LECTURES ON LIE GROUPS
J FRANK ADAMS
university of Manchester
W A BENJAMIN, INC
New York 1969 Amsterdam

LECTURES ON LIE GROUPS
Copyright (Ê 1969 by VV. A. Benjamin, Inc.
All rights reserved
Standard Book Numbers.8053-0116-X(Cloth)
8053-0117-8 (Paper)
Library of Congress Catalog Card number 78-84578
Manufactured in the United States of America
1234R32109
The manuscript was put into production on March 19,1969
this volume was published on June 15,1969.
W. A. BENJAMIN, INC.
New York, New York 10016

CONTENTS
Page
FOREWORD ix
Chapter
1. BASIC DEFINITIONS 1
2. ONE-PARAMETER SUBGROUPS, THE
EXPONENTIAL MAP, ETC. 7
3. ELEMENTARY REPRESENTATION THEORY 22
4. MAXIMAL TORI IN LIE GROUPS 79
5. GEOMETRY OF THE STIEFEL DIAGRAM 101
6. REPRESENTATION THEORY 142
7. REPRESENTATIONS OF THE CLASSICAL
GROUPS 165
REFERENCES 180
vu

FOREWORD
These notes derive from a course on the representation-
theory of compact Lie groups which I gave in the University of
Manchester in 1965 , and in particular from duplicated notes
on that course which were prepared by Dr. Michael Mather.
It may be asked why one who is not an expert on Lie
groups should release such a course for publication. The
answer lies partly in the very limited and modest aims of the
course; and partly, too, in the continued demand for the
duplicated notes , which seems to show that a number of
readers sympathise with these aims. I feel that the
representation-theory of compact Lie groups is a beautiful, satisfying
and essentially simple chapter of mathematics, and that there
is a basic minimum of it which deserves to be known to
mathematicians of many kinds. In my original lectures I
addressed myself mainly to algebraic topologists . If an
algebraic topologist tries to read, for example, Borel and Hirzebruch's
paper "Characteristic Classes and Homogeneous Spaces" [3]

X
he finds that he needs to know the basic facts about maximal
tori, weights and roots of Lie groups. If he tries to read, for
example, Bott's "Lectures on K(X)" [4] he finds that he needs
to know two main theorems on the representation-theory of
compact Lie groups [4, p. 50, Theorem 1; p. 51, Theorem 2],
These theorems appear "in modern dress" , but they go back to
H. Weyl [22]. I have given these examples for illustration,
but they are fairly typical; and they help to indicate a basic
syllabus on Lie groups which may be useful to students of
many different specialities , from functional analysis and
differential geometry to algebra. The object of these notes is to
cover this basic syllabus, with proofs, in a reasonably concise
way. The material on maximal tori, weights and roots appears
in Chapters 4 and 5 . The two theorems on representation-
theory appear in Chapter 6 as Theorems 6.2 0 and 6.41. The
first three chapters allow one to start the proofs more or less
from the beginning.
There is little or no claim to originality; I have simply
tried to assemble those lines of argument which I found most
attractive in the classical sources. There are perhaps a few
small exceptions to this.
(i) In Chapter 3, on elementary representation-theory, I
have proceeded in an invariant and coordinate-free way even
at certain points where it is usual not to do so. Here my
starting-point was a suggestion by H. B. Shutrick for proving
the orthogonality relations for characters without first proving
the orthogonality relations for the components of a matrix
representation (see 3.33(ii) and 3.34 (i) below).

XI
Unfortunately, the usual proof of the completeness of
characters , following Peter and Weyl [15] , makes use of the
orthogonality relations for the components of a matrix
representation. I was therefore forced to rewrite this also in an
invariant way (see 3.46 and 3.47 below).
I have not seen these "invariant" proofs in the sources
I have consulted, but I would be sorry to think they were not
known to the experts .
(ii) In the same chapter, I have laid particular stress on
real and symplectic representations, which are important to
topologists; and I have preferred those methods which apply
simultaneously to the real and to the symplectic case.
(iii) Theorem 5.47 allows one to read off the fundamental
group of a compact connected group from its Stiefel diagram;
the statement is surely well known to the experts , and is
undoubtedly implicit in Stiefel's work, but I do not remember
seeing an explicit statement or proof in the sources I have
consulted.
(iv) It is usual to give a meaning to the words "highest
weight" by ordering the weights lexicographically, in a way
which is somewhat arbitrary; I have preferred to use instead
a partial ordering which is manifestly invariant, and which
seems to me to have some technical advantages (see 6.22 and
6.2 3 below). I hope this departure from tradition may commend
itself to other workers .
I am most grateful to A. Borel, to Harish-Chandra and
particularly to H. Samelson for giving me tutorials on Lie
groups and representation-theory. I have also profited from

Xll
R. G. Swan's "Notes on Maximal Tori, etc." . I am also very
grateful to Michael Mather, who prepared the notes on the
original course. In particular, the trick in the present proof of
2.19 is due to him; it allowed him to slim the original
lectures by removing a good deal of standard material on the
relation between a Lie group and its Lie algebra. He also
removed a good deal of hard work from the proof of 5.55.
Finally, I am grateful to H. B. Shutrick for the suggestion
noted above.

Chapter 1
BASIC DEFINITIONS
1.1 DEFINITIONS. Let V, W be finite dimensional vector
spaces over the real numbers R. Let U be an open subset of
V, fa map from U to W, and x a point of U. Then f is
differentiate at x if there is a linear map f' (x) : V ~* W such that
f(x + h) = f(x) + (f (x))(h) + o| h| .
If f is differentiable at each point of U, we say that f is
differentiable on U. In this case we have a function
f : U - Hom(V/W)/
and we may ask if this is differentiable. We say that f is
smooth (or of class C00) on U if each function f, f , f" , ... is
differentiable on U. (Of course, the definition of each of
these depends on the previous one being defined and
differentiable.)

2 LECTURES ON LIE GROUPS
1.2 DEFINITIONS. If X is a topological space and V a
finite dimensional vector space, a chart is a homeomorphism
to : U - X , where U <= V is open and X cr X is open.
a a a a a
An atlas is a collection of charts {(p } with UX = X.
a a
The atlas is smooth if the functions (fi^ifi , defined on
ß a
to"1 (X Hx ), are smooth.
a a ß
Let X, Y be topological spaces with smooth atlases
{<p 3 and [^ }. Then a map f : X - Y is smooth if the maps
a ß
ù^îcfi , defined on <p-1 (X Df~ Y ), are smooth. Notice that
ß a a a ß
the composition of two smooth maps is smooth, and the
identity map of a space with atlas is smooth.
Two atlases {(p }, [é„} on X are equivalent if the maps
a ß — "
1 :X, [to } -X, {$J
a ß
1 :X, {d } -X, {iß }
ß a
are smooth.
A differential or smooth manifold is a Hausdorff space
with an equivalence class of smooth atlases. This equivalence
class is called its differential structure.
1.3 PROPOSITION. If X, Y are smooth manifolds, then
X x Y can be given the structure of a smooth manifold in a
unique way to satisfy:

BASIC DEFINITIONS 3
(i) irx : X x Y - X and 7T2 : X x Y - Y are smooth maps.
(ii) f : Z - X x Y is smooth if and only if tjxî and 7r2f are
smooth.
Proof. Given charts <p in X, ^ in Y, form the chart <p x^
aß a ß
in X x Y. Do this for each pair, a, ß. The rest of the proof
consists of checking the necessary properties, and will be
left to the reader.
1.4 DEFINITIONS. A Lie group G is
(i) a smooth manifold, and
(ii) a group, with product fi;G xG"G and inverse
i : G - G, such that
(iii) \i and i are smooth.
A homomorphism of Lie groups 6 : G — H is
(i) a homomorphism of groups , and
(ii) a smooth map.
1.5 EXAMPLES
1. R considered as a group under addition, with an atlas
of just one chart given by the identity map.
2 . T = R /Zn (where Zn is the set of points in Rn all of
whose co-ordinates are integers) considered as a quotient

4 LECTURES ON LIE GROUPS
group of R , with charts given by the restriction of the
projection R - T to small open sets.
3. Let V be a finite dimensional vector space over R.
Then Aut V, the set of automorphisms of V, is an open subset
of Hom(V,V) given by det / 0, so Aut V is a smooth manifold,
and is a group under composition. The product map is smooth,
since it is given by polynomials (La..b. ) and the inverse map
is smooth since it is given by polynomials divided by the
determinant. Thus Aut V is a smooth manifold. Aut R is
called GL(n, R).
This also works over the complex numbers or the
quaternions. For instance, Horn (V, V) is a linear subspace
o
of HomD(V, V), and Aut_V = AutV C Horn (V, V). So Aut V is
R 0 R O O
an open subset of Horn (V, V).
1.6 DEFINITION of the tangent bundle of a smooth
manifold X. Let [o : U -X } be an atlas based on the vector
a a a
space V. Take the disjoint union of the spaces X x V over all
a
a and, whenever x 6X OX , identify (x,v) 6X x V with
aß a
(x, fo~l<p )' v) € Xo x V. Call the identification space T(X),
pa ß
and define p : T(X) - X by projection on the first factor of each
product. This is the tangent bundle. It is an invariant of X.

BASIC DEFINITIONS 5
We call p-1x the tangent space at the point x € X,
written X , and a point of X is a tangent vector at x.
Note that T(X) can be made into a smooth manifold in
an obvious way, and p is a smooth map.
Given a smooth map f : X - Y construct a smooth
natural bundle map f. : T(X) - T(Y) as follows. For x € X
a
and f x € Y set fjx, vj^fxYr1^ Yv).
1.7 NOTATION. Let G be a Lie group, with unit e. Then
we write L(G) for G and L(f) for f*|G . Then L is a functor.
We also write f for f ^, in the light of the following example:
1.8 EXAMPLE. Consider example 1.5.1. Then the tangent
space at the origin of R may be identified with R under the
chart. If f : R - R is a smooth map, then f. I R = f under
this identification.
1.9 DEFINITIONS. A smooth vector field on a manifold X
is a smooth cross-section of the tangent bundle. That is, it
is a smooth map X : X - T(X) such that pX = 1.
Let G be a Lie group. For x 6 G define L : G -* G by
L (g) = xg. This is smooth. Then a smooth vector field X on
G is left invariant if the following diagram is commutative for

LECTURES ON LIE GROUPS
each x € G:
T(G) X* >T(G)
G - >G
1.10 DEFINITION. Let G be a Lie group. For each x € G
define A : G - G by A (g) = xgx . This is a smooth auto-
morphism, and hence defines a linear map A' : G - G ; that
x e e
is , A' € Aut G . Hence x - A' defines a map Ad : G - Aut G ,
x e x e
This is a smooth homomorphism.

Chapter 2
ONE-PARAMETER SUBGROUPS,
THE EXPONENTIAL MAP, ETC.
2.1 LEMMA. Suppose given a smooth vector field v(x)
defined in a neighbourhood U of 0 in R . Consider the
following equations for a function f : R1 -» R , namely,
f (t, 1) = v(f(t)), f(0) = 0. Then there is € > 0 for which the
equations have a solution in (-€, c), and this solution is both
unique and smooth.
This is a particular case of the more general:
2.2 LEMMA. Let UcRn and V c Rm be neighbourhoods of
0, y respectively. Let v(x, y) be a vector field in R
depending smoothly on x € U and y € V. Consider the equations
f(0) = 0, f (t, 1) = v(f(t), y), for each fixed y €V, as equations
for a function f : R1 - R . Then there is € > 0 and a
neighbourhood V of y in R such that a solution exists in (-e , c) for

8 LECTURES ON LIE GROUPS
each y 6V, this solution is unique, and depends smoothly on
t € (-e, c) and y € V .
Proof . We refer the reader to [12, p. 94, Proposition 1],
[5, Chapter 2 , Theorem 4 .1] , [2, Appendix, Section II], or
[8, Chapter 9, Theorem 1] .
2.3 DEFINITION. A 1-parameter subgroup of G is a homo-
morphism of Lie groups 6 : R1 - G, where R1 is a Lie group
under addition with an atlas with one chart given by the
identity map.
2 .4 EXAMPLE. In T2 = R2/Z2 set 6(t) = (t, ct) for c any
constant.
2.5 Let 6 be a 1-parameter subgroup of G. Let (0, 1) be
the unit tangent vector at the origin in R1 . Associate with 6
the vector 6'(0, 1) € G . Then:
e
2.6 THEOREM. This sets up a 1-1 correspondence
between 1-parameter subgroups of G and vectors in G .
Proof . We need:
2.7 LEMMA. Let X be a smooth manifold, v(x) a smooth

ONE-PARA M ET ER SUBGROUPS 9
vector field on X, and 6, <p : [a , b] - X two smooth functions
satisfying 6' (t, 1) = v(6(t))
<P'(t, U=v(<p(t))
6(a) = <p(a).
Then 6 (t) = (p{t) for all t € [a, b] .
Proof . Let c be the least upper bound of the set of d for
which 6 (t) = <p(t) on [a , d] . Then 6(c) = (p(c) by continuity.
If c < b we may take local coordinates at 6(c) = <p{c) and apply
2.1, showing that 6(t) =<p(t) in some (c - € , c + c), which
contradicts the definition of c. Thus c = b.
Proof of 2.6.
(i) Uniqueness. Suppose that 6 corresponds to v € G .
The vector (0, 1) can be extended to a left invariant vector
field (t, 1) on R1 , and v can be extended to a left invariant
vector field v(x) on G. Taking the diagram of tangent spaces
corresponding to
6
R >G
L.
Le(t)
6
R —^->G
we see that 6'(t, 1) = L' v = v(6(t)). Thus, by 2. 7, 6 is

10 LECTURES ON LIE GROUPS
unique.
(ii) Existence. Given v € G , extend v to a left invariant
e
vector field v(x) on G. Then the equations 6 ' (t, 1) = v(0(t)),
6 (0) = 0 have a solution for t € (-e , c), by 2 .1.
We will show, firstly, that 6(s)6(t) = 6(s + t) for
|s| < |"€, |t| <j€. Well, for s fixed, 6(s)6(t) and 6(s + t)
are both solutions of <p' (t, 1) = v(<o(t)), ^(0) = 6(s). Thus, by
2.7, 6(s)6(t) = 6(s + t).
Now define tp : R1 -Gas follows. For t € R1 choose a
positive integer N such that |—| < —, and set ip (t) = f 6( — j J
Then ip is well-defined since, if M is another such integer,
(aMnJ/ = 6Cm)' by the previous paragraph,
so
WM
(•(ï))--Wîfe)r-«ïïj/"- — ***
group homomorphism, for, if |—| < —and |—| < — we have
Now ip is also smooth, and extends 6. So $ is a 1-parameter
subgroup and tp ' (0 , 1) = v.
2.8 DEFINITION of the exponential map. Define
exp : G - G as follows. Let v € G and let 6 be the corres-
e e v
ponding 1-parameter subgroup of G. Then exp(tv) = 6 (t).

ONE-PARAMETER SUBGROUPS 11
We need to show that 6 (t) depends only on tv. Well,
for fixed s, 6 (st) is clearly the 1-parameter subgroup
corresponding to sv. Thus e (st) = e (t), so e (s) = e (i).
v svv v sv
2.9 THEOREM, exp is smooth.
Proof . Let v € G . We show that exp is a smooth function
o e
in a neighbourhood of v .
o
Well, 6 (t) is the solution of the differential equation
e^(t, i) = v(ev(t»
= L'e (t)v-
V
Now L' v is a smooth function of x € G and v 6 G . So (2.2)
x e
the solution is a smooth function of t and v for |t| < € and v in
a neighbourhood of v .
Take a positive integer N with — < c. Then
exp v = 6 (1) = ( 6 ( —] ] , which is a smooth function of v
in a neighbourhood of v .
o
2.10 REMARK, exp : G - G induces exp' = 1 : G - G
e e e
and
2.11 PROPOSITION, exp is natural. That is , given a
homomorphism of Lie groups <p : G - H inducing tp ' : G - H ,

12
LECTURES ON LIE GROUPS
the following diagram is commutative:
G >H
e e
exp
<P
exp
■^ H
Proof . Let v € G , and let 0 : R1 - G be the corresponding
e
1-parameter subgroup of G. Then<p0 : R1 - H is the l-para-
meter subgroup of H corresponding to tp'v, since the derivative
is natural. Thus exp p'v = p0(l) = <p exp v.
2.12 EXAMPLE. Let V be a finite dimensional real vector
space, and take G = Aut V, which is an open subspace of
Hom(V, V). We can identify G with Hom(V, V). Let
A € Hom(V, V). Then we assert
A A
exp A = 1 + A + —■ + ... +—7+ ....
2 n!
Ast2 Antn
Proof . Consider 1 + At + —— + .. . + —— + This is
2 n !
easily seen to be a smooth homomorphism from R1 to Aut V,
and is the 1-parameter subgroup corresponding to A. Thus
A2 An
exp A=l+A + — + ... +77+ ....
2.13 EXAMPLE. Consider G = T° = Rn/Zn. Then G = Rn,
e
and exp can be identified with the covering map R - T .

ONE-PARAMETER SUBGROUPS 13
2.14 THEOREM, exp is a diffeomorphism of a neighbourhood
of 0 € G with a neighbourhood of e in G.
e
Proof . This is immediate from 2.10 and the Jacobian theorem.
(See [12, p. 12, Theorem 1].)
2.15 THEOREM. Let G = V, ©V2 , and define <p : G -G
e ^ e
by (fi (vx , v2) = exp v1 exp v2 . Then (p is a diffeomorphism of
a neighbourhood of 0 € G with a neighbourhood of e in G.
Proof . <fi is the composition Vi © V2 *->■ G x G—>G,
and so is differentiable. Further, <fi' is the identity on both Vx
and V2 , and so is the identity on G . We may proceed as in
2.14.
2.16 PROPOSITION. Let Gx denote the identity component
of G, and let S c Gi be a neighbourhood of e. Then the
subgroup generated by S is Gi .
Proo f . Clearly gp{S] cz d . Now gp{S} is an open
subgroup of Gi , so all its cosets are open. Thus gp{S} is also
closed, so gp{S] = Gi .
2.17 THEOREM. If G is connected, a ho mo mo rph is m of Lie

14 LECTURES ON LIE GROUPS
groups 6 : G - H is determined by the induced homomorphism
6' : G - H .
e e
Proof . By 2.11 we have the commutative diagram:
e-
G >H
e e
exp
exp
6
G >H
Thus 6 is determined by 0' at least on the subgroup of G
generated by the image of exp. But this is a neighbourhood of
e in G, so 6 is determined on G.
2.18 LEMMA. Let <p :U -G be a chart on G which
a a a
sends 0 € V to e € G. Then, omitting tp , we can write
a
xy = x + y + o(r) in a neighbourhood of e in G, where r = r(x, y)
denotes the distance of (x, y) from (e, e) in G x G under a
metric.
Proof . Since the product in G is differentiate, there is a
constant vector a and constant linear functions b, c such that
xy = a + bx + cy + o(r). Set x = e and we find that
y = a +cy+o(r) so a = 0, c = 1. Similarly b = 1, so
xy = x + y + o (r).

ONE-PARAMETER SUBGROUPS 15
2.19 THEOREM. A connected Abelian Lie group G has the
a b
form T x R .
Proof. We show first that exp : G -G is a homomorphism,
Well,
/ s \N/ t \N
exp s exp t = fexp -j [exp —J
/ s t \N
= (exp ïï exp-)
since G is Abelian
|N
[^P(n + N+°(n))]
by 2.18 and 2.14, where we consider s ,t fixed and N varying
= exp(s + t + o(l))
= exp(s + t).
Thus exp is a homomorphism, and, by 2.16, exp is onto.
Consider K = Ker exp. By 2.14 , since exp is a
homomorphism, K is discrete. Now a discrete subgroup of a real
vector space is a free Abelian group, with generators
gi,. .. , g which are linearly independent over R. (This is
proved by induction over the dimension of the vector space.)
Extend this to a basis of G . Then K is expressed as the set
e
of points with coordinates (nx , . . . ,n , 0 , . .. , 0), each n. € Z.
Thus G = G /K = TF x Rn"r.
e

16 LECTURES ON LIE GROUPS
2.20 COROLLARY. A Lie group which is compact,
connected and Abelian is a torus.
2.21 EXERCISE. Classify the compact Abelian Lie groups.
2.22 DEFINITION of submanifold .
Let WcVbe a real finite dimensional vector space and
subspace. Let M be a smooth manifold, and N a subset of M.
Then a chert <p : V - M is good if
a a a
(i) M P N =0 (the empty set), or
a
(ii) <p sends V P W onto M PN.
a a a
An atlas is good if all its charts are good.
N is a submanifold if there is a good atlas in the
differential structure of M.
Equivalence of good atlases is defined by the identity
map being smooth, as before.
2.23 PROPOSITION. If N is a submanifold of M , then N can
be given a differential structure as a manifold so that
(i) the inclusion of N in M is smooth, and
(ii) P - N is smooth if and only if the composition
P - N - M is smooth.

ONE-PARAMETER SUBGROUPS 17
Proof is clear, and left to the reader.
2.24 REMARK. It follows that T(N) is embedded in T(M).
2.25 EXERCISE. If Ni , N3 are submanifolds of Mx ,M3then
Nx x N2 is a submanifold of Mx x M3 , and its differential
structure as a submanifold is the same as its differential
structure as a product.
2.26 PROPOSITION. If G is a Lie group, and H is both a
submanifold and a subgroup, then H is a Lie group.
Proof . Apply 2.23 and 2.25 to the maps of pairs
M:GxG,HxH-G,H and i : G,H - G,H.
2.27 THEOREM. A closed subgroup H of a Lie group G is a
submanifold.
Proof . The next three lemmas constitute a proof.
2.28 LEMMA. In 2.27, suppose G has a norm. Suppose
0 /h 6 G is a sequence of points such that exp h 6 H,
n e M n
h -0, and r-—rh - v <E G . Then exp(tv) 6 H for all t € R.

