 0. Lemma 8.3.8 now implies Tq = 0. Continuing in this way
we find that 7], = 0 for all j. Thus T = TF as asserted.
This completes our discussion of the two special cases.
8.3.12. We now begin the proof of Theorem 8.3.5. We prove the result by
induction on dim[g, g]. If dim[g,g] = 0 then g = j and Q = Q' so the
result is trivial in this case. The next possible dimension of [g, g] is 3. These are
the two cases that were handled in the previous numbers. Assume that the
result has been proved if 3 < dim[g,g] < n. We know it for dim[g, g] = n.
The proof of the inductive step follows the same pattern as in the previous
case.
(1) supp(T- Tf)c.r® U.
Suppose that X e supp(T — TF) and Xs is not in j. Set m = g*\ Then
dim[m, m] < dim[g, g]. Since m contains a Cartan subalgebra of g (2.3.1)
it is easy to see that B is non-degenerate on m. Put V = m1 relative to B.
Set m" = {Ye m|det(ad Y\v) # 0}. If m' denotes the regular elements
of m (internally to m) then m"nm' = g'nm. Clearly, lein". Let M =
{ge G\ Ad(g)Xs = Xs}. We show that there exists a neighborhood, mfi,
of X in m" nQ of the type described in 8.3.3 for M, m. (It is this kind of
neighborhood for which the inductive hypothesis is valid.)

308
8. Character Theory
Now, m = 3® $! © [m,m] with 3, = 3(m)n [g, g]. Write Xs = X0 + Xx
with X0 e -5 and A\ e 3,. We note that X„ e [m,m] (8.A.4.6(2)). Let p be the
maximum of dim V and the degrees of the 0,- (8.3.3) used in the definition of ft.
Let 1 = v0, t;!,..., va be a basis, consisting of homogeneous elements, of the
polynomials of degree < p on 3,. We choose a basis 1 = u0,...,ub, consisting
of homogeneous elements, of the M-invariant polynomials on [m, m] of
degree < p. Then
^(z+y) = XflU.A(Z)«i(n
k.l
for all Z e 3,, 7 e [m,m] (here, of course, a-, Jk, are independent of Z and 7).
Now 0XX) = <M-JQ = 0;(^i)- Thus there exists e > 0 such that |0,(*2)l <
r - e for all i. We may therefore choose an open neighborhood, l)l, of Xl such
that C1(L\) is compact and 10,(7)1 < r - e for all 7 6 01(1/,) and all i.
Set C = supZ6„1(Z,.JU|a,,Jk,/«t(Z)|). Set f = e/2Cb. Put mQ(Ul,..., wb) =
{7 6[m,m]||«j(7)| < f}. If Ye mCl(uu..., ub,t) and if 7, el/, then
0,(7, + 7) = Zai.Jk.oM^i) + 11 «,-,u^i)"im
k k 1 >0
= 0^i) + I I a,-.uM^)"i(n
k 1 >0
The first term has absolute value at most r — e. The second term has absolute
value at most bCi = tj2. Hence 7 + ^ e ft. Thus [/©I/, © mn(«1,...) uh,t)
is a neighborhood of X in ft n m.
Set >?(7) = det(ad Y\v) for 7Em. We note that deg r\ = dim V.
Hence ^(7, + 7) = Zu cut;k(71)u1(7) as above. Now \n(Xs)\ = M*,)!. So
^(X^l = A > 0. Let C/2 be an open connected neighborhood of Xl with
compact closure in [/, such that |>y(7)| > A/2 for 7 e l/2. If we argue as above
we may choose 0 < f' < f such that if Yx e C/2 and if 7e mft(u!,..., ub,t')
then 1*7(7, + Y)\ > A/4 > 0. Set mft = [/© t/2 © mft(u,,..., u„,f'). Then mft
is a neighborhood of X in m" n ft.
If g e G and if 7 e mft then set ^(g, 7) = gY. Then 4* is a submersion onto
an open neighborhood W of X in g. Fix a choice of Lebesgue measure on m.
Let mD be the "D" for m, B. Then 8.A.3.5 implies that
dim I(mc)\ri\li2*i">(T)\m^ < 00 and dim C[mD]h|1/24/0(T) < 00.
The inductive hypothesis now implies that T = TF on W. This contradiction
implies (1).
We note that if p e /(gc) then FpT = pFT. Thus supp(p(T - 7») =
supp(pT - pTF) and suppfpT - TpF) are contained in .yV © [/ by (1). This

8.3. Harish-Chandra's Regularity Theorem on the Lie Algebra
309
implies
(2) If p e 7(gc.) then supp(pTF - 7» c JT ® U.
8.3.13. As in the two special cases of the theorem that were proved above we
must calculate DTF — TLF. With the inductive hypothesis in hand and since
dim[g, g] > 3 (because of the two cases above) we can actually prove
(1) OTF=TUF.
Let//!,..., Hs be the Cartan subgroups of G corresponding to r^,..., hs. We
will be using the notation in 7.3.6 and the formula in the proof of 7.3.9. Let
(j> e q°(ft). Then
DTf - TUF = j F(X)U^)(X)dX - j UF(X)<f)(X)dX
9' 9'
= I cj j ej{H)(n}(H)F(H)n<l>Z>{H) - ns{H)UF{H)^{H))dH.
Here e,- is a locally constant function on h" that takes on the values + 1. Let
D; be the element of S(h;) defined as in 7.A.2.9 for □. 7.A.2.9 implies that the
expression that we are calculating is equal to
£cy j £j(H)(nj(H)F(H)n^(H) - nj{njF){H)<b^{H))dH.
Set Qj{H) = £j{H)nj{H)F{H). Then Theorem 8.3.4 implies that Qj extends
to an exponential polynomial on each component of hj. Let Cjmk be a labeling
of the connected components of h'/. We must therefore calculate
Iu = j (£.(//)D,.<I>^(H) - ajQj(H)^(H))DH.
Let y1,...,yper be such that Cjk = {// e h,|y^H) > 0, 0 < i < p). (see
Lemma 8.3.2). Set VUk = {// e I); | ^(/Z) = 0 for 0 < i < p}. Set L^ = Vftk in h,-
relative to B. Let /i,,..., /ipe l/M be defined by y,-(/ia) = <5ia. We also note that if
Hy, is defined by B(Hyi,H) = y,(H)for H e h,then//yi e C/MlLet X,,..., Xfbea
basis of Vs<k such that B(Xa,Xb) = ^a<5a.b with ^a = ±l.SetD} = Z ^X^.Then
D = Z haHv + D}. Since neither Q, nor <t^ has jumps in the directions in V]k
(7.3.8) it follows that
j e/ff)n}*jj'(ff)dff= j n}e;(ff)*jf'(ff)dff.
We therefore find that
h* = I I (e,(H)*flH,. W) - haHyaQj{H)<b$>{H))dH.

310
8. Character Theory
If we use the yjas coordinates on Ujk then Cjk = Vjk x (x p(0, oo)). If 1 < s < p
then set CJJiS = {H e h;|ys(H) = 0 and yf(H) > 0 otherwise}. If u is a
continuous function on C\(Cjk) then define us+(H) = limr^0+ u{H + ths) for
// e C^j. If we integrate by parts twice in the above expression for IJk then we
find that
'm = E4..w.* I {haQj)t{H){<b?)UH)dH
o,h Cj.k.b
+ Z«-.m I (e;)a+(H)(HVao^):(//)rf//
a Cj,k,a
with da.t;.); and eaj-)k constants and d// some choice of Lebesgue measure
on the pertinent hyperplanes. We are now ready to prove (1). Let u e Cco(R)
be such that u(x) = 0 for x < 1, u(x) = 1 for x > 2. Set for e > 0, (E(X) =
u(co(X)2/£2). Since C£ is 0 in a neighborhood of U ® JT, 8.3.12(1) implies that
if/e Q°(n) then n^J) = 0 for £ > 0. We note that
HyiCE = (4y;co/e2)«'(co2/e2)
which vanishes on the set CJkJ. Thus
hAU) = L<w j wH)(fcfle;)ir(H)(*7'tt(H)<*ff
+ Z««.m I ce(H)(ey)fl+(H)(H,a*^):(H)dH.
a CJ,k,a
If X e g and if e > 0 is sufficiently small then (E(X) = 1 if o)(X) is
nonzero. Let x be the characteristic function of the set {X e g | co(X) # 0}. Then
limE_0 CE = X- Our hypothesis on g (dim[g, g] > 3) implies that x is one almost
everywhere on each Cjka. Thus the dominated convergence theorem implies
that limE_0 ljMgJ) = lUk(f). We therefore find that
0 = lim n(U) = Hm X c,/,,k(CE/) = I Cjlhk{f) = M/)-
e-»0 e-0 j,k j,k
This completes the proof of (1).
If p e 7(g) then FpT = pFT. Thus (1) implies
(2) If p is a polynomial in one variable then p(D)TF = TP(D)F.
8.3.14. We are now ready to complete the proof of the inductive step and
hence of the theorem. Let p be a non-zero polynomial in one variable such
that p(D)T = 0. Then
p(D)(T - 7» = p(D)T - p(D)7> = -p(D)7> = - TP(L])F = 0
since p(D)F = 0 and Q' and we are extending by 0. Now supp(T - TF) c
U ® Jf. Hence Lemma 8.3.9 implies that T - TF = 0 as asserted.

8.4. I larish-Chandra's Regularity Theorem on the Lie Group
311
8.4. Harish-Chandra's regularity theorem on the Lie group
8.4.1. The next theorem is one of the most profound results of Harish-
Chandra. After its statement the remainder of this section will be devoted to its
derivation from the main result of the last section. Let d(g) be as in 7.4.11 and
set G' = {g e G | d{g) # 0}.
Theorem. Let T be an invariant Z($)-finite distribution on G. Then there
exists a locally integrable function F = Fr that is real analytic on G' such that
T = TF on G. If H is a Cartan subgroup of G, if h e H and if C is a connected
component of {X e I) | h exp X e H nG'} then X -> F{h exp X)\d{h exp X)\1/2
is the restriction of an exponential polynomial to C.
Set H' = G' n H. Let ip(g,h) = ghgx for g e G and h e H'. Then \jj is a
submersion of G x H' onto an open subset U of G. Let AG H = A and 5G H = (5
be as in 7.A.3.6. We use the notation of 8.A.3.2.
If z e Z(g) then
iP°(zT) = W(T).
Since <5(z) = A_1y(z)A and since t/(l)) is finitely generated as a y(Z(g))-module,
we see that
dim U(\))Atp0(T) < oo.
This implies that there exists a real analytic function, (, on W such that
Ai/>°(T) = T?. Let h and C be as in the statement of the theorem. Set
(,,(X) = i(h exp X) for xeC Then dim U(\))Ch < oo. Thus C is the restriction
to C of an exponential polynomial (8.A.2.10).
There exists a nowhere vanishing, locally constant function, e, on H' such
that A(x) = e(x)|d(x)|1/2 for x e H' (here we have chosen a system of positive
roots for <t(gc,hc)). Set nH(x) = £(.x)-'C(x) for x e //'. Then i/>°(T) = |d|""^h-
Let Hu. ..,Hr be a complete set of nonconjugate Cartan subgroups of G. If
// = Hj then set ^ = nH. G' is the disjoint union of the open subsets
UgsGgH'jg~[. We may therefore define a real analytic function F on G' by
F(g%-') = |dWr,/2»;#)
for h e H'j and g e G. Then T = TF on G'. |<i| 1/2 is locally integrable on G
(7.4.11) and each ns is locally bounded, hence F is locally integrable on G. In
order to complete the proof of the theorem we must prove that T = TF on G.
8.4.2. Let x be a semi-simple element of G. Let M = {g e G\gx = xg}.M is
a real reductive group (see 8.A.4.10). Then we can write g = m © V with V an

312
8. Character Theory
Ad(M)-invariant subspace of g. Set M" = {me M|det((Ad(m) - l)\v) ¥= 0}.
Set ip(g,m) = gmg~x for g e G, m e M". Then ip is a submersion of G x M"
onto on open subset of G. We now show that there exists an open
neighborhood of 0, [/0, in m of the type described in 8.3.3 (for m) such that exp
restricted to U0 is a diffeomorphism onto an open neighborhood C/t of 1 in M
and xl/, is a neighborhood of x in M". Let D0,..., Ai-i be as in 7.3.9 for g.
Then Dj is homogeneous of degree n — j > 0 (in the indicated range of indices.
If U is a connected neighborhood of 0 in j(g) such that exp is a diffeomorphism
of U onto its image then exp is a diffeomorphism of V@Q(D0 ,...,£)„_,, n - 1)
onto its image (7.6.3, notation as in 8.3.3). If 0 < f < 1 then
tn(D0,...,D„.1,n- l)=>n(D0)...,/>„_„r"_1(jt- 1)).
This implies that if 7 e ft(£>0, • • •, £>„_ x, r"~ 1(tt - 1)) and if X is an eigenvalue
of ad Y then \X\ < tn (7.A.1.3, 7.A.1.4). We may now argue as in 8.3.12 to find
a neighborhood, Qr c V® ft(£>0,...,£>„- i,f"~ \n - 1)) of the desired type of
8.3.3 for m. If we take f > 0 sufficiently small then it is clear that l)x = Q, has
the desired properties.
Set ft = >p(G x (xC/,)). Let j be the "j function" for M (see 8.A.3.6). Set
C(fif,«) = ip(g, xu) for g e G and « e Ux. Let C°(T) e D\Vl )M be as in 8.A.3.2.
We note that M" nG' = M" n M' where M' is the set of all regular elements
in M relative to its action on m. 8.A.3.2 implies that
dim^.M(Z(g))C°(T)<oo.
Now,
«5g.m(z)C°(T) = (AG,M)-1yG.M(z)AG>MC0(T).
This implies that
dim Z(m)AG.MC°(T) < oo,
since Z(m) is finitely generated as a yG M(Z(g)-module.
We can now apply Lemma 8.A.3.6 to see that
dim 7(m)j1/2 exp*(AGJWC°(T)) < oo on U0 n m'.
If Cm is the Casimir operator of m then yGM(C) = Cm + XI with XeC. Let
□m be the corresponding "□" for m. 8.A.3.7 implies that
dim C[Dm]i"2 exp*(AG,MC°(T)) < oo on U0.
Theorem 8.3.5 applied to U0, implies that there exists a locally L1 function,
H, on U0, that is real analytic on U0 n m', such that
j1'2 exp*(AG,MC°(T)) = T„.

8.5. Tempered Invariant Z(g)-Finite Distributions on G
313
This implies that (°(T) = (°(7>). Hence T = TF on Q. x is an arbitrary semi-
simple element of G. Hence 8.A.4.11 implies that T = TF on G. This completes
the proof.
8.5. Tempered invariant Z(g)-finite distributions on G
8.5.1. Let TeD'(G). Then T is said to be tempered if T extends to a
continuous functional on #(G). In this section we will prove some extremely
technical results about tempered, invariant distributions on G. To do this we
must introduce some notation. Let v.G-> G0 be as usual. Define (j)(g) =
tr(v(g)Tv(g)) and a(g) = log (j)(g). Then 0 is a norm on G. The topology of
#(G) is given by the semi-norms (7.1.2)
vw(/) = sup9eG El(g)(\ + a(g)Y\L(X)R(y)f(g)\, reR,x,ye [/(g).
We have seen that there exists d > 0 such that
l(\ + a(g))d~.(g)dg<K.
G
This implies that the semi-norms
&.*.,(/) = IK* + a(g)YL(x)R(y)f\\2, reR,x,ye [/(g)
are continuous on (€{G). (We will see that these semi-norms actually define
the topology of ^(G).)
8.5.2. Lemma. Assume that G = °G. Let (n,H) be an irreducible square
integrable representation of G. Then 0^ is tempered. More precisely, there exists
a positive integer m0 and a positive constant M depending only on G such that
|0ir(/)|<rf(7r)-1M||(/ + Qr/ll2-
(Here CK is the Casimir operator of K relative to B restricted to k.)
If y e KA then we have seen that dim H(y) < d{y)2. Let Xy denote the
eigenvalue of CK on any representative of y. Then there exists a positive integer
m0 such that (8.1.1)
X (1 + Xyym° d(yf < oo.
Let vUy be an orthonormal basis of H(y). If / e C™(G) then
®-(/) = 11 f(9Kn{g)vUy,vUy > dg.
1,7 G

314
8. Character Theory
If m > 0, m e Z and v e H{y) then
j ((/ + CK)mf)(gKn(g)v,v)dg = (1 + Xy)m j f{gKn{g)v, v) dg.
G G
This combined with the Schwarz inequality and the Schur orthogonality
relations implies that if v is in addition a unit vector then
Hence
I f(9)<n(g)v, v)dg
!©,(/)! < I
<<%)-'(!+ ^rmH(/ +Q)m/ll2-
S f{9)<n(9)vi.y,vi.y>dg
G
<(Z(H-Ay)-"d(n)-^||(/ + CK)"/||2
< f £ (1 + 2y)-md(y)2)d(ny'||(7 + Q)m/ll2.
The result now follows by taking m = m0 and
7
8.5.3. It is not hard to see that the character of an irreducible tempered
representation is tempered (we will see this later). The above Lemma implies
an interesting property of the characters of irreducible square integrable
representations.
Let %{G) denote the space of all / e CK(G) such that
vr>x(/) = sup9eG(l + a(g)YE(g)-l\xf(g)\ < co
for aH reR and x e l/(g). (Here the action is as left invariant differential
operators, that is by R{x).) Endow %(G) with the topology induced by these
semi-norms. As usual, it is a Frechet space.
The above Lemma implies that
(1) If (71, H) is an irreducible square integrable representation of G then 0^
extends to a continuous functional on %JR{G).
We now begin the proof that this extension property is true for any K-
central tempered distribution on G. For this we will need some preliminary
Lemmas.
8.5.4. Lemma. Let x, ye (7(g) and let j be a positive integer. Then there

8.5. Tempered Invariant Z(g)-Finite Distributions on G 315
exists a positive constant Cxyj such that
\(L(x)R(y)(\ + (T)'){g)\ < C,y.,(l + o{g))>
for all g e G.
If X, Y e g. Then <f>{e,xgesY) = tT(gTe'XTe'*ge're'rT). If we differentiate this
equation in f and s at f = s = 0 then we find that there are universal constants
£p.,.m,„ such that
L(X")R(Y")4>(g)= I Ep^mMix(gT{XT)mX"-'ngy'{YTy-n).
0<m<p.O<n<q
The Schwarz inequality implies that
\L{X>)R{Y')<Hg)\ < Const, ^g)
with "const." depending only on j, p, q.
Using this it is easy to show that
|L(JnR(n0(0)-"| < Const. 0(0)-'
for all p, q, k > 0 in Z.
We note that if (say) q > 0 then
L(X")R(r> = L(X")R(^-')((R(y)0)/0).
This, in light of the above, implies that if p, q > 0 and p + q > 0 then
|L(Xp)R(7>| < Const.
An easy induction on j now shows that
|L(X")R(7')(1 + a)}\ < Const.(l + a)1.
This implies the Lemma since the powers Xp, X e g, span (7(g).
8.5.5. Lemma. Let C denote the Casimir operator of G corresponding to B.
Put A = 2CK — C. If xe l/J'(gc) then there exists a constant C depending only
on x such that
l|x-/||2<C||(l + A)y||2 and
HL(x)/||2 < C||L((1 + A)Vll2-
Let <X, r> = -B{X,0Y) for X, Y e g. Let Xlt...,X„ be an orthonormal
basis of g relative to < , >. Then A = — Z (X,)2. We will also write
\\X\\ = «X,X»1/2. We will prove the first estimate of the Lemma. The
second is proved by exactly the same argument.

316 8. Character Theory
If Y e g and if / e Cf{G) then it is easy to see that (||- • -|| = ||- • -||2)
l|y/ll2<imi2Ll|xjll2.
Thus||yTH2<||y|rih,2 ik\\xir-xikf\\2.
Thus it is enough to prove that
I ||X,,---Xit/||2<CJ|(/ + A)Y||2.
11 ik
We prove this by induction on k. If k = 0 there is nothing to prove. Assume
that the result has been proved for 0 < k < r and that k = r + 1. Now
I \\Xh-Xlkf\\2= I {Xir-Xikf,Xh-X:kf)
■l ik il ik
= I ki<xtl-Lxi],xlky-xlk_j,xtl-xljy
il ik j= 1
+ I <XikXi<--Xik^f,Xi<Xi2---Xikf)
il ik
= I *Z <^1--[x,,xjj--xit_1/,xh--xij>
il Ik j = 1
- I <*i, •••*.* 1/,Xil---Xlt_iA/>
il ik - l
il Ik j= 1
We observe that if e > 0 then
(*) K/,0>l<(l/(2£2))||/||2 + (e2/2)||.9||2.
Let C, = max ||[X,-,^/]||. If we use (*) with e2 < 4^ and the above we have
I \\xtl-xj2A I \\xh---xikf\\2
il ik *■ h 'k
Thus
+ (\ + £'2) I ll^i,--^ik-,/ll2
\^ / "1 'k- 1
+ \ I ||Xli---XJk_1A/||2.
L h ik -1
I \\Xir-XJ\\2< Const. X 11^,-^.,/H2
ii ik »i ifc - 1
+ X ||Xil---Xik_,A/||2.
11 ik -1

8.5. Tempered Invariant Z(g)-Finite Distributions on G
317
The induction hypothesis now completes the proof of the inductive step.
8.5.6. Lemma. If p, q, r are non-negative integers and if f e CX(G) then set
HMAf) = im(l + Q)")R((1 + AHO + ayf\\2. Then f e <tf(G) if and only if
Hp.qAf) < °° for a" P> <?> r e N. Furthermore, the topology of <^(G) is given by
the semi-norms \ip.qr.
We will prove this result by a sequence of reductions. Let Vx be the space of
all / e CX(G) such that
W/) = lid + <x)rL(x)R()0/||2 < «>> r e R, x, y e [/(g),
endowed with the topology given by these semi-norms. Then ^(G) c K, and
the inclusion is continuous. Let
<W/) = ll^(x)/?(y)(l + a)7||2, r e R, x, y e [/(g).
Then K, = {/eC°°(G)| ^iX>y(/)<oo, rsR, x, j/el/(g)} by Lemma 8.5.4.
If we apply Lemma 5.A.3.2 we find that / e Vx if and only if
j \(L(x)R(y)(l + a)rf)(kxak2)\1alpdadkldk2 < oo
A* XK*K
for all x, y e [/(g), r e R. This combined with the compactness of K and
Lemma 8.A.5.7 implies that if / e Vu x, y e [/(g), reR then
™PasA.XusK fl"(l + ff(a))r|(L(x)RO0/)(R,a«2)l < oo.
This implies that <^(G) c Kt. The closed graph theorem now implies that
Vx = ^(G) as topological vector spaces.
Lemma 8.5.5 now implies that (€{G) is the space of all / e C^iG) such that
«P.,.r(/) = 11(1 + *)ri4(/ + A)")R((7 + A)")f\\2 < oo
with the topology induced by these seminorms. Now A = — C + 2Q. So
L(l + A) = — R(C) + L(I + 2CK). This combined with another application
of Lemma 8.5.5 implies the Lemma.
8.5.7. Lemma. Let T e D'(G) be tempered and K-central. Then T extends to
a continuous functional on ^(G).
If / e (^(G) then set f°(g) = \Kf(kgk~x)dk. The preceding result implies
that / h-» f° is a continuous linear map of ^(G) into ^(G). This implies the
result, since T(f) = T(/°).
8.5.8. With the above Lemma in hand we will now prove some (even) more
technical results in preparation for the main theorem in the next section. Let

318
8. Character Theory
HcGbea 0-stable Cartan subgroup of G. Let H = °H • A as usual. Let (P, A)
be a p-pair associated with A. Let <t = <t(gc, hc) and let <t+ be a system of
positive roots for <t compatible with (P, A). Set <t, = {a e <t | a^ is pure imaginary}
and <tR = {a e 01 a!b is real valued}. Set <J>. = 0 - (<tR u O,). Put X°#) =
{h e °H I h" # 1, a e <»,}.
Lemma. // f > 1 then \°H)Af c //'.
Let h = ua,ue \°H) and a e Ar+. If a e 0 - 0, then
\h*\ =a"> f > 1.
If a e <D, then h" = ua # 1.
8.5.9. Lemma. Lef t > \. If g e G, x, y e \°H)Af and if gxg~l = y then
there exist k e NMClK(°H), he H such that g = kh. (Here M is a standard Levi
factor for P.)
The preceding Lemma implies that g normalizes H. Since H is 0-stable,
this implies that g e KH. We may thus assume that g e K. Let x = ua,
y = uxax with u, ux e \°H) and a, ax e A*. Then gxg~l = y implies that
6(g)6(x)6(gyl = 6(y). Hence g6(x)g~1 = 6(y). This implies that
gx9(x)-lg-l=y9(y)-1.
So ga2g~' = a\. This implies that g e P. Now P n K = M n K, so the Lemma
follows.
8.5.10. Fix y e °H. Let C0 c °h be open, convex and such that
exp: C0 -> exp C0 = Cx
is a diffeomorphism. We also assume that C1(C0) is compact and that
y exp (C1(C0)) c '(°H). Let for f > 1
V:G/H0 x yd x/l^C
be given by
4/(g(//0,yc,a) = g(ycag("1.
Lemma. There exists an open neighborhood V of 1 • H° in G/H° such that
(1) x¥(V,yCl,A + ) = CltisopeninG.

