Introduction to the Representation Theory of Compact and Locally Compact Groups

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M. James, Mathematical Institute, 24-29 St ...

Author: Alain Robert

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M. James, Mathematical Institute, 24-29 St Giles,Oxford 1. 4. 5. 8. 9.

10. 11. 12. 13.

15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

General cohomology theory and K-theory, F.HILTON Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British Combinatorial Conference 1973, T.P.McDONOUGH & V.C.MAVRON (eds.) An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P. BOURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillations, K.E.PETERSEN Pontryagin duality and the structure of locally compact Abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F. ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A.BELLER, R.JENSEN & P.WELCH Low-dimensional topology, R.BROWN & T.L.THICKSTUN (eds.)

49. Finite geometries and designs, P.CAMERON, J.W.P.HIRSCHFELD & D.R.HUGHES (eds.) 50. Commutator calculus and groups of homotopy classes, H.J.BAUES 51. Synthetic differential geometry, A.KOCK 52. Combinatorics, H.N.V.TEMPERLEY (ed.) 53. Singularity theory, V.I.ARNOLD 54. Markov processes and related problems of analysis, E.B.DYNKIN 55. Ordered permutation groups, A.M.W.GLASS 56. Journ&es arithm&tiques 1980, J.V.ARMITAGE (ed.) 57. Techniques of geometric topology, R.A.FENN 58. Singularities of smooth functions and maps, J.MARTINET 59. Applicable differential geometry, F.A.E.PIRANI & M.CRAMPIN 60. Integrable systems, S.P.NOVIKOV et al. 61. The core model, A.DODD 62. Economics for mathematicians, J.W.S.CASSELS 63. Continuous semigroups in Banach algebras, A.M.SINCLAIR 64. Basic concepts of enriched category theory, G.M.KELLY 65. Several complex variables and complex manifolds I, M.J.FIELD 66. Several complex variables and complex manifolds II, M.J.FIELD 67. Classification problems in ergodic theory, W.PARRY & S.TUNCEL 68. Complex algebraic surfaces, A.BEAUVILLE 69. Representation theory, I.M.GELFAND et. al. 70. Stochastic differential equations on manifolds, K.D.ELWORTHY 71. Groups - St Andrews 1981, C.M.CAMPBELL & E.F.ROBERTSON (eds.) 72. Commutative algebra: Durham 1981, R.Y.SHARP (ed.) 73. Riemann surfaces: a view toward several complex variables, A. T. HUCKLEBERRY

74. Symmetric designs: an algebraic approach, E.S.LANDER 75. New geometric splittings of classical knots (algebraic knots), L.SIEBENMANN & F.BONAHON 76. Linear differential operators, H.O.CORDES 77. Isolated singular points on complete intersections, E.J.N.LOOIJENGA 78. A primer on Riemann surfaces, A.F.BEARDON 79. Probability, statistics and analysis, J.F.C.KINGMAN & G.E.H.REUTER (eds.) 80. Introduction to the representation theory of compact and locally compact groups, A.ROBERT 81. Skew fields, P.K.DRAXL

London Mathematical Society Lecture Note Series.

Introduction to the Representation Theory of Compact and Locally Compact Groups ALAIN ROBERT Professor of Mathematics University of Neuchatel

CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

80

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521289757

© Cambridge University Press 1983

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1983 Re-issued in this digitally printed version 2008

A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 82-19730 ISBN 978-0-521-28975-7 paperback

CONTENTS

Foreword

Conventional notations and terminology

PART

I

:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

vii ix

REPRESENTATIONS OF COMPACT GROUPS

Compact groups and Haar measures

p. 3

Exercises

11

Representations, general constructions

13

E xeccii e s

20

A geometrical application

21

Exenai.ae4

28

Finite-dimensional representations of compact groups

29

(Peter-Weyl theorem)

38

Decomposition of the regular representation

40

Exehci u

51

Convolution, Plancherel formula & Fourier inversion

53

ExeAciz u

61

Characters and group algebras

63

ExeActis u

76

Induced representations and Frobenius-1Veil reciprocity

78

Ex"ci eis

89

Tannaka duality

90

Representations of the rotation group Exeeci6e,o

95 107

vi

PART

II

:

11.

12.

REPRESENTATIONS OF LOCALLY COMPACT GROUPS

Groups with few finite-dimensional representations

111

Exetcih es

116

Invariant measures on locally compact groups and homogeneous spaces

117

E xencus ez

126

13.

Continuity properties of representations

128

14.

Representations of G and of L'(G)

133

Exe,tci s e s

144

15.

Schur's lemma :

unbounded version

145

ExeAci6e4

150

Discrete series of locally compact groups

151

ExeAcii3 e4

162

The discrete series of S 12 (JR)

164

ExeAci es

171

18.

The principal series of S12OR)

172

19.

Decomposition along a commutative subgroup

179

Appendix: Note on Hilbertian integrals

185

Type I groups

187

Exe ccise4

193

Getting near an abstract Plancherel formula

194

Epilogue

201

References

202

Index

204

16.

17.

20.

21.

FOREWORD

These are notes from a graduate course given in Lausanne (Switzerland) during the winter term 1978-79 (Convention romande des enseignements de 3e cycle en mathematiques). This term was devoted to a self-contained approach to representation theory for locally compact groups, using only integral methods. The sole prerequisite was a basic familiarity with the theory for finite groups (e.g. as contained in the first chapter of Serre 1967). For didactic reasons, I spent the first half of the term discussing compact groups, trying to be more elementary and more complete in this part. In particular, I have given several proofs of the main results. For example, the Peter-Weyl theorem is proved first with the use of the Stone-Weierstrass approximation theorem (p.33) and then without it (p.36). The "finiteness theorem" (irreducible representations of compact groups are finite dimensional) is proved first for Banach (or barrelled) spaces ( (5.8)p.46 ), then in the general case (quasi-complete spaces) ( (7.9)p.69 ) and finally in a more elementary fashion for Hilbert spaces ( (8.5)p.81 )

.

Thus I hope that readers with various backgrounds

will be able to benefit from these notes. My way of introducing the subject has forced me to repeat some definitions in the second part where I gradually assume more from my reader (this is particularly so as far as measure theory is concerned). This part in no way claims to be complete and only has an introductory purpose. I consider that the existing books on the subject more than prove that complete treatises on the subject are heavy going...

Finally, I should mention that the representation theory for Lie groups - starting with the rotation group SO3(]R) and compact Lie

groups - was considered in subsequent graduate courses, but has not been included in these notes since differential methods in the representation

viii

theory of Lie groups are probably more readily available in recent texts.

My presentation of the subject has certainly been influenced by R. Godement whose courses introduced me to this field. I would like

to thank him here. I would also like to thank Sylvie Griener who read the manuscript and helped me to detect various inaccuracies, and my wife Ann for hints on language.

July 1982

Alain Robert Institut de Mathematiques Universite de Neuchatel

Chantemerle 20 CH-2000 NEUCHATEL 7 (Switzerland)

CONVENTIONAL NOTATIONS AND TERMINOLOGY

N =10,1,2,.. .} , 7l, Q ,

IR ,

E

fundamental numerical sets

N real quaternions (skew-field, din (H) = 4,IR-basis l,i,j,k) IFq finite field with q elements ( q = pf for some prime p ) p field of p-adic numbers, p D ap ring of p-adic integers

A" group of units in a ring A (k' = k - {0} if k is a field) IR+ multiplicative group of positive real numbers x > 0 (= neutral connected component of ]R')

¢ empty set, Card(X) = number of elements of X countable: finite or denumerable (equipotent to some part of N)

E Ci F injective (one-one into) map xA characteristic function of a subset A c X

:

= 1 on A, = 0 outside A

f1A restriction of a mapping to a subset A c X C(X,E) space of continuous maps f

:

X - E

C(X) = C(X,T), CR(X) = C(X,]R) continuous numerical functions on X Cc(X) subspace of C(X) consisting of functions with compact support Sup ess If : smallest M with IfI M nearly everywhere (on a measured space) f(x)

= f(x-1)

f*(s) = f(s1)

symmetric function on a group G on a unimodular group (cf. (14.2) p.133 in general)

scalar products are always linear in the second factor

a(x I y)

=

(x I ay)

=

(ax

y)

normal subgroup = invariant subgroup (= distinguished subgroup) commutative = abelian ( = Abelian' )

in sec. 20-21, separable group means locally compact group admitting a countable basis for the open sets (hence has a countable dense subset) In = idn = id.

tg, to

unit matrix (in dim n)

transpose of a matrix (A* = tA

,

g = tg

contragredient)

11n vector space of (complex) polynomials in z and degree < n

XG set of fixed points in a group action of G on X Hausdorff space = T2-space (= separated space)

(p.102)

PART

I

:

REPRESENTATIONS OF COMPACT GROUPS

3

1

COMPACT GROUPS AND HAAR MEASURES

Before starting representation theory, it is certainly appropriate to start with a review of examples of compact groups.

EXAMPLES Some matrix groups Let On(JR) denote the group of real n x n matrices which preserve

the standard quadratic form x1 + x2 + ... + xn . Elements of this group

are real n- n matrices g satisfying the relation g g = In (the columns of the matrix g must constitute an orthonormal basis of Iltn). This group is the orthogonal group :

it is a compact subgroup of the general linear group

Gln(]R) in n real variables. Since the relation

= 1 hence detg = ± 1

t

gg = In implies (det g) 2 =

and both cases occur in 0n(R), we see that this

group is not connected. Its index two subgroup SOn(R)

=

0n(JR) o Sln(R)

special orthogonal group

is known to be connected (Chevalley 1946, Dieudonne 1970 (16.11.7) p.68). The first non-trivial group in this series is the circle group

R/7L

(isomorphism)

s SO2 (1R)

.--.e2nit=a+iby b b) ( a 2 + b 2 = 1)

tmod2Z

.

The next one is the rotation group SO3(lR) (we shall study it in detail). For n >, 3, the (special) orthogonal groups SOn(]R) are not commutative.

We can use the complex field E (instead of ]R) and consider the

hermitian form zlz1 + z

+

z 2

2

"'

+ z z on cn, thus defining unitary n n

transformations g as complex n x n matrices satisfying g*g = 1n (recall that g* = tg). The unitary group Un(cE) is a compact connected (loc. cit.)

subgroup of GIn((E). The first such group is the circle group U1(E) (identified with the multiplicative group of complex numbers of modulus 1).

We shall see later that the quotient of U2(E) by its center is (isomorphic to) SO3(]R). Quite generally, the circle group U1((E) can be embedded

diagonally into Un(E), the image of this embedding

U1(0) " Un((E)

4

being the center of the unitary group Un((E). Thus the center of Un(a) is

connected (but the center of On(R) is finite). Imposing furthermore the determinant 1 condition, we define the special unitary groups SUn((t)

Un(¢) n Sln(()

=

.

(The condition g* g = In in Un(iI) implies Jdet

gl2

= 1 and all complex

numbers having modulus 1 are determinants of elements of Un((E). )

The center of SU(T) consists of the scalar matrices of determinant 1 it is a cyclic group of order n isomorphic to the subgroup of nth roots

of 1 (in f or Ul ((E) )

.

Similar considerations hold over the field 1H of real

quaternions (the involution q ), q being the quaternionic conjugation) with respect to the real bilinear form glgl + g2g2 + ... + gngn . But since this field ]H is not commutative, some care has to be taken with

respect to the representation of H-linear mappings from Hn into itself by quaternionic n x n matrices (one should say left ]H-linear mappings

to be quite precise). Thus one can construct compact connected groups

Un(H) 7 SUn(H), which are also called symplectic groups. The three series SOn(JR)

(n , 3), SUn((E)

(n > 2) and SUn(H)

(n , 1) are the classical groups. Together with five exceptional groups, they exhaust the list of compact connected "simple" groups (more precisely, their center is finite and the

quotient by their center is simple).

Some connected groups (not Lie groups) The simplest example of a compact connected group which is not a Lie group. is certainly the group G = (]R/7L) N (infinite product of

circle groups). This group is commutative and each neighbourhood of its

neutral element 0 contains a subgroup Gn

=

(03 x (]R/7l)[n,coE

(this follows immediately from the definition of the product topology). Thus G contains arbitrarily small subgroups. More generally, let (G i)1

be a family of (non-trivial) compact connected groups. The product G = TT Gi is also a compact and connected group. Since a topological

group is metrizable exactly when there is a countable fundamental system of neighbourhoods of its neutral element, we see that such products are

5

metrizable precisely when the family I is countable. For compact groups, the following properties are equivalent (cf. also (5.11) ) i) G is metrizable, ii) G has a countable basis for open sets.

They imply iii) G is separable (there is a countable family which is everywhere dense in G).

Totally discontinuous groups In any topological group G, the connected component of the

neutral element G° is a closed and normal subgroup. When G° = {e} only connected subsets of G are the points (and the empty set ! )

,

the

and

G is totally discontinuous. In general, G/G° will be totally discontinuous. A locally compact space which is totally discontinuous has the following property :

each point has a fundamental system of open and closed neigh-

bourhoods (Bourbaki 1971, TG II cor. of prop.6 p.32). The simplest example

of totally discontinuous compact group is the cantor group G

=

(2Z/27L)N

Its elements are the infinite sequences a = (an) with an = 0 or 1 (n C IN).

The topological space

underlying this group

is usually obtained by

removing successively from the unit interval I = [0,11, its middle third ]1/3,2/3[ , then removing from each remaining interval its (open) third, etc... (cf. picture below). The intersection of this decreasing family of compact sets is the Cantor set. The topology induced by the real line coincides with the product topology when elements of this set are represented in the dyadic system (as usual).

open closed subgroup defined by ao = a1 = 0 open closed subgroup defined by ao = 0

elements

a = (10 ...) elements a = (1 1 ...)

6

The Cantor group is commutative and is not the most interesting totally discontinuous group. More sophisticated examples of totally discontinuous groups occur naturally in two contexts.

p-adic groups. The topological ring of p-adic integers 7Lp can be defined either as

(here, p is any prime number) completion 2Z(P)

of the local ring I (p) C

(consisting of

fractions a/b with b not divisible by p), or as

inverse limit

7l/pnZZ

(with respect to the canonical

n homomorphisms

7L/pn+17L --* 27./pn7l of finite rings).

The second definition makes obvious the fact that 2zp is compact. The

additive group 7lp and the multiplicative group Zlp are compact commutative groups. The groups

Gln(7lp)

p-adic n xn matrices g with det g e 7Lp

are good examples of totally discontinuous compact groups (for n > 2, these groups are not commutative).

Galois groups. Let k be any field, ks denoting a separable closure of k (in some algebraic closure k of k). The group G

=

Gal(ks/k)

=

Autk(ks)

is a compact topological group with respect to the Krull topology. Recall that this topology on G is defined by taking for fundamental system of neighbourhoods of the neutral element of G (the identity of ks) the subgroups G' = Gal(ks/k') for all finite (separable) subextensions k' of k. These subgroups G' are both open and closed so that G is totally discontinuous. Moreover, with this topology, there is a Galois correspondence (inclusion reversing) closed subgroupst of G

H

=

Gal(ks/L)

(intermediate extensions between k and k'

<-->

L = Fixed field under H

Compact totally discontinuous groups are always pro-finite groups.

7

HAAR MEASURE The main analytical tool in the study of compact groups is the Haar integral. We shall construct this integral (or Radon measure)

for

continuous functions only, assuming that the reader is familiar with the extension procedure to Borel functions and sets (and eventually to measurable sets). However, very little of the abstract integration theory is needed : LP spaces can be defined abstractly as suitable completions of the space of continuous fonctions (l c p < o

, p = 1 and 2 being the

most important cases). Negligible sets can be defined as those sets N for each

which have the following property :

c > 0, there is a continuous

positive function f = fE which has a restriction to N, fN >,,l and integral

smaller than E. Here is the main statement. Theorem. Let G be a compact group. Then, there is a unique linear form

m

:

C(G)

--s

¢

( C(G) :

space of continuous maps G -- cE )

having the properties 1. m(f) > 0

for

f

(m is positive)

0

(m is normalized) , 3. m(sf) = m(f) for sf(g) = f(s-1g)

,

2. m(1) = 1

( s , g e: G)

(m is left invariant) Moreover, this linear form m is also right invariant 4. m(fs) = m(f) for fs(g) = f(gs)

(s , g c G)

The proof of this theorem can be based on the following two classical results.

a) (Ascoli's theorem) Let X be a compact topological space, E a Banach space and

a subset of CE(X) = C(X;E) (Banach

space of continuous mappings X --3 E with the uniform norm). Then 4i is relatively compact (i.e. has a compact closure in CE(X)

all sets

)

if and only if

is equicontinuous and

(x) = {f(x) : f E q51 (x E X) are relatively

compact in E. (cf. Dieudonne 1960, p.137 for the metric case, or Rudin 1973,

.)

b) (Kakutani's fixed point theorem) Let E be a Banach space (or any locally convex topological vector space), K a convex ;4

compact subset of E, and G a compact group acting linearly on E. If the action

,

:

G --6 Gl(E) leaves K invariant

8

i = A(G) is equicontinuous, then there

A(9)K C K (g (E G) and

is a fixed point of G in K. (cf. Rudin 1973,

.)

Here is a construction of the measure m. For f c C(G) (i.e. f is a continuous complex valued function on G) we denote by Cf the convex hull of all left translates sf (s E G) of f. Thus, the

elements of Cf are the functions which are finite sums of the form

L

finite

ai f(six) with

a > 0 and

i

Eai

=

1

Clearly if g c Cf

Maxlf(x)I =

= MaxIg(x)

ItgIf

G

IIf II

G

In particular, all sets Cf(x) _ g(x)

:

g e Cfj are bounded and relatively

compact in C. On the other hand, since G is compact, f is uniformly

continuous on G:

for each c > 0, there exists a neighbourhood V = V.

of

the neutral element e e G with the property

y lx E V

:

If (y) - f (x) < E

Since (s-Iy -ls-lx = Y-1 ss- ix = Y-1 x Isf(y) - sf(x)j < e

,

we shall also have

as soon as

y lx e V

Making convex combinations of these, we also infer Ig(y) - g(x)I < r-

as soon as

y lx e V

(all gE Cf).

This proves that the set Cf is uniformly equicontinuous. By Ascoli's theorem, we conclude that Cf is relatively compact in C(G). Let Kf denote the closure of Cf in C(G): this is a compact convex set. The

compact group G acts by left translations (isometrically) on C(G) and leaves Cf hence also Kf = Cf invariant. Kakutani's theorem asserts then that there is a fixed point of this action of G in Kf . Such a fixed

point is a constant function

sg

=

g

(s c G)

g(s-1)

= sg(e) = g(e) = c (s c G).

Such a constant has the property of being approximated by elements of Cf i.e. convex combinations of left translates of f

:

for each E>0, there are finitely many si e G and ai > 0 with Z ai = 1 and I c - E aif (six) ( < e (x e G)

(1) .

9

Let us show that there is only one constant function in Kf. For this, we start the same construction with right translations of f (we can apply the preceding construction to the opposite group G' of G, or the function fl(x) =

f(x-

)

...) ,

obtaining a relatively compact convex set Cf

with compact convex closure Kf containing a constant function c'. It will be enough to show c = c'

(all constants c in Kf must be equal to one

chosen constant c' of Kf , and conversely !). There is certainly a finite combination of right translates which is close to c'

I c' - Ebjf(xtj) I < r- (for some tj c G, bj

0 with

1).

Let us multiply this equality by ai and put x = si Ic'ai - Z aibjf(sitj)I <

(2).

. ai

Summing over i, we obtain

Icai - E aibjf(sitj) I< ELai i,j

=E

(3).

Operating symmetrically on (1) (multiplying by bj , putting x = tj and summing over j), we find

Cl I- 2 aibf(Sit j j)I <

i,j

Adding (or subtracting ! Since

F-

)

(4).

E

(3) and (4), we conclude Ic - c' < 2 c

can be taken arbitrarily small, we must have c = c' (uniqueness).

From now on, the constant c in Kf will be denoted by m(f)

:

it is the only

constant function which can be approximated arbitrarily close with convex combinations of left (or right) translates of f. The following properties are obvious

m(1 )

=

m(f) > 0

1

if

m(af) = am(f)

(Kf = Z1 j

if f = 1)

f >' 0 if a is any complex number (Kaf = aKf)

m(sf) = m(f) = m(fs)

(by uniqueness)

.

Our proof will be complete if we show that m is additive (hence linear). Let us take f , g c C(G) and start with (1) above with c = m(f). Put

moreover h(x)

=

7- aig(six)

Since h E Cg , we certainly have Ch C Cg whence Kh e Kg . But the set Kg contains only one constant :

m(h) = m(g)

. We can write

10

(m(h)

- 7 bjh(tx) I

<E

for finitely many suitable tj e G , bj > 0 and Z.bj = 1 . Using the defini-

tion of h and m(h) = m(g), we find

Im(g) - 2 aibjg(sitjx) I

< g

i,j

But multiplying (1) by bj ,

replacing x by tjx in it and summing over j

we find

i m(f) - E aibf(sitx) i,j

<e

(6)

Adding (5) and (6), we find

I m(f) + m(g)

- Fi,3 aib j (f + g) (sitjx)

2g

Hence the constant m(f) + m(g) is in Kf+g : but the only constant of this compact convex set is m(f + g) .

q.e.d.

This measure m on G is the (normalized) Haar measure of the compact group G. Instead of m(f), we shall often write f(x) dm(x) or even,more simply

ff(x) dx

J

fG

This notation will also be used for the (regular Borel) extended measure and integrable functions f E L1(G) = LI(G,m). It is countably additive. For example, the invariance under translations implies that points of G

have same measure, and since m(G) = 1, we infer

m(teI) > 0 : G finite group In this case, the (normalized) Haar measure of G is simply given by m((ej) m(f)

=

=

1/n

(n = Card G)

n L f(x)

,

(f in the group algebra of G)

In the opposite case, m(te)) = 0, all points have measure 0 and G cannot be countable (by countable additivity of m, m(N) = 0 for all countable sets in G :

countable sets of G are negligible).

The measure of a subset of G is (by definition) the measure (or integral) of the characteristic function of this set. For measurable sets (or more simply Borel sets), m(A) is at the same time the suprenum of the measures m(K) of the compact subsets of A, or the infimum of the measures m(U) of the open neighbourhoods U of A.

11

EXERCISES

1. Let G be a compact group and m its Haar measure. Show

that m : CG) -+ 1? is continuous (CG) is the Banach space with respect

to the sup norm ). Hint : use -//f// < f(x) < lift/ (f EC.R(G), x E G). 2. Take for G the Cantor group. What is the measure of the subgroup defined by a0 = 0 ? Same question for the subgroups defined by Deduce from the preceding observations an expression

ai = 0 for i < n .

for the Haar integral of a locally constant function. This gives a good "feeling" for the Haar integral of any continuous function (any such function is uniformly continuous hence uniformly approximable by locally constant functions and ex.l can be applied). Let now I = [0,1] be the unit interval and consider G as embedded (topologically and metrically) in I. Show that G is a Lebesgue negligible set in I. The measure m on I (with support G) is not absolutely continuous with respect to Lebesgue measure (m is singular with respect to Lebesgue measure). The function f on I defined by f(x) = m([0,x])

is increasing and continuous. It is continuously differentiable in I - G (remember that G is negligible) with f'(x) = 0 for x 6t G. However, f is

not constant in I ! (One should recall that if f is differentiable outside a countable subset A of I with f'(x) = 0 for all x 4C A, then

f is constant. One cannot replace the assumption A countable by A negligible.)

3. Let p be any prime number and consider the Haar measure m of the additive (compact) group 7l P

subgroups pna P

.

What is the measure of the

?

Show that the restriction of m to 76 " is (proportional to) P a Haar measure of this multiplicative group. Hint: observe that 1 + p7l P is a subgroup of index p-1 in the multiplicative group M " . Observe

P is a subgroup of index pn-1 of the multiplicative

then that 1 + pn72

p group 1 + p7L

P

(similarity with :

pna is a subgroup of index

pn-l

P

in the additive group p7Lp , and a subgroup of index pn in Mp ).

More precisely, show that the restriction of to 7dp

is the normalized Haar measure of this multiplicative group.

12

4. Show that the Haar measure of SO2(R) is given by (the

absolute value of) the differential form db/(2na) = -da/(2nb)

in terms of the parameters a and b of x = ( b a ( a2 +b

2

= 1

Show similarly that the Haar measure of U1(0) is given by (the absolute value of) the differential form (this is a real 1-form)

dz/(27tiz)

in term of the parameter

z E 2",

Izi = 1

.

(Remark. Although very few books insist on this point, a measure is canonically associated to the absolute value of a real exterior differential form (on an oriented manifold). For example, Lebesgue measure 2

dxdy in 1?

is the absolute value of the exterior 2-form dx A dy :

= IdxA dy/.

dxdy

The relation dy A dx

theorem : dxdy

=

= - dx A dy is thus compatible with Fubini's

dydx. )

5. Let m be the (normalized) Haar measure of a compact group G. Show that m(f) = m(f) (remember that f(x) = f(x-1) ) for f E C(G)

or f E L1(G). This equality is usually written

f f(x) dx

=f

G

f(x1) dx

G

v

(Observe that f y m(f) is a Haar measure on G and use the uniqueness part of the theorem on Haar measures.)

13

2

REPRESENTATIONS, GENERAL CONSTRUCTIONS

If E is a (complex) Banach space, we denote by Gl(E) the group of continuous isomorphisms of E onto itself (if u is linear, conti-

nuous and bijective, u 1 will automatically be continuous, but we can require more simply u and u-1 to be continuous... and we should do it if E were a more general locally convex topological vector space...). A representation 7[ of a compact group G in E is a homomor-

phism

it

:

G

--*

Gl (E)

for which all maps

G-) E

s

> 7c (s) v

(v GE)

are continuous. (I sincerely hope that the choice of Greek letter it

for

representations will not induce any confusion with the number n = 3.14... even

when both appear, e.g. for representations of rotation groups,

where angles have to be introduced !)

The space E = EX

in which the representation takes place

is called representation space of s. A representation -rt of a group G in a vector space E canonically defines an action (also denoted it)

it

:

G.E

E

(s,v) H X(s) v The definition requires that this action is separately continuous. The action is then automatically globally continuous (this implication holds quite generally for barrelled spaces, as is explained in

Bourbaki 1963 INT. Chap.VIII

§ 2

Prop.l p.130) .

One should require global continuity of the action in the definition for general locally convex topological vector spaces.

We say that the representation it is unitary when E = H is

a Hilbert space and each operator it(s) (s e G) is a unitary operator (thus each it(s) must be isometric and surjective). Thus it is unitary

14

when E = H is a Hilbert space and

,rc(s) *

=

Tt(s)-1

=

IC

(s-1)

(s e G)

The representation it of G in E is said to be irreducible when E and (0} are distinct and are the only two closed invariant subspaces under all operators it(s) (s E. G) (topological irreducibility).

Two representations

it and sc' of a same group G are called equivalent

when the two spaces over which they act are "G-isomorphic", i.e.

when there exists a (continuous) isomorphism A : E --> E' of their respective spaces with

A (TC(s)v)

(s G, vEE)

i'(s)Av

=

.

More generally, continuous linear operators A : E -s E' satisfying

all commutation relations A Tr(s), = n' (s) A ( s c G) are called intertwining operators or G-morphisms (from it. to

Tc') and their set

is a vector space denoted either by

or by

HomG(E,E')

Hom(it, it' )

The following two propositions are relatively elementary. The first one does not even use compactness of G whereas the second one uses the Haar measure of G for averaging over G (the reader who has some familiarity with the representation theory of finite groups will have no difficulty in recognizing how the algebraic proof has been adapted to our analytical context). (2.1) Proposition. Let it be a unitary representation of G in the

Hilbert space H. If H1 is an invariant subspace of H (with respect to

of H1

all operators i(s) , s E G) , then the orthogonal H2 = H1 in H is also invariant.

Proof. We have to show that if v a H is orthogonal to H1

,

then all

ic(s) v are also orthogonal to H1 (s e G). This is obvious since for

any x e H1 we can write (XI X(s)v)

=

(rz(s)*x v)

_

(m(s-1)x I v)

=

0

by assumption (ic(s-1)x also lies in H1

q.e.d.

(2.2) Proposition. Let it be a representation of a compact group G in a

Hilbert space H. Then there exists a positive definite hermitian form which is invariant under the G-action, and which defines the same topological structure on H.

15

Proof. Since the mappings s --o

s

--p

rt.(s)v are continuous, the mappings

(v,w c- H)

(TL(s)v I X(S)W)

will also be continuous (continuity of the scalar product in H x H). We can thus define

=f

-T (v,w)

(Tc(sjv t' it(s)w) ds G

using Haar integral. It is clear that T is hermitian and positive. Let us show that it is non-degenerate and defines the same topology on H. Since G is compact, it(G) is also compact in GL(H) (strong topology in this

space). In particular, Tt(G) is simply bounded whence uniformly bounded

(Banach-Steinhaus theorem: Rudin 1973 pp.43-44). Thus there exists a

positive constant M > 0 with

(s EG, v .H)

II7t(s)v1I < MNvll

.

We thus see that 11v11 = II Tt(s-1)TC(s) V II < MNtt(s) vii 5 M211vI1 and thus

M 111vIi , Iixt(s) v1I 5 MIIvJ Squaring and integrating over G, we find

m-2

2 11v 11

4 p(v,v) 6

M211v112

.,

Thus q(v,v) = 0 obviously implies lvll = 0 and v = 0. At the same time,

we see that (Q and

11

112 induce equivalent topologies (equivalent norms)

on H. Finally, invariance of T comes from invariance of the Haar measure

(f(T[(t)v,n(t)w) =

= $G (Tt(st)v 7t(st)w) ds =

f f (st) ds = f f (s) ds = j f (s) ds = G

G

t

This shows that it is p -unitary as desired.

(f (v,w) .

G

q.e.d.

From these propositions follows that any representation of a compact group in a Hilbert space is equivalent to a unitary one, and any finite dimensional representation (the dimension of a representation is of course the dimension of its representation space) is completely reducible (i.e. direct sum of irreducible ones).

16

When E is still finite dimensional, say E = fin, we know that

any hermitian form p can be put in diagonal form. Hence, the unitary

group U(p) with respect to p is conjugate to the standard unitary group Un((E) in Gln(U). In particular, the second proposition shows that any compact subgroup G of GLn(E) is contained in a conjugate of Un((E)

(apply that proposition to the identity representation G

Gln(cm) ).

We see thus that the conjugatesU(p) of Un(E) are the maximal compact One can show similarly that the orthogonal group

subgroups of Gln((E) .

0n(JR) is a maximal compact subgroup of Gln(JR), and all maximal compact

subgroups of Gln(R) are conjugate to this one.

GENERAL CONSTRUCTION OF REPRESENTATIONS The main interest of the notion of representation of (compact) groups comes from the fact that each such group has indeed some canonical representations. We shall even construct faithful representations of any compact group G (n (s) = idE = 1E only for s = e neutral element of G ). For 1 < p < oo let LP(G) = LP(G,m) denote the Banach space obtained by

completing the space C(G) normed with

Ilfllp

=

JG

If(s)I' ds

In particular L2(G) is a Hilbert space since its norm derives from the

scalar product

(f I g)

=

JG

f(s)g(s) ds

In any space of functions on G, we define left translations by (Q(s)f)(x)

= f(s-lx)

(if we do not want to identify elements of LP(G) with functions, or classes of functions, we can simply extend translations from C(G) to

LP(G) by continuity). Thus we have

I (s) Q (t)

=

(s, t E G)

C (st)

and we get homomorphisms

t : G -* Gl(E) with any E = LP(G)

,

,

s r- e (s)

1 < p < °°. Let us check that these homomorphisms

are continuous (in our representation sense). For fixed 8 > 0 and f e LP(G),

we can find a continuous function h e C (G) with [If - h 11 < & (the norm is the p-norm of LP(G) ). For s and t in G, we have

17

Q(s)f - t(t)f II < 11t (s) (f - h) II

+

IIt(s)h - £(t)h II

+

IIt(t) (h - f) II

But, by invariance of the Haar measure, all operators e(s) are isometric

and thus IIQ(s)(f - h) II

IIt(t)(f - h)II

=

If - h II < E

=

.

On the other hand, since h is continuous and G is compact, h is uniformly

continuous and

Q(t)h - .(s)h

uniformly when t -+ s

A fortiori, L(t)h -- t (s)h in the norm of LP(G) and we can find a neighbourhood Vs of s such that

tEVs ==+ lI$(s)h-Q(t)hii< E We shall then have II G(s)f - 2(t)f II < 3 E . This proves that all mappings G

--+

LP (G)

t(s)f

s

,

are continuous and the homomorphisms

t

( f E LP (G) )

are representations. These are

the left regular representations of G. The right regular representations of G in the Banach spaces LP(G) are defined completely similarly with

(r(s)f) (x)

( f E LP(G)) .

f(xs)

=

With this definition of right translate, one has indeed r(st) = r(s)r(t). One can also consider the biregular representations

R, x r of G x G in

LP(G)

R x r (s,t) (f) } (x)

=

defined by

and its restriction to the diagonal

f (s-lxt)

(f E Lp(G) )

G -+ G h G , s H (s,s)

which is the adjoint representation of G. It is defined by

{Ad(s) f } (x)

=

f (s-1xs)

( f c LP(G) )

.

The regular representations are faithful (for any couple of distinct points s,t in G ,

there is a continuous function f with

f(s) = 0 and f(t) # 0) , but the adjoint representation can be trivial (this is the case when G is commutative !).

18

There are other general examples of representations. They are obtained by some functorial constructions. We have already alluded to the direct sum of representations (when we mentioned complete reducibility of finite dimensional representations of compact groups). Here are some examples of these constructions.

Let

TL :

G -- Gl(E) and

Tt'

G' -- Gl (E') be two represen-

:

tations. We can define the external direct sum representation of G x G'

in E $ E' by

rd? Tt'

(s,s')

=

it(s) a 7t(s')

(s r= G, s' E G')

.

When G = G', we can restrict this external direct sum to the diagonal G of G x G ,

obtaining the (usual) direct sum of it and it'

G -s Gl (E ®E' )

it tB Td

s

'--301

it(s) ® re (s)

One can also define the external tensor product r ®7t' as a representation

of G x G' in E x E' (let us assume that one of the two spaces E , is finite dimensional ,

E'

so that this algebraic tensor product is complete :

in general, some completion has to be devised). It is defined by M ® TC

(s,s')

(s 6 G, s' E G')

= 7t(s) ® rc! (s')

.

The (usual) tensor product of two representations of the same group G is again the restriction to the diagonal of the external tensor product (G = G') and is simply given by TC ®T!!

(s,t)

( s, t E G )

='it(s) 07C(t)

This is a representation of G acting in E ®E'.

7t'

= TC , one can define TCOD 7I

=

.

In particular, taking

rot and by induction, all (tensor)

powers

Tton

= 7C® Ti ® ... 0-11

(n terms)

.

These powers can be restricted to symmetric (resp. anti-symmetric)

tensors. We thus obtain definitions of the symmetric powers s7t

(resp. anti-symmetric powers /fin 7C) of 7C . When it: C - * Gl(E) is a given representation of G, we can also define the contragredient representation

c of it. This representation

acts in the dual E' (Banach space of continuous linear forms on E) of E,

19

and is defined by

tr (s-1)

=

n(s)

(s EG)

(since transposition reverses the order of composition of mappings t(A B) = tB to , it is necessary to reverse once more the operations by taking the inverse in the group :

in this way ic(st) = *(s)n(t) as is

required for a representation !) . When E = H is a Hilbert space, we can define the conjugate it of it as a representation acting in the conjugate

H of H. Recall that this space H has same underlying additive group as H, but the scalar multiplication in H is twisted by complex conjugation. More precisely, the external operation of scalars in H is given by (a,v)

a dot in H).

1

The scalar product (. .)

(v I w)

_

of H is defined by

(v i w)

(= (w i v) )

This suggests that an element v of H is written v when we consider it as element of the dual Hilbert space H. With such a notation, we have

av

=

(a e (E) and (v w)-

The identity map H --* H ,

_

(v I w)

v b-+ v is an anti-isomorphism. The conjugate

of it is simply defined by E(s)

= tr(s) in H .

Since the (complex vector)

subspaces of H and H are the same by definition, it and it are reducible or irreducible simultaneously. However, it is important to distinguish these two representations (in particular, they are not always equivalent).

Any orthonormal basis (ei) of H is also an orthonormal basis of R, but a decomposition v

V

= L

= E viei in H gives rise to the decomposition (complex conjugate components) in H .

Thus the matrix representations associated with 7r and n in the bases (ei)

(=

(ei) ) of H (resp. H) are complex conjugate to one another.

When it is unitary, the contragredient it and the conjugate r of it are equivalent (ex.2 below). There is another general procedure for constructing representations of G :

it is induction starting from a representation of a

(closed) subgroup of G. We shall give its definition in sec.8 below

20

EXERCISES

1. Show that the left and right representations f and r of a group G (in any LP(G) space) are equivalent.

(Hint: consider the mapping f '-o f defined on any space of functions on G where f(x) = f(x-1).)

2. Let it be a unitary representation of a group G. Show that

7r and is are equivalent.

(Hint : Riesz' theorem asserts that H --) H' , v .'---s (v f . ) is a complex isomorphism.) 3. If it and

rr' are two representations of the same group G

(acting in respective Hilbert spaces H and H'), show that the matrix coefficients of rr ® rC'

and (e. 0

(with respect to bases (e.) of H, (e'.) of H' 3

of H 0 H' ) are products of matrix coefficients of it and rr'

(Kronecker product of matrices).

4. Let 1n denote the identity representation of a group G in dimension n ( the space of this identity representation is thus an and In (s) = idan for every s E G) . Show that for any representation

7C of G, 7C ® 1n

is equivalent to rr® rz ® ... ®7r (n terms)

.

5. (Schur's lemma) Let k be an algebraically closed field, V a finite dimensional vector space over k and

y5 any irreducible

family of operators in V (the only invariant subspaces, relatively to all operators belonging to P , are £Oj and V) . Then, if an operator A commutes

with all operators of

45 , A is a multiple of the identity operator (i.e.

A is a scalar operator). Hint: take an eigenvalue a (in the algebraically closed field k) of A and consider A - al, which still commutes with all

operators of

C3 . Show that Ker(A - al)

(

(01 ) is an invariant subspace.

21

A GEOMETRICAL APPLICATION

3

In this section, we shall answer the following

uec

stion :

Which are the bounded symmetrical star-shaped (with respect to the origin) bodies in]R3 having all plane sections through the origin of

equal area? Obviously spheres (with center at the origin) have the required property. We shall show that there are no others... Let K be any bounded star-shaped (with respect to the origin) subset of n3. We shall assume that K is closed (hence compact) and describe this body by the function on the unit sphere

f(r)

Sup{. >0: Xr 6K I

=

(117", I= 1).

Since we want to evaluate areas, we assume that f is measurable hence integrable (it is true that this follows from the fact that K is closed, but our object is not to insist on these measure theoretic questions...). Areas can be evaluated in polar coordinates according to the well known formula

f 2n

P(p) 2

0

dr

In our case, if Urll = 1

i.e. I on the unit sphere S2 c Il23, we denote by

,

Af(r) the area of the planar section (of K) orthogonal to r. Using the

circle

r

= { sY

:

Af (r)

sY 1 r , =

Us U = 1] c S2, we can express this area by

zf

f (1) 2 ds

(ds = U dI U ).

Thus our assumption on K means that Af is a constant function (and we must show that f is constant when it is even,i.e. K is symmetrical). Let us study more systematically the operator

J

:

f -. Jf

,

Jf (r)

_ pp

f (s) ds

pT We shall show that J is one to one on even functions (in other words, the kernel of J consists precisely of odd functions). As a consequence, the

22

only functions f such that if = constant, are the constant functions f (describing spheres).

Even more precisely, we shall determine all eigenvalues of the operator J in L2(S2), or in L2

even

L2(S2/±l)

(S2). =

=

L2(IP2)

Thus1P2 represents the real projective plane. Technically, we need a first lemma.

