Real Reductive Groups II
NOLAN R. WALLAO:
Real Reductive Groups II
This is Volume 132-11 in
PURE AND APPLIED MATHEMATICS
H. Bass, A. Borel, 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 II
Nolan R. Wallach
Department of Mathematics
University of California, San Diego
La Jolla, California
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
Boston San Diego New York
London Sydney Tokyo Toronto
This book is printed on acid-free paper. @
Copyright © 1992 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
transmitted in any form or by any means, electronic
or mechanical, including photocopy, recording, or
any information storage and retrieval system, without
permission in writing from the publisher.
ACADEMIC PRESS, INC.
1250 Sixth Avenue, San Diego, CA 92101-4311
United Kingdom Edition published by
ACADEMIC PRESS LIMITED
24-28 Oval Road, London NW1 7DX
Library of Congress Cataloging-in-Publication Data
(Revised for vol. 2)
Wallach, Nolan R.
Real reductive groups.
(Pure and applied mathematics; v. 132- )
Includes bibliographical references and indexes.
1. Lie groups. 2. Representations of groups.
I.	Series. II. Series: Pure and applied mathematics
(Academic Press); 132)
QA3.P8 vol. 132, etc. [512'.55] 86-32199
[QA387]	510 s
ISBN 0-12-732961-7 (v. 2: alk. paper)
Printed in the United States of America
92 93 94 95 96 97 EB 9 8 7 6 5 4 3 2 1
Contents
Preface	ix
Introduction	xi
Chapter 10.	Intertwining	Operators	1
Introduction	1
10.1.	The intertwining operators	2
10.2.	The proof of Theorem 10.1.5	17
10.3.	Limit formulas	28
10.4.	A generalization of L. Cohn’s determinant formula	32
10.5.	The Harish-Chandra ^.-function	39
10.6.	Notes and further results	47
10.A.	Appendices to Chapter 10	49
10.A.	1. Some constructions related to finite dimensional
representations	49
10.A.	2.	Some results related to Sterling’s	formula	55
10.A.3.	Miscellaneous results	56
Chapter 11.	Completions	of Admissible (fl, A)-Modules	59
Introduction	59
11.1.	Some results on Weyl group invariants	60
11.2.	A lemma of Kostant	67
V
vi
Contents
11.3.	Representations with small K-types	69
11.4.	The automatic continuity theorem	77
11.5.	Completions of (g, K)-modules	84
11.6.	Analysis of completions of (g, K)-modules	88
11.7.	The proof of the main theorem	96
11.8.	The action of ^(G) on admissible representations	103
11.9.	Poisson integral representations	105
11.10.	Notes and further results	110
11.A.	Appendices to Chapter 11	111
11.A.1	. Some results on the action of a compact group
on a symmetric algebra	111
11.A.2	.	Small K-types	113
11.A.3	.	Some results on Verma modules	126
11.A.4	.	Some functional analysis	128
Chapter	12.	The Theory of the Leading Term	133
Introduction	133
12.1.	Characters of principal series representations	134
12.2.	The modules |Go.(„	139
12.3.	The leading term	144
12.4.	The dependence of the leading term on parameters	149
12.5.	The leading term and intertwining operators	158
12.6.	The main inequality	167
12.7.	Wave packets	182
12.8.	The Harish-Chandra transform of a wave packet	191
12.9.	Notes	200
12.A.	Appendices to Chapter 12	201
12.A.1	.	Traces of certain kernel operators	201
12.A.2	.	Some inequalities	205
12.A.3	.	The topology of induced representations	213
Chapter	13.	The Harish-Chandra Plancherel Theorem	215
Introduction	215
13.1.	The Eisenstein integral	216
13.2.	The leading terms of Eisenstein integrals	228
Contents	vii
13.3.	Wave packets of Eisenstein integrals	235
13.4.	The Harish-Chandra Plancherel theorem	239
13.5.	The calculation of р.(ш, v) for the fundamental
series	247
13.6.	The intertwining algebra of IP(rtiv and the
irreducibility of	the	fundamental series	249
13.7.	Groups with one conjugacy class of Cartan
subgroup	256
13.8.	The Plancherel theorem for L2(G/K)	258
13.9.	Notes and further results	260
Chapter 14.	Abstract Representation	Theory	263
Introduction	263
14.1.	The basic theory of C* algebras	265
14.2.	The C* algebra of a locally compact group	273
14.3.	Quotients of C* algebras	275
14.4.	Density theorems	279
14.5.	Representations of C* algebras and positive
functionals	283
14.6.	The topology on the unitary dual of a C* algebra	294
14.7.	The topology on the unitary dual of a locally
compact group	306
14.8.	Direct integrals and Von Neumann algebras	312
14.9.	Direct integrals of representations of C* algebras
and locally compact groups	326
14.10.	Decompositions of representations of CCR C*
algebras and locally compact groups	329
14.11.	The Plancherel formula for CCR locally compact,
unimodular groups	340
14.12.	The Plancherel formula for real reductive groups	349
14.13.	Notes and further results	354
14.A.	Some functional analysis	355
Chapter 15.	The Whittaker Plancherel Theorem	363
Introduction	363
15.1.	The support of certain induced representations	364
15.2.	Some asymptotic expansions and estimates	368
viii
15.3.	The Schwartz space for L2(N \ G; x)
15.4.	The holomorphic continuation of the Jacquet
integral
15.5.	First steps for the holomorphic continuation
15.6.	The completion of the proof of the holomorphic
continuation
15.7.	Cusp forms revisited
15.8.	The first steps for the Plancherel theorem for
generic x
15.9.	The Plancherel theorem for L2(A0 \ G; x)
15.10.	Some examples of the Plancherel theorem for
generic x
15.11.	Notes and further results
15.A.	Appendix to Chapter 15
Bibliography
Index
Contents
375
381
383
393
405
412
422
426
430
435
439
451
Preface
This book is the second volume in the series Real Reductive Groups and it
is intended to be read as a continuation of the first volume. As in the first
volume, much of the material in this one is based on lectures and seminars
given at Rutgers University over the past 10 years. I thank the students
and colleagues that have suffered through those lectures over the years,
especially Roe Goodman for his help with Chapter 14 of this book. The
reader should note that in the time interval between the two volumes I
have changed my academic affiliation. I would like to take this opportunity
to thank Rutgers University for presenting me with the exciting research
environment that allowed me to spend so many years in the writing of
these volumes. I thank Bruce Ramsey for having developed his wonderful
program, “Leo,” which was used in the final draft of this manuscript. I
would also like to thank the students of the University of Tokyo for their
list of typographical errors and possible gaps in the first volume. Finally, I
thank the National Science Foundation for the summer support during the
preparation of this volume.
There is a possibility of a third volume in the series; however, there are
no forward references to “volume three” in this work. In that volume,
several topics that are missing from the volumes at hand would probably
be included. If the pursuit of the unitary dual is completed while I am still
Preface
in possession of my faculties, then Volume 3 is certain. In addition, I
would include the Arthur-Campoli Paley-Wiener theorem (and later
developments) and the Plancherel theorem for semi-simple symmetric
spaces (Oshima and his school).
This volume (as is the first) is dedicated to my mother, Pauline Wallach,
with all my love.
Introduction
This is the promised second volume of Real Reductive Groups. In this
book, the emphasis is on the more analytic aspects of the theory. However,
the beauty of the subject is (to this author) the interaction between
algebra, analysis, and geometry. The serious reader can approach the book
in several ways. Chapter 14 is an almost self-contained introduction
to abstract representation theory (locally compact groups, C* algebras,
Von Neumann algebras, and direct integral decompositions), which can
(except for the last section) be read independently of the rest of this
volume. The reader who is mostly interested in the Plancherel theorem for
real reductive groups would do well to read this chapter first. Chapter 10,
which contains an exposition of an approach to intertwining operators due
to Vogan and the author, is the bedrock on which most of the rest of the
book rests and is therefore a prerequisite for the other chapters. From
there the reader could then move on to Chapter 11, which is an exposition
of the work of Casselman and the author on C" vectors of admissible
representation of real reductive groups, or jump right into Chapter 12,
which lays the analytical basis for the Harish-Chandra Plancherel theorem
(Chapter 13). Chapter 15 depends on all of the previous chapters. Its
subject matter is more specialized than the earlier chapters and the style
of exposition in it is (even) more terse than the rest of the book. The
xi
xii
Introduction
reader would do well to fight his (or her) way through it (after reading
Chapters 11, 12, and 13) in order to test his grasp of the earlier material.
Throughout this volume there are references to Volume 1. A reference
to, say, Theorem 5.3.4 means the theorem in Section 5.3 of the first
volume labeled, 5.3.4. References to the present volume will involve x.y.z
with x > 9. Thus, we will look upon the two volumes as one 16-chapter
book. The bibliography in this volume is a superset of the bibliography of
Volume 1 and the numbering is consistent with that volume. There are
therefore some places where the numbering for a specific author is not
chronological (cf. the bibliographic references to Jacquet, H.).
As indicated, Chapter 10 is basic to all but Chapter 14 of this volume. It
contains an exposition of intertwining operators that is strongly influenced
by the work of L. Cohn [1]. However, the methods are quite different from
those of L. Cohn. As with other parts of this book, the scope of the theory
developed goes far beyond the needs of the rest of the book. This was also
true of the first volume, where for example the unitarizability theorem
(Vogan [3], Wallach [4]) for Zuckerman derived functor modules is given
in its full generality (Theorem 6.7.5) even though the “ЛЙ(А)” are all that
are used in the rest of that (and this) volume. Also, as with other parts of
this book there are interesting problems for further research based on the
material in this chapter. The most obvious is the determination of the
polynomials in Theorem 10.2.2. In principle, the method of the proof of
Theorem 10.2.2 is constructive so, at least for some examples, there should
be an algorithm for the calculation of the ingredients in the theorem.
Chapter 11 contains a detailed discussion of C°° vectors of admissible
Hilbert (more generally, Frechet of “moderate growth”) representations
of real reductive groups. The main theorems are due to Casselman and
the author. The main thrust of the results is that the algebraic structure of
the underlying (g, /О-module completely determines the C“ vectors of
any Hilbert (more generally, Frechet of “moderate growth”) realization.
This result implies that most algebraic constructions in the theory of
admissible (g, Ю-modules are “automatically continuous.” Except for the
first section of this chapter, the only serious use of this chapter in this
book is in Chapter 15.
Chapter 12 is the most difficult in the two volumes. It contains a theory
of what Harish-Chandra called the “constant term.” We have opted to
stray from the Harish-Chandra lexicon and call it the theory of the leading
term (the reason becomes evident in the definition). Our approach to the
subject is representation theoretic rather than through differential equa-
Introduction	xiii
tions. Although the theorems in this chapter are completely analogous to
(and for the most part “cribbed” from) those of Harish-Chandra [14, 15],
the difference of approach allows for a substantial simplification of the
theory. Here we come to the difficult analysis of this chapter armed with
an a priori theory of intertwining operators and thus the Harish-Chandra
C-functions are initially defined in terms of intertwining operators rather
than vice versa in Harish-Chandra’s approach. This is analogous to our
approach to the discrete series, where the material of Chapter 6 gives an
a priori construction of the representations.
Chapter 13 is an exposition of Harish-Chandra’s masterpiece (Harish-
Chandra [16]). Although our approach differs in detail, the main line is
fairly true to the original, which we hope will be read by the reader of this
book. As in the original, we complete the basic theory of “cuspidal”
intertwining operators using the Harish-Chandra Plancherel theorem
(13.6). This application is ample evidence that Harish-Chandra’s theorem
goes much farther than merely calculating the abstract Plancherel mea-
sure. If that is not enough evidence, Chapter 15 should be a “clincher.”
As mentioned before, Chapter 14 could have appeared earlier in this
volume, and for the most part it can be read independently of the other
chapters. As indicated, Harish-Chandra did more than compute the
Plancherel measure for real reductive groups. However, there is also a
sense that he did less. In the end of this chapter we give a (not completely
trivial) proof of the fact that Harish-Chandra’s Plancherel theorem is the
Plancherel theorem. Also, the reader will find motivation for the form of
Harish-Chandra’s Plancherel theorem (in terms of tempered representa-
tions) in this chapter. It is an interesting paradox of the representation
theory of real reductive groups that it can be studied without any real
knowledge of abstract representation theory. Thus, the younger generation
of researchers in the subject are for the most part ignorant of the abstract
theory. This is analogous to the generation of researchers in enveloping
algebras that have no understanding of or interest in Lie groups. We hope
that this chapter will be a not too painful introduction to this beautiful and
important theory.
Chapter 15 has been included in this book for two reasons. The first is
that the main theorem (15.9) is important in its own right and the second
is that it makes use of almost every important part of the earlier chapters
of this work. It also contains a decomposition theorem analogous to the
Plancherel theorem where there are no obvious normalizations of the
measures that come into it. Even so, there seems to be a natural measure
xiv	Introduction
that appears (the restriction of the Plancherel measure to an open subset
of the unitary dual). A similar phenomenon occurs in the case of “semi-
simple symmetric spaces.”
The author is very gratified with the comments and suggestions that
have been made by the readers of the first volume of this work. The most
common complaint was that the book was too “tightly written.” We have
endeavored to make the line of reasoning in the main results of this
volume more transparent. However, if anything, this volume is even more
difficult than the earlier one. The last sentences in the introduction to
Volume 1 are even more appropriate to this volume. You can’t just read
this book if you wish to master it. You must work through it.
Intertwining Operators
Introduction
In Section 5.3, we introduced intertwining integrals for a class of induced
representations and in 5.8.4 we promised to implement an analytic contin-
uation of these integrals. This chapter fulfills that promise, but it goes
much further than the scope of Section 5.3. It contains a general theory of
intertwining operators for parabolic induction on real reductive groups,
which will form the bedrock on which most of the rest of this volume rests.
This theory is based on a difference equation (Theorem 10.2.2) due to
Vogan and the author that sharpens and extends a similar result of
L. Cohn. The equation is a b-function type formula analogous to the
method of Bernstein [1] in the meromorphic continuation of complex
powers of polynomials. The difference equation allows us to carry out the
meromorphic continuation of the intertwining integrals in much the same
way as the Г function is classically continued to the complex plane. The
critical difference between the results of this chapter and the vast litera-
ture in the subject (see the notes in Section 10.6) is that we give a
meromorphic continuation in the context of C" induction rather than
/Cfinite induction. This will be used in the next chapter in a proof of the
results of Casselman and the author on the structure of the space of C"
1
2
10. Intertwining Operators
vectors of an admissible representation. Another by-product of the differ-
ence equation is a determinant formula (generalizing one of L. Cohn for
Harish-Chandra’s C-functions) for the intertwining integrals on A>iso-
typic components. An outgrowth of this formula is an a priori proof that
the Harish-Chandra Plancherel density is tempered. This fact leads to a
significant simplification of Harish-Chandra’s theory of wave packets
(Chapter 12) and thereby of the proof of the Plancherel theorem
(Chapter 13).
As indicated, the theory developed in this chapter is more general than
is needed for the applications in this book. However, in the theory of
automorphic forms, the leading term of the Eisenstein series is a product
of two terms: The first is a scalar given in terms of the discrete group in
question; the second is an intertwining operator, which in principle is in
the generality of this chapter.
Section 1 is devoted to the general theory of intertwining integrals. It
contains a proof of the meromorphic continuation (based on Theorem
10.1.5, which is proved in Section 2) and the product formula for the
intertwining operators. Most of the general results (for /Cfinite induction)
can be found in some form in the literature (see 10.6.1). As just men-
tioned, the critical (for our purposes) difference equation is inspired by the
results of L. Cohn [1] on the Harish-Chandra C-functions. Section 2
contains the proof of the difference equation. Section 3 contains some
limit formulas that extend a result of Harish-Chandra [15]. These results
are used in Section 4 to give a generalization of L. Cohn’s determinant
formula for C-functions. In Section 5, we define and analyze generaliza-
tions of Harish-Chandra’s p-functions. As has been mentioned, the key
result is that they are tempered. In Chapter 13, we will see that the p
function is essentially the density in the Plancherel formula.
10.1. The intertwining operators
10.1.1. We begin this section by introducing the class of induced repre-
sentations that will be studied. We will then begin the theory of intertwin-
ing operators. Let G be a real reductive group. Fix K, a maximal compact
subgroup of G. Let P = ° MAN be a parabolic subgroup of G with a given
(standard) Langlands decomposition. Let (<r, Ha) be a Hilbert representa-
tion of M( = ° MA) and let be the space of C" vectors for a. We
denote by /“ the space of all C" functions f from К to //“ such that
f(mk) = a(m)f(k) for m с M П К and к e K. If f e /“ and if v g a£,
10.1. The Intertwining Operators
3
then we set
Pfv(nmak) = a''+pa(tna)f(k), n^N, m^°M, a^A, k&K.
Here, p = pP. If P is understood we will use the notation fv. Then, fv is
a smooth function from G to Я".
We endow with the usual topology. That is, if x g f7(mc) then we
set qx(v) = ||<r(x)f||. The topology is given by the semi-norms qx, x g
U(mc). For simplicity, we will assume that the К П M - C" vectors for a
are the same as the M - C" vectors. This is no assumption if dim Ha < oo
or if a is irreducible and unitary. Indeed, if dim Ha < oo this is clear. If a
is irreducible and unitary then the argument in the proof of Lemma 8.5.5
(which uses only the unitarity of the left or right regular representation)
implies that if x g (7y(mc), then
< c||o-(l + Д)'уII
for v g H“. Since a is irreducible and unitary, Schur’s Lemma implies
that o-(C) = kl. Thus, ||<r(x)i>|| < C'||o-(1 + CK)yr||. Thus, the semi-norms
qx for x g L7((t П m)c) suffice to define the topology on Я“. If /g
then we define Ц/Ц» = 8ир^еК ||/(Л)||. We define rra(k)f(x) = f(xk). We
will also write тга for the corresponding action of U(tc) on 7“. If /g 7“
and if x g U(tc), then we set px(f) = l|7ro.(x)/|L. We endow 7” with the
topology given by these semi-norms. It is an easy matter to see that, with
this topology, 7“ is a Frechet space (see 1.6.4).
If /g 7“ and if v g a^, then we set TTp,a,v(g)f(k) = f„(kg) for к g K,
geG. Also, if X g g, then set
d
^P.a.Ax)f(k) = -fv(kexptX)^0.
With this action we have a representation of gc on 7“ for each v g a£.
Lemma. (тгРа1,, defines a smooth Frechet representation of G.
We fix v and write for trP a v.
(1)	If X g g then there exist y}- g U(tc), j = 1.........d (depending on v
and X) such that ||-7r„(A')/,||e E Py(f)-
4	10. Intertwining Operators
Indeed,
/„(fc exp tX) = a(k exp tX)v+po-(m(k exp tX))f(k(k exp tX)).
Thus, if we set
<Px(k) = -ra(k exp tX)\t+=o,
at
then
H(*)/IL £ sup 1^(^)111/1100 +||77_p(X)/||oo.
We may therefore assume that v = -p. Set tt = ir_p. Set
q = {У g ш|0У = -У).
Since M = (exp qXAf П K) with unique expression, we may choose m(g)
= exp(y(g)), with У a smooth function from G to q. We may also assume
that k(g) defines a smooth function from G to K.
With these normalizations, we note that
Y(kexptX) = tZ(k) + O(t2)
and
k(k exp tX) = к exp(tW(k) + O(t2)),
with Z(k) g q and W(k) g t. Thus,
^X)f(k) = a(Z(k))f(k) + ^W(k))f(k).
Let Zj be a basis of q. Then Z(k) = 'Lui(k)Zi with g С“(КЭ. Further-
more, the assumption on a implies that there exist m(7 g L7(t П m) such
that
k(Z;)/(fc)||< LMm17)/(fc)||.
But
о-("»,7)/(Л) = L(m£)/(fc) = R(Ad(k)~lmu)f(k).
Thus, if nij g Ur(V) for all j, and if is a basis of Ur(V), then there exist
continuous functions Д,77 on К so that Ad(£-1)m(7 = Ej3//Z(fc)z(-. Hence,
||<r(m(7)/(£)|| < CLpz(f ). The assertion now follows.
(2)	The topology on Ц is given by the semi-norms ||тгр a P(x)/IL for
X G U(q).
10.1. The Intertwining Operators
5
To prove this we first show by induction on n that if Yv..., Yd g I and
if Xlt..., Xn g g, then
YdXt  Xn=^ZiUiX2 ••• Xn
for some Z, g g and u, g U(l). Indeed, if n = 0 this is clear. Assume this
is true for n - 1 > 0. We prove the assertion for n. We note that
••• YdX{   • Xn = - ЕУ1	••• Ydx2 Xn
i<d
+ XxYx  YdX2--- Xn.
Since the second term of the right hand side of this equation is of the
desired form, we concentrate on the first. This term can be written as
-	•••	••• Ydx2 ••• xn
i<,d
- E yt  [[^.y,],^]	^-Л+1 Ydx2 xn.
i<i<d
As before, the first term is of the desired form. Clearly, after a finite
number of steps this procedure will terminate.
This assertion combined with an obvious induction using (1) implies that
if x g [/(g) then the semi-norm qx(f) = ||тгр(x)/||„ is continuous on
This clearly implies (2).
To complete the proof we must show that the function
S ir„(g)f
is of class C“ for f g /“. We first show that the map is continuous. To see
this, we note that if x g [/(g) then irv(x)Trv(g)f = 7rp(g)7rp(Ad(g-1)x)/.
Since the map g, к -»/p(^g) is uniformly continuous on compacta, it is
clear that
lim ||ir,(x)(7r,(g)/-/)||„ = 0.
g-»i
That g >-> ir„(g)f is of class C“ now follows by exactly the same argument
used in 1.6.4.
10.1.2.	Let P = MN be the opposite parabolic subgroup to P. Fix an
invariant positive measure dn on N. We will normalize our measures in
10.1.7. If v g and we use P in place of P, we can also define a
6
10. Intertwining Operators
representation тгр a v of G on We form
(1)	= l-fAnk)dn.
Here, the integral is the obvious vector valued generalization of the
Riemann integral (the so-called Bochner integral). Of course, there is no
reason to believe that the integral converges.
Lemma. There exists c = ca such that if Re(p, a) > c for a g Ф(Р, A),
then the integral defining (1) converges absolutely on compacta of
(p g a£|Re(p, a) > с) x K.
Furthermore, there is a constant Ca such that
< CJI/L, /G Re(p,a) > c„.
In the indicated range, we have
JP\p(^)^P,a,Xg) = irp<,'V(g)Jpip(v).
Finally, if Re(r>, a) > ca then Jp^tv) is not the zero operator on I™.
f„(hk) = a{hf+p<r{m{nk))f{k{nk)). Thus,
II fXnk) || < a(n)Re 1'+р||о-(т(пЛ)) || ll/IL.
Fix a norm ||    || on G (2.A.2.3). If we use the argument in the proof of
Lemma 5.2.8, we see that
|| m(nk) || < Const ||n^||d = Const ||n||d.
Also, ||<r(/n)|| < Const ||m||r (2.A.2.2). Thus, ||o-(zn(nA:))|| < Const ||n||s for
some j. 4.A.2.3 implies that there exists Ago* such that ||n||s <
Const a(n)-A. Hence, we have
||Л,(п£)|| < Const a(n)Rep+p л.
Thus, if Re(r> a) > (A + p, a) for a g Ф(Р, A), then
||/„(«£) || < Const a(n)2'’||/||oo.
This implies that if Re(p, a) > (A + p, a) for a g Ф(Р, A), then
|| f„(nk) || dn <, Const f_a(n)2p dnWfW^.
10.1. The Intertwining Operators
7
We set ca = тахцеф(Р /1)(Л + p, a). Since the integral on the right con-
verges (see either Lemma 2.4.5 or Theorem 4.5.4) the lemma will be
proved when we prove the last two assertions. We set
jp\p(v)f(g) = f_fXng)dn.
JN
We observe that if g = vmak with v g N, m a g A, and к g K,
then
LfXng) dn = (fv(nvmak) dn
JN	JN
= ( fv(nmak) dn = av+pa\m) [ f„((ma) lnmak} dn
JN	JN
= av~pff(m) f_f„(nk) dn.
This implies that if Re(r>, a) > ca then the integral defining jP\P(v)f(g)
converges, and that P(JPiP(v)f)v = JP!P(v)f. In light of this, it is now clear
that
Jp\p(V)'!rP,<r,Ag) = 7ГР,а,1'(^)^Р|р(Р)-
The last assertion is proved in exactly the same way as 5.3.1 (2).
10.1.3.	Lemma. Letf^ 1“ then the map of {r> g a£|Re(p, a) > ca] into
1“ given by v Jp\P(v)fis continuous and holomorphic on the interior.
This is a direct consequence of the uniform convergence assertion in
10.1.2.
10.1.4.	We now assume that a has an infinitesimal character. Since G is
of inner type there exists p g a£ such that o-(a) = apI for a & A. Thus,
without any loss of generality we may assume that p = 0. The /^-finite
vectors for 7TP a<v define the (g, K>module IP<„tV as studied in 5.2.1.
Here, we identify a with <Г|0М.
Our next task is to prove, under our new assumptions, a meromorphic
continuation of the operators Jp|F(p). We will prove this using a variant of
the method of Bernstein polynomials by deriving a difference equation
satisfied by these operators.
8
10. Intertwining Operators
If И is a vector space over C and if <p is a function from a£ to V then
we say that <p is a polynomial if there exists a finite dimensional subspace
W of V such that the image of <p is contained in W and <p is a polynomial
map of a£ into W.
10.1.5.	Theorem. Assume that G is of inner type. There exist polynomial
maps ba and Da from a£ to C and U(qc)k, respectively, such that if
Re(p,a) > ca and iff g 7“, then
bAv)Jp\p(v)f = Jp\p(v + 4p)-D-F>or „+4p(Dor(p))/.
We will prove this result in the next section. A similar theorem with
Da(v) having a slightly different range of values can, in the case when a is
square integrable, be derived from the results of L. Cohn [1] on
Harish-Chandra’s C-functions. The result as stated is due to D. Vogan
and the author.
10.1.6.	We now derive the desired meromorphic continuation using the
previous result. We assume that G is of inner type.
Theorem. Let f g . Then the map v >-» JP\P(v)f, initially holomorphic
from
(p g a£|Re(p, a) > ca, a g Ф(Р, Л)}
to I“, extends to a meromorphic map of a£ to I". Furthermore, if
Re(r>, a) > ca - 4k(p, a) for a g Ф(Р, A), then
k-l
ГИК 4jp) X JP|P(p)/
7 = 0
is holomorphic.
We call the statement for к > 0, (l)ft. If Re(p, a) > ca - 4(p, a) for
a g Ф(Р, A) then we may use Theorem 10.1.5 to define JP^P(v)f, so that
(1), is true. If we have implemented the continuation into the range
Re(r>, a) > ca - 4k(p, a) for a g Ф(Р, A) such that (1)л is true, then we
can define JP^P(v)f in the range Re(p, a) > ca - 4(fc + lXp, a) using
Theorem 10.1.5, so that (l)fc+1 is true.
10.1.7.	Up to this point we have used an arbitrary invariant measure on
N. We will now give some normalizations of invariant measures that will
10.1. The Intertwining Operators
9
be used in the rest of this book. Fix В a non-degenerate, Ad(G)-invariant,
symmetric, real valued bilinear form on й such that
<Х,У> = -B(X,0Y)
is positive definite. If I is a Lie subalgebra of й and if L is the connected
subgroup of G with Lie algebra I, then we define dL to be the left
invariant measure on L that comes from a left invariant dim(I)-form ы on
L such that if X{,...,Xr is an orthonormal basis of I relative to < , >
then
|w(X1,...,Xr)| = l.
On К we always use the normalized invariant measure.
Let P = °MAN be a minimal parabolic subgroup of G with given
Langlands decomposition. We choose da = dA, dn = dN. We normalize
dg so that (See 2.4.1)
(1)	ff(g)dg= f a~2pf(nak) dndadk.
JG	JNXAXK
We note that, since all minimal parabolic subgroups of G are conjugate
with respect to K, which acts by isometries relative to < , ), this normal-
ization of dg is independent of the choice of P.
If Q = °MqAqNq is a parabolic subgroup of G with given standard
Langlands decomposition such that AQ c A, NQ cN, then we set *Q =
°MQ П P = °MA*qN*q a minimal parabolic subgroup of °MQ. We choose
an invariant measure on °MQ using *Q in the same way as we used
P to choose dg. On AQ, NQ, and N*o we take dAQ, dNQ, and dN*Q,
respectively.
Lemma. The map NQ X N*Q N given by n, *n >-» n(*n) is a surjective
diffeomorphism and
I <p(xy) dxdy = / <p(n) dn.
JNqXN,q	jn
Furthermore, if f is integrable on G then
[ f(g) dg = [	a~2pof(nmak) dndadmdk.
JG	JNqx°MqXAqXK
n = nG Ф n»0. If h e a is such that a(h} > 1 for a e Ф(Р, A) and if
at = exp th, then Ad(a,)nG c nQ and Ad(a,)n*G c n»0. Thus, 4.A.2.1
10
10. Intertwining Operators
implies that
dN = cdNQdN*Q
for some real positive constant c. Let (resp., w2) be a left invariant
form on Nq (resp., N*Q) defining dNQ (resp., dN*Q). Since (nG, n*Q) = 0,
л w2 defines dN. Hence, c = 1 and the first assertion follows.
The second is a direct calculation which we will now indicate. To
simplify notation we write KM = Q П K, U = NQ, V = N*Q,°M = °MQ.
f	a~ 2pQf( nmak) dn dm da dk
•'NqX°MqXAqXK
= I	b 2pQa 2p*of(uvambk) dudvdadmdbdk.
JUxVxA*QxKMxAQxK
Now, ka = ak for a &Aq, к g К b~2pQa~2p*Q = (ab)~2p if b g
Aq and a ^A*Q, and d(mk) = dk for k, tn g K. This implies that the
right hand side of the formula agrees with the first formula in the
statement of the theorem.
10.1.8.	We now develop some of the results on roots that will be
necessary to our study. Our discussion follows Harish-Chandra [16]. Let
а с p(= {X g g|0A” = -X}) be such that exp(a) = A is a standard split
component of a parabolic subgroup of G. Then A is called a special vector
subgroup of G. Fix a special vector subgroup A of G. Let £?(A) denote
the set of all p-pairs (P, A) in G. If P g &(A), then set 2(P) = 2(P, A)
= (a g Ф(Р, A)lca & Ф(Р, Л) for 0 < c < 1}. If Pt, P2 g ^(Л), then
set 'Z(P2\P1') = S(P2) П KP^. We set d(P{, P2) = |S(P2|PX)|. Here, if F
is a finite set then |F| denotes the cardinality of F.
Lemma. If P, P{, P2 g ^(Л), then the following assertions are equivalent:
(1) d(Px, P2) = d(Px,P) + d(P, P2>,
(2)	S(P2|PX) э S(P|Pj);
(3)	X(P21Pj) is the disjoint union of S(P2|P) and 'Z(P\P1).
If a g a*, then set Qa = {X g g| Ad[Zi, X] = a(h)X, h g a}. Put
Ф(й, a) = {a g a*|a =# 0 and Qa # 0}. Let S(g, a) = {a g Ф(й, a)|ca <£
Ф(й, a) for 0 < |c| < 1]. If P g ^(Л), then X(g, a) is the disjoint union
of S(P) and 2(P). We now prove the lemma.
10.1. The Intertwining Operators
11
Assume that (1) is true. The preceding observation implies that
S(P2) A ^(P^ is the disjoint union of S(P2) A SfPj) A 2(P) and
S(P2) А УАР^ A 2(P). Now,
S(P2) А ДЛ) nS(P) с ЦР2) n 2(P)
and
S(P2) А 2(Л) A S(P) cS(P)n ^Pf).
Thus, (1) implies that both inclusions are equalities. Hence, ’£(P\P1) c
S(P2|Pj). This is the assertion of (2).
Assume that (2) is true. (2) says that S(P) П S(Pj) c X(P2) n S(Pt).
So 2(P) A "ZiPf) c S(P2) A "ZiPi). Now, S(P2) A 2(Pj) is the disjoint
union of S(P2) A ^(Pf) A 2(P) and S(P2) А SfPj) A 2(P). On the other
hand, S(P2) А 2(Л) A 2(P) = ЯЛ) A 2(P) by (2). Also,
S(P2) A 2(P) = S(P2) A £(Pj) A 2(P) U S(P2) A 2(P) A
But
S(P2) A S(P) А 2(Л) С ЦР2) A (S(P2) А 2(Л)),
by the preceding. Hence, S(P2) A 2(P) A 2(Pt) = 0. Thus, (2) im-
plies (3).
Since it is obvious that (3) implies (1), the lemma follows.
10.1.9.	If , P2 g ^(Л) then we say that they are adjacent if d(P1, P2)
= 1.
Lemma. Let PX,P2& 0(A) be such that d(P{, P2) = d > 0. Then there
exist Qx,...,Qd g 0(A) such that Px and Qt are adjacent, Qt and Qi+l
are adjacent, and Qd = P2. Furthermore, if 2(£>( + 1|£>() = {a,}, i =
l,...,d - 1, and ifiaj = KCJPj), then ^(P^Pf) = {aj,..., aj.
If Pj and P2 then the lemma follows with 2i = Pi-
We now assume the result for d - 1 and prove it for d.
Consider 2(Pj|P2) = £(Pj) A S(P2). If jSj,... ,j3r G 2(Pj) A S(P2)
and if Д1 +    +Pr g 2(й, a), then /3j +  •  +pr g y^Pf) n S(P2). We
assert that there exists a g 5(P]) A S(P2) such that a is simple in S(P2).
Suppose not. Let H g a be such that S(P2) = {/3 g Х(й, а)|Д(//) > 0}.
If p g 2(P|) A S(P2), then our supposition implies that there exist
|3j,..., Pr g S(P2) with r > 2 such that p = |3j +    +Pr. But then
12
10. Intertwining Operators
Д,(Н) < 0 for some i. Hence, say, /3, g S(P2). Continuing in this way we
arrive at a contradiction. So let a g П S(P2) be simple in S(P2).
Define Qd_x by S(Od-i) = (S(P2) _ {«}) u {“«}. Then, Qd_x and P2
are adjacent. Furthermore,	э S(0d-ilF2). Thus, Od-i is be-
tween Pt and P2. Hence, d(Pt, Qd_t) = d - 1. The result now follows
from the inductive hypothesis.
The last assertion follows from the proof of the first.
10.1.10.	Fix a special vector subgroup A of G. If P1( P2 g ^(Л), then
we set
W*')= f- aP^YP' + V dn
JNi C\N2
for v g a*. Then, since the integrand in this formula is positive, it follows
that 0 < 1р2\р[г) < °0.
Lemma. There exists a constant CA < » such that if (v, a) > (pP , a) for
a g S(P2|P1), then Ip2\p[v) < CA.
We first look at the case when P{ and P2 are adjacent. Let ^(Р2, Р{) =
{a}. Set n“ = Есг1йса and let № be the corresponding connected
subgroup of G. Set n“ = 0(n“), № = 0(№), and й“ = n“ ®°m Ф RHa
Ф n“. Then, й“ is a reductive subalgebra of й- Let Ga be the correspond-
ing connected subgroup of G. Then PY П Ga = Pa = MaAaNa (Aa =
exp(RHa)) is a parabolic subgroup of Ga with = °M° and P2 П
Ga = Pa. Let pa be the “p” for Pa. Then
/p2|Pi(1') = b'jpjSv + Pp,)|rh - Pa)-
The last expression is less than or equal to Ca < » if О + p^, a) >
2(pa, a). We assert that (pPi,a) = (pa, a). Indeed, let na =
Ep^tpp-ia} Йса- Then ПР1 = n“ Ф na. Since a is simple in ХСРД
Ad(g_a)na c na. Thus, tr(Ad Haln) = 0. Now,
2(pPi,a) = 2pPi(Ha) = tr(Adtfa|n) = tr(AdHa|n.) = 2(pa,a).
This proves the assertion, and, hence the result if d(P{, P2) = 1.
We now prove the lemma by induction on d(Pl,P2). Assume it for
d(Pt, P2) = (/-1^1 with a majorizing constant Q_1. We consider the
case when d(P1, P2) = d. Let QY be as in Lemma 10.1.9 for P2, PY. Then
d(Qt, P2) = d - 1 and d(Plt Q2) = 1. We set Q = Qt. Lemma 10.1.8
10.1. The Intertwining Operators
13
implies that
=/_	_ aP(xy)Pp’+v dxdy.
JNQr>NP2xNP)nN0
Let S(P2I(?) = {«}• Then a g 2(Pj), so if Pa is as before we see that
Pa c Pt and Na = Npt П Ga. We also note that NQ П Np2 = Aa. If x tNa,
then x = nmaa(x)ka(x) with aa(x) g exp(RHa) and ka(x) g Ga П K.
Thus,
aP,(xy) = aa(x)aP[ka(x)y) = aa(x)aP[ka(x)yka(xyl).
If к g Ga П K, then Ad(fcXnPi П ne) = (nPi П ne). Thus, d{kxk~x) =
dx on NP} П Nq . We therefore find that
M") = faa(x)Pp' + v dxf	ap(y)pp'+’' dy.
JNa	JNP,nN0
These integrals converge if (v, Д) S: (pPi, Д) if Д g Х(0|Р,) U {a} =
X(P21Л) s*nce d(Q< P\) = d - 1. Also, under this condition the product is
at most CaCd_f The result now follows.
10.1.11.	Let (<r, Ha) be an admissible, Hilbert representation of °M,
having an infinitesimal character. If P g ^(Л), / g and v g a£, then
we set
Pfv(nmak) = al’+ppo-(m)f(k), n^NP, aeAP, m g °Mp, k&K.
If P^Pi g &(A\ then we consider
Jp2ipf<v)f(k) = L ptU”k)<ln.
JNtnN2
Here, dn stands for d(N{ П N2). We note that if P2 = Pj then our
notation is consistent with the earlier usage.
Lemma. There exists ca> 0 such that if Re(p, a) > ca for a g X(P2, Pt),
then the integral defining (1) converges uniformly in
{p g a£|Re(p, a) > ca) x K.
Furthermore, there exists a constant Ca such that
14
10. Intertwining Operators
forf^ /", and Re(p, a) > ca for a g S(/,2|/’1). Finally, in the indicated
range
= ^P^a.Asyjp^P), g^G.
We are looking at
f_ aP(n)Pp' + vff(mP(hk)}f(k(hk)) dh.
jn1hn2 1	1
As in the proof of Lemma 10.1.2, the integrand is dominated by
M«)'’/’,+ReX"I4^))ll 11/11» c>Pi(«)^+Re,"Aii/iL.
Here, A is as in 10.1.2. Since
[ aP(n)pp' + Re,"A dh = /(Rev - A),
jn1hn2 1
we see that the first assertions follow from the previous lemma with
Ca = C'aCA and ca as in 10.1.2.
We write
jptip2(v)f(x) = f_ Pf,(hx)dn
JNlnN2
for x g G. To prove the last assertion it is enough to prove that
jp2\p^)f(nmak) = v(m)av+p?2jP2]Pi(v)f(k)
for n g Np2, a g A, m g °M. To this end consider the map
A\ A N2 X N{ A N2 -» N2, x, у >-» xy.
The result in 4.A.2.1 implies that this map defines a surjective diffeomor-
phism. We therefore have a diffeomorphism
ф: П N2-» (Nx HN2)\N2.
Since Ad(Aj A N2) consists of unipotent matrices there exists a smooth,
non-zero, differential form w on A, A N2 \ N2 that induces an invariant
measure under the right action of N2. Now ф*ы = hd(Nl A N2) with
h g С"(А[ A N2). Since ф*ы is right (N{ A A2)-invariant, h must be a
constant. We may thus choose w such that ft = 1. We write dn for this
measure on (N{ A N2) \ N2. We have
jp2\pt(v)f(x) = f (pf„)(nx)dn.
JNlON2\N2
Clearly, jP2\P(.v)f(nx) = j p2\P(.v)f(x) for n g N2 and x g G. If
m g °M, then the map n >-» mnm ~1 preserves the invariant measure on
{N2C\Nx)\N2. Hence, JP2lP(.v)f(mx) = (r(m)jP2\P(.v)f(x) for m e°M
10.1. The Intertwining Operators
15
and x g G. If a g A, then
jp2\p^f(ax) = f (М(пах) dn
JNlnN2\N2
= av+ppi ( (P fv)(a'lnax) dn.
JNlnN2\N2
If a g A, then we write p.(a) for the action of a on n2/(nt A n2) induced
by Ad(a). Then, on (Aj A A2) \ A2,
(/(ana-1) = det(^(a)) = a2'>'12(det(Ad(a)|ninn2)).
The lemma will follow if we show that if H g a, then
(*) pFi(H) + 2pPfH) - tr(Ad(H)|llinn2) =pP2(H).
Now,
2pp2(A) = tr(Ad( Н)|П1пп2) + tr(Ad(//)|П1пп2),
so
2Pp2(H) - tr(Ad(H)|n,nn2) = -2pPl(H) - tr(Ad(H)ln,nn2.
Hence,
2Pp2(H) = -2pPi(H) + 2tr(Ad(H)|n,nn2).
This implies (*). The proof of the lemma is now complete.
10.1.12.	If P{, P2, P g &(A), then we say that P is between Px and P2
if any one of the equivalent conditions of Lemma 10.1.8 are satisfied.
Lemma. If P is between Px and P2 and if Re(p, a) > ca for a g
0(Pt, A), then Jp^piv) and JP\P[v) are defined and Jp2ip(v)Jpip[v)f =
jp2^fforf^i:2.
We use the notation of the proof of the previous lemma. Lemma 10.1.8
(3) implies that
(Ар A Apjf/Vp, A AP) = Ap, A Ap2.
We therefore have
Jp2\p(v)JplPl(v)f(k) = ( Jpip^)f(nk) dn
JNPnNp2
= L L (ptfP)(»1»2k) drii dn2
JNPnNP2JNPinNP
= L (pM^k) dn = Jp2lPl(v)f.
NPl П Np2
16
10. Intertwining Operators
10.1.13.	Suppose that P1 and P2 are adjacent. Let {a} =	Let
a“ = (H e a\a(H) = 0}. Put “m = {X e й| [H, X] = 0 for H g a“},
aM = {g g G|Ad(g)H = H for H g a“}. Then, aM is a real reductive
group in our sense. We set Q, = P, П aM, i = 1,2. Then, and Q2 are
parabolic subgroups of aM with standard split component A. Further-
more, Q2 = Ci.
If f g then we define f{k\m) = f(mk) for к & К, m &aM И К.
We note that M c aM. We use the notation for the space defined in
the same way as 1“ with G replaced by aM. Then, f(k) g for all
к & К. Furthermore, f g /“ for v g aJ and i = 1,2. If v g a?.,
Ц.<г, Ицл,,
then we write v = av + va with v{H) = 0 for H g a“ and av(Ha) =
v(Ha\ va(Ha) = 0, and va(H) = v{H) for H g a“. Set “A = exp(RHa)
and Aa = exp(a“).
Set P = aMPl = aMP2. Then, P g 0>(Aa). Set *£, = P, C\°MP for i =
1,2. Then, *Cj is a parabolic subgroup of °MP with standard split
component “A and *02 = *2i- We also note that = °MPi. If
pg a*, then we denote by the representation of °MP corresponding
to I,Q <,v. Then, the correspondence /•-»/ defines a (g, КЭ-module
isomorphism of IPi a v onto Ipt„aa v». With this interpretation we will use
the notation H(f) = f. The obvious calculation now yields
(1)	H(jP2lP^)f)(k) =J^Ql(av)(H(f)(k)).
10.1.14.	Theorem. If f g 1“ and if P{, P2 g &(A\ then v >-» JPiiP(.v)f
has a tnerotnorphic continuation to а£.
This result is an immediate consequence of 10.1.13 (1), Lemma 10.1.9,
and Theorem 10.1.6.
10.1.15.	We note that 10.1.13 (1) implies a product formula for the JPiQ.
We will make this more precise in Section 10.4. We should mention that
formulas analogous to 10.1.13 (1) have appeared in many papers that we
will discuss this in the notes at the end of Section 10.6.
10.2. The proof of Theorem 10.1.5
10.2.1.	We assume in this section that G is of inner type. We maintain
the assumptions and notation of the previous section. We assume that
10.2. The Proof of Theorem 10.1.5
17
(a, has an infinitesimal character. Let (p, F) be an irreducible finite
dimensional representation of G such that:
(1)	FK + 0;
(2)	acts trivially on F\
(3)	z(qc) acts trivially on F.
We note that the Cartan-Helgason theorem (10.A.1.4) implies that (2)
implies (1).
We fix such an F and we denote by A = kF the action of a on Fn.
Example. Let G act on А‘й by A Ad. Let p = dim n. Let F be the
G-cyclic space of Ap n ® Ap n in Ap й ® Лр й- Then Fn = Ap n ® Ap n.
Since °M acts on Apn by a real valued character it must have order at
most 2 (°M has no non-trivial characters with values in (0, <»)). So it acts
trivially on Fn. This implies that F satisfies the above assumptions with
Af = 4p.
The purpose of this action is to prove the following generalization of
Theorem 10.1.5.
10.2.2.	Theorem. There exist polynomials ba A and Da A with values in C
and U(q)k, respectively, with ba A * 0 and such that
(1)	^<z,a(p)^?|p(p) = Jp\p(v + ^)7Гр,<г,и-л(^<г,л(р))/
for f g /“ and Re(p, a) > ca for a g Ф(Р, A).
In light of the example in 10.2.1 this theorem implies Theorem 10.1.5.
We now begin the proof.
10.2.3.	We first observe that both sides of 10.2.2 (1) are continuous in f
in the indicated range of the v parameter. Thus, if we prove (1) for f right
/^-finite then the result will follow by continuity. Let Ia be the space of all
right tf-finite elements of Let, as usual, IPt„tV (resp., IPt(TtV} be the
(й, ЛЭ-module Ia with the action corresponding to тгР (resp. irP a v).
Fix f)0 a Cartan subalgebra of °mc then t)0 Ф ac = t) is a Cartan
subalgebra of gc- We will use (resp., t)*) to parameterize the infinites-
imal characters for й (resp., °m) via the Harish-Chandra isomorphism.
Thus, if a has infinitesimal character MXx then IPt„tV and IPt<TtV have
infinitesimal character Xx + v •
18
10. Intertwining Operators
10.2.4.	Let v0 g Fn be fixed and non-zero. Let v{,..., vd be a basis of F
and let v*,...,v* be the dual basis of F*. Define a,(g) = f*(ju(g)_1fo)
for i = 1,..., d. Set b, = a,.r. If n g N, m g°A7, a g A, then
afnmag) = a~Aat(g), g G G.
It is also obvious that
M(g)"4 = 'Lai(g)vi for gGG.
Define a linear map T from to /” ® F by
r/ =
i
Lemma. T is injective and T(ttp a v+A(g')f) = (TTPcri,(g) ® n(g))Tf for
/g/“. Furthermore, spanr*eF*(/ ® о*\Т(1^У) =
If Tf = 0, then bjf = 0 for all i. Thus, in particular, E \bfk)\2f(k) = 0
for all к & К. But E |b,(£)|2 > 0 for all к g K.
Clearly,
^p,a,v+>kg)fW = a(kg)ATrPa:V(g)f(k)-
Also,
n(kg)~lv0 = a(kg)~An(k(kg))~lv0.
This implies that
«(*g)AEb,(^)y, = ^(g)'Ebi(k(kg))vi-
Thus,
EM>,<7,,+a(s)/® = Ebi(A:(-g))7rp (7 p(g)/'® M(g)f,
= Y^p,<r,Xg)bif® F(g)Vj
= (^P,a,Xg) ® F(g))Tf.
We also note that (I ® r*XF/) = btf. Let w* g (F*)k be such that
tv*(w0) = 1 (such an element exists by the Cartan-Helgason theorem). Let
w* = Ec,t>*. Then (I ® w*\Tf) = Lc^f. But
Hcibi(k) = w*(n(k)'lv0) = n*(k)w*(v0) = w*(v0) = 1.
Thus, (I ® w*\Tf) = f.
10 J. The Proof of Theorem 10.1.5
19
10.2.5.	Let v* g (F*)n be such that v*(v0) = 1 (clearly such an element
exists). We define a linear map 5 from /" ® F to /“ by
S(f®v)(k) = v*(n(k)v)f(k).
If /g then we set pf^nmak) = av~p(r(m)f(k) for n g N, a g Л,
m e°M, к g K. If g = nmak for n g N, tn G°Af, a g A, and к g K,
then we set ap(g) = a, mp(g) = tn, kp(g) = k.
Lemma. 5 is surjective, S((ttp a „(g) ® fi(g))(f ® r)) = irp,a,v+A(g) X
S(f ® v) for f g /“ and v g F. Furthermore, if S(f ® v) = 0 for all
v g F, then f = 0.
Let w g FK be such that r*(w) = 1. Then S(f ® w) = f. So 5 is
surjective. Clearly, this also implies the last assertion.
We now prove the intertwining assertion:
S(Trp,a,Xs)f^ n(g)v)(k) = v*(fi(kg)v)pfv(kg)
= ap(kg)Av%(p.(k(kg))v)pf,,(kg)
= pS(f® v)v+A(kg)
= irp,„MS(f®v)(k).
10.2.6.	Lemma. If Re(r>, a) > ca, a g Ф(Р, A), then the following dia-
gram is commutative:
Jpht -₽(p)® /
/ “ ® f ——>	® f
r|	|S
Jbi/>(p+ Л)
TOO	1	TOO
The obvious calculation yields
(1)	5((/Р|Р(р)®/)(Т(/)))(Л)
= I ^aj(hk)a(hk)AfXhkyv^^kyVi) dn.
JN
Now, г*(д.(п)х) = v*(x) for x g F and n g N. And
= v^fi(nk)fi(nkylv0) = 1.
Thus, the right hand side of (1) is
f a(nk)Afv(nk) dh = (_fv+A(nk) dh = Jp\P(v + k)f(k).
JN	JN
This is the content of the lemma.
20
10. Intertwining Operators
10.2.7.	If x is a homomorphism of Z(gc) to C and if И is a g-module,
then we set Vх = {w g И| (z -	= 0 for all z g Z(gc) and some
k}. Then
Ip,<,.„	= © (/pi<T.,®P)*.
X
Let P,(p) be the corresponding projection of Ipt<TtV ® F onto
(IP a v ® Р)*л+а+«'. Similarly, let Qfv) denote the projection of Ipt<rtV ® F
onto (Ipt(T,v ® р)*л+А+*'. We look upon P^v) and Q^v) as linear maps of
Ia ® F into itself. Note that T and 5 have no dependence on v but,
clearly, P,(p) and Qfv) depend on v.
Lemma. There exists a non-zero complex polynomial <p on a£ such that if
<p(v) + 0 then T: Ia-> Pfy\Ia ® P) and 5: Qfy\Ia ® F) -> Ia are lin-
ear bijections.
We will use the notation of 10.A.1.1. Let
Р, d F2 d • • • d Fr d pr+1 = (0)
be a Jordan-Holder series for P as a P-module. We assume (as we may)
that Fr is the one dimensional P-module, with °MN acting trivially and A
acting by A. If V is a (°m, К П Af)-module and if p g а£, then we denote
by Vv the (p, К П M)-module V with n acting trivially, °m acting as it did
on V, and a acting by v. Then, each P,/P, + 1 is of the form with Vt
an irreducible finite dimensional (°m,P n M)-module. Thus (10.A.1.7),
IP a v® F has a composition series
IP „ „ ® P =	 эМгэ Mr+l = (0)
with
Now, each (На)кпм ® V\ has a Jordan-Holder series with intermediate
quotients of the form V^. Thus, IPt<TtV ® P has a composition series with
intermediate quotients IPtV..tV.+v. Here, p, is a weight of the action of а
on P and if p, = A, then i = r, there is only one j, j = 1, and =
(Ha)KnM. Let A,y be the infinitesimal character parameter for Vtj. Then
Хло+У(+ЛС) - Ал+л+ДС)
= (Ai;, Л(7) - (Л, Л) + (v + vt, v + vt) - (v + A, v + A)
= (Л0»Ли) “ (Л’Л) +	~ (A>A) + 2(p’pi - A) =
10.2. The Proof of Theorem 10.1.5
21
The preceding implies that =# 0 for i < r. Set
V = П <Рц
i<r,j
If <p(v) Ф 0, then
{IP<a<v ® F)Xr'+,'+>'= IP a v+A.
Similarly, if <p(v) =# 0 then
(h,a,.®F)x^ = IP^+A.
Since T is injective and 5 is surjective the lemma now follows.
10.2.8.	Let «5: F ® F* -> C be defined by 8{v ® v*) = v*(v). We define
U: I„®F*^ Ia
by U = (I® 8)°(T® /).
We define
V: Ia^Ia® F*
by И(/) = Е5(/®ц)®^.
Lemma. U is surjective and V is injective. Furthermore, U is a (q, K)-mod-
ule homomorphism of IP a v+A ® F* to IP <T P and V is a (й, K)-module
homomorphism of IP a v to IPt<T,v+A ® F*.
We note that U(f ® v*) = (I ® 8\T(f) ® v*) = (I ® v*\T(f)). Since
spant*eF*(/ ® t>*)(T(/o.)) = Ia,
(Lemma 10.2.4) we see that U is surjective. If V(f) = 0, then Y,S(f ® vf)
® v* = 0. Thus, S(f ® v) = 0 for all v g F. Hence, Lemma 10.2.5 implies
that f = 0. The intertwining assertions follow from the corresponding
intertwining assertions for T and S.
10.2.9.	Let P2(v) be the projection of 1р,<г,?+л® F* onto (IPav+A
®F*y^+“ and let Q2(v) be the projection of IPt<TtV+A ® F* onto (fp>(7>y+A
)*л+".
Lemma. There exists a non-zero complex valued polynomial <рг on a£ such
that if <pfv) =# 0 then U: P2^v\IPt<rv+A ® F*) -»IPt<TtV and V: IPt<TtV -♦
Q2^\IP atV+A ® F*) are bijective.
This is proved in precisely the same way as Lemma 10.2.7.
22
10. Intertwining Operators
10.2.10.
phisms:
(1)
At this point we have the following diagram of (g, A)-homomor-
Jpip(p + A)®/
P2(Ia ® F*) —----------> Q2Ua ® F*)
"!	к
Our next task is to see how far (1) is from a commutative diagram. In
order to do this we will use a result in 10.A.1.1.
10.2.11.	Let x = Xx for A g t)* be a homomorphism of Z(gc) to
C with its given Harish-Chandra parameter. Let = f7(gc)ker^A-
Let ^A = be as in 10.A.1.5. Let PA be the projection of
X ® F onto (X ® F)*A+A for X g ^A. We define, for X g ^A, £x(x) =
{I ® <5)E, PA(m ® r() ® v*. Then, £ is a natural transformation of the
identity functor in the category ^A. Thus, Lemma 10.A.1.5 implies that
there exists zA g Z(qc) such that gx acts on by zA. Hence, by the
scalar Xx(za)- We set y(p) = Xa+v(za+v). We note that y(v) is a compli-
cated function of v. We will therefore make no general assertions about it
at this point.
We now study the diagram 10.2.10 (1). Let x = P2(vXf ® v*) for
and v* g F*. Set J{v} = J^P{vY Then,
(J(p + A) ®/)(x) = (J(p + A) ®Z)(F2(p)(/® y*))
= QAv)(J(v + X)f®v*).
If cp^v) Ф 0, then Q2(v\J(v + A)/® v*) = V(g) for some g g Ipa v.
Lemma 10.2.6 implies that
J(p + A)(/)=5(J(p)®Z)(T(/)).
Thus,
K(g) = 02(p)(5(J(p)®Z)(T(/)®/;*)).
We write T(/) = E, T//) ® r(. Let P3(v) be the projection of
p№lP,a,v ®F)®F* onto	® F) ® F*V^. Then,
V(g) = (S ® /)(J(p) ® /) EP3(p)(Fi(p)(7;(/) ®	® v*)-
i
Now, assume that <p(v) =# 0. Then,
Fi(v)(Ip,a,v ® F) =	
10.2. The Proof of Theorem 10.1.5
23
Since <?|(p) =# 0, we see that
®F^ = IptVtV.
We next assume that y(v) * 0. Then, £lp ° is bijective since it equals
y{v)I. We note that
= C1 ® 5)(^Л(^)(и ® ",) ® ",*)
= (1 ® а)^Р3(р)(Р!(р)(м ® vt) ® r*)j.
Hence,
^з(’?)(Л(’?)(/р.а,р ® F) ®F*) = (Ел(")(« ® Vi) ® ц!*|и e ZF>(7 Л.
' i	'
So, if y(v) # 0 then there exists f' e IP a „ such that
W) ® ",) ® И =	®	® v*.
i	i
The preceding observations imply that
y(V)f' = (I® 8)l^P3^)(PM(Ti(f) ®	® r*))
i
= (/®а)(^л(^)(7;(/)®^)®^)-
i
But LP^Titf) ® v^ = Pi(T(f)) = T(f) = UTSf) ® Vi. Thus,
y(v)f' = (I® 8)(X(Ti(f) ® v^ ® r*) = (/ ® r*)(T(/)).
We therefore see that
y(v)V(g) = (5 ® I)(J(v) ® I)(T(f))
= И(/® r*)(J(p) ®I)(T(f)).
Hence, y(v)g = (I ® v*\J(v) ® I\T(f)). So,
y(v)(J(v + A) ® I)(P2(v)(f® v*))
= VJ(v)U(f®v*) = VJ(v)UP2(v)(f ® v*).
We have thus proved:
24
10. Intertwining Operators
Lemma. If <p(v) # 0, tp^v) # 0, y(v) # 0 and if Re(p, a) > ca for
a g Ф(Р, A), then the following diagram is commutative:
yMJpiplv + Л)®/
P2(Ia ® F*) -----!------> Q2(Ia ® F*)
"!	к
Jp^pM
10.2.12.	We must now study the function y(v). We assume that A =# 0, so
dim F > 2. We will show that there exists a non-zero rational function
/3(r>) of v that agrees with y(v) if <p(p) =# 0. For this we must analyze the
natural transformation for X^jFx+v in more detail. Proposition
6.A.3.13 implies that X ® F is a direct sum of g-modules with generalized
infinitesimal characters of the form Л'л+к+м with M a weight of F relative
to t), and that if r(p) is the multiplicity of the weight p in F then
(2-AA + ,+M(2))r<M>(^®F)ZA—= 0.
If <p(v) =# 0, then we have seen that if p is a weight of F and if p + A
then Ал+х+/С) * Ал+х+л(С)- Let Л/л + „ =	Then, if 11(F)
is the set of weights of F with respect to t),
Mx+v®F = © (Ma + p®F/a+^.
д. ^n(F)
Set
П (C-Xa....(C))w,
Men(F)-{A)
Set zv = Xs+v+^ti^^u,,. Then, the projection of Mx+v®F onto
(Mx+v ® f)*A+^+A is given by the action of zv on Mx+v ® F. Thus,
Here, x is the projection of x g L7(gc) into Mx+v. Hence, Lemma
10.A.1.5 implies that z„ = QF(zv) with QF defined as in 10.A.1.1. Set
j3(v) -	=	‘Pt’') * 0-
The following result is critical to our discussion.
Lemma, fl is a not identically 0.
Let Ф = Ф(йс, Ю and let Ф+ be a system of positive roots of Ф
compatible with P. Let 8 denote the half sum of the elements of Ф+. Let
10.2. The Proof of Theorem 10.1.5
25
Л/(Л + v) be the Verma module with highest weight A + v - 3 relative to
Ф+. We choose vx,..., vd such that hvt = Hi(.h)Vi for h e t) and such that
= A. We set k(v) = X\+Suv)^v)- Let £„ denote 1 ® 1 in M(A + v)
and let £* e Af(A + p)* be defined by £*(£„) - 1 and £*(t>) = 0 if v is a
weight vector in Af(A + v) for a weight other than A + v - 8. Then
Ф) = E(C ® ц*)(П(с -хл+,+ДС))(^ ® r,)V
,	4>1	>
Let Xa e (gC)a be chosen such that B(Xa, X_a) = 1. Fix Hv.. .,Ht an
orthonormal basis of t). Set L -	and T - 2Х,а^ф+Х_аХа.
Then [L, T] = 0. Hence
П (C - XA+,+JC))(^ ® vt) = n (T + <A,y(p))(t ® Vi)
J>1	J>1
with ^(v) = Хл+„+^С) - Ал+к+м/О- И ° к is the ^-th elementary
symmetric function in d - 1 variables then the right hand side of the
above equation is given by
d-1
E <rd-i-k(il>i2(v),---,>l>id(v))Tk(^ ® v^.
k = 0
We note that
(2)	If к > 0 then deg,((£* ® vf\Tk{^v ® r,)) < k.
Indeed, Tk{^v ® v,) is a C-linear combination of terms of the form
X-aXai • • • X_aXa^tv ® Vi) - X_aXai    X_ak(£lv ® XakVi)
with aj e Ф+. Thus (£* ® y?XT*(£„ ® f,) is a C linear combination
of terms of the form £*(ХШ1 • • • Хш ^,Jv*(XTi • • • ХТ2к_ц) with w,,
т, e Ф, r < 2k - 2, and w1 + • • • +wr = 0. It is easily checked that
deg, £*(*<„, • • • Хш v) < [r/2] < к for such wp..., wr. This proves (2).
We note that deg, |Д,7(1г>) < 1. So, if к > 0 then
deg, trd-t-k^M, • • • ^iAv))Tk^v ® Vi) < d - 1 - к + к = d - 1.
This implies that
x(fp) =	‘ ‘ ‘ <M^) + ^(t)
i
with deg, h(t) < d - 1. Now <Д„ = 0. And
M'") ••• tid(tv) = td~l - M,) +g(0
i = 2
with deg, g(t) < d - 1. This implies that k(v) is not identically 0 and
hence that Д(р) is not identically 0.
26
10. Intertwining Operators
We note that if <p(v) =# 0 then the commutative diagram in the state-
ment of Lemma 10.2.11 is the same as the commutative diagram

F2(/a®F*)
"I
I,
Q2{Ia ® F*)
|и
Jp^pM
We set
{p g a£|Re(p, a) > ca, a g Ф(Р, Л)}.
If <p(v) # 0 then Lemma 10.2.9 implies that U and V are bijective linear
transformations. The above commutative diagram therefore implies that if
v g fl is such that <p(v) =# 0 and j3(r>) = 0 then Jp^v) = 0. This implies
that if j3(v) = 0, <p(v) =# 0 and v g fl then we would have a contradiction
to the last assertion of Lemma 10.1.2. We have therefore proved the
stronger result that Д(р) =# 0 for all v g fl such that <p(v) =# 0. The
lemma now follows.
10.2.13.	Let p be the rational function in the preceding number. If we
combine the two preceding lemmas, we have:
Proposition. If <p(v)<pfv) Ф 0 and if Re(r>, a) > ca, then the following
diagram is commutative:
J3(p)Jbid(p + a)®/
P2(4®F*) ----------!-------> Q2(Ia®F*)
И	1K
Jp\p(v)
*	Ar
Here, p a non-zero rational function of v defined for all v with <p(v)<pfv)
* 0.
10.2.14.	We are now ready to prove Theorem 10.2.2. We first invert U
and V. Let, as before, w g Fk and w* g (F*)k be such that и'*(г0) = 1
and r*(w) = 1 (this can be done in light of the Cartan-Helgason theo-
rem). Set X„(f) = P2^fp+A ® ”'*) f°r f e f?- Recall that P2 is the projec-
tion of fp><TiP+A ® F* onto (fpt<7>y+A ® F*)*A+p. We define, for f g Ia and
v* G F*, R(f ® v*) = v*(w)f. ’
(1)	If f g Ia then UX„( f)=f and RV{ f) = f.
10.2. The Proof of Theorem 10.1.5
27
Indeed,
UX„(f) = t/(P2(/®w*)) = C/(/®w*) = (I ® 8)(T(f) ®w*)
= (/®w*)(7’(/)) = En'*(r,)V-
But
L^^b^k) = ^w^(v,)i^(^(k)~lv0)
= w^utky'vo) = w*(v0) = 1.
Also, V{f) = Ег*(т,)/® v*. (Here, v*(-v) is the function whose value
on к e К is v*(ii(k)v}) Thus, RV(f\k) = ’Lv*(w)v*(p.(k)vi)f(k) =
v*(fi(k)w)f(k) = v*(w)f(k) = f(k).
In light of (1), if <p(v) Ф 0 and if (p^v) Ф 0, then
Xv = (.U\P2(^lp.a., + ^F*))
and
RlQ2^lp.<r.. + ^F*) =VX-
Here, V is looked upon as a mapping into QiivYJp a v+K ® F*).
Thus, if <f^v) Ф 0 and if <px(v) Ф 0, then
(*)	Jpip(r)f = P(V)R(JplP(V + X)
We now expand the right hand side of (*). As before, we set
„ _ j-r C ~ Лл + р + л-ДО
£еП(Г) Ал + ЛО — Ал + к + Л-ДО
If <p(v) and tp^v) are non-zero, then P2 is given by (тгР a ,„+л ® д.Хг„).
Thus, if A(z„) = Ex,(v) ® y,(v) (see 10.A.1.1 for Д), then we have
Jp\p<v)f = Д(^) Е(м*(У,(^))и'*(и’))/?|Р(р + Л)7гл<71,+л(х,(р)/).
i
Let w'*(w) = i. Then, in the notation of 10.A.1.5,
^(M*(y,(v))w*(w))x,(v) = LqF»(zvY
i
Hence, we have shown that if (pMtp^v) Ф 0, then
( * *) Jp\p(. v)f = j3(v)iJ?lp(v + A)7rPiO.iP+A(9F*(zJ)/.
28
10. Intertwining Operators
In light of 10.A.1.5, there exists a non-zero polynomial on a£, ba ^v),
such that if we set Da = ba A(v)i@(v)qF*(zv) then Da A is a polyno-
mial on a£ with values in t/(gc)K. Hence, (* *) implies that
ba,MJpip(v)f = JP\p(v + ^p,«,v+x(D«,x(v))f-
This completes the proof of Theorem 10.2.2.
10.2.15.	We note that ba A and Da A are given in a constructive manner.
Thus, in principle, they are computable.
10.3. Limit formulas
10.3.1.	This section is devoted to some results on the behavior of Jp^tv)
for v large. These results will be used in the next section in our proof of a
determinant formula for the restriction of Jp^tv) to an isotypic compo-
nent. We retain the notation of the previous section. Let A be a special
vector subgroup of G. Let P g 0(A) and let N = Np. If vea*, then we
set
cP(v) = fa(nY+p dh.
JN
Here, we use dN (10.1.7) for dh. If P is understood we will write c(v)
rather than cP(r>). Lemma 10.1.10 implies that if Re(v, a) > (p, a) for
a g Ф(Р, A), then the integral defining c(v) converges absolutely and
uniformly on compacta in the indicated range. Theorem 10.1.6 implies that
c has a meromorphic continuation to a£.
10.3.2.	Lemma. Let p = dim n. There exists a constant C > 0 such that
if v g a* then if (v, a) > (p, a) for all a g Ф(Р, A), then
cp(p) > C(1 + Ill'll) P.
The Schwarz inequality implies that (v, a) < ||a|| Ill'll. Set
q = max ||a||/( p, a ).
«еф(/1, A~)
Then, (v, a) < tflMKp, a). Hence, 3.A.2.3 implies that
c(v) = [_а(пУ+р dh > fa(h)i<lM+v>p dh.
Jn	JN
10.3. Limit Formulas
29
It is therefore enough to prove the lemma for v = xp with x > 1. Set
<p(x) = f_a(n)XP dn
for x > 1. Set F = spanc{Ap Ad(g) Ap n|g g G} and set p.(g) =
Ap Ad(g)|F. Then, (p,,F) is a representation of G. Let (X,Y) =
-B(X, 0Y) for X, Y g g and extend < , > to a Hermitian inner product
on gc. On F we use the restriction of the inner product on Лрйс
corresponding to ( , ), which we also denote by ( , ). Then (p.(g)e, f)
= (e, p.(0(g))-1/> for all g g G, e, f g F. Let || •  • || denote the corre-
sponding norm on F. Let v g Fn be a unit vector. Then
=11м(й)-М-
If X g n, then (see 4.A.2.4 (1))
||M(exp A)< = 1 + Q(X),
with Q a polynomial in X without constant term. Thus, there exist
constants C\ and r such that
i + 0(*) <(1 + ^11*11)'.
Let Xl,...,Xp be a basis of n such that (Xi; X^ = 8^, with (X,Y) =
-B(X,0Y). Then X = ExjXi; and we have
1 + Q(X)< n(i + c1W)r.
Now,
r	-	/
<p(x) = C2/(l + Q(X)) xdX>cA (1 + Q) rxdt\ .
\ Jo	I
The lemma now follows if we directly evaluate the last integral.
10.3.3. For each e > 0 we set (a*)e+= {p g a*| (v,a) > б||р||(р, a), a g
Ф(Р, Л)}. If <p is a function on a*, then we write
lim <p(v) = L
p —» 00
p
if
lim <p(v) = L
IMI-.00
i»g(a*)e+
for all c > 0.
30
10. Intertwining Operators
The results that follow are extensions similar ones in Harish-Chandra
[15], p. 47.
Let ||    || be a norm on G (2.A.2.3).
Lemma. Let <p be a measurable function on N, continuous at 1, such that
there exist C* and s > 0 such that |<p(n) I < Cj|n||i for h g N. Then
r “№V+P
i™ f_ r t \ ^n>> dn = i^1)-
V-,COJN Cp(v)
p
Set Nr = {« g N\a(n)~p < 1 + r}. Then, Nr is compact for each r > 0
and ПГ>ОАГ = {1} (4.A.2.3). If n g N - Nr, then a(n)p < (1 + r)-1.
3.A.2.3 implies that if p g (a*)E+ then
a(nY/2 <а(пУмр/2.
Thus, if v g (a*)e+ , n g N - Nr, then
a(n)1' = а(пУ/2а(пУ/2 < a(hy/2(l + r)~eM/2.
The preceding lemma now implies that if v g (a*)f , n g N - Nr, then
< C-'(l + ||p||)₽(1 + гуеМ/2а(пУ/2+р.
Thus, if /(«) = <p(n) - <p(l) then
< 2C C-1(l + ||p||)p(1 + r)-W f _a(ny/2+p^s dn.
JN-Nr
4.A.2.3 implies that ||n||’s <	for some A g a*. If ||p|| >
2(1 + ^||А||)/б (q is as in 10.3.2), then (p/2 - A - p, a) > 0 for
a g Ф(Р, A\ So
I^2C C-'(l + ||p||)p(1 +r)-eM/2f _a(h)2pdn
JN-Nr
<Const(l + M)p(l+r)-*M/2
Thus, if 3 > 0 is given and if r > 0 is fixed, then there exists R(r) such
10.3. Limit Formulas
31
that if ||p|| > R(r) then I < 8/2. Fix r > 0 so small that
|<p(n) - <p(l) | < 3/2, n^Nr.
If ||p|| > R(r), then
r <n)V+P
I- r ( л <p(n) dn - <p(l)
,«(«)
f- e (v\	~
JN Cp(V)
Г «(«)
<	(---^~\<p(n) - <p(l)|dn
JNr Cp{V)
a(n)v+p
N-N,
f(n) dn
<	8/2 + 8/2 = 8.
10.3.4	. Corollary. If p. Ga£, then
Cp(v + n)
hm -------—— = 1.
p->00 Cp(l')
p
We have
cP(v + д.)	, а(пУ+р
Cp(v)	JN Cp(v)
with <p(n) = a(n/. <p satisfies the growth condition of the previous lemma
(see the proof of Lemma 10.3.2).
10.3.5	. Let (a, Ha) be as in 10.1.1.
Lemma. Iff& /" and if ц g a^., then
.. Jp\p(v +
hm ----------—-------
= f(k)
weakly in Ha.
Let w g Ha. Then
/ Jp\p(v + ^)fW \ r {fv+p(nk),w) dn
-------r-;-----, w = / ---------—-------
32
10. Intertwining Operators
Fix к К and set <p(n) = а(пУw>. Then,
<p(n) dn.
Lemma 10.3.3 applies, and so
lim 7(p) = <p(l) ={f(k),w).
p —♦ 00
p
10.4.	A generalization of L. Cohn’s determinant formula
10.4.1.	In this section we will derive formulas for the determinants of the
intertwining operators on K-isotypic components. Before we begin the
main material of this section we study a refinement of Frobenius reciproc-
ity. Let К be a compact Lie group and let Kx d K2 be closed subgroups
of K. If т e К (resp., ц e k{), we fix VT e r (resp., g д,). Let И be a
finite dimensional (continuous) A2‘m°dule- Let 3 be the natural mapping
from
(*)	© НоткМ > П ® Hom4vr’^)
to НотК2(Ит, И) given by 8(T ® S) = TS.
Lemma. 8 is a linear bijection.
As a /^-module,
K= © K(M).
m <=k2
Here (as usual), Ут(ц) is the ^.-isotypic component of VT. By its very
definition, it follows that
dim Ит(д.) = dim НотК1(Ит, И^) dim .
It is therefore clear that the dimension of (*) is equal to dim НотК2(Ит, V).
Thus, in order to prove the lemma, it is enough to show that 3 is
surjective. To this end, let
К(м) = Ф^,м,
J
10.4. A Generalization of L. Cohn’s Determinant Formula
33
with Vj e д.. Let M be the projection of VT onto corresponding to
the direct sum decomposition К = Ф V,,,. Let Q, „ be a non-zero
element of
Hom4^,f< jGHom^.K)-
Let e НотК2(Ит, H^) be such that QJifiSiia = 8^^. S-exists
and is uniquely determined. If T e НотК2(Ит, И), then
T- T.TPI.,-’E(TQI.,)SI,,.
j. м	Л м
TQj e НотКг(И^, V) and e НотК2(Ит, И^). So 3 is indeed surjec-
tive.
10.4.2.	We now assume that И is a ^2-module, and that it is a direct sum
of irreducible continuous /f2-modules with finite multiplicities. We can
define 3 as in 10.4.1 (*) in this more general situation since the sum in (*)
is finite. As before, we have:
Lemma. 8 is a linear bijection.
Indeed, if r is fixed then there exists a finite dimensional /f2-submodule
V of V such that Нот^2(Ит, V') = Нот^2(Ит, И) and such that
Horn^/W;, V) ® НотК1(ИТ, W^) = Horn^fW;, Г) ® НотК1(ИТ, И;)
(take the sum of the /^-isotypic components of V that correspond to
K^-types that occur in FT). The result now follows from the preceding
lemma applied to V'.
10.4.3.	We now return to the notation of Section 10.1. Let A be a special
vector subgroup of G (10.1.8). We set °MA = °MP for any P e 0(A)
(10.1.8) and KA = °MA П K. Let (a, Ha) be an admissible Hilbert repre-
sentation of °MA and let Ia be the space of all К-finite vectors of /“. For
each у e К we choose once and for all (ту,Уу) g y. Let Ia(y) be the
y-isotypic component of Ia.We look upon
нот^и,,(«,)«,) er,
as a /(-module with К acting only on the second factor. If T g
HomK(Vy,(Ha)KA) and v g Vy, then set fTv(k) = T{ry{k}v} for к g K.
Then, clearly, fTv g Ia(y).
34
10. Intertwining Operators
Lemma. The linear map
T ® v •-» fT'V
is a К-module isomorphism of
onto Ia(y\
This is a reformulation of Frobenius reciprocity.
10.4.4.	Suppose that Pl,P2& 0(A). We calculate Jp2^p(.v)fT l in the
range of absolute convergence of the corresponding integral. If g e G,
keK, then
gk = nPt(g)aPi(g)mPi(g)kPi(g)k,
with the usual ambiguity. We will use this equation in the course of our
calculation, (p = pPi, r = r7.) We have
Jp2\pl(v)fT,v(k) = L	(fr,v)A^k) dh
JNPiONp2
= /_	aPi(hy+pa(mPl,h))T(T(kPi(h))T(k)v)dh.
Np, П NP2
We isolate a part of the last expression. If v is as before and if T g
Hom^fKy, Ha), then set
(1)	ap2\p^>t>^)T= f aP(hy+p<T(mP(h))TT(kP(h))dh.
JNPlONP2
Notice that this integration is actually taking place in a finite dimensional
space. Then one has
(2)	JP2\Pl( V^T, v = fAP^Pl(a , t, v)T, v *
10.4.5.	In this section, we will study the meromorphic functions
(1)	Cp2|Pi(o-,y,p) = det(jp2|Pi(p)|/(r(7)).
10.4.4 (2) implies that if т = t7 then
(2)	Cp2|Pi(o-,y,p) = detpF2|Fi(0-,T,p))‘/<1'>,
with d(y) = dim Vy, as usual.
We use this formula to analyze the case when P{ and P2 are adjacent
(10.1.9) with Х(Р21Л) = We use the notation of 10.1.13. Let P be as
10.4. A Generalization of L. Cohn’s Determinant Formula
35
in that number and set = К П P, K2 = К П P{ (= К О P2). Let 3 be
the map
© Нот^Ир (Я<т)к)®НотК1(Ит,	НотК2(Ит,(Н<,)к)>
defined as in 10.4.2. The formula 10.1.13 (1) translates into
лр2|р1(°’>т>’')5(7’ ® s) =	® 5)
for Те Hom^/Wp(Ha}K\ S e НотК1(Ит,Ир.
It is convenient to use the notation Qa for and Ka for Kx. We also
write
(у:м) = dim Hom Ki( Ит, И;)
(recall т = t7) and (p: cr) for dimHompwpfHPp. Then, we have
detAP^a,y,v) = П	M)-
This formula combined with (2), Lemma 10.1.12, and Lemma 10.1.9 gives:
Proposition. Let Plf P2 e A). Then
Cp2\P^,y,v) = П П det(^|Ga(o-,M,“p))<7 M).
10.4.6.	The preceding result reduces the calculation of the functions
Cp2|Pi to the calculation of det( А^0(а, p, p)) for Q e A) and dim A =
1. Let (Q, A) satisfy this condition and let {a} = 2(0)-
Theorem. There exist a/cr),..., ar(a) g C and b/cr, p),..., bs(a, p) g
C, with r — ria) depending only on a and s = rip: a), such that
,	,	nyi)r(((p,a)/4(p0,a)) - д,(о~))(М g)
et( &Q(a,p,p))	n;(=<T1x'i:<T)r(((p,a)/4(pG,a)) -b/o-.p))'
Here, Г is the classical gamma function.
The proof of this generalization of a theorem of L. Cohn is involved and
will take up the rest of this section. Before we begin the proof we first give
a corollary.
10.4.7.	Corollary. Let A be a special vector subgroup of G and let
Pi,P2e 0>(A) and let a be as before. For a g 2(P2|Pj) = S let Qa be as
36
10. Intertwining Operators
before and set pa = pQa. Then there exist for each a g I complex numbers
aaj(a'), i = 1,..., r(a, a), depending only on a and for each у g К such
that (y : a) =# 0 there exist complex numbers ba fxr, y),i = 1,..., r(a, a) X
(p.: a) such that
Cp2\p^,y,v)
= п п ПУГ)Г(((р,а)/4(Ро,а))-а,..а(а))(м:<гХ7:м)
This is an immediate consequence of Proposition 10.4.5 and Theorem
10.4.6.
10.4.8.	We define a representation of t/(°m) ® 17(1) on Нотс(Ит,
by (m ® k)T = a(m)Tr(kT). If к g К П °M, m®ue
iU(°m) ® 17(f), then we set k(m ® u) = Ad(£) m ® Ad(£) u. If
и g ((7(°m) ® U(t))Kn °M and if T g HomKn »м(Ут,(На)Кп oM), then uT
g HomKnoM(VT,(Ha)KnoM\
Consider the linear map
Ф- 17(nc) ® t/(°mc) ® U(ac) ® U(fc) Цйс)
given by ф(п ® m ® a ® k) = nmak. Then P-B-W (0.4.1) implies that ф
is surjective. If T g HomKnaM(VT,(Ha)Kn oM) we consider rrp(g)/rr for
V g VT and g G (7(йс)К- Also, TrJig)fT v(k) = R(.g\fT,v)v(.k) =
L{gT\fTv)v{k). If и = nmak, then L(gT) = L(kT)L(aT)L(mT)L(nT).
Here, gT is defined (as usual) as the involutive antiautomorphism of
17(йс) with lr = 1 and XT = -X for X g йс. The following assertions
are obvious.
(1)	If n g 17(nc), then L(nT)(.(fTflfAk) = e(n)fT v(k). Here, e is de-
fined (as usual) as the homomorphism of 17(йс) to C with e(l) = 1 and
е(й«7(й)) = 0.
(2)	If a g U(ac\ then L(aT\(fT ДХЛ) = (p + p\a)fT v(k).
(3)	If к g 17(fc), m g t/(°mc), then UkT)L{mT\{fT v)v\k) =
If g = E«A(n( ® mt: ® ak ® k,), then we set
QAs) =	+ p)(aTk)e(ni)mi ® kt.
10.4. A Generalization of L. Cohn’s Determinant Formula
37
We have
Q.-U(Qc)K^ (Ц°тп) ®
and we have proved that
irAg)fr,v =fQv(g)T.v for g e ЦйсЛ
We note that it is also clear that if g e (7;(йс)к then v •-» Qv(g) is a
polynomial map of degree j from a£ into (t/(°m) ® t7(t))Kn °M. Theorem
10.1.5 now implies:
Theorem. If T e HomKn oM(VT,(Ha)Kn oM) then
ba(v)A(<r,r,v)T = A(o-,t,v + 4p)Qv+4p(Da(v))T.
10.4.9.	Set aa T(z) = det(A(a-,r,4zp)) and
da,r(z) = det(04(z+l)p(f)<7(42P))|HoInKnllM(KT,(H„)KnoM))-
Then,
b<T(4zp)(T:<T)a<TT(z) =«<z,t(2 + !Xt(z)-
Now, ba(4zp) and da T(z) are polynomials in z. They can therefore be
written as follows:
M4zp) = иаГ1(г - a,(cr)),
i = l
da,r(Z)=Va,r П (Z - Й,(О-,Т)).
> = 1
Hence,
aa.r(Z+ 0

Lemma. va T = s(a,r) = г(сгХт : a).
Lemma 10.3.5 implies that
1 = lim + 1)/с0(4хр)(т:а)
X —» + 00
= lim a<T T(x)/cG(4xp)(T <T).
x —» + 00
38
10. Intertwining Operators
Thus,
-- ^.ТПЙТ’(Х-М<7,7)) •
This clearly implies the lemma.
10.4.10.	Let
z 4 ч ПГ=:Г(<г)Г(г - Ь,(а,т))
’'",<2) “’-т<2) пстг(2-«,(»-)),т“'> '
Then ya T(z + 1) = yatT(z), where both sides are defined. Theorem 10.4.6
will follow if we can show that ya T is constant.
Let Wj, w2 be such that yaiT(w;) is defined for i = 1,2 and ya T(w2) # 0.
Then
V<r,r(W2)
+ Л)
lim -----------—-
(7:<z)r(<7)	+ W1 _ b^ff, t))
lim П
->+00	, = 1
T(fc + w2 - Ь((ат))
'<£) / T(fc + w2 - д,(о-)) j T:a a<,,T(k + w1)c0(4^p)
,Ц \ T(fc + w2 - a,(o-)) ) ce(4kp)aa T(k + w2) '
Sterling’s formula (cf. Whittaker-Watson [1]) implies that, as к -» +<»,
Г(х + к - a) e (x+k a\x + к - a)x+ ° '
V(x + к - b) e-^k~b\x + Л - ьу+к-ь~х/2
hh (1 + (x - a)/k)x+k a l/2
-a-bnb-a 2.___':_____ ’____________
(1 + (х_й)Д)-^-^
~ ea-bkb-a	(l + (x-q)A)*
(1 + (x - b)A)^b’1/2 (1 + (x - b)/k)k
~ ea~bkb~aex~aeb~x ~ kb~a.
Thus, the previous limit is equal to
lim (Ли'1_и'2)'’(<тХт:<т)(Ли'2-и'1)г<<тХт:<т) = j
We therefore conclude that = y(w2). The theorem now follows.
10.5. The Harish-Chandra p-Function
39
10.5.	The Harish-Chandra ji-function
10.5.1.	In this section, we introduce a generalization of the Harish-
Chandra fi-function and derive several properties of it that will be basic to
our proof of Harish-Chandra’s Plancherel formula. We retain the notation
of the previous sections. Let (P, A) be a p-pair. Let (a, Ha) be a Hilbert
representation of °M such that ||<r(/n)|| < C for m g °M (that is, a is
uniformly bounded). We begin this section with a simple (but more
general) version of 5.3.4.
If c > 0 is given then we set ал+= {H g a\a(H) > e||H|| for a g
Ф(Р, Л)}. If <p is a function on A, then we write
lim <p(a) = L
a —» 00
P
to mean that, for all e > 0,
lim <p(expH) = L on af.
ЦНЦ-.00
Set p = pP.
Lemma. If v g а£ is such that Re(p, a) > (pP,a) for a g Ф(Р, A) and
iff, g^Ipta,v, then
lim ap-v{TTPta v(a)f’g) = ( Jp\p(v)f(l), g(l))•
a —♦ 00
P
In the proof of 5.3.4, we saw that if a g A then
= av~p у a(ana~l)P * (о-(т(апа~1У) 1 f(n), g(k(ana~1)^dn.
If we write n = n(n)m(n)a(n)k(n) with n(n) g N, m(n) g °M, a(n) g A,
and k{n) g K, then this expression is equal to
av~p f_a(ana~l)P va(n)p+v
JN
X (a{m(ana~l) ltn(n)}f(k(n)), g(k(ana~l))^dn.
The result will therefore follow if we can justify the interchange of limit
and integration in the first formula. To do this we use the second formula.
40
10. Intertwining Operators
The integrand in the second formula is dominated by a constant multiple
of
a(ana~l)P RePa(n)p+Rei'= a(n)2p(a(ana-1) *a(n))	< a(n)2p,
by 3.A.2.3. Since the last term is integrable on N (Lemma 2.4.5) the
Lebesgue dominated convergence theorem gives the justification for the
interchange.
10.5.2.	Corollary. Assume that in addition to the preceding condition
{a, Ha) is irreducible. If Re(t>, a) > (p, a) for a e Ф(Р, A) and if f e
7^ 0. „ is such that Jp^p{v)f Ф 0, then f is a cyclic vector for тгРа<1,-
Let V denote the closure of span {ttp a p(g)/|g g G}. Set W = {g g
Ha’v\(u,g} = 0 for all и g И}. Then, W is a closed invariant subspace of
Ha'v under the conjugate dual representation тг* of ttp a v. Now тг* =
'1Гр,а*,-й (see the proof of 5.2.4). Hence, if W + 0 then W П IPt„tV =
W°° =# 0. Let g g W°°. Then for all m e °M, к e K,
0= lim ap-v{irP^v(amk)f’8) = {<r(m)JP]P(v)f(k), g(lf).
a —♦00
P
Thus, the irreducibility of a implies that g(l) = 0 for all g e 1У“. But
then 0 = 7r*(it)g(l) = g(k) for all к e K, g g W°°. Hence, W~ = 0, so
W = 0. This implies that V = H"' v as asserted.
10.5.3.	The next result asserts the generic irreducibility of the IPt<TtV-
Although it is quite crude, it suffices for the purposes of this section. In
Chapter 13, we will give a precise theorem on irreducibility in the case
where a is square integrable.
Lemma. Let Ka = {y e K\Ia(y) Ф 0}. There exists collection (fy Д,
of non-zero holomorphic polynomials on a £ such that if v e a£ and if
Д>) # 0 for all у, т e Ka, then IPt„tV is irreducible.
Let <%\K) be algebra of /(-finite smooth functions on К under convo-
lution. <№(K) is a f-bimodule under the left and right regular representa-
tion, and the elements of f act by derivations. If (т, И) is a (f, /Q-module,
then we define an action of <^{K) on V by
r(f)v = f f(k\r(k)vdk.
JK
10.5. The Harish-Chandra ц-Function
41
Let L denote the left regular action of К on ^(K). Set
U(Qc)
with the algebra structure given as follows: If
(Ad ® L)(f)(y ®g) = En;®g;,
J
then
(x ® f)(y ® g) =	® gj.
j
We note that if x = у = 1, then (1 ® /XI ® g) = 1 ® (f*g) (here, ★
denotes the usual convolution of functions). Let ttv = irP v v and ir =
'Trp,a,v\K> which is independent of P and v. Then a direct calculation
shows that IPav is an ^module under the action тгр(х ® /) = 7гр(х)тг(/).
IP a v is irreducible as a (g, КЭ-module if and only if it is irreducible as an
«^module. If Re(p, a) > (p, a) for a g Ф(Р, A) and if f g is such that
JPiP(v)f =# 0, then the previous result implies that тг/а^)/ = Ia.
If у K, then set ay = d(y)xy, with xy the character of у and d(y)
the degree of y. Then, Ia(y) = ir(ay)Ia and Ey = ir(ay) is the projection
of Ia onto its y-isotypic component relative to the direct sum decomposi-
tion Ia= //у). Fix у g Ka. Then Lemma 10.3.4 implies that
there exists v0 with Re(f0, a) > (p, a) for a g Ф(Р, A) such that
det(EyJP[P(v0)Ey) =# 0. Thus, we see that -rr^c^yf = for all / =# 0,
/ g Ia(y). We note that ay * ay = ay. Thus, (1 ® ауУ^(1 ® ay) = <^7
is a subalgebra of Ж and тг^а^7)/ = 1а(уУ for all f =# 0, / g Ia(y).
Hence, 7r„|)(<^’7) = Endf/^fy)) (cf. Lemma 10.A.3.1). Let Xlt..., Xd be a
set of elements of <^7 such that тг^Х^ = У1(..., Trv£Xd) = Yd is
a basis of Endf/^fy)). If v g а^, then тгДА',) = 'Ljaji{v)Yj, with atj a
holomorphic polynomial in v. Put fy<y(v) = det([a/,(i')]). Then, fyy(voy =
1. Obviously, if /7 7(p) =#= 0 then тгр(^7) = Endf/^fy)).
Now, let у, т e Ka. Fix v0 g a£ with Re(p0, a) > (p, a) for a G
Ф(Р, A) such that fyyy(v0) # 0, /t,t(f0) # 0, and det(EyJP)P(v0)Ey) =# 0.
If /g /а(у) and if / =# 0, then тгРо(ат<^’а7)/= /а(т). Since тт^^7) =
End(r(y)),	= Endfrfr)) and
rrj(l ® ат)^(1 ® а7)) D 77j^)7rJ(l ® ат)^(1 ® а7))^о(^7),
we see that
тгД(1 ® ат)^(1 ® a7)) = Homc(/-(y), /"(r)).
42
10. Intertwining Operators
We now proceed as before to define /T 7(p). Let
X{,...,Xd ^a^ay
be such that TtAXf) = Y, defines a basis of НотДГТу), Г(т)\ Then
j
with a}i a holomorphic polynomial in v. Put
fr.y(v) = det( [«/;( ")])•
Then, fTt7(v0) = 1. If fT,7(v) * 0, then
^(ат^а7) = Нотс(Г(г),7-(т)).
It is now clear that if fT y(v) =# 0 for all у, r g Ka and if f g Ia, then
= Ia.
10.5.4.	Lemma. There exists a meromorphic function <pP a on a*c such
that
Jp\p(v)Jp\p(v) = <Pp,„(v)I,
wherever the left hand side is defined.
We use the notation of the previous number. Let у g Ka. Let v be such
that f7t7(v) # 0. Then
7Гк(х) EyJp\?(v)Jp\p(v) = EyJp\p(v)Jp\p(v)Ey'jrXx)
for all x g Thus, Schur’s lemma implies that
E7Jpi?(p)Jpip(p)E7 = <py(v)Ey.
Since
dim(/<T(y))^p7(p) = tr(E7JP|?(p)J?|p(p)E7)
for such v, <py extends to a meromorphic function in v. If now fTt7(v) * 0
for all у,т g Ka, then Schur’s lemma (3.3.2) implies that
jp\p(v)jp\p(v) =
We conclude that if fT-7(v) * 0 for all у, r g Ka, then <p7(v) - <pr(v) = 0.
Hence, Lemma 10.A.3.2 implies that <py = <pT. Put <pP a = <?y for some
(any) у, т g Ka. The lemma now follows.
10.5. The Harish-Chandra ц-Function
43
10.5.5.	Lemma. <pP a = <pp,a.
Let у & Ka, and set n = dim Ia(y). Then,
n<Pp,Av) = tr(E7JPlp(v)JplP(v)Ey) = tr(E7JP\p(v)E7E7JP'P(v)E7)
= tr(EyJp|P(i')Jp|p(i')E7) = n<pP a(v).
Hence, <pP a = <f>p ,r.
10.5.6.	We denote by a* the conjugate dual representation of M on Ha.
We assume (as we may) that а|КпЛ/ is unitary. Thus, o-^nM = а\К(ЛМ.
Hence, /„» = Ia. We write J0lP(<r, v) if the dependence on a is not
understood. We also write fPav for what we called fv. On Ia, we have
the inner product
(Л«) = I (f(k),g(k)}dk.
If T is a linear operator on Ia such that TIa(y) c la(y) for all у e K,
then we define T* on la by {y) = (T^,)* for у g К (here, we use the
preceding inner product on each of these finite dimensional spaces).
Lemma. JP\P((r, v)* = JP\P(o-*, — v).
Let f, g g Ia. Then we assert that
(1)	(M) =
JN
= I (fp.a.Xn), gp.v* ,-₽(«)) dn.
JN
Clearly, it is enough to prove the first equation (since the second follows
from the first if we replace P by P). To this end, we write
(Л«) = I <f(k),g(k)')dk = f_a(v)2p(f(k(v)),g(k(v))}dv
JM\K	JN
= La(v)2p{a(v)va(m(v))f(k(v)),
JN
a(v)~v	g(k(v))} dv
= (Afp.v.Xnl’gp.^.-An^dn.
JN
44
10. Intertwining Operators
We now prove the lemma. Since both sides of the assertion are mero-
morphic in v, it is enough to prove it for Re(i>, a) > ca for a g Ф(Р, A).
Under this condition,
(Jp|P(o-,p)/,g) = j fAfp,„,Xvk),g(k))dvdk,
JMC\K\KJN
with the integral on the right converging absolutely.
Recall that (see 10.1.2)
(JplP((r,v)f)p a v(g) = f-fp'^tvgydv.
This combined with (1) implies that
(J?ip(o-, v)f, g) = f {fP,a,v(vn),gp<a^^(n)} dvdn
JNxN
= L {fp,a,Avn), gAdvdn
JNxN
= / Jfp,a,v(nv),gpt(r^^(nv))dndv
JNxN
(Lemma 10.A.3.3)
= f Sfp,a,Av)'Sp.^,-Anv))dndv
JNxN
= (f,JPlp(a*,-v)g)
by the previous observations. The lemma now follows.
10.5.7.	For the more delicate properties of the intertwining operators we
will need to assume that a is unitary.
Theorem. Assume that a is unitary.
(i)	If P,Q g &(A) then <pP a = <pQ a. We can therefore define a func-
tion, meromorphic in v for each a, by p(a, v) = 1 /(f>Pa(v) for P g 0>(A).
(ii) If Re v = 0, then pfcr, • ) is holomorphic at v and p(a, v) > 0.
(iii) If a g 2(P, A), let Q G ^(Л) be such that a is simple in Ф(£?, A).
Let a{a) = {H g ala(H) = 0}. Set = (m G|Ad(m) H = H, H g
Put *Q = °M{a}n Q and	= l/<f*Q a(v,vKAn*Q)). Then,
10.5. The Harish-Chandra p-Function
45
g„(a, v) depends only on (v, a) {and а), р._а{(т, v) =	v), and
ae2(P. A)
(iv) Let к g К be such that Ad(k)a = a. Set
ka(tn) = a{k~lmk), kv(H) = v(Ad(k)~ 1
for tn g °M, H g a. Then, p{ka, kv) = p{a, v).
The function p{a, v) is essentially Harish-Chandra’s “^.-function”.
We now begin the proof of the theorem. Let Qlt..., Qd be as in
Lemma 10.1.9 for P and P. Then
Jp\p(v)f = ]Qd\Qd-Av) ’ ’ ’
If Re v = 0, then Jp^p(v) = (JpiP{v))* by Lemma 10.5.6. Thus, if Re v = 0
then
‘ ‘ ‘ •^2d|Gd_l(1') ^Qd\Qd-(.v) ‘‘
Let PltP2 denote Qd~i,Qd, let {a} = 1t(Pl\P2), and let *QX be as in
10.1.13. Let H be as in that number also. Then
Thus
Jp\p(v)Jp\p(v)f = <P*Qlt<r( V)^Q,Ip(V) •^Cd_2|(2d_1(1')
X JQd-2\QdSv^ ‘ ‘ ‘
Now <p*Qlta{av) = <P*Ql,<r(“t')- So this expression only depends on ±a. We
denote it by <pa a(av)- We have thus shown that
(0	<Pp,Av) = П <Pa,A“v)-
aeS(PIP)
This proves (iii).
If Q g &>{A\ then
S(OIO) U (-S(OIQ)) = S(P|P) и (-S(P|P)).
Since both of these unions are disjoint and <patA?v) = <p~ata{“v\ (*)
follows for Re v = 0, and hence for all v.
46
10. Intertwining Operators
We now prove the second assertion. If Re v = 0, then
4>p,AvV = (Jp\p(v)y Jp\p(v)-
Thus, if v is not a pole of <pP a then <pP a(v) > 0. But if <pP(T(v) = 0, then
Jpip(p) = 0. Thus, the content of (ii) is that if JP^P(v) is defined for
Re v = 0, then it is non-zero. For this, we proceed as usual. Let h g
C”(N). If fi g a£, v g H", then we set
Дшпап) = ap+pa(m)h(n)v,
for n & N, m g °M, a g A, and n g N. If g g G is not of this form, then
set h^ v(g) = 0. Then, д. >-» h^ v}K = v is a holomorphic map of a£
into Thus, fi >-» /р|р(д.)мм is a meromorphic mapping of a£ into
Now,
= [_h(n)dnv
JN
if Re(p, a) > ca for a g Ф(Р, A). The formula is thus true by analytic
continuation for all fi. It is now clear that JPiP(fi) * 0 where it is defined.
We now prove assertion (iv). Let Ш):	„ -»	be defined
by L(^)/(x) = f(kx). If Rep = 0, then L(k) is a unitary operator. We
also note that if Re(p, a) > 0 for all a g Ф(Р, A), then
h?k-'\kPk-Akv)L(k)f(u) = j kPk-'fkv(k lnu)dn
JkNk~'
= ( fXnk '“) dn = L(k)JPiP(v)f(u).
JN
Hence,
JkPk-llkPk~l(v) = L(k)JP\p(v)L(k)
So, if Re v = 0,
JkPk-'lkPk-'iv) JkPk-'lkPk-'tj') = L(k)JP\p(v) JP\P(v)L(k)
= <pp,Av)Uk)Uk)~l = <pp,Av)-
10.5.8. If 3 > 0 is given, then we set a£(3) = {p g a£| ||Re p|| < 3}.
Theorem. Assume that a is unitary. Then there exists 8 > 0 such that
filo, v) is holomorphic on a^{8) and there exist constants C, r > 0 such
10.6. Notes and Further Results
47
that
< C(1 + Him р||/
for all v g a£(3).
By the product formula in the previous number it is enough to prove
this result in the case when 2(P|P) consists of one root. If we fix у g Ka
and apply the determinant formula (10.4.6, see also 10.4.5), we find that
there exist a,, bt, i = 1,..., m, ct, dt, i = 1,... ,m, such that
т-r Г«р,а)/4(р,а) - а/)Г(-(1',а)/4(р,а) - c;)
цда,p) 7	'7 = II ----------------------------------------------.
12/<m Г((1',а)/4(р,а) - bi)r(-(v,a)/4(p,a) - dj)
Now, Lemma 10.A.2.2 implies that an estimate of the desired form is
satisfied for some fixed <5 > 0 and all v g a£(3) with ||p|| > T for some
T > 0. Since the set {p g a£| ||Re i'll < 3, ||Im i'll < T} is compact and
since p(a, v) is holomorphic at all v with Re v = 0, the result now follows.
Note. Since a „ is generically irreducible, Vogan’s minimal K-type
theorem (Vogan [1]) implies that there exists у g К with multiplicity one
in Ia. This implies that we have an expression for pa((r, v) of the form
П
l<i<m
l<j<n
r((i',q)/4(p,a) - д,)Г(-(1',а)/4(р,а) - c;)
T((i', a)/4(p, a) - bi)r(-(v,a)/4(p,a) - d-)
10.6.	Notes and further results
10.6.1.	The literature on the intertwining integrals is vast and no doubt
we will do a disservice to some researchers in the subject. The path
breaking work was done in a series of papers of Kunze-Stein [1, 2, 3],
which dealt with minimal parabolic subgroups. As opposed to this chapter,
Kunze-Stein worked in the so-called non-compact picture and looked
upon intertwining operators as being special cases of singular integral
operators. The product formula for minimal parabolic subgroups was
given in the form of the material of Section 1 by Schiffman [1]. His work
involves a generalization of the method of Gindikin-Karpelevic [1] for the
Harish-Chandra c-function. In this method, the calculation is reduced to
reductive groups of R-rank 1 rather than to parabolic subgroups with one
48
10. Intertwining Operators
dimensional split components. In essentially the same generality it can be
found in Knapp-Stein [2] and Harish-Chandra [16]. Our development of
the root theoretic aspects follows the latter reference rather closely. The
map H in 10.1.13 is a variant of several constructions of Harish-Chandra.
This chapter is an exposition and an expansion of the joint paper
Vogan-Wallach [1]. For a more complete discussion of the history of the
subject we suggest that the reader consult Knapp [1].
10.6.2.	The difference equation in the form of Theorem 10.4.8 for a
square integrable is a sharpening of the result of L. Cohn [1] in light of the
results in Chapter 13 relating the C-functions to intertwining operators.
Theorem 10.4.6 for the case of square integrable a is due to L. Cohn.
Cohn’s method of deriving the Г-function formulas was the inspiration for
the method in Vogan-Wallach [1] and hence of the method of this
chapter. The basic difference between our method and Cohn’s is that we
use the limit formulas of Section 10.3 and Cohn uses an ingenious method
of integration by parts that only applies to the case when a is square
integrable. The algebra (t/(fc) ® (7(ac))Knp was introduced by L. Cohn
[1]. In that reference, he called the algebra the C-algebra.
10.6.3.	As is pointed out at the end of Section 2, the methods of that
section are constructive. Thus, in principle the functions ba A and Da A
are computable. Obviously, it would be very useful to have more results
about them. Calculations for SL(2, R), SL(2, C), SL(3, R) and 5(7(2,1) are
given in L. Cohn [1]. A variant of Lemma 10.2.11 can be found in the
thesis of Chen-bo Chu [1] with F an arbitrary finite dimensional represen-
tation. In this work, Chu follows the method of Vogan-Wallach [1] with
some simplifications for the case of regular infinitesimal character.
10.6.4.	The material in Section 10.A.1 involving the maps “QF” deserves
further study as does the “strange function” y(i>) that appears in Lemma
10.2.11. In Chu [1], there is a formula (due to the author) for /3(p) (10.2.12)
in the case when IP a v has regular infinitesimal character. It is given as
follows: Let t) be a Cartan subalgebra of gc and fix a system of positive
roots for Ф(йс,1)),Ф+- Let A(p) be the Harish-Chandra parameter for
IP a v. Then if A is the highest weight of F with respect to Ф+, we have

(A(p) + A,a)
Дф+ <A(p),a>
10.А.1. Some Constructions Related to Finite Dimensional Representations
49
If P is cuspidal and if <r has a singular infinitesimal character then IP a v
will have a singular infinitesimal character for all v. This is the reason for
the complicated proof that y(y) is not identically 0 in Section 2.
10.A. Appendices to Chapter 10
10.A.1. Some constructions related to finite dimensional representations
10X1.1. Let G be a real reductive group and let К be a maximal
compact subgroup of G. Let (p., F) be a finite dimensional, irreducible
representation of G. We look upon (7(gc) as a g-module under left
multiplication. We can thus form the g-module G(gc) ® F. Fix B, an
Ad(G)-invariant, non-degenerate, symmetric bilinear form on g. Let C
denote the Casimir operator of G corresponding to В (that is, if {2f,} is an
orthonormal basis of gc relative to B, then C = E,(A;)2). Let A denote
the eigenvalue of C on F. If g g L7(gc) and f g F, f* g F*, then set
(/ ®/*Xg ®/) = f*(f)g. Let {/Jibe a basis of F and let /,* g F*
be defined by /,*(/,) = 8ir We define a linear map QF from L7(gc) to
L7(gc) by the following formula:
As usual, we set ZG(gc) = {x g (7(gc)| Ad(g)x = x, g g G}.
Lemma. If z g ZG(gc), then QF(z) g ZG(gc).
We recall some notation in 6.A. 1.1. Let
A: G(gc) G(gc) ® G(gc)
be the diagonalization homomorphism. We look upon L7(gc) ® F as a
L7(gc) ® (7(gc)-module in the obvious way. Let z g ZG(gc). Then
z(l ® /) = A(zXl ® /). Let и = Qf(z). If g g G, then
Ad(g)u = E(Z®/*)(((Ad(g) ®Z)A(z))(l ®/;))
=	® Ad(g-1))(Ad(g) ® Ad(g))A(z)(l ®/,))
=	Ai(g)/,*)(((Ad(g) ® Ad(g))A(z))(l ®M(g)/,))-
Now, T,^(g)f* ® n(g)fi = Y.f* ® ft and (Ad(g) ® Ad(g))A(z) = A(z).
Thus, Ad(g)u = u, as asserted.
50
10. Intertwining Operators
10.A.1.2. We also need some “explicit” formulas for QF in Section 10.2.
Let v g F, v* e F*. Let Xx,..., Xn be an orthonormal basis of gc with
respect to B. For к = 0,1,2,..., set
zk,F(v,v*) = Lv^XhXi2  Xikv)X-tXi2 Xik.
Here, the sum is over all indices i{, i2,..., ik.
Lemma. There exist for 0 < i < j < к constants a -t , k depending only on A
such that
(I ® v*)(Ck(l ® v)) = E aij,kC,~'zi,F(v>v*)-
O^i&j^k
Furthermore, the a( j k satisfy the following recurrence relation {ai Fk = 0 if
i, j, к do not satisfy 0 < i <j < k):
ao,o,o=
ai,j,k + l = ai,j-l,k + ^ai,j,k + 2ai-l,j— 1, к 
If к = 0, then the formula is clear. Assume the formula for k. We prove
it for к + 1. We first note that Д(С) = C ® 1 + 1 ® С + 2E A", ® A",.
Since С ® 1 and 1 ® C are both in the center of L7(gc) ® Wflc) f°H°ws
that EA'; ® X: commutes with &(U(qc)\ We now begin the proof of the
inductive step:
(I ® r*)(C*+1(l ® v))
= (Z ® у*)(Д(С*)Д(С)(1 ® v))
= (Z® у*)(Д(С*)(С® 1 + 1 ®C + 2^xi ®A;)(1 ®r))
= (C + A)(Z®r*)(C*(l ®r))
+ 2£(/ ® r*)((2f; ® А',)Д(С<:)(1 ® r))
= (C + A)(/®r*)(C<:(l ® v))
~ 2Е^,((/®А',г*)(Д(С<:)(1 ® r)))
= (C + A) E
t><.i<.j<.k
-2E E ahJ.kC^XmzhF(v, V).
m 0<,i<j<,k
10.А.1. Some Constructions Related to Finite Dimensional Representations
51
Now
- E*mZ,. A^Z?*) = +
m
This implies that
(Z® y*)(C*+1(l ®г)) = E а(.,7ЛС + 1-%. F(v,v*)
0<.i<j<.k
+ A E
0<i<jnk
+ 2 E au,kC^zi+l^(v,v-).
0&l<l<k
Thus, if we define the ai/k+l by the recurrence relations given in the
statement then the lemma follows.
10.A.13. Set z, F = E; ztF(f j, f*). The previous lemma implies that:
к
(1)	QF(Ck)=^ E aFp^4C^‘zFF.
j-0 0£i<.p£k-j
We now assume that if X g <?(g) (the center of g), then XF = 0. If
g e Z7(gc), then we set trF(g) equal to trace of the action of g on F.
(2)	z1F = 0.
Indeed, Zj F = E; trF(A’l)Arl. By our assumption, trF(2f) = 0 for X g
Se-
lf X,Yeg, we set BF(X, У) = trF(AT). Set CF = BF(X,, X;)
XjXj. Then, it is clear from our definitions that:
(3)	Z2,F = CF-
10.A.1.4. We now assume that FK = (v & F\kv = v, к & К} Ф 0. Fix
v g FK, w =# 0. Let v* g (F*)K be such that v*(v) = 1. The following
result is usually called the Cartan-Helgason theorem (Helgason [3]).
Lemma. Let Po = °MAN be a minimal parabolic subgroup of G. Let
v0 g F" be such that Hv0 g Cr0 for H g a. Then v*(v0) * 0. In particu-
lar, this implies:
(1) dim Fn = 1 and °M acts trivially on F";
(2) dim FK = 1.
52
10. Intertwining Operators
If F is a finite dimensional representation of G such that (1) holds for F,
then (2) holds.
Since G = KAN we see that F = span{^.(G)r0} = span{/Cr0}. If v*{v0)
= 0, then v*{Kv^ = {0}. Hence, v* = 0, contrary to our assumption.
Now, Fn is irreducible as an °M-module. Thus, °M must act trivially on
Fn. So dim Fn = 1. The preceding argument implies that the pairing
between Fn and (F*)K given by evaluation is non-degenerate. This proves
all but the last part of the lemma.
To prove the last assertion, it is enough to show that
w = f p,(k)~lvodk =# 0.