8.5. Tempered Invariant Z(g)-Finite Distributions on G 319
(2) 4*: V x yCl x A* -> Qt is a diffeomorphism.
4* is everywhere regular. Thus fir is open for all choices of V. Thus we need
only show that we can choose V so that 4* is injective. If
y¥(gH°,yc,a) = 4/(xH°,yc',a')
then gycag'1 = xyc'a'x-1. If we set u = x~lg then 8.5.9 implies that u = kh
with ke M r>K and he H°. But then a = a' and fcafc~' = a. Now this implies
that mod °H° the possible "fc's" vary in a discrete set. Since gH° = xkH° it is
clearly possible to choose V so small that k e H°.
8.5.11. Fix K, C0, y, t > 1, ft, as above. Let a e Q°(K) be such that
J a(x)dx = 1.
Here we have chosen a G-invariant measure, dx, on G/H°.
Let yfyC,/!;'-) denote the space of all / e C»(yCxA+) such that
(1) supp / c co/ls+ with co c yC,/lr+ compact and s > 1.
(2) v,ifl(/) = supfc(l + ffM^ID/MI < oo for all d > 0 and all D e [/(h).
We say that a net /, -> / in this space if
(3) There exists co c yC^A* compact and s > 1 such that supp /, c co/l + for
all large r\ and vd>fl(/„ - /) -> 0 for all d > 0 and D e [/(h).
We define S:£f(yCxA + ) ->• Cx(ft,) by
S(/)(«P(x,yc,fl)) = «(x)/(ycfl).
Lemma. // / e y{yCxA + )and if we extend S(f)byOtoG then S{f) e C^G).
We use the notation xyx~l = yx and {yx\x e S} = ys. Let £cG be
compact and such that E • H° = supp a, Then supp S(f) = (supp f)E. (Here
S(/) is looked upon as an element of C^ft,).) Now supp f ^ co- (C\(A+,)) for
some f' > f. Thus supp S(f) is closed in G and the result follows.
8.5.12. Lemma. Let P(f) = \D\~ 1/2S(f) for f e ^{yCxA+t\ Then 0{f) e
y?R(G) and the map
P:Se(yC1At)^%l{G)
is continuous.

320 8. Character Theory
LetR = C[(l -fc-«)|«e«] c C^yC^t). Put
0(/i) =
n a-*-":
It is an easy exercise to show that if D e (7(h) then D(j> = (j)rf with r a
positive integer and / e R[/Ta, conj(/Ta) | a e <t+] = R". We also note that
\DihT1 = a-"(/)(/i) for h = yea, ceCuaeA+,
We will now be using notation and results in 7.A.3. If g e (7(g) and if
h eyCxA + then
0*« = r„(X V, ® ft,) with /i; e R- \ at e (/(g) and bj e I/(h).
Thus if x e £ (see the previous number) then
S-'fxfoT'Xi + a{xhx-l))m\g ■ P(f)(xhxl)\
< Const. 3(/.)-'(l + o(h))m X |/#)l|fl«a(x)||ft/fc^/)(fc)|
< Const. (1 + a{h)T I |«fr(fc)||^rA/WI
with D; e (7(h), u;r e R". Here the "consts" depend on E but not on /. This
proves the result since the elements of R"[0] are easily seen to be bounded on
yCtA? forf > 1.
8.6. Harish-Chandra's basic inequality
8.6.1. The following theorem of Harish-Chandra is the key to the study
tempered, invariant eigendistributions.
Theorem. // Te D'(G) is central and Z(q)-finite then T is tempered if and only
if there exist constants C and d such that if H is a 6-stable Cartan subgroup of G
then
\d(h)\ll2\FT(h)\ < C(l + a(h))d
for heH,
The sufficiency of the condition is an easy consequence of Theorem 7,4.10
and will be left to the reader. We now begin the proof of the necessity.
If z e Z(g) and if heH' then (7.A.3.7)
z-FT(h) = A(hyly(z)(AFT)(h).
Here we have used A corresponding to a choice of positive roots in <t(gc, hc).

8.6. Ilarish-Chandra's Basic Inequality 321
This implies that
(1) dim [/(I)) • A(Fr|H,) < oo.
Let / = A(Fr|H,). Then (1) combined with 8.A.2.10 (more precisely proof of
that result) imply that there exist A,,..., Ar e f)£ and an integer d > 0 such that
if y e H' and of 0 e U c h is open, connected and such that y exp U c H' then
(2) f(y exp h) = Y. Pi,7(h)eA,[h) with p,., a polynomial of degree at
most d depending only on y and T.
Let H = °HA, as usual. Set A" = {a e A \a* # 1 for ae$-<D,}. Let
{Pi,A), i=l,...,p, be the p-pairs with split component A. Set tA + =
{ae/l|aA> 1, a e ®(R,A)}- Then A" = (Jj/1 + . Put '(°H) = {h e °H|fc« # 1
for all a e <!>,}. We have seen that
(3) H" = \°H) ■ A" c //'.
Clearly, H" is open and dense in H. Let C be a connected component of
\°H). Then C • ,A+ is connected. Thus if jix,..., fiq are the distinct restrictions
of the Aj to a then
(4) /(era) = X 0yk(c)Pj(log a)a"\ ceCandae tA+ with 0ljt
a function that extends continuously to C1(C) and pj a polynomial
of degree at most d.
Thus to prove the theorem we must show
(5) If Y, 0;jk(c)Pj(l°g a) # 0 then Re nk is non-positive on ;a+.
8.6.2. We now prove (5) above. Fix a p-pair (P, A) [A as above). Let y e °H,
C0, C,,/l+ and /?as in 8.5.12. We write(A+ =j/l+)0ij)i(c) = (j)Jk(yc) (notation as
in (4) above).
Let (j)eC^{yClA + ). Then it is clear that 0E,9"(y C1/4;1') for some f > 1. Now
T{p{<t>)) = SFr(9)P(<t>)(9)dg
G
j \d(yh)\FT(yh)\D(yh)\-ll2<x{x)<l)(yh)dhdx
CiA* x G/H°
= j \d(yh)\'l2FT(yh)<f)(yh)dh.
C,A*

322 8. Character Theory
Set v(h) = \d(h)\ l/2/A(h) for h e H'. Then \v(h)\ = \forhe H'. We have
(*) T()8((A)) = X j (t)Jk(yc)Pj(\oga)v(yca)a"k(l)(yca)dcda.
j,k CiM*
Now (j) \—► T(P(<f>)) is continuous on ^(yC!/l+). So (*) implies (5) above and
hence the theorem.
8.6.3. Corollary. Let T e D'(G) be central and Z(g)-finite. Then T is
tempered if and only if there exist constants C > 0 and d>0 such that
\d(X)\l'2\FT(X)\<C(l+G(X)Y
for x e G'.
The preceding theorem implies the sufficiency of this condition. We must
therefore prove the necessity. Since there are only a finite number of conjugacy
classes of Cartan subgroups in G, it is enough to prove the inequality on
G[H'~\ = {ghg~l \g e G, he //'} for a fixed 0-stable Cartan subgroup of G. We
note that d(ghg~l) = d(h). Thus the result will follow from the previous
theorem if we can show that there exists a norm ||- • -|| on G such that if g e G
and if he H then
ll^-'II^PII-
To prove this it is enough to observe that if h e GL(n, C) is diagonal and if
g e GL(n, C) then
tr(ghg-l(ghg-l)*)>tr(hh*).
We leave this exercise in linear algebra to the reader.
8.6.4. Corollary. Let T be a compact Cartan subgroup of G. Let S e D'(G)
be a tempered, central Z(Q)-finite distribution. If f e 0(£(G) (7.6.3) then
S(f) = w j A(t)Fs(t)Fj(t)dt.
T
Let fj e C?(G) be such that lim^ fs = / in (€{G\ Let T, H2,...,Hr be a
complete set of representatives for the conjugacy classes of Cartan subgroups
of G. Then
S(fj) = w j A(t)FT(t)FTf.(t)dt + £ c; { AMFrWFl^dh
by the Weyl integration formula and Harish-Chandra's regularity theorem.
For simplicity of notation we set Hx = T.

8.7. The Completeness of the nT
323
Theorem 8.6.1 implies that
lim j At{h)FT{h)F%ih)dh= j ^{h)FT(h)Ff{h)dh.
This limit is 0 for i > 2 since / e 0<#(G) (7.5.4). This formula with i = 1 implies
the result.
8.6.5. In the next section we will show how the above Corollary can be
used to complete our discussion of the irreducible square integrable
representations.
8.7. The completeness of the jit
8.7.1. We continue with the assumption that G is of inner type and that
G = °G. We assume in addition that there exists T c K a Cartan subgroup
(which we fix). We will use the notation of 6.9.
If te TA setA(r) = Ar be as in 6.9.3. Let h = tcandset<I> = <t(gc,h).Tissaid
to be regular if (A(t), a) # 0 for all aeO, If t is regular then P = P(t) =
{a e <t |(A(t),a) > 0} is a system of positive roots. Set (nz,HT) = (nPr,HPz) in
the notation of 6.9.4, Then (nr,Hr) is an irreducible square integrable
representation of G.
The purpose of this section is to prove
Theorem. If a> e G* is the class of a square integrable representation then
there exists a regular element t e Ta such that nz e co. Furthermore, nz is
equivalent with nz, if and only if there exists se WK such that z' = ts,
The proof will take some preparation which we now begin.
8.7.2. Fix a regular t e T\ Put P = P(z), Set V = (HZ)K. We first calculate
&Kz (see 8.2), For this we must calculate mv{y) for y e KA, In the notation of
6.9.3 and 6.9.4
F = Ind^n M(b,£A),
and
r"1M(b,£A) = Xt®Z)P,A(r).
Let y0 e (K°)\ Then Theorem 6.7.6 says that (A = A(t))
(1) dimHomKo(Fyo,Z)riA)= X det(s)p„(s(Ayo + pk) - (A + p„)).

324 8. Character Theory
If y^lK1)* is given by (r®y0 where z(z) = (r(z)7 for zeZ then
dim HomKI(Vn,r"M(b, £rA)) is given by the right hand side of (1). If y e KA
then y = Ind£i(yt) with yx e(K1)A, Thus we see that if y, y! and y0 are related as
above then
(2) mv(y) = X det(s)p„(s(Ayo + pk) - (A + p„)).
Suppose that /eC°°(K) and that y = Ind^) with y^iX.1)*. We
calculate
\f{k)rly(k)dk = irzy{f).
K
Since we have realized zy as an induced representation it is clear that
(ry(/)0)(x)= j fix'k^T^k^mdkdk,.
Now dim Fy < oo so it is easily seen that
(3) trry(/)= j /(kktk-^^k^dkdkt.
Kl = ZK° and riyi(zk) = tz(z)riya(k) for z e Z and fc e K°, Thus the right
hand side of (3) is
(1/|Z|) X j f{kzk0kl)r,yo(k0)!:z{z)dkdk0.
zeZ K*K°
We now apply the Weyl integration formula which yields
(4)
\ f{k)yly(k)dk = {]Z\\W{K,T)\Y' X j |AK(f)|2 \ f{kztkl)nya{t)Ut)ddkdt.
K zeZ r° K
Here we have chosen a system of positive roots <&t c <t(fc, h) and AK is the
corresponding Weyl denominator. We will assume (for the sake of simplicity)
that we have gone to a covering of G so that (f -> tPk) e (T°)A,
The Weyl character formula combined with the fact that conj(Ak) =
(- \)"Ak with n = \<&t\ implies
(5) {-\r\Z\\W(K,T)\\f(k)ny(k)dk
K
= X I det(s) j AMfikztk-'KW^+^dtdk.
zeZ seW(K.T) T° * K
Here ky = XyQ. Let jiyx be the character of the representation of T given by Z
acting by T|z and T° acting by tXy. Then we have proved
(6) J/(fc)»,,(fc)dfc = (-!)« j Ak{t)^(t)f{ktkl)dtdk.

8.7. The Completeness of the nT
325
This in turn implies that
(7) 0*.k(/) = (-!)" I I det(s)pB(sUro + pt) - (A + p,))
yoe(K°)A seTO.T)
• j hk{t)nya,At)j\ktkl)dkdt,
T x K
In 6.5 we have seen that the expression on the right hand side of (7) is
given by
Ak(t) X de«*) ^(z'{t))
(- D" J SS,WA' I f{ktk-l)dkdt.
This implies (finally!)
(8) ©*,„(/) = (-1)"| W(K, T)\ j A*(f> t.'",T(t) j f{ktkl)dkdt.
8.7.3. Let / e 0(€(G). Choose for each c> 0, e6£ e ^{G) with supp e6E e Gte
and limE_0 e6£ = / in (<i(G'e),
Then (see Theorem 8,2.2)
©K(«fe) = ©K.K(%.) = (-l)"|W(K,T)|
x j |7 - det(Adf|p)|(A,(f)/A„(f))| tr(r(0) j <t>t{gtgl)dgdt
T G
= (-l)'p'|H/(K,T)| Jtr(T(r))Fj,(r)dt.
7'
As in the proof of 8.6,4 we have lim£_0 0^(e6E) = &v(f)- Thus
(1) ®K(/) = (-l)W\W(K,T)\(FTfy(z*) withr*
the dual representation of z.
Theorem 7.7,2 now implies that
(2) If f s0(€(G) and if 0„t(/) = 0 for all regular z e Ta then /(l) = 0.
8.7.4. We can now complete the proof of the main theorem in this section.
Let co e GA be a class of an irreducible square integrable representation of G.
If nz £ co for all regular z then if / is a K-finite matrix coefficient of co then (2)
above implies that /(1) = 0 (we already know that /e°^(G)). This is
ridiculous. Thus there exists z e TA with z regular such that n, e co.
If nz ^ 7rr. then the equality of K-characters implies that z' = sz' for some
s e W(K, T) = W{G, T). This completes our determination of the irreducible
square integrable representations of G.

326
8. Character Theory
8.A. Appendices to Chapter 8
8. A. 1. Trace class operators
8.A.I.I. The purpose of this appendix is to develop the elementary aspects
of the theory of trace class operators. Let H be a separable Hilbert space with
inner product < , >. An endomorphism, T, is said to be compact if T maps
bounded subsets of H onto subsets of H with compact closure. In the
literature, the term completely continuous is also used for compact. It is obvious
that a compact operator is bounded.
If T is a bounded operator on H with finite dimensional image then T is said
to be of finite rank. Obviously, an operator of finite rank is compact. We set
L(H) equal to the algebra of bounded operators on H, Then L(H) is a Banach
space relative to the operator norm
imi=sup|N| = 1||Tt;||.
Let K(H) denote the space of compact operators in L(H),
Lemma. K(H) is a closed ideal in L(H).
It is clear that K(H) is an ideal in L(H). Let {7}} be a sequence of
operators in K(H) that converge to T in L(H). We show that T e K(H).
Let {/„} be a sequence in H with ||/„|| < C. The diagonal process yields a
subsequence {u„} such that T}u„ converges for each j. Let e > 0 and let r be
such that || T — T}|| < e for j > r. Let N be such that if m, n > N then
||Tr(u„ - wjll < e. If m, n > N then
nn«„-«jn
= \\(T - Tr)(«, - um) + Tr(Un - «JII <\\T- Tr\\2C + e < (2C + l)e
The result follows from this.
8.A.I.2. If TeL(H) then we define T* by (Tv,w> = (v,T*w) for
all v, w e H. T is said to be self-adjoint if T = T*. The following result is
standard.
Lemma. Let T be bounded and self-adjoint. Then T is compact if and only if
there is an orthonormal basis [vj\ of Ker T1 such that
(1) Tvj = XjVj with Xj e R,

8.A.I. Trace Class Operators
327
(2) lim k} = 0.
j-*
We may assume that Ker T = 0. We first prove the sufficiency. Set
PN{v) = Sj<N (v,Vj)Vj. Then TPN = PNT for all N. Let e > 0 be given and
let N be such that |^| < e for j > N. Then \\{T - PMT)v\\ < e\\v\\ for all
M > N. Hence \\T - PMT\\ < e for M > N. Since PMT is of finite rank, the
sufficiency follows from the previous Lemma.
We now assume that T is compact and self-adjoint. Let H0 be the span
of the eigenvectors of T. We show that Hq = {0}. Assume the contrary.
Let m be the norm of T as a bounded operator on Hq. There is a
sequence {vj} of unit vectors in Hq such that lim (Tvj, Tvj} = m2. Since T
is compact, we may assume that lim Tvj exists and is a vector ue Hq. Let
Q denote the restriction of T to Hq. Then ||u|| = m = \\Q\\. If m = 0 our
assertion follows. We therefore assume that m > 0. Set x = «/||«||. Then
we have
lien > HQxii = iim iie^ii/iieii > nm <e2i^.>/iieii = lien.
Thus HQxIl = HSU. Hence, HQH2 = <Q2x,x> < ||e2x|| < ||<2||2. Schwarz
inequality implies that Q2x and x are linearly dependent. Thus Q2x =
||Q||2x. We conclude that if Hq # 0 then T has an eigenvector in Hq.
This contradiction implies our assertion.
Since T is compact, the eigenspaces for T are finite dimensional (we are
assuming ker T = 0). We can therefore find an orthonormal basis for H
that satisfies (1) in the statement. If (2) were not satisfied then there would
be an infinite sequence {«,-} of unit vectors in H such that Tuj = [i-jui and
lAf/l > C > 0. Hence T would not be compact.
8.A.I.3. If Te K(H)then T*Te K(H). Let 1^} and {Xj} be as in the previous
Lemma for T*T. We define an operator \T\ as follows
|T|(KerT*T) = 0, \T\vj = U;)1'2^.
The preceding Lemma implies that \T\ e K(H).
Let V = \T\(H), W = Cl(TH). We define a mapping U of V to W by
U\T\v = Tv for heH. Then U is linear and ||C/i;|| = ||i;|| for v e V. We
extend [/ to // by setting [/(ker |T|) = 0. U is a so called partial isometry. We
have proved the following standard decomposition
(1) T=U\T\
8.A.I.4. Let T e L(H). Then T is said to be summable if there exists

328
8. Character Theory
an orthonormal basis {w„} of H such that
Zl<Twj>w*>l < co.
Lemma. Let T e L(H). If {vj} is an orthonormal basis of H then
El|T«,||<:Xl<7V*>l-
J j.k
Let Wj be a unit vector for each index j. Then
IK^>VI=I
Y,<Tvk,Vj)(\Vj,vk)
<Y,\<Tv},vk)\.
j.k
Choose wj = vj if Ti>,- = 0 and w, = Tt^/HTfjH if Tuj # 0. The inequality now
follows.
8.A.I.5. Let T e L(H). Then T is said to be of trace class if there exists an
orthonormal basis {v„} of H such that
Il|Tt;„||<co.
n
The above Lemma implies that if T is summable then T is trace class.
Lemma. // T is trace class then T is compact.
Let {v„} be an orthonormal basis of H such that X \\Tv„\\ < oo. Define
Pkv = X;<k <i;, fj)^-. Let 0 < e < 1 be given. Then there exists N such that
I HT«;||<e2.
U k> N then
||(T - TPk)v\\ =
I <».«>;>Ti>;
<
v\\( I II^11
J>N
1/2
If j > N then UTi^H < 1, hence \\Tvj\\2 < \\Tvj\\. We therefore conclude that
||(T-Tn)«||<e||i;||.
Hence lim TPk = T. Lemma 8.A. 1.1 now implies that T is compact.
8.A.I.6. If Te L(H) then we say that T is of Hilbert-Schmidt class if
there exists an orthonormal basis {vj} of H such that
Ill^||2<o).

8.A.I. Trace Class Operators 329
(1) If A is of Hilbert-Schmidt class then so is A*. Furthermore, if {vj} and
{wj} are orthonormal bases of H then
T\\Avj\\2=1\\AWj\\2.
Indeed, \\AWj\\2 = Zt \<AWj,vk)\2 = I, KW],A*vk)\2. Thus
j.k
(1) now follows.
(2) If B e L{H) and if A is of Hilbert-Schmidt class then BA and AB are of
Hilbert-Schmidt class.
Indeed, let {e,} be an orthonormal basis of H. Then E||ft4e;||2<
||B||2 I ||/ley||2. Thus BA is of Hilbert-Schmidt class. Now AB = (B*A*)*.
So the second assertion follows from (1).
Lemma. Let T e L(H). Then T is of trace class if and only if T is compact
and if kj are as in 8.A.1.2/or |T\ then Z k-s < oo.
Let {vj} be an orthonormal basis of (kerlTl)1 such that Tvj = k-Svr
Define |T|1/2 = S by S(ker \T\) = 0 and SVj = {k})ll2v}. Clearly
IH^II2 = I^.
This implies that S is of Hilbert-Schmidt class if and only if
X kj < co.
j
Suppose that T is of trace class. Let [ef] be an orthonormal basis of H
such that 11| Te;\\ < oo. Then
I ||7e;|| = X \\V\T\e,\\ = I lll^kyll > I <\T\eJte^ = X ||Sej||2.
This implies that if T is of trace class then S is of Hilbert-Schmidt class.
The lemma now follows since it is clear that ker T = ker | T\ and
Eii™,ii = Eiim«,n = zv
8.A.I.7. Lemma
(1) Let T be trace class. If {e„} is an orthonormal basis of H then Z <Te„,e„>
converges absolutely and is independent of the choice of {e„}. We set
X<7e„,e„> = trT.
(2) Let Te K(H). Then T is of trace class if and only if for each choice of a

330 8. Character Theory
pair of orthonormal bases {e„}, {/„} of H
Ll<Te„,/„>|<oo.
Furthermore, if T is of trace class then the supremum of such sums is
tr | T\ < oo.
Let {x„} be an orthonormal basis of H such that Z || Tx„\\ < oo. Let {e„} be
another orthonormal basis of H. Then
Xl<rx„,em><em,x„>|<fxi<rx„,em>|2y/2 = ||rx„||.
m \ m J
This implies that
Z \(Txn>em>(em,X„y\ <Zll^JI-
m.n n
Now E<Tx„,x„> and Z (Tem,em} are both rearrangements of the
absolutely convergent series
m,n
This proves (1).
We now prove (2). 11< Ten, f„}\ = 11<[/| T\en, fn)\. Let {x„} and {2„} be as
in 8.A.1.2 for \T\. Then the above formula implies that
Zl<re.,/.>|=Z
by Schwarz's inequality.
Z;-m<^*m,/„><<VXm>
<Z^
8.A.I.8. Lemma. Let T e L(H). Then T is of trace class if and only if T
can be written in the form AB with A, B of Hilbert-Schmidt class.
Assume that T is of trace class. Let T = U\T\, as usual. Then \T\ = S2 with
S of Hilbert-Schmidt class (see the proof of Lemma 8.A.1.6). Clearly, US is
also of Hilbert-Schmidt class. Set A = US,B = S.
Suppose that T = AB with A, B of Hilbert-Schmidt class. Let {e„} and {/„}
be orthonormal bases of H. Then
Z \<Jen, /„> = Z \<ABen, fn}\ = £ \{Ben, A%)\
<&l(\\Ben\\2 + \\A*fn\\2)<cv.
The result now follows from the previous Lemma,

8.A.2, Some Operations on Distributions
331
8.A.I.9. Set LX{H) equal to the space of all trace class operators on H. If
TeLx(H) then set \\T\\X = tr|T|. Lemma 8.A.1.7 implies that ||- - -||x defines
a norm on LX(H) and that
(i) imi<imii.
We leave it to the reader to prove
Lemma. Ll(H) is a Banach space relative to ||- • •\\l.
8.A.1.10. Lemma. If T e Ll(H)andif Ae L(H)then AT,TA,T* e LX(H).
Furthermore, tr AT = tr TA.
Write T=XY with X and Y Hilbert-Schmidt, Then AX and YA
are also Hilbert-Schmidt (8,A.1,6(2)). Now, AT = (AX)Y, TA = X(YA) and
T* = Y*X*. The first assertion follows from 8.A.I.8.
Let {e„} be an orthonormal basis of H such that Z || Tet\\ < oo. Then
X KAep,en){Ten,ep)\ = £ \{ep, A*en)(Ten,ep)\
p,n p.n
<Xl|A*e„||||Te„||<||A*||Xl|7e„||<a).
Since
ZZ (Aep,eny(Ten,epy = X <^c/,.c/)>
and Z„ Zp (Aep,eny(Ten,epy = £ </17e„,e„> the second assertion now
follows.
8.A.2. Some operations on distributions
8.A.2.I. Let M be an n-dimensional orientable smooth manifold. Fix, ojm, a
volume form on M. Let CC{M) denote the space of all compactly supported
functions that have r continuous derivatives in each coordinate chart of M.
We recall the topology on CC{M). Let {Uj, tpj} be an atlas for M and let {</>;} be a
partition of unity subbordinate to {U,}. If to is a compact subset of M then
denote by C(co) the space of all C functions on M with support in a>. If
/ e C(ct}) then we set
v,.o,(/)=Z I sup 10,.(.x)8'/•</,-'OAM)!-
J |/|<r
We are using standard multi-index notation. If we choose a different atlas and
partition of unity then we would have an equivalent norm. We endow Cr(a>)

332
8. Character Theory
with the topology given by this norm. We endow CC(M) with the topology it
inherits as the union of the spaces C(o}). Finally, C™(M) is given the topology
of the intersection of the spaces CC(M).
As is usual, we write D'(M) for the space of all continuous functionals on
C™(M) and D'r(M) for the space of those elements of D'(M) that extend
to continuous functionals on CC(M). We will call these the distributions of
order r.
Let T e D'(M). We say that xeMis not in the support of T if there exists
0 e C?(M) such that 0 = 1 in a neighborhood of x and T(0/) = 0 for all
/ e C™(M). This defines the support of T.
8.A.2.2. If / is a function on M then we say that / is locally integrahle if for
each volume form to on M and each compactly supported continuous function
(j) on M, (j)f is integrable with respect to a>. It is clear that this condition need
only be checked for ojm .
Set L\0C(M) equal to the space of all locally integrable functions on M. If / is
locally integrable then we define 7} e D'(M) by
TA<t>) = I </>M,
M
for (j) e C?(M). It is clear that 7} depends on the choice of (aM.
8.A.2.3. Recall that if N is another smooth manifold and if ip is a smooth
mapping of M onto N then ip is called a submersion if dij/p is surjective for all
pe M. Fix M, N and a submersion \\i of M onto N.
The implicit function theorem implies that if pe M then there exists an open
neighborhood, U, of p in M and local coordinates xx,..., xm on [/ such that
1^(1/) = V is open in A? and there are local coordinates y u..., y„on V such that
(i) Xj = yj°ip for j= l,...,n.
(ii) IfF(p) = (x1(p),...,xm(p)) forpE[/thenF([/) = {xERm||x;|<l
for j= l,...,m}.
Assume that A? is orientable and fix, caN, a volume form on N. Let p e M
and let x e [/, Xj, y, be as above. Then
oiM\v = vdxx A--- \dxm and
«;vk = ^^iA---Adj;„.
Thus
•A*wnu> = n° *//dxlA---Adxn.
We set (coM/coN)|(, = (v/n- ip)dxn + 1A--- Adxm.

