Lectures on Lie Groups

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

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LECTURES ON LIE GROUPS

J FRANK ADAMS University of Manchester

W A BENJAMIN, INC New York

1969

Amsterdam


Copyright ~ 1969 by \N. A. Benjamin, Inc. All rights reserved

Standard Book Numbers. 8053-0116-X (Cloth) 8053-0117 -8 (Parer) 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

jx

Chapter 1.

BASIC DEFINITIONS

1

2.

ONE-PARAMETER SUBGROUPS THE EXPONENTIAL MAP ETC. I

I

7

3.

ELEMENTARY REPRESENTATION THEORY

22

4.

MAXIMAL TORI IN LIE GROUPS

79

5.

Gr:OMETRY OF THE STIEFEL DIAGRAM

101

6.

REPRESENTATION THEORY

142

7.

REPRESENTATIONS OF THE CIASSICAL GROUPS

165 180

REFERENCES

vii

FOREWORD

These notes derive from a course on the representationtheory 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 rna y be asked why one who is not an expert on Lie

groups should relea se 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 represen-

tation-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. raic topologist tries to read

I

If an algeb-

for example, Borel and Hirzebruch's

paper" Characteristic Classes and Homogeneous Spaces" [3J

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, Botti s

II

Lectures on K(X)" [4J 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 a s Theorems 6.20 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 orthogona lity 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

inva riant wa y (see 3.46 and 3.47 beloW). I have not seen these

II

invariant

ll

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 t!1e 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) weight

It is usual to give a meaning to the words ll

II

highest

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 technica 1 advantages (see 6.22 and 6.23 beloW).

I hope this departure from tradition may commend

its elf to other workers. I am most grateful to A. Borel, to Haris h-Chandra and particularly to H. Samelson for giving me tutoria Is on Lie groups and representation-theory.

I have also profited from

xii R. G. Swan's .. Notes on Maxima 1 Tori

I

etc.".

I am a Iso 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;

I

it allowed him to slim the original lec-

tures 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

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, f a map from U to W, and x a point of U.

Then f is differ-

entiable at x if there is a linear map fl (x) : V - W such that f (x + h)

= f (x) +

(fl (x)) (h) +

0

I hi.

If f is differentiable at each point of U, we say that f is

differentiable on U.

In this case we have a function

fl : U - Hom (V , W) , and we may ask if this is differentiable.

We say that f is

smooth (or of class Ceo) on U if each function f, fl, fll, .•. is differentiable on U.

(Of course, the definition of each of

these depends on the previous one being defined and differentiable. )


2

1.2

DEFINITIONS.

If X is a topological space and V a

finite dimensional vector space, a chart is a homeomorphism ~

a

: U

a

-t

X , where U c V is open and X c X is open. a a a

An atla s is a collection of cha rts [ep } with UX a

a

= X.

The atlas is smooth if the functions ep-1ep , defined on f3 a ~-1

a

(X nx ) a

f3

I

are smooth.

Let X, Y be topological spaces with smooth atlases (epa) and {l/l f3}.

Then a map f : X

~-lfep ,defined oncp-1 (X nf-1y),

f3

a

a

a

f3

-t

Y is smooth if the maps are smooth.

Notice that

the composition of two smooth maps is smooth, and the identity map of a space with atlas is smooth. Two atlases {epa}' {l/lf3} on X are eguivalent if the maps 1 :X, [G, IJ. h ecomposltlonV1'07 . . 2

and so is differentiable. and V2

= V1

e

Further

I

~

I

is the identity on both V1

and so is the identity on G • We may proceed as in e

2.14.

2.16

PROPOSITION.

of G, and let S

C

Let G1 denote the identity component

G1 be a neighbourhood of e. Then the sub-

group generated by S is G 1

•

Pro 0 f.

G1

Clearly gp{S1

C

•

Now gp[S} is an open sub-

group of Gl , so all its cosets are open. Thus gp{S} is also closed, so gp[S}

2 .17

THEOREM.

= G1 . If G is connected, a homomorphism of Lie


14 groups 9

1 :

G

e:G - H • e

e

Proof.

Thus

H is determlned by the induced homomorphism

By 2. 11 we have the commutative diagram:

e is

determined by 9 at least on the subgroup of G 1

generated by the image of exp. e in G, so

2.18

e is

determined on G.

LEMMA.

Let

~

sends 0 E V to e E G. xy = x + y +

0

But this is a neighbourhood of

a

: U

a

- G

a

be a chart on G which

Then, omitting

~

a

,we can write

(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 differentiable, there is a

constant vector a and constant linear functions b, c such that xy

=a +

bx + cy +

0

(r).

Set x

=e

and we find that

y = a + cy +

0

(r) so a = 0, c = 1. Similarly b = I, so

xy = x + y +

0

(r) •

15

ONE-PARAMETER SUBGROUPS

2.19

THEOREM. a

form T

Proof.

A connected Abelian Lie group G has the

b xR •

We show first that exp : G

e

- G is a homomorphis m.

Well, exp s exp t = (ex p

~)N (ex p ~)N

t s ( exp N exp N

)N

since G is Abelian

by 2.18 and 2.14, where we consider s , t fixed and N varying

= ex p (s

+

t

+

0

(1 ))

= exp(s + t).

Thus exp is a homomorphism, and, by 2.16, exp is onto. Consider K = Ker exp. morphis m, K is dis crete.

By 2.14, since exp is a homo-

Now a dis crete subgroup of a rea 1

vector space is a free Abelian group, with generators gl, ••• , gr which are linearly independent over R.

(This is

proved by induction over the dimens ion of the vector space.) Extend this to a basis of G • Then K is expressed as the set e of points with coordinates (nI, ••. ,nr,O, •.• ,0), each n n r r Thus G = G /K = T x R - • e

i

E Z.

16


2.20

COROLIARY.

A Lie group which is compact, connec-

ted and Abelian is a torus.

2.21

EXERCISE.

2 .22

DEFINITION of submanifold .

Classify the compact Abelian Lie groups.

Let W c V be a real finite dimensional vector space and subspace.

Let M be a smooth manifold, and N a subset of M.

Then a chart (i)

M

(ii)

~

a

a

r

~

a

N

: V

a

=¢

sends V

a

-+

M

a

is good if

(the empty set) , or

nW

onto M

a

n N.

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

I

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 is clear and left to the reader.

Proof

I

2.24

REMARK.

2.25

EXERCISE.

Nl

X

17

It follows that T(N) is embedded in T(M).

If Nl

N2 are submanifolds of Ml

I

N2 is a submanifold of Ml x M2

I

1M2

then

and its differential

structure as a submanifold is the same as its differential structure a s a product.

PROPOSITION.

2 .26

If G is a Lie group

submanifold and a subgroup

Proof.

and H is both a

then H is a Lie group.

Apply 2.23 and 2.25 to the maps of pairs

Ii : G x G H x H I

2.27

I

I

-+

THEOREM.

G Hand i: G H I

I

-+

G H. I

A closed subgroup H of a Lie group G is a

s ubmanifold •

Proof.

LEMlvtA.

2.28

oI h

n

h ->

The next three lemmas constitute a proof.

n

In 2.27

I

suppose G

e

has a norm.

EGis a sequence of points such that exp h e

1 0, and P h h I nn I n

->

Suppose n

E H,

v E G . Then exp (tv) E H for all t E R. e


18 t Ih

I h n - tv and Ih n I - 0 so we rna y choose inten gers m such that m Ih I - t. Then exp m h - exp tv. But n n n n n Proof.

I

exp m h = (exp h) n n n

mn

I

E H

I

and H is closed. So exp tv E H,

as required. Let W be the set of such tw in G • Then exp We H. e

2.29

LEMMA.

Proof.

W is a vector subspace of G • e

Clearly w E W implies tw E Wall t E R.

So

I

suppose WI w 2 E W I

I

10.

and suppose WI + W2

We will show that WI + W2 E W. Consider exp(twI )exp(tw2). This is in H.

For t suffi-

ciently small we can write exp(twI )exp(tw2) = exp (f(t))

I

where

f (t) is a smooth curve in G and f (0) = O. e Now exp(twI )exp(tw 2 ) - exp t(w 1 + w 2 ) so

~f(t)

hn

= (~),

- WI + W2 as t - O.

= oCt)

I

by 2.18

Thus we may apply 2.28 with

for n sufficiently large, and v

= Iw,

! w,1 (w, + w, l,

and deduce that w 1 + W2 E W.

2 .30

LEMMA.

Proof.

exp W is a neighbourhood of e in H.

Split Gas WI e

e Wand

cons ider the diffeomorphis m

cP (Wi w) = exp (Wi )exp (w) between a neighbourhood of 0 in G I

I

e

19

ONE-PARAMETER SUBGROUPS and a neighbourhood of e in G (2.15). not hold.

Suppose the lemma does

Then there is a sequence of pa irs

exp(w )exp(w ) E H, exp(w )exp(w ) n n n n l

l

-+

(Wi , W

e, and

n

Wi

n

n

)

such that

10.

Since

exp(w ) E H,exp(w l ) E H. Then we can find a subsequence of n n 1 Wi such that - - Wi' -+ Wi E WI, for some such Wi, and n Iw~1 n Iw II = 1. It follows from 2.28 that

Wi

E W, which is a contra-

diction. 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. any other point of H.

2 • 31

Left translation gives a good chart round This completes thE: proof of 2.27 •

EXAMPLES. O(n)

C

GL(n, R)

U(n)

C

GL(n, C)

SP (n)

C

GL(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 consists of the matrices X such that

XT

= -X.


20 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 subgrouP. Then f (t) Tf (t) = I, by definition of the subgroup. That is,

-T

(1 + tX

+ o(t)) (l + tX + o(t))

= 1.

Thus xT+X=o. (ii)

-T

= -X.

Suppose X

Then (exp tX) T=

(~

= ~t =

00

o

n

t n Xn In!

)T

by 2.12

n

(-X) In!

(exp tX)-l.

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 a

closed connected Abelian subgroup

2 .33

PROPOSITION.

I

then H is a torus.

A closed connected subgroup H of a

Lie group G is determined by its tangent space at e.

Proof.

See 2. 1 7 .

21

ONE-PARAMETER SUBGROUPS

2.34

DEFINITION.

closed subgroup. cosets gH.

Suppose G is a Lie group and H a

Then the quotient space G/H is the set of

We have the projection p : G

-+

G/H, and give G/H

the quotient topology.

2.35

EXERCISE.

2.36

PROPOSITION.

G /H is Hausdorff.

If H is a closed subgroup of a tie

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.

SplitG

e

asW'E9WwhereW=H

e

as before.

be a small neighbourhood of 0 in W', and define l/J : U by U

-+

W'

-+

G

e

exp!» G ~ G/H.

with a neighbourhood of eH.

Proof.

PROPOSITION.

H

See [17, 1.7.5J.

-+

G/H

Then l/J is a homeomorphis m

The rest of the proof is left as an

exercis e for the reader.

2 .37

-+

LetU

G

->

G/H is a fibration.

Chapter 3


In this chapter we set up elementary representationtheory.

The ba sic definitions a nd constructions occupy 3. 1

to 3. 13. Then we introduce integration. the usual consequences

I

including complete reducibility (3.15

to 3.21). Then comes Schur's Lemma quences

I

From this we draw

I

some of its conse-

and the definition of the representation ring (3.22 to

3.28). Then traces

I

characters and the orthogonality relations

(3.29 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


3.1

DEFINITIONS.

23

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

e:G

- Aut V.

(Such a V is also called a representation of G over A or a Gspace over A.) Alternatively, for each g E G and v E V we are given gv E 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

e 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 E Q, v E V).

Here the conjugate of a quaternion is defined as usual:

if


24 q

=a

+ bi + cj + dk, then

q=a

- bi - ij - dk.

Let V and W be I\.G-s pa ces. A G-map is a function f : V - W which commutes with the action of G, that is, f (gv)

= g (fv).