JK
Let v* g (F*)n be such that Го(го) = 1. Then
z^(w) = v^f^(k)~1vodk^ = f^(fi(k)~1v0) dk
= lJ_a(fi)2pv^p.(k(h))~lv0) dn,
by Lemma 2.4.5. Now, n = n(h)a(h)k(n). So
^(м(Л(п)_1)г0) = я(«)Ч(м(«)'^о) = «(«)Л-
Thus,
r*(w) = fa(n)2p+Adn > 0.
JN
Hence w =# 0, as desired.
10A.1.5. If и g 17(йс)*, we define qF(u) = (I ® r*X«(l ® v)).
Lemma. Ifu& U(qc)k, then qF(u) g 17(йс)*.
The proof is almost identical to that of Lemma 10.A.1.1. Let к & К.
Then
Ad(к) qF(u)
= (I® r*)((Ad(fc) ® /)Д(м))(1 ® v)
= (I0 г*)((/ ® Ad(fc~'))(Ad(fc) ® Ad(fc))A(u))(l ® v)
= (I ® r*)((f ® Ad(£~'))A(u))(l ® v)
= (I ® kv*)( Д(м))(1 ® kv) = qF(u).
10.А.1. Some Constructions Related to Finite Dimensional Representations
53
10.A.1.6. The next result is a useful observation of Vogan. Let У be a
two sided ideal in (7(gc). Let denote the category of all g-modules V
such that .XV = 0.
Lemma. Let X -» Tx be a natural transformation (6.2.2) of the identity
functor of Then there exists z = zT& Z(gc) such that Tx(x) = zx for
x g X g
Note. In the proof a formula for z will be given that will be used in our
application of this Lemma.
Let M = U(qc)/.X. Let x -» x be the natural projection of I7(gc) onto
M. Set и = TMd). Let z0 g L7(ec) be such that z0 = u. If X g and if
x g X, then define a g-module homomorphism 8X from M onto X by
8x(m) = mx. Then
Tx(x) = Tx{8x(i)) = 8X(TM(1)) = 8x(u) = zox.
In particular, if X = M then TMfm) = (zom) *s a Й-module homomor-
phism. Thus, и is central in M. Since the center of the algebra M is the
projection of the center of L7(gc), we can choose z0 central.
10A.1.7. We conclude this appendix with some results about tensor
products of the modules IPat, (see 10.1.1) and finite dimensional (g, Ю-
modules. Let (a, Ha) be a smooth Frechet representation of P. We set /“
equal to the space of all smooth functions f from К to Ha such that
f(mk) = abn)f(k) for к g К and m g К П P. If f g set fPa.{pk) =
a(p)f(k). Then, fP a is a smooth function from G to Ha with fP(T(pg) =
trip) fig) for p g P and g g G. We define ttp a(g)f(k) = fPa(kg) for
g g G, к g K, f g We endow /“ with the topology induced by the
semi-norms qx(f) = 8ир^еК qirrP aix)f), for q a continuous semi-norm
on Ha and x g Uik). Then, we have a smooth Frechet representation
7ГЛо. ОП
Let ip, F) be a finite dimensional representation of G. Then we can
form the representation Tf F with action ttp a ® p. Let, for f g Ipa
and v g F, T(f ® rXg) = fig) ® pig)v. Then, clearly, T(f ® v) g
If	then define e ГР.<г ® F as foHows: Let ц.
be a basis of F and let A, be the dual basis. If w Ha ® F and if A g
F*, let I ® A(w) g Ha be given by ЕА(г,)м, if w = Eu, ® r, . We set
54
10. Intertwining Operators
S(f) =	® и,, with
fi(s) = (I ® A;)((/ ®
It is easy to see that
T(^p,a(s) ® M(g)) = irp,<r^P(g)T,
S^P,a^^P(g) = (^P,a(g) ® H(g))S,
and that TS = ST = I. We have thus proved:
(1)	T% a® F and /р><7вд are equivalent as smooth Frechet representa-
tions.
Suppose now that И is a closed invariant subspace of Ha such that
V there exists a closed (К П P)-invariant subspace W of Ha with Ha =
V Ф W. Let o-^p) = a(p\v and let a2(p) be the induced action of P on
W by projecting off of V. Then (<rj, И) and (o-2, IF) are smooth Frechet
representations of P. Let Ц be the space of all f g Ipa such that
/(g) e И for all g e G. Let I2 be the space of all f g Ip a such that
f(k) g W for all к g K.
(2)	and 12 are closed in Ip a, Ц is G-invariant, and I2 is /^-invariant.
Furthermore, Ip a = Ц Ф I2.
This is clear since f g Ipa is completely determined by /|K.
Assume now that the underlying (p, К n P)-module to (a, Ha) is admis-
sible and finitely generated as a [/(°m)-module. Let IP a denote the space
of /^-finite vectors of a. Since the /С-finite vector functor is exact on the
category of smooth Frechet representations of K, we have:
(3)	IP a ® F and Ip'V®^ are equivalent as (q, КЭ-modules.
Let now V a (Ha)KnP be a (p, К n P)-submodule such that n acts
trivially on V. Then C1(F) is a P-invariant subspace of Ha, and if FK is a
(К П P)-module complement for V in (Ha)Krip then Cl(IF) is а (К n P)-
invariant complementary space for С1(И) in Ha. Let o\ be the representa-
tion of P on C1(D and let a2 be the representation of P on Cl(FK) as
before. Then (2) implies (by taking К-finite vectors) that:
(4)	We have a (q, КЭ-module exact sequence
0 “*	“* Ip,<r	Ip,<r2 0.
10.А.2. Some Results Related to Sterling’s Formula
55
We now assume that (a, Ha) is a representation of °M as in 10.1.1. Let,
for v g a£, av be the representation of P on Ha given by (rv{nam) =
av+p(r(m). Let F = Fo о F{ о • • • Fd э Fd+l = (0) be a Jordan-Holder
series for F as a representation of P. Then Fj/Fi+l is irreducible as a
representation of M, with N acting trivially. Let A act on F;/Fi+l by av‘I
and let °M act by д., . If we apply the preceding results, we have:
Lemma. Ip^v ® F has a (g, K)-module composition series IP a ® F =
Mo d Mj d ••• d Md Md+X = (0), with Mt/Mi+X isomorphic with
^P, v + Vj'
10.A.2. Some results related to Sterling’s formula
10.A.2.1. Let Г denote the classical gamma function. Then Sterling’s
formula (cf. Whittaker-Watson [1]) implies that if 0 < 3 < тг, then
T(z) = (27r)1/2e(2~1/2)logZ~2(l + E(z)),
with E(z) < C5|z|~1 for |Arg(z)| < 3.
10.A.2.2. Lemma. Let a,b g C and a g R, a > 0. Then there exist
R > 0, C > 0 such that if I Re z\ < a and |Im z\ > R, then
T(z + a)
T(z + b)
< C|Im z|Re(0-b).
This result is well known. However, it plays such an important role in
this volume that we will give a proof. Let z = x + it, with x, t g R and
|x| < a. Then
4	( ir/2 as t -> + oo,
Arg(x + it)	,
6V 7 тг/2 as t -> - oo.
Since logfx + it) = log |x + i7| + i Argfx + it), Sterling’s formula implies
that
|T(x + it)| = (2тг)1/2|х + |7|х-1/2е-'Аг8(х+")(1 + r(x,t)),
with r(x, t) < C/\t\ for t » 0. Hence,
|T(x + <7)| = (2ir)l/2]t\x~'/2\x/t + <Г1/2е~' Arg(*+">( i + r(x,t)).
56
10. Intertwining Operators
Now,
lx/t + ir1/2 = (x2/t2 + l)x/2~1/4
<	(x2/t2 + i)“/2-1/4
<	1 + ct~2,
for some c > 0 and |t| » 1. Hence,
|Г(х + <7)| = (27r)1/2lf|x-1/2e-'Arg(j:+")(l + s(x,t)),
with |j(x, t)| < Ca|t|for |x| < a and |t| » 1. Now
Arg(x + it) = Arg(x/|t| + isgn(t)) = в.
So
sinfl = sgn(t)(x2/t2 + 1) 1/2 = sgn(t) + u(x,t),
with |u(x, t)| < C|t|~2. Thus, в = (sgn(t))Tr/2 + v(x, t) with |r(x, t)| <
C'|t|~2. Thus, if R > 0 is sufficiently large then
(*) |Г(х + <7)| = (27r)1/2|t|x-1/2e~l'|ir/2(l + <r(x,t)),
with (r{x, t) < C|t|for |t| > R.
We now prove the lemma. Write a = c + id, b = f + ig, z = x + it.
Then, if |x| < a and |t| > R for R > 0 sufficiently large, (*) implies that
T(z + a)
Г(г + Й)
<, Ctc~f\l + d/t\x+c 1/2|1 +g/t\ x f+l/2
x e-v/2Ut + d\-\t+g\)_
Now, | |t + d\ - |t + g| | < \d - g|. So, if |t| > R and R is sufficiently
large, this implies that
T(z + a)
T(z + b)
< Ctc~f.
10.A.3. Miscellaneous results
10.AJ.1. In this appendix, we prove several (well known) results that are
used in this chapter. The first is a theorem of Bumside.
10.А.З. Miscellaneous Results
57
Lemma. Let V be a finite dimensional vector space over C and let srf be a
subalgebra of End(L) such that srfv = V for all v g V, v =# 0. Then
st= End(L).
Set n = dim V. We assume that n > 0. Suppose that -6 is a left ideal in
srf. We observe that if v g V then either -tfv = V or -tfv = {0}. Indeed, if
-tfv =# 0 then V = stf-tfv c -£v.
Let be a left ideal in л/ of minimal positive dimension. By the
preceding observation, there exists vr g V such that -£xvx = V. Set «0^ =
{X g iQ/IA’zjj = 0}. Then, П is proper in hence, it must be 0.
We therefore see that dim -ёх= n and that Л = j/j Ф j/j is a left
ideal in srf. If = 0, then we are done with the construction that
follows. Otherwise, choose a left ideal in srf contained in of
minimal positive dimension. As before, there exists t>2 such that =
and if = {X g ^1^2 = 0}, then л/2 is a left ideal in л/ and
л/2 Ф Ф ^2. If = 0> then we are done with this procedure.
Otherwise, we choose a left ideal of л/ contained in ja/2 of minimal
positive dimension, etc. This procedure must stop after, say, m stages.
Thus, s#=	Ф  • • Ф -ёт, with a left ideal in A of dimension
n, and there exists g V such that = V. Thus, dim mn.
If 1 < i, j <m, define a linear map Qh from to 6i by Q^X} = Y
if XVj = Yvt. Then, QtjA = AQ:J for A g stf. If T g stf and if x g
then xT = E; Tjfx) with Tjt a linear map of to Clearly, TjtA = ATjt
for A g sf. Schur’s lemma implies that Tu = ри(Т^и, with Рц(Т) g C.
Set p(T) = [pti(.T)]. Then, p. is a linear map from s# into Afm(C). If
д.(Т) = 0, then srfT = 0. Thus, s#Tv = 0 for all v g К By our hypothe-
sis, this implies that Tv = 0 for all v g V. So T = 0. This implies that p is
injective. Hence, dim m2. This combined with the previous inequality
implies that m2 > mn. So m > n, since m > 1. Thus, dim n2. The
lemma is now obvious.
10.A.3.2. The next results are of a different nature. We record them here
so as not to interrupt the flow of the body of the chapter.
Lemma. Let p be a meromorphic function on C" and let {ffi be a
sequence of non-zero holomorphic polynomials on C". Assume that if
x g R" is such that f^x) =# 0 for all j, then p(x) = 0. Then p is identi-
cally 0.
Let Vt = {x g R"|/((x) = 0}. Then V\ is closed in R" and contains no
interior. The Baire category theorem implies that U, Vt has no interior.
58
10. Intertwining Operators
The set of all x e R" such that p.(x) =# 0 is open and non-empty if д. =# 0.
Our assumption implies that this set is contained in U, V,  The lemma now
follows.
10.A.3.3. Fix P = MN, a parabolic subgroup of G. Let P = MN be the
(standard) opposite parabolic subgroup. Fix invariant measures dn and dn
on N and N, respectively.
Lemma. Let h be a function on G such that h(tng) = h(g) for m & M.
Then
[ _h(rin)dndn = f_ h(nn)dndn.
JNxN	JNxN
This means that if one side converges absolutely, then so does the other side
and the two sides are equal.
Let dg and dm be fixed choices of invariant measures on G and M,
respectively. We note that there exists a constant c > 0 such that
I f(g)dg = cf _ f(mnn) dmdndn.
JG	JMxNxN
We note that d(mnm~l) - det(Ad(m)|n)-1 dn and d(mnm~l) =
det(Ad(m)|n) dn. Hence,
f f(g)dg = f f(g~l)dg = cf f((mnn)~l) dmdndn
JG	JG	JMxNxN v	'
= cl _ f(nnm) dmdndn = c i _ f(mnn) dmdndn.
JNxNxM	JMxNxN
So
/ _ f(mnn) dmdndn = f _f(mnn) dmdndn.
JMXNxN	JMxNxN
Thus, if
(*)	= f f(mg)dm,
JM
then
f_ h(hn)dndn=f h(nn)dndn.
JNxN	JNxN
Since every compactly supported function h on M \ G is of the form (*),
this completes the proof of the lemma.
11 Completions
of Admissible
(g, Ki -Modules
Introduction
The main purpose of this chapter is an exposition and extension to
non-linear groups of work of W. Casselman and the author on the
structure of the space of C“-vectors of an admissible finitely generated
representation of a real reductive group. Let G be a real reductive group
and let К be a maximal compact subgroup of G. If V is an admissible
(g, /О-module, then there are many inequivalent (say) Hilbert representa-
tions (tf, Я) of G with HK equivalent with V. However, the results of this
chapter imply that if (тг,, Ht), i = 1,2, are admissible finitely generated
representations of G on a Hilbert space (indeed more generally, as we
shall see) such that is equivalent to (Я2)к as a (g, /О-module, then
(HXT is equivalent to (Я2)“ as a smooth Frechet representation. This
somewhat surprising result has important consequences in the theory of
automorphic forms. Our exposition of this work follows the general outline
in Wallach [3], including an early version of an argument of Casselman to
complete the line indicated in those lectures. It is suggested that the
reader consult Casselman [1] for a complete account of his approach to
this problem.
The only part of this chapter that will be referred to in Chapters 12-14
is the first section. Thus, if the reader is in a hurry to get to the Plancherel
59
60
11. Completions of Admissible (0, К )-Modules
theory for real reductive groups he (she) can skip over the rest of the
sections without missing anything critical. To the reader of this chapter we
suggest that on first reading it is assumed that the groups appearing are
linear. In this case the only “small К-type” necessary is the trivial
representation (see Section 2). Therefore, the complicated second ap-
pendix is unnecessary. We also recommend Chapter 5 in Warner [1] as a
good reference for the theory of C"-vectors (in addition to the terse
treatment in Chapter 1 of this opus).
We have included this material in this book since it uses and extends the
theory of intertwining operators developed in Chapter 10 and it plays an
important role in Chapter 15. The proof of the main theorem also makes
serious use of the Langlands classification (Chapter 5) and the asymptotic
properties of matrix coefficients of admissible representations (Chapter 4),
and gives a hint of the essential richness of the representation theory of
real reductive groups (we will see more evidence of this in the later
chapters). The theory developed in this chapter also demonstrates the key
role that the space of functions cZ(G) (7.1.2) plays in the theory of
representations of real reductive groups (there are hints of this in Chap-
ter 8).
As just indicated, Section 1 is an exposition of the theory of Weyl group
invariants, and parts of this section will be used in Chapter 12. Section 2
contains a result of Kostant [4] that is a refinement of Proposition 3.7.1.
Section 3 is an extension of much of the theory of the spherical principal
series to principal series containing a small К-type (e.g., a one dimensional
К-type). This material plays an important role in our extension of the
automatic continuity theorem of Casselman to nonlinear groups (the
content of Section 4). In Section 5, we introduce the smooth Frechet
representation completions of admissible finitely generated (g, КЭ-mod-
ules that are the subject of the rest of the body of the chapter. Sections 6
and 7 are the heart of the chapter and contain the theorem just alluded to.
Section 8 contains a surprising theorem about the algebraic irreducibility
of the action of ^(G) on irreducible admissible representations of G. In
Section 9, we show how the results of the chapter can be used to prove
Poisson integral representations of classes of functions on G.
11.1. Some results on Weyl group invariants
11.1.1. In this section, we will prove several theorems about Weyl group
invariants that will be used in this chapter and in Chapter 12. Let G be a
11.1. Some Results on Weyl Group Invariants
61
real reductive group of inner type. Fix К a maximal compact subgroup of
G, and (P, A) a standard minimal p-pair. Let В denote an invariant
non-degenerate form on g such that В(вХ, X) 0 for X eg. We denote
by ( , ) the Hermitian extension of B|a on ac. Let W = WXA) be the
Weyl group of G with respect to A. This group is generated by the
reflections about the root planes A = 0, with A e Ф(Р, A). Set 5 = 5(ac)
and let I denote the subalgebra of JP-invariants in S. The notation will be
as in 11.A.1, except that we will write ws for Sk(w)s, for w g W and
j g Sk. If j g S, we write Av(j) = I IP| “1 Ew G w ws. Then Av is a projec-
tion of 5 onto I. Furthermore, if и g I and if s gS, then Av(us) =
и Av(j).
The main result on Weyl group invariants is the following theorem of
Chevalley. The exposition that we give is strongly influenced by (the very
similar) presentations of Helgason [2] and Carter [1].
Theorem. Let r = dim a. Then there exist algebraically independent homo-
geneous elements ul,...,ur of I such that I is the polynomial algebra in
uv...,ur.
The key result in the proof of this theorem is:
Scholium. Let xl,...,xm g I be such that x j £ E; > j fx,. If	G
S are such that is homogeneous and Ei a;1 x,i>, = 0, then vx g I+S.
We prove this by induction on the degree d of . If d = 0, then is a
scalar. In this case, we must show that = 0. If 14 =# 0 then x1 =
~(.vl)~1Li>lxtvi. Thus, x, = Av(xj) =	Avfy,). This is
contrary to our hypothesis. Now, assume the result for d - 1 > 0.
Let A g Ф(Р, A). Then 0 = sA0 = E xtsAVi. Hence, E x,(t>; - JAr,) = 0.
We note that if v g S then v - sAv g HaS. Indeed, if x g a then xk -
sAxk = xk - (x - (2A(x)/(A, А))ЯЛ)\ Expanding this (using the binomial
formula) yields the assertion since the elements xk for x & a, к gN span
5. Thus, v,- sAvt = HAtt. Furthermore, tj is homogeneous of degree
d - 1 and Ex,t, = 0. The inductive hypothesis implies that tj g I+S.
Hence, - sAVt g I+S for all A g Ф(Р, A). Since I+S is IP-invariant
and the sA generate W (2.1.10), we see that r, - wvY g I+S for all
w g W. Hence, r, - AvG,) g I+S. Since d > 0, Av(i>j) g I+ . This proves
that g I+S.
62
11. Completions of Admissible (g, К)-Modules
We now begin the proof of the theorem. Let iq,..., ud be a minimal set
of homogeneous generators for the ideal I+S, with ut e I. Then we note
that Uy..., ud generate I as an algebra. Indeed, let J be the subalgebra
of I generated by the u;. We show that Ik = I A Sk cj for all k. If
к = 0, this is clear. Assume that V c J for j < к and к > 0. If и e Ik
then и e I+, so и = E ци,- with each r, of degree equal to к - degfw,) <
k. But then и = ’LAx(vi)ui and the inductive hypothesis implies that
Av(t>,) e J.
Suppose that Uy...,ud are not algebraically independent. Then there
exists a non-zero polynomial p in indeterminates xb...,xd of minimal
positive degree with p(uy ...,ud) = 0. If we assign each x, the degree
di = deg(u;) > 0, then we may assume that p is homogeneous of a fixed
weighted degree. Let Zy..., zr be linear coordinates on a*. Our minimal-
ity assumption on p implies that if p, = (д/дх^р and if p^Uyud) = 0,
then Pi = 0. By rearranging the x; (if necessary) we may assume that
Py...,pm are the non-zero p,. Set qt = p^Uy..., ud). Then, is homo-
geneous. We assume after another possible rearrangement that qv..., qn
generate the ideal in 5 generated by the qt, i = 1,..., tn, and that qt is
not in the ideal of 5 generated by {q^j + i}. If we differentiate the
p(uyud) with respect to z;, then we have
m ft
= 0.
1 °Zi
Now, there exist uj: e S, j > n, i < n, such that qs =	with
deg(t>,7) + deg(<7;) = deg(<7;). We thus have the equation
n l д	d \
Notice that the terms in the parentheses are homogeneous. The scholium
therefore implies that
д	д
dzl t>n dzi
with degfufc) + deg(wj4) = deg(iq) - 1 and wik e I+S. Thus, we have
д	д
= L>ukwik-
dzi ,>n dzi k-2
If we multiply this expression by z;, sum over i, and use the homogeneity
11.1. Some Results on Weyl Group Invariants
63
of the иj, then we find that
d
deg(Uj) «1 + L deg(u,) vltut e £ Suk •
t>n	k=2
This implies that m1 is in the ideal generated by u2......ud, which is a
contradiction.
We have therefore shown that the u; are algebraically independent. We
now show that d = r. This can be proved by a simple transcendence
degree argument. However, we give the following elementary argument
instead. We note that we have shown in 3.1 that each z, satisfies an
equation of the form
Pi-t
гГ + E Ar(«i.........ud)z' = 0,
k = 0
with p, > 0. Let c = max Let 5;(Cr) =	5'(Cr). Then the preced-
ing equation implies that
dim 5;(Cr) < I Пр, I dim 5;(Cd).
\/“i I
Since the span of the monomials of degree j in the u, is contained in Sci,
we also have
dim5;(Cd) < dim5c;(Cr).
But dim S/C") is a polynomial in j of degree q with leading coefficient
\/q\. Thus, the preceding inequalities imply that d = r.
11.1.2. Theorem. Let H be as in ll.A.l. Then dim H = IWI and the
map I ® H -» S, и ® s us is a linear bijection. In other words, S is a free
I-module on |1У| generators under multiplication.
That the map is surjective is the content of Lemma 11.A. 1.1. Let
hl,... ,hd be linearly independent homogeneous elements of H. Suppose
that r, g I and E vihi = 0. We must show that w, = 0 for all i. So we
suppose not and look for a contradiction. Let m1 ur be as in the
previous number. Set di = degfw,). We write uJ = u{' • • • uf and w(J) =
djj + +drjr for a multi-index J. Then the previous result implies
that
j
64
11. Completions of Admissible (g, К )-Modules
Set У = {J\ajj # 0 for some i}. Let Jo g У be such that w(J0) < w(J)
for J g ST. Then m7° is not contained in the ideal generated by the uJ,
with Jg У, J ф Jo. Now,	aj:ihj)uJ = 0. Hence, every homoge-
neous component of this sum is 0. The scholium preceding now implies
that each homogeneous component of ’LiaJo ihi is in I+S n H = (0).
Thus, E, а7о ,Л, = 0. But then , = 0 for all i. This is the desired
contradiction.
Let dt = deg(u,). To prove that dim H = |IF|, we use the following two
formulas.
(1)
^det(l-tw)-
Indeed,
£ dim(Z*)f*= |ИТ* £ det(l-tw)-1
k 1> 0	W
by Lemma 11.A. 1.3. Now,
dim(/*) = |{i?| Y.dih = ^}|-
Thus
E dim(Z*)f* = fl(l - t*')"*-
*20	<-l
(1) now follows by multiplying both sides by | JTl (1 - t)r.
(2) \W\=dl ••• dr.
Indeed, lim,.,/! - tdi)/(l - t) = dt. Thus, the limit of the left hand
side of (1) as t -» 1 is IJTl/dx •   dr. We note that if w has at least one
eigenvalue that is not equal to 1 then lim,.,/! - t)r/det(l - tw) = 0.
Thus, the limit of the right hand side of (1) as t -» 1 is 1.
We now complete the proof of the theorem. By the first part, we have
E dim(Z*)f* £ dim(Hn = (1 - t) r.
''*20	-'''*20	'
So
r 1 - td‘
E dim(H П Sk)tk = П _
*20	1-1 1 f
11.1. Some Results on Weyl Group Invariants
65
The limit as I -> 1“ of the right hand side of this equation is dl  dr = | H^l.
Thus, for each n, and each 0 < t < 1, |Ж| > EOsJts„dim(H n Sk)tk,
which has limit at 1,	dim(H n Sk). This implies that
dim(H n Sk) = Ofor к sufficiently large. The evaluation of the preceding
limit now implies that
|^| = £ dim(H П Sk) = dim H.
0<k
11.1.3. We look upon 5(ac) as the polynomial functions on a* as in
11.1.1. Let u1,...,ur be as in 11.1.1. Let zl,...,zr be linear coordinates
on a*. Let 2(P, A) be the set of all A g Ф(Р, A) such that tA <£ Ф(Р, A)
for all 0 < t < 1. If A g S(P, A), let HA be as in 11.1.1.
Lemma. det((5/5z,)w;) = сПле 2{Л A) HA.
Set w = Пле2{Р Л)НЛ. Then, we have noted in the proof of 7.A.2.9
that ra = det(j) nr for j g W.
(1) Let g g a* - {0} be such that g W. Then there exists t g Rx
such that tn g £(P, A).
Indeed, s^w = - nr. Thus, nr vanishes on the hyperplane = 0. So w
is divisible by . Thus, д. is a multiple of an element of 2(P, A).
(2) ВД-1)= |S(PM)|.
Indeed, if s g 1У then set n(s) = dim{H g a\sH = H}. Then,
(1 - t)r/det(l - ts) vanishes to order r - n(s) at 1. If n(s) = r - 1
and j =# 1, then j is a reflection about the hyperplane perpendicular to its
-1 eigenspace. Thus, s = sA for some A g X(P, A). For such s,
(1 - t)r/det(l - ts) = (1 - t)/(l + t). Thus, we have, from 11.1.2 (1),
W П --------T = 1 + (1 _ OlS(F, Л) |/2 + higher order in (1 - t).
i = i 1 - t 1
If we calculate the derivative of the right hand side of the preceding
expression at 1, we get (by the obvious method)
'	- l)/2
-I^IE ... ..k--
1 = 1	^r)
Since | IF| = dj    dr (11.1.2 (2)) the assertion follows.
66
11. Completions of Admissible (g, К)-Modules
We now prove the lemma. Set p = det((d/dzjM;). Then, deg(p) =
'L{dl - 1) and sp = det(j) p. The argument proving (1) implies that HA
divides p for all A g £(P, A). Thus, (2) implies that p = сот with c g C.
We must show that c =# 0. This follows from general facts about algebraic
independence. However, we will give a direct argument (which is essen-
tially the one used to prove these “facts”). Let, for each i, Qt be a
non-zero polynomial in variables x0,Xj,...,xr of minimal degree such
that Q/z,, мр..., м J = 0 (such a polynomial exists for each i since 5 is
integral over I). Then
d
4'
d	4 д	d
=	+ L — 0,(z,mJ—к,.
OX0	f=l 0Xt	0Zj
Let A,; = (.d/dx^Q^Zi, ulf.... mJ, T,7 = (d/dzjM;, = -8^(д/дх0) X
QitZj, Mb..., mJ. Set A = [Л,7], Г = [Г,7], and D = [Z>,7]. Then, AT = D.
The definition of the Qt implies that (d/dx0)Qi(zi,u1,..., mJ # 0 (as a
polynomial in the z;). We conclude that as a polynomial in the z(,
0 =# det A det Г. Since det Г = p, p =# 0, c =# 0.
11.1.4. The rest of the material in this section will be used in Chapter 12.
Let g be a reductive Lie algebra over C. Fix B, a symmetric invariant
nondegenerate form on g. Let t) be a Cartan subalgebra of й and fix й„ a
real form of й such that is negative definite (it may be necessary to
replace В by - В in order for this to be possible) and gu П t) is maximal
abelian in gu • Let r denote conjugation in g with respect to qu and set
<X,Y> = ~B(X,tY) tor X.TGfl.
Let W = Mg, Ю (the Weyl group of g on t>). Set I = 5(^)HZ,
5+(t)) = ®;>0 S'dj), I+= I A 5+(t)). Let E denote the harmonics for I
in 5(t)) (denoted H in 11.A.1).
Theorem, dim E = I FK|. The map I ® E -» 5(t)), i ® e >-» ie, is a linear
bijection.
We look upon g as a real reductive Lie algebra. Then т is a Cartan
involution relative to Re B. Let a = i(gu n t)). Then S(fi) is isomorphic
with 5(ac) as a IF-module under the obvious linear isomorphism of ac
11.2. A Lemma of Kostant
67
onto t), with W acting as ТИ(д, a) on a. The result now follows from
Theorem 11.1.2.
11.1.5. We return to the original situation in this section. Let A be a
special vector subgroup of G. Let P g &(A). Then M is a real reductive
group of inner type. Let t) be a Cartan subalgebra of mc. Let E be as in
the previous number. Let Em = EW(mc^\ The following result will play an
important role in the next chapter.
Theorem. The map ® Em -> 5(t))’r(mc,f|) given by i ® e >-» ie is a
linear isomorphism. Furthermore, dim Em = I Wz(0c,l))l/lMmc, 1))|.
Set I = ЗДЛ J =	Let E = Em Ф E' as a Wm =
WTmc, t))-module. Then
E se = 0, e e E'.
Let e1,...,er be a basis of Em and let er+1,..., ew be a basis of E'. If
h g J, then the previous theorem implies that
w
h = E uiei, ui e I-
i=i
Thus,
1 '
h = rjTrr E и,^( = E uiei 
1 "ml seWm,i	>' = 1
Hence, the indicated map is surjective. Since it is the restriction of an
injective map it is bijective. Let Em be the space of harmonics for J. Then
the previous theorem implies that the map J ® Em^> 5(t)), j ® e -»je, is
a linear bijection. We therefore see that 5(t)) is a free 7-module on
generators V = EmEm and that the map Em ® Em -» V, e ® e' ее', is a
linear bijection. Hence, the previous result implies that dim Em x \Wm\ =
I W\. This completes the proof.
11.2. A lemma of Kostant
11.2.1. In this section, we give a refinement of Proposition 3.7.1 that will
play an important role in the next section. Let G be a real reductive group
68
11. Completions of Admissible (g, К )-Modules
of inner type. Fix К and В as in the previous section. Let (P, A) be a
minimal standard p-pair. Let в be the Cartan involution associated with К
and let p denote the -1 eigenspace for в. Let N be the unipotent radical
of P. We identify the universal enveloping algebra of ac with 5(ac). Let
W, I, and H be as in the previous section. The lemma of Kostant of the
title of this section is:
Lemma. The map L7(nc) ® 5(pc)K ® H ® (7(tc) to U(qc) given by
n®p®h®k^> n(symm( p))hk
is a linear bijection. Here, symm is the usual symmetrization map of S(qc)
onto U{qc) {see 0.4.2).
11.2.2. We first prove the analogous result for the symmetric algebra.
The argument that we will use is essentially the same as that of 3.7.1. We
will therefore be a bit less detailed in the proof. Let q be the projection of
Й onto p corresponding to the direct sum decomposition g = I ® P- Then
p = q(n) Ф a. We note that this decomposition is orthogonal with respect
to B. We therefore see that the map 5(^(n)c) ® S(ac) to 5(pc) given by
n ® a na is a degree preserving linear isomomorphism onto 5(pc). Let
r be the orthogonal projection of p onto a. If we identify 5(pc) with the
polynomial functions on p£ then the extension of r to 5(pc) is just the
restriction map from p£ to a£. Thus, the Chevalley restriction theorem
(3.1.2) implies that r is a graded isomorphism of 5(pc)K onto S(ac)w.
Since Ker r is the ideal in 5(pc) generated by q(n), we have:
(1) The map 5(#(n)c) ® 5(pc)K ® H to 5(pc) given by n ® p ® h >-+
nph is a linear isomorphism onto 5(pc).
Let 'P be the map of 5(nc) ® H ® 5(pc)K ® 5(tc) to S(gc) given by
ЧЧп ® h ® p ® k} = nhpk.
(2) 'P is a linear isomorphism onto 5(йс)-
We note that if p g 5(gc), then
P + ^(йс)^с = #(p) + ^(бс)^с-
Hence, 5(пс)Я5(рс)к + 5(gc)tc = S(pc) + 5(gc)^c by the preceding
observations. This, obviously, implies that 'P is surjective. Since 'P pre-
serves the obvious gradings, a count of dimensions of graded components
shows that 'P must be bijective.
11.3. Representations with Small K-Types
69
11.2.3.	We now prove the theorem. We note that U(flc)» Wnc)> anc*
t7(tc) are filtered by the standard filtration. We filter 5(pc) and H by the
filtration corresponding to the degree gradation. On t/(nc) ® H ® 5(pc)K
® Udc), we put the tensor product filtration. Let Г be the map in the
statement. Then Г preserves filtrations. We identify Gr U(Qc) with S(qc)
as usual. Let F1 be the jth level of the filtration of (7(nc) ® H ® 5(pc)K
® U(lc). We note that F° = C1®1®1®1 and Г(1 ® 1 ® 1 ® 1) = 1.
We assume that Г is an isomorphism from F1 onto (7;(йс)- Now, it is
easily checked that if n g 5'(nc), h g Hr, p e S'fPc)*, к g 5m(tc), and
if i + r + I + m = j + 1, then
r(symm(n) ® h ® p ® symm(£)) + (7;(йс)
= symm(4,(n ® h ® p ® £)) + (7;(йс)
The result now follows from 11.2.2 (2).
11.2.4.	We note that in the course of the preceding proof we also showed
that the map 5(^(nc)) ® 5(pc)K ® H -» 5(pc) given by n ® и ® h >-»
nuh defines a graded bijection. Let JF" be the space defined in ll.A.1.1 for
the action of К on p (it was denoted by H there). Then Lemma ll.A.1.1
implies that as a graded vector space <&= 5(pc)/5(pc)/+ (i.e., the
dimensions of the graded components are equal. This implies that
dim <&’ = dim( £ ^*(^(nc)) ® H'}-
\k+i=j	1
Hence, we have
dim 5;(pc) = dim( £	® 5'(pc)K
U+z-/	'
Now Lemma ll.A.1.1 implies:
Theorem. The map 5(pc)K ®	5(pc) given by и ® h -» uh is a linear
isomorphism.
113. Representations with small X-types
11.3.1.	Let G be a real reductive group of inner type. Let К be a
maximal compact subgroup of G and let (P, A) be a minimal standard
70
11. Completions of Admissible (g, К)-Modules
p-pair. Then an irreducible representation (т, Ит) of К is said to be small
if T|0M is irreducible. An obvious example of such а /Муре is the trivial
representation of K. In 11.A.2, we have shown that many of these /Mypes
exist for non-linear G. We begin this section with a generalization of
Theorem 3.6.6. We will use some of the techniques in Section 3.5.
P-B-W (0.4.1) implies that the map ¥ of I7(nc) ® U(ac) ® U(lc) to
(7(йс) given by ЧЧп ® a ® k) = nak is a surjective linear isomorphism.
Thus,
Цйс) = ^(^(aC) ® Ц*с)) ® пЦйс)-
As in Section 3.5 we look upon U(ac) ® £7(tc) as an algebra with the
tensor product algebra structure. If g g U(flcX then we define q(g) g
U(ac) ® by g g 4(q(g)) + n[/(gc)-
Lemma. If g g (7(йс),	^(Ad(m)g) = (I ® Ad(m))q(g) for m g
°M. Also, ifx, у g (7(йс)К> q(xy) = q(y)q(x).
We observe that Ad(wt) n c n. We let °M act on (7(nc) ® U(ac) ® U(t c)
by
Ad(wt) (n ® a ® k) = Ad(wt) n ® Ad(wt) a ® Ad(wt) k.
Then, clearly, 'PfAdfm) x) = Ad(m) 'P(x). The first assertion is now
immediate. The second assertion was proved in 3.5.6.
11.3.2.	Let 7) be the automorphism of U(ac) given by -q(H) = H +
р(Я)1 for H g a. Let т) ® r denote the homomorphism of U(ac) ® U(fc)
to t/(ac) ® End(KT) given by (tj ® rXa ® k) = 17(a) ® r(£) (here,
U(ac) ® End(KT) is given the tensor product algebra structure).
Lemma. There exists a homomorphism yT of U(gc)K to U(ac) such that
(V ® r)(<7(g)) = yT(g) ® I.
The previous lemma implies that if g g U(gc)K, then
(17 ® J)<7(g) = L (I ® Ad(m))(i7 ® I)q(g) dm.
J°M
If к g U(tc), then
f r(Ad(m) k) dm = d(r)~l tr(r(k)) I,
JoM
11.3. Representations with Small K-Types
71
since т restricted to °M is irreducible. Thus, (tj ® rX<?(g)) g U(ac) ® I.
Thus, (rj ® rX<z(g)) = a ® I, with a g U(ac). Since q is an antihomo-
morphism of f7(gc)K and г? ® г is a homomorphism of U(ac) ® t7(tc),
the lemma follows.
11.3.3.	Let JT = Ker т c U(lc).
Lemma. yT(f7(gc))K c f/(ac),r. The following sequence of algebra homo-
morphisms is exact:
О Цйс)К А Цйс)/Т Цйс)* U(acf 0.
yT restricted to symm(5(pc)K) is a linear isomorphism onto U(ac)w.
We first prove that ут(17(йс)к) c Ifac)* Let £ = T|<>M. If v g a£,
then Frobenius reciprocity implies that IP f v(t) is irreducible as a К-mod-
ule. It is thus an easy matter to see that if g g £7(йс)к and if f g Ip f „(т),
then тгр f v(g)f= v{yT{gf)f. Let j g W and let у g К be such that
Ad(y)|n = s. Let Q = yPy~l. We define As(v)f(k) = T(y)~40\P(v)f(yk)
for к e К and v such that is holomorphic at v. Then, one checks that
As(v) defines a (fi.K)-module homomorphism of IP tv to IPt sv. Thus,
for v as in the preceding, 17(йс)к> and f e we have
= ^s(p)77p f „(g)/
= »Mg))As(v)f.
If / =# 0, then the set of all v for which As(v)f is defined and non-zero
has interior (Lemma 10.3.5). Hence, sv(yT(g)) = p(yT(g)) for all g a£
and all g g (7(йс)к. This clearly implies that yT(f7(flc)K) c U(ac)w.
Lemma 3.5.9 implies that Ker yT = U(qc)k П U(qc)Jt. Thus, to complete
the proof we must show that yT(symm(5(pc)K) = U{af)w. To see this, we
first note that yT(symm(5;(pc)K) c U'(ac)w. Furthermore, if p g 5;(Pc)
then yT(symm(p)) - Resp/a(p) g t7;-1(ac). We now prove that
yT(symm(5'(pc)K) = U’\a*c)w,
by induction on j. If j = 0, this is clear. Assume this is true for j - 1.
Suppose that a g t7;(ac),y. The Chevalley restriction theorem (3.1.2)
72
11. Completions of Admissible (g, К)-Modules
implies that there exists p e 5;(pc)K such that
Res₽/a(p) - a e U’~\ac)w
Thus,
yT(symm(p)) - a e U’~\ac)w.
The inductive step now follows.
113.4.	If v e a£, then we define а 17(йс)к^с)'то<1и1е structure on VT
by letting g e (7(йс)к act by v(yT(g))I. We denote this module by VT v.
We set
Y’ ’ - t/(ec) ®ияс)«и,с) V,„.
We look upon YT'V as a (I, /О-module with й acting by left multiplication
and К acting by k(g ® v) = Ad(£) g ® т(Л)г. Since Z(gc) c (7(gc)K,we
see that if z e Z(gc) then z acts on УТ1' by yT(z)/. Hence, Corollary
3.4.7 implies that УТ1' is admissible.
Lemma. As a U(nc)-module, YT,V is free on I W'ldfr) generators. Further-
more,
yr,Y/nj + iYr,Y s ([/(nc)/nT7(nc)) ® H ® VT
as a °M-module.
Let 7j„ be the natural (й, /О-module homomorphism from
^0c)®Wc) Kontoy-
given byg®r>-»g®r (this means that the two sides should be appropri-
ately interpreted in the corresponding non-commutative tensor product).
Set
Л = U(Qc)K/U(gc)K n (Цйс)кегт).
Then the preceding lemma implies that /T is an abelian algebra, which we
can consider to be an algebra structure on 5(pc)K. Lemma 11.2.1 implies
that as a °Af-module and as a (l/(nc), /T)-bimodule (the first factor acts on
the left, the second on the right),
Цйс) ®Wc) К s l/(nc) ® H ® К ® IT.
113. Representations with Small X-Types
73
Let C„ be the /T-module C with action v ° yT. Then
yT’^(t/(0c)®Wc)
The result now follows.
113.5.	We next calculate the formal character (ll.A.3.1) of the Jacquet
module (4.1.4, 4.1.5) of У7'1'. The previous lemma implies:
(1)	dim Ут’1'/пУт’1' = |WW).
We note that the proof of Lemma 11.3.3 implies that we have for each
j e W a (g, КЭ-homomorphism of У7’1' into lP (	= T|0M) given as
follows. Fix a /^-isomorphism T of VT onto /?(т). Then we set
® v) = rrPsv(g)T(v). Set 8S V(v) = r)s „(rXl). Then, Ker«5J PD
пУт1' and Im 8S v = VT. We also note that 8sv(Hv) = (sv + p)(H)8s V(v)
for H e a. Thus, if we set (Ут *'/пУт ,')л equal to the generalized weight
space for a with weight Л, then
dim(yT’l'/nyT’l')sp+p > d(r).
We conclude that:
(2)	If v e a£ is such that sv Ф v for j e W - {1}, then (У^/пУ71') is
a direct sum of the weight spaces (YT'l’/nYT’v)st,+p, s c IF and each is of
dimension d(r).
With these observations in hand, we can prove:
Theorem. сЬл/(/(У7’1')) =	chM(M(^*, -sv)). Here, chM and
M(£*, — v) are as in ll.A.3.1 and j is the Jacquet module functor
(4.1.4,4.1.5).
Let Ho e a be such that a(H0) = 1 for a simple in Ф(Р, A). Set
(7(nc){;) = {« e f/(nc)|ad(H0)n = jn}. Put (7(nc)(,) =	(7(nc){;).
Then:
(3)	U(nc) = t/(nc)<‘> Ф n'+1(7(nc).
We note that if a, еФ(Р,А) then [na,np] = na+/J (cf. Wallach
[1; 8.11.3]). Let Xv..., Xd be a basis of n such that ad H0Xj = mjXj,
j = 1,..., d. Then e Z and > 0. We assume that = 1 for
j = 1,..., r and > 1 for j > r. Then the preceding observation implies
74
11. Completions of Admissible (g, К )-Modules
that Xlt...,Xr generate I7(nc). If n g n' + 1t7(nc), then n is a sum of
terms of the form Xjt • • • -V, +1«'. This implies that n g ®y>. t7(nc){;).
Thus,
n, + 1t7(nc) c © t/(nc)0).
i>‘
If n g U(nc)( ), then n is a sum of products of elements of the form
Xit  • • Xt with ik < r, since Xlt ...,Xr generate n. Thus, n g n;t7(nc).
Hence,
n'+1t7(nc) = ® t/(nc)0)
j>i
and (3) follows.
Consider the (m,°Af)-modules Ут’1'/п; + 1Ут’1'. Since A is simply con-
nected they integrate to representations v of M. Let B^v) be the map
from t7(nc)o) ® H ® VT to Ут’1'/п'' + 1Ут’1' given by
® h ® v) = nh ® v mod п; + 1Ут’1'.
Then (3) implies that B^v) is bijective for all j. Set
Then, fij v is a representation of M and v\oM = Ad|0M ® I ® Т|0М. Let
hl,...,hw be a basis of H. Let J = Resp/a5(pc)K = U(ac)w by the
Chevalley restriction theorem. If Л g a, then
= YjijPjiih),
i
with p^h) g J. Thus, since
symm(p) = Resp/a(p) mod(nt7(8c) + Цйс)Кегт),
we have
Л(Л, ® r) = J2p(yT(symm(p;,(/i)))j/i; ® v mod пУт’1'.
j
This implies that д.о „(m) is holomorphic in v for all m e M. Let r)T be
the character of T|0M. Now, (2) implies that if -qT is the character of T|0M
then:
(4) tr ц0 v{ma) = T7T(m)Ese(y asv+p for a g A and tn g °M.
11.3. Representations with Small K-Types
75
We now observe that:
(5) tr^; „(wt) = tr^0 „(wt)tr(Ad(wt)|t/{1Ic)o>) for m g M.
To prove this assertion, we consider the exact sequence
0	п; + 1Ут’1'/п; + 2Ут’1'	yr.yn; + 2yr,x ут,упЛ-1ут,х 0
If n g t/(nc)o+1), v g VT, and h g a, then
h • nhi ® v = [Л, п]Л, ® v + J2i/(yT(symm(p;7(^))))wA; ® v
J
mod п; + 2Ут’1'
as before (5) follows from this and the observation that tr fiJV(m) is
independent of v for m g °M.
Since
chM(j(y-9) = E сЬм(п'Ут’7п'+1У-')*,
(5) implies that
chM0(y^)) = ch»Mr* • ( E • E «'AchoM([/(«)*).
'jelF ' Лео*
ll.A.3.1 implies that this expression is the asserted one in the theorem.
11.3.6. We now prove the same result for IP^V. The product formula
(Lemma 10.1.12), 10.1.13 (1), and Lemma 10.3.5 imply that there exists
c > 0 such that if Re(p, a) > c for a g Ф(Р, A) then det JP\P(v)\lp {p{T) +
0. Hence, Corollary 10.5.2 implies that f y(t/(g))ff(r) = for these v.
Thus, under this condition we have a surjective (g, АЭ-module homomor-
phism of YT-V to IPtitV- Let v = Kerp.^,v. Then we have an exact
sequence of Jacquet modules
0 - i(iP,(, J — J(yT’")	- 0.
If Re(p, a) > c then }р^Р(и)Ц(т) # 0. Thus (in the notation of 11.3.3),
As(v)^(t) =# 0. Thus, if sv =# v for s =# 1, then (in the notation of 11.3.4)
8SV is a non-zero multiple of 3 ° As(v)° tj1u, (8(f) =/(l)). This implies
that j(ii£ v)(j(IP ^v)n) = j(YT'v)n. If we apply ll.A.3.2 (2), then we find
that there exists a sequence {<p;} non-zero holomorphic polynomials on a£
76
11. Completions of Admissible (g, A)-Modules
such that if tpj(v) =# 0 then the (g,°Af)-modules M(£*, -sv), s g W, are
all irreducible. Thus, Theorem 11.3.5 implies that if <pk{v) =# 0 for all k,
then
j(YT,v) =
seW
We therefore see that if <pk(v) =# 0 for all k, then
Hence, j(Z( v) = (0). So 4.1.5 implies that Zfj„ = (0). We have therefore
shown:
(1)	There exists a sequence of non-zero holomorphic polynomials on a c,
<p; , such that if Re(p, a) > c for а g Ф(Р, A) and <p;(p) =# 0 for all j, then
v is bijective.
In ll.A.l, we have seen that if is the subspace of 5(pc) that was
denoted by H in that section then, as a /С-module under Ad,
5(pc) = ^®5(pc)(5(pc)Kn5 + (pc)).
Theorem 11.2.4 implies that as a /С-module, YT'V is isomorphic with
<^® VT. If we now apply (1) using т = 1, then we find:
(2)	As a /С-module is isomorphic with Ц.
This implies in particular that the /^-module structure of YT,V is
independent of v. If Re(p, a) > c for all a g Ф(Р, A) and if tpk(v) Ф 0
for all k, then we have seen that ytv is bijective. This implies that УТ1'
and are isomorphic as ^-modules for all v. Since these modules are
admissible and v is surjective if Re(p, a) c for all a g Ф(Р, A), we
have:
(3)	If Refp, a) > c for a g Ф(Р, A), then УТ1' is isomorphic with IP % „
Now, if у g К then we note that, as a function of v for fixed g g C7( й c),
^(g) = tr(E7g|y,.,(7))
is a polynomial function in v. Indeed, if x g symm(5(pc)) and if g g
17(йс), then gx = E, Pikt with pt g symm(5(pc)) and kt g U(Ic). Thus,
in YT'V we have
g(x ® v) =
11.4. The Automatic Continuity Theorem
77
for v g Ит. p,• = Y.jhtjZj, with ft,7 g Ж and z; g 5(pc)K. Thus,
g(x ® v) = Е1'(Уг(г/))ло ® т(Л,)г.
i
This implies that
Eyg(x9v) = £»>(ут(2;.))Е7(Л17 ® т(Л,».
i
Our assertion is a simple consequence of this. Also, <pp(g) =
\xЕутгРл v{g)Ey is a polynomial in v for fixed g e (7(gc) (see 1LA.4.9).
Since <p„ = for Re(p, a) > с, a g Ф(Р, A), we see that <pv = fiv for all
v. But this implies that YT,V and IP^tV have the same distribution
character (see the proof of 8.1.4). Thus, in particular, IP^tV and УТ1' have
the same irreducible subquotients with the same multiplicities. The exact-
ness of the Jacquet functor (4.1) now implies:
Theorem. chM(j(IP ( v)) = chM(M(f;*, -sv)) for all v g a£.
113.7.	Let
= (i> g OcIjp =# v for j g W, s =# 1, and Af(£*, -sv) is irreducible}.
Then ll.A.3.2 implies that is open and dense in a£.
Lemma. If v g then j(IP („) = ®sG(y Af(£*, -sv).
This follows from the previous result and 6.A.3.7.
11.4. The automatic continuity theorem
11.4.1. We retain the notation of the previous section. The purpose of
this section is to prove:
Theorem. Let (cr, Ha) be a finite dimensional representation of P. Let IP a
and IP a be respectively the corresponding К-finite and C°° induced represen-
tations of a from P to G. If A g j(IP a), then A extends to a continuous
functional on IPa.
The proof of this result will occupy the rest of this section. We will give
a short history of it in 11.10.4. We begin the proof by reducing the
78
11. Completions of Admissible (a, A)-Modules
theorem to the case when G is connected and semi-simple. Let G° be the
identity component of G and let G{ be the commutator subgroup of G°.
Since G is reductive, Gj is connected and semi-simple. Let Kx = К П Gj.
(1) K = °MKl.
Let к K. Since G is of inner type there exists k{ g Kx such
that Ad(£j) a = Ad(£) a (see 2.1.9). There exists k2^Kl such that
Ad(^2)|a = Ad(fcf lk)\a (2.1.10). Thus, k2 lk g °M. This proves (1).
(1)	implies:
(2)	G = PGl.
Set Pj = Gj П P and (rt = trlpj.
(3)	The map f f^G defines a topological isomorphism of Ipa onto
(2) implies that this map is injective it is clearly continuous. If f g Ip a ,
then set T(fXpg) = a(p)f(g) for p g P and g g Gj. Then, T(f) g I" a
and T’(/)|G1 = f. So the map is surjective. The open mapping theorem now
implies (3).
Now, N c Gj. Thus, j(IPa) = j(IPb<T1). This implies that the theorem
will be proved if we can prove it in the case when G is connected and
semi-simple. We will therefore assume that G is connected and semi-sim-
ple throughout the rest of this section.
We begin by studying a very special case.
11.4.2. Let (t, Vt) be a small К-type and let £ = T|0M. Let c > 0 be such
that if Re(p, a) > c for all a g Ф(Р, A), then Ipft, = YT’v (see the
previous section). Set
(a*)c+= (p Ga*|Re(p,a) > c, a g Ф(Р, Л)}.
Lemma. If v g (a£)* and if A ej(Ip ( v), then A extends to a continu-
ous linear functional on Ip ( v.
Let = {v g	-sv) is irreducible for all j g W, and sv Ф v
for j g W, s + 1}. Set n; c = n (a£)c+. If j g W, f g and p, g
Hf, then we set 8s{p,v\f) = p(As(v)f(l)). Then, 8s(p,v) g ( „У,
11.4. The Automatic Continuity Theorem
79
8s(p, vXnlpл v) = (0), and
H8s(p,,v) = -(sv + p)(H)8s(p,v).
11.3.7 implies that if v g O^c, then
Klp,^)= L U(nc)8s(H* ,v).
Let T„’. U(nc) ® H ® Ц(т) -> I( be defined by Tv(n ® h ® f) =
irP'itV(nh)f. If v g (flc)c > then Tv is bijective. Furthermore,
Ц = K(U(nc)w Zf(r)) Ф ni+llp^v.
Let Wj,..., wd be a basis of (7(nc){;) ® H ® /f(r). Define
by
If f g I( then it is easily checked that м,(рХ/) is a rational function of
v holomorphic on 5 = (a£)c+ (see ll.A.4.9). Now, let n;, j =
and д., , i = 1,...., n, be such that if v g then ni8s(pl, v) is a basis of
(fpjfj>,/n;+1ZF f J*. Fix v g S. Let e > 0 be such that v + zp g n 5
for 0 < |z| < e (Lemma ll.A.3.1). Then
«,(»' + zp)= L	+ zp), (1)
with at j k s holomorphic for 0 < |z| < e. Lemma 10.1.11 combined with
the definition of 8s(p, v) implies that there exist elements ..., xp g
(7(gc) and C, k s > 0 such that
IM.Xm*,»' + zp)(f)\ Chk,s S||^. p + zp(
i
for all l,s,kf g I(. If we argue as in 10.1.1 (1), (2), then we see that there
exist a constants Dt k s and d such that if |z| = e/2 then
\n,8s(pk,v + zp)(n\<Dl<ktS\\^((I + CK)d)f\i.
This implies that there exist constants E, such that
|M,.(p + zP)(f)\< фди + cK/)/L
80
11. Completions of Admissible (a, A)-Modules
for all f gand |z| = e/2. The maximum principle now implies this
inequality for z = 0. If A g j(IPtiv), then there exists j such that A g
(/p f v/ni+lIP f v)* for some j. Thus, A is a linear combination of the
ufv). The lemma now follows.
11.4.3. The next step in our proof involves a slight extension of the
preceding lemma. Let (£, be an irreducible finite dimensional repre-
sentation of °M. Let (a, Ha) be a finite dimensional representation of
M(=°MA) whose irreducible subquotients are all of the form (£0,Hf)
with £0(та) = g(m), m and a G A. We will call a a unipotent
extension of £.
Lemma. If (%, Hf) is an irreducible finite dimensional representation of
and if (a, Ha) is a unipotent extension of %, then there exists a unipotent
extension (p,	of the trivial representation of °M such that a is equivalent
with £0 ® p.
As a °Af-module Ha is isomorphic with HomoM(H?, Ha) ® H( under
the map ЧЧТ ® v) = Tv. If a g A, T g HomoM(Hf, Ha), then
and
0-(a)4,(7’ ® v) = У(а(а)Т ® v).
Set	Ha). We define a representation of M on by
p(ma)T = a{a)T, m g °M, a g A. Then, p is a unipotent extension of
the trivial °Af-module. Then ¥ defines an intertwining operator from
£0 ® p to a.
11.4.4. Lemma. Let (r, Vf) be a small К-type, let = and let
(a, Ha) be an unipotent extension of If v G (a£)c+ and if A g j(IP a v),
then A extends to a continuous functional on 1“.
Let Tv : U(nc) ® H ® 1а(т) -»Ia be defined in exactly the same way as
before.
(1) If v g (a£)c+ then Tv is a bijection.
We prove this by induction on the length of a. If a is irreducible, then
(1) has already been proved. Assume that <x has length r + 1. Let V a Ha
11.4. The Automatic Continuity Theorem
81
be an irreducible submodule of Ha. Then the restriction of a to V is
equivalent to £. Let aj be the representation on Ha/V. Then we have the
exact sequence of (g, K>modules
0 Ip,£,v Ip,<r,v Ip,<rt,v O'
The definition of Tv implies that the following diagram has exact rows and
is commutative:
0	Ip,t,v	Ip, <r,v	“* Ip, <zi, v	0
|r„	|r„
0 -> Mnc) ® H ® //r) -> t/(nc) ® H ® Ia(r) -> Wnc) ® H ® /^(т) -> 0
(1) now follows by induction.
The integral formula used to define the As(v) converges absolutely for
g (a£)c+ and makes sense for unipotent extensions of £ (see Lemma
10.1.2). Thus, we may define the analogue of the 8s(p,,v) with д. g H*.
As in the previous case, it is enough to prove that if v g then
Е1/(пс)5,(Я:,р) =J(/p,<7,J.
This follows from the exactness of the Jacquet functor and an induction
completely analogous to the proof of (1). We leave the details to the
reader.
11.4.5. Lemma. Let (a, Ha) be a finite dimensional representation of P.
Then there exist, for i = 1,..., r, r, small К-types, unipotent extensions of
Ti\aMi vt e (acV > and Fi finite dimensional (g, K)-modules, such that a
is topologically equivalent with a subquotient of a. ® Ff).
Before we prove this lemma, let us show how it implies Theorem 11.4.1.
We first note that
(see the proof of 6.A.3.8). Clearly,
W, J ® r -	• f,)'-
We therefore see that the elements of ® Ff) extend to elements
of (Ip,ahVi ® Fff (И.4.4). If A g j(JPД then A extends to an element д.
82
11. Completions of Admissible (g, К )-Modules
of ДФ( IP ® Fj) (4.1.5). By the preceding, ц extends continuously to
® Ft). Thus, the restriction of ц gives the continuous extension
of A.
11.4.6. We are left with the proof of the lemma. We will prove it several
steps.
(1) Let (£, яр be an irreducible finite dimensional representation of °M.
Then there exists a small /С-type (r, VT) and a finite dimensional (g, K)-
module F such that is isomorphic with a °Af-submodule of VT ® F.
Let л/ be the space of all functions on °M that are linear combinations
of functions of the form tr(T(m)T)tr(7r(m)5), with (r, VT) a small K-type,
(tt, F) a finite dimensional representation of G, and T g End(Kp, 5 g
End(F). Let Z be as in ll.A.2.1. We note that if Ti,t2 are small tf-types
and if t,|Z = xj f°r 1 = 1,2, then there exists a small К-type r3 such that
T3]z = XiXtJ- This implies that Tj ® t2 is a subrepresentation of t3 ® r4,
with t4 a representation of КR. Since every representation of КR can be
imbedded in a finite dimensional representation of G, this implies that
л/ is closed under multiplication. By taking complex conjugate representa-
tions we note that л/ is closed under complex conjugation. We also note
that srf is invariant under left and right translation by elements of °Af. Let
D = {m	= f(l),/G^}.
If m g D then tr(-n-(m)5) = tr(5) for all finite dimensional representa-
tions (tt, F) and all 5 g End(F) (since (m >-» 1г(тт(т)Т)) g л/). This
implies that m g Z (the subgroup of G defined in ll.A.2.1). Hence,
D c Z. If г is a small /С-type and if m g D, then r(m) = I (since
(m >-+ 1г(т(т)Т)) g srf). Hence, Theorem ll.A.2.1 implies that m = 1.
So D = {1}. The Stone-Weierstrauss theorem implies that the uniform
closure of л/ is C(°M). Thus, л/ is the space of all °Af-finite functions on
°M. If (f, яр is a finite dimensional irreducible representation of °M then
the matrix coefficients of £ are contained in л/. Thus, £ is equivalent to a
subrepresentation of a direct sum
И, Ф • • • ® Vd
of representations of the desired form, and hence to a subrepresentation
of one of the summands. (1) now follows.
11.4. The Automatic Continuity Theorem
83
Let (a, H) be as in the statement of the lemma. It is enough to show
that (a, H) is a subquotient of ®( (H ® F,) with a,, vt, and Ft as in
the statement of the lemma. If A g a£, set WA equal to the A-generalized
weight space of H. If t) g °M, then let HA(rf) be the 17-isotypic compo-
nent of HA. Then, H =	H/17). So Я is a quotient of ©^ U(pc)HA(rf).
We may thus assume that H = (7(рс)ЯА(т7) for some A and 17. Now,
HA(rf) is Af-invariant. Thus, H = U(nc)HA(r)). Since H is finite dimen-
sional, there exists к > 0 such that пкНА(т)) = 0. We may therefore
assume that H is of the form
(l/(nc)/n‘l/(nc))®HA(4),
with n acting by left translation on the first factor and trivially on the
second, and M acting by Ad(m) on the first factor and by the given action
on the second.
We note that HA(if) = CA ®	® E, with CA the one dimensional
Af-module with °M acting trivially, A acting by A, having action i70,
and E a unipotent extension of the trivial representation of °M (Lemma
11.4.4). It is therefore enough to prove that a representation of the form
(l/(nc)/n‘l/(nc))®CA®H,
is a quotient of a module of the desired form.
Let I) be a Cartan subalgebra of mc such that I) э a. Fix Ф+, a system
of positive roots for gc with respect to I) such that if n+ = ®aG4+(gc)a,
then n+z> n. Let ..., a, be the simple roots in Ф+. Fix e, , a non-zero
element in the a, rootspace. If FA is an irreducible gc-module with
highest weight A relative to Ф+, then, as a b = I) Ф n+ module,
FA= (E^(n+)/t/(n+)e"") ® Сл,
' i
with m, = 2(A, a,)/(a,, a,) + 1 and Сл is the one dimensional b-module
with n+ acting by 0 and b acting by A (9.1.3).
We assume that the preceding indices have been chosen so that the set
of i such that (gc)a c mc is the set i > r. Let A be chosen with
(A, a,) = 0 for i > r and » 0 for i < r. Then, for appropriate such A,
FA is a (g, КЭ-module and (FA)n is one dimensional (see 10.A.1.4). From
this, we see that, for an appropriate choice of A, FA has
(U(nc)/nkU(nc)) ® as a P-module quotient, with a one dimen-
sional P-module, with A acting by v g a* with (v, a) > r for a g Ф(Р, A)
and r > 0 as large as we please.
84
11. Completions of Admissible (g, К )-Modules
We may therefore assume that a is of the form F ® with F a
finite dimensional G-module, £ an irreducible °Af-module, and Re(p, a) as
large as we please for a g Ф(Р, A). Now, (1) implies that F ®	„ is a
P-module subquotient of a P-module of the desired form. The proof of
the lemma and hence of Theorem 11.4.1 is now complete.
11.5. Completions of (0, K)-modules
11.5.1. Let G be a real reductive group. We fix a maximal compact
subgroup К of G, a minimal standard p-pair (P, A) and a norm (2.A.2)
|| • • • || on G. If (tt, V) is a smooth Frechet representation of G, then we say
that V has moderate growth if for each continuous semi-norm A on И
there exists a continuous semi-norm vA on V and dA g R such that
A(Tr(g)t>) < llgll^^f)
for g G G, v g V.
Lemma. If (тг, H) is a Banach representation of G then (тг, H°°) has
moderate growth.
If A be a continuous semi-norm on H" then, by the definition of the
topology on FT, there exist elements uv ..., ud g U(gc) such that A(z?) <
E, ||тг(м,)гII for all v g H". Let j be such that u, g G;(0c) for all i. Let
Xj,..., x„ be a basis of G;(0C). Then
Ad(g) щ = lL<f>ki(s)xk.
к
Since each of the functions <p, is a matrix coefficient of a finite dimensional
representation of G, we see that there exist constants C and r such that
Wg)l < Cllgir. Thus,
A(Tr(g)r) < ЕкЖ«Н
i
= E II7r(g)7r(Ad(g-1) M,)r||
i
EI<p*r(^-1)l
i.k
< </C||g||r£||7r(g)7r(Xft)l7 ||.
к
11.5. Completions of (g, g)-Modules
85
Now, Lemma 2.A.2.2 implies that there exists a real number 5 and a
constant Ct such that ||-n-(g)|| < CjHgH5. Set рл(у) = dCC\ Y.k ||тг(хл)у||
and dA = r + s.
11.5.2.	We denote by 5^od(G) the category of all smooth Frechet repre-
sentations of G having moderate growth, with morphisms the continuous
G-intertwining operators.
Lemma. If (тт, И) g 5^od(G) and if W is a closed G-invariant subspace
of V, then W g 5^od(G). If W is a closed G-invariant subspace of V, then
H/^G^mod(G).
If A is a continuous semi-norm on W then A extends to a continuous
semi-norm on V. So it is clear that W g ^^(G). Let p denote the
canonical projection of V onto W. If A is a continuous semi-norm on V/W
then A ° p = p is a continuous semi-norm on V. Thus,
|m(77(g)V)I < Ilgll^Cv).
Let a be the induced action of G on V/W. Define
p(t>) = inf у (и).
p(u ) “ <’
Then, v is a continuous seminorm on V/W and
A(a(g)t>) < llgll^(r).
11.5.3.	Theorem. Let V g 5^od(G) be such that VK is admissible and
finitely generated. Then if A g j(VK\ A extends to a continuous functional
on V.
As in 4.3.3, it is easily seen that (VK)~ = (И')к. Also, the proofs of 4.3.5
and 4.4.3 go through virtually unchanged to this context. We will therefore
use these results in the course of this proof. We prove the theorem by
induction on dim a. If dim a = 0, then P = G, dim V < », and j(V) =
V* = V. Assume the result for all G with dim a < r - 1. We now do the
inductive step. If 5 is the standard split component of G and if 5 =# {1},
then dim а П °g < r. Since К c °G, we see that И as a representation of
°G is in 5^od(0G) and VK is admissible and finitely generated. Also
n c °g, and hence j(VK) is the same space of functionals on VK if we look
86
11. Completions of Admissible (g, К )-Modules
upon VK as a (°0, Ю-module. The inductive hypothesis now implies the
result in this case. We may thus assume that G = °G.
Let Д = {a,,..., ar] be the simple roots of Ф(Р, A). Set F, = Д - {a;}
and (P;, Л,) = (PF,AF^ (2.2.7). Then, dim Л, = 1 and dim°m, n a =
r - 1. Thus, the inductive hypothesis applies to °Af, , and by the preceding
argument to Mt. Consider the representations of Mt on k = И/С1(п*И).
Since V is of moderate growth as a representation of Mit Lemma 11.5.2
implies that Vt k g 5^od(Af;) for each i and k. Clearly, (Vitk)KriM. is
admissible. The inductive hypothesis now implies that if r g J(kVifk)KnM),
then т extends to a continuous functional on V\ k. We note that И, k has a
natural structure of a smooth P,-module. We denote the corresponding
representation of P, by (тг,д, Vik). тг1к]ом is of moderate growth.
We now apply the results in 4.3 and 4.4 with P replaced by P. Let
H; g a, be chosen so that a^Hj) = 8^. If A g (Vk)~ = then (in
the notation of 4.4)
А(тг(ехр tH,)r) ~ £ ехр(1м(Н,)) £ ехр(1Л)р,. мД(1Я,.; A,r)
Me£°	k = 0
as t -oo. The uniqueness of such an expansion implies that for each
p,,k there exists n such that р,-д ft(fH(; Л, п"И) = 0. Since pitlJL<k is
continuous on V, we see that Pi^^tHi', А, С1(п?И)) = 0. Let Ti k be the
natural continuous G-intertwining operator from V into Ipv given by
Ti,k(v)(g) = Pilots)
with pik the natural projection of V onto Vik. Set k = Ker Tik. Set
IF =	If w 6 IF, then Pi k(tHi, A.,w) = 0 for all i and ail ц, к.
The proof of 4.3.5 now implies that if A e (И')# and if d > 0, then there
exists a continuous semi-norm vd A on W such that
|А(тг(а)и') | ||a|rdpd A(w) for w g W, a^Cl(A~).
Let, for w g W, w К-finite, fA<w(g) = A(Tr(g)iv). Then fkw g ^A(G) is
right and left К-finite and Z(g)-finite. The argument at the end of the
proof of 3.8.3 implies that fkw = 0. This implies that (V')K(W) = (0).
Since the К-finite vectors in W are dense in W, we conclude that W = (0).
Set Tk = Ф, Tik. Then we have shown that Ker Tk = (0). Since VK
has finite length, there exists £0 such that TA() is injective.
The inductive hypothesis applies to each тг,1Ао. Let *Pi = PC]M,.
Then, there exists m such that (И/С1(п*°И))Кп M. injects into
/*р„((И/а(п?ои))лпл//)/*пГ(И/скп?ои))лпол/)-
11.5. Completions of (g, g)-Modules
87
The inductive hypothesis implies that this injection extends to a continu-
ous injection of И/С1(и*°И) into the corresponding C" induced represen-
tation (note that the inducing space is finite dimensional). Let cr, denote
the finite dimensional representation
(И/С1(п,^И))КпЛ//*п"'(И/С1(п^И))КпЧ
of P. If we apply C" induction in stages, we find that we have a
continuous injective G-intertwining operator T from V into Ip a, with
a = Фа(. We thus have the (g, K>module exact sequence
0^VK^Ip,a^IP,a/T(VK)^0.
If A g j(VK), then Theorem 4.1.5 implies that there exists A' g j(IPt„) such
that A' ° T = A. Now, Theorem 11.4.1 implies that A' extends to a continu-
ous functional fi on IP a. Thus, p,°T gives the desired extension of A.
11.5.4.	Corollary. Let V g 5^od(G) be such that VK is admissible. If
{(t, Ha) is a finite dimensional representation of P and if T g
Home K{VK, IP a), then T extends to a continuous intertwining operator
from Vto Ip a.
If p. g H*, then set Ам(г) = д.(Т(гХ1)). Then, Ам е/(Ик). Thus, the
previous theorem implies that AM extends to a continuous functional on V.
If v g V, then we define T(rXg) g Ha by fi(T(vXgj) =	for
g g G. Then, T(v) g 1% a and it is clear that T is continuous.
11.5.5.	Let Ж be the category of all admissible finitely generated
(g,К)-modules. Let HgcF, Let, for each Л =1,2,..., ok be the
representation of P on V/nkV and let Tk be the (g, K>homomorphism
of V into Ip ak as in 4.2.3. As we have seen in 4.2.3, if к is sufficiently
large then Tk is injective. In 4.2.4 we saw that Cl(Im Tk) is G-invariant in
Hak (here, we are using the notation of that number) and {Hak)K = Im Tk.
Set Vk be the space of C" vectors in Cl(Im Tk).
Lemma. If Tk and Tm are injective, then the natural (g, K}-module
isomorphism Im Tk to Im Tm extends to a continuous isomorphism of Vk
onto Vm.
Lemma 11.5.1 implies that Vk and Vm are of moderate growth. If
m = k, then there is nothing to prove. So assume that m > k. Let 5 be
the natural P-intertwining operator from V/nmV to V/nkV. Then 5
88
11. Completions of Admissible (a, A)-Modules
induces a continuous intertwining operator T from Ip a to Clearly,
Т(Ит) c Vk. On the other hand, 11.5.4 implies that fm °(Tk^K)~l extends
to a continuous G-intertwining operator from Vk to Ipt<r , with image
contained in Vm. This implies the result.
11.5.6.	Let V g Let Vk be as before, with Tk injective. If v is a
continuous semi-norm on Vk then we set pv(v} = v(Tk(v)). Then, the
completion of V with respect to the seminorms pv is a Frechet space V,
with Tk extending to a linear homeomorphism of V onto Vk. We pull back
the action of G to V and obtain a representation (тг, И) of G with
VK = V. The content of the previous lemma is that V is independent of k.
We call V the maximal completion of V of moderate growth. The following
result explains the terminology.
Theorem. If V g 5^,od(G) is such that VK is admissible, then the identity
map from VK to VK extends to a continuous G intertwining operator from V
to (VK). The correspondence V -» V defines a left exact functor from the
category Ж to the category 5^,od.
The first assertion is a direct consequence of Corollary 11.5.4 and the
definition of V. Let V, W g Ж and let A g Homfl К(И, IF). Let <rk,Tk
be as before for W. Then we may identify W with Cl(Tft(IF)) in T^„k.
Since V has moderate growth, Tk ° A extends to a continuous G-intertwin-
ing operator from V into Ip,ak- This defines A. With this definition we
have a functor.
Consider the exact sequence
О ->Л В C -> 0
in Then (Ker a)K = Ker a = 0. So a is injective. (Ker Д)к = Ker /3.
Thus, Ker/3 = С1(а(Л)) in В. Now,
«1^) - a(A) ->A
extends to a continuous intertwining operator from С1(а(Л)) to A by the
first assertion. Thus, a(A) = С1(а(Л)) = Ker/3. This is the content of the
second assertion.
11.6. Analysis of completions of (g,K)-modules
11.6.1.	We retain the notation of the previous section. If V g Ж, then
we define F^od to be the set of all A g V * such that there exists dA g R
11.6. Analysis of Completions of (а, К)-Modules
89
and for each v g V there exists an analytic function /A v on G with the
following two properties:
(1) xfA r(k) = X(kxv) for x g G(gc), k&K (here, as usual, xf =
R(x)f).
(2) |/A.,.(g)| < CA.J|g||^ for all g g G.
Throughout this section, we will assume (as we may) that ||g|| = ||g-1||
for g g G.
Lemma. P^od is a subspace of V*. IfVe ^^(G) and if VK g Ж, then
V с V*
' |ИА 'mod 
If A g KXd, then clearly cA g for c g C. If A, p g p^od, then set
= max(dA,(/M) and fA+lll. = fA l. + f^v for v g V. The second as-
sertion is an immediate consequence of the definition of 5^,od(G) (see the
proof of Lemma 11.5.1).
11.6.2. Let V g Ж and let (тг, Я) be a realization of V such that the
space of К - C" vectors of (тг, H) is equal to the space of G - C"
vectors (the realization in 4.2.4 has this property). Let V be the conjugate
dual (g, КЭ-module.
Proposition. Let (тг, H) be the conjugate dual representation of тг. If
(НГ = (HK)~ then (Н-У}Нк = (HK)*od.
Although this result is not terribly difficult, it is intricate. Let ( , )
denote the G invariant sesquilinear pairing of H with H (i.e., the original
Hilbert space structure on H). We will identify V with HK.
Let A g be fixed and non-zero. Let d > 0 be fixed and so large
that there exists C > 0 such that:
(i) ||77(g)|| < CHgir for g G G.
(ii) |/A.,.(g)| < CAi,..||g||d for g G G, V G V.
Here, we assume that the norm || • • • || on G is given as in 8.5.1 (there it
was called <p).
Let W be a finite dimensional /С-invariant and Z(gc)-invariant sub-
space of V such that L7(gc)LK = V. Let .., vm be a basis of W. Fix d0
such that
90
11. Completions of Admissible (g, К )-Modules
Set, for w g H,
l|w||l = E /" |(t’;,7r(g)w')|2||gir2‘*~‘*0</g.
i = l •'G
(1)	If g g G and if w g H, then
||7r(g)H’||1 < ||g||d + do/2||w’||1.
Furthermore, if v g H then ||r|li < CJMI.
Indeed,
H(g)y||i = E f |(y, ,ir(Ag)w)|2|Wr2‘*_‘A)tZr
i = lJG
= E [ |(y;,7r(x)w')|2||xg-1ir2‘i_‘iotZr.
; = 1 JG
Since ||xg-*|| > ||x|| Hgir1, the first assertion now follows.
As for the second,
Ml = E f , тг( g) w) |2||g||~2rf_rf° f/g
i-lJG
< m(max ||t>J|2)С2|Ы|2Д|1gird° dg,
since | (vt, 7r(g)w)| < ||?r(g)|| ||f,|| ||w||.
Let Ht denote the Hilbert space completion of H with respect to
||•  -||i. Then, (1) implies that 77(g) extends to a bounded operator rr/g)
on Hi for each g g G. In light of (1), it is an easy matter to prove:
(2)	(-П-!, Hj) is a (strongly continuous) representation of G.
We now prove:
(3)	H°° сЯ“, and the (g, КЭ-action on H°° induced from the (g, K)
action on is the original action.
We note that if к g К then both -rr(k) and тг/Л) are unitary operators,
which agree on the dense subspace V. We may thus concentrate on the
11.6. Analysis of Completions of (a, К )-Modules
91
action of g. Let X g g. Then, if v g H°°,
(Vi,TT(g exptX)v) = (t>;,7r(g)t>) +
+ t2(r;,ir(g expcX)tt(X)2v}/2,
with c between 0 and t depending on g, v, and i. Hence,
II ir(exp tX)v - v	||2
(* *) = ||----------------X)v||^
™ f к Tr(g exp tX)v - Tr^gyu	\2
= E f (ц-,-------------;------------ir(g)i7(X)z>| llgll 2 d° dg
= E [ l(^,ir(gexpcX)iT(X)2t))|2||gir2rf'rforfg,
4 i~lJG
with c a function of g (and i) taking values between 0 and t. The obvious
estimation yields
(**) < C2|t| f ||girdMg,
JG
with C2 = m(C(max ||r,||2)||7r(Ar2)r|Kmax|;|sl ||тг(ехр tAr)||))2 for |t| < 1.
Thus,
lim
/->0
ir(exp tX)u - v
- tt(X)v
= 0.
i
t
So if v g H°°, then g >-» 7r(g)v is of class C1 as a mapping into HY. (3)
follows by the obvious iteration of this argument.
We note that, by the definition of H1, V is dense in H\. We therefore
see that = V with the original (g, КЭ-module structure. We have
proved:
(4)	(tt j, Hj) is a realization of V.
Let CK be (as usual) the Casimir operator of f corresponding to B.
(5)	The topology on Я“ is given by the norms v IK7 + CK)"r||i,
n = 0,1,2,... .
92
11. Completions of Admissible (а, К )-Modules
This result will take some preparation. If v, w g H, then set
i JG
Then, ||* • • Hi is the norm associated with < , >1.
(a)	HCrlli < Cjlrlli for v g V.
Indeed,
IICy||2 = £ f |(y,^(g)^(C)r)|2||gir2d-‘iMg
(=i7g
= E /" l(^(Qy<,'£(g)y)|2llgir2‘y~‘y°dg.
Now, 7r(C)i>, = E; with au g C. (a) now follows from the Schwarz
inequality with C, = (Е,- y |a(/|2)1/2.
Let <X, Y> = -B(X, BY) for X, Y g g.
(b)	Let Xp..., Xd be an orthonormal basis of g with respect to < , >.
Set Д = Е,- X?. (Note that Д = C - 2CK). Then,
E<X(.y,X(.y>1<C2(|<Ay,y>1| + ||r|li).
i
Indeed,
^<Х^,Х^>1= £ I (t>,.,'ir(g)A'/t>)(t>,.,7r(g)A;.t>)||gir2‘i_‘i,)(/g
= - Е/ (f,,-n-(g)Ar;2t>)(«j,ir(g)«)||gir2‘i_‘i,)(/g
U G
~^f (vi^g)Xj'v')(vi,Tr(g)v)
i,j G
x^- Hgexpr^.ir^Mg.
at ,^o
11.6. Analysis of Completions of (g, К )-Modules
93
The argument in the proof of 8.5.4 implies that
d
IIg exp tXj11
2d-d0
< C3||gir2d-d».
We therefore have
<|<Дг,г>1| + C3^U,.r||il|r|li.
We observe that if e > 0 and if a, b > 0 then ab < j(ea2 + b2/e). We
use this observation with a = b = llflli, and e = Cj"1. Then, we
find
£ <Xtv, XiVh < 2| <Дг, r>i | + C4||r||i.
i
This proves (b).
We conclude (as in 8.5.5) that:
(c)	The topology in Я“ is induced by the semi-norms 11Д"vII1 for n =
0,1,2,... .
Now, Д" = E; (j	Thus, in light of (a),
||Д"г|| < Const £ IIQflli-
0<.j<.n
(5) now follows.
If у e К then set Ay = A ° Ey (here, Ey is, as usual, the projection of V
onto И(у) relative to V = Ф И(у)). Then, Aye V (here, if p, g V* then
p,(v) = д.(г)). We assert that ЕуАу converges in . Indeed,
A,,(g) = EAr,(g)
У
is the /^-Fourier series expansion of fAv relative to the left action of K.
Now,
E j IW^ Wr2''*1* - Е/ |A„.,(s)I llsll-"-""*,
( = 1 G	i,y G
94
11. Completions of Admissible (0, К )-Modules
by the Schur orthogonality relations. Thus,
m
”> E f\f^g)\ \\g\\~2d~d°dg
= Е/|ли*)|211*11'2^0^
i, у G
= Е/ [(77(g)г,.,лу)I iigir2d'd°</g
f,y
= Е/|(t’;,^(s)Ay)| llgir2d~d° dg = EllAjli.
i,v G	У
(1) implies that we have a continuous G-intertwining operator i from H
into Нг extending the identity map from V to V. On the other hand,
H" = (HK)~ (11.5.6). Thus, 11.5.6 implies that the identity map extends to
a continuous G-intertwining operator j from Я" into H°°. On V, i = j = I.
Thus, ij = ji = I.
This implies that the identity map of И to И extends to a continuous
isomorphism of Я“ onto H*. In light of (5), this implies:
(6) There exists r > 0 such that ||w|| < IK/ + CK)rw'Hi for w g V.
Notice that I + CK is invertible on V. (6) implies:
(7) IK/ + CK)~rAj|2 < 00.
(7)	implies that A extends to a continuous functional on H°°. This is the
content of the theorem.
11.6.3.	Corollary. IfVe J^then there exists V g 5^,od(G) with VK = V
and V'y = L^od. Furthermore, Vis uniquely determined by this condition.
Let (тт, H) be a realization of V such that H" = (V)~. Let (тг, H) be
the conjugate dual representation of тг. Then тг is a realization of V. We
may assume that the К - C“ vectors of (тг, H) and (тг, H) are respec-
tively equal to the G - C" vectors. Indeed, let (a, Ha) be a finite
dimensional representation of P such that there exists an injective
(g, К)-module homomorphism T of V into IP a. Then (И) is isomorphic
with СК7ХИ)) in ГР>„. Set H = СКПИ)) in HPa. Now, (HP ^ = HP^.
Let Z = {/ g HPi(f| </, T(V)> = 0). Then, H = H = HP &/Z is a realiza-
11.6. Analysis of Completions of (g, К )-Modules
95
tion of V. These realizations have the^ desired property. The preceding
result implies that (Я“)^ = F^od. Set V = H°°. Let U e 5^,od(G) be such
that UK= V and = L£od. The identity map from И to И induces
continuous G-intertwining operators from V and U into V. We identify
these spaces with their images in V. Let W = И_п U with the topology
induced by the continuous semi-norms of both V and IL_ Then W is a
Frechet space. Denote by В the inclusion map from IF to К and by^C the
inclusion map from W to U. Then (in the notation of 11.A.4), BT(V')y =
CT(U')y = L^od э W[v. Thus, Theorem ll.A.4.2 implies that В and C are
surjective. Thus, Z = V and Z = U. This proves the result.
11.6.4.	Lemma. Let (тг, H) be an admissible Hilbert representation of G
such that = (HK). Let (тг, H} be the conjugate dual representation of
77. Then (HKY = (HK)= .
This is a direct consequence of the previous results.
11.6.5.	Lemma. Let V, W e aY. Let T be a surjective (g, K)-homomor-
phism from V to W. Then T extends to a continuous intertwining operator
from V onto W.
Let (tt, H} be a realization of V such that H" = (V)~ (as in 11.6.3). T
induces an injective (й, К)-homomorphism, T, of W into V. Hx = C1(7W)
is a realization of TW with action тг,. Also, = C1(7W) in Я“. Thus,
Hi = (TW)~. Let (тг, H} be the conjugate dual representation of_Tr. We
identify HK with V under the natural isomorphism. Then H" = V. Now,
H|W| is the conjugate dual space to Hx. Thus, we have a continuous
surjective G-intertwining operator of H onto (тг1, Hf). We therefore have
a surjection from H* onto H°f = ((TW)A)= by the preceding results.
Since T is an isomorphism from {TWY onto W, the result follows.
11.6.6.	We now come to the main theorem in this direction.
Theorem. IfV& then V = V.
This involves a new idea. We will prove it in the next section.
11.6.7.	We record the following implications of the preceding theorem.
96
11. Completions of Admissible (g, К )-Modules
Theorem.
(1) If V e 5^od(G) and if VK e Ж, then the identity map of VK to VK
extends to an isomorphism of V onto V.
(2) Let V, W e 5^od(G) be such that VK,WK e JP. If T e
Home К(УК, WK) then T extends to a continuous G-intertwining operator
from V to W, with closed image that is a topological summand of W.
Let T be the continuous extension of the identity map on VK to a
G-intertwining operator of V to V. Then TT((VK)')^ = L^od. So
TT((VK) ) = V since V}'yK c L^od. Thus, T is surjective by ll.A.4.2. Since
T is clearly injective, (1) follows.
Since W = T extends to a continuous G-intertwining operator
from V to W. Let Z = C1(7V) in W. Then, Z\TVk = (7VK)*od (Z
= (^к)= (^k) ) by (1) and the previous theorem. As before, we have
TV = Z. Let (tt, Я) be a realization of WK such that the space of К - C"
vectors of H is equal to the space of G - C" vectors of H. Then, by (1),
we may assume that W = H°°. Set = C1(Z) in H. Then, Я, is G-
invariant, hence in particular /С-invariant. Thus, the space of К - C"
vectors of Hl equals Z. As a representation of K, H\ is a summand.
Thus, Z is a summand of W.
11.6.8.	Let JO^XG) denote the category of all V e 5^od(G) such that
VK is admissible and finitely generated and such that the morphisms are
the continuous G-intertwining operators whose images are topological
summands. If Heathen V e SW. If V,W e and if T e
Нош^ЛИ, И0, then let T denote the extension of T in 11.6.7 (2). Then,
T g Нош^г^ЛИ, И0 by 11.6.7 (2). Thus, V -» V is a functor from to
We note that, since V -» V is left exact and V -> V is right exact,
V -> V is an exact functor. 11.6.7 (1) implies that V = W in if and only
if V = W in <$O^(G). We therefore have:
Corollary. The functor V -» V defines an equivalence of categories between
and & Ж.
11.7. The proof of the main theorem
11.7.1.	The purpose of this section fe to give a proof of Theorem 11.6.6.
We say that V c is good if V = V (the notation is as in the previous
11.7. The Proof of the Main Theorem
97
section). We now collect some properties of “goodness”, with the eventual
goal to prove that all V g are good. If V g let be as in
11.6.1.
Lemma. Let V g <&. Then V is good if and only if V'y = Here (as
usual), if W is a topological vector space then W' denotes the space of
continuous linear functionals on W.
The^identity map of V to ^extends to a continuous G-homomorphism
5 of V into V (11.5.6). Since V'y = k^od, Lemma 11.6.1 implies that
Зг(Й|и=Ит%эЙ' = Ит*О£1
(see ll.A.4.1 for ST). Thus, ll.A.4.2 implies that 5 is surjective.
11.7.2.	Lemma.
(1)	IfV&JYis good and if Wis a (g, K)-module summand of V, then W
is good.
(2)	If V g and if Wis a good submodule of Vsuch that V/Wis good,
then V is good.
(3)	If V g JY and if every irreducible subquotient of V is good, then V is
good.
(4)	If V g aY is good then V (the conjugate dual (g, K)-module) is good.
To prove (1), we observe that if p is the (g, КЭ-module projection of V
onto W and if A g H£od, then A°p g Km*od. Indeed, take /Aop>/. = fyPiL-
Thus, if A g W^od then A ° p extends to a continuous functional on V.
Since CRH7) in V is W, this^ implies that A extends to a continuous
functional on W. Thus, W = W by the previous lemma.
We now prove (2). We may realize V as H°° with (тг, H) a realization of
V such that the space of К - C" vectors of H is equal to H°°. Then
Cl(LK) in is a topological summand of H°°. We may identify CRH7) in
/Г with W. If A g R^od, then A|(y g (W7)*^ Thus, since W is good, A|(y
extends to a continuous functional /3 on W. Since W is a topological
summand of V, /3 extends to a continuous functional 3 on V. Let
= A - З^. Then £ g L^od and t/w = 0. Thus, £ induces and element
v g (L/W7)*^. Since V/W is good, this element extends to a continuous
functional 77 on (V/WY. Let q be the natural projection of V onto V/W.
Then q extends to a continuous G-intertwining operator from V to
98
11. Completions of Admissible (g, К )-Modules
(V/W) . Clearly, r)°q^ = £. Thus, A = (rj°^ + S)^. This implies that
>> = ^*od- Thus, V is good.
We prove (3) by induction on the length /(И) of V. If /(И) = 1, then
V is irreducible and (3) is obvious in this case. Assume (3) for all V with
/(И) < r - 1. Suppose that /(И) = r. Let Vt с V be a non-zero irre-
ducible submodule of V. Then /(F/Fj) = r - 1, so V/Vx and Vx are
good. Hence, (2) implies that V is good.
We now prove (4). Let (тг, H) be a realization of V such that the
К - C“ vectors of (тг, H) and (it, H) are^ equal to the G - C" vectors
and such that H°° = (Й)-. Then H°° = V by 11.6.2. Since V is good,
V = V. Thus, 11.6.2 implies that Я“ = (K)=.
11.7.3.	Lemma. If (P, A) is a p-pair and if W g ^(от, К n M) is
good, then Ip w,v is Sood for v g aj.
Let (Q,B) be a minimal p-pair for °M. Then there exists a finite
dimensional representation a of Q such that we may identify W with a
submodule of IQ a. The closure of W in T£ a is therefore W. Now Ipw v
imbeds in	under induction in stages with closed image. Thus,
= Up.w, j” Also, Up,w,-г)" = Ip,w,v- Let (тг, Я) be a realization
of W such that the space of К n M - C" vectors of (тг, H) is equal to the
space of °M — C" vectors.
Since = ^p,w,-p)-’ И-6.2 implies that Ip = (IP>WtV)= . The
result now follows.
11.7.4.	Proposition. If V Ж is irreducible and square integrable, then
V is good.
Let A|/ = A be as in 4.3.5. Let (тг, H) be a realization of V let d be as
in Theorem 4.3.5. Let А ей (the admissible dual). Then Theorem 4.3.5
implies that there exists a continuous semi-norm pa on H°° such that
| A(тг(а) v) |	(1 + log ||a||)daAi'A( v)
for v g РГ, a g С1(Л+). Proposition 5.1.2 (combined with the proof of
5.1.3) implies that if v ^H°°, then fA v(.g) = X(ir(g)v) defines a square
integrable function on G. Furthermore, ||/A> J|2 < CvA(v). Fix A g V - {0}.
Let T(v) = fAu. Then T maps V into L2(G), and we have seen in 5.1.3
that Hx = C1(7V) in L2(G) is G-invariant and defines a realization of TV.
By the preceding, T extends to a continuous G-intertwining operator from
11.7. The Proof of the Main Theorem
99
H" to Hx. Since H" can be chosen such that №° = V, we have that
Hi = (TV)-. But Hi = (Hj)A. Thus, Proposition 11.6.2 implies TV is
good. Hence, V is good.
11.7.5.	Corollary. If V Ж is tempered, then V is good.
If V is tempered then every irreducible subquotient of V is tempered
(5.1.2, 5.5.1). Thus, in light of 11.7.2 (2), we may assume that V is
irreducible. Proposition 5.2.5 implies that there exists (P, A) a p-pair,
(cr, Ha) an irreducible square integrable representation of °M, and v g a*
such that И is a summand of IPta<iv. The result now follows from 11.7.4,
11.7.3, and 11.7.2 (1).
11.7.6.	To complete the proof of the main theorem we must introduce
some new ideas due to Casselman. We first give some preliminary results.
Fix (P, A) to be a standard p-pair, a an irreducible tempered representa-
tion of °M, v g a£ with Re(p, a) > 0 for all a g Ф(Р, A). As in 10.1, we
will look upon the operators T(p.) = and L(p.) = as acting
on the fixed space Iх = 1“. Let <p(z) = (pPa(v + zp) (10.5.4) for z g C. If
c > 0 set De = {z| |z| < e}.
Lemma. There exists e > 0 such that (p is holomorphic and nowhere zero
in the punctured disk De - {0} and <p has a worst a pole at z = 0. Further-
more, J(v + zp) is bijective for z g De - {0}.
The formula 10.5.7 (1) in the proof of Theorem 10.5.8 implies that there
exists c j > 0 such that the first assertion is satisfied. We now prove that
there exists e2 > 0 such that the second assertion is true for e2. Let у g К
be such that Ia(y} # 0. Set Jy(z) = J(v + zp)|^(7) and let L7(z) =
L(v + zp)|W?). Then, 10.5.4 implies that
L7(z)T7(z) = <p(z)T
on Ia(y). We therefore have the equality in the sense of meromorphic
functions,
det(T7(z)) = <p(z)m/det L7(z).
We assume that c, has been chosen so small that Re(p + zp, a) > 0 for
a g Ф(Р, A) and z g De>. Thus, if 0 < |z| < 6j then a zero of det(T7(z))
must be a pole of det L7(z). Now, 10.4.7 implies that the poles of
100
11. Completions of Admissible (g, К )-Modules
det( L(p)) lie on hyperplanes of the form (p, a)/2(p, a) = aaл + к, with
aai contained in a fixed finite set depending only on a and к = 0,1,2,... .
Thus, if det Jy(z) = 0 for some у g К with Ia(y) =# 0, then we must have
z = 2aa i + 2k - (p, a)/(p, a) for some к = 0,1,... and some aa , in a
fixed finite set. Clearly, there are at most a finite number of such z with
|z| < £p This implies that there exists e2 < c, such that if 0 < |z| < c2,
then det(J7(z)) =# 0 for all у g К with Ia(y) =# 0. If we take e = c2, the
lemma follows.
11.7.7.	We are now ready to introduce Casselman’s idea. We will use the
notation in 11.A.4.
We write I" for ^(/“). If f g I", then we write f(z, k) = f(z)(k), where
defined. Put Pf(z, nmak) = av+il+z)pa(m)f(z, k) and
pf(z,nmak) = av+(z l)pa(m)f(z, k)
for n g N, n g N, a g A, m g °M and к g K. Set (-7rP(g)fXz, k) =
Pf(z, kg) and (-n-p(g)fXz, k) = pf(z, kg) for g & G, к & К. The observa-
tions in 11.A.4 imply:
Lemma. ttp and ттр define (strongly) continuous representations of G.
11.7.8.	Let c > 0 be as in Lemma 11.7.6 and put A(z) = Jp^P(v + zp)
for |z| <6. Let к be the smallest non-negative integer such that z >-»
zkJP\p(v + zp)f is holomorphic on De for all f^I™. Put B(z)f =
zkJP\p(v + zp)f. If z g De, then
A(z)B(z) = B(z)A(z) = p(z)/,
with p, a holomorphic function on Df and p(z) # 0 for z g De - {0}. The
reader should not confuse this p with the p-function in Chapter 10.
If f g I", then we define Af(z) = A(z)f(z) and Bf(z) = B(z)f(z). The-
orem 10.1.6 implies that A and В are continuous linear operators on I“.
Furthermore,
AtTP(g)f = tTp(g) Af,
BtTp(g)f = 77p(g)Bf,
for f g 1“ and g g G.
11.7. The Proof of the Main Theorem
101
11.7.9.	We note that A(z*F) c z*F and B(z*F) c z*F. Thus, A and В
induce continuous maps
Ак:Г/гкГ ->Г/гкГ,
Вк-.Г/гкГ Г°/гкГ°.
The key result of Casselman is:
Proposition. Let p. be as in the previous number. If p has a zero of order
tn > 0, then A2m has closed range.
Define M: 1“ -> 1“ by Mf(z) = p(z)f(z). Then, MI“ c z"T“. Let M2m
be the operator induced by M on W = i“/z2"T“. Set IP, = 2">r/22">i“ c
W. Then, A2mB2m = B2mA2m=M2m. We assert
(*)
Note that the assertion implies the proposition. Indeed, IT, is a topolog-
ical summand of W, hence closed. B2m is continuous. So (*) implies that
/l2m(Wz) is closed.
We are left with the proof of (*). Let F: z"T" -»1“ be defined by
Ff(z) = (^.(z)/z'”)-1z_'”f(z). Then, F is continuous and FM = I, MF = I.
Let F2m be the map corresponding to F on zmV‘/zlmV‘ = Wx.
F2m\ -»Г°ДтГ° is a topological isomorphism. Let
z, = /;®z/; ©... ©z"1-1/;
and
z2 = zm/; © zm+1z; ©... © z2m~7“.
Then 1“ = Zj © Z2 © z2mI", with Zj and Z2 closed and Z2 © z2mI" =
z"T". Let E: l“/zml“ -»i“/z2mI“ be defined (using the preceding de-
composition) by E(x + zmI") = zmx + z2mI" for xeZj. Then, E is
continuous injective with closed range. We have M2mEF2m = I on JPj. If
z e Ker M2m, then z = Zj + z2, with z, = E(u), и e la‘/zml°°, and z2 e
W\. 0 = M2m(z) =	= M2mE(u). So 0 = F2mM2mE(u) = E(u\
Thus, KerM2m =
Let v	Then B2m(v) e So M2mEF2mB2m(v) = B2m(v).
Hence, B2mA2mEF2mB2m(v) = B2m(v). So if we set w = EF2mB2mv, then
B2m(A2mW - v) = 0. If B2mx = 0, then A2mB2mx = 0. Hence, M2mx = 0.
But KerM2m = ITj. So A2mw - v = M2my, у e W. Hence, A2mw - v =
102
11. Completions of Admissible (g, К )-Modules
A2mB2my. So v = A2m(w - B2mv\ Thus, В;'(Н\) c A2m(W). If w =
A2mv, veW, then B2mw = B2mA2mv = M2mv g Wx. Thus, A2mW c
This proves (*).
11.7.10.	We are now ready to prove that if V g is irreducible then V
is good. In light of Lemma 11.7.2 (3), this will prove the theorem. If
V g Ж is irreducible then set Ak = (Ли + p)0 (see 5.5.3), the Langlands
parameter of V. Let x be a homomorphism of Z(gc)to C and let 5 be
the set of isomorphism classes of irreducible V g Ж with infinitesimal
character x- Then 5 is finite. We show that the elements of 5 are good by
induction on their Langlands parameters relative to the partial order given
in 5.A. 1.1 (see 5.4.5). If V g S has a minimal Langlands parameter then
let (P, a, v) be Langlands data for V (5.4.1). If W is an irreducible
subquotient of Ip^v other than V, W must have a strictly smaller
Langlands parameter according to Corollary 5.5.3. Thus, V = IP a v. Since
<r is tempered, 11.7.5 and 11.7.3 imply that V is good. Let A = Аи for
some V whose class is in S. Suppose that we have shown that W is good
for A,y strictly less than A. We show that V is good. Fix (P, a, v),
Langlands data for V. We use the notation of the previous numbers. If p
is as in 11.7.8 and if p(0) Ф 0, then /1(0) = Jp^pty) is surjective. Hence,
V = IP a so V is good. Hence, we may assume that m > 0. Proposition
11.7.9 says that X = А2т(Г/г2тГ) is closed in I“/z2mI“. Set Y =
zl"/z2ml" and U = X + Y in 1“Д2т1“
(1)	UK/XK and (,r/z2mr)K/XK are good.
Indeed, X contains 2m - 1 copies of V in its Jordan-Holder series
since V = A(0)Ip a (r>/z2mr°)K has a composition series
м DN2D  DA2m_, DN2ffl = (0),
with Nj/Ni+l = Ipt<JtV (see ll.A.4.8). Thus, every irreducible subquotient
of
(F/z2mF)K/XK
has strictly smaller Langlands parameter than A. Hence, every subquotient
of
(T/z2m\^K/XK
is good by the inductive hypothesis, so 11.7.2 (3) implies (1).
11.8. The Action of 5(G) on Admissible Representations
103
(2)	XK is good.
Indeed, (Г°/г2тГ°)к is good by 11.7.3 and ll.A.4.8. Thus,
A2m(r/z2mF) = (A2m(r/z2mr)K)~ .
Since this space is closed in P/z2"1!00, it is also equal to
(A2m(r/z2mr)K)~.
Thus, XK is good.
(3)	YK is good.
Indeed, YK = (гГ/12аГ)к = (Г/г2т~1Г)к, which is good by 11.7.3.
(4)	UK is good.
Indeed, XK is good (2) and UK/XK is good (1), so UK is good by
11.7.2 (2).
We can now finish the inductive step. To do this, it is enough to show
that U/Y is closed in (T°/z2mr°)/Y. Indeed, if this is so, then as before
С/к/Ук is good. But,
UK/YK = Л,(Г/гГ)к = J-P]P(V)Ip^v V.
Now, (UK)~ = U and (YK)~ = У by (4) and (3) respectively. Thus, U is a
topological summand of r°/22mi“ and У is a topological summand of U.
That is,	= z © U, with Z closed in Г°/г2тГ°, and U = W © У,
with W closed in U. Let q be the canonical projection of i“/22mi“ onto
(y°/z2m\°°)/Y. Then the closed graph theorem implies that q~(ZS)W} is a
topological isomorphism of Z © W onto (la>/z2ml“)/Y. Thus, q(W) =
U/Y is closed in (r°/z2mr°)/Y. The induction is now complete.
11.8. The action of eZ(G) on admissible representations
11.8.1.	We retain the notation of the previous section. Let ^Z(G) be the
Frechet convolution algebra defined in 7.1.2. Let (тг, V) g 5^,od(G). Let v
be a continuous seminorm on V. Then there exists d > 0 and another
continuous semi-norm on V such that v(7r(g)r) < C||g||dgp(r). For
each r, there exists a continuous seminorm on ^Z(G) such that
104
11. Completions of Admissible (g, X)-Modules
l/(g)l < ^(/)||gll r. Thus, the integral
lGf(g)Tr(g) dg = ir(f)
converges and defines a continuous operator on V, with
|p(7r(/)r)|<Qd+4/)M,(r)
(here, d0 is as in 11.6.2 and
C= [Ilgir^Mg <00).
JG
Furthermore, rr(f*g) = тг( f)-rr(g). Hence, тг is a continuous representa-
tion of e/(G).
11.8.2.	Theorem. Let (тг, V) g 5^,od(G) be such that VK g If v g
V, then Tr(c/’(G))r is a closed G-invariant subspace of G. VK is irreducible
as a (g, K)-module if and only if V is irreducible as an (algebraic) ^Z(G)-
module (i.e., the only abstract subspaces of V that are ^(G)-inuariant are V
and (0)).
Let W = C1(7t(^Z(G))l>). We note that Tr(L(g)/) = 7r(g)7r(/) for g g
G. Thus, FK is a G-invariant closed subspace of V. To prove the first
assertion it is enough to show that V = tt(^(G))v if Cl(7r(^/(G))f) = V.
Let T(f) = ir(f)v. Then T is a continuous linear map of ^Z(G) into V.
We must prove that T is surjective. To do this, we must show that TT is
injective and that Im(Tr) is weakly closed in ^(Gf (ll.A.4.1). Since Im T
is dense in V, TT is injective. Let Aa be a net in V such that Tr(Aa)
converges weakly to p g <У(СУ. Let Z = Ker T. Then Tr(AaXZ) = (0).
Thus, p(Z) = (0). Hence, p defines a linear functional, £, on Im T.
(1) 1тТэИк.
Indeed, if у g К let ay be as in 1.4.6. Since Im T is dense in V and VK
is admissible, Vk = тг(а7) Im T. Now,

11.9. Poisson Integral Representations
105
(here, the expression in the braces denotes convolution over K). Since
ay*Kf^(G),
(Theorem 7.1.1 (3)), (1) follows.
Let a = We assert that a g (FK)*od. Assume this for the moment.
Then, Theorem 11.6.6 implies that a extends to a continuous functional
on V and that (Тт(а) - д.)|£ ^(G) = 0 for all у g К (see 7.1.1 for nota-
tion). Since к Ey^(G) is dense in ^Z(G), this implies that TT(a) = ц.
This would complete the proof of the first part of the theorem.
We are left with the proof of the assertion. Note that the proof of
Theorem 7.1.1 implies that (L, ^Z(G)) g ^nod(G). This implies that if, for
и g VK, we choose w g eZ(G)K be such that T(w) = и and set fa Ss"> =
then fau satisfies 11.6.1 (1), (2). So a g (FK)*od as asserted.
The second assertion of the theorem is now easy. Assume that VK is
irreducible. Fix v g V, w =# 0. Then, 7r(^/(G))f is closed, G-invariant,
and M^(G))v)K * (0). So (тг(с/(С))у)к = VK. Hence, ir(S(G»v = V.
Thus, the representation of ^Z(G) on V is algebraically irreducible. If the
representation of ^(G) is algebraically irreducible and if W is a non-zero
(g, КЭ-submodule of VK, then ir(c/’(G))W с С1(И0. So W = (0) or
1P = VK.
11.8.3.	The preceding result is (as can be seen from its proof) a reformu-
lation of Theorem 11.6.6 that was suggested to the author by J. Bernstein.
11.9. Poisson integral representations
11.9.1.	The purpose of this section is to prove a generalization of the
Poisson integral representation theorem for an important class of func-
tions. We first introduce the class. As usual, G(g) will be thought of as the
algebra of left invariant differential operators on G (i.e., it acts by right
differentiation). We denote by srfmod(G) the space of all f g C“(G) such
that f satisfies the following three properties.
(1)	f is right /Gfinite.
(2)	f is Z(g)-finite.
(3)	There exists d (depending on f) such that if g g U(g), then there
exists Cg < oo such that
|gf(x)| <С>|Г, xgG.
106
11. Completions of Admissible (g, К )-Modules
This class of functions appears in many contexts in the theory of real
reductive groups. For example, if Г is a discrete subgroup of G such that
T\G has finite invariant volume then s/mo(fG) П C“(T\G) is usually
called the space of automorphic forms on G.
11.9.2.	We fix a minimal p-pair (P, A).
Theorem. If f g £/mod(G), then there exists a finite dimensional represen-
tation of P, A g (/£ <,/, and v g Ip a such that
k^p,As)v)=f(g), g^G.
Set Vf = span{(7(gc)Kf) (here, kf(g) = f(gk)). Then 11.9.1 (1), (2)
combined with 3.3.2 imply that Vf is an admissible finitely generated
(g, К)-module. Define 8(u) = u(l) for и g We assert that 8 g (Pp*od.
Indeed, if и = k^fig), ki g K, Xj g L7(gc), then set f8u = u. Then,
clearly, fSu satisfies 11.6.1 (1). The definition of J3^nOd implies that fs u
satisfies 11.6.1 (2). This proves the assertion.
Let (a, be a finite dimensional representation of P such that there
exists an injective (g, КЭ-module homomorphism of Vf into IP~. Then
there exists a (g, КЭ-module homomorphism T of IP a onto Vf. Thus,
«5°Tg	Thus, 8°T extends to an element A of (/^J (11.7.3,
11.6.1)	. Let v g Ip a be such that Tiv) = f. Then А(тгР>o.(g)r) = fig),
g g G. This completes the proof.
Note. The proof of the preceding theorem uses only the fact that IP a =
ilp a)= (11.7.3), not the more delicate result proved in 11.8. We also note
that any function given as in the statement of the preceding theorem is in
11.9.3.	Now we apply the preceding technique to a more special class of
functions. Let (r, VT) be a small tf-type (11.3.1). Let ^.,mod,/G) denote
the space of all functions f g 33/mod(G) such that:
(1)	R(K)f spans an irreducible representation of К equivalent with r.
(2)	xf = v(yT(x))f for x g G(gc)\
Let a = t]om. Let „ be the (g, КЭ-module homomorphism of YT,V
into /р><т>р defined in 11.3.6. Fix a ^-module isomorphism T from V, onto
^Р.о-./т).
11.9. Poisson Integral Representations
107
Theorem. If д.т „ is surjective, then the map PT v given by
pT,A*- ®	= ^p.a,As)Tv),
from (Ipi<T,vy ® VT to ^.,mod,p(G), is a linear bijection.
Let /g .Q^mod>„ and let Vf= span(7?(t7(gc))7?(K')/) (as before). Then
(1), (2) imply that there exists v g V. and a (g, /module homomorphism
Sf of У7,1' onto Vf such that 5y(l ® u) =/. Let <5 g V* be defined by
8(u) = u(l) (as before). Then, the argument in the proof of the previous
theorem implies that 3 ° Sf g (Ут,1')*(к1. Hence, there exists a continuous
functional p on (Ут,1,)= such that = 8° Sf. Now, if p7 v is suijective
then it is bijective (see 11.3.6 (2) and the argument in 11.3.6 (2)). Thus,
11.6.3 implies that pTV extends to a topological isomorphism W of (YT v)=
onto (1Р <Г1,У • Lemma 11.3.6 implies that (fp><7>J“ = Ip,a,v- Thus, A =
p ° W~l e(J”vvy. Let v 6 Ит be defined by Ml ® u) = T(v). Then (as
before), /(g) = А(тгр „ v(g}T(v}}. This proves that PT v is suijective. We
now prove that the map is suijective. Let vl,...,vn be a basis of VT. Let
Aj,...,A„ e (1р,а,„У, and assume that
£P^(A,®r,.) =0.	(3)
i
We must show that A, = 0 for i = 1,..., n. Burnside’s theorem (Lemma
10.A.3.1) implies that for each i there exists u, g t/(tc) such that
т(м,)г; = 30f,-. Now, (3) implies that (тг = тгР a v)
LA,(7r(g)7’(r()) = 0	(4)
i
for all g g G. If we differentiate (4) by uf (on the right) then we find that
0 =
i
= EA,(7r(g)7’(T(M;)r,.)) = Ay(7r(g)7’(r;)).
i
If we differentiate this expression by all of L7(gc) and translate on the
right by K, then we find that
A;(span(7r(I7(gc))7r(K')7’(r;)) = (0).
Our hypothesis says that
span(Tr((7(gc))7r(K')7’(r;.)) = IP^V.
Since this space is dense in Ip a v we see that A; = 0, as was to be proved.
108
11. Completions of Admissible (g, X)-Modules
11.9.4.	Example 1. The simplest example of a small A^-type is the trivial
one dimensional representation of K. In this, a is the trivial °Af-type and
/“ = C°°(°M \K). Also, in Kostant [5] it is proved that if
then the constant functions in C"(°M\K) form a cyclic space for IP<atV.