(3.1) Lemma. Assume that the operator J is given on continuous functions f by the above integral formula. Then J is continuous in quadratic mean L2(1P2),

and extends continuously to the space L2(IP2). Moreover, if f e_

the integral expression of J is still valid for nearly all r Proof. Let el be the first element of the canonical basis of1R3 and 11 denote the orthogonal circle to el (on S2). Take an arbitrary rotation

= r, so that T = g 'l. We can make the isometric

g e S03(R) and put g el

change of variable (transformation with Jacobian 1) g1 = s in the integral for Jf(r)

if (r')

f

_

ds

(f continuous)

f (g s?1) dsl

=

yr'

1

Thus

<

Iif (T)I

f,

If(g

11)1 ds1

and using the Cauchy-Schwarz inequality in L2(y1) for the scalar product of the two functions

constant 1

and 1 H f (g sl)

IJf(gel)12

=

we conclude

IJf(r)I2

if (g ,1)12ds1

1I1U2

1 11 Integrating moreover over g c SO3(JR), using the invariant measure coming from the spherical measure (we onlly(f use the formula

G

F(g rl) dg

4nJJS2 F(r) dQ

(G = SO3(R) )

for the normalized measure of the rotation group... the Haar measure of this rotation group G will be derived in sec. 9 below) we obtain

((Jf((S2 2 (we have used 11111y1

=

41C f (Jf(ge1)(Zdg

47C2n fdsl fGdgF If (g l)IZ 1

27C and Fubini's theorem for an integrable function!).

23

As the value

4n fG dgIf(gsl)J2

=

IIf12

is independent of sl , we conclude

(2R)2 Ufu2

I1Jf112 <

Hence J can indeed be continuously extended from C(S2) to L2 (S2) with

IJ ll C 2n (in fact, IIJ II = 2it since J(1) = 27t )

q.e.d.

.

To find invariant subspaces of J in L2(S2) or in L2OP 2)

we observe that J commutes to rotations (hence is a G-morphism). We define indeed rotations of functions by the usual formula for left translations, namely

(g) (f)

gf (g r) = f (r') or

gf

=

gf ('r)

= f (g r)

Then

if (r)

=f

f (s) ds

=

f yr

ur

g

f (g s) ds

is computed using the isometric change of variable t = g -s

if (r)

e gy =

gT

gf (t) dt = J(gf) (g r)

=

g 1-r But the left term is

if (r)

=

g(Jf) (g r)

J(gf)

=

g(Jf)

so that

(g EG=SO3(]R)).

,

This is the intertwining property and it is certainly appropriate to decompose the left regular representation of G in L2(]P2). For an integer n e 1N, define the subspace

Mn

= { p l

2

:

p is a homogeneous polynomial of degree n }

S

of the space of continuous functions over the sphere S2. Since the restriction of the homogeneous polynomial x2 + y2 +z

2

of degree 2 on S2

is identically 1, we see that Pn c Mn+2 (n e 1N). It is also clear that

even n's give spaces of even functions and thus subspaces of L2(1P2). Let us define

Vn = orthogonal supplement of Dn-2 in Mn ( c L2P2)) .

24

We shall then have (for n even)

n® n-2 6 n-4

= Vn ® n-2

Mn

(Vo = (E) .

Vn O n-2 ® ... ®V2 O V0 It is also clear that

U M n even '0 n

®

=

(orthogonal sum)

V

n even >,0

n

is a subalgebra of the algebra of continuous functions on]P2 (algebra of continuous, even functions on S2). This subalgebra contains the constants, and is stable under complex conjugation. Moreover, since the restrictions to S2 of

x2,y2, z2,xy,yz, zx already separate

(in M2)

points of the compact space P2 = S2/±l, the subalgebra

of even polynomial functions is dense in C(]P2) (Stone-Weierstrass theorem). In particular,

© n is dense in

L2(JP2)

n even >,0

and the completion of this orthogonal direct sum (Hilbert sum) gives

a decomposition of L2(]P2) L22

ven(S2)

L2(IP2)

=

n even > 0

n

Each n is obviously finite dimensional and G-invariant (with respect to the left regular representation) :the same must hold for Vn (cf.(2.1)).

(3.2) Lemma. The dimension of n is 2n + 1

(n E AI)

Proof. The homogeneous monomials n x, xn-l

xn-lz

y,

,

xn-2y2, xn-2yz xn-2z2

yn , yn-1 z , ... constitute a basis of M n

.

, z

n

The number of elements in this basis is

1 + 2 + ... + (n+l) = 21(n+l)(n+2)

Consequently, dimVn = dimMn - dimMl_2 = 2 (n+l) (n+2) - 2'n(n-1) = 2n + 1

q.e.d.

25

We intend to prove that the n's (even n's) are irreducible G-modules (i.e. the left regular representation is irreducible in each such subspace of L2(TP2)) . To see this, let us consider the unit sphere

S2

as G (= SO3 (]R)) -orbit of el (for the canonical action in IR3) .

The

stabilizer of this vector 1 for this action is the closed (hence compact) subgroup K = SO2(1R) (operating in the orthogonal plane to e1). We make identifications

G --b S2

g gK

G/K

g

e1

L2 (S2) 2IN

L(G/K)

(3.3) Lemma. Any non-zero G-invariant subspace V of L2(G/K) has some non-zero K-fixed vectors VK

={f

:

:

f (k) f = f (k r- K) }

#{0} .

Proof. The K-fixed vectors f E L2(G/K) correspond to functions over the unit sphere S2 which are invariant under rotations around the These functions depend only on x (remember

i-axis.

y2 + z2 = 1 - x2 on S

).

Take any 0 # f E :V and replace (if necessary) f by a suitable gf in order to have f(el)

0. Define fK

.

VK by averaging along circles parallel to

Y1

f dk = 1)

= f f (k r) dk

fK(r)

K

K

As all left translates kf (k E K) are still in the G-invariant subspace V, we see that fK E V. Moreover, fK(el) = f(e1) # 0 implies fK # 0 One checks without difficulty (using invariance of that

.

the Haar measure of K)

0 # fK E VK .

q.e.d.

Observe that the functions f on S2 which are invariant under

K-rotations are identified with functions over K\G/K. Thus

VK

=

V ( L2(K\G/K)

(3.4) Lemma. The Vn (n even >O) are irreducible subspaces of L2QP2). Proof. In any direct sum decomposition

Vn

= Vn 49 Vn

VK

=

(V' and V"

we must have

n

V, K ® V, K

n

n

G-invariant subspaces of Vn),

26

If we can show that dim Vn = 1, we shall necessarily have V,K = Vn'K =

VV =

0 .

0

By the preceding lemma (3.3) it would follow Vn =

0 or

0 or

whence the irreducibility. Thus we shall prove that Vn has

dimension 1. But elements of this space Vn are functions which depend only on x. In Dn , such polynomial functions must be linear combinations

of

xn,

xn-2 (y2 + z2) =

xn-2

(l - x2)

...

,

A basis for Mn is thus constituted of polynomials

xn , xn-2 , ... , x 2 Hence dim Mn = Zn + 1 (recall that n is even ! ) and also

dim VK

=

dim Dn

-

dim Mn_2

=

1

This completes the proof, but we observe that the formula dim Mn = Zn + 1

corresponds to a decomposition

rn

=

Vn ® Vn-z ® ... ® VZ ® Vn

(Vn = Vo = ¢ )

where each VK is a line (i.e. has dimension 1).

q.e.d.

From the analysis just made, it follows that the intertwining operator (or G-morphism) J must leave each Vn subspace (n even >, 0)

invariant and operate like a scalar in each of them (Schur's lemma, cf. ex. 5 of sec.2). We have to determine these constants (to show that

none of them is 0! Jn

)

= JI= An idsn

(n even > 0).

Vn

(3.5) Lemma. The scalar product of two functions fl, f2 which depend only on x on the unit sphere S2 (i.e. fi

(f11 f2)

= JJS 2

.

L2(K\G/K))

f1f2 dT

n =

is given by

1

2Ti0f f1(cos 9) f2(cos 9) sin 9 d8 = 27E lf fl(x) f2(x) dx

The proof of this lemma is obvious (exercise !).

From this lemma follows

that we obtain a sequence of orthogonal polynomials Pn E Vn (n even > 0 )

by orthogonalization of the sequence of even powers 1, x2,

... for the

usual scalar product on [-1,+1] ("isomorphic" to K\G/K up to the t sign). We know that we can take the Legendre polynomials (of even index). These

27

polynomials are given by the Rodrigue's formula (cf. Robert 1973)

Pnx) W

n d n(x2

=

n

n.

Successive integrations by parts show indeed that these nth derivatives

define a sequence of polynomials with deg(Pn) = n

Pn orthogonal to xm for m < n

and

.

On the other hand, Leibniz' formula for the nth derivative of a product can be used with

(x2

- 1)n

(x + 1)n(x - 1)n

=

and we thus see easily that Pn(1) = 1 (this is the usual normalization condition, justifying the introduction of the factor (2nn!) -1 in front

of the nth derivative). Thus we have Vn

=

(E Pn

(n even >,O)

and the computation of the eigenvalue an of the operator Jn = JIV

n

reduced to finding the value of J(Pn) at one particular point:

n

=

nPn(1)

=

is

J(P0)( 1)

But by definition of J, we have

J Pn( 1 ) = f

Pn(sl) dsl

xl

2n =

J0

Pn(0) dT =

2n Pn(0)

(we identify Pn to a function on S2 in the obvious way...) .

It

only remains to determine the value of Pn at the origin (this value is 0 for odd n's since Pn has the same parity as n, and only even n's are interesting for us). This value P (0) is the constant term of the n polynomial Pn , in the nth derivative; this term comes from the nth power of x in (x2 - 1)n .

of xn

The binomial formula gives this interesting coefficient

it is (-1)zn (ri/2

Summing up, we find that Pn(0) (n even >,O

is the product of (2nn!)-1 (normalization coefficient in Rodrigue's formula), n! (coming from the nth derivative of xn) and (-1)n/2(nn2)

(binomial coefficient of xn in (x2 - 1)n)

Finally,

n

(-1)n/2 2x (nn(n(2)3) ...4321 =

# 0

(n even >0)

)

28

EXERCISE

Generalize the results of this section for the canonical Sn-1 C 1Rn as follows. action of G = SO OR) on the unit sphere n

1) Denote by Me(n) the space of homogeneous polynomials

of degree e on B2n

.

Prove

dim M (n)

=

B

(e+n-1)! e!(n-1)!

(Here is G. B. Folland's explanation for this. The number of monomials

of degree

.e is the number of sequences (i1'...,in) of non-negative

integers with i1+...+in = e

. Line up e black balls in a row and

divide them into n groups of consecutive balls with cardinalities ill ... ,in

.

To mark the division between two adjacent groups, interpose

a white ball between two black balls :

for this purpose, we need n-l

white balls. The number of ways we can make such a configuration is the number of ways of taking e+n-1 black balls and choosing n-i of them to be painted white! ) 2) By restriction to the unit sphere, we write 2 (Sn-1) Me-2(n) + Ve(n) C L MZ(n) = with an orthogonal complement Vi(n) of dimension

dim V (n)

=

(2e+n-2)

(n+1-3)! e!(n-2)!

(One can show that this space Ve(n) consists of restrictions of the harmonic polynomials in Me(n).) 3) Prove that the natural representation of SOn(R) in VQ(n) is irreducible. (Embed K = SOn-l(it) in G = SOn(]R) by means of the last

n-1 coordinates and prove that the space of K-fixed vectors in VC (n) has dimension 1.) Conclude that the decomposition of L2(Sn-1) into

irreducible components is given by the Hilbert sum ® VZ(n) e3 a

4) Identifying K-invariant functions with functions of x = xl = cos a with e = angle between r"; and

1,

show that V4(n)K consists of

multiples of the Gegenbauer (or ultraspherical) polynomials

Con - 1

i

v

(Recall that the system of polynomials (C't ) is orthogonal with respect to 2

the density (1 -x )

on the interval [-1,1] .)

29

FINITE-DIMENSIONAL REPRESENTATIONS OF COMPACT

4

GROUPS (PETER-WEYL THEOREM)

In sec. 2, we have shown that all compact groups have faithful representations. For that purpose, the regular representations were constructed and examined. However, these last representations are infinite dimensional in general (i.e. when G is infinite). In particular, this does not prove the existence of irreducible representations

(different from the identity in dimension 1) for compact groups. This section is devoted mainly to the proof of the following basic result (and its consequences). Theorem (Peter-Weyl). Let G be a compact group. For any s

e in G, there

exists a finite dimensional, irreducible representation n of G such that

ir(s) # id.

.

Since certain compact groups have no faithful finite dimensional representations (groups with arbitrarily small subgroups are in this class when infinite), this result is the best possible. This theorem is sometimes stated in the following terms : all compact

groups have enough finite dimensional representations, or: all compact groups have a complete system of (irreducible) finite dimensional representations. As we have already seen that all finite dimensional representations of compact groups are completely reducible, the theorem will already be proved if we show that for s

e in G, there exists

a finite dimensional representation 7t with 7r(s) # id.

.

The proof of the preceding theorem is based on the spectral properties of compact hermitian operators in Hilbert spaces. Let us review the main points needed.

Lemma 1. Let G be a compact group , function and K

:

L2 (G)

(Kf) (x)

---

k

:

G x G -> E

be a continuous

C(G) be the operator with kernel k

k(x,y) f(y) dy

= fG

Then K is a compact operator, and if moreover k(y,x) = k(x,y) (identically on G - G ) K is hermitian as operator

L2 (G)

-) C(G) C

L2 (G)

.

30

Let us quickly recall the proof of this lemma. Denote by kx the functions y ---- k(x,y) on G. Thus we have

(Kf) (x)

=

fGkx(y) f(y) dy

=

(kx I f)

Since x H kx is a continuous mapping G

L2(G) (k is

C(G)

uniformly continuous on the compact space G x G) , we deduce that all functions Kf (f r-- L2(G)) are continuous :

Kf

.C(G). Moreover, the

Cauchy-Schwarz inequality gives

I Kf (x) I

<

I l k xll Of ii

<

Max I k I

hit

G>, G

In particular, when f remains in the unit ball Of If S 1

x is fixed, the numbers Kf (x)

of L2(G) and

remain in a bounded (i.e. relatively

compact) subset of CE . On the other hand, the set 4' of all Kf for Ilf 11 4 1 is equicontinuous :

IKf(s) - Kf(t)I < Ilks - ktill IlfII < Ilks - ktll is arbitrarily small for s sufficiently close to t (independently of f in the unit ball of L2(G)) . Ascoli's theorem (cf. sec.l) shows that the set of Kf for Ofll < 1, i.e. the image of the unit ball of L2(G) by

the operator K, is relatively compact in C(G) (with the uniform norm in this last space). A fortiori, the operator K is a compact operator of L2(G) into itself. Finally, a formal computation shows that the adjoint K* of K is also an integral operator, with kernel k*(x,y) _

= k(y,x)

.

Lemma 2. Let K be a compact operator (in some Hilbert space H). Then the spectrum S of K consists of eigenvalues and possibly of 0. Each eigenspace H'A

with respect to a non-zero eigenvalue A E S is finite-dimen-

sional. When moreover K is hermitian,

IIKII =

S C]R

Sup IAI AcS

and eigenspaces associated to distinct eigenvalues are orthogonal

H,LH for A#µ inS-{0} We shall not re-prove this classical theorem which can be found in any functional analysis book (cf. Rudin 1973 pp. 97, or Dieudonne 1960. Chap.XI where the more general case of Banach spaces is considered).

31

The finiteness of the dimension of the Ha (for A# 0) is however so easily shown that I recall the argument. Let B denote the (closed) unit ball of H and B. = B n H9 the (closed) unit ball of HA . Then, sinceK is compact, K(B,) must be relatively compact

in H, . But this

set

K(BA)

'X B A Ha

=

is the (closed) ball of radius

In particular, HA must be

ICI in H

it is finite dimensional.

a locally compact space :

We shall use lemma 2 above in the following form: then K has a non-zero eigenvalue A

If K # 0 is compact and hermitian,

and the corresponding eigenspace HA is finite-dimensional. We can now prove the main theorem (stated at the beginning of this sec.). Assume that s # e in G and take an open symmetric neighbourhood V = V-1 of e in G such that s ot V2 .

There exists a positive

continuous function f on G such that

f (e) > 0

V

f (x) = f (x -1 ) = f (x) , Supp (f) C V

,

(Supp(f) denotes the support of f :

it is the complement of the largest

open set in which f vanishes). Replacing if necessary f by f + f, we can assume that f = f . Let us examine the function

= f * f

defined by

T (x)

=

f

f (y) f (y lx) dy

G

Obviously, the support of this new function tP is contained in V2 and

p(s)

=

(s

0

A fortiori we see that t(s)c

V2) , y(e)

=

HfIII > 0

.

# . But the operator K with kernel

k(x,y) = f(y lx) is compact (lemma 1) and the convergence of

f

=

fo

+ F fi

fi e Ker(K- AiI) = Hi (A i e Spec(K) )

(in quadratic mean) implies that

= T Kfi = Dai fi

( fi = Kfi e Im(K) cC(G) ). 0 (with a uniform convergence). Since t(s) 9 l', we must have t(s)fi # fi for one index i (at least). But the definition of the kernel k

9=

Kf

shows that k(sx,sy)

=

k(x,y)

=

f(y 1 x)

(s,x and y in G).

32

The consequence of these identities is the translation invariance of all the eigenspaces Hi of K. The left regular representation restricted to a suitable finite dimensional subspace Hi (any i with t(s)f i # fi will do)

will furnish an example of a finite dimensional representation

It

q.e.d.

with it(s) # e .

The corollaries of the main theorem are numerous and important.

(4.1) Corollary. A compact group is commutative if and only if all its finite dimensional irreducible representations have dimension 1. Proof. Since Gl1((E) = G

is commutative, any dimension 1 representation

is trivial on the commutator subgroup [G,G] of G .

If all finite dimen-

sional irreducible representations of G have dimension 1, the theorem implies that all commutators must coincide with the neutral element of G and thus, G must be commutative. Conversely, assume that the (compact) group G is commutative, and take a finite dimensional irreducible representation it of G .

Since any n(s) commutes to all it(t)

(t c- G),

Schur's lemma (Ex.5 of sec.2) implies that rz(s) is a scalar operator

(for all s e G). Thus the whole action of G on H = H, is by dilatation: all subspaces of H are G-invariant. The irreducibility of n requires that H has dimension 1.

q.e.d.

(4.2) Corollary (Peter-Weyl). Any continuous function on a compact group is a uniform limit of (finite) linear combinations of coefficients of irreducible representations.

Proof. Let it. be a (finite dimensional) irreducible representation of the compact group G and take a basis in the representation space of it in order to be able to identify Tt:

G -+ Gln(C), the coefficients

of 1t being the continuous functions

ci

:

g

i--s c (g)

=

(ei I n(g)e

on G. In fact, more generally, if u and v F -H we can define the (function)

coefficient cu of it on G by

g - cu (g)

=

(u I Tt(g)v)

(these functions are obviously finite linear combinations of the

previously defined matrix coefficients ). Introduce the subspace of C(G) spanned by the c

, or equivalently by all cv

(u,v e Hz).

V(it)

33

Observe that the subspaces of C(G) attached in this way to two equivalent

representations n and

coincide: V(n) = V(ie). Thus we can form the

algebraic sum (a priori this algebraic sum is not a direct sum) =

AG

c C (G)

V,7C

where the summation index it runs over all (classes of) finite dimensional irreducible representations of G. The corollary can be restated in

the following form AG is a dense subspace of the Banach space C(G) (uniform norm).

But this algebraic sum AG is a subalgebra of C(G) (the product of two continuous functions being the usual pointwise product) :

the product

of the coefficients

of it

cu

and Yt of

o'

is a coefficient of the representation

u'

(the coefficient of

this representation with respect to the two vectors u ® s and v ® t)

Taking it and o

to be finite dimensional irreducible representations

of G , nor will be finite dimensional, hence completely reducible and all its coefficients (in particular the product of cv and fit) are

finite linear combinations of coefficients of (finite dimensional) irreducible representations of G. This subalgebra AG of C(G) contains the constants, is stable under complex conjugation (because 7t is

irreducible precisely when

is irreducible) and separates points of G

by the main theorem. The Stone-Weierstrass theorem furnishes the q.e.d.

conclusion.

Observe that on a compact group the Stone-Weierstrass theorem can be proved by convolution (regularization). Thus, the preceding proof can be made in a more elementary way. We give this alternative at the end of the section (after corollary (4.5)) . (4.3) Corollary. For each neighbourhood V of the neutral element in G, there exists a finite dimensional representation with kernel contained in V (this kernel is a closed invariant subgroup of G). Proof. We can assume that V is open, hence G -V is compact. For each x e G - V, let us construct a representation

X(x) # id.

.

By continuity of x ,

we

as in the main theorem :

x shall still

have X(y) # id.

34

for all y in a neighbourhood x of x. The interiors of the V('s make up

an open covering of the compact space G -V. We can select a finite sub-covering (Vi) = (V

) corresponding to a finite number of points xi. i

Let us denote by it Tri(y)

The direct sum

TC

the representation corresponding to xi id.

= m Tc

for all

y e Vi

.

of the Tci's will be such that Tt(y) # id.

i

for all y a UVi .

Hence the kernel of T[ is contained in the neighbourhood

V of e.

q.e.d.

The next corollary is due to J. von Neumann. It gives an.

answer to Hilbert's fifth problem in the case of compact groups (the general case has obtained a definitive answer by the joint work of Montgomery and Zippin). (4.4) Corollary. Let G be a compact group. The following conditions are equivalent.

i) There is a neighbourhood V of the neutral element e of G containing no closed invariant subgroup different from tel. ii) There is a faithful finite dimensional representation of G

and G is isomorphic to a closed subgroup of a unitary

group UnX) . iii) G is a real Lie group (with a finite number of connected components).

Proof. The implication i) = ii) follows immediately from (4.3) above. To see that ii) ==> iii)

it is enough to remember that a closed subgroup

of a Lie group is a Lie group, and to apply this result to the real Lie group Un((E) (also observe that since G is compact, any continuous

injective map G - Un(T)

is a homeomorphism into). In this case,

one could even see that G is an algebraic group. Finally, the implication

iii) - i) is known classically. In our case, we can use the exponential map Mn(]R)

-- Gln(R)

,

A *-* exp(A) = L An/n! n>0

This map is a local diffeomorphism in the neighbourhood of 0 E Mn(R): there exists a neighbourhood V of the zero matrix of Dn(]R) for which e x p : V - - + exp(V) (this is a neighbourhood of In in Gln(R) ).

As

exp(nA) = (exp A)n , a subgroup containing a non-trivial element s E exp(V)

35

contains all its powers, and these powers cannot stay in a prescribed neighbourhood W c exp(V) of In in Gln()R).

q.e.d.

(4.5) Corollary. Any compact group is an inverse limit of a family of compact Lie groups (hence a closed subgroup of a product of compact Lie groups).

Proof. Let us take a fundamental system (Vi) of neighbourhoods of the neutral element in G (we can take a countable such system if G is metrizable). For each i, choose a finite dimensional representation tci

of G with kernel contained in V.

.

Since inclusions V. c V. do not

necessarily lead to inclusions Ker(ni) c Ker(7cj) (and since the index

set can be very large...) we proceed as follows. Take as new index set J =

set of finite parts of the preceding index set I,and for any tie J, define

v

7r,,

iEL

=

I

1

Vi

1EL

With respect to inclusion (in I), the family (irt)J is an inverse system

of representations of G

: fort and 1! in J, there exists an index % > ti ti'

(take the union of i. and G/Ker(7Cti)

--+

t.'

in I) and transition homomorphisms G/Ker( K)

(canonical projection)

since Ker( 1) D Ker( z). Each G/Ker(rc,') can be identified to the

compact Lie group ir,,(G) (closed subgroup of some Ud ((E) z By construction, the continuous homomorphism G

--->

luim G/Ker(Tti)

=

1

i(G)

,

d., = dim ic,' ) .

(canonical)

is one-to-one onto. Since both groups are compact, it is also a topological homeomorphism.

q.e.d.

Since Card(J) = Card(I) in the above construction, we see that if G has a countable basis for open sets, we can take I (and also J) to be countable so that G can be embedded in a countable product of Lie groups. This shows that G is metrizable (this application was already alluded to in sec.l, p.5). One should also point out that each 7(.,,(G)

G/Ker(ry) can very well be a finite group (finite groups are Lie groups of a very particular type...) in which case G is a profinite group. The reader will certainly have understood that we can obtain many variations on this theme...

36

Let us come back now to a more elementary proof of corollary (4.2) above. Start with a continuous function f e C(G) and a positive

e > 0. Choose an open symmetric neighbourhood U of the neutral element of G such that

y1x EU

)f (Y)

- f(x)I < E

(f is uniformly continuous on the compact group G). Take then a continuous

function Y on G such that

'=4

0 ,

Supp (p) C U ,

,

dy =

SG

1

T (Y)

Thus we can write

f (x)

f

f (x)

=

y (y lx) dy

G

and coming back to the operator K

L2(G) - L2(G) with kernel k(x,y)

:

= 4 (ylx), we see that Kf(x) = fG f(y) T(ylx) dy

.

Thus

=f

Kf (x) - f (x) I

I f (Y)

(Y lx) dy

- f (x) I

<

G

< f e.

(y lx) dy

=

E

U On the other hand, the L2-expansion f

f0 +

=

(convergence in quadratic mean)

Z_ fi

with some continuous functions fi (i # 0) associated to the non-zero eigenvalues ) i of K , leads to a uniformly convergent expansion

Kf

=

7- Ai fi

(fo e Ker fcL2 (G) )

(recall that K is even a compact operator from L2(G) into C(G) ).

Taking a sufficiently large finite partial sum, we shall have II Kf -

L

i4 N

Summing up, we shall have

ifi 11f

-

E

aifiII < 2 E. It

(uniform norm). only remains to prove

itN that the functions f

are finite linear combinations of coefficients

of (finite dimensional) irreducible representations. This follows from the next lemma.

37

(4.6) Lemma. Let V c L2(G) be a finite dimensional subspace which is invariant under the left regular representation of G . Then V consists of (classes of) continuous functions and each f E V can be written

f (x) = Tr(A iF(x)) = Tr (A ic(x)) (here is

for some A E End (V)

denotes the restriction of the left regular representation to V).

Proof. Let us take an orthonormal basis ()(i) of V and introduce the

coefficients ci of it. They are defined by

1i(x) xi = Z ci (x) X j

(x e G).

J

If f

we have

7r(x) f= I al n (x) X i = 2 alci (x) hence

f (x)

=

(e(x 1) f) (e)

_ Z ci (x

ai

(with a

=

2(j

Since n(x 1) = 1t(x)*, we have ci (x-1) as claimed

= c (x)

and f (x)

=

Tr (A V(x) )

(cf. ex.2 of sec. 2, p.20).

The reader will check that if n denotes now the restriction of the right regular representation on a finite dimensional invariant

subspace V c L2(G), then any f e V can be written f(x)

=

Tr(A n(x))

(for some A E End (V) ).

For this reason, we shall often prefer to work with the right regular representation.

38

EXERCISES

1. Let G be a compact totally discontinuous group. Show that AG is the algebra of all locally constant functions on G. (Observe that a locally constant function on G is uniformly locally constant, hence can be identified with a function on a quotient G/H where H is some open subgroup of G. Conversely, any finite dimensional representation of G must be trivial on an open subgroup H of G.)

2. Let G be any compact group. Show that AG consists of the

continuous functions f on G for which the left and right translates of f generate a finite dimensional subspace of C(G). In particular, if G1 and G2 are two compact groups, any continuous homomorphism h :

G1 -p G2 has a transpose th : A2 -l A

(Ai = AG

1

)

defined

i

by

(a priori this transpose is a linear mapping

th(f) = t

C(G2) - C(G1)

h

3. Let G = U (¢') with its canonical representation n

in V = 2n

.

Since

7E

it is unitary, we can identify 9C with the contragre-

dient of 7Z: it acts in the dual V* of V. a) Let Ap denote the space of linear combinations of q coefficients of the representation

Ap q

=

7C®p

0 7Oq

(V*) Op

in

aS)

VOq = Tp(V) q

Prove that the sum of the subspaces Ap of C(G) is an algebra A

q

_

(show that Aq Ar C Aqs ) stable under conjugation (show that Aq = Ap s

which separates the points of G. Using the Stone-Weierstrass theorem, conclude that A is dense in C(G).

b) Show that A = AG (use part a to prove that any irreducible representation of G appears as subrepresentation of some >

other words, can be realized on a space of mixed tensors).

4. Let G be a closed subgroup of U (Q). Using the fact that n

any finite dimensional representation of G appears in the restriction of some finite dimensional representation of U

n

(¢')

(this is a consequence

39

of the theory of induced representations: cf. sec. 8), show that G is

a real algebraic subvariety of U (C). n

G C--'r

(The transpose of the embedding

n(¢') is the operation of restriction on polynomial functions

and is surjective. Hence AG is a quotient of the polynomial algebra

A of n(2). By the exercise 3, A is generated by the coordinate functions

Un(Q)

F-- xl - + ¢ , x = (xl) J 3

and their conjugates.)

40

5

DECOMPOSITION OF THE REGULAR REPRESENTATION

In this section, we establish a connection between finite dimensional representations of G and subrepresentations of L2(G). Eventual-

ly, we shall be able to prove that any irreducible representation of a compact group is finite dimensional. The possibility of constructing a link between any finite dimensional representation and a subrepresentation of L2(G) is suggested by lemma (4.6). To avoid complex conjugations, we work with the right regular representation. Let (ic,V) be any finite dimensional representation of the

compact group G. For any endomorphisms A E End(V), we define the corresponding coefficient cA of it by cA(x)

The right

=

translates of these coefficients are easily identified

(p(s) cA) (x)

=

cA(xs)

=

Tr(Att(x)x(s))

Tr(Bit(x))

=

=

=

(B =

cB(x)

Thus we consider the representation of G in End(V) defined by

Qn(s) (A)

=

(s e G , A E End (V)).

7c(s) °A

Then the above computation shows that A -- cA c

(intertwining

:

End(V) Q,R

is a G-morphism

-+ C(G) c L2(G)

and -it).

Let us study this morphism more systematically. By lemma (4.6)

we already know that its image contains all subrepresentations of (L2(G), ?) which are equivalent to it.

The situation is, of course, mach

simpler when it is irreducible, and we shall make this assumption from now on.

(5.1) Definition. Let it be a finite dimensional irreducible representation

of a compact group G. We denote by L2(G, n) the sum of all subspaces of the right regular representation which are equivalent to

it.

According to lemma (4.6), the subspace L2(G,it) is finite

dimensional (and of dimension

n2 if n = dim(V ) ) . A function f c L2(G)

41

belongs to this subspace L2(G,IC) precisely when the right translates of f generate an invariant subspace (of the right regular representation) equivalent to a finite multiple of n (a finite sum of subrepresentations equivalent to 7t). We say that L2(G,it) is the isotypical component of 1t

in L2(G). Obviously, this subspace L2(G,ic) only depends on the equivalence class of X. We shall now prove that the dimension of an isotypical component L2(G, 7C) is exactly (dim 7t)2 (this result is quite similar to

the finite groups case, but since L2(G) is infinite dimensional when G is infinite, we see that infinite compact groups should have infinitely many classes of finite dimensional irreducible representations... in order to exhaust L2(G)). There are two methods for doing this: the first one is based on a purely algebraic lemma, while the second is analytical. As they both have important applications, we present them in succession. (5.2) Burnside's theorem. Let G be any group and it any irreducible

representation of G in a finite dimensional vector space V over an

algebraically closed field k. Then a) all operators A E End(V) are (finite) linear combinations of the operators rc(s)

(s E G),

b) for an operator A E End(V), the relations Tr(A n(s)) = 0

(for all s c G) imply A =

0 .

Proof. Let us denote by U the enveloping algebra of the representation it, namely, the space generated by all n(s)

UL = u1 =

{ A E End(V)

=

Tr(B 7t(s) A)

Tr(BA) = 0 for all B E ti(,

:

E U

Since B E U implies B' =

(s e G) in End(V). Also define

(s e G), we have

Tr(B'A)

=

for A C

0

As above, let us consider the representation Q,n

:

Gl((R)

---p'

G

,

Qn(s) (A)

=

Evaluations at elements a e- V of

:

(L --

are thus G-morphisms

V,

A

i -9

Aa

:

a(Qn(s) (A))

= a (7t(s)-A) =

it(s) (Aa)

('2(s) A) (a) =

7c(s) (a (A))

_

42

Let us show that a, # 0 leads to a contradiction. In this case, we could

take an invariant irreducible subspace ao c (X

(since we are working

over a field of arbitrary characteristic, we cannot invoke complete reducibility, but we just take (L G-invariant, minimal among the nonzero subspaces of Ci ). Take then 0 # A E (10 and a E V with A a # 0 .

The corresponding evaluation operator a is not zero and is a G-morphism

from ((to, ) to (V, Tc) . This implies that of is an isomorphism ( Ker(o( ) and Im(o() must be invariant subspaces, and Ker(o() # V , 0 ). Let us take a k-basis (ei) of V with el = a and denote by all o(i the corresponding evaluation operators. Then the operators Im(oc)

ai

must commute to the irreducible G-action in ()(.o. By Schur's lemma,

they must be scalar operators

'k1=1). To be able to compute traces, we introduce the dual basis (ei) of (ei)

(it is defined by &i(ej) = ij = With our choices 0

0 if i # j and 1 if i = j).

0), we have

(e1 = a, A a

=

Tr( t (s) A)

=

7- (Ei, Aim(s) al(A)> =

Tt(s) ai (A)>

_ E<Ei, 7C (s) A e i >

_

(s EG).

But A a # 0 implies that the vectors Tt(s) A a generate V (this space is

irreducible under G!) and we infer

contradicting

'A1 = 1

i -0

'

0

1A i E i

q.e.d.

!

(S.3) Corollary. Let it be a finite dimensional irreducible representation

of G (in V). Then, the representation cx z of G ,' G in End CV) defined

by

Q-x t (s,t) (A)

=

is irreducible. Proof. Let W denote the G - G -module End(V) and take any G - G -morphism W

---s

W . We have 7L (g)A)

=

Tr(g) (A) hence 4P (Tc(g))

=

>z(g)

whence j5 (A)

=

A.

(I)

for all A 6 W

=

End(V)

(by the theorem). But we must also have Ji (7c (g))

= q5 (I)Tr(g)

and thus

43

the operator 4(I) commutes to all operators it(g) (g E G). Since irreducible, Schur's lemma again implies that

it is

k(I) = *A I is a scalar

operator. Consequently, as we have seen, t(A) = a A. In particular, there is no projector in W other than 0 and 1 which is a G x G -morphism :

there are no invariant subspaces in W other than 0 and W (# 0) itself.

The corollary is thus completely proved. The fact that the G-morphism c

:

End(V ) 0 L2(G,1t) A

i--*

( c C(G) c L2(G) )

cA

is injective (hence an isomorphism on the isotypical component of it) follows from Burnside's theorem (5.2.b). Alternatively, one can check that c is a G ' G -morphism : using the notations of the corollary

(=v,

p _ t) we indeed have

cA(sxt) =

=

Tr(A 7L (s-1) 7r (x) it(t))

Tr( it(t) Alt(s)1n(x))

=

_

c0-, r(t,s) (x)

.

Since c # 0 and End(V ) is irreducible under r x z , c must be an isomorphism into L2(G).

It is interesting to observe that Burnside's theorem admits a natural generalization for k = T and it unitary (in any Hilbert space). The density theorem of von Neumann asserts that if

7c is any unitary

(topologically) irreducible representation of any group G, then the set of operators it(g) (g s G) is dense in

e(.(H) = End cont(H) for the .

strong, or the weak topology on this set of operators. Let us turn now to the analytic study of the G-morphism c

A

:

End(V) -+ L2(G)

,

A -3-- cA = Tr(Ait). The fact that cA # 0 for

0 can be deduced from a computation of the quadratic norm of these

coefficient functions. It is easier to start with the case of rank < 1 linear mappings. To bring these special maps into the limelight,

we use the isomorphism

-

Vv ® V

(V , = dual of V)

End (V)

defined as follows. If u e Vv and v E V, the operator (corresponding to)

u0v is u0v

:

x r. u(x) v

= v

.

44

The image of u ® v consists of multiples of v and u ® v has rank 1 when u and v are non-zero (quite generally, decomposable tensors correspond to operators of rank 4 1). The coefficient cA with respect to the

operator A = u ® v coincides with the previously defined coefficient

cv

=

cu 0 v (x)

=

(cf. first exercise at the end of this section). (5.4) Fundamental lemma. Let it and o 'be two representations of a compact

group

G and A : V1 A4

o V,

be a linear mapping. Then

= f a-(g) A n(g) -1 dg G

is a G-morphism from V, Lo Vr :

HomG(Vr, Vim)

A4

The proof of this lemma is solely based on the invariance of the Haar measure :

A41c(s) =

f r(g) A lv(g)-ln(s) dg

=

f o'(g) A iC(s-lg)-1 dg

and replacing g by sg (i.e. s-1g by g) A4it(s)

=

J

o'(sg) A n(g)1 dg = a(s) A4

q.e.d.

.

Thus the averaging operation (given by the Haar integral) of

lemma (5.4) leads to a projector

Hom(Vn, Vr) -* HomG(V1, Vr )

4

,

A -- A4

(this point of view is expanded in the second exercise at the end of this section, where an alternative proof of (5.4) can be found). In particular, when it and o are disjoint, i.e. HomG(Vn, Vr) = 0, we must have A4 = 0 This is certainly the case if n and a- are non-equivalent irreducible

representations (Schur's lemma). Another case of special interest is it = r finite dimensional and irreducible. Schur's lemma gives

HomG(Vn,Vr) = t and thus A

= AA id.

is a scalar operator. It is not

difficult to give the value of AA as a function of A in this case. (5.5) Proposition. If it is a finite dimensional irreducible representation

of the compact group G in V, the projector End(V)

--*

EndG(V)

= a id.

,

A H A4 =

Aid.

45

is given explicitly by the following formula :

A4

=

JG

x (g) A 7t(g) -1 dg

Tr(A) idV dim

=

Proof. Since we know a priori that the operator A4 is a scalar operator RAid., we can determine the value of the scalar AA simply by taking traces in the defining equalities

AATr(idV)

Tr

=

JG Tr

...

=

dg

fG =

Tr (A)

q. e. d.

.

(5.6) Theorem (Schur's orthogonality relations). Let G be a compact group and it, a- two finite dimensional irreducible representations of G.

Assume it and o- unitary. Then a) if T and a- are non-equivalent, L2(G,7t) and L2(G,a') are orthogonal in L2(G)

,

b) if it and

a- are equivalent, L2(G, it) = L2(G,(') and the

scalar product of two coefficients of this space is given by

(cu I cy)

J G (u I i(g) v) (x I 7r-(g) Y) dg

=

c) more generally in the case TC= tr

,

= (u I x) (v I Y) /dim V ,

the scalar product of

general coefficients is given by

(cA I cB)

=

JG

Tr (Ait(g))Tr (B it(g)) dg

=

Tr (A* B) /dim V

Proof. a) follows from the fundamental lemma (5.4) and b) follows similarly from the proposition (5.5). It will be enough to show how b) is derived. For this purpose, we consider the particular operators V ® y

(V E Va = V ,

y E V) and apply the result of the proposition

lt(g) V M

1

n( 6

)-1 dg

=

Tr(V 0 Y) Aim\T

1JIT

(v

)

.aim IT

1 "I1

(cf. ex.l at the end of the section for the computation of the trace of the operator V ® y) . Let us apply this operator to the vector u, and take

the scalar product with the vector x

(x I f TC(g) V ®Y n(g) But we have

udg) =

d

I Y)

(x I u) _ (u I X)

(y I Y)

46

M(g) v 0 Y n(g)

1

u

(n(g)v I u) n(g) Y

=

n(g) (v I n(g 1) u) Y

=

=

=

(u I lt(g) v) lt(g) Y

hence

(x I f ... ) G

=

f

(u I n (g) v) (x ) n(g) y) dg

=

(cv I cy)

G

as expected. Finally, c) follows from b) by linearity since the operators

v 0 y (of rank 6 l) generate End(V).

q.e.d.

