Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34 9780691223803, 069108498X

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LIE GROUPS, LIE ALGEBRAS, AND COHOMOLOGY

LIE GROUPS, LIE ALGEBRAS, AND COHOMOLOGY

by

Anthony W. Knapp

Mathematical Notes 34

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

1988

Published by Princeton University Press, 41 William Street. Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester. West Sussex Copyright © 1988 by Princeton University Press All Rights Reserved Library of Congress Cataloging-in-Publication Data

Knapp, Anthony W. Lie groups, lie algebras, and cohomology. (Mathematical notes ; 34) Bibliography: p. Includes index. 1. Lie groups. 2. Lie algebras. 3. Homology theory. I. Title. II. Series: Mathematical notes (Princeton University Press) ; 34. QA387.K57 1988 512\55 88-2514 ISBN 0-691-08498-X (pbk.) Printed in the United States of America Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources The Princeton Mathematical Notes are edited by Luis A. Caffarelli. John N. Mather, and Elias M. Stein 5 7 9 10 8 6

To My Family With Love

CONTENTS

Preface Chapter I. 1. 2. 34. 56. 78.

1. 2. 3. 4. 5. 6.

45 48 56 64 73 75 86 92

Representations of Compact Groups

Abstract theory Irreducible representations of SU(2) Root space decomposition for U(n) Roots and weights for U(n) Theorem of the Highest Weight for U(n) Weyl group for U(n) Analytic form of Borel-Weil Theorem for U(n)

Chapter IV.

3 7 10 15 18 25 31 36

Representations and Tensors

Abstract Lie algebras Tensor product of two representations Representations on the tensor algebra Representations on exterior and symmetric algebras Extension of scalars - complexification Universal enveloping algebra Symmetrization Tensor products over an algebra

Chapter III. 1. 2. 3. 4. 5. 6. 7.

Lie Groups and Lie Algebras

S0(3) and its Lie algebra Exponential of a matrix Closed linear groups Manifolds and Lie groups Closed linear groups as Lie groups Homomorphisms An interesting homomorphism Representations

Chapter II. 1. 2. 34. 5. 6. 78.

ix

99 111 114 118 124 129 138

Cohomology of Lie Algebras

Motivation from differential forms Motivation from extensions Definition and examples Computation from any free resolution Lemmas for Koszul resolution Exactness of Koszul resolution

153 l6l 166 175 188 190

viii

CONTENTS

Chapter V. 1. 2. 3. 4. 5. 6.

Homological Algebra

Projectives and invectives Functors Derived functors Connecting homomorphisms and long exact sequences Long exact sequence for derived functors Naturality of long exact sequence

Chapter VI.

Application to Lie Algebras

1. Projectives and injectives 2. Lie algebra homology and cohomology 3. Poincare duality

4. Kostant's Theorem for U(n) 5- Harish-Chandra isomorphism for U(n) 6. Casselman-Osborne Theorem Chapter VII.

(g,K) modules The algebra R(g,K) The category C (g,K) The functors P and I Projectives and injectives Homology, cohomology, and Ext Standard resolutions Poincare duality Revised setting for Kostant T s Theorem Borel-Weil-Bott Theorem for U(n)

Chapter VIII. 1. 2. 3. 4. 5. 6. 7. 8.

266 282 288

292 301 317

Relative Lie Algebra Cohomology

1. Motivation for how t o construct representations

2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

199 210 230 238 245 258

325

334 340 360 365 373 384 386 396 4o2 4o8

Representations of Noncompact Groups

Structure theory for U(m,n) Cohomological induction Vanishing above the middle dimension First reduction below the middle dimension Second reduction below the middle dimension Vanishing below the middle dimension Effect on infinitesimal character Effect on multiplicities of K types

417 428 437 448 46o 468 473 479

Notes

490

References

500

Index of Notation

503

Index

506

PREFACE

This material is based on a one-semester course given at SUItfY Stony Brook in Fall 1986. The audience consisted largely of graduate students knowledgeable about geometry, acquainted with tensor products of vector spaces, and having little or no background in Lie groups. The objective was to go in one semester from the beginnings of Lie theory to the frontier in algebraic constructions of group representations. The course consisted of much of the first seven chapters of the present book, done in a slightly different order. Actually the course was designed backwards from a key algebraic computation (7-79) that yields the Borel-Weil-Bott Theorem, and it ended up including whatever seemed appropriate as preliminary material. Chapter VIII was added to indicate how the computation (7-79) leads to the frontier. The topic of interest here is the representation theory of compact Lie groups and of their natural noncompact analogs, the noncompact semisimple Lie groups. Special linear groups, symplectic groups, and isometry groups of quadratic forms give examples of noncompact semisimple Lie groups. Group representations are homomorphisms of a group into invertible linear transformations on a complex vector space, possibly infinite-dimensional and possibly with some continuity assumption. Understanding of group representations allows one to take advantage of symmetries in various problems in analysis and algebra. For a compact group, the irreducible representations (those with no nontrivial closed subspaces invariant under the representation) are finite-dimensional, in the case of any compact connected Lie group, the Borel-Weil Theorem gives a way of realizing all such irreducible representations in

-ix-

x

PREFACE

spaces of holomorphic functions, and a generalization due to Bott gives alternative realizations in spaces of differential forms having only

cfz Ts present.

In 1966 Langlands conjectured that a version of the Borel-Weil-Bott Theorem should provide a realization of "discrete series" representations of noncompact semisimple Lie groups.

"Discrete series" are certain irreducible infinite-

dimensional representations that are known to play a fundamental role in the construction of all irreducible representations.

Over a period of some years ending in the

mid 1970 T s, Schmid proved the Langlands conjecture and several variants of it. In 1978 Zuckerman found that he could bypass a number of difficult analytic problems of SchmidTs by phrasing the Langlands conjecture in algebraic terms and using constructions in homological algebra.

He found that similar

algebraic constructions yielded other interesting representations for a variety of groups.

His technique, now known as

cohomological induction, has become a fundamental tool in representation theory. Our goal is to start with elementary Lie theory and to arrive at the definition, elementary properties, and first applications of cohomological induction, all the while developing the computational techniques that are so important in handling Lie groups.

A byproduct is that we are able to

study homological algebra with a significant application in mind; we see as a consequence just what results are fundamental and what results are minor.

A person who wants to

study further may wish to read the general theory of roots and weights from any number of books (e.g., Khapp [1986], Chapters IV and v) and to study cohomological induction further from the book by Vogan [1981]. Prerequisites for reading the present book are a knowledge of metric spaces, an advanced course in linear algebra, and a passing acquaintance with topological groups. Also invariant integration for compact groups plays a role in Chapters III, VII, and VIII.

PREFACE

xi

Chapter I is a quick introduction to Lie groups of matrices. The reader who already knows beginning Lie theory may skip this chapter except for the last section (representations). Representations of groups are the objects of interest in these notes, but the approach is by means of algebra. What we therefore study is representations of Lie algebras that are related to representations of Lie groups; the sense in which we can actually pass from the Lie algebras to the Lie groups is not addressed here very seriously. Since the group representations are the objects of interest, we constantly need examples of them in order to motivate what happens with Lie algebras. The first half of Chapter II contains two kinds of topics alternately: a devlopment of multilinear algebra and the imposition of representations on various spaces of tensors. Most readers will be able to skip at least part of the material in §§1-4. The second half of Chapter II constructs some algebraic objects that will be used repeatedly throughout the later chapters

universal enveloping algebra,

symmetrization, and tensor products over an algebra. Chapter III treats the representation theory of compact groups, with emphasis on the unitary groups as examples. This material is included at this stage mostly for motivation and is not used until the latter part of Chapter VI. Chapters IV and V, as well as the first half of Chapter VI, develop homological algebra as it applies to Lie algebras and their representations. The motivation that is provided comes largely from geometry and starts with the analytic Borel-Weil Theorem in §7 of Chapter III. A second line of motivation begins in §1 of Chapter IV. The two lines merge in § 1 of Chapter VI and continue in §1 of Chapter VTI. Chapter VII makes a modification in the theory of the previous three chapters, so that the setting matches better what is needed for representation theory. The ingredients of cohomological induction are assembled in this chapter. Section 11 is the most important and illustrates how the theory can be applied in the ideal setting of a compact

xii

PREFACE

connected Lie group.

This section contains the key

computation (7.79) mentioned earlier. Chapter VIII indicates the kinds of adjustments that need to be made when working -with noncompact semisimple Lie groups, rather than compact groups.

Specific illustrations are given

for the unitary groups of indefinite quadratic forms. At the end of the book is a chapter entitled "Notes," •which gives historical comments, amplifies some notions, and points to the list of references at the end. I am grateful to David Vogan for his advice in presenting this material and to the members of the class at SUNY Stony Brook for a number of comments and suggestions.

Clifford

Earle helped with some of the motivation, Chin-Han Sah helped with some of the bibliographical material, and Leticia Barchini helped with checking the mathematics.

Some of the

material in Chapters VII and VTII is new, and part of it is taken from the unpublished notes Knapp and Vogan [1986] and from other unpublished work joint with Vogan.

This research

was sponsored by the National Science Foundation under one or more of the grants DMS 85-01793, DMS 87-11593, and DMS 85-04029. A.W.K. December 1987

LIE

GROUPS, L I E ALGEBRAS, AND COHOMOLOGY

CHAPTER I LIE GROUPS AND LIE ALGEBRAS

1. .SO(3) and its Lie algebra We denote by

GL(n, 3R) the real general linear group

consisting of all nonsingular n-by-n real matrices with matrix multiplication as group operation.

Similarly

GL(n, C)

denotes the group of nonsingular n-by-n complex matrices. Let

G

be the set of matrices

S0(3) = {x€GL(3,B)

SO(3)

defined by

| x x t r = I and det x = 1} .

This is the set of 3-by-3 rotation matrices and "will be an example of a Lie group.

For now we can notice that

G

is a

group (being closed under multiplication and inversion within GL(3>IR) ) and is also a compact subset of

3Er

(being closed

under entry-by-entry limits and having all entries absolute value).

< 1

in

The group structure and the topological

structure are related in that multiplication and inversion are continuous (being given by nice formulas). Of more interest, however, is the fact that a kind of differentiation is available.

Let us agree for now that a

smooth curve

is a

image is in c(t)

in

c(t) SO(3)

S0(3)

in

S0(3)

for each

with

t.

c(0) = I .

C°° curve into

3Er whose

We consider smooth curves One such is

I:

LIE GROUPS AND LIE ALGEBRAS

c(t) =

[ -sin

For any such, we can form matrix.

With

t

cos

t

c'(o) , which -will be some 3-by-3

c as above

'-sin t c ! (t) = [ -cos t

cos t

0\

-sin t

0

0

0

/0 ,

cT(0) =

0/

-1 \ 0

1

0

0

0 j .

0

0^ (1.1)

cT (0) tells us the direction in

Here

extends from describing

I.

G

in which the curve

Think of it as telling how a parameter

G is to start out from the identity.

Let

3 = {c!(0) | c(t) = smooth curve in G with c(o) = 1} . Then

g

is a subset of 3-by-3 matrices, not necessarily

invertible. Let us identify space.

g.

First of all,

g

To see closure under addition, let

be in g .

Forming

is a vector cT(0) and b!(o)

c(t)b(t) , we have = c(t)b'(t)+c'(t)b(t)

=Q

Thus

g

= c(o)b'(o) + c'(O)b(o) = b'(o) +c'(0) .

is closed under addition.

scalar multiplication, let IR . Forming ^

!

To see closure under

c (0) be in

g

and

k

be in

c(kt) , we have c»(kt)k,

-^[c(kt)] t = 0 = kc'(o) .

1. SO(3) AM) ITS LIE ALGEBRA Thus

g is closed under scalar multiplication. Next let us find additional concrete matrices that are

in

g.

Two such are 0

c!(0) = | 0

0

1\

0

0

-10

/ cos t

0

sin t

\ -sin t

0

cos t

from c(t)

0/

and

from c(t)

Taking into account (1.1) and the vector space operations, we see that '0

a

£ {( ~a

°

c

-c

0y

} = (skew-symmetric real matrices} . (1.

Actually equality holds in (1.2). c(t)trc(t) = I ,

In fact, if

then c(t)trc'(t) + c'(t)trc(t)

= 0

I t r c ' ( 0 ) +c»(0) t r I = 0 c ' ( 0 ) t r + c'(0) =

0.

Thus

g = {skew-symmetrie real matrices} . The group structure of structure.

Indeed,

G forces

(1*3)

g to have additional

G is closed under group conjugation

6

I: LIE GROUPS AM) LIE ALGEBRAS

x -» gxg through at

. I

Thus if at

t = 0,

c(t)

t = 0,

is a smooth curve in

then so is

we see that

gc T (o)g

closed under the linear maps g

into

3

gc(t)g~ . is in

3 .

Ad(g) , for

G

passing

Differentiating Thus

3

is

g € G , that carry

and are given by Ad(g)X = gXg" 1 .

Next let in

G

with

X

"be in

c(0) = I .

function into

3

and let

Then

(1.4)

c(t)

be a smooth curve

t -> Ad(c(t))X

3 , as we see "by substituting

(1.4) and writing matters out.

is a smooth g = c(t)

into

By definition we have

^ A d ( c ( t ) ) x | t = Q = lim 1 [Ad(c(t))X-X] , and we know that the left side exists.

(1.5)

The right side

involves vector space operations and a passage to the limit, all on members of of

3 .

Since

3 , as a 3-dimensional subspace

JPr , is closed topologically, the limit is in

3 .

Let us

calculate this limit. We need a preliminary formula, namely

A

(1.6)

To see this, we differentiate

c(t)c(t)~

= I

by the product

rule, obtaining c(t)"1] = 0 , and (1.6) follows.

Therefore

2. EXPONENTIAL OF A MATRIX

= c»(t)Xc(t)" 1 - c(t)Xc(t)""1c^(t)c(t)'1. Putting

t = 0

and taking (1.5) into account, we see that cl(o)X - Xc'(O)

i s in

g.

We conclude that

g

i s closed under the

Lie "bracket operation [X,Y] = XY-YX .

(1.7)

Of course, we could have calculated directly that closed under (1.4) and (1.7) once we had identified explicitly as in (1.3)«

g

is

g

But the point is that these closure

properties did not depend upon our explicit knowledge of

g.

We shall return to this matter in §3, considering it in a wider context.

2. Exponential of a matrix It is possible also to go "backwards from §1.

g

to

G

The tool for doing so is the exponential map. If

A

is an n-"by-n complex matrix, then we define exp A = e A =

I * AN . N=0 N '

This definition makes sense, according to the following proposition.

in

8

I: LIE GROUPS AND LIE ALGEBRAS Proposition 1.1.

is given "by a convergent series (entry-by-entry). eXeY = e X + Y

(a)

A , e

For any n-by-n complex matrix

if

X

and

y

Moreover

commute

•yr

I

(b)

e

(c)

t -> e

at

is nonsingular GL(n, C)

is a smooth curve into

that is

t = 0 (d) ^ ( e t X ) = X e t X (e)

det e X =

(f)

X -» e

^

n

)

is a

C°° mapping from matrix space

into itself.

Proof.

For any n-by-n matrix

M,

put

M . = sup Il where

|jx|| and

||Mxi|

refer to the Euclidean norm.

I f wr^ < l and the right side tends to infinity.

0

Then

2

as

Hence the series for

e

entry, and it must be convergent.

N-j_ and

N2

tend to

is Cauchy, entry~byThis convergence is good

enough to justify the manipulations in the remainder of the proof.

(a) eXeY = ( Z r£> Xr/)( I S A Ys) = Z V r=0

oo

s=0

N

N=0 k=0

'

'

r,s

2. EXPONENTIAL OF A MATRIX

•i*£«> £ ( )

(b)

N=O JN# Y = -X I n ( a ) , and use

Take

(cd) £(•«) = £ = E

N=O

(e) Moreover

If

X

y det e

e° =

I

£ (txf - J £

J ^

iN#

is upper triangular, then so is

eX .

depends only on the diagonal entries of

Y

e

, which depend only on the diagonal entries of X . Thus X Tr Y det e = e in this case. A general complex matrix is of the form

gXg"

det e ^ (f)

with

X

upper triangular, and then we have

= detfgeV 1 ) = det e X = e T r

X

= eTr

This follows from standard facts about term-by-

term differentiation of series of functions. Let us return to

and its

g

given by (1-3).

Then the exponential map actually carries

g

into

fact, if

X

G = SO(3)

is skew-symmetric, then (ex)(eX)tr = eXeX

r

= e X e" X = e° = I ,

by (a) in the proposition, and det e X = e

T r

»

= e° = 1 ,

G.

In

10

I: LIE GROUPS AND LIE ALGEBRAS

by (e) in the proposition. Thus (c) says that, for each curve in tx e has X

in

at

I

X

in

t = 0•

G

that is

at

X

as derivative at

tx g, e

From (d) we see that

t = 0.

Consequently for any

g , there is a ""best" smooth curve in

t= 0

and has derivative

X

is a smooth

that is tx t = 0 , namely e .

at

G

Moreover this family of curves varies nicely as

X

I

varies.

We shall see that the exponential map recaptures a number of local properties of

G,

even though

g

depends only on

infinitesimal data near the identity.

3. Closed linear groups The critical property of § 1 is that

S0(3)

SO(3)

for the developments in

is closed topologically as a subset of

GL(3,C) when GL(3>C) is regarded as an open subset of p. Q 2 3R D . We shall refer to any closed subgroup of some GL(n,C)

as a closed linear group.

The reason that

SO(3)

by polynomial equations.

is closed is that it is defined

The polynomials in question are the

entry-by-entry relations in the 9 coordinates that amount to xx

r

= I

and

det x = 1 .

Here are some more examples of

closed linear groups; they too are closed because they are defined by polynomial equations: SO(n) = {x€GL(n,]R)

| x x t r = I and det x = 1} tr

U(n) = {x€GL(n,C) SU(n) = {x€U(n) |

| xx

= i}

det x = 1}

3- CLOSED LINEAR GROUPS

11

SL(n,3R) = {xeGL(n,IR)

| det x = 1}

SL(n,C) = {x€GL(n,C)

| det x = 1} .

Further examples may be found in pp. 4-6 of Knapp

[1986].

Let us generalize the calculations in § 1, seeing the extent to which they can be carried through for all closed linear groups.

For any closed linear group

of smooth curves in T

c (0)

G , and -we let

for all smooth curves

3

G , we can speak

"be the set of matrices

c(t) in

G

such that

c(o) = 1 .

We cannot instantly write down any nonzero members of 3 in the way that we did in (1.1).

But we can see Just as in § 1

that

The same argument as in §1

3

is a real vector space.

shows that

3

is closed under the linear maps

given as in (1.4). 3

Ad(g) ,

geG,

And the same argument as in § 1 shows that

is closed under the Lie bracket operation (1.7).

Thus 3

is a Lie algebra of matrices in the sense that it is a real vector space of matrices that is closed under the bracket operation [X,Y] = X Y - Y X . Let

3I (n, ]R) and

31(11, C)

complex n-by-n m a t r i c e s .

be t h e spaces of a l l r e a l and

For t h e f i v e examples above, we f i n d

t h a t t h e c o r r e s p o n d i n g Lie a l g e b r a s a r e given by So (n) =

{X 6 gl (n,3R)

|

X+Xtr =

u(n) =

{X€gI(n,C)

|

X+X * = 0}

{X €31 ( n , C )

I

X+X

{ X e g l (n,3R)

I

Tr X = 0}

*u(n) = • I (n,IR) =

tr

=

0}

0

a n d T r X = 0}

12

I: «I(n,C) =

LIE GROUPS AND LIE ALGEBRAS

{X e 9 I ( n , C )

The Lie algebra X = c'(0) I

at

g

|

Tr X =

consists of the set of matrices

for smooth curves

t = 0.