A I\. G-map is a G-map which is I\.-linear: mostly we deal with such.

The set of such I\.G-maps is written HomI\.G(V, W), or

sometimes simply Hom tor space over R if I\.

G

N, W)

= R or

if I\. is understood.

Q, over C if I\.

It is a vec-

= c.

A I\.G-isomorphism is a I\.G-map which has an inverse. As usual, we say that two I\.G-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 E C)

I

and

= ±1.

(ii)

j2

3.3

EXPIANATION.

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


2S

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 j2 = -I, we can clearly reconstruct a G-space over Q. Similarly, it is often convenient to regard a G-space V over R as being equiva lent to a G-space V lover C provided with a structure map such that j2

= +1.

To pass from V to VI

we take V I = C ®R V provided with the obvious operations and structure maps: Z (Zl ~

v) =

g(z ®v)

ZZI

=z

®v

(z , Zl E C)

® gv

j (z ® v) = z ® v.

To pass from V

I

to V we split V into the +1 and -1 eigenI

spaces of j; these are G-spaces over R which are isomorphic under i. These operations are clearly inverse to one another,


26

up to isomorphism.

3.4

DEFINITION.

Given I\G-spaces V and W, we can form

the direct sum of the two vector spaces

V (f)W and make G

I

I

act on it by g(v w) = (gv gw). I

I

Equivalently we may take two G-spaces V and Wover I

C with structure maps j ,j

v

w

such that j2 = j2

w

v

I

and put on

V ffi W the structure map \, (f) jw • The next five operations start from a G-spa ce V over 1\ and construct a G-space over some 1\'.

are displayed in the following diagram

I

The possibilities

which is not commuta-

tive.

3.5

DEFINITIONS.

(i)

If V is a G-spa ce over R define cV = C 0 R V regarded I

I

as a G-space over C as in 3.3. (ii)

Similarly

I

if V is a G-space over C

I

define

qV = Q ®C V and regard it in the obvious way as a G-space I


27

and a left module over Q. (iii)

If V is a G-space over Q

let c V have the same under1

I

lying set as V and the same operations from G but regard it I

as a vector space over C. (iv)

Similarly

I

if V is a G-space over C

I

let rV have the

same underlying set as V and the same operations from G

I

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

I

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

I

and

they act by forgetting the structure map. All these constructions are natural; given a J\G-map

f : V- W

I

we can construct maps cf

I

qf

I

c' f

I

rf and tf.

All these constructions commute with direct sums

3.6

PROPOSITION. rc

=2

cr

= 1 +t

qc' = 2

~.

28

LECTURES ON LIE GROUPS c'q

=1 + t

tc

= c

rt

=r

tc'

c'

qt

=q

t2

= 1.

These equations are to be interpreted as saying that rcV';: V ~ V crV ~ V Ei) tV

for each V over R 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 0 with C acting on the first factor and G on the second.

R

V

Let C

act on the first factor of C ®R C, and let C ®R C act on C

~

V in the obvious way.

Then C ®R V is a G-space over

the C-a Igebra C ®R C. We will now split the unit 1 ® 1 of C ®R C into orthogonal idempotents, and so obtain a splitting of C ®R V. detail, let e1 =

t(l

® 1 + i ® i)

In

ELEMENTARY REPRESENTATION THEORY e2

=

Then ei = e 1

t(l ,

29

® 1 - i ® i) •

e~ = e 2 , e 1 e 2 = 0 and e1 + e2 = I, as required.

So

where the isomorphism is a G-isomorphism over C.

Further,

v ~ e 2 (C ®R V)

by, for instance,

v"'" e 2 (1 ® v)

tV ~ e 1 (C ®R V)

by, for instance,

v ..... e 1 (1 ® v) •

and

Thus crV ~ V ~ tV.

3.7

DEFINITION.

Given G-spaces V and W over C, we

can form the tensor product of the two vector spaces, V ®C W, and make G act on it by g (v ® w) = gv ® gw. Suppose now that V and W admit structure maps such that j~ = map j

= jv

f.

V'

j~= f.

\r,jw

Then V ®C W admits a structure

W

® jw such that j2

= f.Vf. W •

We can separate three

cases. (i)

f.

V

= f. W = +1.

sentations is real.

The tensor product of two real repre-

The construction amounts to taking two

G-spaces V, W over R and forming V ®R W. (ii)

f. V = + I, f. W

= -1.

The tensor product of one real

representation V and one quaternionic representa tion W is

30


quaternionic.

The construction amounts to taking V ®R Wand

making Q act on it by q(v

~

w)

=v

® qw.

The case (V = -I, (W = + 1 is similar. (iii)

(V

= (W = -1.

The tensor product of two quaternionic

representations V and W is real. construction in terms of V

«b W.

It is natural to interpret the

For this to make sense we

must consider V as a right module over Q and VV as a left module over Q. If we use the resulting structure maps, then the -1 eigenspace of the structure map jv ® jw on V 0

C

W

coincides with V ®Q W.

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 EB.

3.9

DEFINITION.

Given G-spaces V and W over the same

field A, we can form Hom from V to W.

A

(V, W), the set of A-linear maps

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

= g(h(g-lv))

(h E Hom

A

(V, W))

31

ELEMENTARY REPRESENTATION THEORY or eq 11ivalently

(Note that Hom AN, W) is covariant in Wand contravariant in v.) The subspace of elements in Hom

A

(V,W) which are in-

variant under G is precisely Hom I\G N, W). We may also proceed as in 3.3, 3.7.

Let V and W be

G-spaces over e which admit structure maps jv,jw such that j~

= f. V ' j ~

= f.

W

• Then Home(V, W) admits a structure map j

given by

(Note that Home(V, W) is covariant in Wand contravariant in V.)

= f.Vf. W •

We have j2

(i)

f.

real.

V

= f. W

= +1.

We can separate three cases.

The Hom of two real representations is

The construction amounts to taking two G-spaces V, W

over R and forming Hom (V , W). R

In fact, if we form

Home (cV, cW), then the +1 eigenspace of j may be identified with Hom

R

tv, W);

thus

Home(cV, cW) ~ cHom (ii)

f.

V

= +1,

f.

W

= -1.

RN

, W).

The Hom of a real representation

into a quaternionic representation is quatemionic. f.

V

= -I, f.

W

= +1

is similar.

The case

We leave it to the reader to

32


interpret the construction in these cases along the lines of 3. 7 (ii). (iii)

f.

V

= f. W

= -1.

tations is real.

The Hom of two q aternionic represen-

The construction amounts to taking two G-

spaces V, W over Q a nd forming HomO (V, W).

In fact, if we

form HomC(c'V,c'W), then the +1 eigenspace of j is HomO (V, W); thus HomC(c'V ,c'W) ~ cHom

O

N , W).

3.10

COROLLARY.

(i)

If V and Ware two G-spaces over R then

dimCHom (ii)

CG

(cV , cW)

= dimRHom RG (V, W).

If V and Ware two G-spaces over 0 then

dimCHom

CG

(c' V, c' W) == dimRHom

OG

(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 Hom's are compatible with the maps c, c' •

(ii)

Hom is bilinear over the direct sum ffi. A particular case of 3.9 is important.

33


3.12

DEFINITION.

Given a G-space V over C, we define

its dual V* by V* = HomC(V, C). Here the target space C is given the trivial operations from G: gz = z for all g E G and z E 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 quatemionic. The general case 3.9 may be reduced to the special case 3.12.

3.13

LEMMA.

We have an isomorphism

HomC(V,W} ~V* ®C W commuting with the action of G and with the structure maps j (if any).

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


real number

r

JG

f =

r

f(g)

.)gEG

so as to satisfy the following conditions. (i)

S ha s the usual properties of an integra I, tha t is,

it

G

is a pos itive linear functiona 1. (ii)

~

1 = 1. G

(iii)

The integral is invariant under left and right transla-

tions; that is, for each x E G we have

=

\ f(xy) , yEG

SyEGf(y)

SyEGf(yx) = SyEGf(y) • Similarly

I

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 disc 1SS this here, but refer the reader to [13,14,20]. 1

The first function which asks to be integrated is

e:G

- Hom AN, V) •

35


3. 15

e:G

PROPOSITION. - Hom/\ (V, V). I

= SeE

S uppos e gi ven a repres enta tion

Then

Hom /\ (V V) I

G

= I)

is idempotent (I2

and its image is VG' the subspace of

elements invariant under G.

Proof.

For each fixed v E V, the function Hom

(V V) - V I

given by h - h (v) is linear (over R); so it commutes with integretion.

That is, Iv =

S

gv.

gEG

It is now clear that 1m (I) c VG;

for

g' (Iv) = g'S gv gEG =

S

g'gv

(since 9 acts linearly)

gv

(invariance of integration under

gEG =

S aEG

left translation)

iv; so Iv E V . Also we have I\V G G

= v. The propos ition follows.

= 1;

for if v E V ' then G


36

Propositions 3. 16 and 3. 18 may be viewed as applications of the principle embodied in 3. 15; they could equa lly easily be proved directly.

PROPOSITION.

3 .16

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. over, if V carries a structure map j

I

More-

we can choose H so that

H(jv,jw) = H(v,w). The reader who wishes to do so may check that if V has a structure map j

I

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

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)

=S

H ( g-1 V , g-l w) .

gEG

If we start by choosing H to be positive definite

I

then K is

37

ELEMENTARY REPRESENTATION THEORY 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 22 or 24 group generated by j; the formula is K(v, w) = t(H(V, w) + H(jv ,jw)). This form ha s the required properties. If we impose on V an invariant Hermitian form, then

we can choose in V an orthonormal basis.

e:G U (n).

Thus we can regard

Aut Vas taking values not merely in GL(n,C), but in We then speak of a unitary representation.

Similarly

for orthogona land symplectic representations in the cases A = Rand Q.

3.17 Proof. form H.

COROLIARY.

If G is compact and A = C, then V*?!!tV.

Impose on Van invariant positive definite Hermitian To be explicit, suppose tha t 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


PROPOSITION.

3 .18

G-space V is projective.

If G is a compact group, then every

That is, suppose given the following

diagram of AG-maps, in which

~

is onto.

x

!~

V~y

Then there is a AG-map y : V - X such that the following diagram is commutative.

P fO

0

3.9.

f.

Consider HomA (V, X), made into a G-space as in

By 3. 15, if we take any A-map 0 : V - X and integrate

gO, we get a A-map y which is invariant under G, that is, a It is given by

A G-map.

y =

S

(9 g) 0 (BVg-l ) .

gEG

x

We can choose 0 to be a A-map such that have

~y

=

~s

(9 g)0(9 g-1 )

gEG =

S gEG

=

S gEG

x

v

~ (9xg) 0 (BVg- 1 ) (9yg)~0 (9 Vg-1 )

~O

= a.

Then we

39


= a.

3. 19

DEFINITION.

A non-zero G-space V is reducible if

some proper subspace 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 -

viS -

0

splits, so we have a /\G-isomorphism V ~ S ~ VIS.

Alternatively, if /\ = 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=SffiT.

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 wi 11 show tha t 3.20 does not hold for

groups which are not compact.

Embed R' in R2 as the subspace of vectors be the subgroup of GL(2, R) which stabilises Rl.

G is the set of matrices

[~l

Let G

Equivalently,

[~ ~] with ac '10.

Then R2 is a reducible G-space. However, no other proper subspace of R2 is stable under G, so R2 does not split as the direct sum of irreducible G-spaces. Alternati vely, to get a

II

minimal" counter-example,

. [10 b]1 .

take the group G to be the set of matrices

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).

(i)

If f : \' - W is a I\G-map and V, Ware irreducible then

Let G be any topological group.

f is either zero or an isomorphis m.


= C,

(ii)

If A

then fv

= AV for

41

f : V - V is a CG-map and V is irreducible

some constant A E C.

(In the second case we may write f = A.)