In particular we see that if 0 # A E End(V),

and the mapping c

:

#0

Tr (A*A) /dim V

=

II CAII 2

End(V) -> L2(G,it) is one-to-one (onto). The

dimension of this isotypical component is thus (dim V)2 (S.7) Corollary. The Hilbert space L2(G) is the Hilbert sum of all isotypical components L2(G)

= m

L2(G,it)

(the summation index TC runs over equivalence classes of finite dimensional irreducible representations of the compact group G). Proof. We have already seen (in the theorem) that the isotypical subspaces L2(G,rt) are mutually orthogonal to each other. Thus, our

corollary will be proved if we show that the algebraic sum

= ® L2(G, TO G C(G)

AG

is dense in the Hilbert space L2(G). But AG consists of coefficients of finite dimensional representations of G (we have indeed proved that all finite dimensional representations are completely reducible), and the Peter-Weyl theorem (4.2) has shown that AG is dense in C(G) for the uniform norm. A fortiori AG will be dense in L2(G) for the quadratic norm.

q.e.d.

(5.8) Corollary.

(continuous, topologically) irreducible representation

of a compact group G in a Banach (or barrelled) space is finite dimensional.

Proof. Let

a-

:

G --> Gl(E) be such a representation, and let E'

denote the (topological) dual of E

:

E' is the Banach space (or locally

convex space) of continuous linear forms on E (by the Hahn-Banach theorem, for each 0 # x s E, there is a continuous linear form x' E E'

47

with

<x',x> = x' (x)

#

0 )

.

For u r. E' and v

.

E, we can consider the

corresponding coefficient of 6cu o E

g H cu (g)

:

_

Letting v vary in E, we get a linear mapping Q

E

:

--)- C (G) C L2 (G)

,

v '--- cu

Since G is compact and the mappings g H o(g) v are continuous (by the definition of continuous representations), the sets 6(G)v are compact hence bounded in E (v e E). By the uniform boundedness principle (Banach-Steinhaus theorem), the set 6(G) of operators in E is equiconti-

nuous and bounded

sup IIo(g) II

=

gEG

M<

Hence

0QvII =

Sup 1Cu(g) (

SupJI

=

g EG

G

< M pull

SupOw(g) vII

OUR

E'

E'

E

G

IIvII E

This proves that Q is continuous from E into C(G) (equipped with the uniform norm): its kernel is a proper and closed subspace F # E if u # 0 (in this case, u(v) # 0 for some v e E and thus c'(e) = = u(v) # 0).

Take v E E with Q(v) # 0, and apply the orthogonal Hilbert sum decomposition of the preceding corollary to Q(v)

L PP(Qv)

=

Qv

#

0

with Pm

= orthogonal projector from L2(G) onto L2 (G, it)

This implies that there is a tation of G) with

.

'ir (finite dimensional irreducible represen-

PnQ v # 0. For this it , we consider the composite

P

E

and its kernel

-q, L2(G) - 0 L2(G,1t) which is a proper closed subspace F7C # E. But Q is a

G-morphism (intertwinning T and the right regular representation)

implies Q(6-(x) v) = ?(x) Q(v)

=

cv(gx)

48

Since P,, is also a G-morphism, the kernel F.m of the composite PPQ

must be an invariant subspace of E

.

But E is irreducible by assumption,

so that FTC must be (01, and the composite P,rQ is one-to-one (into):

dim E < dim L2(G,it)

q.e.d.

(dim V)2

=

The preceding finiteness result can be proved independently (cf. sec.7 below for a

from the theorem (not using its corollary (5.7) )

proof in the category of quasi-complete vector spaces, and sec.8 for a direct proof in Hilbert spaces). In this second proof, we still use the intertwinning operator Q

---

E

:

L2 (G) ,

v

i--' cu

which must be one-to-one (into) when non-zero (u # 0) by irreducibility assumption on E.

Let us show that E is a strict morphism (the inverse

of E, defined on the image of E is also continuous) :

with the above

definition of M,

IIVII

=

IIr(g-I)C(g) v11 <

11 a- (g-1)II

110-(g) VII s M 116(g) v11

implies

IItr(g) v11 >, M-1 11 v H Since E is metrizable and complete (this proof works for Frechet spaces), Im(Q) = Q(E) will also be complete, hence closed in L2(G). Since the image of Q consists of continuous functions (and the Haar measure of a compact group is bounded), the conclusion follows from the next lemma. (5.9) Lemma (Godement). Let X be a locally compact space, m a positive bounded measure on X, V a closed subspace of L2(X,m). If V C L7(X,m), then V is finite dimensional. Godement's proof. Taking the spectrum of the commutative von Neumann algebra L°°(X,m) (this is a compact stonian space...), we are reduced

to the case where V consists of continuous functions (anyway, this is the case in our application ! ). Replacing X by the support of m, we can

assume moreover that V c Cbounded(X)

L2(X,m)

is one-to-one (into)

.

Since m is bounded, this embedding is continuous and V is a fortiori closed in Cbounded(X) (equipped with the uniform norm). The Banach

homomorphism theorem (Rudin 1973, p.47) asserts then that the uniform

49

norm and quadratic norm are equivalent on V :

(*)

(f EV) .

< Of II2

II f II

Hence evaluation forms (at points x e X) are continuous on V equipped with the quadratic norm, and are given by scalar product (Riesz' theorem)

f(x)

Ex (f)

=

(kx I f)

=

(f E V)

.

Taking an orthonormal basis (ei) of V, we shall have

kx

= L (ei i kx ) ei = 7 ei(x) ei

(with a convergence in L2(X,m) ). Parseval's identity gives I1 kxlI2

= Z lei(x)l2

But (*) shows that it was possible to choose the vectors kx with IlkxII '< c

(anyway, the kx make up a simply bounded set:

they must be uniformly

bounded...) and thus Iei(x)j2

c2

The measurable function x H 7- Iei(x)I2

(Egorov's theorem is applicable)

is bounded,hence integrable and we can write

dim(V)

_

ZIleill2

=

= f L lei(x)

l2

f lei(x) 12 dm(x)

_

dm(x) : c2 m(X)

< ao

Hormander's proof. It still uses the inequality (*) written in the form

Sup ess `57ciei(x) 12

<'

c2 7 I cil2

for any finite orthonormal system in V, hence

(**)

Icil2

17- ciei(x) l2 < c2

outside a negligible set N(cl,...,cn). The countable union N of the sets N(cl,...,cn) where the ci (finite in number) have rational real and imaginary parts is still negligible and the above inequality is true outside N for all those finite families. As both sides of (**) are obviously continuous in the ci e E, it will still hold for all ci (but x 0- N). Take ci = ei(x) ! We obtain

lei(x)12)2

<

c2Zlei(x)I2

and

lei

(x), 2

c2

(x 4 N) .

50

Integrating, we find

n

f 2

=

ei(x)l2 dm(x)

1 16i<, in

< cZ M(X)

whence finally dim V

=

Sup n

c2 m(X)

q.e.d.

Coming back to representation theory, we give a definition. (5.10) Definition. The dual a of a compact group G is the set of equivalence classes of irreducible representations of G. Let it be an irreducible representation of the compact

group G, 'a = [it] its equivalence class (7 E G). We say that it

is a

model of Z1 in this case, i.e. when 7c E tiS . The dimension of 0 is defined as dim(7C)

= dim(V7.) independently from the model it chosen.

Similarly, the isotypical component of a`7 (in the right regular representation) is defined by L2(G,V) = L2(G,lt) independently from the model 7r chosen for U. By the finiteness of the dimension of the irreducible

representations of a compact group

(5.8) and (5.7), we can write

L2(G,t7)

L2(G)

VEG In this way, the dual G of G is in one-to-one correspondence with a subset of closed subspaces (even finite dimensional subspaces) of L2(G). In particular, this dual G is a set... (mathematicians only rarely

consider

such trivialities ! ). Instead of 7t E 4' E G we shall usually

write more simply it E G (hoping that this notation will not lead to any confusion).

(5.11) Proposition. Let G be a compact group. Then the following properties are equivalent. i) The dual a is countable.

ii) L2(G) is separable (countable Hilbert dimension). iii) G is metrizable. Proof. The equivalence between i) and ii) is obvious since L2 (G) is

the Hilbert sum of the finite dimensional isotypical components L2(G,Li)

over the index set G. Moreover, G can always be embedded in a product

TT G

U(n)

with U(it) " Udim(TC) ((E) (metrizable group).

51

Since any countable product of metrizable topological spaces is

metrizable, we see that i) = iii). Finally, the implication iii) ==# ii) is a classical application of the Stone-Weierstrass theorem (group theory is not involved in this point, cf. Dieudonne 1960, (7.3.1) p.131

and (7.4.4) p.134) .

EXERCISES

1. Let V be a finite dimensional vector space, V" its dual

denote by u® v the operator x H u(x) v as

and for u e V" , v E V

defined in this section. Show a) Tr(u 0 v)

_

b) (u 0 v) . (x ® y) c) tA u ® B v

u(v) (intrinsic definition of Tr),

= x 0 v

,

B E End(V)

=

Moreover, identifying the dual of V"' ® V to v 0 V' (in the obvious

canonical way: ), show

d) t(u ®v)

v(Du

=

Assume now that V is a representation space for a group G . Using a)

and c) prove e) cu ® v

=

u

cv

(i.e.

v))

=
2. Let G be a compact group and (mV) be a finite dimensional representation of G. We denote by VG the subspace of invariants of V. VG

= {vEV: 7t(g)v =vforall gEGJ .

a) Check that the operator P = f 7C(g) dg is a projector from G

V onto VG. If 7C is unitary, P is the orthogonal projector on this subspace.

b) For two finite dimensional representations m and 0- of G, consider Hom(V7r,VIr) as representation space of G via the action

Observe that

Hom(Vm,Vr)G

=

HomG(Vm,VQ)

and deduce a proof of the fundamental lemma (5.4) from this observation.

52

c) Using the G-isomorphism

V 0 V,. (V' 0 V being equipped with the representation e® r), conclude that

the projector 4 : V" ®

VLr

-+ (V,0 V(r)G is given by f (iv® o-) (g) dg . G

3. Rewrite Schur's orthogonality relations (5.6.b) without

assuming that

it is unitary. Hint: Compute the integral cvu

J

1) cy(g) dg

(observe that cu(g-1)

v

-

cu(g)

v

,

cu is the cv-coefficient of 7C

v

u

)

.

53

6

CONVOLUTION, PLANCHEREL FORMULA & FOURIER INVERSION

and ends.

In this section, we gather up odds

CONVOLUTION On a compact group G, the convolution of two continuous functions f and g is defined by

f * g (x)

=

Defining f*(x) = f(x 1)

f * g (x)

JG ,

f (Y) g(Y 1x) dy

we can also write

=

f f (xY) g (y-') dy

=

f

=

f f (xy 1) g (Y) dy

=

f*(Yx-1)

g(Y) dy

= f (e(x

f*J g)

The Cauchy-Schwarz inequality gives thus

I f * g (x) I < of-I2 IIg112

=

If112Igh12

whence

Sup If*g(x)I

6

IfRI21Ig'2

xEG

Consequently, the convolution product can be extended by continuity from C(G) to L2(G) and by definition,

*

:

L2(G)

K

--s

L2(G)

C(G) ,

(f,g) )-+ f * g

is a continuous bilinear mapping (still satisfying the above inequality).

On the other hand, the preceding formulas show

(f I g)

=

f* * g (e)

,

If112

=

f* * f (e)

Since the group G is compact, the constant 1 is in L2(G) and for any f E L2(G), the product f =

must be in L1(G)

:

L2(G) C L1(G). The

convolution product can even be defined in L1(G) by the same integral formula, but this integral will not converge for every x e G and the result will no more be continuous. To see what happens, let us take

f and g in L1(G) . I claim that x . is also an integrable function .

f * g (x) = f f (y) g(y 1x) dy

We apply Fubini's theorem as follows

54

If *g(x)Idx < f dxfdY If(ygfy lx)I = f dYlf(Y)I fdxlg(Y lx)I =

HfIl

1

Hglll < o

=

=

VgVl f If(Y)I dY

=

.

These inequalities prove that f f(y) g(y Ix) dy

converges absolutely for

nearly all x in G (all functions considered are measurable...) and the result

f * g e L1(G) satisfies

IIf * gHl < IIfH1IIgVl This convolution product is associative (cf. infra.) and LI(G) is an algebra for convolution. This algebra has no unit element in general (more precisely, it has no unit when G is not discrete, i.e. G not finite). We shall also see that this algebra L1(G) is commutative exactly when G is commutative.

INTEGRATION OF REPRESENTATIONS Let us assume first that 7t is a unitary representation of the compact group G in a Hilbert space H. We can "extend"

it to a

representation of L1(G) by the formula

7r (f)

= f f(x)r(x) dx

(f e LI(G) ) .

G

These integrals converge absolutely in norm : pr(x)11

f IIf(x)ic(x)II dx

=

JIf(x)Idx

=

= 1

implies

IIf Ill

Thus we even have IIX1(f)II

(feL1(G))

Ilful

Although G is not really embedded in L1(G) (when G is infinite), we consider tc as an extension of r ,

as in the finite group case. Later

on we shall even drop the index 1, writing 7c instead of

7CI. The

association 7C :

G -p Gl (H)

can even be made when

:

LI (G)

--*

End (H)

TC is a representation in a Banach space since

7[(G) is always a bounded set of

is uniformly bounded.

7tl

End(H) :

being weakly compact, it

55

Let me introduce here some comments on vector integration. The definition of 1tl is obtained by integration of the vector functions

G - End(H) , g H f (g) ir(g)

(f e L1(G) ).

Thus our definition was based on a strong integral (converging in norm)

defined for functions with values in a Banach space. This theory is expounded

for example in Bourbaki (Integration, 1965, Chap.][) and

uses for instance the following approximation lemma if f is continuous with compact support and takes its

values in a Banach space, for every E> 0, there exists a partition of unity (Ti) with continuous functions

and suitable points xi e Supp(ri)

f(x)

-

finite

cfi(x) f(xi) }I

E

(pi

with

(loc. cit. p.44 )

There is another notion of weak integral, extending the preceding one (loc. cit. Chap.III). If f

:

G -+ E (locally convex space) is weakly

continuous with compact support, the integrals

f <x' , f (x)> dx are well defined for each linear form x' E E'. Obviously

x' --+ f <x' , f (x)> dx is a linear form on E'

(this linear form is not necessarily continuous

on E') hence an element of the algebraic dual (E')* of E'. This element of (E')* is simply denoted f f(x) dx .

From its definition as

linear form, we have

<x'

,

f f (x) dx> = f <x', f (x)> dx

A general theorem about (locally convex topological vector spaces) asserts that (E')* can be canonically identified with the a'(E,E')-weak completion

of E. But when E is quasi-complete (i.e. each closed bounded set in E is complete), one proves easily that

f f (x) dx e E C (E') *

(f e C M)

(loc. cit. Prop.7,p.79).

For this reason, all results obtained by integration of representations are valid for representations in quasi-complete topological vector spaces ...

56

COMPARISON OF SEVERAL NORMS Let us start by recalling the definition of the Hilbert-

Schmidt norm of operators in finite dimensional Hilbert spaces V (for general Hilbert spaces, cf. sec.S below). Let A E End(V) be any endomor-

phism in V. Take any orthonormal basis (ei) of V and assume that A is represented by the matrix

in the basis (ei). Obviously

2 Jai 12

IIAV2 =

i,j

defines a norm on End(V) (a priori, this norm depends on the choice of orthonormal basis (ei)) . If B is another endomorphism, represented

by the matrix (b

(in the same basis), a small computation shows that

(matrices are to be multiplied line by column)

= 2 a b

Tr(A*B)

i,j

This shows that the Hilbert-Schmidt norm is derived from the HilbertSchmidt scalar product

(Al B) 2

=

ij

a b

=

Tr(A*B)

on End(V), and is in particular independent from the choice of orthonormal

basis (ei) of V. Now, we come back to a compact group G and a unitary irreducible representation rr e G in some finite dimensional Hilbert space V = Vj The spaces VV (Y V ,

End(V) ,

L2 (G, 7r)

are G-isomorphic. We shall give explicit isomorphisms between them,

keeping track of the various norms involved. We have introduced the coefficients of

c' x) _

(u6Vv, v E V)

and the more general coefficients (linear combinations of the preceding ones)

cA (x)

=

Tr (A n(x) )

with the relation

cA

=

cv

for A = u®v

(A E End (V) )

,

57

For fixed u e V , the linear mapping

Cu

.

V -> L2 (G, -rc)

v rs cu

,

is a G-morphism from 7tto p (right regular representation). Similarly, if v e V is fixed, =

=

c'(s-lx)

< u,n(s-1)a(x)v>

(t(s)cu)(x)

=

cv(s)u (x)

,

u

_

shows that the linear mapping

Vv-* L2(G,7t)

cv

---+ cu

is a G-morphism from 7C to t (left regular representation). Summing up,

. VV -

c

L2(G,7t)

u®v r-, cu

,

to t x P (biregular regular representation). The action of G x G on End(V) (cf. sec.5) is defined by is a G x G - morphism from

7t ®7c

(txp(s,t))cA (x) =

=

cA(s-lxt)

Tr(7r(t) A n(s)7t(x))

=

=

Tr(A7t(s_l)7t(x)n(t))

cn(t) A

it(s)-+

_

(x)

hence is given by

(s, t) - A

'rz(t) A 1t(s)-1

=

In the following diagram of G-morphisms, Vv ® V is equipped

with the scalar product defined by

(u®vIx®Y)

(u Ix) (vIY)

=

=

(uIx)(vIY)

(we still use the Riesz' isomorphism between V" and V). This scalar

product corresponds to the Hilbert-Schmidt norm

(u®vlx®y)2 = Tr((u®v)*x®y) Here is a diagram summing up the whole situation.

A

End(V)

c

A

Vv ®V

L2(G,it) c L2(G) D L2(G,FE) - V®V V

u U WV 1 - ' cv

complex conjugation

End(V-)

58

Schur's orthogonality relations (5.6)

(cu v (cx) y

=

1

dim C

(dim n= dim V)

(u x) (v 1 y)

show that

c ( u ® v i- cv ) is

isometry map

(dim 7c)

The inverse of c is nearly the extension 7C1 of it. Let us compute Trl(cv). For this purpose, we apply this operator to a vector y and compute the

scalar product of the result with a vector x

(cv cy)

= f cp(s) (xi n(s)Y) ds =

(x 17r(cv) Y)

This quantity will vanish when n is not equivalent to Tr (5.6.a) It would have been more appropriate to compute ir(E)

!

Indeed, Schur's

relations give then

(x 1 Tr(EV)Y)

Tc)-1(x

(dim

_

(c, I cy) = (dim 1r)Fu I x) (v I Y)

=

I (v I Y) u )

_

_

( x (dim 7C)-ly ®u (Y) )

which implies

it (cv)

-1 v ® u

_

(dim 7c)

_

(dim 7c) -l A*

and by linearity 7r (cA)

Since the coefficients cA are the coefficients of FL, we see that

on L2 (G, 7L) ,

f*

o

n(f) is of the form

it

=

(dim Tr)

2

isometry map

L2 (G, 1C ) Th e compos it e

End(V)

-3

A F- s

conj. L2(G,7t)

L2(G,ic)

cA

cA =

--i

f

End(V")

- 4. Ttl (f)

can be identified with

(dim 7c) 1. ( A

F- A*

_

isometry map

(dim 7C) -1

The Plancherel theorem (and Fourier inversion

theorem)

follows immediately from the preceding computations. Here it is.

Let G be a compact group. For It e G, denote by Pr

:

L2 (G) -+ L2 (G, 7c)

the orthogonal projector on the isotypical component of it, and for f E L2(G), let

f

f, converges to f

= P7r (f) so that the series G

59

in quadratic mean (i.e. in L2(G)). Then

a) fn(x) b) II f'112 2

=

c) 1f12

=

2

(f(x) = f(x-1)),

dim n Tr(7t(f)7t(x))

=

(Hilbert-Schmidt norm)

dim it II7t(f)112

2

(Parseval equality)

dim it 1174f)112

2

7[EG

(In particular, all components fn

are continuous functions cf.(4.6).)

Proof. The orthogonal projection ff of f in L2(G,7c), is the element cA

of this space having same scalar product with all elements cB of

L (G,1r). (Burnside's theorem shows that the endomorphism A is completely

determined by the condition cA = fn cf.(5.2).) Let us determine A as function of f. We must have

(cB I

f)

=

(cB I

f)

=

(cB I

fn)

(cB I cA)

=

=

(dim

7t)-l

Tr(B*A) .

But

=

J

cB(x) f(x) dx = f Tr(B7t(x)) f(x) dx

f Tr(rc(x 1)B*)f(x) dx

=

=

Tr(B*fTC(x 1) f(x) dx) _

Tr(B*1t.(f)) Comparing the two results obtained for all B, we indeed find

A=

dim 7t

7r(f)

This gives

fn(x)

=

cA(x)

=

Tr(Aa(x)) =

as asserted in a). Moreover, Schur's relations show that

frll2

=

II cAII2

=

(dim 7t)-1IAIl2

=

dime

IIn(f)112

Thus b) is proved and c) follows from the observation that f and f have the same quadratic norm (Ex. S

of sec.l): we can permute f and f

Also observe that the dimensions of it and of it are the same'

q.e.d.

Let us observe explicitly that this theorem does not

imply that the series Tf of continuous functions

converges for

nearly all x in G (convergence in quadratic mean does not imply simple convergence outside a set of measure zero).

Even when f is

continuous, simple convergence of this series Efc outside a negligible set is a deep statement (proved when G is the circle group, cf. infra.).

60

EXAMPLE Let now G be the circle group ]R/

. As this group is

commutative, all its irreducible representations have dimension 1 (dimension 1 representations are always irreducible and are also called characters :

a more general definition of characters will be given in

next sec.7). The dual G of this circle group is thus

G={ n:

n c a

i

7cn(x) =

where

e2ninx

The Fourier coefficients cn of a continuous function f (or f c- L1 (G) )

are given by the usual formulas 1

cn

=

cn(f) 1

J0 e

=1

2xinx

=

f (x) e-27
f(-x) dx

=

71n(f)

Thus the Fourier series of f is identical with the series T- fir

given

in the Plancherel theorem

cn(f) e2 ninx _

dim 7[n

= Z n(f) 7Cn(x)

_

Tr(7tn(f)7cn(x) )

This series converges in quadratic mean, and it is only recently (1966) that Carleson has been able to prove convergence for nearly all x (cf. Fefferman 1974, p.105). This answered a question raised by Lusin in 1915.

When f is continuous (and periodic of period 1), Fejer's

theorem asserts that the Cesaro means converge uniformly to f. But for a compact group G, a has no natural ordering and Fejer's theorem has no generalization in this direction.

61

EXERCISES

1. Let G be a compact group, f and g E L1(G). For any representation

a- of G (in a Banach space), prove

a-(f * g)

=

If iris unitary, prove also

<"(f*) =

d-(f)*

(recall that f*(x) = f(x1) ).

2. Show that the "extensions" of the regular representations

of a compact group G are given by

t(f)(q)

f*

=

where f and g e L1(G)

,

P(g)(go)

and P 6 L2(G)

=

gv

S° * V

(recall that q(x) =

g(x-1) ).

Conclude from this that

If

(p 112 <

ll f l/1 I //2

Moreover, using 1), deduce the associativity

(f

'g)A,sP

(f, g E L1(G),

=

E L2(G)

f it(git 5 )

or all three functions in C(G) !)

.

Also check that for any representation a- of G

= 0(x) a-(f)

Lr(,e(x) f )

,

ir(p(x) f)

=

r(f)0(x1)

3. Let G be a compact group and denote by L1

.inv

= L1

inv (G)

the closure in L1 (G) of the subspace of continuous functions f satisfying f(xy) = f(yx)

(or equivalently, f(y 1xy) = f(x)) for all

x and y in G. Show that L1 is contained in the center of L1(G) (as inv 1 convolution algebra . prove f * g = g iF f for f E Linv and g 6 L 1(G) ). For any irreducible representation

TC(f)

where

_

jr

: G --j Gl(V) prove that

(dim 7t)-1 <x,f> I V

x (g) = Tr 7C(g)

(f Einv L)

and ('x,f> = J X(g) f(g) dg.

(Hint: use Schur's lemma to prove that 7C(f) is a scalar operator and

then take traces to determine the value of the constant in this multiple of the identity.)

62

4. Let 6': G

Gl(V) be a unitary representation of a 6'(1) = P (1 = constant function 1 in L1(G)

compact group G. Check that is the orthogonal projector V of V.

(Hint: show that

1 * f = f * 1

-- *

VG on the subspace of G-invariants

1 * 1 = 1 and 1* = 1 in L1(G) ; more generally,

is the constant function ff(x) dx .)

5. Show that the "extended" left regular representation t =

el

:

L1(G)

-->

has trivial kernel (Oj (Hint: Let 0 sequence (gn) C C(G) with

gn

0,

End(L2(G))

f E L1(G) and construct a

#

fgn(x) dx = 1 , and t(f)(gn) =

= f * gn - f # 0.) Conclude that if 0 # f E L1(G), there exists a

Jr in G such that

7C(f) # 0 (use the decomposition of the regular

representation given in sec.5). Finally, prove that L1 (G) commutative

4==:)-

G commutative

6. Let G be a compact group,

7C E G and consider the

adjoint representation of G in End(V) (V = V.) defined by the following composition Ad :

G - G x G

--0 End(V)

s H (s, s) H (A h-T 7C(s) A 'R(s) _l) (s,t) 1--) (A H 1C(t) A7t(s)_l) Prove that the multiplicity of the identity representation in this adjoint representation is 1 (this identity representation acts on the subspace of scalar operators:.Schur's lemma).

7. The decomposition of the biregular representation

of a compact group G in L2(G) gives the decompositions of the left (resp. right) regular representation simply by composition with it

G --i G x G

(resp. i2 : G -1 G x G

s H (s,e)

s 1-0 (e,s)

Conclude that

2

®

1 ® 7C

(Compare with Ex.l of sec.2.

dim

dim fr. n

dim7t'-X

)

63

7

CHARACTERS AND GROUP ALGEBRAS

Let (ir,V) be a finite dimensional representation of a compact

group G. The character x = xn of 7c is the (complex valued) continuous function on G defined by

X(x)

Tr (R (x))

=

.

(This is the function cA for A = id. E End(V), cf. sec.6.) When dim(V) = 1,

x and 7t can be identified :

in this case, x is a homomorphism.

Quite generally, since the trace satisfies the identity Tr(AB) = Tr(BA), we see that the characters of two equivalent representa-

tions are equal. Moreover, characters satisfy

x(xy)

X(yx)

=

or X(y lxy) = x(x)

(x , y E G

.

Thus characters are invariant functions

x E Cinv =

{ f E C (G)

:

f (y lxy) = f (x) , x and y in G

We shall also have to use 1

= closure of Cinv in

Linv

L1 (G)

= closure of Cinv in L2(G)

Linv

(cf. Ex.3 of sec.6). Invariant functions are also called central functions (they belong to the center of L1(G) with respect to convolution). Still quite generally, the character of the contragredient V

of 7t is given by x,,(x)

hence XT = x to it) and

Tr ir(x)

= .

=

Tr t7t(x 1)

=

Tr 7t(x-1)

When 7r is unitary, it(x 1) = X(x)*

=

x(xl)

( it is equivalent

is the complex conjugate of X. One can also check without

difficulty that for two finite dimensional representations it, a' of G

X Wo-

_

2(ir + x,r , X rt

y- =

xn' x°"

When it is irreducible, Schur's lemma shows that elements z in the center Z of G are mapped on scalar operators by

7r

:

7t(z) = A 1V so that

64

V .

X(z) =

Thus the restriction of (dim V)X to the center Z

is a homomorphism Z

:

This is the central character of

it

it is independent from the special

:

model chosen in the equivalence class of it. In particular if A(Z) is not contained in t±l}, it and 7t are not equivalent (their central

characters are different). Also observe that X(e) = dim V ( = dim it). (7.1) Proposition. Any continuous central function f e Cinv on a compact

group G is a uniform limit of linear combinations of characters of irreducible representations of G.

Proof. Let 6 > 0. By the Peter-Weyl theorem (4.2), we know that there is a finite dimensional representation ((r,V) and a A E End(V) with If(x)

- Tr(A6(x))

< E

(x E G)

In this expression, replace x by one of its conjugates yxy 1

.

I f(x) - Tr(AT(yxy 1))) = If (x) -Tr(T(y l)A6(y)o(x) I < £

Integrating over y, we get f (x) - Tr (B T(x)) I < £

where

B

=

J

o(y) A r(y) dy

By invariance of the Haar measure, the operator B commutes to all

operators 0(x). Hence, if we decompose T into irreducible components (or rather, into isotypical components)

ic®ln

n,, 7-c

n

the operator B will have the form B

= O ldimn: ® B,r

(cf. Ex.1 of this sec.)

and

B 6(x)

0(x) B

=

Tr(Ba'(x))

_

= O 7C ® BJ

anZx)

(a. = TrB .

This shows

f (x) -

E arX,(x) finite

I

<£

(7.2) Theorem. Let i and 0- be two finite dimensional representations of

a compact group G with respective characters xn and xo

(Xn xa )

=

dim Homy( n,V,)

Then

)

65

Proof. From (5.4) (or alternatively, Ex.2.c) of sec.5), we know that the integral

f n(x) ® o(x) dx is an expression for the projector

-- (V;r® Vr)G

Vn

Hom(V,,Vr)

HomG(VI,Vd)

The dimension of the image space is the trace of this projector. Thus

(.X,r 12(,)

=

f 7(,(x) Xo(x) dx

dimHomG(VV ,VV) .

=

q.e.d.

As in the case of finite groups, there are many important corollaries of this theorem. Their proof is immediate. (7.3) Corollary 1. Let n be a finite dimensional representation of G. Then

it is irreducible <==4

(7.4) Corollary 2. Let n, o- E (x,t lx")

IX,rII2

=

2

(XI I X,t)

=

1

Then

= 91ra. (= 1 if n equivalent to r , 0 otherwise).

(7.5) Corollary 3. Let o' be a finite dimensional representation of G

and o = (9na n (summation over a finite subset I c G) be a decomposition into irreducible components. Then a)

n, =

b) IIJ12 Proof. Let

=

()C,l)

is well determined

Z. n 2

(summation over I)

V. _ ® (V® ® (Ent). Since each G-morphism V..--> V., must

vanish on all isotypical components V., ® (Ent

where z is not equivalent

to It', we have HomG (VV , VV)

HomG (V,t ® e., v,,)

=

= e"®HomG(V,,y

=

=

(Enr

Assertion a) results from this. Moreover, b) follows from Pythagoras

theorem (and (7.4) above ! ). (7.6) Corollary 4. The set of characters (xn)

rE G

is an orthonormal

basis of the Hilbert space Linv (closure of Cinv in L2(G) ): every invariant square summable function f on G can be expanded in a series

66

f = c(X.If)-X.

(convergence in L2(G) )

.

G

(7.7) Theorem. Let G be a compact group. For x E

let

P

denote the

P,

projector L2(G) -+ L2(G,ic) onto the isotypical component of

the normalized character P,r

$,c

f y f,c

:

dim 7r. xc :

=

Pnf

=

Proof. We have already seen (part

that

is (in the

is given by convolution with

right regular representation). Then P,.

f *r 4 .

=

of Plancherel's theorem, sec.6)

a)

f,c(x)

=

dim n Tr(ir(f)ir(x))

f,C(x)

=

dimirTr f f(y 1)7C(y)n(x) dy

Thus,

dimly f f(y) Tr 7r(y 1x) dy

=

=

=

f * i9,C (x)

q.e.d.

Still denoting by P the right regular representation,

f * ,r =

we can write

L2(G)

P,c

P

(,9n) (f)

This also shows that

.

-+ L2(G, it)

is also given by

P,c

= P (e.) where en = dim 7r Xz = dim i X.

Since ,S,c is a linear combination of coefficients of Ic , hence belongs to L2(G,n), we have

4

= P(4)

_

19,r3,Sty

By conjugation, we also have e,r * e,

are not equivalent,

1.9, * t9

=

0

=

en . Moreover, if 7c and o E G

and en * e r

=

0

Let us also observe that if we had worked with the left

regular representation, the isotypical component of type 7r in this representation would have been L2(G,iF)

and the projector onto this

component could have been written

Q?r(f)

Q,r

=

f, = f* 'OL

=&*f

= t (ex) is the projector L2 (G)

= 2()f ---r L2 (G, t)

More generally, we have the following decomposition theorem.

67

(7.8) Theorem. Let (6,V) be a representation of a compact group G in a Banach space (or more generally in a quasi-complete locally convex

s ac). Then Vn

1)

6(en) V is a closed subspace

=

Or 6 G) ,

2) YVn = © V, is a direct sum in V 3) ®V1, is dense in V 4) v -> (v,,) = (d(e,) Moreover,

,

v)

:

V -+ TTV,. is injective

any a E Vn generates a G-invariant subspace of dimension

G (dim n)2 . When a is unitary (V being a Hilbert space), V is the Hilbert sum 4 V,,

(completion of the orthogonal direct sum 6 Vn)

in this case, we thus have v

= L vn with a convergence of the series

in V.

Remark. Taking for G the circle group and for V the Banach space C(G) of continuous functions (with the uniform norm), we see that the series Z.

v,t

cannot be expected to converge in V = C(G) (Example at the end of

sec.6). The series is thus only considered in the Hilbert space case. Proof. Projectors in a vector space are characterized among linear operators by the identity P2 = P. For such an operator, the image of P is the kernel of I - P, hence is closed if P is continuous. Point 1)

of the theorem results from this observation with P

=

a'(e,)

=

O (ec

- ex)

=

o(e,r)

To prove 2), we observe that in a finite sum

o-(e,)

L,vt = 0

subset I of G ) of elements v., ca V. , we can write vt

P2

=

(sum over a finite =

6(e..) vz

hence 0

= T(e6(er) v., _

o'(e.M

e.) vti

=

vn

Thus the subspaces V, are linearly independent and their sum is direct. The other points will result from the fact that L2(G,it) is irreducible

under the biregular representation l x p of G x G ,

hence is generated

by the left and right translates of any of its non-zero elements. For example, the left translates of xn already generate L2(G,ir ) (since

st)

=

)(i(ts), there is no distinction between left and right

translates of xrd . Then, for f e C(G), u E V and v e V we have T

=
68

hence

_
(f ICU)

To prove 3), we apply the preceding equality to f = p(s-1) (p(s-1)xn

I c') _

x ft

_

(1t E

cu L L2(G, w) cu whence u = 0 . theorem.

=

0

,

in particular =

cu(e)

=

(v E V)

0

Density of ® A results then from the Hahn-Banach

To prove 4), we consider f = e(s) x,

P(s-1) xn)

(e(s)x, I cu) = _ .

Hence if v E V is such that all components vn = 0 (for all tt E G) , we have

cv 1 e (s) xn

,

cu 1 L2 (G, iv)

(all

=

cu(e)

it E G )

,

hence cv

=

0

,

in particular

=

0

(for every continuous linear form u E :E1 on E). Again v = 0 by the

Hahn-Banach theorem. Finally, take a E Vn. The relation a = d(ex) a can be written a

= f 0'(x) e. (x) a dx

,

hence

a(y) a

= f e,(y lx) C'(x) a dx

= f e , , , (x) u-(yx) a dx

.

As the translates e(y)e,, generate the subspace L2(G,FC) of dimension equal to (dim n)2, we see that the a'(y) a

remain in a subspace of dimension

(dim 7t)2. More precisely,

f

a

ff(x)r(x)adx

=

T(f) a

is a G-morphism L2 (G, it)

-p- Vi

:

1r ®1 = e 12

-- U- I V

L (G, 9t ) The image Vn(a) of this G-morphism is thus a sum of at most dim it

69

representations equivalent to it and

LrIV

is isotypical of type it

n (as sum of the isotypical representations acting in all V,%(a)

, a E Vn).

This concludes the proof of theorem (7.8). (7.9) Corollary. Any irreducible representation of a compact group G in a quasi-complete locally convex vector space is finite dimensional. Proof. Let V be the space of the irreducible representation

a- and take

0 # a e V . There must exist a 7CE G (hence 7r finite dimensional) with a n

0 (part 4 of the theorem (7.8)). The corresponding invariant

subspace V. is not t0}, hence V = Vn = V..(a) by irreducibility. But we have seen that dim V,,(a) c (dim 1C)2 is finite.

AN APPLICATION: STONE'S THEOREM and consider a

We take for G the circle group ]R/7L ,

unitary representation U of this group in a Hilbert space H : U :

--0

]R

Gl(H)

'-s

t

,

Ut

satisfies

Ut+l

Ut

=

Ut+s

,

=

(s , t c ]R)

Ut us

The structure of this one parameter periodic subgroup t -* Ut is as follows.

We already know that the dual of the circle group is

= Orn

G

Hence

en = dim

Tr(

rtn(t)

n E 7Z

:

n)

is given by en(t)

= =

e

2rrint

e-2wint

and Pn = U(en)

is the orthogonal projector on the isotypical component of type itn

For v e H, we have thus

= 7-vn = F-Pn(v)

v with

vn

=

Pn(v)

and Ut(v)

e 21tint

=

= 7- Pn(v)

e21rint

Ut(v) dt

(the integral being computedJover 1R/2l or simply over the interval [0,1J). Thus we can write U

t

=

e

taint

with a simple convergence (

P

n .e

2nint

Pn(v) converges in V for every v E V ).

This result can be stated in a slightly different way. Put

70

Ex

7

=

Pn

2

Then we can write

U

t

j

=

eiAt

eiAt

=

A

with a certain self-adjoint (unbounded) operator A

= f A dEA . The non-

periodic case of a one parameter unitary group t .-s Vt is easily reduced to the periodic case by the definition

Ut The

=

Vt V1t

.

definition of the power Vlt requires some spectral theory for the

unitary operator V1

:

for these classical notations, and for the

treatment of one parameter groups of unitary transformations, including the reduction to the periodic case, one can consult Riesz-Nagy 1975, pp. 375-380

.

Characters of irreducible representations of compact groups can also be characterized by an integral equation. (7.10) Theorem (H. Weyl). Let

x

be the character of an irreducible

representation of a compact group G. Then

x(s) x(t) = x(e) Conversely, let

(

rG

J

x(xsx lt) dx

# 0 be an integrable function satisfying

p(s) ap(t)

= fG (xsx 1t) dx

Then there is a (unique) 7c in G with 9' = x, /dim 7r Proof. We have seen (5.5) that for 7r e A4

= f ir(x) A n(x) -1 dx

is a scalar operator in V = V,r

f 7G (x) A n(x) -1 B dx

Take A=7t(s) and B=n(t) f 7t(xsx lt) dx

and A E End (V,) ,

= (dim tt) -1 Tr (A) idV

Consequently, for any B E End(V)

.

=

(dim 7t) -1 Tr (A) B

:

=

(dim n)-lx(s) 7t (t)

whence the result by taking traces ( dim 7t= )((e) )

.

Conversely,

suppose 0 # rp e L1(G), and

satisfies the integral equation. There

exists a 7G E G with 7r(q)

0 (cf. Ex. 5 of sec.6) . With this 7r

71

we can write

= f cp(s)r(t)n(t) dt

p(s) 7r(.?)

=

f dt nt(t) f'dxV(xsx-lt)

= f dx f dt n(t) 9(xsx lt) = f dx f dt 7r((xsx 1)-1 t) 9,(t)

_

f dx f dt y(t)1t(xs-lx-lt) = f dx n(x)n(s-1)n{x)-l7r((P But now

f dx n(x)ir(s-l) n()-l

(dim 1C)-l Tr 7r(s-l)

=

=

X(S)

x(e)

Hence

ti(s) n(9)

%(e)-1 7(.(s) n(f) E End(V)

_

This proves that z(e) f(s) = xs and T = Xj /X (e)

q.e.d.