18 LECTURES ON LIE GROUPS
Proof . ——; h -tv, and In | - 0, so we may choose inte-
|hn| n
gers m such that m In I - t. Then exp m h - exp tv. But
n n1 n1 n n
mn
exp m h = (exp h ) € H, and H is closed. So exp tv 6 H,
n n n
as required.
Let W be the set of such tw in G . Then exp W c: H.
e
2.29 LEMMA. W is a vector subspace of G .
e
Proof . Clearly w 6 W implies tw 6 W all t 6 R.
So, suppose wx ,w2 € W, and suppose wx + w3 / 0.
We will show that wx + w2 6 W.
Consider exp(twx )exp(tw3). This is in H. For t
sufficiently small we can write exp(tw1)exp(tw3) = exp (f (t)), where
f(t) is a smooth curve in G and f(0) = 0.
Now exp(tw!)exp(tw2) - exp t(wx + w2) = o(t) , by 2 .18 ,
so —f (t) - wx + w2 as t -* 0. Thus we may apply 2.28 with
h = f( — ), for n sufficiently large, and v = (w, + w2),
n \n / ' wx + w2 \ l 2/
and deduce that wa + w2 6 W.
2.30 LEMMA, exp W is a neighbourhood of e in H.
Proof. Split G as W © W and consider the diffeomorphis m
e
(p(w' ,w) = exp(w1)exp(w) between a neighbourhood of 0 in G

ONE-PARAMETER SUBGROUPS 19
and a neighbourhood of e in G (2.15). Suppose the lemma does
not hold. Then there is a sequence of pairs (w1 ,w ) such that
n n
exp(w' )exp(w ) € H, exp(w' )exp(w ) - e, and w' / 0. Since
n n n n n
exp(w ) 6 H,exp(w' ) 6 H. Then we can find a subsequence of
n n
w' such that -. r w' ' - w' € W , for some such w ' , and
Kl n
|w'|= 1. It follows from 2.28 that w' 6 W, which is a
contradiction.
Thus exp W is a neighbourhood of e in H.
It is now clear that exp provides a good chart for a
neighbourhood of e. Left translation gives a good chart round
any other point of H. This completes the proof of 2.27.
2.31 EXAMPLES.
O(n) cGL(n,R)
U(n) cGL(n,C)
SP(n) cGL(n,Q)
are closed subgroups, and so submanifolds , of Lie groups.
Thus they are Lie groups.
In each case, the tangent space at e of the subgroup
-T
consists of the matrices X such that X = -X.

20 LECTURES ON LIE GROUPS
Proof.
(i) Suppose X is in the tangent space at e of the subgroup.
Take a smooth curve of the form f (t) = 1 + tX + o(t) in the sub-
— T
group. Then f (t) f (t) = 1, by definition of the subgroup. That
is,
(1 + tXT + o(t))(l + tX + o(t)) = 1.
Thus
XT + X = 0.
-T
(ii) Suppose X = -X.
Then
(explx) T= ( L00 tnxn/n ! ) T by 2 .12
= Ltn(XT)n/n!
= Ltn(-X)n/n!
= (exp tX)-1.
Therefore exp(tX) lies in the subgroup, and X in the tangent
space.
2.32 EXAMPLE. If G is a compact Lie group, and H is 5
closed connected Abelian subgroup, then H is a torus.
2.33 PROPOSITION. A closed connected subgroup H of a
Lie group G is determined by its tangent space at e.
Proof . See 2.17.

ONE-PARAMETER SUBGROUPS 21
2 .34 DEFINITION. Suppose G is a Lie group and H a
closed subgroup. Then the quotient space G/H is the set of
cosets gH. We have the projection p : G -* G/H, and give G/H
the quotient topology.
2.35 EXERCISE. G/H is Hausdorff.
2.36 PROPOSITION. If H is a closed subgroup of a Lie
group G, we can give G/H a differential structure as a
manifold so that
(i) p is smooth,
(ii) f : G/H - M is smooth if and only if fp : G - M is
smooth.
Proof . Split G as W © W where W = H as before. Let U
e e
be a small neighbourhood of 0 in W, and define ^ : U -* G/H
byU-W'-G eXp> G -^-> G/H. Then ^ is a homeomorphism
with a neighbourhood of eH. The rest of the proof is left as an
exercise for the reader.
2 .37 PROPOSITION. H - G - G/H is a fibration.
Proof . See [17, 1.7.5] .

Chapter 3
ELEMENTARY REPRESENTATION THEORY
In this chapter we set up elementary representation-
theory. The basic definitions and constructions occupy 3.1
to 3.13. Then we introduce integration. From this we draw
the usual consequences, including complete reducibility (3.15
to 3.21). Then comes Schur's Lemma, some of its
consequences , and the definition of the representation ring (3.22 to
3.28). Then traces, characters and the orthogonality relations
(3.2 9 to 3.37). Then the Peter-Weyl theorem and the
completeness of characters (3.38 to 3.49). Then the usual material on
real and symplectic representations (3.50 to 3.64). Next
comes the behaviour of the representation ring for products
(3.65 to 3.67) and coverings (3.68 to 3.70). Finally we have
the representation-theory of the torus (3.71 to 3.78).
22

ELEMENTARY REPRESENTATION THEORY 2 3
3.1 DEFINITIONS. Let A be one of the classical fields R
(the real numbers), C (the complex numbers) or Q (the
quaternions). Let G be a topological group. Then a AG-space is a
finite-dimensional vector space V over A provided with a
continuous homomorphism
6 : G -Aut V.
(Such a V is also called a representation of G over A or a G-
space over A.)
Alternatively, for each g € G and v 6 V we are given
gv 6 V, and the following conditions are satisfied.
(i) ev = v and g(g'v) = (gg')v.
(ii) gv is a A-linear function of v.
(iii) gv is a continuous function of g and v.
By choosing a base in V we can regard 6 as taking
values in GL(n,A). We then speak of a matrix representation.
In the case A = Q, if we wish to write our matrices on the left,
it will be prudent to arrange that V is a right module over Q.
Fortunately we can make any left module over Q into a right
module over Q, and vice versa , by the formula
qv = vq (q € Q, v 6 V).
Here the conjugate of a quaternion is defined as usual: if

24 LECTURES ON LIE GROUPS
q = a + bi + cj + dk, then q = a - bi - ij - dk.
Let V and W be AG-spaces. A G-map is a function
f : V - W which commutes with the action of G, that is ,
f(gv) = g(£v).
A A G-map is a G-map which is A-linear; mostly we deal with
such. The set of such AG-maps is written Horn (V,W), or
sometimes simply Horn (V,W) if A is understood. It is a vec-
G
tor space over R if A = R or Q, over C if A = C.
A AG-isomorphism is a AG-map which has an inverse.
As usual, we say that two AG-spaces are equivalent if they
are isomorphic.
3.2 DEFINITION. Let V be a G-space over C. A structure
map on V is a G-map j : V - V such that
(i) j is conjugate-linear, that is,
j(zv) = z(jv) (z € C), and
(ii) ja=±l.
3.3 EXPLANATION. If V is a G-space over Q, we may
regard it as a G-space over C with a structure map such that
j2 = -1. Actually we may do so in two ways. On the one hand
we can take the C-module structure given by i acting on the

ELEMENTARY REPRESENTATION THEORY 25
left and the structure map given by j acting on the left. On
the other hand we can take the C-module structure given by i
acting on the right (-i acting on the left) and the structure map
given by j acting on the right (-j acting on the left). It makes
no difference which we take , because we can define an
automorphism a : V - V taking one structure into the other, for
example a(v) = kv.
Conversely, given a G-space over C with a structure
map such that j3 = -1, we can clearly reconstruct a G-space
over Q.
Similarly, it is often convenient to regard a G-space
V over R as being equivalent to a G-space V over C provided
with a structure map such that j2 = +1. To pass from V to V
we take V = C ®D V provided with the obvious operations and
R
structure maps:
z(z' ®v)=zz' ®v (z,z' € C)
g (z © v) = z ® gv
j (z © v) = z ® v.
To pass from V to V we split V into the +1 and -1 eigen-
spaces of j; these are G-spaces over R which are isomorphic
under i. These operations are clearly inverse to one another,

26 LECTURES ON LIE GROUPS
up to isomorphism.
3.4 DEFINITION. Given AG-spaces V and W, we can form
the direct sum of the two vector spaces , V © W, and make G
act on it by
g(v,w) = (gv,gw).
Equivalently, we may take two G-spaces V and W over
C with structure maps j ,j such that js = j3 , and put on
v w v w
V (BW the structure map j (£ j
V w
The next five operations start from a G-space V over
A and construct a G-space over some A'. The possibilities
are displayed in the following diagram, which is not
commutative.
R t Q
c
3.5 DEFINITIONS.
(i) If V is a G-space over R, define cV = C ® V, regarded
R
as a G-space over Cas in 3.3.
(ii) Similarly, if V is a G-space over C, define
qV = Q ® V, and regard it in the obvious way as a G-space
c

ELEMENTARY REPRESENTATION THEORY 27
and a left module over Q.
(iii) If V is a G-space over Q, let c'V have the same
underlying set as V and the same operations from G , but regard it
as a vector space over C.
(iv) Similarly, if V is a G-space over C, let rV have the
same underlying set as V and the same operations from G, but
regard it as a vector space over R.
(v) Let V be a G-space over C. We define tV to have the
same underlying set as V and the same operations from G, but
we make C act in a new way: z acts on tV as z used to act on
V.
Let us adopt the viewpoint of 3.3; then both c and c'
act on a G-space over C provided with a structure map j, and
they act by forgetting the structure map.
All these constructions are natural; given a AG-map
f : V - W, we can construct maps cf, qf , c'f, rf and tf.
All these constructions commute with direct sums <£.
3.6 PROPOSITION.
re = 2
cr = 1 + t
qc' = 2

28 LECTURES ON LIE GROUPS
c'q =
tc -
it -
tc'
qt =
t3 =
= 1 + t
= c
= r
c'
= q
= l.
These equations are to be interpreted as saying that
rcV = V © V for each V over R
crV = V ® tV for each V over C,
etc.
Proof. Most of this can safely be left to the reader; we
show that cr = 1 + t.
Let V be a G-space over C. We want to study C ® V
R
with C acting on the first factor and G on the second. Let C
act on the first factor of C ® C, and let C ® C act on
R R
C &L V in the obvious way. Then C ® V is a G-space over
the C-a Ige bra C8LC.
R
We will now split the unit 1 ® 1 of C ® C into ortho-
R
gonal idempotents, and so obtain a splitting of C ® V. In
R
detail, let
ex = j(l ® 1 + i ® i)

ELEMENTARY REPRESENTATION THEORY 2 9
e3 = j(l ® 1 - i ® i).
Then ei = e1 , e2 = e2 , ex e2 = 0 and ex + e3 = 1, as required.
So
C ©R V = ex(C ©R V) G e2 (C ©R V),
where the isomorphism is a G-isomorphism over C. Further,
V = es (C © V) by, for instance, v -* e2 (1 ®v) and
R
tV = ey (C ®D V) by, for instance , v - e1 (1 ® v).
R
Thus crV = V e tV.
3.7 DEFINITION. Given G-spaces V and W over C, we
can form the tensor product of the two vector spaces , V ®_ W,
and make G act on it by
g(v ©w) = gv © gw.
Suppose now that V and W admit structure maps i ,j
W
such that j2 = e j2 = € Then V ® W admits a structure
V V W W O
maP j = j,, ® jw such that j2 = ee, We can separate three
cases.
(i) 6 = 6 = +1. The tensor product of two real
representations is real. The construction amounts to taking two
G-spaces V,W over R and forming V ® W.
R
(ii) 6„ = +1, 6W = -1. The tensor product of one real
representation V and one quaternionic representation W is

30 LECTURES ON LIE GROUPS
quaternionic. The construction amounts to taking V ® W and
R
making Q act on it by
q (v ® w) = v ® qw.
The case €„ = -1, e,.r = +1 is similar.
V W
(iii) € = € = -1. The tensor product of two quaternionic
representations V and W is real. It is natural to interpret the
construction in terms of V © W. For this to make sense we
Vf
must consider V as a right module over Q and W as a left
module over Q. If we use the resulting structure maps , then
the -1 eigenspace of the structure map j ® j on V ® W
coincides with V ® W.
Vf
3.8 PROPOSITION.
(i) All our tensor products are compatible with the maps
c ,c' .
(ii) The tensor product ® is bilinear over the direct sum ©.
3.9 DEFINITION. Given G-spaces V and W over the same
field A, we can form Horn. (V,W), the set of A-linear maps
from V to W. It is a vector space over R if A = R or Q, over
C if A = C. We can make G act on it by
(gh)v = gCMg"1 v)) (h € HomA (V, W))

ELEMENTARY REPRESENTATION THEORY 31
or equivalently
gh= (ewg)h(evgJ).
(Note that HomA(V,W) is covariant in W and contra variant in
V.) The subspace of elements in Horn. (V,W) which are
invariant under G is precisely Horn (V,W).
AG
We may also proceed as in 3.3, 3.7. Let V and W be
G-spaces over C which admit structure maps j,,/^ such that
j2 = c , j 2 =ew Then Horn (V,W) admits a structure map j
V V Vv VV \s
given by
Jh = jwh)v •
(Note that Horn (V,W) is covariant in W and contravariant in
V.) We have j3 = € € . We can separate three cases.
(i) €„ = €,Ar = +1. The Horn of two real representations is
V W
real. The construction amounts to taking two G-spaces V,W
over R and forming Horn (V,W). In fact, if we form
R
Homç(cV ,cW), then the +1 eigenspace of j maybe identified
with Horn (V,W); thus
R
Horn (cV,cW) =cHom (V,W).
O R
(ii) € = +1, ç = -1. The Horn of a real representation
into a quaternionic representation is quaternionic. The case
e = -1, € = +1 is similar. We leave it to the reader to

32 LECTURES ON LIE GROUPS
interpret the construction in these cases along the lines of
3.7(ii).
(iii) € = ç = -1. The Horn of two q aternionic
representations is real. The construction amounts to taking two G-
spaces V,W over Q and forming Horn (V,W). In fact, if we
Vf
form Horn (c1 V^c'W), then the +1 eigenspace of j is
Horn (V,W); thus
Vf
Horn (c'V,c'W) = cHom (V,W).
3.10 COROLLARY.
(i) If V and W are two G-spaces over R then
dimcHomCG(cV/CW) = dimRHomRG(V, W).
(ii) If V and W are two G-spaces over Q then
dimcHomCG(c'V/c,W) = dimRHom (V,W).
This follows immediately from 3.9(i) and (iii), by
looking at the subspaces of elements invariant under G.
3.11 PROPOSITION.
(i) All our Horn's are compatible with the maps c,c' .
(ii) Horn is bilinear over the direct sum ©.
A particular case of 3.9 is important.

ELEMENTARY REPRESENTATION THEORY 33
3.12 DEFINITION. Given a G-space V over C, we define
its dual V* by
V* = Horn (V,C).
Here the target space C is given the trivial operations from
G: gz = z for all g 6 G and z € C. The G-space C is real; it
follows that the dual of a real representation is real, and the
dual of a quaternionic representation is quaternionic.
The general case 3.9 may be reduced to the special
ca s e 3.12.
3.13 LEMMA. We have an isomorphism
Horn (V,W) = V* ® W
commuting with the action of G and with the structure maps j
(if a ny).
Proof . The isomorphism sends V* ® W into the map h,
where
h(v) = (v*v)w.
In dealing with compact topological groups , one of
our best weapons is integration.
3.14 INTEGRATION. Let G be a compact topological group.
Then for each continuous function f : G ~* R we can define a

34 LECTURES ON LIE GROUPS
real number
W = Ç f(g)
JG Jg€G
so as to satisfy the following conditions.
(i) \ has the usual properties of an integral, that is, it
JG
is a positive linear functional.
(ii) ^1 = 1.
JG
(iii) The integral is invariant under left and right
translations; that is , for each x 6 G we have
\ f(xy)=C f(y)
Jy<EG Jy€G
\ f(yx)-Ç f(y).
y<EG Jy<EG
Similarly, we may integrate functions which take
values in any finite-dimensional vector space over R so as to
obtain values lying in that vector space; and if we do so,
integration commutes with linear maps.
If G is a Lie group then the integral is slightly easier
to construct than if G is a more general topological group. We
will not discuss this here, but refer the reader to [13,14,20].
The first function which asks to be integrated is
6 : G -HomA(V,V).

ELEMENTARY REPRESENTATION THEORY 35
3.15 PROPOSITION. Suppose given a representation
6 : G -HomA(V,V). Then
I = V 6 € HomA(V,V)
JG
is idempotent (I2 = I) and its image is V , the subspace of
G
elements invariant under G.
Proof . For each fixed v € V, the function Horn (V,V) - V
given by h - h(v) is linear (over R); so it commutes with inte-
gretion. That is,
Iv = \ gv.
Jg£G
It is now clear that Im (I) c V ; for
G
g' (Iv) = g'\ gv
g€G
= \ g'gv (since g acts linearly)
'g£G
\ gv (invariance of integration under
left translation)
iv;
so Iv 6 V Also we have l|V = 1; for if v 6 V then
G G G
I\ Ç w
-I v
Jg6G
The proposition follows.

36 LECTURES ON LIE GROUPS
Propositions 3. 16 and 3. 18 may be viewed as
applications of the principle embodied in 3.15; they could equally
easily be proved directly.
3.16 PROPOSITION. Let G be a compact topological group
and let V be a G-space over C. Then we can give V a positive
definite Hermitian form H which is invariant under G.
Moreover, if V carries a structure map j, we can choose H so that
H(jvjw) = H(v,w).
The reader who wishes to do so may check that if V has
a structure map j, then a Hermitian form with the property
stated amounts to a Hermitian form over A = R or Q according
to the case. The statement we have given avoids separating
cases, and is convenient for later use.
Proof. Consider the space L of Hermitian forms H on V.
This is a vector-space over R, and G acts on it by
(gH)(v,w) = Hfo-'v^w).
By 3.15, if we take any Hermitian form H and integrate gH,
we get a Hermitian form invariant under G, given by
K(v,w) =^ Hfg-^g-'w).
Jg<EG
If we start by choosing H to be positive definite, then K is

ELEMENTARY REPRESENTATION THEORY 37
positive definite.
Now suppose that V has a structure map j, and that we
begin by choosing a positive definite Hermitian form H
invariant under G. Then we can construct a new form by integrating
over the Z3 or Z4 group generated by j; the formula is
K(v,w) = j(H(v,w) + H(jv,jw)).
This form has the required properties.
If we impose on V an invariant Hermitian form, then
we can choose in V an orthonormal basis. Thus we can regard
6 : G ~* Aut V as taking values not merely in GL(n,C), but in
U(n). We then speak of a unitary representation. Similarly
for orthogonal and symplectic representations in the cases
A = R and Q.
3.17 COROLLARY. If G is compact and A= C, then V* = tV.
Proof . Impose on V an invariant positive definite Hermitian
form H. To be explicit, suppose that H(v,w) is conjugate-
linear in v and linear in w. Then we can define
a : tV - V*
by
(av)w = H(v,w),
and a is a G-isomorphism over C.

38 LECTURES ON LIE GROUPS
3.18 PROPOSITION. If G is a compact group, then every
G-space V is projective. That is, suppose given the following
diagram of AG-maps, in which ß is onto.
X
Then there is a AG-map y : V - X such that the following
diagram is commutative.
y V
V—^ Y
Proof. Consider Hom^ (V,X), made into a G-space as in
3.9. By 3.15, if we take any A-map Ö : V - X and integrate
go, we get a A-map y which is invariant under G, that is, a
AG-map. It is given by
y = V (eyg)à(e..g^).
Jg6G X V
We can choose 6 to be a A-map such that ßo = a. Then we
have
ßy = ßV (eYg)6(6 g"1)
Jg€G X V
= $ ß(exg)6(evg-)
gtG
= $ (6Yg)ß6(evg-1)
g€G

ELEMENTARY REPRESENTATION THEORY 39
Jg6G
3.19 DEFINITION. A non-zero G-space V is reducible if
some proper subs pa ce of V is a G-space; otherwise irreducible,
3.20 THEOREM. If G is a compact group, every G-space
V is the direct sum of irreducible G-spaces.
Proof. By induction over dim^V; so assume the result
true for G-spaces W with dim . W < dim. V. It will now be
sufficient to show that if V is reducible, then it is the direct
sum of two subspaces of less dimension. Suppose that V has
a proper subspace S which is a G-space; then 3.18 shows
that the exact sequence
o-s-v-v/s-o
splits, so we have a AG-isomorphism
V = S © V/S.
Alternatively, if A = C we may complete the argument
by imposing on V a Hermitian form H which is invariant under
G, and taking T to be the orthogonal complement of S; then

40 LECTURES ON LIE GROUPS
V = S @ T.
If V has a structure map j, and S is closed under j and H is as
in 3.16, then T is closed under j.
3.21 EXAMPLE. We will show that 3.20 does not hold for
groups which are not compact.
Embed R1 in R2 as the subspace of vectors I . I. Let G
be the subgroup of GL(2,R) which stabilises R1 . Equivalently,
G is the set of matrices I J with ac / 0.
Then R2 is a reducible G-space. However, no other proper
subspace of Rs is stable under G, so R2 does not split as the
direct sum of irreducible G-spaces.
Alternatively, to get a "minimal" counter-example,
take the group G to be the set of matrices I I.
Next we shall need to know to what extent the
decomposition of a G-space into irreducible summands is unique
(3.24). For this purpose we need the following classical
result.
3.22 (SCHUR'S LEMMA). Let G be any topological group.
(i) If f : V - W is a AG-map and V,W are irreducible then
f is either zero or an isomorphism.