Since
= a(kg)v+p,
we define, for b g °M\ К and x g G/K, p„(b, x) = a(kg)v+p if b is the
coset of к and x is the coset of g. Also, we define C“od V(G/K) to be the
space of all f g j2/mod(G) such that f(gk) = f(g), к g K, g g G, and
x/ = "(YtrivC^))/ f°r a" x e G(gc)^- In this case, we have:
Theorem. The map from C“(°Af\K7 to C“od U{G/K) given by T >-»
(g >-> T(p„(-, g)) is bijective if (1) is satisfied.
Note. Since v ° ytriv = sv ° ytriv, it follows that C“od V(G/K) =
C^od^/G/K), and we can choose v so that Re(p, a) > 0 and hence so
that (1) is satisfied.
11.9.5.	Example 2. In this example, we concentrate on G = SL(2, R).
We take К = SO(2) and P the upper triangular matrices in G. Since К is
abelian every r g К is one dimensional, hence small. If в g R, then we
set
v(a\ = cos в	sin в
' [-sin0 cos 0 *
If к g Z, then we define тк{к(.бУ) = e‘ke. If к = 0, then r0 is the trivial
representation of K. We therefore concentrate on the case к =# 0. Set
Then, we identify a£ with C under v >-+ v(H\ We set YTk,v = Yk,v. If we
use the calculations in 6.1, then we find that if к is even and Re v > 0,
then Yk'v = Ip i'V if v £ {1,3,..., |£| - 1}. If к is odd and a g °M is
defined by a(-/)= -1, and if Rei^O, then Yk'v = IP a v if v
{0,2,..., |£| - 1}. We note that if к is odd and v = 0, then Yk’° is never
11.9. Poisson Integral Representations
109
isomorphic with Ip,a>0- Indeed, the latter representation is not cyclic for
any isotypic component. We leave it to the reader to find a finite dimen-
sional representation (of lowest dimension) £ of P such that У* 0 is a
quotient of IP {. The other exceptional values behave similarly.
Note. If к is odd then the module Yk-° depends only on the sign of k,
and if к > 0 then it has a unique non-zero quotient Dg ; if к < 0 it has a
unique non-zero quotient Dg . Also,
Л*,<г,0 S Dg © Dg .
Thus, if, say, к > 0, Yk'° gives a non-trivial extension of Dg by Dg .
11.9.6.	Example 3. If К is not semi-simple (e.g., G =
SU(p, q), SO(n, 2), Sp(n, R)), then К has many one dimensional represen-
tations. These are, of course, small. The preceding example implies that
this case is quite delicate. Also, if К is semi-simple but G is non-linear,
then there are always small /С-types that are not one dimensional. The
next example gives small /С-types for a class of linear groups.
11.9.7.	Example 4. Let G, = SO(2n, 1)°. Then we may take =
SO(2n) (imbedded with a 1 in the 2n + 1 diagonal position) and we may
take Afj = SO(2n - 1) (imbedded with a 1 in the 2n diagonal position).
We take G to be the spin group corresponding to signature (2n, 1)
(ll.A.2.7), which we look upon as a two fold covering of G,. Then we may
take К = Spin(2n) and °M = Spin(2n - 1). If r is either of the half spin
representations of К then the restriction of r to °M is the spin represen-
tation of Spin(2n - 1). Thus, т is small. The previous result therefore
applies. This implies a Poisson integral formula for the eigenvalues of the
Dirac Laplacian on even dimensional hyperbolic spaces. It should be easy
to do something analogous for SO{2n + 1,1), since in this case the spin
representation of К splits into the two half spin representations of °M (so
it is pretty small).
Note. In the case of Spin(2n, 1), the representation /Лт|Од/>0 splits into
two irreducible pieces corresponding to the unique non-trivial quotient of
Ут 0 for т one of the two half spin representations of K. Since Spin(2,1)
= SL(2, R), this example is a direct generalization of situation in the note
in Example 2.
по
11. Completions of Admissible (g, К )-Modules
11.10. Notes and further results
11.10.1.	The exposition in 11.1 is strongly influenced by the treatment in
Helgason [2]. We have included a complete exposition of these standard
results since they will play an important role in the next chapter. The
theorems in 11.1 are all due to Chevalley except for Theorem 11.1.5, which
is due to Harish-Chandra.
11.10.2.	Lemma 11.2.1 is due to Kostant [4]. Theorem 11.2.4 is a due to
Kostant-Rallis [1] (see also 11.10.4).
11.103.	Theorem 11.3.6 is a special case of a theorem of Hecht-Schmid
[2] (Osborne’s conjecture). It is quite possible that one can derive (the
critical for our purposes) Lemma 11.3.7 from this theorem without using
the theory of the УT’ v modules.
11.10.4.	Theorem 11.4.1 was first proved by Casselman under the condi-
tion that a is irreducible. Kashiwara, at about the same time, showed that,
using the theory of hyperfunctions one could derive the result for irre-
ducible a from an unpublished result of Rader. The result as given and
the method of proof (at least for linear groups) is due to Casselman and
the author (с/. Wallach [3]). We note that in the course of the proof of
that theorem we have also proved the following theorem of Kostant-Rallis
[1].
Theorem. Let g = t © p be a Cartan decomposition of g. Let be the
subspace of 5(pc) corresponding to the action Ad|K of К on p in 11.A.1
(denoted H in that number). Let I = 5(pc)K. Thenthemap ® I -> 5(pc)
given by h ® i >-» hi is a linear bijection. Furthermore, if у e К and (т , Vy)
e у, then
dim Нотк(И7, <^) = dim VyM.
The first assertion is Theorem 11.2.4. The second is Frobenius reciproc-
ity combined with 11.3.6 (2).
11.10.5.	The material in 11.5-11.8 is an outgrowth of a long term project
of Casselman and the author. One should consider this to be joint work
(we have indicated material that should be attributed only to Casselman).
Casselman’s version of this theory can be found in Casselman [1]. There is
1 l.A.l. Some Results on the Action of a Compact Group
111
another general theory due to Schmid [3] that constructs functorial com-
pletions of elements of Ж. In these completions (essentially the spaces of
analytic vectors and hyperfunction vectors of our completions) the algebra
C^(G) plays the role of <yTG). Schmid developed his theory in order to
study hyperfunction solutions of invariant differential equations on quo-
tients of real reductive groups.
11.10.6.	Theorem 11.8.2 is analogous to a theorem of Howe [1] for
nilpotent groups. Let (тг, И) e 5^,od(G) be such that VK is irreducible and
V = for (тг, H) a Hilbert representation of G such that тг (K is unitary.
Note that the main theorem implies that V is up to isomorphism the
unique element of 5^,od(G) with VK = V. We assume that dim V = ». Let
{t>J be an orthonormal basis of VK consisting of elements that are each
contained in some /^-isotypic component of V and such that the eigen-
value of the Casimir operator of К on vt is increasing in i. If x g И, then
x = E„ x„v„. The map x >-» (x,, x2, ..) (which we denote 5) maps V to
the space of all rapidly decreasing sequences (i.e., n*|x„| -> 0 as n -> oo
for all k). One can show (using the methods of du Cloux [1] and the main
theorem) that 5 is bijective. Let W be the algebra of all rapidly decreasing
oo x oo matrices ((« + m)kanm -> 0 as n + m -> oo). Then du Cloux [2] has
shown that 57r(^/(G))5-1 = W. In du Cloux [1], the interested reader can
find a complete exposition of the structure and representation theory of
the algebra W. In du Ooux [2], a generalization of the notion of minimal
completion is given for Lie groups that are finite coverings of open
subgroups of affine algebraic groups over R.
11.10.7.	The method of proof of Theorems 11.9.2 and 11.9.3 is taken
from Wallach [3]. At that time, the author was unaware of the important
work of Oshima-Sekiguchi [1], which (in particular) gives a proof of
Theorem 11.9.4.
11.A. Appendices to Chapter 11
11.A.1. Some results on the action of a compact group
on a symmetric algebra
ll.A.1.1. Let И be a finite dimensional vector space over C with (Hermi-
tian) inner product < , ). Let U be a closed subgroup of the unitary group
of ( , ). We put the usual inner product on the Л-fold symmetric power of
112
11. Completions of Admissible (a, A)-Modules
V, Sk(V). Then, the symmetric power action of U is by unitary transfor-
mations on Sk(V). We put an inner product on 5(И) as the orthogonal
direct sum of the Sk(V) = Sk. The action of U on 5(И) is by algebra
automorphisms. Let I denote the (/-invariants of 5(И) = 5. Set 5+ equal
to Ф^>05<:(И) and I+ = S+ П I. Then, SI+ is a proper graded ideal of
5(И). Set H = ®kHk with Hk = {s e 5*| <5, SI+ n Sk> = (0)}.
Lemma. 5(И) = I+S(V) Ф H. The map I ® H S given by и ® h •-» uh
is surjective.
The first assertion is clear. Denote the mapping in the second assertion
by ЧС Clearly, 1 e Im’F. Suppose that Im Ч' э for j < к, к > 0. Then,
Im4< э Sk n I+S. Thus, if 5 e Sk and if <s, Im 40 = (0), then 5 e H.
This shows that Sk c Im Ч'.
ll.A.1.2. Lemma. If A g End(K), let Sk(A) be the action of A on
Sk{V). Let r be the maximum of the absolute values of the eigenvalues of A.
Then the series Xk^otr(Sk(A)) tk has radius of convergence at least 1/r
and is equal to det(/ - t4)-1 where it converges.
Write A = s + n with i semi-simple and n nilpotent, with [i, n] = 0.
Let x = n, y, h be a TDS in End(K) such that [i, u] = 0, и = x, y, h
(8.A.4.1 applied to {Xe End(K)| [X, i] = 0}). tr(5*(X)) = p(X) is a
polynomial function on End(K) with p(gXg~l) = p(X). Thus, if t g R,
we have
p(s + n) = p(e~'adh(s + «)) = p(s + e~2,n).
If we take the limit as t -> +oo, then we find that p(A) = pts). Thus, if
Aj,..., A„ are the eigenvalues of A counting multiplicity, then
tr(5*(^))=	£	А‘Г--А^.
<i+ ••• +>„=*
Thus, |tr(5*(^))| < dimfS^K)) rk < C(1 + k)nrk. Thus, the assertion
about the radius of convergence is clear. The sum of the series is now
clearly equal to Ilf.jd - tA,)-1 = det(/ - t4)-1.
ll.A.1.3. Let du be the normalized invariant measure on U. Set
Pk= ( Sk(u) du.
Ju
Then, Pk is the orthogonal projection of Sk onto Ik. Thus, tr Pk = dim Ik.
11.A. 2. Small «-Types
113
Lemma. E*.>0 tk dimf/*) has radius of convergence at least 1, and in the
range of convergence it is equal to
f det(/ - tu)' du.
Ju
dim Ik < dim Sk < C(1 + k)n. Also, if и g U then the radius of con-
vergence of Y.k >o tk tr5*(u) is at least one. Thus, if |t| < 1, we have
f det(/-tu)~' du = £ tk f irSk(u)du= J^t^trP^.
JU	k>0 Ju	k^O
11.A.2. Small K-types
ll.A.2.1. Let й be a semi-simple Lie algebra over R. Let йс be the
complexification of й and let Gc be the connected, simply connected Lie
group with Lie algebra g. Let GR be the connected subgroup of Gc with
Lie algebra g. Let G be a covering group of GR with covering homomor-
phism p. Let Z be the kernel of p. Fix K, a maximal compact subgroup of
G (note that Z с K). Let U be a compact form of Gc such that
Gr П U = KR = p(K}. Let (P, A) be a minimal (standard) p-pair in G,
P = °MAN a standard Langlands decomposition. The purpose of this
appendix is to give a proof of:
Theorem. Let x e Z- Then there exists an irreducible representation (т, V)
of К such that TjZ = xl and т^м is irreducible.
The representations r as in the statement of the theorem are the small
/(-types of the title of this appendix. D. Vogan has informed us that he has
a direct proof of this result. We will however give a case by case argument
since the nature of these small /(-types becomes fairly transparent during
the proof. We should warn the reader that some of the arguments below
are somewhat sketchy for the E-type groups. To fill in all of the details the
reader will need to have an understanding of the tables in Chapter X of
Helgason [1]. Also, we recommend that he use the tables in Tits [1]. It
should also be pointed out that this result is related to the theory of
models in Bernstein et al. [2].
We first give a reduction to the case when g is simple. Let
Й= Й1 ® Й2®	® в
114
11. Completions of Admissible (g, К )-Modules
be a decomposition into simple ideals. Then
Sc = (Si)c ® (вг)с ® ’" ® (вг)с-
Let G, c be the connected simply connected Lie group with Lie algebra
(в,)с. Then
^l.C X ^2,C ’ ’ ’ X Gr C
is the connected simply connected Lie group with Lie algebra gc. Also, if
G, R is the connected subgroup of G, c with Lie algebra fl,, then
^l.R X ^2,R X X ^r.R
is the connected subgroup of flc with Lie algebra fl! hence, it is GR. If
G, is the connected, simply connected covering group of G, R, then
Gj x • • • x Gr is the connected, simply connected Lie group with Lie
algebra fl- Oearly, the covering homomorphism of G onto GR is just the
product of the covering homomorphisms of G, onto G, R. Let Z, be the
kernel of the corresponding covering homomorphism. Then, Z = Zj
x • • • xZ,. Also, if Кi is the corresponding “K”, °Mi the corresponding
“°M”, then К = x • • • x Kr and °M = °Ml x • • • x °Mr. If x e Д
then x = Xi " ' Xr with Xi e Z,-. Thus, if t, is a small /С-type for x, (as
before), then Tj ® • • • ® rr is one for x- If 6 were some intermediate
covering then it would be clear that if x e Z^, then r would push down
to the corresponding K. This completes the reduction to the case when fl
is simple. We now begin the proof in this case.
ll.A.2.2. If Z = (1), then we can choose r to be the trivial representa-
tion of K. This simple observation implies the theorem in the following
cases:
(1) G = K.
(2) flc is not simple over C.
Indeed, in case (1), G = К = GR. In case (2), g has a complex structure
and G = Kc = Gr.
In light of this, we assume for the rest of this appendix that gc is simple
and that К ¥= G.
ll.A.2.3. We now study the simplest cases not covered by the previous
observations. We assume that t is not semi-simple.
11.A.2. Small «-Types
115
Lemma. Under the preceding assumptions, if t is not semi-simple then
t = Rh Ф tj, with tj = [t, t], and Ad h has eigenvalues 0, i, —i on йс-
Furthermore, t =
For a proof of this result see, for example, Helgason [1].
Let G be the simply connected covering group of G. Let q be the
covering homomorphism of G onto GR. Set К = q'(K). Then К =
R x Kx, with Kx the connected, simply connected Lie group with Lie
algebra I,.
(1) q is injective on (0, KJ.
Under our assumption, there exists a maximal abelian subalgebra of t,
t, that is a Cartan subalgebra of й (cf. Helgason [1]). Let t) be the
complexification of t. Put H = ih. Choose a system of positive roots
Ф+с Ф(йс, I)) such that if a e Ф+ then a(H) > 0. Let a1,...,a/ be the
simple roots in Ф+. We may assume that a^H) = 1. If afH) > 0 for
some i > 1, then Д = La; g ф+ and Д(Я) > 1. This is impossible since
the eigenvalues of ad H are 0,1, -1. Thus, afH) = 0 for i > 1.
Let t[ = i( Фу> jRHa ). Then tj is maximal abelian in tj and {a2,..., az}
is a simple system for a system of positive roots Ф^ for Ф((11)с,(11)с).
Let A( g fj* be defined by
2(A,, a;)/(a;, a J =	,	1 < i, j < I.
Let Ft be the irreducible finite dimensional representation of U with
highest weight A, relative to Ф+. If i > 1, then the cyclic space of the A,
eigenspace for Ft under Kx is the irreducible representation of Kx with
highest weight A,1t . Thus, every irreducible representation of Kx pushes
down to q(Kj. The Peter-Weyl theorem now implies that q is injective
on Kx.
Let Z = Ker q. Then the exact sequence of group homomorphisms
- ч
1 Z КKK 1
induces
1 -> Z -> K/Kv -> KR/KX 1.
Qearly, K/Kx is abelian and KR/KX is isomorphic with the circle group
51. We may thus look upon K/KY as the additive group of R and Z as Z.
116
11. Completions of Admissible (0, К )-Modules
Let q' be the covering homomorphism of G onto G. Set Z' = Ker q'.
Then, Z = Z/Z'. If x e Д then x lifts to a unitary character of Z and in
turn extends to a unitary character of K/Kx. This gives a one dimensional
representation of K, r, but Ker т э Z'. Hence, r pushes down to K.
Clearly, t(z) = ^(z)for z e Z.
Under the hypothesis of this number, we have proved Theorem ll.A.2.1.
ll.A.2.4. In light of the preceding results, to complete the proof of
Theorem ll.A.2.1 we may assume that t is semi-simple. Thus, the simply
connected Lie group with Lie algebra t is compact (Weyl’s theorem).
There will therefore be no loss of generality if we assume that G is simply
connected. We therefore make this assumption and also assume that t is
semi-simple.
Lemma. Under the above assumption Z is either trivial or isomorphic
with Z2.
This result will take some preparation. We first assume that there is a
maximal abelian subalgebra t of t that is a Cartan subalgebra of й- Let
I) = tc and let Ф = Ф(йс,I))- Set Фк = Ф(1С, t)). Then, it can be shown
that there exists a system Ф+ of positive roots for Ф such that if a1;..., a;
are the simple roots for Ф+, then <£ Фк and a, g Фк for i > 1.
Furthermore, if /3 is the largest root of Ф+ then /3 = E n,a, with = 2,
and if у = Em,a, g Ф then у g Ф if and only if m, = ±2. Set a0 = -Д.
Finally, there exists a system of positive roots PK for Фк such that
a0, a2,..., a, is a system of simple roots for PK. For a proof of this result
see Borel-de Siebenthal [1]. We replace the Killing form by the positive
multiple that has the property that the shortest root(s) have length 2.
We now prove the lemma in this case. We first assume that all of the
roots have the same length (which has been normalized to 2). Let TR be
the kernel of the restriction of the exponential map of GR to t and let Гк
be the kernel of the restriction of the exponential map of G to t. Then,
Z s rR/TK (see, for example, Helgason [1], Chapter VII, Section 7). Also,
in this reference one sees that
TR = 2-пч’[ ® ZHa ]
''isjs/ ’>
and
= 2ttIZH & 2iri( ® ZW |.
K “°	Ч<7</ и
Since the coefficient of a1 in /3 is 2, it follows that TR/rK s Z2.
11.А.2. Small К-Types
117
Suppose that there are two root lengths. We note that a0 must be long
(the highest root must be of maximal length). The maximal ratio of root
lengths is either 2 or 3 (cf. Bourbaki [2]). We first assume that the maximal
ratio is 3. Then, йс *s °f type 62.
Д = 2at + 3a2.	(“i,“i)=6,	(a2,a2)=2.
So
TK = 27n(z((2/3)H„, + Ha2)] Ф 2mZHa2.
TR = (2ТП/3)ZHai Ф 2iriZHa2.
This implies that rR/TK = Z2 in this case. We are left with the cases
where the largest ratio of root lengths is 2. We use the tables at the end of
Bourbaki [2]. We look at then case by case. We will use the labeling of
roots given as in this reference, so the root previously labeled will in
general have another label. We also use the notation involving extended
Dynkin diagrams.
1.	B„ I > 3.
“o
О
I
о   О  О “ • • • “ о	о
«1	«2	а/-1	а1
(3=0^+ 2(а2 + ... 4-а,).
2.	С,, I > 2.
о => О --- О “ • • • “ о <= о
а0	а2 at_i at
fl = 2(at + ... +a/_j) + а/.
3.	F4.
О --- О ---- о => о о
«О “1	«2	«3	«4
fl = 2«[ + За2 + 4а3 + 2а4.
We now look at the cases that correspond to 1,2,3.
118
11. Completions of Admissible (g, X)-Modules
1.	We can choose for the “a/’ any a;- with j > 2. If 2 </ < I, then it is
clear as before that Гк/Гк s Z2. We look at the case j = I. Then,
I'd = rril © ZH | Ф 2iriZH„ .
4^/-i	'/
rK = iriZi(Hai + 2(Ha2 + • • • +Hai)) Ф TTi( © ZhA
' 1 <j <Л-1	'
So TR = Гк in this case.
2.	In this case, we may choose any a; with 1 < j < I - 1. We leave it to
the reader to check (as before) that in these cases TK = TR.
3.	There are two choices for j: 1 and 4. If j = 4 one finds that TK = TR,
and if j = 1 then TR/rK s Z2.
ll.A.2.5. We must now prove the lemma in the cases when a maximal
abelian subalgebra t of t is not a Cartan subalgebra of й, so we assume
this. We will use the table in Helgason [1], p. 518, and label the cases as
they are given therein. The pertinent cases are A I with n > 3, A II, BD I
with both p and q odd, E I, E IV. We now handle the cases one at a time.
A I. Gr = SL(n, R), n > 3. KR = SO(ri), К = Spin(n) so Z = Z2.
A II. In this case, KR (as given in the table) is Sp(n) which is simply
connected. So Z is trivial.
E I. In this case, KR is locally isomorphic with Sp(4). According to the
material on pages 252 and 253 of Bourbaki [2], there are only two
possibilities for KR: Sp(4) or Sp(4)/(±1). We will see shortly that the
latter case prevails. In any event, Z is either trivial or isomorphic with Z2.
E IV. In this case, КR is locally isomorphic with the compact form of F4.
Since the adjoint group of F4 is simply connected (Bourbaki [1], pp.
273,273), Z is trivial.
We are left with the case BD I. For this we must recall some results
about spinors, which we will begin in the next number.
ll.A.2.6. We now begin the exposition of the results that we will be using
about spinors. A good reference for this material is E. Cartan [1]. Let Vc
be an «-dimensional vector space over C with a non-degenerate symmetric
bilinear form ( , ). Let T(VC) be the tensor algebra on Vc and let I be the
11.А.2. Small А-Types
119
two sided ideal in T(VC) generated by the elements x ® x + (x, x)l. Set
Wc,( , )) = T(VC)/I.
If ex,.,.,en is a basis of Vc with (e,,e;) = 3,7, then СН(КИс,( , )) is
isomorphic with the universal associative algebra over C generated by
el,...,en with relations ej = -1 and	if i # j. We will use
the notation Cliff„ for this algebra. It is usually called the Clifford algebra
on n generators over C.
Suppose that И is a real form of Vc such that (V, И) c R. We will say
that И is a real form of (Vc,( , )). Let ClifK И,( , )) denote the subalge-
bra over R of ClifK Ис, ( , )) generated by V. If ( , ) is understood, we will
drop it from our notation. Then ClifKИ) is a real form of OifKLc).
If ,vk g Vc, then we look upon ••• vk as an element of
ClifK Vc). It is easily seen that the elements	• • -eik with < i2 < • • •
< ik and к <n form a basis of ClifK Ис) (that they span is fairly clear, that
they are linearly independent can be derived from the existence of the
space of spinors corresponding to V = ©Re,; see 9.2). We define an
anti-homomorphism x -»xT of T{VC) by ® • • • ® vk vk ® • • • ® Vj.
Clearly, IT = I. Thus, we have an anti-automorphism of ClifKLc), also
denoted xT. It is clear that if И is a real form of (Ис,( , )) then
СНК(И)Г c ClifKИ).
If v g Vc is such that (v, v) = 1, then v • v = -1 in ClifKVc). Thus, v
is an invertible element. Let 5ргп(Ис) denote the subgroup of the invert-
ible elements of СНЖИс) generated by 5 = {t; g Ис| (v, v) = 1}. Let
Spin(Vc) denote the normal subgroup of products of even numbers of
elements of 5. Note that if g g Spin{Vc), then gT = g~l.
If v g S, then sv(x) = v • x • v for x g Vc is the linear transformation
of Vc that acts by -1 on Cv and the identity on t'x = {x|(x, v) = 0}.
Thus, sr (the reflection about the hyperplane is an isometry of Vc.
We therefore have a group homomorphism 3: 5ргп(Ис) SO(VC) given
by 8(g)x = gxg~l for g g Spin(Vc) and x g Vc. Now, O(Vc) is generated
by the reflections sr, and SO(VC) is the subgroup consisting of products of
even numbers of reflections (if the reader hasn’t seen this before, it is a
good exercise). Thus, 3 is surjective.
We endow Spin(Vc) with the subspace topology in ClifKИс) (thought of
as C2"). Then we assert that Spin(Vc) is connected. Indeed, let ..., v2k
g S. Since 5 is connected, there exist continuous curves 2 < i < 2k,
with values in 5 such that crz(0) = and a((l) = vt. Set git) =
vx(r2{t) • • • <r2k(t). Then, g(0) = (i^)2* = 1 and g(l) = v} • • -v2k. We also
120
11. Completions of Admissible (0, К )-Modules
note that Ker«5 = {±1} (a not too difficult exercise). Thus, Spin(Vc) is a
two-fold, connected, covering group of SO(VC\ Thus, it is a Lie group.
Furthermore, if dim Vc > 3 it is simply connected (cf. 9.2).
ll.A.2.7. We now look at groups associated with real forms of (Fc,( , )).
So fix a real form V. Let SpiniV) denote the subgroup of Spin(Vc) that
consists of all products v}  •  v2k of elements e V with (vit i?,) = 1 or
-1, and there are an even number of with (ц, ц) = 1. Then, as before,
if SO(V)° is the identity component of the special orthogonal group of
(V,( , )), then SpiniV) is a connected, two-fold covering group of SO(V)°
(Cartan [1]).
For the purposes of this appendix, the most important cases will be
V = R" and p > q > 3 with p + q = nA , ) on V will be given by
(*,?) = E-V/У/ -
i <,p	i >p
In this case, we write Spin(p,q) for SpiniV). Let	be the
standard basis of R". Let W be the real subspace of Kc with basis e, ,
i < p, and iej, j > p. Then Spin(W) is the simply connected covering
group of SO(W) = SO(n) and hence it is a compact form of Spin(Vc). Let
т denote conjugation in Cliff „ corresponding to the real form ClifKlF).
Then, r(ClifKF)) = Cliff(F) and ASpinip, q)) = Spinip, q). Denote by в
the restriction of r to Spinip,q).
We now look at the situation Gc = Spin(Vc), GR = Spin(p,q) with
p > q > 3 and в given as above. Then KR = Spinip, q) П SpiniW). We
have a homomorphism of Spinip) X Spiniq) onto KR given as follows.
Let be the span of ex,..., ep and let IF2 be the span of ep+1;..., en.
Then, we have Spin(W{) x Spin(W2) mapping into Spin(p,q) under
x, у >-»xy. The image of the map is KR. The kernel of the map is
Z = {(1,1),(— 1, —1)}. This completes the proof of Lemma ll.A.2.4.
11A.2.8. We now complete the proof of Theorem ll.A.2.1. We prove the
result by a case by case check using the table on p. 518 in Helgason [1].
A I. In this case we need only look at GR = SL(n,R) with n > 3 and
KR = SO(n). Then, G is the twofold covering group of GR and К =
Spinin). We choose P to be the group of upper triangular matrices in
SL(n, R). °MR is the group of all diagonal matrices
d = diag(d1,...,</„)
11.A.2. Small «-Types
121
with d; e{l,-1} and П d}•= 1. On R", we put the standard inner
product ( , ). Then,
m;. = ((-l)4...,(-l)5"')=V
Thus, °M is the group of all even numbers of products of the in Spinin).
If x is the trivial character of Z = {1, -1} c Spinin), then choose r to
be the trivial representation. If x is the non-trivial character and n is odd,
choose т to be the spin representation (9.2.2) of K. If n is even, choose r
to be either of the half spin representations (9.2.2). Now, C[°Af ] modulo
the ideal generated by the elements z - ;t(z)l, z g Z, is the subalgebra
of Cliff „ spanned by the even numbers of products of the e; . Thus (see
9.2.2), т restricted to °M is irreducible. The theorem has thus been proved
in this case.
A II. In this case, we have observed that Z = {1}.
A III. Here, t is not semi-simple. This case is therefore covered in
ll.A.2.3.
BD I. Here, g is $oip, q), p > q > 1. If q = 1 and p > 3, then one
checks that Z = {1}. If q = 2, then t is not semi-simple. We therefore
assume that q > 3. We use the notation and results in the previous
number.
We take for a the subalgebra spanned by the etep+i, i = 1,..., q. Then,
°MR is the subgroup of
(Spin(p) X Spin(q))/{!,(-!,-1)}
of all elements of the form
+ ^i, ’ ’ ’ ei2kei{+p ' ' ' ei2k+P ’
with
g G Spin(p ~ q) = 5pin(span(e1?+p...,ep)).
Let p2 be the projection of К = Spinip) X Spiniq) onto Spiniq). Then,
p2CM) is the subgroup described as in case A I.
Z = {1, (— 1, - 1)} in Spinip) X Spiniq). If x is the trivial character of
Z, then choose т to be the trivial representation. If x is non-trivial, then if
q is odd let a be the spin representation of Spiniq) and if n is even let a
be either of the half spin representations. Choose r = ap2. Then, as in
case A I, we see that r restricted to °M is irreducible.
122
11. Completions of Admissible (0, К)-Modules
D III, С I. These cases are covered by ll.A.2.3.
D II. In ll.A.2.4, case 2, we have seen that in this case Z = {1}.
We are now left with the exceptional groups. For these it is convenient
to go through the table from G down to E I.
G. In this case, К = SU(2) x 517(2). GR is just the split adjoint group of
G2. The action of К on pc is the tensor product of the two dimensional
representation of the first factor and the four dimensional representation
of the second factor. To see this, consider the discussion of G2 in
ll.A.2.4. We take the first factor to correspond to a0 and the second to
correspond to a2. The highest weight of pc is —a1.
This implies that KR = (5(7(2) x 5(7(2))/{l,(-l, -1)}. Since GR is
split over R, it is a simple matter to see that °MR = (Z2)2. If we examine
the root system of G2 on p. 274 of Bourbaki [2] we see that we have an
imbedding of g/(3, R) into й- We assert that the corresponding homomor-
phism, 8, of 5L(3, R) into GR is injective. Indeed the adjoint representa-
tion of й restricted to 3/(3, R) is easily seen to contain the standard three
dimensional representation. We may thus take MR to be contained in the
image of 5L(3, R). We may also choose 8 such that 3(50(3)) is the image
of the diagonal group on 5(7(2) x 5(7(2). Thus, the image of the lift of 3
to the two-fold cover of 5L(3, R) contains M.
Let a be the standard two dimensional representation of 5(7(2). Let p{
be the projection of 5(7(2) x 5(7(2) onto the first factor and put т = <rpl.
Then, as in the case A I, the restriction of r to °M is irreducible and т is
the desired small /С-type for the non-trivial character of Z.
F II. We have seen in ll.A.2.4 that in this case Z = {1}.
F I. In this case К = 5p(3) x 5(7(2). The highest weight of the action of
К on p is -Oj (here, we are using the notation of ll.A.2.4). Thus, (-1,1)
and (1, -1) both act on p by -I. Hence, Z = (-1, -1). As before,
AfR = (Z2)4 xA. If we look at the information on p. 272 in Bourbaki [2],
we see that we have a homomorphism 3 of Spin(5,4) into GR. The adjoint
representation of й restricted to Spin(5,4) splits into the adjoint represen-
tation of Spin(5,4) and the spin representation. Thus, 3 is injective. We
can choose 3 so that 3 maps (Spin(5) X 5p(n(4))/{l,(-l, -1)} into
(5p(3) x 5(7(2))/{l, (-1, -1)}. Now, Spin(5) = Sp(2) and Spin(4) =
SU(2) x 5(7(2). We see that the lift of 3 to Spin(5) X Spin(A) must be
given by the obvious map 5p(2) x 5p(l) into 5p(3) (here, we note that
5p(l) s 5(7(2)). The image of Spin(5,4) contains the split Cartan subgroup
of Gr. Thus, °M is isomorphic with the °M for Spin(5,4). Let p2 be the
11.А.2. Small К-Types
123
projection of 5p(3) x 5p(2) onto the second factor. Let a be the standard
representation of 5(7(2) on C2. Then, ap2 is one of the preceding choices
for Spin(5,4). Hence, it is irreducible when restricted to °M. Thus, if x is
the non-trivial character of Z then choose т = ap2.
E IX. This case will be studied later.
E VIII. In this case, К = Spin(16). Using the material on p. 269 of
Bourbaki [2], we see that the highest weight of К on p is given by -a,.
Thus, this action is one of the two half spin representations of 5pzn(16),
say s+. Hence, Z = Kers+. As before, there is a non-trivial homomor-
phism 3 of 5L(9, R) into GR. Since 5L(9, R) has trivial center, 3 is
injective. As before, we may assume AfR is contained in the image of 3
and 3(50(9)) c XR. Now, 3 lifts to a homomorphism, 3, of Spin(9) into
К = Spin(.16). Let -л- be the 16 dimensional representation of Spin(16)
gotten by using the covering homomorphism of Spin(16) onto 50(16).
Then, 77 ° 3 = fi is a 16 dimensional representation of Spin(9). Using the
Weyl dimension formula, one sees that Spin(9) has three irreducible
representations of dimension at most 16: 1 (the trivial representation), a
the 9 dimensional representation corresponding to the covering of 50(9),
and i the spin representation. Since p is non-trivial there are two
possibilities: p = 7 • 1 Ф a, or fi = 5. In the first case, the map 3 is the
standard imbedding of Spin(9) into 5рш(16). But then the composition
p ° 3 is injective on Spin(9). However, it must push down to 50(9). So this
case is not possible. Hence, p = s. From this we see that °M is isomorphic
with the “°M” for 5L(9, R). If x is the non-trivial character of Z then we
choose г = 77. Then, 77 restricted to °M is irreducible by the previous
result for 5L(9, R).
E VII. In this case t is not semi-simple, so the result follows from the
material in ll.A.2.3.
E VI. This case will be studied later.
E V. This case corresponds to К = 5(7(8). KR = 5(7(8)/{l, -1}. From
the diagram on p. 265 in Bourbaki [2], we see that we have a non-trivial
homomorphism 3 mapping 5L(8,R) into GR. If we go to the complexifi-
cations it is an easy matter to see that 3 is injective. We may thus assume
that aMR is contained in the image of 3, and 3(50(8)) c KR. Let 3 be the
lift of 3 to a homomorphism of Spin(8) into 5(7(8). The corresponding
representation of Spin(8) cannot factor through 50(8). Thus, it must be
one of the half spin representations, say s +. We also look at 3 as a
124
11. Completions of Admissible (g, Kl-Modules
homomorphism of the simply connected cover of SL(8, R) into G. Then, <5
is injective on the “°M” for this group. Thus, if x is the non-trivial
character of Z then we may choose r to be the standard eight dimensional
representation of 5(7(8).
E IV. In this case we have already seen that Z = {1}.
E III. Here, t is not semi-simple.
E II. This case will be studied later.
E I. This is the split case over R. К = 5p(4). We imbed GR in the “GR”
for E V as the identity component of °MQ, for Q a parabolic subgroup of
Gr. This homomorphism, 3, maps KR into 5(7(8)/{1, -1}. The lift to К
must be the standard eight dimensional representation. Hence, Z =
{1, -1}. (This resolves the previous ambiguity in this particular case; see
ll.A.2.5 E I). Let z be the non-trivial element of Z. Then С[°Л7 ]/(l + z)
is isomorphic with Oiff6 (here, we use the results for E V). The lowest
dimensional non-trivial representation of Cliff6 is eight dimensional. Thus,
if x is the non-trivial character of Z we may choose r to be the standard
eight dimensional representation of K.
ll.A.2.9. We are left with the three cases E IX, E VI, E II. We now
prove some general results that simplify the proof of the theorem at hand
for these three groups. We assume that t э t, a Cartan subalgebra of fl.
Let t0 be a maximal abelian subalgebra of °m. Set I) = (t0)c Ф ac. Then,
t) is a Cartan subalgebra of gc. Set Ф = Ф(йс> and Фи = {a e Ф|ск(1
= 0}. We choose Ф+, a system of positive roots for Ф, such that if
ala e Ф(Р, A) then a e ф+. Let § = {X e fll [X t0] = 0}. Then flc d I)
and Ф(дс, Ij) = Фи. Set fii = [fl, fll- Then, one can show that fl = t0 Ф fli •
We assume in addition that Z = Z2, that G is simply connected, and
that tj = t n fl j is semi-simple. Let Gt be the connected subgroup of G
with Lie algebra flt.
(1) Gt is simply connected.
Indeed К is compact and simply connected. We may assume that
t d t0. We set Kx = К П Gt and 1\ = Kx n T. We can identify Ф(К\,
with {a e Ф(К, T)!»^ = 0}. We can thus choose a simple system of roots
Дк for Ф(К, T) such that Дк П Ф(К\, Т[) is a simple system of roots for
Ф(Х1,Т1). But this implies that every irreducible finite dimensional
trmodule integrates to a representation of Kx. Hence, Kx is simply
connected. So G, is simply connected.
11.А.2. Small А-Types
125
We also assume that Gt is simple. Then Gt corresponds to one of the
cases that we have studied previously. Let Zt be the “Z” for Gt. Then a
study of our earlier work leads one to conclude that Zx = Z2. Let
p\ G -> GR be as before. Then, piG^ is the “GR” for Gt. Thus,
Ker p c Gj. So Zj = Z.
ll.A.2.10. We now use these observations to handle the three last cases.
EIX. Let K2 be the simply connected compact Lie group with complexi-
fication E7. Then, К = K2 x 5(7(2). Let zx be the non-trivial element in
the center of K2. Then, we assert that Z = {1,(z1; -1)}. Indeed, one
checks by observing that (in the notation of Bourbaki [2], p. 269 that the
highest weight of the action of К on p is -a8. This representation is
faithful on each of the factors. Since we have seen that Z = L2, the
assertion follows.
Using the material in ll.A.2.9 and the table on p. 534 of Helgason [1], it
is not hard to see that G1R = 5рш(4,4). Since E8 has only one root length
one sees that [°m, gj = 0- One checks that °m s §o(8). We therefore
have a local injection of Spin(8) x 5pin(4) x 5рш(4) into K. Also, the
material in ll.A.2.9 implies in this case that Z с 1 x Spin(4) x Spin(4)
and that the non-trivial element of Z is 1 x (-1, -1). We note that
Spin(4) = SU{2) x 5(7(2). Thus, we must have a local isomorphism of
Spin(8) x Spin(4) x 5(7(2) into K2. This observation combined with the
preceding results for Spin(4,4) implies that if p2 is the projection of К
onto the 5(7(2) factor, then p2(°M) is not abelian, and if z is the
nontrivial element of Z then p^z) = -1. Thus, if x *s the non-trivial
character of Z we may take т = <тр2, with a the two dimensional
irreducible representation of 5(7(2).
E VI. Here, К = SpiniXl) X 5(7(2). In this case, we have seen in ll.A.2.4
that Z is of order 2. Let a be the standard two dimensional representa-
tion of 5(7(2) and let p2 be the projection of К onto the second factor. If
X is the non-trivial character of Z then choose r = ap2. The details are
similar to the case E IX. As in the previous case, one has Gj is the
two-fold covering group of Spin(4,4) with Z c GP °M is locally isomor-
phic with 5(7(2) x 5(7(2) x 5(7(2). The rest of the argument is essentially
the same as the previous case.
E II. In this case, one again finds that G{ is the two-fold covering of
Spin(4,4). К = SU(6) x 5(7(2). Let a be as before and let p2 be the
projection of К onto the 5(7(2) factor. If x is the non-trivial character of
126
11. Completions of Admissible (g, A)-Modules
Z, then we choose т = ap2. The details are essentially the same as the
previous two cases.
ll.A.2.11. Although we will not use the following consequence of the
previous arguments, it is instructive to point it out. We pose the problem
(possibly first suggested by Vogan): Give a direct (not case by case) proof
of it.
Theorem. If й й split over R, then GR always has a connected 2-fold
covering group.
11.A.3. Some results on Verma modules
ll.A.3.1. Let G be a real reductive group of inner type. Let К be a
maximal compact subgroup of G and let (P, A) be a minimal standard
p-pair. Let (f, Hf) be an irreducible finite dimensional representation of
°M. If v g cie, then we form the (p, °Af)-module structure H( v on Hf
with °M acting by £, n acting trivially, and a acting by v + p. We set
М(£, v) = 17(йс) ®i/(pc) Щ.» the S'module structure correspond-
ing to left multiplication and the °M-module structure m(g ® v) =
Ad(m) g ® ^{m)v.
If И is an (m, °Af)-module and if A g aj, then set = {r g V\ (h -
k(h))kv = 0, h g a for some k}. Then, VK is a °Af-module. Assume that
V = Ф. V. with dim V. < » for all A. Then we define
A G a J А	A
СМИ) = l>AchoM(K).
Л
Here, choM is the usual character of a finite dimensional representation of
°M. We will deal with these characters formally as in 6.A.3.3. The
calculation in the proof of 6.A.3.3 yields
chM(Af(^,p)) = chM(t7(nc))el'+p choM(H^).
ll.A.3.2. Fix a Cartan subalgebra I) of mc. Fix ф +, a system of positive
roots for mc with respect to I). Let A^ be the highest weight of £ and let
pm be the half sum of the elements of Ф*. Set A(£, v) = A^ + pm + v.
Then, М(£, v) has infinitesimal character Xx^, V)  Let Ф+ be a system of
positive roots for йс with respect to h such that Ф+эф+ and Ф+ is
compatible with n. Let n+= ®аеф+ (ficV
11.А.З. Some Results on Verma Modules
127
Lemma. The set of all v g a £ such that М(£, v) is irreducible is open and
dense in a£. Furthermore, if v g a£ then there exists e > 0 such that if
t g C and 0 < |t | < e, thenMCg, s(v + tp)) is irreducible for all s g 1У(Л).
Suppose that X = v) is reducible. Then there must be a non-zero
proper submodule Y of X. Since (7(n+)m is finite dimensional for each
m g X we see that Уп+¥= 0. Thus, there must be a non-zero n g Y and
A g t)* such that n+n = 0 and hn = A(h)n for h g t). As an mc-module,
X = I7(nc) ®Hi v. Thus, the mc highest weights that can occur in X
are of the form Л(£, p) -	- • • • -Дг, with Д, g ф+- ф+. Thus, A
must have an expression A = A(£, p) - Q, with Q =	+ • • • +(3r and
/3, g Ф+- ф+ . Since Y Ф X and the A(£, p)-weight space generates X,
r > 0. Since У is a submodule of X we must have Хл = Л'ла.к)- But then
in particular we have 2(A(£, p), Q) = (Q, Q). This implies that
2(A(£,Rep),(?) = (Q,Q) since (Q,Q) is real. Set Q = Еаеф+_ф+ Na
(here, as usual, N is the set of non-negative integers). We have thus shown:
(1) If p) is reducible then there exists Q g Q - {0} such that
2(A(£,Rep),0) = (Q,Q).
Now, if
2(A(£, Re p), 0) = (Q,Q),
then
non2 < 2||a(^,p) ii non.
So
IIOII < 2||A(£,Re p) ||.
This implies that if ы is a compact subset of a£, then there are a finite
number Qlt ...,Qr of elements of Q - {0} such that if p g ы and if
2(A(£, v), Qi) # (Qi,Qj) then v) is irreducible. Now,
2(A(^, p),0J = 2(A, + pm,Qi) + 2(p,0,)-
We therefore see:
(2) If ш is compact in I), then there exist Qlt...,Qr^ g Q - {0} and
a, g R such that if p g ы and if 2(p, 0,) * then M(l-, v) is irreducible.
Obviously, (2) implies the first assertion of the lemma.
We now prove the second assertion. Let 5 = {p + tp\ |t| < c}. Put
ш = Uje(Fs5. Let Op • • •»Qd e Q _ (0) and flp- • •,ad e R be such that
128
11. Completions of Admissible (0, К )-Modules
if д. g ы and if 2(/x, QJ ¥= a(, then M(s£, p) is irreducible. We calculate
2(s(v + tp), Qt) = 2{sv,	+ 2t(sp, Qj). Since W(A) is finite, the set of t
such that 2(s(v + tp), Qt) = at, for some i and some 5 g W(A), is finite.
The second assertion of the lemma now follows.
Note. The proof of this result contains an extremely crude irreducibility
criterion for the MQj, v). If °M is connected, there is a necessary and
sufficient condition for irreducibility due to Jantzen [1].
11.A.4. Some functional analysis
11A.4.1. The purpose of this appendix is to record several results in
functional analysis that are used in this chapter. The first is a theorem of
Banach (с/. Treves [1], Theorem 37.2, p. 382). Let V and W be Frechet
spaces. Let и be a continuous linear map of V into W. Define
W' -> V by мг(А) = A ° u.
ll.A.4.2. Theorem, и is surjective if uT is injective and uT(W') is weakly
closed in V'.
This result has an immediate corollary.
ll.A.4.3. Corollary, и is bijective if and only if uT is bijective.
If и is bijective, then it is clear that uT is bijective. Assume that uT is
bijective. Then the preceding theorem implies that и is surjective. If
v g Ker u, then V'(v) = W"(u(v)) = 0. Thus, the Hahn-Banach theorem
implies that v = 0. Thus, и is injective.
ll.A.4.4. In the rest of this appendix, we study variations of Frechet
spaces. Let И be a Frechet space. If 8 > 0, then we set Ds =
{z g C| |z| < 8}. Put Ds = C1(DS). Let 0S(V) denote the space of all
continuous functions from Ds to V that are holomorphic on Ds. If p. is a
continuous semi-norm on V, then we set ps(f) = supzeDs p(f(z)). Then,
0S(V) endowed with the topology induced by the semi-norms ps is a
Frechet space. If 8 < r, then the restriction map &S(V) -> &T(V) is
continuous. We define 0(V) to be the strict inductive limit as 8 -> 0 of
the spaces 0S(V) {cf. Treves [1], p. 126). Then, €АУ) is a locally convex
space (LF space), which we look upon as Us> 0 ^(F) using the preceding
restriction maps as inclusions. A convex subset U of 0W) is a neighbor-
hood of 0 if U П &S(V) is a neighborhood of 0 for each 8 > 0.
11.А.4. Some Functional Analysis
129
If W is a locally convex topological vector space and if T is a linear map
from to W, then T is continuous if and only if is continuous
for all 3 > 0 (с/. Treves [1], Proposition 13.1, p. 128).
ll.A.4.5. If /е^8(И), then we set Tk(f) = (l/k'.)f(k4O) (the Ath
derivative of f at 0). Then, Tk is defined and continuous (use the Cauchy
integral theorem) for all 3 > 0. So it defines a linear map Tk:	V.
Lemma. If к > 0, then the subspace zk0(V) is closed in
zk0s(V) = Ker Tk. Hence, zk0s(V) is closed in Since
(^(F) -z^(H) А ^(И) = ^(F) - z‘^(F),
the lemma is clear.
ll.A.4.6. We define Д(/) = (To(/),..., Tk_ff)) for f e 0(V). Then jk
is a continuous surjection from ^(F) to Xk V (with the obvious Frechet
space structure). Ker jk = zk0(V). Thus, the closed graph theorem for LF
spaces (cf. Treves [1], Proposition 17.4, p. 173) implies that jk is an
isomorphism of 0(V)/zk0(V) onto Xk V.
ll.A.4.7. Let G be a real reductive group and let К be a maximal
compact subgroup of G. Let (P, A) be a standard p-pair and let (a, W)
be a smooth Frechet representation of °M. Set V = /", the space of all C“
maps f of К to W such that f(mk) = a-(m')fik') for т e °M П К and
к e K, with the C°° topology. If v e a£ and if f e V, then set (as usual)
ffnmak) = av+pa(m)f(k\ Put a „(g)/U) = f„(kg). Set тгР a v = ttv.
Then, is a smooth Frechet representation of G. We will now
define a representation of G on If f g then we write
f(z,k) = f(z)tk). We set (-rr(g)f)(.z, k) = Trv+zp(g)f(z, k). Then, rr(g)
defines a (strongly) continuous representation of G on Thus, we
have a strongly continuous representation of G in ^(F), which we also
denote by тг. We note that -n-fgXz^fF)) c zk0(V) for all к = 0,1,... .
Thus, we have a representation ttv k of G on €AV}/zk0{V\ We now
describe this representation as an induced representation.
ll.A.4.8. Let к be a positive integer. We define a representation ok v of
M on Xk W as follows:
^,Д"’а)(г0,...,^_1) = (и0,
130
11. Completions of Admissible (a, A)-Modules
with
u, = av+p £ P jp(log a)₽z?;_p
for m g °M, a g A. Then, ak vGn) = Xk afm) for m g °M. Thus, Ipak ?
is as a space Xk V. Set, for g g G, и g K,
= ^k<Xtn(us)a(ug))(f0(k(ug)),...,fk_l(k(<ug))).
Then, Ip ak * is equivalent to (тг^ „, Xk И).
Lemma. jkMg)f) = irk v(g)jk(f) forfe 0(V). In particular, the rep-
resentation of G on 0(V)/zk0(V) that is the quotient representation of тг
is equivalent to Ip,aku-
This is clear from the definitions.
ll.A.4.9. The last result in this appendix is of a different nature. If И is a
finite dimensional vector space over C and if A' is a vector space over C,
then a polynomial map of И to A' is a function f from V to X such that
there exists a finite dimensional subspace of X,W, such that /(И) c W
and f is a polynomial map from V to W (i.e., if A g IT* then A ° f is a
polynomial).
Lemma. Let (P, A) be a standard p-pair. Let (a, Ha) be a smooth
representation of P on a Frechet space Ha such that (Ha)K nP is admissible.
If v g cic, we set av(man) = avff{man). Let Ip<a<v be the C“ induced
representation of a„ from P to G. Let f g IPt<ryV = (Ip<aiV)K- If g e U(ScK
then
g -*
is a polynomial map from a J to IP<a<v-
We define a sequence of polynomials рк(хх,xk) recursively by
pQ = 1 and
fc-l Q
Pk(xi,...,xk) = £

11.А.4. Some Functional Analysis
131
If f e C”(R), then
dk
= Pit f.f
With this in hand, we prove the lemma. It is enough to prove it in the
case g = Xk with X g g. We write a(g) = eH(g\ with H a C°° map of G
to a. Then,
d1
' l ;\ dr
= E К 77	e^k^lX4P><r>0(Xk-p)f(k)
r=0 V) dt |z=o
± / I I d
= E ! A ” T H(^exptV)
r=0 \r)	\ \ dt i(=o
I dr	П
X J — H(k exp tX) \'n'p,(r<o(X,~p)f(k).
The result is now obvious.
This Page Intentionally Left Blank
12 The Theory
of the Leading Term
Introduction
This chapter contains most of the analysis necessary for our proof of the
Harish-Chandra Plancherel formula. It is the most complicated chapter in
the book and, perhaps, the reader should, on first reading, go on to the
next chapter (which is one of the more beautiful chapters in the two
volumes) to motivate a serious onslaught into this one. The theory devel-
oped in this chapter is what Harish-Chandra called the “theory of the
constant term.” We have opted to call it the “theory of the leading term”
since it is the first term in an asymptotic expansion. We hope that this
slight change in the Harish-Chandra lexicon will not irritate too many
experts. In Harish-Chandra’s development of the theory, it followed from
differential equations. Our approach is representation theoretic. The
differential equations are replaced by our theory (in Chapter 4) of asymp-
totic expansions of matrix coefficients. Also, in Harish-Chandra’s ap-
proach, the theory of intertwining operators is an outgrowth of his theory
of the constant term. Chapter 10 yields a theory of intertwining operators
that allows us to bypass some of the complications in Harish-Chandra’s
original approach. Most notably, we are able to incorporate his “p,’’-func-
tion at the outset in our approach to wave packets. This was not possible
in his method.
133
134
12. The Theory of the Leading Term
The basis of our approach to the leading term is in Section 2, which
gives the algebraic version of it. This section is strongly motivated by
Harish-Chandra’s approach to the case of reductive groups over p-adic
fields (which in turn was influenced by Jacquet [1]). The material in
Section 1 is used in order to identify the terms of the algebraic leading
term. Section 3 contains the definition of the leading term of a tempered
Z(g)-finite, right /^-finite function. This notion is an analytic version of the
algebraic definition in Section 2. In Section 4, we study its analytic
dependence on parameters. This section is the heart of the chapter. We
feel that our representation theoretic approach makes the development
more transparent. Section 5 relates the constant term to intertwining
operators. It is here that the main difference between our development
and the original development of Harish-Chandra becomes most apparent.
Section 6 is the most difficult of this chapter, it lays the groundwork for
the theory of “wave packets” of principal series (Section 7) and the
calculation of their Harish-Chandra transforms (Section 8). The main
result is an inequality, uniform in certain parameters, for the difference of
a tempered matrix coefficient and its leading term. The material in Section
8 is, as we shall see, the essential step (beyond the theory of the discrete
series) in the proof of the Plancherel theorem.
12.1.	Characters of principal series representations
12.1.1.	Let G be a real reductive group and let К be a maximal compact
subgroup of G. Let (P, A) be a standard p-pair and let (a-, Ha) be an
admissible finitely generated Hilbert representation of °M such that o’|KnM
is unitary. Let v g aj and let a „ be the character of IPy(TjV (5.2.1,8.1.3).
In this section, we will prove several results about these characters that
will be applied in the rest of the chapter.
If f e C“(G), let fp(m) be as in 7.2.1. Then fp g C“(M). If v g a*,
then set
(//’)A(p)(m) = f fp(ma)avda
JA
for m g °M. Here, we have chosen da as in 10.1.7. All measures in this
chapter will be taken as in 10.1.7. In particular,
f	a~2pf(namk) dndadtndk = f f(g) dg.
JNxAx°MxK	JG
12.1. Characters of Principal Series Representations
135
We first prove:
Proposition. &P a „(/) = 0Д/р)л (p)), with
f(g) = fKkgk-')dk.
Let 77 = ttp a If <p g HP a and if v g а£, then we set <p„(namk) =
av+pff{m)<f>{k) for n g N, a e A, m g °M, к g K. If f e C“(G), then
= fGf(g)<PXks) dg = fGf(k~'g)<pv(g) dg
for <p g Hp’<r'v. The preceding integration formula implies that
7г(/)^(Л) = f a~pf(k~'nmaki)al’o-(tn)<p(k^dndadmdk.
JNxAxoMxK
Set
Uf(kt, ma, k2) = f a~pf(k2'nmaki)dn
JN
for к,,к2е K, tn g °M, and a g A. Then, uf g C^(K x M x K). A
direct calculation, using the fact that M П К is compact and normalizes
N, yields
(1)	uf(miki ,ma,m2k2) = u^k],m^mm2a,k^
for m,, m2 g К П M.
Set
Ту(Л|, k2) = т(£| ,k2) ~ I uf(k,,ma,k2)av(r(m)dadm.
jax°m 1
Then, (1) implies that
т(/И|Л| ,m2k2) = a(mi)T(ki ,k2)a(m2Y'
for /И|, m2 g /С n Л/ and kx, k2 g K.
It is now an easy matter to see that the hypotheses of 12.A. 1.1 are
satisfied with (in the notation of that number) M replaced by M П К and
a replaced by a]KnM (8.1.1). Hence, Proposition 12.A. 1.2 implies that
tr-rr(f) = f tr(r(£,£)) dk.
JK
136
12. The Theory of the Leading Term
We will now show that this is just a rewriting of the formula we are
proving. We first note that tr ir(f) = tr тг(/) (cf. 8.1.2 (1)). We therefore
replace f by f. Then,
uf(k, m, k) = fp(m)
for к g К, m g M. The definition of (fpf' (v) now implies that
т(к,к) = cr((fpy(v)).
The result now follows.
12.1.2.	We now derive a direct consequence of the preceding result.
Lemma. Fix (a, Ha) as in the preceding. Then, the function
v ~ ®P.a,Af)
is holomorphic on a£ for f g C“(G).
Indeed, f >-» fp is continuous from Cf(G) to Cf(M). If <p g C“(Af),
then set
<p(v)(m) = J <p(ma)av da
for tn g °M, v g a£. Then, v >-» <p(v) is holomorphic from a£ to C“(°M).
The lemma now follows from the previous result.
12.1.3.	We will now use the notation in 10.1.8. We will also assume that
G is of inner type. Let 1У(Л) = {j g GL(a)\ there exists к g К such that
Ad(&)|a = j}. We will call the “Л” in the definition a representative of j. If
j g W(A) and if (a, Ha) is as before, we define ka(tn) = a(k~'mk) for
к & К, Ad(^)|Q = j. Then, the equivalence class of ka depends only on j
(if Л| is another representative, then ktk~l g M П K). If v g a£, then
set sv = v ° №1 (as usual).
Proposition. Let P,Q g 0(A), s g W(A), and к be a representative ofs.
Then,
®P,<r,v = ®Q,k<r,sv
for all v g dp.
12.1. Characters of Principal Series Representations
137
We note that if V g Ж then depends only on the isomorphism
class of V. Let P, Q be in £?(A). Let {/;} be a collection of non-zero
polynomials on a£ such that IP(rv and IQ a „ are irreducible if f^v) =# 0
for all j (10.5.3). For these v, JQ\P(v) defines an isomorphism where it is
defined and non-zero (another countable set of non-zero polynomial
conditions). Thus, off of that set ©PiO.,p = 0GiO. Lemma 12.1.2 (com-
bined with 10.A.3.2) implies that if f g C?(G), then 0P,o.,p(/) = ©с.а.Л/)
for all v.
Let j g JV(A), and let к g К be a representative for j. Let Q g ^(Л).
Then, kQk~' g &>(A\ If /g C“(G; then set Uk)f(g) =f(k~'g).
If f g 1^, then we set a v =ofv in the notation of 10.1.1.
(1)	Uk)fQ^v = (L(k)f\Qk-tk<rsv for f el”.
Indeed, if nx g NkQk-i, tnt g °MkQk-t, ax g A, и g K, then
= fQ^^(k^'nxkk~'mxkk-}axkk^}u)
= katm^a^oftk-'u)
= (L(k)f)kQk-\k<TtSV(nimiaiu).
(1)	implies that
L(*)7rG,<7,,(g) = ^kQk-\ka,sXs)L(k).
Thus,
®Q,a,v ~ kQk~', ka, sv ‘
The first part of the proof implies that &kQk-\kaySV = ®Q,ka,sv The
proposition now follows.
12.1.4.	We next prove a converse to the preceding result if (a, Ha) is
irreducible and square integrable.
Theorem. Assume that (a,	are square integrable and irre-
ducible representations of °M and that v and g g ci*. Let P,Q g ^(Л).
Then ®Р,а,,, = ®е.ч.м tfand only if there exists s g W(A) such that p = sv
and г) is unitarily equivalent to к a, with к a representative of s.
In light of the preceding result we may assume that P = Q and we need
only prove the “only if” part of the assertion. Since we are assuming that
°M has square integrable representations, Theorem 7.7.1 implies that
there exists a Cartan subgroup T of °M with TcK. We fix such a T. Set
138
12. The Theory of the Leading Term
H = TA. Then, H is a Cartan subgroup of G. Let H' be the set of all
regular elements of H and put GH, = {ghg~'\g g G}. Then, GH> is open
in G. We will compute the indicated characters more explicitly for certain
/gQ(G„,).
Let у g T and let A be as in 6.9.3. We may assume that (a, Ha) =
(irP y, Hp,y) with A , P-regular and dominant (see Lemma 6.9.4, we apol-
ogize to the reader for the two uses of P). Let A(t) = ts ПаеР(1 - t-“)
(here, 8 is the half sum of the elements of P). Let W(K, T) be the Weyl
group of К with respect to T. If <p g	) the Harish-Chandra
Schwartz space, 7.1.2) and if <pj = 0 (notation as in 7.4.8 (1)) for all Cartan
non-compact subgroups of °M, then (up to a constant of normalization)
®Af) = d(y)fTir(y(t))<Pf(t)dt
(see 8.7.3). If we combine this with 7.4.10 (2) and Proposition 12.1.1, we have
®р,«,АП = d(y)JT^a4r(y(t))FfH(ta) dtda.
Let H" be as in 7.4.10. Let U be an open subset of H" such that
gUg-1 c U for g g NG(H) = {geG\gHg~l = H}. Let <p g QW) be
such that <p(gug~l) = <p(u) for и g U and g g Ng(H). Let 'VigH, u) =
gug~l for g g G and и g U. Then, ¥ is a covering map onto an open
subset of G with deck transformations given by NG(H)/H = WG(H).
That is, 'VigH, u) - ^(gjH.Uj) if and only if gjg-1 g Ng(H). Let a g
C~(G/H) be such that
f a(x) dx = 1.
JG/H
Here, the invariant measure on G/H and on H are normalized so that
(	( 0(gh) dhdg = ( 0(g) dg.
JG/HJH	JG
Set f(gH, u) = a(gH)<p(u) for g g G, u g U. Then, f is invariant under
the deck transformations of ¥ so f defines an element of C“(G). We have
\Wc(H)\Ff»(h) = bH(h)<p(h)
for h g H". Let £ g T be related to rj in the same way as у is related to
12.2. The Modules KG|/>
139
a. Then we conclude from the preceding considerations that
d(y) f avtr(y(t))<p(ta)bH(ta) dtda
JTxA
= d(£) a*lr(£(t))<p(ta)tsH(ta) dtda
JTxA
for all <p as described in the preceding. We may choose representatives for
IVG(H) in K, so we conclude that
d(y) £ det(j) as,,tr(5y(t)) = d(^)	det(s) as/i tr(s£(t))
s^Wc(H)	s^W^H)
for all ta g H' = {h g H\ &H(h) =# 0}. Since both sides of this equation
are continuous on H, and H' is dense in H, the preceding equation is
true for all t g T and a g Л. The theorem now follows.
12.2.	The modules KC|Pir
12.2.1.	In this section, we assume that G is a real reductive group of
inner type and that G = °G. We denote by Int(gc) the subgroup
of Aut(gc) generated by the automorphisms of the form eAd* X g ec.
Fix a maximal compact subgroup К of G. Let (P, A) be a standard p-pair.
Fix (a, Ha), an irreducible square integrable representation of °M. Let
(Q,Aq) be another p-pair. Let v g a*. We set V = IP,a,iv/nQIPa iv.
Then V g ^(т0, К П Q). Thus, as an aG-module, V = Vf, with
V( = {r g K| (Я -	= 0 for some к and all H g aG}. We set
у = Ф I/
r Q\P,a,lv	'ip+pQ-
Proposition. Pgip a iv is a tempered (°mG,£) П Ю-module. If IPt<Tit, has
a regular infinitesimal character and if VQ\P a iv + (0), then there exists
к g К such that kAQk~1 c A.
Letfgj, ^G)and(P|, Ap) be minimal stand ardp-pairs such that Q f c Q,
An ^>An, P. cP, AP z>A. There exists к & К such that kP,k~} = Q.,
140
12. The Theory of the Leading Tenn
kApk 1 =Aq . We may thus assume that Qt = Pt, AQi = APi. Set
AP< = A]. Let fi g aG be such that Vip+PQ + 0. Set *Q = °M П P}. Then,
Vip +PQ/ * n Q^ip +PQ = ®	, л +P|( P i = Ppt) >
fe(°A/r
with H( a the ^-isotypic component of (l</i+pe/*nGl<>+pe)<T. If Hf,A+P1
=# (0) then A|Qe = ifi.
Let a,,..., ar be the simple roots of Ф(Р|, А Д Let ft be defined by
(Pi, a;) = 8и. Since IPf<rjv is tempered, if H(A+pi =#0 then ReA =
E, eF x,a, with Xj >0 for i g F c {1,..., r} (5.1.1). Now, (Q, AQ) =
((P^P^iA^p) for some Ft c {1,..., r}. So a% = EjeF| Rft. Re A|ae = 0
by our assumptions. Hence, (Re А, ft) = 0 for i F\. Thus, F c F,. We
thus conclude that Vip +pQ is tempered. This proves the first assertion.
Let A be a non-zero irreducible quotient of Vip+Po- Since X is
tempered, there exist a p-pair (Q2, A2) in °MQ with Q2 ^>*Q and A2 c
*Aq, (<t2,H2) an irreducible square integrable representation of °Mq2,
and v2 g a*2 such that X is equivalent with a submodule of Iq2<„2jV2
(Proposition 5.2.5). Set Q2 = Q2Nq and Л3 = A2AQ. Frobenius reciproc-
ity (Lemma 5.2.3) implies that there exists a non-zero (q, A)-module
homorphism from IP a iv to /Оз,а2,,„2+(>.
Let f с I n °m be a Cartan subalgebra of °m and let с f n °тСз be
a Cartan subalgebra of °mGi (such Cartan subalgebras exist by Theorem
7.7.1). Set t) = t Ф a and I), = t, Ф aQ}. Then, t) and I), are Cartan
subalgebras of q. Put I)R = it Ф a and (Ij^r = it1 Ф aGj. Let Aa g (ft)*
(resp., g (it,)*) be a Harish-Chandra parameter for a (resp., a2).
That is, the infinitesimal character of a is given by x.\ • Then there exists
g g lnt(qc) such that gt)j = t) and
л<72 + i(v2 + M) = (Ao- + «>) ° g-
Since (t)R)* ° g = ((I)i)R)*, we conclude that Aa ° g = A^ and v ° g =
v2 + fi. Since t Ф ia and t, Ф iaQ} are maximal abelian in qu = t Ф
i(X g q| вХ = —X] (a compact form of qc) we may assume that g g
lnt(q„).
We note that вЛ.а = Ля and flA^ = A^, 0v = -v, e(v2 + fi) =
-(i>2 + fi). Hence, Aa ° 6g6 = A^, v ° 6g6 = v2 + fi. This implies that if
we set x = g(ege)~l, then A„"x=A„, v°x = v. Our regularity as-
sumption now implies that x g exp(Ad(t + ia)). If we set t = x~l, then
12.2. The Modules Ие|/> „	141
6g8 = tg. We conclude that
(1)	g''ocoej.
Indeed, if H g a then 0(g~ 'Я) = -6g~ l8H = -g~lt~'H = -g~ 'H.
Thus, g~'a c (aGi)c. But the roots of gc with respect to (I)|)c are real
on g^'fl. SogHac a0
Similarly,
(2)	g-'tct,.
(1)	and (2) imply (by a dimension count):
(3)	g~'a = aQ} and g~'t = tP
We need the following lemma (see Helgason [1; Chapter 7, 8.7] for a
proof).
Lemma. If c is a subspace of {X g g| вХ = -X} and if g g Int(gu) is
such that gc с (X g g| 6X = —X}, then there exists к g К such that
Ad(k)H = gHfor all He c.
We now complete the proof. By (3), g~la = aQ} с {X e g| вХ = -X}.
So there exists к e К such that Ad(&)H = g~'H for H e a. Hence,
Ad(fc)~'aG c Ad(fc)~'aG3 = a, as was to be proved.
12.2.2.	We continue with the previous notation. We assume as before
that IP a iv has a regular infinitesimal character. If (Q, AQ) is a standard
p-pair such that ZG|P „ iv # (0), then we may assume (after conjugation by
к e K) that AQ a A. We assert that aQ acts semi-simply on a iv. To
prove this, we need a digression using some results related to those in
11.1.
12.2.3.	Let й be a semi-simple Lie algebra over C. Let t) be a Cartan
subalgebra of g, let Ф be the root system of g with respect to t), and let
W be the Weyl group of g with respect to fi.
Lemma. Let M be a ^-module such that there exists A g t)* such that if
h g Utt)}*', then hm = A(h)m. If A is regular (i.e., sA + A if s g IV - {1})
then M = ®5еИ, MsN, with = {m g M\Hm = £(Н)т for Het)} for
^et)\
142
12. The Theory of the Leading Tenn
Let G be a connected, simply connected Lie group with Lie algebra g.
Let t be a compact form of g and let К be the corresponding connected
subgroup of G. Then, К is a maximal compact subgroup of G. We can
apply the results in Section 11.1 to this situation using a = it, with t
maximal abelian in t. Theorem 11.1.4 implies that (7(t)) is a free (7(t)),y-
module on | JTl-generators, elt..., ew. This implies that
A\ = G(^)/ E t/(Ij)(ft - A(ft))
is a w-dimensional ^-module. If H g t), then since G(t)) is integral over
there is a relation
p-1
Hp + E = °, with u, e U(^)w.
J=o
Thus, on Na, Hp + EfJo1 A(u;)H' = 0. If j g W, then (sH)p +
Y.pJo(sHyuf = 0. Thus, if /x is a generalized weight of t) on A\ then so is
sfi. As in 3.2.4, the generalized weights of t) on A\ must be of the form
jA for some s eW. Hence, every у A is a weight of A\. Our hypothesis
implies that {уЛ|у g W} has order w. Thus, Nx = CsA.
Let tn g M. Then, is a quotient of A\. So (7(t))/n splits into
weight spaces with weights of the form sA, у g W. This implies the
lemma.
12.2.4.	We now return to the situation in 12.2.1. If v g a£, and (Q, AQ)
is a p-pair with AQ сЛ, then set V = Ip a v/nQIP a v. We assume
that IP<T'V has a regular infinitesimal character. Suppose that V is not
semi-simple as an nG-module. Let P, and *Q be as in the proof of
Proposition 12.2.1. Let Vp+PQ * 0. Assume that there exists H g aQ such
that (H -	+ РеХЯ))Им+Ре * 0. Then (4.1.5 (1)),
(h-(m+Po)(h))p;
+pe/*nG (Я-(м+ре)(Я))Им
+ PQ 0’
Now, [(H - (ц + pQ\H)),*nQ] c*nG. Thus,
(H- (M	*0.
We note that I^+Po/*nGI^+pe is a finite dimensional *AfG^G-module.
Let t)2 be a Cartan subalgebra of I = (*mc Ф aG)c and let Ф* be a
system of positive roots for I with respect to t)2. Let Z be the direct sum
of the generalized highest weight spaces for t)2 with respect to ф + in
I^+Pe/*nGI^+pe. Let Ф+ be a system of positive roots for Ф(йс, l)2)
12.2. The Modules Ие|Л„	143
containing Ф*. Let S be the half sum of the elements of Ф+. Let
17(H) = H - 8(H). Then, Z is a (7(1) |)-module, and if x.\ is the infinites-
imal character of IPaiv then t)(m)z = A(u)z for zgZ. Since A is
regular, the preceding lemma implies that the action of f7(t)j) on Z is
semi-simple. This is a contradiction.
We have proved:
Lemma. If IP a v has a regular infinitesimal character, then the action of
aQ on
IP,<r,	a, v
is semi-simple for all p-pairs (Q, AQ) with AQ c A.
12.2.5.	Fix x a homomorphism from Z(gc) to С, x = Ал with A regular.
Let a* = {i> g a*| if a g <f2(°M) and if has infinitesimal character
X, then it is irreducible for all Q g 0(A)}. Note that the condition “for
all” in the definition of a* can be replaced by “for some” by Proposition
12.1.3. We also note that there exist a countable number of non-zero
polynomials, {/J, on a* such that a* d {p g a*\ffv) =# 0 for all i}.
According to the previous lemma, if v g a*, if P,Q g &(A), and if
* 0, then the latter splits into a direct sum of irreducible
(°m, К П Af)-modules and a acts by (ip + pQ)I.
Lemma. Assume that (^Q\P,ajv)ip+PQ * 0, with v g a*. Then there exists
s g W(A) so that p = sv and as an (°m, К П M)-module, and
^Q\p,<r,ivXp+pQ ® irreducible and isomorphic with the (M П K)-finite vec-
tors in (ka, Ha), with к g К a representative of s.
Let W be an (°m,/f П M) summand of (VQ\P<a<iV)iP,+PQ. Then Frobe-
nius reciprocity (5.3.2) implies that there is a non-zero (g, A)-homomor-
phism T from IP,a,iv to IQ,w,ip. Since W is square integrable and p = sv
for some j g W(A) (see the proof of Proposition 12.2.1), the definition
of a* implies that T is an isomorphism. Theorem 12.1.4 implies that
there exists j g W(A) with representative к such that W = (ka, Ha)KnM
and p = sv. If X and У are irreducible (g, K>modules, then
dimHom0 tK(X, Y) < 1. If kt,k2 are two representatives for j, then
Ad(£|)|Q = Ad(fc2)|a. Thus, и = k2'kt g °M. Hence,
k2a(m) = a^k2'mkf) = a(uk^'mk\U~'} = а(и)(к}а(т))а(и)~'.
144
12. The Theory of the Leading Term
This implies that (kta, Ha) and (k2a, Ha) are unitarily equivalent. The
lemma now follows.
12.2.6.	Set (а*У = {p e a*I (v,a) * 0, a e Ф(Р, Л)}.
Lemma. If v g (a*y and if Q g &(A), then Ip,a,iV/ttQIpiaiiv =
^Q\P,<r,iv
We note that if v g (а*У then IPt<riv has regular infinitesimal charac-
ter. Indeed, let t) be as in 12.2.3. An infinitesimal character parameter
for Ip a<iv is Aa + iv. If a g Ф(йс, I)) and if (Aa + iv, a) = 0, then
(Ло.,а) = 0 and (v, a) = 0. The second equation implies that aG
Ф(тпс, I)). But Aa is regular for Ф(тс, I)). This proves the assertion.
Set V = IP'a,it,/nQIp a iv. Suppose that Им+Ре # 0. Let Y be an irre-
ducible (°m, К П M)-submodule of Vp+PQ- Let A be an infinitesimal
character parameter for Y on tc. Then there exists sg ИЧйс>^)) such
that s(Aa + iv) = Л + p,. Hence, sAa = A + Re p and sv = Im p..
Lemma 12.2.1 implies that there exists к & К such that kv = Im p. We
may assume that Ad(£)t = t (use the conjugacy under Ad(/f П M) of
maximal abelian subalgebras of t П m). Let t = Adffc)^. Then t~'sAa =
t^'A + Re/i and t_|j|Q = /. Thus, t~'s g W4°m, tc). Since v g (а£У,
ta = a and t-1ja = a, so ft = t and t_|jt = t. Thus, t~]sAa g it* and
t-lA g it*. So Re д. = 0, which is the assertion of the lemma.
Note. Without the assumption, the assertion of the lemma is false even if
h.a.iv has regular infinitesimal character. The reader should check this for
SL(2, C) using the material in Section 5.7.
12.3.	The leading term
12.3.1.	Let (тг, H) be an admissible finitely generated Hilbert representa-
tion of G. Then we say that (тг, H) is tempered if (тг, H) satisfies the weak
inequality (see 5.1.1). We recall that this means that there exists d > 0
such that if A g (H°°fK, then there exists a continuous semi-norm ал on
such that
|A( rr(g)t>) | < crA(f)(l + log llgll/H(g)
(here, S is as in 4.5.3).
12.3. The Leading Tenn
145
Let (тг, H) be a tempered representation of G and let A g (Я“УК. Fix
(P, A) a standard p-pair, P = °MAN, as usual. Let L+=L£ =
^аеФ(р.л) Na- Choose (Po, Ao), a minimal standard parabolic subgroup
of G such that Po с P and A c Ao. Theorem 4.4.3 says that if H g a + =
{H g a\a(H) > 0, a g Ф(Р, Л)} then, as r -> +<»,
(1)	A(77(exp tH)v) ~	E	А, у).
Here, the notation is as in 4.4.3. We recall that f(H; X,v) is a
polynomial in H and linear in A and v. Furthermore, there exists a
continuous semi-norm л on H" such that
|рМ1ДЯ;А,тг(а)у)| < (1 + ||Я||/(1 + ||log а||/алДм,^л(у)
for v g H°° and a g С1(Лц) П °M. Here, A = A.Hk (4.3.5). In light of
5.5.1 and 4.5.3, we have:
Lemma.
А,тг(т)у)| < Дм.£.л(у)(1 + IIH||)"(1 + log ||m||/coM(m)
for m g °M, H g a, £ e L+, A G (Я“)'к, v g H”.
12.3.2.	Let A g v g Я“. Set (p = pP)
<pP(a, X,v) = ap Y, aM+fPM,f(log a; A, v).
pf=E°
feL+
Re /x + £ = — p
Lemma.
(1)	There exists a continuous semi-norm on H°° such that
|<pp(a; А,тг(т)у)| < Дл(у)(1 + ||log a||/(l + log ||m||)dSoM(m)
for m g QM, a g A, v g H°°.
(2)	If X g n, У g n, then <pP(a-, XX, v) = <pP(a-, А, тг(У)у) = 0 for all
A g (Н°УК, v(=lF,a^A.
(3)	If H g a+, then there exists e > 0 and a continuous semi-norm H
on fP° so that
|e,p<//)A(7r(exp 1Я)тг( m)y) - <pP(exp 1Я; A, Tr(m)y) |
Ул.н^’К1 + Н1°в «H/U + 1°8 llw||)dHoM(m)e-"i'’(H).
146
12. The Theory of the Leading Tenn
We note that (1) is just a rephrasing of what we have already observed.
To prove (2), we note that
ХХ(тг(а)и) = -Х(тг(Х)тг(а)о) = -A(ir(a)7r(Ad(a-1)
So if X g na, then
XX(tr(a)v) = а~аХ(тг(а)тг(Х)и).
Similarly, if У g n_a then А(тг(а)7г(У)г) = -а_“УА(тг(а)г). (2) now
follows from the uniqueness of the expansion in 12.3.1 (1). (3) is an
immediate consequence of the derivation of 12.3.1 (1) (see 4.4.3) and the
definition of tempered representation.
1233. Lemma. Let v g H°°, A g (Н°°Ук. Suppose that vt,..., vr g a*
and that pt,..., pr are polynomials on a such that
lim e,p(H)X(TT(exptH)v) -	=0
t —» + ao	•
for all H & U an open non-empty subset of a+. Then,
Yleiv><H',pj{H) = <pP(exp H;X,v)
j
for all H g a.
The previous lemma implies that
lim ^е'7,'/н)р/(1Я) - <pP(exp tH\ X,v) =0
t —» + OO .
for H g U. Thus, 4.A. 1.2 (1) implies that
= <pp(exp H; A, v) for H G U.
i
Since both sides of this equation are real analytic in H, and U is open and
non-empty, the lemma follows.
123.4. We will be devoting most of the rest of this chapter to the study
of the properties of <pP(a- A, v), which we will call the leading term of
А(тг(а)г).
Before we begin the (difficult) analysis of the leading term, we will
introduce a slightly more general construction in the /С-finite case.
123. The Leading Tenn
147
Let f g C“(G). Then we say that f g £fw(G) if f satisfies the following
three conditions:
(1)	dim Z(gc)f < oo.
(2)	dimspanc(7?(/f)L(/f)/) < oo.
(3)	There exists d such that if x, у g U(gc), then
\R(x)L(y)f(g)\<Qy(l + log||g||)</S(g)
for g g G.
We note that (1), (2) just say that f g ^(G) (see 7.7.5, where it was
denoted A(Gf). Fix f g ^/w(G). Set
Vf = R(U(gc))spanc(R(K)f).
Then, Vf is an admissible finitely generated (q , К)-module. Let (тг, H) be
the admissible representation of G constructed as in 7.7.5 with HK = V^.
Then, as in that construction, there exists v{ = f) g Hk, A g (Я“Ук such
that A(Tr(g)r) = /(g), g g G.
Condition (3) combined with Theorem 5.5.2 implies:
Lemma, (тг, H) is tempered.
12.3.5.	We continue the discussion of the previous number. Let (P, A)
be a standard p-pair. We set fP(ma) = <pP(a; A, tr(a)v) for a g A, m g
°M. The following result is the basis of Harish-Chandra’s “theory of the
constant term”.
Proposition.
(1)	fp g ) is independent of the construction of (тг, H\
(2)	If H g a+, then there exists e > 0 such that
|e,p(H)/(m exp tH) - fP(m exp tH) |
< Сне e'₽(H)HoM(/n)(l + log ||m exp tH||/
for t > 1, m & °M.
(3)	If vt,.. .,vre a* and if <pr g C“(Af) are such that H >-»
<f>j{m exp H) is polynomial in H and if U is an open non-empty subset of a+
148
12. The Theory of the Leading Term
and if, for each H g U,
lim e lp(M}f(m exp tH) - ^e'tl'’(H)(f>j(m exp tH) = 0>
t —» + 00	j
then
E aiVi<f>j (ma) = fP(ma),
j
tn G °M, a g Л.
(4)	If <p g srfw(M) and if, for some open non-empty subset U of a+,
lim \e'p(-H}f(m exp tH) - <p(m exp tH) | = 0
t—> +00
for all H g U and m g °M, then <f> = fP-
That fP g srfw(M), and assertions (2), (3) are special cases of Lemmas
12.3.2 and 12.3.3. (4) clearly implies the second assertion of (1). Thus, to
complete the proof we need only prove (4). To prove (4), we need only
show that <p has an expression as in (3). Since <p g srfw(M), 12.3.4 (1)
implies that dim R(U(ac))<p < oo. This implies that there exist vu..., vr,
distinct elements of a£, and functions <pfm;H), i = l,...,r, C" on
°M x a with H (pfm; H) a polynomial, such that
<p(ma) =	fl)-
i
But <p g srfw(M), which implies that Re vs = 0 for all j.
12.3.6.	If f g s/w(G), Harish-Chandra dubbed fP the constant term of f
in the direction P, by analogy with theory of automorphic forms. We will
however call it the leading term of f in the direction P. The next result is
what Harish-Chandra called the transitivity of the constant (leading) term.
Proposition. Let (P, A) be a standard p-pair and let (Q, AQ) be a
standard p-pair such that Q с P and AQ dA. Let f g &fw(G) and let
*Q = QnM. Then (fP)*Q = fQ.
In light of Lemma 12.3.3, it is enough to prove that there exists an open
subset U of a q such that
lim \etp(H}f(m exp tH) - (fP)*Q(mexp tH)\ = 0
»-♦+<»
12.4. The Dependence of the Leading Term on Parameters
149
for H g U and m g °MQ. Set *aQ = aQ П °m. Then, aQ = *aQ Ф a.
Let *flg, H2 g a+. Then, there exists e > 0 such that, if (say)
t > 1,
|exp exp _ fp(m exp tH] exp sH2) |
< CH2e~esaoM(mexp tHf)(l + |t|/(l + |s|/(l + log ||m||)<
Also
|e'p*°(H1)/P(zn exp tHexp sH2) - (fP)*Q(m exp tHt exp sH2) |
<CH2e^‘a0MQ(m)(l + |t|/(l + |s|/(l + log 1М1/
for j > 1. Thus,
|e'₽e(H,+H2)/(/nexp(f(H1 + H2))) - (/P)*G(mexp(t(H1 +H2)))|
< Ce~s‘aoMQ(m)(l + log ||m||/,
for some <5 > 0 and all t > 1. Take U = {H g a^\H = Ht + H2, Ht g
*flg, H2 g a+}. The result now follows.
12.4. The dependence of the leading term on parameters
12.4.1.	Let (P, AP) be a fixed standard p-pair. Let (a, Ha) be an
irreducible square integrable representation of °MP. Let /“ be (as usual)
the C" induced representation of а]КпМр to K- If v e (ap)c> then we
have an action ttp a „ = ttv of G on If v g (aP)c, then the represen-
tation TTiv of G on the Hilbert space completion Ha of is a unitary and
tempered. Thus, the material of the previous section applies. We note that
if a is the dual representation of °MP then (I^)'K can be identified with
(/pK under the pairing
(f\g) = ftf(k)\g(k))dk.
Let (Q, A) be a standard p-pair. If A g (1“Ук = then we write,
for v g 1“, v g a*, a A, <pQ\P(.iv, a; A, v) for the expression denoted by
<pQ(a-, A, v) in 12.3.2.
Lemma. If iriv has a regular infinitesimal character and if
4>Q\P{iv,  ;  ,  ) + 0, then there exists k g К such that kAk~' cAP.
150
12. The Theory of the Leading Term
Lemma 12.3.2 (2) implies that, under the hypothesis of the lemma,
Pgip _iP =# (0). Since TTP t -iv has a regular infinitesimal character, the
result follows from Proposition 12.2.1.
The main result of this section is the following reinterpretation and
strengthening of a basic result of Harish-Chandra [15].
Theorem. Assume that A cAp. For fixed A, <pQ\P(iv, a; A, v) is continu-
ous in v, a, v and real analytic in (v, a).
The proof of this theorem will occupy the rest of this section.
12.4.2.	We begin with some simple observations about the action of nQ.
For this, we need some notation. As before, we will look upon = I&
as (ГаУк. We therefore have for each v g (ap)c a (g, КЭ-module structure
тг_„ = ttPj&j-„ оп(7^Ук such that ir_v(X)X(v) = -X(ttv(X)v) for A g I&,
v I*, and X g g.
For each y^K we fix (ry,Vy) g y. If F с K, then set F' =
{y g К]Нотк(Иу, ® gc) =# 0 for some д. g F}. We define F; recur-
sively by F,+l = (F7)1 U F’. We note that if F is finite then Fl is also.
If И is a (g, К)-module and if F с K, then we set V(F) = Фуе P F(y).
With this notation, we observe that tt_v(q)I&(F) c Is(F').
Lemma. Let F be a finite subset of К. Let p g I&{F), X g g. Then
v тг_„(Х)ц
is a polynomial map from (aP)c into ^(F1) of degree at most 1.
d
p_v(hexptX)
t = 0
d
dt
(aP(kexptX) *p0(k exp tX)}
/ d	\
= -V — log aP (k exp tX) ki(fc) + тг0(X)p(k).
\ dt (-o	/
This proves the lemma.
12.43.	Let F be a finite subset of K. Set s(F, v) = dim tt_ v(n0)I-(F).
Then s(F, v) < dim I&(Fl). Let s(F) = max{s(F, p)|p g (аР)с). Put
a* F G = {p e (aP)*c\s(F, p) = s(F)}.
12.4. The Dependence of the Leading Term on Parameters
151
Lemma.	°Pen (flp)c- Zf po e ac,F,Q> (^en ^ere exists a
polynomial <p on (aP)c such that <р(и0) =# 0, {м|^(м) =# 0} c ofc F Q and
such that there exist v, G X, g nG, i = \,...,s(F\ such that if
ufv) = tt_v{X^Vi then ufv\us(Ffv) is a basis of Tr_v(nQ)I-(F) for
each v such that <p(v) + 0.
Let Xi,...,Xn be a basis of nQ and let {t>,} be a basis of /^(F). Let, for
t = (i,f), Zfv) = TT_lfXi)vi. Then £, is a polynomial of degree at most 1
in v with values in I&(F'). Let ti,...,ts be such that {£,O0)} is a basis of
7r_,,0(no)Aj(F). Set ui = - Then,
v ~ ufv) A • • • л us(v) = ф(и)
is a polynomial map of a£ into AV^fF1). Let д.р...,д.т be a basis of
\SI&(F'). Then
i
with <pj a polynomial in v. Since ф(и0) =# 0, there exists i such that
<pf.vQ) =# 0. Set <p = <Pj. Then, the conditions of the lemma are satisfied by
<p and {mJ.
12.4.4.	We set
t(F,p) =dim{(/J(F) +7?_JnG)/-(F'))/7?_JnG)/.(F')}.
Let t(F) = max{t(F, m)|m g (ap)£}. Set (a£ F Qf = {i> g a£ FiG|f(F, v)
= t(F)}.
Lemma. Let v0 g (ajF and let <p, ufv) be as in the previous lemma
for v0. Then, there exists a polynomial ф on a £ such that ф(и0) =# 0, and
l>p ..., t>,(F) g/^(F) and such that vu... ,vt(F),ufv),... ,us(F>fv0) is a
basis of I&(F) + 77_p(nG)/J(Fl) for v such that <f>(v) =# 0, ф(у) Ф 0.
Let Гр..., r,(F) g /^(F) such that гр..., r,(F), ufvQ\..., us{FifvQ) is
a basis of I^tF) + 7r_„o(nG)/(?(Fl). Then, as before,
^(м) = Г| A • • • Л Г; A M|(m) A • • • A us(v)
is a polynomial in v with values in /\,+sI&(F2). Let {p.,} be a basis of
h‘+sId(F2). Then,

152
12. The Theory of the Leading Term
Since £(v0) # 0 there is an i such that ф^Рд) # 0. Set ф = ф1. Then ф,
{г,} satisfy the conditions of the lemma.
12.4.5.	We now make some observations of a different nature. As usual,
we consider the direct sum decomposition
Цйс) = ^((mo)c) ® (п£?^(йс) + Цйс)й0).
Let pQ be the corresponding projection of L7( gc) onto W(tnG)c). Then,
pQ restricted to Z(gc) is a homomorphism into Z((mG)c). Set IT =
IT(gc,t)),	and d = |!T/lTm|. Theorem 11.1.5 says
that:
(i)	(7(1) У*"1 = Ф Ud))wUi, with 1 = и|,..., ud g Ud)Wm.
Let у (resp., ym) be the Harish-Chandra isomorphism (3.2.3) from
Z(gc) to (resp., Z((mG)c) to (7(^)’y"1). Set z, = y~'(u(). If we
apply у ~1 to (i), we have:
(ii)	Z((m0)c)= ®y-1y(Z(ec))z/.
If A 6 a*, then we denote by p.A the isomorphism of f/((mG)c) defined
by p.A(X + H) = X + H - А(Я)1 for X g °mG and H g a. The proof of
3.2.3 implies that:
(iii)	у~ 'y(z) = mp/pg(z)) for z g Z(gc).
Thus:
(iv)	Z((mG)c) = ®nPQ(p0(Z(gcy)Zj.
Set e, = n-.p^Zj). If we apply to (iv), we have:
(v)	Z((mG)c) = ®pG(Z(gc))e,.
In particular, since a c Z((mG)c), this implies that if h g a then
he, = LPQ^i^h^ej,
j
with z(7(/i) g Z(gc).
If v g ас, then denote by qQ(p) be the canonical projection of Id(p)
onto I&(p)/nQI&(p). If z g Z(gc), then <7g(pX7t_/z)A) = pQ(z) 
qQ(vYX) for A g I& (here, the dot means the canonical action that de-
pends on v on the quotient). In particular, this implies that if z g Z(gc)
then pQ(z) acts on I&(v)/nQI&(v) by a scalar.
12.4. The Dependence of the Leading Term on Parameters
153
Let r = min{j | {et,..., ed} c UJ((me)c)}. Let A g Id(F), with F a finite
subset of K. The preceding observations imply:
(1)	U{ac) • qQ{v\k) C qQ(Vyj&(Fry).
(2)	dim U(ac)  qQ(v\k) < d.
12.4.6.	Let F с К be a finite subset and let A g Id(F) - {0}. Set
w(k,v) = dimf7(ac) • qQ(k).
Then, w(k, v) < d. We also write
w(A) = max{n'(A,p)|p g (a£,Fr+i,G)'}-
Set
a£(A) = (p G(a* F,+I 0)'|h-(A,p) = w(A)).
Lemma. a£(A) is open and non-empty. If v0 g a£(A), then there exists
S c {1,..., d], |S| = w(A), and a polynomial q on a£ such that q(.v0) =# 0
and if 7)(p) =# 0 then {e,  qQ(vXk)}i<ES is a basis of t/(ac) • ^g(pXA).
Let S be such that (e,- • <?Q(v0XA))j e s is a basis of U(ac) • qQ(v0)(k).
Let <p, ф, {о), {Uj(v)} be as before for Fr+ ’. Then,
= ЕУ/;(>')«/+ EM")"/")’
J	J
with y0 and <5(; rational in v for <p(v) =# 0, &(v) =# 0. Set
/
We relabel so that S = {1,..., w}, w = w(A). Then,
^(p) =z,(u) л ••• az„(p)
is a polynomial in v with values in AwId(Fr+l) and gtvg) =# 0. Now argue
as before to complete the proof.
12.4.7.	Let v0 g a£(A) and let <p, ф, vf, ut,q, S be as above. We relabel
the indices so that ex = 1 and S = {1,..., w(A)}. By definition, if H g a
then
H e, -q0(u)(k) = Y1bi](H,v)eI - qQ(v)(k),
154
12. The Theory of the Leading Term
with bjj linear in H if <р(р)ф(У)т](р) # 0. We note that
)	i
with yif and rational and defined for <р(р)ф(и) =# 0. Also (and this is
the reason why we used Fr+l in the previous lemma),
/	i
with a/; and p,;/ linear in H, rational in v, and defined for <р(г)ф(г) =# 0.
Set z;(p) = L}yji(v)vf. Then,
j	i
This implies:
(1)	bjj(v, H) is rational in v linear in H and defined if <р(и)ф(р)т)(р) Ф 0.
We also note:
(2)	TT_v(H)Zi(v) = bjjtH, vyZjtv) + w^H, v) with z/p) = A and
Wj{H, v) = ’LfPjjiH, v)Uj{v), with PjjtH, v) linear in H, rational in v, and
defined if <р(р)ф(р)т](р) Ф 0.
Let lTG|p = {s g ИЧЛР)| j|Q = I}. Let j| = 1,..., sq be a set of repre-
sentatives for the right cosets of JV(Ap) with respect to W^p.
Let а^СлУ be the set of all v g aJ(A) such that (v, a) =# 0 for a g
Ф(Р, Ap) and SjV^ =# i + j. We note that if iv g а£(АУ, v e a*,
then Ip'&'~iv also has regular infinitesimal character. Also, the condition
that v g а£(ЛУ is given by the non-vanishing of another polynomial,
Thus, aJ(Ay П ia* is non-empty and hence is open and dense in ia*. If
iv g cic(Ay П ia*, then Lemma 12.2.6 implies that B(H,iv) is semi-sim-
ple with eigenvalues of the form (~isfv + p\H). This implies that there
exist natural numbers d^v) such that
det(B(H,iv) - tl) = П ((“»/»' + p)(H) - t)^.
/=•1
Fix v0 such that iv0 g а^АУ П ia*. Let HQ g a be such that SjVgiHg) =#
SfV0(H0) for i Ф j. Then there is a connected neighborhood Ut of iv0 in
ар(АУ П ia^ such that s,p(H0) SjvtHg) for i Ф j and v g Ut. If 5^(Н0)
12.4. The Dependence of the Leading Term on Parameters
155
=# stv(H0), then we set
Д(Я, y) + (sjv - p)(H0)
siv(Ho) ~ siv(Ho)
If v g Ut, then dt(v) = tr P^v). Thus, d[v) is a constant, dit on Ut. Since
det(B(H, v) - t) is rational in v and polynomial in H and t, we have:
я
(3)	det(B(H,p) -fZ) = П((-^+р)(Н) -t/' forpea*(A)'.
i-1
12.4.8.	We are now ready to begin the proof of Theorem 12.4.1. To carry
it out, we return to the basic argument in 4.4, keeping track of the
dependence on parameters. Let v g a£(A). Set, for v g H g a,
zt(v)(irv(exptH)v)
F(v,t,H,u) =
G(v,t,H,v) =
wt(v, Я)(тгДехр tH)v)
vvw(p, Н)(тгр(ехр tH)w)
and
Then
d
—F(v,t,H,v) = -B(H,v)F(v,t,H,v) + G(v, t,H,v).
dt
This implies that
F(v,t,H,v) =e-‘B(‘H'v}F(v,G,H,v)
+	f'eTB(HfV)G(v, t, H, r) dt.
Jo
12.4.9.	We note that in the definition of the m,(p) in 12.4.3 we may
assume that Ad(a) = aa‘Xt with az g Ф((), A). Recall that u^v) =
156
12. The Theory of the Leading Term
Hence,
i	/
If H g a+, we set /3(H) = min{a g Ф((), Л)|а(Я)}. If H g a+, then
и',(»',Н)(7гДехрЯ)1>) = ^3/,(р,Я)тг_р(А;.)А1/(7гр(ехрЯ)1>)
/
= - Е^,(»'>^)М/(7гДА;)7г,(ехрЯ)г)
/
= - ЕМ^Юв~“/НЧ(^ЛехрН)1гДХ;)и).
j
Thus, if ы is a compact subset of а^(АУ, then there exists a constant Сы
such that
|и',(»',Н)(7гДехрЯ)г)| < Сш||Я||е-/3(Н)^|м;(7гДехр H)ir„(Xf)v)\.
i
If we apply Lemma 5.2.8 (noting that in that lemma ft need only be a C"
vector), we have
|и'/(»', Я)(7г„(ехрЯ)г>)| < C;(l + ||H||)''<7(L>)SRep(expH)e"/3(//),
with С’ы depending only on ы and q a continuous semi-norm on /".
Fix fl, a compact subset of a+ with non-empty interior. Let
U2 = (p G(flp)*| |Rejp(H)| < Д(Я)/4 for H G fl, j g И^(Лр)}.
We now assume that ы c U2 A a£(A)'. If v g ы, then the projection Pj(v)
of Cw onto the (-S/V + p)iQ eigenspace of B(v, • ) is given locally by an
expression of the form 12.4.7 (*). Thus, it is continuous and B(v, • ) =
E;(-5;p + p)\aPf(v). Combining this with the previous inequalities, Lemma
3.6.7, and Theorem 4.5.3 we conclude that if H g fl, t > 0, then
(*)	||e'B<H’,'>G(p,t,H,r)|| <;	+t||H||)^(r)
<C,e-'^3q(V),
with C depending only on w, C, depending on ы and fl. This implies that
reTBiH’v)G(v,T,H,v) dr
Jo
converges absolutely and uniformly for H g fl, v g w.
12.4. The Dependence of the Leading Term on Parameters
157
Set
F°(v,t,H,u) = e~,B(v'H)\ F(v, t, H,v) + ГетВ(Н’^С(р, т, H, v) dr
\	Jo
Define
The preceding estimates imply that if v g a* is such that iv g w, Я g fl,
then
<pG|P(n', exp tH; A, v) = elp(M)e(F0(iv, t, H, t>)).
If v g Ы, H g fl, then set
^0|F(i/,exp tH; A, v) = elpwe(F°(v, t, H, r)).
(1)	There exists constants C, d depending only on ы such that if v g w
and if H g fl then, if t > 0,
|е'р<//)А(тгр(ехр tH) v) - Ф0]р(р, exp tH; A, v) | < Ce~lB(H)/'2q(v).
Thus, our assumptions imply that
||£?'P("VB(H,,)|| < Ce^(H)/4(l + ИНЯН)’*'.
So
\\e,p(H\F(v,t,H,v) - F°(v,t, H,v))\\
= T H dT
< С3е’Ке^Н)Ге-^н^Чт^(и) < Ce-'^H^'2q(v).
12.4.10. We are now ready to complete the proof of Theorem 12.4.1. Fix
v0 g a*. Let h be a non-zero polynomial function of v such that if
h(v) =# 0 then v G ас(АУ. Let g a* be such that h(iv0 + =# 0. Then,
there exists e > 0 such that h(iv0 + z£) =# 0 for 0 < |z| < e. Let fl be a
compact subset of a+ with non-empty interior. Fix 0 < 3 < e such that if
H g fl, then 3|(Re e‘es£(H)\ < p(H)/4. Then, h(iv0 + 8eieO > c > 0
158
12. The Theory of the Leading Term
for в g R. There exists a compact neighborhood W of vQ in a* such
that h(ip + 8е'в£) > c/2 for в g R, p g W. Let w = {ip + 8е‘в£\р g W,
6 g R}. If H g П and if p G IV, then (12.4.9 (1))
7r(>+Se^(exp - il/Q\p(jp + 8eieg, exptH; k,v
2 77
Since A(7r„(a)r) is holomorphic in v, we have
1 2ir
— f 1ГА(7ГФ + 5еМДеХР tH)V) de = ^il^P
Z.TT Jo
If p g W, then we set
e'p(H) 2ir
—-— f ^iliQipfJp + 8e‘e^, exp tH; A, w) d6 = y(ip, exp tH; A, v).
2tt •'0
Then, y(ip, exp tH; A, v) is real analytic in p and has an expression of the
form given in Lemma 12.3.3. Since
|е'р(Н)А(7г(м(ехр tH)v} - y(ip,exp tH; A, u)| Ce~‘^H}/'2q( v),
Lemma 12.3.3 implies that y(ip, exp tH; A, v) = <f>Q^P{ip, exp tH; A, v). We
have (finally) proved the theorem.
12.5. The leading term and intertwining operators
12.5.1.	In this section, we will relate the leading term that has been
studied in the last two sections to the intertwining operators of Chapter
10. We first need a simple lemma. We maintain the assumptions and
notation of the previous sections.
Lemma. If v g a* and if (v, а) Ф 0 for a g Ф(Р, A), then sv Ф v for
all s e W(A), s * 1.
Let (Po, Ло) be a minimal standard p-pair with A c Ao and Po с P.
Set *a = a0 П °m. Then a0 = *a Ф a. We extend v to a0 by p(*a) = 0.
Let у g IV(A) be such that sv = v. Let к & К be a representative for у
(12.1.3). Then there exists и <= К П °M such that Ad(fcX*a) = Ad(uX*a).
We replace к by u~'k. Put 5| = Ad(£)|Qo. Then, 5| g IV(AO) and yI)Q = s.
12.5. The Leading Tenn and Intertwining Operators
159
Let H = Hv g a be defined by B(Hv,h) = v(h) for h g a0. Since a(H)
* 0 for all a g Ф(Р, A), we see that mc = {X g gc| [A', H] = 0}. Since
we are assuming that G is of inner type, this implies that if g g G and if
Ad(g) H = H, then there exists m g Int(mc) such that tn = Ad(g). This
implies that s1|Q = I. So j = 1.
12.5.2.	Set (a*)' = {p g a*| (v, a) =# 0, a g Ф(Р, A)}. We note that (a*У
depends only on A and not on the choice of P g 0(A). Indeed, if
Q g 0(A) then <&(Q, А) с Ф(Р, А) и (-Ф(Р, A)). Fix P g 0(A), and
let (a, Ha) be an irreducible square integrable representation of °M. In
12.2.6, we have seen that if v g (а*У then IPt<T<iv and IPt&t-iv have
regular infinitesimal characters. Let Q g 0(A). 12.3.2 (2), Lemmas
12.2.4,12.2.5, and the previous lemma imply that if v g (а*У, A g J.,
v g and Q g 0(A), then
<p&P(iv,a; A,y) =	£ csaisv.
We set
<Pq\P,s(°Av\^v) = cs-
Lemma. If A g I&, then (pQiP s(tr,iw, k,v) is continuous on (а*У x /“
and real analytic in v g (а*У.
Let v0 g (а*У. Then sv0 =# v0 for s g JV(A) - {1}. There exists H g q
such that sv0(H) Ф tvQ(H) for j, t g W(A), s * I. We label W(A) as
{$!,..., jw}. There exist tt,..., tw g R such that if
y(v) = det[e"P‘HH)],
then y(vQ) =# 0. Let U be a neighborhood of v0 in (а* У such that
y(v) Ф 0 for v g U. Set, for v g U,
[«*;(»')] =
Then, if v g U, we have
E«a(i')<PG|p('i'> exp tkH; A, v) = <peiP<S)((r, iv,X,v).
к
The lemma now follows from Theorem 12.4.1.
160
12. The Theory of the Leading Tenn
12.5.3.	Proposition. If v g (а* У then is holomorphic at iv and
(pPlP,i(<r,iv;X,v) = ypI(Jp|P(tp)r(l)|A(l)).
Here, we are using the normalization of measures as in 10.1.7 and
yP = f_aP(n)2pp dn.
JN
We return to the notation in 12.4.9. Then, if v g w, H g fl, and if
t > 0,
|e,p(H)A(7rp(exp tH)v) - i[iP^P(v, exp tH; A, v) | < Ce~m/l2q(v).
On the other hand, if Re(p, a) > 0 for a g Ф(Р, A), then Theorem 5.3.4
implies that
Нт^е'(р-,,ХН)А(7гДехр1Я)г>) = yf 1(7?|Р(р)г>(1)|А(1)).
Here, the yP arises as follows. In 5.3.4, the normalization of measures is
such that
f f(x) dx = f_aP(n)2ppf(k(n)) dn.
J0MHK\K	JNp
Here, dx is normalized so that the total mass of °M П К \ К is 1. Thus, in
our new normalization we must put in the factor yf1.
We may assume that if v g ы, then IP^_„ has regular infinitesimal
character and that sv Ф v for j g 1У(Л), j Ф 1. Thus, as before,
tyP[P(y,a;k,v) =	£ а!рфР[Р s(v; A, t>),
>eW(A)
with	A, v) holomorphic in v g ы. This implies that
<pP|P /a, iv; A, v) has a holomorphic extension to ы as t[iP\P1(v; A, v).
If v g ы and if Re(p, a) > 0 for a g Ф(Р, A), then the preceding
considerations imply that
iAp|PiI(p;A,r) = ур1(7р|Р(р)г(1)|А(1)).
This implies that
<pP|PiI(o-,jp;A,r) = ур’( JPIP(iv)v(l)\A(l)),
wherever the right hand side is defined. But the left hand side is real
12.5. The Leading Tenn and Intertwining Operators
161
analytic in v g (a*)'. So the right hand side must be also. The proposition
now follows from Theorem 10.1.6.
12.5.4.	The following result is Harish-Chandra’s generalization of a re-
sult of Bruhat [1] for minimal parabolic subgroups.
Corollary. If v g (a*)', then ttpa iv is irreducible.
Let v g Ia - {0}. Assume that A g Ц is such that
A(7rP.<7,I,(t/(Sc))span(77p a,iv(K)r)) = 0.
If we show that A must be 0, then we will have shown that г is a cyclic
vector for IP a . Since v is an arbitrary non-zero element of IP a lv, this
would prove the irreducibility.
Since v is an analytic vector for irP a iv = iriv (3.4.9), 1.6.6 implies that
А(тг^(А:Г1)7г/„(а)-п-/„(т)7г^(Л:2)у) = 0,
for , к2 g K, a A, m g °M. By the definition of the leading term, we
therefore have
<Pp|p,i(°’>Л^1)^,7Г,Л"’)77,ч,(Л2)г) = 0.
The previous proposition now implies that
(о-(т)/?|Р(^)г(Л2)|А(Л1)) = 0
for all m g °M, kltk2^ K. Thus, if we can show that JP\P(iv)v Ф 0 then
the irreducibility of a would imply that A = 0. We therefore complete the
proof by proving that JPiP(iv)v =# 0.
The preceding proposition implies that Jp|F and JP]P are holomorphic
at iv. Theorem 10.5.8 therefore implies that
= <pPa(iv)v,
and that <pP a(iv) > 0. Thus, if JPiP(iv)v = 0 then v = 0. This completes
the proof.
12.5.5.	We now study other ramifications of Proposition 12.5.3. In partic-
ular, we will give a meromorphic continuation of the <Pq]P s to a£ by
explicitly calculating it in terms of intertwining operators. The formulas
that we will derive were first discovered by Arthur [1] and are equivalent
162
12. The Theory of the Leading Tenn
to Harish-Chandra’s functional equations for his C-functions. We begin by
extending the theory of the Harish-Chandra ^.-function. We will use the
notation in 10.5.8. Set, for P,Q c &(A),
^Q,p{a’v) = П
a<=X(P\Q)
Then, as in 10.5.8, we have:
(1)	p{(t, v)JQ^p{v)Jp^Q{v) = I.
It will be necessary to indicate the dependence of JQ^P{v) on a by
writing JG|P(o-, v) = JqiP(v). We now will now make some simple observa-
tions about the leading terms. We first observe that
= x(JQ\p(aAv)TTpaJv(a)^'
This implies that if v e (а*У, then:
(2)	(pQlp /а, iv, A ° JG|P(o-, iv), v) = ‘Pq\q,s(°'’ tv, A, Jq\p^, iv)v).
For the next observation, we recall that we are identifying (Г£УК with Ц
under the standard pairing. We may thus write L(k)k(x) = k(k~lx) (see
12.1.3). We have (L(k)v\L(k)X.) = (r|A) for k g K. In 12.1.3, we ob-
served that
= ^kQk->,ka,sXs)L(k).
This implies that
= K^^kPk-'.kaJsX^K^v).
Hence:
(3)	(pQ^p s(a, iv, A, v) = (pQikPk~l, isv; L(k)k, L(k)v).
Now, (2) implies that
<pQikPk-1,i(k°'’isv'’ L(k)t;°JQikPk-'(.k<r, isv), L(k)v)
=	L(k)^,J&kPk-,(ka,isv)L(k)v). (*)
Proposition 12.5.3 implies that
(pQlQ,i(ka’ ^v; L(kH, JQ\kPk-l(k<r’ isv)L(k)v)
12.5. The Leading Tenn and Intertwining Operators
163
If we replace £ by kPk-i(ka, iv)X. ° JkPk-tiQ(ka, iav) in (*) and (* *),
then (1) implies that:
(4)	<P&p,s(a’iv’K’V) = fLQlkPk-i(k<T,iv)yQ1
x (J^ka, isv)JQ\kPk-\(ka, isv)L(k)v(l)\
x(L(^)A°J^-i|G(^o-,isp))(l)).
If v g a^, then we set
<р<э|Л5(°’>1';а,у)
= *01кРк-'(к(7, v)yQ\jQ[Q(ka, sv)JQ^kPk-\(kff, isv) L(k) v(l)\
xL(k)k ° JkPk-<lQ(k<r, 5i>)(l))
for P,Q g 0(A), A g I&, v g j e JV(A), к a representative of j, and
all v for which the right hand side of the equation is defined.
We also extend <pP|G to a£ using the formula
<pG|p(0-,p,a; A,t>) =	£ asv(pQiP s(ff,v;X,v).
seHU)
We have proved:
Theorem. If P,Q g 0(A), A g I&, then <p0\P(<r,  ,a;k, ) has a weakly
meromorphic continuation toaf
12.5.6.	In the last three numbers the constants yP, P g 0(A), have
played an important role. We conclude this section with an observation
about the yP.
Lemma. If P,Q g 0(A), then yP = уQ.
We first prove the result in the special case when A is a maximal special
vector subgroup. In this case, if P g 0(A) then (P, A) is a minimal
standard p-pair. If Q g 0(A), then there exists j g W(A) and к a
representative of j such that Q = kPk~'. Thus, NQ = kNPk~l. If g g G,
g = nPmPaPkP, nP&NP, mPe°M, apcA, kP&K, then kgk~l =
knPk~ikmPk~1kaPk~1kkPk~1. Thus, а0(к~^к) = kaP(g)k~1. Let
Int(fcXx) = kxk~x for x g G. Then, in light of our normalization of
164
12. The Theory of the Leading Term
measures on NP and NQ, Int(£) is a measure preserving diffeomorphism
of NP onto Nq. Thus,
yQ = aa(n)2pQ dn = f_aQ(knk~l)2pQ dn
jnq	jnp
= f_ (kaP(n)k~l^pQ dn = [aP(n)2s 'pQ dn
JNP	JNP
= Lap(n)2pPdn = Ур-
JNP
This proves the result in the special case.