0} .

c(t)

in

G

that pass through

As with

S0(3) > it will turn out that there tx is a "best" such curve, namely e . Proposition 1.1 shows

tx that

e

is a smooth curve with the correct derivative at

t = 0 , but we need to see that

e

lies in

G.

A little

computation shows this to be the case for our five examples, and we establish it in general in Proposition 1.4 below. Lemma 1.2. gl(n,C) that

There exist an open cube

and an open neighborhood

exp : U •» V

V

of

U I

about in

0

in

GL(n, c)

such

is smooth (= C°°) and one-one onto, with a

smooth inverse. Proof.

Proposition 1.If says that

exp

is smooth.

By

the Inverse Function Theorem, it is enough to prove that the derivative matrix at Let

{E.}

X=0

be the standard

gl (n, C) , and let

x.

coordinate function.

of

X -> exp X

is nonsingular.

2n -member basis over

B

of

be the corresponding coordinate or Then

j j

X.

°)|t=o

x=o = xj_(E-e

tE. J )[t==0

since x^f*) is linear

= x.( E .) = 6 . . . So the derivative matrix is the identity, and the lemma follows.

3- CLOSED LINEAR GROUPS Lemma 1-3c(0) = I

If

c(t)

is a smooth curve in

c T (o) = X ,

and

t

in the domain of

Remark.

GL(n,C)

with

then

lim cfp) for all

13

= exp tX c.

The prototype of this lemma is as follows:

We

identify the additive reals with a closed linear group by t -» (e ) . have to

One curve in this group is

c(t/k)

= (1 + t/k)

c(t) = 1 + t , and we

, which is well.known to converge

e* . Proof.

e > 0

Choose

so that

c(t)

U

and

V

as in Lemma 1.2, and choose

is in

V

for all

t

with

|t| < 2e .

Using the lemma, we can form a smooth curve Z(t) = exp" 1 c(t) Now

z(0) = 0

for

|t| < 2c .

clearly, and the Chain Rule gives Z'(0) = ( e r p ^ O ) ) " 1 ^ ^ ) = X .

Thus Taylor's formula gives Z(t) = tX + 0(t 2 ) , where

0(t )

is a term that is "bounded for

remains bounded near t

by

t/k

t=0

and regarding

|t| ^ e and p t . Replacing

when divided by t

as fixed gives

«

$

•

•

14

I:

LIE GROUPS AND LIE ALGEBRAS

Thus

kz(|) = tX + oQ)

for any

|t| < c ,

and (|) k = (exp z(|)) k = exp Letting

k

tend to infinity and using the continuity of exp ,

we obtain the conclusion of the lemma for For some other the a"bove argument. integer t/N .

N

with

t

in the domain of

with

c , let us modify

First we choose and fix a positive

|t/N| < e .

Instead of replacing

t/(Nk+t)

|t| < c .

The above estimates apply to t

0 < -t < N - l .

by

t/k,

we replace

Again regarding

t

t/N

by

as fixed,

we obtain

Thus

and

Letting

k

tend to infinity, we obtain the conclusion of the

lemma for this value of Proposition 1,4. is in its Lie algebra

t.

If

G

is a closed linear group and

g , then

exp X

X

is in G. Consequently

4. MANIFOLDS AND LIE GROUPS 9 = {Xegl (n,C) Proof.

Let

curve in

G

is in

for

G

X

with

i exp tX

be in

is in G for all real t} .

g , and let

c(o) = I

and

c'(o) = X .

is a closed set) if the limit exists.

exp tX .

Thus

exp tX

t

is in

G

t .

for

"be a smooth Then n

c(t/n) n

(since

G

Lemma 1.3 says that the

in the domain of

follows by raising to powers that real

c(t)

n > 1 , and so is the limit on

limit does exist for all

15

|t|

exp tX

c

and is

small, and it is in

G

for all

The rest is clear from Proposition 1.1c.

Actually

exp

maps

g

onto a neighborhood of

I

in

and this fact accounts for the strong connection between and

G.

G,

g

To get at this fact, however, requires a digression.

4. Manifolds and Lie groups In the terminology of advanced calculus, the difficulty with handling the exponential map as well as we would like for a group like

SO (3)

or one of the later examples is that

the group is defined implicitly, and we need to define it parametrically (at least locally) in order to work with it better.

Actually it is not so important to know a

p aramet rization concretely; we just have to know that one exists.

The parametric form is the one in the definition of

Lie group below. First we define the notion of "smooth manifold."

Other

kinds of manifolds are important for Lie groups, as well, but we shall not introduce them here.

In what follows we shall

16

I: LIE GROUPS AND LIE ALGEBRAS

use the terms "smooth" and Let

M

connected.

be a separable metric space, not necessarily An n-chart on

open subset of subset

"C°°" interchangeably.

cp(u)

M

and

cp

3Rn .

of

M

is a pair

a homeomorphism of

Two charts

(U1,cp1)

smoothly compatible if the mapping cp-^U-jfiUp) inverse.

u*

cover

Having a M

to

cpgocpT

cp2(U-, HlJp)

with U

M

an

(Up>cp2)

are

from the open set

is smooth and has a smooth

is called a

(U-,cp.)

for

C°° atlas.

C°° atlas allows us to define smooth functions IRn .

by referring matters back to

function

f : E -» B

on an open set

for each

x

there is some chart

in

such that- x

U

onto an open

and

A set of smoothly compatible n-charts

which the

on

Bn

in

(U,cp)

E

is in

U

and

f*cp~

E

of

For example, a M

is smooth if

(U,cp)

is smooth on

in the atlas cp(uHE) .

The compatibility of the charts makes it so that this smoothness persists for all charts about We say that

M

as above, when endowed with a

is a smooth manifold of dimension technical point here:

x.

n .

c°° atlas,

(There is a small

A different atlas that leads to the

same smooth functions on open sets is to yield the same smooth manifold.

To handle this, one can observe that the set

of all charts smoothly compatible with a maximal

C°° atlas.

Two

C°° atlas is a

C°° atlases lead to the same smooth

functions exactly if their corresponding maximal atlases are the same.

So, technically,

is endowed with a maximal

M

is a smooth manifold when it

C°° atlas.)

4. MANIFOLDS AND LIE GROUPS Examples.

(1)

Any open subset of

manifold in a natural way.

lRn

17 is a smooth

Only one chart is needed, and cp

can be taken as the identity map. (2)

The unit sphere

Sn

smooth manifold of dimension

3Rn

in n

can be made into a

by using two charts.

One is

n+i defined on

t^ = S n - { (0,..., 0,1)} , and the other is

defined on

U 2 = S n - { (0,..., 0,-1)} .

(3)

The closed linear group

SU(2) , defined in §3, is

a smooth manifold in a natural way since the formula SU(2)={(

a

_ £ ) | a e c , P ^ C , |a| 2 +|p| 2 = l}

-JB

a

identifies it with the sphere

S .

Every connected component of a smooth manifold is an open subset.

Since the underlying metric space is assumed

separable, there are at most countably many components. If f : U -> V f

u

and

V

are open subsets of smooth manifolds and

is a one-one smooth map with a smooth inverse, then

is called a diff eomorphism of

U

onto

V.

We come to the definition of "Lie group."

A separable

topological group is a separable metric space that is a group in such a way that multiplication and inversion are continuous.

A

Lie group

G

is a separable topological group

18

I: LIE GROUPS AND LIE ALGEBRAS

that is a smooth manifold in such a way that multiplication and inversion are smooth. from

Gx G

into

G,

(Her

ultiplication is a mapping

and we understand

GX G

to be a smooth

manifold whose charts are products of charts in the factors.) The term analytic group is used for a connected Lie group.

In a Lie group

G,

the identity component

G Q , which

we know to "be open, is an open closed subgroup and is an analytic group. Both

GL(n, 3R)

and

GL(n, C)

are Lie groups.

In fact,

each is an open subset of Euclidean space and is a metric space for free.

One chart suffices to give an atlas.

turns out to have two connected components, whereas is connected.

GL(n,IR) GL(n, C)

The formula for multiplication is given by

polynomials and is smooth; the formula for inversion is given by Cramer's rule, using determinants, and is smooth. GL(n,3R) and

GL(n,C)

triangular subgroup in

are Lie groups.

Thus

Similarly the upper

GL(n,3R) is a Lie group, and so is

the subgroup of the upper triangular group with ones on the diagonal. But for the most part, it is not so obvious how to exhibit a group as a Lie group.

It turns out that all closed

linear groups are Lie groups automatically.

This is a hard

result whose proof we give in the next section.

5. Closed linear groups as Lie groups A relatively easy case that gives some insight into why closed linear groups are Lie groups is the case of

5- CLOSED LINEAR GROUPS AS LIE GROUPS

G = SL(2,]R) = {(y This

G

^)

real

I

entries and

19

wz - xy = ij .

is the zero locus of the single polynomial P(w,x,y,z) = w z - x y - l ,

and the derivative of

is the l-by-4 matrix

P

P ! = (z which is nowhere

0

-y

-x

w) , P=0.

on the locus where

Thus the

Implicit Function Theorem allows us to solve locally about each point of

G

other three.

for one of the variables in terms of the

(We can see this explicitly here.

for example, we can solve for and

w = z

z

or

w:

About

I,

z = w~ (1+xy)

(1+xy) .)

To be concrete, let us fix attention on a neighborhood of

I . • The map

X

(™

J •* (w,x,y)

because its inverse is

defines a c h a r t . -> (x,y,z)

is a local homeomorphism

(w,x,y) •* \

V

,

x

1

).

w" (l+xy)

/

^+x^^

x

Thus it

A second choice f o r a c h a r t i s

with inverse

(x,y, z) -» ( z

j .

The

composition of the inverse of the first followed by the second is (w,x,y) » (w

x

V w

and is smooth.

Thus the charts are compatible.

this way for every point of

Arguing in

G , we obtain an atlas.

Let us check that multiplication is smooth near each point.

For

Ix I , we are to check on

20

I : LIE GROUPS AND LIE ALGEBRAS

1

This is smooth.

We see that the coordinates that arise are

compositions obtained from multiplication and our function produced "by the Implicit Function Theorem; thus -we did not need to make the explicit calculation to see the smoothness. By a similar argument, multiplication is smooth near all other points. And in similar fashion, inversion is smooth.

Thus

SL(2,IR) is a Lie group. If we try to proceed similarly with a general closed linear group

G , the first problem is that the group need not

b e defined by polynomial equations.

Even so, suitable entries

of the matrices do look promising as local coordinates for the group.

The difficulty is to decide what the open sets should

be that yield the charts.

To handle the difficulty, we shall

work with the exponential mapping rather than the matrix entry functions; the advantage of the exponential mapping is that Proposition 1.4 gives us a relationship between Theorem 1.5.

If

G

g

and

is a closed linear group, then

G. G

(with its relative metric) becomes a Lie group in a unique way such that

5. CLOSED LINEAR GROUPS AS LIE GROUPS (a)

the restrictions from

GL(n, C)

to

G

21

of the real and

imaginary parts of each entry function are smooth and (b)

cp : M -> GL(n, C)

whenever

smooth manifold

M

is a smooth function on a

such that

cp (M) ^ G , then

cp : M -> G

is smooth. Moreover the dimension of the Lie algebra dimension of the manifold

G .

open neighborhoods

0

U

of

in

exp : U -» V

(V,exp

is a compatible chart.

Remark.

equals the

And, In addition, there exist

such that )

g

g

and

V

of

1

in

G

is a homeomorphism onto and such that

Part (b) explains our definition of smooth curve

in §1 and §3Lemma 1.6.

Let

gl (n,C) = Q © b . in

a

and

neighborhood

U2

a

and

b

be real subspaces with

Then there exist open balls about

V

of

0 I

in

in

U^

about

0

b , as well as an open GL(n,C) , such that

(a,b) -> exp a exp b is a diffeomorphism from Proof. a

and

Let

U^x U 2

X nA., ... , X» r

onto

and

V.

Y-, j-, •. • , Y_ s

be bases of

b , and consider the map

(u 1 ,...,u r ,v 1 , ...,vg) -» exp" 1 {exp(Zu i X i )exp(S v^.Y^)} defined in an open neighborhood of

0 , with the result

written out as a linear combination of Y

.

(1-8)

X 1 , ... , X

, Y^ , ... ,

We shall apply the Inverse Function Theorem to see that

this map Is locally invertible.

Since

exp

is locally

22

I: LIE GROUPS AND LIE ALGEBRAS

invertible, the lemma will follow. Thus we are to compute the derivative matrix at (1.8).

0

of

In computing the partial derivatives, we can set all

variables but one equal to

0

before differentiating, and

then we see that the expression to be differentiated is linear.

The derivative matrix is thus seen to be the

identity, the Inverse Function Theorem applies, and the proof is complete. Proof of uniqueness in Theorem 1.5. T

G

are two versions of

i : G •*• GT

G

as a smooth manifold.

be the identity function.

t : G -> GL(n, C)

Suppose that

G

and

Let

The function

is smooth because smoothness of this map is

detected by smoothness of each real or imaginary part of an entry function, which is given as (a). smooth.

By ("b)>

t" : GT -> G

By the same argument,

Proof of existence in Theorem 1.5. subspace

«

of

gI(n,C)

such that

i : G -> Gf

is smooth.

Choose a real vector

gl(n,€) = g © s , and

apply Lemma 1.6 to this decomposition, obtaining balls and of

Up I

about in

0

in

GL(n, C)

U2 •

and

U-, x Up

For each integer

X

is in

V±

and

Y

k

in

U

l

and

exp Xfc exp Y^

Y

k

in

in

G.

V.

Let

2c

k , exp X exp Y

is in

Assume the contrary. X

(X,Y) -> exp X exp Y onto

(k+l)"1^

Then for every

(k-fl)" 1 ^

U-,

and an open neighborhood

with

(k+l)~n[J2 .

unless

Yk ^ 0

By Proposition 1.4,

is a

cannot be in

k > 1

V

be the radius

k > 1 , form the set

The claim is that for large if

«

such that

diffeomorphism from of

g

is

G

Y=0. we can find

and with

exp Xfe

is in

G.

5- CLOSED IZItfEAR GROUPS AS LIE GROUPS Thus

is in

exp Y k

choose an integer

G

for all such that

nfe

k.

Since

23

Y k ^ 0 , we can

e/2 < n^IlYjJl < e .

passing

to a subsequence if necessary, we may assume that converges, say to

Y . k

Then

Y

is in

8

and

Y ^ 0.

is in

G

and

G

is closed,

Let us show that

exp £ Y is in

G

for all integers

q

Write

Since

n

exp n k Y k = (exp Y k ) in

Then

with

q > 0.

-J= Y, -> 0 since Q

Since

k

with

p

O^r, I , lim exp m, Y v

q

K

2. Y * However, exp 2. Y * However, is closed, and thus

exp m Y exp m kk Y kk exp 2 Y

In other words, G

Also

JS.

exp —

Since

n,p = m 0 .

iv

(exp m k Y k ) (exp ^

is closed,

definition of

g , Y

exists and equals m k = (exp Y = (exp Y kk) is in G and is in G •

exp tY

is in

G

for all rational

exp tY

is in

G

for all real

is in

g .

Thus

Y

G t .

t.

By

is a nonzero member

g na , and this fact contradicts the directness of the sum

g ©8 . in

is

G.

and

of

exp Y

G

We conclude that for large if

X

is in

U-, and

Y

k,

is in

exp X exp Y (k+1) ~ Up

cannot be unless

Y = 0 . Changing notation, we have found open balls about in

0

in

gl (n, C)

morphism of in

G

g

and

8

such that U-,x u 2

only if

an open set in

(X,Y) -> exp X exp Y

onto

Y = 0. G

and an open neighborhood

V

and such that

U-j_ and V

exp

I

is a diffeoexp X exp Y

In view of Proposition 1.4,

such that

of

U2

is

VnG

is

is a homeomo rphi sm from U-j_

24

I:

onto

V0 G • We take

G .

LIE GROUPS AND LIE ALGEBRAS

(Vfl G , exp"" )

About the point

g

(L (Vfl G) , exp" °L~ ) g :

translation by

in

as a chart about the identity in G, we shall use also

as a chart, where

L_(x) = gx .

U

that

UT

and

exp" L

1

are open subsets of © exp : U -> U

T

is left

Let us show that this

system of charts is smoothly compatible.

Here

L

The picture is

g •

is smooth.

We are to check This is Just the

restriction to a lower-dimensional Euclidean space of the smooth map

exp" © L

T «exp : UX U~ -» $1 (n, C) , and hence it

is smooth.

Thus

is a smooth manifold.

G

The same reasoning shows that multiplication and inversion are smooth.

For example, near the identity, what

needs checking in order to see that multiplication is smooth is that, for a small open neighborhood

U

about

0

in

g,

(X,Y) € UX U -> exp"1(exp X exp Y) is smooth into map on

g .

But this map is a restriction of the same

(ux u 2 ) * (Ux U 2 ) > where we know it to be smooth.

6. HOMOMORPHISMS

25

In addition, this reasoning readily proves (a) and (b). Since

G

have

as a manifold has open subsets of

dim G = dim g . Corollary 1.7,

f : G -> M

If

G

is a closed linear group and

is a function into a smooth manifold that is the

GL(n, IR) or Proof. G

as charts, we

Thus the theorem is completely proved.

F : U -» M , where

restriction of a smooth map

of

g

into

GL(n, C)

and

G ^ U , then

We can write

U

is open in

f : G-» M

is smooth.

f = F©i , where t is the inclusion

GL(n, IR) or

GL(n, C) .

Then

%

is smooth by

Theorem 1.5a, and hence

f

is a composition of smooth maps.

Corollary 1.8.

G

is a closed linear group and

If

is its Lie algebra, then component Proof. G .

Tnus

exp g

GQ

generates the identity

Gn . By continuity, exp g c &

.

exp g

is a connected subset of

Theorem 1.5 says that

contains a neighborhood of of

g

I

in

GQ .

The smallest subgroup

containing a nonempty open set in

Corollary 1.9.

If

G

and

G1

exp g

GQ

is all of

GQ .

are closed linear groups

with the same Lie algebras (as sets of matrices), then the identity components of Proof.

G

and

G1

coincide.

We apply Corollary 1.8.

6. Homomorphi sms Suppose that

G

and

H

are closed linear groups.

We

26

I: LIE GROUPS AND LIE ALGEBRAS

shall be especially interested in the case that but -we do not make such an assumption for now*

H = GL(n,C) , Let 3 and t>

be the Lie algebras of G and H , and suppose that smooth homomorphism of G

into

H.

T is a

Our objective is to

associate to IT a map drr : 3 -» % . Before considering examples, let us comment on the case that

G or H

is the Lie group

IR . We can always regard

B

as a closed linear group, say as the set { (e )} of 1-by-l matrices.

We use this convention throughout.

Examples.

(1) Regard

u

by

t *-* (e ) .

in

§ , then

into

3R as a closed linear group, as

If H is any closed linear group and X is

t -> exp tx is a smooth homomorphism of JR

H. (2)

linear group within

GL(1, C) , and t -» e

homomorphism of 3R into (3)

S 1 = [z € C | |z| = 1 }

The circle group

is a closed

is a smooth

S .

The triangular group

G=

/l

0 \0

x z\ 1 y 0 1/

of real

matrices is a closed linear group, and the map that sends the indicated matrix into x into

is a smooth homomorphism of G

3R . We return to the general setting.

If X

in g is

given, let c(t) be a smooth curve in G with cT(0) = X .

(For example, we can take

the composition TT(C(O)) = I ,

t -> 7r(c(t))

and we define

c(o) = 1

and

c(t) = exp tX .) Then

is a smooth curve in H with d7r(X) = (TTOC) * (0) .