It is also possible to express the Plancherel theorem (sec.6)

with characters of irreducible representations. Let us recall that the Dirac measure (or distribution) on a compact group G is given by the

linear form :

--*

C (G)

f

,

U

f--*

of evaluation at the neutral element.

,If q2

For f e L2(G), we have seen that

U L(f)12

dim

=

f (e)

By polarization (replace f by f t g

and by f t i g and add the results),

we also have

= L dime Tr(7t(f)* 1C(g))

(f I g) But

(f

g)

= f* * g (e)

=

cu(e)

.

(with f*(x) = f(x l) and

= f* * g

continuous on G), hence

n(f)*rr(g) T(e)

=

7t(f*)7t(g)

= n(f* * g) = n(y),

= 7- dim n Tr it(y)

for all continuous functions T of the special form f* * g with some f and g E L2(G). Moreover, the sum giving T(e) converges absolutely (since the sum giving (f 1g) converges absolutely). Writing

Tr 7r(9)

=

Yr f 7r(x)cf(x) dx

= f x(x)?(x) dx =

(x, ,>

we see that

(e)

= F dim 7r <,X,, , ,>

(sum over a) .

72

In a certain sense, we can thus write

S' = 2 dim n 'x,.

(sum over G) .

Replace the special function cP by one of its right translates (if

cP =

f* * g, this amounts to replacing g by the corresponding right translate:

P(S)T = f** p (s)g ) y(s)

=

P(s) 9 (e)

_

dimir Tric(p(s)T)

= E dime Tr(7C((') n(s-1))

(by ex. 2 of sec.6 )

and s by s-1

Replace now cP by cp(s)

_

_

T(s-1)

E

_

IG(s))

We already knew that this formula holds (part a) of the Plancherel theorem, sec.6), but we have now proved an absolute convergence at each point when cP is a finite linear combination of continuous functions

of the special form f* * g

(f and g in L2(G)). (Observe that left

translations would have led to the same result since cp(s) = Z(s-1)q,(e)

XWs-1) cP)

=

n(s-1) w(c') has same trace as n(9') ie(s-1) .

)

When G is the circle groupRR/2Z, the special functions f* * g are those which have an absolutely summable sequence of Fourier coefficients. It is not possible to characterize these functions as simply in the general case (i.e. when G is any compact group).

ALGEBRAS ATTACHED TO A COMPACT GROUP Since several algebras connected with a compact group have

played a role in our theory, we review them and introduce another universal construction. The first group algebra of G is the convolution algebra L1(G). In a certain sense, it plays a role completely analogous to the group

algebra of a finite group. We have seen that if 6- is a unitary 1

representation of G, and f EL (G), the operator 6'(f) has a uniform norm bounded by the L1-norm of f

0TU)II G IIfill Let us define a new norm on L1 (G) by

IIfL

=

Sup IIO'(f)II

73

(the supremum being taken over all unitary representations 6 of G). Thus we have

. We also denote by C*(G) the Banach algebra

AfU* 4 AfII1

obtained by completion of the convolution algebra L1(G) with this new norm (the elements of C*(G) cannot be represented by functions over G). By definition, any unitary representation of G can be extended canonically to a representation of C*(G) (extending the corresponding representation of L1(G)) . This algebra C*(G) is a stellar algebra

:

the

involution f '-'r f* ( f*(x) = f(x 1)) extends to an involution

a r-* a* of C*(G) and

Iia*aII*

=

(a e C*(G)).

IIaII*

We shall now give a model (faithful representation) of the algebra C*(G) For this purpose, we extend the

by means of operators in L2(G) = H .

left regular representation and obtain C*

:

C*(G) - End(H)

(H = L2(G)).

I claim that L* is an isomorphism onto a uniformly closed subalgebra of operators of the Hilbert space H (the involution of End(H) being the operation of taking the adjoint of an operator). By the decomposition theorem (7.8), we have II f A*

Sup

=

T unitary

IT(f)II

iC e G

Since the left regular representation all it E. G

Sup_ u,r(f) II

=

Q

has a decomposition in which

have a positive multiplicity (sec.5), we also have

Sup- A it(f) II

n6G

=

IItG(f) II

=

p fit-

Hence

IIftL

=

IIL(f)N

(f a L1(G))

and C*(G) can be identified to the closure (uniform norm) of the

operator algebra .(L1(G)) acting in the Hilbert space H = L2(G). (7.11) Proposition. The image of the stellar algebra C*(G) by the left regular representation of a compact group G consists of the decomposable operators (Ax (D 1)

of the Hilbert sum tD(V. ® VV) _

L2(G,n )

with A rr e End(Vrz) and 11 An 11 -- 0 (for is --) oo in the discrete space G) .

74

Thus the stellar algebra C*(G) can be identified with for the uniform norm.

= completion of

End(V,) G

Proof. For f E L2 (G) we have L dim e . I n(f) 12 I7[(f) p

19

II iG(f) I2

-p-

0

=

II f 112 < oo , whence

for 7c -- co

in G (discrete).

Consequently we must also have

-+ 0 for 7C -; ao -P 0 for it -- oo

I7c(f) II 117c (a) I

when f e L1(G)

,

when a e C* (G)

.

Thus e*(C*(G)) is contained in the indicated algebra of decomposable operators. Conversely, it is enough to check that the image of Q* contains the algebraic sum 9 End(V,r) .

But the character of a 7GE G

gives rise to an idempotent e, = dim it.

E(en)

E C(G) for which

(A,01)

_

with v equivalent to it

id E End(Vrz)

if

0 E End(Vt)

otherwise

Ay Taking finite linear combinations of translates of the e., we get all

decomposable operators belonging to the algebraic sum

®End(V7r) . q.e.d.

One can also consider the closure of e(L1(G)) for the

strong topology on End(H), H = L2(G) (a sequence n --) A for the strong topology when An(x) -s A(x) in H for all x e: H). This strong closure contains the identity operator 1 E End(H) (indeed, the series

Z P,

converges strongly to 1). We obtain thus the von Neumann algebra

generated by the left regular representation. =

t4(G)

(C(G))"

=

(c(L1(G))"

=

(e*(C*(G)))"

(if O is any set of operators in a Hilbert space H, we denote by a the set of operators W C End(H) which commute with all elements of CL,

and CC' = (W) I

;

if Ck is stable under A -+ A*, Ot" is the von

Neumann algebra generated by UC ).

This strong closure can be identified

to the algebra of decomposable operators

(AX(P 1)

with Ax E End(Vg) and Sup, 11 As I < oo 7[E G

,

75

and is thus the "product" of the von Neumann algebras End(V,) (n 6 G)

.

The use of the right regular representation p instead of C would lead to symmetrical results. For example

'U(G)

=

p(G)'

= U(G)' = TTE ® End(V1)

The center of both U(G)

D(G)') and r(G) (= It(G)

1T T o T

= U(G) n 7/'(G)

is

CM(G)

All these "products" of von Neumann algebras are characterized in the

Cartesian product by

Sup 11An

11 < c

In the following table, we list all functional algebras attached to a compact group. The smallest one is the algebra AG consisting of linear combinations of coefficients of finite dimensional representations (with the usual product: pointwise multiplication of values). On the vector space AG , AG

where

0

one also considers the co-product

AG ® AG

f(st) = Efi(s)gi(t) .

AG

= ®

:

f

E f i ® gi

' --)-

With this co-product, AG is a Hopf algebra.

L2(G,a )

Hopf algebra (co-algebra)

n C (G)

n L2(G)

Hilbert algebra

n

convolution

L1 (G)

Banach

C* (G)

stellar algebra

algebra

n 1L(G)

von Neumann algebra

I

I

composition of operators in LZ(G)

76

EXERCISES

1. Let H1 and H2 be two Hilbert spaces. Prove that any operator A in HI ® H2 which commutes to all operators r 1M 1

(T E End HI)

can be written in the form 1 0 B for some B E End H2 . (Hint: Introduce an orthonormal basis (ei) of H1 and write A as a matrix

of blocks with respect to this basis

A(e. ® x)

_ Ee i®AIx J

(A1 EEnd H ). J

i

Using the commutations (P. (V 1) A = A (P. ® 1)

J

2

where P , is the ortho-

J

gonal projector on ¢e. , conclude that AJ = 0 for i # j. Finally, using the commutation relations of A with the operators U.. ® 1 72

Uji(ei) = ej

UJi(ek) = 0 for k # i

,

conclude that Ai = B EEnd H2 is independent of i.

2. Generalize the result of the preceding exercise in the following context. Let (V.) and (W.) be two sequences of finite dimensional Hilbert spaces and let H be the Hilbert sum

(Vwi)

H=

Prove that all operators B E End(H) which commute to all decomposable

operators

(A . 0 1) (Ai E End (Vi) and Sup 11A, I(<°°) are decomposable

and have the form B

= ® (1 ® Bi )

for some B.

End (W ) with Sup I Bi p <00

3. Check that the formula

f = Z (xti I f) xs

(cf. (7.6) )

coincides with the Fourier inversion formula (end of sec.6) for

invariant functions. (Observe that <x , f > _ (x / f) 4. Prove that L2(G) * L2(G) D AG .

check that f* * g tation.)

=

cg

(Hint: if

(x I f)

.)

f, g E L2(G,7r),

as coefficient of the right regular represen-

77

5. Let G be a compact group and K a closed subgroup of G. We denote by AK the subspace of L1(G) consisting of functions f such that f(k1gk2) = f(g) (ki e K : subspace of K-biinvariant functions). a) Show that AK is a convolution subalgebra of L1(G). b) Let (7C,V) be a representation of G. Show that the

subspace V of K-fixed vectors in V is invariant under all operators

(f E AK) .

7C(f) Show that if V is irreducible and V

y6 0j, then VK is irreducible for

T[(AK) (observe that we can assume tCto be unitary; then if Vu = WI e W2 put V. =flL G

and check that these spaces V. are invariant under

with VI J W2 , V2 1 W1 )

.

c) We assume now that A

K

(hence finite dimensional) and v

is commutative, V irreducible 101. By part b, dim VK = 1 and

we fix a v r-_ V with JJvfJ= 1. In particular, we must have

7C(f) v

=

af v for some a f

(f E AK) .

ET

Show that

af = ff(x) (p (x) dx = (( f f) with the function

O(x)

(f E, AK defined by

=

(v

7c(x) v)

.

If x is the character of 7C, show that

?(x)

=

f x(xk) dk K

(let P denote the orthogonal projector of V on VK : P

= f 7C(k) dk,

K

and use

(0(x)

=

Tr(P 7[(x) P)

=

Tr(TC(x) P)

)

78

INDUCED REPRESENTATIONS AND

8

FROBENIUS - WEIL RECIPROCITY

We have already used the Hilbert-Schmidt norm of operators in finite dimensional Hilbert spaces (sec.6). We need some analogous results in arbitrary Hilbert spaces. Let thus H1 and H2 be two Hilbert spaces and A E Hom(H1,H2) be a continuous linear operator H1 --) H2 Let us take an orthonormal basis (ei) of H1 .

Then

Z IIAeill2 E [O,m) is independent from the choice of

orthonormal basis (ei)

in H1 .

Take indeed an orthonormal basis (ej) in H2 and write Parseval's identity

for each vector Aei e: H2

IlAeip2

= L I(ej I Aei)I2

As the operator A is continuous, it has an adjoint A* (A*6j lei)

:

Aei)

_

and thus

pAeill2

i

f I(A*ej I ei)I2

=

i,j

=

IIA*Ej1I2

The last series is obviously independent from the choice of (ei) (8.1) Proposition-definition. An operator A E Hom(H1,H2) between two Hilbert spaces is a Hilbert-Schmidt operator if for one (hence for all) orthonormal basis (ei) G H1

,

A e Hom2(H1,H2) ,

ZIAei112 < oo. In this case we put IIAI12

=

2IIAeiII2

.

Let us still fix the orthonormal basis (ei) G H1 and denote f

by H1 the dense subspace of H1 consisiting of (finite) linear combinations of basic vectors ei .

For any linear operator A : Hi -> H2 satisfying

JIAeig2< oo , we have Ax

=

L xi Aei

for every x

= Z xiei (finite sum),

hence 11 Ax 11

Ix.I IIAeiII 5

_ Ixil 2) 2 (f NAeiII 2) 2

79

Thus

(x E Hi )

NAxi 4 NxNlAIL 2

and A is bounded (continuous) with a continuous extension to H1 still satisfying

IIAxN 6

IIAII ( DAN2

,

IIA921Ix11

From this we infer that there are isomorphisms

Hom2(H1,H2) -

4)- H

=

I

A

i--0

(I)

P

H

(I = index set of (ei))

2

(Aei)

As a consequence, Hom2(H1,H2) is a Hilbert space (it is complete). Its

scalar product is given by

(A J B) 2

=

2 (Aei I Bei)

(and this series converges absolutely when A and B E Hom2(H1,H2)) Choosing an orthonormal basis (Ej) in H2 , we could still identify

this Hilbert space to C2(I x J) by Hom2(H1,H2) A

r-

Q 2(I x J)

((Ej I Aei)) I x J

(8.2) Lemma. Hilbert-Schmidt operators are compact operators. Proof. Take A E Hom2(H1,H2) . As

LUAeiD2 < oo

, the set 10 of indices

i E I with Aei # 0 is countable and we can decompose H1 into the orthogonal with Ho c Ker A . Replacing H1 by Ho ,

sum Ho ® Ho

we can assume H1

separable if we like. For any finite subset F C I, let us denote by PF the orthogonal projector of H1 on the subspace generated by the ei

,

i e F. The continuous operators APF have finite rank, hence are

compact. But

IIA-APFII2

A-APFII2

=

7-

I-F

IIAeiii2

is arbitrarily small when F is sufficiently large. This proves that A is a limit (in uniform norm) of a sequence of compact operators, hence is also compact.

(8.3) Lemma. For A E Hom2(H1,H2), T E End(H1) and S 6 End(H2) we have

SAT EHom2(H1,H2)

and ISATQ2 4 NSNIITIIIIAII2

.

80

Proof. a) In the special case T = 1, the lemma follows from the definitions IISAI12

=

flISAeiII2 < USII2EIIAeill2

= IIS112 I(All2

b) But we have noticed (just before (8.1)) that IB*I2 = IIB12 so that, quite generally,

IISATII2 = IIT*(SA)*Il2

UT*IIIISAII2
(a) c) The result follows now from the well known equality IT*I

= IITII

q.e.d.

Let us denote by Homf(H1,H2) the vector space of continuous

operators H1 -* H2 with finite rank. We have already used the isomorphism

H1 ®H2 -i Homf(H1,H2)

u®v H (x I-4 (uIX)V) The Hilbert-Schmidt scalar product corresponds to the scalar product

E Xi ® yi

F X! ®

E3 (Xi I X!)1(yi 11)2

(all the sums are finite sums since we consider algebraic tensor products).

By completion, we obtain H1 ® H2

-=)-

Hom2(H1,H2)

By choice of an orthonormal basis (ei) in H1 ,

can be represented in a unique way

ei 0 yi

with

elements of H1

H2

as

Ilyill

2

< «0

In fact, the element L e® ® yi s Hi ii H2

corresponds to the operator

A E Hom2(H1,H2) defined by Aei = yi . From (8.3) and (8.2) we see that (8.4) Lemma. For any Hilbert space H, End2(H) = Hom2(H,H) is a two-sided ideal of End(H). Moreover idH e End2(H)

4==

dim H < °°

As a first application of this theory, we prove once more the basic finiteness result for the dimension of irreducible representations of a compact group. The following proof is the most elementary one. It does not use the Peter-Weyl theorem, nor the Stone-Weierstrass theorem, and is applicable to unitary representations (in Hilbert spaces).

81

(8.5) Proof of the finiteness theorem (5.8), (7.9) in the unitary case (Cartier, Godement). Take a unitary irreducible representation 7r of G in some Hilbert space H. For A E End(H), define

A = f n(x) -1 A lc(x) dx

f 7r(x) * A x (x) dx

=

as usual by

J = f (7r(x)u I Air(x)v) dx

(u I Av)

I claim that if A is a Hilbert-Schmidt operator, A is also a HilbertSchmidt operator. To prove this assertion, let (ei) be an orthonormal basis of H and compute sums over finite subsets F c I F_ II T Bill

2

= T (ai I A ei) _

= f dx (with ai

=

(ir(x) ai I Ax(x) ei)

Aei and the function f

f(x)

=

f (ic(x) ai I Ax(x) ei) dx

=

=

(1 1 f)

=

II If 112

fF defined by

E (ic(x)aiI A1{x)ei)

Using the Cauchy-Schwarz inequality several times

(EIIAeiII2) 2 4

t

If 112

=

f dx IZ(rc(x)ai I Arz(x)ei) 12

j f dx _Tpn(x)ai112. YAi(x)eII2
After simplification by ElIAeill2 IIAeuII2

F-11a i112 we are left with

=

IIAI12

To finish the proof we observe that we can take for A the orthogonal projector on a line 1v generated by a v H with IIvII = 1

x

:

then, the positive function

t--i (ic(x) v I A ir(x) v) Thus A # 0 in this case. But by

does not vanish at the origin .

construction, the operator A commutes with all 7r(x) (x r. G). Since m is assumed to be irreducible, Schur's lemma (its infinite dimensional is a non-zero scalar

version given in (8.6) below) shows that A =

operator. As A must also be a

Hilbert-Schmidt operator, we see that

H must be finite-dimensional.

c End(H) a set of

(8.6) Schur's lemma. Let H be a Hilbert space,

operators containing the adjoints of all its elements. If -P is (topologically) irreducible in H, the only operators commuting to are the scalar operators :

I' =

in End(H)

.

82

Proof. (Since the proof of this lemma involves no group theory, we only sketch it. For more details on spectral theory, cf. S. Lang 1975, Appendix 1, Th.4 p.362 .) Let T e End(H) be an operator commuting to all

operators belonging to ci

.

The operators

and

I(T + T*)

Z(T - T*)/i

(because A E

are hermitian and still commute to

implies A* E

Denote by A resp. B these operators, so that T = A + B. Since the spectral projectors of A and B are weak limits of polynomials in A (resp. B), these spectral projectors still belong to the commutor 5' of

(this commutor 4' is the intersection of the kernels of the weakly

continuous mappings

S i-+ ST - TS , T e (P ). By irreducibility assump-

tion of t , these projectors can only be 0 or 1, and A (resp. B) is a multiple of the identity operator. Observe that ' = E idH characterizes (topological) irreducibility of

in H (orthogonal projectors on

invariant subspaces commute to t).

q.e.d.

INDUCTION Let G be a compact group and K a closed (hence compact) subgroup of G. We are going to decompose the left regular representation

of G in L2(G/K) (this representation will turn out to be induced from the identity representation of K) as an introduction to the study of more general induced representations. Observe that we studied such a problem in sec.3 with G = SO3(It) and K = SO2(R). The results of sec.5

allow us to generalize the situation of sec.3 as follows. The right

regular representation of G admits a decomposition of the form

(1 ® 1C)

P

in

L2(G)

= N7r®Vn)

Since the subspace L2(G/K) e L2(G) consists of the functions which are fixed under right translations coming from elements of K,

L2 (G/K)

(V' ® V.V)

In this isomorphism, left translations of G still act through the first factors of the tensor products PK

= ®i[® 1VK

in

L2 (G/K)

(V W Vn)

83

Thus the multiplicity of the representation n in eK is equal to the dimension of VK_ (subspace of K-invariants in V,u). But the restriction n1K of 7C

in Vn can be written as ® milrj with a sum over the unitary

dual K of K and thus nIK = ® min with the same multiplicities mi . In particular for the identity representation in dimension 1 of K, we get mo

=

dim VK

Thus the multiplicity of

= 7L

dim VK in eK is m

0

and replacing 7L by it. we can

state

the multiplicity of n in Q K is equal to the multiplicity of idK in X1K Before giving the general definition of induced representations, we have to review some results on measures. The canonical

projection G -4 G/K is a proper map and the Haar measure ds of G has an image dx on G/K characterized by

µ.(f)

f(x) dx

=f

=

fG f(p(s)) ds

( f e C(G/K))

G/K

(8.7) Lemma. Negligible sets of G/K (relative to the measure dx) are those sets N for which p-1(N) is negligible in G (relative to the Haar e C(G) (or by extension any

measure ds of G). Moreover, for any

f E L1(G) )

f

dx f f (xk) dk

G/K

The measure dx

G

is invariant under left translations from G in G/K

Proof. By definition of dx ,

f

f f (x) dx

=

K

we have

dx f f (xk) dk = f f f (xk) dk dx

G/K

K

G

K

Using Fubini's theorem, this integral is also

f dk f f (xk) dx K

G

=

f dk f f (x) dx K

G

=

f

f (x) dx

G

The other statements of the lemma result from this formula (take for f the characteristic function of a negligible set N C G/K or replace f E C(G) by one of its left translates).

q.e.d.

84

Let now (6,V) be a unitary representation of K. We define the Hilbert space L2(G,V) as the completion of the space C(G,V) of continuous

functions G -* V with the norm 1f112

=

IIf(x)12 dx

J

G

The elements

f E L2(G,V) such that f(kx)

=

6(k) f(x) for all k e K

constitute a subspace H E L2(G,V). Since pf(x)JI only depends on the

coset Kx of x

for f in H ((- is assumed to be unitary), we define

f

,If 112

Il f (x) 12 dx

K\G

(f I g)H

=

(f (x) I g(x)) dx

J K\G

(for f and g in H : norm and scalar product under the integral sign

being computed in V). The induced representation P = IndK(6) is by definition the right regular representation of G in H = H e c L2(G,V):

f CH = p(s) f E H

(where p(s)f (x) = f(xs) )

.

This induced representation is unitary. For example, if T- is the identity representation of K (in dimension 1), V = E ,

L2(G,V) = L2(G)

and H is simply L2(K\G).

(8.8) Proposition. 1) The linear map HG -->. VK given by f i f(e) is an isomorphism of vector spaces. 2) Let it be a unitary representation of G. Then there is

an-equivalence

7t ®

Indj(6)

IndK(lrIK®

(r) given by v 0 f i--k (p with T(x) = 7[(x)v ® f (x)

H7c® H -- H

Proof. The elements of HG (G-fixed elements in H) are the functions f :

G - V which are (equal nearly everywhere to a) constant f(x)

=

f(ex)

=

?(x)f(e)

=

f(e)

As

p(k)f(e)

=

f(k)

=

f(e)

,

we have f(e) E VK and part 1) of the proposition results from this.

85

To check 2), let us first show that the functions T (as defined in the proposition) belong to the space of the induced representation

IndK( 7q ®o ) T(kx)

=

it(kx) v ® f (kx)

=

n/K ®a- (k) (f(x))

Next, we show that v ® f f-+ T

=

n(k) ic(x) v 0 6(k) f (x)

is mapped on the function (x)

=

IndK(fK ® T-) )

n(s)v ® p(s)f

defined by

= 1L(x)7t(s)v ® (p(s)f)(x)

_

n®p

is a G-morphism (intertwining

and the right regular representation p =

x®P (s) (v ® f)

_

=

7C(xs)v 0 f(xs) _

(P(s)(P) (x)

as desired. Now, we check that v ® f H (p

is isometric (hence injective).

If (ei) is an orthonormal basis of Hjr, every element of H, 6 H can be

written uniquely T. ei ® fi with Z IIfiII2 < 00 , and such an element has for image the function Cf = E dpi x H L 7C(x) ei ® fi (x) having norm :

II

f

f ZIlfill2

1G

Ilfix)II2 di

=

Ilk eiOD fi112

=

(the third equality is justified by the fact that (rz(x)ei) is also an

orthonormal basis of H 7r since tt is unitary). Finally, to see that

v 0 f H (P

is onto, it is enough to see that all continuous functions

N E H belong to the image. But write the (unique) expression of t(x) using the orthonormal basis (TC(x)ei) of Hr (x)

II

= Lit(x)e®®fi(x)

(x) I12

=

(fi(x) E V)

F-If(x)i!2

Therefore

i[(kx)ei 0 fi(kx)

= (kx)

= 1C(k) ® a-(k) . I (x)

_

= L it(k)lt(x) ei ® a-(k) fi(x) The uniqueness of the decomposition gives fi(kx) _ r(k)fi(x), i.e. fi E H and this concludes the proof.

86

Let us observe that if we consider T as trivial G- or K-space the isomorphism of the preceding proposition can be written as HomK(E,V)

HomG((E,H)

.

This form admits the following generalization. (8.9) Reciprocity theorem (Frobenius-Weil). Let (Tr,Hx) be a unitary

representation of a compact group G and (6,V) a unitary representation

of one of its closed subgroup K. Put P = IndK(O) and H = H

Then

there is a canonical isomorphism

MorG(Hf,H) =

MorK(H,,V)

representation spaces, the

where we take for morphisms between two Hilbert-Schmidt morphisms.

Proof. We have already seen (sec.6) that in the identification

-

Hx ® H P

Hom2(Hn,HP)

.

into the representation

the representation n ®P is transformed

A -+ p(s)

(s E G, A .E Hom(H,,H,)) ,

Art(s)-1

At least, we have seen this for finite dimensional representations. But for infinite dimensional unitary representations, we can complete the algebraic tensor product of Hilbert spaces and get an isomorphism with the space of Hilbert-Schmidt operators (observation after (8.3))

Hn® HP "

Mor(H,, ,HP

HP correspond to G-invariants in H H

Thus G-morphisms Hn -Thus we have

(H. ® HP)

MorG(H2,HP) and since H/

H

=

H

G

can be identified with the space of the

r

induced representation

TcIK ®a- (of K in V) we infer

G(Hn ® HP)

HG

-VK

=

(Hn ® V)

K

The conclusion follows then from the similar identity

(H'®

V)K ..

(8.10) Corollary. Let

q.e.d.

MorK(Hn,V)

and re K .

Then

multiplicity of win IndK((r) = multiplicity of 6 in 7LIK

87

Proof. Let us still denote by H the (infinite dimensional in general) space of

IndK(a). Any G-morphism HX -* H must send H7r into the iso-

typical component of it in H : this isotypical component is isomorphic

to a ®I Hn whence

MorG (Hm, (B I Hr)

= ® I MorG (H., Hr)

=

®I

(by Schur's lemma). On the other hand, every K-morphism HX --+ V must vanish on the isotypical components of H

IC

which are not equivalent to

(or disjoint from) 0 . Let us write the isotypical component of tr in

Hnas V®& : MorK(H,,V)

=

MorK(V ® U',V) - Mor((,MorK(V,V))

The last isomorphism associates to A : V ® e --'- V the morphism

v

'-rr

restriction of A to the subspace V ® Ev ' V

Since MorK(V,V) _ (C 1V (Schur's lemma), we have

MorK(Hn,V) = dual of E The isomorphism of the theorem implies equality of the dimension of the spaces. They are respectively Card(I)

m

=

=

multiplicity of in in

multiplicity of 6 in ttIK

q.e.d.

Observe that the statement of the theorem can be written

MorG(9c ,

IndK(o') )

=' MorK('cIK '

°- )

In the language of categories, the functors "induction from K to c" and "restriction from G to K" are adjoint from each other.

GEOMETRIC INTERPRETATION OF INDUCTION Let us just indicate how induction can be interpreted with vector bundles (for details on this theory, cf. Dieudonne 1970, Chap.XVI). If K is still a closed subgroup of the compact group G, a representation (6, V) of K defines a vector bundle

over B = K\G as follows. The total

space of this bundle is the set of K orbits in V x G with respect to the K-action

k (v,x)

=

(6(k) v , kx )

88

Here is a diagram of this fibration E

=V

B

=

KX G

9 *

K\G

= V x G/, where (v,x) N (6(k)v, kx )

(equivalence class for x ^.'kx)

Sections of this bundle are identified with functions

f

:

x ' -->

f (x) = v

with

f (kx) = a'(k)v = 6(k) f (x)

These are precisely the elements of the space of the induced representa-

tion. In other words, the space H of the induced representation IndG(6) is the Hilbert space of square summable sections of the bundle

E -* K\G associated to (TV) and the induced representation acts by right translations. On the basis K\G, the action is x -- k(x) = xk 1 and on sections (identified to functions G --> V) the equivariant action is sf(x)

f(s-1(x))

=

=

f(xs)

=

P(s)f (x)

.

Of course, one could define the induced representation (P, H) of (6,V) by using

f e L2(G,V) satisfying

functions f(xk)

6(k)-I(f(x))

=

With this other definition of induced representation, we would consider

the vector bundle with total space

G XK V Denoting by this "product" to

=

G

V / N where

(x , v)

(xk 1, 6(k) v )

[x,v] the equivalence class of (x,v) ,

has the typical property

tensor products over K 1) .

we see that

v (comparable

89

EXERCISES

1. Let H be a Hilbert space and I any index set. Prove that 2

2

Q (I) ® H and e2(I) are canonically isomorphic. More generally, if X is any measure space, prove that L2(X) S H and L2(X,H) are canonically isomorphic.

2. Give a proof of the second part of (8.8) along the following lines. Using Ex.l above and sec-5, write

L 2(G,V)

L2(G) $ v =

=

® Vi ® V,. 0 V LEG

Consider this space L2(G,V) as a representation space of G with A(s)f (x)

=

0(s) f(s-1 x)

p(t)f

=

f(xt)

(x)

Then the space of IndK((-) consists of the

1(K)-invariants in L2(G,V).

Write this space of invariants for IndK( n/K 0 a) as

® Hom(Vt , Vn 0 V) ® Vt

t

and conclude by showing that in this expression, the space VIC can be "pulled in front of the sum".

3. Let (it,H) and (7t',H') be two unitary representations of a group G. Assume that there is a (continuous) bijective operator

A : H - H' such that A 'QC(s)

=

7t' (s)

A

for all s E G .

Prove that there is a unitary operator B : H --p- H' (i.e. B is a

bijective isometry) giving a unitary equivalence of hand W. (A*A)-4

(Hint: Define T =

Borel 1972, (5.2) p.46 .)

and put B = A T. This exercise is taken from

90

TANNAKA DUALITY

9

Let G be a compact group. We consider the category (complex) finite dimensional representations of G

:

l°G of

its objects are the

finite dimensional representations (7t, V) of G and its morphisms

between two such representations are the G-morphisms

Mor(fr,r)

=

HomG(V1r, Vr)

.

Now consider a fixed element s c G and the corresponding collection

of operators 7t(s) when it or (tr, V) runs over all objects of tG . By definition of G-morphisms, for 'r and TE eG , A E Mor(7r,0 ), the following diagram is commutative

V

Ir ,

IA

s*

W

V I A

(V = V, , W = VV).

W

Still by definition

7t ®r (s) 7c(s)

=

=

1r(s)

n(s) ® T'(s)

for it unitary in

L°G

We call representation of the category (°G any family (or collection of endomorphisms

(XV)

(y E End (V,) )

or (y,,)

(parametrized by the class of objects of LOG) having the above three

properties. Axiomatically, representations of EG are collections (YV) satisfying

1. For A E

V = VIC and W = V. the following

diagram is commutative V

YV ---s

IA W

V

IA vW*-

W

91

2.

(YV) is multiplicative in the sense that

W =

(X,0-C- CP

3. For it unitary in

'G ,

0-9 =

it

Let us draw a few consequences from these axiomatic properties of representations of L°G

.

a) Let (xCo,f) denote the identity representation of G in

dimension 1. Then Yo

=

= X[

ida

0 Indeed,

YO must be given by scalar multiplication by a certain A E T

and property 2 above gives

hence 2,2 = 'A

oo® Yo

aiio®7r.

Yo

and

A=1.

b) If (It, V) and (0-, W) are two elements of L°G YV®

YV®W

,

then

'

OW

To prove this, consider the G-morphisms V -i V ® W and W --' V49 W and the corresponding commutative diagrams

--

W -- W

YV

V

YW

V

V®W-* VOW

VOW-4 VOW

,

New

NOW

Commutativity implies that Y V®W leaves V

invariant and induces yV

c) For ?C E 'G

,

(resp.

V e {Oj (and similarly W)

YW) in it. Thus Y

V®W = IV ® Y W

O jC - O 1C

Let us identify Vn 0 V,, to End(Vrz) in the usual way. Then the represen-

tation r®1t is transformed into the representation A r-> This representation leaves the scalar operators fixed, whence a G-morphism from the identity representation (o,(E) in dimension 1 into (1C ® is , End V.) furnishing commutative diagrams as shown on next page.

Using property 2 for

X n etc

= Yn 0 Yn

AH

acting by

(Ex.l.c of sec.5)

Commutativity of the following diagram gives id = 1 }

(A = id. of VV_ corresponds to 1 E E )

G).

92

v

y

t O;i

o qC

End V7r --

End V r

VV is n

V"S

(Alternatively, if (ei) is a basis of V = V. and (Ei) the dual basis of Vv, the G-morphism the commutativity of

--s

It ® 9L

o the diagram

sends 1 e O on

Ei ® ei and

requires that

Ei 0 ei = yt V Xic EEi ® ei =

Xn (Ei) ®

But the dual basis (iii) _ (jfn(Ei)) of (y,.(ei)) is characterized uniquely

by

Ei ® ei = hence

41-

(Ei) _

v

Y%(ei)

d) If IC is unitary,

V

VV

V

y

rc a End(V,)

By axiom 3,

Y7 =

(Ei) and thus

Yn is also unitary. .

But in the usual identification

(given by Riesz' theorem), we have i[ = Ire (Ex.2 of sec.2).

Hence

0

/

01

vv01r ,

t

O 1C

O7C

Now the set of representations Rep( e.)

of eG is a

group with respect to composition of the endomorphisms yV and we have a homomorphism G

--Y Rep('eG)

:

s H (1t(s))ICE`Z G

In fact, with the topology of simple (pointwise) convergence, Rep(P- G) is a topological group. Since any finite dimensional representa-

tion of G is unitarizable and a sum of irreducible representations,

there is a continuous homomorphism

Rep(CG) -- T U(V7C)

(Y

x) °G

H ()

>t E G

(points b and d above). The image is a closed, hence compact subgroup

93

of the product with which we identify Rep( e°G). The continuous homomorphism G

---*.

Rep( leG)

s

:

#--*

=

s

IV)

(X(s))

=

is injective by the Peter-Weyl theorem (first theorem of sec.4) hence a homeomorphism onto its image. In fact, it is surjective Theorem (Tannaka). The canonical homomorphism

Rep( r°G)

G

where Y V

= 1[(s)

sH

:

ys

( V = Vim) is an isomorphism of topological groups.

To prove this theorem, we need a lemma.

Lemma. Let V be the space of a finite dimensional representation it of G 1c(s) v = v for all s e G), then

If v E VG is a fixed vector (i.e.

V(v)

v

=

for all

X E Rep(tG)

.

Proof of the lemma. By hypothesis v c VG, we can define a G-morphism from the identity (TO ,¢) in dimension 1 to (7c, V) by sending 1 on v.

Thus we get a commutative diagram

iad J

J

V

VEV

0

V

whence the conclusion

Proof of the theorem. The elements of the image of G --+ Rep(eG) are the Y such that there exists an s E G with n = Tc(s) (all Tc6eG). By negation

tt- Image(G) <-=>

for every s e G there exists (Tt,V) E tG

Isuch that

YV

TC(s)

.

We show that the existence of such a y 4 Image(G) leads to a contradiction. But a condition YV

Tc(x) defines an open set in G containing

the element s. Since G is compact, we can find a finite set of elements si such that the corresponding open sets cover G. But these elements si correspond to a finite set of representations (Tri,Vi) of G. We consider

W = ®Vi (finite sum of finite dimensional representations) and XW = ® XW. (point b above). The property Y $_ Image(G) would imply the existence of 6 =(H7i with

W

$_ 6(G) C G1 (W)

.

The two compact sets

94

6(G) and 6'(G) y W would be disjoint and it would be possible to construct

a continuous function taking value 0 on the first and value 1 on the second set. Approximating such a function uniformly by a polynomial function P on End(W), we would find a polynomial with

I PI ( 1/3 on T (G)

,

P > 2/3 on 6'(G) XW .

Averaging P on G using the Haar measure, we would thus construct the polynomial Q (of degree smaller or equal to the degree of P)

Q(A)

P(o-(t) A) dt

=

.

fG

This polynomial Q would have the properties

Q(A)

=

Q(0'(s) A)

(all s e G),

Q(A)

#

Q(d'WA)

But the finite dimensional representation of G in the space of polyno-

mials

on End(W) of degree smaller or equal to deg(P) would contradict

the assertion of the lemma (with the fixed vector v = Q ! ).

q.e.d.

95

10

REPRESENTATIONS OF THE ROTATION GROUP

In this section, we consider the compact group G = SO3(R) of proper rotations in]R3. Its elements are the 3 x 3 real orthogonal matrices with determinant 1. We first have to give its Haar measure and study various classical isomorphisms.

We start by reviewing the parametrization of this group

by Euler angles (like in any book on mechanics... ). Any rotation can be decomposed into :

a rotation around the z-axis, a swing around the

line of nodes (image of the x-axis under the first rotation) and finally a rotation around around the new z'-axis (cf. picture below). Let us denote by K = subgroup of rotations around the z-axis, A = subgroup of rotations around the x-axis (both subgroups are isomorphic to the circle group SO2(]R): observe that

the subgroup now denoted by A was called K in sec.3). Thus we write

g

g

g? hB gq,

?,e,`Y

with

g? e K , he = gy,

= he

gY gy,h9-1

eg91

E g,A

E h9 K

gil

,

he-1

z

nodes

96

Multiplying throughout, an elementary computation shows that

g

g

=

g?he g, E KAK

(Observe that the order of the angles

.

T and V is thus reversed, and we

have now rotations around fixed axes Ox in A, resp. Oz in K In particular, we have a decomposition

G

= KAK .

Since the Haar measure dx of G satisfies

f f (x) dx

=

G

J G/K

with the invariant measure dX

dX f f (xk) dk K

on G/K identified to the normalized

invariant measure diL on this sphere, we have

dg?,G,y

4n diL

=

8n2 sin9d?d9dy

(, , 9 , i') E (0,2rc)

(O,n)

(0,21r)

UNIVERSAL COVERING OF SO3 AND CLASSICAL ISOMORPHISMS Before we start representation theory, we study the universal

covering of the rotation group G = SO31R). This group G can be identified to the group SU2((E) consisting of 2 x 2 complex matrices g with g* g

=

1

,

det(g) = 1

.

As is easily seen, this group consists of matrices

( g

=

v

u)

lull

,

+

lvJ2

=

(u,v6E).

1

The two complex parameters u and v of g are called the Cayley-Klein parameters.

We shall establish the following isomorphisms

(G = ) 2:1

( G

= )

S3

=

N

H1

SU2(cC)

(b

]P3(]R)

(Cayley-Klein)

/(c) SO3(R)

(Euler)

Here, we have denoted by H1 the group of (real) quaternions with norm 1: obviously, this set is identified with the unit sphere in ]R4

= H .

The other identification (homeomorphism)]p3(I R)

"--

3(R)

can be seen as follows. Let g be a rotation, w its axis (Uwll = 1)

97

and 95 < n- .

its angle (counted with the corkscrew law!) so that

In this way, the rotation is identified with a point

of the ball of radius w (antipodal points being identified). But this (full) ball can be deformed on the northern hemisphere of a sphere S3 in]R4 as suggested in the picture.