Now let A be an arbitrary special vector subgroup of G. Let P g A).
Let (Po, Ло) be a minimal standard p-pair such that A cA0 and Po с P.
Set *P = Po П M. Then, (*P, Ло) is a minimal (К П M)-standard p-pair
in M. Let *N = 6{N*P), as usual. Then (p = pPo, *p = p*P),
Yp = [ a(*nn)2p d*ndn.
J,NxNf
Now, *n = n*a(*n)k(*n), with *a(*n) g Ao П °M, n g *N. So
Yp = [_ _a(*n)2pa(k(*n)n)2p d*iidn
J,NxNf
= f_ _a(*n)2 Pa(k(*n)nk(*n)~1} P d*ndn
J*NxNp	'
= [_ _a(*n)2 pa(n)2p d*ndn,
J*NxNp
since Int(/c) preserves dNP for к g К П M. We therefore have:
(1)	Ур0 = У*рУр-
If Q g £Р(А) and if Qo g 0>(Ao) is such that Qo c Q, then *Q is a
minimal standard parabolic subgroup of M. Hence, the special case of the
result that we have already proved implies that y*q = У*р an£I Yp0 = Yq0-
Hence, (1) (for Q) implies that yP = yG.
12.5.7.	We now consider the more general situation: (1°, Ap) is a stan-
dard p-pair and (Q, A) is a p-pair with A cAP. Let W(A, Ap) =
{j g L(a, aP)| there exists к g К with Ad(£)|a = 5} (here, L(a, aP) is the
12.5. The Leading Tenn and Intertwining Operators
165
space of all linear operators from a to aP. If 5 g W(A, AP) and if к g К
is such that Ad(£)|Q = 5, then, as before, we call к a representative of 5.
Oearly, МЛ, A) = W(A).
Lemma. W(A, AP) is a finite set.
Let 5 g W(A, AP) and let к К be a representative for 5. Then,
Ad(£)Ce(a) Cfl(aP). Thus, Ad(£)mG mP. Let fj0 be a Cartan subal-
gebra of mP such that f)oz>aP. Then, E), = Ad(/c)~1E) 0 is a Cartan
subalgebra of mG with fjj z>a. Up to conjugacy relative to MP (resp.,
MQ\ there are up to inner conjugacy only a finite number of Cartan
subalgebras of mP (resp., mG). Thus, to prove the lemma we need only
show that if fj0 (resp., is a Cartan subalgebra of mP (resp., mG), then
S = {5 g lV(A,AP)f s has a representative к with Ad(^)^ = fj0} is
finite. If F is a subset of g, set NK(F) = {£ g K| Ad(£) F = F}, CK(F) =
{k g K| Ad(fc) f = f, feF], Then, |S| < \NKO),)/CK(^,)| •
|CK(fj0)\./VK(fj0)|. Since NK(f))/CK(f)) is finite for t) a Cartan subalgebra
of g, S is finite.
12.5.8.	If v g (aP)* and if 5 g W(A, Ap), define sTv(H) = v(sH) for
Hgq. Then sT: (aP)c aj. Fix (a, Ha\ an irreducible square inte-
grable representation of °MP. Let, for a & A, v g a*, A g I&, v g 1^,
(pQ^dv; a; A, v) be defined as in 12.4.1. Then (12.4.1), <pQIP(iv; a; A, v) is
real analytic in v g a*. If v g (a*)', then Lemma 12.2.6 combined with
Lemma 12.3.2 and the proof of Proposition 12.2.1 imply that if A g I-,
v g then
<p0lP(iv-,a;A,v) =	£ csa,sTv
ic W(A, Ap)
for a g A. As in 12.5.2, we set
4>q\p,s(^AV, A,f) = cs.
The following lemma is proved in exactly the same way as Lemma 12.5.2.
Lemma. If A g I-, s g WlA, Ap), then (pQ\p,s(<r; iv; A, v) is continuous
on (a*)' x /" and real analytic in v g (a*)'.
12.5.9.	Our next task is to relate <pG|p,J(o’; iv; A, v) to intertwining opera-
tors. We first note that MP c MQ. Set *P = MQ П P. We set Qx = *PNQ.
166
12. The Theory of the Leading Tenn
Then, Qi c &>(AP) and °M*P = °MP. Set *aiv equal to the representation
тг»л<7>/ with *a = aP П °mG. If v e a*, then *<riv is tempered. We
note that *<riv\KnMQ is independent of v. We denote this representation of
MQ П К by *a. We define a linear map T from Iq to Iqx a by
7Г(Л)=ЛЛ)(1).
T is clearly continuous. We define S from Iqi(T to by
W)(m) = f(tnk).
Then, TS = ST = I. Thus, T is a linear isomorphism. We note that
—	iv( s) T
and
If Pt, e &(At) and if д. is an admissible tempered representation of °MPi,
then set JP iP[ti,v) equal to the corresponding family of intertwining
operators.
We are now ready to carry out the calculation of leading terms following
the method of the special case that we have already treated. Suppose that
v = JPiq[<t, iv)w. Then,
Thus,
iv'> A> ») = <PQiQ„s(<r, iv, A ° JP|Gi(o-, iv), w).
The following result is proved in exactly the same way as Proposition
12.5.3.
Lemma. <pG|Gi /a, ip; £, w) =	ip|a)Sn'Xl)lS£(l))-
As before, fip^a, iv)JQ^P{a, iv)v = w. So we have:
(1)
xSA°JP|Gi(o-,ip)(l)).
12.5.10.	Now let j g W(A, Ap), and let к К be a representative of j.
Then, L(k~i)7rPaiv(g) = 7rk-tPkk-iaisTv(g)L(k~l). So, as in 12.5.5,
12.6. The Main Inequality
167
we have:
(1) <pQik-'Pk,^~i<T> ifc-1»'; L(&-I)A, Щ-1)г) = <pG|piJ(o’, iv; A, v).
If we combine this with 12.5.9 (1), we have the complete formula:
(2)
= Ув iilk~'Pk\k~'Qlk(^ 1O’,ik ’ v) ( (jk-lQk\k~'Pk(*	1(rik~'v >
xSL(k ')A ° Jk-'Q^k-'Pkt^ l°~Ak Ii')(l)j.
We will study some ramifications of this formula in Section 12.7.
12.6.	The main inequality
12.6.1.	We retain the notation of the previous sections. In this section,
we will derive some estimates (depending on parameters) on the differ-
ence between a matrix coefficient and its leading term. In order to carry
out these results, it will be necessary to make some of the material in
Section 4 more explicit. Let (Q, A) be a standard p-pair and let t c °m be
a Cartan subalgebra. Set 1) = tc Ф ac. Then, f) is a Cartan subalgebra of
Йс. We have seen that if wt = |1У(йс> Ь)1/1Мшс, ^)l and if p = pQ is
defined as in 12.4.5, then there exist ex = 1, e2,. ., eWl e Z(mc) such that
Z(mc) = ®p(Z(flc))e,.
i<.wt
If A ef)*, let Xx (resp., rjA) be the corresponding homomorphism of
Z(gc) (resp., Z(mc)) into C (3.2.4). If p e t)* then r)M(^(z)) = Xx(?) if
and only if there exists у e Мйс, Ю such that p = sA + pQ. We note
that i7jA+Pe = т?м+ре if and only if there exists t e WTmc,f)) such that
p = tsA.
If и e Z(mc), then uet = q(Zij(u))ej, with z,7(u) uniquely deter-
mined elements of Z(gc)- We set b,7(A, u) = ^a(z,7(m)) and B(A,u) =
[&,7(A,u)]. Then, B(-,u) is a polynomial map of I)* into MW[C). If
у e 1И(йс> Ь) = Wc, then we set
+p^(	1)
^(A) =
168
12. The Theory of the Leading Term
Now, if г) = vs\+Pq> then 7)(ме;) = тДиЭтДе,) and тДие,) =
7?(Е, ^(z./m))^) = E, TitqtZjjtujyqtej) = 'Ljbij{S.,u)i^e1}. We therefore
have:
(1)	B(A, n)t>5(A) = т75Л+ре(и)^(Л).
We also note that r/A) = z^/A) if t g Wm = 1У(тс, fj). We label the
cosets Wm \ Wc as (the identity coset), s2,..., sW). Let
И(Л) = [^(Л),...,%(Л)].
Let Ф+ be a system of positive roots for Ф(йс> Ь).
Lemma. There exist a constant c and, for each a g ф+, pa g
N( = {0,1,2,...}) such that
det(K(A)) = с П (Л,а)Р“.
аеф+
Since e, = 1 and T7jA+Pe(l) = 1 for all s, us(A) * 0 for all 5. If sA =# tA
for all j, t g 1УС, then ys.A+P(? # Vs.a+Pq for i *j. Thus, the t>5(A) are
joint eigenvectors for distinct joint eigenvalues of the matrices B(A, u),
и g Z(mc). So, if (A, a) =# 0 for all a g Ф+, then det(HA)) =# 0. The
lemma will now follow from the following assertion.
(2)	If f is a polynomial on fj* such that /(A) = 0 implies that there
exists a g ф+ such that (A, a) = 0, then /(Л) = с Паеф+ (A, a)4a for
some c g C and qa g N, a g Ф+.
We prove (2) by induction on deg f. If deg f = 0, the result is obvious.
Assume the result is true for deg f = tn - 1 > 0. We consider the case
deg f = tn. Set V = {A g b*|/(A) = 0} and let Va = {A g fc*| (A, a) = 0}.
Then,
v= U (WU-
оеФ +
Thus, there exists a g ф+ such that V n Va has interior in V. Now the
regular set of V has dimension I - 1 and dim Va = I - 1. Thus, V n Va
has interior in Va. Hence, f(Va) = {0}. So /(A) = (A,a)g(A), with g a
polynomial of degree tn - 1. If g(A) = 0 then /(A) = 0, so the inductive
hypothesis implies that g is a product of the desired form.
12.6. The Main Inequality
169
12.6.2.	If и g Z(mc), then we denote by D(A, u) the diagonal matrix
with diagonal entries rjSiA+pJ.u),..., -qSw s+PQ(u). If (A,a)=#0 for all
а g Ф+, then
K(A)'’b(A,h)K(A) = D(A,m), u g Z(mc).
Set HA)"1 = [^(A)]. Then,
^(A) П (Л,а)Р“
аеф +
is a polynomial for all i, j. If у g y., then set
ej(A) = U(A)e;.
j
We note that:
(1)	Lj ^,](A)b]k(A, u) = vSiN+P£u)£ik(X).
12.6.3.	Let M be a gc-module with infinitesimal character We
assume that det(HA)) =# 0. Let q be the canonical projection of M onto
M/nQM. Then, if у g y.,
q(ues(A)m) = ues(X)q(m) = ^и^к(А)ек
к
=
kJ
=	= T?jA+Pe(M) Е^(А)еу<?(т)
kJ	j
= ^a+pc(«)^(A)^(w) = Vss+Po(4)q(es(A)m).
12.6.4.	Set ИЛ) = к,7(Л)]. Then,
Ец7(А)^.(Л) = et.
j
Thus, in particular, we have:
(1)	Lj V^PQ(ej)es(A) = 1.
170
12. The Theory of the Leading Tenn
12.6.5.	We will need one more such “tautological observation”. For
convenience, we will write £sj for if j g If и g Z(mc), then
j	i-k
=	+ Efs/(A)(p(z/t(H))-z/t(H))et.
j, к	j, к
Let	be a basis of a and let X{,..., Xn be a basis of nQ such
that [h, A',] =	, with a, g Ф(£), A) for all h g a. Then,
р(МЯ*)) -г^(Нк) = Y,Xauaijk,
a
with uaijk g t7(gc). If H = LjhjHj, then set
uja(A,H) = E ЦЛ)йрив/4/4.
j,k,p
We have:
(1)	Hes(A) = Y.jtk £sj(X)zjk(H)ek + E; Xtusi{A, H) for H g ac.
Thus, if M is a gc-module with infinitesimal character Xx then:
(2)	He5(\)m = (sA + pQXH)es(A.)m + Ef Xtusi{A, H)m for H g ac,
m g M.
12.6.6.	We now return to the situation and notation of 12.4, where we
dealt with two parabolic subgroups P, Q. We will assume that (P, AP) is
a standard p-pair and that AcAP. The preceding constructs are all for
Q. Let f)0 = t) П (°mF)c. Then, f)0 is a Cartan subalgebra of (°mP)c and
fj = fj0 Ф (aP)c. If A g (resp., A g (aP)c), then we extend A to t) by
setting A(ap) = 0 (resp., A(f) 0) = 0). Let (a, Ha) be an irreducible square
integrable representation of °MP and let a have infinitesimal char-
acter corresponding to Aa g ()* relative to the Harish-Chandra paramet-
erization of infinitesimal characters. Then	has infinitesimal
character ^_(Л +й). Let (а*Ус a = {i> g (aP)£| Aa + v is regular}. Then
(арУс,<г n ,а* ° «(а*У. If v g (aP)£, then we set es(v) = es(-Aa - v),
usi(v, H) = usi( — Aa - v, H), and Vs v = v-s(xtr+l.)+PQ-Then, we have:
(1)	If A g/^, then
TT_,(H)Tr_,(eJ(i'))A = -(5(Aa + v) - ре)(Н)7г_й(е5(р))А
+ E*-,(Ar,.)77_p(MJi(»',H))A.
12.6. The Main Inequality
171
And we have:
(2)	E, •>?!,„(e,)eJ(i')A = A.
Let p0(H) = min{a(H)| a g Ф(£), A)} for H g a.
Lemma. If s g Wc, then exactly one of the following alternatives holds:
(I)	sAa(H) > 0 for some H e Qq= (H e al a(H) >0, a g
Ф(0, A)}.
(II)	= 0.
(Ill)	There exists 8 > 0 such that sAa(H) < —8[}q(H) for all H g
Cl(a+).
If sA.a does not satisfy (I), then sA.a(H) < 0 for all H g a+. Thus, if
aj,..., am are the simple roots of &(Q, A) then there exist non-negative
real numbers x, such that sA.a = -E, х,аР If all of the x, = 0, then we
are in case (II). If x, > 0 and if H g Q(a + ), then sA.a(H) < -XjafH)
< —х^0(Н). Take 8 = x,.
12.6.7.	We preface our analysis with two simple inequalities.
(1)	If x > 0, b > 0, r g N, and t > 0, then
j e~XT(l + rb)r dr < Cre~‘x(l + h) (l + tb)r(l + x)r/xr+I.
Indeed,
ex'j“e~XT(l + тЬУ dr
= f e~XT(l + (t + t)b)r dr
Jo
<	(1 + tb)r f e~XT(l + rb)r dr
Jo
=	(1 + to/x-'-1 f e~T(x + rbY dr
Jo
<	(1 + th)rx_r“I(l +x)(l + ЬУ f e~T(l + r)r dr
Jo
<	(r + 1)!(1 + th)rx-''-I(l + x) (l + ^У
172
12. The Theory of the Leading Tenn
(2)	If x > 0, b > 0, r g N, and if t > 0, then
f eXT(l + тЬ)г dr < e‘xt(l + tb)r.
Jo
Indeed, the mean value theorem implies that there exists 0 < в < t
such that the integral is equal to teex(l + 6b) < te'x(l + tb).
12.6.8.	We will now analyze the leading terms corresponding to each
A5(p) = Tr_„(es(v))X. Set W\ = {s g 1Ус| alternative I of Lemma 12.6.6
holds}, Wn = (s g 1ИС| alternative II of Lemma 12.6.6 holds}, and Wm =
{y e IFC| alternative III of Lemma 12.6.6 holds}.
Fix у e 1ИС and A g I-. Set
fs(v,t,H,v) = e,p^H}Xs(v)(Trv(^ tH)v)
and
gs(v, t, H,v) =
for H g a, v g
Let (Po, Ло) be a minimal standard p-pair with Po c Q and A cA0.
Put °M+ = (kja0k2l kj,k2 e К П M, a g С1(Л^) П °M}.
(1)	There exists a continuous semi-norm, q = qK, on /“ depending only
on A, and r > 0 depending only on a, and £ > 0 such that if H g а£,
m g °M+, and t > 0, then
\gs(v,t,H,ir„(m)v)\
< e~^о<">||т||Л|КеИ|(1 + dlHll)r(l + M)'
xHHllI П (Aa + i', a)-P“|e'(l|Re,'lll|H|l)SRep(m)^(r).
1«г£ф+	I
Indeed, we note that
|^(Aa + p)| < Cl П (Aa + lSa)-₽“|(l+M)e
1«г£ф+	I
for some a > 0. Also,
— pl	>
V к	> i
12.6. The Main Inequality
173
with A, g I- fixed and aajpi(v) a polynomial in v. Now, tn = kia0k2 with
aQ g С1(Ло ) and k}, k2 g К П M. Set a, = aQ exp tH,
e~'pv(H)gs(v, t, H,
= ~ Y,e~‘a*H)ir-Xusi(v> H))A(7r„(exp 1Я)7г„( А^ттД^ао^г)
i
= - E «,““'6у(Ла + v)hpir_v x
I lLUijkpek )^(7Гк(Ad(fcj) 1 Xi')Trv(ao')^k2)v'}.
V к	’
Let, for a g Ф(£), А), Да = {/3 g Ф(Ро, Ло)| Д|а = a}. Let Xltj be a basis
of na. such that Ad(u) Xtj = u^X^ with g Да . Then, Adf^)-1^, =
E;T0('m,7.So
(*)	e~'p<^H)gs(v, t, H, irv(m)v)
= - E е~'“'("К^(л<7 + v)hpaijpk(v)
i,j,k,p
X	Ad(fc2) ”)•
Let Pj,..., vr be linear coordinates on a£. Then,
= ir0(Xi)v + Е"Л^>
j
with 7]j a continuous operator on I™. If ц g I-, w g then
|(тг,(а,)и'1м)| < (1 + IHog а,Н)ГНКеЛа,)к(и')
for a g A, with к a continuous seminorm on /“ (see the proof of Lemma
5.2.8). (1) now follows by estimating each term in (*), using (Lemma 3.6.7,
Theorem 4.5.3)
SRe,(«,) <	+ tllHH + log ||w||)deiiRe>'il<'l|H|l+il,og«oll))g(w)!
and choosing £ > 0 so that ||a0||f > el|log °o11.
We now begin the advertised estimation. As usual, all the estimates are
based on:
(2)	f!(v,t,H,v) =s(X^V)(H)fs(v,t,H,v) +gs(V,t,H,v).
174
12. The Theory of the Leading Tenn
And hence
(3)	=e'siA‘’+vXH)fs(v,0,H,v)
+ е«(л<г+1'Х«) f e~TS(‘x<’+v^'H^gs{v, т, H, v) dr.
Jo
Let j e and let H 6 be such that sAa(H) > 0. Let v с (а^Уа
be such that Де(Я)/4 > ||Re i'll l|H||. If т > 0, then
|e~T,!(A‘'+,'XW)g^l,) T; /f)	|
<е-т^ю/2(1 + т||н||)г(1 + М)Г||Я||| П (Л, + м)Ч(4
1аеФ+	।
Thus,
f e~'rs(N,r+v^H}gs(v, т, Н, и) dr = b(v, H)
Jo
converges absolutely, and if we set a(v) = f(ff) and
f°(v, H, t) = e,s^+^H\a(v) + b(v, Я)),
then f°(v, H, t) is a continuous function of H in the indicated range and
\fs(v,t,H)-f°(V,H,t)\
£ Cre-‘^/2(l + ||Я||)Г(1 + t||H||)r<7(v)(1 + Ill'll)'
х|аП (Ла + p,«)^|l|H||/(/3e(H)/2)r+1.
If Re v = 0, then |/(t)| < C(1 + t)r (C depending on v). Thus, if
П (A^ + naf^O
«еф+
and sAa(H) > 0, then
|e"A<'(")(fl(I>) + b(v,H))\ < C(1 + t)r.
We conclude that a(v) + b(v, H) = 0, where defined, for Re v = 0. Since
these functions are meromorphic in v, we see that a(v) + b{v, H) = 0,
where defined, if sAa(H) > 0. Fix such an H. Then, the Lebesgue
dominated convergence theorem implies that limu_ +00 b(iv, uH) = 0.
12.6. The Main Inequality
175
Hence,	= 0. Since v is arbitrary, this implies that As(iv) = 0.
But v >-» А/p) is rational in v with values in I&(F) for an appropriate
finite subset F of K. Thus, A/p) = 0.
We have therefore proved:
Lemma. Ifse then A/p) = 0.
12.6.9.	We now look at the case when у g Wul. Let 3 > 0 be as in
Lemma 12.6.6 (III). Let e = min{l/2,8/2}. We have:
Lemma. If s g H e Oq, if ||Re HI \\H\\ < 8(3(H)/2, and if tn g
°M, then
| e ,pQ(H ’A 5 (v) (7r„ (exp tH) ir„ (m) v) |
<	e-««">/2||miri|Re,'ll(l + ||Я||)Г(1 + t||H||)^(r)(l + IK
x| П (Aa + P,a)-P“| (1 + log ||m||)rS(m).
1«еф+	I
Set /3 = Pq. If we estimate 12.6.8 (3) using 12.6.7 (2), we find that if
||Re HI ИНН < 0(H)/4, then
| e,p°(H’A5( v) (7r„(exp tH)тг„( m) v) |
<	ezRei(A„+l<XH)(e-'Re5(A„ + xXH)-^(H)/2 + 1)||m||fl|Re*'ll( 1 + ||Н||)Г
x(l + t||H||)r(l + ||p||)r^(t>)(l + log ||m||)r
x| П (Л.7 + »'»«) P“|S(m),
1«еф+	I
for appropriate q and r. If
HRep|| ИНН < 8/3(H)/2,
then
ezRej(A„ + pXH) < e-tSP(H)/2_
The lemma now follows.
12.6.10.	We are now left with the main case of interest, s g JFn, that is,
sA.a(H) = 0 for all H g a. Let Я g a£. As before, we look at v g a£
with ||Re HI IIH|| < /3(H)/4. We begin with 12.6.8 (3), as usual. This time
176
12. The Theory of the Leading Term
f° is not necessarily 0. We set ff(v,t,H,X,v) = e‘s(X<'+v*H)(a(v) +
b(v, НУ), with the definitions given as in 12.6.8. Then, we have:
Lemma. If s g Wu, H g aG, m °M+, t > 0, and if v g a£ is such
that ||Re p|| ||H|| < /Зв(Н)/4, then
|e'MW)Aj(p)(7rl,(exp tH)ir„(tn)v) - fs°(v, t,H,X, 7r„(zn)t>)|
< e-'M«>/2||m|ri|Re-'l|(i + \\h||)r(l + t||Я||)^(r)(l + IMI)'
x(l+ log||m||)rS(m)| П (Аа + р,а)Ч/де(Я)г+1.
I аеф+	'/
12.6.8 (3) implies that, under the hypothesis on H, tn, and v,
e'pQ^>Xs(v)(TTy(eiqptH)TTv(m)v) - f°(v,t, H,irv(m)v)
=	e-Tsv(H)g^v^ T> ц Trv(m)v) dr.
The result now follows from 12.6.8 and 12.6.7 (1).
12.6.11.	We are now ready to put all three cases together to give a
preliminary form of the main inequality. We note that
A = Е171,Л^)\(р).
i
Thus, the preceding considerations combined with Lemma 12.3.3 imply
that (see the proof of Theorem 4.4.3)
<pG|P(p;exptH;A,t>) = Vi Xei)fs°(vJ, H, A,t>),
if ||Re HI ШН < Д(Я)/4. Let 6j be the minimum of the “S’s” for j g WIU
and set e = minUj, 1/2}. We have:
Proposition. If H g aG, tn g °M+, t > 0, and if v g (аРУ£. is such that
HReHI ИНН < бДе(Я)/2, then
|е"’е(//>А(тгр(ехр tH)7r/m)i>) - tpQlP(v; exp tH; A, 7r„(zn) v) |
< е-в^о<">||т||Л|КеИ|(1 + log 1Ы1)Г(1 + HH||)r(l + t\\H\\)r
X«(p)(l + M)rg(m)| П +
'аеф+	'/
with q a continuous setni-norm on I* depending only on A, r g N, and
£> 0.
12.6.12.	We are now ready to give the most difficult part of the main
inequality.
12.6. The Main Inequality
177
Theorem. Let (P, AP) and (Q, A) be standard p-pairs with A cAP. Let
(a, Ha) be an irreducible square integrable representation of°MP. There exist
r > 0, c > 0, and, for each A g I-, a continuous seminorm q on I™, such
that if a g C1(^g), m g °Mq , and v g then
|ap<?A(77P.iv(ma)v) - <pQlP(iv; a; A, irP^,„(m)v) |
< (1 + IMI)r(l + log l|w||)r(l + lllog a||)re”tp2<loga)5oM(w)<7(i>)
for v g a*.
Let Ф„+= {a g ф+ (Aa, a) =# 0}. We note that
ф+эФ + п Ф((тР)с,Ь).
Let tj = rninfKA^, a)| |a g Ф/}. Let Ux = {p g a*| |(p, a)| < tj/2,
a g Ф+}. If U c a*, then set УШ) = {p g (aP)*|Rep g U}.
(1)	There exists C > 0 such that if v g У/Ц), then
I ПаефЧЛ. + V, а)РЛ > С | П«еф + -фД t', «И 
Indeed, if a g Ф/, v g Ult then KA^ + v, a)|2 = (Aa + Re v, a)2 +
(Im v, a)2 > tj2/4. (1) is now obvious.
We set X = Ф + - Ф/. If a g X, then (v, a) defines a non-zero linear
functional on a£. Let H g be such that 1/2 < ||H|| < 1 and set
U2 = {t' g Ц| ||i>|| < e/3e(H)/2}. Proposition 12.6.11 implies that if v g
t > 0, then
I П (v> а)Р"{е"’°(Н)А(ттДехр tH)rrv(m)v)
laeX
-<PG|p(»'; exp tH\ A, TTV(|
< C1e-£'^<H)||m||f||Re,'ll(l + t||H||)r
x^(r)(l + Ill'll)'S(m)/j3G(H)r+1.
This implies that
П (v, а)Р“{е"’е(Н)А(ттДехр tH)TTv(tn)v)
aeS
-(pQlp(v; exp tH; A, tt„(m) v)}
is holomorphic on Fix m g and set U = {p g (72| ||Re i'll <
l/£ log ||/n||}. Let 3 be as in Lemma 12.A.2.8. Then
S'1 <C(1 + log ||m||)/j3G(H),
178
12. The Theory of the Leading Tenn
with C a constant depending only on £. If p g У(и), then
I П (^,а)Р“{е'Ре<//)А(тгр(ехр tH)77l,(w)Li)
I a gS
-<PG|p(1'; exP tH; A, 77p(m)t>)}|
£ C2e-e,^H\l + t||H||)^(r)(l + ||p||)rS(m)/pQ(H)r+\
Thus, Lemma 12.A.2.8 implies that if v g a*, then:
(2)	|e"’e(H)A(ir,„(exp tH)iTv(tn)v) - <pQ]P(iv; exp tH; A, 7r„(m)t>)|
#
£ C2e~e,WH\l + log ||m||)r(l + t||H||)r
Xq(v)(l + М)гЗ(т)/Де(Я)“ + 1
for appropriate C\,u,r (independent of H, v, m).
Let Л g be such that ||Л|| < 1/4. Set His) = H + sh. Let к = poih).
Then, 0QiH + sh) > ks and e~e'0c<H<5» < e-etpQ(H) for о у 1. Set
<p(y) = е''’е(НО))А(тг,1,(ехр1Я(5))7г1,(т)г>)
- <pQ]P(iv;exptH(s); A,7r„(m)t>).
Then, <p(kKs) is a linear combination, with coefficients depending only on
h, of terms of the form
1*е"’е(НО))А(тг(Дехр tHis^ir^m)^^^)
~ <pQ]P(iv;exp tH(s); K^^tn)^^)1 v),
with j < k. Thus, if 0 < у < 1 then
I <₽<*’( s) I <; C3e~e,WH\l + log Hm||)r(l + t||H||)r+‘
X^(r)(l + ||p||)r+*S(m)/y“+1,
with qx a continuous semi-norm on Ц and C3 depending only on h, A.
Hence, Scholium 7.3.4 implies that
|<p(0)| < C4e-^<w’(l + log IMI)r(l + t\\H\\)r+u+2
X^1(r)(l + IMI)r+u+2S( m).
12.6. The Main Inequality
179
We note that a(m) < C(1 + log Н/пНУ'НодД/п), and thus if a g A + then
a = exp tH, with H g a+, ||H|| = 1, and t > 0. We have thus proved
|а*’оЛ(тг iv(ma)v) — (pQ\p(iw, a', A,7r,„(/n)f)|
< (1 + M)r(l + log Ilw||)r(l + lllog a||)re-£'’e<108‘')SoM(m)9(r),
for appropriate r and q and all a g A+. The theorem now follows from
the continuity of both sides of this inequality.
12.6.13.	We note that in the course of the proof of the preceding
theorem we have also proved (in light of the proof of Lemma 12.A.2.8):
Theorem. Let P,Q,<r be as in the previous theorem. There exists 8 > 0
such that ifU={vea*\ Ill'll < 3}, then <pQ\P(v, a\ A, v) is holomorphic in v
for v g	{p e (aP)*|Rep e U}\
12.6.14.	We now derive a result that is a consequence of the analysis that
we have used to prove the theorems (so far) in this section. We set
5p(3) = {i> g (aP)J| ||Re p|| < 3}. We use the notation of Section 12.5.
Theorem. If A g I., then there exists 3 > 0 such that
]~[ (Aa + v,a)<pQlP<s(a,v;k,-)
аеф+
is weakly holomorphic from ^(8) to {Ifff. Furthermore, there exists a
continuous semi-norm, <?A, on and r > 0 such that
п (Ла + v,a)<p&P ^,v; A,77„(m)y)
!аеф+	1
< (1 + IMI)r(l + log Hm||)rSRe,(m)?A(V)
for m G MQ, v g У(3).
Let sg W(A, AP). Set ^„(s) = b, g 1Уп|(s,.|e)~1 = s}. Let 5A(p) =
Then, Паеф+(А<Т + v, a\X(v) is a polynomial map
of (a)£ into I&(F) for an appropriate F. The preceding analysis implies
that if a g A +, then
= (p&P s(iv, A,z?).
Since
‘Pgip.X1'; a> ^(fl)y) = asTv<pQ\p,Av'> A> ^)>
180
12. The Theory of the Leading Tenn
Theorem 12.6.12 implies the desired result for m g °MqA. We now show
that °MqA = Mq. Once this is done the proof will be complete. If
m MQ, then m = kxak2, with a & Ao and a(log a) > 0 for aG
Ф(Р0 П Mq, Ло) = *Ф. Choose h g a such that a(h + log a) > 0 for all
a g Ф(Р0, Ло) - *Ф (this can clearly be done). Then а exp/г g G(Aq).
Thus, m exph g °MqA. Hence, tn g °MqA.
12.6.15.	We now consider the case when (Po, Ло) is a minimal standard
p-pair with A c Ao, Po c Q, AP cA0 but there does not exist any к g К
such that kAk~l cAP. We fix f), Cartan subalgebra of йс with t) z> (a0)c.
Set fj0 = t) П (°mP)c. Then, as before, t) = fj0 Ф (aP)c. Fix (a, Ha) an
irreducible square integrable representation of °MP and let g be as
before. We set up the notation as in 12.6.6. We set, for A g^, 5 g JFC,
A/p) = TT_„(eJ(i'))A. Lemma 12.6.6 is true in this case, with the same
proof. Set IFC = Wl и Wn и IFni, as before. Let fs and gs also be as in
12.6.8. 12.6.8 (1) is true, as are Lemmas 12.6.8 and 12.6.9, with exactly the
same proofs. However, Lemma 12.6.10 is slightly different so we will give
some details. Define fs°(v, t, H, v) as in 12.6.10. Then, arguing as in
12.6.10 one finds that if H g and if ||Re p|| \\H|| < pQ(H)/4, then
|e'MW>AJ(»')(7rl,(expt)r) - fs°(v, t, H, A, r)|
<	+ ||H||)r(l + t||H||)^(г)( 1 + |MI)r
X(1 + log ||m||)r| П +
1<»еФ+	'/
The proof of Theorem 4.4.3 (specifically the material in 4.4.3 between
(XV) and (XVI)) combined with Lemma 12.3.3 implies that
f°(iv, t, H, A, v) = <pQiP(iv; exp tH; A, v)
for v g (а*У. Lemma 12.4.1 implies that <p0\P(iv; exp tH; A, v) = 0 if
+ iv is regular. We have thus proved:
Lemma. If s g Wu, H g a^, tn g °M+, t > 0, and if v g a£ is such
that ||Re i>|| ||H|| < /3Q(H)/4, then
Ie,p&H}Xs(v)(7r„(exp tH)trv(m)I
<e-'^<"’/2||m||fl|Re,'ll(l + ||H||)r(l + t||H||)^(r)(l + М/
X(1 + log ||m||)rS(m)| П (ЛЛм) 4//3e(H)r+1.
1<»еФ+	I/
12.6. The Main Inequality
181
12.6.16.	As before, we can now combine all of the cases to find (e is
defined as in 12.6.11):
Proposition. If H g a^, m e °M+, t > 0, and if v g (aP)c is such that
HRe p|| \\H\\ < e0Q(H)/2, then
|e'MW)A(7rp(exp 1Я)тгДт)г)|
< e~e'^<"’||m||fM(l + log ||w||)r(i + 11/711)71 + 111Я11)'
X <z(r)(l + ||p||)rS(m)| П (Ла + v, a) ~Pa\ IpQ(H)r+l,
|аеф+	'/
with q a continuous semi-norm on I™ depending only on A, r g N, and
C> 0.
12.6.17.	The proof of the following result is now identical to that of
Theorem 12.6.12 (if we replace <pQiP by 0).
Theorem. Let (P, AP) and (Q, A) be standard p-pairs with AP, A c Ao,
and PQ c Q, and such that there does not exist к g К such that kAk ~1 c A P.
Let (a, Ha) be an irreducible square integrable representation of°MP. There
exist r > 0, c > 0, and, for each A g Ц, a continuous semi-norm q on /"
such that if a g СКЛ^), m g °Mq , and v g then
\аРок(ттРaiv(ma)v)\
< (1 + M)'(l + log ||m||)r( 1 + ||log a\\ye~epo('°gaV2oM(m)q(v)
for v g a*P.
12.6.18.	The preceding theorem has as its immediate consequence:
Corollary. If {P, AP) and (Q, A) are standard p-pairs and if <pQ^P{(x, iv;
•;•,•)=# 0 for some v g (aP)* and some irreducible square inte-
grable representation (a, Ha) of °MP, then there exists к К such that
kAk~l cAP.
12.6.19.	If we put Theorems 12.6.12 and 12.6.17 together we have proved
(compare Harish-Chandra [15], Lemmas 10.8, 14.5):
Theorem. Let (P, AP) and (Q, A) be standardp-pairs. Let (a, Ha) be an
irreducible square integrable representation of °MP. There exist r > 0, e > 0,
182
12. The Theory of the Leading Tenn
and, for each A g a continuous semi-norm q on 1“ such that if
a g C[(Aq), m g °Mq , and then
\apQX(TrPcrii,(ma)u) ~ 4>Q\P(iv; a; A, ttp<„Дm)v)|
< (1 + IMI)r(l + log l|w||)r(l + lllog a\\')re~epQ(loga)’SloM(m)q(v)
for V g a*.
12.7. Wave packets
12.7.1.	We retain the notation of the previous section. Let (P, A) be a
standard p-pair and let (a, Ha) be an irreducible square integrable
representation of °M. If v g a£, then let pfo-, v) be as in 10.5.8.
Let У(й*) be the (usual) Schwartz space of a*. We recall that this
means that if	is a basis of a and if we set xfv) = v(H), then
«Z= У(а*) is the space of all C" functions f on a* such that
= sup I x1 dJf( V) I < oo.
pea*
Here, we use standard multi-index notation:
x1 = xj1 • • • x{',	|Z| = Jj,,
/! = /!!•• i,\,
5|/|
dx{‘ • • • dx\‘ ’
We endow with the topology induced by the semi-norms pz j.
If a g <Z(a*) and if A g I-, v g Ia, then we set (irv =
(0	= ( Li(o-,iv)a(v)X(TTiv(g)v)dv.
ja*
Here, we normalize dv as the Euclidean measure corresponding to an
orthonormal basis of a. At this point, there is little evidence that the
integral in (1) converges. We now state main result of this section.
Theorem. If a g >Z(a*), then the integral in (1) converges and defines an
element of Z(G) (7.1.2). Furthermore, TA u defines a continuous map of
^Z(a*) into ^(G).
12.7. Wave Packets
183
This result of Harish-Chandra [15], Theorem 13.1; [16], Theorem 26.1, is
a critical step in his proof of the Plancherel formula. Our proof (which is a
modification of that of Harish-Chandra) will take the rest of this section.
We first need some lemmas. Let У(3) = {i> g a£| ||Re p|| < 3} (as usual).
12.7.2.	Lemma. Let и be meromorphic on У(8) for some 8 > 0, and
such that there exist non-zero elements hx,...,h g a (we allow p = 0)
such that П, v(h,)u(v) is holomorphic on 3^(8). Assume that if v0 g a*
then there exists a neighborhood, U, of iv0 in ia* and Cb > 0 such that
Iw(i')I < Cv for v g U such that u(v) is defined. Assume, in addition, that
there exist constants C > 0, m > 0, a > 0 such that
|Пр(Л,.)м(р)| < C(1 + ||HI)'”ee"Re*'", v g 3^(8).
Then, и extends to a real analytic function on ia*, and for each multi-index
J there exists Cj depending only on hl,...,h and 8 (but not on и or a)
such that
I^M(jp)I < CjC(l + IMI)m(l + a)171, for V G a*.
Let g(v) = П, и(Ь/)и(и). We first prove the result for g.
Let С\ > 0 be such that if v g a£, e > 0, and |p(H,)| < Сге for
i=l,...,l, then HpII < c. Let 0 < 3t < min{3, l/a}/Cl. Then, 3f1 <
С\(1 + a)/min{3,1]. If v0 g a* and if v g a£ is such that |p(Hp -
iv0(Hj)\ < 31; then v g У(3). Let r); g a* be defined by tj/H,) = 3I;.
The Cauchy integral formula implies that
2I'/|JI	,	, I 5,	\
=	/. ’'( T’+  M'-
Our assumptions imply that if we set
с^л/^Лзуг)171),
then
\dJg(iv0)\<C'jCf2^ 	+
7 о A) \
Si _
vo + у E(sin6„)T7n
Z n
d
dOt  de,
< QC(1 + ll^oll/,
with C'j = Cj(2tt)'. Now C'j < (1 + a)|7|C7, with Cj depending only on J
and 3. This proves the lemma for g (i.e., for p = 0).
184
12. The Theory of the Leading Tenn
Clearly, it is now enough to prove the lemma with p = 1. We assume
that hx= Hx (this can be implemented by a linear change of coordinates).
We write /(x1,..., xz) for fiiLj Xj-qf). Then,
rt d
g(x1,...,x/) =xj —g(tt1,x2,...,x/) dt + g(0, x2,...,xz).
•'0 0*1
If x = (x2,..., xz) g R/-1 is fixed, then lim,_0 g(t, x) = 0 by our as-
sumption on u. Hence,
,1 d
g(x1,...,xl) = xJ —g(tx1,x2,...,xl)dt.
J0 0X1
Thus,
r1
m(x1;..., xz) = / —g(tx1,x2,...,xz) dt.
J0 0X1
The lemma now follows from the estimates on g.
12.7.3.	If g g G, the we write g = °gs(g) with °g g °G and s(g) g S(G)
the standard split component of G (2.2.2). Set °a = a П °g, and if v g a£
then set °v = р^д.
Lemma. There exist m > 0, £ > 0, and q, a continuous semi-norm on /"
(depending on A), such that
|j(g) ^(rr^g)t>)I < (1 + log ||g||)m||°g||fl|Re0,,|l/7(t>)H(g)
for v g a£, all multi-indices J, v g 7^, g e G.
We note that ir^(g) = s(g)*'irv(°g). We may thus assume that G = °G.
As we have pointed out in the proof of 12.6.8 (1), there exists m > 0 and a
continuous semi-norm on I* such that
I^W?)")! (1 + log llgll)m<7(y)SRe,(g)-
Fix (Po, Ло), a minimal p-pair such that A cA0, Po с P. We write g =
kiak2, with a g QU0+). Then, SRe,(g) = SRe,(a) < ..................
Choose £ > 0 such that el|log0|1 < ||a||f.
12.7.4.	In order to prove Theorem 12.7.1, it will be necessary to intro-
duce two auxiliary spaces of functions. We introduce the first space (which
12.7. Wave Packets
185
is motivated by the condition //(A) of Harish-Chandra [15]). Let Fx, F2 be
finite subsets of K. We denote by TF1 F2(a) the space of all real analytic
functions ф on ia* x G satisfying the following condition:
(1)	There exist	g a* - {0}, 3 > 0, A,,..., Ao g /^(Fj),
,..., vb g Ia(F2), and Ujj holomorphic functions on У(3) such that
there exist C > 0, q > 0, and
|niy(p)|^C(l + МГ for PG ^(3),
and if v g a*, g g G, then
= E um„(ip)Xm(Triv(g)v„).
' j	'	m,n
We denote by ф(и, g) the corresponding “function” on У(3) x G.
Lemma 12.7.3 implies that:
(2)	If x, у g U'(z), then
< c(i + i|p|i)2/(i + log iiM)riiogiifl|Re%lls(g)
for g g G, v g У(8), and £ as in 12.7.3.
We now collect some obvious properties of TF[ F2(a).
(3)	TF[ F(jr) is a vector space under scalar multiplication and pointwise
addition.
(4)	If x, у g t7y(ec) and if Ф e ,F2(o-), then Ь(х)Н(у)ф g TF> F>(a)
(see 12.4.2 for F;).
(5)	If klfk2 g К and if ф g TF> F(.a), then Цк^Щк^ф g TFF(,ff).
12.7.5.	Lemma. Fix A g and v g Ia. Set ф(г, g) = X(ir„(g)v) (or
fi(a, v)X(rrv(g)v)). Then, ф g TFi F(xr) with p = 0 in 12.7.4 (1) and appro-
priate Fx,F2.
This follows from Lemma 12.7.3, Theorem 10.5.9, and the fact (ll.A.4.9)
that if x g (F(g) then v tt_v(x)X (resp., tt„(x)v)) is a polynomial map
of degree at most j from a£ to (resp., Ia).
186
12. The Theory of the Leading Tenn
12.7.6.	Let ф g TF) F2(o-). Let A,,..., ka g ^(F,) and let r,,..., vb g
f(7(F2) be as in 12.7.4 (1). Then,
/ \-1
</'(»', s) = n"(M E
' j ' m,n
where defined. Let (Q, AQ) be a standard p-pair with AQ cA. Set, for
tn &MQ, s g W(Aq, A),
'WX1'-"1) = ГЬ(М E
' j ' m,n
where the right hand side is defined.
If (Q, Aq) is a standard p-pair and if W(AQ, A) + 0, then fix j0 g
W(Aq, A) and let k0 be a representative of j0. If AQ a A, we choose
= 1 and k0 = 1. Set Q = IcqQIcq and Aqi IcqAqIcq . then, Aq* c
A. Furthermore, W(AQ, A) = W(Aq>, A)s0. If <p g ^/w(G) (12.3.4) and if
m^MQ, then <р0Актк~1) = <pQ(m). Thus, if we set Ф^р,^, m) =
Фо^р ,(v, komkg *), then we have defined ф0,Р s for all 5 g W(Aq, A). If
we write = ф(у, g), then we have
lA(ip)G(m) = E Ф&р,А iv’m)
s^W(AQ , A)
for tn g Mq, v e a*.
If F с K, then let F' = {£ g (К n M)" ] [£: y] =# 0 for some у g F}. If
F is a finite set, then so is F'. We will use the notation TP G F F(.a) if P
and G are not clear from the context.
Lemma. If ф g Tp g f> F£cr), then ЩефЛА,. + v, а)Раф0\Р s(v, tn) de-
fines an element of Tk-iPkonMQ Mq f; p'2(kgl(r) with the same h!,...,hp
for 12.7.4 (1).
We note that Theorem 12.6.14 combined with Proposition 12.5.3 implies
that if A g I&, then there exists 8k >0, mA > 0, and qk a continuous
semi-norm on 1“ such that if v g 1^, v e У(3А), then
П (Л.+
аеф+
p,a)%7P1/,(p)r(l)|A(l)) <; (1 + IM)4(")-
Now, v(k) =	(notice that trv(k) is independent of v for
к К). Let A g/rf(F) and let A,,...^ be an orthonormal basis of
I^tF). Then, TT_v(k)X = Ljhj(k)A.j, with a smooth function on K.
Thus, if q(v) = supk^Ki q^tr^kyv), 8' = min;3A., ш = тах;тл, we
then have (after estimating the by constants and integrating over K),
12.7. Wave Packets
187
for V G ^(8),
I П (Ла + р,а)Р“рр|Р(р)г|А)кс(1 +M)m<7(r).
1«еф+	I
Let rj be as in 12.6.12 (1). Then, if we choose 3 = min{3', rj/2} and use
12.6.12 (1), the formulas in 12.5.10, Theorem 10.5.9, and the product
formula 109.5.8 (1), the lemma follows.
12.7.7.	We say that ф g ТД F2(<r) if •Aqip,/1') e TF, F<2(kg l<r) for all
p-pairs (Q,Aq), s g W(Aq,A\ We note that Proposition 12.3.6 implies
that if ф g ТД F(.tr), then	g Т/.	*0-). The main reason for
introducing these auxiliary spaces is the following result.
Theorem. If ф g T^ F(<r) and if a g <У(а*), then set
fa,M = j a(v^(iv,g) dv.
Ja*
Then, the integral converges absolutely and uniformly, fa ill g ^(G), and the
map a fa i/l is continuous from <У(а*) to ^(G).
We note that 12.7.4 (2) combined with Lemma 12.7.2 implies that the
integral defining fa ill converges absolutely and uniformly in compacta of
G and defines a C“ function on G. We will prove the theorem shortly. We
first note that once it is proved Theorem 12.7.1 is reduced to proving:
Proposition. If A g I&, v g Ia, then the function ф(и, g) =
pfa, v)k(trv(g)v) is an element of TF[ for appropriate Ft, F2-
We are thus left with the preceding theorem and the proof of the
proposition.
12.7.8.	We first prove the theorem. In order to prove it, we need the
following result:
Lemma. If ф g	then there exists e > 0 such that if a g С1(Л^)
and ifm^ °Mq , then
ароф(п>,та) -	£	ww)
s^W(Aq , Л)
£	+	+ log ||W||)rHoMc(m),
for an appropriate r and all v g a*.
188
12. The Theory of the Leading Term
Let /ij,..., hp be as in 12.7.4 (1) for ф and ft1>s,..., hn s be as in 12.7.4
(1) for iAG|p,s. Let hi,..., hn be the union of all of these h’s counted with
the maximum multiplicity that each appears. Then, Theorem 12.6.19
implies
ГЬ(Л,) 1ар°ф(1Р,та) -	£	ф&Р!(1Р,та)\
j	'	s<=W(Aq , A)	I
<	+	+ logHznll/Ho^zn).
The result now follows from the argument in the last part of the proof of
Lemma 12.7.2.
12.7.9.	We will prove Theorem 12.7.7 by induction on dim G. If dim G =
0, the result is trivial. We assume that the result is true for all G with
dim G < n and that dim G = n. We now prove it for G. Let (Po, A 0) be a
minimal standard parabolic subgroup of G with Po а P and A c Ao.
(1)	It is enough to prove that if a g Q(Ag), then for each j > 0 there
exists a semi-norm on У(й*) = such that
|л,^(«)|	+ mog «и) "•'a-'’0.
Indeed, if x, у g U(flc) resp., or if x, у g K, then Ь(х)Р(у)ф is a
function of the same type as ф.
We will thus prove (1) for G. We next assume that S(G) < {1}. Set
S = S(G), °Ao=Aon°G and °A = A Ci °G. Set § = Lie(5). Then,
О(Лц) = 5 • C1(°/1q ). Also, ф(р, sg) = s^(v, g) for g g G, s g S. If
v e (a)c, then we write v = p, + v2 with g extended to a by
^(“a) = 0, and v2 g (°a)£, extended to a by p2(§) = 0. We choose the
Hi,...,Ht so that H1,...,Hjg§. Let J be a multi-index such that
jp = 0 for p > y. Lemma 12.7.2 combined with 12.7.4 (2) implies that if
/3 g eZ(§*) and if
f 5'/(Д(А)'А(г’А + V2,SZ) dk = ф2 p(s-,v,g),
then, for fixed s g S, ф2р(х; • ) g T6g PnoG Fi p(xt). And (using the
usual integration by parts argument to show that the Fourier transform of
a Schwartz function is rapidly decreasing) we have, for v g (°a)£ n У(3),
12.7. Wave Packets
189
ge °G,
|	v> s) |
^,(Д)(1 + lllog s||)"'(l + M)r(i + log ||g||)r||glla|Re*'llH(g),
for appropriate r (here, qj is a continuous semi-norm on ^Z(§*)).
This (using the fact that the span of all functions of the form a(p1 + v2)
= p(vt)y(i'2), p g <Z(3*), у g Д°а*), is dense in .Z) reduces the
proof to the case when G = °G, since dim°G < dimG if 5(G) # {1}. We
therefore assume that G = °G.
Let U = {H e d(aj)| ||Я|| = 1}. Fix Ho e U and let F = {a g
Ф(Р0, Л0)|а(Н0) = 0}. Let (Q, AQ) = ((P0)F,(Z0)F). Then, Ho g (aG)+.
Fix a neighborhood Uo of Ho in U such that a(H) > a(H0)/2 for all
a g Ф(Р0, Ло), H g Uo. Let e, r be as in 12.7.8. Let 8 = epo(H0)/4. Set
q(a) = sup„Go*(l + ||i'l|)r+2/. If t > 0, then 12.7.8 implies that
e'pQ(H}fa^(^PlH) -	£ fa(v)$&Ps(iv,exptH)dv
s^W(AQ,A)a*
< Q(a)(l + t)re-s'SoMe(exptH).
Since G = °G and Ho =# 0, dim MQ < dim G. Thus, the inductive hypoth-
esis applies to the integral involving (/,G|F s. We therefore find that for
each j there exists a continuous semi-norm qUo . on «Z such that
|/а.Дехр1Я)| < ^о.,(а)(1 + t)Je~'Po(H)
for H g Uo. Since a finite number of the Uo cover U, the proof of (1) is
complete. This completes the proof of the inductive step.
12.7.10.	We now prove Proposition 12.7.7. We first recall some notation
from Section 12.5. It is enough to prove that pXcr, iv)(pQiP s(.iv; A, Triv(m)v)
defines an element of TM FeM F> F>(<r) if AQ c A (see 12.7.6). We will
therefore prove this. Let к g К be a representative for 5. Let *P be as in
12.5.9. Then (*P, A n °MQ) is a standard p-pair for °MQ and °M*P = °MP
= °M. Set *A = A n °Mq. If v g (*a)p, then we write p,oMQ(<r, v) for the
“p.” function, with °MQ replacing G.
In light of Lemma 12.7.6 combined with Lemma 12.7.2 and 12.6.12 (1),
to prove Proposition 12.7.7 it is enough to show:
(1)	If ы is a compact subset of a* and if A g I-, v g Ia, then there
exists Сш A r such that |д.(а, iv)(pQiP s(iv; A, y)l < сш л t. for v g w.
190
12. The Theory of the Leading Term
To prove this we will use 12.5.10 (2), which is an explicit formula for the
right hand side of the assertion. We will in fact prove a more precise
result. Let к be a representative of 5. We first note that it is easily seen
(using the intertwining operator L(k~1)) that:
(2)	p(k~la, ik~lv) = p((r,iv).
Also, if we use the product formula for the intertwining operators
(Lemma 10.1.10) then we find that (notation as in 10.5.9,10.5.10)
(3)	p(k~la,ik~lv) = fi(*k-l<Tk-4v,isTv)fik-l0MQk^k~la,ik-lv^ay
We now come to the critical point of the argument. Fix A g I&.
(4)	If m g °Mq and if v e a*, then
A,Tr;„(m)y)|
^(1 +IMI)^(p)(l + log||m||/S(m),
for appropriate d. Here, q is a continuous semi-norm on /" depending
on A.
Indeed, Lemma 10.5.6 combined with the definition of the p function
implies that if ((J,, Лх) is a standard parabolic subgroup of G, if (rj, HJ is
an irreducible unitary representation of °Ml, if v g a*, and if Q2 g
then
P£?2l£?i(7b'I')/|r = VQwtvAvy'WfW2.
This implies that:
(5)	\\p(i)Av)l/2J02l0[i),iv)\\ = 1.
We now apply this observation to 12.5.10 (2). According to that formula,
|, isTv)<p&P iv-,k, Triv(m) v)|
< (1 + log ||т||)‘УЕ(т)р.(*к“1о-А.-1,р,й7'р)1/2
^'P'k-'Qiklk-'Pki^ '(T,lk *»') SJk-'Qik\k~'Pk(k 1<r > & 1p)
XL(k-1)Kl))}|{sUfc-1)A»fik-lpk|k-l01^fc-1a,ik-^)1/2
Х^к-‘Рк1к-'О1к(^ lO’,lk 1»')}(l)ll-
12.8. The Harish-Chandra Transform of a Wave Packet
191
Here, we have used p.Gi(G2 = an£J Qi is a continuous semi-norm on
I™. If we apply Lemma 12.A.3.2 and (5), we have (z as in that lemma)
\p(*k-lffk-4v,isTv)<pQlP
< C/i(*/:“1o-(t-i,p,is7'p)1/2<71(7r(z)Li)||7r(z)A||(l + log ||m||/S(m).
Theorem 10.5.9 now implies (4). Clearly, (4) implies (1).
12.8. The Harish-Chandra transform of a wave packet
12.8.1.	We retain the notation of the previous section. In this section we
will study the functions TA v(a)° (the notation is as in 12.7.1 (1), 7.2.1). As
we shall see (in the next chapter), this is a critical step in the proof of the
Harish-Chandra Plancherel formula. The calculations are somewhat intri-
cate. In order to carry them out, we will need some preliminary results. Fix
(P, AP) a standard p-pair, and (a, Ha) an irreducible square integrable
representation of °M. Let A g I&, v g Ia, a g ^A(a*), and let Tk „(a) be
as in 12.7.1 (1). Then, TA v(a) g if(G) (Theorem 12.7.1). Let (Q,A) be a
standard p-pair. If к g К and if tn g Mq, f g ^(G), then
fk 'Qk(k~lmk) = hQ(m), with Л(х) = f(k~lxk). Let тг (resp., тг) denote
77,.P)K (resp., 7Г_,ИК). If /= rAiP(a), then f= T^k)k^k)v(a). Thus, with-
out loss of generality, we may assume that there exists (Po, Ao), a minimal
standard p-pair with Po с P n Q and А и AP c Ao. We may also assume
that either A cAP or that for every к g K, kAK~l is not contained
in AP.
If f g -tf(G), g g G, set
H(f)(g) = f f(gn)dn.
Jnq
Then, the material in 7.2 implies that H(f) g C“(G) and that H(f\am)
= a~pQfQGna) for m g °M, a ^A(d(ana~l) = a~2po dn). We also note
that if x g (7(йс)> then тг_р(х)А = E^=1 Uj(v)kj, with Aj,..., Xd g I-
and иj a polynomial function in v. Thus, Theorem 12.7.1 implies that
TA,v(x>a)(s) = f a(v)p(<7Jv)(t_J.x)A)(Tr^g)u)dv
defines an element of ^(G), and that the map a >-» Tkv(x, a) is continu-
ous from У(й*) into ^(G).
192
12. The Theory of the Leading Tenn
Lemma. Ifxe U(qc), then
L(x)H(TKv(ay)-H(7\v(x,ay).
It is enough to prove this in the case when x = X g g. If t > 0, then the
mean value theorem implies
){H(7Ua))(exp(-tX) g) - H(TA>l,(a))(g)}
JNQJa*
ХА(тг;1,(ехр( ~0(t, g, n)X) gn)v) dv dn,
with 0 < 0(t, g, n) < t. Let ш be a compact subset of G. We note that if
у g co then
ITX,v(X> “)(Уп) I <P(«),
with tp an integrable function on NQ (depending on w). We can thus apply
dominated convergence to conclude that
lim f f a(t')pL(o-,iv)Tr_iv(X)X(TTi„(exp(-0(t,g,n)X) gn)v)du dn
JNQJa*P
= f lim f a(v)n(a,iv)ir_iv(X)
jNq‘->0 Ja*
XA(Tr;i,(exp( -0(t, g, n)X) gn) r) dv dn
= f f a(v)fi(<T,iv)ir_it,(X)X(7riv(gn)v)dv dn.
jnqjo*
Here, we have made another obvious use of dominated convergence.
12.8.2.	We now fix A g I-, v g Ia. Let ex = 1,..., eWi be as in 12.6.1.
We set
7r_„(q)A(7r„(g»
F(v, g) =

If /3 is a Wj x wj matrix with matrix coefficients in У(й*) (i.e., an
12.8. The Harish-Chandra Transform of a Wave Packet
193
element of Mw(.^A(a*))), then we set
TA,,(/3)(g) = f n((r,iv)0(v)F(iv, g) dv.
Ja*P
We note that if (3 = al with a g cA(a*), then the first component of
TA, t.(/3) is 7\ ,(a). Also, the preceding material implies that the compo-
nents of Тл t,(/3) are in i^(G).
We now use the notation in Sections 12.6.1 and 12.6.14. Let H g a.
Then
d
—F(iv,exp( — tH) g)
dt
= B(-Aa - iv, H)F(iv,exp(-tH) g) + G(iv, exp(-tH) g),
with the entries of G linear combinations of functions of the form
А')Л,(тг<ч,(ехр( — ^/7) g)v')
with X g n, A' g , v' g Ia, and h a polynomial on a£. Thus, if we apply
Lemma 12.8.1 we find that if a g A, m g °M, then
(1)	£(Я)Я(Тл>1.(Д))(«т) =H(T^B(-Aa-i- ,H)))(am).
Fix H g ciq such that s^tH) =# 0 for s, g Wt и Wm (see 12.6.8,
12.6.14). We now use the notation of 12.A.2.9 with V = a and B(A, h) in
that number replaced by B(-A,h) - pQ(h)I. Let P^ H be as in 12.A.2.9.
Set Pf,v) = L^^qP^hHv) and P0(v) = I - Pfv).
12.8.3.	Lemma. If A g Is, v g Ia, and (3 g Mw(.^A{a*)), then
Н(ТМ1.(рУКт) = Н(Тм,,(рР0Жт) for m g M.
We note that ер^Л)Я(Тл г(ДР,)Хехр(й)т) is in ^Z(a) as a function of
h g a for i = 0,1, tn g °M. We must prove that Я(ТЛ „((ЗР^Хат) = 0
for a g A, tn g nM. We note that if p g a*, then 12.8.2 (1) implies that:
(1)	-	+ PQ)(H)f a^H^fpP^atn) da
JA
+ [ a^+poH(TA ,(/3B(-X,r-i-, H)Pl))(am) da = 0.
Set L(v) = P0(iv) + (B(-A.a - iv, H) - pQ(H))Pfiv) and S^fv) =
(L(v) - ip(H)Pl(iv))~l (this exists in light of our definitions). Then, the
194
12. The Theory of the Leading Tenn
material in 12.A.2.9-12.A.2.10 implies that if /31 g Mw(.^A{a*)) then so is
Thus, if we apply (1) with p = p^, then we find that
f a^H^^P^am) da = (}
J A
for all ц g a*. Thus, Я(ТЛ	= 0 for all a g Л, tn g °M by
Fourier inversion. This proves the lemma.
12.8.4.	We are now ready to set up the notation to state the main result
of this section. If A g I-, v g 1^, then 12.3.2 (2) implies that:
(1)	<p&P(iv, а; А, 7г,р(й)у) = <pQlP(iv, a; A, v) for n g Nq , a g Л.
If g = nmak, n g Nq, m g °M, a g A, к g K, then we set
^Q{P(iv,k,v,g) = <pQ[P(iv,a-,k,Triv(tnk)v).
The following result is essentially [Harish-Chandra [15], Theorem 13.2].
Theorem. If m g M, A g I-, v g Ia, a g У(а*), then
TA v(a)Q(m) = j a^n) pQ f a(v)p,(<r, iv)^Q,P{iv, A, v, mn) dv dn
’	JNQ	Ja*P
for m g M.
Part of the preceding assertion is that the integral converges absolutely.
The reader should note the order mn rather than nm in the integral. In
the proof of this result, we will maintain the notation preceding the
statement. We first note that 12.8.2 (1) implies that if a g/1, m g °M,
then
-rf f a(v)p.((r,iv)e~'B(~A'’~il''H}P0(v)
dt JNQJa*p
x F(jp,exp(-tH) amn) dv dn = 0.
So Lemma 12.8.3 implies that
f f a(v)[i(ff,iv)e~tB(~A,’~'l’ H}P0(v)F(iv,exp(-tH) amn) dv dn
JNQJa*
= H(TAtV(aI))(am).
12.8. The Harish-Chandra Transform of a Wave Packet
195
12.8.5.	We set
We will derive a transformation rule for J. If £ea*, then will
denote the automorphism of (7(mc) defined by p,f{X + h) = X + h -
£(h) for X g °mc, h g ac. We now prove:
d
(1)	—<pG|p(ip,exp(-tH) a; A,t>)
= <pG|P(ip,exp(-tH) а;7?_,„(Мре(Я))А,и).
If a0 g A, then (see 10.5.1 for notation)
!im {(aa0)pQk(TTiv(aa0)v) - <pQlP(iv, aa0; A,t>)} = 0.
Thus,
lim {(a)PQ^iXaao)v) ~ aoPQ(pQ\p(iv^ aa0; X., v)^ = 0.
°e“
So
<pQlP(iv,aa0; A,t>) = a^<p&P(iv, a; A, iriv(a0)v).
Hence,
d
—<pQlp(iv, a exp(tH); X, v)
at
= pQ(H)<pQip(iv,aexp(tH);X,v)
+ <pQlp(iv,aexp(tH);X,Triv(H)v).
Now,
lim [apoX(Triv(a)Triv(H)v) - (p&P(iv,a; X,Triv(H)u)} = 0.
a —»»
Q
Since
A(7r,.l,(a)7rh,(H)i') = -7?_,p(H)A(77,p(a)t>),
196
12. The Theory of the Leading Tenn
we find that
<pe|p(zp,a;A,7r,p(H)i>) = -<p&P(iv,a-,^_iv(H)k,v).
(1)	now follows.
We also note that the material derived in the course of the proof of (1)
implies that
(2)	tQ\p(iv-,b,v,g) = aQ(g)pQ(pQlP(iv,l-X,TTiv(g)v).
(1)	implies that if X g m, then
d
(3)	—iliQlp(iv,\,v,exp(-tX)g)
= tl/Qlp(iv;^_i^n_PQ(X)y,v,&xp(-tX) g).
We will now use the notation in 12.4.5. As in that number, e, = p._Pe(z,).
So
v, g) = Ь(г})фв[Р(1Р-k,v, g).
12.4.5 (ii) implies that if h g Z(mc), then
(4)	V
j
with g Z(gc) linear in h. As we have observed, y“*y(z) =
LLp£Pq(z)) for z g Z(gc). Thus, if we apply H-Pq to both sides of (4), we
have
(5)	М-Рс(Л)е, = У}ре(м,7(Л))е;.
j
This implies that
(6)	Мл) = и,7(мРе(Л)).
If we combine the preceding remarks, we find that
(7)	ЦНг^ф0'р(1р-, k,v,g) = 'LL^y-ly(uij{h))zj^Q}P(iv, k,v, g).
j
By the very definition of ф01р(1р-, к, v, g), it follows that
Ь(У)ф0\Р(1Р-, A, v, g) = 0 for Y g nQ.
12.8. The Harish-Chandra Transform of a Wave Packet
197
If we combine this with (2), we find that if z g Z(gc) then, since
z - pQ(z) g U(Qc)nQ,
L(a4(^£?(z)))'/'£?|p('1'; a> u’s) =	v> S)-
This, in light of (7) and the fact that p-PqPq = PPqPq’ implies that
(8)	L(№,)iAG|P(ti';A,r,g)
= ЕХ-ла-Д«м(Я))£(г;),/'0|р(/1'5 A>y> S)-
j
(6) implies that	= Ь^-Аа - iv, H) + pQ(H)I. We
are now ready to give the advertised transformation rule:
(9)	J(/p,exp(-tH) g) = е'(В(-л„-1,,н)+рс(Н))70-1,) gy
Indeed, the preceding remarks imply that
d
—J(iv,exp(-tH) g)
at
= [B(-Aa - iv,H) + pQ(H)I)J(iv,exp(-tH) g).
(9)	follows from this differential equation.
We next note that the definition of J(iv) implies that
(10)	P0(v)J(iv) = J(iv).
12.8.6.	Our next step in the proof of Theorem 12.8.4 is to estimate
(*) ||e_'B(_A<'_n'’H)P0(p)F(ip,exp(-tH) g) - a^g) P°J(iv, g) ||.
We note that
e-'B(-A'_'r’H)P0(i')F(n',exp(-rH) g) -aG(g) PQJ(iv,g)
X^e‘p^H)F(iv,exp(-tH) g) - a^g) pQJ(iv,exp(-tH) g)}.
198
12. The Theory of the Leading Tenn
Set B0(p) = P0(v){B(-Aa - iv, H) + pQ(H))P0(v). Then,
So
( *) = || e~tB<*v)P0(v)^etp<^H)F(iv, exp( -tH) g)
-aQ(g)~P°J(iv,exp(-tH) g)}||.
Lemmas 8.A.2.4 and 8.A.2.9 imply that
||e-'zW0(p)|| < C(1 + M)P(1 + tf1’1.
Theorem 12.6.9 (in light of 12.8.5 (2)) implies that if w is a compact subset
of G, then there exists a constant Сш such that
\e'p<*H)F(iv,exp(-tH) g) - a^g) pQJ(iv, exp(-tH) g)}||
Z Сш(1 +
with c > 0 and q chosen appropriately.
We conclude that
(1)	lim e~‘B(~Aa~‘,'’H)P0(v)F(iv, exp(-tH) g)
t—> +co
= a^g)~PQJ(iv,g).
12.8.7.	We now complete the proof of Theorem 12.8.4. Theorem 12.7.7
(and its proof) combined with Theorem 4.5.4 imply that if we set
h(v,t,m,n) = e~,B(~H)PQ(v)F(iv,exp( - tH) mn)
- ag(mn) p°J(iv,mn)
and if w j is a compact subset of M, then if tn g M, t > 0,
j a(v)p(cr,iv)h(v, t, m, n) dv
<Рш,(п),
with <рШ[ integrable on Nq. We can thus apply dominated convergence to
find that
lim / / a(v)p(<r,iv)h(v, t,m, n) dv dn
+°° JNQJa*P
= f lim j a(v)p(<r,iv)h(v, t, m, n) dv dn.
jNq +°° •'aj
12.8. The Harish-Chandra Transform of a Wave Packet
199
Lemma 12.7.8 implies that we may use dominated convergence again to
conclude that
lim / a(v)fi(a, iv)h(v, t, m, n) dv
+» •'a?
= [ a(v)fi(a,iv) lim h(v, t, m, n) dv = 0.
•'a*	(->+»
The upshot is
/ / a(v)n(a, iv))F(iv, amn) dv dn
JNqJo*p
= f f a(v)n(a,iv)e~tB(~fi'r~"''H)P0(v)
JNQJa*
XF(iv, exp)( -tH) amn) dv dn
= I I a(v)n(a, iv)a^{man) p°J(iv,man)dn.
}NQJa*
Thus, taking the first components of both sides of this equation and
multiplying both sides by ap°, we have
(*) apQ I I a(v)iv)X.(ttiv(man)v) dv dn
}NqJo*p
= apQ I aidman) pQ I a(v)ii(o-, iv)tyQ\P(iv, A, v, man) dv dn.
JNq	Ja*P
We have observed that
TA (a)Q(ma) = aps [ [ a(v)p,(o-,iv)X.(Triv(man)v) dv dn.
JNQJa*p
Since a^fman) = aa^n), (*) is precisely the formula asserted in Theo-
rem 12.8.4.
12.8.8.	Corollary. If there does not exist к g К such that kAk~l cAP,
then 7\ p(a)e = 0 for all A G I-, v g Ia, and a g ^A(ap).
This follows from Theorem 12.8.4, since under the hypothesis of this
result il/Q\p(iv; k,v, g) = 0 (Theorem 12.6.17).
200
12. The Theory of the Leading Term
12.9. Notes
12.9.1.	As with most of this chapter, the material in Section 12.1 is due to
Harish-Chandra. More detailed formulas of this type given in 12.1.4 can
be found in Hirai [1].
12.9.2.	The modules VQ[P a iv in Section 12.2 are related to a construc-
tion of Harish-Chandra [17].
12.93.	The notion of leading term and srfw(G) in Section 12.3 are due to
Harish-Chandra [14], where he used the terms constant term and
respectively. Our approach is also influenced by Harish-Chandra’s theory
for p-adic groups.
12.9.4.	The material in Section 12.4 was one of our motivations for the
development of the theory in Chapter 4 in this book. There is a related
paper of Trombi [2] that uses similar methods to study the full asymptotic
expansion.
12.9.5.	The relationship between the constant term (in his sense) and
intertwining operators was well understood by Harish-Chandra (as we
shall see in the next chapter). The formula in 12.5.10 is due to Arthur [1].
(Similar formulas were derived by the author in unpublished notes on
Harish-Chandra’s completeness theorem.) Corollary 12.5.4 is Harish-
Chandra’s generalization of a theorem of Bruhat [1]. A sharper irreducibil-
ity theorem will be given in the next chapter.
12.9.6.	The “main inequality” of Section 12.6 is a mixture of several
theorems of Harish-Chandra [15]. It is in this section that one can see how
our asymptotic theory and theory of intertwining operators lead to signifi-
cant simplifications to Harish-Chandra’s original methods.
12.9.7.	The results of Sections 12.7 and 12.8 are due to Harish-Chandra
[15, 16]. One critical difference between our approach and the original is
that we can include the ^.-function in the definition of wave packet, since
we already know that it is tempered. In our development, we were also
influenced by the work of Trombi-Varadarajan [1] in the spherical case.
12.А.1. Traces of Certain Kernel Operators
201
12.A. Appendices to Chapter 12
12.A.1. Traces of certain kernel operators
12.A.1.1. Let К be a compact Lie group and let M be a closed subgroup
of K. Let t and m be respectively the Lie algebras of К and M. Fix B, a
negative definite Ad(K9-invariant symmetric bilinear form on t. Let C and
CM be respectively the Casimir operators of К and M with respect to B.
If у g К (resp., £ g M), let Ay (resp., t)() be the eigenvalue of C (resp.,
CM) on any representative of у (resp., £). Let (a, be a unitary
representation of M such that there exist constants C and r so that
dim Ha(£) < C(1 + V(Y for all £ g M.
Let т: К x К -> L(Ha) be a C" mapping (L(Ha) is, as usual, the space
of all bounded operators on Ha with the operator norm) such that
т(т1Л1 ,m2k2) = <T{m\)T(kx,k2}a(m2}~x
for ml, m2 g M, k{, k2 g K. Let (ira, denote the induced representa-
tion of <r from M to K. That is, Ia is the space of all measurable
functions f from К to Ha such that f(mk) = a(tn)f(k\ tn g M, к g K,
such that
ll/ll2 = f \\f(k)\\2dk<<»,
JK
and ira(k)f(x) = f(xk\
12.A.1.2. We define an operator T = TT on Ia by
Tf(k) = f т(ЛД1)/(Л1)^1.
JK
Let LfHa) be the space of all trace class operators on Ha with the trace
norm (8.A.1.9). The purpose of this appendix is to prove:
Proposition.
(1)	and т is a continuous mapping from К X К into
(2) T is trace class on Ia and tr T = fK tr(r(£, kf) dk.
This result is a special case of much more general theorems about
kernel operators. However, we will not need anything more general in this
book. We now begin the proof.
202
12. The Theory of the Leading Tenn
12.A.1.3. Let Ь^хУцОс, kJ = ц(х~1к, kJ and Ь2(х)ц(к, kJ =
ц(к, x~lkj for x, k,kt g К, д. g C°°(K x K; Let L, also denote
the corresponding representations of k. Let {r,} be an orthonormal basis
of Ha such that each vt g HJ^J for some i. Then, each vt is a C" vector
for M and we note that
(Li(l + CM) L2(l + CM) т(кх, k-JVj, uj
= 0	+ V(j)P{T(ki,k2)vitVj).
Set тр q(kl,kJ = Lj(l + CM)PL2(1 + См)чт(кх, kJ. Then,
|(т(£1Д2)г;,г,.)| = (1	+^(i) P|(W*1,*2)y-',y/)|
^(1 + т?б) 71+ч) pllw*i>MI-
Set CP,4 =	Then,
\(r(kl,k2)vi,vj)\^Cp^l + ^"’(l + tjJ P.
Thus,
Е|(т(л1 ,л2)ц-,^)|
U
^Ср.ч E dim(H„(f))dim(H„(M))(l + T,j’\l + T,j’P
^cP,4c2 E (i + ^r+r(i + ^)-₽+r<«,
if p, q are sufficiently large (see 7.A.4.1 and 8.1.1). This implies that
т(кх,к2) is trace class (8.A.1.4). Similarly, if we set
Cp1<?(&i Дг> ^з> ^4) = ||Tp,9(^i > ^2) _ TP,q(^3> ^4) ||>
then
E|((r(fci,fc2) - т(Л1,Л2))г1.,г7)|
i, j
^c^tk^k^k^kjc2 E (1 + ^)’9+r(i + Vp)~p+r-
f, p.
So 8.A.1.4 implies that т is continuous from К x К to L^HJ. This
completes the proof of 12.A.1.2 (1).
12А.1. Traces of Certain Kernel Operators
203
12.A.1.4. That T is a bounded operator is clear. Indeed,
IIWJU /т(лд1)/(л1)^1
JK
< sup ||т(и,у)||/ И(Л1)||^1
u,v€K
/ ?/2
< sup ||r(u,r)||И ||/(Л1)||2^1	.
u, и	\ К	/
We now prove that T is of trace class (indeed, summable 8.A. 1.4). Fix,
for each у g K, (^y,Vy) y- Let, for у g K, Ti y be a basis of
Нотм(Иу,Ha) such that trT*yTj y = d(y)8i j. Let vi y be an orthonor-
mal basis of Vy. Set /->Лу(Л) = T,y(p.y(k)vjy). Then, {fiJ<y} is an or-
thonormal basis of Ia. Let
t, _m = L,(l + C)'L2(1 + С)тт.
Set Tl m = TT/ m. Then,
= (1 + A?)'(l +Л3)т<Т/,л?,/г.,3>.
Thus,
(*) E \<TfiJ<y,fr<s<s>\
i,j,V,r,s,8
< E (1 +Ay)-'(1 +А5)’т<7’Лт/„Лу,/г.,,5>.
i,j,y,r,s,8
Now,
dim Hom M(Vy,Ha) = £ (£:?)(£: o’) < С E (у :£)0 + VeY-
t<=M
We note that if (y: £) + 0 then rj^ < Ay. Indeed, let t = m Ф V orthogo-
nal direct sum. Let {A’,} be a basis of V such that B(Xt, = — 3,,.
Then, С = CM - ЕДА',)2. If v g Vy(^) is a unit vector, then this implies
that
Ay = V( +
i
This proves our assertion. Also, since : y) < d{y), we have
dim Hom M(Vy,Ha) < C</(y)(l + Ay)r.
204
12. The Theory of the Leading Term
Thus, if we use the preceding observations to count the terms in (*), we
have
E |<г/;.Лу,Д,,8>|
i,j,y,r,s,8
< С||ТЛт|| E d(y)d(3)(l + Ayf'+r(l + As)-m+r.
7.8
Since this sum converges for tn and I sufficiently large (see 8.1.1), T is
indeed trace class.
12A.1.5. We are left with the calculation of the trace of T. Let, for each
£ g M, E( be the orthogonal projection of Ha onto	Let
Д2) = Е^т{кг, k2)Efi. Let T( M = TT(^. Then, it is easy to see that
trT-Etr(T{.{).
4
This observation implies that we may assume that dim Ha < oo as we
complete the proof of 12.A.1.21 (2). Let 5 be the operator on L2(K; Ha)
given by
Sf(k)= ( r(k, kfjftkf) dklt
JK
Then, SL2(K', Ha) c Ia and since /^is a closed subspace of L2(K\ Ha), S
is trace class and tr5 = trT. Let zjj,...,^ be an orthonormal basis
of Ha. Let g EndfH^) be defined by Е^ик = 8jkVi. Then, r(klf k2) =
^ij Ti№i,k2)Eij- If <p e C°°(K X K), then define 5^ on L2(K) by
SJ(k) = jv^kjftkjdk!.
Then, as before, Sv is trace class. At this point, we have shown that
trT= £tr5Tw.
i
Thus, the result will be proved if we prove
tr\ = l^p(k,k) dk.
So let {f;} be an orthonormal basis of L2(K) such that /, g L2(/(X%) for
some % g K. Then, using the preceding argument with the Casimir
12.А.2. Some Inequalities
205
operator, we find that
<₽(*., *2) =
Л/
with
< «о.
i,j
Now, SJj = a^. So
This completes the proof of the proposition.
12.A.2. Some inequalities
12.A.2.1. In the first part of this appendix, we will prove some inequali-
ties for exponentials of matrices. If Л g Afn(C), then set MHhs =
(и(Л*Л))1/2 = (E, ; |a,y|2)1/2. Let ||Л|| be the usual operator norm. Then
Mil < MIIhs < n1/2|| Л ||.
12.A.2.2. Lemma. Let A g Mn(C) have eigenvalues At,..., An (counting
multiplicity) with Re At > Re A2 > • • • > Re An. Then there exists a uni-
tary matrix U such that UAU~l = D + N, with D diagonal having diagonal
entries Aj,..., A„ (in that order) and N upper triangular with zeros on the
main diagonal.
We prove this (well known fact) by induction on n. If n = 1, the result is
obvious. Assume it for n - 1. Let e{,..., en be the standard basis of C".
Let v g C" be a unit vector such that Av = Ajf. Let Ц g U(n) be such
that l^v = e^ Then Л(71-1е1 = Лг = A^. So UlAUile1 = A^j. This
implies that
UXAU;X =
with Л1 g Mn_l(C) having eigenvalues A2)..., A„. Let U2 g U(n - 1) be
such that U2AXU2X has the desired form. Set
At *
0 A{
and U = VUl. Then, UAU 1 is as asserted.
206
12. The Theory of the Leading Tenn
12.A.2.3. If t > 0, then we set
eRm|r>Zi> > tm > 0}.
Lemma. Let D g Mn(C) be diagonal and let N g Mn(C) be upper triangu-
lar with zeros on the main diagonal. Set N(t) = e~,DNe,D. Then, if t > 0,
n — 1
e-tDe'(D + N) = j + £ Г	... N(t])dtl dt,.
Set u(t) equal to the right hand side of the preceding equation and set
n(t) equal to the left hand side. We note that:
(1) If Nj,..., Nn are upper triangular matrices with zeros on the main
diagonal, then
M • Nn = 0.
Also,
«7+1(0 = f W('i) •••	••• dtj+l
Jsj+lM
= j‘N(s)j NfJt) ••• N^dti dtjds.
Jo JS1ls)
Thus,
d
^«7 + 1(0 = W(0«j(0-
We conclude that
d
—u(t) = N(t)u(t)-Nltju^t).
But N(t)un_l(t) = 0 by (1). Hence:
d
(2)	— u(t) = N(t)u(t),	t>0 and u(0) = I.
dt
On the other hand,
d
-fn(t) = -e~,DDe,(D+N) + e~tD(D + N)el(D+N}
= e-‘DNeHD+N> = N(t)n(t).
12.А.2. Some Inequalities
207
Qearly, n(0) = I. Thus, the existence and uniqueness theorem for ordi-
nary differential equations implies that nit) = u(t) for t > 0.
12.A.2.4. We now come to the first main result of this appendix, which is
a sharpening of Lemma 60 in Harish-Chandra [8].
Lemma. If A Afn(C) has eigenvalues Aj, . .. , A„ and if и =
max t ss n Re A,, then
IHIIHs<n1/2^(l +11/111™)"’*.
According to Lemma 12.A.2.2, there exists U g U(n) such that UAU~l
= D + N, with N upper triangular with zeros on the main diagonal and
D diagonal with diagonal entries Aj,..., An with Re A, >  • • > Re An
(here, if necessary, we reorder). Clearly, HeL,'4L' Hhs = Hc^IIhs,
ЦСЛ4С7-1 Hhs = ||AHhs, and UAU~X and A have the same eigenvalues with
the same multiplicities. We may thus assume that A = D + N. We will
now use the notation of Lemma 12.A.2.3.
We note that e~‘D = e-“'mDe~tReD and that e~“lmD is unitary. So
I|A(0IIhs =lle-'D№'DllHS < ||e-'ReD№'ReDH.
Now, if N = [niy] then = 0 if j < i, so
He-'Re D№'Re Xs = Le2'(ReA>-ReA-% l2-
i<j
If t > 0, then e'<Re A,“ReA<> < i for i <j. Thus,
l|e-'ReD№’ReD||HS^llAllHS^IU||HS.
This combined with Lemma 12.A.2.3 implies that, if 0 < t < 1,
||e-'De'<D+/v’llHs £ 1 + MIIhs +	+!№'
< (1 + НЛ||hs)
Thus,
IHIIhs =kD(e-DeD+w)||Hs
< lkDllHslk-^D+/VllHS < n1/2lleD||(l + IMHhs)"’1.
Since ||eDll = eM, the lemma follows.
208
12. The Theory of the Leading Term
12.A.2.5. Corollary. Let the notation be as in the preceding lemma. If
t > 0, then
lM|HS<n1/2^(l + Мне)"-1.
This is just Lemma 12.A.2.4 for tA.
12.A.2.6. In this number, we sketch another, somewhat more elementary
proof of Lemma 12.A.2.4 (with nl/2 replaced by e"~ln1/2). We assume, as
we may, that A = D + N as in the beginning of the proof of 12.A.2.4.
Then,
eA = lim (eD/meN/m)m.
We write X(m) = eN/m - I. Then, X(ni) is upper triangular with zeros
on the main diagonal. Thus,
= eD/m(I + X(rn))eD/m(I + X(m)) eD/m( I + X(m))
= eDe-(m~1)D/m(I + X(m))eim~l)D/me~im~2)D/'n(I + X(m))
Xe(m-2)D/m ...	+ X(m))eD/m(I + X(m)).
Set X(m, t) = e~‘DX(m)e‘D. Then, we have
(eD/meN/m)m
= eD(I + X(m,l - l/m))(I + X(m,l - 2/m)) •(/ + X(m,0)).
We note that if Хг,..., Xn are upper triangular with zeros on the main
diagonal, then XrX2  •  Xn = 0. Also, as in the proof of 12.A.2.4, we
have
l|A'(m,f)||Hs II-V(zn)||hs fortSiO.
Thus, if m > n then
||(eo/^»/»)"||HS s „/=к»|Г£	+1 j||X(m) Hi,,.
j=0 \ J I
But
X(m) =N/m + (A/m)2/2 + • +(A/m)n 1 /(« - 1)!.
12.А.2. Some Inequalities
209
Assume that m > ||N||Hs • Then
II X(m) Ls ellNllHs/w e|U||Hs/w.
This implies that
l|(eD/me'v/'”)'"l|HS <n1/2lkDll E Г” +7 |(e||X||Hs)'/m<
j-o \ } I
Since limm^„J j/mJ = 1/jl, we have
lim П(е^те^т)т||HS < en-1n1/2lkDll(l + MHhs)"’1.
m -»»
12.A.2.7. We now give a refinement of another lemma of Harish-Chandra
(see also Trombi-Varadarajan [1]). If U c Rn, then we set &~(U) = {x +
iy |x g U, у g R"}. Let zt,..., zn be the usual coordinates on C".
Lemma. Assume that U is open in R" and that h is holomorphic on
У(и), satisfying
|ft(z)| <C(1 + ||z||)r,	z6^(l/).
Let If be open and such that СКЦ) compact and contained in U. Then
there exists a constant C{ depending only on Ux such that if f(z) = h(z)/zx,
where defined, and if f is defined and holomorphic at z g then
|/(z)| < ОД1 +||z||)r.
Furthermore, Cx < max{6  2r, 3 • 2r/8}, with 8 = inf{||x - u|| |x g Ux,
U G dU}.
Let z g S7\Ux). Set c(z) = inf{||z - w|| |w g dS^lU)]. If x g Ux, then
set 8(x) = inf{||x - u|| |u g dU}. Then, clearly, c(z) > 8(Re z). Since
Q(CJj) is compact and contained in U, 8(x) > 8 for all x g Ux. Thus,
c(z) > 8 for all z g ^(Ц). Let 0 < 8! < min{l/6, 8/3}. Let 5 = {z g
^(Ц)| |zj < 8J. If z g ^(Ц) - S, then, clearly,
|/(z)| £ |/t(z)|/81^(C/81)(l +||z||)r.
If z g S, then write z = (zt, w). We have
rlh^,w)
O  /	r
27Г1 J\(\=28x C - z
210
12. The Theory of the Leading Tenn
Now, 1 + Hwll > 1 + ||z|| - 33,	(1 + ||z||)/2. So
|/(z)| ^(2730^1 + ||z||)r.
12.A.2.8. Lemma. Let A,,..., Am g (Cn)* - {0}. Let U be a neigh-
borhood of 0 in R". Let h be holomorphic on &\U) and assume
that (A,  Am)-1/i = f is holomorphic at every iy, у gR". If 8 =
inf{||u|| |m g dU}, then set Ct = max{6  22,3  22'/3}. Then, there exists a
constant C2 depending only on A,,..., Am such that if
|ft(z)| <;C(1 + ||z||)r,	zg^(U),
then
\f(iy)\ <C2C[C(1 + ||y||)r, yGR".
We prove this by induction on m. If m = 1, we may assume that
A, = z,, and the assertion follows from the previous lemma. Assume the
assertion for /и = Л - 1	1. If m = k, then, as before, we may assume
that A, = Zp Let Ux = {x g R"| ||x|| < 8/2}. Set Л, = zf lh. We assert
that Л, is holomorphic on УК/,). Indeed, if z g У([/,) and if z, =# 0,
then Л, is holomorphic in a neighborhood of z. Since f is holomorphic at
each iy, у g R", we see that Л(0, iy) = 0 for у g R"-1. Thus, Л(0, и) = 0
for all (0, u) g У([/,). This implies that if и is fixed then h/z, u) =
z~lh{z, u) is holomorphic at z = 0. Hartog’s theorem (cf. Hormander [2])
now implies our assertion. The previous lemma now implies that
|fc,(z)l <C,C(1 + ||z||)r,	zg^-(C/,).
If we replace U by Ux, h by ft,, then we have replaced m by m - 1. So
the inductive hypothesis prevails.
12.A.2.9. In the last part of this appendix, we will prove two more results
about matrices. Let W be a finite dimensional vector space over R and
assume that V and U are real subspaces such that W = V Ф U. If v gK*
(resp., Uq ), then we extend v to W by vfU) = 0 (resp., p(F) = 0). Let
В:1Г*ХИ^МЛ(С)
be linear in V and polynomial in Wc*. We assume that there exist
y,,..., sr g GL(W*) such that the eigenvalues of B(A, v) are of the form
i,A(r). Fix A, g U*.
12.А.2. Some Inequalities
211
We fix linear coordinates {x,,..., xm} on V and we will use the usual
multi-index notation for derivatives. We define an inner product on V
(hence on И*) by (v, w) = E x/tOx/w).
If д. g R, v g И*, and v g И, then we will use the notation P^lv)
for the projection onto the sum of the (generalized) eigenspaces for
B(Aj + iv, v) with eigenvalues having real part equal to p.
Lemma. Рд>(, is real analytic on V*. Furthermore, if II •  • II denotes the
operator norm on Mn(C) then there exists p such that
G(1 + ||p||)p(1 + Ilr||)".
If s,Aj(tO =# p for all i, then ^(p) = 0 so there is nothing to prove. If
we reorder the s,, we may assume that s^tv) = ... = SjA.f.v) = p and
SjA^v) # p for j > m. Let 0 < e < |minI>m l^-Ajft?) - p\. Let v0 g V*
and let R{v0) = 2max{|.sIp0(i>)| li m}. Then, R(v0) < C||p0||. Let fl =
{p g V*\ |i,p(r)| < R(v0), i < m}. Let I be the rectangle in C with center
p, width 2c, and height 47?. Let C be the boundary of I, oriented
counterclockwise. If v g fl and if i,(A1 + ivKv) g I, then i < m and
^(Aj + ivXv) C. Thus,
(1)	Р^(У) = ^-/(B(Al+iv,v) -ziy'dz.
ZlTl Jc
This implies the first part of the lemma.
We now prove the inequality. Let ш = {г g V*\ |^р(г)| < e/4, i < r}.
Set flj = {pj + iv2\vl G co, p2 g fl}. If p g flj, then set
Q(v) = f (B(Ai + »,») ~ zl)~l dz.
ZTTl Jc
Then, Q is holomorphic on fl, and Q(iv) = P^ffv) for v g fl. We will
now estimate Q(v). So fix p g fl,. We write B(A, + p, v) = U{D + N)U~l,
with U unitary, D diagonal, and N upper triangular with zeros on the
main diagonal. We may assume that B(A, + v, v) = (D + N) (for the
purpose of our estimation). If z g C, then
D +N-zI= (D -zl)(l+ (D -z/)-1N).
212
12. The Theory of the Leading Tenn
So
(*)	(D + A-z/)“‘ = (/-(D~z!)~lN + 
+ (-l)n-1((D -	-zl)~l.
Our assumption on v implies that if z g C, then
(**	)	||(D-zZ)-*||<2A.
Also, as in 12.A.2.5, 12.A.2.6,
HNll £	+ p,r)|| £ C(1 + MfIM.
Thus, if z g C, then
||(D + N - zl) "* || < (2/e)(l + C(1 + IMD’llrllA)"-1.
Hence,
HO(p)|| < ((8R(p0) + 4e)/e)C(l + IMI)9<n-1)(l + Hrll)"’1.
Now, R{v0 < CHpoII. We have therefore shown that if p g ft,, then
110(^)11 < C(1 + IM)(1 +	+ llrll)"-1.
If we use the argument in the first part of the proof of 12.7.2, we find
ke(n,0)|| < cz(i + iM/(n-1,+1(i + iirii)"-1.
The lemma now follows.
12.A.2.10. We end this appendix with one more inequality.
Lemma. Let A g Afn(C) be such that if p, is an eigenvalue of A then
I Re p, I > 8 > 0. Ifx g R, then
||(Л - ixl) ’* || <; (1 + nl/2\\A\\/8)n~'/8.
As usual, we may assume that A = D + N, with D diagonal and N
upper triangular with zeros on the main diagonal. The inequality follows
by estimating the individual terms in (*) of the previous lemma and using
(* *) therein.
12.А.З. The Topology of Induced Representations
213
12.A.3. The topology of induced representations
12.A.3.1. In this section, we relate the C" topology on an induced
representation relative to the L2-norm to the C“-topology relative to the
L“-norm for induced representations. Let К be a compact Lie group and
let M be a closed subgroup of K. Let (a, Ha) be a unitary representation
of M such that if £ g M and if (д., И) g £, then dimHomM(F, Ha) <
Cd(£). Let 1“ be (as usual) the space of all f: К -> such that f is of
class C°°. If к g K, f^I™, then we set Tra(k)f(x) = f(xk). Set, for
x g L7(fc), px(f) = supkEK ||7г<т(х)/(Л)|| and qx(f) = ||тго.(х)/||. Here,
H/ll2 = fj\f(k)\\2dk,
as usual. We have seen that if we endow /“ with the topology induced by
the semi-norms px (or the semi-norms qx) x g U(tc), then (тта, I™) is a
smooth Frechet representation.
12.A.3.2. The purpose of this appendix is to prove:
Lemma. There exists z G Z(t) such that px(f) < qx(tra.(.zx)f).
Note. Since it is clear that qx(f) < Px(f) for all x, this lemma implies
that the two topologies are equal.
Clearly it is enough to prove that there exists z g Z(t) such that
Pi(/) < qz(f\ Fix B, a negative definite Ad(KT-invariant form on k. Let
CK be the Casimir operator of К relative to B. If у g K, fix (ry, Vy) g y.
If T g HomM(Fy, Ha), then set fT<v(^ = T(ry(k)v). If T g
HomM(I/, H„), S g HomM(I/, H„), v g Vy, w g V^, then
= f{T(Ty(k)v),S(rtl(k)w))dk
К
= 3y.^(Y)“1tr(5*7’)<r,w>.
If T g HomM(Fy, Ha), then set ||T||2 = tr T*T.
Let Tj>y be a basis of Нотм(Иу, with tr(7}*y7} y) = 8^. Let vjy be
an orthonormal basis of V. Then,
f.j.y = d(y)l/2fTiy,Vjy
is an orthonormal basis of the Hilbert space completion of We note
214
12. The Theory of the Leading Tenn
that
|/,7.у(Л)| £ ^(у)1/2117}.у11 • ll^.yll = d(y)l/2.
If f e £(y), then f = E;>а^,у. Thus,
НЛЛ)Н< LMII/UMII
/	\l/2/	U/2
dW2 DW*)H2
'«j	'	''t.i	'
<d(y)1/2(dim Ш))1/2ИЛ-
Now, if is a fixed representative of £ g Al, then
dim £(y) = (dim HomM(l/,//„))</(?)
= <f(y) E dim Hom	• dimHomM(W^, Ha)
< d(y) £ G/(£)dimHomM(f;,n9 = G/(y)2.
Thus, if f g /”(y), then
\\f(k)\\ < Cd(y)3/2||/||.
Let D = I + CK. Let, for у g K, ry(CK) = XyI. If then f=Lyfy,
with fy g Г^у\ Since ||/||2 = Ey ||/y||2, we see that
11Д1Щ1 +АУ)’Х(ЯГШ
for all r g N. Thus,
Wf(k)\\ = EHfy(k)\\ C£d(y)3/2(1 + Ay)’r||7r<T(Dr)f\\.
У	У
Let r 0 be so large that Ey d(y)3/2(l + Ay)~r = (^<00 (8.1.1). Then,
\\f(k)\\ CC^D^fW.
Set z = CCiDr.
The Harish-Chandra
Plancherel Theorem
Introduction
The main results in this chapter are the Harish-Chandra Plancherel
theorem and Harish-Chandra’s refinement that gives a decomposition of
the Harish-Chandra Schwartz space into a direct sum of ideals associated
with conjugacy classes of special vector subgroups. These theorems go far
beyond the abstract Plancherel theorem (as we shall see in the next two
chapters). We then (following Harish-Chandra) derive his completeness
theorem for intertwining operators and his upper bound for the intertwin-
ing number of a unitary principal series. Another application that we give
of this theory is a sharpening of the Langlands classification in the
(simplest) case of one conjugacy class of Cartan subalgebra and the
decomposition of L2(K\G).
As in the previous chapter, our approach differs from Harish-Chandra’s
in its emphasis on representation theory. Harish-Chandra [16] looks upon
the Eisenstein integral as the basic object of harmonic analysis. We
consistently use the relationship between the Eisenstein integral and
matrix coefficients of principal series. We hope that the reader of this
chapter will also study Harish-Chandra’s original paper.
215
216
13. The Harish-Chandra Plancherel Theorem
Section 1 gives an exposition of Eisenstein integrals and their relation-
ship with matrix coefficients. In Section 2, we use the theory of the
previous chapter to calculate the leading term of the Eisenstein integral.
This leads to a definition of the Harish-Chandra C-function, the (critical)
Maass-Selberg relations, and the functional equation of the Eisenstein
integral. In our approach, the C-functions are secondary to the intertwin-
ing operators. As we have indicated earlier, Harish-Chandra derived his
theory of intertwining operators from his theory of C-functions. Section 3
is the heart of the chapter. The main theorem 13.3.2 calculates the
Harish-Chandra transform of a wave packet of Eisenstein integrals. Sec-
tion 4 contains the Plancherel formula (13.4.1) and Theorem 13.4.7 (due,
of course, to Harish-Chandra), which is the more basic theorem. In
Section 5, we give a calculation of the Harish-Chandra ^.-function for
fundamental series. Section 6 contains a proof of Harish-Chandra’s com-
pleteness theorem for intertwining operators and (thereby) his bound for
the intertwining number of a unitary principal series. In particular, this
bound implies the irreducibility of the fundamental series. In Section 7, we
apply this theory to the simplest case, the groups with one conjugacy of
Cartan subgroup. In Section 8, we derive Harish-Chandra’s earlier decom-
position of L2(G/K) from the Plancherel Theorem.
13.1. The Eisenstein integral
13.1.1. Let G be a real reductive group. Fix a maximal compact subgroup
К of G. Let (P, A) be a standard p-pair for G with Langlands decompo-
sition ° MAN. If g e G, then we write g = nm(g)a(g)k(g) with n g N,
m(g) g °M, a(g) gA, and k(g) gK. Here, as usual, there is ambiguity
in the definition of m(g) and k(g) but a(g) is a well-defined function
of g.
Let (t,V) be a finite dimensional representation of К x К. Then we
write r{k)v = f(k, l)t> and vr{k) = f(l, k~v)v. (т, И) will be called a
double representation of K. If (f, И) is a unitary representation of К X К,
then (т, И) is called a unitary double representation of K. The basic
example we have in mind is when (r, FK) is a finite dimensional represen-
tation of К, V = End(WO, т(Л] ,кг)о = т^к^итСк^’). Then, in this case,
the notation is consistent. If (r, W) is unitary, then we put the
Hilbert-Schmidt inner product on End(WO, making (r, End(lF)) into a
unitary double representation.
13.1. The Eisenstein Integral
217
Let (т, И) be a double representation of K. If <p g С“(°Л/; И) is such
that tp{kxmk2) = т{кх)<р{т)т{к2) for kx,k2 g KM = К П Aland m g °M
then we set
<p(g) = <p(a(g)m(g))r(k(g)).
We note that the transformation rule of <p implies that this extension of <p
to G is defined and of class C°°. If v g a£, then we set
E(P,<p,v)(g) = f a(kg)v+pT(k)~x<p(kg) dk.
JK
Following Harish-Chandra, we call such a function an Eisenstein integral.
13.1.2. We now show how these functions relate to the matrix coefficients
of induced representations. Let (a, Ha) be an admissible Hilbert represen-
tation of °M. We set equal to the span of the K^-finite matrix
coefficients of a. If (т, V) is a double representation of K, then we set
t) equal to the space of all C" functions <p on °M satisfying the
following two conditions:
(1) (ptkpnkj = т^к^^тУт^к^ for kx,k2 g KM and m g °M.
(2) If A g V*, then A ° <p g ^(a).
The next result implies that if <p g л/’Са, t) then E(P,<p,v)e
Lemma. Let {(т,НаУ be an admissible representation of °M, (т, И) a
double representation of K, and let <p g r). If p g V*, then there
exist A, g Ip a, Vj g Ip a, i = 1.d, such that
d
/x( E(P, <p, v)(g)) = ЕШ^Ш
i = i
for all g g G.
We may assume that f is irreducible. Then there exist (т,, V,), i = 1,2,
irreducible representations of К with V = V\ ® V2 and
,A2)(z?i ® r2) = r^k^Vi ® r2(k2)v2.
Let щ.....ud and .......vm be bases of V\ and V2, respectively. Then,
<p(m) = EM"1)"/ ® y/>
218
13. The Harish-Chandra Plancherel Theorem
with g J3/(cr). Thus, there exist A.kiJ e (На)Км, wki! g (На)Км, with
k
We now note that
<p(tn) = [	т(к1)~1(р(к1тк2)т(к2)~1 dkxdk2.
jkmxkm
We may thus assume that there exist A g {На)Км, w g (На)Км, и g ,
v g V2 such that
<p(m) = /	Л(0-(Л1/иЛ2)и')т1(Л1)-im ® r2(k2)vdk1 dk2.
jkmxkm
Continuing in this way, we may assume that there exist д.,- g V*, i = 1,2,
such that ц(и 8 v) =	Thus,
(*) M(E(P,<p,p)(g))
= f av+pk(ff(klm(kg)k2}w)p,l(Tl(klk)
jkxkmxkm	v
X^2(r2(fc(fcg) -1fc2)z?) dkdkt dk2.
Define
f(k) = f ц2(т2(к~ lm)v)(r(m)vdm
KM
and
ф(к) = J ftl^Tl(tnk)~ln>jcr(m)~1kdm.
KM
Then, f g Ia, ф e Iand (*) implies that
n(E(P,<p,v)(gy) = j-a(kgy+f^(k)(f(k(kg))) dk.
К
The lemma now follows.
13.13. Let (<r, Ha) be an admissible representation of °M such that 0-|Кдг
is unitary. Let F be a finite subset of K, let Ia(F) = ®y eF ^(y), an£l let
13.1. The Eisenstein Integral
219
Ef be the orthogonal projection of Ia onto Ia(F). If T g End(/a(F)),
then we look upon T as a bounded linear operator on HP<rv for all
v g йс by the formula 7EF. Set VF = Endf/^fF))*. If kx, k2 e K, then
we write rF(kx, k2\k\T) = k(k2'Tkx). Then (rF, KF) is a representation
of К X К. Let (tf, Vf) be the corresponding double representation of K.
Let fx...fd be an orthonormal basis of I^F). Then we set
= Z<^(m)4(l),	//>
ij
for m g °M and T g End(Ia(F)). Then, a direct calculation shows that
% G tf\
Lemma. E{P,^F,v\g\T) = triT-тгр a v(g)) for T e Endf/^F)), ge
G, v g a£.
Indeed,
E(P,V)(g)(T)
= I a(kg)v+p'irF(m(kg))^k(kg)Tk~i>) dk
К
= Lf «(*g)‘'+'’{a(/n(Fg))/;.(i),/z.(i)X*(*g)^-y/,4)^
ij K
i,j K
= Lf a(kg)v+P{(r(m(k8)')k(kg)f](l'),kT*fi(l))(f,,f]')dk
i,i K
= Ef a(kgy+p{a(m(kg))fj(k(kg))^fi(k))8iJdk
UJk
= Ef {^Xs)fi(k),T*fi(k)')dk
i jk
= Lf <TTrP,a,v(g)fl(k),fl(k))dk = tr(T^><7t,(g)).
i jk
13.1.4.	We now assume that (a, Ha) is an irreducible admissible repre-
sentation of °M such that is unitary. Let (т, И) be a double represen-
220
13. The Harish-Chandra Plancherel Theorem
tation of K. We will now give a description of the space r). Let f be
the corresponding representation of К x К on V. Let F be a finite subset
of К such that V = Ф ,eF V(y ® y') relative to f. If у e K, let Vy g y.
Set FM = {£ e KM\Vy(O * 0 for some у g F}. Set Ha(FM) =
and let PF be the orthogonal projection of Ha onto
H^fFay). We define a double representation of KM on EndfH^fF^)) by
using the left and right action of <г\км- If («, IF),(/3, U) are double
representations of KM then we denote by НотКдг(1У, U) the space of all
linear maps from W to U that intertwine the left and right actions of КM.
If T g Hom^M(End(H0.(FM)), И), then set <pT(mj = T(PFa(m)PF).
Lemma. <pT g r). Furthermore, the map
T >-» <pT
is a Unear bijection of
Hom^End^Fa,)),!')
onto t).
It is clear that <pT g .ofta, r). Since a is irreducible and admissible,
spanc{PFa(m)PF\m g °M} = End(Ho.(FM)).
Thus, if tpT = 0 then T = 0. Thus, to prove the lemma we need only show
that the map is bijective. Let be the space of functions и g
such that the Fay-isotypic components of the К ^cyclic spaces of и with
respect to the regular representation are in FM. Let S: End(Ha(F)) ->
be defined by 5(tXm) = tr(tP(r(jn)P\ Since a is irreducible, 5 is
bijective. Let g r). Define <p(AXm) = A(<p(m)) for A g V*,
m g °M. We look upon V* as a double representation of К using the left
and right contragradient actions. Qearly, <p g НотКдг(И*, ^(a)F). Thus,
S-^GHom^.End^Fa,))).
Let
<7: End(H<7(FAy))-> End(H<7(FAy))*
be defined by q(t\s) = tr(ri). Then,
qS~\p g НотКд,(И*,EndC^Fay))*).
13.1. The Eisenstein Integral
221
Finally, let д. be the natural map from
НотКм(И*. End(Ha(FM))*)
to
Hon^End^^)),!')
defined by
л(м(Т)(О) = Т(л)(О.
Set 7^ = fiqS~l<p. Then, a direct “unraveling” yields ipT^ = <p.
13.1.5.	We continue with the situation of the previous number. If
5 <^KM, then we set S* = {£*l£ e 5}. We identify End(74.(FM)) with
Ha(FM) ®	in the usual way ((v ® AX«) = А(м)г). We identify
Ha(FM)* with H&(Fm) using the canonical pairing of H„ with H& = H'„.
We therefore look upon End(Ha(FM)) as H„(FM) ® H&(F^). The repre-
sentation of Км x KM corresponding to the double representation on
EndfH^fF^)) is 0|Кд, ® cr^M- Similarly, we identify the double representa-
tion of К on Endf/g.fF)) with the double representation of К on Ia(F) ®
I6(F*) corresponding to тг„ ® тг&. If T e HomKjEnd(H„(FMj), V), then
we define i(T) e Hom^fEndf/^fF)), И) by
i(T)(/®/) = f r(kl,k2)~lT(f(kl) ®f(k2))dkxdk2
KxK
for f e Ia(F), f&Iv(F*). Here, as usual, f is the representation of
К x К corresponding to r.
Proposition, i is a linear bijection between
Hom^EndC^F^)),^)
HomK(End(/0.(F)), И).
Furthermore, if
TeHom^EndC^F^)),^),
then
E(P,<pT,v)(g) = l(T){Efttp „ v(s)EF)
for g g G, v e a£. Here, EF is (as usual) the orthogonal projection of I„
onto I„(F).
222
13. The Harish-Chandra Plancherel Theorem
We will prove the first assertion in two steps. First, we show that
dimHomKM(End(H<T(FM)),F) = dimHomK(End(/<T(F)), И).
Then, we show that t is injective. As for the first step, the previous
discussion implies that it is enough to show that
dimHom^^jH^F^) ®^(F£),F)
= dimHomKXK(Z<7(F) ®/ДГ*),И).
Now, under f, V splits into a direct sum of К x К modules of the form
(tj , Fj) ® (t2, V2). We therefore may assume that V is of this form. It is
easily seen that Hom^x^JH^fF^) ® Hd(F^\. V) is naturally isomorphic
with
Hom^H^F^),^) ® Hom^H^),^)
and that
HomKXK(/<T(F) ®^(Е*),И)
is naturally isomorphic with
HomK(/0.(F), Fj) ® HomK(/j(F*), K2).
An application of Frobenius reciprocity implies the first step. We now
prove the second. As before, we may assume that V = Vt ® V2. Suppose
that i(T) = 0. Let e V*, i = 1,2. Then ® ^(W ® /)) = 0.
Thus, if f e I„(F), fe I&(F*) then
0 = f Vi ®	® t2(F2))'iT(/(Fi) 0f(k2))dkl dk2
KxK
= У	® 72(F2);i2)(/(Fi) ®f(k2)}dkx dk2.
(Here, T'(£) = £ ° T.) Frobenius reciprocity for induction from Км X KM
to К x К now implies that F'(mi ® М2) = 0 f°r a" Mi > М2 as before.
Thus, T = 0. The first assertion of the proposition is now proved.
We now prove the second assertion by the obvious method of calculat-
ing both sides and seeing that they are equal. We may, as before, assume
that V = V\ ® V2. We may also assume that T = 7\ ® T2, with Tt e
and T2 e НотКд,(Н^),И). Let gfei;*, i =
1,2.	Let vt.vd be a basis of Ha(FM) and let A].Xd be the dual
13.1. The Eisenstein Integral
223
basis in Hd(F^). Then,
Mi ® F2(E(P,<pT,v)(g)')
= f a(kg)v+PiLi ® M2(Ti(M ® т2(л(^)) _1
JK
X T(PFa(m(kg))PF) dk.
Now, T(.PF<r(m)PF) = ’Ll {(<r(tn)vj\A.i')vi ® A.j. Thus, the expression we
are studying is
EI a(kg)l'+P(^(fn(.kg))Vj\ki)p.l(Tl(k)'Ti(ui))
i,J K
X M2(t2(^(^))”I7’2(a/)) dk.
We now observe that T* e Нот^НУ*, H/F^)). We define fT? s(k) =
T*(r*(k)8) for 8 e И*. Then, fT* s g I&(F*). With a similar construct
for T2, our expression becomes
E / <ksY+P^{m(kg))vj\Xi']fTrt^k)(vi)fT^^k(kg)){ks)}dk
>,j K
= Е/«(^)’'+7г,*,ДЛ)(^)Л?.4л(^))(й;(т(^))А<)‘/л
К
= Lf fTf.4kY^)(^P,a,V(S)fT}.4kY(XY)dk
j K
= f	д *)))<**
К
= (fTr,F.l\('!rp,<r,v(s)fT}^2)-
We now do a similar calculation for the right hand side of the formula
that we are proving. Let /, be a basis of I„(F) and let /• be the dual basis
of fj.(F*). Then
EFTrp,„,v(S)EF = ^(^P,a,v(S)f}\fi)fi ® fj-
So
i(T)(EFrrP.<T„(g)EF) = Е(^,,.Д«)//Й(П(Л ®Z)-
224
13. The Harish-Chandra Plancherel Theorem
Thus,
Mi ® ^2^(T)(Efttp a V(S)EF))
ij
X Мг(*г2(^2 1)^2^'(^2))	^2
ij
= (^Р.а.ЛвУ/т^т^ТГ.щ)-
The proposition now follows.