Let us see that this definition is indeDendent of the

6. HOMOMORPHISMS choice of value

I

c.

If

c, (t)

and

27

c 2 (t) both have starting

and starting derivative

X , then we compute

Oroc2)'(0) = = -gg 7r(c2(t)c1(t)~ )7r(c-L(t)) ._Q

since ir is a homomorphism

Thus it is enough to prove that the curve

c(t) = c2(t)c-.(t)

which has

(TTOC) T (o) = 0 .

c(o) = I

and

c'(0) = 0 ,

has

, We

now refer matters to local coordinates, using the exponential map.

(See Theorem 1.5.)

The local expression for 7r©c

is

exp" ©7TOC , which we write as exp

©TT©C = (exp

©7T©exp) (exp" ©c) ;

here the two factors on the right are local expressions for ir and

c.

has

(exp

Thus

However, the second factor is a curve in ©c)T(0) = 0 by the Chain Rule, since

(exp" ©7roc)T(0) = 0

"by the Chain Rule.

g , and it

c T (0) = 0.

Applying the

exponential map and using the Chain Rule once more, we see that

(iroc) T (0) = 0 .

Thus our definition is independent of

the choice of ' c . Now we can imitate some of the development of §1 and §3* First of all,

dir : g -» $

is linear.

In fact, let

c^(t) and

28

I : LIE GROUPS AND LIE ALGEBRAS correspond t o

Cg(t)

d

X and Y ,

and l e t

k

be i n

Then

JR.

/

and

d7T(X+Y) = £ = d7r(X) by the product rule for derivatives. Now let and

g be in G and let

c'(0) = X .

t =0 .

Then

gc(t)g" 1

c(t) in G have

has derivative

Ad(g)X at

Hence

ehr(Ad(g)X)

=

4 t^(gc(t)g"1) 1

1 = 0

= ^

ir(gMe(t)

) 7 r (

= ir(g)dTr(x)Tr(g)"1. If also

c(0) = 1

Y

1

(1.9)

is in g , then this formula says d7r(Ad(c(t))Y) = T(c(t))d1r(Y)T(c(t))-1.

Differentiating at

t = 0

and using the fact that

dir

is

l i n e a r , we obtain d7r[X,Y] = d7r(X)d7r(Y) - d7r(Y)d7r(X) .

(1-10)

The right hand side of (1.10) is the definition of [d7r(X),d7r(Y) ] . of

d?r ,

Thus (1.10), in the presence of the linearity

says that

dir is a Lie algebra homomorphism.

our smooth homomorphism

IT : G -> H

Thus

leads to a Lie algebra

6. HOMOMORPHISMS homomorphism

dir : g -» § .

Examples, c o n t i n u e d . d7r(l) = X . has

29

(To s e e t h i s ,

r

c (0) = 1 ,

(1)

If

ir(e ) = exp tX ,

we u s e t h e c u r v e f

and we compute

(TTOC) (o)

d7r(l) =

then

c(t) = e ,

which

from P r o p o s i t i o n

l.ld. (2)

If

ir^)

=

ext ,

If

/l irO \0

x 1 0

z\ y = 1/

then

(i)

as a 1-by-l

matrix. (3)

(e x ) ,

then

/o a c\ irlo 0 b = \0 0 0/

(a) .

The fundamental relation for dealing with homomorphisms is the formula that relates Theorem 1.10.

ir , d-rr , and the exponential map.

If 7r:G->H

i s a smooth homomorphism

between closed linear groups, then Proof. and

C

2^

Fix X ^e ^

7roexp = exp©d7r .

in the Lie algebra of G , and let c 1 (t)

e s m 0 0

^ curves of matrices (actually in H )

given by c x (t) = exp(t dir(x)) Then

and

c2(t) = 7r(exp tX) .

c1(o) = c2(o) = I and J L c^t) = dir(x)exp(t dT(X)) = d7r(X)c1(t)

by Proposition l.ld. •are C2^

Also

= m r ^(exp(t+h)x)| h=0 = -*- T(exp hx)ir(exp t x ) | h = 0

tx) =

30

I: LIE GROUPS AND LIE ALGEBRAS

by definition of c 1 (t)

and

d7r(x) .

c2(t)

j t h columns of both

Since the

solve the initial value problem for the

linear system of differential equations y(0) = ( j t h column of I) ,

§f = d7r(x)y,

the uniqueness theorem for systems of ordinary differential equations says

C

T("^)

= C

2^) •

^

e

"theorem follows by taking

t = 1 . Corollary 1.11.

Let

TT : G -> H

be a smooth homomorphism

between closed linear groups, and let

s

algebras.

x=I

If the map

x -> TT(X)

near

and

^

be the Lie

is referred to

local coordinates relative to the exponential maps of H , then the corresponding map is exactly is linear.

Hence

identity when

w

G

and

dir : g -> § , which

ctor is also the derivative of

T

at the

is referred to local coordinates by the

exponential maps. Proof. Y = exp

The map in local coordinates is

(Tr(exp(x)) near

is the same as linear.

X = 0, and Theorem 1.10 says this

Y = exp" (exp d7r(x)) = dir(x) , which is

A linear map is its own derivative, and the corollary

follows. Corollary 1.12.

If

T*

and

ir^ are smooth homo-

morphisms between two closed linear groups that

dir^ = d-n-g , then Proof.

For

X

in

TT 1 = ir^ on

G

GQ .

g , Theorem 1.10 gives

and

H

such

7. AN INTERESTING HOMOMORPHISM 77^ (exp X) = exp d ^ X ) By Corollary 1.8,

= exp d7T2(X) = 7r2(exp X) .

TT1 = ir2 on

Corollary 1.13.

Let

GQ .

ir : G -» H

be a smooth homomorphism

bet-ween two closed linear groups, and let corresponding Lie algebra homomorphism.

of

I

(a)

dir

onto implies

(b)

d-ir

one-one implies

in (c)

on

31

ir is onto

dir : 9 -» §

be the

Then GQ

TT is one-one in a neighborhood

G d-rr

one-one onto implies

ir is a local isomorphism

GQ. Proof.

Parts (a) and (b) are immediate from Theorems

1.10 and 1.5.

Part (c) carries with it a statement about

smoothness of

rr~ , which follows from Corollary 1.11 and the

Inverse Function Theorem.

7« An interesting homomorphism This section gives an optional example of a nontrivial smooth homomorphism and illustrates a number of the points in §6.

The map

IT will carry

SU(2)

into

SO(3) .

In §4 we gave particular charts to make spheres into manifolds.

If we specialize to the case of

S

and use

complex variables in the notation, then the statement is that X-, + ix o 2 = -i 2. l-x3

(1-11)

32

I : LIE GROUPS AND LIE ALGEBRAS

maps S CU{co}. w

2

"3 ( i n 1RJ ) one-one onto t h e extended complex p l a n e Meanwhile SU(2) a c t s on CU{»} by

= g(z) =

a

l

+

if

?_

- j3z + a

is

g = / * M

in

It is easy to check that this is a group action. We shall 2 reinterpret this action as an action on S by the identification (1-11) and shall see that each g in SU(2) 2 acts on S by the restriction of a linear transformation of 3R . It follows that each of these linear transformations is 2 an orthogonal transformation (since S is preserved); since 2 "3 we have a group action on S , we have a group action in IRJ This means that the map 7T : g in SU(2) — » effect of g on IP? is a group homomorphism of

SU(2) into

0(3) = {x | x x t r = 1} . Since

SU(2) is connected and ir is continuous, the image

must be contained in SO(3) • Let us construct

ir .

Afterward we shall compute d-rr

and examine what is happening. x + ix A l *X A 2 1 -Xo

TT

and we compute ( y ^ y ^ y * ) 2 2 2 x ^ + X p + X o = 1 , we have

Thus we write

az+6 - jSz + a

T7

^ terms of

v 4- iv ^1 + ^ 2 1 -y(x1,x2,x^) .

Since

7. AN INTERESTING HOMOMORPHISM 1-x-

Then

| z| 2 - 1 = 2x 3 /(l - Xg)

33

x

and | z| 2 + 1 = 2/(1 - x-)

Consequently

A similar computation solves for y, in terms of w , and we find W+W

l + lwl 2 ' To compute

y2

_

W- W

"^77Tm^'

y

3

y^ in terms of x.. , Xp , and x- , we write :

gz

+ 2Re(gzg) + l g | :

Thus

_ \v\f-l _ (lal2- |g|2)(lzl2-l) Iwl + 1 Iz| + 1

4

Ui 2 +1 = 2(Re To compute y 1

and

2/(l-x3) a| 2 -

yg , we write

+4Re(azg)

.

(1.12)

34

I : LIE GROUPS AND LIE ALGEBRAS

2

2 Re w

\ - gz + a] _ 2 Re[(qz+6)(-gz+q)]

l+|w|2

la-pzl'^l+lwl2)

l+NI*

_ 2 Re[(az+j3) (-gz'+a)]

=

2Re(a2z-/z)

M*

+

± - U\2

ITur

2 Re

2 2 = Re[a (x 1 + ix 2 ) - P ( x 1 ~ i x 2 ) ]

- 2x^ Re ajS

= x± Re(a 2 -jS 2 )

- 2x^ Re ap

- x 2 Im(a 2 +p 2 )

(1.13)

and

2 imf ^

2 Im w

2

\ - gz + a / _ 2 Im[ (az+p) (-gz+a) 1

l+|W|a

1+lwl^

|a-1

2 Im[ ( a z + g ) ( z|2 2 l m ( a 2 z - B 2 z)

2

= x± Im(a2 - p 2 )

x

\ )=

Hence

2

2

2

= Im[a (x 2 + ix 2 )

Combining (1.12),

l>|z]

- p ( x ] _ - i x 2 ) ] - 2x^ Im + x 2 Re(a 2 +p 2 )

- 2x^ Im

(1.13), and (1.14), we see that

/Re(a 2 -p 2 )

-Im(a 2 +p 2 )

-2 Re ap \ /x x

Im(a 2 -p 2 )

Re(a 2 +p 2 )

-2 Im ap ) ( xo \ .

\ 2 Re dp -2 im dp |a|2-|p|2 'a p\ irl-g - I i s the 3-by-3 matrix in (1.15)-

(1.15)

35

7. AN INTERESTING HOMOMORPHISM

We compute

d-rr on the basis

^ J) , (°

(^

by using the curves cos t

sin t

cos t

i sin t

-sin t

cos t

i sin t

cos t

So, for example, /cos 2t

-sin 2t

0^

= -^r- sin 2t

cos 2t

0

t=0

\

o

o

i, t=0

Similarly

d7T

V-l

It is clear that

d-rr maps

0

8u(2)

is connected (because

1

=

\^ 0

-2 0 So (3) -

one-one onto

S0(3)/S0(2) ^ s 2

S0(2)

are connected), Corollary 1.13 tells us that

SU(2)

onto

T

and maps

and is a diff eomorphism in a neighborhood

Actually

check directly that

0

0

S0(3)

of the identity.

0

(0

Since

SO (3)

0

CVJ

dir

i\

^\"0

everywhere two-to-one.

rr is not globally one-one; "we can _ T )= ^ *

^

e

k° momo:r T?ki sm

^

^s

36

I:

LIE GROUPS AND LIE ALGEBRAS 8, Representations

Let

G

be a topological group.

representation of

G

A finite-dimensional

is a homomorphism

IT of

G

into the

group of invertible linear maps of a finite-dimensional complex vector space map of

Gx V

When

G

into

V

V

into itself, such that the resulting is continuous. GL(n, C) , then

is given to us as a subgroup of

the identity map on representation of

G G

gives us what is called the standard

on

Cn.

It is reasonable to ask "why

one should study representations of a group that is already represented as matrices.

The answer is that representations

are often forced on us by some outside area of mathematics, and we have to deal with them. Examples. (1)

G = SO(n) , V = all polynomials in

n

real

variables with complex coefficients and all terms of degree N

(i.e., homogeneous of degree

N ),

and

x = P(g-1

ir(g)P

x. (2) z

G = U(n)

or

SU(n) , V = all polynomials in

l ' * *' ' z n 9 ~*1 ' " ' '^n

(3)

G = U(n)

or

hom

°geneous

of

degree

N , and

SU(n) , V = all members of the

8. REPRESENTATIONS A Cn

subspace

A basis over

37

of the exterior algebra of C

e. A .../v e. 1-L ik

of A C

with

(see §11,4).

is all alternating tensors

i, < ... < jL , where 1 k Cn

standard basis of

n

Cn

over

C.

{e.} 0

is the

If we define

77-(g) (e. A ... A e. ) = ge. A ...A ge. , 1 x x 1 \ l k then

extends to a linear map of A C n

ir(g)

ir is a representation.

into itself, and

(We shall see this a little more

systematically in Chapter II.) An invariant subspace for such a vector subspace such that

7r(g)U c u

this case we get representations of the obvious way.

tions

0

7T on

and V

g

in

for all G

on

U

g

in

and

v/U

in

ir is

Two finite-dimensional representa-

7rT

and

G; in

and if ir has no invariant sub spaces

V.

on VT 1

E : V -» V

an invertible linear for all

is a (complex)

A finite-dimensional representation

irreducible if • V ^ 0 other than

T

are equivalent if there is such that

E7r(g) = TTT (g)E

G.

Examples. (1)

G = SO(n) , V = all polynomials in n

variables homogeneous of degree 7T as before.

N

real

with complex coefficients,

Define the Laplacian operator by

This operator reduces degrees by two, but the operator E(P) = |x| AP maps

V

into V-

Moreover this operator

38

I: LIE GROUPS AND LIE ALGEBRAS

commutes with in

SO(n) .

image of

IT in the sense that

Eir(g) = 7r(g)E

E

are invariant sub spaces.

With some effort, one N

if

2. (2)

in

g

This commutativity implies that the kernel and

can show that the kernel is irreducible for each n>

for all

G = SU(n) ; V = homogeneous polynomials of degree

z^ , ... , z , "z^ , . . . , "z_ ; ir as before.

holomorphic polynomials (those with no subspace.

N

The subspace of

I's ) is an invariant

With some effort, one can show that this subspace

is irreducible. (3)

G = SU(2) , v = homogeneous holomorphic polynomials

of degree

N

in

z,

and

holomorphic polynomials in

z^, F z

VT =

as before,

of degree

N

with

\j3z + a E : V -> V !

Define !

and

7r

by

EP(z) = p(^) .

Then

E

exhibits IT

as equivalent.

Our interest will be in the case that particularly when

G

G

is a closed linear group.

is a Lie group, If

V

is a

finite-dimensional complex vector space, then we can identify V

with

where

Cn

and the invertible linear mappings with

n = dim V ;

GL(n, C) ,

all we have to do is choose a basis.

If we

choose a different basis, then the two identifications are related by an invertible linear mapping, which preserves all the Lie group structure of interest. representations on C on

V.

Thus a result about

gives us a result about representations

The connection between representation theory and Lie

8. REPRESENTATIONS

39

groups comes from the following theorem.

We state it for

closed linear groups, but it is valid for all Lie groups. Every finite-dimensional representation IT

Theorem 1.14.

of a closed linear group is a smooth mapping. Lemma 1.15.

If G

is a closed linear group, then there

exist arbitrarily small open neighborhoods in

G

such that

U5 V

U

and V

and each element of V

of I

has a unique

square root in U . Proof. open ball I

in

By Theorem 1.5 we can choose a sufficiently small B

about

0

G such that

in

5 and an open neighborhood

exp is a diffeomorphism of B

B T = iB. and U = exp B T .

Put

On U

the map

V of

onto

V.

x -» x , when

referred to local coordinates by the exponential map, is just and is a diffeomorphism of B ! onto B . Therefore the 2 map x -» x is a diff eomorphism of U = exp B' onto 21

V = exp B . Proof of Theorem 1.14. F i r s t suppose

Let

G = IR «—> { (e )} .

can choose open neighborhoods of

I

GL(n, C)

in

such t h a t

Choose

e > 0

Then we can write e

IT(e '

)

and

so t h a t 7r(e ) =

€

7r(e /

2

0

gl (n, C)

in

U is a ball,

k

) = exp (e/2 )Y

V is a

V have unique square roots in

v(e ) exp eY

i s in

V

for some

I t e r a t i n g t h i s fact, for a l l

and

exp : U -» Y

for Y in

exp ^Y are both square roots of

V and hence are equal.

be given.

By Lemmas 1.2 and 1.15 we

U of

diff eomorphism, and members of V.

ir : G -> GL(n, C)

k > 1.

|t|

< e .

e" U.

exp eY

Now

within

we see t h a t

Raising to powers

40

I : LIE GROUPS AND LIE ALGEBRAS

shows that

ir(e ) = exp tY

continuity of

ir(e ) =

ir ,

i s smooth "when

G,

let

the previous paragraph,

7r(exp

exp tY

for a l l r e a l

t .

t .

By

Thus

ir

G = 3R .

For general

t -> exp tY.

for a l l diadic r a t i o n a l

for some U

3_X1 ' "

X-, , — , X. t -> 7r(exp tX.)

Y.

e x p u X

in

d d^

gt(n,C) . =

be a b a s i s of

g.

By

i s of the form Since

7r ex

( P u-^X^)* • *Tr(exp u^X^)

= (exp u ^ ) • • • (exp u d Y d ) , the map (u 1 , . . . , u d ) -» Tr(exp ^ 1 X 1 * - - exp u d X d ) i s smooth from

3R

into

GL(n, C) .

I f we can show t h a t

(exp u 1 x 1 )*--(exp u d X d ) is locally a diffeomorphism into then

x -» TT(X)

G

(1-16)

near the origin of 3R ,

will be smooth near the identity, and the

theorem will follow. We shall apply the Inverse Function Theorem.

In local

coordinates, (1.16) is given by (u1,...,ud) -> exp"1!(exp u 1 X 1 )---(exp u d X d )} , and the derivative matrix of this at the origin is the identity (since we can set all but one u . equal to 0 before differentiating).

The Inverse Function Theorem applies, and

the proof is complete. As a consequence of this theorem, a representation

8. REPRESENTATIONS IT : G- -> GL(n, €) morphisin

4l

automatically gives us a Lie algebra homo-

d7r : g -> 3 I (n, €) .

To make the terminology match, we

say that a finite-dimensional representation algebra of matrices is a linear mapping of space

End c V

vector space

cp 9

of a Lie

into the vector

of linear maps of a finite-dimensional complex V

into itself such that )

=

(Hom(L A ,®L B ,,L c )(cp))(a'®b')

= f(HQm(L A ,®L Bt ,L c )cp)(a')(b') . This proves the proposition. By way of motivation for the tensor product of two Lie algebra representations, let G be a topological group, and let

a

and p be representations of G on finite-dimensional

complex vector spaces representation

V

and w , respectively.

tr of G on V ® P W

Define a

by means of (2.8) as

Tr(g) == a(g) ® p(g) . Relation (2.9) assures us that can use bases to see that indeed a representation.

w i s multiplicative, and we

ir i s continuous; thus Now l e t us think of

r

is

G as a closed

linear group (as in Chapter I) and differentiate to find We will be replacing

g

by

c(t)

and differentiating.

product rule will then give us two terms.

dir . The

Accordingly we make

5*-

II: REPRESENTATIONS AND TENSORS

the following definition. Let

3

be a Lie algebra over

be complex vector spaces, and let tions of 5

on V

ir = a ® p

product

B or C , let V a

and W

and p be representa-

and ¥ , respectively.

Define the tensor

on v®« W by

———————

i/

(a® p) (X) = a(X) ® I + I® p(X) . A little calculation with (2.9) shows that

(2.14)

a ® p is a

representation. Again by way of motivation, let us consider group actions on a

Hom c (V, W) .

Thus let

G be a topological group, and let

and p be representations of G on finite-dimensional

complex vector spaces representation

V and ¥ , respectively.