Proof. (i)

Since V and Ware irreducible, Ker f is V or 0 and 1m f

is 0 or W. (ii)

The res ult follows.

Consider f - A : V

-+

map is singular for some A.

V, where A runs through C.

This

By (i), f - A is then zero.

Thus

f = A.

3.23

COROLLARY.

(i)

If Vand Ware inequiva lent then Hom

(ii)

If V and Ware equivalent and A = C, then

dimCHom (iii)

CG

Let V and W be irreducible AG-spaces. AG

(V, W)

= o.

(V, W) = 1.

If V and Ware equiva lent and A = R or Q, then

dimRHom AG (V, W) ~ 1.

Proof.

For (iii) we observe that Hom AG (V, W) contains at

least one isomorphism. For the next proposition, let G be any topological group, and let V. run over the inequivalent irreducible 1

AG-spaces (as i runs over some set of indices I).

Let m

i

I

n be i

42


non-negative integers, of which all but a finite number are zero.

Let m. V. be the direct s urn of m. copies of V., and simi1 1 1 1

larly for n. V.. 1

THEOREM.

3.24 m.

1

1

= n.1

If

m.V. is equivalent to ~n.V., then ill ill

(B

for all i.

Pro 0 f.

Suppose

= EBn.V ..

(Bm.V. ill

ill

Then HomAG(V. ,E8m.V.) ~ Hom \G(V, ,ffin.V.), J i 11 i Jill that is, ~m.HomAG(V"V.) ~ G)n.HomAG(V., V.l.

i

1

J1

i1

J

1

Using 3.23(i), we get m.HomAG(V., V.) ~ n.HomAG(V., V,). J J J J J J Taking the dimension of both sides and using 3.23 (ii) or (i'ii), we get m.

J

= n .. J

If G is compact and A = C we can express the situation

which arises here in the following way (which we need for later use).

For any G-space V over C we can form

This is a finite sum, since HomCG(V , V) is zero for all but a i

43

ELEMENTARY REPRESENTATION THEORY finite number of i, by 3 .20 and 3.23 (i).

IJ. : ~HomCG(V, V) ®CV. 1

1,

1

-t

We ca n define

V

by evaluation:

IJ. (h. ® v.) 1

1

= h.1 (v.)1 .

We make G act on ~HomCG (Vi' V) ®C Vi by g (h. ® v.) = h. ® g v .. 1

1

1

1

Then IJ. is a G-map over C.

3.25

LEMI'v1A.

Assume G compact and A = C.

Then the map

is an isomorphism.

If V is irreducible the result is immediate by 3.23.

Proof.

Pass to direct sums and use 3.20.

3.26

DEFINITION.


Then K/\ (G) is the free abelian group generated by the equivalence classes of irreducible G-spaces over A. Tnus an element of K A (G) is a formal linear combination I; n.V., in which the V. are the equivalence classes of ill

1

irreducible G-spaces over A, and the n. are integers (positive, 1

negative or zero) which are zero for a II but a finite number of i.

44


By 3.20 and 3.24, the equivalence classes of G-spaces over

A are in 1-1 correspondence with those elements l:n.V. in ill

KA (G) such that n 2: 0 for all i. i An element of KA (G) is called a virtual representation or virtua 1 G-space. The operations c/c· ,r,q and t of 3.5 induce homomorphisms of abelian groups as displayed in the following diagram, which is not commutative.

The equations of 3.6 continue to hold.

3.27

PROPOSITION. c

: KR(G)

c· : KQ(G)

-0

-0

The maps

KC(G) KC(G)

are mono.

Proof.

rc

= 2,

qc' = 2 and KR (G), KO (G) are free abelian.

We shall normally regard KR (G) and KO(G) as embedded in KC(G) by c and c' .


45

3.28

COROLLARY

(i)

If V and Ware two G-spaces over R such that cV

~

cW,

then V ~W. (ii)

If Vand Ware two G-spaces over Q such that

l c V ~ c'W, then V ~

W.

This follows immediately from 3.27, but in ca s e 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 jv,jw such that j~ CG-isomorphism f : V

--+

= j~.

We suppose given a

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

I

or if, and integrating over the 22 or

24 group generated by j. f' = 1.(f + j 2

f"

= ~ i (f

The formulae are

Jrl)

W- V

- j

vl j ~ ).

We have fl - ifll = f.

So det (fl+ z fll) is a polynomial in z which

is not identically zero (for it is non-zero for z

= -i).

fore there is some real x for which det (fl + Xf")

10.

fl + Xf" is a CG-isomorphism which commutes with j.

ThereThen

46


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 KC (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 U'es in KA (G)

A = R or Q, and y lies in K 1\' (G)

C

C

KC(G), where

KC(G), where 1\

I

= R or Q,

then the product xy behaves as described in 3.7. The standard method of studying KC(G), and indeed the standard method of proving 3.24, is the study of characters. To define these, we need the

3.29

DEFINITION.

~.

Let V be a finite-dimensional vector

space over C, and let f : V - V be a linear map. define Tr f, the (i)

~

of f, in two ways.

Take a base of V, so that f corresponds to a matrix M ..• IJ

Set Tr f = I:M ... i

This is invariant under change of base, since

11

I: T., M 'k (Ti,j,k IJ J (ii)

Then we may

1

)

k'

1

= j,k 1: 1. k M' k = I: M ... J J j JJ

(Bourbaki) We ha ve an isomorphism a : V* ® V - Hom

C

(V, V)

given by (a (v* ® w»v = (v*v)w, as in 3. 13.

47

ELEMENTARY REPRESENTATION THEORY We have an evaluation map ( : V* ®V

-+

C

given by ((v* ®w) = v*w.

Define Tr f =

1

(a-

f.

It is easy to check that the two definitions are equiva-

lent.

The principal properties of the trace are as follows.

3.30

PROPOSITION.

(i)

Tr : HomC(V, V)

(ii)

Consider V

(iii)

Consider ~ ffi Y : V

L

-+

W

C is a linear map.

.L V.

Then

Tr(~y)

W

-+

V E9 W.

®Y : V ®W

-+

V ® W.

Ef)

=

Tr(y~).

Then Tr(~ ffi y) = Tr~ + Try. (iv)

Consider

~

Then Tr(~ ® y) = Tr~ . Try. (v)

Given ~ : V -+ V, define ~ * : V*

as usual. (vi)

Tr~ *

Then

Given

~

: V

-+

=

-+

V* by (~*v*)v =v* ((3v),

Tr~.

V, let

t~

: tV

-+

tV be as in 3.5 (v).

Tr(t~) = Tr~.

(vii)

If

~

: V

-+

V is idempotent, then

The proof may safel y be left to the reader.

Then

48 3.31


DEFINITION.

Given a G-space over C, we define its

character Xv : G - C by XV(g) = Tre g. It is clear that Xv depends only on the equiva lence

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 clV as the case may be.

(In the case A = R it would be equivalent to consider the

trace over R, but this doesn l t work so well for A = Q.)

3.32

PROPOSITION

(i)

Xv : G - C is continuous.

(ii)

Xv (xyx-1 ) = XV(y).

(iii)

XVffiW(g) = XV(g) + XW(g)·

(iv)

XV®Vj.g) = XV(g) . XW(g).

(v)

X * (g) = Xv (g-l ) . V

(vi)

XtV(g) = XV(g); if v is real or quaternionic then

XV(g)= XV(g)· (vii)

XV(e) = di m C (V) .

Each part follows from the corresponding part of 3.31; the second half of (vi) uses also the equations tc = c tc l = c l I


49

from 3.6.

3.33

PROPOSITION.

(i)

Xv(g-l.) = XV*(g) = XtV(g) = XV(g).

(ii)

SgEGXV(g) = dimCVG , where VG is the subspace of

Assume G compact.

elements of V invariant under G.

Proof. (i)

See 3. 17 .

(ii)

Since Tr is linear, we ha ve

SgEGTrB

9

= Tr =

SgEG B 9

Tr I

(s ee 3. 15)

= dimCIm I

(see 3.30 (vii))

=dimCV

(s ee 3. 15).

G

3.34

THEOREM.

(i)

Let G be compact and let V, W be G-spaces over A.

(Orthogona lity relations for characters. )

Then

SgEGXV(g)XW(g) = dim Hom AG =d

say

(V, W)

I

where the dimension is taken over C if A = C ,over R if A = R or Q.

50 (ii)

LECTURES ON LIE GROUPS If V and W

Now assume that V and Ware irreducible.

are inequivalent we have d = O.

If V and Ware equivalent and A = R or

A = C we have d = I. Q we have d

~

If V and Ware equivalent and

I.

Pro 0 f .

(i)

By 3. 10 the cases A = Rand Q follow immediately from

the case A= C.

So suppose A = C, and consider

H = HomC(V, W). dimCHom =;

We have CG

(V, W) = dimCHG

r

X (g)

by 3. 33(ii)

')9EG H

=S

gEG

=

S

Xv*®w(g)

by 3.13

Xv(g) Xw(g)

by 3.32 (iv),

3.33(i).

gEG

(ii)

See 3.23. Let us choose one irreducible AG-space V. in each I

equivalence class, as in 3.24, and let X. be its character. I

Then the functions X. are orthogonal, and therefore: I

3.35

COROLIARY.

The functions )(. are linearly independen I

ThIS fact can evidently be u'sed to give a second proof

ELEMENTARY REPRESENTATION THEORY of 3.24.

51

If

@m.V. ~ EBn.V., ill ill

then their characters are equa I, so I:m.x. = I:n.x., ill ill

and m. = n. for each}. 1

1

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 CI(G) for the set of class functions.

We

make CI (G) into a ring by pointwise addition and multiplication of functions. Characters are cIa s s functions, by 3. 32 (i) and (ii). We can define a homomorphism of abelian groups

by x(!;n.V.) = l:n.x.· ill ill

For every G-space V we have

xCV) by 3.32 (iii).

=

XV' X is a homomorphism of rings by 3.32 (iv) .

52


3.37

PROPOSITION.

Pro 0 f.

X: KC(G) ..... CI (G) is a monomorphism.

See 3.35. The image of X is called the character ring of G.

It is

natural to ask how large a part of CI(G) it is; and we will see that it is as large as could be hoped (3.47).

For this purpos e

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 rep1J

res entations 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 proj ection onto the (i, j)th component.

We therefore introduce the following

lemma.

3.38

LEMMA.

The vector-space dual to HomC(V, W) is

HomC(W, V), where the pairing between a E HomC(V, W) and

f3 E HomC(W,V) is given by

V is a G-map, then Tr(a9 (g)) is a

class function.

Proof.

Tr(a9(xyx- 1

» = Tr(a(9x)(9y)(9x=

1

))

Tr( (9 x-1)a (9 x)(9 y))

(3. 30 (ii))

58

LECTURES ON LIE GROUPS Tr(a9 (y))

=

since a is a G-map. There is a converse to this remark.

3.46

PROPOSITION.

Let G be a compact group.

every class function f : G

--t

Then

C can be uniformly approximated

by functions of the form Tr(j39 (g)), where 9 runs over representations 9 : G

->

Hom

C

(V, V) and j3 runs over Hom

CG

(V, V),

that is, j3 runs over G-maps .

Proof.

By 3.39 we can find 9: G

--t

HomC(V,V) and

a E Hom C (V ,V) s u c h t ha t \ f(x) - Tr( a 9 (x)) \ ~ (. If f is a clas s function we can substitute y-l xy for x and get

\ f (x) - Tr( a 9 (y-l xy)) \ ~ ( . Arguing as in 3.45, this gives \f(x) - Tr(9Y)a(9y-l)(9x)) \ ~ (. Integrate over y; we find \f(x) - Tr(j3(9x)) \ $. ( where j3 ::

S

(9 y) a (9 y-l ).

yEG But as in 3. 18, j3 is a G -rna p .


3.47

THEOREM.

59


Then every class function f : G - C can be uniformly approximated by a linear combination

~

A.X. of irreducible complex

ill

characters.