13.1.6. The preceding two results imply that there is no essential differ-
ence between matrix coefficients of principal series and Eisenstein inte-
grals. However, the added algebraic structure of Eisenstein integrals is
sometimes useful. We will now study the case when (a, Ha) is an irre-
ducible square integrable representation of °M. Let (т, K) be a double
representation of K. Then we say the (т, V) is unitary if К is a (finite
dimensional) Hilbert space and if both the left and right actions of К on V
are unitary. We retain the notation of the previous number. Assume that
(т, V) is unitary. Let dm be a fixed choice of invariant measure on °M. Let
d(a) be the formal degree of a with respect to dm (1.3.4). If <p, 17 e
•q/(<t, t), then we set
<ф, i?> = f {(P(m),r](m)') dm.
Jom
The integral converges since a is square integrable. On End(Ha(FM)) we
put the Hilbert-Schmidt inner product (<t, j) = tr(ts*)). If
T,S eHomc(End(H<7(FM)),K)
and if tj is an orthonormal basis for End(Ho.(EM)), then we set
<T,S) = UT(ti),s(ti)').
i
Lemma.
(1) IfT,Se HomKjEnd(H<7(FM)), K), then <<pT, <ps) = d{aY\T, S>.
(2) IfT,Se HomKM(End(Ha(EA/)), V), then (T,S> = <i(T), i(S)>.
13.1. The Eisenstein Integral
225
Note. (2) can be used to give a second proof of the first assertion of
Lemma 13.1.5. In the next section we will implicitly give a third proof.
(1) is a simple exercise using the Schur orthogonality relations (1.3.3).
We now prove (2). For this we may assume that V is irreducible as a
representation of К x K. If we take into account the steps in the first part
of the proof of Proposition 13.1.5, then (2) is reduced to the following. Let
(77, ИО be an irreducible representation of K. If T e YiomK^Ha,W),
then we set, for f e Ia,
JK
Then, i(T) e HomK(4, IV). It is enough to show that
tr(i(S)*i(T)) = tr(T5*).
We note that S* & Hom^-JK, Ha). Thus, if Aj,..., Ar is a basis of
Hom^JF, Ha) such that МДЛ*) = 3,7, then:
(i) trfTS*) = E< trOAJtriSAi).
We now begin the calculation of tr(i(5)*i(T’)). For this we write out an
orthonormal basis of Ia(y). If A s HomKjF, Ha), v e. V, then set
fA<v(k)=A(v(k)v).
Then, fA s I (y). Furthermore, if В e Нотк (V, HA w e K, then
= f <A(V(k)v),B(V(k)W))dk
JK
= J (т](к)~1 B*A(if](k)v), w^dk
К
= d(y)~l tr(B*A) <r,w>
by the Schur orthogonality relations. Thus, if A, are as above and if
г j,..., vd is an orthonormal basis of W, then
tr(i(5)*i(T)) =d(y)^(T)fAitVj,l.(S)fA^.
226
13. The Harish-Chandra Plancherel Theorem
Now,
^T)fAt„ = jv(k) 1TA(r)(k)v) dk = d(y)~l tr(TA) v.
Thus,
tr(i(5)*i(T)) ^d(y)Z<vJ,i’J)d(y)-2tr(TAi)^(SA^
= Ltr(Z4(.)tr(£4(.).
i
(2) now follows from (i).
13.1.7. We now show that Eisenstein integrals are adjoint to the Harish-
Chandra transform. We recall (7.2) that if f e if(G) and if (P, A) is a
p-pair, P = ° MAN, and if аеЛ, me °M, then
fp(ma) = a~Ppf f(nam) dn.
JN
If f e C“(G) and if v e a£, then we set
fp(m) = f avfp(am) da.
JA
If (г, K) is a finite dimensional unitary double representation of К and if
f e C“(G; K), then we use the preceding formulas to define fp and fp.
We set C“(G;t) equal to the space of all /eC“(G;K) such that
f(kxgk2) = rUjI/XglrUj) for g e G, kx, k2 e K. Then, fp e
Lemma. Let <p s C*(°M; V) be such that <p(k1mk2) = т(кх)<р(т)т(к2)
for me°M,kx,k2^KC\ °M. Iff e C”(G; r), then
f (E(P,<p,u)(g),f(g))dg = f (<p(m),ff(m)) dm.
JG	J0M
Here, we have normalized the invariant measures so that dg =
a~2pp dn da dm.
13.1. The Eisenstein Integral
227
This is a direct calculation. Indeed,
/ (E(P,(p,v)(g),f(g))dg
JG
= f f a(kg)‘'+PF(r(k)~I<p(kg),f(g))dkdg
= [ [ a(kgy+PF{<p(kg),r(k)f(g))dkdg
J G К
= f [ a(kg)v+PF{<p(kg), f(kg)) dkdg
J G К
= f a(g)v+PF{<p(g),f(g))dg
JG
= (	а1'~Рр{<р(т)т(к), f(nmak)} dndadmdk
JNXAX°MXK
= j a" Pp{<p(m), f(nma)) dndadm
JNxAxaM
= L {<P(m)’fv(m))dm.
J0M
13.1.8. We now use the preceding result to derive a variant of induction
in stages for Eisenstein integrals. Let (P, A) be a p-pair and let (P^AJ
be a p-pair with PX<^P, A cAt. Let P = ° MAN, PY = °M1A1N1 (as
usual). Set *P = P{ П °M = °M*A*N. Then *PAN = PY. We normalize
dnx =d*ndn (see 4.A.2.1). Let (т, V) be a finite dimensional, unitary,
double representation of K. Let <p e V) be such that (p(klmk2)
= т(к1)<р(т)т(к2) for k{,k2 e К П °MX and m e 0М^.
Lemma. Let v e (aj)c- *v = v}»a, v* = v]a. Then,
E(P,E(*P,<p,	=E(Pl,<p,v).
Let f e C“(G; r). Then the previous lemma implies that
[ {E(*P,E(*P,<p, ^),^)(g),f(g))dg
JG
= Jo
M1
228
13. The Harish-Chandra Plancherel Theorem
We note that
(/£)?(т) =	f	*av-p,fF{f^(*n*am) dnd*a
J*AX*N
=	[	*a*p-p*pf av*~ppf(na*n*am) dndad*nd*a
J*AX*N	JAXN
=	f	*a*v-P*pf av*~Ppf(n*na*am) dndad*nd*a.
J*AX*N	JAXN
Since dAd*A = dAi and d*ndn = dnx (on Nx), *pPt = p*P, pP)IA = pP,
we have:
(1)	(/Д)*f(w) = f ap-pF>fp‘(am) da
This implies that
[ (E(P,ECP,<p*V)^W>,f(^}dg
JG
J°Mi '	'	J0Ml
= f {E(P1,<p,v)(g),f(g)')dg.
JG
The lemma now follows.
13.2.	The leading terms of Eisenstein integrals
13.2.1.	We retain the notation of the previous section. In this section, we
will translate the results of Chapter 12 on leading terms into correspond-
ing theorems for Eisenstein integrals. We will then use the results to prove
a variant of Harish-Chandra’s Maass-Selberg relations and his functional
equation for Eisenstein integrals. Let т be a unitary double representation
of K. Let (P, A) be a standard p-pair. Let (a, Ha) be an irreducible,
square integrable representation of °M. Then the space r) depends
only on the equivalence class ы of a. Thus, we can write r) for
£/(a, t). As usual, we write ^(G) for the set of equivalence classes of
13.2. The Leading Terms of Eisenstein Integrals
229
irreducible square integrable representations of G. We set
Ш G )
If M is understood then we will write -Z(°Af, r) = -Z(r).
Lemma, dim _Z(°A/, r) < oo.
This follows from Lemma 13.1.4 and Corollary 7.7.3.
13.2.2.	If К is a finite dimensional vector space over C, then we denote
by snfw(G\ K) the space of all f e C°°(G; V) such that if Л s V* then
A ° f e jrfw(G) (12.3.4). We set jtfw(G, r) = {/ e <(G; V)\f(klgk2) =
T(^i)/(g)r(^2), kx,k2 K, g e G}.
Lemma. If <p e _Z(°A/, r), then E(P, <p, iv) e ^W(G, r) for all v e a*.
This is a direct consequence of Lemma 13.1.2 and Lemma 5.2.8.
13.2.3.	If fe <srfw(G; V) and if (Q,Aq) is a standard p-pair, then we
define fQ e <n/w(MQ;V) as follows. If A e V*, then in 12.3.5 we have
defined (A ° f)Q. If m e MQ, then we set
Then, em s V**. Thus, there is a unique element <p(m) e V such that
A(<p(m)) = em(A). If we write <p out with respect to a basis of V, it is clear
that if we set fQ = <p then fQ s £/w(MQ; V). The following is also clear.
Lemma. Iff e <&W(G, r), then fQ e <q/w(Mq, t).
We will also use the asymptotic properties of fQ (see 12.3.5), which are
also clear.
13.2.4.	The next result is just Lemma 12.4.1 phrased in terms of Eisen-
stein integrals.
Lemma. Assume that v e (а*У (12.5.2). If there exists no к s К such
that kAQk~l c A, then E(P, <p, iv)Q = 0 for all <p e _Z(r).
230
13. The Harish-Chandra Plancherel Theorem
13.2.5.	If f e	then we say that f ~ 0 if </, <p> = 0 for all
<p G _Z(°Af, t).
Theorem. If (Q,Aq) is a standard p-pair such that there exists k & К
such that kAQk~1 is properly contained in A, then E(P, <p, 1?)@10Мо ~ 0 for
all v g a*, <p g _Z(°Af, r). If kAok~l =A, then E(P,<p,iv)0oM g
By Lemma 13.1.2 combined with Theorem 12.4.1, we may assume that
v g (а*У. The result now follows from 12.5.10 (2) and 12.5.8.
13.2.6.	We now consider the case Q g 0(A) in more detail. If s g W(A)
and if ы g °M, then define sa> g °M as follows. Let (a, Ha) ш, and let
к e. К be a representative for j. Set ka(m) = a(k~1mk) for m g °M and
let sa> be the class if ka.
Theorem. Let QetAA). Then there exist for each s e W(A) a mero-
morphic mapping CQ^P(s, v) in v with values in End(_Z(°Af, r)) such that
E(P,(p,iv)Q(ma) = a‘s,,CQ\p(sAv)<p(m)
seW(A)
for m g °M, a g A. CQ\P(s, • ) is holomorphic on i(a*f. Furthermore, if
a> G d’2(°M), then
Cq\P(s, v): л/(ш, t) -» <Qf(sa>, r)
for v g such that CQ\P(s, v) is defined.
Note. The CQ^P(s, ) are usually called the Harish-Chandra C-functions.
Let g g K*. Then Lemma 13.1.2 implies that if ф g £/(ы, t), a g ы g
d’2(°Af), then
d
p(E(P^,iv)(g)) =
i=i
for appropriate . We define E(P, ф, v)Q s for m g °M by
13.2. The Leading Terms of Eisenstein Integrals
231
in the notation of 12.5.5. We note that the formulas in 12.5.5 imply that
E(P, ф, v)q,s e £/(sa>, t).
Since E(P, ф, v)q<s is clearly linear in ф, we define Ce|P(j, v) by
cq\p(s’ *)Ф = E(p> Ф* ^)g.s-
All of the assertions now follow from the properties of intertwining
operators and 12.5.5 (4).
13.2.7.	The following result is essentially Harish-Chandra’s Maass-
Selberg relations. Let (т, K) be a unitary double representation of K.
Theorem. If P, P', Q, Q' g &(А\ s,t g W(A), if <p g _Z(r), and if
v g (a*X, then

The proof will take some preparation. In the course of the proof we will
have given fairly explicit formulas for the C-functions in terms of inter-
twining operators. We note that, in light of the last assertion of the
previous theorem, it is enough to prove the result under the assumption
that <p g t) with ш g d’2(°Af). Let yA denote the common value of
yp for P g &(.A) (Lemma 12.5.6). Fix (<r, Ha) g ш. Let g(<u, v) denote
the common value of рьр^ш, v) for P g £P(A). Then, we will actually
prove:
13.2.8.	Theorem. IfP, Q g &>(A\ s g W(A), <p g r), v g (a*)',
then
||C0l/>G.'l'M|2 = Уа 2p(to,iv) 1H<pll2-
Clearly, this theorem implies the previous one. In the proof we will use
the notation of Section 13.1. Let F be associated with V as in that section.
Let T g Hom^jEndfH^fF^)), K). If v g V, then we define f(v) g
Endf/gfF)) implicitly by (see 13.1.5 for i)
tr(f(r)X)=<i(T)(X),l;>
for X g End(/O.(F)). Then, f g HomK(E, End(/O.(E))). Proposition 13.1.5
232
13. The Harish-Chandra Plancherel Theorem
combined with Lemma 13.1.3 implies that:
(1)	(E(PttpT,v)(g),v) = E(P,^F,v)(g)(f(v))
for all g e G and v e V. To indicate its dependence on a we write a.
Now, 12.5.5 (4) is exactly the same statement as:
(2)	E(P,WF,iv)Q'S(m)(X) = fjL^pkdkaAvyy^^p^mXXCsAv)),
with
X(s,iv) = jQlQ(ka,iv)JQlkPk-i(ka,siv)
xL(k)XL(k) 1JkPk-^Q(ka,isv).
If we apply 12.5.5 (1) and the results of 10.5.8, we have:
(3)	tr( X(s, iv) X(s, iv)*) = g(<u, iv) 1 tiQ^kPk-i(ka,iv) 2trXX*.
We therefore see that if vl,...,vd is and orthonormal basis of V, then
E /^£(^>^,^)е.Л™)(Йц))Г dm
= уа2[1(ы,1р)~^(ш)~1 22 tr(f(г,)/(!;,.)*)
i
= Уа 2g(<w,iv)~1||<p7-l|2.
The last equality is a consequence of Lemma 13.1.6 (1), (2). This completes
the proof of the theorem.
13.2.9.	Following Harish-Chandra, we define
Cq\p(s’ v) = Sv) Cq\p(s>v)-
Then, °Ce|/>($, v) is meromorphic in v. The next result is what Harish-
Chandra called the functional equation for the Eisenstein integral and the
functional equation of the C-function.
Theorem. Let P,Q e &(A) and let s e W(A). If <p s -Z(r), then
e[Q,°Cq\p(s,v)<p,sv) = E(P,q>,v)
for all v e etc for which both sides of the equation are defined.
// P,Q,Q' e &(A) and if s,t s W(A), then we have the equality of
13.2. The Leading Terms of Eisenstein Integrals
233
meromorphic functions
Cq'\q(J>Sv) Cq\p(s>v) = °Cq,\p(Js>v) 
In light of the meromorphy of both sides of the equation in v it is
enough to prove the first formula for v e z(a*y. If we use the material in
the proof of Theorem 13.2.7, it is enough to prove that if F с К is a finite
set, then
E(Q,°CQ^s,v)^F<a,sivyg)(X) =£(P,%,iv)(g)(X)
for X s EndUgXF)). To this end, we use 13.2.8 (2) and Lemma 13.1.2,
which imply that (in the notation of 13.2.7) the left hand side is equal to
УАУА1^о\кРк '(к^^^)1г^о(ка,18р)~1Х(5,^)ттО ка isv(g)y
Now,
}&0(ка,1зр) lX(s,iv)
= JQ\kPk-fk(T^iV)L{k)XL{kYijkPk-'\Q(k(T^SVY
Since
P-Q\kPkYk(T’tSv)JQ\kPk~'(ka’’ S^V) = JkPk~'\Q(ka’’iSV) ’
we have
E[Q,0CQ]P(s,v^F<a,siv)(g)(X)
= tr(-4™-iie(£a,i^)_1L(£)AL(£)-1
^kPk- ^ka,isv)TTQ
,k(r,siv(g )j
= tr^Jkpk-^Q(ka, isv) 1L(k)XL(k)~1
^'7TkPk-l,ka,ikp(.8)JkPk-llQ(.ka’
= tr(L(k)XL(kYlTTkPk-l<k(r<ikv(g))
= tr(L(k)X^iv(g)L(kyl)
= tr(XirPtativ(g)) = E(P,VF>")(g)(X)-
This completes the proof of the first formula.
234
13. The Harish-Chandra Plancherel Theorem
If we take the leading terms of both sides of the first formula at iv,
v e (а*У, in the direction Q and equate coefficients of altsv, then we
have
(*)	Cq^pIJs, iv) = Cq^q(1, sv) Cq\P(s,v).
Clearly, (*) implies the second formula.
13.2.10.	Let Px, P2 e £P(A) and let (Q,Aq) be a p-pair such that
Aq cA and PX,P2 c Q. Set *P, = °MQ П for i = 1,2, and *A =
°MQ nA. Set *IL= {j e W(A)\sa = a, a ^AQ}.
Lemma. If v e Ka^)' and ifse*W, then
Cp2\p[S’V) = 0^*pt\*p2(s\*A
If j e *W. then e W(*A). We will identify j with this restriction.
We set *v = jya for v e a£. We also write v* for p|Oo. The previous
result implies that
Е(*Л , <p, i *V) = e(*P2 ,°C^pt(s, i*v)<p,is*v).
Thus, Lemma 13.1.8 implies that
E(Px,<p,iv) = E(Q, E(*Px,<p,i*v),iv*)
= E(Q, E[*P2,°C.p2^pt(s, i*v)q>, is *y), iv*)
= e(p2> C»p2i»p£s,i*v)<p,iv).
Proposition 13.1.5 now implies the result.
13.2.11.	Theorem. °Ce|/»(s, v) is meromorphic in v and holomorphic for
v^ia*. Furthermore, 0CQ\P(s,iv) is a unitary operator on _Z(r) for
v e a*.
The meromorphy of °Ce|/>($, v) is clear from the definition. If v e (a*)',
then °Ce|/>($, iv) is a unitary operator by Theorem 13.2.8. The second
formula combined with 13.2.8 (2) implies that it is enough to prove that
°Ce|/»(s, • ) is holomorphic on ia* in the case 5 = 1 and P and Q are
adjacent. But, in this case, the previous lemma implies that °Ce|/»(l,r')
depends only on (v, a), {a} = 1<(P\Q). Thus, since °Ce|P(l, v) is meromor-
13.3. Wave Packets of Eisenstein Integrals
235
phic in v and unitary at iv, v e a*, no iv e i(a*) can be a singularity.
The result now follows.
13.3.	Wave packets of Eisenstein integrals
13.3.1.	We retain the notation of the previous section. Let (т, K) be a
unitary double representation of K. Let (P, A) be a standard p-pair. If W
is a finite dimensional vector space over C, then we define ^Z(a*; IK) to
be the space of all smooth functions f on a* such that A°/e ^Z(a*)
(the usual Schwartz space) for all A e W*. We endow ^Z(a*; IV) with the
topology induced by the semi-norms qk(f) = q(X ° f) for A e IV* and q a
continuous semi-norm on ^Z(a*). Then C“(a*; _Z(r)) is a dense subspace
of ^Z(a*; _Z(t)). We set tf(G, r) equal to the subspace of C°°(G, r)
consisting of those f such that A ° f e -tf(G), for A s V*, endowed with
the topology given by the semi-norms j3A(/) = /3( A ° f) for A e V* with (3
a continuous semi-norm on -^(G).
If a e ./(a*; -Z(r)), then a(v) =	aa(v), with aa(v) e
&/(ы, t). We set
<I>(P,a)(g) =	22 f'E(P,a„(v),iv)(g)fi(a),iv) dv.
The functions Ф(Р, a) are the wave packets alluded to in the title of this
section. The following result is Theorem 26.1 in Harish-Chandra [16].
Theorem. а^»Ф(Р,а) is a continuous map from ^(a*; _Z(r)) to
^(G, t).
We may assume a = аш. Fix (a, Ha) e ы. Let F с К be associated
with т as usual. Then, the material in 13.2.8 implies that there exists a
smooth function T on a* with values in HomK(K, End(/a(F))) such that
if Tj is a basis of End(/a(F)) and is an orthonormal basis of V, then
П«')(",) =
i
with a,7 e ^Z(a*) and
{E(P,a(v),iv)(g),v} = tr(T(v)(v)TTP^y(g))-
236
13. The Harish-Chandra Plancherel Theorem
Thus,
= E f a^tr^v^TTp^^g)) ^,iv) dv.
i W
The result is now a direct consequence of Theorem 12.7.1.
13.3.2.	Our next task is to calculate the Harish-Chandra transforms of
the wave packets. Here, as usual, we extend the notion to the vector
valued case by composing with linear functionals. If (P, A) is a standard
p-pair then we set, for v e. a*, f e -^(G, r),
fp(m) = f fp(ma)a~iv da
Ja*
for m e °M. We note that since fp e r), fp e т^(°Л/, r). Let cA be
defined by
cj1 f f u(a)a~,v dadv = u(l)
for и e С;(Л).
The following theorem is Corollary 26.1 in Harish-Chandra [16]. As we
will soon see, it is essentially the Plancherel theorem.
Theorem. Let a e У(а*; Then, if P,Q e ^(A),
¥(P,a)£ = cAyA 22 0Се|р(5,^)а(5-^).
We first reduce the proof to a special case. Assume that we have proved
it for a of compact support contained in (a*)'. We set TQ p(a) = У(Р, a)£
for a e ^A(a*; -гТ(т)). Then, TQ v(a) is continuous in v and a with values
in _Z(t). We choose a 0-stable Cartan subalgebra fi of m containing a. If
ш <f2(°M), then we write Рш for the orthogonal projection of _Z(r) onto
.o/(w,-r). Set ТШ1 Ш2 p = PaTQ'„РШ2. Let t)m = I) Ci °m and let Л,- denote
the Harish-Chandra parameter of a>i with respect to (t)m)c. Then, we set,
for z e Z(gc), pfv) = xA +,p(z). Then, we have:
(1)	Tat.ai,SP2^ = PiMTUt Ш2 Aa\
Indeed, гУ(Р, Рша) = У(Р,р2а). If we extend ТШ{ Ш2 v(a) to f on G
by setting f(nQmak) = apo+ivTait	then zf = pfv)f. Since
zf is the extension to G, as before, of гУ(Р, РШга)9, the assertion
follows.
13.3. Wave Packets of Eisenstein Integrals
237
(2)	If v g (a*)', then supp ТШ1>Ш2>Р (as a distribution on C“(a*, _Z(t))) is
contained in the set {$Я s g W(A)}.
Indeed, if v0 is in the support of T ш , then we must have P2(^o^ =
Pj(v). Since z g Z(gc) is arbitrary, this implies that there exists j g
WXgc, fic) such that s(Aj + iv) = (Л2 + iv0). Thus, Lemma 12.2.1 and the
related arguments in 12.2 imply that v0 g W(A)v.
We now prove our contention. If a g C“(a*,_Z(r)) and if v g (a*)',
then choose /3 g C“((a*)') identically equal to 1 in a neighborhood of
W(A)v. The assumption that the result is true if supp a c (a*)' combined
with (2) implies that
тШ1.Шг.Ла) = тШ1,Ш2,АРа)
= саУа E
seIV(A)
= СаУа E /’Ш1°Се|р(5,5“1^)РШ2а(5“1^).
seIV(A)
Thus, by continuity in a, we find that:
(3)	Ф(Р,а)° = cAyA 22 *CQ\P(s,siv)a{s-xv)
setfU)
for all a e ^A(a*; -Z(r)), v e (a*)'.
If a is fixed, then both sides of (3) are continuous in v. This completes
the proof of the reduction to C“((a*)', -Z(r)).
Now, assume that a g C“((a*)', -Z(r)), a = аш. We may thus assume
that there is
T G Cc“((a*)',HomK(r,End(UF)))
such that
<a(v)(m),t>) = VF(m)(J(v)(v)).
If <p g _Z(t), v g a%, then define <pOiP(g) by
p(nmak) = ар+ре<р(т)т(к)
238
13. The Harish-Chandra Plancherel Theorem
for n e Nq, m e °M, a e Л, к e K. Theorem 12.8.4 implies that
Ф(Р, a)Q(ma)
= E f f (C&p(s,iv)a(v))Uisv(man)p.(a>,iv) dvdn.
s^W(A) N0 a*
Fix j e W(A). We now calculate
' Nq'cl*
(тап)р(ы, iv) dv dn
= lim
s—* +0
(Ce|P(i, iv)a(v))Q ^_Ер(тап)(1(ш, iv) dv dn
since the inner integral is absolutely convergent for e > 0. If we now use
the formula in 13.2.8 (2) and the definition of Je|g (isv - ep), we find that,
if v e V,
( [ (C0.P(s,iv)Q,isv-ep(man) dn,v)
Vnq	I
= VA'^,ka(™)(JQ\Q(ka,isv - epQ)S(iv)),
with
S(iv) = JQ\Q(ka, siv)Jkpk~'\Q(k<y, isv) 1
X L(k)T(v)(v)L(k)~lJkPk-l]Q(ka,isv).
Here, we have used 12.5.5 (1). Applying 12.5.5 (1) and 13.2.8 (2) again, we
have
+ePab cqA
Vnq
a
s, iv)a(v)Q,isv-ep0(man) dn,v
= iv - es XyA(ce|e(l, siv - epQ) 1
^-Cq\p(S’ ^v)a(v))Q,isv-ep0(m), u'j.
13.4. The Harish-Chandra Plancherel Theorem
239
Thus,
Г?/ / C&p(s,iv)a(v) QtiSV(man)fi(a), iv) dv dn
JNQJa*
=	siv ~ £PqY'
e-* +0 •'a*
g(<u, iv)
X CQ\P(s,iv)a{v)(m)—-------::-------- dv
p[a>,iv — lespQ)
= ( a'st’C&Q(l,siv)~1C&P(s,iv)a(v)(m) dv,
•'a*
since a has compact support contained in (a*)'.
We have therefore shown that
Ф(Р,а)в(та) = yA £ f °CqIP(s, iv^v^m)^'’dv
s^W(A) a*
= Уа E I °C&p(s,is~1v)a(s~lv)(m)aivdv.
s^W(A) a*
Hence
Ф(Р,а)?(/и)
= yA 22 f [ °CQ\p(s,is~lX)a(s~lX)(m)a‘x dX (Г"'da
seH'XA)'4* a*
= саУа E ^CQ\p{s,is~xv)a{s-xv){m).
s<eW(A)
This completes the proof of the theorem.
13.4.	The Harish-Chandra Plancherel theorem
13.4.1.	In this section, we will show how the results of the preceding
section can be used to prove the Harish-Chandra Plancherel theorem.
Although the proof of the result will take the rest of this section, we first
240
13. The Harish-Chandra Plancherel Theorem
give its statement. Fix (Po, Ao), a minimal standard p-pair. We fix dg as
in 10.1.7. If (P, A) is a standard p-pair, then we write (P, A) > (PQ, Ao) if
P э Po and A c Ao. If (P, A) > (Po, Ao), then we fix dn = dN, da = dA,
and dm on °M as in 10.1.7. Then (Lemma 10.1.7),
f f(g) dg = f	a~ 2pf( namk) dn da dm dk.
JG	JNXAXUMXK
Set CA = (\W(A)\yAcA)~l. Let 0Лш>/р be as in 12.1.1.
Theorem. Let f g t^(G) be left and right К-finite. Then
f(g) E CA L d(a>)
(P, A) >(P0, Ao)
X f ®P,,o,iu(R(g)f)P(M’iv) dv-
Ja*
The proof takes some preparation. We start with some preliminary
material on cusp forms on G, which constitute a part of Harish-Chandra’s
“philosophy of cusp forms” (we will see the “philosophy” again in Chap-
ter 15).
13.4.2.	We assume that G = °G. Let (т, K) be a unitary double represen-
tation of K. If f g tf(G, t), then we say that f ~ 0 if
</,<P> = f (f(g),<₽(g)) dg = 0
JG
for all <p g _Z(G, t).
Theorem. If f g if(G, t) and if R(a)fpMp ~ 0 for all standard p-pairs
(P,A) > (PQ, Aq) (including G) and all a g A, then f = 0.
We prove this result by induction on dim G. If dim G = 0 or 1, then the
only p-pair is (G, 1) and if(G, r) = -Z(G, r). So the result is obvious in
this case. Assume the result for all G of dimension less than n. We now
do the inductive step.
(1) fp = 0 for all proper p-pairs (P, A) > (Po, Ao\
13.4. The Harish-Chandra Plancherel Theorem
241
Indeed, let (Q,Aq) be a standard p-pair in MP with (Q, AQ)>
(Po П MP, Ao). Then (QNP, Aq) is a standard p-pair in G, (QNP, Aq) >
(Po, Ao), and °MQNp = °MQ. Set Q = QNP. We note that (fp)Q = fQ'. By
assumption, Ь(а)/& ~ 0 for all a eAq. Thus, L(a)(fp)Q ~ 0 for all
a ^Aq. Since P is proper, dim°MP < dimG. The inductive hypothesis
now implies (1).
(2)	If fp = 0 for all proper standard p-pairs (P, A) > (P0,A0), then
f £ -Z(G, t).
Indeed, it is enough to show that (L(x)R(y)f)p = 0 for all proper
parabolic subgroups of G and all x, у e G by the definition of cusp form
(7.2.2) and Theorems 7.2.2, 7.7.6, and 8.7.1. We note that (R(y)f)p(m) =
L(y~l)fp'(y~lmy), with P' = y~lPy. Thus, it is enough to show that
(L(x)f)p = 0 for all proper parabolic subgroups of G. If P = ° MAN is a
standard Langlands decomposition of the proper parabolic subgroup P
and if x = nj/njajAZ], nx e N, mx e °M, ax e A, e K, then a direct
calculation using the definition of fp (7.2.1) yields
(L(x)/)P(w) = a?₽T(Az)“7/’(("’1a1)_1>n).
Thus, it is enough to show that fp = 0 for all proper P. Now, let (Q, AQ)
be a standard p-pair. Then there exists к е К such that (kQk~l, kAQk~x)
> (PQ, Ao). Since
fQ(m) = (L(k)R(k)f)P(k~1mk) = т(к~1)/р(к~1тк)т(к),
the assertion now follows.
We can now complete the inductive step. In light of (1) and (2),
f e -Z(G, t), so f ~ 0 implies that </, f) = 0. Hence, f = 0.
13.4.3.	Let A be a special vector subgroup of G (see 10.1.8). We set
^(G, r) equal to the space of all f e if(G, r) such that fp ~ 0 for all
standard p-pairs (P, AP) such that Ap is not К-conjugate to A.
Theorem. Let P e £?(A) and a e ^(a*, -if(0MP, r)). Then Ф(Р, a) e
<(G, t).
242
13. The Harish-Chandra Plancherel Theorem
This follows from Lemma 13.2.4, Theorem 13.2.5, Theorem 12.8.4 and
Corollary 12.8.8.
13.4.4.	Let F с К be a finite set and let (P, A) be a standard p-pair. If
ы e d’2(°Af) and (<т, Ha) e ш, then we set ^(a*,End(Ia(F))) equal to
the space of all T e C”(a*, End(/a(F))) such that (v ~ tr(T(v)S)) e
^(a*) for S e End(/a(F)). If T e ^(a*,End(/a(F))) and if 7} is a basis
of EndC/oX-F)), then T = with u( e ^Z(a*). We note that
ЕФ(Р,и,%.>a)(g)(Tj) = f tr(T(p)TTP ir iv(g)) ii(a>,iv) dv
i	Ja*
= f(P,T)(g).
We use this formula to define f(P, T). The results of the previous section
imply that f(P, T) e if(G) and if Q e ^(Л), then:
а) (ллt))₽ = yAcAL^W(A)°as,s-^v^Gxns-^)).
We now fix for each j e W(A) an element ks e. К such that
Ad(^j)|a = j.
Lemma. Iff e C“(G)f and if T(v) = irp^tf), then
(f(P,T))? = yAcA E W)(W/))-
13.2.8(2) implies that
°С(5,5-Чр)^><7()(Т(5-1р)) = %<ksA)(S(v)),
with (k = ks)
s(v) =JkPk-'\Q(k^Av)~1L(k)TTPaJs-tv(f)L(li)~lJkPk-'\Q(k(T,iv)
~ JkPk-,\Q(ka’ tv) 7rkPk-',ka,lv( f)JkPk-'\Q(k<T’ tv)
1Г(2, ka,iv( f) ’
The result now follows.
13.4.5.	We now introduce a specific unitary double representation of К
that was extensively used by Harish-Chandra. Let F be a finite subset of
13.4. The Harish-Chandra Plancherel Theorem
243
K. Set VF equal to the space of all smooth functions и on К x К such that
if у g F, then
E ^(У)^(У')/ xy(k1)x7'(k2)u(xkil ,k2y)dkldk2 = u(x,y)
/**
for all x, у g K. We set
(u,v) = / u(x, y)v(x, y) dxdy
JK*K
for u,v g And т f(x)ut F(y\k x ,k2) = u(kxx, yk2). If /gC“(G)f,
then we set Д1Ш(^)(х, у) = /(Р,тта(х)ттр a ^/)тта(у)\ Observe that, as
indicated by the notation, the preceding expression depends on only a>.
We also set f(g)(k1 ,k2) = f(k1gk2). For the rest of this section, F will
be fixed and r = rF.
Proposition. Set 1У(ш) = {$ g Ифю = w}. Ifs g W(A), <p G tf),
and if Q g ^(Л), then
We may assume that
<p(m)(x, у) = ^ ^(•)(л-<7(у)Т’л-а(х)),
with T e End(/CT(F)). Then, in the notation of 13.1.3,
(J JKXK
X L fS(m)(x, у){ка(т)^(1), dmdxdy
J0M
= Ef Жл>
j j ]KXK
x L /£(т)(х,у){ка(т)/^у)^х-1)} dmdxdy
M
= Lf
i i'KxK
ft?(m)(x 1,y/ka(m)fj(y),fi(x)} dmdxdy.
JaM
244
13. The Harish-Chandra Plancherel Theorem
Now,
= f f(s)-rrP a il,(g)u(x)dg
= /	<lg
JG
= f	a~2pf(nam)(x~1, k)ap+,va(m)u(k) dndadmdk
JNxAx°MxK
= f (f)Q(m)(x~l, k)<r(m)u(k) dmdk.
J°MXK
Hence,
</£,?> = ЕО/ К^.-Д7)Л(х),/((х))Л;
i,j	K
ij
i
We now compute
(УаСаГ1 ((Л.ш),?. <p)
= (УасаУ1 f o (fA,a>)^(m)(ki’k2)<p(rn)(ki^2)dkldk2dm
= E f „ Ъ,к1ЛГп)^к1ЛУ)^О,к1а,1Л/)^к,а(х))
,eW(A)K*K*°M
td> =зш
Х<р(пг)(х, у) dxdydm,
by Lemma 13.4.4. We note that if ta> = sa>, then
^F, k.a^k.ai У ) ^Q, k.a, iV( f ) ^k,a( *) )
= %-, k^k'A У ) ^Q, ksa,iX f ) 1?ksA X ) ) •
13.4. The Harish-Chandra Plancherel Theorem
245
Hence, the expression that we are calculating is (k = ks)
,1/1Р1/кхкхом
x( 77^( у )T^a( x)fp, fq) {ка(тЩ1), /,(1)>
x{ka(m)fq(l),fp(l)} dxdydm
= l^(")l E (	„
iJ.P.q KXKX M
x{ka(m)fq(y),fp(x~1)') dxdydm
= d(a>) X\W(a))\ E j
i,j,p,qJKxK
*<Tfp,fq>{fj( y),fq( y)}{fp(x),fq(x)) dxdy
= J(n>)->(n>)| E
= d(a)) l\W(a))\ E ^p,k.a, Л)<ЯР-Л>
iJ,P,q
= d(a>)->(") I E< irPM( f)fi > ТМ
i
= 4<И)->(Ш) | и(Т*77л^,p(/))
= J(n>)->(W)| tr(77p>^>ip(/)T*).
This completes the proof.
Let f e . Then we set
fA = cA E <!(<») fA,„-
We note that if 5 e И"(Л), then fA sa) =fAta).
246
13. The Harish-Chandra Plancherel Theorem
Theorem. f= £а<2а0/а- Here, the sum extends over all special vector
subgroups A of A 0.
Set g = Елсл0/л- In light of Theorem 12.8.4 and Corollary 12.8.8, if
(Л AP) > (Po, Ao) then g? ~ (fAp)f„. The preceding result now implies
that if (P, A) > (Po, Ao), then
with Рш the orthogonal projection of r) onto the closure of
t). Hence, gfv ~ f? for all (P, A) > (Po, Ao). Theorem 13.4.2 now
implies that g = f.
13.4.6. We now show that Theorem 13.4.5 implies Theorem 13.4.1.
Indeed, f(x) = /(xXL 1). We calculate (in the preceding notation)
g(x)(l,l) = E	E d(a>)fAai(x)(l,l).
А с?4о	«eZ/MJ
Now, if P e &(A), <r e then
Z4,O>(^)(1,1) = f tr(77p a jp(/)77p a ,.p(x)) p(a),iv) dv
Ja*
= f ^TTp aAv(R(x)f))p(a>,iv) dv
Ja*
= J ^p^,iv(R(x)f)li(oi,iv) dv.
Thus, Theorem 13.4.5 does indeed imply Theorem 13.4.1.
13.4.7. Fix A, a special vector subgroup of G. Let -^(G)FA be the space
of all f e ^(G)F such that fQ ~ 0 for all p-pairs (Q, Aq) such that AQ is
not conjugate to A. Let -£{G)A be the closure of Ufc/< ^(G)F A in
^(G).
Theorem.
(1)	^(G) = ФЛсЛо J(G)a.
(2)	^(G)a is a two-sided ideal in ^(G).
13.5. The Calculation of p.(<o,v) for the Fundamental Series
247
Let /e -£(G)A П with A and A' special vector subgroups of
G such that A =#= A and A, A c A. If F с К is a finite subset and if aF =
^7 e f d^y^Xy j then set ctF * f * a F = g = Ff g ^(G)^ a A ^(G)^ A>.
Now, gQ ~ 0 for all Q such that AQ is not conjugate to A, and for all Q
such that Aq is not conjugate to A. Thus, gQ ~ 0 for all Q. Theorem
13.4.2 now implies that g - 0. If Fj, j = 1,2,..., are finite subsets of К
such that
Г,сГ2с---, ил = к>
j
then
lim Ff = f-
j~>x 1
Thus, f = 0.
If /e if(G), then Ff e -£(G)F, so Theorem 13.4.5 combined with
preceding limit argument implies (1).
We will now prove (2). We note that f e -^(G)F A if and only if
for all Q such that Aq is not conjugate to A, and all g e G, ш e d’2(°Me),
v e a*. If <p e tT(G)f, then
= f<p(g)®Q,a>.iXL(g)f)dg
J G
and
0е.ш,и/*ф) = f <p(g-l^Q^iv(R(g)f)dg.
J G
These two equations clearly imply (2).
13.5.	The calculation of |x(w,v) for the fundamental series
13.5.1.	In this section, we will show how one can use the Plancherel
formula and the results of Chapters 7 and 8 to give a more explicit formula
for v) for a particular class of parabolic subgroups. We first give the
statement of the formula and then we will use the rest of the section to
prove it. First, we need some notation. Fix (Po, Ao) a minimal standard
p-pair of G. Let (P, A) = ((P0)F,(A0)F) (2.2.7). Then (P, A) is said to be
248
13. The Harish-Chandra Plancherel Theorem
fundamental if HF (2.3.5) is a fundamental Cartan subgroup. Fix (P, A) a
fundamental p-pair. The representations TrPaiv, <r е ш e <f2(°M), v e
a*, are called the fundamental series. Set H = HF and I) A °m = t. Then,
t is a Cartan subalgebra of tn °m and t) = t®a. If ы e then
Лш will denote a Harish-Chandra parameter for ш (Лш is determined up to
the action of tc)). Fix Ф+, a system of positive roots for Ф(дс, t)c).
Theorem. There exists a positive constant, CG, such that
p.(a>,iv) = CGd(a>)l\ П (a,Aw + z>)|.
13.5.2.	Let ы e d’2(°M) and choose (a, Ha) e ш. Let у e К be such
that Ia(y) #= 0. If a e C“(a*), consider Ta(v) = a(v)Ey. Let f(P,Ta) be
as in 13.4.4. If (Q,Aq) is a cuspidal parabolic subgroup of G, then
Corollary 12.8.8 implies that f(P, Ta)Q = 0 if (Q, AQ) is not fundamental.
Hence, 7.4.10 (2) implies that (in the notation of that number)F^PT > = 0
for H a Cartan subgroup of G that is not fundamental. We therefore see
that formula 7.6.6 (1) implies (as in 7.6.7 and in the notation therein)
/(/>,T„)(l) = cfj(h)F(h)^-^—F»P,Га)(ехрЛ) dh.
If we now argue in exactly the same way as we did in 7.6.8,7.6.9, then we
have (in light of Theorem 7.5.2):
(1)	MG^PiTa)(l) = /(P,T„)(1),
with MG a constant depending only on G and ет = ПаеФ+ На.
Let T be the Cartan subgroup of °M corresponding to t. Then H = TA.
If Л e !)£, then set ет(Л) = ПаеФ+ A(H„). If we apply the Peter-Weyl
theorem for T combined with the Fourier integral theorem and 7.4.10 (2),
we have that
mFfU>,Ta)(A) = L f ®(AT + iv)(FliPTxY(T)d(v).
'Г *U*
reT
Here, Лт/ is the differential of r (/ = the identity operator). Note that
the terms in the preceding sum corresponding to non-regular r (8.7.2) are
0. For regular r, it is an exercise (using the material in Section 6.9) to see
that </(t) = N independent of r. 8.7.3 (1) implies that if r is regular and if
<aT e ^(°Af) is the corresponding square integrable representation of °M
13.6. The Intertwining Algebra of	249
(8.7.1), then
with Ct depending only on the choices of invariant measures. If we apply
Lemma 13.4.4, we find that ,//(/’, Ta)) = 0 if ыт =#= su for some
j e W(A). Since ®PtSlOtiv = ®Ptl0,iV for j e W(A), we conclude that
mF^Jl) = CGf СТ(ЛШ +iv)0P.aMP,Ta))dV,
Ja*
with ш = шт. The Plancherel formula implies that
®P,u,iv(f(P’Ta)) = ^(<w)dim(/a(y))C2 E Ol(sv).
s^W(A)
We now assume that supp a A W(A)v = {p}. Applying this discussion to
(1), we have:
(2)	f(P,T)(l) = C3dim(/O.(y))d(<u) f a(v)^(Aa + iv) dv.
Ja*
On the other hand, the definition of f(P,T) in 13.4.4 implies that:
(3)	f(P,T)(l) = dim(/o.(y))f a(v)/i(o,iv) dv.
Ja*
Since a is arbitrary beyond the support condition, (2) and (3) imply
Theorem 13.5.1.
13.6.	The intertwining algebra of IP,„,iv and the irreducibility
of the fundamental series
13.6.1.	In this section, we will prove a version of Harish-Chandra’s
completeness theorem, give an estimate on the intertwining number of a
principal series representation, and apply this estimate to the fundamental
series. Fix (P, A), a standard p-pair. Let ш e d’2(0M), and fix (<т, Ha) e ы.
If г- e a*, then we set JV(u, v) = {j e W(A)lsa> = a>, sv = v}. Let Кш =
{-у e k\Ia(y) =# 0}. Let, for each j e W(A), ks e К be chosen such that
Ad(^)la = s.
Theorem. Fix v e a*. Then, for each s e W(u, v), there exists a unique
Aj 6 Иотй K(IP a iv, IP a iv),
250
13. The Harish-Chandra Plancherel Theorem
up to a scalar of norm one, such that:
(i)	As extends to a unitary operator on Ha.
(ii)	If F e Ka, then
°cPlP(s,iv)^F,A)(T) =
for T e End(/a(F)).
Furthermore,
^omg K(IPtaJv,IP<(rtiv) = 8рапс{Л> e №(ш, у)}.
If we apply 13.2.8 (2), we find that if s e W(A), sa> = a>, and if
Л e (a*)', then
°CPlP(s,iX)^C)(T) =
with r]SiPW an algebra homomorphism of Endf/gfF)). Theorem 13.2.11
implies that i7J>F(A) is defined for all A e a* and is real analytic in A. Set
Vs,f = Vs,f^- Theorem 13.2.11 also implies that r)s F is an automorphism
of End(/a(E)) that is unitary with respect to the inner product (X,Y) =
trXY*. Thus, for each F c Ka there exists a unique, up to scalar
multiple, unitary operator Bs F on Ia(F) such that r]sF(X) = Bs FXB~F.
13.2.8 (2) implies that:
(1)	If /e if(G)F, then
F^FFF^P.aJvif) FfBs F = EFTTP a iv( f )EF .
We also note:
(2)	If FcT cK„, then
Bs,F'\I<AF) = CF',F,sBs,F >
with cF>Fs e C and |cr f>J| = 1.
Let F c Ka be such that
irPiaJACAG)K)IAF)=Ia.
If F' э F, then set As F, = Cp\F SBS F. Then, (2) implies that if F c F' c
F", then AsiF»|z (f') =As F,. We can thus define As by setting AS^(F^ =
As F,. Then, (2) implies that
Homg IP,jv, Ip,a -lv) •
The uniqueness assertion follows from the uniqueness of Bs F.
13.6. The Intertwining Algebra of	251
The last assertion of the theorem is the hard part. Before we get to the
more serious aspects of the argument, we note that the uniqueness
assertion for the As combined with the second assertion of Theorem
13.2.11 implies:
(3)	If s, t e W(a),v), then ASA, = c(s, t)Asl with c(s, t) e C, |c(s, t)|
= 1.
13.6.2.	The preceding theorem rests on:
Theorem. Let F c Ka be a finite set. Then
{X e End(/a(F))|ЛSX = XAS, 5 e p)}
= {EFirP<a<iv(f)EF\f^C^G)K}.
That the right hand side of the equation is contained in the left hand
side has been proved in the course of the arguments of the previous
number. Since
dim End(/O.(F)) <
we have
ЕР^а^(С)к)ЕР = ЕР^а^(С)к)ЕР.
Let X e {У e М/а(Г))|ЛЛ = YAS, s e W(a>, p)}. Let a e C?(a*) be
such that a(v) = 1, a(tA) = a(A) for Леа* and t e v), and a(sv)
= 0 if j e Wb4) and sv =# v. Set T(A) = a(A)T. Then 13.4.4 (1) implies
that
= yAcA<*(v) £ °C(s,iv)^F,A)(T)
se 1¥(ш, p)
= yAcA E
sfEWtu.v)
= X4C4I v) -)(T).
This combined with Proposition 13.4.5 implies that
Т=С77Ла>,.р(/(Р,П)
for some с e C. The theorem now follows.
252
13. The Harish-Chandra Plancherel Theorem
13.63. Let @F be the span of	e W(a>, iv)}. Then 13.6.1 (3)
implies that £8F is a subalgebra of End(/O.(F)). Furthermore, if we set
fiF(s) = As^F), then 13.6.1 (3) implies that is a projective representa-
tion of W(a>, iv) (with a unitary cocycle). Mashke’s theorem implies that
31 F is a semi-simple algebra. Thus, the double commutant theorem for
semi-simple algebras implies that:
(1) {X g End(la(F))\XEFirPait,(f)EF = EFirP^iv(f)EFX,
f^(G)K}
= spanc{^J|Uf)| j g W(u, zp)}.
The main assertion of Theorem 13.6.1 follows from (1) if we take F such
that Ia(F) generates IP,ativ as a (g, A)-module.
13.6.4	. We will now develop a few results to give a refinement of
Theorem 13.6.1. If a g X(P, A), then let /ia be as in 10.5.7.
Lemma. If v g a* and if fia(a>,iv) = 0, then (v, a) = 0 and sae
W(a>, v).
Let Q g &(А) be such that a is a simple root in &(Q, A). Set
a{o) = {H g a\a(H) = 0}. Let M{a} = (g e G|Ad(g) H = H for H g
a{a)}. Set A{a) = {0 g ®(Q, A)\0(a{a}) * 0}. Put n{a) = ®ЭеЛ{а) Qp. Set
N{a} = exp(n{a}). Q{a} = M{a}N{a} is a parabolic subgroup of G. If A{a} =
exp(a{e)), then A{a} is a split component of M{a}. We set (Pl,Al) =
(Q{a)M{a}). Let PY = °MlAlNl (as usual). Set *Q = Q П °Ml. Then,
*Q = °M*A*N. Let (т, V) be a finite dimensional, unitary, double repre-
sentation of К such that -Z(zu,t) =# 0. Suppose that sa <£ W(A). Set
*p = P|*a- If <f> g _/’(a>, t), then
E(*P, <p, i *v)*P(ma) = a,v+poC*P\*P(l,i *v)<p(m)
for a g *A and m g °M. Thus, Theorem 12.4.1 implies that C»P|»P(w, •)
is holomorphic at i*v. This implies that ца(ы, i*v) =#= 0 (10.5.7). This
contradiction implies that sa g W(A). If (v, a) =#= 0, then *v =# 0 and thus
*v g ((*a)*)'. This implies that i*v is not a pole of C*P|*P(zu, •), which
also contradicts the assumption that ga(<u, iv) = 0. Now,
E(*P,<p,i*v)*P(m) = C*P|*P(1, i *v)<p(m) + C*P[*P(sa, i *v)<p(m).
13.6. The Intertwining Algebra of
253
If sao) =# ы and if ф g _Z($a<w, t), then
( Е(*Р,<p, )*р,ф) — (C*p|*p(l, • )<p, i/r).
Theorem 12.4.1 implies that the left hand side of this equation is holomor-
phic at i*v. This again contradicts the assumption that ga(<u,iv) = 0.
Hence, sa g W(a>, v).
13.6.5	. Lemma. Let v0) = {a g 2(P, А) 1ра(ш, iv0) = 0}. If a g
v) and if s g W(a>, v), then sa g SJw, v0).
Let 2(P, A) = {a = a1(..., ar}. We assume that sa = ±aj. Fix v e. a*
such that (v, a) = 0 and (v, a,) =# 0 for i > 1. Then, ра(ш, iv) = ра(ы, 0)
= ра(ы, iv0) = 0. Also, 0 = р,(ш, iv) = П,г=1 ga(<w, iv). Now (10.5.7),
fi(s<i), isv) = iv) = 0. Thus, there exists an index i such that
Pa(w, isv) = 0. If i =# j, then a, =# +sa. Hence, (v, af) =# 0. Thus, i = j.
This implies that 0 = ра(ы, isv) = р,а(ы,0) = pa(o>,iv0). We conclude
that aj g XJw, v).
13.6.6	. Let v) denote the subgroup of W(a>, v) generated by
{sja g SJw, v)}. The preceding result implies that И^(ы, v) is a normal
subgroup of W(u, v).
Theorem. Ifs g W^(w, v), then As is a multiple of the identity operator. In
particular, dimHoma K(IP a iv, IP a iv) < \W(a),v)/W„(a),v)\ for v g a*
and a g ш.
It is enough to show that ASa is a multiple of the identity for a e
XJw, v). We choose Q g &(A) such that a is a simple root for (Q, A). In
light of the definition of ASa, it is enough to show that Ceie($a, iv) is the
identity map on _Z(<u, r) for all finite dimensional, unitary, double repre-
sentations of K. Let Pt = Q{a} (as in 13.6.4). Lemma 13.2.10 implies that if
Ф g т), then Ceie(ia,z»iA = C*Ql*Q(sa,i*vty = C*e|*e(ia,0)i/z (by
Lemma 13.6.5). We are thus reduced to the case when dim A = 1. We
assume this now, that is, that Q = *Q and v = 0.
Let ф e -/’(ш, т), ф =# 0. Since р(ы, 0) = 0, Ceie(i, га)ф has a pole at
z = 0 of order к 1, for j g WfA). We write the first two terms of the
Taylor expansion of z*Ceie(s, га)ф as ф5 0 + гф5л + • • • . Let H g a be
254
13. The Harish-Chandra Plancherel Theorem
such that a(H) = 1.
zkE(Q, ф, za)Q(m exp(tH))
= («А1.0 + &a,o) + {'Al, 1 + ^„,1 + ^('Al.O “ K,o)}z + °(z2)-
Since к 1, Theorem 12.4.1 implies that i/q 0 +	0 = 0. If к =#= 1, then
we would have фх j + j + и(ф1 0 _	= 0 f°r a" t e R- This would
imply that <Ai,o = •As ,o = contradicting the assumption that the pole is
of order k. Thus, к = 1. We therefore see that
zE(Q,i//, га)в(техр(Ш))
= 2zztiAli0(m) + z(iA1>1(m) + <AJ<tjl(w)) + O(z2).
We conclude that
E(Q^,0)Q(mexp(tH)) = 2Иф10(т) + ф1Л(т) + ф5а1(т).
Since ф10 =# 0, this implies that the map ф -> E(Q, ф,0) is injective. Let
C = °Ceie(sa, 0). Then Theorem 13.2.9 implies that E(Q, ф, 0) =
E(Q, Сф, 0). Thus, E(Q, ф - Сф, 0) = 0. Hence, Сф = ф. This completes
the proof of the theorem.
13.6.7	. We will now apply the preceding theorem to prove the irreducibil-
ity of the fundamental series (13.5.1).
Theorem. Let (P, A) be a fundamental p-pair and let (a, Ha) be an
irreducible square integrable representation of°M. Then is irreducible
for all g a*.
Obviously, this result has content only if P =#= G. We therefore assume
this. Since = lQ,ajv for Q g £?(A), we may (and do) assume that
(v, a) 0 for all a g Ф(Р, A). In light of Theorem 13.6.6, it is
enough to prove that if a g ш then WJjv, iv) = W(a>, iv). Let Ф(г0 =
{a g Ф(Р, Л)| (v,a) = 0}. Let СЦ = {H G a\a(H) = 0, a g Ф(Р, Л)}.
Let M, = (g e G|Ad(g) H = H, H g a J. Set Aj = {0 g Ф(Р, Л)| ^(сц)
=#= 0} = {/3 g Ф(Р, Л)| (/3, v) > 0}. Put Uj = ®0еЛ1П0 and set =
exp(n1). If we set PY = MxNlt then (P1,A1) is a p-pair. Furthermore, if
j g IV(o), iv) and к g К is such that Ad(&)|a = j, then к g Mx n K. Set
*P = °Ml П P. If we set *A = А П °MX, then CP,*A) is a fundamental
p-pair for °Ml. Since S«,(a>, v) = XJco, iv[*a), the preceding observations
reduce the burden of proof to the case when v = 0.
13.6. The Intertwining Algebra of fP„(,	255
Let T be a Cartan subgroup of °M with T с. K. Then TA is a funda-
mental Cartan subgroup of G. Let fi = tc®ac.fiisa Cartan subalgebra
of gc.LetWc bethe Weyl group of (gc,fi) and let = {$ g Wcls& = 6s
and there exists g g G such that Ad(g)|f) = j}. Let Wo = П lT(mc, fi).
(1)	The map j -> s|a defines an isomorphism of Wl/W0 onto W(A).
Suppose that j g iV(A). Let к g К be such that Ad(&)|a = s. Then
kMk~x = M. Thus, there exists кх g M П К such that kxkTk~xk^1 = T.
Set и = kxk. Then, Ad(u)fi = fi and Ad(u)0 = 0Ad(u). Thus, =
Ad(u)|f) g Wi and s1|a = j. This implies that the map is surjective. The
injectivity is clear.
We will identify W(A) with W1/Wo. Let Л be the Harish-Chandra
parameter of ы (see 13.5.1). Let HA g it be such that B(h,HA) = A(h)
for h g fi. Set gA = {X g g| [X, Ял] = 0}. Set W(A) = {s g Wtl sHA =
Ял}. We note that t ® a is a Cartan subalgebra of gA and that Ф((дл)с, fi)
contains no real or imaginary roots. Indeed, if there were a real root then
TA would not be fundamental. Since the imaginary roots of ((дл)с, fi) are
roots of (mc,fi) and Л is regular, there are no imaginary roots. This
implies that t ® a is a maximally split Cartan subalgebra of дл. Let
GA = (g g G| Ad(g) ЯА = Яд}. Then W(A) = {s g H>Ha = Ял} =
{$ g >F((gA)c, fi)| there exists к g Ga П К with Ad(fc)|f) = $}. We note
that W(A) П IFo = {1} since Ф((дл)с, fi) contains no imaginary roots. We
also note:
(2)	The image of W(A) in W(A) is W(Ga,A) (hence generated by
reflections sa, a g Ф(дл,а)). Furthermore, if we identify W(A) with its
image in IV(A), then W(A) э ^(«,0).
If a g 2(дл, a), then
J(o>)M^0) = C п 1(Л,Д)|,
£ еф«т(а))с, b)
with C > 0. There exists
P e Ф((0л)с»Ь)
with /3|а a non-zero multiple of a. Thus, 0 g Ф((тм)с, fi). So
ga(w,0) = 0.
Lemma 13.6.4 now implies that W(A) c Wj.a>,0). Since WTw, 0)c
W(A\ H'J.u, 0) = W(a>, 0). The proof is now complete.
256
13. The Harish-Chandra Plancherel Theorem
13.7.	Groups with one conjugacy class of Cartan subgroup
13.7.1.	In this section, we show how the irreducibility of the fundamental
series can be used to give an explicit classification of irreducible admissible
representations of a real reductive group with one conjugacy class of
Cartan subgroup. We fix G, a real reductive group of inner type such that
all Cartan subgroups of G are conjugate. If we apply the material in
Section 2.3, this assumption means that every Cartan subgroup of G is
both maximally split and fundamental. We fix a minimal (standard) p-pair
(Po, Ло) of G. Our assumption implies that if (P, A) is a cuspidal p-pair
for G, then (P, A) is conjugate to (Po, Ao).
Let (P, A) be a p-pair of G such that P э Po, A c Ao.
Lemma. If P = nMAN is a Langlands decomposition of P, then °M has
exactly one conjugacy class of Cartan subgroup.
If Hl is a maximally split Cartan subgroup of °M, H2 a fundamental
Cartan subgroup of °M, and H1, H2 are not conjugate in °M, then H1 and
H2 have split components of different dimensions. This implies that the
Cartan subgroups HXA and H2A of G are not conjugate. This is contrary
to our assumption.
13.7.2.	Lemma. Let (a, Ha) be an irreducible tempered representation of
°M. Then there exists an irreducible (finite dimensional) representation
(Ij, H£) of°M0 and p e (°m П a0)* such that (a, Ha) is unitarily equiva-
lent to (ттРоП»м<(<^,Н^.
As a (°m, К П °Af)-module, (Ha)Kr, oM is equivalent to a summand of
/pon f°r appropriate £ and p s (°m П a0)*. This follows from the
previous observation that Po П °M is the unique cuspidal parabolic sub-
group of °M (up to conjugacy), by Proposition 5.2.5, Theorem 5.5.4, and
Theorem 7.7.1. Theorem 13.6.7 implies that /pnoMf/ is irreducible.
The lemma now follows.
13.73.	Let (P, a, v) be Langlands data (5.4.1), with (P, A) as before.
Then there exists (£, ЯД an irreducible finite dimensional representation
of °MQ, and p e (°m П a0)* such that (Ha)KnoM is equivalent to
/pon OM.t.ip- Thus, induction in stages (see 10.1.13) implies that IPtCr>v is
equivalent to IPoi(,ill+v.
13.7. Groups with One Conjugacy Class of Cartan Subgroup
257
The main result of this section is:
Theorem. If (£, Hf) is an irreducible, finite dimensional representation of
°M0 and v s (а0)£ & such that Re(i,, a) > 0 for all a s Ф(Р0, then
IP has a unique non-zero irreducible quotient J(„. Furthermore, if %' is
an irreducible finite dimensional representation of°MQ and if v'e (а0)£ is
such that ReO', a) > 0 for all a s Ф(Р0, Ao), then v, is equivalent to
J^ v if and only if there exists s e W(A0) such that = s£, v' = sv. If V is
an irreducible (g, K)-module, then Vis equivalent to Ji v, for such £ and v.
The previous discussion proves the last assertion of the theorem.
Let £ and v be as in the first assertion. Let F = F(v) = {a e
Д(Р0, Ло)| Re(p, a) = 0}. Set (P, A) = (PF, AF) (2.2.7), P = ° MAN (as
usual). Let (a, Ha) denote the representation (тт>оП nom)' We note
that у, n om takes purely imaginary values. Thus, (a, Ha) is tempered.
Theorem 13.6.7 implies that (a, Ha) is irreducible. We also note that
Re(r,J a) > 0 for a e Ф(Р, A). Thus, (P, a, v|a) is Langlands data. Induc-
tion in stages implies that IPo is equivalent with IP,a,v  The first
assertion now follows from Theorem 5.4.1 (2).
If v and Jf, are equivalent, then Theorem 5.4.1 (3) implies that
F(v) = F(v'),	and ^|a = ^a- Theo-
rem 12.1.4 now implies the second assertion.
13.7.4.	The preceding refinement of the Langlands decomposition ap-
plies to the following class of groups (the notation is as in Helgason [1],
p. 518, Table V).
1.	G connected semi-simple and g has the structure of a Lie algebra
over C. The connected Lie subgroup of G with Lie algebra t is a maximal
compact subgroup of G. The conjugacy of maximal torii in К implies that
G has exactly one conjugacy class of Cartan subgroups.
2.	A II. This is (up to covering) a the real form of 5L(n,C) classically
denoted SU*(2n). К is isomorphic with the group of quaternionic unitary
matrices.
3.	SO(2n + 1,1)° (or a covering group).
4.	E IV. In this case, G is a real form of E6 and К is isomorphic with the
compact form of F4.
258
13. The Harish-Chandra Plancherel Theorem
13.8.	The Plancherel theorem for L2(G/K)
13.8.1.	In this section, we show how the Harish-Chandra Plancherel
theorem implies Harish-Chandra’s earlier theorem for G/K. We first
prove:
Theorem. If G is a non-compact, real reductive group of inner type, if y0
is the class of the trivial representation of K, and if (a, H„) e ш e <f7(G),
then Ha(yo) = {0}.
We note that if °G =# G then <^2(G) = 0. We may thus assume that
°G = G. Suppose that (<r, exists with H^y^f) =#= 0. Let Z denote the
center of G. Then Z с K, whence <r(Z) = {1}. We may thus assume that
Z = {1}. We first assume that G is connected. Theorem 7.7.1 implies that
there exists a Cartan subgroup T of G with TcK. Theorem 8.7.1 implies
that there is a system of positive roots P for (gc, tc) and g e T such that
(Ha)K is (g, ЛЭ-isomorphic with DP , with
(g, a) > 0, a e P
(see 6.7.6 for notation here and in the following). Proposition 6.5.4 (3)
combined with Theorem 6.7.6 implies that if Ha(y) =#= 0, then the highest
weight of у with respect to Pk = Ф(!с, tc) is of the form g + pn - pk +
Q, with Q a sum of elements of P. This implies that if H^y^ =#= 0, then
there exists Q, a sum of elements of P such that
0 = M + pn - pk + Q.
Hence,
2pk = M + P + Q-
By our previous assertion for g, g - p is P-dominant. Thus, g + p + Q
= 2p + %, with (p, £) 0. We therefore have
4(P*>P*)	4(p,p).
On the other hand, p = pk + pn and (pn,pk) 0 (9.3.1). Thus,
(p,p) S> (pk,pk) + (p„,P„)-
If we combine the two inequalities, we conclude that pn = 0. But this
implies that g = t. That is, G = K. The result now follows in this case.
If G is not connected, then <r|Go (G° the identity component of G) is a
direct sum of a finite number of square irreducible square integrable
13.8. The Plancherel Theorem for L2(G / K)
259
representations of G°. If (Ha\y0) =# 0, then at least one of the con-
stituents, say, (o\, of <r|Go must have the property that H^y^o) =# 0.
So the first part of the proof implies that G° = K°. Since G = KG°, the
result follows.
13.8.2.	Let (P, A) be a minimal (standard) p-pair with P = MN, as
usual. If v e , then we set
c(r-) = fa(n)y+p dn.
JN
Here, we initially take the domain of c to be those v for which the integral
converges absolutely. We also normalize dn as in the rest of this chapter.
We will now give an interpretation of the preceding integral in terms of
intertwining operators. Let £0 be the class of the trivial (one dimensional)
representation of °M. We set = irv and Hp’^’v = Hv. Let 1 s Hv
be the constant function identically equal to 1. Then,
c(p) = (W-1)-
Lemma 5.3.1 implies that the integral defining c(v) is absolutely conver-
gent for Refy, a) > 0, a s Ф(Р, A), and Theorem 10.1.6 implies that c
has a meromorphic continuation to .
We note that g(£0,iv) = l/c(iv)c(-iv) by 10.5.7. We denote by
if(G/K) the space of all right /С-invariant functions in ^(G). Then we
have (in the notation of 13.4.1):
Theorem. Iff^^(G/K),then
r	dv
Ks} - (1»'(Л)|7^) /a.tr(Ms)^(/))c(l„)c(_—r
If (Q, Aq) > (P, A) and if ы e ^Mq), (a, Ha) e a>, then we note
that since G is unimodular, 0е,ш,,р(Л(^)/) = tr(Tre><7>,p(g)_1TrG>a>,p(/))
= 0e,Wi,p((L(g)-1/). Thus, if any of these expressions is non-zero, then
* 0- This implies that a must contain the trivial (К П Afe)-type.
Thus, the preceding theorem implies that °MQ is compact. This implies
that Q = P. The result is now a direct consequence of Theorem 13,4,1.
13.8.3.	We set ^(K\G/K) equal to the space of all f e ^(G) that are
right and left K-invariant. Let S„ be the zonal spherical function corre-
sponding to v e etc (3.6,1), Then,
S„(g) ={n,(g)l,l).
260
13. The Harish-Chandra Plancherel Theorem
Also, if f e ^(K\G/K), then
tr(77,p(g)-1Tr,p(/)) = S,p(g)	dg.
We set
/(У) = J f(g)^-iv(s) dg.
JG
Then, we have:
Theorem. If f if(K\G/K), then
-i r -	dv
M - (l^(^)l^)
13.9.	Notes and further results
13.9.1.	The Eisenstein integrals of Section 13.1 were introduced by
Harish-Chandra to play the same role for G that Eisenstein series play for
T\G, In Harish-Chandra’s approach the double representation of К
could be infinite dimensional. The main case that he had in mind is
described in 13.4.5. The key general result about Eisenstein integrals
is Lemma 13.1.3, which seems to have been motivated by a lemma of
Arthur [1],
13.9.2.	The results in Section 13.2 are an exposition of Harish-Chandra’s
theory of the Eisenstein integral (Harish-Chandra [15,16]). Our exposition
differs in detail from the original in its direct use of intertwining operators,
which is motivated by Arthur’s approach [1] and an early unpublished
exposition of Harish-Chandra’s work by the author. In Harish-Chandra’s
development the C-functions are the basic objects of study and the
properties of intertwining operators are derived from properties of C-
functions. For example, the meromorphic continuation of the intertwining
operators is completed in Harish-Chandra’s approach simultaneously with
the completion of the proof of the Plancherel formula.
13.9.3.	Theorem 13.3.2 is the essential step in the proof of the Plancherel
theorem. Our approach to this theorem differs in detail and philosophy
since our development of the intertwining operators yields an a priori
proof of the temperedness of the g-function.
13.9. Notes and Further Results
261
13.9.4.	Theorem 13.4.1 is Harish-Chandra’s Plancherel theorem. In
Chapter 14, we will relate this theorem to the abstract Plancherel theorem
for “Type I” groups of Segal [1]. Obviously, the proof of this major
theorem of Harish-Chandra is the culmination of a great deal of difficult
mathematics. The essential simplicity of this proof is a tribute to the
powerful machine that Harish-Chandra constructed to apply to the proof.
Our approach is a mixture of Harish-Chandra’s method in the real and
p-adic cases [17]. An exposition of the Plancherel theorem can also be
found in Trombi [2]. There is also a very useful exposition of how all of the
parts of the proof fit together in Varadarjan [2]. Alternate approaches to
the Plancherel formula can be found in Herb-Wolf [1] (which proves the
theorem for a larger class of groups) and Duflo-Vergne [1].
13.9.5.	The material in Section 13.6 is the culmination of work on
intertwining operators initiated in Kunze-Stein [1] and Bruhat [1]. Many
authors contributed special cases of Theorems 13.6.1 and 13.6.6, notably
Knapp-Stein [1] and Arthur [1]. A formulation of the “completeness
theorem” (13.6.6) similar to ours can be found in Knapp-Stein [2].
Theorem 13.6.6 has been refined by Knapp-Stein [2] and Knapp [2]. For
example, their results imply that if G is a connected linear semi-simple
Lie group then the inequality at the end of the statement of Theorem
13.6.6 is an equality. Also, in Knapp-Stein [2] and Knapp [2], a subgroup
of WXa), v), which they call the “7?-group”, was introduced that is comple-
mentary to W/ы, v). This group and its analysis (and generalization) plays
a critical role in the Knapp-Zuckerman [1] classification of tempered
representations. The final result of the section is Harish-Chandra’s proof
of the irreducibility of the fundamental series. He looked upon this result
as a natural generalization of results of Zelobenko [1] and the author. An
alternate proof of this result can be found in Enright-Wallach [3]. A far
reaching generalization of the irreducibility result can be found in
Speh-Vogan [1].
13.9.6.	Theorem 13.7.3 is a generalization of Zelobenko [2], which gives a
classification of irreducible (g, K)-modules for g a semi-simple Lie alge-
bra over C and Lie(K) a compact form of g.
13.9.7.	Theorem 13.8.3 is due to Harish-Chandra [9] (modulo a conjec-
ture equivalent to Theorem 13.8.1, which he proved in Harish-Chandra
[13]). A proof that does not use the theory of the discrete series can be
found in Gangoli [1].
This Page Intentionally Left Blank
14 Abstract
Representation Theory
Introduction
This chapter (up to Section 12) can be read independently of the rest of
this volume. Its purpose is to give a fairly self-contained treatment of
“abstract representation theory” and to relate that theory to Harish-
Chandra’s Plancherel theorem. By abstract representation theory, we
mean the representation theory of C* algebras and locally compact
groups. This subject is vast and we have opted to develop only those parts
of the theory that have a bearing on locally compact, separable groups and
more specifically to real reductive groups. Thus, for the deeper theorems
on C* algebras we impose the simplifying assumption that the algebra is
separable. Also, rather than treating the general “tame” case (i.e., Type I)
we study CCR algebras (liminaire in the sense of Dixmier [1]). Since real
reductive groups satisfy the CCR condition, for the purposes of this book
this is no real restriction. Because of these assumptions the theory be-
comes manageable and many of the more dedicate results have simpler
proofs. The reader interested in the general theory and a more encyclope-
dic account should consult the books Dixmier [1,3,4]. The historical notes
in Fell-Doran [1] are also a useful reference for a reader who would like to
understand the evolution of this beautiful theory. The main theorem (in
263
264
14. Abstract Representation Theory
the context of this book) is Segal’s Plancherel theorem for locally compact,
separable, CCR topological groups (which is true in the more general
context of Type I groups). This theorem is derived as a consequence of a
general theorem that gives decompositions of unitary representations into
irreducible representations (again in the context of CCR rather than
Type I). We will see an example of this in the next chapter.
We have opted to give a self-contained treatment for two reasons. The
first is that the new generation of researchers on the representation theory
of reductive groups is largely ignorant of the general theory. This is
reasonable since the development of the field has been mainly indepen-
dent of the abstract theory. Our second reason is that we feel that the
theory of C* algebras could play an important role, even in the case of
reductive groups. A hint in this direction is that the unitary dual of a CCR
group is in bijective correspondence with the primitive ideals of its C*
algebra (as an abstract algebra over C). In the theory for reductive groups,
a representation is usually analyzed in terms of the decomposition of its
restriction to a maximal compact subgroup. The theory of semi-simple
symmetric spaces and of Whittaker models (see the next chapter) suggest
that one might gleen important information from the decomposition of
restrictions to certain “large” non-compact subgroups.
It is surprising to the author that the notion of C* algebra (due
essentially to Gelfand and Naimark) is so powerful since it is so simple. An
algebra £ over C is a C* algebra if £ is a Banach space over C with norm
|| • • • || and £ admits a conjugate linear involutive isometry x -> x* such
that ||xy|| < ||x|| ||y||, (xy)* = y*x* for x, у in if (i.e., it is a *-algebra),
and (the critical condition) ||x*x|| = ||x||2. If Я is a Hilbert space and if
End(H) is the algebra of all bounded operators on H, then under the
operator norm and with T* being the usual adjoint, End(H) is a C*
algebra. Clearly, any closed (relative to the operator norm) *-invariant
subalgebra of End(H) is a C* algebra (e.g., the algebra of compact
operators). One of the more striking theorems (Gelfand-Naimark) is that
every C* algebra is * and (hence isometrically) isomorphic with a closed
*-invariant subalgebra of End(H) for some Hilbert space H (Theorem
14.5.9). If G is a locally compact, separable, topological group with a fixed
choice of Haar measure dg, then under convolution Ll(G) is a *-algebra.
There is a canonical norm || • • • || (see 14.2), on Z?(G) such that 0 < ||/||
<, 11/111 for f s Ll(G), /=# 0, and \\f* ★ f\\ = ||/||2 (this is not true for the
ZAnorm). The completion of Ll(G) with respect to || • • • || is called the C*
algebra of G. Since unitary representations of G unambiguously corre-
spond to *-representations of the C* algebra of G, one can derive the
14.1. The Basic Theory of C* Algebras
265
basic theory of unitary representations of locally compact groups from the
*-representation theory of C* algebras. This is the approach that seems to
be the most standard in “abstract representation theory.”
Our exposition of the general theory is strongly influenced by the
standard references Dixmier [1,3]. In Sections 11 and 12, there are some
results that are not quite so standard. We base our approach to the
tempered part of the unitary dual of a real reductive group to an ingenious
result of Cowling, Haagerup, and Howe [1]. This theorem could be taken
as strong motivation for the form of Harish-Chandra’s Plancherel theo-
rem. In the last section, we relate the Segal theorem to the Harish-Chandra
theorem. In doing this, we also (essentially) determine the topology of the
tempered representations. In particular, the results of that section imply
that the tempered part of the unitary dual (with the “hull kernel topology”)
is the primitive spectrum of an appropriate C* algebra, which is a
completion of the Harish-Chandra Schwartz space.
14.1.	The basic theory of C* algebras
14.1.1.	A Banach algebra is a Banach space (^, || • • • ||) that is also an
algebra over C such that ||xy|| < ||x|| ||y|| for x, у g SB. The examples that
we initially have in mind are:
1.	Let X be a compact topological space and let C(X) denote the space
of all complex valued continuous maps on X. If f g C(X), then we set
ll/ll = sup{|/(x)| |x g X]. Then, C(X) is a Banach algebra under point-
wise multiplication.
2.	Let В be a Banach space and let End(B) denote the space of all
bounded linear operators of В to В with Ill’ll = sup{||7x|| |x g B, ||x|| = 1}.
Let SB denote a Banach algebra with identity element 1. Let denote
the set of invertible elements of SB. The following result is standard.
Lemma. SB0 is open in SB.
If x g SB and ||x - 111 < 1, then the series E„s0(l “ *)" converges
absolutely in SB to an element y. Since multiplication is continuous in SB,
xy = £ (x - 1)(1 - х)"+У = У - E(l-x)" = l-
nSO	n>l
Set В = {у g SB\ ||y|| < 1}. If x g SB0, then x(l + В) c
266
14. Abstract Representation Theory
Corollary. If J? is a maximal proper, two sided (resp., right, left) ideal in
31, then J is closed.
If eZn =# 0, then 1 g Z This would imply the contradiction that
= eZ. Hence, Zc 3 - 3°, which is closed. So the closure of .Z,
C1(.Z), is contained in - 3°, which is proper. Since CK^Z) is an ideal
of the same type as .Z and contains ^Z, it must equal .Z by maximally.
14.1.2.	If x g 3, then set spec^(x) = {Л g Cl (x - Л) £ 3й}.
Proposition, spec^(x) is compact and non-empty. Furthermore,
lim„_oo llx"||1/n exists and is equal to sup{|Л I |A g spec^(x)}.
Note. The common value in the last assertion is called the spectral radius
of x in and denoted specrad^(x).
If A g C and |A | > ||x||, then (x - A)= A(A-1x - 1) g (see the
argument in Lemma 14.1.1). Thus, spec^(x) с {A I |A| < ||x||}. The map <p
of C to given by <p(A) = x - A is continuous, spec^(x) = <p~l
(3 - 3°), which is closed. Hence, spec^(x) is compact. Suppose that
spec^(x) is empty. Let g g 3' (continuous linear functionals). Then
A -> g((x - A)-1) is bounded and entire in A with limit 0 at oo. Thus,
g((x - A)-1) = 0 for all g g 3'. So (x - A)-1 = 0 for all A. This is
impossible. Hence, spec^(x) =#= 0.
We will now prove the last assertion. The radius of convergence of the
series
L znxn
n^O
in z is given by l/limsupd|x"||1/n). Where this series converges, its value
is
(1-zr)-1.
Thus, by the definition of spec^(x) the radius of convergence is equal to
specrad^x)-1. Hence, limsup ||x"||1/n = specrad^(x). Thus, to prove
the desired formula we need only show that ||x"||1/n has a limit at + oo.
For this, we set f(n) = log ||x"||. Then, f(n + m) < f(n) + f(m). Fix n
and write m = pn + q, with p an integer and 0 < q < n - 1. Then,
14.1. The Basic Theory of C* Algebras	267
/(w) < p/(n) + f(q\ Thus, if m £ n,
f(m)/m = f(n)/(n + q/p) + f(q)/m.
This implies that
limsupm(/(w)/w) ^f(n)/n.
But then lim supm(/(w)/w) = inf(/(n)/n). This clearly implies that the
limit exists.
14.1.3.	Theorem. If every element of	- {0} is invertible, then £ё = Cl.
If x g ёё, then there exists Л e C such that (x - A) £ ^°. Thus,
x - A = 0.
14.1.4.	We now assume that ёё is commutative and that it contains a unit
element. Let spec(^) denote the set of algebra homomorphisms A of
into C such that A(l) = 1. Notice that there is no a priori assumption of
continuity.
Lemma. If A g spec(^) and ifx g then A(x) g spec^(x). In partic-
ular, |A(x)| < ||x||, so ||A|| = 1 (since A(l) = 1). If g g spec^(x), then
there exists A g spec(^) such that A(x) = g.
We assume that A =# 0. A(x - A(x)l) = 0. If (x - A(x)l) g &°, then
there would be у g ^withy(x - A(x)l) = l.Thus, 1 = A(y)A(x - A(x)l).
This is ridiculous. Hence, A(x) g spec^(x). In the proof of 14.1.2, we saw
that if c g spec^(x) then |c| < ||x||.
If g g spec^(x), then (x - g)^ с @ - ^°. Zorn’s lemma implies
that there exists a maximal proper ideal J containing (x - р)дё. Since
3)/^ is a field and a Banach algebra under the quotient norm, дё/^=
C(1 + cZ). Define A(y) = c if у + c(l + Then, A g spec(^)
and A(x) = g.
14.1.5.	On 3' we put the weak* topology (14.A.5). Then spec(^) is a
closed subset of the unit ball in дё' and hence is compact (see Theorem
14.A.5). We map 3 into the space of all continuous functions on spec(^)
by defining x(A) = A(x) for x g дё, A g spec(^). If x g 3, then |x(A)|
= IA(x) | < ||x||. So the map x -> x of 3 into C(spec(^)) is continuous,
with ||x|| < ||x||. x will be called the Gelfand transform of x.
14.1.6.	If 3 is a Banach algebra and if x >-> x* is a conjugate linear
endomorphism of 3 such that (x*)* = x, (xy)* = y*x*, and ||x*|| = ||x||,
268
14. Abstract Representation Theory
then (^, *) is called a Banach *-algebra. If, in addition, we have
llx*x|| = ||x||2,
then (^, *, || •  • II) is called a C* algebra.
Note example 1 in 14.1.1 is a C* algebra if we use f* = f. If the Banach
space in example 2 is a Hilbert space and if T* denotes (as it usually does)
the adjoint of T, then End(B) is a C* algebra. To see this, we must show
that if A g End(B) then ||Л*Л|| = ||Л||2. Before proving this we will make
some observations. Suppose that T = T*, and that <7x,x) > 0 for all
x g B. The Schwarz inequality implies that
I {Tx, y) |2 <7x,x><7y,y>.
This implies that if x, у g B, then
|<Tx, y>| < ||x|| ||y||sup{{Tz, z>|z G B, ||z|| = 1).
If we apply this inequality with у = Tx, then we find that
IlTxll ||x|| sup{ {Tz, z>|z G B, llzll = 1).
Hence,
Ill’ll < sup{<7z, z>|z g B, ||z|| = 1} < IITII.
We apply this to the case when T = A*A. Then,
M*^|| = sup{U*,4x,x>|||x|| = 1} = sup{||y4x||2| Hxll = 1} = ||Л||2.
Theorem. Let S be a commutative C* algebra with unit. Then the
Gelfand transform defines an isomorphism of дё onto C(spec(^)). Further-
more, (x)* = (x*)A.
We first observe that if h g дё is such that Л* = h, then Л(Л) g R for
all A g spec(^). Indeed, consider e“h = E„s0(z7A)"/n! g @L Then
(е"л)* = Thus, 1 = ei,h(ei,h)*. The C* condition implies that 1 =
||е"л||2. So ||е"л|| = 1. If A g spec(^), then 1 > |А(е"л)| = |е"А<л>| for all
t g R. This is only possible if А(Л) g R.
If x g then we write x = (x + x*)/2 + z(x - x*)/2z‘. If A g
specif), then A(x) = A(x + x*)/2 + z’A(x - x*)/2z. Hence, A(x*) =
A(x)*. This implies the last assertion.
If x g and if x = x*, then ||x2|| = ||x*x|| = ||x||2. Hence, ||x2"||1/2"
= ||x||. Thus (14.1.2), specrad(x) = ||x||. Lemma 14.1.4 implies that ||x|| =
specrad(x). Thus, if x = x* then ||x|| = ||x||. If x g then (x*x)* =
(x*x) and so Hxll2 = ||x*x|| = IKx*x)A|| = ||x||2. Thus, ||x|| = ||x||.
14.1. The Basic Theory of C* Algebras
269
The Stone-Weierstrass theorem implies that {x|x g S} is dense in
C(spec(^)). Thus, since x •-» x is an isometry it is surjective.
14.1.7.	Lemma. Let S be a C* algebra with unit and let £ be a closed
*-invariant subalgebra with unit. Ifx g then spec^(x) = spec^(x). Let
x g S be such that xx* = x*x. Let 3 be the closure of the subalgebra of S
generated by x, x*, and 1. Then x is a homeomorphism of spec(^) onto
spec^(x).
Let 3 be the closure of the subalgebra generated by 1, x, x*. We first
show that if a g 3 is invertible in @ and if a = a*, then a-1 g 3).
Indeed, if t g R, t ¥= 0, then it £ spec^(a) (Lemma 14.1.4 combined with
the previous result). Set, for t g R, f(t) = (a - it)~l. Then, f is a real
analytic map of (0, <») into &. If t > ||a||, then
/(0 = L(it)-nan^^.
n>0
Let t0 = inf{t|/(t) g 3}. Then t0 < ||a||. Since 3 is closed and f is
continuous, /(t0) g 3. Thus, /(t) g 3 for t t0. But then all derivatives
of f at t0 are in 3. Hence, there exists e > 0 such that f(t) g 3 for
lit _ toll < Hence, t0 = -oo. This implies the assertion that a~l g 3.
If a g 3 is invertible in S, then a*a is invertible in 3 and (a*a)~l g
3. Thus, a~l = (a*a)~la* g 3. This implies that if x - A is invertible in
S then it is invertible in 3. So spec^(x) = spec^(x). This proves the
first assertion. We now prove the second. So assume that x and x*
commute. In light of the first assertion, we may assume that 3 = Si.
Lemma 14.1.4 implies that x(spec(^)) = spec^(x). x is continuous. Thus,
to complete the proof we need only show that x is injective. If x(At) =
x(A2), then A/x) = A2(x). Thus, A/x*) = Aj(x)= A2(x)= A2(x*). Thus,
A j = A2 on the algebra generated by 1, x, x*, which is dense in 3. Hence,
Ai = A2.
14.1.8.	We now remove the restriction that Si contains a unit. We
assume that S is a C* algebra that has no unit. Set Si = S ® Cl, the
algebra obtained by adjoining a unit. We set (x + Al)* = x* + Al. If
x g Si, we define Lx g End(^) by Lxy = xy. Then, ||LX|| < ||x||.
Lemma. ||LX|| = ||x||.
||Lxx*|| = ||лх*|| = ||x*||2 = ||x||2 = Hxll ||x*II. Thus, ||LX|| £ ||x||.
270
14. Abstract Representation Theory
14.1.9.	We define ||x + Л1Ц = ||LX + A/|| (the operator norm on ^).
Lemina. with the given norm and *-operation is a C* algebra.
Suppose that ||x + Al|| = 0. Then xy = -Xy for all у e <0. If A =#= 0,
then — A-1x = z is a left unit for S. Since zy = у for all у e yz* = у
for all у e Thus, zz* = z and zz* = z*. Hence, z is a unit for Si. But
we have assumed that Si has no unit. Thus, A = 0. Hence, x = 0.
To complete the proof, we must show that IKx + A)*(x + A)|| =
||x + A||2. It is obvious that IKx + A)*(x + A)|| ||x + A||2. We assume
that ||x + A|| = 1. Let 0 < r < 1 and let у e Si be such that ||y|| = 1 and
IKLX + A)y||2 > r. Set z = (x + A)y. Then, ||z*z|| = ||y*L(x+A).(x+A)y|| <
И^(х+л)*(х+л)УИ- We note that z e S. Hence, ||z||2 < llL(x+A),(x+A)y||.
Now, Hzll2 = IKLX + A)y||2 > r. This implies that HL(x+A).(x+A)y|| > r.
Since r is arbitrary subject to 0 < r < 1, this implies that IKx + A)*
(x + A)|| £ 1.
14.1.10.	If Si is a commutative C* algebra without unit, then Si is a
commutative C* algebra with unit and Si is a maximal ideal in S. Let Ая
be the element of spec(^) defined by Хя{х + cl) = c for x e S, с e C.
Set
C0(spec(^)) = {/e C(spec(^))|/(Ae) = o).
Then Theorem 14.1.6 implies that under the restriction of the Gelfand
transform of Si to Si, Si is isomorphic with C0(spec(^)).
14.1.11.	In general, if Si is a C* algebra then we will write Si = Si if Si
has a unit, and if Si doesn’t contain a unit then Si will be as before. We
note that if Si =# Si, then 0 s spec^x) for all x s Si.
If Si is a C* algebra and if x e Si is such that xx* = x*x, then we set
Six equal to the closure of the subalgebra of Si generated by x and x*
(this algebra has no unit if Si has no unit). Lemma 14.1.7 implies that x
defines a homeomorphism of spec(^x) onto spec^(x). The following
result is the “continuous functional calculus” for C* algebras.
Lemma. Let f s C(R) be such that /(0) = 0. Then there exists a unique
element fix) e. Six such that
о(*|£х)-1 = /
This result is a direct consequence of the preceding observations and
Theorem 14.1.6.
14.1. The Basic Theory of C* Algebras
271
14.1.12.	If x g SB, then x is said to be positive if x = x* and spec^(x)
c [0, oo). We write x > 0 if x is positive. If x is positive, then (14.1.11)
x1/2 > 0 exists in SBX. This implies:
(1)	x > 0 if and only if x = y*y for some у g SB.
If x g SB, x = x*, then |x| > 0 is defined in SBX. We have:
(2)	If x e then x = x+— x~ with x+= (x + |x|)/2 > 0 and x~ =
(|x| - x)/2 > 0.
We set SBh = {x g ^|x* = x}. We note that SBh is a closed real
subspace of SB such that as a real vector space SB = SBh Ф iSBh. Put
^+= {x g SBh\x > 0}.
Lemma. SB+ is a closed convex cone in SBh with S+C\(- S+) = {0}.
By replacing SB with SB, we may assume that 1 g SB. We note that:
(3)	{x g SB+\ Hxll < 1} = (xe SBh\||x|| < 1, ||1 - x|| < 1}.
Indeed, let x g SBh. Let 3> be the closure of the subalgebra generated
by 1 and x. Then, under the Gelfand transform, 3) is isomorphic with
C(spec^(x)) (14.1.7) and x corresponds to the function /(t) = t restricted
to spec^(x). If s g R, then |s|	1, |1 - s| < 1 if and only if 0 < j <, 1.
Obviously, if x g SB+ then tx g SB* for all t > 0. Suppose that x, у g
SB\ We must show that |(x + y) g SB*. If we multiply both x and у by a
sufficiently small positive real number, we may assume that ||x|| < 1,
llyll < 1. Then (in light of (3)), ||1 - |(x + y)|| < |||1 - x|| + |||1 - y|| < 1.
So (3) implies that |(x + y) g SB\
If x g ^+n(- SB+\ then spec^(x) = {0}. Thus, applying the Gelfand
transform as before, we find that x = 0.
14.1.13.	If SB is a C* algebra then a *-representation of SB is a pair
(тг, H), with H a Hilbert space and тг a homomorphism of SB into
End(H) (14.1.1 (1)) such that тг(х*) = тг(х)*. (Notice that we have made
no assumption of continuity.)
If (тг, H) is a representation of SB then we say that тг is non-degenerate
if tt(SB)H is dense in H.
Lemma. Let SB be a Banach * -algebra and BaC* algebra. If тг is an
algebra homomorphism of SB into with тг(х*) = тг(х)* for x e Si, then
14. Abstract Representation Theory
272
||ir(x)|| < ||x|| for x e 3. If, in addition, 3 is a C* algebra and tt is
injective, then tt has closed range and ||тг(х)|| = ||x|| for x e 3.
We set 3 = 3 Ф Cl, with ||x + Л|| = ||x|| + |Л I and (x + A)* = x* +
A. Then, is a Banach *-algebra. Let ё be as in 14.1.11. We extend tt to
3 by setting tt(1) = 1. We may thus assume without loss of generality
that both 3 and ё have units.
Suppose that x s 3, x* = x. Then тг(х)* = тг(х)*. Thus (14.1.6),
specrad^(ir(x)) = |l'”’(*) II-
But if (x - A)-1 exists in 3, then (тг(х) - A)-1 exists in ё. Thus,
spec^(x) э spec^n-Cx)).
This implies that
II-rr(x)|| < specrad^(x) <. ||x||.
Hence, if x =x* then ||ir(x)|| < ||x||.
If x s 3, then ||x||2 > ||x*x|| ||ir(x*x)|| = ||тг(х)*тг(х)|| = ||тг(х)||2.
Thus, ||ir(x)|| <. ||x||.
We now assume that tt is injective and that is a C* algebra. To
prove the last assertions, it is enough to prove that ||ir(x)|| = ||x|| for
x e 3. To prove this, we may assume that x*=x (l|x*x|| = ||x||2,
||ir(x*x)|| = ||tt(x)||2). Let = 3X and F= ёх. Extend tt to by
-rr(l) = 1. Then tt induces a continuous map tt* of spec(X) to specC1^) by
tt*(A) = A ° tt. We assert that tt* is suijective. If not, since the two spaces
are compact there exist f,ge CfspecC^)) such that f =# 0 and
/(77*(sPec(r)) = {0}, gk*(spec(r)) = 1 and fg =# 0. Now, there exist u, v s
such that u = f, v = g. But тг(м)тг(г) = 0, while uv =# 0. This contra-
dicts the injectivity of tt. Hence, tt* is suijective.
We now use 14.1.2 and 14.1.7 to see that
||x||= sup |x(A)| = sup |(тг(х))л (A) | = ||ir(x)||.
Aespect^)	AespectX)
14.1.14.	The preceding lemma implies that a *-representation of a C*
algebra is automatically continuous in the norm topology. Let ё be a C*
algebra. We consider the category of all *-representations of ё with
objects representations and morphisms homomorphisms, that is, T: (тг, H)
-> (a, V) if T is a bounded operator from H to V such that Ttt(x) =
a(x)T. We will say that (тг, H) and (a,V) are unitarily equivalent if there
exists a unitary bijective homomorphism of (тг, H) with (a, V).
14.2. The C* Algebra of a Locally Compact Group
273
14.2.	The C* algebra of a locally compact group
14.2.1.	Let G be a locally compact, separable, topological group. We fix a
left invariant measure dg on G. Let 3 be the modular function of G
(0.1.1). Let Ll(G) denote the space of all measurable functions f on G
such that
H/Hi= f \f(g)\dg<<°.
JG
If f,g g LX(G), then we set
= f f(y)g(y~lx)dy.
JG
We also put, for f g Ll(G), /*(х) = /(х_1)3(х)_1. Then:
(1)	(f*gy =g* *f*, ll/*glli < ll/llillglli and \\f*\\i = 11/111.
Thus, with this *-operation LX(G) is a Banach *-algebra (14.1.6). It is
not, however, a C* algebra.
If g g G and f e l}(G), then we set L(g)f(x) = f(g~xx). We note
that (L, L*(G)) is a (strongly continuous) representation of G and that:
(2)	L(gX/*A) = (L(g)/)*A.
14.2.2.	Let S^{G) denote the set of unitary equivalence classes of unitary
representations of G. If (тг, H) is a unitary representation of G and if
f g Ll(G), let tt(/) be as in 1.1.3. Then ||ir(/)|| depends only on the
unitary equivalence class of тг, so if ы g ^~(G) and if (тг, Я) g ш, then
we set H/IL = ||ir(/)||. We note that:
(1)	IW/)II < ll/lli (see 1.1.3).
(2)	77(/)* = 1г(/*).
(3)	ir(/*g) = ir(/)ir(g).
We define a new norm on L*(G) by ||/|| = 8ирше ||/||ш < ||/|li.
Lemma. If ||/|| = 0, then f = 0 as an element of Ll(G).
If H/H = 0, then тг(/) = 0 for all unitary representations of G. We
consider the representation (L, L2(G)), L(x)f(g) = f(x~xg). Then L(f)g
= f*g for f g Z?(G) and g g Z/(G) A L2(G). Thus, if L(f) = 0 then
/*g = 0 for all g g Ll(G) A L2(G). Let Uj be as in 1.1.3. Then
lim^.,, f*Uj = f. Since u; g L2(G), this implies that f = 0.
274
14. Abstract Representation Theory
Note. We will call the sequence {u;} an approximate identity for Ll(G).
14.23.	We set C*(G) equal to the completion of Ll(G) with respect to
the norm [| • • • |[ of the previous section. Then the inclusion map of Ll(G)
into C*(G) is continuous and 14.2.2 (1),(2),(3) imply that f -> f* extends
to a *-operation on C*(G). C*(G) is called the C*-algebra of G.
Lemma. C*(G) is a C* algebra.
Let f g L*(G), II/* */II = 8ирше^-(С)||/* */||ш. If ш g ^~(G) and if
then ||тг(/* */)|| = ||7г(/)*7г(/)|| = ||7t(/)||2. Thus, II/**/11 =
suPwe^-(G)ll/* */IL = suPc^e^G) ll/ll« = H/Ц2- Since L*(G) is dense in
C*(G), the lemma follows.
14.2.4.	Lemma. If A is a countable dense subset of G and if {иД is an
approximate identity for L\G), then {L(y)Uj\y e A, j = 1,2,...} spans a
dense subset of C*(G).
That {L(y)uj\y g Л, j = 1,2,...} spans a dense subset of Ll(G) fol-
lows from the construction of dg in Weil [1] and the fact that
lim/* Uy =/.
/-.00
Since Z?(G) is dense in C*(G), the result follows.
14.2.5.	Theorem. If (тг, H) is a unitary representation of G, then f •-»
tt(/) (/ g Ll(Gf) extends to a non-degenerate representation of C*(G).
Every representation of C*(G) is of this form, and this correspondence is an
equivalence of categories.
Let (тг, Я) be a unitary representation of G. If {uy} is an approximate
identity for Ll(G), then liniy^ tt(uj)v = v for all v g H. Hence, тг is a
non-degenerate representation of C*(G).
Suppose that (тг, H) is a non-degenerate representation of C*(G).
Then since ||ir(/)|| ll/ll < ll/lli for / g Ll(G), tt(L1(G))H is dense in
H. Suppose that /, /' g Ll(G), v g H, and rr(f)v = rr(f')v. Then
rr(L(g)f)v = lim Tr(L(g)(uy */))г = lim rr(L(g)Uy * f)v
J —* 00	j —*00
= lim Tr(L(g)uATr(f)v = lim ir(L(g)u)ir(f')v
/ —> 00	j —» oo v
= 4L(s)f')l>.
14.3. Quotients of C* Algebras
275
Similarly, we find that there is a well defined linear map <r(g) of
spanfHfAG))//} to itself given by
r	r
Indeed, a(g)u = limy_«, Tr(L(g)uy)u for и s spanfH/AG))//}. We note
that if и e spanfirCZAG))//}, then
||<r(g)u|| < limsup ||тг(Л(^)му)и|| < limsup ||uy|lil|u|| = ||u||.
Hence, a(g) extends to a bounded operator on H. Since L(xy) =
L(x)L(y), we have a(xy) = <r(x)<r(y). It is also clear that <т(1) = I.
We also note that
{<r(g)ir(f)v,<r(g)ir(f')v'') = {ir(L(g)f)v,ir(L(g)f')v')
Thus, the operator a(g) is unitary for g s G. That (a, H) is a unitary
representation of G follows from the strong continuity of ir^G) an<^ the
strong continuity of (L, Ll(GJ).
Finally, we note that if f, и s Ll(G), then a(f)ir(u)v = ir(f*u)v =
тг(/)тг(и)и. So a(f) = tt(/).
The equivalence of categories is now straightforward.
14.3.	Quotients of C* algebras
14.3.1.	In this section, € will denote a C* algebra with a countable
dense subset {xy}. The results of this section actually apply in more
generality; however, this case is somewhat simpler and is adequate for our
applications.
Lemma. There exists a sequence {uy} in with u* = u-, satisfying the
following three conditions:
(1)	spec c. [0,1];
(2)	limy_«, UjV = v for all v s
(3)	uj+1 - Uy > 0.
276	14. Abstract Representation Theory
Note. We will call such a sequence an approximate identity for -ё.
Set Vj = Li£j XjX*. Then, Vj = v* and Vj > 0. Thus, (vj + 1/7) “1 exists
in -ё. Set Uj = Vj(Vj + I//)-1 e ё. We note that = /p), with /y(t) =
r(t + 1/j)-1. Thus, spec^(uy) c/;(spec^(uy)) c [0,1]. Hence, 1 - u; > 0.
Thus, {u;} satisfies (1). We now prove that it satisfies (2). Consider
E ((«; - !)*<)((“) “ O**)*
i-i
= E («, -	- 1)
i=l
= (“, -	- 0
1 f n-2 i
+	=]2<РЛ^)
with <py(t) = t(t + I//)-2. We note that max0<( <00 <py(t) = j/4. Hence,
This implies that
- E ((«> - i)*i)((«, - ip)* o-
which implies that if i < j, then
7- ~ ((«> - !)^)((“/ - i)*.)* °-
We therefore see that
lim («у - l)x( = 0,
for all i. Since ||uy - 1|| < 1 and {xy} is dense, a “3e” argument implies
that
lim (uy - l)x = 0
for all x e ё.
14.3. Quotients of C* Algebras
277
We now prove (3). We note that
t
= (t + 1/; + 1)(, + 1/;);(; + 1} •
So >fj(t) for t > 0. Thus, fj+l(vf) -	> 0. On the other
hand,
d	t
—fdt) =-------------2~	°-
dt 1 (t + 1/J) J
Thus, fj is increasing. So /, + 1(г, + 1) -/, + 1(r,) > 0. (3) now follows since
W = «,•
Corollary 1 (to the proof). {иД can be chosen such that {up is also an
approximate identity for e Z, p > 1.
Let Uj be as in the preceding proof. Clearly, {up satisfies (1). Also,
II (uf - l)x|| = ||(иД~' + ••• +Uj + l)(u, - l)x|| <p||u,x - x||.
So {up satisfies (2). We note that if t > 0, then
=	+Л+1(рр-2Л(Р +
and fj+i(t)p is monotone increasing for t > 0. Thus, (3) follows as in the
last part of the proof of the lemma.
Corollary 2. Let L be a closed left ideal in Then there exists a sequence
{иД in L with u* = u} and:
(1) spec^(uy) c [0,1];
(2) lim,	UjX = x for x e L.
Note. We will call {иД an approximate projection for L. We note that
lim, ^ xuj = x for x e L. Indeed,
||(x(u, - l))*(x(u,. - 1))|| =||(uy - l)x*x(u; - 1)11
<2||(uy - l)x*x||.
Since x*x e L this implies that lim, -,», ||x(u, - 1)|| = 0.
278
14. Abstract Representation Theory
Let {x„} be a countable dense subset of L. Set Vj = LisJx^xt. Then,
v* = Vj. The closure of the subalgebra of ё generated by v} is contained
in L. Thus, if /e C(R), /(0) = 0, then /(r;) e L. Set	(as
before). Then,
E ~	- 1)) = (Uj - l)Vj(Uj - 1).
is;
The rest of the argument to prove (1) and (2) is the same as that of the
lemma.
143.2. If В is a Banach space and if И is a closed subspace of B, then
on B/V we put the norm ||x + И||В/|/= inf„eK ||x + r||. Since V is
closed in B, this formula does indeed define a norm.
Lemma. Let be a closed two sided ideal in ё. Then:
(1) ./* = J.
(2) Set (x + vX)* = (x* + ^). Then, with this *-operation and the
quotient norm, ё/^ is a C* algebra.
Let {uy} in be as in 14.3.1, Corollary 2. If v e .J, then (note 14.3.1)
lim vuj = v.
This implies that
lim UjV* = v*.
i->“
Hence, v* e since UjV* s and is closed.
We will call {иД an approximate projection for We note that the
preceding implies that
||x + ^|| = ||x* +^||.
We set J = x + It is easily checked that ё/Л is a Banach *-algebra.
Thus, to complete the proof we must show that ||х*х||^/>-= ||x||^/ j. It is
clear that llx*x||^/>-< ||x||^/wr. We now prove the opposite inequality.
For this, we need:
(i)	If x e ё, then lim, ^ |K1 - u;)x|| = ||
Indeed if у s then
lim ||x — Ujx\\ = lim 11(1 - )x II
j —» 00	j —» 00
= lim ||(1 - uj)(x +y)|| <, ||x + y||.
j-KX>
14.4. Density Theorems
279
Thus,
lim ||x - u x|| < ||x||^/>-.
Since the opposite inequality is clear, (i) follows.
If z e we therefore have
llx||^/wr= lim ||(x - Ujx)(x - u x)*||
= lim ||xr* - xx*Uj - UjXX* + и;хг*и-||
;-»00
= lim ||xr* + z - UjZ - xx*Uj - UjXX* - zuj + UjZut + и.хг*и;||
= lim 11(1 - Uj)(xx* + z)(l - Uj)|| < ||xr* + z||.
If we take the inf over all z e then we have
l|j||^/ur< ||xx*||^/ur.
If we replace x by x*, then this says
\\x\\2^^< llx*Jk/^.
So the equality holds and the lemma is proved.
14.4.	Density theorems
14.4.1.	In this section, we will prove two density theorems that of Von
Neumann and that of Kaplansky, which will play important roles in the
later sections.
Let H be a Hilbert space. If sf is a subset of End(H), then we set
comm(ja/) = {T e End(H)|7M = AT for A s sf}. The following is the
Von Neumann density theorem.
Theorem. Let srf be a subalgebra of End(H) such that if T s srf then
T* e and such that I e stf. If T e comm(comm(j/)) and if {x„} is a
sequence in H such that E“ = 1 ||x„||2 < », then if e > 0 there exists A s ssf
such that E“_, IKT - Л)х„||2 < e.
280
14. Abstract Representation Theory
This result was proved in an important special case in 1.2.2. As in that
case, the critical observation is (see 1.2.2):
(1)	If v g H, then comm(comm(<Q/)) v c srfv.
Set H°° equal to the space of all sequences {x„} in H such that
E„al llx„||2 < oo. Then, H“ is a Hilbert space with inner product
<{*„},{уп}> = E <yn,y„>-
nzl
We let srf act on by {a(a)xn} = {ox„}. Let £B = <r(&O. Then, £B* = £B
and I g SB. Let Pn: H°° -> H be defined by P„{xm} = xn. We also define
Qn - H -♦ H” by Q„(x) = {8m nx}m^i. Then, Qn is the adjoint of Pn.
Suppose that T g comm(^). Let a g st/. Then PmT’<r(a){xn} =
Pmo-(a)T{x„} = aPmT{xn}. Also, TQmax = Ta(a)Qmx = a(a)TQmx. Thus:
(2)	PnTQm g comm(ja/) for all n, m if T g comm(^).
Let pn be the orthogonal projection of H“ onto QnH°°. Then pn g
comm(^). Thus, if T g comm(comm(^)), then p„T = Tpn for all n.
Hence, TQntT c for all n.
(3)	If T g comm(comm(^)), then PmTQm = PnTQn = S g comm(^)
for all n, m.
To prove this we introduce the elementary operators Rn <m{xq} = {yj,
with yr = 0 if r <£ {n, m}, yn = xm, and ym = xn. Then, Rn m g comm(^).
Thus, TRn m = Rn mT for all n, m. This is the content of the assertion.
To complete the proof we apply (1) to SB. So
comm(comm(^)) {xj c^{x„}.
Let В g comm(comm(ja/)), and define T{y„} = {By„}. Then, given e > 0
there exists a g j/ such that
||T{x„} - {ax„}||2 <£.
That is,
22 HSx„ - axn\\2 < e.
nal
This completes the proof.
14.4.2.	The strong topology on End(H) is the topology induced by the
semi-norms pv(T) = ||Tu|| for v eV. The following result (which is an
14.4. Density Theorems
281
immediate corollary of 14.4.1) is also known as the Von Neumann density
theorem.
Corollary. Let srf be a subalgebra of End(H) such that <£?* = <0/ and I
is in the closure of srf in the strong topology. Then comm(comm(ja/)) is the
closure of srf in the strong topology.
14.4.3.	We now give the Kaplansky density theorem.
Theorem. Let srf be a subalgebra of End(H) such that = зэ/, and
such that srf is dense in End(H) with respect to the strong topology. Then
= {A g ja/'l ЦЛЦ < 1}
is dense in {T g End(H)| Ill’ll < 1}.
We may assume that <0/ is closed in the norm topology of End(H). So
without loss of generality we may assume that <0/ is a C* subalgebra of
End(H) and hence the (continuous) functional calculus applies.
We first show that if T g End(H), T* = T, and Ill’ll < 1, then T is in
the strong closure of {5 g = S*, ||5|| < 1}. To this end, consider the
function /(t) = 2r(l + t2)-1. Then, -1 <, f(t) < 1 and /(0) = 0. Thus,
f(T) is defined for every self-adjoint T, and if Те зэ/ then f(T)e j/.
Also, if S is self-adjoint and ||5|| 1, then S = f(S') with [5, S'] = 0, and
if S g stf then S' can be chosen to be in srf. Let T' be such that
T = f(T'). Let v g H. We choose S' g srf such that
l^i + cny^-^i + ^-^hr
and
US'Tt- - T'TvW < e.
The first inequality implies that
h'(l + (r)2)-|Tr|<e.
We note that
-S'(l + (5')2) ' + ^'(l + (T')2)"'
= (1 + (Г)2)“'(Г - S')(l + (5')2)1
+ (1 + (Г)2)-1Т'(5' - T')5'(l + (5')2)"1.
282
14. Abstract Representation Theory
Indeed, if we multiply on the right by 1 + (S')2, then the right side is
equal to
(1 + (T')2)~\T' - S') + (1 + (Г)2)-1Г(5' - T')S'
= (1 + (Г)2)_1{Г - S’ + T’(S’)2 - (T’)2S’}
= — S' + (1 + (Г)2)-1Г(1 + (S')2).
This clearly implies the observation. Set S = 2S'(1 + (S')2)-1. Then,
||S|| 1 and S e j/. Also,
S - T = 2(1 + (S')2)-1(S' - T')(l + (T')2) 1 + |S(F - S')T.
This implies that
||(S - T)v\\ < 2||(S' - T')(l + (Т')2)Лj + |||(F - S’)T|| <
since IK1 + (S')2)-1|| 1.
We have therefore shown that s#x П {T e End(H)|T* = T] is strongly
dense in {T s End(f/)| Ill’ll < 1, T = T*}. We will use this to complete
the proof of the density theorem. Consider the Hilbert space H ® H with
(vx + v2,wx + w2) = (ux,wx) + (uj, w2). We look upon H ® H as col-
umn vectors
.У2. ‘
Then, T s End(H ® H) can be written in the form
Л1 Л2
^21 T22
with Ти e End(H). Let di be the subalgebra of all T s End(H ® H)
such that Tjj s лэ/. Then di is strongly dense in End(H ® H). Let X s
End(H), ЦХЦ < 1. Set
T= ° X
[F O’
Then, Ill’ll <,1, T = T*. Thus, if v s H and if e > 0, then there exists
В e such that
||(T- B)w|| <8,
14.5. Representations of C* Algebras and Positive Functionals
283
with
Thus, |KB12 _ А')г|| < с. Since ||B12|| <. 1, the result follows.
14.5.	Representations of C* algebras and positive functionals
14.5.1.	Let be a separable C* algebra. Let (тг, H) be a *-representa-
tion of ё (14.1.13). Then, (тг, H) is said to be irreducible if the only closed
invariant subspaces of H are H and {0}.