But]P3(]R) is precisely obtained by identification of antipodal points of S3 so that the homeomorphism between SO3(]R) and1P3(l) is now clear. (a) Let us consider the vector space 1H of real quaternions as a

complex vector space with basis l,j (l,i,j,k being the usual basis

of 1H over]R: i2 = j2 = k2 = -1). For this, we consider the scalar

multiplications

q=) q

2 q in H)

and product

( T E CE, q e I-I

Then, for g e-F

g

q H qg

1

qg

=

is CE-linear and defines a representation (with complex dimension 2)

of G = Hl . Obviously, taking successively g = i , j , k ( E Fl ) 6i -

(i 0 `0 -i/

'

and finally for g = u + vj

bj (det

0

=

g

1)

0

k = (i

Cl 0/ '

it

01

= N(g) = gg = 1)

u

g = rv

u

Thus we obtain the first isomorphism G - SU2((E) (b) The group

=1H

1

acts by conjugation on the space

E of pure quaternions (E is a real subspace of dimension 3 of H )

v e(g)v

=

g v g-1

=

v'

N(v')

,

= xi+yj +zk, =

N(v)

x2 + y2 + z2

=

N

N

We obtain in this way a homomorphism G -4- 03(]R), and since G is connected, its image must be contained in SO3(I2). An easy verification

shows that its kernel is the two elements subgroup {tl}

.

.

98

(c) Let V be the real vector space (of dimension 3) of 2 x 2 complex matrices which are hermitian and have zero trace. These

matrices can be written =

X

- det X = x2

(hence

( yXiz y+z )

+

y2

+

z2 ) .

The group SU2(G) acts in this space by conjugation

tg(X)

=

gXg*

= gXg 1

whence a homomorphism

t

:

--*

SU2 ((E)

S03 (ft)

with kernel {tl}. The lifting

SO3(R)

--0 SU2(U)/{±1 C Sl2((E)/{±ll

can also be given in terms of the Euler angles as follows. To determine

the image of a rotation

gf,A,

g

11

rr

gyh9g4' ~' L v u}

±1

v u/

we consider separately the two cases h9 and gT and use stereographic projection (cf. picture at the bottom of this page). 1) The rotation gf is projected on the rotation

H e

of the same angle in the a -plane. This rotation is given by the matrix

i [e 0 2

e 0Z

E Aut hol(U)

=

S1 2((E)/{±1}

']

Stereographic projection from the north pole N of a sphere of radius ' , on

the tangent plane to the southern pole S .

99

2) The rotation h9 of angle 9 around the axis 0 is projected on a homography (or fractional linear transformation) that is completely determined by its action on the imaginary axis Sy in the

Z-plane. If y = tg2o

y'

then

,

tg2(or+A) - tg2« + tg28 1 - tg Za( tg29

=

so that

tg29 -y tg 29 + 1

y y,

iy,

+

cos 29 y + sin 29 i2 sin 29 y + cos 29

_

cos 29 iy + i sin 29

=

isin

29

cos

29

i

cos 29

And the homography corresponding to h9 is given by

cos 29

i sin 29

[i sin 29 cos 29 Combining

two

the

results for g, and h9 , we find by matrix multipli-

cation

_

g

_

g,

rcos 29 e12

i sin29 e12

g be a rotation of angle 4' around a certain

axis. Then Re(u) in terms of the Cayley-Klein parameters,

cos 2,O =

cos 24 = cos 29 cos 2 ((f +y') in terms of the Euler angles

.

Proof. The angle of rotation is invariant under conjugation (in SO3 and in SU2). Consequently, we can find the angle cp by diagonalization.

-

Let us find the eigenvalues of the 2 x 2 matrix with Cayley-Klein

parameters u and v :

ua u--

=

I

a2 - 2 Re(u) A + (uu + vv)

14' whence

a 12

= 1

1,2

=

Re(u) +- i

l - Re(u)2

_

0

,

1

q.e.d.

100

We can use the preceding proposition to determine conjugacy classes in SO3(JR). First in SU2

g

=

vl

(u

is conjugate to I

Thus we can always choose the eigenvalue Im(A1) > 0 and we fix 2 = e12 In this way,

e12' 0

and to

,

/

.

= al = e12o so that

20 x , i.e. 0 < q5< Dr.

with 0

is the unique solution of cos 20 =

(P

e-lb 0

\ 0 e0 e12

\-v u /J

Re(u) with

E [0,2n]. In SO3(]R) = SU2((r)/t±1} we still have

so that

and

4)

2 1r

- 0 lead to conjugate matrices, and we can choose

the solution 0 with 24> E [0,n/2] , namely c4E[0,7E] . We do this by putting Re(u)l

cos 2

(u = ul+iu2)

lull

=

and by choosing the representation

cv

with

V

0 < u1 < 1

U Remarks

1) To go from Cayley-Klein parameters to Euler angles, one

has to solve IuI

=

(unique 9 E [0, 7t [) , sin 29 =Iv

cos 29

J

and one must have

argu

2(f -

+ n)

=

argv

whence Cf

=

argu + argv - ITC

(mod 2 Tt)

t'

=

argu - argv + 21t

(mod 2 TC )

The closed curve T -+ g

(parametrized by p E[0,27L]) is lifted in

SU2((E) in the (non-closed) curve

e'2T `P

I\

(

0

0

with extremities 12 and -12 in SU2(U). This proves that the closed path

(p H g,

is not homotopic to a point in SO3(IR).

101

2) If 1H is regarded as CE-vector space with scalar multipli-

cation given by

A

H1 -> G12((U)

,

the left regular representation

is given by

_v

u

(v u)

LT

g

With the scalar multiplication

2 q

the right regular representa-

,

tion leads to

Tg

Cv u

As we see, to insure (U-linearity, we have to take group action and

scalar multiplication on opposite sides. 3) The subgroup

A = Jgr :

046[0,2x] I C SO3(]R)

is a maximal abelian subgroup. Indeed, any commuting family of rotations (each rotation is a semi-simple operator) can be simultaneously diagonalized, hence has a common rotation axis. In particular, all commutative subgroups of SO3(JR) are conjugate to a subgroup of A,

and all maximal abelian subgroups of SO3(R) are conjugate to A.

REPRESENTATIONS

N

We define a sequence of representations of the universal

covering G = SU2((U) of G = SO3(JR) as follows. For an integer n > 0 (i.e. n 4_=1N), let Vn denote the space of homogeneous polynomials of

degree n in two variables (and complex coefficients)

:

V®n}.

Vo

=

(E

V1

,

=

V ,

...

Vn ={symmetric tensors in

,

We'define the action of

M =

(

v

u) e SU2(E)

on polynomials by right multiplication P

=

P(zl,z2) H PM

=

P(uzl-Vz2,vz1+uz2) We can de-homogeneize by p(z)

=

P(z,l)

(conversely

:

P(zl,z2)

=

z2 p(zl/z2) )

.

102

Thus the above action on polynomials of degree
p

P(z) H PM = (vz+u)n p(vZ+u)

=

p e Trn (space of complex polynomials of deg <. n in z)

In particular, the action of M = -12 is given by p a-r p_1 = (-1)n p of

and we obtain a representation

SU2((E)/{±1}

SO3(R)

precisely when n is even

the corresponding dimension dim Trn = n+ 1

:

is odd. Let us take such an even integer n = 2e and define the

following diagram by commutativity SU2(U)

Gl(Tln)

,

,0

Dn/2

Dn/2

(e = In integer)

SO3 (]R)

We obtain two sequences of representations for

SU2(t)

:

Do = id., DZ , Dl dim Dn/2

for

SO3 (IR)

:

=

,

Dn/2

...

n + 1 > 1

Do = id., Dl , D2 , ... , Dt , .. .

= 2e+1>1

dim De

.

(10.2) Theorem. The character of Dn/2 is given Xn/2(g)

=

0g E [0,n]

where

Un(cos z lg)

is the angle of the rotation g and Un is the nth

z

Tchebycheff polynomial of the second kind. Similarly, for in = e integral, is given by

the character of Dt,

xe (g)

=

U2e(cos

Og ) z

( i Og E [0, Zn] angle of rotation g).

Before we prove this statement, we remind the reader that the

Tchebycheff polynomials of the first kind can be defined by

Tm(cos 9)

=

cos m8

.

103

... is orthogonal on [-1,1] with respect

The sequence To, T1, ... ,Tn

to the density (1 - x2) 2

:

+1 J

dx

M(x) Tn(x)(1 - x2)

=

for m # n

0

1

-1

These polynomials satisfy the normalization condition m(1) = 1.

The

Tchebycheff polynomials of the 2nd kind can similarly be defined by

= sin (m+1)0 /sin @

Un(cos A)

The sequence Uo, U1, ...

(1_x2)4

to the density

Un .

.

... is orthogonal on [-1,1] with respect

2

+1

S-1 m(x) Un(x)

1 - x2 dx

for m

0

=

n

They satisfy the normalization condition m(1) = m+l Proof of the theorem. Let us simply make the computations for SU2((E), taking the basis

1, z, z 2 , ...

, z

n

of 1Tn

consisting of simultaneous eigenvectors for the action of the i2

matrices

0,

( e

0

e-lzf

Such a matrix acts by

l

)

Thus the eigenvalues are

e-in 2

ein 12T

and the corresponding trace is

ei

.V

xn/2(gf)

- Z n)w = e -i 2 np

el(n+l)(p

1-e

o6N)Gn _

1-

e-1(n+l)29' - el(n+l)!?

e-'2' - e"I0

sin (n+l)Ztp

=

Un(cos

q.e.d.

sin I The reader has observed that in the case of SO3(R)

the polynomials U22 with even index appear.

,

only

They form an orthogonal

basis of the space of even polynomials on [-1,1] (with the above density).

104

(of G) and D't (of G) are

The representations Dn/2

To prove it, we shall use the criterion

irreducible for all n , t e 1N .

of (7.3). For this purpose, we have to know how to integrate central

N

functions on G

(or G)

(10.3) Lemma. Take G = SO3QR) and y e Cinv(G) and write

T(g)

N

f(coszg)

=

N

(and f(-x) = f(x) = f(x) for -1Sx,< 0).

Then we have

fG f(cos20g)dg

f0 f(x) 1-x2 dx

=

=

2 7C

=

1

l - x2 dx

r f (x) 1

and

1 f(cos 1$g) dg

=

c

f f(cos 10) sin2'0 d(IO) 2 2 2

G

(The constant in this last formula is easily determined by taking f = 1,

dg = 1.)

using the normalization of the Haar measure

fG Proof. Let us make several changes of variables, with the purpose of ending up with

x

=

cos 24

=

cos ,2

ul = Re(u) )

( =

We start with P

@

,

0

=

2 (9' - i)

z=

,

1(F + y.,) 2

The absolute value of the Jacobian of this transformation is ; sin 29

Thus we have =

(8x2) -1sin9d9dT di

=

(81t2 ; sin Z9)-1 sin 9 dp da dt

ul

=

P cost (= cos 29 cos 2 (lP +') = ± cos Z O )

`u2

=

psint

dg

_ =

7G-2

P dp dcr d2

Then we put

with Jacobian

a

,u2)

a (P,t)

=p

>

(thus we have u = u1 + u2i = P e1z

p dp d2

=

duldu2

). This proves

dg

=

n-2du1du2d r.

105

The domains of variation of these variables are indicated in the following

pictures

U

1

, t 6 C0,Zit] , t fixed

keeping u1 and u2 fixed

tr = t - VE[ t,t-2lt]

(i.e. P and r fixed) T,

f d6

( for all t).

27r

=

t- VG Integrating in u2 (u1 still fixed) we get

J

dug

=

2

1 - u,

1 -uA

Finally,

fG f(cos 210g) dg

=

7C

J1 f(x) 1-x2 dx

fO du1 f(ul)

1-ul

f 1 f(x)

1-x2 dx

J =

0

=

-1

as claimed.

(10.4) Theorem. All representations Dn/2

and D, (of G ,

reams

. G) are

irreducible (n, E F -IN) . More precisely, the dual of the compact

group G consists of the representations

and the

Dn/2 (n s 1N)

dual of the compact group G consists of the representations

Dt (L E N ).

Proof. The Tchebycheff polynomials of the second kind Un (n r. N) form a

total orthonormal system on [-1,11 with respect to the density The even polynomials U2P to the density ,

1 - xZ dx

1

x2dx.

have the same property on [0,1] with respect . Thus this theorem results from (7.3) and

(10.3).

q.e.d.

(10.5) Theorem (Clebsch-Gordan). For E , m E IN, the representation

D, 0 m of S03(]R) is equivalent to Die-mi ® ... a De+m

.

106

(Thus DE (& D

m

is a sum of 2N + 1 inequivalent irreducible representations,

the integer N being the minimum between Q and m.) Proof. Assume for example that m < t. We have seen (10.2) that the

character

is given by

of DC,

xt(gj)

=

e-iey

+

... + el

For simplicity, replace elf by t

t =

:

$

(1 + t + ... + t29) = t o+, t e 1 t2Q+1 = t-

1-t

1-t

We have also seen that the character of a tensor product of representations is the usual product of characters (sec.7), hence

xlk®D,

t -e - tQ+, 1-t

(91

But

t

o+µ

1µl4 m

=

T lµl< m

IN` m

t-

t

1- t

L

(exchange f and µ)

so that we can still write

t-P-'" - tl+p+1

x1)®DM (g) IN <

m

Xt-m + (when m
1-t xE-m+1

+

Xt+m q.e.d.

107

EXERCISES

1. For m _< e r .3V ,

Dt-m ,

DG-m+1

'

compute the sum of the dimensions of ...

. De+m

(Either proceed directly without using (10.5) or use the fact that the dimension of a representation is the value at e of its character.)

2. Determine the dual of the group U2(¢'). (Hint. Denote by

Z = Z2 the center of this group, so that Z /)SU2(Q') = 1il}. Observe that any irreducible representation of U2(0) has an irreducible restriction to SU2(0). Identify finally the dual of U2(0) to the part of the product of the duals of Z and of SU2(a') consisting of representations agreeing

on the matrix -12 .)

3. Let 1t be the canonical (3 dimensional) representation of G = SU3(Q). a) Show that it and ff are inequivalent irreducible represen-

tations of G (use sec. 7).

b) Identify the representation Ad :

G

--1

GL(M) ,

1C4)7C with

M = M3(¢') = End(V) with V = 23

g F-* (A I-* g A g 1) (Use the generators u ® v of M with u and v in V.) c) Give the decomposition of n ®1C into irreducible components. (The subspaces M0 of zero trace matrices, and M1 of scalar matrices are

both invariant under the Ad representation.) d) Let p be the eight dimensional irreducible representation of G. Show that the restriction of p to the diagonal subgroup T of G contains the identity representation of T with multiplicity 2.

(This representation P is used in physics to interpret the symmetry structure of octets of baryons and of mesons.

In particular,

the meson *) - companion of the meson 770- was invented before it was

discovered simply to account for the above multiplicity 2. The threedimensional representations 7C and 7C correspond to triplets of quarks...)

108

4. Compute the following coefficient (up to a multiplicative constant)

(zP / D (aB) ze )

(a

B

B sin e = image of ( cos sin a cos e ) in SO3pa)

of the representation Dl of SO3cR) in the space of polynomials in z of degree G 2 e .

(Observe that since the basis l,z, ...

,

z2e is an

orthogonal basis, this coefficient is proportional to the coefficient

of ze in the expansion of

De (ae) z e =

(z sine + cos 0) It (z cos e - sin e) e . )

In particular, show that this coefficient is proportional to the Legendre polynomial Pe(cos 2e) . (Recall that

aa e(r2 -

2ee.'

1)e

)

5

More generally, show that the coefficient

(z

f+m

ti D't

ze)

is

proportional to the associated Legendre function

(-

PP(cos 29) defined by

m

'm(

y

PPS )

=

(i - 2) fml

lp()

Using Ex.5 of sec.7 (part c), give an integral relation

between Legendre polynomials and Tchebycheff polynomials of the second kind.

.Q )

PART

II

:

REPRESENTATIONS OF LOCALLY COMPACT GROUPS

ill

11

GROUPS WITH FEW FINITE-DIMENSIONAL REPRESENTATIONS

In this section, we show that for non-compact groups, the consideration of finite-dimensional representations is not sufficient. Theorem. Let G be a connected solvable group. Then any finite dimensional irreducible representation of G has dimension 1.

Proof. Denote by G' = [G,G] the closure of the subgroup generated by the commutators of G, and by induction G(i) =

(G(i-1)),

G(i)

Thus

G('-') and since G is solvable, G(h) ={e}

is a closed normal subgroup of

for some integer h E N that we take minimal G D G'

G" ]

G(h)

...

= {el

Let us call this minimal integer h such that G(h) _ }e}, the length of G. We shall prove the theorem by induction on this length h. For h = 1 (i.e. G commutative), the theorem follows from Schur's lemma. Let now h > 1 and assume that the theorem has already been proved for solvable groups of length < h-1. Take any finite-dimensional irreducible representation (ir, V) of G and decompose the restriction of it to G'

into irreducible components (by induction assumption, these irreducible components will have dimension 1). For X E Hom(G',r"), we put

= tv EV : 7G(s) v = ,(s) v

Va

The non-zero subspaces Va

are linearly independent

:

take any linear

dependence relation

aivi

T_

=

(vi e Va )

0

1
i

with k minimal. Then, obviously I al(s) aivi 0

=

= 0 and also

7c (s) Y aivi = Zai(s) aivi

whence by subtraction

Z

2
(AI(s) - Ai(s)) aivi

=

0

contradicting the minimality of k. In particular, the set of -AE Hom(G',(C`)

112

such that VV # {0}is finite. On the other hand, the group G acts on this set of characters (by conjugation) :

1r(S)n(g) v

=

if v

ir(g)rc(g-lsg) v

=

.

VA

1C(gA(g-lsg) v

(s,g c G),

so that with A g(s) = A(g lsg) we have

it(g) V C VI

(g e G). g

The action of G on this finite set of characters is continuous, and

Ag = A for all g e G The preceding argument shows that V. is

G being connected, it must be trivial

Take now any Va # {0}.

:

.

invariant under the action of the whole group G, hence V. =

V by

irreducibility. Thus G' acts by scalar multiplications in V

1iIG,

=

Let us now take an element s e- G and an eigenvector v of 1c(s), say

= t,v

1C(s) v

(v # 0, µ ¢ 0 ) .

From the relation 7t(s) rr(g)

=

lsg)

follows that 1C(g) v is also an eigenvector of tt(s) corresponding to

the eigenvalue T(s-lg-1sg)

µ

= µ a(g)

But the set of eigenvalues of 1L(s) is finite and the action of G on this

set is continuous. As before, G being connected,

µ,a(g) must be

independent of g

µ

P) = µ X(e) = µ

.

This proves that the rt.-eigenspace of 1[(s) is G-invariant (and #

hence equal to the whole space V by irreducibility This is true for any s E G dim(V) = 1

:

:

{0})

1r(s) =

G acts scalarly in V and by irreducibility,

.

q.e.d.

Corollary 1. Let G be as in the theorem and denote by G. the intersection of the kernels of all finite-dimensional irreducible representations of G. Then Go contains the commutator subgroup G'.

113

Corollary 2. Let G be the solvable subgroup of Sl2(1R) consisting of

the matrices

(01/t)

(t>0,uelR)

and U its subgroup defined by t = 1. Then, in any finite-dimensional irreducible representation of G, U acts trivially. Proof. Just observe that G is connected and G' = U. Corollary 3. Any finite-dimensional unitary representation of the group S12(]R) is trivial.

Proof. Let G be the connected solvable subgroup of S12(JR) defined in

the corollary 2. By (2.1), the restriction to G of any finite-dimensional unitary representation can be decomposed as the sum of irreducible ones.

By Cor.2 above, the matrices

(1 u) 0

0)

(1 u

1

_

0

1

1

-1 )(1 -u 0

0

0 -1)-1 1

1

0

act trivially in all these irreducible components, hence act trivially.

Now, if a matrix

(c

Sl2(IR)

a)

has a coefficient c # 0, it can be written a

b

c

d

_

1

0

(a-1)/c 1

1

0

1

(d-l)/c

c

1)

0

1

hence acts also trivially. Finally, if c = 0

a b) 0

=

d)

-b

a) (0 -1)

(-d 0) (1

0)

must also act trivially as we have just seen (product of two matrices with c

0).

Corollary 4. The finite-dimensional unitary irreducible representations

of the Heisenberg group G consisting of the matrices

/1 a c I0 1 b 0

0

(a, b, c E ]R)

1

have dimension 1 and correspond to the characters of the abelianized group Gab = G/G'

]R 2

114

Proof. In our case, the derived subgroup G' consists of the matrices with a = b = 0 (and coincides with the center of G). Thus G' acts trivially in any finite-dimensional irreducible representation. In particular,

G' will also act trivially in any finite-dimensional completely q.e.d.

reducible representation.

It is possible to show that for any non-trivial character

,(: Z = G' -* T

of the center of the Heisenberg group G, there is

one and - up to equivalence - only one (infinite dimensional) represen-

tation it of G in a Hilbert space inducing the character x on Z. Moreover, it

is equivalent to a unitary representation precisely when

its central character x is unitary ( Ex. 2 at the end of this sec.) . The above corollary 3 admits the following generalization. Theorem. Let G be a connected semi-simple real Lie group having no invariant compact subgroup / tel. Then the only finite-dimensional

unitary irreducible representation of G is the trivial representation (in dimension 1).

Sketch of proof. It is enough to prove this theorem when G is simple (the center Z of G is then a discrete subgroup and G/Z is simple in the abstract algebraic sense). We use the following properties : a) there exists a one-parameter subgroup ]R = H c -G and

elements gn e G such that gn h

gnl

-- e for all h e H

b) if H is a non-trivial one-parameter subgroup, the union of all conjugates of H contains a neighbourhood of e in G. From these, we infer that for any finite-dimensional representation

(7t,V) of G 7t(gn) 7r(h) rr(gn)-1

- 1V

(h E. H)

whence

d

=

dim V

=

Tr 1V

=

lim Tr 7t(gn) 7t(h) ir(gn) -l = Tr ir(h)

Fix h E H and put 7t(h) = A = R+ i S with R= i (A + A*) and S = 2i (A - A* ) both hermitian. Then d = Tr A = Tr R + i Tr S so that Tr S

But it is unitary

=

0

,

Tr R

=

d

(a priori, these traces are real).

115

IIAII

= Ir(h)N

=

1

IIRII

=

112 (A+A*)II < 1

.

Since R is a hermitian operator, we infer that

R41

1-R>0,

and

so that

= Trl-TrR = d-d =

Tr(l-R)

0 .

But the positive operator 1 - R can have zero trace only if it is zero R

But A is unitary 1

=

,

A

= 1+iS

AA*

=

(1+iS)(1-iS)

1

=

:

SZ

This proves that

tn(h)

0

= =

S

=

0

=

1 + S2 A

=

1

1 for h e H. Since Kent is a normal subgroup

of G, it contains all conjugates of H, and by b) above, contains a neighbourhood of e e G. But G is connected, hence generated by any neighbourhood of its neutral element.

q.e.d.

116

EXERCISES

1. Let G be a compact solvable group. Show that the neutral connected component of G is commutative (this neutral connected component

is solvable: one can use the first theorem of this section and (4.1) In particular, any connected compact solvable group is commutative But there are some non-commutative finite solvable groups: Also show that if p is any prime integer, the subgroup of Gl2(Wp)

consisting of the matrices

a

b) with a E ap and

0

b E ZLp

is solvable (and compact: ).

2. Let G be the Heisenberg group and x : I?

¢'"

a

character. We consider the space V. of functions f on G satisfying

/1 x ff0 1

z y

0

1

0

J

=

!n (x) x(z)

for some Co L2(BZ).

With the scalar product of the space L2(P),

VA is a Hilbert space.

a) Show that the right regular representation of G acts on V. (it leaves VX invariant). b) The action of the center of G on Vx is given by the character x .

c) The representation V.

is unitary precisely when 'T is.

d) Assume x unitary and

# id. so that the characters

/Zb

:

x H It (bx)

represent all unitary characters of B2. Irreducibility of Vx

(b EJR ) is then

equivalent to irreducibility of L2(I2) under translations and multiplica-

tion by unitary characters.

117

12

INVARIANT MEASURES ON LOCALLY COMPACT GROUPS AND HOMOGENEOUS SPACES

Let G be a locally compact group and Cc(G) the vector space of (complex) continuous functions on G which vanish outside some compact subset of G. A Haar measure on G is a positive continuous linear form m

Cc(G) - C

:

,

f t-+

m(f)

=

fG f(x) dx

invariant under left translations m(sf )

=

m(f)

(for f E Cc(G), s e G, sf(x) = f(s-lx) ).

Continuity refers to the following property: for each compact subset K of G, there exists a constant CK such that

jm(f)j 6 CK Suplf(x)l for all f E Cc(G) vanishing outside K. X Positivity simply means m 0 and

0 (f e Cc(G)) ; m(f)

f

0

The existence of a Haar measure on any locally compact group is a standard (technical) result that we are going to take for granted (cf. Weil, 1953 p.34 or Bourbaki 1963, Int. Chap.7 p.13). Uniqueness up to a positive multiplicative constant is however easily proved as follows. Let m and m' be two Haar measure, and define the measure n by n(f)

(f E Cc(G) )

m'(f")

=

Thus the measure n is invariant under right translations. We show first that n has a density with respect to m : for f and g r= Cc(G) m(f) n(g)

=

m(f) f g(ts) dn(t)

= f dn(t) fdm(s) f(s) g(ts)

f dm(x) Jg(x)

= f dm(s) f(s)f dn(t) g(ts)

= f dn(t) fdm(x) f(t-lx) g(x)

f dn(t) f(t lx)

.

J

From this follows that n = w m with a density function w(x)

=

f f (t lx) dn(t) /m(f)

independent of f provided m(f) # 0 (e.g. f > 0 and 0 # f E Cc(G) ).

_

_

118

Observe that this density function w is continuous (it is given by an integral depending continuously in the parameter x) and the support of

n is the whole group G, so that w is the unique continuous function with n = w m . Take now x = e in the definition of w

w(e) m(f)

n(f')

=

=

for all f > 0, 0 # f E Cc(G) .

m' (f)

This proves m' = w(e) m proportional to m as claimed. From the relative uniqueness of Haar measures follows that if

G -o- G is a (continuous) automorphism of G,

f -* m(fa)

fa(x)

where

=

(f EC(G) )

f(o(-1(x) )

is proportional to m, say m(fa)

=

modG(a) m(f)

With the integral notation, this relation can be written

fG f (0(-l(x)) dx

(dx = dm(x)).

mod G (0r) fG f (x) dx

=

In particular when of = Int(s) is the inner automorphism produced by an element s e :G , we put

AG

:

AG(s) = mode (Int(s))

G - 1R+

modular function of G

,

By definition, we have thus r

(12.1)

f f(sxs-1) dx G

= Q G(s) f f(x) dx

f(xs-1) dx

= J G

G

Extending the Haar measure m to Borel or measurable subsets of G, the preceding relations can be written in the perhaps more natural form

m(o((A) )

=

modG(o() m(A)

m(slA s)

=

m(As)

(12 . 2)

The modular function e G

:

=

AG(s) m(A)

G -*. R+

is obviously a homomorphism.

It is continuous since by (12.1) we can write it

4(s)

f(xs-1)dx/ fG f(x)dx

= fG

for any f e Cc(G) with m(f) # 0 for s in the center Z of G,

.

In fact, since obviously A G(s) = 1

the modular function can be regarded as a

function over G/Z: (12.3)

QG

=

1

when

G/Z is a simple group ,

or G/Z compact .

119

(12.4) Lemma. Let G be a locally compact group, dx = dm(x) a Haar

measure on G. Then with A = Q G

m(fg)

rG f(x 1) dx =

=

r

JG

(X) dx

m(f/0)

=

for f E Cc(G) (or f 6 L1(G)...). More simply d(x1) = A(x)-1 dx Proof. One checks immediately that the two measures d(x-1) and A(x)-1 dx are invariant under right translations, hence proportional

d(x)

c p(x) -1 dx

=

Replacing x by x 1 in the preceding equality, we have dx = c A(x) d(x)

and substituting the expression for d(x)

dx

=

c p(x) c A(x)-1 dx

=

c2 dx

Since all these measures are positive, c > 0

c = 1

and

q.e.d.

(12.5) Proposition. Let G be a locally compact group and H a closed

normal subgroup of G. Then the modular function G1H

of H is the

restriction to H of the modular function AG of G. Proof. Since H is closed, G/H is a Hausdorff space. Moreover, since the canonical projection p

:

G --- G/H is an open map by the definition

of the quotient topology, for any compact neighbourhood V of e in G, p(V) is a compact neighbourhood of 6 in G/H .

In

particular, G/H is a

locally compact group. Let fn denote a Haar measure of G/H. Then the

linear form

f '-}

I

dm (x) f f (xh) dh

G/H

is a Haar measure m on G .

H

(f ccc (G)

Replacing f by a right translate fs (for s r. G

we put fs(x) = f(xs) ) we obtain immediately when s e- H d G(s)-1

=

AH(s)-1

q.e.d.

We say that a locally compact group G is unimodular when its modular function is trivial: A G = 1. Unimodular groups are those having a left and right invariant (Haar) measure. (12.6) Corollary. Let G be a locally compact group. Then the kernel of

the modular function QG is the largest closed invariant unimodular subgroup of G.

)

120

HOMOGENEOUS SPACES Let H be a closed subgroup of the locally compact group G.

We are going to investigate when there exists an invariant measure on the homogeneous space G/H (G acts by left translations on this space). If dx and dh are Haar measures on G and H respectively, any invariant

measure m (or dm(x)) on G/H should satisfy

(*)

f f(x) dx

f

=

dm(x) f f(xh) dh

G/H

G

(f E Cc(G)) .

H

Applying this equality to a right translate of f by an element s E. H

gives the condition 6 G(s) = 6H(s).

Thus, an invariant measure on G/H

can exist only if

AH =AGIH We are going to show that this condition is sufficient. Quite generally,

let us define a function fH on G/H when f E Cc(G) by fH(z)

=

f f(xh) dh H

Thus, (*) can be rewritten

f f(x) dx

=

f

dm(x) fH(x)

G/H

G

In particular, we see that if an invariant measure m on G/H exists, (**)

fH

=

f f(x) dx

0

=

(f E C(G) ).

0

G

This is the crucial consequence of (*). Indeed, if (**) is satisfied, the value of m on the special functions

f E Cc(G))

cf =

fH E Cc(G/H) (for some

is given by

m(T)

=

m(fH)

= fG f(x) dx

independently from the choice of f

fH

=

:

gH = (f - g)H fG (f - g) dx

=

(*)

=

0

0=

fdx

=

rG gdx

fG

According to these preliminary observations, the strategy is as follows

a) prove that all 16 Cc(G/H) can be written (for some f E Cc(G)

)

,

=

fH

121

b) show that AH = 6G I H

= (*) holds

.

For reference, we formulate these two steps as follows.

(12.7) Lemma. The linear mapping Cc(G) -- Cc(G/H)

, f

fH

is surjective.

Proof. First of all, letting p

:

G --> G/H denote the canonical projec-

tion, we show that every compact subset A of G/H is the image A = p(K) of some compact subset K of G. Indeed, when U is an open relatively compact set of G, p(U) is also open and relatively compact in G/H. These sets cover A so that we can find a finite number of open relatively compact subsets Ui of G for which the union of the p(Ui) cover A. Thus,

we can take for K

P (A) () UUi Then, it is enough to show that a positive function T E Cc(G/H) with

compact support A = p(K) in G/H is of the form fH for some positive continuous function f with support in K C G .

Cc(G)

9

,

But there exists a

9 > 0 on K (hence 9 > 0 in a neighbourhood of K).

The obvious formula

(T e)

H

= T eH

-

shows immediately that T =

(9

(?/9H )H

=

fH with

f = 9 T/9H Since 9 has compact support, f also has compact support and since 9H > 0

on the support of T,

T / 9H is well-defined and continuous.

q.e.d.

(12.8) Proposition. Let H be a closed subgroup of the locally compact

group G. An invariant measure exists on G/H if and only if AH dG

IH

Such an invariant measure m on G/H is characterized by

ff(x) dx

=

fG/H

rH f(xh) dh

fH(x) dm(X)

=

fG/H for all f E Cc(G)

.

Proof. It only remains to show that the assumption on the modular

function of H implies that (*) holds. But when f and g E Cc(G) ,

Fubini's theorem

(for continuous functions on compact spaces) gives

122

f f(x) gH(x) dx = f dx f(x) r dh g(xh) G

f dh f dx f (x) g(xh)

=

By assumption, and since d(h-l)

f dh f dx f (th-1) g(x)

= =

_

JH

G

AG(h)-1

AH(h)-1 dh (12.4), we can still

write the preceding integral as

dx ('f(xh-1)

dH(h)-1

g(x)

dh

=

f dx f f(xh) g(x) dh

f f (x) g(x) dx

_

G

Consequently,

fH

JG

implies

0

=

f (x) gH(x) dx

=

0

for all g e C

(G)

c

But by the preceding lemma (12.7), there exists a function g for which gH(x)

1 for all x r- Supp(f)

=

f f (x) dx

=

.

.C(G)

Thus

0

G

as was to be shown.

q.e.d.

One should observe that when H is a compact subgroup of G, the condition of the proposition (12.8) is automatically satisfied.

In this case, p t

p

:

.

G -+ G/H is a proper map and its transpose Cc(G/H)

---9-

Cc(G)

(composition with p)

allows one to define the invariant measure m on G/H simply by composition of the Haar measure of G with tp : m is then the image of the Haar measure of G by the proper map p (cf. proof of (3.1) and (8.7) ).

When a quotient G/H has no invariant measure (cf. examples below), it may still be useful to consider relatively invariant measures on this homogeneous space. By definition, these measures are those which are multiplied by some constants under translations.

More generally, quasi-invariant measures on G/H are those which are multiplied by functions under translations. According to the RadonNikodym theorem, the quasi-invariant measures are characterized by the fact that their family of negligible sets is invariant under translations.

123

EXAMPLES 1) The Heisenberg group consisting of matrices 1

x

z

(0

1 0

1

`0

y)

(x, y, z e IR )

is unimodular. Indeed, the measure

dx dy dz is both left and right

invariant. 2) Let K be a locally compact (non-discrete) field and dx a Haar measure on the additive group of K. For any a e K",

x H ax

is an automorphism of this additive group and thus d(ax)

modK(a) dx

=

=

I a I

K

dx

a H IalK

for some positive constant IalK . Moreover,

homomorphism K"

is a continuous

On the multiplicative group K", d"x =

lxjK'dx

is a Haar measure.

3) Let G denote the subgroup of upper triangular matrices

of Sl2 (R). The identity a b

x y) =

(0 a-1/l(` 0 x l)

ay+bx 1

ax

l

a-lx 1/

`0

shows that

x 2dxdy is a Haar measure on G (observe that d(ay + bx 1) = d(ay) = laldy)

On the other hand,

1x y lra

b l)

=

0 x-1/\ 0 a 11

(ax x ly+bx)

a-1x1 /

0

shows that

(a b1)

0 a

=

1/a2

.

In particular, this solvable group is not unimodular. The reader will check that a Haar measure on the group G1 consisting of matrices

(0

1)

is given by x 2 dx dy. Here,

a

b

jai-' and W-1 dx dy = d"xdy

is a right invariant measure. Similar considerations hold when the field R is replaced by any locally compact field K.

124

4) Let K be a locally compact (non-discrete) field. We are for

going to determine a Haar measure on the group Gln(K). Write g =

an element of this group. The product measure

® dg i,j

is not invariant, but it is easy to compute its transforms under trans-

lations. Let V = Kn, a = End(V) =

VV® V

,

(ei) the canonical basis

of V and (e1) its dual (canonical basis of Vv). We also denote by Pi = el ® ei the projectors on the lines Kei generated by the basis

vectors. Then

OL = ®(xPi x

O+

i = L el

i

V

xi i-+ (xi) --

T

(Tei)

is a left (t-isomorphism (A

el ® xi) (x)

= L e'(x) g(xi)

=

=

A T el(x) xi

=

L el ® g(xi) (x)

From this follows that left multiplication by a g ca Gln(K) in

a determinant equal to (det g)n dg

=

CL

has

and consequently

Idet g, Kn ®

i,3 dgJ

is a Haar measure on Gln(K). Similar computations hold with respect to right translations, hence Gln(K) is a unimodular group. Its invariant subgroup Sln(K) is thus also unimodular. S) Take now in particular the unimodular group G = S12OR). It is useful to let G act on the upper half-plane y = Im(z) > 0 z = x + iy e (E

g

=

by fractional linear transformations

b

(c

)

acting by

cz+d

This action is transitive (the subgroup with c = 0 already acts transiti-

vely) and the stabilizer of the point i is a compact subgroup K

ai+b ci+d Thus

_

i 4==; ai + b = id - c 4- a = d

K = SO2 (]R) c G = S12 (R)

and

g r-s g i

&

:

b = -c

gives an isomorphism

125

of the homogeneous space G/K with the upper half-plane. We first determine an invariant measure on the upper half-plane. Let

ab

az + b

g

= c d)

w

=

Ic z + dl-2 (a z + b) (c z + d)

=

Icz +

w

g. z

=

cz+d

so that

dl-2

_

(... + i(ad-bc))

Im(w)

=

dw

(cz + d)2 ( a(cz + d) - c(az + b)) dz

Ic z + dl -2 Im(z)

and =

_

=

(cz + d)-2 dz

In particular

d Adw = Icz + dI-4 d2Adz Im(w)2

dl-4 lm(z)2

Icz +

=

and

Im(z) -2 d! Adz is invariant under fractional linear transformations. Since dz n dz

=

2i(dx A dy), a positive invariant measure on the upper half-plane is

given by

y

-2

y-2

l dxAdyl =

dxdy

The formula SG f(g) dg

fG/K dm(g)

=

shows that with the identification

f f (g) dg

gK y

fy = Im z > 0

=

G

fK

(f E Cc(G)

f(gk) dk

x + iy we can write

y 2 dx dy f

f (gk) dk

SO2 (]R)

Of course, the Haar measure of the circle group K = SO2(JR) is

Zn d9

if

g

=

g8

-

sing

8 E (0,2x[).

(sin 9 cos B

Observe however, that if B denotes the subgroup of upper triangular matrices in S12(IR), the modular function A B # 1 does not coincide with

the restriction of the modular function A G (example 3 above) and thus there is no invariant measure on the homogeneous space G/B

.

126

EXERCISES

1.Let G be a locally compact group and H a closed subgroup of G. Assume that there is a relatively invariant measure m or dm(i) on GIH By definition, there are identities

JG/H (P(s-1 i) dm (k) = x(s) JG/H go(k) dm(ic)

( fv E Cc(G/H)).

a) Show that x is a character (i.e. a continuous homomor-

phism) G - i 2 c

.

b) The

map f i-i

f

dm(i) r f(xh) dh

G/H

(f E Cc(G))

H

is a relatively invariant measure /t on G (we have denoted by dh a Haar measure on H). More precisely,

sf(x) = of G.

f(s-1x)

c) The measure V /% : f Thus we can write At = x dx and

fdm(i) f f(xh) dh G/H

H

= x(s),(f)

a (sf)

=--,'

=

(f/x)

(s 6 G)

is a Haar measure dx

r f(x)x(x) dx

(f E Cc(G)).

JG

d) Replacing f by one of its right translates ft for some t E H ( ft(x) = f(xt) ), conclude from the preceding equality that 16H

= x.dG QH has an extension xdG : G

in particular,

Q'

which is a

character.

e) Let G = Gl2OR), H = upper triangular subgroup in G.

Show that there is no relatively invariant measure on G/H. (Observe that any character of G12(R) is trivial on the subgroup S12QR and thus

QH

cannot be extended as character of G. Remember that Sl2(Bt) is

the commutator subgroup of G.)

f) Let

G and H be as in e). Identifying the quotient G/H

with the projective line via c

d )

mod H H line generated by (a) in ]R2

or x = a/c C - 5

v{ooj,

show that the Lebesgue measure dx of.I? gives a quasi-invariant measure on the homogeneous space G/H.

.