8.A.2. Some Operations on Distributions
333
It is easily checked that (coM/(DN)p is independent of the choices used in its
definition. We have thus defined a smooth m — n form coM/a>N on M.
Let x e N, x = ip(p). Set Fx = ip'Hx). Then Fx is a closed submanifold of
M and if W = Fxn U then xn+1,...,xm restricted to Wgive a system of local
coordinates on W whose image in Rm" is the cube {(tl,..., tm_„)| |tj| < 1}.
Let i denote the canonical inclusion of Fx into M. Set
vx = i*(o)NlmM).
Then vx is a volume form on Fx.
8.A.2.4. Lemma. If fe C?{M) then set for x e N
<M/)(x) = I /v*.
Then ip%(f) has compact support contained in ip(suppf). Furthermore, tp^, is a
continuous linear map of C[(M) onto C'(N) for all r > 0.
The result is clear if M = U and if N = V as above. To prove it in general
one uses a partition of unity. We leave the details to the reader.
8.A.2.5. Lemma. If f e C*{M), F e Ccr{N)then
(1) f/F *<»M = SK(f)F<»N-
M N
Furthermore, tp^.f is uniquely determined by this formula.
As above this is obvious locally. To prove it globally one uses a partition of
unity.
8.A.2.6. We define
iP*:D'r(N)^D'r(M)
by
(1) r(T){f) = mj).
The following assertions are obvious.
(2) ip* is injective.
(3) supp^* T) c ip ~'(supp T).
(4) If F is locally integrable then \p*TF = TF„^.
If F is a smooth function then wc set ip*(F) = F <> \p. The above observation
implies that this notation is consistent.

334
8. Character Theory
8.A.2.7. If D is a differential operator on M than there exists a unique
differential operator DronM such that if /, g e C™(M) then
J (Df)goJM = J f(DTg)coM.
M M
In local coordinates £)r is defined as follows. Let xu...,xm be local
coordinates on an open subset, U, of M. Then on U, D = Z a7 3' and coM =
0 dx i A ■ • • A dxm. If u is a smooth function on M looked upon as a differential
operator under multiplication then uT = u. We set
(3/3xj)r = -9/8x, - 0"1 30/3xf.
Finally we insist that (£>!£)2)r = (D2)T(Dl)T. This defines the operation
locally. Globally we can piece it together with a partition of unity.
It is clear that DT depends on the choice of o)M. If T e D'(M) then we set
DT(f)= T(DTf) for f e C?(M).
Lemma. Let D and E be differential operators on M and N respectively.
Assume that
D<P*(f)=<P*(Ef)
for all f e C?(N). Then
Dip*(T) = ip*(ET)
for all T e D'(N).
This is an immediate consequence of the definitions.
We are now ready to apply these ideas to G-spaces.
8.A.2.8. Theorem Let G be a Lie group and let dg be a choice of left
invariant measure on G. Let T e D'(G) be such that T • L(g) = T for all g e G.
Then T is a constant multiple of dg.
Let Xu...,Xn be a basis of g. Set D = I L(Xt)2. Then D is an elliptic
operator on G and DT = 0. Thus there exists a smooth function, F, on G such
that T = TF. The left G-in variance implies that F(g ~lx) = F(x) for all x, g e G.
Thus F is a constant.
8.A.2.9. Suppose that A and 8 are smooth manifolds. Set M = A x 8. We
define a mapping, C, of C?(A) (g) C?(B) into C?(M) by C(f®g)(x,y) =
f(x)g(y). We will write / ® g for £(/ (x) g). We also recall the corresponding
tensor product operation on distributions. Let TeD'(A) and let Se

8.A.2. Some Operations on Distributions
335
D'(B). If feC?(M) then set h(a) = S(b -► f{a, b)). Then heC?(A). Set
(T ® S)(f) = T(h). We note that (T ® S)(f ® g) = T{f)S(g). This equation
determines T ® S since
(1) q°(/l)® C7(B) is dense in C?(M).
Let G be a Lie group and let N be a smooth manifold. Set M = G x N. Let
G act on M by g(x,y) = (gx, y). If / e C^(M) and if x e M then we set
M.9)/W = /(0_1x).
Lemma. If T e D'(M) is such that T • L(g) = T /or all g e G then
T = dg®S
with S e D'(N).
If feC?{N) then set kt(h) = T{h® f) for heC?(G). Then ^ is an
L(g)-invariant distribution on G for each g e G. Thus the preceeding Lemma
implies that
lf{h) = S(f)\h(g)dg.
G
We leave it to check that S{f) defines an element of D'(N).
8.A.2.10. We conclude this appendix with two results about distributions on
R". If v e R" then we look upon uasa vector field on R" in the usual way
(vxf = dfx(v)). This correspondence extends to an algebra homomorphism of
S(R") (the symmetric algebra on R") into DO(U) for each open subset, U, of
R". This gives the usual identification of S(R") with the constant coefficient
differential operators on R".
Lemma. Let U be a connected open subset of R". Let T e D'(U) be such that
S(R")T is finite dimensional. Then there exist jj, /ipe (R")* and Fe
C[xu..., x„, e"',..., e"p1 such that T= TF on U. Here we use Lebesgue
measure for coRn.
Note. We will call a function such as F above an exponential polynomial.
Let if = S(R")T. Then dim <f < oc. This implies that 3/3x,,..., 3/3x„
define commuting operators on the finite dimensional space if. There exist
HueC, i = l,...,n,j = \,...,pt, such that if S e $£ then
II (9/3xf - nu)" • S = 0.
j

336
8. Character Theory
Set D = S, iij (8/8xj - nLj)2q. Then £> is elliptic and DS = 0 for S e if.
This implies that if S e if then there exists a real analytic function F = Fs on
[/ such that S = TF. We may thus identify Jif with a space of real analytic
functions. If F e if then let F = F,,..., F, be some enumeration of the
functions 8'F for |/| < q(L pt). Set
F =
>i"
for F s ££. Then there exist d x d matrices I^,..., T„ such that
9/9x(F = T,F for i = 1,..., n.
Fix x0 e C/ and let r > 0 be such that
{x eR"||(x0), - x,| < r, i = l,...,n} c U.
Then the above differential equations imply that
F(x1,...,x„) = eEU'-(Xo)l)r'F(xo)
on this cube. Since U is assumed to be connected and F is real analytic the
above formula is true for all x e U. This proves the Lemma.
8.A.2.11. For our applications we need a slight generalization of the above
Lemma. If p < n then we look upon R" as Rp x Rn~p.
Proposition. Let A be open and connected in Rp and let B be open in R"~p.
Let T e D'(A x B) be such that dim S(R'')r < oo. Then there exist
exponential polynomials Fu..., FdonRp and Su..., Sde D'(B) such that
T = ^TFi®St
on A x B.
Let i^ = S(RP)T. Then as above, the operators d/dxt, i = \,...,p, can be
put into simultaneous triangular form on if Let nu..., nd e(Rp)£ be the
(generalized) joint eigenvalues of these operators on if Let F1,...,Fq be a
basis of the space of elements in C[xu..., xp, e"',..., e"d] of degrees (as an
element of Cft,,..., tp + d\) at most dim if If g e C™(B) then set Tg(f) =
T(f ® g) for / e Cf(A). Then Tg(f) = I S,(g)TF.(f) by the previous Lemma.
Here S, are linear functionals on C?(B). We leave it to the reader to check
that they are continuous.

8.A.3. The Radial Component Revisited
337
8.A.3. The radial component revisited
8.A.3.I. Let Nbea smooth orientable manifold and let U be an imbedded
submanifold. Let G be a Lie group acting on N. Set M = G x U, which we
look upon as a G-space with the action of G being left multiplication in the
first factor. We assume that there is a G-invariant volume form coN on N and
that U is orientable. We fix a volume form, c%, on U. We set caM = dg Ac%
with dg a fixed choice of left invariant measure on G. Finally, we also assume
that the map tp of M to N given by ip(g, u) = gu, g e G, u e U is a submersion.
If TeD'(N) then we define ip*TeD'(M) as in 8.A.2.6. We will denote by
L(#) the action of G on D'(N)and on D'(M). That is, L(g)T(f) = T(L(g~x)f).
Set D'(M)G (resp. £>'(N)G) equal to the space of all T such that L{g)T = T for
all c/ e G.
Lemma. If T e D'{N)G thenip*T e D'(M)G.
It is clear that coM/(aN (8.A.2.3) is a G-invariant differential form on M.
8.A.2.5 implies that ^{L(g)f) = L(g)^(f) for all / e Cf(M) and all g e G.
The lemma now follows from the definition of ip*(T).
8.A.3.2. The above Lemma combined with 8.A.2.9 implies that if T e D'(N)
then there exists a uniquely defined element S e D'(l/) such that
(1) ^*(T) = dg®S.
Lemma. Assume that dg is bi-invariant. Let L be a closed subgroup of G
such that L • U = U. Assume that iov is L-invariant. Then the S in (1) above is
L-invariant.
Let L act on M by a(\)(x,u) = (x\~l,\u) for x e G, u e 1/ and 1 e L. Then
i//((t(1)x) = i/f(x) for 1 e L and xeM, Our assumptions imply that coM is
invariant under a (I) for 1 e L. Let t(1)/(x) = /((7(1"1)x) for /6C*(M),
x e M and 1 e L. Then i/> *T • t(1) = dg® L(\)S for 1 e L. The Lemma follows.
For lack of any standard notation, we will write
(2) i//°(T) = S if S is related to T as in (1) above.
It is clear that ip°(T) depends on the choices of dg, u>v and coN.
8.A.3.3. We now return to the situation in 7.A.2. Let G be a real reductive
group and let H be a closed subgroup of G. As in the above mentioned sec-

338
8. Character Theory
tion we will assume that g = h ® V with Ad(H)V = V. We also assume that
h" = {h e h | det(ad fy J # 0} #0.
Lemma. Ad(G)h" is open in g. Furthermore, the map tp(g,h) = Ad(g)h is a
submersion of G x h" onto its image.
If he h", geG, xeV, Ye\) then
d^h(X,Y) = Ad(g)([_X,K\ + Y).
Hence dipgh is surjective for all geG, he h" so ^ is a submersion.
8.A.3.4. We now assume that G and H are reductive and that G = G+ (2.2.1),
H = H+. We also assume that we have an invariant non-degenerate
symmetric bilinear form, B, on g. such that B restricted to h is non-degenerate.
We choose Lebesgue measure on g. and on h corresponding to B and B\
respectively. In this context we have (as in 8.A.3.2) ip° = ipGH with
iP°:D'(Qf^D'(\)")".
If we trace through the definition we see that
(1) If F e LUd)G then i/,°(7» = TFW, and Fw. e Llloc(l)").
Let 8:DO(q)g -ȣ>0(h") be as in Lemma 7.A.2.5. We write 8GM if it is
necessary to indicate the dependence on G and H.
Lemma. If D e DO(q)g and if T e D'(q)g then
^g,h(d)Kh(T) = Kh(DT).
This is an immediate consequence of the definition of DT and Lemma
7.A.2.5.
8.A.3.5. The above Lemma implies that if DuD2e £>0(g) and if T e £>'(g)G
then
(1) ^(DtDMl.AT) = SG,H{D1)SG,H(D2)^„{T).
We now come to the main result of this appendix. Set
r,(X) = r,m(X) = |det(ad A"|„)| for X e h.
Lemma. Let p e S(q)g (looked upon as a constant coefficient differential

8.A.3. The Radial Component Revisited 339
operator on g). If T e D'(q)g then
iP°(pT) = r1-il2p(r1il2iP0(T)).
Here p is defined as in 7.A.2.9.
In light of Scholium 7.A.2.9 and the argument thereafter it is enough to
prove that if to is as in 7.A.2.9 then
(1) <5G.,,M = >T1/W/2.
The proof follows the same line as in 7.A.2.8. We set h' = g' n h. Then
h' c h" and both subsets are open and dense in h. Fix X e I)' and set b =
{Ye g | [X, 7] = 0}. Then b is a Cartan subalgebra of g contained in h. Set
<t = <t(gc,bc). Set <&H = <t(hf,bc) and <bv = $ - <tH. Fix, <t+, a system
of positive roots for <t and set <tj, = <t//n<I)+ and <bv = <t+ n<&v. Put
n = nae<1,+ a, nH = nae<1,+ a and nK = n^,,,^ a. Then n = nHrv
We can (and do) choose <t+ such that for each connected component C
of b' there exists a complex number, /.ic, such that
(2) r\il2 = ncYlv on C.
If we repeat the calculation in 7.A.2.8 in this context, we find that
8(co)x = 65+2 X <*(*)"'#* and
with Hu..., Hr an orthonormal basis of bc and the £;, are as in 7.A.2.8. This
implies that if / e C°°(h') then
r,-112^1'2/ = lr,-ll2Hfnll2f + 2n-112 £ £.£-.»/1/2/-
0IE<I>jj
If X e b1 n h then Xr\ "2 = 0. Thus
,rl'2dy/2/ = >r1/2Ltf,V/2/ + 2 I £.£_./.
The result now follows.
8.A.3.6. We now study the pull-back of differential operators from G to g
via the exponential mapping. If X e g then set /(X) equal to the Jacobian
of exp at X. That is,

340
8. Character Theory
One calculates that if h c g is a Cartan subalgebra then (h = exp H)
(1) \j(H)\l>2 = £(H)h-»A(h)/7z(H)
with c a locally constant function on h', A is as in 7.A.3.6, n is as in 8.A.3.5
and A, n and p are computed using the same system of positive roots.
If / e Cr(G) is such that figxg'1) = f(x) for g, x e G and if z e Z(gc)
then 7.A.3.7 implies that z/(h) = (A"'y(z)(A/))(/i) for h e //'. Let for z e Z(gc),
A(z) e /(gc) = S(gc)G be defined by
X(z) = p if y(z) = p.
Let fi0 be an open Ad(G)-invariant subset of g such that exp is a diffeomor-
phism of fi0 onto an open subset Q of G (which is Int(G)-invariant). We can
thus define exp*:D'(Q) -» D'(Q0) as in 8.A.2.6.
Lemma. If z e Z(gc) and if T e D'(Qf then
exp*(zT) = |jr1/2(A(z)(|./|1/2exp* T) on Q'.
Let h be a Cartan subalgebra of g such that h' n Q0 # 0. If /i = exp H
with H e h' n fi0, if / e C* (Q)° and if z e Z(g) then the lemma follows from
the following calculation
zf {exp H) = A '(%(z)A/(exp H) = Al{h)n{h)8(k(z))(n-X exp* Af)(h)
= U(H)r1'W(z))(|./|1'2 exp* /)(//)
= (Ur1/2Wz)(|./|1'2 exp*/))(//).
8.A.3.7. Lemma. Lef C i? the Casimir operator of G. If T e £>'(fi)G then
exp*(CT) = |)r"2/(C)|./|"2(exp* T).
Let U be an open subset of g such that exp is a diffeomorphism of [/ onto
an open subset V of G. Fix a basis Xx,..., X„ of g such that det(B(Xj, A^)) > 0.
If X e [/ and if 7e g then we set T(X)Y equal to the element of g such that
d exp^fy) = T{X)YexpX. Set gtj{X) = B{T{X)Xi, T(X)Xj). We assume that U
is connected. Then g(X) = det(gu{X)) > 0 for X e U. Set u(X) = g(X)1'2 for
X e U. On [/ we take the linear coordinates x, defined by the equation
X = I x,^)*,-. Then if [cT] = [c/J1 we have
(1) (exp*)C(exp*)1 =«-'£Ag»MA.

8.A.3. The Radial Component Revisited 341
This formula is a direct consequence of the standard formula for the
Laplace-Beltrami operator on a pseudo-Riemannian manifold (see any book
on Riemannian geometry).
The standard formula for the differential of exp implies that
(2) rWy = (___jy.
We note that
(3) g(X) = j(X)2.
Indeed, j(X) = det T(X). Clearly, g(X) = (det T(X))2.
We now assume (as we may) that B(XhXj) = eft^ with e, = ± 1. We set
D = (exp*)C(exp*)1 - j-l'2MC)j1'2. Then
D = F' I e,e, ^ B((T(X)T( - X))~' X„ X.)j ~ - f "2 £ e,-^ 2jil2.
(4) D\ = 0.
The previous result implies that D annihilates the G-invariant smooth
functions on g.
»> (?"5?V'1"a
Indeed Bl--r"!('l8,|^)/"2.
Thus
D = j-lJ£er£s^B((T(X)T(-XYlXr,Xs)j^
= r1Jiere.-^-(B(((T{X)T(-X))-1 - I)Xr,Xs)j)^-.
Now if F is a smooth function from U to g then Zs esB(F,Xs)-— is the
3xs
vector field on U corresponding to F.
9
Thus I.sesB{{(T{X)T{-X))-1 - I)Xr,Xs)— is the vector field corre-

342
8. Character Theory
sponding to the function ((T(X)T(- X))-' - I)Xr. We note that this function
can be written in the form [X, Gr(X)] with Gr an analytic function on U. We
have therefore shown that
-IV 3
(6) D = j L er ^~~ JK with Vr the vector field corresponding to
[X,Gr(X)].
The Lemma now follows from
Scholium. Let Gbe a unimodular Lie group, let U be an open \d(G)-invariant
subset of q and let Y be a vector field on U of the form X t—► [X, G(X)] on U.
Then YT = 0 for all Te D'(Uf.
Let Xu..., Xn be a basis of g. Then G(X) = 'Lgl{X)Xl. Thus Y = Eg,. 7,
with 7, the vector field corresponding toln [X,X,]. Thus YT = I g-J-J.
Thus we may assume that G(X) = Z e g. If / e C°°(l/) then 7*/ = d/dtI = 0
/(Ad(exp(-tZ)X). Thus YT = - Y. Hence
0 = -/- T(/ o Ad(exp tZ)) = 7T(/).
This completes the proof.
8.A.4. The orbit structure on a real reductive Lie algebra
8.A.4.I. Let G be a real reductive group with Lie algebra g. We continue to
write gX for \d(g)X for g e G and X e g. If X e g then we say that X is
semi-simple if ad X is semi-simple on gc. If X e [g, g] and if ad X is nilpotent
then we say that X is nilpotent. If X e g then ad X can be written uniquely in
the form ad X = S + N with [S, AT] = 0 and S semi-simple, N nilpotent
(Jordan canonical form). It is easily seen that S and N are derivations of g.
Thus there exists X„ e [g, g] such that ad X„ agrees with N on [g, g]. Set
Xs = X — X„. Then ad Xs is semi-simple. We have proved
(1) If X e g then X can be written uniquely in the form Xs + X„ with
[XS,X„] = 0 and Xs is semi-simple, X„ is nilpotent.
The key to the orbit structure of the action of G on g is the following Lemma
of Jacobson, Morosov (c.f. Jacobson [1, Lemma 8, p.99].
Lemma. // X is a non-zero nilpotent element of g then there exist H, Y e g
such that [H,X] = 2X, [//, 7] = - 27 and [X, 7] = H.

8.A.4. The Orbit Structure on a Real Reductive Lie Algebra
343
We first look at the case when g = s/(n, R). If X e g is nilpotent then X is
nilpotent as an endomorphism of R". The Jordan decomposition implies that
there exists a basis if R" such that X is the direct sum of Jordan blocks
"0 1 0 ... 0"
0 0 1 ... 0
0 0 0 ... 1
0 0 0 ... 0
It is now an easy exercise to prove the existence of H and Y in this case. In
the general case, we may clearly assume that [g, g] = g. If we choose a basis
of g then ad defines an isomorphism of g onto a subalgebra of s/(n,R) = a,x
such that the form B(X, Y) = tr XY is nondegenerate when restricted to it.
We identify g with this subalgebra of g^ Let V = {X s a,x\ B(X, g) = 0}. Then
9l=9©Kand[9,K]cK
Let X be a nilpotent element of g. Then there exist Y', W e g! such that
X, Y', W have the desired commutation relations. We write Y' = Z + Yj and
H' = H + Hx with Z, H e g, Yx,Hxe V. It is easily seen that \_H,X] = 2X
and that [Z, X] = H. Set g* = {y e g | [X, y] = 0}. We assert that
(2) ad H + 21 is invertible on g*.
Let us show how one completes the proof of the Lemma using (2). A direct
calculation shows that [X,(ad H + 2/)Z] = 0. Thus there exists W e qx such
that (ad H + 2l)W = (ad H + 2l)Z. Set Y = Z - W. Then (ad H + 2l)Y= 0
and \_X, Y] = H. We are thus left with the proof of (2).
We note that (ad X)mg = 0 for some m > 0. We set (ad X)° = I.
(3) (ad H - jI)(Qx n (ad X)'a) c9xn (ad X)>+ 'g.
Indeed, if j = 0 then ad H(g*) = ad X ad Zg* c g* n ad Xg. If j > 0 and
if yeQx n(adX)JQ then y = (ad X)J'Twith Teg. Hence
ad//j; = adXadZ(adX)jr
= -adxT^ (ad X)'ad H(ad Ar)J''1r +(ad X)j+'ad ZT)
= X 2(i + l)(ad XyT-j ad H(ad X)JT + (ad X)i+ladZT
= j(j + l)(ad X)JT- j ad H(ad X)JT+ (ad X)J+ 'ad ZT.

344
8. Character Theory
This implies that (j + l)(ad H - jl)y = -(ad X)i+l[Z, T]. Which
implies (3).
(3) implies that the eigenvalues of ad H on qx are contained in the set
{0,1,..., m - 1}. (2) follows from this.
8.A.4.2. Let j be the center of g. For the remainder of this appendix we will
assume that \d(g)X = X for g e G and X e j. Let 7(g) denote the algebra of
all G-invariant complex valued polynomials on g. Set 7+(g) be the subalgebra
of elements that vanish at 0.
Lemma. Let Jf denote the set of all nilpotent elements of g. Then
J" = {Xe&\l+(Q)(X) = 0}.
9 = 5 © [9.9]- If A. e 5* then extend X to g by setting 2([g, g]) = 0. Thus
5* c I+(g). It is now clear that if X e g and if I+ (q)(X) = 0 then X e [g, g].
We may thus assume that g = [g, g].
(1) If X e sl(n, R) and if tr X1 = 0 for j = 1,..., n then X is nilpotent.
This is well known and left to the reader.
Let n = dim g. If we choose a basis of g then ad g c s/(n, R). The
polynomials fj(X) = tr(ad X)J are in 7+(g) for j > 0. Thus (1) implies that if
I+(q)(X) = 0 then X is nilpotent.
If X is nilpotent then there exists //eg such that ad HX = 2X. Thus if
/ e /+(g) then f(X) = /(Ad(exp(-tH))X) = f(e-2'X) for all t > 0. If we
take the limit as t -» + 00 then we see that f(X) = 0.
8.A.4.3. Our next goal is to prove a basic result of Kostant. The proof will
use the following Theorem of Whitney [1].
Theorem. Let f\,---,fm be polynomials on R". Then {x e R"| jf(x) = 0 for
i = 1,..., m} has a finite number of connected components.
This theorem has an immediate corollary.
Corollary. Let f be a nonzero polynomial on R". Then U = {x e R" | f(x) # 0}
has a finite number of connected components.
If x e U then set F(x) = (x,/(x)_1). Then F defines a homeomorphism of
U onto {(x, t) e R" +' | tf(x) =1}. This reduces the corollary to the Theorem.