Theorem. Let (тг, H) be a *-representation of ё such that тг(ё) =#= 0.
Then the following are equivalent:
(1)	(тг, H) is irreducible.
(2)	сотт(тг(т^)) = CI.
(3)	If v e H, v =# 0, then тг(ё)и = H. In other words, (тг, H) is alge-
braically irreducible.
We prove that (1) implies (2) implies (3). Since it is obvious that (3)
implies (1), the theorem will then follow.
If T e сотт(тг(тГ)), then Т*тг(с) = T*ir(c*)* = (тг(с*)Т)* =
(Ттг(с*))* = тг(с)Т*. Непсе, Т* е сотт(тг(т^)). Thus, to prove
сотт(тг(т^)) = CI it is enough to show that if Г e сотт(тг(т^)) and
T* = T, then T = XI. Let h such that h* = h. Then specEnd(H)( itOi)) c R.
Let be the family of spectral projections corresponding to T (cf.
Reed-Simon [1], p. 234]). Then е^)р^е-и^н) = p^ for all , e R and all
fl. This implies that ei,irWPav = Paei,vWv for all t e R, h e ё, h* = h,
fl measurable in R, and v e V. If we differentiate this equation at t = 0,
then we find that ir(h)Pa = Рптг(й) for all fl and h e ё, h* = h. This
implies that s сотт(тг(т^)) for all fl. Now, the rest of the proof is
identical to that of Lemma 1.2.1.	________
We now prove that (2) implies (3). If v s H, then W = тг(ё)о is
invariant and, for some v, non-zero (тг(ё) =#= 0). Let P be the orthogonal
projection of H onto W. Then P e сотт(тг(т^)). Hence, P = XI. So
P = I. This implies that тг(ё)Н is dense in H. This combined with
Lemma 14.3.1 implies that I is contained in the closure of тг(ё) with
respect to the strong operator topology. Indeed, if v = тг(х)и, и e H,
then limy-,», IK/ - тг(му))тг(х)и|| = 0. Now use a “3e” argument. (2) im-
284
14. Abstract Representation Theory
plies that comm(comm(Tr(Tf)) = End(H). Thus, Corollary 14.4.2 implies
that тг(т^) is strongly dense in End(H). Let <X= Ker it. We replace ё by
ё/,^ (14.3.2). We may thus assume that tt is injective. Lemma 14.1.13
implies that we may assume that ёc End(H) and that ё is closed in
End(H) relative to the Banach space topology induced by  ||. Theo-
rem 14.4.3 implies that the unit ball in ё is strongly dense in the unit ball
of End(H).
Let и g H, ||u|| = 1. If v g H, we must show that there exists c g ё
with cu = v. If w g H, set Tw(x) = (x, u)w. Then, Tw(u) = w and HTJI =
||w||. Let To = Tv. Then, since ёх is strongly dense in End(f/)j, there
exists c0 g ё, ||c0|| < ||f||, and ||Tou - cou|| < ||r||/2. Thus, ||r - cou|| <
llfll/2. Set Tx = Tll_Coll. Then, there exists Cj g ё with Hqll < ||r||/2 and
IK^ - Cj)u|| < Hfll/4. Thus, ||r — cou - qull < ||u||/4. Continuing in this
manner, there exist, for n = 0,1,2,..., cn g ё such that
Hr - cou - • • • -C„u|| < IIr||/2"+1
and
llc„|| < ||r||/2".
The last inequality implies that, since ё is closed in End(H), E„a0 cn
converges to c g /. The first inequality implies that cu = v.
14.5.2.	Let ё' denote the space of all continuous linear functionals on
ё. Then f g ё' is said to be positive if /(x*x) is a non-negative real
number for all x e. ё. The main example is gotten as follows. Let (тт, H)
be a *-representation of ё. If £gH, then set f(x) = (tt(x)^, for
x g ё. Then, f is a positive functional on ё. Before the end of this
section we will have shown that this example describes all positive func-
tionals on ё.
Lemma. Let f be a (not necessarily continuous) linear map from ё to C
such that f(x*x) g R for x g ё. Then f(x*) = f(x) for x g ё. If, in
addition, f(x*x) 0 for allx g ё, then
\f(y*x)\2 Zf(x*x)f(y*y)
for all x,y g ё.
If x g ё and x* = x then x = x+- x~ with x ±= (y ±)*y * (14.1.12
(1), (2)). Thus, f(x) = f(x+) - f(x~) g R. Now, if x g ё then x = Xj +
14.5. Representations of C* Algebras and Positive Functionals
285
ix2 with x* = Xj, j = 1,2. The first assertion follows. We now prove the
second assertion. Set (x, y)f = f(y*x) for г, у e /. Then, ( , )f is Her-
mitian by the first assertion and (x, x)f > 0 for x e ё by assumption. The
Schwarz inequality implies that
|(x, y)z| < (x,x)}/2(y,y)}/2.
This is the second assertion.
14.5.3.	Lemma. Let f be as in the previous lemma, with f(x*x) > 0 for
all x g ё. If ё has a unit, then f is a positive functional on ё (that is, f is
continuous) and ||/|| = /(l).
If x g ё and x* = x, ||x|| < 1, then 1 - x > 0. Thus, 1 - x = y*y for
some у g ё (14.1.12 (1)). Thus, /(1 - x) > 0. So f(x) </(l). This im-
plies that if x* = x then \f(x)| < ||x||/(l). Thus, f is continuous and
ll/ll </(l). Hence ||/||=/(1).
14.5.4.	Let ё be a C* algebra without unit. Let ё be as in 14.1.10. If /
is a positive functional on ё then we extend / to a functional f on ё by
setting /(1) = H/Ц.
Lemma, f is a positive functional on ё.
We may assume that ||/|| = 1. Let {uy} be as in Lemma 14.3.1. Let
x g ё. Then
|/(x)|2= lim |/(u„x)| </(x*x) sup/(u*u„),
by 14.5.2. Since l|u*u„ll < 1, this implies that
|/(x)|2 </(x*x).
Now,
/((x + A)*(x + A)) =/(x*x) + A/(x) + A/(x) + |A|2,
since /(1) = 1. 14.5.2 implies that 2Re(A/(x)) = A/(x) + A/(x*). Thus,
/((x + A)*(x + A)) > |/(x) |2 + 2Re(A/(x)) + |A|2
= |/(x) + a|2 > 0.
286
14. Abstract Representation Theory
14.5.5.	The next result implies that positive functionals exist and, in a
sense, separate points.
Theorem. If xQ g (14.1.12), x0 =#= 0, then there exists a positive func-
tional f such that f(x0) > 0.
According to Mazur’s theorem (14.A.6), there exists a real linear contin-
uous functional f0 in ёь (14.1.12) such that f^-x^) > 1 and f0(x) < 1
for all x g if + . Set f = -f0. Then, f(x0) = f0( -x0) > 1 and, if x g if+,
then f(x) = -f0(x) £ -1. Let x g if+ ; if t > 0, then te e / + . Thus,
tf(x) > -1 for all t > 0. Hence, f(x) > 0. Now, extend f to ё by
f(x + iy) = f(x) + if(y) for x, у g ёь.
14.5.6.	We now begin a discussion of what is usually called the
Gelfand-Naimark- Segal (GNS) construction, which implies that the ex-
ample in 14.5.2 describes all positive functionals. Let f be a positive
functional. Set = {x g ^’|/(y*x) = 0 for all у g ё}.
(1)	is a closed left ideal in ё.
Since f is continuous, is a closed subspace of ё. Indeed, if x g
and if z g ё, then /(y*(zx)) = /((z*y)*x) = 0 for all у g ё.
In light of Lemma 14.5.2, (x, y)f = f(y*x) defines a positive semi-defi-
nite Hermitian form on ё. The radical of this form is Thus, ( ,
induces a pre-Hilbert space structure ( , )/ on ё/^. We set || • • • ||y
equal to the corresponding norm on ё/Л^. Let лу denote action of ё on
ё/Л^ induced by left multiplication.
(2)	If x, у g ё, then \f(y*xy)\ < l|x||/(y*y).
Set g(x) = f(y*xy). Then, g is a positive functional on ё ||g|| = g(l)
= /(у*у).
Lemma. Let Hf denote the Hilbert space completion of ё/^ff with respect
to < , )f. If x g ё, then тт^(х) extends to a bounded operator on Hf with
||лу(х)|| < ||x||. Furthermore, irf(x)* = irf(x*).
If x, у g ё, then
(xy,xy)/ = /(y*x*xy)	||x*x||/(y*y) = ||x||2(y, y)f.
14.5. Representations of C* Algebras and Positive Functionals
287
This implies that ||i7y(x)z||/ < ||x|| ||z||y for z s ё/Л^. Hence, лу(х)
extends to a bounded operator on Hf, with the indicated bound on the
operator norm. If x, u,v e ё, then
(xu,v)f = f(v*xu) = f((x*v)*u) = (u,x*v)f.
Hence, тт^(х)* = ir^x*).
14.5.7.	In light of the preceding lemma, (irf, Hf) is a *-representation of
ё. Suppose that ё does not contain a unit. Let ё and f be as usual.
Since = {x e ё\/(х*х) = 0}, ё = Thus, we have unitary
imbedding of ё/^ into ё/J^f.
We assert that the image is dense in Hf. To prove this, we will use the
following observation. Let {u„} be an approximate identity for ё as in
Lemma 14.3.1.
(1)	lim„_00/(u2) = ll/H.
We note that
||u„xu„ - xll = ||u„xu„ - xun + xun - xll
<	Ilu„xu„ - xu„|| + ||xu„ - xll
<	||u„x -xll ||u„|| + ||xu„ - xll
||u„x - x|| + ||u„x* - x*||.
Hence, limn_00 unxun = x. This implies that
lim f(unxun) = f(x).
П -» 00
Now, |/(u„xu„)| < ||x||/(u2). Thus, limsup„f(u2) > ||/||. Since ||u2||
< 1, we have limsupn_00/(u2) = ||/||. We may (and do) assume that {u2}
is also an approximate identity (Corollary 1,14.3.1). Thus, u2 + l - u2 > 0.
Hence, f(u2) <f(u2+l). Thus, lim„_00/(u2) = ||/||.
(2)	lim„ _„/(«*) = ||/||.
Indeed, Corollary 1, 14.3.1 implies that we can replace {мД with {u2}
in (1).
We now prove that ё/jYj is dense in ё/^. Indeed, (u2 - 1, u2 — l)y
= f(uj) - 2/(u2) + Ц/H. Thus, lim;^00(uJ2 - 1,u2 - l)y= 0. This implies
that limJ_00(uJ2 + ^) = 1 +
288
14. Abstract Representation Theory
We therefore see that Hf = Hf. Let £ e Hf correspond to 1 + Л[. If
ё has a unit, set £ = 1 +
(3)	<лу(х)£,	= /(x) for all x g ё.
Indeed, <лу(х)£, £>/ = (x, Dy = /(x) = /(x).
14.5.8.	We summarize the preceding in:
Theorem. Let f be a positive functional on ё. Then there exists a represen-
tation (тг, H) of ё and Ij g H such that fix) = (тг(х)^, for x g ё. If
ia, V) is a representation of ё and if v g V is such that а(-ё)о is dense in
H and fix) = (aix)v, v) for x g ё, then ia,V) is unitarily equivalent to
iirf,Hf).
All but the last statement has been proven. If x g then 0 = /(x*x)
= (aix*x)v,v) = iaix)v,aix)v). Thus, aix)v = 0. Hence, there exists
a linear map T of ё/^ into V such that Tix + ^Kf) = aix)v. Clearly,
T-n-fix) = aix)T for x g ё. Also,
(T(x + ^),T(x +	= (<r(x)r, a(x)v) = (<r(x*x)r, v)
= /(x*x) = <x+^,x + ^)/.
Thus, T extends to a unitary operator from Hf onto the closure of а{ё)о,
which is H.
14.5.9.	The next result combined with 14.1.13 implies that every C*
algebra is (* and norm) isomorphic with a closed *-invariant subspace of
End(H) for some Hilbert space H.
Theorem. Let ё be a C* algebra. Then there exists an injective *-represen-
tation irr, H) of ё.
If c is a cardinal we choose Hc, a Hilbert space of dimension equal to c.
Let S be the set of all representations of ё on some Hc. If tt,tt' g S,
then we write тг < тг' if Ker тт э Ker tt'. If T is a linearly ordered subset
of S then Ф^ e т тг on Ф^ e T is equivalent to some element a of S. a
is an upper bound for S. Ibis implies that S has a maximal element тг. If
x g Ker tt and x =# 0, then x*x g Ker тт Ci ё^ and x*x =# 0. Hence,
Theorem 14.5.5 implies that there exists a positive functional f with
14.5. Representations of C* Algebras and Positive Functionals
289
f(x*x) > 0. We note that ir Ф лу > ir but Ker (л- Ф лу) =# Ker тг. This
contradiction implies that Ker тг = 0.
14.5.10.	Let f be a positive functional on Then we will write f > 0. If
f, g are positive functionals on € and f - g > 0, then we will write f > g.
If f > 0, then f is said to be pure if whenever 0 < g < f there exits
0 < A < 1 such that g = А/.
Proposition, f > 0 is pure if and only if (-ny, Hf) is irreducible.
Suppose that 0 < g <f. Then if x e if, g(x*x) <f(x*x). This implies
that c . We can therefore define a linear map T from to
-&/^g by T(x +	= x + Also,
<x + ,x +	= f(x*x) > g(x*x) = (x + ^, x + ^)g.
Thus, T extends to a bounded operator from Hf to Hg with Ill’ll < 1. It is
clear that Tirfx) = тгg(x)T for x e
We now prove the result. Assume that тг^ is irreducible. Then T = 0 or
Ker T = 0. If T = 0, then g = 0 and the result is clear. So we may assume
that T is injective. We note that T*T e comm(^(if)). Thus, Theorem
14.5.1 implies that T*T = А/ for some A with 0 < A < 1. Hence, A/(x*x)
= g(x*x) for all x e The material in 14.1.12 now implies that g = А/.
Hence, f is pure.
Assume that f is pure. Let IK be a closed invariant subspace of Hf. Let
P be the orthogonal projection of Hf onto W. Set g(x) = (,Trfx)P^, Plf)f.
Then,
g(x*x) =||77/(X)P^||2 =||P77/(X)^||2 <Ь(ХИ||2 = f(x*x).
Thus, since f is pure, there exists 0 < A < 1 such that g = А/. This
implies that ||Pr|| = A1/2||r|| for all v e Hf. If A > 0, then this implies that
A = 1 and P = I. If A = 0, then P = 0. Hence, W = {0} or W = Hf.
Hence, TTf is irreducible.
14.5.11.	If f > 0 and is such that ||/|| = 1, then f is called a state. Set
£(if) = {/ > 0| H/H < 1}. Then, £(if) is a closed convex subset of
(continuous real functionals on -£h with the weak* topology).
If C is a convex subset of a real vector space X then a subset S of C is
said to be extreme if whenever x = Ay + (1 - A)z with y, z e C, 0 < A
< 1, y,z e S. x e C is said to be extreme if {x} is extreme.
290
14. Abstract Representation Theory
Lemma. The extreme points of Е(ё) consist of 0 and the pure states.
If f, g £ 0 and if А/ + (1 - A)g = 0 with 0 < A < 1, then Xf(x*x) +
(1 - A)g(x*x) = 0. Thus, f(x*x) = g(x*x) = 0 for all x e Hence
(14.1.12), f = g = 0. Thus, 0 is extreme. Suppose that f is a pure state and
that f= Xg + (1 - А)й, with g,h s Е(ё) and 0 < A < 1. Then f > Xg
0. Thus, there exists 0 g 1 with Xg = pf. If ё contains 1, then
ll/ll = /(l) = Ag(l) + (1 - А)Л(1) = A||g|| + (1 - A)||A||.
If f doesn’t contain 1 and if {иД is as in 14.5.7, then
ll<pll = lim <p(u?)
for <p 0. Thus,
ll/ll = A||g|| + (1 - A)||A||.
Since 0 < A < 1, this implies that ||g|| = ||й|| = 1. Hence, A = g. So g = /.
Thus, h = /. This proves that if / is pure then / is extreme.
Suppose now that / is extreme and f #= 0. If ||/|| < 1, then / = ||/||g +
(1 - ||/||)Л with g =//11/11, h = 0. Thus, ||/|| = 1. Now, assume that
0 < g < /. Then / - g s Е(ё\ and since / = g + (/ - g) the preceding
argument implies that
1 = llgll + ll/-gl|.
Thus, if 0 =# g =# / then ||g|| < 1, II/ - g|| < 1. We can therefore write
Since / is extreme g/llgll = /. Hence, / is pure.
14.5.12.	We now come to the crux of the matter. Set Р(ё) equal to the
set of all pure states.
Theorem. Let L be a closed, extreme, non-empty, subset of Е(ё) with
L =# {0}. Then L contains pure states. IfL is a closed, convex, extreme subset
of Е(ё) then L is the closed convex hull of L П (P(i^) U {0}). In particu-
lar, Е(ё) is the closed convex hull of Р(ё) U {0}.
14.5. Representations of C* Algebras and Positive Functionals
291
£(if) is compact. So L is compact. Let Q be the set of all compact
non-empty extreme subsets of L ordered by inclusion. If S is a linearly
ordered subset of Q then Ko = ПКеХ К s S defines a lower bound.
Zorn’s lemma implies that there exists a minimal element M s S. We
assert that M consists of exactly one element. Suppose that x, у e M and
x =#= y. Then there exists <p s ((т^УУ such that <p(x) =# <p(y) (Hahn-Banach
theorem). Let Mx = {z s M\<p(z) = infueA/ <p(u) = g}. Then Mx is com-
pact, non-empty and Mx =#= Л/. If u,v s £(^) and if Au + (1 - A)r s Mx
for some 0 < A < 1, then u, v s M (since M is extreme). Now, g = A<p(u)
+ (1 - A)<p(r) and since u,v s M, <p(u) £ g, <p(v) > g. Hence, <p(u) =
<p(r) = g. Thus, u,v e Л/j. Hence, Mx is extreme and Mx is properly
contained in M. This contradiction implies that M consists of exactly one
element, f. Since f =# 0, Lemma 14.5.11 implies that f is a pure state.
Let L be the closed convex hull of £ П (P(^’) U {0}). Suppose that
L =#= £. Let x0 e L - L. Then Mazur’s theorem (14.A.6) implies that
there exists s ((^УУ such that <p(x0) > 1 and <p(x) < 1 for all x e L.
Set g = supxe/ <p(x). Then, g > 1. Let KQ = {z s L\<p(z) = g}. Then,
as before, Ko is closed, extreme, and non-empty, and doesn’t contain 0.
The first part of this proof implies that Ko contains a pure state, z. Thus,
z & L. But then 1 < <p(z) < 1. This contradiction implies that £ = L.
14.5.13.	Corollary. Let x € -ё, x =# 0. Then there exists (тг, H), an irre-
ducible * -representation of -ё such that тг(х) =# 0.
Let z = x*x. There exists f > 0 such that f(z) > 1. Let g =
sup/en/(z). Then g > 0. Let £ = {/ e E(-£)\f(z) = g}. Then £ is
closed, extreme and 0 £ L Thus, there is a pure state f s £. (irf, Hf) is
an irreducible representation of ё and g =/(z) = <яу(х*х)£,	=
||'пу(х)£||2.
14.5.14.	Theorem. Let G be a locally compact, separable, topological
group. If g e G, g =# 1, then there exists an irreducible unitary representa-
tion (tt,H) of G such that ir(g) I.
Let Uj be as in 1.1.3. Let U be an open neighborhood of 1 in G such
that g £ U. Let W be an open neighborhood of 1 in G such that
xy~' c. U for all x, у e W. Let j be so large that suppuy c W. Set
<p = L(g)uj - Uj. Then, <p e Ll(G) and <p =# 0. Hence, there exists an
irreducible representation (it, H) of C*(G) (14.2.3) with тг(<р) =# 0. Since
ir(<p) = ir(L(g)uj) - tt(uj) ~ Tr(g)Tr(Uj) - tt(uj) =# 0, ir(g) =# I.
292
14. Abstract Representation Theory
14.5.15.	We will now look at an important example of a C* algebra. Let
H be a separable Hilbert space and let СС(Я) denote the algebra of
compact (completely continuous) operators on H (this algebra was de-
noted K(H) in 8.A.1). Then СС(Я) is the closure with respect to the
operator norm of the operators on H of finite rank. Hence, СС(Я) is a
C* algebra under the norm and *-operation of End(H). Let L^H) be the
subalgebra of CC(H) consisting of trace class operators (8.A. 1.9).
Lemma. IfT^ L^H), thenfT(X) = tr TXdefines an element of СС(ЯУ.
The map T fT is a linear, isometric, bijection of LfH) onto СС(ЯУ.
fT> 0 if and only if T > 0. fT > 0 is pure if and only if there exists v g H
such that T(x) = {x,v)v (i.e., fT(X) = {Xv, v)\
If T g Lt(H), then \fT(A')|	||X|| ||T||i (notation as in 8.A.1.9). Thus,
fT is continuous and ||/r|| < ||T||i. If v, w g H, then set Tv w(x) = (x,v)w.
Then, fT(Tv w) = (Tw, v). Thus, if fT = 0 then T = 0.
Now, let HS(H) denote the space of all operators of Hilbert-Schmidt
class (8.A.1.6). Then HS(H) is a Hilbert space with inner product (T, S)
= tr(S*T). Since HS(H) contains the finite rank operators on H, HS(H)
is dense in СС(Я). Thus, if f g СС(ЯУ then f is determined by its
restriction to HS(H). Since {T, T) > Ill’ll2, we see that f induces an
element of Н8(ЯУ. This implies that there exists Tf g HS(H) such that
f(X) = tr(7}X) for X g HS(H). If we can show that Tf g then we
will have proved surjectivity.
If f g ССХНУ, then define f*(X) = f(X*). Then f = (/ + /*)/2 +
i(f - f*)/2i. To complete the proof of surjectivity, we need only show
that if f*=f then T = Tf is trace class. So assume that T g HS(H),
T* = T, and f(X) = tr(7X) for X g HS(A'). There exists an orthonor-
mal basis {r„} of H such that Tvn = A„r„. Let P„(x) = <x, vn)vn. Set
= 1 if Ап > 0 and en = -1 if An < 0. Set Xn =	Then,
f(Xn) = Lisn |AJ. Since ||X„|| = 1, this implies that |A;|	||/||.
Hence, T is trace class, so f = fT.
We must now show that ||/r|| = ||T|li. Let {e„} and {/„} be orthonormal
bases of H. Let g C be such that ||£„|| = 1 and £n{Ten,fn) > 0. Let
Xn(x) = E/s„	Then, U„|| < 1 and fT(Xn) =	I <7^, Л> |.
Thus, ||/r|| > Lisn \{Tei,fi)\. 8.1.A.7 now implies that ||T||i < ||/r||.
If T g LfH) and T > 0, then T = S2 with 5* = S and S g HS(H).
Thus, if X g СС(Я), fT{X*X) = Ш2^*^) = txfSX^XS) S: 0 since
SX*XS > 0. If fT > 0, then {Tv, v) = fT(Tv v) > 0 so T > 0.
14.5. Representations of C* Algebras and Positive Functionals
293
If fT > 0, then let {r„} be an orthonormal basis of H such that
Tvn = A„r„. Then, f(X) = E„al kn(Xvn,vn). Thus, if f is pure there can
be only one non-zero term. If v s H, f(X) = (Xv,v) for all X s СС(Я),
and if 0 < g < f, then g(w) = 0 if (w, v) = 0. Thus, g = Л/. So f is pure.
14.5.16.	We call the representation тг(Х) = X the identity representa-
tion of СС(Я).
Corollary. If (тг, Hn) is an irreducible * -representation of СС(Я) then it
is equivalent with the identity representation.
Let v e H„, v*0. Set f(X) = (tt(X)v, v), X e CC(H). 14.5.1 (3)
implies that tt(CC(H))v = H. 14.5.8 implies that (tt, is unitarily
equivalent with (тту, Hf). Thus, f is pure. This implies that there exists
и e H such that f(X) = (Xu,u). Since it is obvious that the identity
representation of СС(Я) is irreducible, the preceding argument implies
that тг is equivalent with the identity representation.
14.5.17.	Theorem. If & is a closed, ^-invariant subalgebra of CC(H)
acting irreducibly on H, then & = СС(Я).
On End(H) we define the ultra strong topology to be the topology
induced by the semi-norms (Е||ТЬ„||2)1/2 for El|f„l|2 < ». Theorem 14.4.1
implies that Si is ultra strongly dense in End(H), hence in СС(Я).
If f s СС(ЯУ and if /* =f, then there exists an orthonormal basis
{e„} of H and AneR such that ElA„| < » and f(X) = E„ Xn(Xen,en)
(14.5.15 and its proof). Thus,
|/(X)| < E|A„|||XeJ= LlA„|,/2|A„|1/2||Xe„||
n	n
/	X1/2/	\1/2
EIA„I EIAJUM2 .
'л	7	' n	'
Thus, f is continuous in the ultra strong topology on СС(Я). Thus, if
f(&) = 0 then f = 0. This implies that 31 = СС(Я) by the Hahn-Banach
theorem.
14.5.18.	Corollary. The only closed two sided ideals in CC(H) are CC(H)
and {0}.
294
14. Abstract Representation Theory
If «X is a two sided ideal in СС(Я) and if v s H, then ,^v is
СС(Я)-invariant. Thus, either Л = H or = 0 (the action of СС(Я)
on H is algebraically irreducible). Let V = {v e H\^v = 0}. Then, V is
CC( H)-invariant. Thus, V = H or V = {0}. If V = H, then ,^= {0}. If
V = 0, then acts irreducibly. Thus, the previous result implies that if
jr* 0 then CC(H).
14.5.19.	Theorem. Let ё be a separable C* algebra. Let (тг, H) be an
irreducible representation of ё. Then H is separable. If тг(ё) c CC(H),
then тг(ё) = СС(Я). Furthermore, if is a closed two sided ideal in ё
properly containing Ker тг, then ё.
Let {*„} be a countable dense subset of ё. Let v =#= 0, v s H. Set
S = {тг(хп)о}. Since ||тг(х)г|| < ||r|| ||x||, S = тг(ё)о. Thus S is closed
and invariant. If тг(ё) = 0, then H is one dimensional. Otherwise, there
exists v such that тг(ё)о =#= 0. Thus, S = H. So H is separable. Lemma
14.1.13 implies that тг(ё) is closed in End(H). Thus, since тг(ё) c
СС(Я), тг(ё) is closed in СС(Я). 14.5.17 implies that тг(ё) = СС(Я).
Let J’ be a closed two sided ideal in ё that contains Ker тг. If
Ker тг, then there exists v & H such that rr(^F)v is non-zero. Since
tt(.^)v is invariant under тг(ё), we see that ir(^)v = H (14.5.1) if
tt(^F)v =# 0. Let V = {v e H\tt(^F)v = 0}. Then И is a closed subspace
of H that is clearly тт/т^ ^invariant. Thus, since V =#= H, V = 0. Thus,
тг(^) acts irreducibly on H. Since is *-invariant (14.3.2), 14.5.17
implies that tt(^F) = СС(Я). But then tt(^F) = тт(-ё), Кегтг c^, so
ё.
14.6.	The topology on the unitary dual of a C* algebra
14.6.1.	Before we introduce the topology on the unitary dual we will first
derive a rather surprising result. As usual, ё will denote a separable C*
algebra.
Theorem. Let (p,V) be an irreducible representation of ё as an abstract
algebra (i.e., Vis a vector space over C, p is a homomorphism of ё into the
algebra of all linear maps of V to V, and if v s V, v =# 0, then р(ё)о = V).
Then there exists an irreducible * -representation (тг, H) of ё and a bijective
linear map T of V onto H such that Tp(x) = тг(х)Т.
14.6. The Topology on the Unitary Dual of a C* Algebra
295
This result will involve some preparation. First of all, we must study the
relationship between closed left ideals and positive functionals.
14.6.2.	Lemma. Let L be a left ideal in ё and let f be a positive
functional. Then the following are equivalent :
(1)	f(L) = 0;
(2)	f(L A <,) = 0 (14.1.12);
(3)	L (14.5.6);
That (1) implies (2) is obvious. Assume that f(L A ^,) = 0. If x e L,
x*x e L so /(x*x) = 0 and thus x e Hence, (2) implies (3). Suppose
that L c Then if x e L, u*x e L and thus /(u*x) = 0 for all и e ё.
Let {и,} be an approximate identity for ё (Lemma 14.3.1). Then
/(x) = lim f(u x) = 0
since и* = иj. Thus, f(L) = 0. Hence, (3) implies (1).
14.6.3.	Proposition. Let L', L be closed left ideals in ё with U c. L.
Assume that whenever f is a positive functional on ё with f(L') = 0,
f(L) = 0. Then L = L.
We may assume that ё= ё (14.1.11) since f has a unique extension to
ё for each f > 0.
(1)	If L А ё+ = L А ё + , then L = U (notation as in 14.1.12).
Indeed, let {uy} be an approximate projection for L (14.3.1). Then
Uj & L C\ ё+ . Hence, e L. Let x e L. Then lim^^ xu} = x. Thus,
x e U. So L c U.
Let a e L А ё+ , a =#= 0. We will show that a e L. Let e > 0 be fixed.
Put
5 = {/>0|/(l) = 1}
and
= {/> 0|/(l) = 1,/(a) > e}.
Then, Se is compact in the weak* topology. If / e Se and f(L) =#= 0, then
f(L') =# 0 by our hypothesis. Hence, there exists xf e L such that f(xf) =#=
0. Set bf = x*xf. Then, f(bf) > 0 and bf e L'. Set
Uf={geSe\g(bf)>0].
296
14. Abstract Representation Theory
Then, Uf is open in Se and
U Uf=Sf
f*se
Since SB is compact there exist j\,..., fn such that
U
isisn
Set b = Ei£„ bf,. Then, f(b) > 0 for all f e Se. Let
A = min{/(b)|/e 5e},
g = max{/(a)|/e 5e}.
Then A > 0 and g e. Let r > 0 be such that rA > g. Then f(rb) > f(a)
for all f e Se. Set ce = rb.
(2)	/(ce) + e > f(a) for all f e S.
Indeed, if f(a) < e this is clear. If f(a) > e, then it is true by construc-
tion.
Let (тг, H) be an injective *-representation of ё (14.5.9). If v e H,
||u|| = 1, then define fv(x) = (ir(x)v,v). Then, fv e S. Thus,
(ir(ce +e)v,v) =f„(ce) +e>fv(a) ={ir(a)v,v).
This implies that ce + e a. We will now prove that this implies that
a e L'.
Since ce > 0 and L' is a closed left ideal containing ce, it follows that
<p(ce) e U for all <p e C(R+), <p(0) = 0 (14.1.11). Set te = (ce)1/2. Then,
2	2
II+ »,/!)_|<. - HI - »ll«,/!('.+•l/!)’,ll
since ||u*u|| = ||u||2.
Now,
(te +	+ Е^У1 < (te + E^yXt* + £)(te +	L
So
||((,	+ .I'T'|| 1.
14.6. The Topology on the Unitary Dual of a C* Algebra
297
We conclude that
Hence,
ai/2 = lim a1/2(te + e1/2)-1te e L'.
e -»0
Thus,
a e L'.
14.6.4.	Corollary. Let L be a closed left ideal in ё. Then
L- П
/(/.)-()
/ao
f pure
14.5.12 implies that
n	Л Л-
/(L) = 0	/(L) = 0
/ao	/>0
f pure
Indeed, the set {/ > 0| ||/|| < 1, f(L) = 0} is extreme (14.5.11). Let
= А -Л •
/(L) = 0
/SO
Then Lx is a closed left ideal and L c Lx by Lemma 14.6.2. Hence,
L = L, by the previous result.
14.6.5.	We now complete the proof of Theorem 14.6.1. We may assume
that ё has a unit. Let v e V, v =# 0. Set L = {x s ё\р(х)и = 0}. Then,
L is a left ideal. Since p is irreducible, it follows that L is a maximal left
ideal. Hence, L is closed (Corollary 14.1.1). The preceding corollary
implies that
А -Л= A
/>0	/>()
f pure	f pure
But L is maximal, and hence there exists f >0, f pure, such that
L = Since f is pure, Theorem 14.5.1 implies that Hf = •ny(^)^ =
298
14. Abstract Representation Theory
-ё/Л^. Thus, the map T given by
T(x +	= p(x)r
defines a linear bijection of Hf onto V. Clearly, Tttj(x) = p(x)T for all
x e ё. This completes the proof.
14.6.6.	Let s& be an associative algebra over C. If is a two sided ideal
in srf, then J is said to be primitive if there exists an irreducible,
non-zero, representation (тг, И) of лэ/ such that Ker tt. Set Рпш(лэ/)
equal to the set of primative ideals in лэ/.
If S is a subset of Ргпп(лг/), then we set
/(5)= QA
>eS
We set S = {/e Рпт(лз/)|/(5) c ^/}. We note that 1(0) = лэ/, so
0 = 0.
Lemma.
(1)	0 = 0.
(2)	£ c S_ _
(3)	S = (S) .
(4)	(J S2 = 5i U S2.
(1)	has already been noted. (2) and (3) are obvious since I(S) = I(S). To
prove (4), we note:
(5)	If Рпт(лз/) and if I, J are two sided ideals in лэ/ such that
IoJcJ, then I c or J с.
To prove this, let (тг, V) be an irreducible *-representation of лэ/ such
that Ker тг = Suppose that I is not contained in If v e V, then
rr(I)v is лэ/invariant and there exists v such that rr(I)v =#= 0. Thus,
tt(I)V = V. If J is also not contained in then tt(J)V = V. This implies
that tt(JI)V = V. But JI c This contradiction implies that J c.
We now prove (4). If </e 7\ U T2, then I(1\) П I(T2) and hence
I(T\) or I(T2). Thus, J’e or УеТ2. Hence, 7\ U T2
c Tj U T2. Since I(T\ U T2) э I(I\ U T2), it is clear that T\ (J T2^>
Tj U f2 .
14.6.7.	The previous lemma implies that there is a unique topology on
Рпт(лэ/) whose closed sets are those S such that S = S. This topology is
14.6. The Topology on the Unitary Dual of a C* Algebra
299
usually called the Jacobson topology. If S c stf, then set Ts =
Prim(ja/)|5 сУ).
(1)	If S c then Ts is closed in Prim(ja^). Furthermore, every closed
subset of Prim(<o/) is of this form.
This is obvious.
Lemma. Prim(ja/) is a To topological space with respect to the Jacobson
topology (i.e., if x,y e Prim(ja/), x =#= y, then there is a neighborhood of
one that doesn’t contain the other).
Let x =# y, x, у e Prim(ja^). Suppose that x is not contained in y. Then
у £ Tx. Thus, у s Prim(ja/) - Tx, which is open.
14.6.8.	We now assume that € is a separable C* algebra. Let ё be the
set of unitary equivalence classes of irreducible *-representations of ё. If
ы e ё, then define к(ы) = Ker tt for тт e ш. Then, 14.5.1 and 14.6.1
combine to imply that:
(1)	k(-&) = Primfif) (the latter is as an abstract algebra).
We pull back the topology of Prim(if) using к to define a topology on
ё. This is the topology that will be used throughout the rest of the
chapter. If (тг, H) is an irreducible ^representation of if, then set [тг]
equal to the unitary equivalence class of tt.
(2)	ё is To if and only if к is bijective. That is, if tt, tt' are irreducible
representations of ё such that Ker tt = Ker тг’, then [тг] = [тг'].
This is also clear.
We now introduce a class of C* algebras such that к is bijective.
14.6.9.	ё is said to be CCR (completely continuous representations) if
whenever (тт, H) is an irreducible *-representation of ё, тг(ё) с CC(H).
This class is also called liminaire in Dixmier [1].
Lemma. If ё is CCR, then к is a bijection.
Let (тт,, Я,) and (tt2, H2) be irreducible representations of ё. Assume
that Кегтт, = Кегтт2. Since ё is CCR, 14.5.19 implies that тт^ё) =
300
14. Abstract Representation Theory
СС(Я,) for i = 1,2. Thus, since ё is separable, -if/Кегтг, = СС(Я) for
i = 1,2 and H is a fixed separable Hilbert space. 14.5.16 now implies that
тг is equivalent with тг2-
14.6.10.	As a consequence of Harish-Chandra’s basic theorem (3.4.10),
we have:
Theorem. Let G be a real reductive group. Then C*(G) is CCR.
Let AT be a maximal compact subgroup of G. Let Ll(G)K denote the
space of right and left К-finite functions in Ll(G). Then Ll(G)K is dense
in Ll(G). Indeed, C“(G) П /ЛОд- is dense in C“(G) (see the proof of
Theorem 7.1.1) and C“(G) is dense in Z?(G).
Let ш g C*(G)a . Then ш gives an element of d’(G) (14.2.5); here
d’(G) is, as usual, the set of equivalence classes of irreducible unitary
representations of G. If (тг, H) g ш, then Theorem 3.4.10 implies that тг
is admissible. Thus, rr(f) is of finite rank for all f L1(G)K. Since
tt(L\G)k) is dense in tt(L1(G)), which is dense in tt(C*(G)), we see that
tt(C*(G)) is contained in the operator norm closure of the finite rank
operators on H. Thus, tt(C*(G)) с СС(Я).
14.6.11.	In the next section, we will study the preceding topology in more
detail for locally compact topological groups. In this section, we will derive
relations between the topology and positive functionals. We will use this
relationship to derive more properties of the topology in the abstract
setting. In the next section, these results will be used to give an intrinsic
definition of the topology for locally compact groups.
If Q с £(тГ) (14.5.11), then set Q = {f\f g Q} (see 14.5.4 for /). We
put Р(ё) equal to the set of pure states on ё (14.5.11).
Lemma. Let Q be a subset of Е(ё) such that if x g ё, x = x*, and
f(x) > 0 for all f g Q, then x > 0. Then:
(i)	The weak* closure of Q contains Р(ё)~.
(ii)	The weak* closed convex hull of Q (J {0} is Е(ёУ.
We may assume that ё = ё. We look upon Е(ё) as a subset of
ё'к = V. Then V = ёь. Let be as in 14.A.7. Then x g Q° if and only
if f(x) < 1 for all f g Q. Hence, x g if and only if /(1 - x) > 0 for
14.6. The Topology on the Unitary Dual of a C* Algebra
301
all f g Q. Thus, Q° = {x g -£h\x < 1}. This implies that
(Q°)° = {/g <;|/(x) < 1 if x <1}=E(^).
This proves (ii). Thus, 14.A.7 implies that the closed convex hull of
Q U {0} is £(if). Since the pure states are extreme, Q d P(if).
14.6.12.	If (тг, H) is a representation of and if v g H, then we define
fv g by /,,(x) = (tt(x)v, v). If ы g # and (тг, f/J g ш, then set
£(w) = {/„If g Hn, ||l>|| < 1}. If S c #, then define £(5) = 11шеу E(to).
Theorem. Let S be a subset of -<?. Let ы g and (tt, Ю g a). Then the
following are equivalent :
(i)	ш g S.
(ii)	There exists rGH^JIul^l, such that fv is in the weak* closure of
E(SY.
(iii)	If v & Ill'll = 1, then f, is in the weak* closure of E(S)'.
(iv)	There exists v g Hn, Ill'll = 1, such that fi: is a weak* limit of linear
combinations of elements of E(S)~.
(v)	If v g Hv, then f. is a weak* limit of linear combinations of
elements of E(S)~.
We may assume that -£= 0?. We first note that (ii) implies (iii). Indeed,
let v g Hv, Ill'll = 1, and fv g £(5). If и g then и = ir(y)v for
some у g since тг is algebraically irreducible. Hence, fu(x) =
(тт(х)и,и) = <тг(х)тг(у)г,ir(y)v) =fu(y*xy). If /g £(5), then (x ->
f(y*xy)) g E(S).
The same argument shows that (iv) implies (v). Hence, (ii) and (iii) are
equivalent and (iv) and (v) are equivalent. It is therefore enough to show
that (i) implies (iii) implies (v) implies (i).
Assume (i). Then, Ker тг d Kero-. Fix (тг,,, Ha) g a. Let HT be
the Hilbert space direct sum of the Ha and let r be the action given by the
direct sum of the тта. Set srf= -&/P\a^s Kero-. Then, r is an injective
homomorphism of s# into End(K). Thus, 14.1.13 implies that т(зэ/) is
closed and т is an isomorphism of srf onto r(<o/). If f g E(S), then
Ker / э По-еу Kero- and thus we may look upon E(5) as contained in
. Suppose that x g and /(x) > 0 for all / g E(5). Then тга(х) > 0
for all a g S, and hence r(x) > 0. Thus, x > 0. The previous lemma now
implies that E(S) э P(srf). Thus, (i) implies (iii).
302
14. Abstract Representation Theory
Clearly, (iii) implies (v). We complete the proof by showing that (v)
implies (i). So suppose that E(a>) is contained in the closed convex hull of
E(S). This implies that if f g E(a>), then /(Пстеу Kero-) = 0. Hence, if
x g ПаKero-, then <ir(x)r, v) = 0 for all v g Hence, tt(x) = 0.
So Ker тт э X e s Ker a-
14.6.13.	Lemma. Let x g -£h and let S be a closed subset of R. Then
Z = {ш g ё\ 8ресЕш](Нш)(л-ш(х)) c
is closed.
We may assume that ё= ё. Let ш g Z. Assume that there exists
a g spec(irw(x)) with a S. Then, there exists a continuous, real valued
function f on R with /(a) =# 0 and f(S) = 0. Now, /(ттш(х)) = ттш(/(х)).
Since irp(/(x)) = 0 for all p g Z, f(x) g QpeZ Kerp. The definition of
the topology on ё implies that /(x) g Ker w. Thus, ттш(/(х)) = 0. This is
a contradiction. Thus, there is no such a. So ш g Z.
14.6.14.	A topological space X is said to be quasi-compact if it satisfies
the usual definition of compactness but is not necessarily a Hausdorff
space.
Theorem. « is quasi-compact.
Let ш g ё. Let x e ё + with ||тгш(х)|| > 0. Then the previous result
implies that Z° = {77 g /1 ||ir4(x)|| > |||irw(x)||} is open. Set a
= 111тгш(х)||. We now show that Z = {77 g ё\ ||тгш(х)|| > a} is compact.
Let I be a directed set and let Zt be a relatively closed non-empty subset
of Z for i g I, with Zj э Zj if i < j. Set
J,= A Ker 77.
Then, J, is a closed two sided ideal in ё and f c Jj if i < j. Let pt be the
canonical projection of ё onto the C* algebra . If у g ё, then set
llylli = llp(-(y)ll^/j(. Then, llyllj < llyllj if j > i. On the other hand, if
77 g Zj then ttv induces a representation of ^/Jj. So a < ||тг (дс)|| < ||x||;.
Let J be the closure of the union of the Jt. Set ||y|L = Нт;^«, ||у||;,
у g /. Then, || • • • IL is a continuous semi-norm on ё and || JjlL = 0 for
all i. This implies that ||J|L = 0. Since ||x|L > a, x £ J. We note that
14.6. The Topology on the Unitary Dual of a C* Algebra
303
lly*lloo = llylloo for all у g < ||ur||«, < ||u||«,||i;||oo, u, v g < ||u*u||«, = ||u|£
for all и g ё. Thus, if J' = {у g ё\ Hyll», = 0}, then J' is a proper, closed
two sided ideal in ё and, with respect to || • • • |L, ё/Г is a C* algebra.
Let ё' = ё/J'. There exists /3 > a, fi g spec^-(x + J'\ Thus, (x + J')
- /3 is contained in a maximal proper left ideal L in ё'. ё' /L defines an
irreducible (abstract) representation of ё', thus an irreducible *-represen-
tation (тг', H) of ё’. We pull this representation back to ё and denote it
by тг. Then, ||ir(x)|| > /3 £ a. Since тг is irreducible, [тг] g П, Z,-
14.6.15.	For the remainder of this section, we will assume that ё is
CCR.
Proposition. If is a closed two sided ideal in ё and if (тг, H) is an
irreducible * -representation of	then тг extends to a * -representation of ё
on H. In particular, is CCR.
Let v g H, v =# 0. Set L = {x g ^\rr(x)v = 0}. Then, as an abstract
representation of (тг, Я) is equivalent to the action of left translation
on S/L (14.5.1).
(1)	If x g ё, then xL c L.
Indeed, xL c Let {мД in L be an approximate projection for L in
(14.3.1). Thus, if у g L then xy = lim^», xyu; g L.
We define an action a of ё on by <r(xXy + L) = xy + L. Then a
is an irreducible abstract representation of ё, so 14.6.1 implies that there
is an irreducible representation (ttj, Hf) of ё as a C* algebra equivalent
with a. Since ё is CCR, тг,(^) = CC(Hf). Thus, ^/Кегтг = CC(Hf.
This implies that тг = тг,^(14.5.16). We may thus assume that ir1|>.= тг.
14.6.16.	Set ёР = {x g ё1тгш(х) is of finite rank for each ш g #}. It is
clear that ёР is a two sided ideal in ё.
Lemma.
(1)	ёр contains a countable dense subset fl of ё.
(2)	If S is a closed two sided ideal in ё, then = ёРС\ J.
304
14. Abstract Representation Theory
Let S be a countable dense subset of If x g S, then x = u(x) + iv(x)
with u(x), v(x) g -£h. Let T = {y|y g {u(x\ r(x)}, x g 5}. If x g -£h and
if тге to e then тг(х) is compact. Thus, the only point of accumulation
of spec(Tr(x)) is 0. Let <p„ be a continuous function on R such that
0 < <p„(t) < 1, <p„(t) = 0 for |t| < 1/n, <p„(t) = 1, |t| > 2/n. Set /„(t) =
Then,
|/„(t) — t| < 2/n
for all t g R. Thus, the uniform limit of the fn is the function /(t) = t. If
x g -£h, then /„(t) has finite rank. Set
v= U e t}.
n
Then, fl = {x + zy|x, у g K} is dense and contained in .
Let be a closed two sided ideal in Let x g ,^f. If wg / and
tt g ы, then TT|^= 0 or тг^ is irreducible. Hence, тг(х) is of finite rank.
So Уу c -&F . If x g -&F n and if to g У, тг g to, then tt extends to a
representation of tt-j (14.6.15). Thus, тг(х) is of finite rank.
14.6.17.	If ы g -%, tt g ш, x g -£f, then set
0ш(х) = trir(x).
We will call 0Ш the character of ш. We look upon 0Ш as a linear
functional on Notice that -&F has not been endowed with any
topology.
Theorem. If tou..., to„ are distinct elements of -io, then 0,,,..., 0,, are
J 1’	1 П	J >	Ш j’	’ Шп
linearly independent.
We prove this by induction on n. If n = 1, then we must show that
0Ш =# 0. If 0Ш = 0, then
0 = ®ш(х*х) = 1гтг(х)*тг(х) = ||тг(х) ||hs
for all x g -&F and тг g ш. Thus, тг(^) = 0, тг g ш. Thus, тг(^’) = 0,
тг g ш. Assume the result for n - 1 > 1. Fix тг, g Set = Ker тг;.
(1)	77,1^ is irreducible for all 1 < i <, n - 1.
Indeed, if тг,(^) = 0 then c 14.5.18 implies that	But
then a>i = ы„ (14.6.9). Thus, тг,(^) ¥= 0 for i < n - 1.
14.6. The Topology on the Unitary Dual of a C* Algebra
305
In particular, this implies that ir,^ is irreducible for i < n - 1. Since
ir,(~^) c CC(H~ ), this implies that тг,(^) = СС(ЯТ.) for i < n - 1.
(2)	['’’Vj * for 1 < i * j < n - 1.
In fact, if [ir^] = then 14.6.9 and 14.6.15 imply that П =
Hence, /cJ' or	But this is impossible
(14.5.18).
The inductive hypothesis implies that there exist x„..., x„_, e ^F =
-£F such that 0ш(х,) = 3,7. If я,0Ш1 + • • • +я„0Шп = 0, then since
0ш (x,) = 0, at = 0, i = 1,..., n - 1. Hence, the case n = 1 implies that
a„ = 0.
14.6.18.	If X is a topological space and if f is a real valued function on
X, then f is said to be lower semi-continuous if for every a e R, {x s
X\f(x) > a} is open. Lemma 14.6.13 implies that if x e £, then the map
w -> ||тгш(х)|| is lower semi-continuous on $ (apply that lemma to x*x).
Theorem. If x e -£F, x > 0, then ы -> 0ш(х) is lower semi-continuous
on ъ.
Let a e R and let w() e # be such that 0Ш(1(х) > a. Let (тг0, Ho) e ш0.
Then
'”()(*)= E atpi’
>=i
with P, an orthogonal projection of finite rank, a, > 0, and P,P7 = 0 if
i ¥= j and dj ¥= if i =# j. If n, = dim PtHQ, then
« < <MX) = Ем,-
i
Select <p,, a continuous function on R, such that 0 < <p,(t) < 1 for all t,
<f>faf) = 0 if j ¥= i, and there exists 3 > 0 such that <p,(t) = 1 for |t - a,I
< a. Set /,(t) = t<p,(t). Then, тт0(/{х)) = aft. Set u, = (l/a,)/((x).
Let vi X,...,vin, be an orthonormal basis of P,H0. Let be the
orthogonal projection of Ho onto Ct\y. Choose q^ e € with тг0(<70) =
Ptj. If Zii = q^qi), then 77()(z0) =Ptj. Let, for each n = 1,2,..., wijn
= г^(ги + l/n)-'. Finally, set pij n = uy2wij nu'/2. Then, 7r0(p17,„) =
(n/(n + 1))Fi7. Choose e > 0 such that Еп,а, - me > a. Let n be so
306
14. Abstract Representation Theory
large that (n/n + Da, > a, - e. Fix ptj = plhn- Then,
X >	Y,aiPij-
i i,j
Let
{a-e\
ui /
Then is closed (14.6.13) and <u0 £ Z,; for all i, j. Let U = £- U,, Z(>.
If ш g U, then
fli@a>(Po) * «i|M Pij) || > ai ~ £-
Thus,
®ш(х) £ £«,©„(«,) > 2>,Ш) > E«, -me>a.
14.7.	The topology on the unitary dual of a locally compact group
14.7.1.	Let G be a locally compact, separable, topological group. We fix a
left invariant measure dg on G. Let C*(G) be the C* algebra of G
(14.2.3). Then C*(G) is a separable C* algebra (14.2.4).
If и g C(G) (continuous complex valued functions on G), then и is said
to be positive definite if, whenever gv..., gn g G, zlt..., zn g C,
'g,) 2: 0.
i,j
(1)	If f is positive definite, then f(x *) = /(x) and |/(x)|	/(1) for all
X G G.
Indeed, if n = 1, g1 = 1, z1 = 1, then the definition implies that /(1) >
0. Also, if gj = 1, g2 = x, then the matrix
/(1) /(*)
A*-1) /(1)
is Hermitian and positive semi-definite. Thus, f(x *) = /(x) and
/(I)2-|/(x)|2 0.
14.7. The Topology on the Unitary Dual of a Locally Compact Group
307
We now do a variant of the GNS construction (14.5.6). Let и be a
positive definite function on G. We assume that и ¥= 0. Let
V = spanc{f?(g)u|g e G}.
Here (as usual), R(g)f(x) = f(xg).
(2)	If v = v^g/fu, w = Ef=1 w,Жg,)u, then a = L^jW^jUtg^gj)
depends only on v, w.
Indeed, a = Ef=, w,(L(g,)i>Xl). So it depends on v (and not the
expression for v). Since [a,J = [«(gf1#,)] is Hermitian (1), a =
E?=1 Df(L(g,)wXl).
We define, for v, w s V, written in the form in (1), (v, w) = a with a as
in (2). Then, ( , ) is a positive semi-definite Hermitian form on V. Let
Z = (z e И|(г,Ю = 0). Let < , ) be the corresponding inner product on
V/Z. Then < , ) defines a pre-Hilbert space structure on V/Z. We set H
equal to the Hilbert space completion of V/Z. Let p be the canonical
projection of V onto V/Z.
(3)	If v e V then (f?(g)r, R(g)v) = (u, v).
If v = E/=1 viR(gi)u, then R(g)v = Ef=1 r,J?(gg,)u. Hence,
(R(g)v, R(g)v) = 'LiJvivju(g/xg~xgg) = (v,v).
(3)	implies that R(g)Z c Z for g e G. Thus, f?(g) induces an operator
on V/Z, which extends to a unitary operator on H for g e G. If v, w е V,
then <Tr(g)p(r), p(w)) = 'L^=iwi(L(gi)v')(.g). Thus, (тг, H) is a unitary
representation of G (i.e., it is strongly continuous (1.1.2)).
(4)	(Tr(g)p(u), p(u)) = u(g\
This is clear from the formula in (3).
We have proved:
Proposition. If и is a non-zero positive definite function on G, then there
exists a unitary representation (тг, H) of G and v e H such that u(g) =
(rr(g)v,v).
308
14. Abstract Representation Theory
14.7.2.	If и is a positive definite function on G and if <p g LX(G), then
we set
/и(ф) = / u(g)<p(g)dg.
If (тг, H), v are as in Proposition 14.7.1 for u, then we can define
fu(x) = (тг(х)г, v) for x e C*(G) (here, тг also denotes the extension of
тг from L4G) to C*(G)). Then, the two definitions are consistent. We
have
|/„(x)| < <r,f>||x|| = u(l)||x||,
with || • • • || the C* algebra norm.
Let ^(G) be the set of all positive definite functions on G. Let
^(C*(G)) be the set of all positive functionals on C*(G).
Lemma. The map и -> fu defines a linear bijection between £P(G) and
W?*(G)).
If fu = 0, then
[ u(g)<p(g) dg = 0
JG
for all <p g Ll(G). This implies that и = 0. If f e ^(C*(G)), then there
exists a representation of C*(G), (ir, H), and v g H such that
{tt(x)v,v) =f(x)
for x g C*(G). We also denote by тг the corresponding unitary represen-
tation of G. The function
“(g) = <ir(g)u,u)
is clearly positive definite and fu=f (see Section 14.2).
14.7.3.	Set
E(G) = {/g^(G)|/(1) <1),
endowed with the topology of uniform convergence on compacta. Set
E(C*(G)) = {f g ^(C*(G))|/||/H 1},
endowed with the weak* topology in (C*(G)')'. If f g E(C*(G)) and
f = fu, и g E(G), then set T(f) = u. We will use the notation of 14.6.11.
14.7. The Topology on the Unitary Dual of a Locally Compact Group
309
Lemma. T is a homeomorphism of E(C*(G)) onto E(G).
We first show that T is continuous. Let и e E(G), Q c G compact, and
e > 0. We must find a weak* neighborhood U of fu such that if /’ s U,
/Д1) < 1, then |u(x) - r(x)| <« forxefi.
Let 7] > 0. Let V be an open neighborhood of 1 with compact closure
in G such that V = V~1 and
|U(X) - U(l) | < 7]
for x e K. Let x be the characteristic function of V. Set
« = IIa'IIi = [ dg < <x>.
Jv
If v s E(G), then
AU) = /v(x) dx.
Jv
Let (/, = {/£ C*(G)'|/ > 0, \(f -fM\ < ar)}. If v e E(G) and fv e
Ut, then
u(l) - v(x)) dx
< 7]a + 7)a < 2t]O.
Now (x(t) = x(t ’)),
a 1 / v(xt)x(j) dt - v(x)
JG
= a
I (v(xt) - v(x)) dt
v
There exists a unitary representation (тг, H) of G and s H such that
310
14. Abstract Representation Theory
v(x) = (тт(х)£, О for x g G. This implies that
k(y) - ^(*)|2 = К'”(>’)£,£>	£>l2
= K('"’(y) - '”(*))£.£>12
Н£||2(||'п-(*)£||2 - 2Re(-n-(x-1y)£, + IIir(у)£||2)
= 2r(l)(r(l) - Rer(x-1y)).
Hence,
|r(xt) - г(х)I (2r(l))1/2(r(l) - Rer(t))1/2.
Thus,
l«-1o*Ar)O) - f(x)l
<, a~l(2v(l))l/2 f (r(l) - Rev(t))l/2dt
(	\l/2l	\l/2
< (2r(l))1/2a-1 [id Н(р(1) - Rer(t)) dt\ .
Vk	/ IV	/
Let U2 = {f s E(C*(G))| |/(1) - u(l)| < -tj}. Then U2 is open in
^(C*(G)) and if fv g G1 П U2, then
Ia~4v *x)(x) ~ f(x)l (2(1 + v))1/2a~1at/2(2i]a),/2
= 2(1 + v)172^2.
We assert that there exists a neighborhood U3 of fu in the weak* topology
such that if /” g U3 and ||/J| 1, then
|a-1u*x(x) - a-1r *^(x) | < r)
for all x g fl. Indeed,
= f v(t)x(t~lx) dt = f v(t)x(x lt) dt =fv(L(x)x)-
JG	JG
The set L(fl)^ is a compact subset of Ll(G). We show that there is a
neighborhood IK of 0 in С*(СУ such that if f s W, then |/(L(x)^)l < e
for all x s fl. If not, then there exists a net {/J in C*(G7 such that
lima/a = 0, ||/J| 1, and a net {xj in fl such that l/a(L(xa)^)| > e.
14.7. The Topology on the Unitary Dual of a Locally Compact Group
311
Since fl is compact, we may assume that lim„ xa = x. We now note that
\UL(XM ~fa(L(x)x)\	-L(x)x||1.
Thus, there is a0 such that if a > a0, then
|Л(М Xa)x) -/„(L(x)x)| < l)/2.
But then |/„(L(x)^)| > 17/2 for all a > a0. This is impossible.
Set U3 = W + fu.
We now assume that fv e t7, П U2 П U3. If x e fl, then
|u(x) - v(x) I ^|u(x)
+ |a“'u*x(x) - tr1i>*A'(x)| +|e-,u*A'(x) - f(x) I
< 4(1 + т)У‘/2т){‘/2 + г/.
Choose 77 so that 4(1 + t])1/2t]'/2 + r) < e.
We now show that T~l is continuous. Let {u„} be a convergent net in
E(G) converging to u. If <p e Cc(G), then
lim/J?) =№
Let x e C*(G) and let e > 0 be given. Let <p e Cc(G) be such that
||<p - x|| < e/3. Let a0 be such that if a > a0, then \fu(<p) ~ fu(<p)\ < e/3.
If a > a0, then
\fUa(x) -/«(*)I
-fUa(<p)\ +\fUa(<p) -fu(<p)\ + \fu(<p) -fu(x)\
< u„(l)||x - <p|| + e/3 + u(l)||x - <p|| < e.
14.7.4. Let d’(G) denote the set of equivalence classes of irreducible
unitary representations of G. Then we have seen that each ы in d’(G)
determines a unique element of C*(GV, and conversely. We may there-
fore pull back the topology on C*(G)’ to d’(G). If (тг, H) is a unitary
representation of G and v e H, then set cv v(g) = (rr(g)v, v) for g s G.
If to e ^(G), (тт, H) g ы, then set Е(ш) = {cr J ||r|| < 1}. If S c d’(G),
then set
E(S) = U E(a>).
The following is just a reformulation of Theorem 14.6.12.
312
14. Abstract Representation Theory
Theorem. Let S c d’(G), ш g Then the following are equivalent :
(1)	Ш E S.
(2)	Е(ш) c £(5).
(3)	Е(ы) A E(S)* {0}.
(4)	There exists f g E(a>), f ¥= 0, such that f is a limit of linear combina-
tions of elements of E(S).
14.8. Direct integrals and Von Neumann algebras
14.8.1.	We begin this section with the definition and basic properties of
direct integrals of Hilbert spaces. Let (5, g) be a measure space (14.A.8)
such that:
(a)	S = U”=i 5;, with 5, measurable and g(5,) < °°.
(b)	There is a countable set, , of measurable subsets of S such that if
is the «т-algebra generated by and if A is a measurable subset of 5,
then there exists В g such that p.(A - А A B) = 0 and g(B - А A B)
= 0.
A family of Hilbert spaces over S is an assignment of a Hilbert space Hs to
each 5 g S. A section of the family {Hs}s<=s is a correspondence у -> v(s)
of v(s) g Hs for each s e S. If there exists a set of sections F of {Hs}
satisfying the following three conditions, then {НДеу is called a measur-
able family of Hilbert spaces.
(i)	If x, у g then у -> <x(s), y(s))5 is measurable;
(ii)	If z is a section of {Hs} and if s -> (z(s), x(s)))s is measurable for
all x g У, then z g
(iii)	There exists a countable subset {хД“=1 of & such that if у g S,
then {xy(y)\j = 1,2,...} is dense in Hs.
The main example is given as follows. Let H be a separable Hilbert
space. Take Hs = H for all s g 5. Let be the space of all functions x
from S to H such that 5 -> <x(s), v) is measurable for all г g Я. Let {хД
be a countable dense subset of H looked upon as a subset of 9r.
Returning to the general case, we have the following observations.
(1)	If v,w g then (y -> v{s} + w(y)) g
(2)	If f is a measurable function on S and if v g & then (s -> /(y)r(y))
G ST.
14.8. Direct Integrals and Von Neumann Algebras
313
Indeed, if w g & then (f(s)v(s), w($)), = f(sXv(s),w(s))s. Now use
(ii).
14.8.2.	Suppose we have a measurable family of Hilbert spaces {ЯД and
as before. We say that x g is square integrable if
INI2 = II x(s) ||2 dp(s) < oo.
We identify x, у g if
fi({s g S| ||x(s) - y(s) H, > 0}) = 0.
Modulo this identification, we use the notation
( Hsdy.(s)
Js
for the space of all square integrable elements of If x, у are square
integrable, then we set
<X,y> = ^<x(5),y(5)> dll(s).
With this inner product, fsHs dp(s) is a pre-Hilbert space. If Hs = H for
all 5 g S and is given as in the previous section, then we will also write
fHsdpL(S) = L2(S, H', y.).
Js
If Я = C, then L2(S, C; g) will be written L2(S; y.).
Lemma. fsHs dy.(s) is a separable Hilbert space.
We note that if {/Д is a sequence in & such that there exists a subset Z
of measure 0 so that
Ell/,(*)!! < 00
i
for 5 g S - Z, then
i
defines an element of Indeed, if v g 3^ and s £ Z, then
(f(s),v(s))s = lim E (/,(*)>
314
14. Abstract Representation Theory
This defines a measurable function on S, so 14.8.1 (ii) implies f e The
rest of the argument to prove that
/ Hsdp.(s)
Js
is a separable Hilbert space is exactly the same as the standard argument
in the scalar case (cf. Lang [1], Dym-McKean [1]).
14.83.	The space fsHs dpts) will be called the direct integral of the
family {Hs}. Notice that the definition involves {Я5}, g, and If S is a
countable set and if g is the counting measure (g(C7) is the order of U),
then fsHsdp(s) is just the Hilbert space direct sum of the spaces
Hs, denoted e 5 Hs.
Lemma. If S = U7« i Ц with U, measurable and Uj n Uj = 0 for i ¥= j,
then
[Hsdp(s)= ® f Hsdp^ (s).
JS	‘ JUi
Here, = [f^.\f s 3^} and we also take the restrictions of the xt.
This is clear from the definitions.
Note. If (X, a) is a measure space and if N is a subset of X with
a(N) = 0 and if S = X - N, g = <r|5, then we will write fxHx dp(x) for
[SHS dpfs). The preceding lemma implies that this convention is consis-
tent.
14.8.4.	We now give two constructions that imply that the preceding
example (L2(S, H; g)) is up to direct sums the general example. We
assume that Hs ¥= 0 for all s e S. Set, for each j = 1,2,...,
5[j] = (se 5|det[{xfp(s),x(?(s))]p ^.+1 = 0 for all
Then, 0 = 5[1] с 5[2] c • • •, and 5[ j] is a measurable subset of S. If
5[N] = S for some N < oo, then set No = minN(5[N] = S). If no such N
exists, then set No = oo. If i < No, set 5(z) = 5[z] - 5[z - 1]. Set S(N0) =
S - Ui</v0S[j]. Then, S is the disjoint union of the measurable subsets
5(z), 1 < z < No.
We note that if s s S(j) then dim Hs = j. Lemma 14.8.3 implies that
[Hsdp(s)= ® f Hsdft^(s).
Js	1 JS(J)
14.8. Direct Integrals and Von Neumann Algebras
315
We define a sequence Ц} in & as follows:
If x, = x2 = • • • = xn_, = 0 but xri is not zero, then set r, = x . If
v ,(5) л *,($) = 0 for r, < j < r2 but A xr(,s) =# 0 for some 5, then
set v7 = x, , etc.
We note that 8рапсЦ(.у)} = Hs for all 5. We now construct {e,} in &
with:
(1)	spanc{e,(s),..., <?„($)} = spanjr/s),..., vn(s)} for s e S.
(2)	(e^s), e}(s)) = 0 if i =# j.
The algorithm is as follows: Set ei=vi. Assume that el,...,en_l
have been found. Set e,7(s) = <e,(s), ey(s)\ = 3,7<e,(s),e,.(s)\ for 1 < i,
j<n-\. Set Sn m = {s g 5|rank[a17] < m} = {5 g S| at most m of
the e^s) are non-zero}. Then, Sn m is measurable and S„t„cSe<m+l,
sn,n-i = s- On sn,o> set en,o(s^ = vn(s^ if 5 £ sn,o> set en,o = °- Then,
en 0 is the product of the characteristic function of Sn 0 with vn. Hence,
0 g If 5 g Snm - S„m_,, then set I(s) = {z| 1 < i < n - 1, et(s) =#
0}. Then, |/(s)| = m. If I c {1,..., n - 1} and |/| = m, then set Sn<m I =
{5 g 5„ m|/(s) = /}. Then, Sn m j is measurable and Sn m is the disjoint
union of the Sn>mI. If s g SnmI, then put
"(S) ,?,<e,(j),«,(S)>e'(S)
Extend en m , to S by setting en<mQ(s) = 0 for s £ Sn m<]. Then, en m I g
We set ’
n- 1
^n,0 +	^е,т,Г
m = 1 |/| = m
14.8.5.	For our next construction, we assume that S = 5(«>). We will
leave the analogous construction for S(j), j < <», to the reader. We will
now construct {u„} in such that {«„(5)} is an orthonormal basis of Hs
for all 5 g S. Fix n = 1,2,.... If 5 g S, set
Nn(s) = min{A|dimspanc{e1(.s),..., eN(s)} = n}.
If N > n, then set Sn N = {5 g 5|A„(.s) = N}. Then,
5 = u Sn,N
N^n
316
14. Abstract Representation Theory
and Sn N is measurable. If P c {1,..., N} and |P| = n, then put
$n,N,P = {$ e $n,N\ei(s) * 0, i e P}-
Then S is the disjoint union
U $n,N,P'
N2H, |P| = n
We now proceed as before to define the u}. For u,, we note that
S = U
N
disjoint union. On N {N) put u^s) = ew(s)/||ew(s)IL. Now, “piece
together” ux as before. Assume that we have constructed u15... ,un_l.
We now construct un. If Sn NtP =# 0, then N e P. If s Sn N P, then
5 e Sn-i,N',p-(N} with = ntaxl/lj e P - {N}}. Set
= eNW/h(*)ll5.
This completes the construction.
14.8.6.	If 1 j < oo, then set HJ equal to CJ with the usual Hilbert space
structure. Let Z2 be the space of all sequences {z„} with
||{г„}||2 = Lk„l2 <
n
Then Z2 is a Hilbert space with inner product
({*„},{4.}) = ILZnWn-
n
We set H°° = Z2.
Let
5= U $0),
i s j s°°
as before. Fix j and let {u„} be as before (u„ s F and {«„(5)} is an
orthonormal basis of Hs for all 5 e 5(J)). We define
by
(TiV(S))n =М*)>мп(*)>г
14.8. Direct Integrals and Von Neumann Algebras
317
Lemma. Tj is a unitary bijection for each j.
We may assume S = S(j\ If v g 3f= jsHsdp(s), then
llty’ll2 = f El (v(s)’Un(sy}s\2 dfi(s)
S n
= /5И*)Н^м(*) = IWI2-
If <p g L2(5( j), H1-,	then set
= £,<f>n(sjun(s).
n
Then, v g 3^ and ||r|| = ||<p||. Furthermore, TjV = <p.
14.8.7.	An assignment 5 -> Bs g End( Hs) is called an operator field. If,
for each v g (s -> Bsv{sf} g F then В is called measurable.
(1)	If В is a measurable operator field then 5 -> Bf is measurable.
Indeed, if v, w g then
{Bfv{sj,w(s}} ={Bsw(s),v(s)},
which is measurable.
(2)	If A, В are measurable operator fields then 5 *-*AsBs is a measur-
able operator field.
Indeed, <AsBsv(s),w(s))s = {Bsv(s), A*sw(s))s. Thus, if v,w g 5^, then
A*w(.s) defines an element of so 5 -> (AsBsv(s), w(s)\ is measurable.
We say that a bounded operator В on 3f= fsHs dp(s) is decomposable
if there exists a measurable operator field Bs such that, if v g 3?,
Bv(s) = Bsv(s)
for g almost every 5.
If f g L“(5; g), then we define an operator Mf on 3f by
(Afzi;)(s) =f{sjv(sj.
318
14. Abstract Representation Theory
Lemma. ||Л/у|| = 11/11», M* = M?.
The second equation is clear. As for the first,
||lW/u||2 = [y(s)v(s)\\2sdp(s) <||/||2M2.
Thus, ||Afz|| < Ц/ll». To prove the reverse inequality, we may assume that
S = S(j) and that Ж= L2(S, g) (Я = H>\ Fix e > 0. Let A be a
measurable subset of S with 0 < д(Л) < » and |/(s)| > II/IL - e, s = A.
Let v s H be a unit vector and let x be the indicator function of A. Set
ф(5)=А'(5)р/(д(Л))1/2. Then, HAf^ll2 > (H/IL - e)2. Thus, Ц1И/1 >
11/11» - 8.
14.8.8.	Let be the algebra of operators of the form Mf,f^L“(S;p).
Let 31 be the algebra of all decomposable operators.
Proposition.
(1)	If В is decomposable and s -> Bs is a corresponding measurable opera-
tor field, then (s -> ||B5||) s L°°(5;g) and ||B|| = ||(s -> ||B5||)Ц».
(2)	comm(ja/) = 3 and comm(^) = sf. (See 14.4.1 for the definition of
comm.)
We first prove (1). We may assume that 5 = S(j) and <%?= L2(S, H; p).
If v,w e H, ||w|| = 1, and if A is a subset of S of finite positive measure,
then
f (Bsv,w) dp(s)
JA
< IIBII 1М1м(Л).
Thus, 14.A.13 implies that ||Bsv|| <, ||B|| ||r|| for g = a.e. seS. Let {r;} be
a countable dense subset of H with v} =# 0 for all j. Then ||B5|| =
supJIfi^H/Ht'JI. Hence, <p = (s -> ||2?J|) is measurable and ||<p||«, < IIM
On the other hand, if v s then
IlJSrll2 = f || !?/>($) ||2 dg(s)
Js
± /j|B5||2h(.)||2dg(.) <||<p||2M2.
Thus, ||fill 11^11». This completes the proof of (1).
14.8. Direct Integrals and Von Neumann Algebras
319
We now prove (2). We may assume that S = S(j) (as usual). Suppose
that T is a bounded operator on Ж such that TMf = MfT for all
/g L°°(5;g). Let Xj be the indicator function of Sj (14.8.1(a)). We may
assume that 5, c 5, + 1 (replace Sj with U,syS;) Then x,f e L2(5;g) if
f g L°°(5;g). Set = 1W . Then, Pj is an orthogonal projection on
and TPj = PjT. Let {d;} be a countable dense subset of H, with Vj =# 0 for
all j. Let Rjj = {s g S\(TxjVj)(.s) is not defined}. Then g(f?;,) = 0. Set
R = ицКц- Then, u(R) = 0. If 5 £ R and if Xjts) = 1, then set Tsvt =
(TxjV^sY Since TPj = PjT for all j and, if i <J, PjPj = Pit Tsv{ is well
defined on S - R. We also note that
||7>,. - TsVj\\ < ||T|| ||r,. - u,.||.
This implies that if 5 g S — R, then Ts extends to a bounded operator on
H. If v g H and if lim^ ^tr. = v, then lim^ ^ TSV; = Tsv for all 5 g
S - R. Thus, s <-» Ts defines a measurable operator field. Since it is clear
that Tv = (s -> Tsv(sf), we have shown that comm(ja/) = SB.
Since <q/g SB we see that comm(^) c (^). Thus (in light of (1)), we
need only show that if T g comm(^), then Ts is a scalar multiple of I for
ft = a.e. 5. Let {nJ be an orthonormal basis of H. Let Tpq(s)un = 8qn up.
Then Tpq g Set unj(s) = Xj(s)un. Then, Tpqunj = 8qn upJ. We have
Tup j = TTpquq j = TpqTuqj. Thus, if s g Sj, then Tsup = (Jsuq,uq)up.
If we apply this with p = q, then we find that Tsup = (Tsup,up)up. Thus,
(Tsup,up) = (Tsuq,uq) for p = a.e. 5 and all p, q. Set f(s) = (Tsul,ul).
Then, Ts = f(s)I for p = a.e. s gS.
14.8.9.	Let H be a Hilbert space. Then a *-invariant subalgebra srf of
End(H) is called a Von Neumann algebra if comm(comm(<o/)) = <0/. In
light of the Von Neumann density theorem, a *-invariant subalgebra of
End(H) is a Von Neumann algebra if and only if / g ja/ and <0/ is closed
in the strong operator topology (14.4.2). The algebras <0/ and SB of 14.8.8
are examples of Von Neumann algebras. We next show that if H is
separable and if sf is a commutative Von Neumann algebra in End(H),
then up to “unitary equivalence” sV is given as in the previous example.
This will take some preparation, which we now begin.
Let X be a compact Hausdorff space and let C(X) be the space of
all continuous functions on X with ||/|| = supxe x |/(x)| and /*(x) =
/(x). Then C(X) is a C* algebra.
Lemma. If C(X) has a countable dense subset, then X has a countable
basis for its topology.
320
14. Abstract Representation Theory
If C(X) has a countable dense subset then clearly
C(X;R) = {/eC(X)|/(X)cR}
does also. Let {/,} be a countable dense subset of C(X;R). Let
Ц = {x eX| |/y(x) - 1| < 1/4).
Then Uj is open in X for each j. We assert that Ц is a basis for the
topology of X. Let U be open in X and let x e U. We must find j
such that x e Uj c U. Urysohn’s lemma (cf. Hocking-Young [1]) implies
that there exists <p e C(Y;R) with 0 < <p < 1 such that <p(x) = 1 and
<p(X - U) = 0. Set V = {y s X\<p(y) > 1/2}. Then, V c U. There exists j
such that ||<p -fj\\ < 1/8. If у s Ц, then
<p(y) = <р(у) -fj(y) +fj(y) -i + i
> 1 -|<p(y)-Цу)\~\fj(y) - 1| > 1 - 1/8 - 1/4 > 1/2.
Thus, у e К Hence, Ц c U. Also,
|/;(x) - 1| =\fj(x) - Ф(Х)| \\fj - Ф|| < 1/8.
Thus, x e Uj.
The converse also follows from Urysohn’s lemma.
14.8.10.	Let X be a compact, separable topological space. Let C(X) be
the corresponding C* algebra (as before). Let g be a Radon measure on
X (14.A.14) and let C(X) act on L2(X;g) as a subspace of L“(X;g).
That is, if f e C(X), Mf<p(x) = f(x)<p(x) for x s X. Then (M, L2(X; g))
is a *-representation of C(X).
Lemma.
(1)	Iff e C{X\ then HM/I = Il/H.
(2)	The closure of {Mf\f e C(X)j in the strong operator topology is
{Mf\f & L“(X', p)}.
We note that the constant function, 1, identically equal to 1 is in C(X),
and hence 1 s L2(X; p). Also, <o/= (Mf\f s L“(X; g)} is the space of all
decomposable operators (14.8.7) on L2(X;g) looked upon as a direct
integral (14.8.2). Hence, зэ/ is strongly closed in End(H) (H = L2(X; g))
by Proposition 14.8.8 (2). The Von Neumann density theorem (14.4.2)
14.8. Direct Integrals and Von Neumann Algebras
321
implies that comm(comm( AfC(X))) is the strong closure of MC(Xy Let
T g comm(AfC(X)). Then T(<pf) = <pT(f) for f g H, <p g C(X). Thus, if
f = 1, then T(<p) = T(D<p. T(l) g L2(X-, p), so it is measurable. Now,
||Т(ф)|| <||П ||ф||2
for all <p g L2(X;p). Hence,
||Т(1Ы2<||П1И2
for all (p g C(X). Thus, this is true for all <p e L2(X;/i). So T is
decomposable and hence T = Mf for some f e L°°(X; g) (Proposition
14.8.8).
14.8.11.	Let H be a separable Hilbert space. Let sf be a *-invariant
subalgebra of End(H) with I g £/.
Lemma. H can be written as a direct sum H = ®. Hj of Hilbert spaces
such that for each j, srfHj c Hj and there exists a unit vector x}- g Hj such
that s/Xj is dense in Hj.
Let {еД be an orthonormal basis of H. Let x, = e, and let Hx be the
closure of sVex. Set Я1 = (Я1)± and let Px be the orthogonal projection
of H onto H'. Since зэ/* = зэ/, Hx and H' are ^invariant. Put mx = 1.
If Pxej = 0 for all j, then Я, = H and we are done. Otherwise, let m2 be
the first index such that Pxe„2 =# 0. Set x2 = Plem2/\\Ple„2\\, H2 = <o/x2,
H2 = (Я, Ф H2)x , P2 the orthogonal projection of H onto H2. Since
is *-invariant, H2 and H2 are ^invariant. If P2Cj = 0 for all j, then we
are done. Otherwise, let m-, be the first index such that =# 0, etc.
14.8.12.	Lemma. Let H be separable and let srf be an abelian
Von Neumann algebra in End(H). Then there exists x g H such that
comm(ja/) x is dense in H.
Let H = Hj and be as in the previous lemma for <я/. Let Qt be
the orthogonal projection on H onto Hj. Then Qj e comm(ja/). Set
x= E0/2J)xj.
;>i
Since srf is abelian, зэ/с comm(ja/) so comm(ja/)x z> ^/QjX = srfXj,
which is dense in Hj. Thus, comm(ja/)x is dense in H.
322
14. Abstract Representation Theory
14.8.13.	Lemma. Let be a Von Neumann subalgebra of End(ff)
(H a separable Hilbert space). Then there is a strongly dense, countable
dimensional, * -invariant subalgebra of srf containing 1.
Let Hj and be as in Lemma 14.8.11 for comm(ja/). Set у,- = 2_Jxy.
Let H°° be as in 14.4.1. We define a linear map <p from jaf to H°° by
<Р(Т)={ТУ/}.
Then, T is injective since if Ту, = 0 for all j, then T сотт(зэ/) х}- = 0. But
then THj = 0 for all j. Hence, T = 0.
Let зэ/j = {T g srf\ Ill’ll 1}. Then ip(jaZj) is contained in the unit ball
of №°. Let Z be a countable dense subset of H™. If z g Я°°, then set
B(z; r) = {w g H°°\ \\z — w|| < r}. Set Zn = {z g Z|B(z; 1/n) П (p(sfi)
¥= 0}. If z g Z„, choose xz „ g such that <p(xz n) g B(z; 1/n). Let
5 = (xz n\z g Z, n = 1,2,...}. Then S is countable, and by construction
(p(S) is dense in <p(<Q/j). We assert that S is strongly dense in s£x. Indeed,
let x g . Then there exists a sequence {z,} in S such that lim, <р(г})
= <p(x). Hence, lim,^ z,y; = xy, for all i. Hence, lim,^ Tz,y(- = Txyx for
all T g comm(ja/). Hence, lim,^ z,Ty(- = xTyi for all T e comm(^/).
This implies that
lim ZjU = xu
for all и in the algebraic sum V of the comm(<Q/)xl. Suppose that v g H,
and let e > 0 and и e V be such that ||u - u|| < e/3, and let n be such
that if j > n then ||z,u - xu|| < e/3. If j n, then
WzjV - xr|| \\ZjV - Zfu\\ + ||z, w - xu|| + IIXU - Art’ll
< 2\\u - v\\ + e/3 < e,
since ||z,-|| < 1 and ||x||	1.
Take 31 be the subalgebra of srf (algebraically) generated by
{1} U 5 U 5*.
14.8.14.	With these preliminaries completed, we can now state and prove
the main result of this section.
Theorem. Let srf be a commutative Von Neumann subalgebra of End(H),
H a separable Hilbert space. Then there exist a compact, separable, Haus-
dorff space X, a Radon measure p. on X, a measurable family of Hilbert
14.8. Direct Integrals and Von Neumann Algebras
323
spaces {Hs}s<eX, and a unitary bijection
T- f Hsdfi(s)
Jx
such that TsrfT x = {Mf\f g L”(X; g)}.
Let be a countable dimensional, strongly dense, *-invariant subalge-
bra of <sa/ containing I (see the previous lemma). Let be the closure of
3) in the operator norm topology in End(H). Then € is a commutative
separable C* algebra containing I. Let X = spec(if) (14.1.4) with the
topology as in 14.1.5. Then, X is a compact Hausdorff space and the
Gelfand transform x -> x (14.1.5) defines an isomorphism of € onto
C(X) (as C* algebras by 14.16). Since Я is countable dimensional, C(A')
is separable, so X is separable. Let V be the inverse mapping to the
Gelfand transform. If v, w g H, then set
Put vt, = vv v. We note that
k.,(/)IMH/)lliMi mhi/iuh iwi.
Thus, the Riesz-Markov theorem (14.A.14) implies that there exists a
unique complex Radon measure w on X such that
= / f{x)dpv<w(x).
Jx
Set pv = p.VtV.
Let x g H be such that comm(ja^) x is dense in H (Lemma 14.8.12).
The key to the proof of the theorem is:
(1)	gt, is absolutely continuous with respect to gx for every v g H.
We first prove an estimate for v e coming) x. So v = Tx, with
T e coming). Thus,
M/) ={V(f)Tx,Tx)
for f g C(X). Suppose that f > 0. Then /1/2 g C(X). Thus,
M„(/) ={V(f'/2)Tx,V(f'/2)Tx) =(TV(f^2)x,TV(f^x)
= ||tv(/1/2)x||2 < linW^M2
= \\T\\2(V(fl/2)x,V(fx/2)x} = ||T||2< V(f)x, x) = l|T||2gx(/).
324
14. Abstract Representation Theory
Let v g H - {0} and f 0, f g C(X), / =# 0. Let e > 0 be given and
let v' e comm(^) x be such that
Hr - u'll < min{e/(6||/|| Hull), Hull}.
If 0 <>h <f, h g C(X), then
|МЙ) -МДЙ)| =|<И(Л)и,и> — <И(Л)и',и'>|
<|<Г(й)и,и> -< K(/i)u,u'>|
+ |<И(Л)и,и'> +<К(й)и',и'>|
= |<К(й)и,и - и'>| +| <Г(й)(и - и'),и'>|
<||И(А)|| Hull ||и - и'|| + ||К(Л)|| ||и'|| Ни -и'||.
Our assumption implies that ||и'|| 2||и||. Thus, since ||И(Л)|| < ||К(/)|| <
11/11,
|мДЛ) - nv'(h) \ < 3||и|| 11/11 Ни - и'|| < е/2.
Now, let и' = Tx as before, and let / ^ 0 be fixed. Let e>0. IfO</i</
and if gx(/i) < e/2||T’||2, then
дЛ) < nv.(h) + e/2 < \\T\\2nx(h) + e/2 < e.
Thus, 14.A.16 implies (1).
We set g = gx. (1) combined with the Radon-Nikodym theorem im-
plies that if v g H, then there exists /i„ g L‘(X;g) such that dp.v = hv d/i
with £ 0. By polarization we see that if v,w g H, then there exists a
unique element hvw g Ll(X; fi) with
<K(/)u,w> = f f(s)hVtW(s) dfi(s).
Jx
We note that:
(2)	hv<w = hwfor v,w g H.
(3)	If /g (XX), v,w g H, then hV{f)v w =fhv w.
(2)	is clear. To prove (3), we note that
={V{gf)v,w') = fxg(s)f(s)hv<w(s) d^(s)
14.8. Direct Integrals and Von Nenmann Algebras
325
and
Thus, (3) follows from uniqueness.
Let fl be a countable dense subset of H. Let N be a subset of
g-measure 0 in X such that hv w(s) is defined for all 5 g X - N, v, w g fl.
Using (2) and (3), we see that hv w(s) is defined for all 5 g X - N and
v,w g V(C(X))H. Let H be the linear span of L(C(A'))fl (in the
algebraic sense). Then hv w(s) is defined for all s&X-N and all
v,w g H. H is K(C(A'))-invariant and (2), (3) apply for v,w g H.
If 5 g X - N and v, w H, then set (n,w)5 = hv w(s\ Then, ( , )5
defines a positive semi-definite form on H. Let Rs be the radical of ( , )5.
Then ( , )5 pushes down to a pre-Hilbert space structure ( , \ on H/Rs
for j g X - N. Let Hs be the Hilbert space completion of H/Rs. Let ps
be the natural projection of H onto H/Rs. If v e. H, then set v(s) = psv.
Then, {НДе x-n is a family of Hilbert spaces over X - N and v is a
section of the family for each v g H. Since <x(s), х($))5 = 1 for all
j g X - N, Hs =# 0 for all j e X - N. Let & be the set of all sections v
such that j -» <r(i), v(s))s is g-measurable for all v g H.
Set S = X - N. We note that the constructions in 14.8.4, 14.8.5, and
14.8.6 can be done using И„е!1 = fl and the sections {u„} are in {£)„е/р
Thus, if v g 5^, then v(s) =	(v(s), un(s))un(s) for s g S(j). Thus,
each ue ^isa pointwise limit of elements of (iV/i- Thus, & satisfies
14.8.1 (i), (ii). If we enumerate fl, then 14.8.1 (iii) is also satisfied. Thus,
{НДе$ is a measurable family of Hilbert spaces over S. Let
dp(s\
If v g H, then set Tv(s) = v(s). Then,
(Tv,Tw) = j (v(s),w(s))sdp.(s) = jhv w(s) dp.(s)
= {V(l)v,w) = (v,w).
Thus, T extends to a unitary map of H into Ж. By our construction, the
image of T is dense. Thus, T is a unitary bijection.
If v g H, then	= (s ->/($)£($)). Thus, TV(f)T~l = Mf for
f g C(X). Thus, T£T~X = (Mf\f g C(A')}. Now, comm(comm(^)) = st
and thus TstT~x is the strong closure of {Mf\f g C(A')} on Thus,
TstT~x = [Mf\f GL“(X;g)}.
326
14. Abstract Representation Theory
14.9.	Direct integrals of representations of C* algebras and locally
compact groups
14.9.1.	Let if be a separable C* algebra. Let (S, p.) be a measure space
(as in 14.8.1) and let {НДеу be a measurable family of Hilbert spaces. Let
<%?= fsHs dp.(s). Let irs be a *-representation of ё on Hs for each s e 5,
such that j -> тг/х) is a measurable operator field for each x e ё
(14.8.7). If x e if, then the decomposable operator тг(х) corresponding to
{тг/х)} defines a *-representation of -ё. We denote this representation by
[iTsdp(s), (Hsdp(s)
\Js	Js
and call it the direct integral of {(тг5, Hs)}seS.
14.9.2.	Let G be a locally compact, separable, topological group and let
{НДе5 be as before. Suppose that for each s e S we have a unitary
representation (irs, Hs) of G such that for each g s G, s -» тг/g) is a
measurable operator field. Let, for g s G, Trig) be the corresponding
decomposable operator on Лё.
Lemma, (тг, Лё) defines a (strongly continuous) representation of G. We
call this representation the direct integral of {(тг5, Hs)}s(=s and denote it by
[tts dp(s), f Hsdp(s)
vs	Js
Furthermore, under the correspondence between unitary representations of G
and non-degenerate * -representations of ё, direct integrals correspond to
direct integrals.
If v, w e Js dp(s), then
{ir(g)v,w) = f {Trs(g)v(s),w(s)')sdp(s).
Js
It is clear that rr(g) is a unitary operator on The dominated conver-
gence theorem implies that
lim <Tr(g)r,w) = {rr(x)v,w} .
g-»x
Thus, Lemma 1.1.3 implies that (тг, ^) is a representation of G.
14.9. Direct Integrals of Representations
327
If f e L\G\ then
(rr(f),v,w) = f f(g) f {irs(g)v(s),w(s)}sdp.(s)dg.
JG	JS
Fubini’s theorem implies that we can interchange the order of integration,
and we find that
(ir(/)U,W> = jT <ir,(/)f(f), w(^)>, </g(^).
By the construction of Haar measure, one sees that {irs(f)}seS is a
measurable operator field for f s Ll(G). We extend tts to C*(G) as in
14.2.4, and since each x s C*(G) is a limit of a sequence in Ll(G),
{тг/х)}1е5 is a measurable operator field for each x. It is now clear that
the corresponding representation of C*(G) is just the direct integral of
the representations of C*(G) corresponding to the tts.
14.9.3.	As in 14.8.3, if (X, a) is a measure space, if N с X with a(N) =
0, and if S = X - N, g = <т|5, then we will sometimes write fxrrx da(x)
for fsirs dp(s). In other words, the representations tts need only be
defined for а-almost every x e X.
The notion of direct integral of Hilbert spaces is a natural generaliza-
tion of that of direct sum. It gives a formalism that allows one to
decompose every unitary representation of a locally compact group (more
generally, every non-degenerate *-representation of a C* algebra) into
irreducible constituents. The first such theorem is:
14.9.4.	Theorem. Let € be a separable C* algebra. Let (тг, H) be a
non-degenerate * -representation £ on a separable Hilbert space. Then there
exists a separable, compact, Hausdorff space X, and a Radon measure g on
X such that (тг, H) is unitarily equivalent to (Jxirx dp(x), jxHx dp(x))
with (тгх, Hx) irreducible for p-almost all x £ X.
Let st be the strong closure of тг(^) in End(H). Then S is a
Von Neumann algebra.
(1)	There exists a (strongly) closed *-invariant, abelian subalgebra Si of
comm(ja/) such that if T s S and [T, S] = 0, then T e
Indeed, let S be the set of all *-invariant abelian subalgebras of
comm(ja/). We order S by inclusion. If S is a linearly ordered subset of
328
14. Abstract Representation Theory
cf then the union of the elements of fT is in c/'. Zorn’s lemma implies
that с/ has a maximal element Since & g c/', we see that is
closed.
We fix & as in (1). We note that @ is maximal abelian in comm(ja/).
Indeed, if [T, 0] = 0 then [T*, 0] = 0. So T, T* g Theorem 14.8.14
implies that there exists a compact, separable, Hausdorff space X, a
Radon measure g on X, a measurable family {Hs}s e x of Hilbert spaces,
and a unitary bijection T from H to <&= fxHs dfi(s) such that T^T~l =
{Mf\f g L°°(X; fi)}. We therefore assume that H = and that & =
{Mf\f g L”(X; fi)}. We have
tt(Z) c^/= comm(comm(<&)).
Thus, tt(Z) c comm(^). Hence Proposition 14.8.8 (2) implies that tt(Z)
(and more generally, s&) is contained in the space of decomposable
operators. Thus, if x g if (resp., T g srf), then тг(х) (resp., T) is given by
{тг(х)ДеА- (resp., {ТДеХ), a measurable operator field. Let fl be a
countable dimensional, *-invariant (abstract) subalgebra of if such that fl
is dense in if. Let N be a subset of X such that fi(N) = 0 and such that
if j g X - N = S, then:
(2)	tts(x) is defined for all x g fl;
(3)	тг/х*) = тг/х)* for all x g fl;
(4)	||ir,(x)IL < tt(x) for x g fl;
(5)	tts defines a representation of fl as an abstract algebra.
If j g S, then we define irs(x) for x g if as the obvious limit of тг/у)
for у g fl. Then, {тгД eJ defines a measurable family of representations
of Z, and if x g if then irs(x) = ir(x)s for g-almost every s g S. This
clearly implies that tt = fxirs dfits).
To complete the proof, we must prove that tts is irreducible for
g-almost every j g S. Let be the strong closure of the algebra
generated by tt(Z) and Then
comm(^T) = comm(.2/) n comm(^).
Since & is maximal abelian in сотт(зэ/), сотт(зэ/) П comm(^) = 31.
This implies that comm(comm(^D) = consists of all decomposable
operators on H. Hence, commfcommfTr/if)) = End(HJ) for g-almost all
j g S. This completes the proof.
14.10. Decompositions of Representations
329
14.9.5.	Corollary. Let G be a locally compact, separable topological
group. Let (tt,H) be a unitary representation of G with H separable. Then
there exists a compact, separable Hausdorff space X, a Radon measure p on
X, and a measurable family of unitary representations (irs)s£X such that tts
is irreducible for almost every s s X and (тт, H) is unitarily equivalent to
firsdp(s), f Hsdp(s)
\JX	Jx
This is a restatement of the previous result for C*(G) (in light of
14.9.2).
14.9.6.	Although the preceding results seem to completely settle the
question of decomposing representations into irreducible ones, they are
far from satisfactory answers. First of all, there is no assertion of unique-
ness (in fact, there are examples to the contrary). Secondly, the space X is
completely unknown and the parametrization is in no way explicit. In the
next section, we will prove a result in the same generality with irreducible
replaced by “factorial,” and show how to get a “good” parametrization in
the case of CCR algebras (groups).
14.10.	Decompositions of representations of CCR C* algebras and locally
compact groups
14.10.1.	Let H be a separable Hilbert space. A Von Neumann subalge-
bra sf of End(H) is called a factor if
Z(^) = {Те ^|[T, ^] = 0} = CI.
If -6 is a C* algebra and if (тт, H) is a non-degenerate *-representation of
if, then tt is said to be factorial if the strong closure of rr(if) in End(H)
is a factor. If G is a locally compact, separable, topological group and if
(тт, H) is a unitary representation of G, then tt is said to be factorial if
the corresponding representation of C*(G) is factorial.
In this section, we will first give a natural decomposition of a *-represen-
tation of a C* algebra (or a unitary representation of a locally compact
group) into a direct integral of factorial representations. The rest of the
section will be devoted to sharpening the decomposition in the special
case that the C* algebra is CCR.
330
14. Abstract Representation Theory
14.10.2.	Theorem. Let ё be a separable C* algebra (resp., G a locally
compact, separable, topological group) and let (тг, H) be a non-degenerate
* -representation of -ё (resp., unitary representation of G) on a separable
Hilbert space. Then there exists a compact, separable, Hausdorff space X, a
Radon measure p on X, a direct integral in representation of -ё (resp., of
G), (fx'rrs dp(s), fxHs dp(s)), and a subset Z of X with p(Z) = 0 such
that:
(1)	(tts, Hs) is factorial for s g X - Z.
(2)	HomB(Hs, Hs,) = 0 for s, s’ X - Z. Here, В = ё (resp., G).
(3)	(тг,Н) is unitarily equivalent to
[ tts dp(s), [ Hsdp(s)
Vx	Jx
It is enough to prove this result for ё since the result for G will then
follow by taking ё = C*(G). Let srf be the strong closure of тт(ё) in
End(H). Then &Z is a Von Neumann algebra. Let Z(s&) be the center
of srf. Then Z(srf) = commit/). So Z(&Z) is a commutative
Von Neumann algebra. In light of Theorem 14.8.14, there exists a com-
pact, separable, Hausdorff space X, a Radon measure g on X, a measur-
able family {Hs}sEX of Hilbert spaces over X and a bijective unitary
operator T from H to <%?= fxHs dp(s) such that
TZ(stf)T~' = {Mf\f GL”(X;g)}.
We therefore assume that H = and
Z(srf) = [Mf\f&LT(X-,p.)}.
As in the proof of Theorem 14.9.4, there exists a subset Z of X with
g(Z) = 0 and for each j g X - Z a representation tts of ё on Hs such
that тг = fxirs dp(s).
Let & be the algebra generated by зэ/и comm(ja/). Then
comm(^) = .й^П comm (лэ/) = Z(&/).
This implies that the strong closure of ёё is the algebra of decomposable
operators. Hence, for j g X - Z, the strong closure of the algebra gener-
ated by тг5(ё) and comm(Tr/if)) is the algebra of all bounded operators
of Hs. Thus, tts is factorial.
Let s, s' g X - Z and assume that s =# s'. Suppose that R is a bounded
operator from Hs to Hs, such that R ° тг/х) = tts,(x)° R for x g ё. We
14.10. Decompositions of Representations
331
observe that the strong closure of {(j -» тг/х))|х s -£} = srf. Thus, if
Ге j/ then the intertwining relation extends to RTS = TS,R. Let s
C(X) be such that <p(s) = 0, <p(s') = 1. Then (j -> <p(s)/) = T s <0/.
Thus,
0 = R<p(s)I = RTS = TS,R = <p(s')R = R.
14.103. We now begin the analysis of the preceding theorem in the
special case when ё is CCR.
Theorem. Let ё be a separable, CCR, C* algebra. If (тг, H) is a factorial
* -representation of -£ on a separable Hilbert space. Then there exists an
irreducible *-representation (тг0,Н0), a Hilbert space V, and a unitary
bijection T from H to Ho 8 V such that T ° тт(х) = (rr0(x) ® I)°T for all
x -6. Furthermore, tt0 and V are determined up to unitary equivalence by
this relationship.
For the proof of the first assertion, we may (and do) assume that тг is
injective if we replace -£ by -£/ Ker тг. We therefore assume that if is a
closed subalgebra of End(H) with respect to the operator norm and that
-ёН= H (since тг is factorial, the strong closure of ё contains /). Let st
be the strong closure of ё in End(H). Our assumption is that Z(3sf) = CL
(1)	If is a closed two sided ideal in -£, then ,^= 0 or ,^= ё.
Let V = {r e V\.^v = 0}. Set W =	. We assert that JW is dense
in W. Indeed, suppose that w e W and (w,^W) = 0. Since is
*-invariant (14.3.2), this implies that if v e W and x e then (xw, v) =
0. The *-invariance of implies that W is ^-invariant. Thus, xw e IV
for all x e Thus, xw = 0 for all x e Hence, w e W A V = {0}.
Since is a two sided ideal in -£, V and hence W are ^-invariant. Let
Ц} be an approximate projection for J (14.3.1). If v e K, then UjV = 0
for all j. If w e ,^W, then w = xv for some v s W and hence
lim UjW = lim UjXV = xv = w.
j —* 00	j —*00
Since ||uy|| < 1, this implies that if P is the orthogonal projection of H
onto W then limJ _00u>r = Pv for all г e Я. But [P, -ё\ = 0. Hence,
P e Z(ja/) = Cl. So P = 0 or P = I. If P = 0, then V = H so 0. If
P = I, then W = H. This implies that if x e -6, v s H, then limy_«, UjXV
= xv. We now show that this implies that -ё.
332
14. Abstract Representation Theory
If v g H, then set f,.(x) = (xv,v) for x g ё. Then, Д is a positive
functional on ё. Set Q = {/„I г g H, ||l>|| < 1}. If x g ё and if fix) > 0
for all f g Q, then x > 0. 14.6.11 implies that the weak* closure of Q
contains P(#). Hence, if f g P(if), then Нту_«, fiujx) = fix). This in
turn implies that if (a, K) is a (non-zero) irreducible ^representation of
ё, then <t(^) =# 0. If ё, then ё/.^ must have an irreducible
(non-zero) ^representation (14.5.13). We therefore have a contradiction.
This completes the proof of (1).
Since ё is CCR, (1) (combined with 14.5.19 and 14.6.9) implies that
-& = {<u0}. Fix (тг0, Hq) g <a0. Then (14.5.19), о\ё) = CCiHQ) and
(14.5.13) тго is injective. Hence, ё is isomorphic with СС(Я0) as a C*
algebra. Fix v g Ho, ||p|| = 1. Let Pviw) = {w,v)v for w g Ho. Then
Pv g СС(Я0) and P2 = Pv. Let x g ё be the unique element of ё such
that 77q(x) = Pv. Then x2 = x and x > 0. Let /(у) = <тт0(у)1л v). Then
f is a pure state associated with <u0.
(2)	If у g ё, then xyx = fiy)x.
Indeed, rr0(xyx) = тт0(х)тт0(у)тт0(х) = Р^^уУР^. = <rr0(y)r,	=
/(у)'П’0(х). Since тгq is injective, (2) follows.
Set Hx = xH.
(3)	ёНх=Н.
Set V = ёНх, W =	. Then, V, W are ^-invariant and Hx с V. But
then xW с V A W = {0}. Let тг/у) = у|И, for у g ё. Assume that W =# 0.
We now derive a contradiction. Let P be the orthogonal projection of H
onto W. Then Р(ёН) = ёРН = ё\¥. Thus, if Pif И7 = {0} then P = 0,
and hence W = 0. This implies that Ker ttj =# ё. Hence, Ker тгх = {0} (by
(1)). However, tt/x) = 0. This is the desired contradiction. This completes
the proof of (3).
Let {y g if |/(z*y) = 0 for all z g ё} (as usual). Then =
{y g ё\тт^у)и = 0)c{y e ё\ух = 0}. Hence, if у g then yHx = 0.
We therefore have a linear map
T: (ё/^)
given by T((y + jp'f) ® u) = yu. Since (rr0, Ho) is irreducible, ё/^ with
the pre-Hilbert space structure given as in 14.5.6 is actually a Hilbert
14.10. Decompositions of Representations
333
space, Hf (Theorem 14.5.1 (3)). And Hf) (14.5.6) is unitarily equiva-
lent with (tt0, Hq). On Hf ® Hi we put the tensor product pre-Hilbert
space structure.
(4)	If y, z e Hf ® Нх then <7y, Tz> = <y, z>.
Let {uj be an orthonormal basis of Нг such that ц = xu,, u, e H (this
is possible by the Gram-Schmidt process). Then ц = хц (x2 = x). Let
yP..., yd s ё. If z = E/yy + ^) ® Vj, then
<7z,7z> =
' i i ' ij
=	= YAxyfyiXUi’Vf)
ij	iJ
=	= Ъ/(у*У.) = (z,z).
This completes the proof of (4).
(4) combined with (3) implies that T extends to a unitary bijection of
Hf ® Hi onto H. Clearly, T °(лу(у) ® I) = yT for all ye/.
To complete the proof we must prove the uniqueness assertion. In the
course of the proof of the first part of this theorem we showed that Ker тг
is primitive. Since -ё is CCR, the class of тг0 is therefore uniquely
determined. In the notation of the first part of the proof, let x0 e / be
such that tt(x0) = x. Then tr тг(х0) = dim Hx. Thus, Нг is determined by
тг up to unitary equivalence.
Note. If if is a C* algebra such that the conclusion of the preceding
theorem is true (i.e., factorial representations are multiples of unique
irreducible representations), then ё is said to be of Type 1.
14.10.4. With the previous two results in hand we can state the basic
decomposition theorem for CCR C* algebras.
Theorem. Let ё be a separable, CCR, C* algebra. Let (тг, H) be a
non-degenerate *-representation of ё. Then there exists a Borel measure т on
334
14. Abstract Representation Theory
-f and a direct integral of representations (}^ттш dr(u), dr(o))) such
that:
(i)	(тт, H) is unitarily equivalent to (}^тгш dr(to), [#НШ dT(a)f).
(ii)	There exists a subset N of if such that t(N) = 0, and if ш if — N
then (тгш, HJ is equivalent to (ттш ® I, Нш ® Кш), with (тгш,Нш) e ш
and Уш a Hilbert space.
(iii)	If ы e -f, then set n(a>) = dim Уш. Then n is a т-measurable func-
tion from -f to the extended positive axis [0, oo], n is called the
multiplicity function.
We start with the decomposition of Theorem 14.10.2. We will use the
notation therein (X, Z, p, tts, etc.). Jf s e X - Z, then (tts, Hs) is factorial.
Thus, Theorem 14.10.3 implies that Ker tts is a primitive ideal in -f. Let
f(s) = Ker irs for j e X - Z.
(1)	If S c Prim(if) (14.6.6, 14.6.7) is closed, then f~l(S) is g-measur-
able.
Let I(S) =	Then S = (Уе Prim(^)|<XD I(S)}. Let {u;} be
an approximate projection for I(S) (14.3.2). Let {u;} be a dense subset of
<^= fxHsdp(s). Set, for seX-Z, fjkm(s) = (TTs(uj)vk(s\vm(s))s.
Then, fj k m is a g-measurable function on X - Z.
If f: k = 0 for all j, k, m, then ir.(«,) = 0 for all j. Hence, if
x e I(S), then
тг5(^) = lim irs(xuf) = lim тгДх)тгДм;) = 0.
j —* 00	J —* 00
Thus, I(S) c Ker tts. Since the converse assertion is clear we see that
f~l(S) = {5 e X - Z\fjkm(s) = 0 for all j, k, m},
which is a g-measurable set.
Let k: £ -* Prim(if) be as in 14.6.8. Then к is bijective (14.6.9). Set
<p = k~l ° f. Then, у defines an injective map of X - Z into -f such that
if S c -f is a Borel subset, then <p-1(5) is g-measurable. Let, for each
e > 0, Ue be an open subset of X with p(Ue) < e and Ue э Z. Suppose
that S с X - Ue is closed. Let
P| Kerw.
w etp(S')
Let {u;} be an approximate projection for Define f}<k<m as before. Let
3 > 0 be given. Lusin’s theorem (14.A.17) implies that for each j, k, m
14.10. Decompositions of Representations
335
there exists Uj km(3), open in X - Ue, such that д(Ц t m(3)) < 2 7 k~m8
and	continuous. Thus, <p(S - S A Ujkm(8)) is
closed in <p(X - Ue). Hence, if
A= U <p(s - n ujtk'm(8)),
j, k, m
then g(5 - <p~l(As)) < 8. This implies that:
(2)	If 3) is the «т-algebra	a Borel subset of and if A is a
g-measurable subset of X, then there exists В s Я such that
fi(A — А A B) = g( В — В ПЛ) = 0.
We define a Borel measure r on if by r(B) = ц(<р~1(В)) for В a Borel
subset of if. Then, t(^- <p(X - Z)) = 0. Set N = % - <p(X - Z). If
ы = <p(s), then set Нш = Hs. Then (2) implies that ([xtts dp,(s),
[XHS is unitarily equivalent with (}^ттш dr(s), ^Нш dr(s)\ This
proves all but the last assertion of the theorem. To complete the proof, we
need:
(3)	Let xef, x > 0. Then j -> trfir/x)) is a g-measurable function
from X - Z to [0, oo].
Set S = X - Z. It is enough to show that for each a > 0, (i e
SltifirXx)) > a} is measurable. Then S = U, SO') as in 14.8.4. We can
assume that S = S(f) and that
(Hsdft(s) = L2(S,Hi-,n)	(14.8.6).
Js
Let {e,} be an orthonormal basis of H1. If j e S and tifir/x)) > a, then
there exists m such that
“m(s) = E <?,•>> a,
i<,m
and conversely if there exists m such that um(s) > a, then trfir/x)) > a.
Set Sm = {$ e S\um(s) > a}. Then, Sm is measurable and
{$ e Sltrfir^x)) > a} = U Sm.
This proves (3).
336
14. Abstract Representation Theory
Let fl be a countable subset of -&F (14.6.16) dense in ё such that if
x e fl, then x* e fl, if x, у e fl, then x + у and xy s fl, and if x e fl,
x* = x, then x1^ fl. Set fl+= {x e fl|x > 0}. Let S" = {je S| there
exists x s fl+, trfrr/x)) = oo}. Then (in the notation of (i), (ii), which
already have been proved), <p(5°°) = {<u e %- A|dimKw = oo}. Indeed,
tifirjx)) = 0w(x)dim and there exists x e fl+ such that 0ш(х) > 0.
This implies that n_1(°°) is measurable.
If j e if- N U S°°, x e fl+, then ttfirjx)) = п(ю)0ш(х). Thus, 14.6.18
implies that n is measurable on if - N U S°°.
14.10.5	. Although the next result is just the previous theorem restricted
to the special case when ё = C*(G), we feel that its importance merits a
complete statement.
Theorem. Let G be a separable, locally compact, topological group. Let
(тг, H) be a unitary representation ofG. Then there exists a Borel measure a
on <f(G) and a direct integral of representation (/^(С)7гш da(u),
da(a>)) such that:
(i)	(тг, H) is unitarily equivalent to (dofto), da(a>')).
(ii)	There exists a subset N of tf(G) such that a(N) = 0, and if
ш s <f(G) - N then (тгш,Нш) is equivalent to (тгш ® I, Нш ® Кш),
with (тгш, Нш) s ш.
(iii)	If ы e. ё(О — N, then set п(ш) = dim Va. Then n is a a-measur-
able function from <f(G) - N to the extended positive axis [0,oo]. As
before, n is called the multiplicity function.
14.10.6	. The preceding results also have companion uniqueness asser-
tions, which we will discuss in the notes at the end of the chapter and
defer to the references therein. We, however, will now prove a result that
says the representations that “appear” in the preceding decomposition are
uniquely determined.
If ё is a separable C* algebra and if (тг, H) is a non-degenerate
*-representation of -ё, then we say that ы e -ё is weakly contained in тг if
Кегю d Ker тг. We set supp(ir) = {<u s ё\ш is weakly contained in тг}. If
G is a separable, locally compact, topological group and if (тг, H) is a
unitary representation of G, then supp(ir) is defined to be {<u s d’(G)!
the corresponding class for C*(G) is in the support of the corresponding
representation of C*(G)}.
14.10. Decompositions of Representations
337
Lemma. Let G be a separable, locally compact, topological group. Let
(тт, H) be a unitary representation ofG. Let ш s <?(G) and let (ттш, Нш) e
ш. Then ы e supp(ir) if and only if the functions
(g	e нш, M i)
are contained in the closure (relative to the topology of uniform convergence
on compacta) of the set
(g -+{Tr(g)v,v) tv e H, Ill'll < 1).