ELEMENTARY REPRESENTATION THEORY 41
(ii) If A = C, f : V -* V is a CG-map and V is irreducible
then fv = Xv for some constant X € C.
(In the second case we may write f = X.)
Proof.
(i) Since V and W are irreducible, Ker f is V or 0 and Im f
is 0 or W. The result follows.
(ii) Consider f - X : V - V, where X runs through C. This
map is singular for some X. By (i), f - X is then zero. Thus
f = X.
3.23 COROLLARY. Let V and W be irreducible AG-spaces.
(i) If V and W are inequivalent then Horn (V,W) = 0.
AG
(ii) If V and W are equivalent and A = C, then
dimcHomCG(V,W) = 1.
(iii) If V and W are equivalent and A = R or 0, then
dim Horn fV,W) > 1.
R AG
Proof. For (iii) we observe that Horn (V,W) contains at
AG
least one isomorphism.
For the next proposition, let G be any topological
group, and let V. run over the inequivalent irreducible
AG-spaces (as i runs over some set of indices I). Let m., n. be

42 LECTURES ON LIE GROUPS
non-negative integers, of which all but a finite number are
zero. Let m.V. be the direct sum of m. copies of V., and simi-
11 11
larly for n.V..
il
3.24 THEOREM. If ©m.V. is equivalent to ©n.V., then
iii iii
m. = n. for all i.
l l
Proof . Suppose
©m.V. = ©n.V.,
iii iii
Then
Horn (V. ,©m V ) = Horn .JV„®nV),
A La Jjll AUr J i 1 1
that is,
©m.Hom 4 _(V.,V.) = ©n.Hom A (V., V.).
i i AG J i ii AG j i
Using 3.2 3(i), we get
mHom (V V.) = nHom (V.,V).
J AG j j j AG j j
Taking the dimension of both sides and using 3.23(ii) or (iii),
we get m. = n..
J J
If G is compact and A = C we can express the situation
which arises herein the following way (which we need for later
use). For any G-space V over C we can form
®HomCG(V.,V) ®CV..
This is a finite sum, since Horn (V.,V) is zero for all but a
Ou i

ELEMENTARY REPRESENTATION THEORY 43
finite number of i, by 3.20 and 3 .2 3 (i). We can define
H : ©Horn (V V) ® V -V
1 L/Lz 1, L/ 1
by evaluation:
W(h. ® v.) = h.(v.).
l l il
We make G act on $Hom (V. ,V)®_ V. by
i OG i Ci
g (h. ® v.) = h. ® gv..
i i i i
Then \i is a G-map over C.
3.25 LEMMA. Assume G compact and A = C. Then the map
\x : ©HomcG(V./V)®cVi-V
is an isomorphism.
Proof . If V is irreducible the result is immediate by 3.23.
Pass to direct sums and use 3.20.
3.26 DEFINITION. Let G be a compact topological group.
Then K. (G) is the free abelian group generated by the
equivalence classes of irreducible G-spaces over A.
Tnus an element of K . (G) is a formal linear combina-
A
tion En.V,, in which the V. are the equivalence classes of
iii i
irreducible G-spaces over A, and the n. are integers (positive,
negative or zero) which are zero for all but a finite number of i.

44
LECTURES ON LIE GROUPS
By 3.20 and 3.24, the equivalence classes of G-spaces over
A are in 1-1 correspondence with those elements Ln.V. in
iii
K (G) such that n. > 0 for all i.
A i
An element of K. (G) is called a virtual representation
or virtual G-space.
The operations c,c' ,r,q and t of 3.5 induce homo-
morphisms of abelian groups as displayed in the following
diagram, which is not commutative.
■ KC(G),
VG)
KR(G)
KC(G)
The equations of 3.6 continue to hold.
3.27 PROPOSITION. The maps
c : KR(G) - KC(G)
c' : KQ(G) - KC(G)
are mono
Proof . re = 2, qc' =2 and K (G), K (G) are free abelian.
K Q
We shall normally regard K (G) and K (G) as embedded
R v
in K (G) by c and c'.

ELEMENTARY REPRESENTATION THEORY 45
3.28 COROLLARY
(i) If V and W are two G-spaces over R such that cV ^ cW,
then V ~ W.
(ii) If V and W are two G-spaces over Q such that
c'V = c'W, then V = W.
This follows immediately from 3.27, but in case it
seems to spring from nowhere we also give a direct proof.
Suppose given two G-spaces V, W over C, which admit
structure maps jw,jTAr such that j ® = j^r . We suppose given a
V W V W
CG-isomorphism f : V ~* W which does not necessarily commute
with j, and we wish to construct a CG-isomorphism which does
commute with j. We can construct CG-maps which do commute
with j by starting from f, or if, and integrating over the Z3 or
Z4 group generated by j. The formulae are
We have f - if" = f. So det(f'+zf") is a polynomial in z which
is not identically zero (for it is non-zero for z = -i).
Therefore there is some real x for which det(f + xf") / 0. Then
f + xf" is a CG-isomorphism which commutes with j.

46 LECTURES ON LIE GROUPS
TRIVIAL EXERCISE. If V admits a structure map j, then it also
admits -j, and clearly forgetting j gives the same result as
forgetting -j. Display a CG-automorphism sending j into -j.
If A = C, we can make K (G) into a ring by using the
tensor product of G-spaces over C; we then call it the
representation ring of G. If x lies in K. (G) c K (G), where
A = R or 0, and y lies in KA,(G) c K (G), where A' = R or Q,
then the product xy behaves as described in 3.7.
The standard method of studying K (G), and indeed the
standard method of proving 3.24, is the study of characters.
To define these, we need the trace.
3.29 DEFINITION. Let V be a finite-dimensional vector
space over C, and let f : V - V be a linear map. Then we may
define Tr f, the trace of f, in two ways.
(i) Take a base of V, so that f corresponds to a matrix M...
ij
Set Tr f = EM... This is invariant under change of base, since
i "
£ T..M., (T-1), . = Z I.. M., = EM...
i,j,k iJ Jk ki j,k Jk jk j jj
(ii) (Bourbaki) We have an isomorphism
a : V* ®V - Horn (V,V)
given by (a(v* ® w))v = (v*v)w, as in 3. 13.

ELEMENTARY REPRESENTATION THEORY 47
We have an evaluation map
c : V* ®V - C
given by € (v* ®w) = v*w. Define Tr f = €a-1f.
It is easy to check that the two definitions are
equivalent. The principal properties of the trace are as follows.
3.30 PROPOSITION.
(i) Tr : Horn (V,V) - C is a linear map.
(ii) Consider V -£-> W ■%-*> V. Then Tr(ßy) = Tr(yß).
(iii) Consider ß©y:V©W-V©W.
Then Tr(ß © y) = Trß + Try.
(iv) Consider ß ®y : V ®W - V ®W.
Then Tr(ß ®y) = Trß • Try.
(v) Given ß : V - V, define ß* : V* - V* by (ß*v*)v=v*(ßv),
as usual. Then Trß* = Trß.
(vi) Given ß : V - V, let tß : tV - tV be as in 3.5 (v). Then
Tr(tß) = Trß.
(vii) If ß : V -♦ V is idempotent, then
Trß = dim Imß.
The proof may safely be left to the reader.

48 LECTURES ON LIE GROUPS
3.31 DEFINITION. Given a G-space over C, we define its
character \v : G ~* C by
Xv(g) = Tr6g.
It is clear that \ depends only on the equivalence
class of V.
If V is a G-space over R or Q, we define its character
to be that of the complex G-space cV or c'V as the case may
be. (In the case A = R it would be equivalent to consider the
trace over R, but this doesn't work so well for A = Q.)
3.32
(i)
(ii)
(iii)
(iv)
(v)
(vi)
PROPOSITION
Xv : G "* C is continuous.
Xv(xyx"x) = xv(y).
*v©w(g)=xv(g) + xw(g)-
xv®\^g)=xv{g) ' %{gK
xv*(g) = xv(g"1).
Xtv(g) = Xyte)* if V is real or quaternionic then
xv(g)= xv(g).
(vii) xv(e) =dimc(V).
Each part follows from the corresponding part of 3.31;
the second half of (vi) uses also the equations 10 = 0,10' = c'

ELEMENTARY REPRESENTATION THEORY 49
from 3.6.
3.33 PROPOSITION. Assume G compact.
(i) xv(g ) = xv*(g) = xtv(g) = xv(g).
(ii) \ Xwfe) = dim V , where V is the subspace of
Jg€G V C G G
elements of V invariant under G.
Proof.
(i) See 3.17.
(ii) Since Tr is linear, we have
Ç Tr6g = Tr C 6 g
Jg€G g€G
= Tr I (see 3.15)
= dim Im I (see 3.30(vii))
= dim V (see 3.15).
C G
3.34 THEOREM. (Orthogonality relations for characters.)
(i) Let G be compact and let V,W be G-spaces over A.
Then
5 xv(g)xw(g) = dim HomAG(v/w)
g€G
= d say,
where the dimension is taken over C if A = C, over R if A = R
or Q.

5 0 LECTURES ON LIE GROUPS
(ii) Now assume that V and W are irreducible. If V and W
are inequivalent we have d = 0. If V and W are equivalent and
A = C we have d = 1. If V and W are equivalent and A = R or
Q we have d > 1.
Proof .
(i) By 3.10 the cases A = R and Q follow immediately from
the case A= C. So suppose A= C, and consider
H = Horn (V,W). We have
dimcHomCG(V/W) = dim^
= (\ XH(g) by3.33(ii)
*g€G
= S *v*®w(g) by3-13
Jg<EG
= [ X^(g)xw(g) by3.32(iv)/ 3.33(i).
Jg6G
(ii) See 3.23.
Let us choose one irreducible AG-space V. in each
equivalence class, as in 3.24, and let x- be its character.
Then the functions \- are orthogonal, and therefore:
3.35 COROLLARY. The functions \. are linearly independen
This fact can evidently be used to give a second proof

ELEMENTARY REPRESENTATION THEORY 51
of 3.24. If
©m.V. = ©n.V.,
ill ill
then their characters are equal, so
i rn i iAi
and m. = n. for each i. But by using 3.34(i), we see that this
proof coincides with the first proof.
Let C(G) be the set of continuous functions f : G "* C.
3.36 DEFINITION. Such an f is called a class function if
f(xyx"1) = f(y).
We write C1(G) for the set of class functions. We
make C1(G) into a ring by pointwise addition and multiplication
of functions .
Characters are class functions, by 3.32 (i) and (ii).
We can define a homomorphism of abelian groups
X : KC(G) -C1(G)
by
X(£n.V.) = En.x. •
A i l l i iAi
For every G-space V we have
x(v) = xv.
by 3 . 32 (iii). x is a homomorphism of rings by 3.32(iv).

52 LECTURES ON LIE GROUPS
3.37 PROPOSITION, x : K (G) - C1(G) is a monomorphism,
Proof . See 3.35.
The image of x is called the character ring of G. It is
natural to ask how large a part of C1(G) it is; and we will see
that it is as large as could be hoped (3.47). For this purpose
we need the Peter-Weyl theorem.
We recall that classically the Peter-Weyl theorem is
stated in terms of component-functions M..(g) of matrix
representations M(g). Clearly such a function is obtained by
taking a matrix representation M : G "* GL(n,C) and composing
with a linear map GL(n,C) ~* C, namely projection onto the
(i,j)th component. We therefore introduce the following
lemma.
3.38 LEMMA. The vector-space dual to Horn (V,W) is
Horn (W,V), where the pairing between a 6 Horn (V,W) and
o c
ß 6 Horn (W,V) is given by
<ß,a> = Tr(aß) = Tr(ßa).
Proof . We have Horn (V,W) = V* ®Wf and therefore its
dual space is
V ®W* = W* ®V=Hom (WfV).

ELEMENTARY REPRESENTATION THEORY 53
To check that the pairing is as claimed is precisely what the
student should already have done in proving Tr(aß) = Tr(ßa)
(3. 30(ii)).
3.39 THEOREM (F. Peter and H. Weyl) [ 15] . Let G be a
compact topological group. Then every continuous function
f : G "* C can be uniformly approximated by functions of the
form Tr(a6 (g)), where 8 runs over representations
8 : G - Horn (V,V) and a runs over Horn (V,V).
The proof will occupy 3.40 to 3.44. Actually our line
of proof will approximate f by functions of the form Tr(a6 (g-1 ));
but this makes no difference, since we can begin by replacing
f with f , where f (g) = f (g_1).
The proof is based on the following ideas. We make G
act on C(G) by
(gf)(x) =f(g-xx).
Then C(G) is an infinite-dimensional representation of G.
However, we can find certain finite-dimensional subspaces of
C(G) stable under G, by using the theory of integral operators
Ç k(x,y)f(y).
Jy<EG
We make G act on C(G x G) by

54 LECTURES ON LIE GROUPS
(gk)(x,y) = k(g-1x/g-1y).
If the "kernel" k is invariant under G, then the integral
operator gives a G-map from C(G) to C(G), and hence its eigen-
spaces are stable under G. In the case at issue they are
finite-dimensional (3.42) and this provides the necessary
representations .
We now start work.
3.40 LEMMA. Let G be a compact group and f 6 C(G).
Then f can be uniformly approximated by functions of the form
v(x)=C k(x/y)f(y)
Jy<EG
with k real symmetric and invariant under G.
Proof . There is a neighbourhood U of e in G such that
|f(x) - f(y) I < € forx"Ve U
and U"1 = U. Let ß : G ~* R be a continuous function such that
/Lt(x) = 0 for x i Ü,
fi(x) > 0,
^(x"1) = fi(x) and
Jx€G
Let
k(x,y) = Ji(x-1y).

ELEMENTARY REPRESENTATION THEORY 55
Then k is real, symmetric and invariant under G. Also
|jü(x-*y)f(x) - MX* y)i{y) \ ÉCjifr^y)
everywhere. Integrating over y € G, we get
|f(x) - v(x) | < €
where
v(x) =[ k(xfy)f(y).
Jy€G
It will now be sufficient to approximate such functions
v(x).
3.41 THEOREM. Assume that k is Hermitian and u € C(G).
Then the function
v(x) = \ k(x,y)u(y)
Jy<EG
can be uniformly approximated by a finite linear combination
of eigenfunctions of k corresponding to non-zero eigenvalues.
The eigenfunctions corresponding to the eigenvalue X
are, of course, the functions w such that
\ k(x,y)w(y)= Xw(x).
y€G
Proof . See [16, p. 117, 127]. (Smithies considers
integral equations on [a,b], but the results are unchanged for
integral equations on a compact manifold.)
Note also that even if we were to consider some class

5 6 LECTURES ON LIE GROUPS
of functions larger than C(G), for example L2 (G), the eigen-
functions would be continuous, since k is continuous.
3.42 THEOREM. Assume that k is Hermitian and \/ 0.
Then the vector space V of eigenfunctions corresponding to X
has finite dimension. Indeed the sum E| X. | 2, in which | X | 2
i 1
is repeated with appropriate multiplicity, is convergent.
Proof . See [16, pp. 48,102,112].
3.43 LEMMA. In 3.42, assume further that k is invariant
under G. Then every element of V can be written in the
required form Tr(a6(g-1)).
Proof . The space V is a finite-dimensional G-space. Let
v 6 V. Define a linear map
ß : Homc(V,V) - C
as follows: if h € Horn (V,V), then
ß(h) = (hv)(e).
Then we have
p(6(g-x)) = v(g).
By 3.38, the element ß corresponds to an element
a € Horn (V,V) such that

ELEMENTARY REPRESENTATION THEORY 5 7
Triefe"1)) = v(g).
3.44 LEMMA. The set of continuous function G ~* C which
can be written in the form Tr(a6 (g"1)) is closed under linear
combinations.
Proof. Suppose given
6' :G-Hom (V'.V), 6" : G - Horn (V" ,V")
a' 6Hom (V\V')f a" € Horn (V ,V")
and X' ,X" € C. Form V = V © V" and consider
a = X' a' © X"a" É Horn (V,V). Then we have
Trfoefo-1)) = X'Trîa-e'Cg-1)) + X»Tr(Q"e" (g"1)).
This completes the proof of 3. 39; any function f (x) can
be uniformly approximated by a function v(x) as in 3.40, which
in turn can be uniformly approximated by a linear combination
of eigenfunctions by 3.41; and this can be written in the
required form Tr(Qe (g-1)) by 3.42-3.44.
3.45 REMARK. If a : V - V is a G-map, then Tr(a6 (g)) is a
class function.
Proof . Trketxyx"1)) = Tr(a(6x) (6 y) (Gx"1))
= Tr((ex-1)a(6x)(ey)) (3.30(ii))

58 LECTURES ON LIE GROUPS
= Tr(Qe (y))
since a is a G-map.
There is a converse to this remark.
3.46 PROPOSITION. Let G be a compact group. Then
every class function f : G — C can be uniformly approximated
by functions of the form Tr(ß6 (g)), where 6 runs over
representations 6 : G -♦ Horn (V,V) and ß runs over Horn (V,V),
G CG
that is, ß runs over G-maps .
Proof . By 3.39 we can find 6 : G - Horn (V,V) and
a € Horn (V,V) such that
|f(x) - Tr(Q6(x)) | < C
If f is a class function we can substitute y-:ixy for x and get
|f(x) -TriaB(y'lxy)) | <€.
Arguing as in 3.45, this gives
|f(x) -Tr{iey)aiey"x)iBx))\ <€.
Integrate over y; we find
|f(x) -Tr(p(6x))| < €
where
ß = <\ (ey)a(ey-1).
Jy€G
But as in 3. 18, ß is a G-map.

ELEMENTARY REPRESENTATION THEORY
59
3.47 THEOREM. Let G be a compact topological group.
Then every class function f : G -* C can be uniformly
approximated by a linear combination ZX.\. of irreducible complex
ru
characters.
Proof . Let 6 and ß be as in 3.46, and let V. and y. be as in
3.25. Then the G-map ß : V - V induces (say)
We have the following commutative diagram.
© Horn (V V) ®V.
1 K-AJ 1 1
^ V
© ß.® 6. (g)
i ! 1
P6(g)
-» V
©Horn (V V) ® V L
i \-j\j l i =
Therefore
Tr(ß6(g)) = Z (Trß.) • Tr(6.g),
which has the required form TX.\.{g).
It is natural to ask for the analogue of 3.47 over R or
Q. By 3.6 we have tcV = cV, tc'V = c'V; so by 3. 32 (vi) the
character of a representation over R or Q is real, and by
3 . 33 (i) it satisfies xfe"1) = x(ç)-
3.48 COROLLARY. Every class function f : G - R such that
f (g) = f (g-1) can be uniformly approximated by an R-linear

60 LECTURES ON LIE GROUPS
combination of characters of representations over R, or by an
R-linear combination of characters of representations over Q.
Proof . Let f : G -* R be a class function such that
f(g) = f(g-1). By 3.47 we can find complex X. such that
|f(g) -£Av (g)| <€.
i 1 *
Since f(g) = f(g-1) we have
|f(g) -EXv.te-MI < e,
i
or using 3. 33(i)
|f(g) -?xxlg")| <€.
1 ! !
Since f is real we also have
|f(g) -£Xxlg~)| <€
i 1 l
|f(g) -EXv(g)| <€.
i * *
Therefore
|f(g) -?j(X. + X)(X.(g) + xlg))| <c.
\ T 1 11 1
But here —IX. + X.) is a real coefficient and v. + y. is the
4 i i Ai Ai
character of the real representation rV. or of the quaternionic
representation qV. (see 3.6).
3.49 COROLLARY. Every class function f : G - C such that
f (g) = f(g-1) can be uniformly approximated by a C-linear
combination of characters of representations over R or by a

ELEMENTARY REPRESENTATION THEORY 61
C-linear combination of representations over Q.
Proof . Approximate the real and imaginary parts of f by
3.48.
We now consider in greater detail which complex
representations are real or quaternionic.
3.50 THEOREM. A representation V over C is real if and
only if there exists a non-singular symmetric bilinear form
B : V ® V - C which is invariant under G.
A representation V over C is quaternionic if and only if
there exists a non-singular skew-symmetric bilinear form
B : V ®V -♦ C which is invariant under G.
Proof . First suppose that V carries a structure map j such
that j2=€ = ±l. By 3.16 we can impose on V a positive
definite Hermitian form H which is invariant under G and
satisfies
H(jv,jw) = H(v,w).
Define
B(v,w) = H(jv, w).
Then B is clearly bilinear, non-singular and invariant under G.
Also we have

62 LECTURES ON LIE GROUPS
B(w,v) = H(jw,v)
= H(v,jw)
= H(jv,j2w)
= €H(jv/W)
= cB(v,w).
So B is symmetric or antisymmetric according to the sign of €.
We now seek to reverse this argument. Suppose given
on V a non-singular bilinear form B : V ®V -* C which is
invariant under G and satisfies
B(w,v) = cB(v,w),
where c = ±1. By 3. 16, we can also suppose that V carries a
positive definite Hermitian form H which is invariant under G.
Then we can define f : V - V by
B(v,w) = H(fv,w).
The map f is conjugate-linear, a G-map and a 1-1
correspondence. The property of B gives
H(fv,w) = B(v,w)
= cB(w,v)
= €H(fw,v)
= cH(v,fw).
Thus

ELEMENTARY REPRESENTATION THEORY 63
3.51 H(fv,w) = €H(v,fw).
We now define another positive-definite Hermitian form
on V by
3.52 K(v,w) = H(fv,fw).
The form K is invariant under G. Substituting fv and fw into
3.51, we get
H(f2v,fw) = cH(fv,f2w),
and taking complex conjugates, we get
3.53 K(fv,w) = cK(v,fw).
The space V now splits as the direct sum of eigenspaces V. for
the pair cf forms H,K. The eigenvalues are positive real
numbers X.; for each such X., V. is the set of v. such that
l il l
3.54 K(v.,w) = X.H(v.,w) forallw€V.
l il
The eigenspaces V. are stable under G. I claim they are also
preserved by f; for we have
K(fv.,w) = cK(v.,fw) (3.53)
= cX H (v., fw) (3.54)
il
= X.H(fv.,w) (3.51).
Thus fv € V. .
l l
We also have

64 LECTURES ON LIE GROUPS
H(f3v.,w) = cH(fv.,fw) (3.51)
= €K(vrw) (3.52)
= cX.H(v.,w) (3.54).
il
So f2 I V. = cX., where X. is real and positive.
1 l l l
Let us now define a map j : V -* V by
j| V. = (X.f*f|v..
1 l l ' l
Then j is conjugate-linear, a G-map and satisfies j2 = €.
Thus V is real or quaternionic according to the sign of €. This
completes the proof.
To sum up, the advantage of structure maps is that
they come normalised by the condition j2 = ±1; the
disadvantage of bilinear maps is that they can be denormalised by a
scalar factor for each summand of V.
3.55 DEFINITION. We say that a representation V of G is
self-conjugate if tV = V. Evidently representations over R and
Q are self-conjugate (either using 3.6 or using the fact that
the structure map j gives an isomorphism from tV to V).
3.56 PROPOSITION. If a complex irreducible representation
V of G is self-conjugate, then it is either real or quaternionic,
but not both.