Define a

ir of G on Hom c (V, W) by -rr(g) = Hom(a(g)"1,p(g)) .

Referring to (2.10a), we see that Homc(V,W)

to i t s e l f .

7r(g)

i s a linear map from

Relation (2.11) shows that

multiplicative and explains the presence of a basis to see the continuity, and thus tion.

Concretely the formula is this:

Homc(V,W) ,

ir(g)

g~ .

is

We can use

TT is a representaIf

cp i s in

then 7r(g)(cp)(v) = p(g)(cp(a(g)-1v)) .

Passing to closed linear groups and differentiating, we are led to make the following definition. Let

g

be a Lie algebra over

be complex vector spaces, and let

IR or C , let V a

and p be

and W

2. TENSOR PRODUCT OF TWO REPRESENTATIONS r e p r e s e n t a t i o n s of IT = Hom(cj,p)

on

g

on

V and

Homc(V,W)

¥,

55

respectively.

Define

by

(Hom(a,p)(X))(cp)(v) =

p(X)(cp(v)) - c p ( a ( X ) v ) .

A little calculation with (2.11) shows that

Hom(a,p)

(2.15) is a

representation. A special case of interest is the case that and

p

write on

is the trivial representation on a* = Horn (a, 1)

V* .

(The use of

C.

¥

is

C

In this case we

for the resulting representation of g "1" here is a reminder that

is not necessarily zero;

Horn (a, 1)

"o" would be a more literal symbol

for the trivial representation.)

Formula (2.15) specializes

to cr*(X)(cp)(v) = -q>(ff(X)v) . The representation

a*

Proposition 2.3> over (over

(2.16)

is called the contragredient of a . Let A , B , and

C

be vector spaces

C , and suppose that representations of a Lie algebra g 3R or

C ) are given on each.

Then in the vector space

identity H o m c ( A ® c B , C ) a Hom c (A,Hom c (B,C)) of (2.12), the isomorphism is an equivalence of representations. Proof.

This is just a question of checking what happens

in (2.13) when we plug in the representation formulas of (2.14) and (2.15).

Dropping the names of all representations

from the notation, we have

56

II: REPRESENTATIONS AND TENSORS 5(38p)(a)(D) = (Xcp)(a®b) = X(cp (a® b)) -cp (X(a® b)) = X(cp(a®b)) -cp(a®Xb) -ep(Xa®b) = X(?(cp)(a)(b)) -?( V

universal property: Ax Bx C

C

Whenever

into a vector space

unique linear mapping

T

of

t

s a vec

"^

or s

P

ace

k.

A triple

over

k "with

having the following is a trilinear mapping of

U

over

k,

V

into

U

there exists a such that the

diagram V t

(fixed)/

( = triple tensor product) \ T

AX BX C

(2.17) > U

t commutes.

It is clear that there is at most one triple tensor

product up to canonical isomorphism.

We shall use triple

tensor products to establish an associativity formula for

3 - REPRESENTATIONS ON THE TENSOR ALGEBRA

57

ordinary tensor products. Proposition 2.4.

(A®kB)®kC

(a)

and

A®k(B®fcC)

are

t r i p l e tensor products. (b)

There exists a unique isomorphism

§

from l e f t

to

r i g h t in (A ®k B) ®k C ^ A ®k (B®k C) such that and

§ ((a® b) ® c) = a ® ("b® c)

for all a € A , b € B ,

c€C . Proof.

(a) Consider

t (a,b) = t(a,b,c) . to a linear = t

t C-j T C p

(AO^B) ® k C .

For c in C , define

be trilinear.

Then

t

+1

and t O

T

= xT .

XC

C

t:AxBxC^U

t : A x B -> U by

is bilinear and hence extends

T« : A ® k B •» U .

C-j

Let

Since

= xt

t

is trilinear,

for scalar

x ; thus

C

XC

uniqueness of the linear extension forces and

(2.18)

T

C

Consequently

l+C2

= T

c

l

+T C 2

t T : (A®kB) X C -» U given by linear

tT(d,c) =

T (d)

i s b i l i n e a r and hence extends t o a

T: (A ®, B) ®, C -» U .

This

T proves existence of the

l i n e a r extension of the given

t .

s i n c e the elements

generate

(a®b)®c

Uniqueness i s t r i v i a l ,

(A® k B)®,C

i s a t r i p l e tensor product.

A ®k (B®fcC)

i s a t r i p l e tensor product.

(b) and

I n (2.17), take

t(a,b,c) =

a® (b®c) .

(A ®, B) ®k C . In similar

V = (A ®k B) ®k C ,

So

fashion,

U = A®k(B®kC),

We have Just seen i n (a) t h a t

i s a t r i p l e tensor product -with

t (a,b,c) =

(a®b)®c .

V

Thus

58

II: REPRESENTATIONS AND TENSORS T : V -» U

there exists a linear This equation means the roles of

(A ® k B) ® k C

and

T.

and existence is proved. (a®b)®c

Ti(a,b,c) = t(a,b,c) .

T((a®b)®c) = a® (b®c) .

two-sided inverse for

elements

with

Interchanging

A ® k ( B ® k C ) , we obtain a

Thus

T

will serve as

$

in (b),

Uniqueness is trivial, since the

generate

(A®kB)®kC.

Just as with Proposition 2.1, Proposition 2.4 carries along with it a certain naturality that we often need to invoke in applications. Proposition 2.5-

A , B , C , A j , Br , CT

Let

be vector

L A : A -» AT , L B : B -> BT , and

spaces over

k , and let

Ln : C -> CT

be linear maps.

Then the isomorphism

$

of

Proposition 2.4 is natural in the sense that the diagram A®k(B®kC)

(AT®kBt)®CI

* A!®k(B!®kCT)

commutes. Proof. ( ( L A ® ( L B ® L c ) M ) ( ( a ® b ) ® c) = (L A ® (L B ® LQ)) (a® (b® c)) L A a® (LBb®.Lcc) L c c)

3. REPRESENTATIONS ON THE TENSOR ALGEBRA

59

= §((L A ® L B ) ( a ® b ) ®L c c) = ( 5 c ( ( L A ® L B ) ® L c ) ) ( ( a ® b ) ® c) . The p r o p o s i t i o n

follows.

P r o p o s i t i o n 2 . 3 showed t h a t t h e v e c t o r space i d e n t i t y (2.12) i s compatible with t h e passage t o Lie a l g e b r a representations.

We now prove a corresponding

compatibility

for the identity (2.l8). Proposition 2.6. over (over

C,

Let

A,

B,

and

C be vector spaces

and suppose that representations of a Lie algebra

3R or

C ) are given on each.

g

Then in the vector space

identity (A®CB)®CC a £®c (B®CC) of (2.18), the isomorphism is an equivalence of representations. Proof.

Let §

be the ismorphism implementing (2.18),

and let us drop the names of all representations from the notation.

Then "we have

§ ( X ( ( a ® b ) ® c ) ) = § ( X ( a ® b ) ® c + (a® b ) ®Xc) = $ ( ( X a ® b ) ® c + (a® Xb) ® c + ( a ® b) Xc) = $ ( ( X a ® b ) ® c ) + $ ( ( a ® Xb) ® c) + $ ( ( a ® b ) ® Xc) = Xa® (b® c ) + a ® (Xb ® c ) + a ® ( b ® X c ) = Xa® (b® c ) + a ® X(b® c ) = X ( a ® (b® c ) ) = X ( ? ( ( a ® b) ® c ) ) .

(2.19)

60

II: REPRESENTATIONS AND TENSORS

Since the elements

(a®b)®c

generate

(A®kB)®kC'

equivalence follows. Remark.

The Lie algebra representation on a triple

tensor product may be motivated by group representations.

In

a triple tensor product, a member of the group is to act in all three factors simultaneously.

The differentiated action

is just (2.19) above, as a consequence of the product rule. There is no difficulty in generalizing matters to n-fold tensor products by induction.

An n-fold tensor product is to

be universal for n-multilinear maps. to canonical isomorphism.

It is clearly unique up

One such tensor product is the

(n-l)-fold tensor product of the first with the n

space.

n-1

spaces, tensored

Proposition 2.^-b allows us to regroup

parentheses (inductively) in any fashion we choose, and iterated application of Proposition 2.5 shows that we get a well defined notion of the tensor product of

n

linear maps.

Iterating Proposition 2.6 shows that representations of a Lie algebra

g

on each factor, when the vector spaces are complex,

lead to a well defined representation on the n-fold tensor product. Fix a vector space n-fold tensor product of we let

T^°'(E)

vector space,

E

over E

be the field

T(E)

k,

with itself. k.

T^n'(E)

be an

In the case

n = 0,

and let

Define, initially as a

to be the direct sum

T(E) =

£©T(n)(E) . n=0

(2.20)

3. REPRESENTATIONS ON THE TENSOR ALGEBRA The elements that lie in one or another homogeneous.

6l

T^n' (E) are called

We define a "bilinear multiplication on

homogeneous elements TW(E)XT(n)(E)

_,T(m+n)(E)

to be the restriction of the above canonical isomorphism T ( m ) (E) ® f e T ( n ) (E) — » T ( m + n ) (E) . This multiplication is associative because the restriction of the isomorphism

to T ^ (E) x ( T ^ (E) x T ^ (E)) T ( ^(E)x (T (m) (E)x T (n) (E)) given by

factors through the map » ( T W ( E ) X T W ( E ) ) X T ( n ) (E)

(r, (s,t)) -> ((r, s),t) .

Thus

T(E) becomes an

associative algebra with identity and is known as the tensor algebra of

E.

T(E)

has the following universal mapping

property. Proposition 2.7* with image I : E -> A

T^'(E)

The one-one linear mapping has the following property:

is a linear map of

E

t : E -> T(E) Whenever

into an associative algebra

with identity, then there exists a unique associative algebra homomorphism

L: T(E) -» A

such that

L(l) = 1

T(E) \

LL

> A commutes.

and the. diagram

62

II. REPRESENTATIONS AND TENSORS Proof.

Uniqueness is clear, since

as an algebra.

For existence we define

E

generates n

l/ '

T(E)

31

on

T' ' (E)

to

be the linear extension of the n-multilinear map

and we let be in

L = Z 9 I, '

T^m'(E)

and

in obvious notation.

v, ® . . . ® v

be in

Let u..® ...®

T^ n ^ (E) .

Then we

have

Hence ... ® v n )

Taking linear combinations, we see that

L

is a homomorphism.

Now let us consider what happens with representations. To orient ourselves, let us think about matters first in terms of groups. can put

If

IT is a representation of

A = T(V)

defined on all of

and

T(V) .

allows us to see that on

T ^ ( V ) , 7r(g)

product

I = ir(g)

G

on

V,

in (2.21) to obtain

The uniqueness of

L

if is multiplicative in

then we ~(g)

in (2.21) g.

In fact,

is nothing more than the n-fold tensor

7r(g) ® ... ® ir(g) .

should expect an action by

Differentiating, we see that we q

involving

n

terms on

T^(v).

3- REPRESENTATIONS ON THE TENSOR ALGEBRA

63

In fact, if we start from a Lie algebra representation IT of

g

on a complex vector space

V , our remarks with n-f old T^n'

tensor products give us canonically a representation S

on

T^ n^(v) .

be trivial.)

(The representation on

T ' 0 ' (V) ^ C

of

is to

It is clear that the formula is

n ( ) 7 T W (X) (v, « ... ® V ) = I (v-,® ... ® V. ,® 7r(X)v.® V. ,-,®...®O , ± n . j J. j-J. j J+-L n

(2.22a) and this agrees with what we' would expect from differentiating a group action. 7T

of

g

Using direct sums, we obtain a representation

on the full tensor algebra

Proposition 2.8. algebra

g

on

If

V

and if

representation on

T(V) ,

TT(X)(V®W)

for all

X

Proof. n

T^ ^(V) .

in

g

T(V) .

IT is a representation of the Lie TT is the corresponding then = TT(X)V® W + v ® T?(X)W

and for all

We may assume

v

v

and

is in

w

in

T^m^ (V)

(2.22b)

T(v) . and

w

is in

Then (2.22a) shows that

and similarly for

ir^ (x)

that

and this is (2.22b).

and

7r^m+n^ (X) .

Then it is clear

64

II: REPRESENTATIONS AND TENSORS 4. Representations on exterior and symmetric algebras Let

E

be a vector space over

tensor algebra. A(E) •

k, and let

T(E)

be its

We begin by defining the exterior algebra

The elements of

A(E)

are to be all the alternating

tensors ( = skew-symmetric if

k

and so we want to force

to be

v® v

has characteristic ^ 2 ) , 0•

Thus we define the

exterior algebra by //two-sided ideal I generated by\ . (E) = T ( E ) / m / I all v ® v with v in T v ; (E) I

(2.23)

This is an associative algebra with identity. It is clear that the ideal

I

satisfies

00

i = E© (mT (n) (E)) . n=0 An ideal with this property is said to be graded.

Since

I

is graded, A(E) = We write A n (E)

I©T(n'(E)/(inTtn)(E)). n=0

for the n

summand on the right side, so

that A(E) = I © A n ( E ) . n=0 Since

IflT^ '(E) = 0 , the map of

elements is one-one onto. denoted A v ^ ® — ® vn a

E

into first-order

The product operation in A(E)

rather than ® .

The image in A n (E)

in T ^ (E) is thus denoted

is in A m ( E )

and

b

(2.24)

of

v^ A ... A V .

is in A n (E) , then

aAb

is

is in

If

4. EXTERIOR AND SYMMETRIC ALGEBRAS Am+n(E) . V

Moreover

1 A •*•* v n

A n (E)

with a11

v

is generated by elements

in

j

Al

^E) -

generated by corresponding elements defining relations for A(E) and

v. J

in A

65

make

E

>

since

T ^ (E)

® vn '

v-,® v. A V . =

is

The

-V.AV.

for

v.

(E) , and it follows that

a A b = (-l)mnbAa

i f a e A m ( E ) and b eA n (E) .

(2.25)

has the following universal property:

Let t

Proposition 2.9. (a) A n ( E ) be the map n

A (E) .

If

Ex...xE

t (v-,..., v ) = v, A I

of

E x ... x E

into

is any alternating n-multilinear map of

into a vector space

linear map

A V

L : A n (E) -* U

U , then there exists a unique

such that the diagram

An(E)

EX...XE commutes. (b) A(E)

has the following universal property:

be the map that imbeds linear map of

E

identity such that

E

as

A (E) c A(E) .

into an associative algebra £(v)

=0

for all

exists a unique algebra homomorphism L(l) = 1

v

such that the diagram

E

A

> A

I

1

is any

with

E , then there

L : A(E) -> A

A(E)

commutes.

in

If

Let

with

66

I I : REPRESENTATIONS AND TENSORS Proof.

I n both cases uniqueness i s t r i v i a l .

For T'n'(E)

e x i s t e n c e we use the u n i v e r s a l mapping p r o p e r t i e s of and

T(E)

t o produce

show t h a t

L

taken as

L,

or

A(E) ,

term in

T(E) .

I f we can

For ( a ) , we have

v®v

L(T^ n ' (E) H i ) =

with

i s thus of t h e form n

T^ '(E) •

v

in

E.

0,

where

A member of

Z a^® ( v i ® v i ) ® b .

with each

Each term here i s a sum of pure t e n s o r s

x, ® . . . ® x

-ft.^ «y

with

or

then the r e s u l t i n g map can be

and we a r e to show

i s generated by a l l

T ^ ( E ) HI

T'* 1 ' (E)

and we a r e done.

L : T^ n ' (E) -> U I

on

a n n i h i l a t e s the a p p r o p r i a t e subspace so as to

A n (E)

descend t o

L

^

r+2 +s=n .

Since

® v. ® v. ® V- ® . . . ® v A

But L ( v ® v ) = £(v)

L

= 0,

L(l) = 0 .

Corollary 2.10. then

vanishes on

is an ideal, it is enough to check that

vanishes on the generators of I . and thus

L(T'n'(E)ni)=O.

n

Hom^(A (E),F)

If

E and F

are vector spaces over k,

is canonically isomorphic (via

restriction to pure tensors) to the vector space of F-valued alternating n-multilinear functions on Ex ... x E . Proof.

Restriction is linear and one-one.

Proposition 2.9a.

It is onto by

4. EXTERIOR AND SYMMETRIC ALGEBRAS Proposition 2.11, space of dimension

Let

N.

n

67

E "be a f i n i t e - d i m e n s i o n a l

vector

Then

(a)

A (E) = 0

(b)

dim A n (E) = (^)

for

n > N .

for 0 < n < N .

ordered basis of E , then

{u. A x±

If [u±] is an

A u. | in < in 1

( i } n

is a basis of A (E) • (c) A n (E*) is canonically isomorphic to A n (E)* by (flA — Proof. The vectors

Af

n)^l""^n)

Let u., , ... , u_. be an ordered basis of E . u. ... ® u. 1

hence the corresponding

are then a basis of T'n'(E) , and

n

u. A ... A u. span A (E) . Using x x l n the skew symmetry in A (E) , we thus see that the elements

u. A — A u. 1

1 ^n proves (a). Let

u^ ,

r1 < — < r 1

Then

t

with , v*

span A n (E) .

i, < ... < i -1

n

be the dual basis of

E* ,

This fix

, and define _ (wj)}

n

for

w^ . . . , w n

i s alternating n-multilinear from

E.

Ex . . . XE into

L : An(E) •*• k .

and extends by Proposition 2.9a to

in

k

If

L(u k A ... A u k ) = I (u k ,... ,ufc ) = det{u* (u^ )} , 1 n I n i J and the right side is 0 unless which case it is 1 .

r^ = k, , —

, rn = k ,

in

This proves that the u A ... A U

are linearly independent in A n (E) . the number of increasing

Then (b) follows, since

n tuples from

{1, ...,N}

is f *M .

68

II: REPRESENTATIONS AND TENSORS For (c) let

f1 , ... , f

be in E* , and define w ) = det{f. (w

' *'*' n

Then £-. ~ is alternating n-multilinear from EX ... x E 1 l'##^1n into k and extends by Proposition 2.9a to a linear Lf

f

:A n (E)->k.

Thus I (f -,,..., f ) = Lr

-

defines an alternating n-multilinear map of E* x ... x E* into A n (E)* .

L maps An(E*)

Its linear extension

into A n (E)* .

The argument in the previous paragraph shows that

L maps

the basis

A...AU

Hence

L

u*

A

... A U

to the dual basis of u

is an isomorphism.

Now let us consider representations. algebra over

B

or C , and let

Let 7r'n^

V.

representations of g

T ^ (v)

and b

TT be the

T(v) .

Let us see It is

F(X) to an element (2.26), which we write in

condensed form as T^fV),

and

and

is an invariant subspace for 7r'n' .

that T ^ (V) HI enough to apply

be a Lie

F be a representation of g

on a complex vector space on

Let g

a®v®v®b in

T^(V).

with a

in

T ^ (V) , v in

By Proposition 2.8,

7?(X) (a® v® v® b) = 7r(x)a® v® v® b + a ® 7 r ( x ) v ® v ® b + a®v®7r(X)v®b + a® v® v® Tr(x)b = 7?(X)a®v®v®b + a® v® v®7r(x)b + a®

( T T ( X ) V + V ) ® (TT(X)V + V) ®b

- a® T ( X ) V ® 7T(X)V® b -

a®v®v®b,

4. EXTERIOR AND SYMMETRIC ALGEBRAS •which is in the span of (2.26).

7r'n' , and

invariant subspace for subspace for

7? .

and

A(V) .

I

T ^ (v) n I

is an

is an invariant

Thus we get well defined quotient

representations, also denoted A n (V)

Hence

69

7r'n'

and

TT for now, on

By Proposition 2.8, we have

~(X) (aAb) = 7?(X)a A b + a A?(X)b .