8.A.4. The Orbit Structure on a Real Reductive Lie Algebra
345
8.A.4.4. The following theorem is the result of Kostant [1] alluded to above.
Theorem. The set of nilpotent elements of g consists of a finite number of
orbits relative to the adjoint action of G.
The idea of the proof is to show that up to the action of Ad(G) there are
only a finite number of choices for the "//-part" of a TDS in g. We will then
show that for each choice of an //, the stabilizer of H has only a finite number
of orbits in the "X-parts".
Let H be an "//-part" of a TDS. Then H is a semi-simple element of H
with integral eigenvalues. GH = {g e G \ Ad(g)H = //} is then a real reductive
subgroup of G that contains a Cartan subgroup of G (2.3.1). Thus H is
contained in a Cartan subalgebra of g, which we may assume (up to conjugacy
by G) is 0-stable. But then it is easily seen that up to conjugacy we may assume
that // e a a maximal abelian subalgebra of p. We can thus choose a
minimal p-pair (P0,A)(A = exp o) such that if ae <&(P0,A) then a(//) > 0. Let
{(*!,...,ar} be the simple roots in <t>(P0,A). We assert that
(1) 0<a<(//)<2 fori = l,...,r.
We note that this will prove that up to the action of Ad(G) there are only
a finite number of such //. Let X, Y e g be such that X, Y, H is a TDS. Then
X is contained in the nilradical of p0- We can thus write X = Zc,E<i>+ X„ with
Xa in the a rootspace. If Xx. # 0 then a,(//) = 2. Otherwise, a,(//) > 0 and if
a;(//) > 0 then [7, g°"] # 0 by the representation theory of a T-D-S. If we
interchange the roles of X and Y we find that there exists a e <&{P0, A) with
a(//) = 2 and a - a, e <&{P0, A). Now a = I m^a,- and mt > 0. Thus m^iH) < 2.
So <Xj(//) < 2 since mt> 1. This proves (1).
Fix H an "//-part" of a TDS. Let gJ = {X e g| [//,X] = jT}. If X is an
"X-part" of a TDS with // the "//-part" then X e g2 and [X,g0] = g2. We
set V = {z e g21 [z,g°] = g2}. Then V is non-empty. We now show that V is
the union of a finite number of orbits under the action of GH. This will
complete the proof of the theorem. Choose bases of g° and g2. If z e g2 then set
/(z) equal to the sum of the squares of the p x p minors of ad z as a linear
map of g° into g2 (dim g2 = p). Then V = [z e g2 \f(z) # 0}. Corollary 8.A.4.3
implies that V has a finite number of connected components. Let U be the
identity component of GH. Then U has Lie algebra g° and thus if z e V the
U • z is open in V. Since two orbits are either disjoint or equal this implies that
the connected components of V are orbits of U.
8.A.4.5. We now study more general orbits under the action of G on g. We

346
8. Character Theory
will use the following Lemma. Let X be a complete metric space. Let A be a
topological group acting continuously on X. We assume that A is a-compact.
That is, A = Uoij with cas c o}]+1 and each cas is a compact neighborhood of 1
in A.
Lemma. // X is a countable union of orbits under the action of A on X and
if xe X then A • x is open in its closure in X. If X is the union of a finite number
of A-orbits then we can label the orbits Ol7...,Ok such that {Jj>m Oj is closed
in X for m= 1,..., k. In particular, Ok is closed in X.
Let p e X. Let Y be the closure of A • p in X. Then Y is a countable union
of orbits of A. Let {<jf,} be a sequence in Y such that {A • q(} is the set of all
orbits of A contained in Y. A • <jf,- = [j a>m • q(. If A • <jf,- has interior then A • <jf,-
is open in Y. Thus, if none of the A • qt are open in Y then we may apply the
Baire category theorem (c.f. Reed, Simon [1, p.80]) and find that \J,A- q{ is
nowhere dense in Y. Since, Y = [J( A • qt this is a contradiction. Hence there
exists q e Y such that A • q is open in Y. If A • q is not equal to A • p then
Y — A • q is closed and contains A • p. This is a contradiction. Hence A • p is
open in Y. We now prove the second assertion.
Let X = {JJ<n Oj. Then X = [j C1(0.). The Baire category theorem implies
that there is an index, j, such that C1(0,) has interior in X. We have just seen
that Oj is open in Cl(0y). Thus O, is open in X. If we relabel the orbits,
we may assume that Ox is open in X. We can now argue as above for
X - Ouetc.
8.A.4.6. UXsq then we set Vx = {Y e g | f(Y) = f(X) for all / e /(g)}. We
note that V0 is the set of nilpotent elements of g.
Theorem. If X e Q then Vx is a finite union of G-orbits.
As usual, we may assume that G is semi-simple. Let h 1?...,hr be a set of
representatives for the G-conjugacy classes of Cartan subalgebras of g.
(1) For each j, \)s n Vx is a finite set.
Indeed, we may choose a basis of gc such that ad H is diagonal for each
H e hj. As usual, we write det(ad X - tl) = I tmDm(X). Then each Dm e /(g)
and if H e hj then Dm(H) is, up to sign, the n — m-th elementary symmetric
function in the diagonal entries of ad H. This clearly implies (1).
(1) implies that up to the action of G on g, there are only a finite number
of semi-simple elements in VX,HX,...,HN. If Z e Vx then we can write Z =
ZS + Z„(8.A.4.1(1)).

8.A.4. The Orbit Structure on a Real Reductive Lie Algebra
347
(2) Z.e[flz",flz'] = 9i-
Indeed, gZs = 3, © g^ If W e 3, then W is semi-simple in g. Write Z„ =
X! + X2 with X, e 3, and X2e a^- Then X2 is nilpotent and Z =
(Zs + Xi) + X2 with Zs + X! semi-simple and [Zs + X1(X2] = 0. The
uniqueness in 8.A.4.1 implies that X! = 0.
Up to conjugacy relative to G we may assume that Zs = Hj for some j.
Then (1) implies that Z„ is nilpotent in gHj. The number of nilpotent orbits
in gHj, relative to the action of G77j, is finite, say, GHj • ZJm for m = 1,..., Mj.
Thus Vx is the union of the orbits G • (Hj + Zjm).
8.A.4.7. We now can apply Lemma 8.A.4.6 to Vx since Vx is clearly
closed in g.
Corollary.
(1) If X e g then G • X is open in its closure.
(2) If X e g then Vx = G • Xx u ■ ■ ■ u G ■ Xk with UJimG- X, closed in g.
With this material in place we can now prove the following basic theorem of
Borel, Harish-Chandra [1].
8.A.4.8. Theorem. Let X e g. Then X is semi-simple if and only if G ■ X is
closed in g.
If X e g, X = Xs + X„ there exists H e g such that [H, Xs] = 0 and
[H,X„] = 2X„ (8.A.4.6(2), Lemma 8.A.4.1). Thus,
lim e'adHX = Xs.
This implies that if X is not semi-simple then G • X is not closed.
If X and Y are semi-simple elements of g with Y e Vx then ad X and ad Y
have the same characteristic polynomials. Hence, in particular, dim Gx =
dim GY. Fix X a semi-simple element. If 7eC1(G-X) then Y e Vx and
hence G ■ Y is open in C1(G • X). Since C1(G • X) c Fx, there exist X! =
X,...,Xk e C1(G • X) such that C1(G • X) is the disjoint union of the orbits,
G • X, and each is open in C1(G • X). Thus each is closed in C1(G • X). In
particular, G • X is closed in C1(G • X). So G • X = C1(G • X) as asserted.
8.A.4.9. We conclude this appendix with several results about semi-simple
elements. Let G be a real reductive group of inner type. If g e G then g is said
to be semi-simple if Ad(g) is diagonalizable on gc.

348
8. Character Theory
Lemma. If g e G then g can be written uniquely in the form g = gs exp X
with gs semi-simple and X e g nilpotent and Ad(gs)X = X.
Let for \x e Cx, (g^ be the generalized eigenspace for Ad{g) on gc with
eigenvalue \i. Then
(1) C(9c)^(9c)v]^(9c),v
Let S be the linear automorphism of gc defined by S|(9c)m = \il. (1) implies
that S is an automorphism of gc. Thus N = S_1Ad(g) is also an
automorphism. Clearly, N — I is nilpotent. Thus log N = D is given by a finite series
and D is a derivation of gc that is zero on the center. Hence D = ad X with
X e [gc, gc] and ad X is nilpotent. Since ad X is a polynomial in N, SX = X.
Let a be conjugation in gc with respect to g. Then Ad(g) = a Ad{g)a =
aSaaNa. Thus the uniqueness in the Jordan decomposition implies that
oNa = N. This in turn implies that aX = X. Hence X e g. Clearly,
Ad(gexp(-X)) = S.
So set gs = g exp( — X). The Lemma now follows.
8.A.4.10. Lemma: If g e G is semi-simple then m = {Xeg! \d(g)X = X}
is reductive and rfc(tn) = rk(q).
We prove this by induction on dim g. If dim g = 0 or 1 there is nothing
to prove since then G is abelian. Assume for all g of strictly lower dimension.
Since G is of inner type 3(g) c m. Hence, if j(g) # (0) then the result follows
from the inductive hypothesis. We therefore assume that G is semi-simple.
We use the notation of 8.A.4.9. Define for se R, 7s, a linear isomorphism of
9o by 7j(flc) = \)i\sI. Then 8.A.4.9(1) implies that for each s e R, T is an
automorphism of gc. Since <x(gc)„ = (flc)^ it follows that oTs = Tsa. Hence 7s
is a one parameter group of automorphisms of g. Thus Ts = eady with Ye g
such that ad Y is semi-simple with real eigenvalues. Now m c qy and g e GY.
Thus if Y # 0 then 2.3.1 and the inductive hypothesis complete the
induction. We may thus assume that the eigenvalues of Ad(g) all have absolute
value 1. With this assumption H = Cl({gk\ke Z}) is a compact subgroup
of G. This implies that there is a compact form of Int(gc), U, such that H c U.
Since U is connected there is a maximal torus 7 of U such that g e T. This
clearly implies that rk(m) = rfc(g). Let u be the Lie algebra of U. Then mc is
isomorphic with the complexification of
{Xeu\Ad(g)X Ad(g)-1 = X}
which is the Lie algebra of the compact Lie group
{ueUWd^uAdig)1 = «}.
Thus m is reductive.

8.A.5. Some Technical Results for Harish-Chandra's Regularity Theorem
349
8.A.4.11. Lemma: // Te D'(G) is central and if for each semi-simple
element g e G there exists an open neighborhood U of g in G such that T\v = 0
then 7 = 0.
If T{u = 0 then T[xUx-, = 0 for all x e G. Let g e G. Write g = gs exp X as
in Lemma 8.A.4.9. Lemma 8.A.4.1 combined with Lemma 8.A.4.10 imply
that there exists a TDS {X, Y,H} with Ad(gs)Y = Y, Ad{gs)H = H. Now
exp{tH)g exp(-tH) = gs exp(e2'X). This implies that if U is an open
neighborhood of gs in G then there exists t > 0 such that e\p(tH)U exp(—tH) is
an open neighborhood of g. Thus our hypothesis implies that T vanishes in
a neighborhood of g. Since g is arbitrary, T = 0.
8.A.5. Some technical results for Harish-Chandra's regularity theorem
8.A.5.I. In this appendix we collect several results that will be used
in Section 8.3. Let H, X, Y be a standard basis for a TDS, u, over C.
Set b = CH + CX.
Lemma. Let M be a u-module such that
(1) dimU(b)m < oo for all me M.
(2) No eigenvalue of H on M is a non-negative integer.
Then the action o/C[7]onM is torsion free.
Let meMbe non-zero. If there exists peC[7] such that p # 0 and pm = 0
then dimC[7]m < deg p < oo. C/(u) = C/(b)C[7]. Hence (1) implies that
dim U(u)m < oo. Now 0.5.5 implies that H must have a non-negative integral
eigenvalue on U(u)m and this contradicts (2).
Corollary. Let M be a u-module such that if me M then dim C[//]m < oo
and such that the eigenvalues of H onM are real and strictly less then 0. Then the
action of C[7] is torsion free on M.
M is a direct sum of the generalized eigenspaces for H acting on M. Let m be
an element of the ^-generalized eigenspace. Then X"m is an element of the
eigenspace for X + In. Thus our hypothesis implies that X"m = 0 for some
positive n. Hence, if me M then dim U(b)m < oo. The corollary now follows
from the preceeding lemma.
8.A.5.2. We now collect a few results about distributions on R". Let p, q e N
with q > 0 and p + q = n. We write R" = R^ x Rq. Let [/, be an open subset
of R^ and let U2 be an open neighborhood of 0 in Rq. Set I) = Ux x U2.

350
8. Character Theory
Theorem. If T e D'(U) and if supp 7 c [/, x 0 then there exists T, e £>'(t/i)
such that T = "L T, ® d'8. Here 8 is the Dirac delta function on Rq supported at
0 (5(f) = /(0)).
For a proof see Schwartz [1, p. 102].
Corollary. Let a> e P(R') be such that co(0) = 0. We extend to to R" by setting
co(x, y) = co(y). If 7 e D'(U) and if supp 7 cz [/, x 0 then there exists k such
that cokT = 0.
As above, 7=7,® D8. Let m = deg D. Let 0 e K c C1(K) c [/2 with K
open and C1(K) compact. Let i/> e Cf (C/2) be such that tp is identically 1 on
V. Extend ^ to U by setting i/^(x, y) = ^(y). If / e Cf (I/) then 7(/) = 7(/) =
7(^/). If r = (r,,..., r„) e N" then set yr = /,' • ■ • /,«. If / e C?(U) and if
f(x,y)= X ar(x)yr + i?m(x,j;)
|r|<m
is the Taylor series around 0 of f(x, •) to order m in y at 0 then
T(f)= I T(ar^/).
|r|<m
Thus, if / vanishes in y to order m at 0. Then T(f) = 0. It is clear that there
exists k such that o/ vanishes to order m in y at 0. Thus o/f vanishes to order
m in y at 0 for all / e Ceo(I/). Thus <o*7(/) = T(<okf) = 0 for all / e Cf(U).
8.A.5.3. We retain the notation of 8.A.5.2. Write D'Vl(U) for the space of
distributions on U supported on [/, x 0.
Lemma. Let Ej = yj d/dyj for j = 1,..., n. Then each Ej acts semi-simply on
D'Vi(U) with eigenvalues of the form —k with k > 0, k e Z.
If f€Cx(V2) then
E}T(f) = -@/dyj(yjf))(0) = -f(0) = -8(f).
Ej8 = -8. If / is a multi-index then [E^a'/Sy7] = -ifi'/dy'. Thus
£J.3,/3j;,^= -(1 +ij)d'/dy'8.
The Lemma now follows from Theorem 8.A.5.2.
8.A.5.4. Lemma. Let as be non-negative real numbers for j = !,...,«. Set

8.A.5. Some Technical Results for Harish-Chandra's Regularity Theorem
351
D = I(aj + \)Ej. Then D acts semi-simply on D'Vi(U) with real eigenvalues
— A such that I > n + £ a-r
Since [£,, E}~\ = 0 for all i, j this follows directly from the preceding Lemma.
8.A.5.5. For lack of a better place to put the following material, we will
conclude this "hodge-podge" of an appendix with it. As usual, let y(R")
denote the space of all / e C*'(R") such that
PrJ(f) = sup (1 + \\x\\y\Q'f(x)/Qx'\ < oo
xeR"
endowed with the topology induced by these semi-norms.
If / e 5"(R") then we write
(1) Ff(x) = (In)-"2 J f(y)e-'<x-»dy.
R"
The following result is standard (cf. Stein, Weiss [1])
Theorem. F is a topological isomorphism of ^(R") onto y(R") with
(F2f)(x) = f(-x).
8.A.5.6. We note that y(R") is dense in L"(R") for all 1 < p < oo. The
Plancherel theorem (cf. Stein, Weiss [1]) says that
(1) I|F/Il2 = ll/Il2
for / e y(R"). Thus F extends to a surjective unitary operator on L2(R").
We also note that if / e ,5^(R") then
(2) HF/IL< 11/111-
Thus F defines a bounded operator from L'(R") into Lco(R").
8.A.5.7. We will be using the following inequality. The argument is taken
from Stein-Weiss [1, Lemma 3.17, p.26].
Lemma. // d'f/dx1 e L'(R") for \I\ < n + 1 then
\\f\L<Cn X Il9,//9x'||1.
|/|<n+l
We note that
(1) F(d'f/dx')(x) = (-i)^x'F(f)(x).

352 8. Character Theory
Also,
(2) (l + ||x||2)("+1"2<(l + |x1|+-'- + |x„|)" + 1<Q I |x'|.
|7|<n+l
Now
|F/(x)|<C;(l+|x|2)-("+1"2f X \x'\)\Ff(x)
= Q(l+|x|2)-("+1"2 £ |(F3'//3x')(x)|
|/|<n+ 1
= c;(i + |x|2)-«"+1,/2 I PWlli-
|7|<n+l
Thus 8.A.5.5. implies that
<c;|(i + |x|2r("+1"2dx. X lieWlli-
R" |7|<n+l
The Lemma now follows.

9 Unitary Representations
and (g, K)-Cohomology
Introduction
Let G be a connected semi-simple Lie group with finite center. Let F be an
irreducible finite dimensional (g, K)-module. In this chapter we give the
Vogan, Zuckerman [1] (Enright [1] in the case when g has a complex
structure) classification of irreducible unitary representations, (n, H), of G
such that H'(% K; HK ® F*) # 0. A consequence of this is the
calculation of the cohomology spaces H'(q, K, HK® F*) with (n,H) irreducible
and unitary.
One of the main applications of this classification is to the study of the
L2-cohomology of locally symmetric spaces, which we now describe. Let Y
be a discrete torsion free subgroup of G such that T\G has finite volume
relative to the quotient measure of Y\G corresponding to some fixed,
choice of invariant measure on G. Let nr be the right regular representation
of G on L2(r\G). If n e GA let (7i„,//J e \i. Then the results in Borel [2]
and Borel-Garland [1] imply that the L2-cohomology of T in dimension i
with coefficients in F is a quotient of
(1) 0 HomG(H,,,L2(r\G))®Hi(g,K;(/gK®F*).
It would take us too far afield to go into any more detail on this application
353

354
9. Unitary Representations and (g, K)-Cohomology
(including the definition of L2-cohomology). However, if T\G is compact
then the L2-cohomology is equal to (1) and to the Eilenberg-MacLane
cohomology H!{r,F) (c.f. Borel, Wallach [1]).
The results of this chapter are an outgrowth of the Kumaresan [1]
determination of the irreducible K-invariant subspaces of Ap that can occur as
K-subrepresentations of irreducible unitary representations of G such that
the Casimir operator of G acts by 0. Kumaresan's main tool is the so-called
"Dirac inequality" and he uses a method based on ideas in Parthasarathy [3].
We have divided the exposition of the results so that the reader interested
only in the vanishing theorems can find the complete proof (including that
they are best possible) by the end of Section 5. The next two sections are
devoted to the actual classification. The basic ideas are not difficult, however
the technical details (mostly results of Vogan) go to the heart of the algebraic
approach to representations of real reductive groups. We have included in 9.6,
several results that could (should?) have been included in Chapter 6, including
Kostant's determination of the Lie algebra cohomology of the unipotent
radical of a parabolic algebra with respect to a finite dimensional
representation (Kostant [2]). 9.7 contains the details of the classification. The proof
follows the outline given in Vogan, Zuckerman [1].
In Section 98 we give some implications of the results of this chapter to the
representation theory of groups of real rank 1. We also give the tabulation of
Enright and of Vogan-Zuckerman of the vanishing theorems implied by the
results in this chapter.
9.1. Tensor products of finite dimensional representations
9.1.1. Let g be a reductive Lie algebra over C. Fix a Cartan subalgebra, h,
ing. Let<5+ be a system of positive roots for <t(g, h). If \i e h*isa<I>+-dominant
integral, then let L(jX) be the finite dimensional irreducible g-module
constructed in 1.7.4.
Let s0 be the element of W{q, h) (0.2.3) such that s0<&+ = -<t+. Then s0n
is the highest weight of L(ji) relative to -<t+. In other words, s0/i, is the
lowest weight of L(n). Let n = ®xe®+ Qx and n~ = ®xe®+ Q-x (as usual).
Set N(n) = {ne U(n)\nL(fi)(s0fi) = 0}. Here, if V is an h-module and if
a e h* then V(a) denotes the a-weight space. The next four results are taken
from Parthasarathy, Ranga Rao, Varadarajan [1]. In that paper some of
the main ideas were also attributed to Kostant.
Lemma. Let a, n, y be dominant integral. Then
dim Horn (L(ff), L(n) ® L(y)*) = dim{t; e L(n)(a + s0y)\N(y)v = 0}.

9.1. Tensor Products of Finite Dimensional Representations
355
As a g-module L(^)®L(y)* is isomorphic with Homc(L(y), L(/x)) with
g acting by (XT)(v) = XT(v) - T(Xv) for X e g, v e L(y). If V is a finite
dimensional semi-simple g-module then V" = {v e V\Xv = 0, X e n} is the
direct sum of the highest weight spaces for the irreducible constituents of
V. Thus dim Hom9(L(<r), V) = dim(Kn)(ff). In our case, (Homc(L(y), L(^))n =
Homn(L(y), L()i)). The above observations now imply
(1) dim HomB(L(ff), L(n) ® L(y)*) = dim Homn(L(y), L(n))(a).
Fix w e L(y)(s0y) - {0}. Then n~w = 0 and hw is contained in Cw. Thus,
since U(q)w = L(y), this implies that U(n)w = L(y). If T e Homn(L(y), L{n)){a)
then set ft(T) = Tw. If Tw = 0 then 0 = U(n)Tw = T(U(n)w) = T(L(y)) so
T = 0. Thus Q is injective.
Set S(n,y,g)={vs L(p)(o + s0y)!N(y)v = 0}. If T e Homn(L(y),L(/i))(<x)
then 0= T(JV(y)w) = N(y)Tw = N(y)Q(T). It is therefore clear that
Q(Homn(L(y),L(/i))((T)) is contained in S(n,y,o).
Suppose that veS(n,y,o). Then N(y)v = 0. Thus we can define T~:
U(n)/N(y) -> L(/i) by T~(n) = nv. The map 0 from U(n)/N(y) to L(y) given by
6(n) = nw is a linear bijection. Set T = T~0 '.Then T e Hom„(L(y), L(n))(a)
and ft(T) = T~(rT'w) = T~(l) = i;. Hence Q(Homn(L(y),L(n))(a)) = S(n,y,a).
This completes the proof of the Lemma.
9.1.2. Corollary, dim Homa{L{o),L(n) ® L(y)*) < L(o)(n - y).
We note that L(y)* « L(-s0y). We have
dim HomB(L(ff), L(n) ® L(y)*) = dim Hom9(L(ir) ® L(y), L(n))
= dim Hom9(L(/4 L(a) ® L(y)) = dim Hom9(L(^), L(a) ® (L(y)*)*)
= dim HomB(L(/i), L(<x) ® L( - s0y)*)
= dim{t; e L(a)(ji - y) \ N(-s0y)v = 0} < dim L{a)(n - y).
9.1.3. If a is a simple root in <t+ fix ^eg, and 7, e g_„ such that
[X„ YJ = ha with a(K) = 2.
Lemma. Let \i be <b+-dominant integral. Set
N'(/i) = ^[/(n)(Ia)-!»*» + 1
the sum over the simple roots in <t+. Then N'(^i) = N(n).
We note that {Xa, Ya,ha) is a standard basis for a T-D-S, ua. It is therefore
easy to see that (Arc,)~so"(',0') + 1 annihilates L(jX)(s0^). So N'(jX) is contained in

356
9. Unitary Representations and (g, K)-Cohomology
N(n). Set
1(H) = N'(n) + V(Q)n- + I I/(fl)(fc - sofi(h)).
We note that if we can show that I(ji) is a left ideal in (7(g) then the Lemma
follows. Indeed, set F = U(q)/I()i). Set b~ = b + n". Then it is clear that F is
a g-module quotient of (7(g) (^)V(b-) CS(M1. Here Csofl is the b"-module C with
n" acting by 0 and b acting by s0n. Set v = 1 + 1{h). Then {YxYSo"ih') + iv = 0.
This implies that if / e F then there exists m (depending on /) such that
(Yx)mf = 0. Thus, if / e F then dim U(ux)f < oo. Hence, W permutes the
weights of F relative to I). (See the proof of Theorem 1.7.4 for details of this
and what comes next. Note that relative to 1.7.4 we have replaced <t+ by
- <t+.) We conclude that dim F < oo. It is now a simple matter to see that F
is irreducible. Hence F = L(jx). But then N(/j) = I(n) n (7(n) = N'(fi),
We are thus left with showing that I(n) is a left ideal in (7(g). It is clear that
nl(n) is contained in I(n). We leave it to the reader to show that j(g)/(/i) is
contained in 1(h) (see the proof of (a) below). Thus, since the Yx generate n"
as a Lie algebra, it is enough to show that YJ(jx) is contained in I(jx) for a
simple. Fix a simple and set Y = Y„. We leave it to the reader to check that
[Y, (7(n)] is contained in (7(n) + (/(n)^. It is therefore enough to show that
(a) M^r,w"*', + 1e/(/i) and
(b) [Y,(X,)-^^ + l-[el(n).
If lie I) then h(Xpr = mP(h)(Xf>)m + (Xf)mh = (mP(h) + s0H(h))(Xp)m +
(Xp)m(h - s0n(h)). If we apply this with m = -s0ji(hfi) + 1, (a) follows.
We now prove (b). If a is not equal to ft then the left hand side of (b) is 0.
Set X = Xx, H = ha. We leave it to the reader to check that
(c) [Y,Xm~\ = mXml(-m- H+ 1).
If m = -s0n(H) + 1. Then -m- H + 1 = -H + s0n(H). Thus (b) is true.
9.1.4. Lemma. Let \i, a be <&+-dominant integral. Let seW be such that
s(/i — a) is <t+-dominant integral. Then
dim Hom9(L(s(/J - a)), L(n) ® L(a)*) = 1.
In the course of the proof of 9.1.2 we showed that
dim Hom9(L(s(/i - a)), L(n) ® L(a)*)
= &\m{v e L(s(n - a))(n - o)\N(-s0o)v = 0}.