127

2. Let G be a locally compact group acting continuously

and transitively on a locally compact space X. Show that if G is countable at infinity, the mappings

G -} X,

s r-- . s- x

(x a X)

are open mappings. In particular, for any x E X, X can be identified to the homogeneous space G/Stab(x) of G. (Observe that if K is any

compact neighbourhood of e in G, G is a countable union of translates will contain an interior point.)

of K. Since x is a Baire space,

3. Let G be a locally compact group and H, M two closed

subgroups with a product HM open in G. We wish to determine the restriction m of a Haar measure dx of G on HM. We assume that H x M

is countable at infinity and let this group act on G by

(h

EH, mEM, x6G)

Thus, the open set HM is the orbit of e a G under this action of HM. We identify HM with the homogeneous space

H x M/A

,

I (a,a)

=

A

:

a EH /) M j C H) M

(Ex.2 above).

a) Show that m is a relatively invariant measure on this homogeneous space. More precisely,

r

,/

f (hxm 1) dx

= Q (m) f f(x) dx

HM

G

(f 6 Cc (HM)

HM

b) Prove that

f f(x) dx HM

=

f

f(hm) AG(m) AH(m)-1 dh dm

HJC M

4. Let G be the locally compact group JR x1? discrete a Haar measure of G, and H the closed subgroup J0JxR discrete Show that

Sup m(K) K compact, K C H

Inf U open, U

,

m

m(U)

H

(In fact, Haar measures are regular Borel measures and equality would hold for all Borel parts H' for which both quantities are finite.)

).

128

13

CONTINUITY PROPERTIES OF REPRESENTATIONS

Let V be a locally convex (separated) topological vector

space and End(V) be the space of continuous linear mappings V --* V We also denote by Aut(V) = G1(V) the group of invertible elements of End(V). Thus A E GI (V) means that A is an invertible linear map V -- V and both A and A I are continuous.

Any homomorphism ttof a group G into G1(V) defines a linear action (still denoted 7c) of G on V 7L

:

G

--*

Gl(V)

,

G x V -- V

,

(s,v) H 7r(s) v

(13.1) Definition. Let G be a locally compact group and V as above.

A representation 7r of G in V is a homomorphism G --- G1(V) for which the corresponding action

G x V -* V is continuous.

It is useful to introduce the notion of separately

continuous linear action when all

S H 7C(S) V

v - ,r(s) v are continuous. The second condition implies n(s) E Gl(V) and thus,

any separately continuous linear action defines a homomorphism rC :

G ---s Gl (V) .

(13.2) Proposition. Let G be a locally compact group, V a barrelled

space and it a separately continuous linear action of G in V. Then 7c is a representation.

Proof. The Banach-Steinhaus theorem shows that i) for every compact subset K of G, the set z(K) c End(V) is equicontinuous.

Since G is locally compact, this property is equivalent to ii) there exists a compact neighbourhood SL of e e G with it(SL) equicontinuous in End(V).

Then take s = sox near so E G and v near vo e V and write

.

129

7r(s)v - 7t(s0)vo

=

7t(so) [lc(x) (v - vo)

+

7r(x)vo - vo

.

Thus continuity of the action follows from separate continuity and equicontinuity of the n(x) (x E A)

q.e.d.

.

A unitary representation it of G in V is a representation for which V is a Hilbert space and each 7t(s)

(s e G) is a unitary

operator. Thus, a unitary representation irof G in a Hilbert space H

is a homomorphism

7t: G --- U(H) into the unitary subgroup U(H) G G1(H)

such that all maps s i--3

rc(s)v (v c H) are continuous on G.

By the above proposition (13.2), any separately continuous linear action of G in a Hilbert space is a representation. But when

all operators 7t(s) (s E G) are unitary, 117r(s)II = 1 shows that 7t(G) is equicontinuous (in fact, U(H) is an equicontinuous set of operators) and the condition ii) of the above proof is trivially satisfied. Thus, the global continuity of the action defined by it follows without using the Banach-Steinhaus theorem. (13.3) Proposition. Let G be a locally compact group, H a Hilbert space

and U(H) the unitary subgroup of Gl(H). Then any homomorphism 7t

:

G

--'r

U(H)

for which all s ,--* (v I rc(s)v) (v e H) are continuous at the neutral element e e G , is a unitary representation of G. Proof. Observe that for s and t e G

V 7c(s)v - 7t(t)v II2 _ where x = t-ls .

( 7c(x)v - v

=

7r(t)-17(s) 11

7t(x)v - v)

=

v - v 11 2

2(v I v)

=

- 2 Re (v I 7t(x)v) q.e.d.

Thus the proposition follows.

REMARKS 1) Since weak and strong topologies coincide on the unitary group of a Hilbert space (cf. Dieudonne 1969, Chap.12, sec.lS p.79), continuity of a map s r-* 7c(s)v certainly follows from continuity of

all s

' -b-

(w 17r(s)v) (w E H). Moreover, on any bounded subset of H,

the weak topology can be defined by any total subset of H. Thus it

would be enough to check the continuity of all

s

i--+

(eil 7c(s)v) where

(e;) denotes an orthonormal basis of H. However, one should observe

130

that U(H) is not closed for the strong topology on the unit ball of End(H) attention, loc. cit.).

(this point seems to have escaped

If (n) C U(H) is a strongly convergent sequence of operators, its limit A = lim An (defined by A(v) = lim An(v) ) is an isometry but is not always surjective (take for example for An the operator defined by

n(ei) = ei+l for 0 4 i < n ,

and n(ei) = ei in H = e2(N)

:

An(en) = eo

for i > n

the strong limit of this sequence is the shift operator ).

2) It is possible to give weaker conditions for a homomor-

phism it :G --> G1(H) to be a representation. For example, if H is a separable Hilbert space, it is enough to check that the mappings

s f--!

(w I n(s)v)

(w, v E H)

are measurable. In this case, the mappings s i-s Tt(s)v

(v E H)

are called weakly measurable, and weak measurability implies strong measurability (Bourbaki 1959, Prop.12 p.21).,For these properties, the reader should consult Gaal 1973 (Th.l p.304, Prop.2 p.305 and Th.3 p.306).

But here is an example where weak measurability is not sufficient to imply measurability (and strong continuity). We take G =]R and H a Hilbert space with orthonormal basis (et)t Tt :

--4

G = lR

U(H) ,

is defined by ic(s)(et) = et+s G

--'-

H

,

s

E ]R

. The homomorphism

s -+ Tt(s)

,

. The mappings

-* tt(s)et = et+s

are not measurable (although they are weakly measurable) and it is not a unitary representation in our sense (H is not separable so that the general results quoted above do not apply). 3) For a discussion of the continuity properties of representations, a general reference is Bourbaki 1963 (sec.2 p.128).

A somewhat easier treatment is given in Borel 1972 (p.16). Irreducibility of representations is defined as in sec.2 (p.14) and

Prop. (2.1) is still valid.

131

EXAMPLES 1) Let G be a commutative locally compact group. Schur's lemma shows that the unitary irreducible representations of G have dimension 1 and can thus be identified with the continuous homomorphisms

x

--+ Ul (U) C X

G

:

.

These are the characters of G. The set a of characters of G is a group (pointwise multiplication) and in fact a locally compact group for the topology of uniform convergence on compact sets. This dual a of G

again has a dual and the Pontryagin

duality shows that

is canonically isomorphic to G

(G)_

.

For example if G =]R, the characters ]R --v U1((E) can all be written

in the form

t y

eist

(s E. ]R)

and the dual a =IR is isomorphic toJR. The regular representation of IR in L2(]R) is a unitary representation (cf. infra., since this group is

commutative, its left and right regular representations coincide). and it is easy to construct non-trivial closed invariant subspaces

of this representation. For this purpose, let us denote by F

--}

L2 (1R)

:

L2 (R)

the Fourier transform (it is a unitary operator defined on L1(IR) t) L2(JR) by the integral formula (Ff) (s)

f f (t) e-ist dt

=

)

lR

Then for any (measurable) subset A C :R, put HA

Thus

=

I f e L2(JR)

:

Ff vanishes outside A }

HA = tOlwhen A is negligible and HA = L2(IR) wheniR - A is

negligible. If fa denotes a translate of f, the formula F(fa)(s)

=

e-ias F(f)(s)

shows that all subspaces HA are invariant under translations. But it can be shown that L2(JR) contains no invariant irreducible subspace.

132

2) Let G be a locally compact group acting continuously on a locally compact space X admitting a quasi-invariant positive measure m,

say dm(s-lx)

(s E G, x c X ).

ICS(x)I2 dm(x)

=

To make integration theory easier to handle, let us assume that X is

countable at infinity (there is a countable fundamental system of compact sets in X). We also assume that the functions cs are continuous

on X and tend to 1 uniformly on compact sets when s -+ e e G . Then the formulas (m(s)f)(x)

cs(x) f(s-lx)

=

define a unitary representation it of G in H = L2(X,m). First of all,

('X I f(s-lx) 12 Ics (x) 12

=

f

= J'If(s'x)12

dm(x)

f(X)12 dm(x) I

dm(s-lx)

=

11f U2

=

proves that 7C(s) is a unitary operator in H for all s e G. Then, the separate continuity of the action associated to it can be verified as in sec.2 p.17, starting with the continuity of the maps

G -* H ,

s

b--*

it(s) f with f e Cc(G)

.

Examples of this kind often arise: homogeneous spaces G/H always admit quasi-invariant measures (with respect to left translations). We shall

have to consider the case of G = Sl2(R), X = upper half-plane consisting of the z = x + iy with y = Im(z) > 0 and positive measures

m=

mk

=

yk-2 dx dy

It is of course possible to generalize this kind of construction (cf. Bourbaki 1963 Ex.13, p.199 for example).

(k

e IN

.

133

REPRESENTATIONS OF G AND OF L1(G)

14

Let G be a locally compact group and L1(G) the Banach space of integrable functions on G with respect to a fixed Haar measure

on G. Thus L1(G) is the completion of the vector space Cc(G) of (complex) continuous functions f : G --s C vanishing outside some

compact set of G, with the norm 11f II

ilflll

=

=

f If(s)Ids

The formula

f * g (t)

(14.1)

=

JG

f(s) g(s-1t) ds

defines the convolution product first on Cc(G) and then on L1(G) by

continuity

:

f

G

I f * g(t)I dt

<

f fG

If(s) g(s-lt) I ds dt

=

xG

f ds If(s)I f dt)g(s-lt) I

=

=

IIgII f dsIf(s)I =

If II lIg II

for f and g in L1(G) by

(first for f and g in Cc(G) and then also

continuity) shows that the space L1(G) is a Banach algebra with respect to convolution

(f, g EL1(G) ).

11f * gII < Of 11 IIgII

There is also an involution f H f* on L1(G). It is defined by (14.2)

f*(x)

f(x

=

1)

aG(x)-1

Indeed, (12.4) shows immediately that

f

=

(14.3)

(f*)* Will

=

(f EL1(G) ).

Of II

Also observe that

f* * g (e)

=

f f*(s)

=

(f I g)

g(s-

ds

=

f f(s

)

g(s-1) d(s-1)

_

for f and g in Ll(G) 0 L2(G)

.

134

When G is not discrete, LI(G) has no unit for convolution. But here is a substitute.

(14.4) Lemma. The convolution algebra L1(G) always has an approximate unit. More precisely, there is a family (ui) c LI(G) with

ui

1 ,

llui * f - f 11 --*

0

,

If * ui - f

0

for each f E LI(G). In fact, w e can even take 0 < ui e LI(G) and Quill = 1.

Proof. Let indeed (Vi) be a fundamental system of neighbourhoods of e 6 G (parametrized by some ordered set I for which j >,i

V c Vi)

and take 0 4 ui E CC(G) with

Iluill

= f ui(s) ds

=

ui vanishes outside Vi

and

1

Then the lemma follows. In fact, if f e Cc(G), ui * f -s f

and

f * ui

q.e.d.

-b.

f uniformly on C.

Let us observe that when the group G is unimodular, the definition (14.1) of the convolution product in L1(G) can also be

given equivalently by

(14. S)

f * g (t)

=

fG f(ts-I) g(s) ds

( G unimodular)

In this case, we also have f*(x)

(i.e.

f(x-I)

=

f*

=

fv)

The presence of the modular function in the definition of f* in general is best explained by the consideration of this involution on the Banach space (algebra) M1(G) consisting

of bounded measures on G.

Using a fixed Haar measure m of G, there is an embedding

LI(G) c-+ MI(G)

f

,

- *

which is an isometry . If n E M1(G) is a bounded measure on G, one

defines a bounded measure Inl by Inl(f)

=

Sup ln(g)l

for

0 < f E Cc(G)

Igl-
(f e L1(G), nEMl(G))

In * n'l <

lnl

*

In'l

(n,n'EMl(G))

135

The Fubini theorem shows that

m(A)

nl (A) < If-ml * Ini(A)

0

=

=

0

for f L1(G) and n e M1(G). Then, the Radon-Nikodym theorem asserts that

n is a multiple of the measure m

n

g E L1(G)

=

Thus, L1(G) can be identified with a

two-sided ideal of M1(G) and

the involution n r--s nv gives, on the image of L1(G)

(f EL1(G) by (12.4).

One of the advantages of working with M1(G) is the possibility

of embedding G in this algebra evaluation measure f i-3

Es * tt

for s g_ G, we denote by S. the

Es(f) = f(s) so that Est

and

G -* Ml (G)

,

--*

s

Es

is an injective homomorphism. However, to remain more self-contained,

we shall only make a limited use of this bigger convolution algebra. For any representation it of G in a (quasi-complete) space V,

we can define the operators

v

i--

9r(f)v = I f(s) 7C(s)vds

(fCCc(G)).

G

When 7C is unitary, one can also define the operators

7t (f)

=

f(s) 7r(s) ds

fG

by the same formula for all f e L1(G) one can define

7C1(n)

X'(f * g) (14.6)

7(f*) 1tl1(Es

=

* f)

=

(more generally, for

7C1 (f) 7i (g)

7L1(f)* , II 7i (f)II =

7t1(

f)

=

II f III

?G(s) tC(f)

for f and g EL1(G) and se G. Thus we obtain a representation

tcl

n E M1(G),

). One checks without difficulty

= f1r(s) dn(s)

:

L1(G) - End(V)

.

136

Moreover with an approximate unit (ui) as defined in the proof of (14.4), for each F-> 0, we can find an index i such that

ui(x)

0 = U t(x) v - v 11

#

(for a fixed v E.V)

<_ £

and thus 117t1(ui)v - v 114 f ui(x)p7[(x)v - vAA dx This proves that for each v e V,

7C1(ui) v i v

E .

.

In particular, the

vectors e(f) v (f E Ll (G) , v E :V) generate V topologically. Quite generally, if A is a *-algebra (* here refers to an involution, not to convolution...)

,

H a Hilbert space and p a

*-homomorphism A - End(H), we have an orthogonal direct sum =

H

Ho $ H1

with

H

0

= intersection of the kernels of the p(a) (a e A),

H1 = closure of the space generated by the p(a)v (a c- A, v e V).

This follows immediately from the formulas

(w I p (a) v) for a e A and v , w E H .

=

(p(a) * w (v)

_

(p (a*) w ( v)

We say that p is non-degenerate when H

0

= to].

Thus, for any unitary representation of G in a Hilbert space H, the *-representation itl of A = L1(G) is non-degenerate. There is a converse to this result.

(14.7) Theorem. Let G be a locally compact group, H a Hilbert space and

p

a non-degenerate *-representation LI(G) -+ End(H)

.

Then there is

a unique unitary representation 7r: G -s G1(H) such that p = 7t1

.

The uniqueness of it follows from the third formula of (14.6)

7cl and 7c (sf) = 7r(s) 1C1 (f) since, by assumption, the vectors generate H .

=-:.+ p(sf)v = 7C(s) P(f)v

7t1(f) v = p(f) v

( f E LI(G), v E H)

At the same time, this formula shows that the only possible

definition of 7r(s)

is an extension of

p(f) v H P(sf) v We have to check that this association is well defined and has all

137

required properties. This is based on the following general result.

(14.8) Lemma. Any *-homomorphism L1(G) -+ End(H) is contracting and hence, continuous.

Proof of this lemma. It is known that if T E End(H) is a hermitian operator, IIT II

is given by the spectral radius of T. By definition,

this spectral radius is

spec rad T

=

l im k Sao

l /k

II Tk II

But End(H) is a stellar algebra (C*-algebra) II T2 Ii

=

IIT*TU

=

IIT112

so that by induction on n, we get is easily

2n IIT

2n II

=

IITII

and the above limit

computed with k = 2n

spec rad T

=

(T hermitian).

IIT II

Now it f = f* E L1(G), T = p(f) is a hermitian operator and thus 11p (f) II =

spec rad p(f) < spec rad f < IIf II

(f -2.l invertible for convolution in L1(G) implies p(f) -,11 invertible

spec p(f) c spec f ). Now when g e L1(G) is

in End(H), and hence

arbitrary, f = g* * g satisfies f* = f as above and

Ilp(g)II2

=

Np(g)*p(g)II = Ilp(f)II < ofIl < Ilg*II hlgll = IIgI12

whence the conclusion. (As the proof has shown, L1(G) could be replaced by any *-algebra A and End(H) by any stellar algebra B in this lemma

cf. Diaonier 1964, (1.3.7) p.7

. )

End of proof of (14.7). By lemma (14.4), for f 6 L1(G) ,

ui * f

f

in L1(G)

,

and the same convergence will hold for left translates sui * f

s(ui * f)

=

Now lemma (14.8) shows that

-s sf

in L1(G)

p(sui) p(f) -- p(Sf) and hence, we can

define

it(s) w = lim p(sui) w

i

for all linear combinations w of special vectors p(f)v (f 6 L(G), v E V).

138

Still by (14.8), IIP(sui)II 4 1 ,

and 7t(s) has a

hence 07c(s)II < 1

unique continuous (contracting) extension to the whole space H. Obviously

ir(e) and thus

7r(s-I)

=

id. , ic(st)

=

ic(s)-1. Now

Iln(s) II K 1

and

= 7r(s) 7r(t)

II7t(s)-III

=

ll it(s-1)II

<-

1

it(s) is unitary (s e G). The homomorphism it : G

implies that

U(H)

is a representation since, by definition s

r--

7c (s) w =

ir(s)p(f) v = p (sf) v

is continuous for a total system of vectors w (take the vectors w of the form p (f) v for some v c H and f E Cc (G), in order to ensure sf -s tf in L1(G), when s -* t in G as on p. 17) . our definition of it,

f*g

=

= p

irl

We have to check that with

. But

f f (s) (Es * g) ds

leads to

p(f)P(g) = p(f * g)

=

f f(s) p(Es * g) ds

= f f(s)ic(s)ds f(g) Since

P

=

=

it1(f)p(g)

is non-degenerate, this implies p(f) = 7t1(f)

.

q.e.d.

If it is a unitary representation of G in F1, the preceding

construction (14.7) can be made in any closed invariant subspace of H. Thus it and >t have the same closed invariant subspaces. In particular (14.9) Corollary. If it is a unitary representation of G in H, it irreducible

7cl

irreducible .

From now on, we can safely write it instead of 7r .

As for compact groups, one could still extend it to the

stellar algebra of G. For this purpose, one should define

IIfiI*

=

Sup

it unitary

Iir(f)If

(f EL1(G))

and define C*(G) as completion of LI(G) for this norm. By definition,

the embedding

L1 (G)

C*(G) has a dense image and any unitary

representation it of G (in some Hilbert space H) has an extension

139

e

:

C*(G)

-- End(H)

One can prove that 'r irreducible = 7t* algebraically irreducible (cf. Dixmier 1964, (2.8.4) p.45 ).

In particular, the left regular representation of G can

be so extended t*

:

End(H)

C*(G)

with H

=

L2(G)

.

The image of this homomorphism is a stellar algebra Cred(G) which is, in general, a proper quotient of C*(G) .

We shall not pursue these

topics any further.

As we have already seen with the regular representation of

the group G =]R, a unitary representation of a locally compact group need not have invariant irreducible subspaces. Here is however an interesting case. (14.10) Theorem. Let 7c be a unitary representation of a locally compact

group G in a Hilbert space H such that the operators

n(ui) are

compact for all elements of an approximate unit (ui) c Cc(G).

Then H is a Hilbert sum of irreducible subspaces with only finitely many of them being equivalent to a given one. Proof. For f E Cc(G), the functions f * ui converge uniformly (hence in L1(G) by compactness of all supports) and thus 7c(f * ui) =

converges to

X(f)7c(ui)

7r(f) in the operator norm. Since any norm limit of

compact operators is compact 7r(f) is a compact operator for every f e Cc(G)

Since 7G(uj

7L(ui) v --- v

0.

.

(v c- H), there is certainly an index j with

Taking f = Z(u + uj)

or

f = -!-(uj 2i

we see that there is a function f = f* E Cc(G) with 0 # ic(f) hermitian.

Since this operator is compact, hermitian and # 0, it must have a nonzero eigenvalue 1 . The corresponding eigenspace

Ha = Ker(7c(f) - A1H)

# 103

140

, we denote by H(a) the closed

is finite-dimensional. For a E H

subspace generated by the n(f) a (f cm L1(G)) . This subspace H(a)

is always invariant under ir. We need a lemma. (14.11)

Lemma. If H1 is a closed invariant subspace of H(a), then the

orthogonal projection al of a in H1 belongs to the eigenspace H. and H1 = H(al). Moreover, for two closed invariant subspaces Hj c H(a)

H1 * H2 4 HI n Hack H2 !1 H'A . To prove this lemma, we notice that the orthogonal projector P1 on H1

commutes to all operators i(f) (f

L1(G)) . Taking for f the preceding

hermitian element of L1(G), we see that

7r(f)al

7G(f)Pla

=

Now, the vectors ,t(f')a

=

P17 (f)a

=

P1'Aa

= Xa1

.

(f'E L1(G)) are total in H (if a # 0) so that

their projections

PIn(f')a

=

ir(f')al

(f EL1(G) )

also make up a total system in H1 = H(al). Finally, if

H1 0 H

a

=

= H2 A Ha , the orthogonal projection a1 of a2 on H1 is equal to a2 a2 = aI and thus H2 = H1 by what we have already proved. Thus the

lemma follows. We can now finish the proof of (14.10). From the lemma (14.11)

we infer that the set of invariant subspaces of H(a) has an artinian property: any decreasing sequence of closed invariant subspaces of H(a) must be stationary (because the same property holds in the finite-

dimensional space H(a) A Ha). In particular, H(a) contains an irreducible subspace (if a # 0). The preceding argument can of course be repeated in any closed invariant subspace of H (the hypotheses of (14.10) will still

be satisfied for the restriction of X to such a subspace): any non-zero closed invariant subspace of H contains an irreducible subspace. Zorn's lemma applies to the set of families of orthogonal

closed irreducible subspaces of H and there exists a maximal such family (Hi), . The orthogonal of the Hilbert sum & Hi is closed and I

invariant. If this subspace were non-zero, we could find an irreducible subspace in it a n d (Hi)I would not be maximal. Thus H

H i is a Hilbert I

141

sum of irreducible subspaces. Select one irreducible subspace Hl and collect all equivalent subspaces Hi function f E Cc(G) with

(j E J). There exists a hermitian

7c(f) non-zero in H1. Thus, we could take a

non-zero eigenvalue of 7c(f) in H1

.

Since the operator 7c(f) induces

the multiplication by A in the corresponding subspace ® is compact, this last space must be finite-dimensional

:

and

the index

set J must be finite.

q.e.d.

The preceding proof shows that any representation of G in a Banach space such that all operators it(f) (f e Cc (G)) are compact

has an irreducible subspace (if # {0)) and hence a generalized composition series with irreducible quotients. Perhaps

one should observe that the conditions of the

theorem (14.10) are satisfied if G is a compact group and Z = e

is

the left regular representation of G in L2(G) (by the exercise 2 of

sec.6, $1(f) = f *

is an integral operator to which lemma 1 of

sec.4 applies). More generally, (14.10) admits the following interesting application.

(14.12) Application. Let 1' be a discrete subgroup of a locally compact unimodular group G .

Then

the left regular representation of G in

L2(G/r) decomposes as a Hilbert sum of irreducible ones, each occuring with finite multiplicities. Proof. We have to check that the assumptions of (14.10) are satisfied.

Since r is discrete, it is unimodular and G/r carries an invariant measure (12. 8) from which the space L2(G/T')

is built. For f r= Cc(G)

and T 6 Cc(G/r) , we have

W) T (x) =

=

=

JG

f(s) T(s-1X) ds

f f(xs1) cp(A) ds

=

f f(s-1) WO ds

=

fc/r d' T(t) L f (x X s-1) je r

Hence t(f) is the integral operator on G1 r with kernel

kf(x,y)

=

2. f(x y 1) 1

r

on

G/r % G/r

=

142

The preceding sum converges :

f(xXy 1)

#

let K = Supp(f) (this is a compact set)

xVy

0

1 E K rl xry l

is compact and discrete, hence finite, so that

But the set K (I x r y 1

for each given couple (x,y), the sum defining the kernel kf is in fact

a finite sum. More generally, if A is a compact neighbourhood of e c G,

SLK If' () xry l is finite and the sum defining kf has only finitely many non-zero terms on S?x -SLY. It is thus uniformly convergent on all compact sets and kf is a continuous kernel. Thus (14.12) follows from the following general result and (8.2).

(14.13) Theorem. Let X be a locally compact space, m a positive measure

on X and k E L2(X K X, m ® m). Then

f '- + Kf ,

(Kf) (x)

the integral operator

fX k(x,y) f(y) dm(y)

=

is a Hilbert-Schmidt operator in L2(X,m) and II KII?

Proof.

=

f fX

Ik(x,Y) 12 dm(x) dm(y) xX

Fubini' s theorem

shows that the functions k(x,.) E L2(X,m)

except for a negligible set of x e X. Thus, for nearly all xE X k(x,.)f E L1(X,m) = L1

(f e L2)

and (Kf)(x) = (f I k(x,.)) is well defined for the same values

of x.

The Cauchy-Schwarz inequality gives

I (Kf)(x) I < hk(x,.)R2 11f112

.

By integration of the square of that quantity, we get

if112

IIKfII2 <

dm(x)

Jdm(y) Ik(x,Y)I2

This shows that

Kf E L2

,

f --=s Kf continuous

L2

-->

L2

and IIKII2 < f Ik(x,y)I2 dm(x) dm(y)

(uniform norm),

but we need much more... Take an orthonormal basis (ei) in H = L2 and denote by (ei) the orthonormal basis with ei(x) = ei(x) all x e X, we have an expansion (in L2)

.

For nearly

143

k(x,.)

= £ ai(x) ei

with coefficients given by the usual scalar products k(x,.))

ai(x)

=

Kei (x)

The Parseval identity is

L IKei (x)I2 Integrating this identity (valid outside a negligible set of x 6:X),

we get

00 >

JJ

dm(x) dm(y) Ik(x,y)12 = TII Keij 2

= II KI12

q.e.d.

144

EXERCISES

1. Show that L1(G) is not a stellar algebra in general. Hint: Show that for f E L1(G), 2

f* * f J) = I f f (x) dx

I

( < )if

u2

)

so that f* * f = 0 as soon as the "average" of f is 0. But

Uf* * f11 =

if f >0 .

If Q2

2. Let K and H be two integral operators in L2 = L2(X,m) given by two kernels k and h on X x X (as in (14.13)) respectively. a) Prove that the scalar product of these Hilbert-Schmidt

operators is given by

f p(x,x) dm(x) where

p(x,y)

=

rk(z,x) h(z,y) dm(z)

is the kernel of K*H. J b) Observe that x t--' p(x,x) = (k(.,x) / h(.,x)) is in L

since it is the product of the two vector valued L2-functions

x h-Y k(.,x) x

,

+--' h(.,x)

(use Bourbaki 1965, Int. Chap.IV, sec.6 Cor.1 de Th.2 p.208).

Remark. This exercise shows that the trace of the (trace class) operator K*H is given by the integral over the diagonal of its kernel.

145

15

SCHUR'S LEMMA

UNBOUNDED VERSION

:

We have to review a few facts on unbounded operators in a Hilbert space. Here are three references for this theory Riesz-Nagy 1975 (Chap. VIII),

Dieudonne 1969 (vol.2, Chap. XV, sec.ll-13), Lang 1975 (Appendix 2).

An unbounded operator T in a Hilbert space H is a linear map DT ---> H defined on a dense subspace DT c H. To be more precise, one

should speak of the unbounded operator (T,DT). In particular, we consider as distinct, two unbounded operators having distinct domains of definition.

An extension S of an unbounded operator T is an unbounded operator = T . In this case, we simply write S T T (15.1) Definitions. a) An unbounded operator T is said to be closed when

(S,DS) with DS :>D T and SID

its graph ( C DT x H) is closed in H >,H. b) The operator T can be closed when it has a closed extension. This second condition is satisfied when the closure of the graph of T is still a functional graph, namely when

y=0

xn -a 0 and T (xn) --s y

.

When T can be closed, the smallest closed extension of T is the operator T having for graph the closure of the graph of T

x e DT

(15.2)

3 (xn) c DT with xn -+ x

and

{ TxnI bounded.

To see this, we first have to introduce the adjoint T* of T. First,

we define DT*

Then

=

{xEH:

y '- (x

Ty) continuous on DT

Riesz' theorem allows us to put T*x = z if

x e DT* and (x Ty)

=

(z

y)

=

(T*x

y)

(y EDT).

Then we have (15.3)

DT* dense

4=4

T can be closed

T

=

T**

146

Coming back to (15.2), if {Txnj is bounded, we can assume -extracting a subsequence if necessary- that Txn converges weakly to an element z. In particular, for all y E DT*

-'> (z I y)

(Txn I y)

whence x e DT** and T**x = z .

,

,

(xn I T*y) - (x I T*y)

Thus (15.2) follows from (15.3).

For any operator T, T* is closed. (15.4) Definitions. Let T be an unbounded operator.

a) T symmetric e= T c.T* (hence DT* dense and T can be closed) b) T self-adjoint 4==)P T = T* (and DT = DT*), c) T essentially self-adjoint 4==> T can be closed and T = T*

.

Thus symmetric operators are characterized by

(y I Tx)

(TyI x) for all x and y in DT ,

=

and essentially self-adjoint operators have a closure which is self-adjoint. If T is an unbounded operator, T*T is canonically defined on

DT*T

= {x EDT : Tx EDT*} =

DT C T 1DT*

.

In fact, one can show that (1 + T*T)-1 is a continuous (bounded) operator: its norm is < 1 and it is positive (spectrum in [0,1] ).

Here is a classical result (15.5)

T closed

=

T*T self-adjoint

.

A normal operator T is a closed operator for which T*T = TT* (this equality requires DT*T T self-adjoint ;

= DT.f*).

For example

T = T** = T closed =

T2 = T*T self-adjoint

T normal

There is a complete spectral theory for normal operators (in particular, for self-adjoint operators). In this theory, it is possible to define functions of T : in particular, (T*T)2 is the absolute value of T and (15.6)

Every closed operator T has a polar decomposition T = UP i

where U is partially isometric (U*U = 1) and P = (T*T)2 is self-adjoint and positive. Here, T is an unbounded operator, but U is necessarily continuous with RU11 4 1. When T is symmetric, T c :T*

T C T*

147

and thus T is still symmetric (with T* = T*). In this case, T can have several closed extensions, but at most one self-adjointextension (by maximality). To analyse this situation, one has to consider the spaces D+

(T + i)DT

=

,

D

=

(T - i) DT

.

(For typographical convenience, we write T + A instead of T +A 1H But (15.7)

When T is closed and symmetric, (T + A)DT is closed in H for every complex ) E (E with Im(A) # 0

(15.8)

When T is symmetric, D+

=

D

=

.

H 4=+ T self-adjoin

In particular, when T is symmetric and both Dt are dense in H, T is

essentially self-adjoint Coming back to the case T closed and symmetric (hence Dt

closed in H), it is easy to check i)xff2 JTxll2

=

11(T +

+

Iixg2

=

1(T -

i)xl2

(x EDT)

hence there is an isometry U : D+ -i D which makes the following

diagram commutative

-

T + i

DT

T

D+

(T + i) DT

U

isometry

D

Self-adjoint extensions of T correspond to unitary extensions of U.

Thus we obtain the following criterion. Define the defect indices m+

=

Codim D+

,

m

=

Codim D

for T closed and symmetric (these cardinals can be finite or infinite). (15.9)

T self-adjoint 4 (m+,m)

(15.10)

T has a self-adjoint extension <=-> (m+,m)

(44 m

= m

=

(0,0)

= m finite or infinite)

,

=

.

We can now turn to applications of this theory.

(m,m)

148

c End(H) a (topologically)

(15.11) Lemma. Let H be a Hilbert space,

irreducible set of operators which is invariant under A '-+ A* and

operator with

(T,DT) a closed (unbounded

A(DT) C DT

AT

for all A e

DT is

c1 -invariant

for all A E .

TAID

=

:

T Then T = a lH is a scalar operator and DT = H

Proof. 1) Let us show that DT* is also

y Ni

-invariant. Take x E DT*

is continuous on DT

(x J Ty)

.

As

(A I TY)

(x I A*Ty)

=

(x I TA*y)

=

since A*6 11 and A*y E DT , the mapping

y H (Ax I Ty) can be factorized as y F--s A*y H. (x I TA*y) and is thus continuous: Ax E DT* and (T*x I A*y)

(AT*x I y)

=

_ I

(x ITA*y)

( y E D

(Ax I TY)

=

proves the expected commutation relation of T* 2) As DT*T = DT fl T-1 DT*

,

T

(T*Ax I y) and A C. t

)

on DT*

this domain of definition is also invariant

under all operators A e I

x E T 1 DT*

4==>

Tx E DT*

ATx E DT*

furnishes for x EDT TAx

=

Ax ET-1 DT*

ATx EDT*

3) The self-adjoint operator T*T has a spectral decomposition with

spectral projectors still commuting to all elements of I. By irreducibility assumption of CI

,

these spectral projectors can only be 0 or 1H

But T*T is a weak limit of linear combinations of spectral projectors.

Thus, T*T = Alfa must be a scalar operator (and A > 0). We infer IITx1I2

=

(Tx I Tx)

=

(T*Tx I x)

= 2Ixb2

(first for x e DT*T ...). Thus T is continuous and the bounded version

of Schur's lemma (8.6) applies: T = ji.l with IH 2 =

.

q.e.d.

149

(15.12) Unbounded Schur's lemma. Let (1C, H) be a unitary irreducible representation of a group G and (T,DT) a closed operator with DT

(s a G) .

7c (G) -invariant and x(s) T = T 7c(s) ID T

Then T = A 1H is a scalar operator (and DT = H). This is the case It = 7C(G) of (15.11).

From this, we deduce the following statement. (15.13) Theorem. Let (7c, H)

0 # T

of a group G ,

and (,',H') be two unitary representations

DT --! H' a closed operator (DT dense in H)

:

with

DT ?c(G)-stable,

le (s) T = T 7c(s) ID

for all s E G .

T If 7c is moreover irreducible, T is a multiple of an isometry (hence is continuous with DT = H) and T furnishes an equivalence of 7L with a

subrepresentation of 7c' Proof. One checks DT*T

=

.

as in part 1 of the proof of (15.11) , that Dr* and

,

DT r T-1 DT* C DT are i(G)-invariant. Since T is closed, T*T (in H, and its domain DT*T is dense). By assumption

is self-adjoint

n,

(s-1)

(se.G)

T = T 7c(s-1) I DT

implies

= IC(s) T*

T* 7e (s) ID T*

and thus x(s) T*Tx

=

T* fe (s) Tx = T*T 9C(s)x

for x E DT*T C DT . Thus the unbounded Schur's lemma (15.12) applies to the unbounded operator (T*T , DT*T) in H, and the irreducible represen-

tation it. Consequently, T*T =A 1H is continuous and its domain is H (a fortiori DT = H and T is continuous by the closed graph theorem, but this last fact can be seen independently as we shall see). Thus,

(Tx I TY)

=

(T*Tx I y)

= A (x I y)

and T is injective (T # 0 = # 0), with T/JX- isometric. This implies that the image of T is closed and T (or T/rA) an equivalence of it

onto 't'IImT

.

q.e.d.

150

EXERCISE

Establish (15..11) with the following weaker assumptions.

Still assume t C End(H) is irreducible in H, (T,DT) is closed, but replace the commutation of elements A E 13 with T on DT by the following.

Let W be a dense subspace of H contained in DT with

w .f -invariant and TAx = ATx for A Et and x E W. Hints: a) Define S = TIW an unbounded operator on DS = W. Thus S has a closed extension (take T) and its closure S = S**.

b) Let the elements A 6 1 "act" in H >cH by

(Ax,Ay). Prove

C Graph(S) hence

C Graph(s) by continuity.

c) Apply (15.11) to

and (S,DS)

Formulate the corresponding generalizations of (15.12) and (15.13).

151

16

DISCRETE SERIES OF LOCALLY COMPACT GROUPS

In this section, G will always denote a locally compact

unimodular group. Let 7t: G - . Gl(H) be a unitary representation of such a group. For each u, v s

.

(u 17t(s)v)

s-s

H, we can form the coefficient cu =

c'(s)

By definition of the continuity of a representation, these functions are continuous. Since 7r is unitary, they are also bounded

I (uI?t(s)v) I <

9u11 p*(s)III1VII

=

AulINvil

.

Left and right translates of coefficients are still coefficients cu(x ls)

cu(sx)

=

(u I7C(x)* 7c(s)v)

(u 17C(s) 7C(x)v)

=

Thus, u - cu (resp. v --+ cu )

=

=

c(x)u(s)

cu7 X)v(s)

is a G-morphism from (7t,H) to

the left (resp. right) regular representation of G in the space of bounded continuous functions. (16.1) Definition. Let of G. We say that 7C

(7t,H) be a unitary irreducible representation

is square summable, or equivalently in the

discrete series of G when it has one non-zero square summable coefficient

0 cv E L2 (G) for some u v e H Similarly, we say that 7t is integrable, when it has a non-zero integrable coefficient

0# cv e L1(G) for some u v e- H Let us observe that if f is a bounded function, say IfI 4 M

then If 12

M If I

so that f E L1

= f s L2

and

7t integrable = 7C in the discrete series. If 7G is irreducible and v # 0, the set {7r(s)v

Thus we see that cu # 0 precisely when u

:

s e G)} is total in H.

0 and v # 0.

152

Let us assume now that 7t is irreducible and in the discrete

series with

cV E L2(G). We define

0

W=

=twEH

Wu

cW E L2 (G) }

:

This is a linear subspace of H containing all w = 7t(s)v, hence dense u in H. Moreover, W is invariant under all 7c(s) (s E G) since the q

7s )w

are right translates of cw . Let us consider the linear operator

T

:

W --s

L2(G)

,

w H cW

(T = Tu)

an unbounded operator with domain DT = W. I claim that this operator (T,W) is closed. Let indeed (wn) be a sequence of elements of W with

(wn , Twn)

-->

(w , f)

in H K L2 (G)

Thus =

CU (s) n

(u I a(s)

n)

(n(s-1)u I wn) +

_

(7t(s-1)u

w)

and the convergence cw -- cW is uniform on G n = cu (s) Wn Extracting a subsequence of (Twn) , -

Icc (s)

cW (s)

=

I(n(S-1)u

w - n)I < auHOw - wd

I

Twn(s)

--

I

we can assume s 0- N (negligible in G)

for

f(s)

n This proves f(s) = ce(s) for si¢ N and thus cw g L2(G) , w E W and

f

=

Tw

:

(w,f) E Graph(T)

Since we are assuming G unimodular, its right regular representation r

in L2(G) is unitary and we can apply (15.13) to T considered as an intertwinning operator between it (on W) and r (on L2(G)). Thus T is a multiple of an isometry and gives an equivalence between it and a subrepresentation of the right regular representation. In particular, cW E L2(G) for all w s W and T is continuous. Similar considerations

would apply to the left regular representation and would show 2

cU CL (G) for all u, v e H We have thus obtained the essential part of the following statement. (16.2) Theorem. Let G be a locally compact unimodular group and it a unitary irreducible representation of G. Then the following properties are equivalent

:

153

i) one coefficient cv # 0 of z is square summable,

cu of 7t are square summable,

ii) all coefficients

iii) 7r is equivalent to a subrepresentation of the right regular representation.