We will (as usual) identify unitary representations of G and non-degen-
erate *-representations of C*(G). Then
supp(ir) = (C*(G)/Ker тг)".
Let Q be the closure of
{x -»(ir(x)l',l')| Ill'll < 1}
in the weak* topology of C*(G)~. If x e C*(G)/Ker тг and x* = x, then
x > 0 if and only if f(x) > 0 for all f e Q. Thus, Lemma 14.6.11 implies
that Q contains the pure states of C*(G)/Ker tt. The lemma now follows
from 14.7.3.
14.10.7	. Lemma. Let £ be a separable, CCR, C* algebra (resp., let G be
a separable, CCR, locally compact topological group). Let (тт, H) be a
nondegenerate *-representation of £ (resp., unitary representation of G). Let
т be as in Theorem 14.10.4 (resp., 14.10.5). If Y = & (resp., ff(G)), then
r(Y - supp(rr)) = 0.
It is enough to prove the result in the case of C* algebras. Let
Ker tt. Let {uy} be an approximate projection of .X (14.3.1). If ы e -6
and <тш e ш and if x e then
<rM(x) = lim аш(хи/) = lim аш(х)<тш(м;).
j —> 00	j —* 00
Thus, Л Кег<тш if and only if <тш(м;) = 0 for all j. Now, ir(u}) is given
(as in Theorem 14.10.4) by the operator field {тгш(му)}, and Ker тгш = Ker ш
for т-almost every ы. Thus, since тг(му) = 0, there is a subset Z} with
r(Zj) = 0 such that e Kerw for ш s -ё- Zj. Take Z = U;Zy. Then,
£ - supp(ir) c Z.
338
14. Abstract Representation Theory
14.10.8	. The following result will be used in the next section. No doubt
there is a simpler proof than the one given. However, it does give an
application of the main theorems in this section.
Theorem. Let and G2 be separable, locally compact, CCR, topological
groups. Then Gt X G2 is CCR. Let (тг, H) be an irreducible unitary
representation of Gt X G2. Then there exist unique (up to unitary equiva-
lence) irreducible unitary representations (rr^Hf), i = 1,2, of G{ and G2,
respectively, such that тг is unitarily equivalent with тг1 ® тг2 (14.A.4).
Furthermore, if (тг, , H;) is an irreducible unitary representation of Gt,
i = 1,2, then tti ® tt2 is irreducible.
We will first prove the second assertion. We will then prove the first,
and finally the last. Let rrL(g) = rr(g, 1), rrR(g) = тг(1, g). We look upon
тг as a representation of C*(G1 x G2), and Trt and rrR as representations
of C*(G). We apply Theorem 14.10.2 to ttl. Thus, there exists a compact,
separable Hausdorff space X, and a Radon measure g on X such that we
may assume that
Я = ( Hsdp(s)
Jx
and that irL is equivalent with fxirs dp(s). Let Z be such that tt(Z) = 0
and (irs, Hs) are defined and factorial for G1. We may further assume that
if s =# s', then HomG(Hs, Hs,) = 0. Now,
77«(C*(G2)) ccomm(rrL(C*(G)).
Hence, ё consists of decomposible elements. This implies that if У is a
measurable subset of X, then
/ Hsdp.(s)
JY
is (G1 x G2) invariant. This implies that if У cA" is measurable, then
either p(Y) = 0 or g(X - У) = 0.
Let её be the set of all compact subsets of У of X such that
p(X - У) = 0. Then X e её. We order её by У > Z if У c Z. We note
that if {}<}“_ i is a sequence in её, then Г\ e её. Indeed, p(X - П Yf)
= p.(Oj(X - Yf)) = 0. Suppose that S is a linearly ordered subset of её.
Let Z = Пу e j T- We assert that Z e. её. Indeed, since X - Z is locally
compact and separable, there exists a countable covering of X - Z, {Ц},
14.10. Decompositions of Representations
339
by open subsets of X - Z such that Vj = Uj is compact and contained in
X - Z. If У A Vi * 0 for all У g S, then П(У A Vt) * 0, which is not
possible. Thus, for each i there exists К g 5 such that Yt A Vt =#= 0. We
assert that П Y, = Z. Indeed,
(ПК) a (x- z) = и(цг> Пк) = 0-
J ' i
Thus, П К c 0- Since П Yt э Z, the assertion follows. Hence, every lin-
early ordered subset 5 of У has a supremum in c/'. Thus, there is an
element У g У such that if Z e У and Z с У, then Z = У. Suppose
that x, у g У, x =# y. If g({x}) > 0, then д(У - {x}) = 0. Hence, {x} g У*.
This is impossible by the choice of У. Thus, g({x}) = 0. This implies that
/а(У - {x}) > 0. Since g is a Radon measure, there exists Yy с У - {x}
with Yy compact and 0. Thus, g(X - Tj) = 0. So Yy g У*. This
contradiction implies that У consists of exactly one element. Hence, if st
is the strong closure of ttl(C*(G1)), Z(<q/) = С/. Thus, ttl is factorial.
We therefore see that there is a unique (up to unitary equivalence)
irreducible unitary representation (ttj, Hx) of G{ such that (ttl,H) is
unitarily equivalent with (irj ® I,HX® H2). But then з/= End(Hj) ® I.
Thus, тгя(х) = I ® rr2(x) (Lemma 14.A.1) for x g C*(G2) and (тг2, H2)
is a ^representation of C*(G2). If W were a close Tr2-invariant subspace
of H2, then Hi ® W would be (Gj x G2)-invariant. Thus, W = {0} or
W = H2. Thus, тг2 is irreducible. Note that in the course of the proofs we
have proved the uniqueness (up to equivalence) for ttj . Since ttr is also
factorial, тг2 is also unique.
We now prove that x G2 is CCR. If f g Ll(Gi), g e Ll(G2), then
we set f ® g(x, y) = f(x)g(y). Then, f ® g g Ll(Gi x G2) and Ll(Gi)
® Ll(G2) is dense in Ll(Gi x G2). To see this, let {u,} be an approximate
identity in L^Gj), Ц} the same for Ll(G2). Then u, ® ц is an approxi-
mate identity for Gj ® G2. Now apply 14.2.4. If ш g (Gx X G2) and if
(тт, H) = (tty ® тг2, Hx ® H2) g a), then tty ® rr2(f ® g) = тг//) ®
Tr2(g). Thus, irUXGi X G2)) с СС(Я).
Suppose that (тг,, H,) is an irreducible unitary representation of G, for
i = 1,2. Set тг = ttj ® tt2. The preceding argument implies that
tt1(C*(G1)) ® tt2(C*(G2)) is dense in the strong closure of tt(C*(Gx X
G2)), which we denote by з/. If T g сотт(лУ), then T g
comm(Tr1(C*(G1)) ® I) A comm(/ ® tt2(C*(G2))). But the strong closure
of tt1(C*(G1)) ® I is End(Hj) ® I. Hence (Lemma 14.A.1), T = I ® У for
some У g End(H2). By symmetry we see that T = I ® У and T = Z ® I
340
14. Abstract Representation Theory
for some У e End(f/2), Z e End(Hj). Thus, T s CI. This implies the
irreducibility (14.5.1 (2)).
14.11.	The Plancherel formula for CCR locally compact,
unimodular groups
14.11.1.	In this section, G will denote a separable, locally compact,
unimodular, CCR, topological group. Since G is unimodular, L2(G) is a
unitary representation of G under both the left and right regular repre-
sentations (L(g)/(x) = /(g-1x), R(g)/(x) = /(xg)). We define a unitary
representation U of G x G on L2(G) by U(x, y)f{z) = f(x~lzy) for
x, y,z e G and f e L2(G). The main result in this section will give a very
precise form of the direct integral decompositions of the last sections for
L, R, and U. We first need some notation and conventions.
If ш e ^(G), then we fix (тгш, Нш) e ш. If (тг, H) is a unitary represen-
tation of G and if H' is the space of continuous linear functionals on H,
then we define a representation on H' by = A ° ir(g)~l. If v e H,
then we set At,(w) = <w, v). Then, the map r from H to H' given by
r(v) = Ar is a conjugate linear bijection of H onto H'. We note that
At(Tr(g)-1w) ={ 77(g)=<W,77(g)r> = A^tv).
Thus:
(1)	т ° Hg) = rr'(g)0 T.
We put a Hilbert space structure of H' by (A,gw) = A(w). With this
Hilbert space structure, {tv, tw) = {w, v). If ы s d’(G), then we note
that (1) implies that тг'ш is also irreducible. We therefore have an involu-
tive map ш -» ш' of d’(G) onto d’(G) defined by тг'ш e ш'. By Нш ® Нш,
we will mean the tensor product of the Hilbert spaces Нш and Н'ш.
ira ® 1тш' will stand for the unitary representation тгш ® тг' of G X G on
Нш ® Нш, (14.A.4).
14.11.2.	Theorem. There exists a Borel measure £ on <?{G) and a direct
integral of unitary representations (	d£(a>), f^(G)Wu d[(<i>)) satisfying
the following three conditions:
(1)	{U, L2(G)) is unitary equivalent with (d^(a>), dlfuf).
(2)	For ^-almost every ы e <?{G), (<тш, is unitarily equivalent with
®	® НШ')-
14.11. The Plancherel Formnla for CCR Locally Compact, Unimodnlar Gronps
341
(3)	If f L2(G) Ci Ll(G), then тгш(/) is of Hilbert-Schmidt class for
^-almost every ш e #(G), and
<f,f>=(
Furthermore, (3) uniquely specifies %. £ is called the Plancherel mea-
sure associated with dg.
Note. Ытгш(/)*тгш(/)) depends only on ш and f and not on the choice of
(тгш, Hoj). The measure [ depends on the choice of invariant measure
on G.
14.113. The proof will take some preparation. If f is a measurable
function on G, then set r](f)(g) = f(g~l). Since G is unimodular rj
defines an isometry of both Ll(G) and L2(G). We assert that extends
(from Ll(G)) to an isometry of C*(G). Indeed, if (тг, H) is a unitary
representation of G and if f e Ll(G), then
ir'(f)A(v) = f f(g)(Tr'(g)A,Ar)dg
JG
= jGf(g)(^^\sYXA^ dg = jj(g-^A,TT'(g)Av}dg
= f	= A('n’(T?(/))i;)-
JG
This implies that:
(1)	hW))ll=k'(/)ll.
Thus, H/H = IH(/)||. We note that if f e L\G), then л(/*) = f. Thus, the
map f -> f extends to a continuous conjugate linear automorphism of
C*(G), which we also denote by x -> J.
We now collect some properties of 77.
(2)	If f, g e Ll(G) A L2(G\ then L(f)g = R(-n(g))f.
Indeed,
L(f)g(x) = f f(y)g(y~lx)dy = f f( y)y(g)(x~ly) dy
JG	JG
= f f(xy)r](g)(y) dy = R(T](g))f(x).
JG
(3)	If X e C*(G), f s L2(G\ then r](L(x)f) = R(x)-n(f).
342
14. Abstract Representation Theory
It is enough to check this for x, f s 0(0 A 0(0. In this case it
follows from the obvious calculation (see (2)).
Similarly:
(4)	If x g 0(0 and if f e 0(0, then (L(x)/)* = R(x)f* (f*(g)
Let sZ (resp., sZL, <q/r) be the strong closure of U(O(G X G)) (resp.,
L(C*(G)), R(O(G))).
(5)	зэ/ is the strong closure of <&L<&R.
We use the notation in the last part of the proof of Theorem 14.10.8.
We saw therein that 0(G) ® O(G) is dense in O(G X G). Since U(f ®
g) = L(f)R(g), the assertion follows.
(6)	comm(ja/) = srfL A srfR = Z(sfL) = Z(sfR).
Let Uj be as in 1.1.3 for G. Then g 0(0 A 0(0 and ||u;||i = 1, so
||u;|| < 1. Thus, lim,-,*, UjX = x for x e 0(0. Let T g comm(<o/). Then
T g comm(<o4) A comm(ja^). If f, g g 0(0 A 0(0, then
T(f*g) = T(L(f)g)=L(f)T(g).
Also
T(f^g) = T(R(-n(g))f) = R(v(g))T(f) = L(T(f))g.
Hence, L(f)T(g) = L(T(f))g. If we apply this to f = u;, then we find
that
lim L(Uj)T(g) = lim L(T(Uj))g.
J —»00	J —»00
Thus, T g srfL. Similarly, T g srfR. (4) now implies that comm(ja/) =
A srfR cZ(srfl) A Z(srfR). On the other hand, (4) also implies that
Z(s^0 and Z(srf^) are contained in comm(ja/). Thus, Z(srfL) = Z(sfR) =
comm(ja/).
Let g be a Borel measure on ^(G) such that
(L,L2(G)) = Ьшс1ц(Ш),{ Уш<1ц(Ш)
V <f(G)
14.11. The Plancherel Formnla for CCR Locally Compact, Unimodnlar Gronps
343
as in Theorem 14.10.4. If we examine the proof of 14.10.4 then we see that
Z«) = {Af^GL“(<f(G);g)}.
This combines with (6) implies that <0/ consists of decomposible operators.
Thus, by the usual argument (14.9.4 (1)—(5)) we have, for g-almost every
cd g a *-representation Uu of G x G on Va. Since comm(ja/) =
Z(<o/), we see (by the argument in the proof of Theorem 14.9.4) that Ua is
irreducible for g-almost every ш. Let N be such that g(N) = 0 and
and Lu are defined for all ы g <f(G) - N. Theorem 14.10.8 implies that
(С7Ш, Уш) is unitarily equivalent with
(тгш ® аш , Нш ® И^),
with (<тш, an irreducible unitary representation of G. The crux of the
matter is:
(7)	(<тш, Нш) is unitarily equivalent with (тг'ш, Н'ш).
To prove this, we first observe that:
(8)	Ker tt' = 7)(Ker тг) if (тг, Я) is a *-representation of C*(G).
Indeed, (1) implies that ||ir'(x)|| = ||тг(,п(х))|| for all x g C*(G). This
clearly implies the result.
We next observe that there exists a set Nx of g-measure 0 containing
N, and a conjugate linear operator on Уш, denoted v -» v*, such that
(/*)„ = (/„)* for all ы g <^(G) - Nx and f contained in an appropriate
dense subset of L2(G). To see this, we first observe that we can define a
multiplication on Уш for g = a.e. ш. Let fl be a countable dense subset of
Z?(G) contained in Z?(G) П L2(G) that is closed under multiplication,
addition, scalar multiplication by algebraic numbers, and under *. Let
N2 N be a set of measure zero such that is defined for all ы g <f(G)
- N2. If f, g g fl, then
(L(f)g)a = LJJ)ga = RMg))fa.
Thus, (L(f)g\, depends only on fa and ga. Furthermore, there is a
subset Nx э N2 such that fi(Nx) = 0 and ||/ш||^ < </,/> for f g fl. We
therefore have (by the obvious limiting argument) an algebra structure on
Уш. We note that if x, y, z g Уш and x = fa with f e Ll(G) П L2(G),
then <xy, z>w = <у,(/*)шг)ш. Thus, (/*)„ depends only on x. So x* is
defined for all x g Уш .
344
14. Abstract Representation Theory
We can now complete the proof of (7). We apply (4) to see that if
x e C*(G) and if v e Уш, ш e <f(G) - NY, f e L2(G) such that fa = v,
then on the one hand
((L(x)/)*)B = ((L(x)/)J* = (LB(x)/X
On the other hand, (4) implies that
((Цх)/)*)в = (Я(х)/*)ш = Яш(х)(/ш)*.
This implies that
Ker = KerLw = ^(KerLJ* = ^(KerLJ.
Since КегЛш = Кег<тш, (7) now follows from (8).
Let if = C*(G). If x e ^F (14.6.16) and if ш e <f(G), then ттш(х) is of
finite rank and hence it is of Hilbert-Schmidt class. On the other hand, we
have the map J: Нш ®	-> HS(HW) given by J(v ® AXz) = A(z)r
(14.A.3). J extends to a bijective unitary operator. We assume (as we may)
that N{ is a Borel set. If ш e £(G) - Nlt then we have a unitary
equivalence Тш of Ua with тгш®тгш,. If x e €F, then JTuLu(x)v =
тгш(х)1Тши. This implies that (see the preceding) JTU defines an algebra
isomorphism of Уш onto HS(HW). Hence, if f e Ll(G) П L2(G), then
тгш(/) is of Hilbert-Schmidt class for ы e <f(G) -	. Schur’s lemma now
implies that if ы e <f(G) - Nj, then there is a positive real number <p(a>)
such that if f, g e Ll(G) П L2(G) then </ш, gu)u =
^(wjtrfjrjg)* ттш(/)). If we apply the argument in 14.10.4 (3), we see that
w -> 1г(тгш(/)*тгш(/)) is measurable on ^(G). Hence, <p is a т-measurable
function on <f(G) -	. We define a Borel measure £ on dXG) - Nj by
d[ = <pdr. If we replace < , )ш with <р(ш)-1< , )ш, then we have our
desired decomposition satisfying 14.11.2 (1), (2), (3).
To complete the proof of Theorem 14.11.2 we must establish the
uniqueness. Let £ be another Borel measure on <f(G) such that:
(9)	</,/> = f tr(^/)=4(/W")
for all f e Ll(G) П L2(G) (see the note at the end of 14.11.2).
Let v be either £ or Let U be an open subset of <f(G). Set
.У= P| Kerw.
o>e<f(G)-G
14.11. The Plancherel Formnla for CCR Locally Compact, Unimodnlar Gronps 345
Let {uy} be as in 14.3.1 for (note u, < u, + 1). If x e -6F, x > 0, then
(clearly):
(10)	77ш(х1/2иух1/2) = тгш(х1/2)тгш(и-)тгш(х1/2).
We note that if ш e U, then = irjjf) = ССХНШ). If ы <£ U, then
1гш(«Х) = 0. Let x = Xu be the indicator function of U. We note that (10)
implies that
1Ta(x1/2UjX1/2) < ТГш(х1/2М; + 1Х1/2),
and thus 0ш(х1/2МуХ1/2) is a monotone increasing sequence with limit
Xt/(<w)0a)(x). Since ы -> 0ш(х) is lower semicontinuous (14.6.18), we see
that it is Borel measurable. Hence, dominated convergence implies that
lim J 0ш(х1/2их1/2) dv(u) = f Хи(ш)®ш(х) dv(a>).
J£(G)	J£(.G)
We rewrite this as
lim ( 1г(тгш(м )тгш(х)) t/p(w) = f 1г(тгш(х)) dp(a>).
J^°° J<S’(G)	JU
(9) and 14.11.2 (3) imply that if f s L2(G) and {/y} is a sequence of
Ll(G) П L2(G) that converges to f in L2(G\ then, replacing {/;} by an
appropriate subsequence, for p-a.e. ы, converges to a Hilbert-
Schmidt operator on Нш, which we also define to be irj/)- Let r- =
u}/2 & J?. Then
lim (л-ш(гу)/,л-ш(гу)/} = lim /	1г(л-ш(иу)л-ш(/)*л-ш(/)) dv(a>)
Ju
This implies that if f e Ll(G) П L2(G), then
J trMf)*^(f))d£(a>) = J tr(^(f)^u(f))d^).
Ju	Ju
Hence, 1г(тгш(/)*тгш(/))^(ш) = 1г(тгш(/)*тгш(/))^(ш) for all /e Ll(G)
П L2(G). Since ы -> 1г(тгш(/)*тгш(/)) is a Borel function on ^(G), this
implies that This completes the proof of Theorem 14.11.2.
346
14. Abstract Representation Theory
14.11.4. We now give a result that is a mild extension of a theorem of
Cowling et al. [1] that gives a growth condition on supp(L, L2(G)) for a
class of groups.
We assume that there exists a closed subgroup P and a compact
subgroup К of G such that G = PK. Let 3 = 8P be the modular function
of G. Let dp be a choice of right invariant measure on P and let dk be
normalized invariant measure on K. We note that dg can be chosen so
that (cf. Lemma 0.1.4)
(1) ff(g)dg=f f(pk)8(p)-ldpdk.
JG	JPXK
If m P A К, then 8(m) = 1 (P П К is compact and 3 is a continuous
group homomorphism into (0, oo) under multiplication). This implies that
we can define a continuous function 30 on G by 8a(pk) = 3(p)1/2. We
define a function on G that is K-bi-invariant by
S(g) = [80(kg)dk.
JK
Theorem. Suppose that
ш e supp(L, L2(G)) = supp(P, L2(G)).
If F e <a(K) and if v s MX?)» с^еп
I	\l/2
I <^(g)i;>i;)| < £ </(y)2	Hr||2H(g).
\ -yeF	f
Assume that ш s supp(P, L2(G)) = S. Then, if ||r|| = 1 the matrix
coefficient cv r(g) = {irj.g)v,v) is in the closure in the topology of
uniform convergence on compacta of the matrix coefficients
9,z(g) = (g-<«(g)/>/»> II/II2<1
(14.10.6). Since К is compact, cv v is in the closure of the space of matrix
coefficients cff, with
E d{y}\~^k}f{-k}dk=f
y<EF	K
and ll/ll < 1 (Lemma 1.4.6). Thus, it is enough to show that:
14.11. The Plancherel Formnla for CCR Locally Compact, Unimodnlar Gronps
347
(2) If f, g g ®? e F L2(GXy), then, for all x g G,
I	\1/2
\<R(x)f, g)| < ||/||2||g||2 E J(y) E(x).
' yf=.F	'
Since ®?ef L2(GXy) A Cc(G) is dense in ®?ef L2(GXy), it is enough
to prove the result for f,ge ®yeF £2(GXy)Vi Cc(G).
If <p g Cc(G), then set ф(^) = sup^eK|<p(g^)|. Then, ф e L“(G) and
supp(^) c supp(<p)K\ Thus, ф g L2(GXy0) (y0 is the class of the trivial
representation of К).
(3) If <p g ®ye F L2(GXy) A Cc(G), then
/	\1/2
Il<pll2 < I Е^(у)2| ПфПг •
' у	’
Indeed, fix g g G and set a(k) = <p(gk). Then,
a(k) = E d(y)l a(ku)xy(u)du.
К
Hence,
/	\l/2/	\1/2
|«(Ac) | < f \a(ku) |2 du И E d(y)d(r)xy(u)xT(u) du\ .
Vk	) \JKyT^F	)
Lemma 1.4.5 implies that
/ \1/2
I a(k) | E 4?)2
Thus,
|<p(gfc)|2 < [ E d(y)2\j\f(gk)\2dk.
\7eF ' %
This implies that
Mi ( E ^(y)2j/ \<p(gk)\2 dgdk = ( E ^(y)2jll<pll2 •
'ye/7	f	\y^p	/
This proves (3).
348
14. Abstract Representation Theory
Now,
\{R(g)f,g)\ = f f(xg)f(x)dx
JG
'g
- JGf(xs)g(x)dx =(R(g)f,g).
Thus, if we show that if /s Cc(G/K) then \{R(g)f,f)\ <, H/IllHCg),
then the theorem will follow.
Let f e Cc(G/K). Then
\<R(g)f,f)\ = f f(xg)f(x)dx
JG
= f 8(p) 1 /(pkg)f( pk) dpdk
JPXK
<J^J?8(p) xf(pkg)f(p)dpidk
/	\ 1/2/	\l/2
< fl [8(p)~I\f(pkg)i2 dpi fs(p)~Ilf(p)l2 dpi dk,
jk\jp	I \Jp	)
by the Schwarz inequality. Thus, (1) implies that
/	U/2
\{R(g)f,f>\<\\f\\2j^j8(py1\f(pkg)\2dpj dk.
We write g = p(g)k(g) (with the ambiguity in M = P fl K). Then,
f(pkg) = f(pp(kg)k(kg)) =f(pp(kg)).
Hence,
I	\1/2
I < R(g)f, f> I	H/II2Д j8( P) - *| /( PP(kg)) |2 dp j dk
I	\1/2
= П/Н2/8(p(kg))1/2{ [8(p)-l\f(p)\2dp] dk
jk	\Jp	)
= Wf\\22f80(kp)dk = \\f\\22Q(g).
JK
This completes the proof of (2) and hence of the theorem.
14.12. The Plancherel Formnla for Real Rednctive Gronps
349
14.12.	The Plancherel formula for real reductive groups
14.12.1.	The purpose of this section is to relate the abstract Plancherel
theorem of the previous section to the Harish-Chandra Plancherel theo-
rem of Chapter 13. Let G be a real reductive group and let К be a
maximal compact subgroup of G. We fix (P0,A0), a standard minimal
p-pair for G.
Theorem 14.6.10 says that G is CCR. Thus, the results of the previous
section apply. We note that if we take the “P” in 14.11.4 to be Po, then
the function H of that section agress with Harish-Chandra’s basic zonal
spherical function (4.5.3). Let d^emp(G) be the set of equivalence classes of
irreducible tempered representations of G (i.e., the underlying (q,K)-
module is tempered; see 5.1). Theorem 14.11.4 implies:
Theorem. supp(R, L2(G)) = supp(L, L2(G)) c ^emp(G).
14.12.2.	We also note (later results suggest that the d(y)3 here can be
replaced by d(y)):
Lemma. If ш e <^етр(С), (тг, H) e ш, у е <f(K), v е Жу), ||u|| = 1,
then
| {ir(g)v, v) I <d(y)3S(g).
Furthermore, d’temp(G) is closed in £(G).
Proposition 5.2.5 (combined with Theorem 5.5.4) implies that we may
assume that (тг, Я) = (irPa iv, FT), with (P, A) a standard p-pair and a
a square integrable representation of °MP. Let ш be the class of a. We set
= SoM . Since	is square integrable, there exists a closed
R(°Af^-invariant subspace V of L2(°AfP) such that (<r, is unitarily
equivalent with (R, V). Let F = {r e <F(P П AT)|(-y: r) =# 0}. Then 14.11.4
(2) implies that if v, w e ®TeF На(т), then:
(1)	|<<t(w)l’,w>| < (EyeFd(r)2) Z Hull ||n'||E/>(m)
< </(y)llu|| ||w||HF(m).
Let f e Ha(y). Then (p = pP)
Wf,f) = f a(kgY+i\a(m(kg))f(k(kg)),f(k))dk.
JK
350
14. Abstract Representation Theory
Now, f(k) = ETef /Д), with fT(K) с Яа(т). Thus, applying (1), we
have
|<a(m)/(^),/(^2)>| <SP(/n)t/(r)||/(^i)|| \\f(k2)\\.
Hence,
\«g)f,f)\^d(y)fa(kg)p\\f(k(kg))\\ \\f(k)\\HP(m(kg))dk.
We note that ||/(£)||2 = <f(k),f(k)) = LYfik^v^ftk)), with {u,}
an orthonormal basis of ETef Ha(r). Let <р,(к) = {f(,k),Vj). The argu-
ment in the proof of 14.11.4 implies that |<p,(&)|	</(y)ll<p(ll2• Thus,
Il/(*)||2 1 WLMI2 = d(y)2\\f\\2.
i
This implies that
\{ir(g)f,f)\ d(y)3f a(kg)pBP(m(kg)) dk = d(y)3E(g).
JK
The last equation is proved using either induction in stages or the obvious
calculation using the definition of S.
We now prove that <Ftemp(G) is closed. Let ш g ^emp(G).If(7T,H)G
у g <F(K), and v g Я(у), Hull = 1,. then <p(g) = (ir(g)v,v') is in the
closure (relative to the topology of uniform convergence on compacta) of
{(g	g ^temp(G), (тгм,ям) Gg, W еям(у), llwll < 1}.
The first part of this result now implies that |<p(g)| < d(y)3S(g). Thus, ы
is tempered.
14.12.3.	We note that the preceding result combined with Proposition
5.2.5 (and the results of 5.5) yields an alternate proof of Theorem 13.4.2.
Let ^usp be the set of (PF,AF) (2.2.7), F с Д(Р0, Ao) and PF cusp-
idal (i.e., °MF has a compact Cartan subgroup). If P g ^.usp and then
we set srfP= <?2(°MP) X (a*X (see 7.7.1 for <f2, 12.2.6 for (а*РУ). We
endow <F2(°Mp) with the discrete topology and (а$У with the usual vec-
tor space topology. We put the product topology on sfP. We let W(AP)
act on by s(<a, v) = (sa>, sv). Notice that W(AP) acts freely. Thus,
<srfp/W(AP) isa locally compact, separable Hausdorff space. We set jrf=
U/>e^cusp <&p/W(AP) (disjoint union), with the disjoint union topology.
14.12. The Plancherel Formnla for Real Rednctive Gronps
351
If (ы, v) g srfp, let Ф((ш, v)) be the equivalence class of ttp a lv for
a g ш (5.2.1). Proposition 12.1.3 implies that Ф($(ю, v)) = Ф((ш, v)) for
j g W(A). Corollary 12.5.4 implies that if (w, v) g then тгР a iv is
irreducible. Thus, Ф induces a map (which we will also denote by Ф) from
srf to d’(G).
Theorem. Ф is a homeomorphism onto its image in Furthermore,
Ф(<о/) is dense in <^етр(О.
We first prove that Ф is injective. Suppose that P,Qe ^.usp and
(oij, i^j) g srfp, (w2, p2) g are such that ^((wj, i^)) = 'P((<u2> r^))-
Then
Home K(/P ai ,Pi,/e ai>,P2) * 0
and
* 0-
This implies that (notation as in 12.2.1)
Vq\p,<t2,iv2 * 0 and VpiQ, * 0.
Hence, Proposition 12.2.1 implies that AQ сЛР and AP ^AQ. Hence,
P = Q. Theorem 12.1.4 now implies that there exists j g W(Ap) such that
•s(wj, i^) = (a)2, г'г)- This concludes the proof of the injectivity of Ф.
We now prove that Ф is continuous. Let S c d’(G) be closed. We must
show that Ф-1(5) is closed. Let g Ф-1(5) and assume that
lim ху = x e stf.
j-,«,
By the definition of the topology we may assume that x e srfP and
x - g srfp. Since <f2(nAf/>) has the discrete topology we may assume that
x = (ш, v) and Xj = (ы, Vj), with
lim Vj = v.
j-хю
Now, ttp aiX for A g a* is a unitary representation of Ha, a Hilbert space
depending only on a\K^P. If v e. Ha, then the map
(g,A) ^{rrPi<T iK(g)v,v}
is continuous. Thus, Ф(х) с	(Theorem 14.7.4 (3)). Hence, 'P(x)
g S so x g Ф-1(5).
352
14. Abstract Representation Theory
We now show that is dense in <^етр(О. Let ш e d^temp(G).
Then there exists P s ^cusp, a s g e <f2(°MP\ v e a*P a subrepresen-
tation (тг, И) of 1гл<7((> with (тг, V) s a>. Let Vj be a sequence of (а£У
such that limy-,», Vj = v. Then if у e v s V(y\ Hull = 1, then
lim^^ <тгЛо. ,p(g)u, u> = (ir(g)v, v) uniformly in compacta of G. Thus,
ш e 'P(jaZ').
We are left with showing that if S с ja/ is closed, then Ф(5) is relatively
closed in Ф(л^). Let x e П Ф(5). We must show that x e 'J'(S).
If (ш,р)ел/Р and if v s Ha(a e a>), then we set сш „ v(g) =
{irP ajv(g)v,v>. If we apply Theorem 14.7.4 (2), we find that if x =
^((w, v)), {ы, v) e £/P, and if a e ы, у e t£\K\ v e Ha(y), ||u|| = 1,
then there exists and ordered set I and a net c „ ||i>„|| < 1, with
lima c v = cavv in the topology of uniform convergence on com-
pacta.
If f s if(G), then we set
Ca(f) = J f(g)Ca,a.,a.Va(g) dg
j G
and
c(/) = J f(g)cUtV,v(g) dg-
JG
(1) If f e C“(G), then lim„ ca(f) = c(f).
This follows from the uniform convergence theorem for integrals.
The key to the proof is the next assertion.
(2) If f e ^(G), then lim„ ca(f) = c(f).
We use the notation of Lemma 5.1.3, which implies that there exists d
such that
M = f S(g)2(l + log||g||)"‘zdg < oo.
JG
On t^(G), we define the continuous semi-norm
v(f) = SUP M/(y)3S(g)-1(l + log||g||)d|/(g)|.
geG
Lemma 14.12.2 implies that if f e -^(G), then
|ca(/)|<^(/) and
14.12. The Plancherel Formnla for Real Rednctive Gronps
353
We now prove (2). Let f s if(G). Let e > 0 and let <p s C“(G) be such
that v(f - <p) < e/3. Let a0 be such that if a > a0, then |ca(<p) - c(<p)|
< e/3. If a > a0, then
lC«(/) - C(/)|	~ Ca(<P)l +lCa(<P) - C(<p)| +|c(<p) - C(/)|
< 2v(f - <p) + e/2 < e.
We now return to the proof that x s S. Suppose that f s if(G)
transforms on the right by y, is left /С-finite and f = fA (13.4.5). Then if
APa =#= AP, ca(f) = 0 (13.4.5). Hence, we may assume that Pa = P for all
a. If we take f as in 13.4.4 and if ыа #= sa> for some j g W(Ap), then
ca(f) = 0- Thus, we may assume that ыа = ы for all a. Finally, let U be a
neighborhood of v in (a£)'. If we take T (as in 13.4.4) such that supp(T)
c W(AP)U and if va <£ W(AP)U, then ca(f) = 0. Thus, there exists a0
such that if a > aQ, then va e W(A)U. lim va = v, and hence x s 5. This
completes the proof.
14.12.4.	We now define a Radon measure A on <0/. On <я/р we take A to
be the product of Lebesgue measure (normalized as in 13.3.2) on (а£У
and counting measure on d’2(°M/,). Then A is invariant under the action of
W(Ap) on <я/р. It therefore induces a measure, also denoted A, on
£/p/W(Ap). We say that X c X= \}PXP, Xp^^p/W{Ap) is A
measurable if each XP is A measurable, and we set A(X) =	A(A'P).
We define a measure on <^,етр(С), A, by A(G) = А(Ф-1(Ю). Let £ be
the Plancherel measure of G with respect to our choice of dg (14.11.2).
Theorem 14.12.1 combined with Lemma 14.10.7 implies that ^'(d’(G) -
<ftemp(G)) = 0.
We define a function on srf as follows: If (w, v) = then (see
Theorem 13.4.1)
Mw(("^)) = (1и/(^/>)1%4с^)1^(")м(">^)-
We define a А-measurable function on <^,етр(С) by jxH = Ph° ^-1-
The relationship between the abstract Plancherel theorem and the
Harish-Chandra Plancherel theorem is given in the following formula.
Theorem. d£ = jiH dk.
Let £ be the measure given on the right hand side of the statement. Let
<p s if(G) be right and left ^-finite. Let f = <p* *<p. Then /(1) = (<p, <p).
354
14. Abstract Representation Theory
Hence, Theorem 13.4.1 implies that
(<p,<p>= f 1г(тгш(<р)*тгш(<р))^(ы).
Since the space of right and left ^-finite elements of ^(G) is dense in
L2(G), a limiting argument as in the proof of the uniqueness part of
Theorem 14.11.2 implies that the preceding formula is true for <p e LX(G)
П L2(G). The result now follows from the uniqueness assertion in Theo-
rem 14.11.2.
14.12.5.	We note that Harish-Chandra’s theorem could be construed as a
calculation of the abstract Plancherel measure. However, the theorem is
much more than that since it also contains at its heart the full analytic
(and a substantial part of the algebraic) theory of tempered representa-
tions of real reductive groups. See also Chapter 15.
14.13. Notes and further results
14.13.1.	The notion of a C* algebra (14.1.6) is due to Gelfand-Naimark
[1] and Rickart [1]. Segal [1] coined the name. The basic theorem on
commutative C* algebras (14.1.6) is due to Gelfand-Naimark [2] (although
many of the critical ideas already appear in earlier papers of Gelfand; cf.
Gelfand [1]).
14.13.2.	Lemma 14.1.11 is the “continuous functional calculus.” It com-
bined with Lemma 14.8.10 implies, in particular, the spectral theorem.
14.13.3.	The critical idea of approximate identity (14.3.1) is due to Segal
[1]. Our treatment follows Dixmier [1].
14.13.4.	Theorem 14.5.9 (as mentioned in the introduction to this chap-
ter) says that the axioms of a C* algebra describe precisely algebras that
are given as closed *-invariant subalgebras of the algebra of bounded
operators on Hilbert spaces. It is due to Gelfand-Naimark [2].
14.13.5.	The definition of a direct integral is as in Dixmier [3] (and [4]).
An alternate definition that is equivalent (in light of 14.8.6) can be found
in Naimark [2].
14.А. Some Fnnctional Analysis
355
14.13.6.	Theorem 14.8.14 is the basis for all “disintegration theorems”
for C* algebras and locally compact groups. Our exposition of this result
follows Dixmier [3] with the simplifications that are entailed in our
assumption of separability.
14.13.7.	Theorem 14.9.4 is usually attributed to Mautner [1]. Although
this theorem seems to settle and problem of decomposing representations
into irreducible components it has not played a significant role in the
theory. A more basic theorem is the decomposition into factorial represen-
tations (Theorem 14.10.2), which is due to Naimark [1].
14.13.8.	Theorem 14.10.4 (and hence 14.10.5) have a companion unique-
ness statement. To give the assertion we will use the notation of 14.10.4.
Suppose that g is another Borel measure on -6 such that (тг, H) is
unitarily equivalent with
\J€	I
We also assume that 14.10.4 (ii) and (iii) are satisfied. Then g is absolutely
continuous with respect to r and the multiplicity functions are equal
т-almost everywhere (cf. Dixmier [1]).
14.13.9.	Theorem 14.11.2 is due to Segal [2].
14.13.10.	Results of the type of Theorem 14.12.3 for the image of s#P
for P minimal or P = G can be found in Lipsman [1]. Theorem 14.12.1 is
due to Cowling et al. [1]. The results in that paper also imply that
^temp(^) = supp(R, L2(G)). An alternate approach, which yields a more
general theorem, is due to J. Bernstein [2].
14.A.	Some functional analysis
14.A.1.	The purpose of this appendix is to prove or to give appropriate
references to several results in analysis that are used in the body of the
chapter. We begin with a discussion of tensor products of Hilbert spaces.
Let and H2 be separable Hilbert spaces. Let Hx ® H2 denote the
algebraic tensor product of Hx ® H2. We define an inner product on
H{ ® H2 by (v ® w, x ® y> = (v, x)(w, y). We note that the bilinearity
356
14. Abstract Representation Theory
in v, w and conjugate bilinearity in x, у imply that this does indeed induce
a Hermitian form on Hx ® H2• If x s Hx ® H2, then there exists an
orthonormal set ux,...,ud s H2 and xx,..., xd & Hx with x e Ej Xj ®
Uj. Thus, <x, x> = E;<x(, x(>. We therefore see that < , > defines an
inner product on Hx ® Я2. We set Hx ® H2 equal to the Hilbert space
completion of Hx ® H2 with respect to this inner product. We call
Hx ® H2 the Hilbert space tensor product of Hx and H2.
(1) If {e-J is an orthonormal basis of Hx and if {/Д is an orthonormal
basis of H2, then {e, ® /Д is an orthonormal basis of Hx ® H2.
Clearly, (e, ® /Д is orthonormal. If x e. Hx ® H2 and <x, ex ® fj} = 0
for all i, j, then <x, Hx ® H2} = 0. Thus, x = 0.
Lemma. If T s End(ffj ® H2) and if [Г, X ® /] = 0 for all X s
End(f/j), then there exists Y s End(H2) suc^ that T = I ® У.
T( Xv ® w) = (X ® I)T(v ® w)
for all v e Hx, w e H2. If Л s H'x, define <pA(u ® w) = A(w)u. Then,
|<<pA(u ® w),<pA(x ®y)>| =|A(w)A(y)<u,x>| < ||A||2||u|| ||x|| ||w||||y||.
Thus, <pA extends to a bounded operator on Hx ® H2. We set, for v s H2,
afw) = w ® v, for w e Hx. Then,
<pfrav(Xz) =<pA(T((X®/)(z®u)))
= <pA((X®/)T(z®u))
= X<pxT(z ® v) = X<pjav(z).
This implies that <pKTav = p(X, v)I. It is clear that H^TaJ < Ill’ll ||A|| ||u||.
Thus |g(A,u)| < Ill’ll ||A|| ||i»||. We therefore see that there exists Уе
End(f/2) with ЦУ|| < Ill’ll and g(A, v) = X(Yv). It follows that for all
A e H'2, v e H2,	-I® Y)av = 0. Hence, T = I ® У.
14.A.2. If Hx, H2 are separable Hilbert spaces, then we say that a
bounded operator T from Hx to H2 is of Hilbert-Schmidt class if, for
some orthonormal basis {e;} of Hx,
(1) IItIIhs = Ell^jll2 <«•
14.A. Some Fnnctional Analysis
357
We set HS(H,, H2) equal to the set of all Hilbert-Schmidt operators from
Hx to H2. As in 8.A.1.6, we have:
(2) If T g HS(H15 H2\ then T* g HS(H2, and (1) is independent
of the choice of orthonormal basis of .
We note that if S, T g HS(H15 H2), then S*T is a trace class operator
(8.A.1.5) on HY. Furthermore, ||T||hs = tr(T*T). We therefore have an
inner product, (T’,5)hs = tr(S*T). As in 8.A.1, HS(H15H2) is a Hilbert
space with respect to < , )Hs, contained in the space of all compact
operators from Hx to H2, and such that the finite rank operators form a
dense subspace.
14.A.3. If Я is a Hilbert space and if v e (H, then we define Xv g H'
(continuous dual) by At(w) = (w, v\ Set T(v) = A„. Then, T is a conju-
gate linear isomorphism of H onto H'. We define an inner product on H'
by (A, A(.) = A(u). If v ® A g H2 ® H'i, then we define
r(v ® A)(w) = A(w)u
for w g Hx. We note that
t(u ® A) g HS(H15H2)
and
(t(u ®A),t(w®/a))hs = {v ® A,w ® g>.
Thus т extends to a unitary operator from H2 ® to HS(Hj, H2).
Lemma, r defines a unitary bijection.
This is clear since the image of r contains the operators of finite rank.
14.A.4. If Gj, G2 are a locally compact, separable, topological group and
if (тг, , H,) is a representation of G, for i = 1,2, then we define, for
(gj, g2) e Gj x G2, (ttj ® ir2Xgj, g2) to be the extension of TTj(gj) ®
rr2(g2) as a bounded operator on ® H2. It is a simple exercise to check
that (ttj ® tt2,Hj ® H2) defines a unitary representation of Gj x G2
(cf. 1.1.3).
14.A.5. We next collect some results on locally convex spaces. Let V be a
topological vector space over C or R. Then V is said to be locally convex if
0 has a neighborhood basis consisting of convex open subsets of V.
358
14. Abstract Representation Theory
Clearly, a Banach space is a locally convex space. If И is a locally convex
space, we set V equal to the space of all continuous functionals on V. The
weak* topology on V is the topology given by the semi-norms /^(A) =
|A(u)|. With respect to the weak* topology, V is a locally convex space.
If И is a Banach space with norm II • • • II, then V is a Banach space
with respect to the operator norm.
The following result of Banach-Alaoglu is critical to the theory of this
chapter.
Theorem. Assume that Vis a Banach space. Then the closed unit ball in V
is weak* compact.
For a proof, see Reed-Simon [1], Theorem IV.21, p. 115.
14.A.6. The following result of S. Mazur is the key to the theory of
positive functionals on C* algebras.
Theorem. Let X be a locally convex topological space over R. Let C be a
closed convex cone in X with O^C.Ifxo£C, then there exists f e X' such
that f(x0) > 1 and f(x) < 1 for x s C.
For a proof see Yosida [1], p. 109.
14.A.7. We will also make use of another result of S. Mazur.
Theorem. Let V be a locally convex space over R. If Q с V, 0 s Q, set
Q° = {A e F'|A(x)	1, x eQ).
Endow V with the weak* topology. Then (Q°)° is the closed convex hull
ofQ.
14.A.8. The rest of this appendix will be devoted to the measure theory
that is used in this chapter. Let X be a set. Then a set of subsets of X
is said to be a a-algebra if it is closed under complements and countable
unions (and hence countable intersections). If X is a set and if Л is a
«•-algebra of subsets of X, then a positive measure of is a function
from to [0, oo] such that if (Uj} is a countable collection of elements of
that are pairwise disjoint, then
/4 U ц) = Ем(Ц).
' J ' j
14.А. Some Functional Analysis
359
A triple (X, ad', g) of a set, a «--algebra of subsets, and a measure on the
«•-algebra is called a measure space. If (X, p) is a measure space and if
S is a subset of X such that for each e > 0 there exists A s Л such that
S c A and g(?l) < e, then S is said to have measure 0. The measure
space is said to be complete if Л contains all subsets of measure 0.
We will sometimes write (X, g) for the measure space (X, p) and
we call Л the set of g-measurable sets.
14.A.9. If stf is a collection of subsets of X, then the «--algebra gener-
ated by <0/ is the intersection of all «--algebras in X containing <0/. It can
be shown that this “definition” presents no set theoretic difficulties (lest
we forget the “set of all sets”). If X is a topological space, then we denote
by &(X) the «--algebra generated by the open subsets of X (which is the
same as the «--algebra generated by the closed subsets of X). Then an
element of £8(X) is called a Borel sets. If (X, p) is a measure space,
and if £8(X), and if for each A s Л there exists a Borel subset such
that g(yl - А П B) = 0 and p(B - В П A) = 0, then g is called a Borel
measure.
14.A.10. If (X, ad', p) is a measure space, then a function f from X to a
topological space Y is said to be g-measurable if for each open subset U
of У, f~ l(U) is the union of a set of g-measure 0 and an element of JK.
Fix a complete measure space (X,	p). Then one defines an integral
corresponding to g in the following way. If f is a measurable function
from X to [0, oo) such that f takes only a finite number of values, then
jf(x)dp(x) = 22 zp(f~\z)).
x	ze[0,°°)
The integral of an arbitrary measurable function on X with values in
[0,’oo) is defined by a standard limiting argument (cf. Lang [1], X, Section
2, p. 235). A measurable function on X with values in C is said to be
integrable if |/| has a finite integral. If f is integrable, then it can be shown
that there exists a sequence {/Д of integrable functions that take only a
finite number of values and a subset S of X of g-measure 0 such that
lim fj(x) = f(x) for x e X - S
j-™
and
f f(x)dp(x) = lim 22 zp(ffl(z)).
Jx	J-™ zec
360
14. Abstract Representation Theory
14.A. 11. If f, g are functions on X, then we write f ~ g if there exists a
subset S of g-measure 0 such that fx-s = 8\x-s- If f ~ S and f and g
are g-integrable, then
f f(x) dp(x) = f g(x) dp(x).
Jx	Jx
As usual, if 1 p < oo, then we define LP(X; p) to be the space of all
measurable functions on X such that |/|₽ is integrable modulo the
equivalence relation ~ . Then, LP(X; p) is a Banach space relative to the
norm
/	\ i/p
f lf(x)lPdp(x) =H/llp.
\JX	I
For this material, see any decent text in measure theory (e.g., Lang [1], XI,
Section 3, p. 237; XI, Section 4, p. 311.
14.A.12. If f is a measurable function on X with values in a Banach
space B, then we say that f is essentially bounded if there exists a subset S
of g-measure 0 such that ||/|| is bounded on X - S. We define ||/|L =
infM(j)=o supxe x-jll/(x)ll. Then, L“(X,B‘,p) is the set of equivalence
classes relative to ~ of essentially bounded functions. It is a Banach
space with respect to 11 • • • IL,.
14.A.13. Let H be a Hilbert space. We will make use of the following
criterion.
Theorem. Let /: X -> H be a measurable function. Let b > 0. Assume
that for each v e H, ||u|| = 1, | jA(f(s), v) dp(s)\ < bp(A) for all measur-
able subsets, A, of X. Then ||/||«, < b.
For a proof see Lang [1], Cor. 5, p. 255.
14.A.14. Our last excursion will be into measures on locally compact
Hausdorff spaces. If X is a locally compact Hausdorff space, then a
positive Radon (regular Borel) measure on X is a Borel measure g, such
that:
(1)	If К is a compact subset of X, then p(K) < <x>.
(2)	If S is a Borel subset of X such that g(5) < oo or S is open, then
g(5) = sup{g(X)|X compact and К c 5}.
14.А. Some Fnnctional Analysis
361
(3)	If S is a Borel subset of X, then
g(5) = inf{fi(U)| U open and U z> 5}.
A map A from &(X) to C is said to be a complex Radon measure if A
can be written in the form A = (A! - A2) + z(A3 - A4), with A} a Radon
measure, i = 1,2,3,4. We will write
/ f(x) dA(x) = f f(x) dAt(x) - f f(x) dA2(x)
Jx	Jx	Jx
+ ilff(x)dA3(x) - f f(x)dA4(x)\.
\Jx	Jx	)
On Cc(X) we put the topology of uniform convergence on compacta. If
v is a linear functional on Cc(X), then v is continuous if for each compact
subset К of X there is a finite, positive constant CK such that
for f g Cc(X), supp(/) с K. We write Cc(ХУ for the space of all contin-
uous functionals on Cc(X). If v e. Сс(ХУ, then we say that v is positive if
v(f) > 0 for f g Cc(X), f(x) > 0 for all x g X. If v g Сс(ХУ, then v
can be written uniquely in the form
(1)	v2)+i(v3-v4),
with Vj positive for i = 1,2,3,4. (cf. Lang [1], Theorem 1, p. 324).
The following is the classical Riesz-Markoff theorem.
Theorem. Let v g Сс(ХУ. Then there exists a unique complex Radon
measure A„ such that if Av = (A! - A2) + z’(A3 - A4), A( positive Radon
measures, and if are as in (1), then, for every open subset U of X
At(U) = sup{p,.(/)l/GCc(X),supp(/) ct7 and 0</<l}.
For a proof (see Lang [1], Theorem 3, p. 327).
14.A.15. If fii,fi2 are (positive) Radon measures on X, then we say that
is absolutely continuous with respect to g2 if whenever S is a subset of
X with pfS) = 0, fi2(S) = 0. The following is the classical Radon-
Nikodym theorem (cf. Lang [1], p. 296).
362
14. Abstract Representation Theory
Theorem. If g1 is absolutely continuous with respect to g2, with pfX) <
oo, then there exists a non-negative function f s Ll(X; g2), suc^ if A is
Borel measurable, then
Mi(^) = fxXA(x)f(x) dp2(x).
That is, dpi = fdp2.
14.A.16. The following simple result plays a role in the basic theorem on
decompositions of abelian Von Neumann algebras.
Lemma. Let AbA2 be positive functionals on Cc(X). Let p,u g2 be the
Radon measures on X corresponding to Ab A2, respectively (as in Theorem
14.A.14). Assume that for each f s Cc( X), f 0, and that for each e > 0,
there exists 8 0 such that if h s Cc(X), Q <,h <f, X2(h) < 8 implies that
Xfh) < e. Then gj is absolutely continuous with respect to g2-
Let Z be a subset of X such that g2(Z) = 0. We must show that
g/Z) = 0. Since X = \JjKj with Kj compact, it is enough to prove that
g/Z П Kf) = 0 for all j. So we assume that Z <^K, К compact. Let
f s Cc(X) be such that f(x) = 1 for x s K. Let, for each e > 0, 3(e) be
as in the statement for f. Let Zb Z2,... be a countable collection of
Borel subsets of К such that g2(Z() < 8(е/2‘)/2 and U, Z, =)Z. Let
c Z( be compact and let ht s Cc(X) be such that Л2(й() < 8(e/2‘),
0 <, hj < f. Then Л/Л,) < e/2‘. Hence, g/AQ < e/2‘. Since Nj is arbi-
trary, subject to compactness, and N/ c Z( , we see that gj(Z() < e/2‘.
Thus, gjdJ/Z/) e. Since e > 0 is arbitrary, Z has gj-measure 0.
14.A.17. The final result that we will record is Lusin’s theorem (cf. Lang
[1], p. 337)
Theorem. Let X be a seperable, compact Hausdorff space. Let g be a
Radon measure on X and let f be a p-measurable function on X. Given e,
there exists g s C(X) and an open subset Z of X with p(Z) < e such that
f\x-z = 8\x-z-
15 The Whittaker
Plancherel Theorem
Introduction
The aim of this chapter is to derive a decomposition into irreducibles of an
important class of induced representations. Let G be a real reductive
group. Let P be a minimal parabolic subgroup of G and let G = MN as
usual. Let be a unitary one dimensional representation of N. The
induced representations that are studied are the unitarily induced repre-
sentations gotten by inducing x from N to G. Let tt* be such a represen-
tation of G. Then the method of Cowling et al. [1] can be used to see that
the support of тгх is contained in the tempered representations of G.
There are two extremes for these characters: x is trivial; x is “generic”.
At the first extreme, the decomposition of tt* is an exercise (carried out in
Section 15.1). The other extreme is the key to the general case and the
study of it takes up a substantial part of this chapter. This case is usually
called the study of Whittaker functions. In 15.10, there is some motivation
for this terminology. The general theorems appear in Section 15.9. We
have included this material in this book for several reasons. First of all, it
makes use of every (major) aspect of the earlier chapters. Secondly, the
reader can test his (her) understanding of the material of Chapters 12 and
13 by reading through this chapter and using the methods of the earlier
363
364
15. The Whittaker Plancherel Theorem
chapters to fill in the places where there is a statement such as “the
argument is now essentially the same as ... .” Thirdly, the theory of
Whittaker functions plays an important role in the study of the Fourier
coefficients of automorphic forms.
We feel that the methods involved in the derivation of the theorem are
more important than the theorem itself. In Section 15.2, we show how the
techniques of Chapter 4 can be used to derive asymptotic expansions for
certain generalized matrix coefficients of admissible representations. Sec-
tion 15.3 contains a rapid development of the theory of cusp forms for the
case at hand. In Section 15.4, we introduce the Jacquet integrals, which
will play the role of matrix coefficients and Eisenstein integrals for the
theory in this chapter. In Sections 15.5 and 15.6, we give an exposition of
our method of proof of holomorphic continuation of this type of integral.
The technique is quite a bit more general than is needed for this chapter.
We will indicate further uses in the notes in 15.11. Armed with the
continuation of the Jacquet integral, we are able to give, in 15.7, a
complete description of the discrete spectrum for the case of generic x-
We should point out that our development of this material uses the
Harish-Chandra Plancherel theorem. 15.8 contains results analogous to
those in 12.8. The main result, which calculates the Harish-Chandra
transform of a “wave packet,” is an application of the analogous theorem
for the usual Plancherel theorem (12.8.4). Section 15.9 contains the actual
decompositions of the induced representations. In Section 15.10, we give
several special cases of the theorem and show how the Lebedev inversion
formula can be derived from the result for SL(2, R). In the notes, we will
indicate how the theorem relates to the solution of the quantized “non-
periodic” Toda lattice and to automorphic forms.
Much of the material in this chapter has not heretofore appeared in
print. The main theorems on the decomposition are unpublished theorems
of Harish-Chandra. We assume that the general line of our approach is
the same as that of Harish-Chandra (although we have never seen Harish-
Chandra’s version). However, we suspect that our use of the Harish-
Chandra Plancherel theorem is different from Harish-Chandra’s. We will
discuss other possible differences in the notes at the end of the chapter.
15.1.	The support of certain induced representations
15.1.1.	Let G be a real reductive group and let (P, A) be a minimal
p-pair, with P = °MAN as usual. Let be a unitary character of N (i.e., x
15.1. The Snpport of Indnced Representations
365
is a Lie homomorphism of N to the circle group 51). Let L2(N\G; x) be
the space of all measurable functions f on G such that f(ng) =
for n g N and g g G, and such that
I ||/(g)||2 d(Ng) < °o.
JN\G
Here, we choose invariant measures on G and N as in 10.1.7 and the
measure on N \ G is chosen so that
/	[ f(ng) dnd(Ng) = f f(g) dg
JN\GJN	JG
for f integrable on G. We define irx(g)f(x) = f(xg) for f g L2(N\ G; x).
Then, (ttx, L2(N\ G; x)) is a unitary representation of G.
Lemma, supply) с <^етр(С) (see 14.12.1 for <ftemp(G) and 14.10,6 for
supp).
The argument is a variant of the proof of Theorem 14.11.4. We first
observe that if p = pP and if f g Ll(N\ G), then
(1)	f f(Ng)d(Ng)=f a~2pf(ak) dadk.
JN \G	JAxK
Set Cc(N\G; x) equal to the space of all continuous functions on G such
that f(ng) = x(n)f(g) for n g N and g g G and such that (g -» |/(g)|)
eCc(N\G).
As in the proof of Theorem 14.11.4, it is enough to show that if у g К
and if f g Cc(N \ G; x) A L2(N \ G; x), then
\{^(g)f,f}\^d(y)\\fW2B(g)
for all g g G (here, E is as in 4.5.1; see also 14.2.1).
Define f(g) = sup{\f(gk)| \k g K}. Then, as in 14.11.4, we have
\{irx(g)f, /)| < d(y){irl(g)f,f],
with 1 denoting the trivial character of N. Thus, to complete the proof we
may assume that x = 1 and that f g Cc(N\G/K). With this assumption,
366
15. The Whittaker Plancherel Theorem
we continue the proof as in 14.11.4. We have
= f f(xg)f(x)dx
JN\G
= f a~2pf(akg)f(ak) dadk
JAXK
<, [ [ a~2pf(akg)f(ak) da dk
}K JA
!	,	\ V2
x| ( a 2p\f(akg) |2 da
/л
<
We write g = n(g)a(g)k(g), with n(g) s N, k(g) s К and a(g) s A (as
usual). Then,
1/2
/	\ 1/2
= f a(kg)p f a~2p\f(a)\2 da] dk = ||/||B(g).
JK \<4	/
The lemma now follows.
15.1.2.	In light of the preceding result and 14.10.5, there is a direct
integral decomposition of (it*, L2(N\ G; x)) of the form
(L	Яш4/о-(ш)к
\4n,p(G)	J^G)	)
with Нш = Нш ® Уш and тгш = тгш ® / with (ттш, Нш) s ш.
15.1. The Snpport of Indnced Representations
367
Our main goal in this chapter is to calculate the multiplicity function
dim Уш and the measure class of <rx. In this section, we give a complete
solution in the simplest case, x = 1 (the trivial character). For most of the
rest of the chapter we will concentrate on the opposite extreme, which will
be introduced in the next section.
15.1.3.	We set L2(N\ G) = L2(N\G-, 1) and тг = тг,. Let P = ° MAN,
as usual. For each £ g °M, fix (£, H() g £ (note the abuse of notation).
Theorem, (тг, L2(N\G)) is unitarily equivalent with the unitary represen-
tation whose total space is
® ( HP ( iv dv ® H?,
е<Баьга*
and the action is the direct sum over £ g °M of
[ ^p,^ivdv ® L
Ja*
Here, dv is a choice of Lebesgue measure on a* (see the proof for the
normalization).
If f g C“(N\ G) and if £ g °M, v &H^,v g a*, set
f^v,v{s) = f a~p-i^(m)~1f(mag)vdmda.
JOMXA
Then, f(l,fnmag) = aP+lv^myf^v,Ss) for n g N, m g °M, a g Л. We
choose dv on a* such that if <p g C“(A), then
<p(l) = f f <p(a)a ,vdadv.
J a* J A
We define, for f g Q(A\ G),
T(,v(f) &Hp^iv®H*
by
(I®v){T^v(n)=f^
for у gH?. A direct calculation using the Plancherel theorem for A and
the Peter-Weyl theorem for °M yields
E dU)f^tV(f),hv(f))dv =
a*
f a 2p\f(ak)\2 dadk.
JAXK
368
15. The Whittaker Plancherel Theorem
Thus, 15.1.1 (1) implies that if we define S(v(f) = d(^)l/2T^v(.f), then
the map
5:C“(N\G)-> ф f HP i,ivdv ® H*
given by
s(f) = E
extends to a unitary bijection. Since
the theorem follows.
15.1.4.	We note that the preceding decomposition is not a direct integral
over <ftemp(G). However, Corollary 12.5.4 implies that the set of (£, v) such
that TTP (iv is reducible has measure 0 in °M X a*. Set
(a*) + = {pea*|(p,a) > 0, а еФ(М)}.
Since iTp't iv is unitarily equivalent to ttp Jsv tors s W(A) (12.1.1), the
actual direct integral can be taken over QM x (a*)+ using the product of
counting measure and the preceding normalization of Lebesgue measure.
The multiplicity function is then
(M ^d(^(A)l.
15.1.5.	In the next section, we will begin the study of the case when x is
a “generic character” of N. In this case, the decomposition will in general
involve more than principal series corresponding to minimal parabolic
subgroups. The multiplicity function will also be more interesting.
15.2.	Some asymptotic expansions and estimates
15.2.1.	The purpose of this section is to give some results that expand
upon the material in 4.3 and 4.4. Let (тг, Я) be an admissible, finitely
generated representation of G. Let (P, A) be a minimal p-pair for G.
Then Л s (H°°y is said to be tame with respect to (P, A) if there exists
15.2. Some Asymptotic Expansions and Estimates
369
3 g a* depending only on A such that
(1)	|мЛ(7т(а)и)| < Cul)as,
for all и e I7(np), v e HK, and a e C1(A~).
It is easily seen that not all elements of (Я“У can be tame for P.
Furthermore, A can be tame for one P but not for another. If (тт, H) is
the contragradient representation of (тт, H) and if A g H°°, then A is tame
for all minimal parabolic subgroups (see Theorem 4.3.5).
15.2.2.	Set V = HK. Let (P, A) be a minimal p-pair in G. Let F c
Д(Р, A) and let (PF,AF) be the corresponding p-pair with PF^>P,
Af c A. Let E(Pf,V) be the set of (generalized) weights of aF on V/nFV
(see 4.4.1). Then E(PF, V) is a finite set. Set LpF = LF=	Na.
We fix B, a non-degenerate, symmetric, bilinear form on g such that
ПУП2 = -B(X, ex) > 0 for X g g, X * 0.
Theorem. Let A g (Н°°У be tame with respect to (P, A). If p g E(PF, V)
and QeLF, then there exists a function pKQ(H;a;v) on aFx
(С1(Л“) П °MF) x V such that is a polynomial in H, contin-
uous on aF x (C\(A~) C\ °MF), real analytic on the interior of aFx
(С1(Л“) П °MF) (in aFx (А П °MF)), linear on V, and such that if
H g (af)+ then
А(тг(ехр(-1Я) m)v) ~	£	е~,*н'> £ e~‘&H)Pxtll.,Q(tH',m;v)
p^E(pf,v)
as t -> +<», for all m g MF.
The proof follows the same line as that of Theorems 4.3.5 and 4.4.3. Let
3 be as in 15.2.1 (1). We may assume that G = °G. Let Д(Р, A) =
{ab..., ar}. Define Hx,..., Hr g a by afHf) = 8U. Define Л e a* by
Л(Я() = min{Re р,(Я,)| p, g E(P,V)}.
(1)	There exists d > 0 such that if v g V, then
|А(тг(а)и)| < CvaA(l + ||loga||)d
for a g Cl(A~). (Here, log a = H if exp H = a, H g a.)
We will give enough detail so that the reader can complete the proof
following the methods of 4.3 and 4.4. Let F = tx(P, A) - {«,}. If a g
370
15. The Whittaker Plancherel Theorem
C1(A“), then a = a'at with at = exp tHi and t <, 0, a' = exp(EJ#l xjHj)
with Xj <, 0. If v g nFV, then v = E; XjVj with Xj g nF and Ad HXj =
PjtHyXj with g Ф(Р, A) and g Ф(РГ, AF). Now,
|A(7r(a)t?)| = Ел(7Г(а)А'л)
E«^A(X.Tr(a)t;.)
i	j
Let Hi = H and let zx,...,zd be the generalized eigenvalues of H on
V/nFV. Let qF be the natural projection of V onto V/nFV. We set
(V/nFV)z equal to the generalized eigenspace for H with eigenvalue z.
Then,
V/nFV = ф (V/nFV)Z/.
i
Let p, be the projection of V/nFV onto (И/пгИ)2 corresponding to this
direct sum decomposition. Set qx F = pt° qF. If v eV, then v = E, v, with
qiF(v) =	and v ~ ^vi e npV- Let “i,i>• • •,ui,dj be such that
qF(Ujj) is a basis for C[H]qx F(v). Then
HqF(uij) = LbijkQF^i.k)’
к
with В, = [Ь(д] a matrix such that В - z,/ is nilpotent. Now,
Set	Hui,i ~ Lbijkui,k = wi,i e nF		V.
	к F(t,a') =	1	1 Э	a4 » • • • x X	у	
and	G(t,d) =	A(Tr(a,a')w,. J	
Then, as in 4.3.5,	d	A(7r(a,a')w/>di)	
—F(t,a') =	+ G(t,a').
15.2. Some Asymptotic Expansions and Estimates
371
Continuing as in 4.3.5, we have
||G(t, a') || < G?(S(H>+1>'(a')s
for t < 0 and a' as above. Also,
||f(0, a') || £ C(a')s
and
||efl,i|| <: C(1 + |j|)deRe2-5
for j e R. We can now use exactly the same argument as in 4.3.5, using
F(t, a') = e‘BiF(0, a') - e‘B‘j°e~sB,G(s, a') ds
for t < 0. The rest of the proof is now almost identical to that of Theorem
4.4.3.
15.2.3.	Let (P, A) be a minimal p-pair for G. Let x be a unitary
character of N = NP. Then we will say that x is generic if for each
a G Д(Р, A), dx(na) * 0.
Lemma. Let (тг, H) be an admissible finitely generated representation of
G. Let A g (H°°f be such that
A(ir(n)v) = x(n)\(v)
for n g N, v g H°°. Then A is tame for every Q g &>(A\ Furthermore, if
g g (7(gc), then gX is tame for (P, A).
Set V = HK. We note that ne = n A ne ® n A nQ. Thus, P-B-W im-
plies that
(7(ne) = G(n A ne)(7(n A ne).
Since A g (Н°°У, there exists 3 g a* such that
| X(ir(a)v) | < Cvas
for a g CIU5), v g V. Now,
G(n A ne)A c CA.
372
15. The Whittaker Plancherel Theorem
We assert that if
g = 22
веД(Р,Л)
with ma g N, then
|A(ir(e)p)| C^a6'"
for a g ClUg). Indeed, choose Xa g n such that [H, XJ = a(H)Xa for
H g a, and d%(Xa) =# 0. Set и equal to the product of X”a in some
order. Then, uA = cA with c =# 0. On the other hand,
иА(тг(а)г) = А(тг(мт)тг(а)г) = а-мА(тг(а)тг(ит)1;).
Hence, if a g Cl(Ag) then
|A(ir(a)u)|
If x g U(n П ne), then x = with g and Ad(a)	. Hence,
|хА(тг(а)г) | < 22 |A('n’(xT)'n’(fl)i;)l =
< c;, 22«M«S‘M = c"as,
for a e C1(Aq) and v g V. This proves the first assertion.
To prove the second, we need only show that if g g (7(gc)A, then
|g(ir(a)f)| < C^„as
for a g A~, v g V. As before, we note that t7(nc)A c CA. If x g (7(n)
and if Ad Hx = -£(H)x for H g a*, then we note that f g L+=Lj.
Hence,
|xA(Tr(a)r) | = |А(тг(хт)тг(а)г)| = |A(Tr(a)Tr(Ad(a-1x))r)|
= af|A(Tr(a)Tr(xT)r)| < a^+&C^xr}v < Cas
for a g С1(Л~). Let g g (Я“У be such that
|g(ir(a)f) | < Cva&
for all v g V, a g С1(Л~). Then, if m g (7(mc),
|/ng(Tr(a)r)| =|g(Tr(?n7’)Tr(a)i;)|
= \^Tr(a)Tr(mT)v)| < CT(mr)pas
15.2. Some Asymptotic Expansions and Estimates
373
for a g С1(Л ) and v g V. Since t7(gc) = t7(mc)t7(nc)t7(nc), the sec-
ond assertion follows.
15.2.4.	Fix, (P, A), a minimal standard p-pair. If V is an admissible,
finitely generated (g, АЭ-module, then let Ли be as in 4.3.5. We assume
that G is of inner type.
Theorem. Let (тг, H) be an admissible, Hilbert representation of G with
infinitesimal character x- Let V = HK and let Л = Ли. Then there exist
d > 0 and continuous semi-norms qx, q2 on and H°° (the C“ vectors of
H with respect to the conjugate dual representation) such that
|<ir(e)u, w>| < ал(1 + ||log )q2(w)
for a g С1(Л+) and v, w g H“.
The proof of this result will be a mixture of the methods in 4.3, 4.4, and
12.4. As in 4.3.5, we start with
(1)	|(тг(а)г, w)| < a2 3 * s||r||||w||
for v,w g H and a g С1(Л+).
Let a g Д(Р, A) and set F = ts(P, A) - {a}. Set (Q, AQ) = (PF, AF).
We will now use the notation of 12.4.5. If H g aQ, then
(2)	He,= ^Р0ЫН^-
J
Here, ex = 1. We note that ziS(H) g Z(gc). Let X} be a basis of nG such
that [h, Xj] = a^hjXj for h g a. Then there exist gkij(H) g L7(gc) such
that
(3)	z,7(H) =pQ(Zij(H)) + LXkgkij(H).
к
Let (тг,Н) be the conjugate dual representation of (тг, H). Set x(z)
= x( zT) f°r z e 2(gc). (2) and (3) imply that if w g H°°, then
TT(H)Tr(ei)w = Ex(z,7(H))ir(eJ)w - ^(XkW(gkij(H))^(ej)w-
j	j, к
Fix H g aQ with a(H) = 1. We will write z/; for ztj(H) (similarly for
gkij). We set
^ki ~	'
374
15. The Whittaker Plancherel Theorem
Then, we have
(4)	тг(Н)тт(е^ = Ytx(zii)ir(ej)w ~ ^(Хк)^(ик^.
i	i, к
We now do the same thing for тт(Хк)тт(е^тг(ик^. We have
•n-(H)7r(A'Jfc)7r(e;)7r(uJfe;)w = ак(Н)тт(Хк)тт(е})тт(ик^
+ £x(zjP)t(Xk)t(ep)Tr(uki)w
p
- ^(Xk)Tr(xq)^(4qj)TT(ukl)w.
«?
Note that ak(H) s{l,2,...}. We can clearly continue this procedure
using (4) on тт(Хк')тт(Хд')тт(ер)тт(ид}')тт(ик^, and then continue indefi-
nitely. Since the next part of the argument is complicated, we will
illustrate it by looking only at the first step. Let а’ еЛ П °MQ and
at = exp tH. Set, for v e H°°, w s Я°°,
F(t, a';v, w) =
(тг(а'а/)г,тг(е1)и')
(ir(e'e,)u,iT(ed)H')
and
< 77 ( a'a,) Г, E * 77 ( X* ) 77 ( u *!) W )
G(t, a'; v, w) =
<77(а'а,)г,Е*77(^)77(и^)^)
Also, put В = [-x(z0)]. Then,
d
—F(t,a';v,w) = BF(t,a’;v,w) + G(t,a';v,w).
We note that if a' e Cl(A+) n °MQ, then
||G(t, a'; v, w) || <	d(max ||-n-( A'*)r»||)(max|| 7t(ua.,)w||).
' к	'' к, i	'
We can therefore proceed as in 4.4.3, using the preceding procedure to
derive an asymptotic expansion
(5)	(7r(a'az)r,w) ~ ^ezi‘ e~n‘Pj.n(Ga'-,v,w)
j n SO
15.3. The Schwartz Space for L2(N\ G; x)
375
as t -> + oo, with z,- -ZjS N for i =# j, Pjn a polynomial in t of fixed
bounded degree, and p,; continuous onRx С1(Л+)п °Afe x H“ x H°°.
An asymptotic expansion of the type of (5) is unique (4.A. 1.2). Thus, the
continuity assertion and Theorem 4.4.3 imply that p., n = 0 if Re z- >
Л(Я). From this, it is a simple matter to prove the theorem using the
method of 4.3.5.
Note. The preceding argument actually implies that the asymptotic ex-
pansions in 4.4.3 are valid for a g H°°.
15.2.5.	If we use the same method and 15.2.2, we have the following
result.
Theorem. Let (тг, H), (P, А), Л and d be as in the previous theorem. If
Л g (H°°)' is tame for (P, A), then there exists a continuous semi-norm qK
on H°° such that
|A(-n-(a)r>) | < ал(1 + ||log a||)^A(r)
for v &1Г, a g C1U+).
15.3.	The Schwartz space for L2(N \ G; x)
15.3.1.	We fix a minimal p-pair, (Po, Ло). If g g G, then we write (as
usual) g = n(g)a(g)k(g), with n,a,k respectively C°° functions from G
to No, Ao, К. Let x be a unitary character on No. We set C°°(N0 \ G; x)
equal to the space of all f g C°°(G) such that f(ng) =	for all
n g No, g g G. If f g C°(N0 \ G; x), x e U(gc), and d g N, then we set
Qx.d(f) = SUP «(^)-₽(1 +ll1°g«(g)ll)‘WU)l-
geG
Here, xf = R(x)f, as usual, and log is the inverse map to exp on a0. Also,
we fix an invariant form В on g such that ~В(Х,6Х) > 0 for X g g,
X + 0. If Xg g, then set ||X|| = (-B(X, 0X))1/2. We put ^(A0\G;a;)
equal to the space of all f g C°(No \ G; x) such that qxd(f) < 00 for all
x g G(gc), d g N. We endow ^(No \ G; x) with the topology induced by
the semi-norms qx d for x g G(gc), d g N. Then it is easily seen that
£(N0 \ G; x) is a Frechet space and that the space C“(A0 \ G; ^) of all
f g C”(A0 \ G; x), l/l e Cc(N0 \ G), is dense in ^(No \ G, x).
376
15. The Whittaker Plancherel Theorem
Lemma. Iff e -^(No\ G;xf then f e L2(N0 \ G’, xf Furthermore, there
exists d0 e N and 0 < C < °° such that ||/||2 < Cqi d(f).
Let d0 be so large that
f (1 + II log a||) ~2d° da = C2 < oo,
JA
with C > 0. If f e if(N0 \ G; x), then
\f(nak)\< ap(l + ||log all)~d°Qi,do(f)-
Thus,
ll/ll2=/’	\f(g)\2dg=f	a~2p\f(ak)\2 dadk
JN0\G	JA0XK
W/)7 (1 +lllog«ll)’2"0^ <C291,J/)2.
Ao
15.3.2.	We now define a variant of the Harish-Chandra transform for the
case at hand. Let (P, A) be a standard p-pair dominating (Po, Ao) (i.e.,
P z> Po, A c Ao). If f e -^(No \ G; x\ then we set, for m e °MP, a e Л,
(1)	fp(ma)=aPp(_f(nma)dn.
JNP
Lemma. Iff s -^(.Nq \ G; x\ then the integral in (1) converges absolutely
and uniformly on compacta in Mp. Furthermore, fp^-^(N0C}MP\
Mp; X\NonMf) and the map f -> fp is continuous from -£(Na\G; x) to
£{Nq A Mp \ Mp, X\n0 n M,)-
We note that
nma = n(nma)a(nma)k(nma).
This implies that if / e -^(No \ G; x\ then
\f(nma)\ <. a(nma)p(l +||loga(nma)||)‘J^1 d(f).
If n e NP, m s °MP, a ^Ap, then we write m = nlalkl with n, eA'on
Mp, aj s Ao П °MP, k{ e К Г> °MP. Hence,
a(nma) = аСпап^) = aa1a(^(an1a1)~1n(an1a1)y
15.3. The Schwartz Space for L2(N\G;x)
377
Thus,
(aaj ₽(1 + ||logaa1||)'z|/(n?na)|
< (1 +1|log a((an1a1)-1n(an1a1))||j
Ха((ап1а1)~1п(ап1а1)^1 d(f).
This implies that if m, a vary in compactum, the integrand in (1) is
dominated by a constant multiple of
(1 +||loga((an1a1)-1n(an1a1))||j a((an1a1)-1n(an1a1))₽.
Now,
(1 +||loga((an1a1)~1n(an1a1))||j a((an1a1)~1n(an1a1))₽ dn
= a-2pP +1| jog а(й) ||)_‘za(n)₽ dn.
JNP
Theorem 4.5.4 implies that the latter integral converges if d is sufficiently
large. Let dY be so large that if d > dY then the integral converges. Then,
if we use the preceding inequalities and do the obvious changes of
variables, we have
|/p(>na)| < Cdqld(f)(l +||loga(wa)||)“Ja(w)'’.
Thus, if x g G((mP)c), then
\R(x)fp(ma)\ < Cdqxd(f)(l +||log a(ma) ||)”Ja(w)₽
for m g °MP, a A. The lemma now follows.
15.3.3.	We say that f g -£(N0 \ G; *) is a cusp form if (R(k)f)p = 0 for
all (P, A), P =#= G, dominating (Po, Ao) and all к g K. We note that if
g g G, g = nmk, n g N, m g Mp, к g K, then
(R(g)f)P(ml) = (R(k)f)P(mim)
for all g MP. Thus, f is a cusp form if and only if (R(g)f)p = 0 for all
g g G and all P =#= G dominating Po.
We set °^(N0 \ G; x) equal to the space of cusp forms. The following
theorem is proved in exactly the same way as Theorem 7.2.2.
378
15. The Whittaker Plancherel Theorem
Theorem. Iff g \ G', дО and if ZG(Qc)f is finite dimensional, then
fe °^N0\G-,X).
153.4.	As we shall see, the space of cusp forms is dense in the
discrete part of L2(N0 \ G; x)- We shall see that if G is non-compact and
if x = 1> then °-^(N0\G-, x) = {0}. The other extreme is much more
interesting. If (тг, H) is a representation of G, then set Whx(H“) =
{Л g (Н°°У\Х(тт(п)о) = x(n)X(v) for v g Я”, n g No}.
Theorem. Assume that x generic. Let (тг, H) e a> e ^(G). If Л g
Whx(H°°), then
TA(v) = (g A(Tr(g)o)) e °^(N0\G;x)
for all v g HK. Furthermore, each Tx extends to a continuous intertwining
operator from H to L2(N0 \ G; xX and
HOmG(H,L2(N0\G-,x)) = {ТлI A G Whx(H°°)}.
Let Л g Whx(H°°). Then Lemma 15.2.3 implies that Л is tame for all
Q g Let Q g 0>(A) and let AG be the “Л” in the proof of
Theorem 15.2.2 corresponding to Q. Since (тг, H) is square integrable,
Ло = pQ + р0 with Pq g +Oq (see 15.1.1 for notation). Also, 15.2.2 (1)
implies that
|Л(тг(а)г)| Cpa₽o+*1o(l + ||loga||)r
for v g HK, a g СКЛд). Now, n0 = n0 П nG ® n0 П nG.
Pq ~ with 8q g Cl(+a*). We therefore see that
Hence, p =
|Л(тг(а)г)| < Cl)ap+flo+so(i + ||loga||)r
for v g HK, a g С1(Ле). If v g HK and if vv..., vd is an orthonormal
basis of span(rr(A)r), then
d
ir(k)v = Y^k^,
with <p, smooth on К and |<pf,k)I < ||u||. Set Cv = d{max.t Cv). Then,
|k(ir(ak)v) | Cpa₽+*1o+so(l + ||log a||)r
for к g K, a ^Aq, v ^HK. Since HK is a t/(gc)-module and Ao =
Uqe^>(A)Aq, it now follows that TA(r) g -£(No\G‘, x) for all v g Hk.
Since (тг, H) is irreducible and admissible, it has an infinitesimal charac-
153. The Schwartz Space for L2(N\ G;x)
379
ter. Thus, Theorem 15.3.3 implies that TA(u) g °-^(N0\G; xf Lemma
15.3.1 implies that Tfu) g L2(N0\G', x) for all v g HK. Schur’s lemma
implies that
<Ta(u),Ta(w))=C(A)<u,w>
for v,w g HK. Thus, TK extends to a bounded operator from H to
L2(N0 \ G; д'). We therefore see that
HomG(H,L2(N0\G;^)) о{Тл|Л e »*/№)}.
Let T g HomG(H, L2(N0 \ G; ^)). Then T maps C°° vectors to C°° vec-
tors and defines a continuous intertwining operator on smooth Frechet
representations. Now, L2(NQ\G; хУ° cC"(A'0\G;^) and evaluation at
1 is continuous on L2(N0\ G; xf°. Define XT(v) = T(uXl) for v g /Г.
Then, Лr g Whx(H“) and T = 1\T. The theorem now follows.
15.3.5.	We now prove a result that relates °Tf(N0\G;^) with the dis-
crete spectrum of L2(NQ \ G; xf We will use it and the previous result to
describe the discrete spectrum in 15.7.
Theorem. Assume that x is generic. Let (тг, H) be an irreducible unitary
representation of G and let T be a continuous G-homomorphism from H to
L2(N0 \ G; x). Then T(HK) c °^(N0 \ G; X).
T maps H°° to L2(NQ \ G; xf°. Thus, if we set A(u) = TXuXl) for
v g Я°°, then A g ИТ^Я”). Lemma 15.1.1 implies that (тг, H) is tem-
pered and Lemma 15.2.3 implies that A is tame for every Q e ^(Ao).
Thus, if Q g ^(Ao) and if v g Hk, then
А(тг(ехр tH)v) ~	22 (ехрГЯ)*1 22 (exp ШУр^ „(Ш; v)
as t -> +oo, for H g (a0)g (as in 15.2.2).
Also, by our assumption,
[	a~2p\X(ir(ak)v)\2 dadk. <
JAflxK
This implies (Lemma 5.A.2.2) that if p^ „ =# 0, then (g + v - p\H) > 0
for all non-zero H g Cl((a0)g). From this it is an easy matter, using the
asymptotic expansions in 15.2.2, to see that TA(u) = (g -> A(ir(g)i;)) g
^(N0\G;x) for veHk. Since тг has an infinitesimal character, 15.3.3
implies that T(HK) c °-^(N0 \G; x)-
380
15. The Whittaker Plancherel Theorem
15.3.6.	The next results are of a different nature.
Lemma. Let <p s ^(No \ G; x) and f e ^(G). Then
f <p(g)f(g) dg =
JG
converges absolutely and there exist continuous semi-norms qx and q2 on
tf(N0\G; x) and ^(G), respectively, such that |(<p, f)\	<71(<р)<7г(/)-
We are looking at
f a~2px(n)<p(ak)f(nak)dndadk.
JNOXAOX К
Now, for each d > 0, we have
\<p(ak) | <9i,d(<p)«₽(l + lUog all)~d-
Thus,
f l<p(g)f(g)ldg
JG
< Qi d(<P) f a~p(l + ||log a||)~d|/(nak) | dndadk.
’ JN0XA0XK
Also, there exists p, a continuous semi-norm on ^f(G) such that
a~p [ lf(na)jdn <p(f)
•’No
(see the proof of Theorem 7.2.1). Take q2(f) = supte K p(R(k)f). We
now have
j \<p(g)f(g)\dg <q1<d(<p)<h(f) f (1 + II log all) ~d da.
If d = d0 is sufficiently large, then the integral on the right converges.
Take qx = qld<j.
15.3.7.	We note that in the course of the proof of the previous lemma we
have also proved:
Lemma. Iff e -6(G), then set
fx(g) = f A'(a)-1/(ng) dn.
No
15.4. The Holomorphic Continuation of the Jacquet Integral
381
The integral converges absolutely and the map f <-» f is continuous from
1?(G) to ^(NO\G;X)-
15.4.	The holomorphic continuation of the Jacquet integral
15.4.1.	In the last section, we developed a concept of cusp forms for the
case at hand analogous to the Harish-Chandra theory for L2(G). In this
section, we give a construction analogous to Eisenstein integrals. It is here
that L2(N0\G; x) becomes more subtle than L2(G). As before, we fix
(P0,A0), a minimal standard p-pair. Let (P, A) be a cuspidal p-pair
dominating (PQ, Ло). Fix ы s d’2(°Af) and (a, Ha) e a>. Let x be a
unitary character of No. Set Xp = X\Np and *x = A>0 n амР • Let A e
Wh*x(H“). If f s then we set, for v s (ctp)c,
(1)	^.,(A)(/) = / X(n)-%fM)dn.
JNP
Here, Pfv(nmak) = a(m)av~PFf(k) for n s NP, m е °MP, a & А, к & K.
These are the Jacquet integrals of the title of this section. The main result
on these integrals is the following.
Theorem. Assume that x « generic. If v e (а£)ё = {v e (a/>)£|Re(i', a)
< 0, a e Ф(Р, A)}, then the integral in (1) converges absolutely for f e /“.
Furthermore,
is continuous on (а£)ёх /“ and holomorphic in v. The map
from (a£)c Ю (ГУ extends to a weakly holomorphic mapping on (a£)c.
Finally, if v (aP)c, then
Jav:Wh*x(Hff)-*Whx(Fpav)
is a linear bijection.
15.4.2.	The preceding theorem is difficult and will be proved in several
steps. In the course of the proof we will have proven a much more general
result analogous to the theory for x = 1 in Chapter 10. We first prove the
convergence assertion of the theorem and that, in the range of conver-
gence, Ja „ is injective.
382
15. The Whittaker Plancherel Theorem
Let f e Then
рЛ(и) = ap(n)-p+va(mp(n))f(kp(n)).
Thus, if A g Wh*x(H*\ then
А(?Л(«)) = ap(n)~p+vX(a(mp(n))f(kp(n))).
This implies that
|А(?Л(«))| = ap(n)"p+Rev\k(a(mp(n))f(kp(n)))\.
Now, Л is tame for all *Q g &(A0 П °MP) so Theorem 15.2.5 implies
that
1АОЛО))1
< ap(n)-p+Reva*P(mp(n))P(l +1|loga*P(mp(n))\\ydqd(f),
with qd a continuous semi-norm on /”. Here, *P = Po П °MP (as usual).
The convergence and continuity assertions now follow from Theorem 4.5.4
applied to (Q, Ao) = (NPP, Ao) and P.
Let v g (a£)_. Fix <p g C^(N) such that
f x(n)'l<p(n) dn = 1.
JNP
If v g H“, then define f g by pf„(g) = 0 if g <£ PN, and
pf„(nman) = av~pa(m)q>(n)v.
Corollary 2.2.11 implies that f exists. We set F(v) = f. It is clear that
4,P(A)(F(r)) = A(r).
Thus, if Ja V(X) = 0 then Л = 0.
The rest will take more work. We begin with a simple assertion that
perhaps should have been proved earlier. We will then begin the proof in
the next section.
15.43.	Lemma. Let x be a character of No. Let V be an admissible,
finitely generated (g, KA-module. Then
dim{A g L*| X(Xv) = d%(X)X(v), X g n0, v g L) < oo.
15.5. First Steps for the Holomorphic Continuation
383
Proposition 3.7.1 implies that V is finitely generated as a t7(n0)-module.
If Л g V* satisfies the transformation rule in the braces in the statement
and if i),,..., vd is a finite set of generators for И as a t7(n0)-module, then
' j ' i
for rij g I7(no). Thus, the dimension in the preceding is at most d < oo.
15.4.4.	Corollary. Let (тг, H) be an admissible, finitely generated Hilbert
representation of G. Let x be a unitary character of No. Then
dim Whx(H°°) < oo.
Since HK is dense in H°°, the map Л -> Л|Нк is a linear injection of
(Н°°У into (HK)*. The result now follows from the previous lemma.
15.5.	First steps for the holomorphic continuation
15.5.1.	We now introduce some algebraic formalism that will be used in
the holomorphic continuation of the Jacquet integrals. We do this algebra
in greater generality than will be needed in our application. Let В be as
usual (see 15.3.1). Let (Po, Ao) be a minimal standard p-pair. Fix (P, A)
dominating (Po, Ao) and let F с Д(Р0, Ao) be such that P = (P0)F. Let
X be a generic unitary character of No. Let P = ° MAN, as usual. If
a g (Д(Р0, Ao> ~ let Ya e (n_a)c be such that dx(X) = B(Ya, X)
for X g na. Set
V- E n-
ae AC/’o’ A))~F
If a g (Д(Р0, Ao) - F), let xa = 6Ya. Then = 0 if a =#= /3 and
[ха,Уа] g a0. Let ca g R be such that if X = Lacaxa, then [X,У] = h
with h g a0 and a(h) = 2 for a g (Д(Ро, Ao) - F). It is easily seen that
this can be done.
Let H g ap be such that a(H) = 0, a g F and a(H) = 2, a F. Let
(gc)2* be the eigenspace for Ad H with eigenvalue 2k. Then (gc)0 = mc
and:
(1) Уе(0с)_2.
(2) Ad У: nc -> gc is injective.
384
15. The Whittaker Plancherel Theorem
Indeed, (1) is clear from the definitions. To prove (2) we note that
{X, Y,h} form a standard basis for a TDS. Since the eigenvalues of Ad h
on nc are strictly positive, there can be no lowest weight vectors for this
TDS in nc. (2) follows from this.
15.5.2. In the rest of this section, we fix (P, A) a standard p-pair and x a
character of N such that if ф = dx, then ф(х) = B(x,Y) with Y s nc
and such that there exists H in the center of mc such that Ad H has
eigenvalues of the form 2k with к e Z, if (0c^2* is the 2^-eigenspace for
Ad H then (gc)2 generates nc and 15.5.1 (1), (2) are satisfied. Let z
denote the complex conjugate of z e gc with respect to g. If u, v e nc,
then we set
(u,v) =в([[0Т,0й],и],у).
(1) ( , ) defines an inner product on nc.
Indeed, (u,v)= -В([0У,0й],[У,и]) = -В(0[УДГ|,[У,и]). This im-
plies that the form is Hermitian and positive semi-definite. If (u, u) = 0,
then [У, u] = 0. So и = 0.
Fix j > 0. Let A\,..., Xd be a basis of (gc)2; such that (Xp, Xq) = 8pq.
Set
Zk = [0Y,ffXk] e(gc)_2/+2.
Then, B([Zp, Xq], Y) = (Xq, Xp) = 8pq. Fix Xo e (gc)2 such that ф(Х0)
= 1. Then:
(2) [Zp, Xq] = 8pqX0 + Xp q, with Xpq e (gc)2 and ф(Хрч) = 0.
We define an element Q} s t/(gc) by
Qj=
i
Then, Qj is independent of the choice of orthonormal basis of (gc)2;.
These operators will play a leading role as this section develops.
15.53.	Let Сф denote the nc-module C with nc acting by ф. If У is a
nc-module, then we identify V with У ® CL^. We set
Vp = (v e У® C_Jnpt> = 0}.
15.5. First Steps for the Holomorphic Continuation
385
Then, Upxj^p is a nc-submodule of V (relative to either action). We
collect some observations about the action of the Qj on V.
(i)	If x g (0c)2j with j > 1, then (x - ф(хУ)Ур c Vp_j.
Indeed, x - ф(х) is a linear combination of commutators of elements of
the form x, - ф(х,), i = 1,..., j, with x, g (0C)2. Since (z - ф(г))Ур c
I^-i, (i) follows.
(ii)	If z g (0C)_2> with j > 0, then zVp c Ур+}+1.
We prove this by induction on j and then on p. If j = p = 0, then the
result is obvious since Vo = {0}. Recall that mc = (0C)O. We prove the
result for j = 0 by induction on p. So assume it for j = 0 and p. If
v e ^p+i> x g nc, z g mc, then
(*)	(x-i^(x))zr = [x,z]r+z(x-i^(x))r.
Now,[z,x] g nc,so[x, z]J^+1 c Vp+l.Also,(x - ф(хУ)Ур+1 c .Thus,
the inductive hypothesis implies that z(x - ф(хУ)и e Ур+1. We therefore
see that (x - ф(х))ги g Ир+1. Hence, zv g Vp+2- This proves the result
for j = 0.
Assume the result for j (i.e., (ii) is true for all p). We now prove it for
j + 1 by induction on p. Let z g (0<O-2j-2- Clearly, zV0 = 0. Assume
that zVp c Kp+7+2.If v g Vp+ j, xenc,we analyze the terms in (*). We
have
[x,z] g 22 (0c)2,>
sz. - j
so[x, z]v g yp+j+2 by the inductive hypothesis, (x - ф(хУ)и e Ур. Hence,
the inductive hypothesis implies that
z(x - lA(x))r G yp+J + 2-
As in the case j = 0, we now see that zv g J/,+j+3. This completes the
inductive step.
(iii)	c Vr for r > 0.
Indeed, we write Qj as in its definition in the preceding. Then Zi e
(0c)-2j+2- (i) implies that (Xj - ф(Х}))Уг c Vr4 and (ii) implies that
Z/r_7 c Vr.
386
15. The Whittaker Plancherel Theorem
Lemma. Let V be an Qc-module. For each j, к > 0 there exist non-nega-
tive integers nikJ,..., ndk kJ such that
dk,i
X\{Q^nhkJI)Vk = Q.
i=i
We may assume that V = UPK- We also set Vp = 0 for p <, 0. Fix
j > 0. We define Vp r to be the space of all v e V that satisfy the following
two conditions:
(1) If xb...,xp+1 e Е,г/0С)2(, then
(*! - iA(xj)) • • • (xp+1 - iA(xp+1))r = 0.
(2) If x, e (0c)2r/with г,- ^1, then
(xj - iA(xj)) ••• (xp - iA(xp))r e Vr_^.(rj_n.
We note that Vp<t c Vpr+i, Vp 0 c Vp+1>0, UPK,o = V, and Vp+1>0 =
Urf$,r- We assert that
(I)
Before we prove this assertion, we first show how it implies the lemma.
Set
fPAT) = Trh(T + ii).
i = 0
We note that Vp+l c Vp 0, and that
^оПК,с UK,-
Thus, (iii) (combined with (I)) implies that
nu»c КПК-1,0 •
Hence,
ПЛ-м(С7)Л-2>1г(е;)К c vr n к-2,0,
etc. Since Vo 0 = 0, (I) does indeed imply the lemma.
We are thus left with the proof of (I). If x e nc, then set x' = x - <Д(х).
Let v e Vpr and let x, e (0c)2r., with r, j and i = 1,...,p. We set
15.5. First Steps for the Holomorphic Continuation
387
m = dim(gc)2j and Q = Qr Then, the expression for Q implies that
Qx\ • • • x'pv = 0.
Thus,
x'i x'pQv
=	••• xl-jxpdx^! ••• x’pV
i = l
= E E*'i x’_l[x/,Zk]X^x'i+1 •x’pV
i=l k = l
+ E E*'i ’ <-i2fc[x,,X;]x',+ 1 x’pV
i = l k = l
= E E*'i • •• x’i_1[xi,Zk]X'kx'i+1 x'pV
i=lk=\
+ E E zkx\ • • • x'f.jx,,x;]x'+1 • • • x’pV
i=lk=l
+ E E E*'i  [xu,zk]x’u_l ••• x'-jx,.,x;]x'z+1 x’pV.
i = 1 к = 1 и = 1
If г,- > j, then [x,, Zk] e (Qc^+^-j) c nc- Thus,
x\ ••• x'i_1[xi,Zk]X'kx'i+1  x’pV
= [Х1,гк]х\  x'i_1Xkx'i+l x’pV
i-1
+ E^'1 ••• [xu,[xt,Zk]] ••• x’l.l^x’i+l  x’pV,
U=1
which is in ИГ_Е/(Г• Also,
x'i ••• [x,, A'Jx' + j ••• x’pV = 0.
If we argue as before, then we find that if ru > j,
x\ ••• [x„,[x,,Zj]    x'i_1X'kx'i+1 ••• x’pV e
388
15. The Whittaker Plancherel Theorem
We therefore conclude that
x\  x'pQv
= Xj	x,._i[Xj, Zk]Aj^x(+]	xpv
r,-j k = l
+ X< X< X< i [xu, Z/j]xu_j • • • x,_j[Xj, Xk]x,+ j	XpV
i=\k=\u=\
rU=j
mod Vr_^rk4)_j.
If r, = j, then [x,., Zk] = ~(Xj, Xk)X0 + ut k with ф(и, k) = 0 (see 15.5.2)
and Uj k e (gc)2. If we argue as before, we find that
X1 ’ ’ ’ (ui,k ~ (xi’ Xk)(X0 ~	‘ xpV S K-Et(rt-j)-l
and
*i • •(«,,* “ (xu’Xk)(Xo - 1)) •’ ’ x;_i[xz,Xjx;+i •• • x’pV
e Vr-Y.k(rk-j)-l 
Set j = (i|r,- = j}. Then, we have shown:
(II)	x'i ’ ’ ’ X'PQV
= -sx\ • • • x'pV
p
~ X* X* Xj xu-lxu + l ’’ xi-1[ xi ’	+ l	’ xpV
i=lIsuSi-l
mod •
If j = 1 and if ru = j, then this says that
x; • • • x'pQv = -x't • • • x'p^x'p+j • • • x'px’uv mod	.
Hence,
X'1 x'pQ2v = -x'i • • • x'pQv mod	.
So
x\  x'pQ(Q +I)v = 0 mod	.
This proves (I) if s = 1. We therefore assume that j > 2. If we observe
15.5. First Steps for the Holomorphic Continuation
389
that [x, y] = [x', y'] for x, у e nc, then the expression in (II) telescopes
to:
(III)	x\ • • • x'pQv = £ x\ ••• x'u_lX'u+l x'px'uv
ru=>
mOd Vr-Urk->)- 1-
In particular, this implies that Qv e Vp r. If we repeatedly apply (III),
we find that x\ • • • x'pQsv is congruent to a sum of terms of the form
z'i 
with Zj e (gc)2a., a, > j, and u, s (gc)2y. we now aPPty same
argument to this expression, we find that
x\	 x'pQsQv = - E £ z'i ’ ’' z'P-su’v.i ' • Q mod	1,
i=l
with vi as in 15.A.7 and the first sum means “a sum of terms of the form”.
We can now apply Corollary 15.A.7 to find that
5-2
x'i • • • x'pQs(Q + si) П (Q +	= 0 mod Vr_^k(rk_j}_x.
1 = 0
This implies (I) and hence the lemma.
15.5.4.	Let d be the largest integer such that (gc)2d * 0.
Lemma. Let j > 1 and let 2 < i < d. Then there exists ptj integers
mi k j > 0, к = 1,..., ptj, such that if
Tj = (0! + jl) П (02 +	• П (0rf +
к	к
then, for every Q-module V,
TjV1+^Vj.
If r > 1, then set
Vf = {r e Vj\ (X ~ ф(Х))о e for X e (0C)2„ 5 > r).
We note that F}1 = Vj_x. We also have:
(1)	0ЛГ+1 c l<r+1 for r 1.
390
15. The Whittaker Plancherel Theorem
Indeed, write Qr = Lk Zk(Xk - ф(Хк)), as usual. If x g (fic^s with
j > r + 1 and if v g ^r+1, then we calculate
(*) xQrv= L[x,Zk](Xk-<KXk))v+ £zk[x,Xk]v + Qrxv.
к	к
Now, [x,Zk] g (0c^2(5-r+i)- Hence, [x, Zk](Xk - ф(Хк))и g
[x, Xk] g (flc^2(5+r)’ Since s + r > r, this implies that [x, Xk]v g
Vj-s-r-i. Thus, Zk[x, Xk]v g by 15.5.3 (ii). Finally, xv g
so Qrxv g by 15.5.3 (iii).
(2)	If r 2, then there exist integers tn, > 0 depending only on j, r such
that
n(Qr + ^i)Vjr+1cv/.
i
Let x g (gc)2r and neV^1. Our assumption on r implies that
ф(Хк) = 0. As before, we analyze the terms in the expression (*) for
xQrv. [x, Xk] g (gc)4r. Since 4r > r + 1, [x, Xk]v g I<_2r_1. Hence,
Zk[x, Xk]v g Vj_r_l. We also note that
^,[x, Zk]Xk = ~-^0x + Euk-Xk,
к	к
with uk g (gc)2 and ф(ик) = 0. Thus,
x(Qr + I)v = Qrxv modL'-r-j.
This implies that for each p > 0, we have
x(Qr + f)₽i; - Qrxv m°d •
Let nlt..., nq, depending only on r and j (Lemma 15.5.3), be such that
n, 0 and
n(Qr + V)^_r = o.
i
Then
+ (n, +	Vj-r-x •
i
This proves (2).
(3)	If rG^thenfQj+jY^GL;..
15.5. First Steps for the Holomorphic Continuation
391
Indeed, let x1;..., x} e (gc)2. We use the notation x' = x - <Д(х) (as
before). Then as in the proof of Lemma 15.5.3, we have
•••	XjQiU = 22 22*'i ••• x'^x^ZMx^ ••• x'jV
i<J к
+ 22 E*i ••	xjx;+1 ••• x’jV.
i&j к
We use the expression [Xj,Zk] = -(x,, Xk)X0 + uik with ф(и1к) = 0.
Thus, the first term of this expression contributes -jx\ • • • x'r. We assert
that the second is 0. Indeed, since [x, , Xk] e (gc)4 we see that since
*p^ + i e we have
x'i '' ‘ x'j-iZ/^Xj, Ajt]x, + 1 • • • XjV e	= Vo = (0).
This implies (3). Now, (1), (2), (3) imply the lemma.
15.5.5.	Corollary. If j > 0, then Tx ••• TjVj + lc:Vl.
15.5.6.	Let F be a finite dimensional irreducible, g-module. Then H acts
semi-simply on F, and nc acts nilpotently on F. Let к be the largest
eigenvalue of H on F. Set F1 = {/e F\Hf = (k - j)f}. If И is a g-
module, then
Vx ® F> с (И® F)r+l
if j = 2r or j = 2r + 1. We fix V with (J, Vj = V.
Lemma. Let f e Fj, v e Vx. If r > 1 and 1 < ix,..., ir < d, then
Q,x - - Qid(v®f)cv® ( 22 d-
We observe that x’(v ® /) = x’v ® f + v ® xf (here, x' = x - ф(х) for
x e nc). We write Qr = Lk Zk.rX'k.r. Then, Q(i • • • Qid(v ® /) is a sum
of terms of the form
(*)	Yx ••• Ydv® Ux ••• Udf,
with each pair (Ij, Ц) in one of the following forms:
(1)	(zkJj,xkJy,
(2)	(X’kJi,ZkJ)-,
(3)	(ZkJX’kJj,iy,
(4)	(I,Zk i.Xk iy
392
15. The Whittaker Plancherel Theorem
We will show that each term in (*) is in V ® (E,<; F‘\ Fix such a term.
Let 5, = {j\(Y},Uf) is as in (i)}, for i = 1,2,3,4- Then Yx • • • Ydv g Vp,
with
P = 1 + E h - E h •
j^S2
Thus, if the term is non-zero, then ij Eje$2 ij- Also, Ц • • • Upf
g Fj+S, with
-2 Eb + 2 E (C--1) - 2|S4I-
yes, ;eJ2
Hence, if the term is non-zero, then j -2|52| - 2|54|. So if s > 0, then
S2 = S4 = 0. But then j -2|5j|. If 51 = 0, then all the terms are of the
form (3). But then the term is 0 since Ydv = 0. The lemma now follows.
15.5.7. We now come to the main result of this section. We use the
notation of the previous numbers. We note that if V is a g-module, then
Vx ® F1 with j = 2r or j = 2r + 1 is contained in (V ® F)r. Let mikj
and ptj be as in Lemma 15.5.4. Set
cr = г! П * 0.
m<. r
k^Pim
We define a map 1} from Vx ® F1 to V ® F by Г/г ® /) = v ® f if
j = 0,1, and if j = 2r or j = 2r + 1, then
г/u®/)	 Tr(v®f).
We define a linear map Г from Vx ® F to V ® F by
ЭД,»/,)
' j ' J
for Vj g Vx and fj g F‘.
Theorem. (1) Г(И} ® F) с (V ® F)x.
(2) Г defines a linear isomorphism of Vx ® F onto (V ® F\.
(1) is just a restatement of Corollary 15.5.5.
We first prove that Г is injective. Let fu.. .,f be a basis of F with
fi g Fr‘ and rx < r2	rq. Let < • • • < sm be the distinct
15.6. The Completion of the Proof of the Holomorphic Continuation
393
Let u e И, ® F. Then и = E, r, ® /, with ц s Vx. We assume that и =# 0.
So we must show that Гм =# 0. Let i be the largest index such that vt =#= 0.
Let r, = sk. Then
и = E vj ® fj mod E И ® PSn-
'Г’к
Lemma 15.5.6 implies that
Гм = 22 vj ® fj mod ЕИ® Fs"-
rj = *k	Sn<Sk
Hence, Гм =# 0.
We now prove that Г is surjective. Let u e (И® F\. Then, as before,
и = Ец ®. If X e nc, then
( X - ф( X)) и = E {( X - ф( X) vt, ® f,,+ vtXft}.
i
Suppose that r, = 0 for i > p, and that vp =#= 0. Set rp = sq. If X e nc,
then
0 = (X - ф(Х))и = 22 ((* “ Ф(ХУ)ц) ® 4 modi'® 22
s"<s«
This implies that s Vx if r7 = sq. As in the proof of the injectivity, we
have
r( 22 yj®/J= E vj ® fj modi'® 22
rj= Sq	rj= Sq	Sn<' Sq
Thus,
и - г( E ®/J e И® E Fs"-
rj = Sq	sn<' sq
This, clearly, implies the surjectivity.
15.6. The completion of the proof of the holomorphic continuation
15.6.1. Fix (Po, Ao), a minimal p-pair in G. Let x be a generic unitary
character of No. Let ф = dx, as in the previous section. Our strategy for
the proof of Theorem 15.4.1 is to first prove the result in the special case
when P = Po and then to derive (a generalization of) it from the special
case.
394
15. The Whittaker Plancherel Theorem
Until further notice, P = Po. Let (£,Hf) be an irreducible (finite
dimensional) unitary representation of °M. We note that PN is open in G,
and G - PN has measure 0 (Corollary 2.2.11). Let U( „ be the space of all
f g Ц such that Р/р|ЛГ has compact support (the notation is as in 10.1.11).
Theorem. Let A g Whx(FpCv). If A(t7f „) = 0, then A = 0. Furthermore,
there exists g g H* such that
A(/) = Jf x(n)-'pfM dn I
\JN	/
foraltfeU('V.
Note. The convergence of the preceding integral is clear.
15.6.2. We will defer the proof of this theorem to the end of this section.
We now show how it can be used to derive Theorem 15.4.1 in the case at
hand. We first observe:
Corollary, dim Whx(Ip( v) <, dim Hf. If v g (etc) , t^n
dimHTiA.(/pf „) = dimHf.
The first assertion is an immediate consequence of Theorem 15.6.1. On
the other hand the injectivity assertion that we proved in 15.4.2 implies
that if v g (a£)~, then
dim	= dim H(.
So in this range, we have
dimlWiA.(Zp>f „) dim H(.
The corollary now follows.
15.63. Lemma. If
dim Whx(Ip ( „) = dim ,
then
dimWhx^Cv+4p) = dimtff.
15.6. The Completion of the Proof of the Holomorphic Continuation
395
Let F be as in the example in 10.2.1. Then F/nF is one dimensional
with °M acting trivially and a acting by 4p. Lemma 10.A. 1.7 implies that
Ip,Cv®f = i^i- э ••• d/;d/;+1 = (o),
with 7“ closed in Fp (v and G-invariant. Furthermore, F has a Jordan-
Holder series as a representation of P:
F = Fx^> F2t> • • • Fd э Fd+ j = (0),
with F2 = nF and Fj/Fi+1 the representation of P with N acting trivially,
A acting by g, , and °M acting by a,. So we may assume that
TOO / TOO	TOO
*i /*i + l = * P,*
Observe that £ ® splits into irreducible components.
We identify (7^ ® FY with (JfY ® F*. We look upon (Т^У as a g-mod-
ule using the action XX = -Л ° ттр („(X). In the notation of 15.5.3,
'?.!.<) * (?)',•
We assume the hypothesis of the lemma. Then, Theorem 15.5.7 implies
that
dim(/^° ® f/j = dim F dim .
On the other hand, we set Wj = {Л e (7^ ® FYl\k(F?+l') = 0}. Then, it is
clear that:
(1) dim Whx(Ip,f,p+4p) > dim И^.
Also, Theorem 15.6.1 implies that dim dim 77f dim H . Clearly,
the dimension of Wd is dim Hf dim F. Thus,
dim Wd_ j > dim H( dim F - dim H( dim Haj.
Similarly,
dim	* dim dim •
So
dim Wd_2 > dim H( dim F - dim H( dim Had - dim H( dim Haj .
If we continue in this way, we find that
dim > dim H( dim F - dim H( £ dim Ha = dim 77f.
i>l
396
15. The Whittaker Plancherel Theorem
Hence, dimR7ix(/P f P+4p) dim Hf. The lemma now follows from
Corollary 15.6.2.
15.6.4.	We now use the formalism of the previous number to implement
the holomorphic continuation. Let g be the action of G on F. Let q be
the natural projection of F onto F/nF. We define a continuous bijection
3 from If ® F to by
3(?>®/)(*) = <p(k) ®p(k)f
for к g K, <p g If, and f g F. We set
T = (I ® q)°8.
Then, T is a continuous map of If ® F into If and
® M(g)) = Trp,(,v+4P(g)T
for all g g G and all г- g a£. We note that Ker T = If in the notation of
the previous number. In other words, If is independent of v. Let тг'
(resp., g') be the action of g on (ГУ (resp., F*).
Fix <p g Cf(N) such that
f x(n)~'<p(n) dn = 1.
JN
We define, for v g , Фр(г) g /” in the same way F(v) was defined in
15.4.2. Then, it is easily seen that:
(1)	v -> Фр(г) is holomorphic from to
Let pg = min{Re(4p, a)|a g Ф(Р, A)}. Note that p0 > 0. Suppose that
we have shown that v „ has a (weakly) holomorphic continuation to
the set (a£)~ a" v e ac such that if a g Ф(Р, A) then Re(ri, a) < q,
q > 0. Let kx,...,kd be a basis of Hf. If Re(p, a) < 0 for all a g
Ф(Р, A), then
(*)	Jf>p(A,)(%(l;))=A/(p).
Thus, by holomorphy we see that (*) is true in the (possibly) larger range.