9.1. Tensor Products of Finite Dimensional Representations
357
Thus to prove the Lemma it is enough to show that this last space is one-
dimensional. Let Q be a system of positive roots such that \i — a is the highest
weight of L(s(n - a)) relative to Q (i.e., s~' <t+). If a e <t+ and if (n - a, a) > 0
then qxL(s(h - a))(n — a) = 0. If a is simple in <t+ and if (ji — a, a) < 0 then
-a e Q. Hence, if m = -(h - a)(hx) then (Xc,)m+1L(s(/i - a))(n - a) = 0.
Now m = — n{ha) + a(ha) < a(hx). Also (—s0( — s0a)) = a. This implies that
N'(-s0a) annihilates L(s(/.i - a))(/i - a), since dim L(s(/i - a))(jx — a) = 1.
The result now follows from 9.1.3.
9.1.5. Let K be a compact connected Lie group with maximal torus T. Let
g = fc and I) = tc. We take B to be negative definite on f. Then B is positive
definite on it. The dual form, ( , ), on (it)* is thus positive definite and Weyl
group-invariant. We maintain the notation of the rest of this section. If y e KA
and if Vy e y then Vy is isomorphic with L(ky) (/v the highest weight of Vy
relative to <t+). The following Lemma is usually attributed to Kostant.
Lemma. Let ^eX' and let ft be a weight of Vy, i=\,...,d. Then
lift + "' + ftll ^ ll^/i + '" + >-M|| with equality if and only if there exists
s e W{K, T) such that sft = 1,, for all i = l,...,d.
Set a = ft + ■ ■ • + ft, fij = k.h and \i = m +••• + Hd- Let s e W(K, T) be
such that sa is dominant. Since sft is a weight of L(/^), sft = \i{ — Qt with Qt
a sum of positive roots. Thus if Q = Ql + ■ ■ ■ + Qd then sa = \i — Q. Now
||(T||2 = ||S(T||2 = (.S'(T,S(T) = (s(T,^-e)
= (sa, h) ~ (sa, Q) < (sa, h) = (h ~ Q, A*)
This proves the inequality. Equality implies that (ji,Q) = 0. Thus \\a\\2 =
\\H\\2 + IIGII2- So Q = 0. But then Qt = 0 for i = 1,..., d.
9.1.6. We now assume (by going to a finite covering of K if necessary) that
p is T-integral.
Lemma. Let h, a, y e KA and let s e W(K, T) be such that s(Xy - X„) is
dominant integral. If WomK(VIJl,Vy®(V„)*) is non-zero then \\k + p\\ >
\\s(Xy — X„) + p|| with equality if and only if x(I = s(/y — /„).
9.1.2 implies that L(Xli)(s(Xy - /„)) is non-zero. Thus Lemma 9.1.4 implies
that \\s(/-y - ).„) + p\\ < \\?. + p\\ with equality if and only if there exists

358
9. Unitary Representations and (g, K)-Cohomology
f e W(K, T) such that s(A„ - X„) = d„ and p = tp.U t e W(K, T) and if tp = p
then f = 1.
9.1.7. The next result is a bit more technical then the preceding ones. It is
taken from Vogan, Zuckerman [1].
Lemma. Let p., a be dominant and T-integral. Let s e W(K, T). Suppose that
s', s" e W(K, T) are such that sp — a is s'<&+-dominant and p — a is s"<5+-
dominant. Then
\\sp - a + s'p\\ >\\p- a + s"p\\
with equality if and only if sp — a e W(K, T)(p — a).
If sp - a = u(p - a) with us W(K, T) then (s'Y' u(p - a) and (s")~ x(p-a) are
both dominant. Hence s"(s'Yxu(p — a) = (p - a). Hence \\sp - a + s'p\\ =
\\(sTlu(p -<j) + p\\ = \\(s")-Hn - ff) + P\\ = ll/i - ff + s"p\\. Thus the "if"
part of the equality statement is true.
We now prove the rest of the Lemma by induction on l(s) (9.A.1.1). Let
ae$+ be simple such that l(sxs) = l(s) - 1. Let f e W(K, T) be such that
s3sp - a is f<5+ -dominant. We show that
\\sp - a + s'p\\ > \\sxsp - a + tp\\
with equality only if
sp - ae W(K, T^s^sp - a).
This certainly implies the Lemma (it actually proves slightly more which we
will indicate in the next number). To prove the above assertion it is enough
to show that if F is the irreducible representation of K with highest weight
sp - a relative to s'®+ then s„sp - a is a weight of F (9.1.5).
In the course of the proof of 9.A. 1.2 we saw that as — s<5+. Thus
m = 2(a, sp)/(ct, a) < 0. Hence sx(sp - a) = sp — a - (2(sp — a, a)/(a, a))a. If
we use the representation theory of the T-D-S corresponding to a we find
that sp - a + jtx is a weight of F for 0 < j < - (2(sp - a, a)/(a, a)). Since j =
- m is included in that interval, s„sp - a = sp — a - ma. is indeed a weight
off.
9.1.8. As we indicated in the proof of the above result we have proved a
slightly stronger result.
Lemma. Let p, a be as above. Let sl,s2,ue W(K, T) be such that s2 = usi

9.2. Spinors
359
with l{s2) = l(u) + l(si). If wu w2 e W(K, T) are such that stn - a is w,<I>+-
dominant, i = 1,2 then
\\SiH - a + Wip\\ < \\s2fi - a + w2p\\
and equality occurs if and only if sxp. — as W(K, T)(s2n — a).
This is what we actually proved if /(«) = 1. The Lemma follows by
"plunking" one simple root at a time.
9.2. Spinors
9.2.1. Let V be a finite dimensional vector space over R with inner product
( , ). Then a space of spinors for (V,( , )) is a pair of a complex vector
space S and a linear map y: V -» End(S) such that
(1) 7(v)2= -{v,v)l, veV.
(2) If Y is a subspace of S such that y(V) Y is contained in Y then Y = 0 or
Y = S.
If (y, S) and (y1, S') are spaces of spinors for (V,( , )) then S is said to be
isomorphic with S' if there exists a linear bijection T of S to S' such that
Ty(v) = y'(v)T for all v e V.
Lemma. Set n = dim V. If n is even then up to isomorphism there exists
exactly one space of spinors, (y, S), for V and dim S = 2"'2. If n is odd then up to
isomorphism there are exactly two spaces of spinors and they are each of
dimension 2["/2).
We first construct a space of spinors for (V,( , )). Extend ( , ) to a
C-bilinear form on Vc. Set k = [n/2]. Let W be a subspace of Vc such that
dim W = k and (W, W) = 0. Let a denote complex conjugation of Vc
relative to V. If w e W - {0} then (w,aw) > 0. We can therefore choose a basis
wu...,wk of P^such that (w^awj) = 5l}. We also note that (oW,(jW) = 0. If
Vis odd dimensional then fix v0 a unit vector in Vsuch that (i;0, W + aW) = 0.
Set S = AW. If w e W and if u e S then set y(w)u = wAu. If x e aW then set
for «!,..., ur e W
y(x)ulA-- Aur = X (_ l)' + 1(x> «i)«i A • • • Au,A-- Aur.
Extend y by linearity to W® aW. It is easily checked that if x e W ® aW
then y(x)2 = -{x,x)l. If n is odd then define y+(tf0) to be (- 1)J7 on AJW,

360 9. Unitary Representations and (g, K)-Cohomology
7"0>o) = (-l)j+1i on A'W. Set y^o + x) = zy*(v0) + y(x) for z e C and
x e W © o-pP. Then y, y+, y~ satisfy (1) above. We show that y satisfies (2).
Let ue S - {0}. Then u = u0 + ■•■ + uk with Uj e A'W. Assume that r is
the minimal index such that ur is non-zero. Then there exist zl,...,zk_re W
such that
zlA---Azk_rAur = iv^ — Awj.
This implies that
Now
y(<Wi)- • • y{° wk) wiA - • • Awk = (-!)*•
Hence, (2) is satisfied.
Let (y', S') be a space of spinors for V.
(i) dim S' < oo.
Let «!,..., u„ be an orthonormal basis of V. If u e S' - {0} then
S' = Zcv'(»,) •••?'(«,•)«•
Now y'(u,)2=-J and 7'(u,)y'(uj) + y'(W;)y'(U;) = 0 if i # j. Thus S' is
spanned by the elements
y'(w,-)•••/(«>)«, 1 < i'i < ••• < i'r <n.
This clearly implies (i).
(ii) Z = jueS" y'(w)u = 0, w e aW) is non-zero.
Since y'(awx)2 = 0, Ker y'{aw{) is non-zero. Since
y'ioWiY/'iowj) = -y'ioWjY/ioWi)
for all i, j it follows that y'((T^) stabilizes Ker y'(awi). Since y'(<iw2)2 = 0,
ker y'(awi) n ker y'(o-w2) # 0, etc.
If n is odd then y'(v0)y'(x) = -y'{x)y'(v0) for xeaW. Thus 7'(i;0)
stabilizes Z. Since y'(v0)2 = —/. Z = Zt + Z_, with Za = {ze Z\y'(v0)z = az).
Assume that Za is non-zero. Fix u0e Za - {0}. Then y'(Cv0 + aW)u0 =
Cu0. Set Sq = Cu0, S'j = y'{W)S'j.i + S)_, for j > 0. Then y'(F)
stabilizes IJS}. Thus [jS'j = S' by (2) above.
Since y'{x)y'(y) = -y'(.y)/M for x,yeW we can define 7: AW -> S' by
T^A---Axr) = y'(x,)- ■■y'ix^UQ. If n is odd and a = i (resp. a = -i) set
y = y+ (resp. y = y"). We leave it to the reader to check that Ty(v) = y'(v)T

9.2. Spinors
361
for v e V. (2) above now implies that T is injective. T is surjective by the
above. Hence T is an isomorphism of spaces of spinors.
9.2.2. We write S(V) for the space that we studied in the previous number.
If n (= dim V) is odd then let y denote one of y±. Recall that 5o(V) =
{X e End(V)\(Xv,w) = -{v,Xw),v,we V}. If v, we Vare such that (i;,w) =
0 then let X(u, v) e End(F) be defined by
X(v, w)u = (u, w)v — (u, v)w, u e V.
Then X(u, v) e 5o(V). Let eu..., e„ be an orthonormal bais of V. Set Xtj =
X(ehej) for i < j. Then Xy, i < j, is a basis for 5o(V).
We define a linear map \i of so(K) into End(S(F)) by
H(XU) =-G)y(eMej).
A direct calculation shows that
MX), MY)] = fiix, y], x, ye S0(K).
This implies that (n,S(V)) defines a module for 5o(V).
Lemma. If n is odd then up to equivalence (n, S( V)) is independent of the choice
of y± and (/i,S(V)) is irreducible. If n is even then we set S+(V) = AevW,
S~(V) = \oMW. Then S£(V) is invariant under \i and each defines an
irreducible representation of so(V).
If n is odd, set V = {v e V[(v, v0) = 0}. Then our construction implies that
S(V) = S(V). If v e V set y'(v) = y(v0)y(v). Then y'(v)2 = -(v,v)I. Let S' be
a non-zero subspace of S(V) of minimal dimension that is invariant under
y'(V). Then S' satisfies 9.2.1(1), (2). Lemma 9.2.1 implies that dim S' =
dim S(V). Hence S' = S(V). So \i is irreducible in this case.
We now relabel the orthonormal basis that we are using. If n = 2k + 1,
let e0 = v0, eh...,e2k be an orthonormal basis of V. If n = 2k then take
eu...,e2k. We assume that wi = e2j-\ — ie2j, l<j<k and that W =
Z Cwj. Set hj = X2j- i.2j, 1 < j < k. Then t = X Rh} is a maximal abelian
subalgebra of 5o(V). Let ^e t* be defined by Hj(hk) = 5}k. A direct
calculation yields
H(h)wi A • • • \wk = (/(/i, + • • • + nk)l2)wx A • • • Awk
and
H(h)y(awj) = - iHj(h)y(aWj)n(h), het.

362
9. Unitary Representations and (g, K)-Cohomology
This implies
(1) The weights of t on S(V) are precisely the linear functionals
'((/Ui + • • • + nk)/2 - (nh +■■■ + n!p), 1 < i, < ■ ■ ■ < i„ < k
and each occurs with multiplicity 1.
Notice that (1) is independent of our choice of y± when n is odd. If n > 2
then 5o(V) is semi-simple and is the Lie algebra of the compact Lie group
0(V) = {g e GL(V)\(gv,gw) = (v,w), v, w e V}. Weyl's theorem implies that
G, the connected, simply connected Lie group with Lie algebra, 5o(V) is
compact. Also, exp t = T is a maximal torus in G and \i integrates to a
representation of G. (1) implies that the character of n is independent of
our choice of y±. So the assertion of the Lemma has been proved in the
case when n is odd.
We therefore confine our attention to the case when n is even. It is
clear that the spaces S±(F) are invariant. Set g = 5o(V). We assume that
dim V > 2 (in the case that dim V = 2 the result that we are proving is an
easy exercise). We set
<t+ = {i(nr - ns)! 1 < r < s < k} u {i(n, + fis)! 1 < r < s < k}.
Then <t+ is a system of positive roots for <t(gc,tc). The only dominant
weights in (1) are -1(^1 "! + A*ik)/2 and -1(^1 ^ + A*ik-i _ A*fc)/2. The
assertion for n even now follows from the theorem of the highest weight.
9.2.3. The module (n, S{V)) for so(V) is called the spin module.
Lemma. There exists a pre-Hilbert space structure < , > onS(V) such that
(y(v)u, w) = - <«, y(v)w), v e V, u, w e S(V).
If n = 1 this is easy and is left to the reader. So assume that n > 2. If n is
even then we may adjoin a unit vector, v0, orthogonal to V. We may thus
assume that n is odd and n > 3. Let G be as in 9.1.2. Then G is compact.
A direct calculation yields
(*) n(exp(tX(u,v))) = yi sin! - lu + cosl - \v lyl -cosl - lu + sinl - It;
(*) implies that n{G) is the set of all products y(ul)--y(u2!,), u1,...,u2p
unit vectors in V. Set G~ equal to the set of all products y(u1)-y(up), with
«!,..., up unit vectors in V. Then G~ contains G as a normal subgroup.
Also, if u e V is a unit vector then it is easily seen that G~ = y(u)n(G) u n(G).

9.2. Spinors
363
Thus G~ is compact. This implies (the unitarian trick) that there exists a
G~-invariant inner product < , > on S(V). If u e V and if (u, u) = 1 then
y(u)2 = — I. Thus y{u)~l = —y(u). The Lemma now follows.
9.2.4. Lemma. Let Q be the natural representation of so(V) on AVC.
(1) // n is even then Q is equivalent with [i ® [i.
(2) // n is odd then Q is equivalent with the direct sum of two copies of n® /i.
This follows from the calculation of the weights of \i in 9.2.2.
9.2.5. The next two results expand a bit on the previous Lemma.
Lemma. Let F be a finite dimensional vector space over C. Let 5 be a linear
map of V into End(F) such that 5(v)2 = -(v,v)l for v e V. If n is even then
there exists a vector space U and a linear isomorphism, T, of S(V) ® U onto
F such that T(y(v) ® I) = 5(v)T for v e V. If n is odd then there exist spaces
U+ and U~ and a linear isomorphism T of S{V) ® U+ © S{V) ® U~ onto F
such that
T(y+(v) ® I+ + y~(v) ® /") = 5(v)T
for v e V (here I+ is the identity map on U+).
We use the notation in the proof of 9.2.1. Let F0 = {f eF\5(aW)f = 0}.
As in 9.2.1 it is easily seen that F0 is non-zero and if n is odd then
•5(^0)^0 = ^o- ^ n is even tnen set U = F0 if n is odd then set U+ =
{f e F0[5(v0)f = if}, U~ = {feF0\5(v0)f= -if}. If / e F0 then let 7>
be the linear map of W into F given by 7}(«iA---Aup) = 5{Ui)---5(up)f. If
n is even then the argument in 9.1.2 implies that Tfy(v) = 5(v)Tf. If n is odd
and if feU± then T^iv) = 5{v)Tf. Set T(sXf) = Tf(s). The result now
follows.
9.2.6. On if v e Fand if u e V then set s(v)u = vAu. If uY,..., ur e V then set
i{v){ulA---Aur) = X(_ l)' + 1(t',«;)«iA---Au,A---Aur.
Set 5+(v) = e(v) + i(v) and 5-{v) = i(r.(v) - i{v)).
It is easily checked that
(1) 5±(v)2 = -{v,v)I and
(2) 5+(v)5.(w) + 5 (w)5+(v) = 0 for v, w e V.

364
9. Unitary Representations and (<j,K)-Cohoinology
Define Q±(X;j) = -(\)d±(e;)d±(ej) for i <j. Then as above Q+ defines
a representation of 5o(V) on AKC.
Lemma. Q(X) = Q+(X) + Q_(X) and [Q+(X),Q_(7)]=0 for X, Yeso(V).
The last assertion follows from (2) above. The first follows from
Q(X,7) = eieMej) + f(e,)e(^)
and the obvious calculation.
9.2.7. The following lemma (although easy) is useful outside of
representation theory. Fix ex,..., e„ an orthonormal basis of V.
Lemma. Suppose that to each 1 < i, j, k, I < n we have assigned a complex
number Rijk such that
(1) R-ijki = R-kuj,
(2) Rijkl = ~~ Rjikli
(3) Rijki + Run + Rjku = 0.
Then Z/jH Rljkl y^M^M^Me,) = 2(Zy R^I.
Set y} = y(ej). In the expressions below all indices are summed. (3) says
X Rtiu7i7 j7k7i " X K^W^i " X Rjknliljlkli = 2 Z R^a7i7j7j7k-
We calculate
Z &uii7i7j7k7i = -Y,Rkiji7i7k7j7i - 2 Z ^tfi7i7/-
Now (1) implies that Kjyi = Kju;. So -2 Z Rjin7,7i = -* Rjiji(7i7i + 7i7i) =
2(1 Rjiji)!. Also (2) implies that,
Z Rkiji7i7k7j7i = "Z Rikji7i7k7j7i = "X Rijki7i7j7k7i-
Set i? = X i?;^;. Then we have
2 Z ^W^i = 2R1 - Z Rjkii7i7j7kyi-
Also, as above
Z Rjkuyiyjykyi = -X fyww*)'/ - 2/?/ = X JW^*y.7i - 4Ri
= X ^wW*?/ - 4^/.

9.3. The Dirac Operator
Hence
as was to be proved.
9.3. The Dirac operator
9.3.1. Let g be a semi-simple Lie algebra over R with Cartan involution 8
and corresponding Cartan decomposition g = i ® p. We will use the material
of Section 9.2 with V = p and ( , ) the restriction of B to p. Let Ck be the
Casimir operator of f relative to B restricted to f. Set fi0(Y) = ad Y\v for
Yet. Then n0 is a homomorphism of f into so(p). Set s(Y) = ix(/x0(Y)) for
Yet. Then (s,S(p)) is a representation of f with a f-invariant inner product
< , > (9.2.3). Let t be a maximal abelian subalgebra of f. Fix Pk, a system
of positive roots for <t(fc, tc).
Let h = {X e q\[X,t] = 0}. Then h is a fundamental Cartan subalgebra
of g. Let <t = <t(gc,hc). Let d act on (hc)* by da(h) = a(dh), hs\). Then
0<b = <t. We say that a system of positive roots, P, for <t is 6-stable if 6P = P.
We say that it is compatible with Pk if it is 0-stable and if a e Pk then there
exists (1 e P such that /?|t = a.
Fix, P, a system of positive roots for <t compatible with Pk. Set o =
{het)ldh= -h}. We identify (tc)* with {a e (hc)*l da = a) and (oc)* with
{ffe(r)c)*!flff = -<t}. If <re(hc)* then write a = a+ + a~ with a+ e (tc)*,
ff" e(oc)*.
Set p(P) = (i) IaeP a, pk = (i) IaePk a, p„(P) = p(P) - Pk. We note that
since OP = P, p(P), pk, pn(P) e (tf)*. Let C(Pk) denote the set of all systems
of positive roots that are compatible with Pk.
If / e (tc)* is Pk-dominant integral then set yx equal to the element of KA
with highest weight k.
9.3.2. Lemma. Let l0 = dim o. Then
PsClPk)
Fix Pe C(Pk). If ae(tc)* then set (pc)a ={Xepc\ [h,X~\ = <x{h)X, he(tc)*}.
Set 1= jae(t(.)* (pc)a is non-zeroj. Put E± =In(±P|t). Set p±(P) =
©«x(Pc).-Then
pc = oc®p+(P)©p-(P).

366
9. Unitary Representations and (g, K)-Cohomology
The analysis in the previous section implies that the weights of tc under s are
of the form pn(P) — a, — ••• - <xm with a,- e Z+(P) and there exists a subset
Q of P such that ax+--- + am= <g>+ (see 9.A.1.5 for <Q». This implies that
p„(P) is an extreme weight.
(1) s(Ck) = cl.
Indeed, let be an orthonormal basis of p. Let y,,..., ym be a
basis of f such that B(yhyj) = — <5y. Set /?yw = B([xi,Xj~\,[xk,xl~\). U yet
then (9.2.2)
My) = Z ([y.^].^i)7(^)7(^)-
Thus 16s(Q) =
= - Z B(3'a,[^i,^])B(3'o.[^.^])7(^)7(^)7(^)7(^)
= Z B([*i.*;M*k>*/])7(X;)7(x;)7(*k)7(X/)
= Z Kijw7(x;)7(Xj)7(x*)7(*z)-
It is clear that i?,-^, satisfies (1), (2) of Lemma 9.2.7. (3) of 9.2.7 follows from
the Jacobi identity. Hence 9.2.7 implies (1).
As we have seen above yPniP) occurs in s. So (1) implies
(2) s(Q) = (||p||2-||pk||2)/.
Suppose that y„ occurs in s. Then a = p„ - <Q>+ with Q a subset of P.
(2) says that ||<r + pfc|| = ||p||. Hence
IIp-<G>+II = IIpII-
Now, p-<Q}+ = (p-<Q))+. So, ||p-<e>+ll<llp-<e>l|. Hence
lip _ <G)II ^ IIpII- P _ <G> is a weight of the finite dimensional irreducible
g-module with highest weight p (9.A.1.5). We therefore see that p — <2> = wp,
for some w e W(qc, hc). Also the inequalities must be equalities, so, 8w= wd.
This implies that wP e C(Pk). We have shown that a = p(wP) — pk. We
leave it to the reader to check that the dimension of the p(P) — pk weight
space is 2[lo/2].
9.3.3. Let G be a connected semi-simple Lie group with finite center. Let
K be the connected subgroup corresponding to K. Then a (g, K)-module is

9.3. The Dirac Operator 367
said to be unitary if there exists a pre-Hilbert space structure < , > on V
such that if X e g, k e K and v, w e V then
(1) (Xv,w)=-(v,Xw),
(2) (ki\wy = (v,k-lw).
Let K be a (g,K)-module, set S = S(p). We now define a K-module
homomorphism, Dv = D from K ® S to V ® S. Let n be the action of g
on V. Then if x,,..., x„ is an orthonormal basis of p set
D = Yd*(xl)®y(xl).
If V is unitary then we put the tensor product pre-Hilbert space structure
on V ® S.
Lemma. D2 = -n(C) ® I - (\\p\\2 - \\pk\\2) + (n® s)(Ck). If V is unitary
then (Dv,w} = (v,Dw}.
In the calculations below all indices will be summed (unless otherwise
specified). Let yx,..., ym be a basis of f such that B(yh yj) = - <5y.
D2 = '£n(xl)n(xj)®y(xi)y(X])= -2>(x,)2®/+ ^ n(xi)n(xJ)®y(xl)y(xj)
= ~7i(C)®I + 7i(Ck)®I+ X x(xtMx,)®y(x,)y(x}).
i*j
Since y(xt)y(Xj) = —y(Xj)y(xj) for i # j we have
D2 = - n(C) ® 1 + n(Ck) ® I + (i) X Jt([x„ xj) ® y(x,)y(xj)
= -tt(C) ® / + n(Ck) ® / - (i) X B([x,.,xJ],yfl)jt(yfl) ® y(xj)y(xj)
= - 7r(C) ® / + n(Ck) ® / - 2 X 7r(ya) ® s(yfl)
= - n(C) ® I + n(Ck) ®I + (n® s)(Ck) - n(Ck) ®I - I® s(Ck)
= - 71(C) ®1 + (7l® S)(Ck) - I ® S(Ck).
The first assertion of the Lemma now follows from 9.3.2(2). The second is
an easy calculation using the definition of D.
9.3.4. Corollary. Assume that V is a unitary (Q,K)-module with
infinitesimal character yA. If (V ® S)(y„) is non-zero then \\a + pk\\ > ||A||.
Indeed, n(C) = (\\A\\2 - \\p\\2)l. Hence
(7r®.s-)(Q) = (||A||2-||P)i||2)/ + Z)2.