Proof. 3.25.

~

Let 9 and

be as in 3.46, and let V. and 1

~

be as in

Then the G-map ~ : V - V induces (say) ~i : Hom

CG

(Vi' V) - Hom

CG

(Vi' V).

We have the following commutative diagram.

Therefore Tr(~9 (g)) =

r

(Tr~i) . Tr(9 g),

i

which has the required form I: A.X. (g). ill 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 rea 1, and by 3.33 (i) it satisfies X(g-l) = X(g) .

3.48

COROLLARY.

Every class function f : G - R such that

f (g) = f (g-l) ca n be uniformly approximated by an R-linear

60


combination of characters of representations over R, or by an R-linear combination of characters of representations over Qo

Pro 0 f f (g)

Let f : G - R be a class function such that

0

= f (g-l) • 1 £(g)

Since f (g)

By 3047 we can find complex A. such tha t 1

- I;

A.x· (g)

o.

Ix. 1

~.I1 ~ (}

for some fixed

Let Co be any cube. Then we will

define a descending sequence of subcubes whose intersection will be a generator. Suppose, inductively, that we have defined Co

~

C1

::l ••• ::l

C m- 1 and that C m- 1 has side 2(.

Then there

is an integer N (m) such that N . 2 ( > 1, so that the image of C m- 1 under multiplication by N is Tk. such that N . C Let g E

m

We ca n find C m C C m- 1

cU.

m

n C . Then g N{m) c U ,so g is a generator m m

m

k

of T .

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.

m

Proof.

k Let t be a generator of T .

j ect to a generator of Z

m

.

Choose u E G to prok

IS divisible, so there is s E. Tk with ms = t - mu. g

=u

+ s. Then mg

= m (u

k

Then mu ETa nd t - mu E T . T

+ s)

= t

I

Take

so the powers of g are

dense in T. Translating by rg, the powers of g arc dense in the coset of T containing ru.

This gIves all cosets.

k

MAXIMAL TORI IN LI£ GROUPS

4.5

NOTICE.

81

From now on G is a compact connected Lie

group.

4.6

DEFINITION.

(i)

a subgroup which is a torus, such that

(ii)

if T

4.7

REMARK.

C

U

C

A maximal torus T

C

G is:

G and U is a torus then T = U .

If G is not compact

I

it need not have any

non-trivial tori.

4. B

PROPOSITION.

Any subtorus of G is contained in a

maxima I torus.

Pro 0 f.

Consider a strictly increasing sequence of subtori

Tl cT 2 c .•• eG.

Then L(Td c L(T 2 )c ... cL(G) is a

strictly increasing sequence, and so is finite.

4 •9

PROPOSITION.

Let T be a max ima I torus of G

a connected Abelia n subgroup of G wIth TeA.

Pro

0

f.

TeA c cl A.

a nd A

Then T = A.

But cl A is a closed connected AbelIan

subgroup, and is therefore a torus (2.20). T = A.

I

Thus T = cl A and


82

4.10 G

e

CONSTRUCTIONS.

by T C G

ric form on G G

e

Ad

~

e

Aut G . e

If T is a torus of G, it operates on Choose a positive definite symmet-

invariant under G

and so under T.

I

Then (3.78)

splits into orthogonal irreducible T-spaces of dimensions 1

and 2.

Those of dimension 1 are trivial.

orthonormal base in those of dimension 2

We can choose an I

and represent T by

T - SO(2).

4.11

DEFINITION.

The integer lattice of L(T) is exp-l (e)

where exp : L(T) - T.

4.12

PROPOSITION.

form Va

e

L(G) = G

e

splits as a T-space in the

~mVi' where T acts on Va trivially dim V.1 = 2 for I

i > 0 and T acts on V. as 1

COS [

Here

211e (t) i

sin 211

e.1 : T --

e.1 (t)

R/Z is given by a linear form

e.I :

integer values on the integer lattice, and no

4. 13

DEFINITION.

e.1

L(T) --- R taking is zero.

If T is a maximal torus, the functions

±BI are called the -roots of G. By 3.24 they are well defIned ---in terms of T.

We will sec that they are Independent of T.


4.14

PROPOSITION.

Proo f. (i) L(T) (ii)

83

T is maximal if and only if Vo = L(T).

It is clear that L(T)

C

Vo.

Suppose Vo = L(T) and T cT'. C

L(T')

C VOl

c

Suppose Vo

Then

, so L(T) = L(T') and T = T·.

Vo

-I L(T).

Then there is X E Vo , X /. L(T).

Now exp (tX) , for t E R is a I-parameter subgroup H of G on I

which T acts trivially, and which is not conta ined 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.

4.16

EXAMPLE.

dim G - dim T is even.

Let G = U(n), and let T be the set of

diagonal matrices: D =

exp 211 iXl

. exp 211ix

n

LU(n) can be decomposed into the following summands. (i)

Matrices

with d. real. J


84

This is L(T). Ma tric es

(ii)

r

M

rs

5

= r

-

-~

5

for r < s. DM

rs

Then D-1 =

w

-w

where w

= exp[21T i(x r

- x ) Jz and 5

ers

= x

r

- x . 5

The matrices (i) and (ii) generate L(G) , so Va = L(T) and T is maximal.

The roots are (x - x ).

4.17

Let G

r

EXAMPLE.

= S U (n).

the previous example are in LSU (n), respect to t of II + tM

rs

I at t

5

Then the matrices M 5

rs

of

ince the derivative with

= 0 is zero.

Similarly, matrices

of type (i) with D:l. = 0 are in LSU (n) • 1

Let T be the set of diagonal matrices

MAXIMAL TORI IN LIE GRO UPS

D

=

85

exp 21Tixl

. exp 211 ix with l:x.

=

1

Va

=

4.18

o.

n

The functlons (x - x ) are still nontrivial, so r

s

L(T), T is maximal, and the roots are (x - x ). r s

EXAMPLE.

Let G

= Sp(n) ,

and let T be the set of

diagonal matrices D

=

exp 211ixl

..

exp 211 ix

n

L Sp(n) splits into the following summands. (i)

Matrices

id 1

with d. real. 1

id (ii)

n r

Matrices M

rs

z

r

s

with z E C.

s

=

-z

86

(iii)


Matrices r

Nr = zj

r

with z E C.

Here

DN D= r exp(21Tix )zj exp(-21T ix ) r r

= exp(41Tix )zj r

(iv)

Matrices r

s

P rs = r

s

with z E C.

Here

zj

zj

87

MAXIMAL TORI IN LIE GROUPS DP

rs

D-1 =

exp 211 i (x + x ) zj r s ex p 211 i (x + x ) zj r s

Thus Va = L(T) , T is a maximal torus, and the roots are ±2x

r

4.19

I

(x - X ) and r s

EXAMPLE.

:t:

(x + X ) for r r s

-I s. We have U(n) c SO(n).

Let G = SO(2n).

Take T to be the image of the maximal torus we had in U (n) • That is, T is the set of matrices

D

where D. = 1

[COS

211X.

sin

211X.

n

-sin 211X i ] cos 211X.

1

1

1

•

Now LSO(2n) splits into the following summands. (i)

L(T)

I

cons ist ing of matrices

h "'o d

n

-dn

0

88

(ii)


The rest of LU (n)

I

cons is ting of matrices s

r M

:::::

rs W

r _WT

s

where

W

Then DM

rs

[

D- 1 = M I with rs

XI]

-SIn

[COS 211 (x r - x s)

YI

So T acts with (iii)

=

ers

S

in 2 11 (x - x )

=

x

r

r

21T(X r

-x s)] [X ]

cos 21T(X -x )

s

r

s

- x . s

Let

s E s

..

o s

o

-1

..

and take matrIces E M E- l s rs s

•

In this caS8, T acts with

y

.

89

MAXIMAL TORI IN LIE GROUP8

ers

=

x + x . r s Thus Va

are (x r

4.20

X

s

)

I

= L(T) , T is a maxImal torus and the roots I

± (x +

r

EXAMPLE.

X

s

)

Let G

t-

for r

=

s.

80(2n + 1).

We have

80(2n) c 80(2n + 1) by letting 80(2n) act on the first 2n coordinates.

Let T be the maximal torus we had in 80 (2 n) •

Then L80(2n + 1) splits into the following summands. (i)

ISO (2 n) •

(ii)

Matrices r

F

r

r

Here D acts by rotatIon through x . r

Thus Va = L(T) ±Xr

I

4.21

(x r

X

s

)

I

T IS a maximal torus

and ± (x + x ) for r r s

THEOREM.

t-

I

and the roots are

s.

Let T c:: G be a maxImal torus.

g EGIS contained In a conjugate of T.

Then any

90


Proof.

(Following A. Wei! [21J; see also [IIJ.) 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 E 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 byA. Dold [61.

(This theorem applies to manifolds rather

than simplicial complexes). We summarise what we need: Let f : X -X be a continuous map, and define J\(f) EZ by taking f*:Hq(X;Q)-Hq(X;Q)andsetting l\(f)=I:(-I)q Tr f*. q

Thenl\(f) depends onlyon the homotopy class of f. If f has no fixed points, then I\(f) =

o.

If f ha s only isolated fixed points

(and so a finite number of fixed points), then I\(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

I

:

X - X • If det(l - fl) > 0, then f has multi-

x

x

pliclty + 1 at x: if det (l - f I) < 0, then the multiplicity is -1. We do not need to discuss the case det(l - fl) = O. To compute I\(f) we may replace f with any homotopic map f o • So we may replace g with any other go E G, since G is path-connected. Take go to be a generator of T (4. 1), and

91

MAXIMAL TORI IN LIE GROUPS let fo be the corresponding map.

Then the fixed points of to

are the cosets nT for n in N(T), the normaliser of T in G (as the rea de=- will easily verify).

Let us examine N(T).

N(T) is a closed subgroup of G, and so is a Lie group (2.27, 2.26), and the identity component N(Th is open and so has only a finite number of cosets. see as follows.

Now N(Th = T, which we

N(T) acts on T by conjugation (i. e. ,

n(t) = ntn-1 ) and Aut T is discrete, so N(Th 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(Th properly contains T it contains a 1parameter subgroup not conta ined in T but computing with T, contradicting the maximality of T.

It follows that N(Th = T,

that T has only a finite number of cosets in N(T), and that fo has only a finite number of fixed points. It suffices to consider just one of these fixed points,

say T, as follows. r

n

Let nT be another fixed point.

Define

: G/T .... G/T by r (gT) = gTn. This is a well-defined diffeon

morphis m, commutes with f o , and takes T to hT. multiplicity at nT is the same as at T •

Thus the

92

L[CTURES ON LIE GROUPS Observe that fa can also be defined as fa (xT)

= goxg~l T.

That is, fa is obtained as a quotient of the map G -- G given by X

-0

goxg~l.

This has the merit that e goes to e.

To obtain a basis of (G/T)T' take a basis for Te' extend It to a ba sis of G , and discard the vectors of T • Then (4.12, e e 4. 14) 1 - f~ has the form sin 217 61 (go)

o

1 - cos 217 61 (9 0 )

o

Therefore det (1 - f~)

m

= III

1 - cos 2178 1 (go)

sin 217 91 (go)

-sin 217 8 1 (go)

I-cos 2178 1 (go)

which is greater than 0 unless cos 2178 (go) = 1 for some r. r

But 9 (go) r

(4. 12).

1

0 mod 1, since 8 is a nontrivial function on T r

Hence the multiplIcity is + 1, and I\(f) =

I N (T)/T I

> O.

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 Proof.

93

Let u be a generator of U.

x C G, and thus U c xTx- 1

•

Then u E xTx-1 for some

But U is a maxima I torus, so

U = xTx-1 • Hence any construction apparently dependent on a choice of T is independent of the choice up to an inner a utomorphism 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, a nd let g E G commute with all elements of S.

Then

there is a torus T conta ining g and S.

Proof.