Proof. We have already proved i) =+ ii) _> iii). We show now that iii) = i). Let (7c,H) be a representation which is equivalent to a subrepresentation of the right regular representation in L2(G). There is no harm in assuming that H G L2(G) and 7c = that

H with

rIH . We have to show

7r has a non-zero square summable coefficient. Take cf and p in (y fi(/)

# 0

.

Since Cc(G) is dense in L2(G), the orthogonal

projection of Cc(G) on H is also dense in H, and we can assume that

y r= H is the orthogonal projection of a To E Cc(G). The corresponding coefficient c? of 71:= rIH is

c,(s)

= f qo(x)

i.e.

c,

_

_

(Tol r(s)V) S o(x) r(s) T(x) dx = J To(x) tp (xs) dx (T 17G(s)i)

lY

=

1(x-ls)

dx

=

o * T = .(?)(')

This coefficient is non-zero since

To *

=

(s)

EL2(G)

c,(e)

0

.

q.e.d.

(16.3) Theorem (Schur's orthogonality relations). Let (x,H) and (7t',H')

be two unitary irreducible representations in the discrete series of G. a) There exists a constant 0 < dr E]R such that

(CV I CV,)

=

(u , u' , v , v' E H) ,

1 (u l u') (v I v')

b) if 7c and X' are not equivalent,

coy)

(cv

=

(u,vEH, x, yEH')

0

This constant do is called formal dimension of 7r. It depends

on the choice of the Haar measure on G

:

d'x = c dx

d = c-1 dr . But

the product d,,dx is independent from the choice of Haar measure dx. If G is a compact group, d, = dim n /m(G) so that di.= dim 7C if the Haar measure is normalized.

Proof. Fix u and u' in H and consider the two (continuous) operators

8

9'

.

H -- L2(G)

,

v -* cu v

(resp. v

'-a.

cvl)

154

Thus we have ,

(cu IcV,)

(9v I e'v')

=

(v 19*e'v')

=

and one checks without difficulty that 9*9' commutes with 7t hence is a scalar operator (bounded Schur's lemma (8.6)). There is a constant

au u, with

(cv I cV,)

=

au,u, (v V')

=

by

(v

,

v' E H).

One proves similarly

(cu

cV )

v, (u

u' )

(one can either compare the left regular representation and

it

, or

transform formally (cu Icu,) = (cv,I cv) using the fact that G is unimo-

v

u

v

u

dular). Hence we can write

( v v ')

=

(u l u ' )

_

"IC

bv,v,

au,u,

is independent of u , u' , v , v' E H. Taking u = u' , v = v' we see that au u > 0 whence d,,> 0 and of course ,

(cu I cV, )

au u, (v v')

=

=

d7, (u I u' ) (v I v' )

This proves a). On the other hand, fix uE H and x

.

H'

and consider

the hermitian form

(v , Y) -+ J(v,Y)

=

(ci I c' y)

The Cauchy-Schwarz inequality gives

IJ(v,Y) I < bcull Ilex II

<<

C NO

by the first part a) already proved. Hence there exists a linear operator A

:

H'

-+>

H

with J(v , y) = (v j Ay) (thus Ay is defined by Riesz' theorem). Taking

v = Ay 4AYII2

=

J(Ay , y) t C IIAYB yll IIAYf 4 CNYQ

Thus A is continuous. But one checks without difficulty that A is a

G- morphism

155

(v I Aye (s) y)

As

J(v,R' (s) y)

=

=

(cv I r(s)cy)

=

J(n(s)*v,y)

=

(r(s-1)cv Icy

_

(n(s)*v I Ay)

_

(cuv I c' , (s)Y)

(cu(s_,)v I cY (v

=

=

7t (s) Ay)

R' is not equivalent to a subrepresentation of 7C, A = 0.

.

q.e.d.

(16.4) Proposition. If a locally compact group G has a finite-dimensional square summable unitary irreducible representation 7t, it is compact and

d 'r = dim 7t / m(G)

( m(G) = fG dx )

.

Proof. Let us take an orthonormal basis (ei) of the Hilbert space H in

which 7t acts and write 1

=

= Z I (ei 17c(s)ej)

n x(s)e) II2

I2

i

= Z IcJ (s)I 2 i

Integrating over G we obtain

m(G)

ds

= G

=

f 2: Ici j gds = Z Dc' M2 i j i j G

=

1/dn = dim 7r / d < ov Thus G is compact.

q.e.d.

(16.5) Theorem. Let 7t E G be a square summable unitary irreducible representation of ,G in H and f E L2(G). Then, the integral

7t(f) =f f(s)7c(s) ds G

converges in the weak sense, defines a Hilbert-Schmidt operator with

IIn(NIZ

=

II Pn(f)II2

where Pn : L2(G) -+ L2(G, 7c) is the orthogonal projector on the isotypical component of type 7t in the right regular representation r.

Proof. By definition of a weak integral, for each u and v e H, we have to show that the integral f f(s)(uI 7c (s) v) ds converges. But

(u,v) H f f( s) (u 17c(s)v) ds

=

(f I cu)

is a continuous hermitian form (it is well defined since f and cV E L

By the Cauchy-Schwarz inequality

I(f I cv')

( IIf12 1 cyp2

d,r afa2auaava

2

(G)).

156

This proves that the operator ic(f) is defined by (uIic(f)v) = (f Icu)

and is continuous with (uniform) norm

1n(?)II < d_;12

pf1I2

To prove that ic(f) is a Hilbert-Schmidt operator, we take an orthonormal

basis (ei) in H . The corresponding coefficients cp(s) _ (eiI u(s)e.)

make an orthogonal basis of the isotypical component L(G,ic) and (d2 cl) >r

j i,j

is an orthonormal basis of this space. Thus

7- I (ei I ic(f)ej) I2

i,j

7- I (f I

=

i,j 1 L I(f I ddci)I2 7t i,j

I2 1

=

=

IIP,fI12

This concludes the proof. Let us denote by .11 the discrete series of G, namely the

set of (classes of) square summable unitary irreducible representations

of G. If it, 7c' E SL are not equivalent, (16.3) shows that L2(G, iC) is orthogonal to L2(G,i ). Thus we can consider the closed subspace Ld(G)

C

L2(G,ir)

L2(G)

and for f s L2(G), we define

fd=

APa(f)

f

_

so that in particular

Ofd 11

2

=

IIPI(f)II2 = F d'II R(fI Z

We can also write this formula (cf. Ex. 1 at the end of this section)

IlfdIt2

=

EIIPP(f)p2

=

Considering the mapping _.Q

it H x(f)

as a Fourier transform of f, we see that we have a "Plancherel formula"

at least if f E L2(G). In fact, the restriction to the discrete series

SL C G of the Plancherel measure is given by the positive masses do (the Plancherel measure is dual to the Haar measure and multiplication

of dx by a positive constant divides the Plancherel measure by the same constant). Thus ,l1, appears as the discrete part of the Plancherel

measure (cf. sec.21) justifying the terminology adopted for it.

157

Assume now that the group G contains an open and compact subgroup K and normalize the Haar measure of G by

Denoting by CPK the characteristic function of K =

TK(s)

1 if s E K and (OK(s)

r ds =

1.

K

0if s0- K,

we see that

TK

TK

=

IK

and

?K * fK

Consequently, if 7r6.fL acts in H , PK

=

x(TK)

:

H -+ HK

is the orthogonal projector on the closed subspace of K-fixed vectors in H. Since TK E L2(G), PK must be a Hilbert-Schmidt operator by (16.5).

Thus HK =

Im(PK) must be finite-dimensional and

dim HK

=

up

42

=

11 Z(?K) If

and

x6 drdim H =

2

N (?K)d1I2

TKQ2

In particular,

d dim HK, < 1

dim HK < 1/di

(16.6) Proposition. An infinite discrete group G = 1'

has an empty

discrete series. Proof. We can take K = {ejin the preceding computations. Thus, if

r'

is discrete and has a square summable (7[,H) in the discrete series,

H

=

HK , dim H

(by the preceding argument) and

/'

=

dim HK

< 00

is compact (16.4). In this case,

/' must be finite.

q.e.d.

(16.7) Proposition. Assume that G contains an open compact subgroup K and a discrete series representation (7[,H) 6 1Z with HK

the normalizer NG(K) of K in G is compact

:

[NG(K):K ] < oo

{0j.

Then,

.

Proof. Let n s N = NG(K). Then 7c(n) leaves HK invariant

7[(k) yc(n)v

= ic(n) 7c(n lkn)v

=

7t(n)7t(k' )v = 7t(n)v

for all k E K (hence k'e.- K). Take an orthonormal basis (ei) of the

finite-dimensional space HK and write

158

1

2 ci(n)I2

=

Integrating this equality on the group N (this group is a union of classes of K, hence is open in G) with respect to the restriction of the Haar measure of G (this is a Haar measure of N)

f

N

hence

do

Ic (n) I 2 d o
= JN IIc1p2

m(N) i

ds ,

= Z 1/dam = dim HK /dX q.e.d.

i

j

(Observe that if G is discrete and if we take K = {e), N = G , so that

we obtain a new proof of (16.6)

-+

G discrete, 11 # yS

G finite . )

(16.8) Definition. A Fell group is a locally compact group G for which L2(G)

L2(G)

=

L2(G,,G) ,rESL

In particular, compact groups are Fell groups, and generally, Fell groups

are the groups having a Plancherel formula

If g2

L d"Ific(f)Il2

=

valid for all f E,L2(G). It is interesting to construct non-compact Fell groups (recall that in this section, all groups are assumed to be unimodular). Before we give such an example, let us make some general

comments when

G contains an open compact subgroup K . We denote

by AK C .IL the set of (classes of) discrete series representations (7r,H) for which HK ' (0)(these representations are sometimes called K-spherical).

(16.9) Proposition. If a Fell group G contains an open compact subgroup K

In

djC > 0

=>

G compact

Proof. Since

d7C dim HI =

1

(with the Haar measure on G normalized by the condition m(K) = 1), our

assumption implies that the set AK is finite. But if G is a Fell group,

L2(G)

_

L2(K\G/K)

(HR®) _ ® (Hn 0 Hie)

(sum over AK)

159

This shows that L2(K\G/K) is finite-dimensional: G is a finite union of compact cosets KsK.

q.e.d.

Let us construct non-compact Fell groups. Let p be a prime and Gp C G12(lFp) be the subgroup consisting of the matrices

a b (0

with a e IFp and b 6 IF

1)

P

Call Kp the diagonal subgroup (b = 0) of Gp and for any set of primes A

consider the restricted product GA

relative

=

TT*' pE.A

G

P

G TT pEA

G

p

to the family (Kp)p E A . By definition, a family s = (sp)A

of elements sp 6 Gp is in GA if sp E Kp

Thus GA

KA

=

(p 6 A) except for a finite set of p

IT K and the topology of GA is defined by

pEA p

GA induces the product topology on KA KA is open (and compact) in GA Obviously, GA is a locally compact group : it is a disjoint union of

open cosets of KA and GA

@4 A finite

compact

To study representations of GA , we have to start with the representation

theory of the finite groups Gp. Since the derived subgroup (GpGp) of Gp is the unipotent subgroup Np (defined by a = 1) the abelianized of Gp is isomorphic to ]Fp and we can thus construct p-1 characters

(1-dimensional irreducible representations) of Gp. As Gp has exactly

p distinct conjugacy classes

X

0

, X1 (a=l,b¢0), Xi (a=i) i = 2,...,p-1

(a=l,b=0)

there is only one other irreducible representation 7tp (all this when

p # 2 :). The equality Card Gp

=

p(p - 1)

=

1+...+1+(p-1)2 p-1

shows that dim 7cp = p - 1

. Since Gp is finite, hence compact and discrete,

there are two canonical normalizations of Haar measures on G P... and we

160

choose the third normalization mp(Kp)

=

1

(hence mp(Gp) = p

and J

In an obvious sense, the product ® mp

= m

= p Gp

1 1

Gp

)

.

is the Haar measure of

A

GA normalized by m(KA) = 1 .

We also construct a representation 7CA of GA as follows.

The left regular representation of Gp in L2(Gp/Kp) decomposes as E ® Vp

1 +

(Vp orthogonal to the constants)

and as Gp is the Punion of the disjoint cosets Kp , KpsKp (where s is the matrix a = b = 1)

dim VKP P

= 1

and we select the unique function xp C Vp which is orthogonal to the constants, both left and right Kp invariant, positive at the neutral element and of norm 1. Relatively to the elements xp g P , construct the restricted tensor product

HA

we can

= O(V

of the Hilbert spaces

P .

This space is obtained by completion of the

vector space generated by the

® yp

,

yp 6 Vp

and yp = xp except for finitely many p's.

(The choices of vectors xp give embeddings of the finite tensor products

over finite parts A' of A and HA is an inductive limit of these finite tensor products.) The unitary representation 7[A of GA in HA is

defined on the total set ® yp by

CA(0 Yp)

= ® 7Gp(Yp

This representation 7tA is unitary and irreducible (cf. Ex.3 at the end of this section).

For each subset B C A, there is a canonical projection

GA -+ GB (forget the components xp for p ¢.B), and the representation ;rB can be considered as a unitary irreducible representation of GA by

composition. It is obvious that

dim 1CB < oo 4=4

B

finite C A

.

161

By our choice of Haar measure on Gp , representation 7cp in

dp

P

=

the formal dimension dp of the

V is

dim

mp(Gp)

_ (p -1)/p

=

1 - 1/P

and similarly, the formal dimension dp of the identity representation in dimension 1 of G

is

p

do

=

1/p

Thus we can expect that dB

7CB belongs to the discrete series when

= iT (1 - 1/p) TT 1/p > 0

.

A-B

B

If this is the case, dB is the formal dimension of 7GB .

(16.10) Proposition. Let us assume that the selected subset A of primes

satisfies c 1/p
A group and the discrete series of GA consists representations 7r

of the unitary irreducible

for which the set

(p E A :

7CfI

G

acts scalarly i= S P

is finite and the formal dimension of such a representation is given by

d,c

= IT 1/p iT (1 - 1/p) S

A-S

The proof of this proposition follows easily from the preceding remarks (cf. also ex.3 below). For more properties of the groups GA (and other examples of Fell groups), the reader is referred to Robert 1968.

More generally, a good reference for the discrete series is Borel 1972.

162

EXERCISES

In these exercises, G will still denote a locally compact unimodular group and .12. its discrete series.

1. Prove 7G' E n 4-+ 7C 6 12, and d, = d in this case. Moreover, if H is a Hilbert space with an orthonormal basis and A is a Hilbert-Schmidt operator in H,

A=

)r(f),

A

= 7C(f)

2 / AI2

7r(f)

=

.

In particular, if

12

2

Conclude,

=Ld

r

Z dR- k7!(f)/I2 AL

= jA

17L(f)a2

2. Let (n,H) be a unitary irreducible representation of G and assume that u, v E H are such that the corresponding coefficient cu E LI(G) is integrable. Show that for any unitary irreducible representation (;V',H') which is not equivalent to (7C,H)

and any coefficient

c'y (x and y E H') of 7r', f cv(s) c'y(s) ds G

=

0

3. With the notations of this section concerning the groups

GA (A subset of primes) and their representations 5B (B (=A), prove the following. a) All 'CB are irreducible (it is enough to show that 7LA is

irreducible, and for this, use a method similar to the one used in (3.4)

b) Show that the coefficient p of >tp with respect to

u = v = x is given by p

p(s)

=

1

cp(s)

=

-11(p-1)

for s E P , for

c) Define the functions cB

s O-Kp

= (&c

Bp

1

and

A-B /cBI2 compute their quadratic norm by integrating the functions over the fundamental system of compact (subgroups)

Ticp

x IT K

(S finite in A) p f-S d) Conclude that ?rB6IL if E1/p
D

S

A

).

163

4. Show that if the discrete series of G is not empty, the center Z of G is compact.

(When the center Z of G is not compact,

one can still define discrete series "mod Z" by requiring that one non-zero coefficient - hence all - has an absolute value in L2 (G/Z).)

164

17

THE DISCRETE SERIES OF S12(]R)

Let (7L,H) be a unitary irreducible representation in the

= d and consider

discrete series of some group G. Fix u E H with IIuII2

the coefficients cv = cu of

7t

. By definition of the discrete series,

they are square summable on G and more precisely (16.3.a) shows that

-o. L2(G)

H

v

,

cv

F--o-

is an isometry which embeds 7c in the right regular representation

cv -* cv uniformly on G

r of G. If a sequence vn -4 v in H ,

n

i cv(s) - con(s) I

=

I (u 17(s) (v - vn)) 1 4

hul I1v - vn11-

In the model of 7t' consisting of right translations in {cv all functions c

v

:

v e H}

are continuous and

cvn -- cv

in

L2(G)

.

cvn -- cv uniformly

(because the assumption is equivalent to vn --!o- v

in

H ). This situation

is very peculiar and reminds one of properties of holomorphic (or harmonic) functions. Indeed, the discrete series of Sl2(U) will be constructed in

spaces of holomorphic (or anti-holomorphic) functions. The group Sl2(dt) acts on the upper half-plane Im(z) > 0,

but it will be more convenient to let it act in the unit disc lwl <1 (conformally equivalent to the upper half-plane) and thus, to use a conjugate SU(1,1) of Sl2(l2) in S12(it). Here is a description of this new group.

The elements g

with

g E Sl2 (E) Writing g = (a

c

d)

. SU(1,1) are by definition the matrices

_

cc - dd)

1

0

(0

-1)

=

bd

Thus

as - bb

_

-1

0 (O1

-1

we find the conditions

as - b6 ac - bd (ac - d6

0 ) g*

g (1

=

1

and

and multiplying the second by a

ac

and

ad - be =

1

16S

aac

=

bad

Bad

=

c + cbb

c

,

b(ad - bc)

=

5

=

Similarly, d = a and thus a) with 1al 2 -lb12 = 1.

SU(1,1) consists of the matrices (b

(17.1)

This group acts in the unit disc by

a b

z -a

=

(b

az + b

$z + a a) z (observe that if IzI = 1, z = ei(P and aelcf + b

e- its

=

fief? + a

ae11P + b

=

e-it' w

ae-lf+ 5

w

is a complex number of modulus 1). The stabilizer of the origin for

this action is obviously defined by

b/a

0

=

hence

,

b

=

0

.

We call T this compact subgroup of SU(1,1) (conjugate to K C S12(]R)

T 3 (e

0

e-lq

0

)

)

For the convenience of the reader, we give a table of correspondance for this conjugation.

12((E

S12OR)

_

g

=

a b (c d)

SU(1,1) 1

-i

1

i i

i ) g (-1 1)

2i (1

= l2'(a+d)+2(b-c)i

Z(a-d)

l Z(b+c)i/

2i(1 1) g(1 Re(a+b) (-Im(a+b)

K

-ii) F- (b

Im(a-b) Re(a-b)

cosy sing

(el'P

(-sins cos

et

O t)

C0

e-t/

Ch t

Sh t

CSht

a)

Cht)

0

E-T 0

0

e

iV

Cht `Sh t

Sht)

(Ch t

-iSh t Cht

(iSht

Ch t

T

166

Before constructing the discrete series of SU(1,1), we have to give the expression of the invariant measure of the unit disc X C T (isomorphic to SU(1,1)/T ). Only an adaptation of the example 5 of sec.12 (p.125) has to be made. For s = (b (17.2)

(fiz +

dz w =

a)-2

b)

we have

and w = j(s,z)-2

=

The chain rule for derivations gives immediately =

j(st,z)2

and one can check directly (or extracting roots: j(s,z) = bz + a does not vanish in the unit disc since (17.3)

j(st,z)

1a12

-

lb12 = 1 implies IaI>Ibl

)

j(t,z)

=

On the other hand, one also shows that 1 -

lj(s,z)I-2

Iwl2

(1 - Izl2)

=

In particular, the 2-form on the unit disc X (1 -

dz Adz

IZ12)-2

(put z = x + iy ) Izl2)-2

2i(l -

=

dx A dy

is invariant under the action of SU(1,1). An invariant measure on X is thus (17.4)

r2)-2

m =

(1 -

r2)-2

dx dy

r dr dT

(1 -

=

(r = 1z I).

Let us look at the (quasi-invariant) measures

mk

=

r2)k-2

(1 - r2)k m =

(1 -

on X and form the Hilbert spaces Hk

=

dx dy

L2(X,mk)

.

The group G = SU(1,1)

acts in a natural way in Hk by (17.5)

j(ts,z)-k

rk(s)f (z)

=

The cocycle condition (17.3) shows immediately that 7Ck(st) = irk(s) lEk(t).

Moreover, since j(s,z) -+ 1 uniformly on compact sets when s i e the continuity of s I-v 7rk(s)f follows (sec.13 p.132) and 7Ck is a

representation of G in Hk . (17.6) Lemma. The representation (1Ck,Hk) of G = SU(1,1) is unitary.

Proof. By definition,

17ck(s)f112

=

(

Iz12)k dm(z)

167

and use the invariance of m

Make the change of variable z' =

= J'Ii(ts 's z) I-2k(1

Ink(s) f'I2

I2 dm(z)

IJ(ts,s'z)J(s,z)I-AIf (z'2(1-Iz'I2)k

=

J

dm(z')

_

= IIfb2

Thus the lemma follows. The representations Xk in Hk are reducible. Indeed, let us Hkol consisting of holomorphic functions

consider the vector subspace in Hk = L2(X,mk).

(17.7) Proposition. The subspace Hhol is closed in Hk and

Hhol

{0} for

k ) 2

Proof. The second assertion is obvious since all polynomials in z belong to Hk when k ) 2 ( mk

=

r2)k-2

(1 -

dxdy and k-2 > 0 ).

To prove the first one, we use the mean value property of holomorphic functions as follows. Let f be holomorphic on a disc Iz-alt r

f

"Iz-al
f(z) dxdy

=

f PdP f2n f(a+peie) dA 0

0

f (a)

21c f Opdp

=

7rr2 f (a)

The preceding computation is valid for any holomorphic f on the disc X provided Ial,<1-r

(say 0 < r < 1). Thus,

7rr2 jf(a)I

< f fz-a I < r If(z)I dxdy

=

r2)2(k-2)I

rf

If(z)(1 -

r2)2(2-k)

(1 -

dx dy

"Jiz-al < r lVz

<(S(r

If(z)I2(1-r2)k-2dx dy/

Iz-aI
<

r 0f11(Jf

Iz-al < r

Vs

(Iz-aI
(1-r2)2-k

\f$

dx

dy}

4

lUs

(1 -

r2)2-k dxdy)

Let us take 0
in the fixed disc IzI4 1- r . Then the integral of (1 -

over this

disc is finite and we can write

If(a)I < Cr 11f II

(

IaI5,1 - 2r ).

This shows that convergence in L2 implies uniform convergence on all

168

compact subsets of X for sequences of Hkol

.

Take now a convergent

sequence (fn) C

Hkol

,

in Hk = L2(X,mk)

--- f

fn

Thus (fn) must also be a Cauchy sequence for the uniform convergence on compact subsets of X and it has a limit g, which is a holomorphic function on X. But extracting if necessary a subsequence of (fn), we

can assume that

fn(z) -+ f(z)

particular, this shows

for z qt N (mk-negligible). In

f(z) = g(z)

(17.8) Theorem. For k

for z qt N , hence f = g E Hk q.e.d.

2, the unitary representations (ck,Hkol)

are irreducible, in the discrete series of G = SU(1,1) and the formal

dimension dk of Irk is

dk = (k-1)/it for the choice of Haar measure

on G given by

f f(s) ds

=

f dm(z) f dt f(st)

G

(

z =

T

X

where dt is the normalized Haar measure of the compact group T. Proof. The proof of (17.7) has shown that the evaluation forms

Hhol -s

bz

are continuous. By the Riesz

T

f H f(z)

,

theorem, they are given by scalar products

with some vectors ez Eol ez E The set

lez

:

Hhol

is defined by is total in

z E X j

(ez I f)

=

(ez1 f) for all f e

f

=

=

0

for all z e X

0 EHk1

By definition of 7tk we have 7Ck(ts) e4(z)

=

j(s,z)-k et(sz)

whence j(s,z)-k

(ez 1 1Ck(ts)

(esz

and

(irk (g) ez for all

I

eZ )

=

(J(s,z)-k

esz

I

e

E X. By the totality of the set of e,, this implies

7(s) ez

Hkol

Hhol

for all z EX =* f(z)

0

=

f(z)

j(s,z)-k esz

169

Thus, for z and Z

in X

j (s,z)k x , ,

,I

7r,

_

(es, I esz)

j (s, )k j (s, z)k (irk (S)e

ez )

7[k(s)ez)

j (s, i)k j (s,z)k (e, I ez) since

Irk(s) is a unitary operator. Now, for any operator A E End(Hk01)

which commutes to all Gk(s) (s a G) we can check that

(e, 1 Aez)

(e, I ez)

and

y

have the same transformation law under the substitution This proves that the function h = hA defined by

h(4 , z)

is invariant

:

= h(4 , z)

sz) But (eI e

(e, I Aez) / (e4 1 ez)

_

ez(4) is holomorphic in

=

)

(s

G)

(and anti-holomorphic in z

z

by permutation of z and Z ). Thus h is meromorphic in and t E T (stabilizer of 0)

h(4 ,0) The points t.

Take

0

:

h(t.

=

.

h(t. ,0)

.

describe a circle centered at 0 (with radius

0).

Since h is meromorphic, it must be constant

h(4 ,0) = c. Now, for

each z E X, there exists a sz E G with

0 whence

h(4

z)

h(sz4 ,0)

=

=

c

.

=

(e, I c ez)

Consequently, h is constant (= c) and

(e, I Aez) This proves A ez

=

c (e I ez)

= c ez for all z E X and A = c l

( 4 , z e X) is a scalar

operator. The only projectors which commute to 7Ck on Hkol are 0 and 1 (rk,Hkol) is irreducible. It is easy to determine a coefficient of

this representation. We take u = v to be the constant function 1 E

Hhol k

and compute c = ci = cu

cu(s)

1z(2)k-2dxdy = JJ u(z) j(ts,z)ku(ts z) (1 -

cl(s) = JJ (bz +

a)-k

(1 - 1zI2)k-2dxdy

if

s

=

(b

a)

170

Using polar coordinates, we have

i[ 2v

1

c1(s)

J

f 2W residues

ei8)-k

(5 + b r

dH

0

0

d9

But the integral J

r2)k-2

dr r (1 -

=

is easily computed with the method of

)k

0(a+ Ae

:

d8 = (iz)-1 dz

z = eie , dz = ieie d9

let us put

(1Z)-1(oc

o2

+Az)-kdz

2zi Residue

zI = 1 =

2X/-Xk

Z=O

I«I>IPI.

when

In our case,

ci(s)

=

ci(a,b)

2k k A

=

1

so dr r (1 -

r2)k-2

X

=

(k-1) a k

From here, it is easy to compute the quadratic norm of this function on G. Since u = v = 1 is T invariant, the function cu = c1 is (left and) right T-invariant and the integral over G reduces to an integral over X (we use the normalization of ds on G, with the normalization of dt on T). We have to consider this function ci on G as a function

on X via the canonical mapping G

But if

s = (b

G/T i X, s

---> a)

b/d

,

=

1a121z12

s- 0

z

and

= 1a12 - 1

Ib12

=

, IaI-2

Ia12(1 -

Iz12)

1

=

1 - IzI2

,

=

This gives

2

ff

012 l

(k-1)s 7ra

lal -2k dm(z)

1 211 2

(k-1)

kl1

l

=

(k-1)L

J (l -

r2)k-2

r dr dO

7r3 3 (k-1)

But it is easy to check that the norm of 1 E Hkol is given by 11102 = k l1

We have thus proved 114112

=

701114

kill

The formal dimension of (TCk,lik°l) is thus dk = !(k-1)

.

q.e.d.

171

EXERCISES

1. Fork 3 2, give the decomposition of the restriction of k,Hk 1

to the compact subgroup T (circle group). (For s =

`0

as = 1 , show that

7Ck(s)

zt _

.)

ak+2,* ze

2. Using anti-holomorphic functions in Hk (k

2), construct

representations in the discrete series of SU(1,1) which are inequivalent to the representations constructed in (17.8). (Calling 2C _k the represen-

tation constructed with antiholomorphic functions, show that the restrictions to T of 7tk and 7Ik are inequivalent: use Ex.l.)

3. Show that (7(,Hko1) is integrable when k > 3

4. Let e.(z) =

zJ-k/1157-k11

, so that (ej)j

orthonormal basis of the representation (7Ck,Hkhol

k is an

).

a) Compute the norm

IIzj112

= 7c j: (m-2): (m+j-1):

=

7c j+m-l m-1/ ( 3 )

b) Prove that (notations of the proof of (17.8)

e (z)

_

e j -o k J

e.(z) J

= jL>k

lzJ//-1J zJ

in particular, e(z) is not a polynomial in z when

74 0 (eo(z) = M-1

is a multiple of 1). c) Compute es 0(z) for s = (a

a)

SU(1,1)

.

172

18

THE PRINCIPAL SERIES OF S12(]R)

We shall need the following general result. (18.1) Proposition. Let X be a locally compact measured space and A a bounded operator in L2(X) which commutes to all multiplication operators MT

f r-k Tf

:

(T 6L°p(X)).

Then A is itself a multiplication operator A = D'Iy, for a certain q'EL°O(X).

This proposition states that the commutative algebra of multiplication operators is a maximal abelian (von Neumann) subalgebra of End(L2(X)).

Proof. For each compact set K C X, denote by

YK

the characteristic

function of K and put

*K E L2(X)

A( TO For K C K', we have

T K

K CfK' whence

A(TK fK,)

=

A(SK)

.

=

fK A( YK')

since by assumption, A commutes with the multiplication operator by

TK E L"(X). From this identity

Y K

, we infer that

Y K 1K'

TK' (restricted to K') extends YKI Hence there exists a function

TIK

=

'4)

K

with

restriction of T. to K

for all compact subsets K of X. Take now f E Cc(X) and call K its

support. We must have

A(f)

=

A(TKf)

=

f A(TK)

=

f'VK

=f

Moreover, as is well known, Sup ess 4(x) xaK

<,IA I

and T e L`°(X). The continuous operator A must coincide with M4, since it

coincides with MW on the dense subspace C,(X) of L2(X).

q.e.d.

173

Let us come back to G = S12(lR). We introduce its subgroups

K (circle group) consisting of the matrices

cos 8 -sin 9

(sin g

(0 C 0 < 2 i ) ,

Cos 9

'

A diagonal subgroup consisting of matrices (0 1/a) (a E ]R")

N unipotent subgroup consisting of matrices 1

n

(0

1

(n r= ]R)

,

P = AN = NA upper triangular subgroup (semi-direct produt). Thus we have =

G

=

KAN

KP

PK

=

=

ANK

=

=

NAK

The product formula G = NAK can be proved for instance as follows. The group G acts transitively on the upper half-plane In z > 0 (z e (E) by

fractional linear transformations

:

the subgroup NA already acts transiti-

vely as the following formula shows a2i + b

b)(o

p

The stabilizer of the point i is K. Thus if g E.G is an arbitrary element, we can put z =

and find n c N, a C A with (na)-Ig e Stab(i)

z. Thus K

=

(na)-1g

and we have found a decomposition g = nak with k =

.

As we have seen in sec.13 (example 2, p.123), x-2

dx dy is a Haar measure on P 9

0

y

1/x

and the modular function on P is

'-P(0

l/x)

=

1/x2

Let us consider the character a of A defined by a (a

l0a) =

a2 so that

= oc(a)(a E A, n E N).

'Ap(an)

We extend the character trivially on N in order to be able to write

A,(p)

=

-(P)-,

(p E P) .

The representations of the principal series of S12(]R) will now be

constructed by an induction procedure from characters of A (compare with sec.8). For the sake of simplicity, we only consider the characters

174

as (leaving as an exercise the consideration of the characters

oc5.sgn(a) ).

The representation ACS acts by right translations in the Hilbert space Hs of (classes of) functions over G satisfying

fIK E L2(K)

:

pfff2

=

Iift)K

f If(k)I2 dk

=

K

(18.2) a

1/0

f(n(a

=

a)g)

°c(0 1/a)s f(g)

a2s f(g)

=

(18.3) Lemma. When Re(s) = zf the representation (S,Hs) is unitary. Proof. Since K is compact, G/K has a G-invariant measure: the quotient mapping G --s G/K is proper, so that by composition it induces a transpose

Cc(G/K) -s Cc(G) and the image of the Haar measure ds of G is such a measure on G/K. By definition,

f f(s) ds

=

G

f

ds f f(sk) dk

G/K

(f 6 Cc(G)).

K

This measure ds on G/K must in particular be invariant under left translations by elements of P, and taking for each element of G/K its representative in P, we see that we can identify ds with dp (Haar measure

of P) and write

f f (s) ds

=

f dp j f (pk) dk . P

G

K

We can now prove the lemma. We have to evaluate the quadratic norm of

a restriction to K

:

k i-'r f(kg) of a right translate of an f e Hs

For this, we write kg =

.

pk and k' depend on g, but this

element is fixed throughout this proof so that only the dependence in k is essential and visible). Thus 117cs(g) f11K

= f Jf(kg)I2dk

=

f If(pkk')Jdk K

K

By definition of Hs , we can write

Ap(Pk)-s f(k') If(Pkk')I2

p(PI)-2Re(s)If(k')I2

= Q

=

p(Pk)-1 Jf(k')J 2 Choose P E Cc(P) with

r

dp T(p)

=

if Re(s)

=

1 and assume fIK continuous. Put

p

IfI2 6 C(K)

,

i

F(pk) = T(p) y(k)

,

175

so that F = (q ® y/ E Cc (PK) = Cc (G). Thus II c(g)f(I2

fK Ap(pk) 4,(k') dk f ((p) dp

=

p

fP x K (p (p) (k') Qpl (pk) dpdk =

=

JK JG

dk

dp F(ppkk')

dk f dp F(pk') aP' (pk) K

_

P

dk fP dp F(pkg)

=

fP

_

fK

ds F(sg)

since G is unimodular !

=

=

=

fG

ds F(s)

We have thus found

f F(s) ds

=

II tcS(g) f IIK

f dp f dk F(pk)

=

JP

JG

f dp T(p) f dk y(k) P

f dk lf (k) 2

=

K

_

K

IIfU;

=

K

The lemma follows by density of the space C(K) in L2(K).

q.e.d.

(18.4) Theorem. The unitary representations

(7csHs)Re(s) group S12(IR) constructed above are irreducible when s

of the z Z

Proof. This irreducibility is best seen on another model. Thus we explain how this other model is obtained from H

.

If

«*

1

0

(0 1/a) (x

1

s

cos 9 -sin 9

_

we have

of

_

(sin 9 cos 9

k9

= 1/cos 0

,

(the representations ITS are trivial on

x = tg 9

the element -1 e S12(R), so that we can consider them as representations

of PS12 (1R)

= S12 (IR) / (±1)

f (k9)

and thus take 9 e [-Z n ,zit [ ) .

0 f ((1/s

=

(cos

_

9

cos 9 )

9)-2s

f

(tg 9

0 1

1 0I

9

1)

Take x = tg9 as a new variable : dx

(tg29 + 1) d9

=

d9

,

=

dx/(x2 + 1)

and put

(P(x)

Tf (x)

=

We have

=

f (1

0

nh

00 >

r JK

If(k)l 2dk = n f If(k9)I2d9 x/z

=

Thus

176

7r

f I(x2 + 1)s _

(x2

Cf(x)I2

1)-1 dx

+

=

(since Re(s) = Z).

(F(x)I2 dx

Thus

f H

_ Tf

defines an isometry in this new model

(1 x

L2(IR,dx). Let us determine the action of 7cs

Hs

:

0)

1/a

0) (a 1) (c

(1 x

=

g

1

bx + d

=

y

b) d

( =

b

y/a =

and

(« i$ ) (1 0 4/a y

)

=

0

1

ax + c

a2

=

(bx +

0) g)

=

f ((0

and

bx + d

a

ax+c bx+d

d)-2

As we have

1Cs(g) f

(X

f ((X

=

0)

=

a2s f(1

)(y

0)J _

0)

the preceding formulas show that

[7CS(g)q'] (x)

(bx + d) -2s

=

In particular, the operators

is('

0)

dc

C (bx

+

)

are translations and the operators

are suitably normalized dilatations :

'rs(0 1/a)

7ls (b lea) T (x) lags T(a2x)12

a2s

=

=

a2

(a2x)12

(since Re(s) _

and thus

17,s(0

1/a)TI2 dx

=

a2 1 T(a2x) 12 dx

12

dx

effect a Fourier transform

Let us now

T(S)

(x)

=

e-ixdx

0),.

The operators

will now be multiplication operators by the

75(1

characters

Xc (1x,I= 1

`-ii

eic t

hence Mx is a unitary operator in L2(IR,dt) ).

'z)

177

The announced irreducibility will now follow from

Schur's lemma.

Any continuous operator which commutes with all 7r (g) will have to

commute with all multiplication operators M

(;(unitary character of 1R

x

But, for any locally compact measured space X, the

in

mapping Lo0(X) --p End(L2(X))

cf H MT is continuous when L°°(X) has

,

the weak topology coming from the duality with L1(X) and End(LZ(X)) has the weak operator topology. Since this weak operator topology is the

weakest for which all mappings

A-

(8

A Vy)

(9

,

lY E L2(X) )

are continuous, it is enough to check that the composites

if y (9

(9 111)

f

=

9T .q dm (x)

are continuous on LO°(X) (equipped with its weak topology). But since

9 ,

yy

g y E Ll (X)

c L2 (X)

,

the preceding assertion follows by definition. We infer from this that if an operator T commutes with all multiplication operators M2(, it will

still commute with all multiplication operators M,

,

cp

being in the

(weak) closure of the space generated by the x E L°(X) (composition of operators is separately continuous for the weak operator topology).

Thus (18.1) will be applicable as soon as we have proved that for any locally compact non-discrete field F, the set of additive unitary characters x, E F is total in LDo(F) for the weak topology

defined by its duality with LI(F). By the Hahn-Banach theorem, it is enough to prove that

f e Ll(F)

,

b'xEF

,

=

==#

0

f

=

0

But this is a classical result of the theory of Fourier transforms since

indeed

=

f f(x) X(x) dx

f

f (x 1)

=

f

=

0

.

From (18.1), we now infer that the operator T which commutes with the representation 7CS in

is a multiplication operator MIF by a

certain function T E 17°(1R,df). Since T also commutes with the dilatations,

T

must be invariant under dilatation

178

r 2s J y a (a

x) e -ixi dx = J (a x) e- ia2x 2

2

a2(s-1)

(t/a2)

j/a2 2s-2

a

d(a2x)

,

T and the commutation relation implies

a2(s-1)(

a2(s-1) A (t/a2) Cf ( ) for all ^EL2 41

' T)

This implies precisely that T is invariant under positive dilatations

T( )

+'

_ /4.sgn

One can see that for s # z , only the functions with operators commuting with >cs

:

T = b4t

/A.= 0 furnish

But for s = z

,

the

multiplication operator corresponding to

Y(E)

i(sgn

=

+ 1)

is a non-trivial projector commuting to 7C, and = z

7C, z

®

7C, z

is a decomposition into irreducible components.

q.e.d.

179

19

DECOMPOSITION ALONG A COMMUTATIVE SUBGROUP

Let G be a locally compact group and A a closed (hence locally compact) commutative subgroup. In order to have an invariant measure on the homogeneous space G/A, we assume that A GIA = 1 (cf.(12.8)). We shall use the duality theory for locally compact abelian groups to give a decomposition of the left regular representation of G in L2(G). Let A be the Pontryagin dual of A : it is a locally compact abelian group having for elements the unitary characters x : A --V.