368
9. Unitary Representations and (g, K)-CohomoIogy
Now, D2 is positive semi-definite and Ck acts on any representative of y„
by ||(T + pk||2 — \\pk\\2- The corollary now follows.
We will refer to the conclusion of the above corollary as the Dirac
inequality.
9.4. (g, K)-cohomoIogy
9.4.1. We retain the notation of the previous section. For simplicity, we
take G to be semi-simple, the identity component of GR and we assume that
Gc is connected and simply connected. If V is a (g, K)-module then let
H'(q,K; V) be as in 6.1. For the next few sections we will be studying these
cohomology spaces. Fix P e C(Pk). Let F be a finite dimensional irreducible
(g, K)-module with highest weight A relative to P. The following result is
usually known as Wigner's Lemma.
Lemma. // V is a (Q,K)-module with infinitesimal character x an^ if
H'(q,K, V® F*) is non-zero for some i then x = Xa + p-
Let gu = f + ip in gc. Let Gu be the connected subgroup of Gc
corresponding to gu. Then Gu is connected and simply connected. Also
GunG = K. Let V' be the i'lh Zuckerman functor (6.2) from C(gu,K) to
C(gu, G„). Let y e G£and let Fy be a representative of y. Then (6.3.2)
r'(F)=©//i(gu,K;F®(Fv)*)®Fr
6.3.3 implies that r'(V) has the same infinitesimal character as V. Thus
H'(qu,K;V®(Fy)*)= H'(q,K;V ®{F7)*) is non-zero only if V and Fy
have the same infinitesimal character.
9.4.2. Let x denote the complex conjugate of 3c e l/(gc) relative to the real
form l/(g).
Lemma. // V is a unitary (Q,K)-module with infinitesimal character x then
(*) X(zT) = X(z) M z e Z(g).
// F is a finite dimensional, irreducible (g, K)-module with highest weight A and
if the infinitesimal character of F satisfies (*) then Q\ = A.
If z eZ(gc), v,weV, then x{z)(v,w) = (zv,w) = (v,zTw) = x(zT)<.v,w).
This proves (*). We now prove the second assertion. Let a denote complex

9.4. (g, K)-Cohomology
369
conjugation in gc relative to gu. Since F is unitary as a (gu, Gu)-module, the first
assertion implies that #F((rzr) = Xf(z) f°r z e ^(9c)- Since ax = 6x for
x e l/(gc), (*) implies that xF(0z) = xF(z) for z e Z(gc). Let Fe be the (g, K)-
module, F with g acting by 6(^)1;, ve F,X e q. Then we have just shown that Fe
and F have the same infinitesimal character. This implies that they are
isomorphic. Since the highest weight of Fe relative to P is 6 A, the second part
of the Lemma follows.
9.4.3. Proposition. // V is a unitary, admissible, (g, K)-module with
infinitesimal character Xa + p then
H'(q,K; V®F*) = HomK(A'p, V ® F*).
Note. H'(g, K; V ® F*) is the cohomology of the complex
C'(g, K; V®F*)= HomK(A'(g/f), V ® F*) = HomK(A'p, V ® F*).
The content of the proposition is that d = 0.
On F we put a Gu-invariant inner product. On A'p put the inner product
corresponding to the restriction of B to p. On (A'p)* use the dual inner
product. Now, C'(g,K; V ® F*) = C = ((A'p)* ®V® F*)K. Set
Di = (Aip)*® F®F*.
We put the tensor product inner product and on D' we restrict that inner
product to C. Since V is admissible, C is finite dimensional.
We will use the following standard result.
9.4.4. Let (C',d) be a complex with dim C < oo. Fix ( , ), an inner
product on each C\ Define d*: Cl -> C'~' by
(d*x,y) = (x,dy), xe C, ye C'~l.
Scholium. The natural map from
Jt"= {ceC'\dc = d*c = 0} = {c e C'\(d + d*)2c = 0}
to H'(C',d) is a surjective isomorphism.
We assert that Cl = dC'~* ®d*Ci + 1 ® Jf' orthogonal direct sum.
Indeed, if (x, dC'~ ' + d*Ci + 1) = 0 then dx = d*x = 0 and conversely. Thus,
jf'= (</(:'-' +d*Ci + 1)-L. If uedCi-\ved*Ci + l then « = dw, v = d*z so
(«,t>) = (dw,d*z) = (d2w, z) = 0. The assertion follows. If z e C and if dz = 0

370 9. Unitary Representations and (g, K)-Cohomology
then write z = dx + d*y + h with h e Jtl. Then 0 = dz = dd*y. So,
0 = (dd*y,y) = (d*y,d*y).
Hence d*y = 0. The first assertion now follows. To prove the second we
note that (d + d*)2 = dd* + d*d. If (d + d*)2c = 0 then
0 = {dd*c,c) + {d*dc,c) = {d*c,d*c) + (dc,dc).
The second assertion is now also obvious.
9.4.5. We now return to the proof of 9.4.3. If x e p define x* ep* by
x*{y) = B(x,y). If u e (A'p)* and if x e p then set e(x)u = x* Au. If x e p
and if m e (A'p)* then set i(x)m(z1,..., z;.,) = «(x,z(,...,zi_1). Relative
to ( , ) on the D\ e(x)* = i(x). Let n be the action of g on V and let a be the
action of g on F*. Then d on C is the restriction of
d = X e(*() ® t(xf) ® / + X e(*i) ® 7 ® a(xi)
on £)'. Here xu..., x„ is an orthonormal basis of p.
We note that a(x)* = a(x) and 7r(x)* = — n(x) for x e p. Thus d* is the
restriction of
d * = - X <'(*/) ® t(^) ® / + Z '(^) ® / ® a(Xj)
on D' to C'.
On D' we have
d + d* = XM*/) ® i(xj) ® ' " «'Z5-(xj) ® 7 ® ff(xj)
in the notation of 9.2.6. Thus, if we apply 9.2.6(2) we find that on D'
(d + d*)2 = (X 8 + (xj) ® n(xj) ® I)2 - (X 8-(xj) ® / ® a(x,))2.
If we combine 9.2.6 with Lemma 9.3.3 then on D'
(d + d*)2=-I®n(C)®I + (\\p\\2-\\pk\\2) + ((fi+°ad\t)®n®I)(Ck)
+ I®I® a(C) - (llpH2 - ||pk||2) - ((n- o ad\t) ® I ® I)(Ck).
Since n(C) and ct(C) act by the same scalar, we find that on D'
(d + d*)2 = {{n+ o ad|,) ® n ® l)(Ck) - ((/i_ ° ad|,) ® / ® a)(Ck).
Thus to complete the proof of the proposition we must show that this
expression isOon (D')K = C. Letyu..., ym be a basis of f such that B(yt,yj) =
— 5ij. Let a.(y) ~ (n+° ad\t)(y) and fi(y) = (jx_ ° ad|,)(y) for y e f. Then

9.4. (g, K)-Cohomology
371
*(y) + fi(y) = &d(y) on (A'p)* (9.2.6). In what follows all expressions will be
looked upon as evaluated on (D')K. We are studying
(*) 2 X «(tt) ® n{yt) ® / + £ / ® n(yt)2 ® /
- 2 I P(yt) ® / ® a(yt) + £ / ® / ® ff(j>,)2.
Now, if ye I then (a + P)(y) ® / ® / + / ® 71(3;) ® / + / ® / ® ff(j/) = 0 on
the K-invariants. If we apply this identity to the above expressions and do the
obvious algebra (which we leave as an exercise to the reader) we find that on
the K-invariants (*) is equal to
I Mtt) - P(yi))(*(y,) + P(yt)) ® / ® / = -(«(Q) - P(Ck)) ® / ® /
since a(y) and P(y) commute for yet This expression is 0 by 9.2.6 combined
with 9.2.5 and 9.3.2(1). This completes the proof.
9.4.6. We now state a result that sums up most of the material of this section.
Proposition. Let P be a fixed Pk-compatible system of positive roots for
<t(gc, hc). Let F be an irreducible finite dimensional (g, K)-module with highest
weight A relative to P. If 8 A # A and if V is an irreducible unitary (g, K)-module
thenH'{Q,K;V®F*) = 0.
If V is unitary with infinitesimal character x and if X^Xa+p then
H-(q,K;V®F*) = 0.
Assume that 6A = A and that V is an irreducible unitary (g, K)-module with
infinitesimal character Xa+p- Then H'(g,K; F® F*) # 0 if and only if there
exists y e KA such that Hom/i:(F),, V ® S) and Hom/i:(Fy, F ® S) are non-zero.
Furthermore, for any such y there must exist Pi e C(Pk) such that A is Pr
dominant and Xy + pk = A + p(Px).
The first two assertions follow from 9.4.1 and 9.4.2. We now prove the
assertions of the last paragraph of the statement. The previous result implies
that
H*(g, K; V® F*) = ((A»* ® V ® F*)K.
On ((A'p)* ® V ® F*) set
D+ = Z <M*i) ® ^(Xi) ® / and
D = £ M*i) ® ' ® *(*!)■
In the course of the proof of 9.4.3 we showed that (£)+)2 - (£)_)2 is 0 on

372
9. Unitary Representations and (fl, K)-Cohomology
((A'p)* ® F® F*)K. Since both (D+)2 and -(£)_)2 are positive operators
this implies that D±((A»* ® K ® F*f = 0.
Suppose that v0 e ((A'p)* ® V ® F*)K - {0}. Let C, be the span of all
elements of the form
(5 + (tt1)®/®/)---(5 + («p)®/®/)(5_(w1)®/®/)---(5_(wr)®/®/)«0
u;, v^ep. Set C2 equal to the span of {(/ ® n(kx) ® a(k2))C11 kx, k2 e K}.
Then C2 is a finite dimensional so(p) x so(p) and K x K-module with action
given as follows: the first so(p) factor acts by \i- ® / ® /, the second acts
by n+ ® I ® /, the first K factor acts by / ® n ® / and the second acts by
/ ® / ® a. All of these actions commute.
If we apply Lemma 9.2.5 we find that
C2= © C^a,^]®^®^
with each C2[a, /?] an so(p) x so(p) module which is a direct sum of tensor
products of spin modules. We therefore conclude that
{V®S®F*®S)K #0.
Furthermore, on (S ® V ® F* ® S)K, £>K ® / and / ® DFj|! act by 0. Thus if
{{V® S)(yi) ® (F* ® S)(y2))' is non-zero then
(*) ll^.+PnllHI^-r-pJIHIA+pll.
This implies everything but the last assertion. Suppose that
UomK(Vy,F®S)
is non-zero and that \\Xy + pk\\ = \\ A + p||. The weights of F ® S with respect
to t are of the form A + p„-<g>+ with Q a subset of P. Thus Xy + pk =
A + p„ + p, - <G>+ = A + p - <6>+. Thus
pv + Pk\\ = ha + p - <e>+n < ha + p - <e>n < ha + pii
by 9.A.I.5. Hence all of the inequalities are equalities. This implies that
<G> = <G>+ and that there exists se W(Qc,hc) such that p - <g> = sp
and sA = A. Since sp = (sp)+, 6s = s8 so sP is 0-stable. Since s(A + p) is
Pk-dominant-sP e C(Pk). Thus 2V = A + p„(sP) as asserted.
9.4.7. In the next section we will give sharper results due to Kumaresan,
Parthasarathy, Vogan and Zuckerman.

9.5. Some Results of Kumaresan, Parthasarathy, Vogan, Zuckerman
373
9.5. Some results of Kumaresan, Parthasarathy, Vogan, Zuckerman
9.5.1. In this section we will be using several 0-stable systems of positive
roots compatible with different systems of positive roots for K. It is thus
worthwhile to recall the relationship between W(K,T) and W(§c,\)c). The
notation will be as in the previous section.
Let s e W{K, T). Then there exists ke K such that Ad(fc)|t = s. Since h =
{leg!IX,t] = 0}, Ad(fc)l) = I). We are assuming that G is connected hence
Ad(fc)|b = s' e W{Qc,t)c). Clearly, s'\t = s. If f e W(Qc,i)c) is such that f|t = s
then t~1s' is the identity in t. Now 1 contains regular elements of g. Thus
f = s'. We have proved
Lemma. If s e W(K, T) then there is a unique element s' e W(gc,hc) such
that s'\t = s.
In light of this we will identify s e W(K, T) with s' e W(§c, hc).
9.5.2. We now continue the discussion initiated in the previous section. Let
F be a finite dimensional irreducible (g, X)-module. If P is a system of positive
roots for <t = ^(Qc,hc) then we write A(F) for the highest weight of F
with respect to P. We assume that if P is 0-stable then 0A(P) = A(P). Let
V be a unitary (g, K)-module with infinitesimal character yL\{P) + p{P). Fix Pk,
a system of positive roots for <&k = <t(fc,tc). Let y e KA be such that
(1) HomK(A>, V(y) ® F*) # 0.
Unless otherwise specified, F, V, Pk, y will be fixed. Let \i denote the highest
weight of y relative to Pk. The following result is due to Kumaresan [1] for
F = C and to Vogan-Zuckerman [1] in general (all of the essential ideas
appear in the case F = C).
Proposition. There exist P\ e C(Pk) and P2 a 8-stable system of positive roots
for <t such that A(Pl) is P2-dominant and
/i = A(P,) + p.(P,) + p.(P2).
We have seen in 9.4.6 that (1) implies that there exists P e C(Pk) such that
(A = A(P), p = p(P))
(2)
HomK{VA+PnlP),V,®S)*0.

374 9. Unitary Representations and (g, K)-Cohomology
Notice that we are denoting Vy by K„. Now S is a multiple of
QsC(Pk)
We therefore must have
HomK(FA+Pn(P),^®(FPn(G))*)#0
for some Q e C(Pk).
Let u e W(K, T) be of minimal length (9.A. 1.1) such that u(n - p„{Q)) is
Prdominant. Then Vu{ll _ Pn(Q)) occurs as a summand in V ® S (9.1.4). The Dirac
inequality (9.3.4) implies that
ll«0i - PniQ)) + Pk\\ > HA + P\\.
On the other hand, 9.1.6 implies
||A + p\\ = ||A + pn(P) + pk\\ > \\u(n - pn(Q)) + Pull-
Thus all inequalities are equalities. This implies (9.1.6)
(3) u(n - Pn(Q)) = A + p„(P).
We rewrite (3) as ufi - pn(P) = A + upn(Q). Let v, t e W(K, T) be such that
f is of minimal length such that u/i - pn(P) is f Pk-dominant and v is of minimal
length such that \i — p„(P) is t;Prdominant. Lemma 9.1.7 implies that
\\un - pn(P) + tpk\\ > ||A - pn(P) + vPk\\.
9.1.4 implies that the irreducible finite dimensional K-module with highest
weight v~l(p. - p„(P)) occurs in V® S. Hence the Dirac inequality implies
that
lit;"'(n - pn(P)) + pk\\> ||A + p\\.
Thus
\\W-pH(P) + tpk\\>\\A + p\\.
On the other hand,
\\W - pH(P) + tpk\\ = ||A + upn(Q) + tpk\\ < ||A + pn(Q) + pk\\
by 9.1.5. Now, A + pn(Q) + pk = A + p(Q). Let w e W{Qc,i)c) be such that
Q = wP. Then || A + wp(P)|| < || A + p(P)||. Hence all of these inequalities are
also equalities. We look at the implications of our new equalities. We first
look at
||A + Wp(P)|| = ||A + p(P)||.

9.5. Some Results of Kumaresan, Parthasarathy, Vogan, Zuckerman 375
9.1.5 implies that there exists r e W(qc,l)c) such that rA = A and rwp(P) =
p(P). Thus rw = 1, so wA = A. We have thus shown
(4) A(P) is g-dominant.
We now look at
HA + upn(Q) + toll = HA + p„«2) + Pk\\.
This implies that there exists r e W(K, T) such that rA = A, rupn(Q) = pn(Q)
and rtpk = pk. Thus r = f"1. We have therefore shown that
(5) rA = A, up„(Q) = tPn(Q).
We use this to prove
(6) uA = A.
If a e Pkn(-tPk) then (A,a) = 0. Thus s.A = A. In light of 9.A. 1.3 it is
therefore sufficient to prove that Pkr\(-uPk) is contained in Pkr\(— tPk). So
assume that a is in Pkr\(-uPk) but not in Pkn(-tPk). (5) implies that
(a,up„(g)) > 0. Also (a,up.) < 0. Hence (ot,u(p. - p„(Q))) < 0. Hence
(*) (a,«(/i-p.«2))) = 0-
Since u was assumed to be of minimal length, 9. A. 1.4 implies that
«-'P, = {/ie4,!(/l,/i-ft(fi))>0}u{/l6Pi!(/i)/i-A(fi)) = 0}.
This says that
Pk = {Pe<bk (P, u(n - Pn(Q))) > 0} u {/i e Pk I ()8, «(/i - p„«2))) = 0}.
(*) now implies that a e uPk. This contradiction implies (6).
(3) implies that n = u~l\ + p„(Q) + u~lpn{P). In light of (6), the Lemma
follows if we take P, = Q, P2 = u~'P (recall our identification in 9.5.1).
9.5.3. If q is a 0-stable parabolic subalgebra of gc (6.4.1), q = lc + u then set
u„ = u n p, uk = u n I. If h e tc then set p„(q)(h) = tr(ad h\uJ/2. We say that
q is Pk-compatible if q n Ic contains bk = tc © ©ae/.k(tc)« = ^c + nk-
(1) If q is Prcompatible then 2p„(q) is Pk-dominant integral.
Indeed, let n = dim u„. Let V = [/(fc)(A"(u„)). Then V is a submodule
of A"p. [uk,u„]cu„ and ad(uk) consists of nilpotent elements, hence
uk • A"u„ = 0. Also I n f stabilizes A"u„. Thus, nkA"u„ = 0. (1) now follows,
since tc acts on A"u^ by 2p„(q).

376
9. Unitary Representations and (g, K)-Cohomology
Theorem. Let Fbea finite dimensional, irreducible (g, K)-module. Let V be an
irreducible unitary (g, K)-module with the same infinitesimal character as F. Let
Ebe a finite dimensional irreducible K-module such that Hom^E, A'p®F)#0
and HomK(E, V) # 0. Then there exists a 6-stable parabolic subalgebra, q, of
qc such that
(1) F" = {v e F\uv = 0} is one dimensional. Let A be the weight of t on F".
(2) // Pk is a system of positive roots compatible with q then E has Pk-highest
weight A + 2p„(q).
The proof of this result (mainly due to Kumaresan [1]) is complicated and
will take up most of the rest of this section.
9.5.4. We use the notation of 9.5.2. In light of the result therein we may
assume that E has highest weight n = A + pn(Px) + pn(P2) with P1, P2 0-
stable systems of positive roots, Px e C(Pk) and that A is both Pr and
P2-dominant. Our first task is to find a system of positive roots P3 such that
p„(P3) = pn(P2\ A is P3-dominant and P3 e C(Qk) with [i (^-dominant. If
a e it* is Pk-dominant set
Pk(a) = {« ePt|(ff,«) > 0} u {«e -Pt!(«,*) = 0}.
Then Pk(a) is a system of positive roots for <S>k (9.A.1.4(1) with Pk replaced
by-Pt).
Set Qk = Pk(n) = Pk(\ + pn(Px) + pn(P2)).
Lemma. Both A and pn(P2) are Qk-dominant.
Let Rk be the system of positive roots for ®k such that P2 e C(Rk).
Suppose that a e Qk is such that (A, a) < 0. Since A is i?rdominant this implies
that xe Qkn(- Rk). Hence (p„(P2),a) < 0. Similarly, a e Qk n(-Pk), so
(pn(Pi), a) < 0. But then (p., a) < 0 contrary to the definition of Qk. The second
assertion is more difficult.
Suppose that ote Pk and (p„(P2), a) < 0. If we show that this implies that
(H, a) = 0 then the second assertion will follow. If (fi, a) is non-zero then it
must be positive. We look for a contradiction. Write p„(P2) = —va with
v e W(K, T) and a a Prdominant form. Let s0 e W(K, T) be such that
*oPk = -Pfc-Then
||p„(P2) -ii + soPk\\ = ||-A - pn(Px) - pk\\ = ||A + p\\.

9.5. Some Results of Kumaresan, Parthasarathy, Vogan, Zuckerman
377
Let r e W(K, T) be such that <r — /x is rPrdominant. Then
l|P.(^2)-/i + S0P*ll^l|ff-/i+»-pJk||
by 9.1.7.
Let a e Pk be such that (p„(P2), a) < 0 and (p,a) > 0. Then
2(P.(P2) " /i,«)/(«,«) = 2(p„(P2),a)/(a,a) - 2(/i,«)/(«, a) < 2(p„(P2), a)/(a, a).
This implies that sxpn(P2) - p is on the a-string of weights in VSo(Pn(P) _ „
through p„(P2) - /*• Also, the above inequality implies that it is not an element
of W{K,T)(p„(P2) - p). It is also easily seen that l(sav) < l(v), hence 9.1.8
implies that if sxvo — p. is r'Prdominant then
\\Pn{P2) ~ H + s0pk\\ > \\s,va - p + r'pk\\ >\\a - p + rpk\\.
Now K_sor-!,„_„, occurs in Vfl ®S we have a contradiction to the Dirac
inequality.
9.5.5. Let Rk be as in the proof of the previous Lemma. Let r e W(K, T) be
such that Qk = rRk. Then we have just proved that both A and p„{P2) are
dominant with respect to Rk and rRk. This implies that
(1) rA = A and rp„(P2) = pn(P2).
Set P3 = rP2. Then pn(P3) = rp„(P2) = p„(P2). This gives
(2) p = A + pn(Px) + pn{P3), A is both Px and P3 dominant,
Pl e C{Pk), P3 e C(Qk) and p is both Pk and (^-dominant.
Lemma. p{P\) + p{Pi) is Pi-dominant.
We first note that
(3) (P* + P«2*),«) = 0 for a e(-Qk)nPk.
Indeed, set u, = ©(fcV the sum over all ft e Pk such that {p,fi) > 0.
Set I, = tc©0(!(.)„, the sum over all (ie<$k such that {p,fi) = 0. Then
[l^u,] c Uj and
(P* + p(Qk))(h) = tr(ad h\Ul) for h e tc.
Since (fc)a is contained in [I^l,], (3) follows.
(4) Uxe(-Qk)nPk then (A + p(P,) + p(P3),oc) = 0.

378
9. Unitary Representations and (fl, K)-CohomoIogy
Indeed, (A + pn(Px) + pn(P3),z) = {n,a) = 0 and
p(Px) + p(P3) = p„{Px) + pn(P3) + pk + p(Qk).
Hence (3) implies (4).
We now complete the proof of the Lemma. Suppose that p(Pi) + p(P3) is
not Pl-dominant. Then there would be a simple root, aeP, such that
(*) (p(P1) + p(P3),oc)<0.
Thus a would be an element of — P3. We now show that this is impossible
by showing that — a. would be P3 simple (if so then 2(p(Pl) + p(P3), a)/(a, a) =
1 — 1 = 0). So we are left with showing that for such an a, —a is P3 simple.
Assume that it exists.
(i) (Qc)x is contained in pc (in particular don = a).
Assume that (i) is false and that 9<x = a. Then (gc)a is contained in fc. Thus a
defines an element of Pk. If a. e Pk n Qk then a e P3 which is contrary to our
assumption. Thus a e (— Qk) n Pk. But then
(PniPl) + PniPl),*) = (Pk + plQkU) = 0
which is also contrary to our assumption. Thus we may assume that (i) is false
and that a. # 8a.
As usual, write a. = a+ + a". Then
0 > (pW + p(P3),a) = (p^) + p(P3),a+).
Let X e (fic)a- Then X + 6X e (fc)a+ - {0} since (gc)a is not contained in pc.
Hence oc+ e Pk. Now (3) and (4) imply that (p(PJ + p(P3),a+) > 0.
This contradiction implies (i).
Set P~ = sxPl. Since da. = a, P~ is 0-stable. a is PX-simple so (i) implies that
P~ e C{Pk) and p„{P~) = pn{Pi) - a. Set a = pn(P3) + a. We assert that a is
an extreme weight of S. Indeed, 5 = A + a + pn(P~) = A + pn(Px) + p„{P3)-
So V6 occurs in V ® S. Now apply the Dirac inequality. This implies that there
exists a 0-stable system of positive roots, PA, such that p„{P") = Pn{Pi) + a>
is both PA- and P~-dominant and pn(PA) + p„(P~) = pn(Pi) + pn{P3). We
can now apply our results for Pl and P2 to P~ and PA to find that p(PA) =
p(P3) + a. (we leave this chore to the reader). Thus sxp(P3) = p(P3) + a. This
leads to our desired contradiction.
9.5.6. We now complete the proof of Theorem 9.5.3. Let Z = (a e Px \
(a,p{Px) + p(P3)) > 0}. Put 0), = {a e P,! {a,p{Px) + p(P3)) = 0}. Set lc = hc®
©ae<i. (9c)a< u = ©aes (9C)«- Then 9 = lc + u is a Stable parabolic sub-

9.5. Some Results of Kumaresan, Parthasarathy. Vogan, Zuckerman
379
algebra of gc. We have seen that of a e <t[ then (A, a) = 0. Thus F" is one
dimensional. Also, 2p„(q) = pn{Px) + p„(F3), so /i = A + 2p„(q), as asserted.
9.5.7. Lemma. Let the notation be as in 9.5.3 then if dim u„ = n we have
((AJp) ® F)U*(A + 2p„(q)) = (AJ-"(l n p))A(A"u„) ® Fu.
Let H e it be such that oc{H) > 0 for a e I (see 9.5.6) and 1 = {X e g !
[*,//]= 0}.
Pc = Pc^1c ©"*©"*• Thus
AjPc ® F = I A"(Pc n Ic) • A'u. • A'u„ ® F
p+q+t-j
Now F = U(u)F". Thus the A(//)-eigenspace for H on F is Fu (which is one
dimensional by the above material) and if \i is an eigenvalue for H on F then
/i < A(Ff).
Let x e Ap(pcn lc), y e A'u,,, z e A'Iip, we F be such that ad Hy = ay,
ad Hz = — fe, ffw = cw. Then H(xAyAz ® w) = (a — b + c)(xAyAz ® w). a =
2p„(H) — m with m > 0 and m = 0 and only if q = n. Also i > 0 and i = 0
only if f = 0 and c = A(H) — m' with m' > 0 and m' = 0 if and only if w e Fu.
Hence A(H) + 2p„(q)(H) = a - b + c = \(H) + 2p„(q)(H) -m-rri - b.
Thus m = m' = b = 0. The result now follows.
Note. We have actually shown that
((A» ® Fr(A + 2p„(q)) = ((A-p) ® F)(A + 2p„(q)).
9.5.8. We conclude this section with a vanishing theorem for (g, K)-coho-
mology (due to Kumaresan [1]) and a proof that it is best possible that are
immediate consequences of the previous results and those of Chapter 6. If F
is a finite dimensional irreducible (q, K)-module then set Q(F) = {q | q = lc © u
a proper 0-stable parabolic subalgebra of gc such that dim F" = 1} (i.e., if
F = C then 2(C) is just the set of all proper 0-stable parabolic subalgebras
of gr). Put c{F) = min{dim u„ | q = lc © n e Q{F)}.
Theorem. // F is a finite dimensional (g, K)-module and if V is a non-trivial
irreducible unitary (g, K)-module then
H'(g, K; V ® F*) = 0 for i < c(F).
Suppose that H'(g, K; F® F*) is non-zero. Then Theorem 9.5.3 and
Lemma 9.5.7 combined with Propositions 9.4.3 and 9.4.6 there exists a

380
9. Unitary Representations and (g, K)-CohomoIogy
0-stable parabolic subalgebra, q, of gc such that dim F" = 1 and i > dim u„.
Also, Lemma 9.5.7 implies that if i >0 then q e Q(F). Now H°(g, K;V®F*) =
{veV ® F*\kv = v and Xv = 0 for k e K, X e g} by the definition of relative
Lie algebra cohomology. Thus H°{q, K, V ® F*) = Homg K{F, V). Since the
only irreducible finite dimensional unitary (g, K)-module is C (the trivial
(g, K)-module) the result follows.
9.5.9. We will now use the modules B„(/i) of 6.10.3 to show that the
Kumaresan vanishing theorem is best possible.
Let F be a finite dimensional irreducible (g, K)-module such that if P is a
0-stable system of positive roots for <t and if A(P) is the highest weight of F
relative to P then 0A(P) = A(P). We fix P e C(Pk). Let q be a 0-stable
parabolic subalgebra of qc compatible with P and such that dim F" = 1. Let
sK e W(K, T) be the element such that sKPk = — Pk. Let sLnK be the element
of W{LnK,T) such that sLnK(Pk n <D((l nf)c,fc)) = - Pk n<t((l nf)c,tc).
Put s0 = sLnKsK. Let k e K be such that Ad(fc)|t = s0. Set q' = Ad(fc)~'q.
Here q = lr ® u. Set p(q)(h) = (tr(ad h\u))/2 for h e 1). Put
A = So1(A + 2p(q)).
We set A„(\) = Bq{/.) (notation as in 6.10.3).
Proposition. Aq(A) is a unitary (g, K)-module. Furthermore
dim HomK(FA + 2pn(q),/lq(A)) = 1.
Let S, e ^(gc,hc) be such that SiP = -P and let s2 e ^(lc,hc) be such
that s2(Pn$(lc,y = -Pn<t(lc,hc). Set s, = 535,. We note that A =
s0'(A+ 2p(q)) = s0'A-s0's,p+ s0'p (use s,p - p = -2p(q)). q' is
compatible with -s^P. If a e <t(hc, Ad(fc)^'u) then a = -Sq'jS with /? e
<t(hc,u) (notation as in 6.4.5). So,
(A - s0'p,a) = (so'A - So's,p,a) = -(AJ) + (p,s~'/?).
Since, s~'<I>(hc,u) cz -P, we have
(i) (/- Sq'p**) <0 for a€0(u',y.
(i) Theorem 6.7.5 combined with Lemma 6.4.5 implies that Aq(A) is unitary.
This proves the first assertion. We note that Aq(A) = rmM(q', CA) with m =
dim uk. (i) also implies that r'M(q', CA) = 0 if i # m. We can thus apply
Theorem 6.5.3 to find that
dim HomK(K„,/lq(A)) = (- l)m £ det(s)P;(A + Pk- stf, + pk)).
stW{K,T)

9.6. u-CohomoIogy
381
Here p'„ is the partition function of Q>(u'n, tr). We note that p'„{a) = p„{-s0a)
(p„ the partition function of <t(u„, tc)). Thus
P'ni>- + Pk~ *(K + Pk)) = Pn(SoS(A„ + Pk) - So(^ + Pk)-
We note that det(s0) = ( — l)m. We therefore have (after the obvious
algebraic manipulation)
dimHomK(^,Aq(A))= £ det(s)p„(s(/„ + pk) - (A + 2p(q) + soPk)).
seW(K.T)
We now assume that A„ = A + 2p„(q) = A + 2p(q) — 2p(qk).
Since, s0pk = -p(qk) + p(Pkn <I>(f( n lr,tc)) we conclude that
/.„ + p* = A + 2p(q) + s0p^.
We must therefore calculate
X det(s)p„(s(/„ + pk) - (A„ + pk)).
We now show that the only term in the above sum that is not 0 is the one
corresponding to 5 = 1. This term yields p„(0) = 1, and the second assertion
would now follow.
Fix H e it such that *{H) > 0 for a e P. Let s e W(K, T) be such that
PM'-n + Pk) - (A„ + pk)) > 0. Then .*(/„ + pk) = (/„ + pk) + Q with p„(g) > 0.
Hence Q(H) > 0. On the other hand A„ + p*. is P^-dominant so (s(/„ + pk) —
(A„ + Pk))(#) < 0. This implies that 2 = 0. Since A„ + p*. is Pk regular this
implies that s = 1. The proof is now complete.
9.6. u-cohomology
9.6.1. In preparation for the proof of the Vogan-Zuckerman theorem on
(q, K)-cohomology we need some results on u-cohomology. For the next three
numbers q will denote a reductive Lie algebra over C. Let h be a Cartan
subalgebra of q and let P be a system of positive roots for <t = <t(g, h).
Let b = b(P) = 1) © ©aeP qa- Let q be a subalgebra of q containing b. Put
<»,= {aE<t|(ga + g Jcqj. Set I = P - <t„ 1 = 1) © 0«e»,g« and u =
0aEi; qa. Then q = 1© u and [l,u] is contained in u. We note that 1 is
reductive and acts semi-simply on u. Set u~ = ©aeE q_a and q" = 1 © u~.
We note that g = u © 1 © u~. Thus P-B-W implies that
l/(fl) = l/(I)©(ul/(9)©l/(9)u-).
Let p be the projection of l/(g) into 1/(1) corresponding to this direct sum
decomposition.