(2.27)

The above discussion makes precise, on the level of the third example of representations given in §1.8. group level, if we assume a representation

TT'11'

V

of

with

On the

is finite-dimensional, we obtain G

on

T'11' (V)

from

IT on

V, n

and this passes to the quotient as a representation A n (V)

g >

7r' '

on

irf n '(g)(v 1 A...A7 n ) = ir(g)v1 A ... A 7r(g) v n .

Its

differentiated version is what we constructed above on the level of

3.

We turn to the symmetric algebra vector space over

k .

S(E) , E

being a

The construction, results, and proofs

are similar to those for A ( E ) .

The elements of

S(E)

are

to be all symmetric tensors, and so we want to force u®v = v®u.

Thus we define the symmetric algebra by

C

two-sided ideal J generated by all u®v-v®u

with u € T l

;

(E) , v € T v

\ ;

(E)y

This is an associative algebra with identity. It is clear that

J

is graded:

J = Z ~ - © (JH T^n' (E)) .

Thus we can write 00

S(E) = EeT tn '(E)/(jnT W (E)). n=0

70

I I : REPRESENTATIONS AND TENSORS S (E) f o r t h e n

We w r i t e

sunmiand on t h e r i g h t s i d e , so

that S(E) = Since

00

Ees n (E) . n=0

JflT^'fE) = 0 , the map of E

elements is one-one onto.

in T^ n ' (E)

The image in S n (E) of

Sn(E) is generated by elements

Moreover

in S (E) ^ E , since

corresponding elements for

is thus denoted

is in S n (E) , then

is in S®(E) and b

v.

into first-order

The product operation in S(E) is

written without a product sign. v,®...®v

S(E) make

T'31' (E) v

v.^®— ®

v^v • = v.v^

and it follows that

(2.29)

n

v-,# • • v

.

If a

ab is in S m + n (E) . v-,#*#v

with all

is generated by •

T rie

^

defining relations

for v^ and v. in S (E) ,

S(E) is commutative.

Proposition 2.12. Sn(E) has the following universal property:

(a) be the map S n (E) .

t (v^,..., v ) = v^* • • v

If £

Ex ... x E linear map

of Ex ... x E

Let t

into

is any symmetric n-multilinear map of

into a vector space L: S n (E) -> U

U , then there exists a unique

such that the diagram

EX . . . x E commutes. (b)

S(E) has the following universal property:

be the map that imbeds

E

as S 1 (E) A

commutes. Proof.

This is completely analogous to Proposition 2.9.

Corollary 2.13* k,

then

If

Hom k (S n (E) ,F)

E and

F

are vector spaces over

is canonically isomorphic (via

restriction to pure tensors) to the vector space of F-valued symmetric n-multilinear functions on Proposition 2.14. N .

space of dimension (a)

be a finite-dimensional vector

Then

E , then

for

0 < n < «> .

{u., •••u__

with

Sn(E*) is canonically isomorphic to

Gn Proof.

if

{u±}

is the symmetric group on (a)

monomials span

Since

S(E)

n

is

j-,+.. .+j* = n}

Sn(E) .

is a basis of

where

E

dim Sn(E) = (^i"1)

an ordered basis of

(b)

Let

Ex ... x E .

S n (E)*

by

letters.

is commutative and since

T ^ (E) , the indicated set spans

S n (E) .

To see its cardinality, we recognize that picking out

N -1

72

I I : REPRESENTATIONS AND TENSORS

o b j e c t s from

n+N-1

t o l a b e l a s d i v i d e r s i s a way of

a s s i g n i n g exponents t o t h e u . T s ; t h u s t h e c a r d i n a l i t y of t h e indicated set is

1

(

Let us see independence. into the polynomial algebra

The map E c ^ ^ -* Z c^X^ k[X-,,... ,X^]

commutative algebra with identity.

of E

is linear into a

I t s extension via

Proposition 2.12b maps our spanning set for

Sn(E)

to

distinct monomials in

k[X-,,... ,XJ , which are necessarily

linearly independent.

Hence our spanning set is a basis.

(b)

This i s proved in the same way as Proposition 2.11c,

and Proposition 2.12a is the tool. Remarks.

The proof of (a) suggests that

S(E)

might be

just polynomials in disguise, but this suggestion i s misleading.

The isomorphism with

depended on choosing a basis. between

S(E*)

k[X-,,. • . ,XJ

in (a)

The canonical isomorphism is

and polynomials on E;

details are left to

the reader. Finally let us consider representations. Lie algebra over of

g

3R or

T^ ^(v)

and

T(V) ,

respectively.

algebra, we find that for

n

7r^

.

Hence

J

g

be a

C , and let • ir "be a representation

on a complex vector space

n

Let

i^ n ' (V) fi J

V.

Form 7r'n'

and ~ on

Just as with the exterior is an invariant subspace

is an invariant subspace for

IT .

Thus

we get well defined quotient representations, also denoted 7r' n '

and

TT for now, on

Sn(E)

and

S(E) •

By Proposition

2.8 we have

£

£

(2.30)

5. COMPLEXIFICATION

73

5. Extension of scalars - complexification For the time "being, extension of scalars "will be a device for taking a real vector space and making from it a complex vector space of the same dimension.

If the original vector

space is n-dimensional real column vectors, the new space may "be regarded as n-dimensional complex column vectors.

In this

case a real matrix gives a linear mapping between real vector spaces originally and then can be regarded as a linear mapping between the extensions to complex vector spaces.

We construct

the new space without using a basis. Let

K

space over

be a field with k,

and we can form

vector space over as a subspace of E®, K

k.

in

is a vector E

The linear map

e -*• e ® l

imbeds

(over the field K

k).

\ c , c1,

in and

is a E

We now make

by defining

/mult by c

= 1 8

E®tK

We easily check that

K

whenever

into a vector space over

(

Then

E® f c K

E®,K

mult by c\

E ®k K

k c K .

for

c€K.

K c2

in

K

and

u, v

in

imply

(i)

c x (c 2 v) = (c 1 c 2 )v

(ii)

c(u + v) = cu + cv

(iii) (c 1 + c2)v = cxv + c 2 v (iv)

1v = v

(v)

c(e®l) = c e ® l

if

These properties show that

c

is in

ES^K

k

and

e

is in

E.

is a vector space over

K

lh

II: REPRESENTATIONS AND TENSORS

and that its scalar multiplication, when restricted to scalars in

k

and the original imbedded

multiplication in

E , agrees with scalar

E.

If we have a linear mapping L ® 1 : E ® k K •» F ® k K

L : E -» F

is linear over

K,

over

k , then

as follows from the

identity

J

mult by\ y\

( j ( /mult by\

L L

/

mult by

® c in K J = = ®(c in KK j= ( ® c in K The mapping

L® 1

X

is the one explained in the first paragraph

of this section. Another kind of extension of linear mappings is available when

L : E •* F

space over E

and

c

is linear over

K . in

In this case, let K.

Then

Moreover

L

*L

and

is already a vector for

e

is bilinear over

k

L: E ® k K -> F -

on the imbedded copy is linear over

F

£(e,c) = cL(e)

I : Ex K -> F

and extends to a k-linear map reduces to

k

E® 1

The map

of

E

in

in

L E®^K.

K -, to see this linearity, we

Just observe that L(c 1 (e®c 2 )) =

The pure tensors

e $ C2

generate

E®^K,

and the K-linearity

follows. Let us now specialize matters to the case K = C.

We denote

E ^ C

c omplexification of Suppose that make

g

C

g

by

k = 1R and

E C , calling it the

E. is a Lie algebra over

into a complex Lie algebra.

1R .

We shall

To obtain a complex

6. UNIVERSAL ENVELOPING ALGEBRA

75

Lie algebra, -we need to define the bracket for two members of 3

.

(This is very easy with bases, but it is not immediately

clear that the resulting definition is independent of basis.) gxCxgxC-»gC

Thus let us form the 4-multilinear map by

(x, r,y,s) -» rs[x,y] . C

g®TR ®7R S ®TD C (g ® _ C) x (g ®

and

given

Extend it to a linear map

restrict the result'to an B -bilinear map

c) •» S .

Using suitable uniqueness conditions

for extensions of multilinear maps, we readily check that this map is C-bilinear from

g xg

to

g , that it is

alternating, and that it satisfies the Jacobi identity. g

is a complex Lie algebra. The ]R-linear map

map

ad X : g -* g

gives us the C-linear

ad X ® 1 : g C -> g C , which is nothing more than ad(X® 1 ) .

We shall write this simply as ad : g -» Endg

Thus

on

c g

carrying

X

ad X . to

The linear map

ad X

is a representation of

C

g .

Suppose vector space

ir is a representation of

g

V .

is real linear into a

Then

ir : g -> End« V

on a complex

complex vector space, and we have constructed a corresponding C complex linear extension from g into End- V . We denote this extension by extended tion of

T g

ir , too.

It is easy to check that the

is a representation of

g

.

Thus a representa-

automatically extends to a representation of

on the same complex vector space.

6. Universal enveloping algebra In this section we suppose that

g

is a complex Lie

g

76

II: REPRESENTATIONS AND TENSORS

algebra of finite dimension

N.

(When we are studying the

representation theory of a real Lie algebra 9 0 *)

If we have a representation

vector space

g , g will be

T of g on a complex

V , then the investigation of invariant

subspaces in principle involves writing down all iterates 7r(X1)7r(X2)'• •7r(Xn) for members of g , applying them to members of V , and comparing the results.

In the course of

comparing results, one might be able to simplify an expression by using the identity

TT(X)T(Y)

= ir(Y)ir(X) + TT[X,Y] .

identity really has nothing to do with

This

T , and our objective

in this section will be to introduce an object where we can make such calculations without reference to ir; to use an identity with the representation

ir, one simply applies

ir

to both sides. For a first approximation to what we want, we can use the tensor algebra of

T(g) .

The representation

g into the associative algebra

algebra homomorphism 7r(Xn)---7r(X1)

T is a linear map

End-, V and extends to an

Ir : T(g) -» Endc V with

TT(1) = 1 .

Then

can be replaced by 7r(Xn® .. - ® X x ) . The

difficulty is that the tensor algebra does not take advantage of the Lie algebra structure of g and does not force the identity and all

7r(X)7r(Y) = T(Y)TT(X) +TT[X,Y] w.

for all

X , Y in g

Thus instead of the tensor algebra, we use the

following quotient of T(g) :

((

two-sided ideal generated by all

U(g)

X0Y-Y® X - [X,Y]

(1) (1)

with X , Y in T[±) (g

is an associative algebra with identity and is known as

6. UNIVERSAL ENVELOPING ALGEBRA the universal enveloping algebra of

g.

77

Products in

U(s)

are written without multiplication signs. The canonical map T^ '(g)

g ^ u(g)

and then passing to

of (2.31),

1

given by imbedding

U(g)

is denoted

into

Because

satisfies

1 [X,Y] = t (X)i (Y) -t (Y)t (X) U(g)

t .

g

for

is harder to work with than

X,Y A(g)

in or

g.

(2.32)

S(g)

because the ideal in (2.31) is not graded, i.e., is not generated by homogeneous elements.

Thus, for example, it is

not evident that the canonical map

t : g -> U(g)

However, when U(s)

g

is abelian (i.e., when all brackets are 0 ) ,

reduces to

S(g) , and we have a clear notion of what to

expect in this case. S(g)

is one-one.

Even when

g

is nonabelian,

U(g)

and

are still related, and we shall make the relationship

precise at the end of this section. Let

U 1 1 ^)

be the image of

Z£=Q T ^ ( g )

passage to the quotient in (2.31)* dimensional subspace of Proposition 2.15* t : g -» U(g)

Then

u(g) , and U(g)

^(g)

under the is a finite-

u(g) = U~ =

and the canonical map

have the following universal mapping property:

Whenever A is a complex associative algebra with identity and ir : g -> A

is a linear mapping such that

7r(X)7r(Y) -7r(Y)7r(x) = TT[X,Y]

for all X and Y in g, (2.33)

then there exists a unique algebra homomorphism such that

•?(!) = 1

and

~ : U(g) -» A

78

II: REPRESENTATIONS AND TENSORS

u(s)

commutes. Proof. t (g)

Uniqueness follows from the fact that

generate

U(s) .

For existence, let

1

and

TT1 : T(g) -» A

be

the extension given "by the universal mapping property of T(s) .

To obtain

if, we are to show that

the ideal in (2.31).

TT-, annihilates

It is enough to consider

ir-, on a

typical generator of the ideal, where we have i r 1 ( i X ® t Y - i Y ® iX - t [X,Y]) = 7r 1 (tX)TT 1 (tY) -7r 1 (tY)7r 1 (tX) - T T ^ I [X, Y] ) = 7r(X)7r(Y) -7r(Y)7r(X) -7T[X,Y] = Corollary 2.16.

Representations of

correspondence with unital left ir -> ~

correspondence Remark. Proof.

apply Proposition 2.15 to

in

U(g)

U(g)

and

1

operates as

v

in

U(g) V .

g

TT : g -> End^ V , module under Conversely if

module, then we can define

implies that

modules (under the

TT is a representation of

becomes a unital left

stand in one-one

of Proposition 2.15).

Unital means that If

U(g)

g

0 .

1. on

V,

and then

V

uv = 7r(u)v V

we

for

is a unital left

TT(X)V = (tX)v , and (2.32)

rr is a representation of

g .

constructions are inverse to each other since

u

The two TTOI = -rr in

6. UNIVERSAL ENVELOPING ALGEBRA

79

Proposition 2.15. Theorem 2.17 ( P o i n c a r e - B i r k h o f f - W i t t ) . i s an ordered b a s i s of

with all

g ,

If

t h e n the monomials

J, y_ 0 , form a basis of U(g) .

canonical map

t : g -> U(g)

Lemma 2.18. a permutation of

Let

X1

, ... ,

In particular, the

is one-one.

Z-. , ... , Z

{1,...,p} .

"be in

g , and let a be

Then

is in I ? " 1 (s) . Proof.

Without loss of generality, let a

transposition of

j

with

j+1 .

Then the lemma follows from

by multiplying through on the left by the right by

be the

(t Z-,) • • • (t Z . ,)

and on

(i Z j + 2 ) • • • (i Z p ) -

For the remainder of the proof of the theorem, we shall use the following notation: I = (i1,

,i )

not, we write

of integers from

Y T = Y. •*-Y- . P 1 }

Lemma 2.19. length

Let Y^ = tX i . 1

Also

to

For any tuple

N , increasing or

i < I

means

The Y T , for all increasing tuples of

< p , generate

\P (g) .

80

II: REPRESENTATIONS AND TENSORS Proof.

If we use all tuples of length

< p , we

certainly have a set of generators, since the obvious preimages generate ^ k / D T^ '(g) .

Lemma 2.18 then implies that

the increasing tuples suffice. Proof of Theorem 2.17. C[z_,...,z_J , and let total degree "^

satisfy P

with

z^.

in

P

for a l l

J

as the union of its definitions on

ir will be a representation by (C ) and will

satisfy (2.34) by (A ). For

-» -n+i

X

TT(XJ) (7r(X i )z J ) +7r[X i ,X 0 .]z J

P^

T

in

i < I

is in

(TT(X, J )Z J )

zT

(2.34).

f o r

so as t o be compatible and t o

(A )

with

p

TT(X) : P

s h a l l d e f i n e l i n e a r maps i n d u c t i o n on

satisfying

ir

8l

p = 0,

Hence we will be done.

we define

^ ( X ^ ) ! = z^ .

Then (AQ) holds,

(B o ) is valid, and (CQ) is vacuous. Thus now assume that for all X (C

in

n) hold.

sequence

I

with

in such a way that (A

We are to define of p

and (C ) hold. to (A ).

g

TT(X) has been defined on

If

TT(X.)Z

P^ T

-j_), (B _ _ ^ ) , and for an increasing

elements in such a way that (A ) , (B ) , i ^ I , we make the definition according

Otherwise we can write in obvious notation

J < i , 3 < J , | j| = p - 1 .

== 7r(Xi)7r(X.)2T

I=(j,J)

We are forced to define

since 7r(X.)Zj is already defined

= 7r(XJ)^(Xi)zJ + T[X i ,X J ]Zj

^

= 7r(X. 5 )(z i z J +'w) +Tr[X i ,X ; j ]z J

with w i n P ^

= z . z i z J + 7T(X^.)^+7r[X i ,X^]z J

by (A )

(Cp) by

(Bp-1)

82

II: REPRESENTATIONS AND TENSORS

So we make this definition, and then (B ) holds. ir(X.)z_

has now "been defined in all cases on

Thus

P , and we

have to show that (c ) holds. Our construction above was made so that (C ) holds if J -X

in

T' '(s) •

Composing with passage to the

quotient, we obtain an antihomomorphism of Its kernel is an ideal.

T(g)

into

To show the map descends to

U(g) .

U(g) ,

it is enough to show that each generator X® Y - Y ® X - [X,Y] maps to

0.

But this element maps in

then maps to to

U(s) •

0

in

U.(g) .

T(g)

to itself and

Hence the transpose map descends

It is clearly of order two and thus is one-one

onto. The map V

u -» u

also as right

a left

U(g)

define

r

U(g)

allows us to regard left modules, and vice versa:

module into a right

vu = u

r

v

for

u

in

U(g)

U(g)

and

U(g)

modules

To convert

module, we Just v

in

V.

Conversion in the opposite direction is accomplished "by uv = vu

.

This conversion will allow us to use tensor

products over universal enveloping algebras in investigating representations.

We take up such tensor products in §8.

7. Symmetrization Let

g

be a finite-dimensional complex Lie algebra.

observed in (2.36) that if Q

and

b

g = 0 ©b

We

as vector spaces and if

are Lie subalgebras, then their universal

7. SYMMETRIZATION enveloping algebras satisfy particular, the canonical.

If

a a

87

U(s) — U(a ) ®~ U(b ) .

type terms and the 'b and

b

In

type terms are

are merely vector subspaces of

we can still choose a "basis of but the linear span of the

a

g

compatible with

g,

g = c ©b ,

type monomials will perhaps

depend on the choice of basis. In this section we introduce a device to get around this problem.

The starting £oint is the remark in §6 that

reduces to the symmetric algebra is abelian.

S(g)

Lemma 2.18 indicates that

rather similar even if

g

U(g)

in the case that U(g)

and

S(g)

g are

is nonabelian, and we introduce a

mapping of the one to the other that captures these similarities precisely. For

p > 1 , define a symmetric p-multilinear map

S p : gx ... X g -» U ( Q )

where

G

by

is the symmetric group on

coefficients

1/pJ

p

letters.

(The

are traditional but are not important.)

By Proposition 2.12a we obtain a corresponding linear map, also denoted

S

, from

of these maps for S : S(g) -» U(g)

The map

S

S^(g)

p > 0

(with

into

U(g) .

The direct sum

$ 0 (l) = 1 ) is a linear map

such that

is called symmetrization.

88

II: REPRESENTATIONS AND TENSORS Proposition 2.22.

isomorphism of Proof. ST

S(g)

onto

P

P

U(g) .

S (Sp(g))

The image

be the composition of

quotient

S

1

U (g )/U ~ (g) .

By Theorem 2.17, p

U (g)/!?"" (s) . §

Symmetrization is a vector space

3"

is in

U p (g) , and we let

followed by passage to the

By Lemma 2.18

maps a basis of

S p (g)

Let us see therefore that

to a basis of

S- is one-one.

carries a linear combination of monomials to

be the largest degree of such a monomial.

0,

Composing

If

let

p

§

on

the linear combination with the quotient

u p (g) - u p (g)/u p - 1 (g) , we obtain

3T

on the homogeneous part of degree

all the monomials of degree coefficient one-one into

0*

Hence

U p (g) .