Set yi(v) =JfyV(kj). Let gp...,gr be a basis of F*. Theorem 15.5.7
implies that there exist pkij g L7(qc) for к = 1,..., dr, i = 1,..., d, j =
1,..., r that depend only on ф and F (and not on v) such that if
® м')(Ръ;)(%(«') ® M;) = ^(^)>
15.6. The Completion of the Proof of the Holomorphic Continuation
397
then ...,7]rd(v) is a basis of (.(Гр^„У ® F*\. Let v0 g (a^ . In
the course of the proof of the previous lemma, we also showed that
dim((/^,J ®	= rd - d = t.
After a possible relabeling, we may assume that
are linearly independent. Thus, there exists an open neighborhood of v0,
U„ , in (a£)7 such that
F0 7	^4
77i(»?)|/2”,---,77((p)|/2”
are linearly independent for p g U„ . Thus, there exist holomorphic
functions a,j, i > t, }<, t, on U„a such that
J
for v e U . Set
p0
=Vt+M - Y/at+i,j(v)Vj(v)
J
for V g UVU. Then, Ф^р), ..., ф/v) form a basis of Whx(Ip ( v+4p) for
v g UVo. Let k^v) g H* be associated to ф,(г) as in Theorem 15.6.1.
Then,
KM(V) = ФМ(%+4Р(»))-
Thus,
j
with a,j holomorphic and	invertible on Uv . Set [а'Чг')] =
[e0(v)]-1. We set
«,>0(p + 4p) =
j
for p g (7 . Set Vvo = U + 4p. Then, v -» a,„o(pX/) is holomorphic on
Ko for f g Ц, and
(**)	aiVo(v)(%(v)) =Xi(v)
for v g and v g V„ . This implies that if v g Vvo n Vv,o, then alfPo(v)
= aifP'0M- We may thus define a, on (ctc)7+ by а,(г-) = aiyV-4p(v).
398
15. The Whittaker Plancherel Theorem
Then, v -> afvjtf) is holomorphic on («cV+ 4p for / e /”. (* *) now
implies that if v e (dc\~ ’then = This completes the proof
of Theorem 15.4.1 in this special case.
15.6.5.	We now fix (P, A) a standard p-pair dominating (Po, Ao). Let
P = °MAN, as usual. Let (a, Ha) be an admissible, finitely generated
Hilbert representation of °M as in 10.1.1 and let Ipav be as in 10.1.1. Set
*X = X|N0n oM. If Л e Wh*x(H”), then we set
(1)	^,(A)(/) = fx(n)-h(Pfv(n))dn
for / e
If q e R, then we set {v e a^lReCr-, a) < q, a e Ф(Р, A)}.
Lemma. There exists ca > -oo such that if v e (ac)c~, ^en integral in
(1) converges absolutely for all f e I™. Furthermore, the map
is holomorphic on for all f e and A e Wh*x(H“). Finally, for
v e (a£)~, Ja v is an injective map from Wh»x(H“) to Whx(Ip a v).
We note that if A e Wh»x(H“), then A is tame for all Q s ^(Ло)
(15.2.3). Thus, Theorem 15.2.5 applies in all Weyl chambers. We may thus
use the argument in 10.1.2 to prove the convergence and holomorphy
assertions, and the argument in 15.4.2 to prove the assertion about
injectivity.
15.6.6.	We look at this construction in the following special case. Let
(£, H() be a unitary irreducible representation of °M0. Let *a = a0 П °m.
Let g s (*ac)* and let	with Q = Po П °M.
Then Ipf(T<v is topologically isomorphic with Ip(>ll+v as a representation
of G. The isomorphism is implemented as in 10.1.13.
Lemma. Let A s Wh*x(H“). Iff s I™, then the map v •-» P(AX/) has
a holomorphic continuation to a^. Furthermore, if v e a£, then Ja v
defines a linear bijection between Wh*x(H“) and Whx(ff).
Let *J( s be the Jacquet integral for Q and *x- Let H be the isomor-
phism of onto IpnQfsV as in 10.1.13. Let A s H*. Then a direct
15.6. The Completion of the Proof of the Holomorphic Continnation
399
calculation shows that
Л0,,8.,(*4,а(Л))=^а+ДА)°Н-1
if 3 + v g ((а*)(Эо and Ле integral defining К0(6,и converges abso-
lutely for all f. This implies that
for (8, v) in an open set. The result now follows by continuing the right
hand side of the equation.
15.6.7.	Theorem. Let (a, Ha} be an irreducible, admissible representa-
tion of°M. Let Л g Wh*x(H“). Iff G then the map v -> Ja P(AX/) has
a holomorphic continuation to a£. Furthermore, if v g a£, then Ja v
defines a linear bijection between Wh*x(H“) and Whx(I“).
Let (£, H() be an irreducible finite dimensional representation of °M0
and let 3 g (a0 n °m)c be such that there exists a surjective (°m, К П
Af)-homomorphism S of IQt^s onto (На)Кг,м- Theorem 11.6.7 implies
that S extends to a continuous °A/-intertwining operator from Iq^s onto
H“. Let V denote the kernel of this extended map. Set p(m) =
ir0>f>e(m)|K. И v e ac> then we have the exact sequence
0	^P,7T0 ftS,v	0
in	(see 11.6.8). In this sequence, the first arrow is given by the
obvious homomorphism given by SffXk) = f(k) since Ис/е{8,
and the second arrow is given by S2(f)(k) = S(f(kji). The point is that the
total spaces and 5j, S2 are independent of v. We therefore have the exact
sequence
$T	ST
(•)0 -> ИД,..) -	- °-
We also have
o -> икдн?) 4 wh^^) -> wh*x(rQ'(^v -> o-
This implies that
d = dimWh»x(H“) = dim - dim
For simplicity of notation we will write irQ t s = rj. Let q{, ...,qd be
400
15. The Whittaker Plancherel Theorem
linearly independent elements of Wh*x(H“) such that q^v = 0. Then
= 0 for v g (a£)c- and hence by holomorphy	= 0 for
all v. Also, 15.6.6 implies that „ is injective for all v. (*) now implies
that:
(1)	dim WhxUp>v) > d for all v g a£.
Let <p g C™(N) be such that
f x(n)~'<P(n) dn = 1.
JN
If v g H“, then define F(v, u) g /” by
pF(v,v)v(mann) = a~p+vr)(m)<p(n)v
for m g °M, a g Л, n g N, n g N, and pF(v, v)v(g) = 0 if g & PN. Let
t = dim - d. Let yit..., у, be elements of Wh*x(H“) such that {
form a basis of Wh*x(H“)^y. We note that if v g V, then F(v,v) g
We have
~yt(v)
for all v. Thus,
dimWiJr(/p ч р)ц- > t
for all v. Now, (*) implies that
dim Whx(Ip a „) < dim Hf - t = d
for all v. Hence, (1) implies:
(2)	dim Whx(I^av) = dim Wh,x(H?) for all v.
If A15...,Ad are the elements of Wh*x(H™) corresponding to the <?,,
i = 1,..., d, then
for v g (a<pc~ • Since Ja v(qt) - S^Cy^v)) for all v, with v -» y/r'K/)
holomorphic for /g/“ and y^v) g Whx(Ip a „), this implements the
desired continuation. (2) now implies the bijectivity of Ja v.
15.6.8.	We are now left with the proof of Theorem 15.6.1. Our proof will
involve standard methods of “Bruhat theory”. We return to the notation
15.6. The Completion of the Proof of the Holomorphic Continnation
401
of 15.6.1. If f g C"(G) and if v g , then we define
W)(g) = fjv(p)~lf(pg)i>drp,
Jp
where drp denotes a choice of right invariant measure on P and Цпта)
= av~p£(m) for n g N, m g °M, and a g A. Then Tv(f) = pSv(f)„, with
SL. a continuous linear map of C”(G) into I’f. We also note that, since
P\ G is compact, Lemma 0.1.3 implies that the map from C“(G) ® to
If given by f ® v -> S^f) is surjective.
Let A be as in the statement of the theorem. If v s Hf, then we define
r,.(/) =A(W))
for f g C"(G). Then, r,. defines a distribution on G (cf. 8.A.2.1). We note
that A(Gf „) = 0 if and only if rv(f) = 0 for all v g and f g Cf(PN).
Also, A = 0 if and only if r,.(/) = 0 for all v g H( and f g C“(G).
Let, for each j g W(A), s* g К be a representative of j. Let s0 be the
element of W(A) such that $0Ф(Р, A) = -Ф(Р, A). Then P =
s*P(s*)-1. Hence, the Bruhat lemma (2.2.10) implies that
(*)	G = U ?S*N>
SEW(A)
and this union is disjoint. We define an action of P x N on G by
(p,n)x =pxn.
(*) implies that there are exactly |W(A)I orbits. Lemma 8.A4.5 implies
that we can label the orbits ^j,..., so that U,is closed in G. In
particular,	is open and hence (2.2.11) = PN and 0W is closed. Set
fly = U* < j • Then, Uj is open in G and is closed in fly. Label the
elements of W(A) so that = Ps*N.
We assume that \(Ufv) = 0. Then тг|П) = 0 for all v g We prove
by induction on j that r,,|n. = 0. Assume that r„|n = 0, j - 1 > 1, for
all и g H^. We must show that т(,|П. = 0 for all v. Now, is closed in fly
and fly = U fly_ [. Thus,
supp(Tr|n.) c
We now begin the classic Bruhat-type argument. Let s = Sj. The critical
402
15. The Whittaker Plancherel Theorem
point is that s =# 1. Let Ф, = {a e Ф(Р, A)|s e Ф(Р, A)} and let
Ф' = Ф(Л A) - Ф5. Set
П, = E 0a-
аеф,
Let Ns be the connected subgroup of g corresponding to n5. Then Ns is
diffeomorphic with n5 under exp. We define ф mapping Ns x to G by
ф(п, x) = nx. Then, ф is a diffeomorphism onto its image U, which is open
in G. Clearly, = <Д(1 x ^), which is closed in U. We can thus find an
open neighborhood W of 1 in Ns such that ф(№ x is open in fly. If
T e ^'(^), then we define, for f s C“(fly), pT(f) = Let m be
the maximum of the orders of the tv . Let ..., xd be a basis of С7т(п5).
Theorem 8.A.5.2 (Schwartz [1], p. 102) implies that if S is a distribution on
fly supported on of order m, then S can be written uniquely in the
form
(1)	S=^L(xi)nSi.
i
We therefore see that
i
with Tt a linear map from Hf to ^'(^). Now,
R(n)rv = x(n)~lTv, n^N.
The uniqueness in the expression (1) implies that
(2)	Я(п)7].(г) =X(n)-1T/(r), n^N.
Let t](x, n) = xs*n, x P, n e N. Then, 77 is a submersion of P x N
onto . If we apply the argument in the proof of 8.A.2.9, we find that for
each i there exists a linear map t, : H( -> ^'(P) such that
Ti(v)(<P) =	x(n)~l<p(-s*n) dn
\JN
The upshot is that if f s C7(fly), then
(3)	Tv(f) = Lti(v)^fN(L(xT)f(-s*n))x(n~1) dn).
15.6. The Completion of the Proof of the Holomorphic Continnation
403
We also observe that
(4)	L(n)rv = rv, n^N.
Since s =# 1, there exists a s Ф(Р, A) such that -sa is simple in Ф(Р, A).
Thus, there exists Yen such that Ad(s*)-1y s n and ^(Ad(s*)-1y) =
i. Let x = exp tY. Then,
тД/) = rv(L(x)f) = YAi(v)[jtf(L(xf')L(x)f')(-s*n)x(n)~l dnj
=	(L(x)L(Ad(x)-1xf)/j(-s*n)^(«) 1 dnj
=	j/^(Ad(x)-1x/’)/)(x“1 •s*n)^(n)'1 t/nj
= EG(i0(^(L(Ad(x)_1x,7’)/)
X(x^ 1 ' XS*((s*) 'xS*j и)х(п) 't/nj
=	(L(Ad(x)-1xf)/)(x-1  xs*n)x(n)~1 dnj.
We define an action of N on P by i(x)y =xyx~1. The upshot of the
calculations is
114(v)^N(L(xI)f)(-s*n)x(nYx dn^
= e“"^i(x)tj(i;)^/J^(Ad(x)_1x/’)/)(-s*n)x(n)_1 dn j.
If we differentiate this expression at t = 0, we find that
=	Xl)f}(-s*n)x(n)~l dn}.
404
15. The Whittaker Plancherel Theorem
Now, there exists p such that
(Ad Y)PUm(Q) = (0).
Therefore, the uniqueness of the expression (1) implies that
(5)	(i(Y) -	= 0 forallz,r.
We now note that if we set a(man) = av+p£(m), then
L(u)tv =	u^P.
We therefore have
r^Af) =	(^(xf)L(u)~1f(-s*n)]x(n~1) dn)
/	\‘,N	/
= Щи)1,(и)^(Ь(А(1(и) xf)f(-s*n))x(n~I) dn).
Thus, if Z e p, then
ra(Z)v(f) = E^(Z)t,(v)lf (L(x[)f(-s*n))x(>Tl) dn)
+ EfXl,)(/^(^(Ad(Z) xf)/(-5*n))x(n-1) dn).
We conclude that
E^(2)t,(y)(^(L(x/’)/(-5*n))A;(n_1) dn)
= -	(^(Ad(Z)	dn) + ra(Z)1,.
। \JN	I
This implies that if Z1,...,Zr is a basis of p, then there exist ct j,
i = 1,..., r, j = 1,..., 2n, such that
E П(^(29) -	J (L(x[)f(-s*n))x(n^') dn] =0.
i,q )	\ N	)
Set
2n
О - E П (Z.(Z,) - C,.,).
ч
15.7. Cnsp Forms Revisited
405
Then, D is an analytic elliptic operator on P and the uniqueness assertion
in (1) implies that £>t,(r) = 0 for all i, v. Thus, t,(r) is a real analytic
function on P. If we argue as before, we find that there exists a positive
integer c such that
Ь(У)СФ) = 0
for all yen. Since n is normal and tf(i>) is real analytic, we also have
Я(у)Ч(г) = 0
for all yen. This combined with (5) implies that
(t(y)-z)%(r) = t(y)ctm(l;) = 0
for all m, v. We therefore conclude that tm(v) = 0 for all m, v.
We have thus proved that т(.|П = 0. This concludes the induction and
hence the proof of the first part of Theorem 15.6.1.
To prove the second assertion, we note that the map r) from P x N to
G given by t](x, n) = xn is a diffeomorphism onto an open subset 0 of G.
Thus (in the notation of 8.A.2.9), we have
^7*(^)(/) = t(v) ®x(n)-'dn.
Now, L(u)t(r) = t(8(u)v) and as before t(v) is real analytic on P. From
this, it is an easy matter to prove the second assertion of the theorem.
15.7. Cusp forms revisited
15.7.1. We maintain the notation of the previous section. Armed with
Theorem 15.4.1 we are now in the position to make a more detailed
analysis of the cusp forms of 15.3.3. Let Q-^(N0 \ G; x)k he the space of all
right К-finite cusp forms. If ы e d’(G), we fix (тгш,Нш) s a>. The main
result is (notation as in 15.3.4):
Theorem. Let x be a generic character of No. Then
°^(N0\G-,x)K= ® span{TA((HJK)|A
15.7.2. Before we begin the proof of this result we note that Theorem
15.3.5 now implies that if д- is a generic character of No then the atoms in
406
15. The Whittaker Plancherel Theorem
the decomposition 15.1.2 are all in ^(G) and the discrete spectrum is
This is, of course, a critical first step in the determination of the full
decomposition.
15.73.	We write p = pP(j. Let x be a unitary character of No. Then we
define sfw(N0 \ G; to be the space of all f e C°°(N0 \ G; x) such that:
(1)	dim Z(gc)/ < oo.
(2)	dim span R(K)f < oo.
(3)	There exists d > 0 such that
|Я(х)/(а*)| 1 Cxap(l +||loga||/
for all x s G(gc), a s Ao, к s K.
If f s pfw(N0\G; x~l) and <p e -^(NQ\G; x\ then
f \f(g)<p(g)\dg = f a~2plf(ak)<p(ak)ldadk
JN0\G	JAt)XK
cidi,m(<p) f (1 + IIlog all)''-"' da.
JA0
Thus,
Tf(<p) = f f(g)<p(g)dg
JNo\G
defines a continuous functional on -£(NQ \G;x^-
Lemma. Let x be generic. Let (тг, H) be a finitely generated, admissible,
tempered representation of G and let Л s Whx(H°°). Then (g -> Л(тг(^)г))
e <s/w(N0 \ G; x) for all v s HK.
If Q e &(A0), then Theorem 15.2.5 (applied to Q) implies that there
exists d > 0 and a continuous semi-norm qK on such that
|л(тг(а)г)| sa₽o(l + IIloga||)у)
for a e C1(Aq) and v &LT. Since ap <. apQ for a e C1(Aq) (see the
proof of Theorem 15.3.4), the lemma follows.
Note. It can be shown that every element of p/w(N0\G;x) is of the
form given in this lemma (see 15.11.6).
15.7. Cnsp Forms Revisited
407
15.7.4.	If f g ^(No \ G; x) and if (P, A) is a standard p-pair dominat-
ing (Po, Ao), then for v e a*, we define
f,Pv(m) = J a~ivfp(am) da
for m g °M. Here, fp is as in 15.3.2 and the integral is absolutely
convergent and defines an element of C\ °M) \°M; X\n n
If |/| g Cc(N0 \ G), then fp(m) is defined for v g a£ and is holomor-
phic in v.
Proposition. Let g w g d’2(°Af). If Л g Wh^Nijr, and if
f e -£(NQ \ G; x), then
f JajM^P,a,iv(s)<p)f(g)dg
JN0\G
= (	„	0	A(a(m)<p(k))(R(k)f)Pv(m) dmdk.
J(Nar,
For all <p g Ia.
Since both sides are continuous in f it is enough to prove this formula
for |/| g Cc(N0 \ G). For such f,
v ~ I da y(A)(TTp a „(g)<p)f(g) dg
JN0\G
is entire in v. Thus,
f dail,(A)(Trpail,(g)<p)f(g)dg
JN0\G
For £ > 0, we have
J d<r>iv_ep/,(A)(irpt<rtiv_ep/(g)<p)f(g) dg
= I A / *(” l)(Piv-ePP(ng)dn\f(g)dg.
JN0\G \JNP	I
If we set *N = No A °M, then No = NP*N (with unique expression).
408
15. The Whittaker Plancherel Theorem
Thus, using the continuity of Л and the assumption on f, we have
(p = pP)
f Ja.iv-'pWfaai (g)(p)f(g) dg
JN0\G
= / I	(ng) dn)f(ng) dg
JN0\GJNP
= f *(<Pi,-ep(g)f(g) dg
J*N\G
= f	a2pa p ,v epX(a(m)<p(k))f(nmak) dndmdadk
JNPX (*N\°M)XAXK
= f X(a(m)<p(k)(R(k)f)?_ep(m) dmdk.
J*N\°MXK
If we take the limit as e -> 0, the proposition follows.
15.7.5.	We now begin the proof of the theorem. We may assume that
G = °G. Let A c Ao be a special vector subgroup (10.1.8). Let -£(G)A be
as in 13.4.7. The key step in the proof is:
Lemma. Iff e 0-^(N0 \ G; x\ and if e ^(G)A, A =#= {1}, then
(*)	jGf(g)<p(g)dg = 0.
Note. Lemma 15.3.6 implies that the integral in the statement is abso-
lutely convergent.
Let (P, A) be a standard p-pair dominating (P0,A0). To prove the
lemma it is enough to prove that if (a, Ha) e e <f2(°Af), a e C“(a*),
v, w e Ia, and if
<f>(g) = f {TTp'a iv(g)v,w)a(v)p.(a>,iv) dv,
then (*) is true (see 13.4.5-13.4.7). For <p given in this way, we compute
= jGf(g)<p(g) dg
for f e ^(No \ G; ^)- We first consider the case when |/| e
15.7. Cnsp Forms Revisited
409
Cc(N0 \ G; %) We begin the calculation:
= f f(g)f x(no)~l<p(ng)dnd(Nog) = f f(g)<Px(g) dg
JN0\G JN0	JN0\G
(see 15.3.7). We now compute <px. We set ir„(g) = irp><7>p(g). We first
observe that if v,w e 1“ then, if v e. a*,
(v,w) = I(v(k),w(k)) dk
= f ap(n)~2p(v(kp(n)),w(kp(n))} dn
JN
= f {V.v(n),wiv(n)} dn.
JN
Here, we have used n = np(n)ap(n)mp(n)kp(n\ so
"(Ы«)) = йр(л)'’"о-("’р(л))-1Цр(л)-
We therefore see that
(ir,p(g)u,w> = f (viv(ng),wiv(n))dn.
JN
This implies that if v, w е , then
C = f x(n0)-1 f (iriv(n0)v,w)a(v)/i(a>,iv) dv
JN0	Ja*
= ( x(no) 1 f [ (.Viv(nno),wiv(n)) dna(v)fi(a>,iv) dvdn0.
JN0	Ja*JN
We note that if e,r] >0, then
( viv-ep(nn0), wiv_vp(n)) = aP(nn0) ~epaP(n) ~vp( viv( nn0), wiv(n) >.
Since
ap(nn0) epaP(n) vp < 1
(3.A.2.3), we may apply dominated convergence to see that
C= lim f x(no)'1 [	(nn0),wi (n))dn
e,iJ->O+^o	Ja*JN
Xa(v)fi(a>,iv) dv dn0.
410
15. The Whittaker Plancherel Theorem
Set Ce equal to the preceding integral. For e,r] >0, the integral
converges absolutely. We may thus perform the integrations in any order.
We therefore have
ce,v = f f X(n)x(no)~l(viv_ep(nno),wiv_vp(n)}dndno
Ja*JNxN0
iv) dv.
We set *N = No П °M. Then, No = *NN with unique expression. Also,
vi„-ep(*nn) = <r(*n)viv_ep(n).
Hence
I x(n)x(n0)~\viv (n0),wiv (n)}dndn0
JNXNO
= f x(n)x(ni)-1x(*n)-1
JNXNX*N
x{<r(*n)viv_ep(n1),wiv_7ip(n)') dnxdnd*n.
If x, у e H™, then 15.2.4 implies that (tn -> (<r(m)x, у » s -£(°M). This
implies that we can define, for у e H“, Ay e Wh**(H“) by
A/*) = [ x(*n)~\a(*n)x,y) d*n = Ax(y).
J*N
If Re(g, a) < 0 for a e Ф(Р, A), then we set
Ja.p.(V) = J	dn.
JN
This integral converges weakly in this indicated range (see 15.4.2), and if
A e Wh*x(I^) then
A(M"))=-MA)(r).
We therefore have
I x(n)x(n0)~Xviv_ (n0),wiv_ (n))dndn0
JNXNO
15.7. Cnsp Forms Revisited
411
We conclude that
TJ U+ Q*
We therefore have
/(/.у) - Um / Я»)/у,.„(л,.^„)(М«)<')
rj > 0
Ха(|')д(ш, iv) dv d(Nog).
Since a has compact support we can interchange the integrals, so Proposi-
tion 15.7.4 implies that
I(f,<p) = lim f [	(Я(к)/)^(т)к^ (w)
V^0+Ja*J*N\°MxK
X(a(m)v(k)) dmdk a(v)p.(a>,iv) dv.
Now,
We can thus take the limit under the integral sign, and we have
(1) /(/»=/ f 0 (R(k)f)t(m)J<rtiv(k<rim)vW)(w) dm
Xa(v)fi(a>, iv) dv.
This formula is continuous in f e Tf(N0\G;^); hence it is valid for all
such f. Thus, if A =#= {1} and if f e a-£(.Na \ G; x\ then I(f, ^A(G)) = 0.
This is the statement of the lemma.
15.7.6.	We will now complete the proof of Theorem 15.7.1. Let f be an
element of °^’(N0\G;^)K. Let FcK be the set of у such that
span{f?(A9/}(y) * 0. Then F is a finite set. Fix, for у e К, ш e. tf2(G), a
choice of Vy e у, (тгш,Нш) s ы. Let <f2,f = {" e ^(GllHom^d^,Нш)
=#= 0 for some у e F}. Then £2 F is a finite set (7.7.3). Let the space of
all right and left К-finite functions <p in °^(G) such that under Я|К,
412
15. The Whittaker Plancherel Theorem
span R(K)<p splits into a direct sum of isotypic components corresponding
to the elements of F. Then, VF is the span of the functions
with ы s <f2tF’ v e w e (MJk (see the proof of Theorem 7.7.6 or
apply the Plancherel theorem). Since №)(1) = nif(G), the previous result
combined with Theorem 13.4.7 implies that if </, <p) = 0 for all <p s TF,
then f = 0. This clearly implies that dim Z(gc)/ < oo. Set Vf =
span Ж17(дс))Я(Ю/Then, Vf is an admissible finitely generated (д, Ю-
module. The preceding characterization of % implies that Vf splits into a
direct sum of invariant subspaces, each equivalent to а (Яш)к for some
ы e ^2,f- This implies that Cl(Lp in L2(N0\G;^) is contained in
E span{TA(Hj|A eWhx(H“)}.
(1)	G j j;
This implies Theorem 15.7.1.
15.8	. The first steps for the Plancherel theorem for generic x
15.8.1	. In this section, we will analyze the calculations in the proofs of
Proposition 15.7.4 and Lemma 15.7.5 in more detail. We retain the
notation of the preceding section.
Lemma. Let (P, A) be a standardp-pair dominating (P0,A0). Then there
exists a continuous semi-norm q on ^(G) such that
f Jf(non)ldnodn <q(f).
JN0XN
If we argue as in the proof of Lemma 15.3.2 we find that, for each
d > 0, there is a continuous semi-norm qd on if(G) such that
/ \f(noak)\dno < qd(f)(l + lllogall)~dap.
•'No
Now, proceed as in the proof of Lemma 15.3.2.
15.8. The First Steps for the Plancherel Theorem for Generic x
413
15.8.2	. Let (P, A) be a standard p-pair dominating (Po, Ao). Fix (a, Ha)
e ы e d’2(°M). If a g <Aa*\ and if v, w e. Ia, then we set
F(P,a,a,v,w)(g) = [ (Trp„,Jg)u,w)a(v)p(M, iv) dv.
J a*
Then, F(P, a, a, v, w) g if(G) (Theorem 12.7.1).
Lemma. If (Q, Aq) is a standardp-pair dominating (PQ, Ло) and if
(F(P,a,a,v,w)x)Q * 0,
then (Q, Aq) dominates (P, A).
Set <p = F(P, a, a, v, w). We calculate, for m g MQ,
(<p )Q(ma) =apo(_ f x(na)~'<p(nQnma) dnQdn.
The preceding lemma implies that we can interchange the order of
integration. We can also assume that a g C“(a*). We therefore calculate
f_<p(nonma) dn = /_ / (irp a iv(nonma)v,w)a(v)p(a>,iv) dv dn.
}Nq	JN0Ja*
We write g g G in the form k(g)m(g)n(g) with k(g) e K, m(g) e MQ,
n(g) g Nq (this is not our standard way of decomposing elements relative
to a parabolic subgroup). Then,
/_ <p(nonma) dn
jnq
= f f (тгр a iv(k(n0)m(n0)nma)v,w)a(v)p(a),iv) dv dn
JN0Ja*
= а0(т(п0))2р° f_ f (ттра^пт(п0)та)о,тг(к(п0)У^)
JN0Ja*
Xa(v)p(ti>, iv) dv dn.
This expression is equal to
«~'’°«е("1(п0))₽ор(Р,<т,а,г,тг(^(п0))	(m(n0)ma).
414
15. The Whittaker Plancherel Theorem
Hence, Theorem 12.8.4 combined with Lemma 12.4.1 and Theorem 12.4.1
implies the lemma.
15.83.	Lemma. If Q =#= P, if is an irreducible square integrable
representation of °Mq , and if p,q e , then
(F(Q,p,0,p,q)x,F(P,<r,a,v,w)x) = 0
for all 0 s
We set f = F(P, a, a, v, w)x, = F{Q, p.0, p, q\ Then, we are looking
at I(f, <p) as in 15.7.5. The formula 15.7.5 (1) combined with the previous
lemma implies that this expression is 0 if A is not contained in AQ.
Interchanging the roles of P and Q implies that we also get 0 if AQ is not
contained in A.
15.8.4.	We will now calculate F{P, p, 0, p, q)x for a s C“(a*). As in
15.8.2, we set <p = F(P,a,a,v,w) and interchange the order of integra-
tion. We use the notation of that number. Then we are looking at
aPf x(n0)~la(m(n0))2p
JN0
x Lf (^,a,4«m(n0)me)u,ir(A;(n0))"1w)
JN Ja*
Xa(v)p(a>,iv) dv dndn0.
We first observe that Theorem 12.8.4 combined with 12.5.5 (4) and the
limiting argument in 13.3.2 imply that
yPap /_ Г \тгр,а iv(nm(n0)nw)v,Tr(k(n0)ylw\a(v)p(a>,iv) dv dn
JNQJa* ' ’	'
= E J aisva(v)ps(a,iv){(ka)(m(n0)ma)(As(iv)v)(l),
sEHU) a*
with
As(iv)f = Jplkspk-^ksa, isv)L(ks)f,
ps(<r,iv) = ppikspk7fks<r,iv),
15.8. The First Steps for the Plancherel Theorem for Generic x
415
and ks is a representative of s. We now use this to calculate the full
integral. The expression that we are calculating is thus
Ур1 E I х(поу' f vs((r,siv)
s<EW(A) N0	a*
If we apply the limiting argument of 15.7.5, then we have (in the notation
therein)
(1)	F(P,a,a,v,w)x(ma)
seM.4) a
Xp.s(a, isv) dv.
This is the expression that we were after.
15.8.5.	Let Ap...,Ad be a basis of Wh*x(H“Y Let wv...,wd be ele-
ments in (На)КпЛ/ such that ЛДи^) = 8,;. If и e Ha, then
Au — ^Au(vVy)Ay — ^Aw(u)Ay.
i	i
We set t]j = Aw . Then, we have the formula
Au = LVj(u)Xj.
J
Lemma. Let a e C“(a*), v, w s Ia. Then
F(P,a,a,v,w)x(g)
= Ef Ja,lv(Vj)(w)Ja>iv(Xj)(irPt<r iv(g)v)a(v)n(a>,iv) dv.
j Ja*
This was essentially done in 15.7.5. Indeed, formula (*) of 15.7.5 implies
that
F(P,a,a,v,w)x(g)
7)	“♦l/+ •'(J*
416
15. The Whittaker Plancherel Theorem
Now,
m	m
We may thus take the preceding limit under the integral sign and the
desired formula follows.
15.8.6.	If a g C“(a*) and if w g Ia, then we set
Л(а ® w)(p) = а(^)Е-/а,,р(^)(и')Лу.
If we endow Ia with the opposite complex structure (i.e., replace i with
— z), then Л is a linear map from C“(a*) ® Ia to C“(a*; Wh*x(H™)) (here,
_ \
X ~ X\NonM'-
Lemma. Л is a surjection from Cjfa*) ® Ia onto C^(a* ;Wh*x(H™y).
We first observe:
(1)	7)j,..., r]d is a basis of Wh»x(H“).
Obviously, we may assume that Wh*x(H“) =# {0}. We first note that if
A g Wh*x(H“), if и g and if v, w g (Ha)K, then
/	X(a(m)u){a(m)v,w) dm = / X(a(m)u)Xw(a(m) v) dm.
J*N\°M	J*N\°M
The argument in 15.7.6 now implies that span{7\ (v)\v,w
is dense in span{TA(r)| v g (На)к, A g Wh*x(H“)}. This implies that
{Ajw g (Ha)K} = Wh*x(H“). In particular, there exists, for each z, ц
such that A„ = A;. We therefore have
A, = Ел/(Ц)ЛУ.
Hence, tj/iz,.) = 8tJ.
We now begin the proof of the lemma. It is enough to show that for
each a g C“(a*), (v -> a(p)A,) g Im Л. Let v g supp a. Then Theorem
15.8. The First Steps for the Plancherel Theorem for Generic x
417
15.4.1 and (1) imply that there exist Wj ..., wd v s Ia such that
det[ja,,„(П;)(и'*,,,)] * 0.
The continuity of Ja „ implies that there exists an open neighborhood Uv
of v in a* such that
detf-W^X^.m)] * 0
for fi e Uv. {t/Jpesuppa is an °Pen covering of supp a. Thus, there exists a
finite refinement of this covering, Ul,...,Up, q>} & C“(Uf) such that
E; (p(v) = 1, v e supp a, and wt k e Ia, 1 < i < d, 1 < к < p, such that
det[ja>,p(77;)("v)] * 0
for v e Uk. We define functions g-j e C“(a*) by g^(v) = 0 if v £ Uk,
and
[^X^)] = фХ^^.ЛХиг*)]
if v e Uk. Then,
Egj-i(t') [ja,iAvP)(wh j] = <Pk(v)8lp
j
for all v e a*. This implies that
EA(g>,n'/>fc)(v) = a(^)Af
i.k
for all v e a*.
15.8.7.	Lemma. There exists an inner product ( , )a on Wh*x(H“) such
that
[	k(a(m)v)'r](a(m)w) dm = (X,7])a(v,w)
J*N\°M
for all u,w s (Ha)K. Furthermore,
The first assertion is a direct consequence of Schur’s lemma and
Theorem 15.7.1. We will, therefore, concentrate on the proof of the
418
15. The Whittaker Plancherel Theorem
second assertion. In the following calculation, all integrals are absolutely
convergent. We will thus integrate in whatever order is convenient.
= £ 0 A„(<r(m)w)Ax(<r(m)y)Jm
= f 0 £
J*N\°MJ*NX*N
x(a(n2m)y, x) dnt dn2 dm
= f	[	И'Мг1) l(o’(nim)w,v){a(n2m)y,x)dnldn2dm
J*N\°MJ*Nx*N
= 1	I	x(ft1)~1(<r(n1n2m)w,v)((r(n2m)y,x)dn2dn1dm
J*N\°MJ*NX*N
( о L	l(a(n2m)w,a(ni)'lv
*N\°MJ*NX*N
X.{a(n2m)y, x) dn2 dnx dm
= L f X(ni) 1(o-(/n)w,<r(n1) 1vVv(<m)y,x}dnldm
= d(a)~l(w,y> f x(nl)~\v,a(nl)~lx)dnl
J*N
= d(a)'\w,y')kv(x).
The second assertion now follows.
15.8.8.	If a & Cc(a*; Wh*x(H^)) and if v & Ia, then we set
Ф(Р,о-,а,г)(£) = f JaJv{a(v))(TrP a iv(g)v)p.(u,iv) dv.
Ja*
We now come to the main result in this section.
Theorem.
(1)	If a a*;	then W, a, a, v) s \ G; x) for all
v ^Ia.
15.8. The First Steps for the Plancherel Theorem for Generic x
419
(2)	The span of the Ф(Р, a, a, v) over all P dominating Po, all a e ш e
tf2(°MP), and over all v s Ia is dense in L2(N0 \ G; ^).
(3)	Let (Q, Aq) be a standard p-pair dominating (P0,A0) and let a\ e
*>i e ^2(°Mq). If a e Cc(a*;Wh»x(Fffi), 0 e C^Wh^^)),
then
(ф(Р, a, a, v), ^(Q, <Tj, j3, w)) = 0
if P * Q °r if P = Q and Wj £ W(A)a>. If P = Q and if a = ax, then
('I'(P, a, a, v),'I'(Q, a, 0, w))
= llV(A)lyA1cA f (a(v),0(v))ap(a>,iv)dv(v,w>
Ja*
with cA, yA as in 13.3.2.
Lemmas 15.8.5 and 15.8.6 imply that Ф(Р, a, a, v) is a finite linear
combination of functions of the form F(P, a, 0t, v}-, wk)x with /3, g C“(a*),
Vj, wk g Ia. Hence, (1) follows from Lemma 15.3.7.
We note that the span of the F(P, a, 0i,v},wk) for (P, AP) dominating
(Po, Ao), aea> e <f2(°MP), 0t g C“(a*), and Vj,wk g Ia is dense in
tT(G) by Theorem 13.4.1. If f g C°(N0 \ G; x) and if |/| g CfNQ \ G\
then there exists <p g C“(G) such that <px = f. Indeed, let и g C“(N0) be
such that
/ x(n)~lu(n) dn = 1.
JN0
Set <p(nak) = u(n)f(ak) for n0 g No, a g Ao, and к g K. Then, <p g
C“(G) and <px = f. Thus, (fx\f g if(G)} is dense in ^(No\ G;^). Lem-
mas 15.3.7, 15.8.5, and 15.8.6 now imply (2).
We now begin the proof of (3). As before, Lemma 15.8.3 implies that
(ф(Р, <r, a, v), ^(Q, <Tj, 0, w)) = 0
if P =# Q. We may thus assume that P = Q. We assume that Wj £ W(A)a>.
As before, it is enough to prove that if 31; 32 s C“(a*), v,wela,
vltw, g I then
{F(P, a, , v, w)x, F(P, <Tj, 82, vx, wj) = 0.
This follows from 15.7.5 (1) with f = F(P, a, 8{, v, w)x and <p =
F(P, al,32,vl,wl) combined with 15.8.4 (1) and the observations in
15.8.5.
420
15. The Whittaker Plancherel Theorem
We are left with the last assertion. It is enough to check the formula for
a = A(8 ® w) and (3(v) = k(v)X with 8, к e C“(a*), w e Ia, and A e
Wh»x(H^Y We first note
(i) (a(vY	= d(a>)~lJa,p(A)(w)S(^)k(^).
Indeed,
(a(^),j3(^))a = а(^к(«')Е-^.,-.,(^)(и')(Л;,Л)(7.
j
We may assume that (A,, Ay) = 3,7. If v s (Ha)K, then
k» =	= 4<u)E(a1,,-»7;)a;
j	j
by 15.8.7. Thus, 15.8.6 (1) implies that tQj = d(to)~xkj. (i) now follows.
We observe that	Л(<3 ® w\ v) = F(P, a, 8, v,w~)x by Lemma
15.8.5. To complete the calculation, we need a few observations. Let
j e W(A) and let k be a representative for j. Let As(v) =
Jp\kpk-i(ka, v)L(k). If A e Wh*x(H%aY then
WAN» Ч.,(ВД))-
Thus, Theorem 15.4.1 implies that there exists, for each v, a linear
isomorphism Ms(v) of Wh»x(H^a) onto Wh»x(H“) such that
Jka,SA*)°As(Sv) = JaAMs(V)A)’
and that Ms is meromorphic in v. This implies that
=j„m
for A s Wh»x{H™Y We therefore see that
Ф(Р,<г,Д,и)(5)
= / Jka,iSV(Ms(iv)~l^(v)^Trpjca isv(g)As(isv)u)fk(a>,iv) dv.
Ja*
We now apply formula 15.8.4 (1), Proposition 15.7.4, the Fourier inversion
formula (as in 13.3.2) and the preceding observation to find that the
15.8. The First Steps for the Plancherel Theorem for Generic x
421
desired inner product (with our choice of parameters) is given by
саУа1 E J 8(х~1р)к(х~1р)/к(ы,1р)
sEH'lA) a*
IN \ /И л Л
ХЛ(^<т(т)As(iv)u(k))fis(a,is~lv) dmdkdv.
We compute each term. So fix j. Let A,, 77, be as before for Hka. Then,
^(ks<r(m)As(iv)i>(k) ~ Y^Ak^m) As(ip)v(k))Aj.
j
Thus,
L 0., Jk^,ivk^(ksa(m)AJiiv)v(k)){^s^v)w)
[N\ MXK
xMs(is~lv) 1 A(&sa(m) As(iv)u(k)) dm dk
= YAs„^j)(a^w') ЦхОмхКч(к'а(т)АА")»(к))
xMs(is~1v)A(ksa(m) As(iv)u(k)) dm dk
= E4Ja,,p(A;)(^J(^)M')(^^(«'1^)-1A)JtjO.<^J(z>)y,^J(z>)M>
j
= d(su)'1JksaJl,(M(is-1v)~1 k)(As(iv)w){As(iv)i’, As(^)u)
= d(5<u)_1J(7i,J-lp(A)(w)<i;,w>gJ(o-,«_1^)_1.
This implies that the formula that we are computing is given by
саУа' E d(sa>)~if 8(s~1p)K(s~1p)/i(M,ip)Jt, is-i„(A)(w) dp.
s^W(A)	a*
Since g(o>, is~1v) = and d(sa>) = d(a>), (3) follows.
422
15. The Whittaker Plancherel Theorem
15.9 The Plancherel theorem for L2(Nn \ G; x)
15.9.1. In this section, we will give a decomposition of L2(N0 \ G\ ^) for
general unitary characters, x- The critical case is when x is generic. This
case will be handled first. We will give explicit unitary intertwining
operators for generic x- In the next section, we will relate the Whittaker
transform to some classical transforms.
Let x he a generic unitary character of . We maintain the notation of
the preceding section. Let (P, A) be a standard parabolic subgroup
dominating (Po, Ao). Let ы e	We fix (<тш, Яш) s ш. We will use
the notation IPav for IPfau,v, 1Ш for irPfUwV for irPf<r^v, etc. We
define a linear map TP a from 1Ш ® C^(aP; Wh*x(H“)) to Tf(N0\G;^) c
L2(N0\G;*)by
TP'U(v®a) = а(Ш)\]¥(АР)Г1уАс^(Р,Ш,а,и)
(see 15.8.8).
We set
g(<u,z^) = d(a>)\W(A)\~\AlyAfi(a>,iv).
On Wh*x(H“) we put the inner product < , >ш. If F с К is a finite set,
then we write IU(F) =	/ш(у). Let ..., vd be a basis of IU(F). If
f is a function on a* with values in IU(F) ® Wh*x(H“), then we can write
f(v) = E; Vj ® atj(v), with atj a function with values in Wh*x(H“). We may
thus look upon
/ш®СГ(а*;^(я:))
as a (dense) subspace of
f Нр,ш,1г ® Wh*x(H“)p.(a>,iv) dv.
•'a*
Here, HP ш iv is independent of v but we will use the notation to indicate
the representation of G for which it is the total space. Thus, TPa is a
densely defined linear operator on this direct integral (14.8.3).
Let d’2(°jWF)/IPC4) denote the quotient space of d’2(°A//>) under the
usual action of W(A). We choose for each class in d’2(0A/p)/H/(^4) an
element ш in the class. We will abuse notation and denote the class by ш.
The Plancherel formula in this case takes the following form.
15.9. The Plancherel Theorem for L2(Na \ G; x>
423
Theorem. The map T= ®(Лл»(/>о.ло) ®ш^Мр)/1¥(А)ТР,ш extends to
a unitary intertwining operator from
Ф	Ф f HPuil,®Wh*x(H“)fi(a>,iv)dv
(P,A)>(Pa,Aa) ue^MJ/WtA)
onto L2(N0\G;x)-
This result is a direct consequence of Theorem 15.8.8.
15.9.2. We will now define the inverse to T on a dense subset of
L2(No\G;^). In the next section, we will give a more explicit form
of the domain (i.e., -£(NQ \G;a)). Let FcK be a finite set. Set
\ G; д-Ху) to be the y-isotypic subspace of -^(No \ G; x) relative to
the right regular action of K. We set
^(N0\G;a)f= Ф ANo\G;x)(y).
yEF
Let (P, A) > (Po, Ло) and let аш e ш e	be as usual. Let Л e
Wh*x(H“) and v s /ш(у). If f s -£(N0 \ G; x)F> then we set
•4\G
We note that 15.7.3 implies that the integral converges absolutely. If
v e Л>(у) with у £ then the integral is zero. Thus, wf V(v ® Л) defines a
conjugate linear functional on /W(F) ®	(/W(F) = ®7ef /ш(у)).
On 1Ш ®	we put the tensor product inner product using the
usual inner product on 1Ш and ( , )ш on Wh*x(H“). Then there exists a
unique element
such that
w/iP(r ® Л) =<И^1Ш(/)(^),г ® A>.
We note that it is clear that ^.„(/Xp) e IU(F) ® Wh**(H“), and that as
a function of v into that space it is of class C°. We will call Ир>ш(/Хр) the
Whittaker transform of fat ы, v.
Theorem. There exists a subspace W of -tf(NQ\G', x\ dense in L2(N0\
G;x), such that W= Uf(^n -£(NQ \ G; (the union over all finite
424
15. The Whittaker Plancherel Theorem
subsets ofk), and such that WPJ.f) is compactly supported for allfe W,
and iff s then
(i) f= E E Ш.Л)).
(P, ЛХЛъЛо) ше^’2(°Л/Р)/И'(у4)
If F с К is a finite set, then we take to be the span of the spaces
TP,a{C°c(^P,Ia{F) ® Wh.x(HZ)))
over all (P, A) > (Po, Ao) and ш e tf2(0MP)/W(A). We take
Uf the union over all finite subsets of K. Then, Theorem 15.9.1
combined with Proposition 15.7.4 and the argument in the proof of 15.8.7
implies that if <p is the right hand side of (1) and if h e then
<Ф,Й> = </,Л>.
Theorem 15.9.1 implies that is dense in L2(N0\G; x). Hence f = <p
and the result follows.
Note. In the notes at the end of this chapter we will indicate how one can
prove (1) in the above theorem with Uf -^{Nq\G',x)f^ uni°n
over all finite subsets of k.
15.93. In Section 15.11, we will give some applications of this transform.
We conclude this section by showing how the Plancherel theorem for
generic x implies a Plancherel theorem for arbitrary x-
Let x be a unitary (one dimensional) character of Nq . Set
^ = {аеД(Ро,Ло)|^|ва*О).
We write for ((Р0)^,(Л0\) (2.2.6,2.2.7). Then, Px = °MXAXNX.
We set *NX = N0C\ °MX. Then, X\*n is a generic character relative to
°MX. We set *PX = Poa°Mx, *AX=AOQ °Mx. Then, (*PX,*AX) is a
minimal (standard) p-pair for °MX. If (Q,Aq) > (*PX,*AX), then we set
(Q, A&) = (QAXNX, AqAx). Then, (Q, A@) is a standard p-pair domi-
nated by (Px, Ax). If ш e <f2(°MQ), then we set, for v e (a£)c, pxGt>, v)
= g0e(<u,^|ao). We set
iix(a>tv) =|n'(°Mf ,Л0)| yQc^px(<o,v).
15.9. The Plancherel Theorem for L2(N0 \ G; x)
In this case, we have:
Theorem. (R, L2(N0 \ G; хУ) is unitarily equivalent with the direct integral
(Q,Ae^(*Px*Ax)a,^^M0)J^	1 °П"е
Note. In the proof of this result we will give an explicit isomorphism in
terms of the Whittaker transform for °MX.
We now begin the proof. If f g -#(N0 \ G; x\ then we set (px = pP)
= a px t f(mak)a ,vda
* A
for m g “M*., v g a*, and к g K. Then, f(v,k) g -£(*Nx \°Mx;
On -£(*NX \°MX;	) we put the usual L2 inner product. If f\,f2 e
tf(NQ\G; хУ then
</iJ2> = / fi(s)f2(g)dg = f f{fl(v,k),f2(2'^)')dkdv.
JN0\G	Ja*JK
Let Жм be the space “X” as in the preceding theorem for °MX and
XyN . We set W equal to the space of all / g -£(Nq \ G; хУ f right
^-finite, and such that f(v, k) g for all v g a* and к g K. Then
(as before), W is a dense subspace of L2(N0\G;^) contained in
Tf(N0\G, x). We define, for (Q, Aq) > (*PX,*AX),

Then, ш р(/) g Iq ш ® Whx^nN{H“). A direct unraveling of the def-
initions implies that if we set

then we have a (well defined) map of span{/?(g)^r} into ®
Wh
X\M0nN0''	<»
If we apply the preceding theorem (to Mx), then we see that
Ф	®	(w^taitVp(a>,iv)dv
(q,aq)>(*px,*ax) iu^^>2(0Mq)/W(Mx,Aq)
426
15. The Whittaker Plancherel Theorem
extends to a unitary intertwining operator onto the direct integral
Ф	® f wh^N(H2)^,iV)dv.
(Q,Aq)>(*Px,*Ax)	ай
This implies the theorem.
15.10. Some examples of the Plancherel theorem for generic x
15.10.1.	Let (Po, Ло) be a minimal (standard) parabolic subgroup of G
with Po = M0N0, as usual. We say that G is quasi-split if Mo is a Cartan
subgroup of G. For the first class of examples in this section, we will
assume that Mo is abelian (for example, if G is linear, of inner type, and
quasi-split). Then we have:
Proposition. Assume that Mo is abelian. Let be a generic unitary
character ofN0. Let (P, A) be a standardp-pair dominating (P0,A0). Let
(a, Ha) be an irreducible, admissible Hilbert representation of °MP. Then
dimWTi (Я”)<1.
Let Q = ®MP П Po and AQ = °MP П Ao. Then (Q, Aq) is a minimal
standard p-pair for °MP. As in the beginning of the proof of Theorem
15.6.7, there exists g e (ctg)c, (£, H() an irreducible unitary representa-
tion of °Mq = °M0, and a continuous surjective intertwining operator S
from Iq £ „ onto H“. Thus, ST defines an injection of Why (H“) into
WTi .,(^5 t A- Since Ma is abelian, dim = 1. Thus, Theorem 15.6.7
implies that
dimUTi (I%( ) £ 1.
Х\№()Г\Мр\
The proposition now follows.
15.10.2.	The preceding result implies that for the class of groups covered
the multiplicity function in Theorem 15.9.1 takes values in the set {0,1}.
Our next class of examples involves general G. We fix (P, A) a minimal
standard p-pair. Let x be a generic character of N. We set if(N \G/K;x)
equal to the space of all f e -£(N \ G; x) such that f(gk) = /(g) for all
g e G, к e K. Then, \ G/K; x) is a closed subspace of if(N \ G; x)-
We endow it with the subspace topology. We define a map Ф from cZ'(a)
15.10. Some Examples of the Plancherel Theorem for Generic x	427
to -£(N\G/K;x) as follows:
У(<р)(пак) = apx(n)<p(\oga).
Lemma. Ф defines a topological isomorphism from c/’Ca) onto
^(N\G/K;x)-
This is a simple matter to prove and will be left to the reader.
15.103.	Let (£0,C) be the trivial one dimensional representation of °M.
Let Л e C* be defined by A(l) = 1. We set /fo>p(A) = J„. We will also
write тгр f „ = тг„ and take the total space for the Hilbert representation
ttv to be H = L2(°M\K). Theorems 15.9.1 and 15.9.2 take the following
form in this context (the material in 13.8 is also necessary and we will use
the notation therein). If a e C™(a*), then we set
T(a)(a) =|И'(Л)Г1с^1у4 f a(v)J_iv(TT_iv(a)l)
Ja*	C(lv)C(-lv)
Then,
«(p) = [ a~2pT(a)(a)Jiv(Triv(a))dv.
JA
We set K„(a) = /р(тгр(а)1)а_₽. We set
(i) ^(«)(«) -1и'(^)Г1^-Х)/а«(-)к_,.(»)с(|[|)^_([|).
Then,
(2) a(v) = f ^(a)(a)Kiv(a)da.
JA
This pair of formulas (in reverse order) define a transform and an
inversion formula for L2(a) with a an / dimensional inner product space
over R. In the next example, we will see that these formulas give general-
izations of the Lebedev transform and its inversion formula.
15.10.4. We will now look at the case when G = SL(2, R). We will use
the notation in Section 5.6. Then g = {x e Af2(R)| tr x = 0}. We choose В
to be given by the formula B(x, y) = tr xy. We observe that if
Л 0 1Г a p'
0 A-1] -p a
a bl _ 1 0
cd x 1
428
15. The Whittaker Plancherel Theorem
with A > 0, a2 + p2 = 1, then
(1)	A = (a2 + b2)1/2
and
(2)	a = a/(a2 + b2)1/2, fi = b/(a2 + b2)l/2.
We look upon а£ as C under the identification of v with v(H). Then
p = 1. If we use (1), we see that
, .	,.-l/2-p/2 ,
c(v) = J (1 + x2)	dx.
A direct calculation yields
Г ( v/2)
(3)	c(v) = Bfv/2,1/2) = 7T1/2	;	'	
v	y v ’	Г(у/2 + 1/2)
We note that yA = c(l) and hence
(4)	yA = tt.
Also, cA = 2tt (by the usual Fourier inversion formula).
For the sake of simplicity, we assume that dxiX) = z. If a = exp tH,
then
4(^(a)l) = f eixa(nxa)~v~p dx.
* — 00
Here,
1 X
nx = exp xX = q .
Thus,
4(тгр(й)1) = У е‘х(е2‘ + e~2lx2) v/2 1/2 dx
= J” e,JC(l + e-4,x2)~v/2~l/2 dx
J — 00
15.10. Some Examples of the Plancherel Theorem for Generic x
429
Here, Kv is MacDonald’s Bessel function of imaginary argument and the
last equation follows from Watson [1], 6.3, p. 185. We therefore have
2^+l^j.l/2
= Г((р + l)/2) Kv^e2'^
We note that we are using the normalization da = dA. Thus, da = 41 dt.
Also, with our identification of a* with R, we are using the measure
(1)/ /2 dv. 15.10.3 (1) now says
,<»	Г((1у + l)/2)
15.10.3 (2) says
2‘'" + 17Г1/2	,00
«(«0 =	^(a)(a)Kiv/2(e21) dt.
1 (H + I)/2)
We set 0(v) = Г((г> + l)/2)2““'+1/2a(^) and
. ,0»	dv
Then,
P(2v) -ir1'2[° ^)(e‘)Kiv(e‘)dt.
J — 00
We note that
тг
-------------- = -zsirnrz.
Г(г)Г(-г)
If we set <p{v) = tt~1/2/3(.2v) and
(5)	_Z(<p)(x) = тг 2j (p(v)K_iv(x)v sinh irv dv,
then
,oo	dx
(6)	<p(v) = J ^(<p)(x)K!y(x)-.
•'0	X
Since Kv = K_v, we see that these formulas hold if <p is an arbi-
trary compactly supported, smooth even function of v. The pair of formu-
las (in the opposite order) correspond to the Lebedev inversion formula
(Lebedev [1]).
430
15. The Whittaker Plancherel Theorem
15.10.5.	We continue with the example G = SL(2, R) and sketch how we
can apply the arguments in 15.10.3 to the space L2(N\G; x)T = {f e
L2(N\ G; x)|f(gk) = f(.g)T(k), geG, к e K}, where r is a unitary
character of K. We define rm by
Tm
COS в
-sin в
sin 9
cos 0
= eima
We look upon rm as both a unitary character of К and an element
of Ia with <r(-f) = (-l)m (here, °M = {±/}). In this case, one has (с/.
Goodman-Wallach [1]) that if a = exp tH, then

(1)
ттгг-^-1)/2
I V - m + 1 \	-m/2,v Ae )•
—
Also, the c-function in this case is
Using this, we can derive an explicit transform using the family iv
in place of the Kiv. There is, however, an interesting difference between
this case and the case of the trivial /С-type. If |m| > 1, then the discrete
series will come into the formula. Hence, for say m = 4, there will be two
contributions from the discrete series. The reader should be able to carry
out the analogous (but complicated) calculations in this case using the
material in Whittaker-Watson [1] on Whittaker functions.
15.11. Notes and further results
15.11.1.	The material in Section 15.2 is based on the asymptotic expan-
sions of generalized matrix coefficients of Wallach [3]. The notion of
“tame” functional appears (implicitly) in those notes. Theorems 15.2.4 and
15.2.5 appear here for the first time.
15.11.2.	The material in 15.3 was no doubt known to Harish-Chandra in
some form since it is completely analogous to results on the Schwartz
15.11. Notes and Further Results
431
space and cusp forms for G. In this analogy, Theorem 15.3.3 corresponds
to Theorem 7.2.2 and is one of the keys to Harish-Chandra’s “philosophy
of cusp forms.” The converse theorem analogous to Theorem 7.7.6 in-
volves different ideas and doesn’t appear until Section 15.7.
15.11.3.	The integral 15.4.1 (1) was first considered by Jacquet [2] for
semi-simple groups over C or split over R and for minimal parabolic
subgroups. Jacquet proved the meromorphic continuation in this context
for К-finite vectors. Some form of Theorem 15.4.1 must have been known
to Harish-Chandra. Theorem 15.6.7, which has Theorem 15.4.1 as a
corollary, is in turn a special case of a more general result that we will now
describe.
Let (P, A) be a standard p-pair and let (<r, Ha) be an irreducible,
admissible, Hilbert representation of °M. Let x be a unitary (one dimen-
sional) character of N. We assume that dx satisfies the conditions in
15.5.2. If A e (H“X, then we set
•U(A)(/) = j X(Pfv(n))x(n)~l dn.
JN
Then, one can show that there exists cA such that if Re(is a) < cA,
a e Ф(Р, A), then the integral converges absolutely and defines a weakly
holomorphic (in v) family of continuous functionals on /“ for v in this
range. One can use the material in 15.5 and a strengthening of the method
in 15.6.8 to prove:
Theorem, v Ja v(.X) has a holomorphic continuation to for every
A e (H“X. Furthermore, if v e a£, then Ja v defines a linear bijection
between (Н“У and
(A e(	J«)/) = x(n)A(f), f e Pp^v, n^N}.
15.11.4.	We use the notation of Section 15.4. If x is a generic character
of No and if V = IPo,^v with £ an irreducible representation of °M0, then
one has:
Theorem. (Kostant [4], Lynch [1]).
dim{A e L*|A(Xr) = dx(X) А(г), X e n0, v e L)
= |!T(/l0)|dimHf.
432
15. The Whittaker Plancherel Theorem
15.11.5.	The material in Section 15.5 is taken from Wallach [5]. The
corollary to Theorem 15.5.7 that asserts that dimfL ® F\ = dim dim F
is due to Lynch [1]. The method in 15.6 is also taken from Wallach [5]. The
material in 15.6.8 follows the line of standard Bruhat theory. A detailed
discussion of this theory can be found in Warner [1], 5.2.
15.11.6.	We suspect that our approach to Theorems 15.7.1 and 15.8.8 is
different from that of Harish-Chandra. It is to be hoped that Harish-
Chandra’s method will eventually appear in print. We will now sketch an
alternative approach to the proof of Theorem 15.8.8 (and hence of the
Plancherel theorem for this case) that is probably more like Harish-
Chandra’s argument.
(1)	Let f e srfw(N0 \ G; x) (15.7.3) and set
Vf = spanR(U(Qc))R(K)f.
Then, Vf is a tempered (g, АЭ-module.
This result is difficult and its proof makes serious use of the asymptotic
results in Section 15.2.
Let (тг, Я) be a tempered representation of G and let Л e Whx(H“).
Let {P, A) > (Po, /l0) and let v e HK. Then Theorem 15.2.2 implies (in
the notation therein) that if H ea+, then
Л(тг(ехр( -гЯ))тг(?п)г)
~ Y, e~‘^Hy £ е”'е(Н)Рл,м,е(^;"’;1;)
Q^l^p
as t +oo. We set
<pP(am-, X,v) = a~p У, aM+fpAiM f(log a; m; r).
p^E(.P,HK)
feL?
Reg+£=p
As in 12.3.2,12.3.3 we can now derive a theory of f -> fp for f s
Let(Q, Aq) > (P0,/l0).Let(7r,H) = (ir2 <7>/p, №’<7’,”),with(<r, HJan
irreducible square integrable representation of °MQ. If Л e Whx^nM{H“),
15.11. Notes and Further Results
433
then we set
^F|e(iv,nw;A,u) = <pP(ma;Jaiv(X),v).
One can prove by the same method as in 12.4.1 that:
(2)	(pp^tiv, та; Л, r) is real analytic in v.
The next steps are completely analogous to the methods used in the
proof of Harish-Chandra’s Plancherel theorem. One proves the analog of
the main inequality (Theorem 12.6.19). This then implies that we can form
the wave packets in 15.8.8 for a g ^Z(a*) (rather than C“(a*)). Now, one
has clear sailing through Harish-Chandra’s method (the Ja iv replacing the
Eisenstein integral) and one finds that in Theorem 15.9.2, the formula
extends the space of all К-finite elements of if(N0 \ G; %)
15.11.7.	In this number, we will sketch how the material of 15.10.3 gives
a solution to the quantized “non-periodic Toda lattice”. We consider the
case when G is split over R and (for simplicity) gc is simple. Let x be a
generic unitary character of No. We note that the map U from C“(A0) to
C°°(N0 \ G/K) given by UtfXnak) = x(n)f(a') defines a linear bijection.
If a g Ф(Р0, Ло), if X e 0a, and if / e C°°(A0), then
R(X)f(a) = dX(X) a~“f(a).
We choose an orthonormal basis Hx,..., Ht of a0 and thereby identify a0
and Ao with Rz. Under this identification, a g Ф(Ро,Ло) becomes a
linear functional on Rz. We choose non-zero elements Xa g ga such that
B(Xa,0Xa) = -1. Then {ХЛеФ(ро,ло) is a basis of n0. We note that if
X g [n0,n0], then dx(X) = 0.
P-B-W (0.4.1) implies that the map
(7(n0 ® a0) ® U(l) - 17(g)
given by j ® a •-» sa is a linear bijection. Thus,
(7(g) = (7(n0 ® a) © (7(g)!.
Let p be the projection of (7(g) onto (7(n0 ® a) corresponding to this
direct sum decomposition. If x g (7(g) and if f g C°°(No\G/K; x), then
R(x)f = R(p(x))f.
[n0, n0] is an ideal in ё = a0 ® n0. We thus have a suijective Lie algebra
homomorphism q of ё onto ё/[п0, n0] = a0 ® n0/[n0, n0]. We set u =
434
15. The Whittaker Plancherel Theorem
n0/[n0, n0], Let Д(Р0, Ao) = {«j,..., at}. We set Xt equal to the image
of Xa, in u. We define an automorphism 17 of U(a0 ® u) by т^Х^ = Xt
and rj(H) = H + p(/7)l for H e a0. We set
-y(x) = 7]°q°p(x).
We write dx(Xa) = icj. We define a representation Tx of a0 ® u on
C“U0) by
Тх(Х^а) = icja~af,
Tx(H)f = R(H)f = Hf.
If x e U(q), then
R(x)f(a) = a»(Tx(y(x))(a~»flAo))(a)
for a e Ao and f e C°(N0 \ G; x)-
As in 3.6.4,3.5.7, у defines a homomorphism of U(qc)k into
G((a0)c ® uc). Thus, у induces a homomorphism of G(gc)K/(G(gc)Ar A
G(gc)t) into G((a0)c ® uc). Let ф denote the homomorphism of a0 ® u
onto a0 gotten by identifying (a0 ® u)/u with the abelian Lie algebra a0.
Then, ф ° у = y0 the usual Harish-Chandra homomorphism (3.6.4). Thus,
у induces an isomorphism of the abelian algebra G(gc)K/(G(gc)K A
G(gc)t) into I7((ao)c ® uc). Let up..., u{ be a set of basic generators for
G(a0)^ (W = W(A0)). We assume that «j = EyHj2. We set D; =
Тх(у(у^*(«,-))). Then, with our identification of Ao with Rz, Я; = d/dxt
and
d2
Dl= ^d^j+2^e~2aj-
This operator is usually called the quantized non-periodic Toda lattice. We
note that the operators Dx,D, commute with D1. We also note that
(notation as in 15.2.3)
D7K1V=(^)(u;)K(.p,
and in particular that
(^)(«i)-----(p»p)-
Thus, 15.10.3 (1),(2) gives a joint spectral decomposition of Dj,...,Dz.
This is the “solution” to the quantized non-periodic Toda lattice.
For more details related to this discussion, we suggest the reference
Goodman-Wallach [2].
15А. Appendix to Chapter 15
435
15.11.8.	In Miatello-Wallach [2,3], the pair of displayed formulas in
15.10.3 for G having R-rank 1 (or a product of such groups) was derived
from the Lebedev inversion formula 15.10.4 (5), (6). In [3], we also showed
how one can use this inversion formula as a first step in deriving a
generalization of Kuznetsov’s trace formula for cusp forms on generalized
Hilbert-Blumenthal groups.
15.A. Appendix to Chapter 15
15.A.1. In this appendix, we will prove a result that will be used in the
holomorphic continuation of the Jacquet integral. It involves the calcula-
tion of the minimal polynomial of a certain element of the group algebra
of the symmetric group. Let Sn denote the group of permutations of
n letters. Let denote the cycle (1,..., z) (i.e., o\ = I, <т;(1) = 2,...,
oj(z - 1) = z, o;(z) = 1, and oj(j) = j for j > i). Let CSn denote the group
algebra of Sn over C. We set
у = E°; e
i=i
The main result of this appendix is:
Proposition, (y - n)n"J02(y _ 0 = 0.
The proof uses several intermediate lemmas.
15.A.2. Lemma. If и e Sn and if и = atat • • •	= o’jO'jr_l '' ‘
with ip p and jp > p, then ip = jp for p = 1,..., r.
We first prove by induction on r that uix = uj\ = r. Indeed, if r = 1 this
is clear since ap = 1. Assume this is true for r - 1. Then, o;-	• • •
= r - 1. Since ir > r, a^r - 1) = r. This proves the assertion. We now
prove the lemma. uix = uj\ = r and hence z\ = j\. Thus, at- • • • ai2 =
ajr • • • aj2. This implies that z2 = j2 > etc-
15.A.3. Lemma. If и e Sn, then и can be written in one and only one
way as
with ij j.
436
15. The Whittaker Plancherel Theorem
The uniqueness follows from Lemma 15.A.2. Also, that result implies
that there are n! distinct elements of the form и = <rin_<rin_2 with
ij >j. This implies the lemma.
15.A.4. We set yk = and g = LalESn<r.
Lemma, (y - y„_2) • • • (y - y2Xy - Dy = g.
This is an immediate application of the previous lemma.
15.A.5. Lemma. Let и =	with ij >j. If t <r, then
atu = aJr ••• ah, jp>p.
We leave it to the reader to check (by direct calculation) that:
(i)	OfO-j =	for 2 < i < j.
(ii)	a2apaj = vJ+1ap for 1 <j <p - 1.
We first prove the lemma in the special case when t = 2. Suppose that
ir > ir_x. Then (ii) implies that
°2M =	•••
This implies the assertion of the lemma in this case. If ir = zr-i = P> then
(ii) implies that (r2(rPa’P-i = ap • So:
(iii)	<T2<rp2 = <rp<rp_x.
Since p > r, the result follows in this case. We now assume that
ir < ir_x. If we apply (i), we have
We now calculate a2u using this and (iii):
-i<VX ‘ ‘ ‘ ai,
= ••• %
=	••• 4t-
This completes the proof in the special case when t = 2. We now prove
the result by induction on t. It is clear if t = 1 and has just been proved
15.А. Appendix to Chapter 15
437
for t = 2. Assume the result for t - 1 > 2. We will now prove it for t. We
note that our hypothesis says that t < r < ir. We first assume that t <ir.
Then, (i) implies that
= ^<-1^-1^., •••
The inductive hypothesis implies that
<r,u =	• • • <rh ,
with jp p. We now apply (i) with j = 2 to find that
<r,u = <r2a/ro;r , • • •	.
The result in this case now follows from the result for t = 2. We are left
with the case t = ir. Hence, t = r = ir. Thus, (ii) implies that
<r,u = <тг\ t • • • <rlt = а2аг(аг_га1г t • • • <r(1).
We apply the inductive hypothesis to the expression in the parentheses to
see that
<r,u = a2araJr t • • • aJt, jk > k.
This part of the inductive step now follows from the case t = 2.
15.A.6. We are now ready to prove the proposition. Lemma 15.A.5
implies that
yj(y - yj-1) •••(?- l)y = j(y - У/-1) • •• (y - l)y-
Suppose that we have shown that
(y - y,-i) • • • (y - l)y = (y - (/ - 1)) • • • (y - i)y-
Then,
y(y - У/-1) • •• (y - l)y
= (у - yj)(y - У/-1) ••• (y - l)y +j(y - У/-1) • •• (y - l)y-
438
15. The Whittaker Plancherel Theorem
We therefore see (by the obvious induction) that
(у - yj • • • (у - l)y = (y - J) • • • (y - i)y
for j = 2,..., n - 2. If we apply Lemma 15.A.4, we therefore see that
n-2
П (y -j) =
J = 0
Since -yg = ng, the proposition now follows.
15.A.7. Let, for i = 1,..., n, vt = (z,..., n). Set r =	+ • • • + vn.
Corollary, (т - n) П-=02(т - z) = 0.
Set s0 equal to the element of S„ such that soi = n + 1 - z. Then,
5оУ5о = T- Hence,
(т - n) П (T “ 0 = 5о(у - «) П (У “ Фо = °-
i=0	Z-0
Bibliography
Arthur, J.
[1] “Intertwining integrals for cuspidal parabolic groups,” preprint (1974)
Atiyah, M. F., and MacDonald, I. G.
[1] Introduction to commutative algebra. Addison-Wesley, Reading, Mas-
sachusetts, 1969.
Bargman, V.
[1] “Irreducible unitary representations of the Lorentz group,” Ann. of Math.,
48(1947), 568-640.
Beilinson, A., and Bernstein, J.
[1] “A generalization of Casselman’s submodule theorem,” Progress in Math.,
40, 35-52. Birkhauser, Boston, 1983.
Bernstein, J.
[1] “Modules over the rings of differential operators,” Fund. Anal, and Appl.,
52(1971), 1-16.
[2] Preprint, 1989.
Bernstein, J., Gelfand, I. M., and Gelfand, S. I.
[1] “Category of 0-modules,” Funckcional Anal. i. Prolozen., 10, Nr. 2, (1976),
1-8. English transl.; Functional Anal. Appl., 10(1976), 87-92.
[2] “Models of representations of compact Lie groups,” Funk. Anal. Pril.,
9(1975), 61-62.
439
440
Bibliography
Borel, A.
[1] “Stable real cohomology of arithmetic groups,” Ann. Sci. de Г Ecole Norm.
Sup., 4e serie, 7(1974), 235-272.
[2] Representations de groupes localement compacts. Lecture Notes in Math., 276.
Springer-Verlag, New York, 1972.
Borel, A., and de Siebenthal, J.
[1] Les sous-groupes fermes de rang maximum des groupes de Lie clos,”
Comment. Math. Helu., 23(1949), 200-221.
Borel, A., and Garland, H.
[1] “Laplacian and the discrete spectrum of an arithmetic group,” Amer. J.
Math., 105(1983), 309-335.
Borel, A., and Harish-Chandra.
[1] “Arithmetic subgroups of algebraic groups,” Ann. of Math., 75(1962),
485-535.
Borel, A., and Wallach, N.
[1] Continuous cohomology, discrete subgroups, and representations of reductive
groups, Annals of Math. Studies, Study 94. Princeton University Press,
Princeton, 1980.
Bott, R.
[1] “Homogeneous vector bundles,” Ann. of Math., 66(1957), 203-248.
Bourbaki, N.
[1] “Integration,” Chapitre 6, Elements de Mathematique. Hermann, Paris, 1959.
[2] Groupes et algebres de Lie, Chapitres 4, 5, et 6. Hermann, Paris, 1968.
[3] Groupes et algebras de Lie, Chapitres 4, 5, et 6, Elements de Mathematiques
XXXIV. Hermann, Paris, 1968.
Bruhat, F.
[1] “Sur les representations inuites des groupes de Lie,” Bull. Soc. Math.
France, 84(1956), 97-205.
Carmona, J.
[1] Sur le classification des modules admissible irreducibles, Lecture Notes in
Math., 1020, 11-34. Springer-Verlag, New York, 1983.
Cartan, E.
[1] The theory of spinors. M.I.T. Press, Cambridge, 1967.
Cartan, H., and Eilenberg, S.
[1] Homological algebra. Princeton University Press, Princeton, 1956.
Carter, R. W.
[1] Simple groups of Lie type. John Wiley and Sons, New York, 1972.
Casselman, W.
[1] “Canonical extensions of Harish-Chandra modules,” Can. J. Math., 41(1989),
315-438.
Casselman, W., and MiLidic, D.
[1] “Asymptotic behaviour of matrix coefficients of admissible representations,”
Duke Math. J., 49(1982), 869-930.
Bibliography
441
Casselman, W., and Osborne, M. S.
[1] “The п-cohomology of representations with infinitesimal character,” Comp.
Math., 31(1975), 219-227.
Сни, C.-B.
[1] Ph.D. Thesis, Yale University, New Haven, 1989.
Cohn, L.
[1] Analytic theory of Harish-Chandra's c-function, Lecture Notes in Math., Vol.
428. Springer-Verlag, 1974.
Deligne, P.
[1] Equations differentialies a points singuliers reguliers, Lecture Notes in Math.,
163. Springer-Verlag, Berlin, 1970.
De Rham, G.
[1] “Solution elementaire d’opeateurs differentiels du second ordre,” Ann. de
I'lnst. Fourier, 8(1958), 337-366.
Cowling, M., Haagerup, U., and Howe, R.
[1] “Almost L2 matrix coefficients,” J. Reine, Angew. Math. 387(1988), 97-110.
Dixmier, J.
[1] Les C*-algebres et leurs representations. Gauthier-Villars, Paris, 1969.
[2] Enveloping algebras. North-Holland Publishing Co., Amsterdam, 1977.
[3] Les algebras d’operateurs dans I’espace hilbertien. Gauthier-Villars, Paris,
1969.
[4] Von Neumann algebras. North-Holland, Amsterdam, 1981.
Dixmier, J., and Malliavin, P.
[1] “Factorisations de fonctions et de vecteurs indefiniment differentiables,”
Bull. Sci. Math., 102(1978), 307-330.
du Cloux, F.
[1] “Representations temperees des groupes de Lie nilpotents,” J. Funct. Anal.,
85(1989), 420-457.
[2] “Sur les representations differentiables des groupes de Lie algebriques,”
Ann. Sci. Ec. Norm. Sup., to appear.
Duflo, M., and Vergne, M.
[1] “La formule de Plancherel des groupes de Lie semi-simples reels,” Adv.
Stud. Pure Math., 14(1986), 289-336.
Dunford, N., and Schwartz, J. T.
[1] Linear operators, Part 1: General theory. Interscience Publishers, Inc., New
York, 1958.
Dym, H. and McKean, H. P.
[1] Fourier series and integrals. Academic Press, New York, 1972.
Enright, T. J.
[1] “Relative Lie algebra cohomology and unitary representations of complex
Lie groups,” Duke Math. J., 46(1979), 513-525.
442
Bibliography
Enright, T. J., and Wallach, N. R.
[1]	“The fundamental series of representations of a real semi-simple Lie alge-
bra,” Acta Math., 140(1978), 1-32.
[2]	“Notes on homological algebra and representations of Lie algebras,” Duke
Math. J., 47(1980), 1-15.
[3]	“The fundamental series of a semi-simple Lie group,” preprint (1976).
Fell, J. M. G., and Doran, R. S.
[1] Representations of*-algebras, locally compact groups, and Banach *-Algebraic
Bundles. Academic Press, Boston, 1988.
Gangoli, R.
[1] “On the Plancherel theorem and Paley-Wiener theorem for spherical func-
tions on semi-simple Lie groups,” Ann. of Math., 93(1971), 150-165.
Garland, H„ and Lepowsky, J.
[1] “Lie algebra homology and the Macdonald-Kac formulas,” Inventories Math.,
34(1976), 37-76.
Gelfand, I. M.
[1] “Normlerte Rlnge,” Mat. Sb., 9(1941), 3-24.
Gelfand, I. M„ and Naimark, M. A.
[1] “Unitary representations of the Lorentz group,” Izvestia Akad. Nauk. SSSR,
Ser. Math. 11(1947), 411-504.
[2] “On the imbedding of normed rings into the ring of operators on a Hilbert
space,” Mat. Sb., 12(1943), 197-213.
Gelfand, I. M., and Shilov, G. E.
[1] Generalized functions, Volume 1. Academic Press, New York, 1964
Gelfand, I. M„ and Kirillov, A. A.
[1] “Sur les corps lies aux algebres enveloppantes des algebres de Lie,” Publ.
Inst. Hautes Etudes Sci., 31(1966), 5-19.
Gindikin, S., and Karpelevic, F.
[1] “Plancherel measure of Riemannian symmetric spaces of non-positive curva-
ture,” Soviet Math. Dokl., 3(1962), 962-965.
Goodman, R., and Wallach, N. R.
[1] “Whittaker vectors and conical vectors,” J. Func. Anal., 39(1980), 199-279.
[2] “Classical and quantum mechanical systems of Toda Lattice type I,” Comm.
Math. Phys. 83(1982), 355-386.
Harish-Chandra
[1]	“Representations of semisimple Lie groups, I.” Trans. Amer. Math. Soc.,
75(1953), 185-243.
[2]	“Representations of semisimple Lie groups, II,” Trans. Amer. Math. Soc.,
76(1954), 26-54.
[3]	“Representations of semisimple Lie groups, III,” Trans. Amer. Math. Soc.,
76(1954), 234-253.
[4]	“On a lemma of F. Bruhat,” J. Math. Pures Appl., 35(1956), 203-210.
Bibliography
443
[5]	“The characters of semisimple Lie groups,” Trans. Amer. Math. Soc.,
83(1956), 98-163.
[6]	“Differential operators on a semisimple Lie algebra,” Amer. J. Math.
79(1957), 87-120.
[7]	“Fourier transforms on a semisimple Lie algebra, I,” Amer. J. Math.,
79(1957), 193-257.
[8]	“Spherical functions on a semisimple Lie group, I,” Amer. J. Math., 80(1958),
241-310.
[9]	“Spherical functions on a semisimple Lie group, II,” Amer. J. Math.,
80(1958), 553-613.
[10]	“Invariant eigendistributions on a semisimple Lie algebra,” Inst. Hautes
Etudes Sci., Publ. Math. No. 27(1965), 5-54.
[11]	“Invariant eigendistributions on a semisimple Lie group,” Trans. Amer.
Math. Soc., 119(1965), 457-508.
[12]	“Discrete series for semisimple Lie groups, I,” Acta Math. 113(1965),
241-318.
[13]	“Discrete series for semisimple Lie groups, П” Acta Math., 116(1966),
1-111.
[14]	“Harmonic analysis on real reductive groups, I,” J. Func. Anal., 19(1975),
104-204.
[15]	“Harmonic analysis on real reductive groups, II,” Inuentiones Math., 36(1976),
1-55.
[16]	“Harmonic analysis on real reductive groups, III,” Ann. of Math., 104(1976),
117-201.
[17]	“Harmonic analysis on reductive p-adic groups,” Proc, of Symp. in Pure
Math., Vol. XXVI, Amer. Math. Soc., Providence, 167-192.
[18]	“Supertempered distributions on real reductive groups,” Studies in Applied
Mathematics, Advances in Mathematics Supplementary Studies, Vol. 8,
139-153, 1983.
Hecht, H. and Schmid, W.
[1] “A proof of Blattner’s conjecture,” Inuentiones Math., 31(1975), 129-154.
[2] “Characters, asymptotics and n = cohomology of Harish-Chandra modules,”
Acta Math. 151(1983), 49-151.
Helgason, S.
[1]	Differential geometry, Lie groups, and symmetric spaces. Academic Press, New
York, 1978.
[2]	Groups and geometric analysis. Academic Press, Orlando, 1984.
[3]	“A duality for symmetric spaces with applications to group representations,”
Aduan. Math. 5(1970), 1-154.
Herb, R.A., and Wolf, J. A.
[1] “The Plancherel theorem for general semisimple groups,” Comp. Math.,
57(1986), 271-355.
444
Bibliography
Hirai, T.
[1] “The characters of some induced representations of semi-simple Lie groups,”
J. Math. Kyoto Univ., 8(1968), 313-363.
Hocking, J. G., and Young, G. S.
[1] Topology. Addison-Wesley, Reading, Massachusetts, 1961.
HoRMANDER, L.
[1] Linear differential operators. Springer-Verlag, New York, 1964.
[2] An introduction to complex analysis in several variables. Van Nostrand,
Princeton, 1967.
Howe, R.
[1] “On the connection between nil potent groups and orbital integrals associ-
ated to singularities,” Pacific J. Math., 75(1977), 329-363.
Hunt, G. A.
[1] “A theorem of E. Cartan,” Proc. Amer. Math. Soc., 7(1956), 307-308.
Jacobson, N.
[1] Lie algebras. Interscience Publishers, Inc., New York, 1962.
Jacquet, H.
[1] Representations des groupes lineares p-adics. Theory of group representations
and Fourier analysis (Proceedings of a conference at Montecatini, 1970),
C.I.M.E., Edizioni Cremonese, Rome, 1971, 119-220.
[2] “Fonctions de Whittaker associees aux groupes de Chevalley,” Bull. Soc.
Math. France, 95(1967), 243-309.
Jantzen, J. C.
[1] Moduln mit einem hochsten Gewicht, Lecture Notes in Math. 750. Springer-
Verlag, Berlin, 1979.
[2] “Kontravariante formen auf induzierten dalsteHungen halbeinfacher Lie-
Algebren,” Math. Ann., 226(1977), 53-65.
Kadison, R. V., and Ringrose, J. R.
[1] Fundamentals of the theory of operator algebras, I. Academic Press, New
York, 1983.
Knapp, A. W.
[1] Representation theory of semisimple groups. An overview based on examples.
Princeton University Press, Princeton, 1986.
[2] “Commutivity of intertwining operators for semisimple groups,” Comp.
Math., 46(1982), 1-38.
Knapp, A. W., and Stein, E. M.
[1] “Intertwining operators for semisimple groups,” Ann. of Math., 93(1971),
489-578.
[2] “Intertwining operators for semisimple groups,” Inventiones Math., 60(1980),
9-84.
Knapp, A., and Zuckerman, G. J.
[1]	“Classification of irreducible tempered representations of semisimple
groups,” Ann. of Math., 116(1982), 389-501.
Bibliography
445
KOSTANT, В.
[1]	“The principal three-dimensional subgroup and the Betti numbers of a
complex simple Lie group,” Amer. J. Math., 81(1959), 973-1032.
[2]	“Lie algebra cohomology and the generalized Borel-Weil theorem,” Ann. of
Math., 74(1961), 329-387.
[3]	“On the tensor product of a finite dimensional and an infinite dimensional
representation,” J. Func. Anal., 20(1975), 257-285.
[4]	“On Whittaker vectors and representation theory,” Inventiones Math.,
48(1978), 101-184.
[5]	“On the existence and irreducibility of certain series of representations,”
Bull. Amer. Math. Soc., 75(1969), 627-642.
Kostant, B., and Rallis, S.
[1]	“Orbits and Lie group representations associated to symmetric spaces,”
Amer. J. Math., 93(1971), 753-809.
KuMARESAN, S.
[1]	“On the canonical f-types in the irreducible unitary д-modules with non-zero
relative cohomology,” Inventiones Math., 59(1980), 1-11.
Kunze, R., and Stein, E. M.
[1]	“Uniformly bounded representations and harmonic analysis of the 2 X 2 real
unimodular group,” Amer. J. Math., 82(1960), 1-62.
[2]	“Uniformly bounded representations III. Intertwining operators for the
principal series on semisimple groups,” Amer. J. Math., 89(1967), 385-442.
[3]	“Uniformly bounded representations, II,” Amer. J. Math. 83(1961), 723-786.
Lang, S.
[1] Analysis II. Addison-Wesley, Reading, Massachusetts, 1969.
Langlands, R.
[1] On the classification of irreducible representations of real algebraic groups
(preprint). Institute for Advanced Study.
Lebedev, N. N.
[1] “Sur une formule d’inversion,” Dokl. Akad. Nauk. SSSR, 52(1946), 665.
Lepowsky, J.
[1] “Algebraic results on representations of semisimple Lie groups,” Trans.
Amer. Math. Soc., 176(1973), 1-44.
[2] “On the Harish-Chandra homomorphism,” Trans. Amer. Math. Soc.,
208(1975), 193-218.
Lepowsky, J., and McCollum, G. W.
[1] “On the determination of irreducible modules by restriction to a subalgebra,”
Trans. Amer. Math. Soc. 176(1973), 43-57.
Lipsman, R.
[1] “The dual topology for the principal and discrete series on semi-simple Lie
groups,” Trans. Amer. Math. Soc., 152(1970), 399-417.
446
Bibliography
Lynch, E.
[1] “Generalized Whittaker vectors and representation theory, Ph.D. Thesis,
1979.
Mautner, F. I.
[1] “Unitary representations of locally compact groups,” Ann. of Math., 51(1950),
1-23.
MacLane, S.
[1] Homology. Springer-Verlag, New York, 1975.
McConnell, J. C.
[1] “Localisation in enveloping rings,” J. London Math. Soc., 43(1968), 421-428;
3(1971), 409-410.
Miatello, R., and Wallach, N. R.
[1] “Automorphic forms constructed from Whittaker vectors,” J. Func. Anal.,
86(1989), 411-487.
[2] “Kuznetsov formulas for rank one groups,” J. Func. Anal., 93(1990),
171-206.
[3] “Kuznetsov formulas for products of R-rank 1 groups,” Israeli Math. Conf.
Proc., 3(1990), 305-320.
MiLidic, D.
[1] “Asymptotic behavior of matrix coefficients of the discrete series,” Duke
Math. J., 44(1977), 59-88.
Moore, С. C.
[1] “Compactifications of symmetric spaces,” Amer. J. Math. 86(1964), 201-218.
Mostow, G. D.
[1] “A new proof of E. Carton’s theorem on the topology of semi-simple
groups,” Bull. Amer. Math. Soc., 55(1949), 969-980.
Mumford, D.
[1] Algebraic geometry I, Complex projective varieties. Springer-Verlag, Berlin,
1976.
Naimark, M. I.
[1] “Decomposition of unitary representations of locally compact groups into
factorial representations,” Sibirsk. Math. Z., 2(1961), 89-99.
[2] Normed Rings. Noordhoff, Groningen, 1964.
Nirenberg, L.
[1] “Pseudo differential operators,” Proc. Sympos. Pure Math., Vol. XVI,
149-167, Amer. Math. Soc., Providence, 1970.
Nouaze, Y., and Gabriel, P.
[1] “Ideaux premiers de 1’algebre enveloppante d’une algebre de Lie nilpotente,”
J. Algebra, 6(1967), 77-99.
Oshima, T., and Sekiguchi, J.
[1] “Eigenspaces of invariant differential operators on an affine symmetric
space,” Inventiones Math., 57(1980), 1-81.
Bibliography
447
Parthasarathy, R.
[1]	“The Dirac operator and the discrete series,” Ann. of Math., 96(1972), 1-30.
[2]	“A generalization of the Enright-Veradarajan modules,” Comp. Math.,
36(1978), 53-73.
[3]	“Criteria for the unitarizability of some highest weight modules,” Proc.
Indian Acad. Sci., Sect. A, 81(1980), 1-24.
Parthasarathy, K. R., Ranga Rao, R., and Varadarajan, V. S.
[1] “Representations of complex semisimple Lie groups and Lie algebras,” Ann.
of Math., 85(1967), 383-429.
Pederson, G. K.
[1] “Measure theory for C* algebras, IV,” Math. Scand., 25(1969), 121-127.
[2] C* algebras and their representations. Academic Press, New York, 1979.
Reed, M., and Simon, B.
[1] Functional analysis. Methods of mathematical physics I. Academic Press, New
York, 1972.
Rickart, С. E.
[1] “Banach algebras with an adjoint operation,” Ann. of Math., 47(1946),
528-550.
SCH1FFMAN, G.
[1] “Integrates d’interlacement et fonctions de Whittaker,” Bull. Soc. Math.
France, 99(1971), 3-27.
Schmid, W.
[1] “Homogeneous complex manifolds and representations of semisimple Lie
groups,” Proc. Nat. Acad. Sci. USA, 59(1968), 56-59.
[2] “On the character of the discrete series,” Inventiones Math., 30(1975),
47-144.
[3] “Boundary value problems for group in variant differential equations,”
Asterique, hors serie (1985), 311-321.
Schwartz, L.
[1] Theory des distributions, Vol. I, Hermann, Paris, 1957.
Segal, I.
[1] “Irreducible representations of operator algebras,” Bull. Amer. Math. Soc.,
53(1947), 72-88.
[2] “An extension of Plancherel’s formula to separable unimodular groups,”
Ann. of Math. 52(1950), 272-292.
Speh, B., and Vogan, D. A.
[1] “Reducibility of generalized principal series representations,” Acta Math.,
145(1980), 227-299.
Stafford, J. T., and N. R. Wallach.
[1] “The restriction of admissible modules to parabolic subalgebras,” Trans.
Amer. Math. Soc., 272(1982), 333-350.
448
Bibliography
Stein, E., and Weiss, G.
[1] Introduction to Fourier analysis on Euclidean spaces. Princeton University
Press, Princeton, 1971.
Thompson, R.
[1] “Inequalities and partial orders on matrix spaces,” Indiana Univ. Math. J.,
21(1971), 469-480.
Tits, J.
[1] “Tabelen zu den einfachen Lie Gruppen und ihren Darstellen,” Lecture
Notes in Mathematics, Springer-Verlag, Berlin, 1967.
Treves, F.
[1] Topological vector spaces, distributions and kernels. Academic Press, New
York, 1967.
Trombi, P. C.
[1] “The tempered spectrum of a real semisimple Lie group,” Amer. J. Math.,
99(1977), 57-75.
[2] On Harish-Chandra’s theory of the Eisenstein integral. University of Chicago
Lecture Notes in Representation Theory, Chicago, 1978.
[3] “Uniform asymptotic expansions of Eisenstein integrals,” Pacific J. Math., to
appear.
Trombi, P. C., and Varadarajan, V. S.
[1] “Spherical transforms on semisimple Lie groups,” Ann. of Math., 94(1971),
246-303.
Varajarajan, V. S.
[1] Harmonic analysis on real reductive groups. Lecture Notes in Math., 576.
Springer-Verlag, Berlin, 1977.
[2] Introduction, Harish-Chandra Collected Papers, Volume I, Springer Verlag,
New York, 1983.
VOGAN, D.
[1]	“The algebraic structure of representations of semi-simple Lie groups I,”
Ann. of Math., 109(1979), 1-60.
[2]	Representations of real reductive groups. Progress in Math. 15, Birkhauser,
Boston, 1981.
[3]	“Unitarizability of certain series of representations,” Ann. of Math.,
120(1984), 141-187.
Vogan, D. A., and Wallach, N. R.
[1]	“Intertwining operators for real reductive groups,” Adv. in Math., 82(1990),
203-243.
Vogan, D., and Zuckerman, G.
[1]	“Unitary representations with continuous cohomology,” Comp. Math.,
53(1984), 51-90.
Wallach, N. R.
[1]	Harmonic analysis on homogeneous spaces. Marcel Dekker, New York, 1972.
[2]	“Representations of semi-simple Lie groups,” Proc. Canad. Math. Soc.
Cong., V)T1, 154-245.
Bibliography
449
[3]	“Asymptotic expansions of generalized matrix entries of representations of
real reductive groups,” Lie group representations, I, Lecture Notes in Math.,
1024. Springer-Verlag, Berlin, 1983.
[4]	“On the unitarizability of derived functor modules,” Inventiones Math.,
78(1984), 131-141.
[5]	“Lie algebra cohomology and holomorphic continuation of generalized
Jacquet integrals,” Advanced Stud, in Pure Math., 14(1988), 123-151.
Watson, G. N.
[1] A treatise on the theory of Bessel functions, Cambridge University Press,
Cambridge, 1966.
Warner, G.
[1] Harmonic analysis on semi-simple Lie groups I. Springer-Verlag, Berlin, 1972.
[2] Harmonic analysis on semi-sample Lie groups II. Springer-Verlag, Berlin,
1972.
Weil, A.
[1] L'integration dans les groupes topologiques et ses applications. Hermann, Paris,
1940.
Weyl, H.
[1] The classical groups, their invariants and representations. Princeton University
Press, Princeton, 1939.
Whitney, H.
[1] “Elementary structure of real algebraic varieties,” Ann. of Math., 66(1957),
545-556.
Whittaker, E. T., and Watson, G. N.
[1] A course of modem analysis. Cambridge University Press, London, 1965.
Yosida, K.
[1] Functional analysis. Springer-Verlag, New York, 1974.
Zelobenko, D. P.
[1] “The analysis of irreducibility in a class of elementary representations of a
complex semisimple Lie group,” Math. USSR. Isvestia, 2(1968), 105-128.
[2] “Representations of complex semisimple Lie groups,” Itogi Nauki i Tekhniki,
11(1973), 51-90; English translation J. of Soviet Math., 4(1975), 295-308.
Zuckerman, G.
[1] “Tensor products of infinite-dimensional and finite dimensional representa-
tions of semisimple Lie groups,” Ann. of Math., 106(1977), 295-308.
[2] “Continuous cohomology and unitary representations of real reductive
groups,” Ann. of Math., 107(1978), 495-516.
This Page Intentionally Left Blank
Index
This index is a combination of the index for Volumes I and II. The page numbers referring to
Volume I are in italics.
Absolutely continuous measures, 361
Adjacent parabolic subgroups, 11
Admissible
(0, X (-module, 81
representation, 81
Affine algebraic group, 42
Algebraicly irreducible, 287
Analytic vector, 34
Approximate identity
for C*-algebra, 276
for L'(G), 274
Approximate projection for a left ideal, 277
AR (Artin-Rees) property, 14
Augmentation homomorphism, 9
Automorphic form, 106
В
Banach
algebra, 265
*-algebra, 268
Borel set, 359
Borel subalgebra, 37
Bruhat decomposition, 52
Cartan decomposition
group, 46
Lie algebra, 43
Cartan-Helgason Theorem, 51
Cartan involution, 42
Cartan subalgebra, 4
fundamental, 57
maximally split, 57
of a real Lie algebra, 56
Cartan subgroup, 59
fundamental, 59
maximally split, 59
CC(H), 292
CCR, 299
Central distribution, 294
Character (of a *-representation), 304
C*-algebra, 268
of a locally compact group, 274
Cheval ley restriction theorem, 75
C“-vector, 31
Clifford algebra, 119
Coefficients (matrix), 22
Compact form, 44
451
452
Index
Constant term
of a function, 147
of a matrix coefficient, 147
Cusp form
(for G), 233
(for G/N), 37
D
Decomposable operator, 317
Dirac
inequality, 368
operator, 368
Direct integral, 314
of representations, 326
Distribution, 332
character, 292
order, 332
Double representation, 216
E
Eisenstein integral, 217
Essentially bounded function, 360
Extreme subset of a convex set, 289
F
Factor, 329
Family of Hilbert spaces, 312
Formal degree, 24
Frobenius reciprocity, 31
Functional equation
for the Eisenstein integral, 232
for the Harish-Chandra C-function, 232
Fundamental
parabolic subgroup, 248
series, 248
G
(0, (-module, 80
equivalent, 80
finitely generated, 80
tempered, 138
underlying, 81
Gelfand-Naimark decomposition, 54
Gelfand transform, 267
Generalized weight space, 108
Generic character (of N), 371
GNS-construction, 286
H
Harish-Chandra
C-function, 230
homomorphism, 93
isomorpism, 78
ц function, 45
Schwartz space, 230
Hilbert space
direct sum, 314
tensor product, 356
I
Induced representation, 31
Infinitesimal
character, 34
equivalence, 81
Infinitesimally irreducible, 81
Integrable function, 359
Intertwining operator,
0-module, 11
group representation, 18
Isotypic component, 28
Iwasawa decomposition
group, 45
Lie algebra, 45
J
Jacobson topology, 299
Jacquet
integral, 381
module, 111
К
Kaplansky density theorem, 281
L
Langlands
data, 149
decomposition, 51
Locally integrable, 332
Index
453
Leading term, 146
Liminaire, 299
Lower semi-continuous, 305
M
Maximal completion of moderate growth, 84
Maass-Selberg relations, 231
Measurable
family of Hilbert spaces, 312
(g) set, 359
Measure
Borel, 359
complex Radon, 361
Radon, 360
regular Borel, 360
space, 359
Modular function, 2
N
Nilpotent element, 342
Norm, 71
О
Operator
compact, 326
Hilbert-Schmidt, 328
self-adjoint, 326
trace class, 328
Operator field, measurable, 317
P
Parabolic subgroup, minimal, 51
P-B-W, 9
Plancherel measure, 341
Positive
element of a C*-algebra, 271
functional, 284
Positive definite function, 306
Primitive ideal, 298
Pure, 289
Q
Quantized Toda lattice, 434
Quasi-compact, 302
R
Rapidly decreasing function, 230
Realization (of a (g, K)-module), 113
Representation
conjugate dual, 20
factorial, 329
(strongly continuous of a) group, 18
Hilbert, 18
identity (of CC(H)), 293
Lie algebra, 11
of moderate growth, 84
nondegenerate, 271
regular, 22
smooth, 18
square integrable, 22
*-, 271
unitary, 18
Representative of a Weyl group element,
136
S
Schur’s Lemma
Dixmier’s, 11
for (g, K)-modules, 80
for groups, 21
Schur orthogonality relations, 23
Schwartz space, 237
Section, square integrable, 312-313
a- algebra, 358
Small representation, 70
Smooth vector, 31
Special vector subgroup, 10
Spectral radius, 266
Split component (standard), 48
State, 289
Sterling’s formula, 55
Strong topology, 280
Submersion, 332
Symmetrization map, 9
T
Tame functional (with respect to a p-pair),
368
TDS, 11
Index
454
Tempered Hilbert representation, 144
Type 1, 333
u
Ultra strong topology, 293
Unipotent extension, 80
Unitarily equivalent *-representations, 272
Universal enveloping algebra, 8
V
Von Neumann algebra, 319
Von Neumann density theorem, 279
W
Weakly contained (in), 336
Weak* topology, 358
Weight
dominant integral, 36
space, 36
Weyl
chamber, 6, 48
character formula, 87
group, 6
integration formula
Lie algebra, 63
Lie group, 63
reflection, 6
Whittaker transform, 423
PURE AND APPLIED MATHEMATICS
Vol. 1 Vol. 2 Vol. 3	Arnold Sommerfeld, Partial Differential Equations in Physics Reinhold Baer, Linear Algebra and Projective Geometry Herbert Busemann and Paul Kelly, Projective Geometry and Projective Metrics
Vol. 4	Stefan Bergman and M. Schiffer, Kernel Functions and Ellip- tic Differential Equations in Mathematical Physics
Vol. 5 Vol. 6 Vol. 7 Vol. 8 Vol. 9	Ralph Philip Boas, Jr., Entire Functions Herbert Busemann, The Geometry of Geodesics Claude Chevalley, Fundamental Concepts of Algebra Sze-Tsen Hu, Homotopy Theory A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations
Vol. 10	J. Dieudonne, Treatise on Analysis: Volume I, Foundations of Modem Analysis; Volume II; Volume III; Volume TV; Volume V; Volume VI; Volume VII
Vol. 11* Vol. 12f	S. I. Goldberg, Curvature and Homology Sigurdur Helgason, Differential Geometry and Symmetric Spaces
Vol. 13 Vol. 14 Vol. 15*	T. H. Hildebrandt, Introduction to the Theory of Integration Shreeram Abhyankar, Local Analytic Geometry Richard L. Bishop and Richard J. Crittenden, Geometry of Manifolds
Vol. 16* Vol. 17 Vol. 18 Vol. 19 Vol. 20 Vol. 21	Steven A. Gaal, Point Set Topology Barry Mitchell, Theory of Categories Anthony P. Morse, A Theory of Sets Gustave Choquet, Topology Z. I. Borevich and I. R. Shafarevich, Number Theory Jose Luis Massera and Juan Jorge Schaffer, Linear Differen- tial Equations and Function Spaces
Vol. 22	Richard D. Schafer, An Introduction to Nonassociative Algebras
Vol. 23*	Martin Eichler, Introduction to the Theory of Algebraic Num- bers and Functions
Vol. 24	Shreeram Abhyankar, Resolution of Singularities of Embed- ded Algebraic Surfaces
* Presently out of print.
+Out of print; please see Vol. 80.
Vol. 25	Frangois Treves, Topological Vector Spaces, Distributions, and Kernels
Vol. 26	Peter D. Lax and Ralph S. Phillips, Scattering Theory, Re- vised Edition
Vol. 27 Vol. 28* Vol. 29	Oystein Ore, The Four Color Problem Maurice Heins, Complex Function Theory R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory
Vol. 30 Vol. 31	L. J. Mordell, Diophantine Equations J. Barkley Rosser, Simplified Independence Proofs: Boolean Valued Models of Set Theory
Vol. 32	William F. Donoghue, Jr., Distributions and Fourier Trans- forms
Vol. 33	Marston Morse and Stewart S. Cairns, Critical Point Theory in Global Analysis and Differential Topology
Vol. 34* Vol. 35 Vol. 36 Vol. 37 Vol. 38 Vol. 39 Vol. 40*	Edwin Weiss, Cohomology of Groups Hans Freudenthal and H. De Vries, Linear Lie Groups Laszlo Fuchs, Infinite Abelian Groups Keio Nagami, Dimension Theory Peter L. Duren, Theory of Hp Spaces Bodo Pareigis, Categories and Functors Paul L. Butzer and Rolf J. Nessel, Fourier Analysis and Approximation: Volume I, One-Dimensional Theory
Vol. 41* Vol. 42 Vol. 43	Eduard Prugovecki, Quantum Mechanics in Hilbert Space D. V. Widder, An Introduction to Transform Theory Max D. Larsen and Paul J. McCarthy, Multiplicative Theory of Ideals
Vol. 44 Vol. 45 Vol. 46	Ernst-August Behrens, Ring Theory Morris Newman, Integral Matrices Glen E. Bredon, Introduction to Compact Transformation Groups
Vol. 47	Werner Greub, Stephen Halperin, and Ray Vanstone, Con- nections, Curvature, and Cohomology: Volume I, De Rham Cohomology of Manifolds and Vector Bundles; Volume II, Lie Groups, Principal Bundles, and Characteristic Classes; Vol- ume III, Cohomology of Principal Bundles and Homogeneous Spaces
Vol. 48	Xia Dao-xing, Measure and Integration Theory of Infinite- Dimensional Spaces: Abstract Harmonic Analysis
Vol. 49	Ronald G. Douglas, Banach Algebra Techniques in Operator Theory
Vol. 50	Willard Miller, Jr., Symmetry Groups and Theory Applica- tions
Vol. 51	Arthur A. Sagle and Ralph E. Walde, Introduction to Lie Groups and Lie Algebras
Vol. 52 Vol. 53*	T. Benny Rushing, Topological Embeddings James W. Vick, Homology Theory: An Introduction to Alge- braic Topology
Vol. 54 Vol. 55 Vol. 56	E. R. Kolchin, Differential Algebra and Algebraic Groups Gerald J. Janusz, Algebraic Number Fields A. S. B. Holland, Introduction to the Theory of Entire Func- tions
Vol. 57 Vol. 58 Vol. 59	Wayne Roberts and Dale Varberg, Convex Functions H. M. Edwards, Riemann’s Zeta Function Samuel Eilenberg, Automata, Languages, and Machines: Volume A, Volume В
Vol. 60	Morris Hirsch and Stephen Smale, Differential Equations, Dynamical Systems, and Linear Algebra
Vol. 61	Wilhelm Magnus, Noneuclidean Tesselations and Their Groups
Vol. 62 Vol. 63*	Francois Treves, Basic Linear Partial Differential Equations William M. Boothby, An Introduction to Differentiable Mani- folds and Riemannian Geometry
Vol. 64	Brayton Gray, Homotopy Theory: An Introduction to Alge- braic Topology
Vol. 65 Vol. 66 Vol. 67 Vol. 68	Robert A. Adams, Sobolev Spaces John J. Benedetto, Spectral Synthesis D. V. Widder, The Heat Equation Irving Ezra Segal, Mathematical Cosmology and Extragalac- tic Astronomy
Vol. 69 Vol. 70 Vol. 71	I. Martin Isaacs, Character Theory of Finite Groups James R. Brown, Ergodic Theory and Topological Dynamics C. Truesdell, A First Course in Rational Continuum Mechan- ics: Volume 1, General Concepts; Second Edition, Corrected, Revised, and Augmented
Vol. 72	K. D. Stroyan and W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals
Vol. 73	В. M. Puttaswamaiah and John D. Dixon, Modular Represen- tations of Finite Groups
Vol. 74	Melvyn Berger, Nonlinearity and Functional Analysis: Lec- tures on Nonlinearity Problems in Mathematical Analysis
Vol. 75 Vol. 76	George Gratzer, Lattice Theory Charalambos D. Aliprantis and Owen Burkinshaw, Locally Solid Riesz Spaces
Vol. 77 Vol. 78	Jan Mikusinski, The Bochner Integral Michiel Hazewinkel, Formal Groups and Applications
Vol. 79 Vol. 80	Thomas Jech, Set Theory Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces
Vol. 81 Vol. 82	Carl L. DeVito, Functional Analysis Robert B. Burckel, An Introduction to Classical Complex Analysis
Vol. 83	C. Truesdell and R. G. Muncaster, Fundamentals of MaxwelTs Kinetic Theory of a Simple Monatomic Gas: Treated as a Branch of Rational Mechanics
Vol. 84 Vol. 85 Vol. 86 Vol. 87	Louis Halle Rowen, Polynomial Identities in Ring Theory Joseph J. Rotman, An Introduction to Homological Algebra Barry Simon, Functional Integration and Quantum Physics Dragos M. Cvetkovic, Michael Doob, and Horst Sachs, Spec- tra of Graphs
Vol. 88	David Kinderlehrer and Guido Stampacchia, An Introduc- tion to Variational Inequalities and Their Applications
Vol. 89 Vol. 90	Herbert Seifert and W. Threlfall, A Textbook of Topology Grzegorz Rozenberg and Arto Salomaa, The Mathematical Theory of L Systems
Vol. 91 Vol. 92	Donald W. Kahn, Introduction to Global Analysis Eduard Prugovecki, Quantum Mechanics in Hilbert Space, Second Edition
Vol. 93	Robert M. Young, An Introduction to Nonharmonic Fourier Series
Vol. 94 Vol. 96 Vol. 97	M. C. Irwin, Smooth Dynamical Systems John B. Garnett, Bounded Analytic Functions Jean Dieudonne, A Panorama of Pure Mathematics: As Seen by N. Bourbaki
Vol. 98 Vol. 99	Joseph G. Rosenstein, Linear Orderings M. Scott Osborne and Garth Warner, The Theory of Eisen- stein Systems
Vol. 100	Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras: Volume 1, Elementary The- ory; Volume 2, Advanced Theory
Vol. 101	Howard Osborn, Vector Bundles: Volume 1, Foundations and Stiefel-Whitney Classes
Vol. 102	Avraham Feintuch and Richard Saeks, System Theory: A Hilbert Space Approach
Vol. 103	Barrett O’Neill, Semi-Riemannian Geometry: With Applica- tions to Relativity
Vol. 104	K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That are Nearly Associative
Vol. 105	Ulf Grenander, Mathematical Experiments on the Computer
Vol. 106	Edward В. Manoukian, Renormalization
Vol. 107	E. J. McShane, Unified Integration
Vol. 108	A. P. Morse, A Theory of Sets, Revised and Enlarged Edition
Vol. 109	К. P. S. Bhaskara-Rao and M. Bhaskara-Rao,	Theory	of
Charges: A Study of Finitely Additive Measures
Vol. 110 Larry C. Grove, Algebra
Vol. Ill Steven Roman, The Umbral Calculus
Vol. 112 John W. Morgan and Hyman Bass, editors, The Smith Con-
jecture
Vol. 113 Sigurdur Helgason, Groups and Geometric Analysis: Integral
Geometry, Invariant Differential Operators, and Spherical
Functions
Vol. 114 E. R. Kolchin, Differential Algebraic Groups
Vol. 115	Isaac Chavel, Eigenvalues in Riemannian Geometry
Vol. 116	W. D. Curtis and F. R. Miller, Differential Manifolds and
Theoretical Physics
Vol. 117 Jean Berstel and Dominique Perrin, Theory of Codes
Vol. 118 A. E. Hurd and P. A. Loeb, An Introduction to Nonstandard
Real Analysis
Vol. 119 Charalambos D. Aliprantis and Owen Burkinshaw, Positive
Operators
Vol. 120 William M. Boothby, An Introduction to Differentiable Mani-
folds and Riemannian Geometry, Second Edition
Vol. 121 Douglas C. Ravenel, Complex Corbordism and Stable Homo-
topy Groups of Spheres
Vol. 122 Sergio Albeverio, Jens Erik Fenstad, Raphael Hpegh-Krohn,
and Tom Lindstrpm, Nonstandard Methods in Stochastic
Analysis and Mathematical Physics
Vol. 123 Alberto Torchinsky, Real-Variable Methods in Harmonic
Analysis
Vol. 124 Robert J. Daverman, Decomposition of Manifolds
Vol. 125 J. M. G. Fell and R. S. Doran, Representations of *-Algebras,
Locally Compact Groups, and Banach *-Algebraic Bundles:
Volume I, Basic Representation Theory of Groups and Alge-
bras
Vol. 126 J. M. G. Fell and R. S. Doran, Representations of *-Algebras,
Locally Compact Groups, and Banach *-Algebraic Bundles:
Volume 2, Induced Representations, the Imprimitivity Theo-
rem, and the Generalized Mackey Analysis
Vol. 127	Louis H. Rowen, Ring Theory, Volume I
Vol. 128	Louis H. Rowen, Ring Theory, Volume II
Vol. 129 Colin Bennett and Robert Sharpley, Interpolation of Opera-
tors
Vol. 130	Jurgen Poschel and Eugene Trubowitz, Inverse Spectral The-
Vol. 131 Vol. 132 Vol. 132-11 Vol. 133 Vol. 134	ory Jens Carsten Jantzen, Representations of Algebraic Groups Nolan R. Wallach, Real Reductive Groups I Nolan R. Wallach, Real Reductive Groups II Michael Sharpe, General Theory of Markov Processes Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex Operator Algebras and the Monster
Vol. 135 Vol. 136	Donald Passman, Infinite Crossed Products Heinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations
Vol. 137	Jean-Dominique Deuschel and Daniel W. Stroock, Large Deviations