ELEMENTARY REPRESENTATION THEORY 65
Proof. Consider V* ® V*, the space of bilinear maps from
V ®V to C. It has an automorphism r defined by
t(v* ®w*) =w* ®v*.
We have T2 = 1. So V* ®V* splits as the direct sum of the +1
and -1 eigenspaces of T. The +1 eigenspace is the space S*
of symmetric bilinear maps; the -1 eigenspace is the space
A* of antisymmetric bilinear maps.
Now we also have
V* ®V* = Horn (V,V*).
By 3.17 we have V* ^tV, and if V is self-conjugate we have
V* = V. If V is irreducible then so is V*, and we have
dimcHomCG(V,V*) = 1
by 3.23. That is, for the elements invariant under G we have
dimcS£. + dim^^ = 1.
Moreover, a non-zero bilinear map B which is invariant under
G corresponds to a non-zero G-map V -*V*, which must be
iso; so such a B is non-singular. We conclude that only two
cases are possible.
(i) dim S* = 1, dim^* =0. In this case V admits a
non-singular symmetric bilinear form invariant under G, but
not an antisymmetric one.

66 LECTURES ON LIE GROUPS
(ii) dim S* =0 , dim A* =1. In this case V admits a
CG CG
non-singular antisymmetric bilinear form invariant under G,
but not a symnetric one.
The result follows by 3.50.
3.57 THEOREM. Suppose given a compact group G. Then
it is possible to choose representations U over R, V over C
m n
and W over Q to satisfy the following conditions.
(i) The inequivalent irreducible representations over R ore
precisely the U , rV and rc'W .
m n p
(ii) The inequivalent irreducible representations over C are
precisely the cU . V , tV and c'W .
m' n' n p
(iii) The inequivalent irreducible representations over Q are
precisely the qcU , qV and W .
m' n p
Proof. We begin by taking the irreducible complex
representations V. First, we can classify them into those such that
tV = V and those such that tV ^ V. The latter occur in pairs
(V,tV),and we choose one V out of each pair. The former are
either real or quaternionic by 3.5 6; we choose U over R, W
m p
over Q so that the cU and c'W give such V. It is clear that
m p
this choice makes 3.57(ii) true.

ELEMENTARY REPRESENTATION THEORY 67
It is also claimed that the representations U ,rV and
m n
rc'W over R are irreducible, and similarly over Q. In fact,
P
U is irreducible because cU is so. We have
m m
crV = (1 + t)V
n n
crc'W = 2c'W
s p p
and neither can be split into real representations because V
n
and tV are not self-conjugate and c*W is not real. Similarly
n p
over 0.
It remains only to prove that there can be no further
irreducible representations over R or Q. For this purpose we
introduce:
3.58 LEMMA. If V and W are inequivalent irreducible
representations over R, then no complex irreducible representation
can occur as a summand both in cV and in cW. Similarly for
c'V and c'W if V and W are over Q.
Proof, dim Horn (cV,cW) = dim Horn (V,W) (3.10)
O OCj R RGr
= 0 (3.23).
To complete the proof of 3.57, it remains only to
remark that all the complex irreducible representations occur
as summands in

68 LECTURES ON LIE GROUPS
cU
m
crV = (1 + t)V
n n
crc'W = 2c'W .
P P
Therefore there can be no more irreducible representations over
R. Similarly over Q.
There is a classical criterion for deciding whether a
complex irreducible representation is real or quaternionic. For
this purpose we introduce the following considerations.
3.59 DEFINITION. Let Vn = V ®V ® ... ®V (n factors). Let XnV
be the summand of V on which the permutation group E acts by
pW = (Cp)w,
where €.p is the sign of p; that is, X V is the space of
antisymmetric or alternating tensors. The G-space X V is called
the nth exterior power of V.
Consider the power-sum
k k k
xi + x2 + • • • + xm
in m > k variables. This can be written as a polynomial
Pi.(tfi /Cr2 /. .. /Cr, ) in the elementary symmetric functions o
of xx ,x2 ,.. . ,x ; the polynomial is actually independent of m,
and the formula is valid even for m < k.

ELEMENTARY REPRESENTATION THEORY 69
3.60 DEFINITION. If V is a complex representation of G,
we define a virtual representation by
<^k(v) = Pk(x1vfxavf...fxkv).
The polynomial is evaluated in the ring K (G).
3.61 LEMMA. If W = <b V, then
*w(g) = xv(g h
Of course it is clear that Xvfe ) is a class function;
after 3.47 it is natural to ask what it is the character of.
Proof . Impose on V a positive-definite Hermitian form H.
Fix g. Then 6 (g) is a unitary map, and we can find in V a
base of eigenvectors v. with eigenvalues X.. Then V admits
a base of eigenvectors v. ® v. ® ... ® v. with eigenvalues
X. X .. . X. , and similarly for xV Hence g acts on X
H la
V
with trace a , the nth elementary symnetric function of the X..
Hence
X^g) = Pk(ai ,cr2/.../Crk)
k k
= X. + X_ + (by definition of P )
= Tr((6vg)k)
= xv(g )•

70 LECTURES ON LIE GROUPS
3.6 THEOREM. Let V be a complex irreducible
representation of a compact group G. Then
II if V is real
0 if V is not s elf-conjugate
-1 if V is quaternionic
Proof . In 3.56, instead of cons idering V* ® V* = S * © A*,
it is equivalent to consider V ® V "= S (£ A. We have
P2(üi /Cr2) = (ax)2 - 2a2
and so
^2 (v) = V2 - 2A
= S - A.
Therefore
[ xv(g2) =\ Xofe) -xAfe)
Jg<EG Jg<EG b
= dimcSG - dimcAG.
This gives the result.
3.63 REMARK. If V is real then \nV is real. If V is
quaternionic then X V is real for n even, quaternionic for n odd.
Proof . Suppose V admits a structure map j whose square is
c. Then V admits a structure map j ® j ®... ® j whose square
is € , and similarly for

ELEMENTARY REPRESENTATION THEORY 71
3.64 REMARK. If V is real then # V is real. If V is quater-
nionic then ip V is real for k even, quaternionic for k odd.
Proof . If we assign weight i to cr., then P, (o1 ,cr2 ,... ,a, )
is a polynomial of weight k. Now use 3.7.
We now move on to calculate K (G x H) in terms of
K (G) and K„(H). Let V be a G-space and W an H-space (over
C). Then we can form V ® W, and make it a G x H-space by
(g , h) (v ® w) = gv ® hw.
This defines a homomorphism of rings
V : KC(G) ® KC(H) - KC(G X H).
3.65 THEOREM. The map v is an isomorphism. More
precisely, the inequivalent irreducible G x H-spaces (over C) are
precisely the products V. ® W., where V. runs over the in-
equivalent irreducible G-spaces and W. over the inequivalent
irreducible H-spaces.
Given the theorem for K (G x H), it is easy to locate
the representations of G x H over R and Q; for an irreducible
representation V. ® W. is self-conjugate if and only if both V.
l j l
and W. are self-conjugate; and then V ®W. is real or
quaternionic according to the nature of V. and W., as in 3.7.
i J

72 LECTURES ON LIE GROUPS
Theorem 3.65 will follow immediately from the next
two results.
3.66 LEMMA. If V is an irreducible G-space and W is an
irreducible H-space (over C), then V ® W is an irreducible
G x H-space.
3.67 LEMMA. Any G x H-space U (over C) can be expressed
in the form £ n..V. ® W.. In particular, the irreducible
G x H-spaces have the form V. ® W..
i J
F irs t proof of 3.66. We have
Thus
*v®w(g'h) = xv(g) ' xw(h)*
= $ xv(g)xv(g)$ x^)x (h)
= 1.
By 3.20 and 3. 34, V ® W is irreducible.
Proof of 3.67. By 3.25 we have an isomorphism over H:
\i : e Hom_. (W., U) ® W. —^> U .
j H J J
Let G act on Horn (W. ,U) by

ELEMENTARY REPRESENTATION THEORY 7 3
(gk)w = g(kw) for k € Hom__(W. ,U).
H J
(It is easy to check that gk is an H-map.) Then \i is a
G x H-ma p. But by 3.20 we have an isomorphism of G-
mocules
HomTT(W.,U) = © n. V..
H j' i lj l
Thus
U = © n..V. ®W..
Finally, if U is irreducible, it is clear that the sum can
contain at most one factor.
Second proof of 3.66. Suppose that V and W are
irreducible and V ® W has a G x H-subspace S. Then by 3.67 we
have
S = £ n..V. ® W..
i,j i) i J
As H-spaces we have
V ® W = (dim V)W;
so the only W. which can have n.. 4 0 is W. Similarly, the
J ij
only V. which can have n.. 4 0 is V. Hence dim S is a multiple
l ij
of (dim V)(dim W), and S = 0orS = V®W.
We now move on to consider the case of a double
covering it : G~* G. That is, it is an epimorphism of topological
groups and Ker 7T = Z2 = {1 ,z} t say. Ker ti is , of course,

74 LECTURES ONXIE GROUPS
normal in G, and even central since Aut Z2 = 1.
3.68 THEOREM. A character x : G - C factors as
~~ 7T X ~
G —> G —>• C for a character \ if and only if \ factors as a
map of sets. Moreover, x is real if an<^ only if )< is real;
similarly, \ is quaternionic if and only if x is quaternionic.
Proof. "Only if" is trivial. So suppose that V is a
representation of G. Then z acts on V and satisfies z2 = 1. So V
splits as the sum of the +1 and -1 eigenspaces of z, say
V = V © V . Since z is central, both V and V are G-spaces.
We have
è(zg) = 8(g) © (-6"(g))
and taking traces ,
X(zg) = x(g) - x~fa).
If x : G - C factors as a map of sets, then
x(zg) =x(g) = x(g) + x'teh
so x~(g) = 0 and V =0. Clearly V = V is then a representation
of G. If it carries a structure map commuting with the
operations of G, then it carries the same structure map commuting
with the operations of G.
3.69 REMARK. This cheorem is also valid for virtual

ELEMENTARY REPRESENTATION THEORY 75
characters.
3.70 EXERCISE. Extend 3.68 to any finite covering,
assuming G compact connected and A = C.
We now turn to consider the representations of the
torus.
3.71 PROPOSITION. If G is abelian and A = C then every
irreducible G-space V is one-dimensional.
Proof . For each g € G consider 6 (g) : V - V. This is a G-
map because G is abelian. By 3.22(ii), 6 (g) is multiplication
by some scalar X(g). So every subspace of V is stable under
G and dim V = 1.
3.72 REMARK. In 3.71, X(g) e C - {0}.
3.73 REMARK. Suppose that G is a compact abelian group
and V an irreducible G-space, so that 6 may be written as
X : G - C - {0}. Then X(G) c S1 c. C - {0}, where Sl is the
unit circle in C.
First proof . If | X(g) | = r > 1, then | X(gn) | = rn - « .
and if |X(g)| =r < 1, then |X(gn)| = r11 - 0.

76 LECTURES ON LIE GROUPS
Second proof. Give V a positive definite Hermitian form
H invariant under G. Then
H(v,v) = H(gv,gv) = | X(g)| 2H(v,v),
so
|X(g)| = 1.
We recall that T1 was defined to be R/Z.
3.74 PROPOSITION. A homomorphism a : T1 - T1 has the
form a(x) = nx mod 1 for some integer n.
Proof . By 2.11 and 2.13, or by the ordinary theory of
covering spaces, a lifts to a homomorphism ß : R - R. Then
ß(l) = 0 mod 1, so ß(l) = n € Z; and ß(a) = na for a G Z, and
bß(a/b) = ß(a) = na for b GZ, so ß(a/b) = na/b. By continuity,
ß(x) = nx for all x € R, and a(x) = nx mod 1.
3.75 COROLLARY. A homomorphism a : T - T1 has the
form
a(xx ,x2 , . . . ,xk) = n1x1 +. . .+ ry^ mod 1
for some nx ,n2 , . . . ,n £ Z.
3.76 COROLLARY. The irreducible complex T -spaces have
the form
X(xx ,x2 , . . . ,xk) = Exp 2 77i(n1x1 +. . .+ nkxk),

ELEMENTARY REPRESENTATION THEORY 77
where Exp z = ez.
This follows from 3.71, 3.73 and 3.75, since T1 is
isomorphic to S1 under x - Exp 2ttîx.
For 1 <J <_k, let p. be the T -space given by
X(xx ,x2,. .. ,xk) = Exp 2ffix
Then p. is invertible, and for any integers nx ,n2 ,. .. , n
k
(positive, negative or zero) pi p2 . . . p, is the T -space
given by
a(xx ,x2 ,... ,x ) = Exp 2ffi(n1x1 +...+ n x ).
3.77 COROLLARY. K„(T ) is the ring of finite Laurent
series in pi,P2,...,p., and so has no divisors of zero.
We have
. n, nP nicx _n, _nP _nu
Kp/p/ ... Pfck) = P1 lPa s ■■■ Pkk-
J«;
Thus the only irreducible representation of T which is self-
conjugate is the trivial representation 1.
3.78 COROLLARY. The inequivalent irreducible real repre-
sentation of T are
(i) the trivial representation 1 of dimension 1, and
(ii) the representations

78 LECTURES ON LIE GROUPS
r/ftni n2 „nk\_r /-ni -na "njA
rfpi p2 ••• ^ J-r(pi p2 ••• pk ;
for (nx ,n3 ,. .. ,n, ) / (0, 0,. . . ,0), which are of dimension2 .
This follows from the above by the discussion of 3.57.

Chapter 4
MAXIMAL TORI IN LIE GROUPS
Not ice. From 4.5 onwards , G will be a compact connected
Lie group.
4.1 DEFINITION. Let G be a topological group and let
g GG. Let H be the subgroup generated by g. Then g is a
generator of G if cl H = G, where cl denotes the closure.
G is monogenic (or monothetic) if it has a generator.
4.2 EXERCISE. Monogenic implies Abelian.
4.3 PROPOSITION. The torus T is monogenic. Indeed,
generators are dense in T .
Proof . Let Ux , U2 ,.. . be a countable base for the open
k k k k
sets of T . Let T = R /Z have co-ordinates (xx , . . . ,x ).
79

80 LECTURES ON LIE GROUPS
Then a cube is a set {x € T ; |x. - £.| < c} for some fixed
point £, an<3 real ç > 0. Let C0 be any cube. Then we will
define a descending sequence of subcubes whose intersection
will be a generator.
Suppose, inductively, that we have defined
C0 ^ Cx => ==> Cm.! and that Cm^ has side 2c. Then there
is an integer N(m) such that N • 2e > 1, so that the image of
Cm„i under multiplication by N is T . We can find CmcCm_i
such that N • C c U .
m m
Let g € H C . Then g c U , sog is a generator
* m m m
ofTk.
4.4 PROPOSITION. Let G be an Abelian topological group,
k k
with T c G such that G/T =Z . Then G is monogenic.
J«;
Proof. Let t be a generator of T . Choose u G G to pro-
k k k
ject to a generator of Z . Then mu G T and t - mu € T . T
m
is divisible, so there is s £ T with ms = t - mu. Take
g = u + s . Then mg = m(u + s) = t, so the powers of g are
dense in T. Translating by rg, the powers of g are dense in
the coset of T containing ru. This gives all cosets.

MAXIMAL TORI IN LIE GROUPS 81
4.5 NOTICE. From now on G is a compact connected Lie
group.
4.6 DEFINITION. A maximal torus T c G is:
(i) a subgroup which is a torus , such that
(ii) if T c u c G and U is a torus then T = U.
4.7 REMARK. If G is not compact, it need not have any
non-trivial tori.
4.8 PROPOSITION. Any subtorus of G is contained in a
maximal torus.
Proof. Consider a strictly increasing sequence of subtori
Ti c T2 c . . . c G. Then L(TX) c L(T2) c . . . c L(G) is a
strictly increasing sequence, and so is finite.
4.9 PROPOSITION. Let T be a maximal torus of G, and A
a connected Abelian subgroup of G with TcA. Then T = A.
Proof . TcAcclA. But cl A is a closed connected Abelian
subgroup, and is therefore a torus (2.20). Thus T = cl A and
T = A.

82 LECTURES ON LIE GROUPS
4.10 CONSTRUCTIONS. If T is a torus of G, it operates on
Ad
G byTCG > Aut G . Choose a positive definite symmet-
e e
ric form on G invariant under G, and so under T. Then (3.78)
e
G splits into orthogonal irreducible T-spaces of dimensions 1
and 2. Those of dimension 1 are trivial. We can choose an
orthonormal base in those of dimension 2, and represent T by
T -SO(2).
4.11 DEFINITION. The integer lattice of L(T) is exp"1 (e)
where exp : L(T) - T.
4.12 PROPOSITION. L(G) = G splits as a T-space in the
e
form V0 © Lj, Vj, where T acts on V0 trivially, dim V. = 2 for
i > 0 and T acts on V. as
l
cos 27r6.(t) -sin 2^6^)
sin 27rG.(t) cos 2 7T0i(t)
Here 6. : T - R/Z is given by a linear form 6. : L(T) -* R taking
integer values on the integer lattice, and no 6. is zero.
4.13 DEFINITION. If T is a maximal torus , the functions
±0 are called the roots of G. By 3.24 they are well defined
in terms of T. We will see that they are independent of T.

MAXIMAL TORI IN LIE GROUPS
83
4.14 PROPOSITION. T is maximal if and only if V0 = L(T).
Proof . It is clear that L(T) c V0.
(i) Suppose V0 = L(T) and TcT'. Then
L(T) c L(T') cV0' cv0( so L(T) = L(T') and T = T'.
(ii) Suppose V0 / L(T). Then there is X € V0 , X / L(T).
Now exp(tX), for t £ R, is a 1-parameter subgroup H of G on
which T acts trivially, and which is not contained in T.
Therefore the subgroup generated by T and H is a connected
Abelian subgroup strictly containing T, so T is not maximal.
4.15 COROLLARY, dim G - dim T is even.
4.16 EXAMPLE. Let G = U(n), and let T be the set of
diagonal matrices:
D = f exp 27TÎX!
exp 2 7T ix
n J
LU(n) can be decomposed into the following summands,
(i)
Matrices
id,
id
n J
with d. real.
J

84
LECTURES ON LIE GROUPS
This is L(T).
(ii) Matrices
M =
rs
-S
for r < s . Then
DM D"*1 =
rs
-w
w
where w = exp[27T i(x - x ) ]z and 6 = x - x .
r s rs r s
The matrices (i) and (ii) generate L(G), so V0 = L(T)
and T is maximal. The roots are (x - x -).
r s
4.17 EXAMPLE. LetG = SU(n). Then the matrices M of
rs
the previous example are in LSU(n), since the derivative with
respect to t of |l + tM | at t = 0 is zero. Similarly, matrices
of type (i) with Ed. = 0 are in LSU(n).
Let T be the set of diagonal matrices

MAXIMAL TORI IN LIE GROUPS
85
D = exp 2 77ixx
"exp 2 TT ix
n J
withLx. = 0. The functions (x - x ) are still nontrivial, so
i r s
V0 = L(T), T is maximal, and the roots are (x - x ).
4.18 EXAMPLE. Let G = Sp(n), and let T be the set of
diagonal matrices
D = exp 2 it ixx
exp 2?rix
n
L Sp(n) splits into the following summands,
(i)
Matrices
r idi
id
with d. real
l
(ii) Matrices
M =
rs
-z
with z € C.

86
LECTURES ON LIE GROUPS
(iii) Matrices
N =
r
zj
with z € C. Here
DND= r
r
exp(2 7Tix )zj exp(-277LX )
exp(47Tix )zj
(iv) Matrices
Prs =
zj
zj
with z € C. Here

MAXIMAL TORI IN LIE GROUPS
DP D"1 =
rs
exp 2ïïi(x +x )zj
r s
exp2 7Ti(x +x )zj
r s
Thus V0 = L(T), T is a maximal torus , and the roots are
±2x , (x -x )andt(x +x ) for r 4 s.
r r s r s
4.19 EXAMPLE. LetG = SO(2n). We have U(n) c SO(n).
Take T to be the image of the maximal torus we had in U(n)
That is, T is the set of matrices
D
'Di
a
D
nJ
w
here D. = T cos 2ttx.
1
I sin 2ttx.
-sin 2ttx.
cos 2ttx.
l J
Now LSO(2n) splits into the following summands.
(i) L(T), consisting of matrices
0
di
-di
0
•.
0
d
n
-d
n
0

88
LECTURES ON LIE GROUPS
(ii) The rest of LU(n), consisting of matrices
M
rs
-W"
W
whore W =
x -y
L y x
Then DM D-1 = M* with
rs rs
cos 2 7T (x - x )
r s
s in 2 TT (x - x )
r s
-sin2î7(x -x )
r s
cos 2 7t(x -x )
r s
So T acts with 0 = x - x .
rs r s
(iii) Let
£ =
s
and take matrices E M E-1 . In this case, T acts with
s rs s

MAXIMAL TORI IN LIE GROUPS
89
e = x + x
rs r s
Thus V0 = L(T), T is a maximal torus , and the roots
are (x - x ). ± (x + x ) for r ^ s .
r s r s
4.20 EXAMPLE. Let G = SO(2n + 1). We have
SO(2n) c SO(2n + 1) by letting SO(2n) act on the first 2n
coordinates. Let T be the maximal torus we had in SO(2n).
Then LSO(2n + 1) splits into the following summands.
(i) LSO(2n).
(ii) Matrices
F =
r
-x -y
Here D acts by rotation through x .
Thus V0 = L(T), T is a maximal torus, and the roots are
±x , (x - x ) and ±(x + x ) for r ^ s .
r r s r s
4.21 THEOREM. Let T c G be a maximal torus. Then any
g G G is contained in a conjugate of T.