§

p

p .

Thus

in the linear combination have

restricted to

Zp

Q

S q (g)

is

But these spaces have the same finite

dimension, again by Theorem 2.17, and thus this restriction of S

is also onto

onto

U p (g) .

Hence

§

S(g)

U(g) . The canonical decomposition of

when

is one-one from

Q

and

U(g)

from

g = Q ©b

b • are merely vector spaces is given in the

following proposition. Proposition 2.23. b

are subspaces of

g .

Suppose

g = aeb

Then the mapping

and suppose

a

and

(a,b) -> S(a)S(b)

1. SYMMETRIZATION of

S(a)®«S(b)

into

U(g)

89

is a vector space isomorphism

onto. Proof.

We argue as in Proposition 2.22 somewhat.

restrict the map to

p

We

m

Z ^ = Q S^a ) ® c S ~ (b ) , with image in

U p (s) , and follow it with passage to the quotient U P (3)/U P " (3) . one-one.

As in Proposition 2.22, this composition is

We then complete the argument as in Proposition

2.22, obtaining an isomorphism onto. Corollary 2.24. subalgebra. into

u(s)

Suppose

Then the mapping

3 = \ u£(p)

I of

is a Lie U(l)® /n S(p)

is a vector space isomorphism onto.

Proof.

The composition (k,p) + (§(k),p) -> S(k)S(p) ,

sending S(!)3S(*>) -> U(!)® c S(p) •* U(s) , is an isomorphism by Proposition 2.23, and the first map is an isomorphism by Proposition 2.22.

Therefore the second map is

an isomorphism, and the notation corresponds to the statement of the corollary when we write

u = S(k) .

We know that a representation a representation of (2.30) holds.

g

on

S(V)

IT of

g

on

V

leads to

such that the product rule

Applying this fact to the adjoint representa-

tion of

g

on

3 , we obtain a representation (also called

ad ) of

g

on

S(g)

such that

90

I I : REPRESENTATIONS AND TENSORS (ad X)(ab) =

for a l l

a

and

TD

in

a((ad X)b) + ((ad x)a)b

(2.37)

S(g) .

The c o n s t r u c t i o n t h a t gave us r e p r e s e n t a t i o n s on symmetric algebras can a l s o take the r e p r e s e n t a t i o n g i v e us a r e p r e s e n t a t i o n on

U(g)

t h e r e i s an e a s i e r "way t o proceed:

satisfying

ad

(2.37).

and But

We simply define

(ad X)u = Xu-uX for

X

in

3 and u

in

U(g) .

Clearly

(2.38) ad

extends the

usual definition on g , and it is easy to check that (2.37) holds.

To see that it is a representation, we compute

(ad X ad Y)u = X((ad Y)u) - ((ad Y)u)X = XYu - XuY - YuX + uYX . Thus (ad X ad Y - ad Y ad X)u = XYu + uYX - YXu - uXY = (XY-YX)u-u(XY - YX) = [X,Y]u-u[X,Y] = (ad[X,Y])u as required. Proposition 2.25. representations on

Symmetrization and the adjoint

S(g) and U(s) are related by (ad X ) o S = S c ad X

for

X

in

Remark.

g . Symmetrization of course does not respect

7. SYMMETRIZATION multiplication i n S(s)

anc

compensates some f o r t h a t Proof.

If

X

T"

# X

91

^ U(s) > and t h i s i d e n t i t y

failure. i s

D

a monomial i n S(g) ,

(ad X)S(X 1 ---X p ) = (ad X) £ S X , ( 1 ) • • - X ,

then (p)

= JT ZZ \ (i) * * ' \ (3-1) [X ' X a (J) ]Xc (j+i)' "*o (p) by (2.37) with (j)=k

(2.39)

Define

X±

if

i ^ k.

Then §(ad X)(X1---Xp) =

E

a Y with

1

y

pT ^

y

k

k

.

#vk

JS. . . v k

J Y a(l) *V(J-l) r k

Y

a(p)

with

•with Since (2.39) and (2.4o) match, t h e p r o p o s i t i o n f o l l o w s .

92

II: REPRESENTATIONS AND TENSORS 8. Tensor product over an algebra Let

R

only that

be an associative algebra over

R

C .

(This means

is a complex vector space with a bilinear

multiplication that is associative.)

In our applications,

R

at first will be a universal enveloping algebra, but later

R

•will be a more complicated algebra that need not have an identity. that

1

"When

R

acts as

does have an identity, we shall assume

1

in all

R

modules; in this case every

R

module is a complex vector space, by restriction of the multiplication to multiples of

1.

identity, we shall assume that an vector space and that the

R

"When R

R

does not have an

module is also a complex

action commutes with scalar

multiplication. For this section let us write is a left R

R

module.

module and

V^

^T

to indicate that

to indicate that R

In the situation

V

V

is a right

R

( V, W), let )

(2.4la)

denote the vector space of complex l i n e a r maps cp : v •*• ¥

that

are R-linear in the sense that f or a l l r e R , V€V.

cp(rv) = r(cp(v))

This definition works compatibly with (2.10).

Namely if

Ham-fVgjWg) 9

and

L± qp

is in is in

actually R-linear from definition

Hom(L^,Lp)

HomR(V1,W]L) , L 2

Hom R (W i ,V 2 ) , then

Y^_ into

W^ •

Hom(L1,L2)cp = Lgocp©!^,

(2.4lb) defined in

is in I^cpoL^

Thus under the

is

8. TENSOR PRODUCT OVER AN ALGEBRA is in S

Suppose that and that the left such a way that

Hom(D(HomR(¥1, Vg) jHorn^V.^ Wg)) .

R module

V

(rv)s = r(vs)

Ham-(V,W)

(2.42)

is another associative algebra over is also a right for all

C

S module in

r€R , s€S, v € V .

* T S to signify this situation.)

(In this section we write Then

93

"becomes a left

S

module under the

definition (sqp)(v) = cp(vs) . The only thing t h a t requires note i n t h i s a s s e r t i o n i s the fact that identity

scp

i s again R-linear, and t h i s comes down t o the

(rv)s =

r(vs) . (RRR, ^T) .

A special case of i n t e r e s t i s the s i t u a t i o n If

R has an i d e n t i t y , then the map

¥

that

veV.

(2.44b)

a r e R - l i n e a r i n t h e sense t h a t cp(vr) =

(cp(v))r

We still use (2.10) to define valid.

In the situation

module under

for a l l r e R ,

Hom(L 1 ,L 2 ) , and (2.42) remains

(SV^,WR) , H o n ^ ^ W )

is a right

S

I I : REPRESENTATIONS AND TENSORS

9h

(cps) (v) = cp(sv) . I n t h e s p e c i a l case

( R ,V^) ,

we o b t a i n an isomorphism jV) 2= v

of r i g h t

R modules "by the map

(2.45)

cp -» cp(l) ,

provided

R

has

an i d e n t i t y . Now we s h a l l make an analogous construction with tensor product in place of

Horn.

Let the s i t u a t i o n be

(v**,RW) .

Then

denotes the vector space quotient of generated by a l l

vr® w - v® rw .

the image in

V®"^

vr®w = v®rw

in

compatibly with HomR(V1,W1)

of

v®w

V®^ W. L-,®Lp

and if

L2

We s t i l l write

in

V^W.

i s in

v® w

for

Then we have

The definition of

i n (2.8).

i n i t i a l l y i s a member of

V®c W by the subspace

Namely i f

V®_W L-,

works

is in

Hom~(V2>W2) > then

L-,®L2

Homc(V1®c Vp» W-i®c Wp) > and i t

passes to the quotient i n the range t o become a member of Homc(V1®cV2,W1®:RW2) .

Since

(L2® L 2 )(v x r® v 2 - v x ® rv 2 ) =

L-L(v1r) ® L 2 (v 2 ) - L-Jv^ ® L 2 (rv 2 )

= L 1 (v 1 )r®L 2 (v 2 ) - L 1 ( v 1 ) ® rL 2 (v 2 ) =

0,

we see that L-L®!^ In the s i t u a t i o n

i s in

Honip^^j^ V2, W1®R W2) .

(S'VR,^)

,

V®RW

becomes a l e f t

(2.46) S

8. TENSOR PRODUCT OVER AN ALGEBRA

95

module under t h e d e f i n i t i o n s(v®w) =

sv® w .

(2.47a)

To make this precise, we define the left action by V8> W by means of (2.46) as L ® I , where in

V.

L is "left by s "

Formula (2.9) shows that this definition makes V ® _ W

into a left V®RW

s on

S module.

becomes a right

Similarly in the situation (V S module under the definition (v® w)s = v® ws .

Special cases of interest are when

(2.47b)

(V,^)

and

R has an identity, and then we obtain the respective

isomorphisms R®~V^V Ju

from the maps

and

V®^ R 2= V

(2.48)

i\

r® v -> rv and v ® r -> vr .

A more general situation where two-sided modules arise is when

R

is a subalgebra of S .

module, and HOHL R (S,V) module.

and S ® R V

Then

S

is a two-sided

make sense if V

R

is an R

Both constructions are sometimes referred to as

extension of scalars, and the notion of complexification in §5 is an instance of this for real associative algebras. The vector space

V®^!^ has a universal mapping

property, given as follows. Proposition 2.26.

in the situation

i : V x W -» V ® ^ W denote the map any complex vector space and

(V^,RW) , let

t (v,w) = v® w .

b : Vx W -> X

If X is

is R-bilinear in

96

II: REPRESENTATIONS AND TENSORS

the sense that b(vr,w) = b(v, rw)

for all r e R ,

vev,

then there exists a unique complex linear map

wew,

B : V ® R W -» X

such that

B VX W

> X

commutes. This kind of argument is by now routine, and we omit it. Our final three results concern associativity formulas. Proposition 2.27. modules

A , B , and

In the situation

(A R , R B S , C S )

for

C , the isomorphism

of (2.12) induces a vector space isomorphism Hom s (A® R B,C) 2= HomR(A,Homs(B,C))

(2.49)

that is natural in each variable. Remark.

"Naturality" refers to the commutative diagram

in Proposition 2.2, which remains commutative when we pass to the quotients and submodules here. Proof.

The map implementing (2.12) from left to right

was *(q>)(a)(b) = cp(a®b)

8. TENSOR PRODUCT OVER AN ALGEBRA

97

"with i n v e r s e Y(t)(a«b) = First we restrict that

$(cp)

$

•(a)(b) .

to cp's that descend to A ® ~ B and see K

i s R-linear:

§(cp)(ar)(b) = cp(ar®b) = qp(a® rb) = $ (cp) (a) ( r b ) = ((§ (cp) ( a ) ) r) ( b ) . So

$

induces a map of

HonufAjHom^B, C)) . $

to

cp's

S-linear

Homc(A£>

B,C)

into

Next we r e s t r i c t t h e induced v e r s i o n of

t h a t a r e S - l i n e a r and check t h a t

§(cp) h a s

values:

$ (cp)(a) (bs) = cp(a®bs) = cp ((a® b ) s ) = (9(a® b) ) s = s(5 (cp) ( a ) ( b ) ) . This gives us our map from left to right in (2.49). We construct a two-sided inverse in the same way from Defining

7($) on elements

? .

a ® b in A S> B defines it on

all of A®_,B as a result of Proposition 2.26.

The remaining

details for proving (2.49) are easy and are omitted. Proposition 2.28. modules

In the situation

(RA,SBR,SC) for

A , B , and C , the isomorphism c

c

c

€

€

c

B

, C))

of (2.3) and (2.12) induces a vector space isomorphism Homs(B®RA,C) 2= HomR(A,Homs(B,C)) t h a t i s natural in each variable.

(2.50)

98

II: REPRESENTATIONS AND TENSORS Proposition 2.29.

modules

A , B,

and

In the situation

(A R , R B S , S C)

for

C , the isomorphism

(A®CB) ® C C a A ®

C

(B8>c C)

of (2.18) induces a vector space isomorphism ( A ® R B ) ® S C sf A ® R ( B ® S C )

(2.51)

that is natural in each variable. Remarks.

Naturality is by Proposition 2.5.

(2.51), we use the map of (2.18) to send A ®-, (B®^ C) . we map

If

A®R(B®CC)

Q : B«> C -> B®« C into

into

check that it descends properly. similarly.

(A®~ B) ®.p C

into

is the quotient maiD, then

A®R(B®SC)

composition sends ( A ® C B ) ® C C

To obtain

by

I®Q.

The

A ® R ( B ® S C ) , and we

The inverse map is obtained

CHAPTER III REPRESENTATIONS OP COMPACT GROUPS

1. Abstract theory We defined representations of topological groups in §1.8. In this chapter we shall assume that

G

is a compact

topological group, and soon we shall specialize

G

further.

Although everything we do will be valid in some form for all compact Lie groups, we shall give statements and proofs only for the unitary groups

U(n) .

In this way we shall avoid the

need for an extensive digression on roots and weights. For this section, let

G

be a compact topological group.

Such a group has a unique Borel measure that is invariant under right and left translation and has total mass one.

We

refer to this measure as normalized Haar measure and write it in integrals simply as A representation $ (x)

dx . $

on a Hilbert space

is a unitary operator for all

case, whenever

U

x

V

is unitary if

in the group.

In this

is an invariant subspace, so is the

orthogonal complement

U x , since

w e U1

and

u €U

imply

(§(x)w,u) = (w,$(x)"1u) € (w,u) = 0 . Proposition 3>1>

If

I

is a representation of

99

G

on a

100

III: REPRESENTATIONS OF COMPACT GROUPS

finite-dimensional

V,

product such that Proof.

Let

§

then

V

admits a Hermitian inner

is unitary.

(•>•)

"be any Hermitian inner product on

V,

and define

= J ($(x)u,v) dx. G

It is straightforward to see that

has the required

properties. Corollary 3- 2finite-dimensional

If V,

?

is a representation of

then

$

irreducible representations. each

V.

G

on a

is the direct sum of

(That is,

V = V-,©

© V.

with

an invariant subspace on which § acts irreducibly.)

Proof.

Construct

invariant subspace

(•,•)

U ^ 0

orthogonal complement

as in Proposition 3-1*

Find an

of minimal dimension and take its

IT1 .

Then 1

Repeating the argument with

IX

IT1

is invariant.

and iterating, we obtain the

required decomposition. Proposition 3.3 (Schurfs Lemma). are irreducible representations of spaces

V

V T , respectively.

and

map such that

§ T (g)L = L$ (g)

one-one onto or Proof.

G If

for all

Suppose

$' and

$T

on finite-dimensional L: V -» V T

is a linear

g

then

in

G,

L

L = 0.

We see easily that

invariant subspaces of

V

and

ker L

and

image

L

are

V 1 , respectively, and then

the only possibilities are the ones listed.

is

1. ABSTRACT THEORY Corollary 3»^» representation of L : V -» V in

G,

Suppose G

$

is an irreducible unitary

on a finite-dimensional

is a linear map such that then

Proof.

L

V.

If

5 (g)L = L? (g)

for all

g

is scalar.

Let

A

be an eigenvalue of

By Proposition 3.3,

L. $ (g)

not one-one onto "but does commute -with G .

101

Then

L - XI

for all

g

is in

L - XI = 0 .

Corollary 3*5 (Schur orthogonality). (a)

Let

$

and

$r

be inequivalent irreducible unitary

representations on finite-dimensional spaces respectively.

and

Let

§

for all u, v € V and u',V € V ! .

be an irreducible unitary representation on

a finite-dimensional

V .

Then (u-.jUp) (V,,

J ( 1 11,v)1)($(x;u ( ( ; 22,v2 2j dx = — ± — f J ( ($(x)u -1

G for all

X

d

u^ , v 1 , u 2 , v^

Proof.

V! ,

Then

J (S (x)u, v) ($ * (xJuSv 1 ) dx = 0 G (b)

V

For (a), let

in

^

±±

£

dim V

V.

I : Vf -» V

be linear and form

L = J $ (x)^| « (x""1) dx . G

(This integration can be regarded as occurring for matrixvalued functions and is to be handled entry-by-entry.) it follows that for all

y

in

!

1

$ (y)L$ (y" ) = L , so that G.

By Proposition 3*3*

Then T

$ (y)L = L$ (y)

L = 0.

Thus

102

III: REPRESENTATIONS OF COMPACT GROUPS

(LvT,v) = o ••

Choose

£(w T ) = (w T ,u T )u, and (a) results.

For (b), we proceed in the same way, starting from I : V -» V

and obtaining

L = XI

by Corollary 3.4.

Taking the

trace of both sides, we find A dim V = Tr L = Tr I , so that

A = Tr £/dim V .

Thus

£(w) = ( w , ^ ) ^ , we obtain (b).

Choosing If

§

is a unitary representation of

coefficient of

$

is any function on

If

on which

acts, then the expressions of

basis are

a matrix

of the form

($(x)u,v) . $

{u^}

G

G,

is an orthonormal basis of the space

i..(x) = ($(x)u.,u.)

§ (x)

in this

and are matrix coefficients.

The linear span of these functions is independent of the basis

{ui} .

Let

$

be unitary on

space as

V

but with multiplication by

multiplication by

V.

Let

-i ; that is, if

V

J

be the same vector i and

respective multiplication-by-i maps, then an inner product in

V

by

$ (x) : V -> V

as maps

§ (x) : V -» V , so that

on

Moreover

$

is unitary on

the complex conjugate of of

$

on

7

$

$

on

V.

J~

are the

J— = - J .

(u,v)== = (v,u) v •

check that the linear maps

V .

replaced by

Define

It is easy to

remain complex linear

is a representation of V.

We call

$

on

G V

The matrix coefficients

are the complex conjugates of the matrix

1. ABSTRACT THEORY c o e f f i c i e n t s of If !

V ,

$

then

$

on

$!

and $&$

T

103

V.

are u n i t a r y r e p r e s e n t a t i o n s on T

i s u n i t a r y on

V®V

V and

w i t h r e s p e c t to t h e

i n n e r product defined by (u®uT,v®vT) =

(u,v)(uT,vf) .

Evidently the matrix coefficients of

? ® $T

are spanned "by

the products of matrix coefficients, one from

$

and one

T

from

$ . We can interpret Corollary 3*5 as follows.

Let

{§'a'}

be a maximal set of mutually inequivalent irreducible unitary finite-dimensional representations of

G.

For each

§^

,

choose an orthonormal basis for the underlying vector space and let

§£?)

be the matrix of

the functions L 2 (G) .

{?L .'(x)}. .

In fact, if

d ^

$^

(x)

in this basis.

Then

form an orthogonal set in denotes the degree of

$^a^

(i.e., the dimension of the underlying vector space), then { (d(a))1//2§(°f) (x)}. . ij

J-^

jj ot

is an orthonormal set in

L 2 (G) .

The Peter-Weyl Theorem below -will say (among other things) that this orthonormal set is an orthonormal basis. If

T

is a unitary representation on a finite-

dimensional space, its character is the function X T (x)

where

{u.}

= Tr T(x)

=Z

(T(X)U±,U±)

is an orthonormal basis.

,

(3.1)

This function is

independent of the basis and lies in the span of the matrix coefficients of

T .

104

III: REPRESENTATIONS OF COMPACT GROUPS To any representation

associate an operator

$ on a Hirbert space, we can

$ (f)

for any continuous

f : G -» C

"by

(*(f)u,v) = J f(x)(§(x)u,v) dx G or more directly by the vector-valued integral §(f)u = S f(x)$(x)udx. G Then

$ (f)

is linear in the

f variable, and we readily

check that §(f *h) = where

f*h refers to the convolution f *h(x) = J fixy^My) G

In addition, if

$

f* (x) = f (x~X)

and

and $ (f )*

f(f») = «(f)*,

(3.3)

is the adjoint of

We shall be especially interested in the operators where

(3.2)

is unitary, then

! U ( f ) ! l < llffii where

dy = J f(y)h(y"1x) dy . G

$ (f) .