Let H be the subgroup generated by g and S.

Abella n, so CI H is a compact Abella n Lie group. identIty component (CI H)l is a torus.

H is

Therefore the

CI H/(CI Hh is finite

and generated by g, so CI I~/(CI H)l ~ Z

m

for some integer m.

By 4.4, CI H has a generator h which lies in some maximal torus T.

Then g

~

S

C

H c CI H

= T•


94

4.26

PROPOSITION.

Let T be a maximal torus of G.

If

TeA c G where A is Abelian, then T = A. That is, a maximal torus is a maximal Abelian subgroup.

Pro 0 f.

Let g EA.

g and T.

But T is maximal so U = T, and gET.

4.27

EXAMPLE.

Then (4.25) there is a torus U containing Thus AcT.

If a E 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 cons ider the set of matrices of the form ±l

±l

These form a maximal Abelian subgroup.

4.29

DEFINITION.

Let T be a maximal torus of G. Then

the Weyl group W (or C:» of G is the group of automorphisms of T which are the restrictions of inner a utomorphis ms of G. This is independent of the choice of T. Any such automorphism has the form t - ntn-1 , n E N(T).

95


N(T) is a closed subgroup of G, and so compact. Let 2(T) be the centraliser of T, that is, the set of z E G such that ztz-1 = t all t E T.

2(T) is also closed, and TCZ(T) C N(T).

Thus N(T) maps onto N(T)/2(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 Xv

is determined by its restriction to T and is invariant under W.

4.31

COROLIARY.

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 correspond-

ence between class functions on G and continuous functions on T invariant under W.

P fO

0

f.

We nave already shown that the correspondence is

mono. Suppose given f : T W.

Extend f to

f: G

-0

-0

Y by

Y continuous and invariant under

f (xtx-1 )

= f(t).

To show that

f is

96


well-defined we need:

LEMlvtA.

4.33

If t1 ,t 2 E T are conjugate in G, then there

is w E W with t2 = wt1 •

T c 2(t 2 ) and, since T c 2(t 1 ), gTg-1 c 2(t 2 ) 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 E H1 such that

T = hgTg-1 h-1 , where Hl is the identity component of H.

But

h E 2(t 2 ) so hgt 1 gh-1 = t 2 . Thus conjugation by hg, which is in W, send s t 1 tot 2 •

Completion of 4.32.

It remains to check that

f

is con-

tinuous. Well, suppose that a sequence 9

f 9 co X

n

-t

• Let 9 X

co

,t

n

n

n

-- 9

ro

f

is not continuous.

Then there is

f 9 n tends to

such that no subsequence of

= x t x-1 and take a subsequence with

-t

nnn

t

for some x 00

a nd so x roro t x-ro1 == 9 ro • Then

ro

,t

ro

f (g n )

which contradicts our hypothesis.

.

Then 9

n

-+

= f (t ) . . . f (t ) n 00

x

t X -1 , rooooo

= f (g ), 00

Thus 4.32 is proved.

97


4.34

LEMMA.

Let N(gh be the identity component of the

normaliser of some 9 E G.

Then N(gh is the union of the

maximal tori of G containing g.

Proof. n E N(gh.

Clearly N(gh contains all such tori.

So let

Then n lies in a maximal torus S of N(gh.

S

commutes with g, so (4.25) there is a maximal torus T of G containing Sand g.

4.35

COROLlARY.

The following two definitions are equi-

valent: (i)

9 EGis regular if it is contained in just one maximal torus, singular if it is contained in more than one maximal torus, 9 EGis regular if dim N (g) = ra nk G,

(ii)

singular if dim N(g) > rank G.

Pro 0 f •

If 9 lies in just one T, then

dim N(g) = dim N(g)l = dim T. If 9 lies in T 1 and T 2 T1

I

T2 , then LTl

dim N(g) > dim T.

I

LTG and LN(g)

=:;

LT1 + LT2 so

,

and


98

4.36

EXAMPLE.

Let G = Sp(l), which is the set of quater-

Iq I =

nions q with

Maximal tori are circles cos

1.

for p any pure imaginary quaternion with

e + p sin e,

I p I = 1.

The singular points are ±l, with dim N(±l) = 3. All other points g are regular, and dim N(g) = 1.

PROPOSITION.

4.37

Pr a of. w EW T

Ad

~

W permutes the roots of G.

(The notation was introduced in 1.10.) For each we must consider two representations for T, namely

Aut G

w

~

and T

e

Ad

~

T

show that these are equivalent. and then G

Ax

e T

Ad

!

~

G

e

Ax

e

It will suffice to

But w = A

x

I T for some x

E G,

is the required equivalence, since

---~)

AutG

Aut G. e

T

!

Ad

~AutG

e

is commutative, where the bottom map is induced from A I . X

4.38

DEFINITION.

Let U = (t E T ; r

er (t)

=:;

0 mod I}.

U is r

a closed subgroup of T of dimension k - 1, where k = rank G. It is clearly monogenic.

instance:

It need not be cO!1nected.

For


4.39 Xl

==

In Sp{l) , 61 = 2Xl and U l is gIven by

EXAMPLE. 0 or

4.40

I

2" mod

I.

LEMlvtA.

dim N(t)

=k

Proof.

If t lies in exa cUy II of the U , then r

+ 211.

Let V c L(G) be the subspace on which t acts a s the

identity. Then, by definition, dim V= k + 211. N(t)

e

99

We show that

= V.

(i)

The elements of N(t) commute with t, so t a cts as the

identity on N(t) and so on N(t) • Thus N(t) c V. e e (ii)

Suppose x E V.

Then t acts trivially on x, and so on

the I-parameter subgroup H corresponding to x. Therefore H c N(t) and x E N(t) . Thus V C N(t) . e e

4.41

COROLlARY.

t E T is regular if it lies in no U , and r

singular If it lies in some U • r

4.42

COROLLARY.

of dimension

~n

The singular elements of G form a set

- 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.

Proof.

Let u bea generator of U • ThendimN(u)Lk+2, r

100


and, if z E 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

r

->

G by f{g, t) = gtg-1

•

Then Imf consists of all points in conjugates of U , f is r

smooth, and dim G/N{u} xU'::; n - (k + 2) + {k - 1} = n - 3. r

All the singular points are obtained with r running over a finite set.

Hence the res ult.

Chapter. 5

GE OM ET RY OF TH E Sn' IEF E L DIAGRA M

(Note:

Not ice.

This is not the DYTllkin-Coxeter diagram.)

Throughout this chapter

G is a compact connected

Lie group, and T is a maximal torws of G.

5.1

DEFINITION.

The infinites imal diagram of G is the

figure in L(T) consisting of the hyperplanes L(U). r

The diagram of G is the fig-ure in L(T) consisting of the hyperplanes given by

er (t)

E Z.

under exp of the singular pOInts of

This is the Inverse image G in T.

102


5 .2

EXAMPLES of didgrams .

(i)

U(2).

Root

Xl

- X2.

~Xl

The integer lattice is marked with asterisks. (ii)

SO (4).

Root s

Xl

± X2.

~Xl

GEOMETRY OF THE STIEFEL DIAGRAM

(iii)

80(5). Roots

Xl

± X2 , Xl,

X2.

---YXI

(i V)

8p (2).

Roots

Xl

±

X2,

2Xl , 2X2.

- - ; ; . Xl

103

104 (v)

LECTURES ON LIE GROUPS SU(3).

Roots

Xl

(01-1)

! 5 •3

PROPOSITION.

Proof.

firs t 1y, Z (G) Now,

Therefore

If 2

er (2)

==

Z (G) =

C

E Z(G)

I

nu r .

Z (T) = T . 2

acts trivially on G and so on G . e

0 mod 1 for ea ch rand 2 E.

nr u r .

105

GEOMETRY Of THE STIEfEL DIAGRAM Conversely

then g acts trivially on G

5.4

nUr

I

and so trivially on G (2.17).

is given by

. centre consIsts

0

(ii)

nU r

SU(n).

+ ••• +

X

wI where w (iii)

e

:= 0 mod 1 for each r

EXAMPLES.

U(n).

Xl

er (g)

if 9 t: T and

I

n

n

Sp(n).

:=

:=:: X

mod 1, so the

n

f matflces . e 21Tix I .

is given by

== 0 mod 1.

=

Xl

;; X

==

Xl

n

mod 1 and

Thus the centre consists of matrices

1.

nur

is given by x. ± x. =- 0 mod 1 all i,j, i.e., J

I

xi == 0 mod 1 all i or xi ==

1

2" mod

1 all i.

Thus the centre con-

sists of matrices ±I. (iv)

SO(2n).

nU

r

is given by x. ± x. I J

==

0 mod 1 for i

I

j.

for n > 1, this is the same a s for Sp (n), and the centre consists of ±I.

Of course, SO (2) is Abelian.

(v)

SO(2n + 1).

r. U is given by r

X

r

:=

0 mod 1 all r.

Thus

the centre consists of just the identity matrix I.

5.5

THEOREM.

Uris then

er

and

es

arelinearlyin-

dependent.

Proof.

U has dimension k - 1. r

We show that


106

=k

dim N((U) ) r

1

+ 2. The result will then follow from 4.40

applied to a generator of (U) • We need two lemmas. r

5.6

LEMlvtA.

Suppose H

1

C

group which is normal in G.

T, and that H is a closed sub-

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 cT.

(ii)

T/H is a compact connected Abelian subgroup of G/H,

and so a torus. Now suppose T/H Then U/H

C

N(T/H)

C

U/H, where U/H is a torus in G/H.

= N(T)/H, so T cUe N(T). Therefore

dim T = dim U, so dim T/H == dim U/H and T/H (iii)

W(G/H) = N(T/H) )'/H = N(T)/H/T/H ~ N(T)/T ~ W(G).

5.7

LEMlvtA.

(i)

n

=1

If dim T = 1 then

and W

= 0,

or

= U/H.

GEOMETRY OF THE STIEFEL DIAGRAM (ii)

n

[Note:

=3

107

and W = 22 .

In fact in (i) G

= S1 ,

and in (ii) G

= SO(3)

If n = 1 then clearly G = T = S1 and W =

Proof.

or Sp(l).]

o.

So sup-

pose n > 1. Take an invariant norm in L(G) and let v be a unit vector in L(T).

Define f : G/f - Sn-

1

L(G) by f(g)

C

=

(Ad g) v .

Then f is well-defined, continuous (even smooth) and is mono

It follows that gIl g2 E T and g1 T = g2 T .

a nd therefore fixes T.

Now G/T is compact and Sn-1 Hausdorff, so f is a homeomorphism of G/T with its image in Sn-l.

But G/T and

sn-1 are both compact rna nifolds of dimension (n - 1), so f is onto.

Then there exists 9 E G such that (Adg)v = -v, and

therefore 9 acts on T by gtg- 1 = t- 1 • automorphisms, so W = 2 2

Now T has only two

•

Let i be the generator of 'lT1 (T). 9 can be joined to e by an arc in G. that is, 21

=

Since G is connected,

So, in 'lT1 (G), i

= -i,

o.

Now we have in fact (2.37) a fibration S1 - G

-+

GI T ~ = S n-1 .

have that 7T2(Sn-l) -

From the exact homotopy sequence we 111

(S1) -

1Tl

(G) is exact. "But


108 TTl (Sl)

-t

17 1

(G) is not mono, since 2i

o.

--+

I

SO TT2 (Sn-l)

0

and n = 3.

Proof U . r

Consider (U) , the Identity component of

of 5.5.

r

1

This is a torus of dimension k - 1.

V..'e wish to show that u

1. U s

for r

Is,

Let u be a generator.

es

for then

will not be

er .

a multiple of

Consider N(U)lo

T is a maximal torus of N(U)l.