90 LECTURES ON LIE GROUPS
Proof . (Following A. Weil [21]; see also [ 11] .)
Consider the left coset space G/T, and let f : G/T-G/T
be induced by left multiplication by g, that is, f(xT) = gxT.
Then a fixed point of f is a coset xT with gxT = xT, that is ,
g € xTx"1 . So we only need to show that f has a fixed point.
We will use the form of Lefschetz's fixed point theorem given
by A. Dold [6], (This theorem applies to manifolds rather
than simplicial complexes). We summarise what we need:
Let f : X -X be a continuous map, and define A(f) £Z by
taking f* : Hq(X; Q) -Hq(X; Q) and setting A(f)=E(-l)q Tr f*.
q
Then A(f) depends onlyon the ho mo top y class of f. If f has no
fixed points, then A(f) = 0. If f has only isolated fixed points
(and so a finite number of fixed points), then A(f) is the number
of fixed points counted with multiplicity, which is defined as
follows. Let X be a smooth manifold and x a fixed point of f.
Consider 1 -f :X -X . If det(l -f) >0, then f has multi-
xx
plicity +1 at x: if det(l - f *) < 0, then the multiplicity is -1.
We do not need to discuss the case det(l - f *) = 0.
To compute A(f) we may replace f with any homotopic
map f0. So we may replace g with any other g0 € G, since G
is path-connected. Take g0 to be a generator of T (4.1), and

MAXIMAL TORI IN LIE GROUPS 91
let f0 be the corresponding map. Then the fixed points of i0
are the cosets nT for n in N(T), the normaliser of T in G (as
the reader will easily verify). Let us examine N(T).
N(T) is a closed subgroup of G, and so is a Lie group
(2.27, 2.2 6), and the identity component N(T)a is open and so
has only a finite number of cosets. Now N(T)a = T, which we
see as follows. N(T) acts on T by conjugation (i.e. ,
n(t) = ntn-1) and Aut T is discrete, so N(T)a acts trivially.
(The reader should verify that N -»Aut T is continuous with this
topology on Aut T. Note that this map arises from the map
NxT - T which is a restriction of the map GxG - G given by
(g,h) -* ghg-1). If N(T)a properly contains T it contains a 1-
parameter subgroup not contained in T but computing with T,
contradicting the maximality of T. It follows that N(T)a = T,
that T has only a finite number of cosets in N(T), and that f0
has only a finite number of fixed points.
It suffices to consider just one of these fixed points,
say T, as follows. Let nT be another fixed point. Define
rn : G//T " G//T by r *gT) = gTn* This is a well"defined diffeo-
morphism, commutes with f0, and takes T to nT. Thus the
multiplicity at nT is the same as at T.

92
LECTURES ON LIE GROUPS
Observe that f0 can also be defined as f 0 (xT) = g0*gö T.
That is, f0 is obtained as a quotient of the map G - G given by
x - g0xg0 . This has the merit that e goes to e.
To obtain a basis of (G/T) , take a basis for T , extend
it to a basis of G , and discard the vectors of T . Then (4.12 .
e e
4.14) 1 - fô has the form
1 - cos 2 TT öj. (g0) sin 2 7T 6X (g0)
-sin 2 7T6! (g0) 1 - cos 2 710! (g0)
m
Therefore det(l - f^) = 1^
1 - cos 2 7T6! (g0) sin 2 tt 8X (g0)
-sin 2vG1 (g0) 1 - cos 2 7T6! (g0)
which is greater than 0 unless cos 2 tt6 (g0) = 1 for some r.
But G (g0) f 0 mod 1, since 6 is a nontrivial function on T
(4.12). Hence the multiplicity is +1, and A(f) = | N(T)/T| > 0.
Thus f has at least one fixed point, and the theorem is proved.
4.22 COROLLARY. Every element of G lies in a maximal
torus, since the conjugate of a maximal torus is a maximal
torus.
4.23 COROLLARY. Any two maximal tori, T,U are conjugate

MAXIMAL TORI IN LIE GROUPS 93
Proof . Let u be a generator of U. Then u 6 xTx""1 for some
x C G, and thus U cz xTx-1 . But U is a maximal torus, so
U = xTx-1.
Hence any construction apparently dependent on a
choice of T is independent of the choice up to an inner
automorphism of G.
4.24 DEFINITION. It follows that any two maximal tori
have the same dimension. This is called the rank of G, and
written k or 1.
4.25 PROPOSITION. Let S be a connected Abelian subgroup
of G, and let g € G commute with all elements of S. Then
there is a torus T containing g and S.
Proof . Let H be the subgroup generated by g and S. H is
Abelian, so Cl H is a compact Abelian Lie group. Therefore the
identity component (CI H)1 is a torus. CI H/(C1 H)x is finite
and generated by g, so CI Ii/(C1 H^ "= Z for some integer m.
By 4.4, Cl H has a generator h which lies in some maximal
torus T. Then g 'J S c H c Cl H^T.

94 LECTURES ON LIE GROUPS
4.26 PROPOSITION. Let T be a maximal torus of G. If
TcAcG where A is Abelian, then T = A. That is , a maximal
torus is a maximal Abelian subgroup.
Proo f . Let g € A. Then (4.25) there is a torus U containing
g and T. But T is maximal so U = T, and g € T. Thus A c T.
4.27 EXAMPLE. If a € U(n) commutes with all diagonal
matrices it is itself diagonal.
4.28 REMARK. It is not, in general, true that a maximal
Abelian subgroup is a torus. For example, let G = SO(n) and
consider the set of matrices of the form
~ ±1
'±1 _ .
These form a maximal Abelian subgroup.
4.29 DEFINITION. Let T be a maximal torus of G. Then
the Weyl group W (or $) of G is the group of automorphisms of
T which are the restrictions of inner automorphisms of G. This
is independent of the choice of T.
Any such automorphism has the form t - ntn-1, n € N(T).

MAXIMAL TORI IN LIE GROUPS 95
N(T) is a closed subgroup of G, and so compact. Let Z(T) be
the centraliser of T, that is, the set of z € G such that
ztz-1 = t all t € T. Z(T) is also closed, and TcZ(T)c N(T).
Thus N(T) maps onto N(T)/Z(T) = W. N(T)/T is finite (see the
proof of 4.21), so W is finite.
Since we are considering G connected, Z(T) = T (4.25),
and W = N(T)/T.
4.30 COROLLARY of 4.21. Let V be a G-space. Then xy
is determined by its restriction to T and is invariant under W.
4.31 COROLLARY. The homomorphism i* : K(G) - K(T) of
(complex) representation rings is mono, and its image is
contained in the subring of elements invariant under W.
4.32 PROPOSITION. Restriction gives a one-one
correspondence between class functions on G and continuous functions on
T invariant under W.
Proof. We nave already shown that the correspondence is
mono.
Suppose given f : T - Y continuous and invariant under
W. Extend f to F : G - Y by f (xtx-1) = f (t). To show that f is

96 LECTURES ON LIE GROUPS
well-defined we need:
4.33 LEMMA. If tx ,t2 6 T are conjugate in G, then there
is w € W with t2 = wtj_ .
Proof . Let H = N(ta) = Z(ts) and let ts = g^g"1. Then
T c Z(t2) and, since T cZ(tj)( gTg"1 c Z(t2) also. H is a
closed subgroup of G, and so a Lie group, and so T, gTg"1 are
maximal tori of H. Therefore there is h é Hj. such that
T = hgTg^h"1, where H! is the identity component of H. But
h 6 Z(t3) so hg^gh"1 = t2 . Thus conjugation by hg, which is
in W, sends tx to t2 .
Completion of 4.32. It remains to check that f is
continuous .
Well, suppose that f is not continuous. Then there is
a sequence g -, g such that no subsequence of f g tends to
Fg . Let g = x t x"1 and take a subsequence with
oo n n n n
x - x , t - t for some x . t . Then g -* x t x""1
n oo n °° oo ' oo ■'ri co oo oo '
and so x t x_1 = g . Then F(g ) = f (t ) - f (t ) = F(g ),
oooooo ^oo n n oo V3oo/'
which contradicts our hypothesis. Thus 4.32 is proved.

MAXIMAL TORI IN LIE GROUPS 97
4.34 LEMMA. Let H(g)1 be the identity component of the
normaliser of some g € G. Then N(g)! is the union of the
maximal tori of G containing g.
Proof . Clearly N(g)x contains all such tori. So let
n € H(g)1 . Then n lies in a maximal torus S of N(g)a . S
commutes with g, so (4.25) there is a maximal torus T of G
containing S and g.
4.35 COROLLARY. The following two definitions are
equivalent:
(i) g € G is regular if it is contained in just one maximal
torus,
singular if it is contained in more than one
maximal torus,
(ii) g € G is regular if dim N(g) = rank G,
singular if dim N(g) > rank G.
Proo f . If g lies in just one T, then
dim N(g) = dim N(g)x = dim T. If g lies in Tx and T2 , and
Tj. ^ T2 , then LTX ^ LTa and LN(g) 3 LTX + LT2 so
dim N(g) > dim T.

98 LECTURES ON LIE GROUPS
4.36 EXAMPLE. Let G = Sp(l), which is the set of
quaternions q with | q | =1. Maximal tori are circles cos 0 + p sin G,
for p any pure imaginary quaternion with | p | = 1.
The singular points are ±1, with dim N(±l) = 3.
All other points g are regular, and dim N(g) = 1.
4.37 PROPOSITION. W permutes the roots of G.
Prop f . (The notation was introduced in 1.10.) For each
w € W we must consider two representations for T, namely
Ad w Ad
T >Aut G and T —>T >Aut G . It will suffice to
e e
show that these are equivalent. But w = A I T for some x € G,
x1
Ax
and then G * G is the required equivalence, since
Ax
T > T
Ad J J Ad
Aut G ^ Aut G
is commutative, where the bottom map is induced from A* .
4.38 DEFINITION. Let U = [t <E T ; 6 (t) = 0 mod 1}. U is
r r r
a closed subgroup of T of dimension k - 1, where k = rank G.
It is clearly monogenic. It need not be connected. For
instance:

MAXIMAL TORI IN LIE GROUPS 99
4.39 EXAMPLE. InSp(l), 6X = 2xx and Ux is given by
xx = 0 or 2" mod 1.
4.40 LEMMA. If t lies in exactly v of the U , then
r
dim N(t) = k + 2v.
Proof . Let V c L(G) be the subspace on which t acts as the
identity. Then, by definition, dim V= k + 2v. We show that
N(t) =V.
e
(i) The elements of N(t) commute with t, so t acts as the
identity on N(t) and so on N(t) . Thus N(t) cV.
(ii) Suppose x € V. Then t acts trivially on x, and so on
the 1-parameter subgroup H corresponding to x. Therefore
H c N(t) and x € N(t) . Thus V c N(t) .
4.41 COROLLARY, t € T is regular if it lies in no U , and
r
singular if it lies in some U .
4.42 COROLLARY. The singular elements of G form a set
of dimension <.n - 3, where n = dim G, in the sense that this
set is the image of a compact manifold of dimension n - 3 under
a smooth map.
Proo f . Let u be a generator of U . Then dimN(u) >. k + 2 ,

100 LECTURES ON LIE GROUPS
and, if z € N(u), z fixes each power of u and so fixes every
element of U .
r
Define a map f : G/N(u) x U - G by f(g,t) = gtg-1 .
Then Imf consists of all points in conjugates of U , f is
smooth, and dim G/N(u) x U < n - (k + 2) + (k - 1) = n - 3.
All the singular points are obtained with r running over a
finite set. Hence the result.

Chapter: 5
GEOMETRY OF THE SHIEFEL DIAGRAM
(Note: This is not the Dymkin-Coxeter diagram.)
Notice. Throughout this chapter G is a compact connected
Lie group, and T is a maximal tonus of G.
5.1 DEFINITION. The infinitesimal diagram of G is the
figure in L(T) consisting of the hypoerplanes L(U ).
The diagram of G is the figure in L(T) consisting of
the hyperplanes given by 6 (t) 6 Z. This is the inverse image
under exp of the singular points of G in T.

102 LECTURES ON LIE GROUPS
5.2 EXAMPLES of diagrams.
(i) U(2). Root xi - xa.
/ /
/
*3
/
/ /
• •
*1
The integer lattice is marked with asterisks.
(ii) SO(4). Roots Xi ±xa.
*3
\
■»Xi

GEOMETRY OF THE STIEFEL DIAGRAM
103
(iii) SO(5). Roots x1 ± x3 , xif x3.
-* • •-
/
\
/
\
\
/
\
/
y \ \ / i \ / \ \
■»x.
(iv) Sp(2). Roots Xi ± xa f 2xi , 2x3.
' i\
*i

104 LECTURES ON LIE GROUPS
(v) SU(3). Roots x1 - x2 , x2 - x3 , x3 - Xi .
5.3 PROPOSITION. Z(G) = nu .
Proof . Firstly, Z(G) c Z(T) - T.
Now, if z € Z(G), z acts trivially on G and so on G .
Therefore 6 (z) = 0 mod 1 for each r and z € H U .

GEOMETRY OF THE STIEFEL DIAGRAM 105
Conversely, if g t T and 8 (g) = 0 mod 1 for each r
then g acts trivially on G , and so trivially on G (2.17).
e
5.4 EXAMPLES
(î) U(n). OU is given by Xj = ... - x mod 1, so the
centre consists of matrices e I.
(ii) SU(n). PU is given by xx = ... = x mod 1 and
x, + ... + x = 0 mod 1. Thus the centre consists of matrices
1 n
col where to = 1.
(iii) Sp(n). HU is given by x. ± x. = 0 mod 1 all i,j, i.e. ,
x. = 0 mod 1 all i or x. = tt mod 1 all i. Thus the centre con-
l l ^
sists of matrices ±1.
(iv) SO(2n). HU is given by x. ±x. = 0 mod 1 for i / j.
Tor n > 1, this is the same as for Sp(n), and the centre consists
of ±1. Of course, SO(2) is Abelian.
(v) SO(2n +1). P.U is given by x M mod 1 all r. Thus
the centre consists of just the identity matrix I.
5.5 THEOREM. If r ^ s then 8 and 8 are linearly in-
r s
dependent.
Proof . U has dimension k - 1. We show that
r

106 LECTURES ON LIE GROUPS
dim N((U ) ) = k + 2. The result will then follow from 4.40
applied to a generator of (U ) . We need two lemmas.
5.6 LEMMA. Suppose HcT( and that H is a closed
subgroup which is normal in G. Then
(i) N(T/H) = N(T)/H.
(ii) T/H is a maximal torus in G/H.
(iii) W(G/H) = W(G).
Proof .
(i) If n preserves T then nH preserves T/H. Conversely,
if n(tH)n-1 c T then ntn-1 c T.
(ii) T/H is a compact connected Abelian subgroup of G/H,
and so a torus.
Now suppose T/H c U/H, where U/H is a torus in G/H.
Then U/H c N(T/H) = N(T)/H, soTcUc N(T). Therefore
dim T = dim U, so dim T/H = dim U/H and T/H = U/H.
(iii) W(G/H) = N(T/H) /T/H = N(T)/h/t/H
= N(T)/T
= W(G) .
5.7 LEMMA. If dim T = 1 then
(i) n=l and W = 0, or

GEOMETRY OF THE STIEFEL DIAGRAM 107
(ii) n = 3 and W = Z3 .
[Note: In fact in (i) G = S1 , and in (ii) G = SO(3) or Sp(l). ]
Proof . If n = 1 then clearly G = T = S1 and W = 0. So
suppose n > 1.
Take an invariant norm in L(G) and let v be a unit
vector in L(T). Define f : G/T - Sn-1 c L(G) by f (g) = (Adg)v.
Then f is well-defined, continuous (even smooth) and is mono
for, if (Adgx)v = (Adgs)v then AdfeY1 g2)v = v, sog71g2 fixes v
and therefore fixes T. It follows that g^gg € T and gx T = g2T.
Now G/T is compact and S Hausdorff, so f is a
homeomorphism of G/T with its image in S . But G/T and
S are both compact manifolds of dimension (n - 1), so f is
onto. Then there exists g € G such that (Adg)v = -v, and
therefore g acts on T by gtg"1 = t-1. Now T has only two
automorphisms, so W = Zs .
Let i be the generator of TT! (T). Since G is connected,
g can be joined to e by an arc in G. So, in v1(G), i - -i,
that is, 2i = 0.
Now we have in fact (2 .37) a fibration
S1 -* G ~* G/T - S . From the exact homotopy sequence we
n~ i
have that 7TS (S ) - rtx (S1) - vx (G) is exact. *But

108 LECTURES ON LIE GROUPS
ti-! (S1) -* tt1 (G) is not mono, since 2i - 0. So ïï2 (S ) / 0
a nd n = 3.
Proof of 5.5. Consider (U ) , the identity component of
U . This is a torus of dimension k - 1. Let u be a generator,
r
We wish to show that u £ U for r / s , for then 6 will not be
s s
a multiple of 6 .
r
Consider N(u)1# T is a maximal torus of N(u)x . The
elements of N(u) fix u and so fix every element of (U ) . We
r x
can apply 5.6 with N(U)x as G, T as T, and (U ) as H. Then
T/(U ) is a maximal torus in N(u)x/(U ) , and
W(N(u) /(U ) ) = WCNCuJi). Now T/(U ) has dimension 1, so
(5.7) N(u) /(U ) has dimension 1 or 3, and N(u) has
dimension k or k + 2. But (4.40) N(u)x has dimension k + 2v where
u lies in exactly v of the U . Hence v = 1 and u does not lie
r
in U .
s
5.8 THEOREM. For each r there is an element <p € W
r
which is not the identity but which leaves every point of U
fixed.
Proof . We use the same proof as 5.5, but with a different
choice of u.

GEOMETRY OF THE STIEFEL DIAGRAM 109
Consider U . We observed (4.38) that U is monogenic
Let v be a generator.
Now consider N(v)i . T is a maximal torus of N(v)x ,
and N(v)x fixes every element of U . We can apply 5.6 with
N(v)x as G, TasT, and U as H. We deduce that T/U is a
maximal torus in N(v)x/U , and Nfv^/U has dimension 1 or 3.
By 4.40, dim NCvJ/U > 3, so dim N(vx )/U = 3 and
W(N(v)1/U ) = Zs . That is, there is n 6 N(v)x which fixes
each point of U and which maps T/U by t - t-1 .
5 . 9 COROLLARY (of the proof). <p is the inner
automorphism induced by an element n which can be joined to e by a
path of which each point leaves each point of U fixed.
5.10 COROLLARY. U has either one or two components .
Proof . o acts on T/(U ) by t ^ t"1 . which has only two
r r i
fixed points , namely 0 and — mod 1. But U /(U ) is fixed by
r
5.11 EXAMPLE. The root 2x of Sp(n) gives U with two
r r
components .
5.12 DEFINITION. For each r let e =±1. Consider the set
r

110 LECTURES ON LIE GROUPS
{t €L(T); crer(t) >0 all r}.
This is either empty or is a non-empty convex set. In the
latter case it is called a Weyl chamber, and its closure is
given by
{t € L(T); crer(t) ;>0 all r}.
So we can say that the hyperplanes of the diagram divide L(T)
into Weyl chambers.
A wall of a Weyl chamber is the intersection of its
closure with a hyperplane L(U ) when the intersection has
dimension k - 1.
W permutes the planes of the diagram and the Weyl
chambers, by 4.37.
For the following theorem, we suppose chosen an
invariant norm in L(G). The word "reflection" is interpreted
by using this norm.
5.13 THEOREM
(i) W permutes the Weyl chambers simply transitively.
(ii) For each r, W includes the reflection in the plane
L(Ur).
(iii) Such reflections generate W.
(iv) More precisely, for any Weyl chamber B, the

GEOMETRY OF THE STIEFEL DIAGRAM 111
reflections in the walls of B generate W.
(v) Let p € L(T) and W be the stabiliser of p. Then W
p * p
permutes simply transitively the Weyl chambers whose
closures contain p.
(vi) W is generated by reflections in the planes L(U )
which contain p.
(vii) More precisely, it suffices to consider those planes
which are walls of a fixed Weyl chamber B such that
o
p 6 Cl B .
H o
Proo f . By taking p = 0, we see that (v) =£> (i), (vi) =^> (iii)
and (vii) => (iv), so we need to prove only (ii), (v), (vi), (vii)
(ii) For each r, W contains an element tp / 1 which fixes
U (5.8), and hence fixes L(U ), and preserves the inner
product in L(T). id can only be the reflection in the plane L(U ).
(v) Firstly, W acts simply. We split the proof into two
lemmas:
5.14 LEMMA. If v € L(T) is fixed by some ip € W, <M 1,
then v € L(U ) for some r.
Proof . Suppose n € N(T), n/T, and n fixes v. Then n
fixes the 1-para met er subgroup H corresponding to v (2.17).