$ (XT) ,

T is an irreducible representation. Corollary 3.6,

Characters of finite-dimensional

irreducible unitary representations

T

and T!

satisfy

X T (x) = X T (x"1) * XTT = 0

*x T = x T .

if T is not equivalent with T !

1. ABSTRACT THEORY Proof. the relation

105

The first equation follows by summing for (Tfxju^u.) = ( T ( X Ju.,^) .

i = J

The other two

equations are routine consequences of Schur orthogonality (Corollary 3-5). Theorem 3«7 (Peter-Weyl Theorem). (a) The linear span of all matrix coefficients for all finite-dimensional irreducible unitary representations of is dense in (b) If

G

L (G) . {$(ah

is a maximal set of mutually

inequivalent finite-dimensional irreducible unitary representations of

G

{ (d^ )1/'2$^ (x)} is a

and if

corresponding orthonormal set of matrix coefficients, then { (d( a )) 1 / 2 §.^ (x)} (c)

is an orthonormal basis of

L 2 (G) .

Every irreducible unitary representation of

G

is

finite-dimensional. (d)

Let

Hilbert space

$

be a unitary representation of

V .

Then

V

G

on a

is the orthogonal sum of finite-

dimensional irreducible invariant subspaces. (e)

Let

Hilbert space tion

T

of

$

be a unitary representation of

V .

G

on a

For each irreducible unitary representa-

G, let

E^, be the orthogonal projection on the

closure of the sum of all irreducible invariant subspaces of V

that are equivalent with

d § (x_) 9 where character of T . then

d

T .

Then

is the degree of Moreover if

E T 5 r , = K. t^r = 0 .

T

and

E T

is given by and X

is the

T * are inequivalent,

Finally every

v

in V

satisfies

106

III: REPRESENTATIONS OF COMPACT GROUPS v = I E^v , T

with the sum taken over a set of representatives

T

of all

equivalence classes of irreducible unitary representations of

G. Proof.

(a) Although this result is valid in general,

we prove it only for unitary groups in question is a subspace of

U(n) .

The linear span

C(G) , the space of continuous

complex-valued functions on

G,

and it is closed under

multiplication (because of tensor products) and conjugation (because of complex conjugate representations). Moreover it contains the constants (because of the trivial representation) and separates points (because of the standard representation on

C n ).

By the Stone-Weierstrass Theorem, it is uniformly

dense in

C(G) .

convergence, it is dense in (b)

o

Since uniform convergence implies

L

L (G) .

The linear span of the functions in question is the

linear span considered in (a). Thus (a) and general Hilbert space theory imply (b). (c)

This will follow from (d).

(d)

By Zorn's Lemma, choose a maximal orthogonal set of

finite-dimensional irreducible invariant subspaces. be the closure of the sum. suppose

U

is not all of

invariant sub space. If then

h

h

Fix

Let

U

Arguing by contradiction, we V.

Then

v j4 0

in

IT1

is a nonzero closed

1

IT .

is a linear combination of matrix coefficients,

lies in a finite-dimensional subspace

S

of

L (G)

1. ABSTRACT THEORY

107

that is invariant under left translation. a basis of this space

S .

Then

g €G

Let

h 1 , ...,h

be

implies

J h(x)?(x)vdx = J h(x)§(gx)vdx G G 1

G

x)$(x)vdX .

= ? C . J h.(x)§(x)vdX, j=l J G J

and hence the finite-dimensional subspace invariant subspace for

$ .

Z . C§(h.) v

is an

Consequently we "will obtain a

contradiction if we show that

$(h)v ^ 0

for some linear

combination of matrix coefficients. To construct

h,

continuous function

we first form

> 0

with

f N (l) = 1

vanishes off an open neighborhood i (f«)v

as

N

is in

1

IT

shrinks to

for every

{1} .

f(fj.)v, where

N

N.

of

such that 1

in

G.

f~ f.,

Then

Let us see that

In fact,

- v = 1 HfNlli1fN(x)i(x)vdx - v G

= J i!fN|!-1fN(x)[f(X)v - V]dX = J l|fNH-1fN(x)[?(x)v - v] dx, N

""

and so

= sup ]| ? (x) v - vR . X€N

is a

108

I I I : REPRESENTATIONS OF COMPACT GROUPS

Since the r i g h t side tends t o

0

as

N shrinks t o

{1} ,

(3-3) follows. I t follows t h a t such an N .

i s

§ (^TJ) V

Now choose

h

no

"t

°

f o r

some

N

•

F i x

by (a) so t h a t

l l ^ - h ! ^ < !!f N -h!I 2 < *U(fN)vfl/jivH .

(3.5)

Then

lU(f N )v-$(hH| = Hi (fN-h)vH < llf^-hUJvH

by (3-3) and (3-5).

Hence

!U(h)v]i > H*(f N H -lli(f N )v-?(h)v|| > lN(f N )v!| > 0. Thus

h

has the required property.

This proves (d) and

also (c). (e)

Put

adjoint of

sj = dT$(xT) .

^

By (3-3) and Corollary 3-6, the

i s given by

and E ^ , = dTdT ,$ (xT * ^ T ,) = 0

Thus for

for T and T ' inequivalent

"El is an orthogonal projection, and T

T

and T

Let

U

EjjElI t = ^v T5J

=

0

inequivalent.

be an irreducible finite-dimensional subspace of

on which $ U i s equivalent with T , and l e t b e an orthononnal b a s i s of U . I f $. . (x) = ( $ ( x ) u . , u . ) > V

then

1-

ABSTRACT THEORY'

m. X^ (x) = Z $.. (x) T =i l x l

and

§(x)u. = J

109

m Z I =i l

Hence Schur orthogonality gives

1 , iv

Thus

KI is the identity on every irreducible subspace of

type

T . For

u

in a space of type

TT

with

T!

inequivalent, we have "Elu = E ! E' T u = 0 . EjI

and T

Now let us apply

to a decomposition as in (d). All terms are then

annihilated except the ones of type on spaces of type Consequently

T ' with

T!

E^T vanishes

T , since

not equivalent with

ELT = K. and v = Z

K,v for all v

r .

in V .

This completes the proof of the theorem. The proof of (e) in the Peter-Weyl Theorem contains information even in the case that

$

representation on L (G) •

is an irreducible unitary

If T

representation and u^ , ... , TI^ the space on which

T

is the right regular

is.an orthonormal basis of

operates, then the span of a row of

matrix coefficients (T(X)U.,U. ) , J

X

i

fixed and

i s an invariant subspace of orthogonality the different orthogonal.

L (G)

J

moving,

of type

spaces, as

i

1 < j < d ,

T.

—

—

T

By Schur

varies, are

In the decomposition of (d) of the theorem, as

made specific in (b), these

d

spaces are the only ones of

110

III: REPRESENTATIONS OF COMPACT GROUPS

type

T , because the proof of (e) shows that

E^.

annihilates

the others. Thus in the case of the right regular representap p tion on L (G) , image K. has dimension d , with T occurring

linearly independent times.

dT

The conclusion of (e) in the Peter-Weyl Theorem implies that the number of occurrences of

T

in a decomposition (d)

is independent of the decomposition.

The number is obtained

as the quotient

We write

(dim image IL)/d

•

this quantity, calling it the multiplicity of Corollary 3>8. tions of suppose

G T

Let

on spaces

$

V

and and

is irreducible.

T V

[§:T] T

in

for $.

be unitary representa, respectively, and

Then

[$:T] = dim Hom G (V ? ,V T ) = dim Ham G (V T ,V*) , where the subscripts

!t !T

indicated actions by

G.

Proof. member of E V$

refer to linear maps respecting the

By SchurTs Lemma and the Peter-Weyl Theorem, any Hom G (v $ ,V T )

annihilates

(^V$)x .

Ihus write

as the orthogonal sum of irreducible subspaces

(d) of the theorem. of the theorem. V

G

to

V

T

Each

V

Thus for each

is equivalent with V

the space of

is at least one-dimensional.

G

maps from

It is at most one-

Then it follows that

[5:T] = dim Hom G (V f ,V T ) . Taking adjoints, we obtain

, by

VT , by (e)

a dimensional by SchurTs Lemma.

V

2. IRREDUCIBLE REPRESENTATIONS OF SU(2)

111

dim Hom G (V ? ,V T ) = dim Hom G (V T , V 1 ) . The corollary follows.

2. Irreducible representations of SU(2) ¥e know from §1.8 that every finite-dimensional representation of

SU(2)

is smooth and hence leads to a

representation of

«u (2) .

Since

«u(2) e i Su(2) = SI (2,C) , we obtain a complex-linear representation of

SI(2,C) .

The

invariant subspaces for these representations correspond, and thus an irreducible representation of

SU(2)

leads to an

irreducible complex-linear representation of

«I(2,C) •

We

shall now classify the latter and see that they all come from representations of

SU(2) .

that the representations polynomials in

z, , z 2

$

As a consequence we shall see of

SU(2)

on holomorphic

homogeneous of degree

n

(Example 2

in §1.8) exhaust the irreducible representations of

SU(2) ,

apart from equivalence. We shall make repeated use of the basis si (2,C)

over

h=

Vo

{h , e , f}

of

f=

Vi

o) •

These elements satisfy the bracket relations [h,e]=-2e,

[h,f]=-2f,

[e,f]=h.

(3-6)

112

III: REPRESENTATIONS OF COMPACT GROUPS Theorem 3*9»

For each integer

m > 1 , there exists up

to equivalence a unique irreducible complex-linear representation

IT of 3 1 (2,C)

there is a "basis

on a space

{v , ... , v

7r(h)vi =

(2)

7r(e)v0 = 0

(3)

Tr(t)v± = v i + 1

(4)

7r(e)vi = i ( n - i + l)v jL _ 1

m.

In V

(n-2i)vi

with

Moreover the representation

vn+1 = 0 with

v_± = 0 .

IT can be realized as the

version of the representation

holomorphic polynomials in Remark.

of dimension

-,} such that ("with n = m - 1 )

(1)

differentiated

V

z^ , z^

$

on

homogeneous of degree

Property (1) gives the eigenvalues of

n.

ir(h) .

Notice that the smallest eigenvalue i s the negative of the largest.

Therefore the largest i s

Proof of uniqueness.

Let

irreducible representation of dim V = m . with

Let

v 4 0

7r(h) v = Xv .

)

0.

IT be a complex-linear «I(2,C)

on

V -with

be an eigenvector for

Then

7r(e)v ,

ir(e) v , . . .

ir(h) ,

say

are also

eigenvectors because 7r(h)7r(e)v = Tr(e)7r(h)v +7r([h, e] )v = Tr(e)Av 4-2ir(e)v = (?w-2)7r(e)v . Since

A , X+2 , ~k-A , ...

independent while nonzero. find

vQ

in V

with

are distinct, these eigenvectors are By finite-dimensionality we can

( \ redefined and)

2. IRREDUCIBLE REPRESENTATIONS OF SU(2) (a)

v

(b)

7r(h)v0 = Av Q

(c)

7r(e)v0 = 0.

0

^ 0

v i = Trff^v

Define

.

Then

7r(h)v^ = ("X-2i)v. , "by the same

argument as above, and so there is a minim"um integer 7T ( f )

n+

\

= 0.

Then

v , ... , v n

(1)

7r(h)vi = (A-2i)v ±

(2)

Tr(e)v0 = 0

(3)

ir(f)vi = v ± + 1

We claim

with

irreducibility.

with

are independent and

It is enough to show

is stable under

7r(e)

because of the

In fact, we show

ir{e)v± = i ( ^ - i + l)v i-]L

with

v^± = 0.

We proceed by induction for (4), the case (2).

n

vn+1 = 0.

V = span{ v Q ,..., v n } .

span{ vQ,..., v }

{h)

113

Assume (4) for case

i .

To handle

i = 0

i + 1,

being

we write

= ir(e)ir(t)v± = 7r([e,f])vi + 7r(f)7r(e)vi = 7r(h)vi + 7r(f)7r(e)vjL

and the induction is complete. To finish the proof of uniqueness, we show

X = n.

have Tr Tr(h) = Tr(7r(e)7r(f) -Tr(f)ir(e)) = 0. Thus

£ i = 0 (A - 2i) = 0 ,

and we find

~K = n .

We

114

III: REPRESENTATIONS OF COMPACT GROUPS Proof of existence.

of the representation z

l 'Z 2

homo

gene°us

Form the differentiated version cp

§

of>

degree

linear and has dimension

n.

Here

m = n +1 . 0 -t

so that

cp (h)

does have

n

V.

all

j.

is complex

d

1=0

e nt 2 n,

as an eigenvalue. cp

is reducible on

=u 0 E u l E • • • E u k = v

U. ^ u-?+i Moreover

and

^n

k > 1.

is

irreducible on

Since

dim U_. -,/U. < n , J+-L

uniqueness argument says on

U. j/U.

for any

eigenvalue

n

__ __n

Then we can find a chain of subspaces 0

such that

cp

Also

Arguing by contradiction, suppose its space

n

on holomorphic polynomials in

on

cp (h)

j .

V,

the

j —

does not have eigenvalue

Therefore

contradiction.

cpn(h)

n

does not have

We conclude

cp

is

irreducible. 3» Root space decomposition for U(n) For most of the remainder of the chapter, we shall work only with unitary groups. 3 0 = u (n)

Thus let

G = U(n) , and let

be its Lie algebra (the set of skew-Hermitian

matrices). working with

We can identify

5=3^

with

gl(n,c) .

In

U(n) , we shall exploit the many naturally

occurring copies of device for using the

SU(2)

that lie within T

SU(2) s

U(n) , and the

is the root space

3- ROOT SPACE DECOMPOSITION FOR U(n) decomposition of

115

g.

Let 0

Define a matrix elsewhere.

For each

= diagonal matrices in g

E- • to be

1

in the (i, j)

Define a linear functional

H

in

§ , ad H

consisting of members of

e.

place and

0

in the dual space

is diagonalized "by the basis of t>

and the

E^.

for

1*3.

g

We

have

So

Ej; - is a simultaneous eigenvector for all

eigenvalue

e. (H) -e.(H) •

eigenvalue is linear. functional on 1 * 39

are

ad H ,

In its dependence on

H,

with the

So the eigenvalue is a linear

§ , namely

called roots.

e. - e . .

The

(e i - e .) fs , for

The set of roots is denoted

A.

We have

9 = ^ e E CE,,, i?^i

which we can rewrite as 9 = ^ 0

E

3

,

(3.7)

116

III: REPRESENTATIONS OF COMPACT GROUPS

where Se

e

= {X€S

| (ad H)X = (e.-e.)(H)X

for all H e^} .

The decomposition (3*7) is called the root-space decomposition of

g with respect to

§ .

The bracket relations are easy, relative to (3-7)• and

jB are roots, we can compute

[E. .,E. , . r ] 3

ij

= Sa+n

[S >Hg] •{ =

0

c y

If a

and see that

j

if ct+jB is a root if cx+jB is not a root or 0

(3*8)

if a+jS = 0 .

In the last case, the exact formula is

All the roots are real on

$-. . We introduce an ordering

on the roots and certain other linear functionals on

§-_ .

The ordering will depend on the choice of an ordered "basis H. for

^

. For example, the elements E± = E±± , 1 < i < n,

form an ordered basis of

$_. . If

f

is in

ij* , we say f

is positive (relative to this ordered basis of $ is real on b_

irC

) if f

and if f (%) > 0

or

f(H1) = 0

and

£r

f(H1) = f(H2) = 0

or ... or

f^)

f(H2) > 0

= ... = fi^j)

and =

f(Hj) > 0 °

and

>

3- ROOT SPACE DECOMPOSITION FOR U(n) If

f

is not

0 "but is real on each

of

f

and -f is positive.

117

H. , then exactly one

The positive elements are closed

under addition and under multiplication by positive scalars. f > g or g < f if f - g is positive. The

We shall say

resulting ordering on the members of b*

is called the

lexicographic ordering relative to the ordered basis H

l '—

' ^n

o f

^TR*

W e d e n o 1 : e fe

y

A +

^he

se

"t °^ positive

roots. The lexicographic ordering obtained from the choice H

i " E ii '

1

^

i

£n '

vji11 b e c a l l e d

the

"standard

lexicographic ordering, " and the corresponding consists of all

e. - e.

with

A + , ^hich

i < J , will be called the

"standard system of positive roots." The trace form BQ(X,Y) = Tr(XY) is complex bilinear on 3 x 9 ,

(3-9)

and its restriction to ^ T O x b_ -IK

i s real-valued and positive definite.

JK

BQ has the invariance

properties BQ((ad X)Y,Z) =

-BQ(Y,(ad X)Z)

(3-10a)

and BQ(Ad(g)X,Ad(g)Y) = on

g

and G , respectively. B Q (E i j ,E J i ) = 1

BQ(X,Y)

(3.10b)

Note also that for all i and 3 .

(3.11)

118

III: REPRESENTATIONS OF COMPACT GROUPS 4. Roots and weights for U(n) Within

U(n) , let T be the diagonal subgroup.

This

subgroup is connected and is maximal abelian in u(n) ; it is often referred to as a maximal torus or a Cartan subgroup. Correspondingly

$

is often referred to as a Cartan

subalgebra of g Q . Let

$

be a representation of G = U(n)

dimensional complex vector space may assume that

$

is unitary.

smooth; let cp : g -> End-, V

t =0

cp .

If X

By Proposition 3*1 we

By Theorem 1.14,

§ is

be its differentiated version.

When we need to, we can extend extension

V.

on a finite-

cp to U(g) , calling the

is in g , then differentiation at

of the identity $(exp tx)$(exp tx)* = I ,

with

(• )*

denoting adjoint, leads to the identity cp(x) + cp(X)* = 0 .

Thus

cp (X)

is skew-Hermitian for all

In particular, each Hermitian.

Hence each

cp(H)

cp (H)

X

for H

in g .

in L

for H in ^

and is diagonable with real eigenvalues. any basis of t)_ , such as H- = E.. .

is skewis Hermitian

Let EL , .. . , H

These matrices commute

and thus the homomorphism property cp[X,Y] = cp(X)cp(Y) -cp(Y)cp(X) says that the cp(H.)

commute.

be

Therefore we can find a

4. ROOTS AND WEIGHTS FOR U(n) simultaneous eigenspace decomposition of V cp (H.) .

Since

119

under all the

cp is linear on $ , this decomposition is a

j

simultaneous decomposition for all of cp(^) , and each eigenvalue is linear.

A typical eigenvalue is

M H ) » H e§,

and the eigenspace is V^ : Vx = {v€V

| cp(H)v = A(H)v

for a l l H € ^ } .

These eigenvalues, which are c e r t a i n l i n e a r functionals on t h a t a r e r e a l on

b_ , a r e called the weights of the -

in.

representation

^

$

or cp , the spaces

V^ are called weight

spaces, and the members of V^ are called weight vectors for the weight

A.

There are only finitely many weights, and we

have an orthogonal direct sum weight-space decomposition V = Z vx . weights We give some examples below.

(3.12)

Of special interest will be

the highest weight; this is the weight that is largest in whatever lexicographic ordering we fix in §3*

To be concrete,

let us use the standard lexicographic ordering in the examples. Example 1.

For G = SU(2) , we can imitate the theory

for u(n), using ,Q = { ( ^ _%)} . Put e^J _° w )=™. If z

$

l 'Z 2

is the representation on holomorphic polynomials in homogeneous of degree

n , the general element of the

representation space is a

n z l + a n-l z l~ l 2 2 + " -

+a

0z2'

120

III: REPRESENTATIONS OF COMPACT GROUPS Cz^? , Cz?