The

elements of N(u) fix u and so fix every element of (U) . r

can apply 5.6 with N(U)l as G, T as T, and (U) r

T/(U) r

1

1

We

as H. Then

is a maximal torus in N(uh/(u) ,and r

1

W(N(U) /(U) ) ~ V..'(N(U)1 ). r

1

1

Now T / (U) r

1

(5.7) N(u) /(U) r

1

has dImension 1, so

has dimension 1 or 3, and N(u)

1

sion k or k + 2. u lies in exactly

1

1

has dimen-

But (4.40) N(U)l has dImension k + 211 where II

of the U. r

Hence

II =

1 and u does not lie

In U • s

5.8

THEOREM.

For each r there is an element

~

r

EW

which is not the Identity but which lea ves every point of U

r

fixed.

Proof.

\;Ve use the

choice of u.

sam~

proof as 5.5

I

but wIth a different

GEOMETRY OF THE STIEFEL DIAGRAM Cuns ider U. r

109

We observed (4.38) that U is monogenic. r

Let v be a genera tor. Now consider N(vh.

T is a maximal torus of N(vh ,

and N(V)1 fixes every element of U.

We can apply 5.6 with

r

N(V)1 as G, T as T, and U as H.

We deduce that T/U is a

r

r

maximal torus in N(vh/u , and N(vh/u has dimension 1 or 3. r

r

By 4.40, dim N(VI )/U 2:. 3, so dim N(Vl )/U = 3 and r

w(N(vh/u) ~ Z2.

r

That is, there is n E N(V)1 which fixes

r

each point of U and which maps T/U by t r

5.9

r

COROLLARY (of the proof) .

cp

r

-t

C1

•

is the inner a utomorph-

ism induced by an element n which can be joined to e by a path of which each point leaves each point of U fixed. r

5.10

Proof.

COROLlARY.

U has either one or two components. r

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 E L(T);

£

e (t)

r r

~

0

all r}.

So we can sa y 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 r

dimension k - 1. W permutes the planes of the dIagram and the Weyl chambers, by 4.37. For the following theorem, we suppose choser! 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(U ). r

(iii)

Such reflections generate W.

(iv)

More precisely, for any Weyl chamber B, the


III

reflections in the walls of B generate W. (v)

Let p E L(T) and W

p

be the stabiliser of p.

Then W

P

permutes simply transitively the Weyl chambers whose closures contain p. (vi)

W

p

is generated by reflections in the planes L(U ) r

which contain p. (vii)

More precisely, it suffices to consider those planes

which are walls of a fixed Weyl chamber B

o

such that

p E CI B •

o

Pro 0 f • and (vii) (ii)

By taking p = 0, we see that (v):=;> (i), ~

(vi)

~

(iii)

(iv) , so we need to prove only (ii), (v), (vi), (vii)

For each r, W contains an element cp

r

I

I which fixes

U (5.8), and hence fixes L(U ), and preserves the inner pror

r

duct in L(T). (v)

o.

= lr;; ~/v lies in B and is fixed by lP. If q

shows that v

I

Let v 1JJ

B.

E:.

Then

11, (5.14)

lies in some L(U ) / which contradicts the hypor

thes is .

Continuation of 5.13 (v)

Secondly, v..'

p

acts transitively on the 'vVeyl chambers

whose closures contain p, as follows. Let B , B o

be Weyl chambers containing p in their

I

closure,andletx

o

(B

0

,Xl

EB'.

Since/forrls,

L(U ) i! L(U ) has dimension k - 2 (5.5) r s path from x

o

to x

I

not

me~ting

ther~

I

is a polygonal

any L(U ) ,', L(U ), not meeting r s

any L(U ) unless it contains p / and m~~tlng each L(U ) transr

versely.

r

~ Take

Suppos~

the path x px I, and move it slightly. ~ o

thIS path crosses (LU

k1

), ... / L(U

k1

)

S

ucccss-

ively to get flom B to B1 ••• to B = B Then ~k .. · 0, 1

where 6 , ... ,6 1

Proof.

s

The fundamental Weyl chamber is given S

are the simple roots.

This is clear from 5.31. So simple roots correspond to walls of the fundamental

Weyl chamber.


124

5 .. 35 is x

EXAMPLE. >x

G = U(n).

... > x .

The simple roots are

n

1:2

The fundamental Weyl chamber

Any other root can be written as a linear sum of these, e. g. , x-x

r

s

=(x -x

r

r-1

)+ ... +(x

S-l

-x)

s

for r < s. And

are linearly independent.

5.36 a =

~IJ.

EXERCISE.

e ,

r r

If a is a simple root and we write

where the IJ. are non-negative numbers and the r

er

are positive roots, then we have wntten a = a.

5.37

DEFINITION.

follows.

The Dynkin diagram is constructed as

Take one node for each simple root a.

Given two

distinct simple roots a, P join the corresponding nodes by II =

0, 1, 2 or 3 bonds, where

5.38

EXAMPLE.

(Xr+1 - Xr-t-2) ,

II

G = U (n).

II

follows 5.26.

Between (xr - xr +1 ) and

= 1. Otherwise

II =

o.

Hence the Dynkin

dIagram is 0---0----0 ... 0---0 with n - 1 nodes.

5.39

LEMMA.

If

er

IS

a simple root then cp permutes the r


125

positive roots except 9 , which goes to -9 . r

r

v..'e give two proofs.

Proof.

Choose a point v of the diagram such that 9 (v) = 0

(i)

r

and 9 (v) > 0 for any other simple root 9 . Then 9 (v) > 0 for s s t any positive root S other than 9 . t r Let S be a spherical neighbourhood of v not meeting any plane 9 t

=0

for t

-I r.

Let w E: S

n (FWC).

Then tPr(w) E. S.

Therefore (ct' 9 ) (w) = 9 (C!) w) > 0 r t t r

for t (ii)

I

Thus cp 9 is a positlve root.

r.

r t

Let 9 , ... ,9 be the simple roots, and let 9 be a 1 s t

positive root. Write

9 = n S + ... + n 9 . t

1

1

S

S

Then

differs from 9 only in the coefficient of S. Therefore cp (9 ) r

t

has at least one positive coefficient if S

t

-I 9r

r

t

and so (5.31

and 5.33) cp (9 ) is a positive root. r t

5.40

DEFINITION.

The fundamental dual Weyl chamber

(F DvVC) is the s~t of points in L(T} * corresponding under i to


126

the fundamental Weyl chamber in L(T). the set of h E L(T)* such that (9 ,h» r

That is, the rDWC is

0 for each simple root

9 . r

5.41

DEFINITION.

Define

p E L(T)* by P

Let 9 , ••• ,9 1

1

= -2 (9

1

m

be the positive roots.

+ ••• + 9 ). This is not necesm

sarily a weight.

5.42

PROPOSITION.

P lies in the fundamental dual Weyl

c h am b ere

' I eroot a. In d eed ' /2(a« , S> ) = 1 f oreac hslmp "a,a

Proof.

Let a

than a. (i)

=

9.

Then cP permutes the positive roots other

r

r

There are three cases: CPr (9 ) = 9 • Then

= 0 so 9

t

contributes 0 to

(a, p). (ii)

9 permutes 9 and 9 , t r t u

-I u.

Then

~

o

r

We have a split extension :> W

1~

W

Define


128 r

is the subgroup of translations.

o

Each one is the transla-

tion by an element of the integer lattice I, so we can regard r

as a subgroup of 1.

o

(It is not necessarily the whole of 1.)

Our next obj ect is to calculate the fundamental group 111

(G) in terms of the Stiefel diagram.

ant

111

The topological invari-

(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

11111

(G)

= 011

,

and some of the subsidiary results used

in its proof use the same condition. going to prove (5.47)

1111

1

(G)

~

However, we are just

I/r II, so it would be possible 0

to rewrite 6.41 with the data in the form II r all is what is used in the proof of 6.41. to use

111

o

= III , which after

Secondly, we propose

(G) to classify the connected covering groups over G,

as is usual in algebraic topology.

For our arguments to pro-

ceed without this (notably at 5.56 below) it would be necessary to construct the double covering Spin(n) of SO(n) without reference to

111 ;

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

129

GEOMETRY OF THE STIEFEL DIAGRAM at too high a price [23]. To continue: we have I ~ I

C

L(T)

to w (1)

-+

T.

The map i : T

THEOREM.

plane

(T), since

111

(T) is Abelian. i

111

(T) ~ 111 (G).

i* is epi and induces I/r ~ o

11

1

(G).

5.48-5.55 will, together, form a proof.

er

Proof.

111

G induces I =

-+

PROPOSITION.

5.48

Consider

Its projection is a closed path in T, and

so represents an element of

Proof.

(T), as follows.

For v E I choose a path w in L(T) from some w (0)

= v + w (0).

5.47

111

= 1.

r

Then r

Let y be the reflection of 0 in the r

is the subgroup of I generated by the y • r

0

contains each y , since reflection in 9 = 0 folo r r

er

lowed by reflection by

= 1 is translation by y .

r

Conversely, we claim that, if y E r, then y(O) = I:n y , r r

whence, if Y E r

, y is translation by I:n y. We prove this o r r

claim by induction on the number of reflections used to build up y. Suppos e suppose () (0)

y = p ()

= I:n s y s •

p (x) = x + (k -

where p is reflection in Now

er (x»

y • r

er

= k, and

130


Therefore p () (0) =

er O:n 5 y 5 )

But

r. n 5y5

er (I; n 5y5 ) y r •

+ kY r

is an integer, since I;n y 5

5

is in the integer

lattice. Therefore p6(0) has the required form.

5.49

EXAMPLES.

(i)

G = U (n) or SU (n) •

x

- x

r

5

Define

ro

r 5 = 1 (r < 5) is the point (0 ••• 0 1 0 ••• 0 -1 0 1T:

I - 2 by

= Ker 1T.

I/r ~ o

(ii)

The reflection of 0 in

1T (x

1

I

••• I

X )

n

= x

0) •

+ ••• + x . Then n

1

For SU(n) we have I/r = O. 0

For U(n) we have

z. G = Sp (n) • The reflection of 0 in 2x = 1 is r

r

r

(0 ••• 0 1 0 ••• 0). We have I/r = O. o

G = SO(2n) or SO(2n + 1). The reflection of 0 in

(iii)

r X

r

- X

5

(r
2. Therefore SO (m) ha s 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). standard maximal torus

T in

maximal torus T in SO(m).

Take as the

Spin(m) the cover of the standard Then L(T) ~ L(T) under the covering

map I though this does not preserve the integer lattices. consists of all (x s ists of all (x

I

I

•••

I

•••

1

,x ) with all x integers n r

,x ) with all x n

I

and I con-

integers and x

r

I

1

+ ••• +

X

n

even. Similarly L(T) * ~ L (T) * I but this does not preserve the lattices of weig hts.

For exa mple

1

I

-2 (x

1

+ ••• + x n ) is not

d

135

GEOMETRY OF THE STIEFEL DIAGRAM weight in SO(m) but is one in Spin(m). Now Ad : G .... SO(n) induces

We distinguish two cases. (i)

Ad* is zero, ar.ld we can lift Ad to get the following

diagram. Spin(n)

G (ii)

~1 Ad

> SO(n)

Ad* is non-zero.

Then Ad defines a double cover G of

G, and we have the following diagram.

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=~(9 1 L.

+ ••• + 9m) (see

5 .41) is a weight.

Pro of.

In 4.12 we split G

e

as a T-space in the form

136


V EB E o

m 1

Choose bases for V , ••• , V ,V , and put them

V..

1 1 m

0

together in this order to form a base for G • Then the come

~ Aut

position T C G

G

e

maximal torus T' of SO(n).

= SO (n)

sends T into the standard

xr Further, if L(T') ~ R denotes

the rth co-ordinate function, then the composition xr

L(T) ~ L(T')

~

R is the root ±9r, for r

~

m, or zero, for

r > m. With the same sign attached to each 9 , we now have r

±9

1

± ••• ± 9

m

= (Ex )Ad. r

Now Ad lifts to Spin(n)

I

and tExr is a weight for

Spin(n), so (tEXr)Ad is a weight for G. is a weight for G, and so is 1 differs from -2 (±9

5.58

LEMMA.