112 LECTURES ON LIE GROUPS
Hence there is a maximal torus U containing n and H (4.25).
Therefore H lies in two distinct maximal tori, soHdjU and
r
v € L(U ) for some r.
* r
5.15 LEMMA. If ^B = B for some Weyl chamber B, ^ € W,
them ip = 1.
Proof. W is finite so ip = 1 some q > 0. Let vtB. Then
1 Q r
v * = — Lx ip v lies in B a nd is fixed by ip. If ip / 1, (5.14)
shows that v' lies in some L(U ), which contradicts the
hypothesis .
Continuation of 5.13
(v) Secondly, W acts transitively on the Weyl chambers
whose closures contain p, as follows.
Let B , B' be Weyl chambers containing p in their
closure, and let x £ B , x ' € B ' . Since, for r ¥■ s ,
o o
L(U ) ■?. L(U ) has dimension k - 2 (5.5), there is a polygonal
path from x to x1 not meeting any L(U ) P. L(U ), not meeting
any L(U ) unless it contains p, and meeting each L(U )
transversely. |_Take the path x px ' , and move it slightly, j
Suppose this path crosses (LU, ), . . . ,L(U ) success-
ively to get fiom B to Bx ... to B = B' . Then tp . . .<p, cp,
o r Kj- Ky Kj^

GEOMETRY OF THE STIEFEL DIAGRAM 113
maps B via B , . . . ,B toB = B' . Thus W is transitive on
o l ' ' r-i r p
the Weyl chambers whose closure contains p.
(vi) Let ti £W and choose B such that p € Cl B . Set
p o o
B ' = ^B . Then B ' = <o . . . cp B , with the notation above ,
O Kj- Kj O
so ip"ld, ... O, B = B . But W acts simply, so
Kj- Kj O O p
^"Vk • • • Vv = 1 or ^ = v. . .. o .
Thus these reflections generate W .
P
(vii) Write q> = tß . . . o, as above, and suppose as an
kr kx
inductive hypothesis that we have written ip . .. cp as a
ks kx
product of reflections in the walls of B . This is trivially
possible for s = 1. Then co, is the reflection in a wall
ks+i
L(U ) of B . But (fi'1 . . . co"1 maps B to B and L(U )
ks+1 s k1 ks so Ks + 1
to, say, L(Um), which contains p. Then
Therefore
Hence V may be written as a product of reflections in the walls
of B which contain p.
o
5.16 COROLLARY. Divide L(T) into orbits under W. Then each
orbit contains precisely one point inthe closure of each Weyl
chamber B.
Proof . Cl B contains at least one point of each orbit, as

114 LECTURES ON LIE GROUPS
follows. Let v € L(T). Then v € Cl B' for some Weyl chamber
B • , and B = wB • for some w € W. Then wv € Cl B.
Cl B contains not more than one point of each orbit,as
follows. Let p,q € Cl B, and p = wq. Then p € Cl(wB).
Since W is transitive on those Weyl chambers whose closure
P
contains p, there is w' £ W such that w'wB = B. Then
w *w = 1 sop = w'p=w 'wq = q.
5.17 EXAMPLES
(i) G = U(n). Ge consists of the skew Hermitian matrices.
Define an inner product on G by <X,Y> =tr(XTY) =tr(-XY). This is
invariant under G. Restricted to L(T) this has the form (up to
a factor 4tt2) x2 + + x2 . This is the 'usual' inner product.
in
so reflection is the 'usual* reflection.
The root 6 = x - x gives the plane x = x for
rs r s r s
L(U ) and reflection in this plane is given by
y =x , ... , y =x , ... ,y = x , ... , y =x .
'i i r s ,ys r' ,yn n
This is indeed induced by an inner automorphism, namely, by
conjugation with

GEOMETRY OF THE STIEFEL DIAGRAM
115
r s
1.
"l
r 0 1
1
1
s -1 0
1
"l
[We write -1 and not +1 for the sake of the next example.]
Thus W is the symmetric group on x ,.. . ,x . The order |w|
of the Weyl group W is n! .
(ii) G = SU(n). The same calculation may be repeated,
and W is the symmetric group on x , .. . ,x . |W| =n!.
(iii) G = Sp(n). This time W consists of transformations
of the form
y = € x /lW . . . ,y = € x . w
i l p(l) n n p(n)
where € = ±1 each r, and p is a permutation. |w| = n!2 .
(iv) G = SO(2n + 1). This gives the same Weyl group as
Sp(n). |W| = n!2n.
(v) G = SO(2n). W consists of transformations of the form

116 LECTURES ON LIE GROUPS
i i p(l) n n p(n)
where € = ^1 each r, fl" ç = +1. and p is a permutation,
r l r
| W | = n!2n_1 .
5.18 DISCOURSE. The roots 6 are real linear forms on
r
L(T), that is, they are elements of L(T)*. W acts on L(T)* by
(wh)(v) = h(w_1v).
We have an invariant inner product on L(T), so we may
identify L(T) and L(T)* by i : L(T) - L(T)*f where we set
(ivjjfva) = (v1 , v2X This commutes with the action of W, so
all results on the action of W on L(T) can be transferred to
L(T)* under the isomorphism i. We put an inner product on
L(T)* by copying that of L(T), that is,
<ivx ,iv2> = <vx ,v2>
or, if you prefer,
< h, iv > = h (v).
This is, of course, invariant under W.
<p acts on L(T) fixing those vectors v for which
6 (v) = 0. Therefore o acts on L(T)* fixing those vectors iv
for which 6 (v) = 0, that is, ^6 ,iv>= 0. Thus o fixes those
vectors perpendicular to 6 , and so tp is reflection in the

GEOMETRY OF THE STIEFEL DIAGRAM 117
plane perpendicular to 6 .
Note that reflection in the plane perpendicular to the
unit vector v is given by w -* w - 2<v,w}v.
5.19 PROPOSITION
<p (h) = h - 2<9r'h>fl m
<er,er> r
This follows from the discourse.
5.20 DEFINITION. A weight is an element of L(T)* which
takes integer values on the integer lattice.
For example, each root is a weight.
W sends weights to weights. Hence:
5.21 PROPOSITION. If X is a weight, then
2<er,x>
r t'
is a weight.
W also sends roots to roots, so:
5.22 PROPOSITION. If 6 is a root, then
s
<p (6 ) = G - 2<er'es> e
*V s' s <er/er) br
is a root ±6 .

118 LECTURES ON LIE GROUPS
5.23 EXAMPLE, tp (6 ) = -6 .
*rx r r
5.24 PROPOSITION. In 5.21 and 5 .22 the coefficient
-2<er>\)
<er,er>
is an integer.
Proof . Choose v € L(T) so that 6 (v) = 1. Then exp v € U .
tp fixes U , so v - (ß (v) is in the integer lattice. Therefore
X(v) - Xp (v) is an integer. That is, X(v) - ((p X) (v) is an
integer; since çT1 = (p , this shows that
2<er/x>
X(V) - X(V) + <ê^>er(v)
is an integer, which is the required result.
5.25 PROPOSITION. Let a,ß be roots with a / £p. Then
either
(0) a,ß are perpendicular, or
(1) a,ß make an angle of 60° or 120° and | a | = | ß | , or
(2) a,ß make an angle of 45° or 135° and their ratio is
v/2, or
(3) a,ß make an angle of 30° or 15 0° and their ratio is y/3.
Proof . We prove this together with:

GEOMETRY OF THE STIEFEL DIAGRAM 119
5.26 PROPOSITION. Let a,ß be roots with a ^ ±ß, and let
k be an integer between 0 and —z 1^~ inclusive. Then
Ka,a/
ß + ka is also a root.
Proofs. The angle between a and ß is given by
cosaa; = x <QV'/>S N < 1.
<a/a><ß/ß>
Therefore
0 < A2<.,pyy-2<p,a>V 4
By changing the sign of a if necessary, we may suppose
<a,ß>< 0. If <a,ß> = 0 then we have case (0) of 5 .25 , and
5.2 6 is trivial. Otherwise at least one of
V<a.a>/ V<ß'ß>/
is 1. If [-}—a'P J = 1, then ß + a is the reflection of ß in
\\a,a> /
the plane perpendicular to a, and 5.2 6 follows in this case.
Since 5.25 is symmetric in a and ß, we may assume now that
-2<ß.a> _ J
<ß,ß>
Let ~^Qijp =Vi v= \, 2, or 3. Then
<a,a>
<ß,ß> Ißl / v Jv
, 'rv = v so -j—7 = Jv, and cos cc = T so cos a; = — .
<a,a> | a I 4 I
If y = 1 we get case (1) of 5.25, and 5.26 has already
been demonstrated. We have the following diagram:

120 LECTURES ON LIE GROUPS
ß a + ß
» a
Example. SU (3).
If y = 2 we get case (2) of 5 .25 . If we reflect a in the
hyperplane perpendicular to ß we get ß + a. If we reflect ß
in the hyperplane perpendicular to a we get ß + 2a, so ß + a ,
ß + 2a are roots.
ß ß + a ß + 2a
^ > a
W contains reflections in two hyperplanes at 45°, and so
contains the dihedral group De .
Exa mples. Sp(2) or SO(5).
If v = 3 we get case (3) of 5.25. Reflecting a in the
plane perpendicular to ß we get ß + a. Reflecting ß and ß + a
in the plane perpendicular to a , we get ß + 3a and ß + 2a .

GEOMETRY OF THE STIEFEL DIAGRAM
121
ß + 3a
> a
W contains reflections in two hyperplanes at 30°, and so
contains the dihedral group D1S .
Example. G3 , which is the group of automorphisms of the
Cayley numbers as an algebra over R. (It is possible to have
fun examining this example explicitly, but we omit this here.)
5.2 7 DEFINITION. Choose a Weyl chamber B in L(T) and
call it the fundamental Weyl chamber (FWC). Alter the signs
of 6T .... .8 so that 6 (v) > 0 for v € B all r. Then
1 m r
{v C L(T); 6 (v) > 0 all r} = B.
The roots 6 .... ,6 are now called positive roots , and
1 m i
-6 , .... -6 negative roots .
1 ' m —2
5.28 EXAMPLE. LetG = U(n). Let the fundamental Weyl
chamber be given by xx > x2>. . . > x . Then the positive roots

122 LECTURES ON LIE GROUPS
are the forms x - x with r < s.
r s
In Sp(n) we take the fundamental Weyl chamber to be
given by x > . . . > x > 0, and similarly for SO(2n + 1).
For SO(2n) we take xx > xs ... > xn_j > x > -xn_x .
5.29 LEMMA. Let the 6 be the positive roots and X .> 0 in
r r
R. Then EX 6 = 0 implies X = 0 for all r.
r r r
Proof . Take v € B. Then (EX 6 )v = 0, so EX (6 v) = 0, so
r r r r
each X is 0.
r
5.30 DEFINITION, a is a simple root if
(i) a is a positive root, and
(ii) we cannot have a = ß + y for ß and y positive roots.
5.31 PROPOSITION. Any positive root Q can be written as
a linear combination of simple roots with non-negative integer
coefficients .
Proof . If a is not simple, then a = ß + y where ß, y are
positive roots. If either is not simple, we may repeat the
process. If this never terminates, since the number of roots
is finite, there is an expression ß = ß + ö +. . .+ Ö
somewhere, contradicting 5.29.

GEOMETRY OF THE STIEFEL DIAGRAM 12 3
5.32 LEMMA. If a,ß are distinct simple roots, then
<a,ß>< 0.
Proof . Suppose<a/ß> > 0. Then , a \ > 0 and is an
<a,a>
integer, so . '^ >. 1. By5.26,ß-aisa root. Therefore
<a ,a>
ß - a or a - p is a positive root, whence either ß = (ß - a) + a
or a = (a - ß) + ß is not simple, contradicting the hypothesis.
5.33 PROPOSITION. The simple roots are linearly
independent.
Proof . Suppose v = £u 6 = T,v 6 where the 6 are simple
r r s s r
roots, all u ,v are non-negative, and the sums run over
disjoint sets of subscripts. Then
<v,v>= Zyiv <6 ,6 > < 0.
N 'r sN r sy
Therefore v = 0 and we may apply 5.29.
5.34 COROLLARY. The fundamental Weyl chamber is given
by 6 (v) > Qi 6s (v) > 0 ,
where 6 ,. . . ,6 are the simple roots.
i s
Proof. This is clear from 5.31.
So simple roots correspond to walls of the fundamental
Weyl chamber.

124 LECTURES ON LIE GROUPS
5.35 EXAMPLE. G=U(n). The fundamental Weyl chamber
is x > x . . . > x . The simple roots are
i s n
xi x2 / xa ~ x3 / • • • »xn-i ~ xn*
Any other root can be written as a linear sum of these, e.g. ,
x -x =(x -x )+...+ (x -x)
r s r r-i s-i s
for r < s . And
xx - x2 , . .. /xn-i " xn
are linearly independent.
5 .36 EXERCISE. If a is a simple root and we write
a = Lu 6 , where the u are non-negative numbers and the 6
r r r r
are positive roots, then we have written a = a.
5.37 DEFINITION. The Dynkin diagram is constructed as
follows. Take one node for each simple root a. Given two
distinct simple roots a,ß join the corresponding nodes by
v = 0, 1, 2 or 3 bonds, where v follows 5.26.
5.38 EXAMPLE. G = U(n). Between (xr - xr+1 ) and
(xr+1 - xr+s), v = 1. Otherwise v = 0. Hence the Dynkin
diagram is 0 0 0 . . . 0 0 with n - 1 nodes.
5.39 LEMMA. If 6 is a simple root then ip permutes the

GEOMETRY OF THE STIEFEL DIAGRAM 125
positive roots except 6 , which goes to -6 .
Proof . We give two proofs.
(i) Choose a point v of the diagram such that 6 (v) = 0
and 6 (v) > 0 for any other simple root 6 . Then 6 (v) > 0 for
any positive root 6 other than 6 .
t r
Let S be a spherical neighbourhood of v not meeting
any plane 6 = 0 for t / r. Let w £ S 0 (FWC). Then <p (w) £ S,
Therefore
(0r6t)(w) = 6 (cow) > 0
for t/r. Thus cd 6 is a positive root,
r t
(ii) Let 6 .... ,6 be the simple roots, and let 6 be a
l ' s t
positive root. Write
e = n e +...+ n e .
tii s s
Then
2<er/et>
v (e ) = e - ya a ' e
*v t' t <er,er> r
differs from 6 only in the coefficient of 6 . Therefore cp (6 )
t r ^r t
has at least one positive coefficient if 6 ¥ 6 and so (5.31
t r
and 5.33) tp (8 ) is a positive root.
5.40 DEFINITION. The fundamental dual Weyl chamber
(FDWC) is the set of points in L(T)* corresponding under i to

12 6 LECTURES ON LIE GROUPS
the fundamental Weyl chamber in L(T). That is, the FDWC is
the set of h 6 L(T)* such that (0 fh^> 0 for each simple root
6 .
r
5 .41 DEFINITION. Let 0 , ... ,0 be the positive roots.
i m
Define ß 6 L(T)* by ß = —(0 +...+ 0 ). This is not neces-
r r 2 l m
sarily a weight.
5.42 PROPOSITION, ß lies in the fundamental dual Weyl
chamber. Indeed, ,—^~ = 1 for each simple root a.
Proof . Let a = 0 . Then <p permutes the positive roots other
than a. There are three cases:
(i) cp (6J = 6.. Then/e , 0/> = 0 so 6, contributes 0 to
*r t t \ r t' t
<a, ß>.
(ii) 6 permutes 0 and 0 . t /u. Then
r t u
<6 ,0^ + 0 >= 0,
N r t u'
so 6 + 8 contributes 0.
t u
(iii) 0=0. This case contributes T<a.t a>to<a,ß>.
Therefore 2/p-'& = 1.
<a/Q>
5.43 EXERCISES. Work out -(0 +...+ 0 ) for the following
2 l m
groups :

GEOMETRY OF THE STIEFEL DIAGRAM 127
(i) SU (3).
(ii) SO(5).
(iü) G2.
5.44 PROPOSITION. In L(T), reflections in the planes
6 = k for k £ Z cover the action of <n on T.
r ^r
Proof . Let v 6 L(T) besuchthat 6r(v) = k. Then the reflection
is given by
x - pr(x - v) + v = <pr(x) - pr(v) + v.
But v maps into U , so tp (v) and v have the same image in T.
Therefore <p (x) and (p (x) - <p (v) + v have the same image in T.
5 .45 DEFINITION. The extended Weyl group T is the group
generated by reflections in all the planes 6 = k, k € Z, of the
diagram.
By 5 .44, F covers the action of W on T. Define
T = Ker (r - W).
o
5.46 DISCOURSE. We have a split extension
r >r > w
/
w

128 LECTURES ON LIE GROUPS
r is the subgroup of translations. Each one is the transla-
o
tion by an element of the integer lattice I, so we can regard
r as a subgroup of I. (It is not necessarily the whole of I.)
Our next object is to calculate the fundamental group
7Ti (G) in terms of the Stiefel diagram. The topological
invariant TTi (G) may be distasteful to some algebraists, and so some
remarks are in order about the use to be made of it. First, one
of the main theorems (6.41) is classically stated with the
condition "ïïj. (G) = 0" , and some of the subsidiary results used
in its proof use the same condition. However, we are just
going to prove (5.47) "it (G) - 1/T " , so it would be possible
to rewrite 6.41 with the data in the form 'T = I" . which after
o
all is what is used in the proof of 6.41. Secondly, we propose
to use 7Ti (G) to classify the connected covering groups over G,
as is usual in algebraic topology. For our arguments to
proceed without this (notably at 5.56 below) it would be
necessary to construct the double covering Spin(n) of SO(n) without
reference to v1; and of course this is possible by pure algebra,
for example, using Clifford algebras. This is an interesting
chapter of algebra , but it involves more work without providing
so much more insight. Sometimes one can buy algebraic purity

GEOMETRY OF THE STIEFEL DIAGRAM 12 9
at too high a price [23 ] .
To continue: we have I = tTi (T), as follows. Consider
I cr L(T) -* T. For v € I choose a path œ in L(T) from some oj(o)
to oj(I) = v + to (0). Its projection is a closed path in T, and
so represents an element of ir1 (T), since tTj. (T) is Abelian.
The map i : T - G induces I = v1 (T) » 7^ (G).
5.47 THEOREM, i* is epi and induces I/T - it (G).
Proof . 5.48-5.55 will, together, form a proof.
5.48 PROPOSITION. Let y be the reflection of 0 in the
r
plane 8=1. Then T is the subgroup of I generated by the y .
Proof, r contains each y , since reflection in 8 = 0 fol-
o 'r r
lowed by reflection by 8 = 1 is translation by y .
Conversely, we claim that, if y € T, then y(0) = En y ,
whence, if y € T , y is translation by En y . We prove this
claim by induction on the number of reflections used to build
up y.
Suppose y = p6 where p is reflection in 6r = k, and
suppose 6(0) = En y . Now
p(x) = x + (k - 8r(x))yr.

130 LECTURES ON LIE GROUPS
Therefore
p6(0) =In y + ky - 6 (En y )y .
^ s s r r s s r
But 6 (Ln y ) is an integer, since En y is in the integer
r s s ss
lattice. Therefore pô(0) has the required form.
5.49 EXAMPLES.
(i) G = U(n) or SU(n). The reflection of 0 in
r s
x - x = 1 (r < s) is the point (0 ... 0 1 0 ... 0 -1 0 ... 0).
r s
Define 77: I - Z by tt(x , ,x )=x + + x . Then
l n l n
T = Ker 77. For SU(n) we have I/T = 0. For U(n) we have
o o
i/r ~z.
o
(ii) G = Sp(n) . The reflection of 0 in 2x = 1 is
r
(0 ... 0 1 0 ... 0). We have I/r = 0.
o
(iii) G = SO(2n) or SO(2n + 1). The reflection of 0 in
r s
x - x (r < s) is (0 ... 0 1 0 ... 0 -1 0 ... 0). The reflec-
r s
r s
tion of 0 in x + x = 1 is (0 ... 0 1 0 ... 0 1 0 ... 0). For
r s
r
SO(2n + 1) the reflection of 0 in x = 1 is (0 ... 0 2 0 ... 0),
which gives nothing new. Define rr : I - Z2 by
77 (x ,...,x)=x + +x mod 2. Then T = Ker n. Thus
i n l n o
i/ro » z2.
In the special case of SO(2), T = 0 and I/r = Z.
' o o

GEOMETRY OF THE STIEFEL DIAGRAM Ï3Ï
5.50 LEMMA. I = tt (T) - it (G) maps T to 0.
1 1 O
Proof . We show that y goes to zero. Well, let co be a
rectilinear path from 0 to y in L(T). Then
exp co (1 - t) = (fi exp co (t) for 0 <. t £ — . By 5.9, we can find
g € G such that tp (x) = gxg~ 1, so that
exp co(l - t) = g exp co(t)g_1, and such that there is a path
from g to e each point of which keeps U fixed. So
exp co(l - t) is homotopic to e expco(t) e~ = expco(t), keeping
t = 0, t = — fixed. Hence exp co (t) for 0 £ t <. 1 is contractible
keeping end points fixed. So y goes to zero in tt (G).
5.51 NOTATION. Let GD,TD,L(T)D denote the sets of regu-
K K K
lar points in G,T,L(T) respectively.
5.52 LEMMA, i. : tt (G„) - ît (G) is an isomorphism.
* i R i
Proof . The complement of G has Hausdorff dimension
R
<. n - 3, by 4.42 and standard Hausdorff dimension theory, and
the result follows by standard homotopy theory.
5.53 LEMMA. Define fD : G/T x TD - Gn by fD(g,t) = gtg"1.
R K K R
Then f is a covering with fibre W.
R

132 LECTURES ON LIE GROUPS
Proof . W acts on the left on G/T as follows. Let ip € W
and let n € N(T) represent <p. Define
<p (gT) = gTn-1 = gn_1T.
W also acts on the left on TD/ and so acts on G/T x T. Let
K K
G/Tx T be the orbit space. Since W acts freely on G/T,
the projection
G/T x TR - G/TxwTR
is a covering with fibre W.
Now f factors through G/Tx T and G/Tx T - G
is a one-one and onto map between manifolds of the same
dimension, and so is a homeomorphism. Hence the result.
5.54 LEMMA. i# : ^(T) - tt x (G) is epi.
Proo f . Consider the map
fR
G/T xTR > GRCG'
where f is a finite cover. Let the components of T be T ;
R RR
then since G/T is connected, the components of G/T x T are
R
G/T x T ; and so each of the following maps is monomorphic.
R
i fR*
tti (G/T x pt) -» irt (G/T x TR) > irx (GR) -> ir^G).
Now the map G/T x t -» G, given by g -* gt g"1, is
nullhomotopic by taking a path from t to e. So t (G/T) = 0.