The weight spaces are respective weights weight is

ne^ .

z~ , ..., Cz^

-ne-L , -(n-2)e 1 , ... , n e 1 .

with The highest

A special feature of this example is that

each weight space has dimension one. Example 2.

For

G = U(n) , let

on holomorphic polynomials in degree

N .

$

be the representation

z, , ... , z

homogeneous of

Each monomial

z "

with

J, +... + j

= N

is a weight vector, and

if

h 1 , ... , h

are imaginary.

- ( ^ h ^ . ••+J n e n ) • Example 3-

Hence the weight is

The highest weight is

For

G = U(n) , let

is only one weight, and it is Example 4.

For

representation on basis of

A C .

If

$ (x) = (det x ) k .

There

k(e,+.. .+e ) .

G = U(n) , let n

-Ne n .

u^ , —

C n , then the vectors

§

"be the usual

, u

is the standard

h. ROOTS ATTD WEIGHTS FOR U(n) u. A u- A ... A u. 1

X

1

with

form a basis of weight vectors of e. +e. +. . .+e. 1 x 1 X2 k

weights being

Example 5.

For

representation cp

acts by

Ad

ad,

and the roots

.

i- < ... . irt

This is clear.

A linear functional satisfying the

equivalent properties in the proposition is said to be • analytically integral. Corollary 3-12. then

A

If

A

in

satisfies the condition

^*

is analytically integral,

124

III: REPRESENTATIONS OF COMPACT GROUPS Proof.

Use (i) in the proposition, noting that

2/|a| 2 = l .

5. Theorem of the Highest Weight for U(n) We continue with §§3-4.

G = U(n)

A linear functional

said to be dominant if

A

the standard one and if whenever

on

^

2/|a|

the set of positive roots.

k. > k.

and with other notation as in that is real on

> 0

for every

a

$_

is A+ ,

in

If the lexicographic ordering is

A = Z k. e^ , this condition means

i < j.

Theorem 3*13 (Theorem of the Highest Weight). lexicographic ordering for

$* .

irreducible representations

I

Fix a

Apart from equivalence, the

of

G

stand in one-one

correspondence with the dominant, analytically integral linear functionals

A

on

^ , the correspondence being that

the highest weight (largest weight in the ordering) of The highest weight (a)

A

A

of

$^

A+

of

(c)

each

$^ .

and not on the particular

lexicographic ordering that yielded the weight space

is

has these properties:

depends only on

(b)

A

V^

for

A+ A

is one-dimensional

a

in

A + , annihilates the members

V^ , and the members of

V^

are the only vectors with

E , for

this property (d) with the

every weight of n

integers

5^

> 0.

is of the form

A -Z

£A+

n a

5-

THEOREM OF THE HIGHEST WEIGHT FOR U(n)

Proof of existence of the correspondence. given.

Let

$

We apply the c o n s t r u c t i o n s of §§3-^ and l e t

h i g h e s t weight.

Then

125 be

A be the

A i s a n a l y t i c a l l y i n t e g r a l , by

P r o p o s i t i o n 3*11* If

a

A+ ,

i s in

a weight.

E

Thus

a

P r o p o s i t i o n 3*10a. Since. f o r each of over

+

A ,

$

v /

S

a

then a n d

0

A+a v € V

A

exceeds imply

A and cannot be cp(E )v = 0 ,

This proves t h e f i r s t p a r t of

is irreducible, in

and l e t

C.

€

V.

Let

E^ , . • . , E^

so i s

cp .

Thus

p 1 , . . . , |5 be t h e b a s i s

by

(c). cp(U(g))v = V

be an enumeration H. = E . .

of

^

By t h e Birkhoff-Witt Theorem, the monomials

E

» • » TT

^1

form a basis of U(g) . monomials to some E 's give

0

XT r

*m

v

Let us apply

±

^m

cp of each of these

in the highest weight space

V^ .

The

r

(by the previous paragraph), the H s multiply

by constants, and the E *s push the weight down (by ~P Proposition 3-10a).

Consequently the only members of v\

that can be obtained by applying vectors

Cv .

Thus

cp of (3.16) to v

are the

V-^ is one-dimensional, and (b) is

proved. The effect of cp of (3-l6) applied to give a weight vector with weight m * - Z i,-/5,, J=l J J and these vectors span

v

in V^

is to

(3.17)

V . ' Thus the weights (3-17) are the

only weights of cp , and (d) follows.

Also (d) implies (a).

126

III: REPRESENTATIONS OF COMPACT GROUPS To prove the second half of (c), let

cp (E )v = 0

a € A+.

for all

*V\ , we may assume

v

has

V.

, and let a € A+,

all

v

0

component in v

V. .

Let

A-

be

has a nonzero component in

be the component. cp(^)v! c cvT .

and

satisfy

Substracting the component in

the largest weight such that T

v ^"VV

Then

Applying

cp(E )vT = 0 cp

for

of (3.16), we

see that V = 1 Cp(E

1 A ) ~P1

Every weight of vectors on the right is strictly lower than X , and we have a contradiction to the fact that

A

occurs as

a

be in

a weight. Finally we prove that +

A ,

and form

1

H , E , and

span a subalgebra of isomorphism carries subspace of

V

is stable under

7\ is dominant.

T

3 H*

E^

h .

cp(H )

For

These vectors

§1 (2,c) , and the v ^ 0

in

V^ , the

spanned by all

*I (2,C) , and the argument for (c) shows it cp(E! )^v .

is the same as the span of all !

as in (3-15)-

isomorphic to to

Let

On these vectors,

acts with eigenvalue

and the largest eigenvalue of 2/|a|

.

cp(HT)

is therefore

On the other hand, this subspace is a

representation space for

SU(2)

and splits as the direct sum

5-

THEOREM OF THE HIGHEST WEIGHT FOR U(n)

of i r r e d u c i b l e r e p r e s e n t a t i o n s of

SU(2) .

By t h e remark

f o l l o w i n g t h e statement of Theorem 3 . 9 , t h e l a r g e s t > 0.

is

Thus

2(X,a>/|a|

2

is

eigenvalue

> 0 , and A i s dominant.

Proof t h a t t h e correspondence i s one-one. "be i r r e d u c i b l e on V and Vr ,

§'

127

Let $ and

r e s p e c t i v e l y , both with

A , and l e t cp and cp! be t h e corresponding

h i g h e s t weight

r e p r e s e n t a t i o n s of

U(g) .

h i g h e s t weight v e c t o r s .

Let v

Form

$®$

and v ' T

be nonzero

on V®Vf .

We claim

that

s = (cpecp where itself.

Since

Ad(x)

cr

is invertible,

Conversely i f

a

such an

Namely l e t

"be

entry,

1 < i 0. Since the highest-weight correspondence is onto when the positive system is representation (A+)o.

$

(A+)o,

whose highest weight is

The weights of

W , and thus

A

$

Then

f

wA

wA

relative to

are closed under the operation of

is a weight of

highest weight relative to weight.

there exists an irreducible

A4" .

$.

We claim

In fact, let

A

is the

AT

be a

is a weight, and part (d) of the theorem

says wAT = wA -

Z n_a = wA -

Z n n w6 = w(A

w/3

Z n O B) .

w0

6. WEYL GROUP FOR U(n) Cancelling A

w , we see that

is the highest weight of

A1

A+ .

Then any member

is conjugate via

W

to a

in

dominant element. Proof. let

Let

A = Z K^e^

(A )

be the standard positive system, and

Q

be a member of are

that is real on

Then the

k^'s

W

is to permute the coefficients

to

A

real.

^*

The effect of applying a member of k^ .

Since

consists of all permutations, we can arrange for the end up nonincreasing, and the result will be For general +

(A )o

dominant.

A+ = w(A+)0.

A + , we start with Choose

$-..

w

A

+

(A ) Q

W k. Ts

to

dominant.

that may be assumed

by Proposition 3*1^ so that

We readily check that

wA

is

A+

dominant,

and the lemma follows. As an illustration of the use of this lemma, we prove the following proposition. Proposition 3*l6.

Fix a positive system

be an irreducible representation of A , and let

\± be any weight of

Proof. assuming that

G

$^ .

A+.

Let

$^

with highest weight Then

|JJL| < |A| .

By the lemma, there is no loss of generality in JI is dominant.

By Theorem 3-13d, we have

13^

III: REPRESENTATIONS OF COMPACT GROUPS H = A -

with all with

n

^> 0 .

E n a

(3.20)

Taking the inner product of both sides

\i , we obtain

the second inequality holding since

\i is dominant.

~h , we obtain

the inner product of both sides of (3-20) with

In |

fore

- I na < |A| 2 ,

- •• ^

the second inequa i . •

-''.;-, Lince

A

is dominant.

(i

j)

in the symmetric group leads

to a Weyl group element with a nice formula.

The

corresponding Weyl group element evidently sends

e.-e. •*•

J

There-

0 , wa > 0} + Jw Z {a | a > 0 , wa < 0}

= £ Z {wa | a > 0 , w a > 0 }

+£Z

{wa | a > 0 , wa< 0}

= £ Z £p | w " ^ > 0 , ]3 > 0} + i Z {Y I w - \ under

> 0,

Y

< 0}

p = wa , Y

138

I I I : REPRESENTATIONS OF COMPACT GROUPS = * 2 U I v~h > 0 , p > 0} - % Z {p | w " ^ < 0 , jB > 0} under

£ = -y •

S u b t r a c t i n g , we o b t a i n 6 - w6 = S {/3 | p > 0 ," w"1/} < 0} =

Z

jB .

jS€A + (w)

7* Analytic form of Borel-Weil Theorem for U(n) This section gives optional motivation for some of the algebraic constructions that occur later.

It assumes

knowledge of the definitions and elementary properties of holomorphic functions and complex manifolds.

Our objective

will be to give realizations of irreducible representations of U(n)

in terms of geometry and analysis.

shall work with

GL(n, C)

But for a while, we

in place of

U(n) ; GL(n, C) 2 complex manifold (as an open subset of C n ) , and

is a

multiplication and inversion are holomorphic. If we identify two nonzero members of

C

when one is a

complex scalar multiple of the other, then the resulting quotient space

CP11"

is called complex protective space and

is a compact complex manifold in a natural way. The group

G = GL(n, c)

acts transitively on

Cn-{o}

multiplication of matrices times column vectors, and the action is holomorphic. classes leading to

The action respects the equivalence

CP11"" , and hence

transitive holomorphic action on

31

G

CP " .

has a natural The isotropy

by

7.

subgroup of

ANALYTIC FORM OF BOREL-WEIL THEOREM.

G

at the class of

all g such that exactly

g f. •. 0 .1

for some

0 A n-1

This is a complex subgroup of

G

CP31"1

A , hence is

since the Lie algebra of i , and

Our group action of

gives us a one-one holomorphic map of

G/Q

G

on

G/Q

CP n ~

Fix an integer homomorphism

N ) 0.

X : Q -> v

G

thus

€P n ~ .

onto

as complex manifolds on "which

X

operates.

Then "we have a holomorphic

given by

(3.22)

gives a holomorphic action of

definition

One

Thus

= A -N

This

Q

becomes a

can verify that the inverse map is holomorphic. G/Q 2= CP11

consists of

1

is closed under multiplication by complex manifold.

in

0 1

139

q(z) = x(q)z .

Q

on

C

by the

We form the quotient space

G X Q C = (GX where

~

GX c .

is the equivalence relation

(gq,z) ^ (g,q(z)) on

The quotient space is a complex manifold, and

holomorphically by The space

g(x,z) = (gx,z)

GxQ c

for

geG,

xeG,

fibers holomorphically over

G

acts zeC.

G/Q. with

140

I I I : REPRESENTATIONS OF COMPACT GROUPS

p r o j e c t i o n map e : Gx c

given by

e ( g , z ) = gQ .

(Actually

the fibering i s a fiber bundle, but t h i s fact w i l l not e x p l i c i t l y concern us.

The bundle

Gx^ C -> G/Q

associated line bundle for

X .)

y : G/Q -> GX c ;

C°° maps such that

these are

i d e n t i t y on

G/Q .

between the

C°° sections

We consider

i s called the

C

sections

ey

i s the

Let us set up a one-one correspondence y

and the

C°° maps

cp : G -> C

such t h a t 9(gq) = X(q)"1cp(g) In fact, if we are given Y (gQ) =

y ,

for

then

y

q€Q,g€G.

(3.23)

must be of the form

(g>¥v ( g ) ) > and t h e image must b e i n t h e same

e q u i v a l e n c e c l a s s i f we r e p l a c e

g

by

gq with

q eQ.

U n r a v e l i n g m a t t e r s , we o b t a i n

Then cp

cp (g) = X(q)cp (gq) , and (3-23) f o l l o w s .

i s given with (3-23) h o l d i n g , we d e f i n e

and o b t a i n a The group

C°°

functions

cp by

y (gQ) =

(g,cp(g))

section.

G a c t s on t h e

(gy)(x) = y(g x) ,

Conversely i f

C°° s e c t i o n s

y

by

and i t a c t s compatibly on t h e above (gcp) (x) = cp (g" x) .

Examples. (1)

Let

P

— be a holomorphic polynomial \2n/ homogeneous of degree N , with N a s i n ( 3 . 2 2 ) , and define

7-

ANALYTIC FORM OF BOREL-WEIL THEOREM

C00 section.

Thus

cp satisfies (3*23) and yields a

Actually

this

cp is holomorphic, and the corresponding section is

therefore holomorphic. (2)

Let cp be as in (1), and let

C°° function.

Define

q^fg) = f(gQ)cp(g) .

satisfies (3*23) and yields a Proposition 3. 20, associated to X

f : G/Q -» C be any Then cp1

C°° section.

For the line bundle

G x Q C -> CP31"

in (3.22), the only holomorphic sections are

the ones that arise as in Example 1 from holomorphic polynomials homogeneous of degree Proof.

N.

Let cp : G -» C be the holomorphic map satisfying

(3.23) that arises from a given holomorphic section. We z

define a function

P r

z1 Z

l

on

n / and let

C n -{0}

/ z1

P |—

n/

is well defined, w e suppose also that 0 \

as follows.

J = cp (g) .

We find

To see this

?.\ IZA

g T jQ =•••!. 1/ \Z n /

Then

142

III: REPRESENTATIONS OF COMPACT GROUPS g 1 = gq

Writing

and applying (3.23), we see that

cp(gT) = cp (g) , so that

P

is "well defined.

Moreover

cp(g)

shows Uiat

P

definition of of

w

T

s

is homogeneous of degre . N . P

Since the

can "be accomplished using a whole open set

at a time, we see that

P

is holomorphic on

c n -{o}. The homogeneity condition implies that near

0.

Hence

homogeneity and

P

P

is bounded

extends to be holomorphic on

€n •

The

C°° behavior on the unit sphere force

|P(z)l < C| Z | N and more generally

I *£>(*) I < C a U I N - | a | for any multi-index

a

and all

z

in

(3.24)

Cn-{0} .

we see from (3.24) that the holomorphic function vanishes at infinity.

Hence it is

(convergent) Taylor expansion of of degree

> N

equal to

0 , and

0. P P

If

|a| > N ,

B^P

Therefore the

about

0

has all terms

is a polynomial.

This

proves the result. In terms of representation theory, Proposition 3.20 says that the natural representation of

GL(n, C)

on holomorphic

7. ANALYTIC FORM OF BOREL-WEIL THEOREM Cn

polynomials on

homogeneous of degree

N can be realized

as the space of holomorphic sections of the line bundle over CP11"

associated to the representation

x

of

Q given in

(3-23). We can, of course, r e s t r i c t from The representation of

U(n)

to

A = -Nen

(see

The concrete realization i s b u i l t from

which i s determined by i t s holomorphicity and i t s to

u(n) .

t h a t we are realizing concretely

i s irreducible and has highest weight Example 2 in §4).

GL(n,C)

U(n-l) x U(l) ;

on this subgroup,

representation with (highest) weight

X

X ,

restriction

is the irreducible

A = -Ne .

I t i s of interest to obtain the above geometric realization of the holomorphic polynomial representation of U(n)

we want to exploit is- that Thus l e t L

0

=

GL(n, C) .

without explicit reference to

Q

° G0 '

GQ = U(n)

t h e

C

°°

m a p

descends to a one-one map i s onto.

u (n) and G

0 "*

C°° map

The fact that

— gi (n,C) .

LQ = U(n-l) x u(l) . G//Q

Siven

b v

Since

g 0 •* S0Q

GQ/L0 "^ G/Q .

(The relevant fact here i s that

Actually this G = GQQ ,

which

follows from G=G 0 B Q , where

(3-25)

B o is the subgroup of the lower triangular group with

positive real numbers as diagonal entries.

Identity (3-25) is

a unique decomposition and is a group-theoretic formulation of the Gram-Schmidt orthogonalization process for

C n . See

Knapp [1986], §v.2.) One can show that the inverse of the map

G Q / L Q ~* G/Q is

144

III: REPRESENTATIONS OF COMPACT GROUPS G^/LQ — G/Q

smooth, so that acts.

as smooth manifolds on which

Go

We can use the isomorphism to transport the complex

structure from

G/Q

to

G Q /L Q .

check for holomorphicity on

(Shortly "we shall see how to

G 0 /L Q

without referring to

G/Q.) To think of on

X

in real terms, we start "by defining

L n = U(n-l) x U(l) .

We extend it holomorphically to

GL(n-l,C)x GL(1,C) , and (3-22) -will still hold. extend it to

Q

X

Finally we

so as to remain a homomorphism, and (3-22)

will still hold. With

X

defined only on

L n , we can form 0 But this is nothing new, since the relation

GA x c. 0 LQ

shows that we have a diffeomorphism

Q

0 L~ 0

that respects the action by GxnC Q

G .

can be transported to

G A x T C , and we seek a way to 0 LQ

detect holomorphicity of sections reference to Let

Q.

g

c omplexif ication of

I .

X

acts on

q

, but

C°°(G0)

C

q

g = 31 (n, C)

without

0

be the Lie algebras of

We can regard

c omplexif ication of

then

y : G Q /L Q -> G Q X L

GL(n,C) .

s 0 , $ , I Q , and

G , L n , and

The complex structure on

GQ ,

as the

is larger than the

As in §11.6, if

X

is in

g ,

as a left-invariant vector field:

7- ANALYTIC FORM OF BOREL-WEIL THEOREM

If

Z

with

is in X

and

3 , then Y

in

Z

acts, too; we write

g , and then

Z

145

Z = X +iY

acts by

Zf = Xf + i(Yf) •

(3.26)

Left-invariant vector fields on matter.

If

Z

is in

3 , then

C^iG)

when

ZF = XF + (iY)F

Z = X + iY

acts on

are quite another C^iG)

by

F(g exp t Z ) | t = 0 .

ZF(g) = ^ On

Z

C°°(G)

with

X

and

Y

(3-27) gQ,

in

but we do not necessarily have

we have

ZF = XF + i(YF).

In fact, this kind of equality is related to the holomorphicity of F €C°°(G)

F:

is holomorphic i(ZF) = (iZ) (F)

for all

Z€g . (3.28)

(To verify (3.28), we have only to realize that gives us a chart about the identity in

G

exp: 9 -» G

compatible with the

complex structure, and thus the right side of (3-28) says that F

satisfies the Cauchy-Riemann equaltions.) Once again we use the roots of

Let

A

+

g

with respect to

§.

denote the standard positive system.

Proposition 3-21.

Let

y : G Q /L 0 -» G Q x L

section defined on an open subset the inverse image of

U

C°° function with domain

in U

Y(g 0 L Q ) =

U

of

G Q , and let

C

be a

G Q / L Q , let cp : G Q -> C

C°° U

be

be the

such that

(g o ,cp(g Q ))

(3.29)

146

III: REPRESENTATIONS OF COMPACT GROUPS

and cpThen

y

* )= X(O"\>(gn) .

is a holomorphic :