1

Thus t(±9

1

•••

±9 ) m

1

p = -2(91 + ••• + 9 m), as this

••• ±9 ) by a sum of positive roots.

m

In this case w ....

W

+ P gives a one-one

correspondence between weights w E CI FDWC and weights W

+ P EFDWC.

Proof (i)

If w is a weight and (w, 9 ) .2:. 0 for all simple roots r

9 then (w + p, 9 ) > 0 by 5 .42. r

(ii)

r

If w is a weight and (w, 9) > 0 for all simple roots r

9 then 2 (w, 9 r ) > 0 and is an integer (5.24), so.2 1. r (9r , 9 r )

Now

137

GEOMETRY OF THE STIEF EL DIAGRAM 2 «(3, 9r ) 2(w - (3, 9r ) = 1 so 2! 0 and (9r ,9r ) , (9r , Sr)

--~

W -

(3 is a weight in

CI FDVvC. We showed (5.24) that, if 2 (9[1w) then (9 9) r' r

5.59

is an integer.

PROPOSITION.

If

W

is a weight and 9 a root, r

We now examine the converse.

2 (9 r ,w) (9 9) is an integer for some

r' r w E L(T)* and all simple roots S , then it is an integer for all r

roots 9 • r

2 (9 p Suppose (9 r' 9 and also for the root r

Pro of.

w) 9)

is an integer for all simple roots

r Let

9 •

s

root 9, and let S =~ (9). r t rs 2

~

r

correspond to some simple

Then (9 ,9) = (9 ,9 ) and so t t s s 2 (9r , 9s )

/

(9 ,9 ) \. 9s - (9 ,9 ) 9r s s r r =

2 (9 s'w) (9 ,9)

s

s

which is an integer. But the reflections and any root 9

s

~

r

generate W (5.34 and 5 • 13 (iv))

can be written as

~

9 for some simple root r

9 and some (/) E W, by considering 9 as the wall of a Weyl r s chamber and throwing this chamber onto the FWC (5.34).

138


Hence the result.

5 .60 some

PROPOSITION.

Suppose

2 (9r ,W)

(9

9) r' r

E L (T) * and each simple root 9.

W

r

is an integer for Then

W

takes integer

values on

ro .

Pro 0 f.

ro is generated by the points y , where 'Y = r r

for any v such that 9 v r

W (y )

r

= 1.

v-

~

r

v

We have

= wv - W (~ v) r = (w - ~ w)(v) r

2 (9 r ,W) 9) 9 r (v) r r

= (9

I

which is an integer.

Thus w is integral on each y

r

and so on

r. o COROLIARY.

5.61

If G is simply connected and

( r

is an integer for each simple root 9

r

Proof.

5.62

I

I

r)

then w is a weight.

r o =I.

THEOREM.

If G is simply connected it has just

k = dim T simple roots 9 1 I ••• 19k I and has weights such that

2 (9r ,W) 9 9

~~:~ e~t) = 6 rt •

binations ~ W

1

+ ••. + nkw

WI

I··· IWk

The weights are then the linear comk

with nr E Z each r.

The rDWC

139


consists of all points I:n w with each n > 0, and the CI r r r

Fnwc

consists of all points I:n w with each n > r r r1 '2(9 1 + ..• + 9 m) = w 1 + ••• + wk·

Pro of.

Suppose there are just k -

II

o.

slmple roots.

the roots lie in a subspace of L(T) * of dimension k -

ra

lies in a subspace of L(T) of dimension k -

ha s rank at lea st

II

which implies

There are elements

II =

II.

.

be written

W =

L;n

W

,

r r

if and only if each n

r

and then

II,

Then

and so

I/ra

2 (B

r,

Wt)

= 0rt'

(Br' Sr)

t

and they are weights by 5 .61.

Then all

O.

In L(T)* such that

W

Also

Every element w of L (T) * can 2 (Sr w)

' = n so w is a weight (Sr' Br) r'

is an integer.

The statements about

FDWC follow from the definition (5.40).

Set (3 = t(S 1

5.63

+ .•. + B ). Then 2 (9 r ,(3) = 1 (5.42), so m (Br,B ) r

EXAMPLE.

Let G = SU(n).

Take w t = x

1

+ ••• + x for 1 < t < n - I. t

--

so

The elements of L(T)* can be written

Then

140


since I;x = i

o.

They lie in rDWC if

Thus they may be written

and lie in rDWC if b I

, •••

,b n-

5 . 64

COUNTEREXAMPLES

(i)

Let G

=

U(2), where

but there is only one root.

TTl

I

> O.

(U(2)) ~ Z. We have dim T

= 2,

The rDWC is a half-plane, which

cannot be expressed in the given form. (ii)

Let G

= SO(4),

where

TTl

(SO(4)) ~

Z2.

The dual diag-

ram is as follows:

rDWC

141

GEOMETRY OF THE STIEFEL DIAGRAM Here the asterisks represent weights. quarter-plane shown.

The weights in the CI FDWC do not form

a free Abelian semi-group, and

are not weights.

The FDWC is the

Chapter 6


Throughout this chapter G is a compact connec-

Notice.

ted Lie group I and T is a maximal torus of G.

6.1

THEOREM.

.

(Weyl Integration Formula.)

There is a

real function u on T such that

r

f(g)

=

.)G

S

f(t)u(t) T

for all class functions f on G. Indeed u (t) =

O=TI

m

j=l

~

e

06/\ W

11 is].

(t)

\

-e

I

where

- 11 i9

o

J

(t))

and 9. runs over the distinct roots of G. ]

Proof.

Define f : G/T x T

-0

G by f(g,t)

Then f factors through G/Txvl:

= gtg-1

•

(See 5.53.)


143

Now c ha s degree 1, since it is a homeomorphism when restricted to G/TxWTR - G , and b is a R covering.

So f ha s degree

I w 1\ . G

f dg =

I Wi,

I Wi-fold

and

SG/TxTf* dg* ,

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 I

fixed.

Second, let v be in a tra nsversal V of T in G. f (gv ,t) = gvtv-1 g-l ,

so f (g, t)(dv) = g(dV)tg-1 - gt(dV)g-l I

so f' (g ,t) = Adg (Adt-1 - I),

Then

144


where we consider the second factor fixed. det f I (g, t)

= det (Adt-1

-

Thus

I).

Now Adt has the form cos 211 9 1 -sin 211 9 1

so Adt-1

-

I ha s the form

cos 211 9 1

-

1

sin 211 9 1

sin 211 91 cos 211 9 1

-

1 00

and I) = nm(cos 2 2119 -2cos2119 + 1 + sin 2 9 ) 1 \: r r r m (4 . 2 9) =rr(11i9 r 11i9 r ) -11i9~ (-11i9 r 1 s In 11 r 11 e - e ') n \ e - e

det(Adt-1

=rr

-

= 06 , where ( 11 i9 r

o =n,e

- e

-11 iB r )

•

Hence the result.

6.2 ~

DEFINITION.

W acts on L(T).

denote the sign of the determinant.

For

~

E W, let sign

Then we say that


145

X E K(T) is a symmetric character if ~X = X for each ~ E W,

and is an anti-symmetric or alternating character if ~ X = (sign~) X.

6.3

EXAMPLE.

Then

o = ITlmC e 11 is r

Suppose Ad : G

-11

- e

-+

SO(n) lifts to Spin(n).

i9 ) r

is an antI-symmetric character.

Pro 0 f .

o = I; (

1

... (

Ex p 1T i (( 9 + • • • + ( 9 ) mIl m m

where (. = ± 1 and there are 2

m

1

1 -2 (( 9

terms.

Note tha t, by 5 .57 ,

+ ••• + ( 9 )

11m m

is a weight, so 0 E. K(T) • Let x E N(T) represent (/) E W. given by g .... xgx-1 T

e

•

Then the action of

This induces a map G

e

~

is

.... G which maps e

to T • On T it preserves or reverses orientation according e e

to (sign~). Also ~ permutes V , ... , V (5.5 and 3.22). 1

m

If ~

maps Vj to V preserving orientation, then it sends 9 to 9 ; j k k and if reversing orientation, then it sends 9 to -9 • If it j k reverses orientation But

CD

II

times then (/) 0 = (- 1)11 o.

preserves the orientation of G , since x may e

146


be connected to e by a path.

Therefore

= +1.

(sign Sr (v)

w(v) -

==

CD W.

such

r

2( 9r , w)

(9

9) 9r (v)

r' r

>

W

(v),

which contradicts the definition of w.

So the result is proved.

The inequality (
=

W

2

f.P C/)

(cp W )(v) 2

w (v) 2

by 6.26

(v).

So the second definitIon holds. (ii)

Suppose, on the contrary, that

convex hUll of the orbit of

W2,

Then there is 'r} E L(T) such that

WI

does not lie in the

and suppose WI

WI

E Cl FDWC.

('r}) > (CD W2) ('r}) all

(lJ;W2)('r}) = ~2 (iP'r}) = ~2 (v),

so the second definition does not hold.

6 • 27

PROPERTIES OF THE RELATION

(i)

Transitive:

(ii)

Given ~2' the number of weights ~l such that ~l ~ ~2

is finite.

Wl ~ W2 ~ ~3

implies ~l ~ ~3, obviously.

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. ] ~I ~

(iii) C[)

W 2

and ~2

E W, as follows.

If ~l

-I W2

~l (v)

-I ~2 (v),

6.28 W2

6.29

if and only if ~l = cp ~2 for some

It suffices to consider ~I '~2 E CI FDWC.

we could find v in any open set of L(T) with contradicting the second definition.

DEFINITION.

::: ~l.

::: WI

Write ~l
m .• J

1

the weight corresponding to mo. 1

1

Let w be

Then in

X(a 1 m 1 + ••• + am) IT r r the only term in S (w) is a.S (w), so a. = 0, which is a 1

1


164 contradiction.

6.40

PROPOSITION

S (n, "', + ••• + nkw ) = X(p n, •.• p:k + lower monomials) k

IT.

We proceed by induction. Write

Proof.

and suppose the res ult is true for all w' < w •

X(p 71 ••• p:k) IT= S(~) + I: m is (~ 1) , where

~.

1

K(G).

vVe define H

7 .1

=:::

Ker(l - t )/Im(l + t).

PROPOSITION.

H is an algebra over 2 2

,

and the

irreducible representations of G which are self-conjugate yield a 22 - ba s e for E. The proof is immediate from Chapter 3.

We may there-

fore mea sure the incidence of self-conjugate irreducIble


166 representations by computing H.

We will also use the following lemma.

LEMlvtA

7.2

For any complex representation V,

V* 0 V ~ Hom(V, V) is real.

It carries the bilinear form

Proof.

Tr(ap) = Tr (f3a) (see 3.38); this form is symmetric, non-s ingular and invariant. Now use 3.50. We now begin to study the groups U(n) and SU(n).

Each

n ha s an obvious representation with V = C : we write Al ,A 2

, ••• ,

An for the exterior powers of this operation.

Let us

write z. = Exp (21T ix-l , J

J

so that the typical element in our maximal torus is

k then the character X(A ) of Ak is the kth elementary symmetric function of z 1

,Z

2

I

•••

I

Z

n

•

(See the proof of 3.61.) The Weyl


167

group acts by permuting z ,z , ••• , z • Thus the character 1

n

2

X ('A k) is the e I ementary symmetric sum S (xl + x 2 + .•• + x ). k

S ince X(A' k) consists k A is irreducible.

0

f a single elementary symmetric sum.

The representation An of U(n) is one-

dimensional, and is essentially det: U(n) -S~. it is invertible.

7•3

In particular,

The restriction of An to SU(n) is trivial.

THEOREM.