GEOMETRY OF THE STIEFEL DIAGRAM 133
Hence, from the homotopy exact sequence of a fibration
we deduce that n1 (T) - itx (G) is epi.
5.55 LEMMA. If v € I maps to 0 under I =i7x (T) -» ttx (G),
then v € T .
o
Proof . We may suppose that, for any y € T , v + y is not
closer than v to the origin in I. Then 6 (v) = -1, 0 or 1 for
each root 6 , for, if 6 (v) > 1, then the reflection of v in
6 = 1 is closer to the origin, and correspondingly if 6 (v) <-l.
Let to be the linear path in L(T) from to(0) = 0 to
co(l) = v. This does not cross any diagram planes, although it
may lie in some, and may meet others at to(0) and co(l). So
there is a linear path to* from to' (0) to to* (1) = to* (0) + v which
is close to co and which meets diagrams planes only close to
W (1).
Consider the diagram
fR
G/T x L(T)R > GR
n
G/T x L(T) f > G .
By taking the identity coset in G/T, the path co* may be

134 LECTURES ON LIE GROUPS
considered as in G/T X L(T). Then fto* is a loop in G which
lies in GD except near fto' (1). By 4.42, we may move this
K
loop slightly near fco' (1) so that it lies in GD, and this loop is
K
contractible in GD. Since G/T x L(T) - Gn is a covering, we
K RR
may now lift the loop to a path to" in G/T x L(T) starting near
R
T x 0. Then to" will be the same as co' except near to' (1).
Further, since we have altered feu' only near e in G, the
projection of to" onto the factor L(T) is close to to' . Now f to"
R
is contractible in GD, so to" is a closed loop in L(T) , and v
R R
is approximately zero. But v is in I, so v = 0.
5.56 DISCUSSION. We have now shown (5.47 and 5.49)
that 7TX (SO(m)) = Z2 for m > 2. Therefore SO(m) has a double
cover called Spin(m). It is clear that the cover of a maximal
torus in SO(m) is a maximal torus in Spin(m). Take as the
standard maximal torus T in Spin(m) the cover of the standard
maximal torus T in SO(m). Then L(T) = L(T) under the covering
map, though this does not preserve the integer lattices. I
consists of all (x ,... ,x ) with all x integers, and I
consists of all (x , . .. ,x ) with all x integers and x +. ..+ x
in r i n
even. Similarly L(T)* - L(T)*, but this does not preserve the
lattices of weights. For example, — (x +. . .+ x ) is not a
a 2 i n

GEOMETRY OF THE STIEFEL DIAGRAM
135
weight in SO(m) but is one in Spin(m).
Now Ad : G - SO(n) induces
Ad^ : TTj, (G) - ff! (SO(n)) = Z2 (for n > 2).
We distinguish two cases.
(i) Ad^ is zero, and we can lift Ad to get the following
diagram.
Spin(n)
» SO(n)
(ii) Ad^ is non-zero. Then Ad defines a double cover G of
G, and we have the following diagram.
G >Spin(n)
v
>SO(n)
For G, (i) applies. By 3.68, the representation theory of G
determines that of G. So, in what follows, we will assume
that (i) applies.
5.57 PROPOSITION. In this case, p = i(ö +...+ 6 ) (see
1 z î m
5.41) is a weight.
Prop f . In 4.12 we split G as a T-space in the form

136 LECTURES ON LIE GROUPS
V $E V.. Choose bases for V .....V ,V , and put them
oil i m o
together in this order to form a base for G . Then the corn-
Ad
position Te G > Aut G = SO(n) sends T into the standard
x
maximal torus T' ofSO(n). Further, if L(T') > R denotes
the rth co-ordinate function, then the composition
xr
L(T) -> L(T') > R is the root ±6 , for r < m, or zero, for
r > m. With the same sign attached to each © , we now have
±6 ±...± 6 = (Ex )Ad.
l m r
Now Ad lifts to Spin(n), and -Ex is a weight for
Spin(n), so (jEx )Ad is a weight for G. Thus ~(±6i ... ±6 )
is a weight for G, and so is ß = —(G +...+ 8 ), as this
differs from ~(±B ... ±8 ) by a sum of positive roots.
2 i m
5.58 LEMMA. In this case co "* to + ß gives a one-one
correspondence between weights co € Cl FDWC and weights
co + ß € FDWC.
Proof
(i) If co is a weight and (co,6 ) >. 0 for all simple roots
6 then <co+ß,6 )> 0 by 5.42.
r ^ r
(ii) If co is a weight and <co,6 ) > 0 for all simple roots
6 then ' r > 0 and is an integer (5.24), so_> 1. Now
(8r, 8r)

GEOMETRY OF THE STIEFEL DIAGRAM 137
2 <ß,er> 2<co - ß,6r>
— = i so — ^ 0 and to - ß is a weight in
<er/er) l' so <er,er> p
Cl FDWC.
We showed (5.24) that, if to is a weight and 8 a root,
2<6r,to>
then —rz—r~r- is an integer. We now examine the converse.
\" $ " )
r r
2<er,to>
5.59 PROPOSITION. If -ig—öT" is an inte9er for some
r' r
to € L(T)* and all simple roots 6 , then it is an integer for all
roots 6 .
r
Proof.
2<6r/to)
Suppose
r r
is an integer for all simple roots
6 and also for the root 6 . Let <p correspond to some simple
r s r
root 6 , and let G = <p (B ). Then <6 ,6 ) = <8 ,6 ) and so
r t *!• s t t s s
2<et,co) 2 / 2<er<es>
"(6,6) \6s "(6,6)
s s r r
2<es<to> 2<er<co> 2<er<es)
<e ,e> <e ,e > \es " (eV) er ' w)
t t s s r r '
<e ,e ) <e ,e > <e fe>
s s r r s s
which is an integer.
But the reflections <p generate W (5.34 and 5.13(iv))
and any root 6 can be written as <ß 6 for some simple root
s r
6 and some <p 6 W, by considering 6 as the wall of a Weyl
chamber and throwing this chamber onto the FWC (5.34).

138 LECTURES ON LIE GROUPS
Hence the result.
2 <er,u>>
5.60 PROPOSITION. Suppose -72—5T is an integer for
\" , " )
r r
some to € L(T)* and each simple root 8 . Then to takes integer
values on r .
o
Proof, r is generated by the points y , where y = v - <p v
for any v such that 6 v = 1. We have
r
w(yr) = cov - aj(0rv)
= (00 -<p w)(v)
2<er^> n , ,
= WJT6r{v)
r r
which is an integer. Thus co is integral on each y and so on
r .
o
2<er/co)
5.61 COROLLARY. If G is simply connected and —r—ö—
<Ver>
is an integer for each simple root 8 , then to is a weight.
Proof. T = I.
o
5.62 THEOREM. If G is simply connected it has just
k = dim T simple roots 6 ,. . . ,6, , and has weights co ,.. . ,üj,
such that 6 r' J- =6 . The weights are then the linear com-
\8r/9r> rt
bina tions n. œ,+...+ n, cc. with n € Z each r. The FDWC
1 1 k k r

GEOMETRY OF THE STIEFEL DIAGRAM 139
consists of all points En to with each n > 0. and the CI
r r r
FDWC consists of all points En to with each n >. 0. Also
r r r
-(6. +. ..+ 6 ) = to, +...+ to, .
2i mi k
Proof. Suppose there are just k - v simple roots. Then all
the roots lie in a subspace of L(T)* of dimension k - v, and so
r lies in a subspace of L(T) of dimension k - v. Then I/T
o o
has rank at least v which implies v = 0.
There are elements to in L(T)* such that ——L_ = Ô .
t <er/er) rt
and they are weights by 5.61. Every element to of L(T)* can
be written to = En to , and then Lü—_ = n , so to is a weight
r r <er,er> r
if and only if each n is an integer. The statements about
FDWC follow from the definition (5.40).
Set ß= |(6 +...+ 6). Then ii?Ii^_ = 1 (5.42), so
21 m <er,er>
ß = tO + . . . + OÎ
k*
5.63 EXAMPLE. LetG = SU(n).
Take to =x +...+ X for 1 £ t £ n - 1. Then
so
<xr-xr+1<cot) =6rt
2<er/tot)
c
<er.er> " rt-
The elements of L(T)* can be written

140 LECTURES ON LIE GROUPS
aixi + •••+ an-xxn-i »
since Ex = 0. They lie in FDWC if
Q1 > a2 > an-1 > 0.
Thus they may be written
bitü! +.. .+ b^iOJ^! ,
and lie in FDWC if b1 ,... »b^! > 0.
5.64 COUNTEREXAMPLES
(i) Let G = U(2), where tî1 (U(2)) = Z. We have dim T = 2 ,
but there is only one root. The FDWC is a half-plane, which
cannot be expressed in the given form.
(ii) Let G = SO(4), where Tfx (SO(4)) ~ Za . The dual
diagram is as follows:
/
ft ft ft ft
/(Xi +X2)
FDWC
ft ft ft #
\(*i -x2)
-fr "Ä- -fr *
\

GEOMETRY OF THE STIEFEL DIAGRAM 141
Here the asterisks represent weights. The FDWC is the
quarter-plane shown. The weights in the CI FDWC do not form
a free Abelian semi-group, and
Wi = j(xx + x2), o>2 = j(xx - x2)
are not weights.

Chapter 6
REPRESENTATION THEORY
Not ice. Throughout this chapter G is a compact
connected Lie group, and T is a maximal torus of G.
*
6.1 THEOREM. (Weyl Integration Formula.) There is a
real function u on T such that
\ i(q)=\ f(t)u(t)
JG JT
for all class functions f on G.
Indeed u(t) = 66/| W | , where
-7rie.(t)\
6=n. Me J -e ] J
m / irie^t)
"Vl
and 6. runs over the distinct roots of G.
J
Proof. Define f : G/T x T - G by f(g,t) = gtg-1 . (See 5.53.)
Then f factors through G/Tx T:

REPRESENTATION THEORY 143
b
/xXwT
Now c has degree 1, since it is a homeomorphism
when restricted to G/TxTA T - G„, and b is a I W I -fold
W R R ' '
covering. So f has degree |w| , and
|W|Ç f dg=C f*dg*,
G JG/TxT
where f*,dg* are the induced function and measure on G/TxT.
If f is a class function, then f* is constant along G/T.
Now we must evaluate det f at a general point (g,t) of
G/TxT. First, let u run through a neighbourhood of e in T. Then
f(g,tu) = gtug"1 = gtg-1 gug"1 .
Therefore f ' (g,t) = Ad g, where we consider the first factor
fixed. Second, let v be in a transversal V of T in G. Then
f(gv,t) = gvtv_1g_1 ,
so
f ' (g,t)(dv) = gfdvHg"1 - gtfdvîg"1
= (gtg-1)(gt"ldvtg_1 - gdvg-1),
so
f(g,t) ^Adg^dt"1 - I),

144 LECTURES ON LIE GROUPS
where we consider the second factor fixed. Thus
det f (g,t) =det(Adt"1 - I).
Now Adt has the form
cos 27r6i sin 2ttB1
-sin 2tt B1 cos 2tt61
so Adt"1 - I has the form
cos 2 7TÖ! - 1 sin InQ-i
sin 2tt 6x cos 2tîB1 - 1
and
det(Adt_1 - I) = nm(cos227T6 -2 cos 2776 + 1 + sin26 )
1 \ r r r/
=nm(4sin^er) =n(e,ri8r - e-ffUfyi(e_,ri8r - e"i60
= 66,
where
/ 7Ti6r -7Ti0r \
6=nfe -e J.
Hence the result.
6.2 DEFINITION. W acts on L(T). For <p € W, let sign
(fi denote the sign of the determinant. Then we say that

REPRESENTATION THEORY 145
X £ K(T) is a symmetric character if (p\ = X f°r each (p € W,
and is an anti-symmetric or alternating character if
(fiX= (sign^)x.
6.3 EXAMPLE. Suppose Ad : G - SO(n) lifts to Spin(n).
Then
is an anti-symmetric character.
Proof .
6 = £c ... c Exp 7Ti (€ 6 +...+ € 6 )
l mil mm
where €. = ±1 and there are 2 terms. Note that, by 5.57,
l
7(c e +...+ € e )
2 i i mm
is a weight, so 6 € K(T).
Let x € N(T) represent <p € W. Then the action of cp is
given by g - xgx"1 . This induces a map G - G which maps
T to T . On T it preserves or reverses orientation according
e e e
to (sign (p). Also <p permutes V , ...,V (5 .5 and 3.22). If tp
maps V. to V, preserving orientation, then it sends G to 6, ;
and if reversing orientation, then it sends 6. to -6, . If it
' j k
reverses orientation v times then <p 6 = (-1) 6.
But o preserves the orientation of G , since x may

146 LECTURES ON LIE GROUPS
be connected to e by a path. Therefore
(sign<o)(-l) =+1.
That is,
00= (sign<p )6.
6.4 PROPOSITION. If a character x (of T) vanishes on U
then it can be written
X= [Exp(2 7Tier) - lty,
where ip is a character.
Proof
(i) Suppose U has just one component. Then we may
take a basis e ,... »ej, of the integer lattice of L(T) as
follows. Let e ,... ,e, be a basis of the integer lattice of
L(U ), and let e be a point of the integer lattice of L(T) for
which 8 (e ) = 1. Let £ ,...,£, be the characters of the basic
r l l ^k
representations of T. Then Exp(2 7ri6 ) = £ , and we can write
\ = T, c E . where each c is a finite Laurent series in
^ n n*i n
£ a / • • • / ik •
On U , £ = 1 so Ec =0. The monomials £ ,...,£,
r ^ l n * 2 ^k
are linearly independent on U . so I c is the zero Laurent
r n n
series. Set

REPRESENTATION THEORY 147
4> = £n(---+ cn+i + cnUi •
This is a finite Laurent series and x = (4 1 ~ 1)^.
(ii) Suppose U has two components. Take a basis as
before, but with 6 (e ) = 2. Then
r i
Exp(27Tier) = £*.
Consider x = £c £ and note that U is given by
ä = 1 and £ = -1. Therefore Zc = 0 and E(-l)nc = 0.
^i *i n n
Thus £ ,,c =0 and E c = 0. and we may argue as
n odd n n even n
before to get x = (£i - 1)^.
6.5 PROPOSITION. If x is an anti-symmetric character,
then
x=nm=1[Exp(27rie.) - ity,
where ljj is a character.
Proof . It is only necessary to show that, if
X= [Exp(2 7rie.) - 1]0
and x vanishes on U for r /i, then ^ vanishes on U , for
we may then argue by induction using 6.4. Well, il> does
indeed vanish on U except possibly on U. OU . But
r J i r
dim U = k - 1 and dim u. H U = k - 2. Therefore tjj vanishes
on all of U by continuity.

148 LECTURES ON LIE GROUPS
6.6 THEOREM. Suppose Ad lifts to Spin(n). Then tf> - tfiö
gives an isomorphism from the additive group of symmetric
characters to the additive group of anti-symmetric characters.
Proo f
(i) Ö is anti-symmetric (6.3), so the map goes where the
theorem says.
1 r _
(ii) J. \ ÖÖ = 1, so Ö / 0 and the map is mono (3.77).
(iii) Suppose x is an anti-symmetric character. Then by
(6.5) x =^ö, where ip is a character. Now
(sign 0)^6= (sign<^)(^^)ö
and<p^(t) =^(t) except, perhaps, where 0(0 = 0, that is, on
UU . Hence by continuity, <p^(t) = ^(0 for all t £ T and ^ is
symmetric. Thus the map is onto.
6.7 DEFINITION. Let h £ L(T)* be a weight, and let Wh
be the orbit of h under W. Then the elementary symmetric
sum S(h) is given by
w£Wh
6.8 EXAMPLE. LetG = SU(n).
Then

REPRESENTATION THEORY 149
s(xx) = ^ +...+ 4n,
where £. = Exp 2ttîx.
S(x+x) = 44 +44 ...+££
+ 4 c ••• + i i
^2^3 ^2 ^n
+ «n-.«n
S(2xi + xa) = Ç=C2+ ... Ç^n
6.9 PROPOSITION. Let h run over a set of representatives
of the orbits. Then S(h) runs over a Z-basis for the symmetric
elements of K(T).
This is obvious.
6.10 EXAMPLE. In 6.9, h may run over the weights in CI
FDWC.
6.11 LEMMA. Let x be an anti-symmetric character, and
h € L(T)* a singular weight, that is, h £ L(U )* for some r.
Then Exp 2?rih occurs with coefficient 0 in \.

150
LECTURES ON LIE GROUPS
Proof. Suppose
X = a Exp 27rih + . . ..
Let cp 6 W be reflection in L(U )*. Then
~X = (P X = a ExP 2 TT ih + ....
Thus a = 0.
6.12 DEFINITION. Let h € L(T)* be a weight. Then the
elementary alternating sum A(h) is given by
A(h) = L r,..(sign ^)Exp 2 7ri<ph.
If h is singular, then A(h) = 0. Otherwise, A(h) contains | W
distinct terms.
6.13 EXAMPLE. Let G = SU(n), and let
h = aj. x^ +.. . + aj-^_^Xj-j_1 .
Then
A(h) = det
ai a
C1 ...r
n-i
'n-i
«n -«n
n-i
1
6.14 PROPOSITION. Let h run over a set of representatives
of orbits of regular weights. Then A(h) runs over a Z-basis
for the anti-symmetric characters.

REPRESENTATION THEORY 151
This is obvious.
6.15 EXAMPLE. In 6.14, h may run over the weights in
FDWC.
6.16 PROPOSITION. Let x be the character of an
irreducible complex representation of G, and let $ = \|T. Then
^i6 = A(h) for some weight h.
Proof. If Ax ,A2 are elementary alternating sums, then
Â\a2 = |W| if Ax =A2
= -| W | if Ax = -A2
= 0 if Aj/iAj
by 3.34, since any weight is a character for T. Now ^6 may
be expressed as £n.A. ,n. € Z (6 .14), so
by 6.1
Thus one n. is ±1 and the rest zero. By a suitable choice of
l
h this can be made +1. Hence the result.
6.17 PROPOSITION. As x runs over the characters of the
distinct irreducible complex representations of G, the
i

152 LECTURES ON LIE GROUPS
corresponding A(h) are all distinct.
Proof . If Ax ,A2 correspond to x »X then
ThS/1*»"KTSr*l5<"8 =$GXiX= = ' "x>= x*
Hence the result.
To give a second proof, let K(T)W consist of the
symmetric elements in K(T) , that is, the elements invariant
under W; and let K(T) consist of the antisymmetric
elements. Consider the following composite.
K(G)-K(T)W-K(T)_W.
The first map is mono by 4.31, the second is iso by 6.6.
6.18 PROPOSITION. Suppose Ad lifts to Spin(n). Then
every alternating sum A(h) arises as ±ipb for some irreducible
representation of G.
Proof . A(h) = QÖ for some symmetric character a in K(T),
and cr= f| T for some class function f on G (4.32).
Now let x be the character of an irreducible complex
representation of G. Then
1 r
SGXf=|W|^a"

REPRESENTATION THEORY 153
= I I \ A(k)A(h), where A(k) corresponds to x
= 0 unless A(k) = ±A(h)
(see 6.16, proof). By the Peter-Weyl theorem (3.47) this is
not zero for all \. Hence A(h) = ±A(k) for some irreducible
representation \ of G.
Summarising, we now have:
6.19 PROPOSITION. Suppose Ad : G - SO(n) lifts to
Spin(n). Then we have a 1-1 correspondence between
irreducible representations of G and elementary alternating sums in
K(T), given by
K(G)-^K(T)W JUK(T)_W.
6.20 THEOREM. If G is compact and connected (but
without the assumption that Ad lifts) then the map
K(G) - K(T)W
is an isomorphism.
Proof . The map is mono, by 4.31.
Now, if Ad does not lift to Spin(n), we have the
following diagram (5.5 6(ii)):

15 4 LECTURES ON LIE GROUPS
->Spin(n)
>SO(n)
Let ||) bea symmetric element of K(T). Then ^TrcK(T) is a
symmetric element, so 077 = x|T for some virtual character \
of G (6.19). Now \\ T factors through G so \ factors through
G (4.32). Thus (3.68) X = Xn *or some virtual character \ of
G, and x|T =0.
6.21 REMARK. Even in the case where Ad does not lift to
Spin, we can define anti-symmetric elements as those x £ K(T)
such that (px = (sign <p)\ and x(xz) = ~x(x) f°r 1 ^ z £ Ker n.
6.22 DISCUSSION. We are going to show that, when
7Tj. (G) =0, K(G) is a polynomial algebra. Classically, the
weights are ordered in a somewhat arbitrary way. When we
propose to prove P = Q + " lower terms" the error must be
"lower" with respect to all choices of ordering. So we will
introduce an invariant partial order with which P = Q + "lower
terms" will have approximately this meaning.
6.23 DEFINITION. Let cüi ,œs be weights in L(T)*. Define

REPRESENTATION THEORY 155
a partial order on the weights in L(T)* by writing toi <.co2 if u^
lies in the convex hull of the orbit of co2 under W. That is,
co < cc if co = L ^,,,c (oco ) for some non-negative coeffi-
1—2 a (p€W ip 2
cients c with £c = 1.
It is clear that coi ^.co2 implies (p i cox ^(p2to2 for any
(fi x ,(fi 2 £ W, that is, we are ordering the orbits. So it will
suffice to consider weights in Cl FDWC.
6.24 ALTERNATIVE DEFINITION. For ccx ,co2 weights in
CI FDWC, write cox < to2 if cox (v) < co2 (v) for all v GFWC. We
may equally take all v £ Cl FWC.
6.25 PROPOSITION. These two definitions are equivalent.
We need:
6.26 LEMMA. If u € Cl FDWC, v £ FWC and (p^W, then
((pu)(v) £u(v) with equality only if (pu = u.
Proof . If (ß u = u then (<p u)(v) = u(v).
Suppose o u ^ u and (<ou)(v) >^u(v). Then, by moving v
slightly in FWC, fo)u)(v) > u(v).
Among the finite number of ^u as $ runs through W,
there is one, say to, such that (co,v) is maximal, and so co/u.