382
9. Unitary Representations and (<(, K)-CohomoIogy
Let H e h be such that ol(H) > 0 for a e I and [//,!]= 0. Set U(q)H =
{ge l/(q)ladH(c/) = 0}. Then as in 3.2.1 (1) we have
(1) l/(fl)Hn(ul/(g) + l/(g)u") = l/(q)H n(ul/(q)) = l/(g)"n(l/(g)ir).
Thus, as in 3.2.1 we find that
(2) p restricted to U(q)" is an algebra homomorphism.
Let V be a q-module with action n. Then C'(u, V) = Homc(A'u, V) is
naturally an 1-module under the action (Xfi)(Y) = X(n(Y)) - //(ad X(Y))
for X e 1 and Y e u. Also, d(Xn) = Xd/i. Hence we have an action of 1 on
H'(u, V) for each i. Also, C'(u, K) is naturally a Z(q)-module under (zn)(Y) =
z{n{Y)),zeZ{$), ye A'u.
If /iel* is such that /i[ 1,1] = 0 then we set q^X) = X - n(X)l for X e 1.
Then q extends to an isomorphism of 1/(1) onto 1/(1). Set p„ = (jf^p.
The following result is due to Casselman and Osborne [1]. The proof below
is due to Vogan [1].
Lemma. If z e Z(g) and if B e H'(u, F) f/ien Zjg = p2p,q)(z)j8. Here p(q)(/i) =
(tr(ad /i |u))/2, as usual.
We prove this result by downward induction on i. If / = n = dim u then
H"(u, V) = A"u * ® V/u V. Thus z acts by / ® p{z). It is also clear that p{z) acts
by(-2p(q)® 7r)(p(z)). Thus(7 ® p{z))P = p2ptq){z)P- This is the result fori = n.
Assume the result for / = r + 1 < n. We now prove it for / = r. Let F be the
q-module l/(q) ® V with q acting by left multiplication. Set a(g ® i;) = gv.
Then a is a q-module homomorphism of F onto V. Let X = Ker a. Then we
have the q-module exact sequence
Now l/(q) is a free l/(u)-module under left translation. Thus H](u, F) = 0
for j < n (6. A. 1.5). So the long exact sequence of cohomology yields the 1 and
Z(q)-module exact sequence
0 -> H'(u, V) -> Hi+'(u, X) -> Hi + '(u,F) ->
This injection implies the result.
9.6.2. We now use the above result to give an especially simple proof of a
theorem of Kostant [2] (Bott [1] for the case when q = b).
Set P, = Pn<t(l,h). Let Wl = {se W(q,ty \sP contains P,}. If \i e h* is
P,-dominant integral then let E„ denote an irreducible finite dimensional
1-module with highest weight \i.

9.6. u-CohomoIogy
383
Theorem. Let F be an irreducible finite dimensional ^-module with highest
weight X relative to P. Then as an {-module
//'(u,F) = ©£sU+p)„„
the sum over se W1 with l(s) = i.
If z e Z(g) then z acts on F by X\+P(z)- Also, z e Z(g) acts on H'(u,F) by
P2P(,)(4 Thus, if z e Z(g) then p2„(,)(z) acts on H'(u, F) by xA+p(z).
Set W, = W(I, I)). We denote by ,y the Harish-Chandra isomorphism of
Z(l) onto \J(\))Wl. Then ,y <-■ pp(q) = y, the Harish-Chandra isomorphism of
Z(g) onto l/Ch)"'.
As 1-modules both F and Au are semi-simple. Thus H'(u,F) splits into a
direct sum of irreducible 1-modules, £y. Let /i(J be the highest weight of £y.
Then Z(l) acts on £y by i^,J+p, (lower left subscript corresponds to objects
defined for 1 in the same way as they are defined for g). This implies that
P-y + Pi + P(l) must agree with 2 + p on 1/(1))^. This implies that there
must exist st] e W such that fitj + p, = sy(A + p). Since \iX] + p, is Pr
dominant and regular stje Wl.
As an 1-module H'(u,F) is a subquotient of A'u*®F. Thus the
weights of H'(u, F) are of the form a — <g> with g a subset of Z and tr a
weight of F Hence, /ifj = ct,7 — <Qy> with au a weight of F and g,7 a
subset of Z. We therefore have, s;j(/ + p) = (t,j + p - <Gy>- Now p - (Qu)
is a weight of a finite dimensional representation with highest weight p
(9.A.1.5) hence au = syA and syp = p - <g,;>. 9.A.1.6 therefore implies
that g,7 = {-sijP)nP Hence /(s) = ;'. Also the multiplicity of this weight
is at most 1. We have therefore shown that as an I-module
H'(u,F) = 0ms£s(A+p)_„
the sum over s e Wl with l(s) = i and ms is either 0 or 1.
The above argument also tells us how to construct the corresponding coho-
mology classes. Set for s e W\ Q = ( — sP) n P. Then the (s/ — <Q»-weight
space in Au* ® F is one dimensional and is contained in A'u* ® F Let p be
a non-zero element of C'(u, F) in that weight space. The dp = 0 and P cannot
be in the image of d. Hence ms = 1 for all se Wl with l(s) = /. This completes
the proof of the theorem.
9.6.3. We now return to the notation of the previous sections. Let q be a
0-stable parabolic subalgebra of gf, q = IC ® u, as usual. Fix h, a fundamental
Cartan subalgebra of g contained in 1. Then h = t + o, as usual. Let H e it be
such that 1 = {X e g | [H, X] = 0} and such that ad H has strictly positive
eigenvalues on u. Clearly, U(qc)K is a subalgebra of l/(gc)H. Hence p is a

384
9. Unitary Representations and (fl, K)-CohomoIogy
homomorphism of U{qc)K into U(lc)KnL. Write u = uk©u„, as usual. Let
R = dim u„ and fix an element f> e AR(u„)* — {0}. Let o> denote the map of
A'(uk)* into Af+Ru* given by o-,(a) = aA/?.
Let V be a g-module. Then at ® I is a K n L module homomorphism of
C'Cun, K) ® A*(u„)* into Cf+*(u, K) which commutes with the pertinent "d's".
It therefore induces a map
Jt,:H'(Uik.H ® A>„)* -► H' + >, K).
We let U(qc)k act on C'(u*, »0 by (zfi(y) = z(/?(}')), ze U(Qcf and ye A'(u*)*-
The following result is due to Vogan [1].
Lemma. If z e U(qc)K and if as H'\uk, V) ® AR(u„)* then
7t|((z ® 7)a) = p2p(q)(z)7r,.(a).
As in the proof of Lemma 9.6.1 we prove this Lemma by downward
induction on i. We first look at i = m = dim uk (the largest index for which there is
anything to prove). Then m + R = dim u so
Hm{uk,V)®\R{un)* = Am+*u*® V/ukV
and
Hm + R(u, V) = Am + Ru* ® V/uV.
Hence nm is given by the natural map
Am+*u*® V/ukV^>\m + Ru*® V/uV.
Thus the result in this case follows in exactly the same way as in 9.6.1. We
now assume the result for i + 1 < m and prove it for i. Let F, X be as in 9.6.1.
Then the g-module exact sequence
0->X->F-> V^O
induces the following commutative diagram with exact rows
W(uk, F) ® A*(u„)* - H'{uk, V) ® A>„)* - H!(uk, V) ® A>„)*
I I I
Hi + R(u,F) *H! + R(u,V) > Hi + 1+R(u,X)
As before, H'(uk, F) = Hi + R(u, F) = 0 for i < m. The result for i now follows
from the result for i + 1 applied to X.
9.6.4. The next result will play an important role in the calculation of
H'(q, K; Aq(A) ® F*). It is a special case of a more general result that is fairly

9.6. u-CohomoIogy
385
easily derived using the derived functor construction of the Zuckerman
modules. Rather than interrupt our exposition to give the more "sophisticated"
result we have opted to give the following cumbersome proof.
Set Pm = Pkn®({tn\)c,tc). Put KWl = {se W(K,T)\Pm is contained in
sPk}. Let s0 be the longest element of KWl. If /i is a Pm-dominant integral
form that is T-integral then let E„ denote an irreducible, finite dimensional
K n L-module with highest weight /i.
Lemma. Let y e KA have highest weight ly and let Vy e y. Then (m = dim uk)
(*) H"(UnK;M(qi,y®(F,n
is zero unless /i = s0(Av + pk) — pk and in this case it is one dimensional.
Lemma 9.4.1 implies that (*) is non-zero only if there exists f e W(K, T)
such that/i = t{Xy + pk) - pk. M0 = M(qk,Ell)®{Vy)* has a(f, Ln K)-moduIe
filtration M0 => M, => ■ ■ ■ Md => Md+, = (0) with MJMi + l x, M(qk, £„„,,.) with
<5,- a weight of Vy. As above the only terms that can contribute to cohomology
are those such that fi — <5,■, + pk - spk with s e W(K, T). For such a term we
write <5, = s/i,. Then t{Xy + pk) = s(|/,- + pk). Since n, is a weight of Vy this
implies that f = s and /i, = Xy. We have therefore shown that
(**) H">(l,LnK;M(qk,EJ®(Vy)*) = Hm(l,LnK;M(qk,E,p„p).
We now show that (**) is non-zero only if f = s0. We prove this by setting
up a "resolution" as in 6.6.2. Let 5 e t£ be Pm-dominant and T-integral. Set
M = M(qk,Ed).Set
D,.= l/(!t.) (X) (A'uk®£,)
(/((lnl)c)
and let S,-:£>, ->£>,-1 be defined as follows:
dj(k ® x1Ax2A--- Axj ® e) = Z (— l)Jfcxj® x^---Ax7A-- Ax,- ® e
+ £ (-l)r+sfc ® [xr,xs]Ax!A--- AxrA-- AxsA- Ax,® e.
r<s
Then as in 6.A. 1.4 we have the (f, K n L)-module exact sequence
0-Dm-D111_1-----D1-Do-M-0.
Let Xj = SjD^X,, = £>m). Then we have the following (f, K n L)-module
exact sequences
0 -> Xl -> D0 ->• M -> 0 and
0->X,+ 1->D,->X.-->0.

386 9. Unitary Representations and (fl, K)-CohomoIogy
These induce cohomology long exact sequences
H'{t,Kn L;D0) -► H'(f,KnL;M)^ H'(f,Kn^X^^ Hi + 1(f,K nL;D0)
and
H'(f, KnL; Dj) -► H'(f, K n L; ^) -► H'(f, K n L; Xj +,) -► Hi + 1(f, K n L; D,)
Now H!(f, K n L; D,) = 0 for i < 2m (6.A.1.5). We therefore find that
Hm(l, KnL;M)*Hm + l(l,KnL; A",),
Hm+i(t,KnL;Xj)KHm+J+1(lKnL;X]+l)
for j + 1 < m. Hence
H"'(f,KnL;M)*H2"M(f,KnL;X„M).
There is still one more long exact sequence
H2ml(l,KnL;Dm^)^H2m~l(lKnL;Xm^)
- H2m(f, K n L;DJ - H2"(f, K n L;Dm„ ,).
Since H2m~ '(f, K n L; £>m„,) = 0, we have the exact sequence
0 -H2m~ '(!, KnL;Xm^)^ H2m(l KnL; Dm) - H2m(f, KnL; Dm. ,)■
NowH2m(f, KnL\Dm) =
((l/(!c) (X) (A"u,0£J)/IU(fc) (X) (A"ut ® E,)))*nL
l>«tnl)c) l'((ln[)c)
which is a quotient of (\muk ® £/"''.
We now look at the case when 3 = tpk — pk. Then (Amuk ® £/nl is
nonzero only if tpk — pk = — 2p(qk). But then t = s0.
We are left with calculating Hm(f, K n L; M(qk, CSoP(t-Pk)). 6.4.5 combined
with 6.5.1 imply that this is C as asserted.
9.6.5. We now turn to the notation in 9.5.9.
Lemma. If y e KA and if Aq(A)(y) is non-zero then Xy = A + 2p„(q) + Q with
Pn(Q) > 0.
Let M0 = M(q',CJ => Mt => M2 => ■■■ be the (f, K nL)-module filtration
constructed in 6.4.4. Then
M,/M, + 1 * Mfoi, £A_Qi) with p^Q,.) > 0.

9.6. ii-CohomoIogv
387
If f? is a homomorphism of Z(fc) to C then let
(Afof = {me M0\(z - ri{z))rm = 0 for some r and all z e Z(fc)}.
The above filtration implies that M0 is the direct sum of the (f, K r> L)-
modules (M0y. Let p be the infinitesimal character of V Then
dim HomK(K„ rmM0) = dim Hm(f, LnK;M0® (V.,)*)
= dim Hm(f, LnK; (M0)" ® (K,)*).
Now, F0 = (M0)" inherits a ./fmre filtration V0 => K, => • • • Fd => Vd+, = 0 with
K/^ +1 * Af (qi, £,-Q,) and A - Q, + pk = s,(/y + pt) for some s,- e W(K, T).
If we use the spectral sequence (9.A.2) corresponding to the filtration
FT(l LnK;V0 ®{Vy)*) = C\i, L n K; V{ ® (V.,)*)
then the El term is the direct sum of the spaces
H\lLnK;(V(/Vi+l)®(Vy)*).
The previous Lemma now implies that Hm(f, LnK\V0 ® (K,)*) is non-zero
only if there exists Q such that p'n{Q) > 0 and /. - Q = s0(/.y + pk) — pk. This
combined with the definition of and q' implies the result (we leave the algebra
to the reader).
9.6.6. We continue with the notation of 9.5.9. So F is a finite dimensional
(g, K)-module satisfying the hypothesis therein. We fix q e Q(F) and / the
highest weight of F relative to a ^-stable system of positive roots compatible
with q.
Theorem. Let R = dim u„. Then
Hi + R(q, K; A„{/.) ® F*) = H'(I, K n L; C).
In light of Proposition 9.5.9, Lemma 9.5.6 and Proposition 9.4.3, it is
enough to show that if q! e Q(F) and if q, is compatible with Pi then
unless /.(P,) + 2pn(qi) = / + 2p„(q).
We note that pc = (pnl)f© u„ © u„. Thus if a is a weight of t on Apr then
a = A + 2p„(q) — B — C with A a weight of A(p n l)r, B and C weights of
Au„. Thus if p. is a weight of Apf ® F then /i = ^i + a as above and 5 a weight
of F. Thus 6 = /. — Q with (? a sum of elements of P. We therefore see that

388
9. Unitary Representations and (g, K)-CohomoIogy
/i = X + 2p„(q) + A — B — C - Q. This implies that
A(P,) + 2p„(q,) = A + 2p„(q) -B-C-6 + /1.
On the other hand AfP,) + 2pB(q,) = A + 2p„(q) + S with p„(S) > 0 by the
previous result. Let H e it be as in the definition of 0-stable parabolic
subalgebra for q. Then if we evaluate the above two expressions on H we find
that 0 < S(H) = -(B + C + Q){H) < 0. Thus S(H) = 0. But then S = 0 and
the result follows.
9.7. A theorem of Vogan-Zuckerman
9.7.1. In this section we complete our discussion of (g, K)-cohomology. If F
is an irreducible, finite dimensional (g, K)-module as in 9.5.9 and if q e Q(F)
(9.5.8) let (F, q) denote the action of 1 on the one dimensional 1-module F".
The Theorem of Vogan-Zuckerman [1] is
Theorem. Let V be an irreducible, infinite dimensional unitary (g, K)-module
such that H'(q, K;V® F*) # 0. Then there exists a 8-stable parabolic sub-
algebra of Qc, q e Q(F) such that V is (g, K)-isomorphic with the irreducible
summand of Aq(X(F,q)) containing the K-type with highest weight X(F,q) +
2p„(q).
Note. This result, combined with Proposition 9.5.9, Theorem 9.6.6 and
Theorem 9.4.6 completely calculates the (g, K)-cohomology with coefficients in
V ® F* for V irreducible and unitary and F finite dimensional. We note that
if we argue as in 6.6.2 using a "resolution" as in 9.6.4, it is not difficult to show
that the Aq(A) are irreducible.
The proof of this theorem will occupy the remainder of this section. We
first give an outline of the proof. Theorem 9.5.3 implies that there exists
q e Q(F) such that V(X(F, q) + 2p„(q)) is non-zero and that V has the same
infinitesimal character as F. Choose q e Q(F) such that ||/(F,q) + 2p„(q) + 2pk\\
is minimal subject to the condition V(X(F, q) + 2p„(q)) is non-zero. Let y
denote the corresponding K-type. We prove that the multiplicity of y in V is
one. Let /i denote the homomorphism of U(qc)k into C that corresponds to
its action on V(y). We show that /i depends only on q and F. Since Aq(X(F, q))
has the properties just used for V (9.6.6) we can apply the above argument to

9.7. A Theorem of Vogan-Zuckerman
389
it as well. Thus U(§C)K acts in the same way on V(y) and on Aq(l(F, q))(y). The
theorem now follows from Theorem 3.5.4.
We will now give the detailed proofs of the assertions made in the course
of the above sketch.
9.7.2. Fix Pk a system of positive roots for <t( fc, tc) such that q is compatible
with Pk. Let bk = tc ® nk be the Borel subalgebra of fc corresponding to Pk.
Set nlk = nk n lc. (q = \c ® u, as usual). Let 7r, be defined as in 9.6.3 and let
R = dim u„.
Lemma. nR defines an isomorphism of AR(n„)* ® V(y)n" onto
HR(u, V)"k(/.(F,q)).
To prove this we analyze the spectral sequences in 9.A.2.3 and 4. We take,
Uj = uk and u2 = u„. Then u, itt and u2 satisfy the conditions of 9.A.2.3. Thus
we have a spectral sequence with abutment H'(u, V) and
£?•« = Hq(uk,Apun®V).
Set A = A(F,q). We prove the Lemma by showing that (E^q)n'-"(k) = 0 unless
p = R and q = 0 and that
(£r)n''"W = (A*(u„)* ® V(y)u"r"(2.).
This will clearly suffice to prove the Lemma. To this end we use the spectral
sequence in 9.A.2.4. This time we have for an fi^-term
/r(uk,F)®(Au„)-ar-fls.
Here H e it is chosen as usual. Let a e KA be such that
(i) (Hr(uk, V(a)) ® (Aull)_flp_fl,)'"--(A) # 0.
Since Hr(uk, V(a)) is a multiple of Hr(uk, V„) we can apply Kostant's
formula. Let /i be the highest weight of a relative to Pk then the K n L-types that
occur in Hr(uk, Via)) have highest weights s(/i + pk) — pk with l(s) = r and
s€KWl = {seW(K,T)\sPk^ Pkn®((lnl)c,tc)}. Thus (*) implies that
(ii) A = s(n + Pk) - pk - <6> +
with Q a subset of I (= <t(u,hc)) and <g>+ is a weight of A(u„) (here <g> +
is, as usual, the projection of <Q> onto (tc)*).

390 9. Unitary Representations and (g, K)-CohomoIogy
Fix P e C(Pk) such that q is compatible with P. Let p„ = p„(P). Then
<6>+ = 2p„ - <e'>+withg' c Pand<g'>+isaweightof t on A"(pc n n(P))
with/; = \Q'\.
Put Rk= {ae<&k\ (a, .s// - p„) > 0} u {a e sPk | (a, s// - p„) = 0}. Then s// -
p„ is /^-dominant (i?k is a system of positive roots for ®k by 9.A.1.4). Let
Rk = tPk,te W(K, T). Set C = {ae sPk |(a,s// - p„) < 0}. Then
spk - tpk= £ a = <c>-
Since C is a subset of Pk n (sPk) there exists a subset, C, of P disjoint from Q'
such that C = {a |, ] a e C'}. Put /I = g' u C. Then
(iii) s// - p„ + tpk = X + p„ + pk - {A) + = X + p - {Ay.
Hence
W -Pn + tPk\\ = \\X + p - {A) + \\ <\\X + p- {A)\\ < \\X + p\\
since p — {A} is a weight of a finite dimensional representation of g with
highest weight p.
Let v e W(K, T) be such that // — p„ is vPk-dominant. Then Lemma 9.1.7
implies that
||s// -p„ + fpk|| > ||// -p„ + vpk\\.
Since the K-type with highest weight /;"'(// — p„) occurs in V ® S (9.1.4) the
Dirac inequality implies that
llA*-A, + »Pikll^P + Pl|.
This implies that all inequalities are equalities. So there exists q! e Q{F) such
that // = A(F, qt) + 2p„(q!). Our hypothesis on q implies that
(iv) ||// + 2pJ| > \\X + 2p„(q) + 2pk\\.
We now show that if // # X + 2pn(q) + 2pk or if // = / + 2p„(<jf) + 2pk and
p < R or q > 0 then we have a contradiction.
Choose a system of positive roots, P,, for ^(Ic.bc) such that if 2plk =
<Pkn<t((fnl)c,tc)> then plk is P,-dominant. Set P, = P, u I. Then P, is a
0-stable system of positive roots for Q>(§C,\)C) compatible with q. Put p,„ =
p(Pt) — pi k. We rewrite (i) as
H + 2pk = s-'(x + 2p„(q) + pk - <G">+) + Pk
with Q" c I and <6">+ is a weight of A*~"(u„). Hence,
li + 2pk = s'l(X + 2p„(q) + 2pk - <G">+ - (P* - SPk))-

9.7. A Theorem of Vogan-Zuckerman 391
Now, <@">+ + pk - spk = <B>+ with B a subset of I and \B\ = R - p + q.
Hence
(v) n + 2pk = s~' (k + 2p„(q) + 2pk - <B> +).
(vi) p„(q) + pk is Prdominant and if a e I then {p„(q) + pk, a) > 0.
Indeed, p„(q) + pk = p(q) + plk = p — p,„. If a e P, then the second
expression implies that (p„(q) + pk,a) > (ptk,oi.) > 0. If a e E is simple then
(P„(q) + Pioa) = (P>a) - (P(.»>a)-
Now 2p, „ = Z m^/J the sum over all P e Pt such that ((lc)/j + 0c)-/i)n Pc is
non-zero and m^ = 1 or \. Since (a,/?) < 0 for p e P,, (p(,„,a) < 0. (vi) now
follows.
If we use (v), we find that
||A + 2p„(q) + 2pJ|2-||/i + 2p,||2 = 2(/ + 2p„(q) + 2p)i,<B»-«B>+,<B>+)
= 2(A + p„(q) + pk,(B» + 2(p„(q) + pk,<B» - «B>+,<B>+).
(vi) implies that the first term in the last expression is strictly positive if
B is non-empty. Thus if we can show that
(vii) 2(p„(«5f) + pk, <B» - «B>+,<B>+) > 0,
we would conclude from (iv) that B is empty and the Lemma would follow.
We are thus left with (vii).
Let C be a subset of P, such that 2p,„ = <C>+. Let
C1 = {aeC!«B)+,a)>0}
and set C2 = C - Q.PutQ, = {a e P, - C|«B>+,a) > 0}.
Let se ^(lc,hc) be of minimal length such that <B>+ is —sP,-dominant.
Then
-St = {ae$(lc,y!«B>+,a)>0}U{a6Pl!((B> + ,a) = 0}
by 9.A.I.4. Thus {-sPt)n P, = C,u C0. We note that since 0(B) + =
<B>+, s0 = 0s. Thus <C0> + <C,> = <C0>+ + <C,>+ and so (p, = p(Pt))
5P, = P,-<C0>+~<C1>+.
This implies
(viii) (2p, - <C0>+ - <C1>+,<C0>+ + <C,>+) = 0.