The complex representation ring K(U (n))

is the tensor product of the polynomial ring generated by A1 , A2

, •••

,A n-l and the ring of finite Laurent series in An.

for Tea h A A I ge b ra H is po I ynomia I on generators A,i,n-i/,n 2

~

2i

~

n.

These generators are real.

There is of course no suggestion that the modules AiAn-i/ An are irreducible; indeed they are not.

Proof

0

f 7.3.

By a classical theorem, the ring of symmet-

ric polynomials in z ,z , ••• , z 1

2

n

is a polynomial ring genera ted

by the elementary symmetric functions x(A 1 ) ,

•••

n ,X(A ).

Now

take any finite Laurent series which is symmetric; by multiplying it with a suitably high power of obtain a symmetric polynomial.

Z1

z

2

•••

zn' we

Hence K(T)W is as described.

168


The res ult for K(U (n)) follows by 6.20. Since we have an obvious pairing

i. n-i n the dual of A IS A /A.

(This also follows from an easy

calculation with characters.)

Hence the conjugate of

is

So

t

permutes the monomials in Al

,

A2

I

•••

I

An; and we eas ily

see that the only monomials which are fixed under t are the polynomia Is in (1

~i ~

in).

These are real by 7 .2, since

7.4

THEOREM.

The complex representation ring K(SU(n))

is a polynomial ring generated by Al ,A 2

I

•

••

n 1 ,A -

•

algebra H is polynomial on generators AlA n-i for 2 .

and, If n = 2m

I

The

~ 2i < n

m i n-i a generator A • The generators A A are m

real; the generator A

is real for m even, quaternionic for

m odd.

Proo f.

The result on K(SU(n)) is a specIal case of 6.41;

REPRESENTATIONS OF THE CLASSICAL GROUPS the identification of the bas ic weights w 1

I

169

•••

I

W

k mentioned

in 6.38 is given in 5 .63. As above

I

i n-i the dual of A is A • Hence the conjugate

of

is II

II

n-2 (A\1) n-1 (\2) A

• ••

(An-1 )1I 1 •

So t permutes the monomials in Ai

I

A2

I

•••

I

An-l; and we

easily see that the only monomials which are fixed under tare polynomials in AiAn-i (l :::. i < in) and Am if n = 2 m. . \ i\ n-i.IS rea I b y 7 •. 2 -''\s 11 f or A\ m resentailon A A

I

The rep-

.. t h e palflng

has

P/\

m

a=(-I) a/\

p;

now us e 3. 5 0 •

7.5

EXERCISE.

Show directly that any representation V of

SU(n) extends to U(n).

(Hint:

irreducible representation;

It is suffIcient to consider an

now consider the action of the

centre of S U (n) .) We take next the group Sp(n).

exterior powers of this representation.

It has an obvious rep-

As we have seen in

170


Chapter 3

I

Ak is real for k even

I

quaternionic for k odd.

If

we take the element

in T, its action on C 2 n is given by

Therefore the cha racter X(A i) of Ai is the ith elementary symmetric function of

7 •6

THEOREM. 1

generctors A

K(Sp(n)) is a polynomial algebra with

2

I

A

I

••• I

An • All the irreducible representations

of Sp(n) are self-conJugate.

PI 00

(i)

f

It

IS

rather easy to see that K(T)w is as stated; now

REPRESENTATIONS OF THE CIASSICAL GROUPS use 6.20. Alternatively (il)

I

171

use 6.41.

It follows from the generators given that the whole of

K(Sp (n)) IS self -conj ugate. Alterna lively

In Sp (n) ea ch

I

element g IS conjugate to g-1 (see 5.17). We take next the group SO(n).

It has an obvious

n n representation on R or C ; we wnte Al

,

A2

I

•••

I

An for the

exterior powers of this representation. All these representations are real. D =

If we take the element 21

in U(n) and embed it in SO(2n)

I

its action on C 2n is equIva-

lent to that of the diagonal matrix

2

n

Therefore the character X(A i) of Ai is ttle lth elementary symmetric function of

172

say


Similarly, if we embed D in SO(2n + 1), its action

0 .• 1

on C 2n + 1 is equivalent to that of the diagonal matrix

1

Therefore we ha ve i

X (A ) = o.1 + o.1- 1· (Here

7 •7

00

is to be interpreted as 1.)

THEOREM. K(SO (2 n + 1)) is a polynomial algebra with 1

n

2

generators A ,A , ••• , A • All the irreducible representations of SO(2n + 1) are real.

Pro of (i)

K(T)W is exactly the same a s 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 Ai have given us all the generators we need.

It is easy to produce arguments to show


173

that for SO (2n) we need something else. (i)

In SO (4n + 2) not every element g is conjugate to g-l.

Therefore it is possible to construct a class function f such that f(g)

-I f(g-l).

Therefore (3.47) SO(4n + 2) has at least

one representation which is not self-conjugate.

But all the

i

A are real. (ii)

Consider the representation An of SU(2n).

already seen that it is self-conjugate.

We have

So its restriction to

SO (2n) is self-conjugate for two essentially different reasons: first because An

IS

self-conjugate on SU(2n), and secondly

beca use ea ch exterior power Ai is rea I on SO (2 n).

But we

have already seen that an irreducillie representation V can have essentially only one isomorphism with V*.

Therefore

the representation An of SO(2n) is reducible. If n is odd this argument is complete in itself; the

representation An of SO (2 n) is both quaternionic a nd 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 An of 0 (2n).

Cons ider an element g in 0 (2 n)

174


such that det(g) = -1; it is easy to see that its action on C 2n is equivalent to that of a diagonal matrix

1 -1 It is now easy to check that the restriction of X =

component of determinant -1 in O(2n) is zero. value of

XX

over SO(2n) be

over O(2n) is

til.

So

II;

til ~ 1

xO, n )

to the

Let the average

then the average value of XX (3.34) and

II

~ 2.

That is , An

must split over SO(2n) into at least two summands. We now amplify argument (ii).

Let us define a non-

singular bilinear pairing

byF(v,w)=vA w. Then F is Invariant under SO(2n); for g E O(2n) we have F (gv, gw) = (det g)F (v, w). Let us define another non-singular bilinear pairing

by

indeed

REPRESENTATIONS OF THE CIJ.\SSICAL GROUPS S(v 1

/\

V2 /\ ••• /\ v n > ~

(Wi /\ W

= ~E:(p)(VI p(I)W 1 )

Here p runs over all permutations inner product in R2 n.

•••

I

2 /\ •••/\

175 W

n

»)

(Vi p(n)w n ).

and v'w is the usual

Then S is invariant under 0 (2n).

Let

us define an automorphism p of ~n(R2n) by setting S (p v, w) = F (v w). I

We easily check that for 9 E O(2n) we have pgv = (det g)pv. We may describe p as follows.

Let vl V2 I

I

••• I

v 2n

be any orthonormal ba sis with determinant + 1 in R2 n; then

n

Thus p2 = (-1) • n

It follows that A (R 2n) splits into the ± 1 eigenspaces

of

P

odd.

if n is even I and into the ± i eigenspaces of p if n is

Of course the latter splitting takes place over C.

Ele-

ments of SO(2n) preserve the two eigenspaces; elements of determinant -1 particular

I

in 0 (2n) intercha nge the two eigenspaces.

In

neIther eigenspace can be zero.

We now enquIre after the characters of the summands (say V and W).

The character X(An) of An is the nth elemen-

tary symmetric function of


176

Let us write

(1

a_

= I; Z l

(2

Z2

(n

•••

Zn

I

(r

=±1

and

(1 (2

••

(n

= -1.

These are elementary symmetric sums (see 5 .17). We have n

X(A ) =

a+

+

a_

+ lower terms.

Since the characters of representations are linear combinations with non-negative coefficients of elementary symmetric sums, we have

where a and bare 0 or 1, and a is a sum of lower terms. Now consider the automorphism 9 of SO(2n) obtained by conjugating with an element 9 of determinant -1 in O(2n), say

,-

9 ='

1 1

1 -1

Its effect on T is to invert zn; thus 9 a + 9a :::: a.

Hence

n

X(A ) = (a + b)(a + + n-) + 2 a

= n-,

9a _

= a+

and

REPRESENTATIONS OF THE CIASSICAL GROUPS and a + b == 1.

177

It follows that we can name the summands of

An so that

7 •8

COROLlARY.

The a utomorphls m

e

of SO (2 n) is not

inner.

Proof.

An inner automorphism takes a representation into an

equivalent representation.

7 .9

THEOREM. K(SO (2 n)) is a free module over the poly-

nomial ring Z[A l , A2 ,

••• ,

n

equivalently 1 and A_).

An] on two generators 1 and A~ (or If n is even all the irreducible

representations of SO (2 n) are real. H -- 22 [1 A ,A\ 2 ,

Proof.

\ n-l ]

••• , A

If n is odd,

•

We have to study K(T)W' that is, the set of finite

Laurent series in z ,z , ••• , z 1

2

n

which are symmetric under

permutations a nd under inverting an even number of the z • r

The set S of such symmetnc elements admits an automorphism

e:

invert an odd number of the z.

explained above.)

r

We have 8 2 == 1.

(Of course,

e arises

as

So over the rationals, S

178


splits as the sum of the +1 and -1 eigenspaces of e: s

1 = 2(1

1

+ e)s +2(1 - 9)s.

The +1 eigenspace is the ring of polynomials in

(as in 7.6, 7.7). eigenspace.

Suppose given an element a in the -1

Then

a == I; c (z ,z , ... , z r

r

1

2

n-1

)z

r , n

where c

-r

=-c

r'

so that n

r

-r

a == I; c (z ,z , ... , z ) (z - z ). 1 r 1 2 n-I n n Thus a == a '(z

- Z-l) .

n

n

By symmetry ~ is divisible by the remaining (Zr - Z~l); so a == a" (z

1

- Z-l) (z 1

2

- Z-l) ••. (z 2

n

- z-l). n

Here a" must be an element of the +1 eIgenspace; so we have a == p (a +

-

a_)

where p is a polynomial in X(A 1 )

, •••

,X (A n).

For a general

element s In S we have

Since a+ + a_ lies In the +1 eigenspace we may write this


179

where q lies In the +1 eigenspace and is integral (since s and pa+ are so). So K(T)W 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 rcal. If n is odd t (A~) == ~, 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 Spln(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 [lJ and the representations of the Clifford algebras out of [7].

In [7 J 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.

REF ERE NO ES

[IJ

M. F. Atiyah, R. Bott andA. Shapiro. ules, Topology, 3

[2]

Bliss.

Clifford mod-

Supplement 1 (1964), Pp. 3-38.

Lectures on Calculus of Variations.

University

of Chicago Press (1946).

[3 J

A. Borel and F. Hirzebruch.

Characteristic classes

and homogeneous spaces I, Amer.

J. Math., 80 (1958),

Pp. 458-538. [4J

R. Bott.

Lectures on K(X).

Mimeographed notes, Har-

vard Univcrs ity .

r5 J

Coddington and Levinson. ential Equations.

[6 j

A. Dold.

Theory of OrdInary Differ-

McGraw-HIll (19~5).

fIxed pOInt index and fixed point theorem for

Euclidean neighbourhood retracts.

Topology, 4 (1965),

PP. 1-8. [ 7J

B. Eckma nn.

Gruppentheoretischer Bew~ls des S2 tzes

von Hunvitz-Radon uber dIe Komposition qUcdrctlscher fonnen.

Comment, rVlath. Eelv., l~ (1 ~42), pp. 35U -3G G.

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Gra vcs.

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G. Hochschild.

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Maximale Toroide und singulare Elemente in

ges chlossen Lleschen Gruppe.

Comment. Math. Helv. ,

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H. Hopf and H. Sa melson.

Ein Satze uber die

Wirkungsraume geschlossener Liescher Gruppen. Comment. Math.Helv. ,13 (1941), Pp. 240-251. [ 12 J

S. La ng.

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Lectures on Lie Groups

Lectures on Lie Groups

Lectures on Lie groups and Lie algebras