15 6 LECTURES ON LIE GROUPS
Then u fi Cl FDWC (5.16), so there is a simple root 0 such
that <0 ,œ) < 0. Consider o cc. We have
r r
(<pru)(v) = hc- <er/er> erJ(v)
2<6r/o;>
= "(v) - <e ,e> er(v)
r r
> üj(v),
which contradicts the definition of w. So the result is proved.
The inequality ((pu)v £u(v) remains true for v € Cl FWC,
by continuity.
Proof of 6.25
(i) If the first definition holds, then œ = Le (ç>(jû ) with
W 1 (p 2 '
0 < c ,Lc =1, and we assume that œ € Cl FDWC. Then for
— (0 <ß 2
all v 6 FWC we have
cd (v) = £c ((pw )(v)
<P 2
< Lc u (v) by 6.26
— <£ 2
= u2(v).
So the second definition holds.
(ii) Suppose, on the contrary, that œi does not lie in the
convex hull of the orbit of 602 , and suppose u^ 6 CI FDWC.
Then there is r? 6 L(T) such that o)j (r?) > (cflO)2)(r?) all <p ew.

REPRESENTATION THEORY 15 7
Write T) = H)(v) with # 6 W and v € Cl FWC. Then
Ux(v) >Wi (T?) (6.26)
> (^^s)(r?) = w2(^r?) = w2(v),
so the second definition does not hold.
6.27 PROPERTIES OF THE RELATION
(i) Transitive: <jôx £602 ^_(jC3 implies oïi £0:3, obviously.
(ii) Given cc2 , the number of weights œx such that cc^ £ o:2
is finite. This is clear from the first definition.
[Note: This is better than the classical ordering,
which allows one to make proofs by induction over the ordering
only for the semi-simple Lie groups. For example, U(n) is not
semi-simple. ]
(iii) <j£i £o)2 and cc2 £ ^>i if and only if a^ =^cos for some
0 6 W, as follows. It suffices to consider ajj. ,œ2 £ Cl FDWC.
If (jC\ / co2 we could find v in any open set of L(T) with
coi (v) / œs{v), contradicting the second definition.
6.28 DEFINITION. Write cüi < w2 if wx < u2 but not
ojg ^ aJx - We then say that cc^ is lower than o)2 .
6.29 EXERCISES, u <v < w implies u < w and u < v £w
implies u < w.

158 LECTURES ON LIE GROUPS
Continuation of 6.27
(iv) If u,v,w € CI FDWC then u + w < v + w if and only if
u <_ v.
(v) If t,u,v,w € CI FDWC and t <u, v < w then t + v <u + w
with equality only if t = u and v = w.
Let ß = —(f) + . . .+ 6 ), and let a; be a weight in CI
^ 2 l m
FDWC.
6.30 PROPOSITION. H Ad lifts to Spin, then
S(o))ö =A(cü + ß) + lower terms ,
that is,
S(cü)ö = A(w + p) + £n A(cc.)
with ce < eu + p.
Proof . (See 6.6.)
S(a:) = L Exp(277ito.)/
where cc. runs over the distinct iptc;
J
6 = L ± Exp(2î7 lu ) ,
where
u. = ^ue ... x6 ].
k. 2 i m
So
S{cc)b = L = Exp 2 T7i (a;. + u )
J k

REPRESENTATION THEORY 159
where uû. runs over those to + u. in Cl FDWC.
l j k
Now, if x € FWC, co.(x) = ((pco)(x) < oc(x) with equality
only if cpto = to (6.2 6), and u (x) £ß(x) with equality only if
u = p. Thus, if co. + u € CI FDWC, a; + u < a: + ß with
K. J K J K
equality only for the term to = to , u = p, which occurs with
J k
coefficient +1.
6.31 PROPOSITION. If Ad lifts to Spin, then
' '—" = S(co) + lower terms.
o
Proof . By induction. Suppose this is true for all to' < to .
Then (6.30)
S(co)ö = A(co + p) + Ln.A(co.)
with to. < co and
l
—— =LmljS(coj)
with co < co.. So
J - i
A(tO -I- ß) 0/ \ <e i \
x K—^ = S(co) - ~ .n m..s(to )
o ij i i) j
with co . < to .
J
6.32 EXAMPLE. If to = 0 we have ^^ = S(0) = 1. That is,
A(p) = Ö.

160 LECTURES ON LIE GROUPS
6.33 THEOREM. There is a 1-1 correspondence between
irreducible complex representations of G and weights œ in
CI FDWC in which xIt = S(cjû) + lower terms.
Proof
(i) If Ad lifts to Spin, then (x|T)ö = A{œ + ß) sets up the
correspondence (6.19 and 5.58) and
^|T _ _i_l—el - s(cjü) + lower terms
(6.31).
(ii) If Ad does not lift to Spin define 77 : G -♦ G as before
(5.5 6(ii)). For G, (i) holds.
For G we have y|t = Zn S (a;.), where a), runs over
/M 11 1
weights of G (6.9), so xtî\t - Dn.S(a;.), a;, being interpreted
as weights of G. If \ is irreducible then x77 1S irreducible,
so X^|T = S(o)) + lower terms. Therefore, by the uniqueness of
such expressions, xIt = S(cjû) + lower terms. This sets up the
correspondence and shows that it is mono.
Now let cc'be a weight for G in CI FDWC, and let x
be a character cf G such that xl'f = ~~^ • Then xlT factors
through T, so x factors through G both as a function and (3.68)
as a character. Since x *s irreducible, so is x-

REPRESENTATION THEORY 161
6.34 DEFINITION. It follows that each irreducible
representation of G has associated a maximal weight, which occurs
with multiplicity one.
6.35 EXAMPLE. Let G = SU(n).
For Lc — Xj we have
Oi + ß = nxj + (n - 2)x2
Then
A(io + ß) = A (co + ß)
6 A(ß)
4n_1Cn"2...C i
a 2 2
i ^n
For to = 2xx we get
1 i<j * J
6.36 PROPOSITION. Let u,v be weights in Cl FDWC. Then
S(u)S(v) = S(u + v) + lower terms.
.+ xn.l
n n-3
1 1
n n-a
2 2

162 LECTURES ON LIE GROUPS
Proof . Let
S(u) = E Exp 2 7Tiu., S(v) = E Exp 27Tiv ,
where u., v run over the distinct ipu ,(pv for (p 6 W. Then
J k
S(u)S(v) = E Exp 2 7Ti(u. + v ).
J k
If x € FWC, then (6 .26) (<ou)(x) < u(x) and (<0v)(x) < v(x)
with equality holding if (pu = u, (pv = v respectively. That is,
(u + vk) (x) < (u + v) (x)
with equality only for the single term u. = u, v. = v. Thus, if
u. = v. 6 CI FDWC, then u. + v. < u + v except in the single
j k j k
case u. = u, v, = v. This gives the result.
j k
6.37 EXAMPLES. Let G = SU(n) (see 6 .8).
(EC.) (EC.) = E£2 + 2.L.É.4.
= ££? + lower terms .
^ l
= E £f £. + lower terms.
Let G = SO(2n + 1).
(EÇ. + ECTMdC^ S«7X)
V *i Si / ^i<j i] i/j i J i<j ! J /
= (E£2 + EC"2 ) + lower terms.

REPRESENTATION THEORY 163
6.38 DISCUSSION. Suppose tït (G) = 0. We know that the
weights in Cl FDWC form a free semi-group generated by
to , ... ,œ, (5 .62). So there are irreducible representations
Pi Pk
of G such that
X(p)|t = S(oj) + lower terms.
Using 6.36 inductively,
X ( pUl . .. p. ) |T = S(n a; +. . .+ n cc.) + lower terms.
6.39 PROPOSITION. If 77 x (G) = 0, then
Z[Pl pR] -K(G)
is mono.
Proof . Let
a m +. ..+ a m =0
ii r r
be a linear combination of distinct monomials m. in the p's
l r
with 0 /a. € Z. Since the monomials are in 1-1 correspondence
with the weights in CI FDWC, we can order the monomials by
reference to the weights. If the linear combination is
nonempty, then it contains an m, suchthat no m. >m.. Let œ be
l j l
the weight corresponding to m,. Then in
X:(aimi +...+ armr)|T
the only term in S(œ) is a.S (to), so a. = 0, which is a

164 LECTURES ON LIE GROUPS
contradiction.
6.40 PROPOSITION
S (n a; +.. . + n oo ) - \ ( p x ... Pk + lower monomials ) | T.
Proof . We proceed by induction. Write
oj = n a; + + n. cc. ,
ii k k
and suppose the result is true for all a) ' < 60 .
x(p^ ... p^k)|T = S(o;) + ZmiS(coi)/
where œ. < œ. By the induction hypothesis,
S(o:.) = x(l°wer monomials)|T.
Therefore
S(c<j)=x(p X ••• Pk + lower monomials) |T.
6.41 THEOREM. Let G be a compact connected simply-
connected Lie group. Then
K(G) = Z[pi ,p2 pR].
Proof . By 6.39,
Zr.p, pk] -K(G)
is mono. By 6.40, the following composite is epi:
Z[px pR] -K(G)^K(T)W.
So the map is iso.

Chapter 7
REPRESENTATIONS OF
THE CLASSICAL GROUPS
In this chapter we will derive the complex representation
rings of the classical compact Lie groups. We will also enquire
if each group has any irreducible representations which are real
or quaternionic. For this purpose we consider the following
maps:
K (G) 1+t> K(G) 1-t > K(G).
We define
H = Ker(l -t )/Im(l + t).
7.1 PROPOSITION. H is an algebra over Z2 , and the
irreducible representations of G which are self-conjugate yield
a Z2 -base for H .
The proof is immediate from Chapter 3. We may
therefore measure the incidence of self-conjugate irreducible

166
LECTURES ON LIE GROUPS
representations by computing H.
We will also use the following lemma.
7.2 LEMMA For any complex representation V,
V* ® V = Hom(V,V) is real.
Proof. It carries the bilinear form
Tr(aß) = Tr(ßa)
(see 3.38); this form is symmetric, non-singular and invariant.
Now use 3.50.
We now begin to study the groups U(n) and SU(n). Each
has an obvious representation with V = C ; we write
i2 n
X ,X ,.. . ,X for the exterior powers of this operation. Let us
write
z. = Exp(2 7Tix.),
so that the typical element in our maximal torus is
D =
k k
then the character \(X ) of X is the kth elementary symmetric
function of z , z ,...,z . (See the proof of 3.61.) The Weyl

REPRESENTATIONS OF THE CLASSICAL GROUPS 167
group acts by permuting z , z ,. .. ,z . Thus the character
X(X ) is the elementary symmetric sum
S(xx + x2 +...+ xR).
Since \(^ ) consists of a single elementary symmetric sum.
k n
X is irreducible. The representation X of U(n) is one-
dimensional, and is essentially det : U(n) -S . In particular,
it is invertible. The restriction of X to SU(n) is trivial.
7.3 THEOREM. The complex representation ring K(U(n))
is the tensor product of the polynomial ring generated by
XX,X2,... ,X and the ring of finite Laurent series in X .
The algebra H is polynomial on generators XX /X for
2 <^2i<_n. These generators are real.
There is of course no suggestion that the modules
XX /X are irreducible; indeed they are not.
Proof of 7.3. By a classical theorem, the ring of
symmetric polynomials in z , z , .. . ,z is a polynomial ring generated
by the elementary symmetric functions x^1) /•••/X(^ )• Now
take any finite Laurent series which is symmetric; by
multiplying it with a suitably high power of z z ... z , we
obtain a symmetric polynomial. Hence K(T) is as described.

168 LECTURES ON LIE GROUPS
The result for K(U(n)) follows by 6.20.
Since we have an obvious pairing
X ® X - X ,
the dual of X is X /X . (This also follows from an easy
calculation with characters.) Hence the conjugate of
[Xl)VliXs)Vs ... (Xn)"n
is
<x>)""-1 (\*)^s ... (xn-l)Vl (xn)"1'1 ""'•••""n .
So t permutes the monomials in X1 , Xs , ... , X ; and we easily
see that the only monomials which are fixed under t are the
polynomials in
'>n M < i <
i n-i. n . 1 .
XX /X (1 £ i £Tn).
These are real by 7.2, since
Horn(X* .X1) = AiXn"i/Xn.
7.4 THEOREM. The complex representation ring K(SU(n))
is a polynomial ring generated by X1 ,XS , ... , X . The
algebra H is polynomial on generators X X for 2 £2i < n
and, if n = 2m, a generator X . The generators X X are
real; the generator X is real for m even, quaternionic for
m odd.
Proof. The result on K(SU(n)) is a special case of 6.41;

REPRESENTATIONS OF THE CLASSICAL GROUPS 169
the identification of the basic weights co , ..., co, mentioned
l k
in 6.38 is given in 5.63.
of
As above, the dual of X is X . Hence the conjugate
(XM'MX3)"3 ... (x11"1)"11"1
is
a« ^(x»)""-»... (Xn-y>.
So t permutes the monomials in X1 , X2 , ... ,X " ; and we
easily see that the only monomials which are fixed under t are
polynomials in X X (1 £ i < rn) and X if n = 2m. The
representation XX is real by 7.2. As for X , the pairing
.m _m ,2m
X ® X - X = C
has
P A a = (-1) a A p ;
now use 3.50.
7.5 EXERCISE. Show directly that any representation V of
SU(n) extends to U(n). (Hint: It is sufficient to consider an
irreducible representation; now consider the action of the
centre of SU(n).)
We take next the group Sp(n). It has an obvious
representation on Q ~ C2n; we write X1 , X2 , ,Xsn for the
exterior powers of this representation. As we have seen in

170
LECTURES ON LIE GROUPS
Chapter 3, X is real for k even, quaternionic for k odd. If
we take the element
D =
?n .,
in T, its action on C is given by
Therefore the character \(X ) of X is the ith elementary
symmetric function of
zl i zl / z2 / Z2 i • ' ' i Zy\'Zi\ '
7.6 THEOREM. K(Sp(n)) is a polynomial algebra with
i2 ri
generators X , X ,...,X . All the irreducible representations
of Sp(n) are self-conjugate.
Pi oof
(i) It is rather easy to see that K(T) is as stated; now

REPRESENTATIONS OF THE CLASSICAL GROUPS
171
use 6.20. Alternatively, use 6.41.
(ii) It follows from the generators given that the whole of
K(Sp(n)) is self-conjugate. Alternatively, in Sp(n) each
element g is conjugate to g-1 (see 5.17).
We take next the group SO(n). It has an obvious
representation on R or C ; we write A1 , A2 , .. . ,X for the
exterior powers of this representation. All these
representations are real. If we take the element
D =
Zl
in U(n) and embed it in SO(2n), its action on C2n is equiva
lent to that of the diagonal matrix
n_l .
Therefore the character \(A ) of A is the ith elementary
symmetric function of

172
LECTURES ON LIE GROUPS
l' l ,Z2' 2 '* * *' n' n '
say a.. Similarly, if we embed D in SO(2n + 1), its action
on Csn+1 is equivalent to that of the diagonal matrix
Therefore we have
XiX1) = a. + at_x.
(Here a0 is to be interpreted as 1.)
7.7 THEOREM. K(SO(2n + 1)) is a polynomial algebra with
generators X1 ,\s , . . . ,X . All the irreducible representations
of SO(2n + 1) are real.
Proof
(i) K(T)\a/- is exactly the same as for Sp(n).
(ii) It follows from the generators given that the whole of
K(SO(2n +1)) is real.
So far the exterior powers X have given us all the
generators we need. It is easy to produce arguments to show

REPRESENTATIONS OF THE CLASSICAL GROUPS 173
that for SO(2n) we need something else.
(i) In SO(4n + 2) not every element g is conjugate to g"1.
Therefore it is possible to construct a class function f such
that f(g) / f(g_1). Therefore (3.47) SO(4n + 2) has at least
one representation which is not self-conjugate. But all the
X are real.
(ii) Consider the representation X of SU(2n). We have
already seen that it is self-conjugate. So its restriction to
SO(2n) is self-conjugate for two essentially different reasons:
first because X is self-conjugate on SU(2n), and secondly
because each exterior power X is real on SO(2n). But we
have already seen that an irreducible representation V can
have essentially only one isomorphism with V*. Therefore
the representation X of SO(2n) is reducible.
If n is odd this argument is complete in itself; the
representation X of SO(2n) is both quatermonic and real, so
it cannot be irreducible. If n is even it is desirable to
amplify the word "essentially" a little, and this will be done
below.
(iii) An alternative argument proceeds by considering the
representation X of 0(2n). Consider an element g in 0(2n)

174 LECTURES ON LIE GROUPS
such that det(g) = -1; it is easy to see that its action on Csn
is equivalent to that of a diagonal matrix
"" zi
zi
zn-i_
zn_i
1
-1 _
It is now easy to check that the restriction of x = x(^ ) to tne
component of determinant -1 in 0(2n) is zero. Let the average
value of XX over SO(2n) be v', then the average value of XX
over 0(2n) is jv. So jv > 1 (3.34) and v >2. That is, \n
must split over SO(2n) into at least two summands.
We now amplify argument (ii). Let us define a non-
singular bilinear pairing
F : Xn(Rsn) ® \Vn) - \2n(R2n) = R
by F(v,w)=va w. Then F is invariant under SO(2n); indeed
for g € 0(2n) we have
F(gv,gw) = (det g)F(v,w).
Let us define another non-singular bilinear pairing
S : Xn(Rsn) ® Xn(Rsn) -R
by

REPRESENTATIONS OF THE CLASSICAL GROUPS 175
Sf(vx A V2 A. . .A vn) ® (wx A W2 A. . .A Wn) \
= £e(p)(v' ,..w,) ... (v" , xw ).
p X^/X p(l) *' p(n) n'
Here p runs over all permutations, and v'w is the usual
inner product in R2n. Then S is invariant under 0(2n). Let
us define an automorphism ß of A (R2n) by setting
S(ßv,w) = F(v,w).
We easily check that for g 6 0(2n) we have
ßgv = (det g)ßv.
We may describe ß as follows. Let vx ,v2 , . . .,v2n
be any orthonormal basis with determinant +1 in R2n; then
ß(Vl A V2 A .. .A Vn) = Vn+1 A Vn+2 A . . .A V2n.
Thus ßs = (-l)n.
It follows that X (R2n) splits into the ±1 eigenspaces
of ß if n is even, and into the ±i eigenspaces of ß if n is
odd. Of course the latter splitting takes place over C.
Elements of SO(2n) preserve the two eigenspaces; elements of
determinant -1 in 0(2n) interchange the two eigenspaces. In
particular, neither eigenspace can be zero.
We now enquire after the characters of the summands
(say V and W). The character \(\ ) of X is the nth
elementary symmetric function of

176 LECTURES ON LIE GROUPS
zi' zi ' z2' Z2 ' • ' "Zn'Zn *
Let us write
a+ = Sz^1 z2s . . . znCn | cr = ± 1 and cx e2 ... cn = +1.
a- = E2i z2 . . . zn | 6r == ± 1 and ex ea . . cn = -1.
These are elementary symmetric sums (see 5.17). We have
X(A ) = a+ + a_ + lower terms.
Since the characters of representations are linear combinations
with non-negative coefficients of elementary symmetric sums,
we have
X(V) = aa+ + ba_ + a
where a and b are 0 or 1, and a is a sum of lower terms.
Now consider the automorphism 6 of SO(2n) obtained by
conjugating with an element g of determinant -1 in 0(2n), say
g=" i
1
1
"I J .
Its effect on T is to invert z ; thus 6a+ = a_ , 6a_ = a+ and
6cr = a. Hence
X(W)= ba+ + aa_ + a,
X(Xn) = (a + b)(a+ + a_) + 2a

REPRESENTATIONS OF THE CLASSICAL GROUPS 177
and a + b = 1. It follows that we can name the summands of
X so that
X(X+) = a+ + a
X(X_) = a_ + a .
7.8 COROLLARY. The automorphism 6 of SO(2n) is not
inner.
Proof . An inner automorphism takes a representation into an
equivalent representation.
7.9 THEOREM. K(SO(2n)) is a free module over the
polynomial ring ZCX1, \2, . .. ,X ] on two generators 1 and X+ (or
equivalently 1 and X_). If n is even all the irreducible
representations of SO(2n) are real. If n is odd,
H = Z2[X1/XS Xn_1].
Proof. We have to study K(T) , that is, the set of finite
Laurent series in z ,z , .. . ,z which are symmetric under
permutations and under inverting an even number of the z .
The set S of such symmetric elements admits an automorphism
6 : invert an odd number of the z . (Of course, 6 arises as
r
explained above.) We have B2 = 1. So over the rationals, S

178 LECTURES ON LIE GROUPS
splits as the sum of the +1 and -1 eigenspaces of 6:
s = j(l + 6)s +j{l -6)s.
The +1 eigenspace is the ring of polynomials in
xftM.xft3) x(*n)
(as in 7.6, 7.7). Suppose given an element a_ in the -1
eigenspace. Then
a = Ec (z ,z ,. . . ,z )z ,
r r l ' 2 n-i n
where
so that
c = -c ,
-r r
n r _r
a = Ec (z ,z ,...(z )(z-z).
l r l 2 n-i n n
Thus
a =a'(z - z_1).
n n
By symmetry a_ is divisible by the remaining (z - z ); so
a = a" (z, - z-1)(z -z_1)...(z -z"1).
i l 2 2 n n
Here a" must be an element of the +1 eigenspace; so we have
a = p(a+ - a_)
where p is a polynomial in \{Xl), ... ,\(X ). Fora general
element s in S we have
s = j(l + 6)s +-p(a+ - a_).
Since a+ + a_ lies in the +1 eigenspace we may write this
s = q + pa +

REPRESENTATIONS OF THE CLASSICAL GROUPS 179
where q lies in the +1 eigenspace and is integral (since s
and pa+ are so).
So K(T) is as claimed, and the result on K(SO(2n))
follows by 6.20.
If n is even all the generators for K(SO(2n)) are real.
If n is odd t(X+) = X_ , and the calculation of H is easy. This
completes the proof.
For lack of time I have not included anything on the
representation-theory of Spin(n). Of course, this is included
as a special case of 6.41; but some may prefer to see the
basic representations arise more directly. I advise such
readers to study Clifford algebras out of [1] and the
representations of the Clifford algebras out of [7]. In [7] Eckmann
actually studies the representations of a certain finite group
G, but the Clifford algebra is an obvious quotient of the group
ring R(G), and so the representations of the Clifford algebra
are easily read off from the representations of G.

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