For f e Cc(G), write fs

A -> U

:

,

T"

a --+ f(sa) and consider its

Fourier transform

fx(s)

fs(x )

=

f f (sa) x(a) da

=

A

One can check without difficulty that (19.1)

fZ(sa)

f%(s) x(a)

=

,

IfxI E Cc(G/A)

On the other hand, the Plancherel formula on A gives =

11fsa2

1IfsQ2

hence

f If(sa)12da

=

f Ifx(s)I2dx

A (dX,

A

is a suitably normalized Haar measure of the group A) . Integrating

the preceding equality on G/A

(19.2)

11 f,12

=f

G

r PA

If(s)t 2 ds

=

dx

f/A

ds Ifx(s)I 2

On the space of continuous functions on G satisfying (19.1), we can

define the norm

II 12 =

J

ds Ifx(s)12 /A

and obtain the Hilbert space L2(G,Z) by completion of this space Cc(G,x).

With these notations, we have obtained a decomposition CC (G)

-3 TTCc(G,x)

:

f F-s (fx)

for which a "continuous Pythagoras theorem" is valid.

180

(19.3)

11f112

d

f .uf xu2

=

L2 (G,9()

A

We can say that the Hilbert space L2(G) has been disintegrated as a

Hilbert integral over A

(19.4)

f

L2 (G)

f f (fx)

dx

^ L2(G, A

A better notation for (fx) would be f fx d X since the Fourier inversion formula for f E CC(G) (hence fs E CC(A)) reads

fs(a)

=

JA

fs(x,) x(a) dx

(Recall that this inversion formula holds in quadratic mean: if one needs pointwise convergence of all integrals, one should take functions f of the form f = fl * f2 with fi E L2(A).) For a = e, we indeed get

(19.5)

f(s)

=

fs(e)

=

J

fs(%) dX = f fx(s) dx . A

All operations on functions have been effected on the right, hence commute with left translations. Taking t-1s instead of s in (19.5)

we obtain f(t-1s)

=

f

fx(t-ls) d

X

whence

e(t)f (s)

=

ftX (t) f X (s) dx

e(t)f = f tx(t) fxdX, or, even more succinctly

(19.6)

t (t)

=

f ex(t) dx

,

t

f

=

dX

This is a decomposition of the left regular representation, parametrized by A. Unfortunately, even when A is a maximal abelian subgroup of G, the representations (IX L2(G,X)) are not necessarily irreducible (even when G is a finite group.). Let us consider some examples. First

(19.7)

G

=

SO3 (JR)

A

=

SO2 (1R)

.

The irreducible representations of this group "are" real and that we can write

L2(G)

(Vr ® Vim)

L2 (G,7,)

(Vn & VIC )

1C = irv so

181

where V,

c n denotes the subspace (having dimension 0 or 1) of vectors

which transform according to the character x, relatively to A. Thus we have

(tx L2(G,x))

=

V"

6),

7r IA?x We see that this decomposition is given by the Frobenius reciprocity (sec.8, especially (8.10))

:

L2(G,x)

is the space of the induced repre-

sentation IndA(x) and the decomposition of this representation "uses" all

x (and in fact all

7G such that 7VJA

7r

except a finite number

of them by sec.10). For G = S12(]R), the situation is no

(19.8)

SO2OR)

better. One can take A =

a "discrete" decomposition of L2(G) parametrized by

whence

the dual 7l of A (the summands of this Hilbert sum must be reducible).

Alternatively, one can take A = diagonal subgroup with positive entries t

e A 9 (0 eot) =

(tE1R).

diag(et,e-t)

The dual A of A is isomorphic to lR by identification of the unitary

character

diag(et,e-t) H Bits to the real number s. In this way, one gets a "continuous decomposition" of L2(G). But for this group, a "decomposition into irreducible components" of L2(G) has both a discrete part (sec.17) and a continuous part (sec.18, cf. Lang 1975). Thus it comes nearly as a surprise that for certain particular groups, the preceding naive decomposition gives a decomposition into irreducible ones. We quote two such examples. (19.9)

Let first G be the free group F2 with two generators a and b.

This group is a discrete group hence is unimodular. We take A

ra

=

=

an

:

n

.

}

The unitary characters of A are identified with complex numbers of modulus 1

9 = x(a),

10I =

1

satisfies x(an)

=

An

Thus, the dual A of A is identified to the unit circle group in L The decomposition of the left regular representation (L ,L2(G)) along A

uses the spaces H9

=

L2(G, 9)

(8 E U(l) = {9 e0 :

IAI

= 1}

182

consisting of functions f over the discrete space G satisfying

f(y an)

f(1) An

=

(n (Z E)

Jf J E e2(G/ a)

,

An example of such a function is

E(Y)

if

0

=

_(an)

ra

= An

In fact, the left translates of this function E with respect to a system of coset representatives of

G/ a make up an orthonormal

basis of He . We intend to show that in the disintegration

2(G)

=

f

H dA ,

all representations

t

d9

=

J 9I = 1

e

101 = 1

.Q8 ,H9) are irreducible and inequivalent. To prove

this contention, let us consider the operator of = function

E = EQ

as eigenvector

a(E) (an) o(( E ) hence o((&)

(if an

E(an-1)

=

=

en-1

= e &(a n)

vanishes outside a

= 6-E

. We even see that the multiples of

only eigenvectors of

it r

satisfies a(f)

if f e H (x 9 and observe that the classes an X

aT'a

..9(a) having the

:

a am X

exist an integer t with

e at

=

'Af ,

take

(n CE) are disjoint

has some elements in common with an-m

=

E are the

a , there must

hence n = m and

= 0). As f -+ 0

at infinity on the discrete space G/T'a , we must have

)nf(1) As

-s 0

=

when Inl-- «>.

0, this implies f(1) = 0 and f proportional to F_. Let us now take

an operator T commuting with the representation $e . Then T(E) must

also be an eigenvector of

a(TE)

of

T(o(E)

(with respect to the same eigenvalue 0) =

T(0 E)

= 9TE

Hence T E = 2 E by the preceding observation. Then,

T(ee(y)E) _ t9()()T(e) = *Ae0()j)E But the elements Yy)E form an orthonormal basis of H0 when ' runs through a system of coset representatives of G /ra, so that we deduce that T = 'A 1 is a scalar operator. In particular, the only projectors

commuting to e0 are 0 and 1 and this representation is irreducible.

183

Moreover, since an equivalence T

8, carries eigenvectors of

Q

:

9

le(a) on eigenvectors of e9,(a) (with respect to the same eigenvalue), we infer that such an equivalence can exist only if 9' = 9. A construction similar to the preceding one can of course be made with any generator of G in place of a, for example with b. We obtain a new disintegration

ez ( G)

f

=

K dZ

e =f

,

As before, the representations (a, ,K Moreover, all

and

dZ

ill = 1

ICI = 1

z

are irreducible and inequivalent.

i9 are inequivalent !

To see this, we show that the

operator (3 = Q9(b) has no eigenvector (whereas AC(b) has an eigenvector

corresponding to the eigenvalue Z ). From an equality (3(f) = we would infer f(b11) _ W nf (1) . But, as the cosets bn a (n E 7L) are disjoint for every

' E G, we should have

µ f (y )

0

and this again implies f(')

for ml --> oo = 0

:

f = 0 .

As this example shows,

two distinct disintegrations can be totally unrelated. The problem of decomposing the left regular representation of a locally compact group does not have a unique solution (not even unique up to isomorphism) and we shall have to introduce a class of groups for which it is well stated. (19.10)

As a second example, let F be a locally compact field and G

the group consisting of matrices

(aEF",

(a b)

together with the subgroup A for which a = 1 (unipotent subgroup of G).

This group G is not unimodular, but the restriction to A of its modular function is 1 (thus coincides with the modular function of A since this group is commutative). For any unitary character x E F we can consider the space of functions on G satisfying (19.11)

W01

b)g)

=

X (b) f(g)

To simplify notations, let us write f(a) instead of f(a

f(a)

=

f((

a0

bl)

=

x(b)-1 f(a

0) so that

b)

Thus we have

f(0 b)

= x(b) f (a)

(a E F').

184

In particular the function f on the group G has an absolute value which is a function on G/A = F" . The condition

IfIEL2(G/A) = L2(F",d"a)

(19.12)

=

L2(F", lal-1 da)

simply means

lal-IIfI2 CL1(F",da)

=

LI(F)

or

IaI2IfI EL2(F) This suggests to put q =

IaI

2 f

and to consider the representation

7t;

in L2(F) which comes from the right regular representation in the space L2(G,X) defined by (19.11) and (19.12). One checks easily that

this representation is given by the formula

1r,(1) 1 (x)

(19.13)

=

((f E L2 (F) )

la 1 2 xb (x) ( (xa)

On this expression, the unitarity of the operators 7!x(a and

7v

( b)

0)

.

(a E F")

(b e F) is obvious. I claim that all these representations

>tx are irreducible provided F is not discrete andx

1

.

To follow the

proof of this assertion, the reader who is not familiar with the theory of locally compact non-discrete fields (i.e. local fields) will have to assume F = E in which case the result is fully significant. First we

have to recall that if x # 1, any unitary character of F"

is of the

form xb (recall ,,.b(x) = x(bx)). Thus any operator T in L2(G, 7 which commutes to all operators nx(1

b) (b e F) has to be a multiplication

operator by a bounded measurable function (as in the proof of (18.4)). If the operator T still commutes with the

(O ) this bounded measurable

function must still be invariant under all dilations x H xa (a e F") and thus be constant. This again proves that the only projectors commuting to 7x are 0 and 1 and 71x is irreducible. Observe that this solvable group G is not unimodular, but trivial modifications will give similar results for the unimodular subgroup G1 = Ker AG consisting of the matrices (0

and bEF .

b) with a E F",

Iat = 1

185

Appendix to sec. 19

Note on Hilbertian integrals.

:

Let X be a locally compact space, m a positive measure on X and x i-

Hx a family of Hilbert spaces on X. A measurable structure

(with respect to m) on this family consists of the data of a vector subspace

TT

.,4, of the product

Hx satisfying

xEX is measurable on X for each f

1) x r , JIf(x)(I H

x

2) (saturation condition) if x

(g(x)If(x))H

measurable on X for all g % , then f E ,(1 , 3) there is a sequence (fn)n {f (x)

n E NJ

:

is

x

C 46 with

e 1N

total in Hx for each x e X

The elements f E 4, are called measurable fields. The Hilbertian integral (with respect to m and

J

Hx dm(x) X

consists of the measurable fields f for which fX IIf(x)IIHx dm(x) < o0 This Hilbertian integral is a Hilbert space with respect to the scalar product

(f I g)

fX (f(x) I g(x))H

=

dm(x)

x

As usual, strictly speaking, the elements of the Hilbertian integral are classes of measurable fields (equal m-nearly everywhere...).

Practically, it may be better to start with a sequence

of fields (fn)n E IN with x '--r 11fn(x)IIHx measurable on X for all n GIN and

{fn(x)

total in Hx for each x E X. Then one defines the

: n 6- IN 3

measurable fields f by

f c 46 1 ,

x -+ (fn (x)

I f (x) )

Hx

measurable for all n E 1J.

Then 4G is a measurable structure on the family 1T Hx (only the first condition needs verification).

The elements f of the Hilbertian integral are also denoted by

f=

JX f(x) dm(x) G

fX H

x

dm(x)

Their "continuous" components satisfy a "continuous Pythagoras theorem" II2

hI f

=

f

X

Iif(x)II 2

dm(x)

186

An interested reader who is familiar with the properties

of measurable functions will be able to show that

f Hx dm (x)

#

t0j

X

provided the subset of X defined by Hx # {0j is not m-negligible.

187

TYPE I GROUPS

20

When a representation p of a group G is an orthogonal (Hilbert) sum of representations all equivalent to some irreducible

representation 7r of G, we say that p is isotypical of type 7r (or equivalently that p is a multiple of 7c ). These representations can be written in the form

p

TC ® 1

=

in

H ® H

for some Hilbert space H. Taking an orthonormal basis of H, the matrices of the operators

p

(x) take a diagonal form with equal blocks on the

diagonal ,7C(x)

7r(x)

p(x)

The algebra

=

?(G)' of operators commuting to all operators of p(G)

is ( 7C(G) ® 1)'

= 10 End (H)

(cf. ex.l of sec. 7) and the operators which commute to p(G) and have"the 11

same form as the p(x) are scalar operators. More precisely, the operators having the same form as the p(x) (x e G) are the operators in the

von Neumann algebra generated by p(G)

p(G)"

_ wk (weak closure of linear combinations of operators p(x), x E G).

This von Neumann algebra is simply p(G)"

= (1 ® End(H))'

=

End(H,,) 0 1

and the elements of this algebra which commute to p(G) are the elements of the center of p(G)' or of the center of p(G)"

:

188

p(G) I n p(G)"

=

(End(I-I) ® 1) = U 1

(1 ® End(H)) /1

We give a definition. (20.1) Definition. A unitary representation p of a group G is called factor representation when the von Neumann algebra generated by p(G)

has a center reduced to the scalar operators

p(G)" np(G)'

=

U1

:

.

In the terminology of von Neumann algebras, p is a factor representation when the von Neumann algebra generated by p(G) is a factor. By definition, isotypical representations of type n (for some irreducible representation 7r) are factor representations. In this case, the factor generated by p(G) has minimal projectors # 0 and this property characterizes factors of type i.

(20.2) Definition. A group of type I is a locally compact group G for which all factor representations are isotypical (all factor representations generate a factor of type I). (20.3) Example. Let G be a discrete group with infinite conjugation classes

{e3(e.g. G = F2 free group on two generators as in (19.9)).

The left regular representation q ,Q2(G)) is a factor representation. Let indeed

'U. = U(G)

=

. (G)

= <e

(G)>wk

be the von Neumann algebra generated by the left translation operators.

Any operator T E U is given by a convolution operator

T(T)

=

fT

=

T( F_ e) E t2(G)

Ee

=

characteristic function of the neutral element e e G.

fT

T

where ,

If we assume that T lies in the center of It, we must have fT

=

(x E G)

Ex * fT * Ex-1

hence fT must be constant on the conjugation classes of G. But fT E e2(G) implies that fT

0 on the discrete space G. Thus fT must vanish on

all infinite conjugation classes of G. In our case, we see that fT = c Ee

must be a multiple of e

e

and T =

is a scalar operator.

189

It can be shown that these factors U(G) are of type II.

Hence, these groups are not of type I. Here is a general construction of factor representations. (20.4) Proposition. Let G = G1 x G2 be the product of two locally compact groups and nr a unitary irreducible representation of G. Then the restriction of 7r to G1 is a factor representation.

Proof. We have to show that if T E ir(G1)" commutes with n(G1), then T is a scalar operator. But x(G1) commutes to 7t(G1)"

still commutes to

7r(G2) hence

7c(G2)

(by the von Neumann density theorem). Thus T commutes with 7r(G1) and n(G2)

:

T commutes with n(G).

Schur's lemma implies that T is a

scalar operator.

q.e.d.

The conclusion of the preceding proposition

(20.5) Remark.

still holds if we only assume the original representation =

7c of G

=

G1 x G2 to be a factor representation. Indeed, T E 7t(G1)" c n(G)"

shows that T is in the center of the von Neumann algebra generated

by 7r(G). The following lemma is to be compared with (14.10). (20.6) Lemma. Let is be a factor representation of a locally compact

group G. If there is a compactly supported continuous function f

with

.

CC(G)

0 # n(f) compact operator, then 7r is a multiple of an irreducible

representation (it generates a factor of type I). Proof. The operator n(f) belongs to the von Neumann algebra n(G)" generated by n(G). The same is true of 7[(f)* and hence of

7c(f)*7c(f)

=

(cf. (14.2))

7c(f* * f)

This operator is hermitian compact and non-zero

7r(f)x

0

==

117c(f)xII2

=

(x I 7G(f)*tt'(f)x)

0

.

Let a > 0 be an eigenvalue of this operator and Pa be the spectral

projector (with finite rank) corresponding to A

PA E {7c(f)*7c(f)jI? C 7c(G)" As P a has a finite dimensional image, and is a type I von Neumann algebra.

7r(G)" has some minimal projectors

190

(20.7) Proposition. Let G = G1 w G2 be the product of two locally compact

groups and it a factor representation of G such that the restriction of 7r to G1 is isotypical (for example, assume G1 of type I) 7r IG ~

7G1 0 1

with

:

1C1 irreducible representation of G1

Then there is a factor representation 7t2 of G2 and an equivalence

7C=

7.1&7C2

If R is irreducible,

7C2 is also irreducible.

Proof. By assumption, the operators belonging to 7E(G1) c 7C(G1)"

can be written in the form 761(x1) ® 1 in a tensor product space H1 0 H2 As 7r1 is assumed to be irreducible, these operators generate the von

Neumann algebra

End(H1) ® 1. The operators 7r(x2) (x2 6 G2 C G1 x G2

must commute to all preceding operators hence can be written in the form 10 7[2(x2). This gives a construction of a representation 7C2 of G2

We have to show that this representation 7C2 is a factor representation (irreducible if it is irreducible). If T r= , (G2)" G End(H2) commutes

with 7C2(G2), 10 T will commute with 7C(G1) = 7Ci(G1) ® 1 and with 1 ® 7C2(G2) hence with

7C(G). But this operator belongs to 1 ® 7c2(G2)" C

c 7C(G)" hence is a scalar operator. The proposition follows

.

(20.8) Proposition. The product G = G1 x G2 of two groups of type I is a group of type I.

Proof. Take a factor representation it of G = G1 n G2 . We have seen that ICIG1 is a factor representation (20.4). As we are assuming that G1 is a group of type I, this factor representation is isotypical, i.e. equivalent to 7VIG1

7Cl 0 1

where 7C1 is some unitary irreducible representation of G1. By (20.7),

we can find a factor representation P of G2 such that

it ° 7C1 0 f

.

Since we also assume that G2 is a type I group, this factor representation

p is isotypical, say f = 762 ® 1 with some irreducible representation n2 of G2

.

Thus we have

it

7E1®1C20 l = To®1

Obviously "o = 7E'1 ® 7C2 is irreducible (7C1 0 7C2(G) = 7E1(G1) ®9L2(G2) generates the von Neumann algebra End(H1 ® H2) ) and the representation

7C ° 7G0 0 1 is isotypical.

q.e.d.

191

When working with factor representations, it is sometimes convenient to identify the representations (ir,H.) and

(7C(S> 1 , Ha ® H),

thus identifying in particular all isotypical representations of the

same type. A way of achieving this identification is to introduce the following equivalence relation. (20.9) Definition. Two factor representations called quasi-equivalent when the mapping

7r(G) C End(H,) ) by

rr(G) and

7r, G of a group G are

7r(x) -ii 0'(x)

(defined on

extends to an isomorphism of the factors generated

0'(G) respectively. The set of quasi-equivalence classes

n

of factor representations of G is called quasi-dual G of G. Of course, each unitary irreducible representation of G defines a quasi-equivalence class and two equivalent unitary irreducible representations of G are quasi-equivalent. Thus we get a canonical

map

from the unitary dual of G (set of equivalence classes of unitary irreduci-

ble representations of G) to the quasi-dual of G. When the group G is of type I, this map is bijective by definition and the two sets can be identified.

To be able to give a general decomposition theorem into factor representations, it is necessary to define a suitable Borel

structure on the set G. Without going into details and proofs, let us just indicate how this can be done (details can be found in Dixmier 1964, where other references can also be found). First, we have to assume that G is separable (i.e. has a

countable basis for the open sets). Let Hn = e2([O,n[) denote the standard Hilbert space of dimension n
= l2([0,-[)

=

e2(N) ). On the set Facn(G) of factor representations

of G in Hn, there is a topology for which 7ri -+ 7r

V v e Hn

,

precisely when

7Gi(.)v --, 7c(.)v on all compact sets of G

On the disjoint union Fac(G) of all Facn(G) (n <eo), we put the sum topology. The Borel structure of the quasi-dual of G is defined by

taking the image structure under the canonical map

Fac (G)

-i Gn

192

(20.10) Theorem, Let 7c be a unitary representation of a separable

group in a separable space H. Then there is a positive measure m on the quasi-dual X = G which realizes a decomposition of 7i into factor

representations in the following sense. There is an isomorphism

H =s

JX

Hs dm(s) leading to an equivalence

7c

f

X

7[s dm (s)

where each (7rs , Hs) is in the class of s C X =G . Moreover, the measure m = mn is essentially uniquely deter-

mined by it. Let us recall that two measures m and m' on X are called equivalent when they have same negligible sets (thus there exists a nowhere vanishing measurable function f with m' =

Then, under

the conditions of (20.10), the class of m is well determined. This

measure m is obtained by disintegration of the commutative von Neumann algebra

'

= center of the von Neumann algebra generated by 7r(G)

n(G)'o7r(G)") . As the example (19.9) has shown, there is no uniqueness for finer decompositions into irreducible components, in general (cf. also (20.3)).

(20.11) Some comments. The theorem (20.10) can in particular be applied to the left regular representation of any (separable, unimodular) locally compact group G, even when G is not of type I. For example, it can be applied to the groups

G

=

GF

= (a

b)

:

aEF", b EF }

where F is an infinite discrete (countable) field. These groups are discrete, hence locally compact and unimodular. But one can show that their left regular representation can be disintegrated into two nonequivalent Hilbert sums of irreducible ones (one can "induce" either

from the diagonal subgroup or from the unipotent subgroup defined by a = 1 to get such disintegrations). In the next section, we shall consider this situation for groups of type I.

193

EXERCISES

1. Let 7G be a factor representation of a group G in some

Hilbert space H. Let H1 C H be a closed invariant subspace of H (with respect to this representation). Show that the induced representation

7r1

:

G -* Gl(H1) is a factor representation (in H1) .

2. Let G and H be two locally compact groups,

7C 4G G and

n

7C1

, 7t2 6 H be irreducible representations such that )T

7C1

and

7c ® 7t2

are equivalent

(as representations of G x H). Show that xl is equivalent to

7G2

194

21

GETTING NEAR AN ABSTRACT PLANCHEREL FORMULA

In this section, G will always denote a locally compact

unimodular group. When necessary, we shall even assume that the topology of G has a countable basis for the open sets (in this case, G is metrizable and has a countable dense subset). As in the preceding section, we motivate a definition by the consideration of a finite dimensional situation. Let 7c be a representation of some group G which is a finite sum of inequivalent irreducible representations. In a suitable basis of the representation space, the matrix form of 7t is given by diagonal blocks

sizes) and

(of varied

Schur's lemma shows that if an operator commutes to

'C,

it is given by a scalar multiplication in each irreducible subspace. In particular, the algebra of operators commuting with 7c(G) is a commuta-

tive algebra

:

it is a subalgebra of the algebra of diagonal operators

in any basis compatible with the decomposition of >t into irreducible components. Conversely, if 7C: G --p- Gl(V) is a finite-dimensional

representation with a commutative commuting algebra n(G)' c End(V),

it is often possible to diagonalize this algebra by choosing a suitable basis of the representation space. This will be the case if 7[(G)'

is stable under the operation of taking the adjoint (e.g.

7C unitary).

In this way, we shall find a decomposition of 7c as a sum of inequivalent irreducible subrepresentations. In general, we give a definition. (21.1) Definition. We say that a unitary representation x : G -*- G1(H)

has a simple

spectrum when its commuting algebra u(G)' C End(H) is

commutative. By definition,

ir has a simple

spectrum exactly when

7C(G)' C 7c(G)" , i.e.

ir(G)'

=

center of the von Neumann algebra generated by n(G).

As an immediate consequence of the definition, we note that (21.2) a factor representation with simple spectrum is irreducible.

195

To be able to prove that the biregular representation of a unimodular group G has a simple spectrum, we need the commutation

theorem of Godement-Segal ( (21.9) below). We first construct the Hilbert algebra of the group G. Let H = L2(G) throughout this section.

(21.3) Definition. A moderate function f on G is an element f E H such

that the operator g -* f * g

CC(G) -> H extends continuously to H.

:

The treatment given in sec.14 shows that all f e L1(G) rl H

are moderate

.

Let us denote by J the unitary operator in H given by

J(f)

=

(where f*(x)

f*

=

f(x-1) )

.

Obviously J(f * g)

=

Jg * Jf

and a function f is moderate precisely when g H g * f (defined on Cc(G)) extends continuously to H. For f and g g Cc(G) first, let us define (21.4)

Uf(g)

f * g

=

=

Vg(f)

(21.5) Lemma. When f is moderate and g E H, one still has Uf(g)

=

f * g

(where Uf still denotes the continuous extension of the precedingly

defined operator Cc(G) --o H) . Proof. a) Take first g e Cc(G). There is a sequence (fn) G Cc(G) which converges to f in H. Thus

fn * g --y f * g

in L2 (G) = H

and fn * g

Vg(fn)

=

-->

V9(f)

=

Uf(g)

proves the assertion in this particular case.

b) Take now g e H arbitrary. There exists a sequence (gn) C Cc(G) with

gn -s g

in H .

Thus

f * gn

=

a)

Uf(gn) - Uf(g)

in H

But the Cauchy-Schwarz inequality shows that gn)(Y-lx)Idy

If * g(x) - f * gn(x)14 f If(Y)(g Ilf II

II g - gn II (in H

so that f * gn -r f * g

:

quadratic norms !)

,

uniformly on G, whence the conclusion.

196

We recall that

(f I g)

f* * g (e)

=

(f , g a H)

Thus, the equality (f * g I h)

(gIf* * h)

=

is an immediate consequence of the associativity of the convolution

(f*g)**h =

f*) * h

(g*

g* * (f* * h)

=

(when all products are defined...) (21.6) Definition. The Hilbert algebra A

A(G) of the group G is the

=

convolution algebra of moderate functions on G

:

If f E A, then f* E A

and

(g* I f*)

(f I g)

=

,

Uf

=

Vg

Uf*

=

Vg*

for f, g EA . By definition, CC(G) C A C H so that A is dense in H Let us prove the following identity (21.7)

J Uf* J

=

Vf

(f e A)

For x and y in A first, we have

(J U f* J x I Y) =

(x * f I Y)

The density of A in H

_

(y* I U f* J x)

(y* I f* * x*)

=

_

= N f x I Y)

together with the continuity of the operators

J Uf* J , Vf gives the equality (21.7).

(21.8) Theorem (Godement). Let us denote by U (resp. V) the von Neumann algebra in End(H) generated by left (resp. right) translations

of G. Then the set

I

=

t Uf : f moderate in H}

is a *-ideal, weakly dense in 1) Proof. For x, y e A, we have UfVx y =

Uf (y * x)

= V Vy f

=

V Uf y

hence Uf Vx

= X Uf C End(H)

if f C A

197

This proves that Uf commutes to V when f is moderate (an operator commutes to 1Y as soon as it commutes to the right convolutions with

elements of C(G)) : Uf C ILP when f is moderate.

Moreover, if T E 7U'', TUfx

TVx

=

f

=

is continuous in x,

Vx T f

proves that T f is moderate and UTf Since Uf

=

=

T Uf GI . Since composition of operators

I is a *-ideal of 11'.

Uf*

is separately weakly continuous, V' is weakly closed and the proof will be complete if we show that V' is the weak closure I" of I. It is enough to show that Jt' c P. Thus, we take any T E V', X E I' and show TX = XT. But we know that for each x E A, T Ux E I. Hence TU

commutes with X

x

:

Taking a sequence xn such that U we conclude T X

X T as

=

T Ux X

=XTU

x

.

--+ 1H (for the strong topology),

claimed

(21.9) Corollary (Godement-Segal).

q.e.d.

We have U' =V, '(f'

=

U..

.

Proof. As U. and V commute, It C i)' and by taking the weak closure ',(," C 7)1. It will thus be enough to show (,(," > V, or by the above theorem (21.8), functions f q- H

.

'L" D I

i.e.

'Lt" -a U f for all moderate

Thus we have to show that all elements Uf (f moderate)

commute to 'U" . Still by the above theorem with left and right exchanged (consider (21.8) for the opposite group...), it is enough to show

Uf commutes to all Vg (g moderate)

(f moderate).

This follows from (21.5) and associativity of convolution. In detail, if h is moderate, y = Vgh is also moderate so that

UfVgh

=

Ufy

=

=

VgVhf

=

VgUfh

This gives the conclusion.

Vyf

=

VV h f g

=

Uf y = Vy f and

Uf(Vgh)

=

(since VgEU')

198

(21.10) Corollary. When G is unimodular, the biregular representation of G in H = L2(G) is unitary and has a simple spectrum. Proof. We have to show that the commuter of the von Neumann algebra

generated by U and u' is commutative. But this commuter is

is the common center of

'LC and 11

.

(21.11) When G = r is a discrete group with infinite conjugation classes different from {e4(

rg

oup ICC as in (19.9)) its biregular representation

is irreducible (cf. (20.3) and (21.2)). There is a general decomposition theorem for simple spectrum representations (compare with (20.10)) . We state it now. (21.12) Theorem. Let G be a (separable) unimodular locally compact

group and p a unitary representation of G in a separable Hilbert space H with simple spectrum. Then, there is a positive measure m

on G and an equivalence

P ->

J® G

,

it dm (7C )

f®

H

H7C dm(it)

G

Again, we refer to Dixmier 1964 for a proof (observe that the decomposition is made according to the commutative algebra P(G)'). Still assuming G to be (separable) and unimodular,

we can

apply the preceding theorem to the biregular representation of G in

L2(G) (cf. (21.10)) and we get thus a positive measure m on (G x G)^ decomposing this representation. When G is of type I,

(G - G) - =

G xG

(cf.(20.7)) and we can write

Ix

r

J

7t®o dm(rr,()

,

L2(G)

fH

H, dm(it,o)

We can say more in this case. The identity (21.13)

J e t x r (s,t) f

is immediately verified :

=

r x t (s,t) Jf

(f e H

= L2(G) )

199

J..Cxr(s,t) f (x) f*(t-lxs)

=

_

Lxr(s,t)f (x 1)

=

f(s-lx-lt)

r l (s,t) Jf (x)

=

Since J is anti-linear, it induces an equivalence

J:

-Lx r --s r x J_

Moreover, the identity (21.7) J Uf* J = Vf gives

JcJ. =

(21.14)

c*

(observe that Uf* = (Uf)*

and Uf = Vf iff Uf E

.

). Thus we have two

decompositions of the biregular representation of G in L2(G):

J

:

f 7<®a- dm(rc,o)

-. f F(D . dm(7C,r)

.

Since there is a relative uniqueness in the decompositim given by (21.12) (just as in (20.10)) , we infer that

Trot

n®a- =

nearly everywhere on the support of the measure m. Thus we can assume = Tt) on the support of m. The support of m is contained in the subset

7CEGJ _ {1L®7L

17L®7C

:

7CEG}

and we can even write

t x r = f A to 7c df4,C) G

for a certain positive measure tu, on G

.

The class of the measure

well determined by the general theorems already alluded to ILt

by f

.

:

. is

replacing

simply amounts to replacing the scalar products

by f 1(T) But since the space of 2®7C can be canonically identified to HC ® HJr (Hilbert space consisting of the Hilbert-Schmidt operators on H 7r), it carries a canonical scalar product. Indeed, the scalar product of H>c is determined up to a positive constant by the requirement that >r is

unitary, and an amplification by a factor 2 > 0 of the scalar product

of H, produces an amplification of a factor 1/2 of the scalar product of H v

,

thus leaving invariant the scalar product of H,Vi6H7. (correspon-

ding to the Hilbert-Schmidt scalar product on operators).

200

For these canonical choices, the corresponding measure µ. on G is the

Plancherel measure

It is well determined up to a positive µPl constant (and completely well determined if a Haar measure on G is .

fixed). By definition, we have

f

.E x r

n® 7v

dpl(n)

JG End2 (Hn) dµpl (?r)

L2 (G)

This last identity can be written IIf112

=

J^

(f e H =L2(G))

Iln(f)12 dfcpl(7t)

Minimal closed invariant subspaces of this biregular representation are

given by the points 7c G for which ftpl(n) > 0. These points are precisely the discrete series representations and as we have seen in sec.16, ,tL, (n)

=

do for these representations. The discrete series

gives the atomic part of the Plancherel measure. The Plancherel formula

for a type I group takes the general form

Ilfll2

d.,,b,c(f)II2

=

+

fG IIn(f)N2 dyjl(7c)

with both discrete and continuous parts (we say that the measure µP1

for which

.1P1(n)

=

0 for all

7C E G

is a diffuse measure) .

This formula generalizes the cases G compact and G =lR (locally compact and abelian).

In general, the support of the Plancherel measure is strictly

smaller than , and one defines the reduced dual of G by

Gred

=

Supp

".P1

CG

For G = Sl2(R), one can show that the reduced dual consists of the discrete series (as in any locally compact unimodular group) and the principal series. But in this example G also contains a supplementary series (cf. Lang 1975).

One can also show that

Gred where p ;

=

nEG : Ker Tt*

Ker Q*3 = {7CE

Ker 7L*

Ker r* }

p* denotes the extension of representations from G to C*(G).

(Recall that the left and right regular representations are equivalent!)

201

EPILOGUE

The following analogue of the Peter-Weyl theorem (p.29) has not been proved.

Gelfand-Raikov theorem. For any locally compact group G and any x # e e G, there is an irreducible unitary representation

i' E G such

that 8(x) # id.

However, this theorem follows easily from the deeper results (20.10) or (21.12) applied to the (bi-)regular representation of G. They show that one can even take

7

in the reduced dual Gred

(For details, cf. Dixmier 1964 or Gaal 1973.)

On the other hand, our introduction of type I groups has been made in an ad hoc way.

A proof

that a certain class of groups

(semi-simple real or p-adic algebraic groups) is of type I is not trivial and usually follows the following pattern. a) Find a large (or maximal) compact subgroup K of G such that 7CIK has finite multiplicities for all 1CE G

b) Prove that

,c(f) is a compact operator for all

and f e Cc(G) (or f 6 L1(G) or even f e C* (G)

)

(groups with this property are called CCR groups).

c) Prove that CCR groups are type I

.

Interested readers will find details in Dixmier 1964: liminaire = CCR ( = completely continuous representations), postliminaire = GCR

--? CCR .

202

REFERENCES Adams J.F.

LECTURES ON LIE GROUPS W.A. Benjamin, Inc. 1969

Borel A.

REPRESENTATIONS DE GROUPES LOCALEMENT COMPACTS Lect. Notes in Math., 276, Springer-Verlag 1972

Bourbaki N. TOPOLOGIE GENERALE Chap. I a IV , Hermann Paris 1971

---

ESPACES VECTORIELS TOPOLOGIQUES Chap. IAV, Masson 1981

-------

INTEGRATION Chap. I a IV, Hermann 1965 INTEGRATION Chap. VI

, Hermann 1959

INTEGRATION Chap. VII et VIII , Hermann 1963

Chevalley C. THEORY OF LIE GROUPS I , Princeton Univ. Press 1946

Dieudonne J. FOUNDATIONS OF MODERN ANALYSIS

-----

--Dixmier J.

Acad. Press 1960

ELEMENTS D' ANALYSE vol . 2, Gauthier-Villars, Paris 1969

Gauthier-Villars, Paris 1970 ELEMENTS D'ANALYSE vol.5, Gauthier-Villars (Bordas)1975 ELEMENTS D' ANALYSE vol. 3,

LES C*-ALGEBRES ET LEURS REPRESENTATIONS Gauthier-Villars, Paris 1964

Fefferman C.

Gaal S.A.

PROC. INT. CONGRESS MATH.

Vancouver 1974 vol.1 (95-118)

LINEAR ANALYSIS AND REPRESENTATION THEORY Springer-Verlag 1973

Lang. S.

S L 2 (IR)

Riesz F., Nagy B.,

Addison-Wesley 1975 D'ANALYSE FONCTIONNELLE

Gauthier-Villars, Paris 1975 (6e ed.)

203

Robert A.

SYSTEMES

DE POLYNOMES Queen's Univ. 1973 (Math.ser.# 35)

EXEMPLES DE GROUPES DE FELL C.R.Acad.Sc.Paris 1978 t.287, pp. 603-606.

Rudin W.

FOURIER ANALYSIS ON GROUPS, FUNCTIONAL ANALYSIS

Interscience Publ. 1962

McGraw-Hill Publ.Co. 1973

Serre J.-P. REPRESENTATIONS LINEAIRES DES GROUPES FINIS Hermann

Weil A.

Paris 1967, 1971

L'INTEGRATION DANS LES GROUPES TOPOLOGIQUES Hermann

Paris 1953

204

INDEX

Approximate unit 134 arbitrarily small subgroup 4 Ascoli's theorem 7

Bounded measures 134 Burnside's th. (5.2) 41

Cantor

group 5 Cayley-Klein parameters 96 central functions 63 character (1-dim. rep.) 60 of a representation 63 central character 64 Clebsch-Gordan th. (10.5) 105 closed operator 14S convolution 53 algebra L1(G) 54 , 72 -

G-action 13

Galois group 6 G-morphism 14 group : Cantor - S Fell - 158 Heisenberg - 113 metrizable - 4 orthogonal - 3 p-adic - 6 profinite - 6

rotation - S03(1R) 95 symplectic - 4 tot. discontinuous - 5 type I - 187 unimodular - 119 unitary - 3 group algebras (table) 75

Haar measure 7 , 10 , 117 Decomposition th. (7.8) 67 discrete series rep. (16.1) 151

-

- of a group :.fl. 156

disintegration 180 disjoint representations 44 dual G (5.10) 50 duality (Tannaka) 90

Enveloping algebra 41 essentially self-adjoint op. 146 Euler angles 95

Heisenberg group (Cor.4) 113 Hilbert algebra 75 , (21.6) 196 Hilbertian integrals 185 Hilbert-Schmidt op. (8.1) 78

Induction of rep. 82 integrable representation 151 intertwining operator 14 invariant function 63 involution f -* f* (14.2) 133 isotypical components

L2(G,it) 40 , 41 V,c (7.8) 67

Factor rep. (20.1) 188 Fell group 158 finiteness th. (5.8) 46, (7.9) 69 ,

Kakutani's theorem 7 kernel operator 29

(8.5) 81

formal dimension of rep. 153 fundamental lemma (5.4) 44 Fourier inversion 58 Frobenius-Weil recip. (8.9) 86

L egendre polynomials 26

205

Matrix

coefficients 32 metrizable group 4

Schur's lemma Ex. S, 20, (8.6) 81 (15.11) 148 , (15.12) 149

model of a class

Schur's orthogonality relations

50 moderate function (21.3) 195 modular function AG 118

Negligible set 7 non-degenerate rep. (of an algebra) 136 normal operator 146

(5.6) 45 , (16.3) 153

simple spectrum (21.1) 194 square summable rep. 151 stellar algebra (= C*-alg.) 73 stereographic projection 98 symmetric operator 146 symplectic group 4

Tannaka duality 90

Operator closed - 145 essentially self-adjoint - 146 Hilbert-Schmidt - (8.1) 78 intertwining - 14 kernel - 29 normal - 146 self-adjoint - 146 symmetric - 146

Tchebycheff polynomials 102 totally discontinuous group S type I group 187

Unimodular group 119 unitary group 3 representation 13 -

Vector integration 55 Peter-Weyl th. 29 , (4.2) 32 Plancherel th. 58

- measure 156

Weyl's theorem (7.10) 70

Quasi

dual G (20.9) 191 quasi-equivalent rep. (20.9) 191 quasi-invariant measure 122

MAIN THEOREMS Reciprocity (Frobenius) (8.9) 86 reduced dual 200 relatively invariant measure 122 representation 13 , (13.1) 128 completely reducible - 15 discrete series - (16.1) 151 disjoint -s 44 equivalent -s 14 factor - (20.1) 188 faithful - 16 globally continuous - 13 integrable - 151 irreducible _ 14,130 non-degenerate - of alg. 136 quasi-equivalent -s (20.9) 191 regular -s 17 simple spectrum - (21.1) 194 square summable - (16.1) 151 unitary - 13 Rodrigue's formula 27

Burnside (5.2) 41 decomposition (7.8) 67 finiteness (5.8) 46 (Banach spaces) (7.9) 69 (general), (8.5) 81 Fourier inversion 58 fundamental lemma (5.4) 44 Frobenius-Weil (8.9) 86 Peter-Weyl 29, (4.2) 32 Plancherel 58 Schur's lemma Ex.5, 20 (8.6) 81 , (15.11-12) 148-149 Tannaka 93 Weyl (for characters) (7.10) 70