392 9. Unitary Representations and (g, K)-CohomoIogy
Now 2p„(q) + 2pk = 2p- 2pln = 2p - <C>+. Thus
(#) = (2p„(q) + 2pk,(B}+) - «B>+,<B>+) = (2p - <C> + - <B>+,<B>+)
= (2p - <Q>+ - <C2> + - <B> + ,<B>+)
= (2p - <C,> - <C0> - <B>,<B>+) + «C0> - <C2>,<B>+)
= (2/)-<BuC0uC1),<BuC0uC1)+)
-(2-<BuC0uC1>,<C0uC1>+) + «C0>-<C2>,<B> + ).
p — <B u C0 u C, > is a weight of an irreducible finite dimensional
representation of q with highest weight p. Hence
||p||2 > ||p -{BuC„u Q>||2 > ||p - (flu C0 u C,> + ||2
= ||p||2 - (2p, <B u C0 u C,> +) + «B u C0 u C,>\ {BuQu Q>+)
= ||p||2 - (2p -{BuC0u Q>+, <B u C0 u Q>+).
Thus
(2p -<BuC0uC, >, (BuC„uC,)+)> 0.
Hence
(#) > -(2p - (Bu C0u C^XQu C,» + «C0> - <C,>,<B>+).
Now, p = p(q) + p,. Hence
(#) > -(2p(q),<C0 u C,» + «B>+, <C0> + <C,»
-(2p, - <C0> - <C1>,<C0> + <C,» + «B>+,<C0> - <C2».
The first and third terms are 0. Hence
(#) > «B> + ,2<C0> + <Q> - <C2» > 0.
This completes the proof of the Lemma.
9.7.4. We can now apply Lemma 9.6.3 to see that the action of U{q)k on
V(>. + 2p„(q)r is given by p2ptq)(x) on H>, V)"'-^).
Let 3i = 30)nt and I, = (3(1) np) © [1,1]. Then I = 3, ©I,. Since,
(2,a) = 0 for aE<t(lc,hc), fnl acts trivially in HR(u, F)n'-"(A). Hence,
Theorem 3.6.6 implies that U(lc)KnL acts on HR(u, V)n'-k(/.) by a
commutative algebra. Thus U(qc)K acts on V(X + 2p„(q)) by a commutative algebra.
Proposition 3.5.4 now implies
Lemma, dim V(l + 2p„(q))n" = 1.

9.7. A Theorem of Vogan-Zuckerman
393
9.7.5. This is the first assertion of our outline. In particular there exists
a homomorphism a of U(qc)K into C such that gv = a(g)v for
v e V(A + 2p„(q)).
We now compute a.
Fix h0, a maximally split fl-stable Cartan subalgebra of 1. Then h0 =
t0 + ci0, t0 = h0nf and ci0 = h0np. Let p0 be a corresponding minimal
parabolic subalgebra of 1, p0 = °m0 + ci0 + n0, as usual. 3, acts on HR(u, V)n'-"(X)
via A|3l. Also, (/((l^c)1'"' acts on this space via v»y0 for some vs
(o0)* (3.6.6). We now look upon A and p(q) as elements of (h0)*.
Notice that (p0)c ® u is a parabolic subalgebra of gc. Let Q be a system of
positive roots for ^{5cAho)c) compatible with this parabolic subalgebra.
Let p denote the "p" for this system of positive roots.
Since V has infinitesimal character Xx + P- 9.6.1 implies that if z e Z(qc)
then p2p(q)(z) acts on HR{u, F)n'k(A) by xx+p(z)I. This combined with the
above implies that there exists s e W(qc, (h0)c) such that
(*) s(X + p(q) + v + Pm) = X + p
here pm is the "p" for <t((0m0 + a0)o(ho)c)n Q-
In particular, (*) implies that v e (a0)*- We may (and do) thus
assume that (v,a) > 0 for a e <t(p0,o0). We rewrite (*) as
(**) / + p(q) + v + p,„ = s'1(a + p).
We note that s~lk = X — Ql and s~'p = p — Q2 with g! and Q2 sums of
elements of Q. Since, (v + pm,A + p(q)) = 0, if we take the inner product of
both sides of (**) with A + p(q) we have
(A + p(q), A + p(q)) = (A + p(q), A + p(q)) - (A + p(q), Q,) - (A + p(q), Q2).
Now, (A + p(q),Gi) > 0 and (A + p(q), Q2) > 0. The above inequality
therefore implies that (A(q),g2) = 0. This says that (-s~1Q)nQ is contained
in <t(lr,(l)0)c)^G = Gi- Hence se W(lc,(t)0)c) (9.A.1.3). We have thus
shown if p0 = ppo then
(v + pm) = ^1(p0 + pm).
s '(Po + Pm) — Po + Pm — S with S a sum of elements of Qx. Hence,
v = po — S. Thus S|t = 0. This implies that S is a sum of elements of ^(p,,, a0).
Also, ||v + pj|2 = ||v||2 + ||pj|2 and
llv + PJI2 = ||5-'(Po + PJII2 = IlPo + PJI2 = IIPoll2 + IIPJI2-

394
9. Unitary Representations and (g, K)-CohomoIogy
So, (v, v) = (p0>Po)- On tne other hand,
(v,v) = (v,p0 - S)< (v,p0) = (p0 - S,p0) < (p0,Po)-
Thus the inequalities are all equalities. In particular, this implies that
(po,S) = 0. Thus (p0,p0) = (v,v) = (p0,p0) + {S,S). Hence, S = 0. Thus,
v = p0.
We have therefore shown that the action of U(qc)k on V(X + 2p„(q))
depends only on F and q. This completes the proof of the steps in the outline
of the proof. Q.E.D.
9.8. Further results
9.8.1. We continue with the notation of the previous section. We note that if
we combine the vanishing theorem of Borel, Wallach [1; V, 3.4], Zuckerman
[2] and 9.5.8, 9.5.9 we have
Lemma. // q is a proper 8-stable parabolic subalgebra of qc then
dim u„ > rkRQ.
Obviously, this result has a direct proof. In fact, there are tabulations the
values of c(G) = cF(G) for F = C. If G is simple and has the structure of a
complex Lie group then the tabulation was first given in Enright [1]. We give
the table. The first column is the classical name (if it exists), the second column
is the name in the Cartan classification and the third is the value of c(G).
We now give the table of Vogan-Zuckerman [1] for G simple over R such
that Gc is simple over C (i.e., G has no structure as complex Lie group). This
Classical group Cartan Label c(G)
SL(n + 1,C) n > 1
SO(2n+l,C) n>2
Sp(n,C) n>3
SO(2n,C) n>4
A„
B„
C„
D„
^6
Ei
£8
f*
G2
n
In- 1
In- 1
2n-2
16
27
57
15
5

9.8 Further Results
395
time we will only include entries for cases when c(G) > rkR{G). In this table
the first column corresponds to the classical label (if it exists) the second
column gives the Cartan label (Helgason [1, p.518]) and the last gives c(G).
Classical group
SU*(2n)n> 3
SU*(6)
SO*(2n), n > 4
Sp(p,q), 1 <p<q
Cartan Label
All
All
Dili
CI I
EI
EII
EIII
EIV
EV
EVI
EVII
EVIII
EIX
FI
FII
G
c(G)
2(n - 1)
3
n- 1
2p
13
8
8
6
15
12
11
29
24
8
4
3
9.8.2. We conclude this section with some results for groups of R-rank one
that are direct consequences of the theory in this chapter and of the
calculations in Borel, Wallach [1; VI, Section 4].
Theorem. Let G = 0(n, 1)° or SU(n, 1). Let V be an irreducible (Q,K)-module
with infinitesimal character %p. Then there exists a 8-stable parabolic sub-
algebra, q, of gc such that V is (g, K)-isomorphic with Aq(0). In particular,
V is the underlying (g, K)-module of an irreducible unitary representation.
Let np(G) denote the set of equivalence classes of irreducible (g, K)-
modules with infinitesimal character xp. In Borel, Wallach [1, op. cit] it was
shown that there is a bijection between n(G) and
S = {yeK* HomK(I/,Apc)#0}.
(This was done using the Langlands classification and by explicitly
decomposing pr as a K-module.) We leave it to the reader to check that each

396
9. Unitary Representations and (g, K)-CohomoIogy
y e S has highest weight Ay = 2p„(q) for an appropriate 0-stable parabolic
subalgebra of gc. Thus the counting argument implies the result.
9.8.3. The connected semi-simple Lie groups of split rank one can be listed
(up to local isomorphism) as 0{n, 1)°, SU(n, 1) n > 2, Sp(n, 1) n > 2 and
FII. Let G correspond to one of the latter two examples. Then the vanishing
theorems imply that if V is an infinite dimensional irreducible unitary (g, K)-
module then H'(g, K; V) = 0. Since there always exists an infinite
dimensional irreducible (g, K)-module with H'(g,K; V) non-zero (Borel, Wallach
[1; V, 4.6]) this implies that the analogue of Theorem 9.8.2 is false for these
groups.
9.A. Appendices to Chapter 9
9.A.I. Weyl groups.
9.A.I.I. The purpose of this appendix is to prove a few results about Weyl
groups that will be used in the body of this chapter. Let g be a reductive Lie
algebra over C. Let h be a Cartan subalgebra of g and let <t+ be a system of
positive roots for <t(g, h). Let W = W(q, h) be the Weyl group of <t(g, h) (0.2.3).
Let A be the set of simple roots in <t+. Then W is generated by the
reflections sx, a e A. If s e W then we define the length of s relative to <t+, l(s), to be
equal to min{r|s = sls2---sr with each s, a reflection about a simple root
hyperplane}. It is clear that l(s) = l(s~l), since reflections are involutive. If
seW then set I (s) = {a e <t+ | sen e -<t+}.
9.A.I.2. Lemma. Let s e W. Then
(1) |Z(s)| = '(s),
(2) s is a product of reflections sa, a e £ (s).
If s = 1 then Z (s) = 0 and (1), (2) are clear. Suppose that we have proved
(1) and (2) for 0 < l(s) < r - 1 and that l(s) = r.
Let A = {«!,..., a,} and set sx = s, if a. = a,. Let s = s,-.■■■sjr be a minimal
expression. Put a = ah. Then sas = s,2 ■ ■ ■ sir is also a minimal expression. Thus
l(sxs) = l(s) — 1. Since a is simple, sa/? e <t+ if /? e <t+ — {a}. This implies
that if p e <t+ and if sxsp e -<t+ then /J e E (s) unless /? = -s_1a. If /? e E (s)
and if s/? is not equal to —a then p e E (sas). This implies that if — s_1a is
not in E (s) then E (s) would be equal to E (sxs). This would imply that

9.A.I. Weyl Groups
397
s<I>+ = sas<I>+ and hence s = sts which is false. Thus
(a) -sMa€l(s) and
(b) Z(vO = Z (*)-{-«"'«}■
This implies that |Z (s)| = |Z (sas)\ + 1. So the inductive hypothesis implies
that (1) is true for s. We note that (2) combined with (b) implies that sxs is
a product of reflections from Z (s). Also, s_sx = sstts~l. Thus sxs(s.sx) =
sas(s'lstts) = s. Hence s satisfies (2). This completes the proof of the Lemma.
9.A. 1.3. Corollary. If s e W then s is a product of reflections about roots
in<b+ n(-s<5+).
By definition <t+ n( — s<5+) = Z (s_1). Thus the previous Lemma implies
that s"1 is a product of reflections about roots in <t+ n( — sQ>+). Since root
reflections are involutive, the Corollary follows.
9.A.I.4. Lemma. Let n e h* fof such that (/i,a) e R, a e <I>(g, h). Let s e W
he o/ minimal length such that (.v/i,a) > 0 /or a e <I>+. Then
(*) s-1<t+ = {ae<D|(,j,a)>0}u{a€O+!(/i,a) = 0}.
We note that
(1) If a e h* is such that (tr, a) e R for all a e 0 then
P„ = (a e <t | (<T, a) > 0} u {a e <t+ | (a, a) = 0}
is a system of positive roots for <t.
We leave this as an exercise to the reader.
(1) implies that the right hand side of (*) is a system of positive roots for <t.
It may thus be written in the form r ' <1>+ with f e W. Let u e W be such that
u\i is <t + -dominant. Then u ' <t+ contains
{a e <D! (//, a) > 0} u {a e<D! (//, a) = 0}.
Hence u_1<£+n(-<£+) contains t~l<&+ n(-<5+). Hence /(«) > /(f). If
/(u) = /(f) then u~x<b+ n(-<t+) = r'fc'1" n(-<D + ). So u = f.
9.A. 1.5. We conclude this appendix with some results related to the
irreducible finite dimensional representation with highest weight p. If Q is a
subset of <t+ then set <g> = IaeQ a.

398
9. Unitary Representations and (g, K)-CohomoIogy
Lemma. Let F be an irreducible finite dimensional Q-module with highest
weight p. Then the weights of F are the linear forms p — <g > with Q a subset of
<t+ and the multiplicity of a weight /i is the number of subsets Qof<&+ such that
n = P-<Q>-
The Weyl character formula says that if ch F is the character of F and if
A = e>Tlxe<l>(l -Othen
ch F = X det(s)e2sp/A.
seW
Now, A = Y.seW det(s)esp. Thus
ch F = e2p n (1 - e-2x)/e" f] (1 - O
= e" n (1 +0= Z e"~<G>-
The result now follows.
9.A.I.6. Lemma. Let Qbe a subset of <t+ and let s e W^. Tfen
<e> = p-sp
if and only if Q = (-s<t+)n<I>+.
We note that 2sp = ^xe<l>*ns^ a + £ae(-<i>+)nS<i>+ a and that
2p = ^ae<I>+ns<I>+ a — ^ae(-<I>+)ns<I>+ a-
The obvious subtraction implies the sufficiency. We now prove the necessity.
The previous lemma implies that p - <g> is a weight of F, an irreducible finite
dimensional g-module with highest weight p. Our assumption says that
p - <g> = sp. Since the weight sp occurs in F with multiplicity 1 the
necessity now follows from the sufficiency.
9.A.2. Spectral sequences
9.A.2.I. In this appendix we collect some material on special sequences
which will be sufficient for the application in this chapter. A detailed account
of spectral sequences can be found in MacLane [1]. Let A be a vector space
over C and let deEnd(A) be such that d2 = 0. Then, as usual, we
write H{A) = H(A,d) = Z(A)/B(A) with Z(A) = ker d and B(A) = dA. If A
is a graded vector space A = @i>0A' and if dA' is contained in A' + l
then we write H'{A) = Z\A)IB\A) with Z\A) = {a e A' da = 0} and

9.A.2. Spectral Sequences
399
B'(A) = dA'+1. We assume that A (resp. A') has a filtration F'A such that
each F'A is d stable (resp. dF'A' is contained in F'A}+1). We also assume
that F'A ^F' + lA, f] F'A = 0 and F'A = A for i < 0.
Put GrA = 0(>oF'k/Fi+U. Then d induces Gr(d) on Gr A. We
analyze H(Gr A, Gr(d)). By definition
Z(GrA)= @^0{aeF'A\daeF'+l}/F!+'A and
B(Gr A) = 0.>o (dF'A + Fi + 'A)/Fi + 1A.
Hence
H(Gr A)= @{ae F'A \daeFi+l }/{dF'A + F'+ lA).
i>0
Set Z\={aeFA\daeF'+i} and E\ = Z\/(dFA + Fi+iA). Then
@E\ = H(Gr A). To establish a pattern for higher terms in the spectral
sequence (which we are both explaining and constructing) we set Z0 = F'A.
Then
E'1=Z'1/(dZJ) + Zo+1).
Set Z'2 = {ae F'A! da e F'+2}. It is clear that dZ' c Zi+' and that
d(dZ{)-i-Z{)+,)cdZ{)+1.
Thus d induces
d1:Ei1->E'i+1.
Let ze£' be such that d,z = 0. Let aeZ' be an element of z (recall
that z is a coset). Then da is an element of dZ'0+2 + Z'0+'. Hence there exists
v e Z'0+2 such that d(a -v)e Z'0+2 = Fi+2A. Hence a - v e Z'2. It is obvious
that Z'0+2 is contained in Z',+'. Hence we have a linear map of Ker^ |£1]) into
Z'2/Z;+1. Set E'2 = Z'2/(dZ\-' +Z\+i). Then the above natural mapping
induces (£, = ©£'i)
TpZ'^J-Ei.
Suppose that Ti(z) = 0. Then if a e z, a e dZ1!-' + Z',+'. Thus
a = dv + u,v e Z\~'
and ueZ\+1 cZ'0+\ Thus Ker dx is contained in B'fE,). Thus T,
induces an injective linear map, S,, of Hi(Ex,dl) into £2. Suppose that ze£2
then there exists aez with aeZ2. Thus oeZ, and dae Zi + 2 = Fi + 2A.
So a defines an element of Zl(Eudx). This proves that St is bijective.

400 9. Unitary Representations and (g, K)-CohomoIogy
This sets the pattern, set
Zir={aeFiA\da€Fi + r} and
El = Zl/(dZ^l+Zlt\).
Then, as above d induces
dr:E^El+\
We note that
(1) Ep = (Zp + Fp + lA)l(dZpZ[+' + Fp + U)
and
(2) Hp(Er, dr) is isomorphic with EP+, under a natural map Sr defined in the
same way as Si.
We now relate these spaces with H(A, d). We note that since each F'A is
d-invariant we have a natural mapping L: of H(FlA,d) into H(A,d)
that assigns cohomology classes to cohomology classes. Similarly, if j > i
then we have a natural mapping L(j of H(F'A,d) into H(FA,d). Obviously,
L;Lij= Li. There is thus a decreasing filtration, F'H(A,d) = H(F'A,d)
of H(A,d).
We assume, for the sake of simplicity that there exists a non-negative
integer, s, such that Fs+lA = 0.
(3) £'' = Z(A) n F'A/B(A) n FA = FlH(A).
Indeed, Zstl={ae FlA da e F;+s+ U} = Z(/l) n FU.
Also
z/-(.+ n+i = {a€Fi-sA\daeF'A} = {aeA\daeFA} = B(A)nF!A.
We say that the spectral sequence EPq has abutment H°(A,d).
9.A.2.2. We now assume that A is graded. So A = @i>0 A1 and dAl c Ai+',
fM' c /!'' and df/l' c FpAi+l. Set
Z?-« = {ze FpAp+,>\dz e Fp+r(/l''+''+1)}>
£?■« = ZrM/(dZrp:[+1'" + '-2 + Z^!1'"-1).
Then £" = 0^ £M and dr maps EPq into £p + >'.''-''+1.
Lemma. Let B be an endomorphism of A such that BF'AP c F'AP, and
Bd = dB. Then BZpq c Zp-q and if Bp is the induced map on Ep then

9.A.2. Spectral Sequences
401
BPdr = drBP. Furthermore, if Fs+1/1 = 0 then BP+1 agrees with the map
induced by B on FPH(A, d).
This is clear from the naturality of the constructions above.
9.A.2.3. We now give some spectral sequences that will be used in this
chapter. These spectral sequences are related to the famous Hochschild-Serre
spectral sequences and to a family of spectral sequences used by Borel in his
study of L2-cohomology.
Let n be a Lie algebra over C. Assume that n, is a subalgebra of n and that
n2 is a subspace such that n = it! © n2, [n,,n2] c n2 and [n2,n2] c n,. Let
M bean-module. Set A' = C(n, M) = Homc(A'n,M)./l = Homc(An, M). Set
FiA = \ueA\u( £ An1-Ajn2J = oi.
Then F'A = A for i < 0 and Fs+ 'A = 0 if dim n2 = s.
Suppose that u e FA n Ap. Let Xu..., Xce n2 and Yu..., yp_(e n,. Then
du^,..., Y„.i,X1,...,Xi+1) = I + II + III + IV +V with (indices
involving only 7's run between 1 and dim n,, those involving X's run between 1
and s)
i = Z(-iV+1yj«(y„...,^,...,irp-„A'1>...>x(),
n= X (-i)-+-M([rr)i;:, Y! yr y, r,,.,.^,,...)
r<s
m = Y,(-iy+p-i+su(lYr,xsiYu...,Yr,...,xu...,xs,...)
lV = (-l)"-,Il(-iy+iXju(Y1,...,Yp_l,X1,...,X],....,Xl)
V= x (-lrwrJjj. ^. ^, ^, 4
This formula easily implies that our filtration is d-invariant. We now
calculate the £, term of the corresponding spectral sequence. In other
words we calculate the cohomology of Gr A. Let u' e Gr* A then modulo
F' + U, u' is represented by u e Homc(An1 ® A(n2, M). In the notation above
du^,..., Yp.i,Xl,...,Xi) = I + 11 + III, since IV and V are0. We note that
(A'n2)* is a ivmodule under the action induced by ad. A simple rewriting of
I + II + III yields
(1) £? = //-(n1,(A"n2)*®M).
9.A.2.4. We continue with the example of the previous number, with an
additional assumption on r^ and n2. Assume that there is a semi-simple

402 9. Unitary Representations and (g, K)-CohomoIogy
derivation, H, of n that stabilizes n, and n2 and has positive eigenvalues. Then
H acts on (A'rtj)* with strictly negative eigenvalues if q > 0. Let 0 = a0 >
— a1>--->—adbe the eigenvalues of H on (An2)*- Set G' = G'(A*n2)*
= Ij^CAttj)*. Then G° = (An2)* and Gd + 1 = 0. Set
F; Hom^A"!!!, Anf ® M) = Hom^An^G'' ® M).
Then F' defines a decreasing d-invariant filtration of Homc(An1, Anf ® M).
We note that n, • G' is contained in G' + 1. We therefore have a spectral
sequence with
£M = Hp(n1,(G',+7G',+'< + 1)® M) = //p(n1,M)®(A"n2)tap_aq.

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This Page Intentionally Left Blank

Index
Admissible
(8,K (-module, 81
representation, 81
Affine algebraic group, 42
Analytic vector, 34
AR (Artin-Rees) property, 14
Augmentation homomorphism, 9
Borel subalgebra, 37
Bruhat decomposition (lemma), 52
Cartan subalgebra, 4
of a real Lie algebra, 56
maximally split, 57
fundamental, 57
Cartan decomposition
Lie algebra, 43
group, 46
Cartan involution, 42
Cartan subgroup, 59
fundamental, 59
maximally split, 59
Central distribution, 294
Chevalley restriction theorem, 75
Coefficients (matrix coefficients), 22
C'-vector, 31
Compact form, 44
Cusp form, 233
Delta sequence, 25
Dirac
operator, 367
inequality, 368
Distribution, 332
character, 292
order, 332
Exponential polynomial, 335
Finitely generated
module for an algebra, 14
Formal degree, 24
Frobenius reciprocity, 31
(3,*0-
module, 80
equivalent, 80
finitely generated, 80
tempered, 138
underlying module, 81
Gelfand, Naimark decomposition, 54
Generalized weight space, 108
Harish-Chandra
isomorphism, 78
homomorphism, 93
Homomorphism
fl-module, 11
G-module, 18
Induced representation, 31
Infinitesimal
character, 34
(ly) equivalent, 81
(ly) irreducible, 81
Intertwining operator,
g-module, 11
group representation, 18

412
Index
Invariant
symmetric bilinear form, 5
subspace (for group representation), 18
Irreducible
group representation, 18
Isotypic component
in a group representation, 28
Iwasawa decomposition
Lie algebra, 45
group, 45
Jacquet module, 111
K-character, 295
Langlands data, 149
Langlands decomposition, 51
Left invariant measure, 1
normalized, 2
Lie algebra
compact form, 8
nilpotent, 14
reductive, 4
Lie group
unimodular, 2
Locally integrable, 332
Modular function, 2
Natural
equivalence, 177
transformation, 177
Noetherian algebra, 13
Nilpotent element, 342
Norm, 71
Operator
compact, 326
Hilbert-Schmidt class, 323
self-adjoint, 326
trace class, 328
P-pair (parabolic pair), 51
cuspidal, 58
Parabolic subgroup (standard), 51
minimal, 51
P-B-W, 9
Rapidly decreasing functions, 230
Real reductive group, 42
inner type, 51
Realization, 13
Regular
element, 4
character, 323
Representation,
conjugate dual, 20
direct sum, 24
(strongly continuous of a) group, 18
Hilbert, 18
Lie algebra, 11
(right) regular, 22
smooth, 18
square integrable, 22
unitary, 18
Root, 4
real, 58
simple, 4
space, 4
system, 5
system of positive roots, 6, 48
Schur's lemma,
Dixmier's, 11
for(s,K)-moduIes, 80
for groups, 21
Schur orthogonality relations, 23
Scwartz space, 237
of Harish-Chandra, 230
Semi-simple element, 342
Smooth vector, 31
Spin module, 362
Split component (standard), 48
Submersion, 332
Support, 332
Symmetric subgroup, 42
Symmetrization map, 9
TDS (three dimensional simple Lie algebra), 11
0-stable
parabolic subalgebra, 184
root system, 365
Universal enveloping algebra, 8
canonical filtration, 9
Unitary (g,K(-module, 367
Verma module, 37
Weight, 36
space, 36
dominant integral, 36
Weyl
chamber, 6,48
character formula, 67
group, 6
integration formula
Lie algebra, 63
Lie group, 63
reflection, 6

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