Unitary Representations and Harmonic Analysis, Second Edition (North-Holland Mathematical Library)

UNITARY REPRESENTATIONS AND HARMONIC ANALYSIS An Introduction Second Edition

North-Holland Mathematical Library Board of Advisory Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B. Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, L. Nachbin, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 44

NORTH-HOLLAND AMSTERDAM OXFORD NEW YORK TOKYO

Unitary Representations and Harmonic Analysis An Introduction Second Edition

Mitsuo SUGIURA Professor Emeritus The University of Tokyo

1990 NORTH-HOLLAND AMSTERDAM OXFORD NEW YORK TOKYO

KODANSHA LTD. TOKYO

Published in co-edition with Elsevier Science Publishers B.V., 1990 Distributors f o r Japan: KODANSHA LTD. 12-2 1 Otowa 2-chome, Bunkyo-ku, Tokyo 112, Japan Distributors outside Japan, U.S.A. and Canada. ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands Distributors f o r the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

ISBN: 0 444 88593 5 ISBN: 4 06 203509 x (JAPAN)

Copyright 0 1975, 1990 by Kodansha Ltd. All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review) Printed in Japan

To the memory of Edward T.Kobayashi

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Preface to the Second Edition

The second edition differs from the first in the following places. In Ch. I, Theorem 5.6 on the Fourier series expansion of piecewise C'-functions has been added. In Ch.11, the proof of Theorem 8.1 has been more fully implemented by adding two Lemmas. In Ch.111, Theorem 1.9 has been replaced by Proposition 1.5. In Ch.IV, the proof of Theorem 5.1 is rewritten using Ch.III.Prop.l.5 instead of Proposition 5.7 which has been deleted. In Ch.V, Q 3, the description of limits of discrete series has been added. In the first edition, we dealt solely with the representations of SU(1,l). In the present edition we have added Q 10 on the realization of irreducible unitary representations of the group SL(2, R). Appendix G on Fatou's theorem is added in connection with the limits of discrete series. Bibliography is tripled in length by adding some recent works and Notes have been rewritten accordingly. The author would like to express his hearty thanks to W.L. Baily Jr. who read through the manuscripts of the 2nd edition and suggested many improvements. He also wishes to thank K. Nishiyama who provided me with an Errata for the first edition.

M. SUGIURA

January 1989

vii

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Preface

The principal aim of this book is to give an introduction to harmonic analysis and the theory of unitary representations of Lie groups. Fourier series and integrals were introduced into mathematics in connection with the problems of vibrating string and heat conduction. They are techniques of representing a vibration as a superposition of harmonic oscillations e'a'. The techniques of harmonic analysis turned out to be powerful tools in mathematical analysis and physics. In the meantime, the group theoretical character of harmonic analysis was discovered. The harmonic oscillation e4=( is an irreducible unitary representation of the additive group R of real numbers. It is a representation of the torus group T = R/2nZ when x is an integer. Thus Fourier series and integrals are deeply connected with the theory of unitary representations of the groups T and R. This connection was fruitfully pursued by many mathematicians and was extended to cover a wider class of Lie groups. In the 19203, H. Weyl constructed the representation theory of compact Lie groups. He determined the irreducible representations, gave an explicit formula for the characters. Moreover he developed the Lo-theoryof Fourier series on compact (Lie) groups with F. Peter. In 1947, two important papers on infinite dimensional unitary representations appeared. One was Gelfand-Naimark's paper on SL(2, C) and the other was Bargmann's paper on SL (2, R). Since then, many interesting works have been done in the field of unitary representatiods of non compact and non commutative Lie groups and of harmonic analysis on them. In the development of the theory, certain difficulties were encountered in connection with the existence of operator algebras (von Neumann algebra) of type I1 and type 111. For example, the uniqueness of (direct integral) decomposition into irreducible unitary representations is lost for representations of type I1 or 111. Fortunately, for a Lie group whose factor representations are always of type I (a type I group), the theory of unitary representations can be constructed along similar lines to the case of compact and abelian groups, although it will generally be more complicated. All the Lie groups treated in this book are type I groups.

X

PREFACE

This book is essentially self-contained. No special knowledge on harmonic analysis, unitary representations or Lie groups is required. However, the reader is expected to have some familiarity with Lebesgue integrals and fundamental concepts of functional analysis. For the reader's convenience, basic facts on these materials are summarized in the A p pendices. The author has preferred a concrete calculation on a given matrix group rather than to develop the general theory for Lie groups or locally compact groups. However, general arguments are adopted without hesitation when they give a clearer insight. I hope that the resultant book will kindle interest for further development of the theory. The following six groups are treated in this book: the onedimensional torus group T,the special unitary group SZ7(2), the 3-dimensionalrotation group SO(3), the n-dimensional vector group R", the Euclidean motion group of the plane M(2) and the real special linear group SL(2, R). In each chapter, all the irreducible unitary representationsof these groups are concretely constructed and harmonic analysis on them is developed. In particular, the Fourier transforms on.the spaces of rapidly decreasing functions, C"-functions with compact supports and square integrable functions are discussed. Chapter I deals with the ordinary Fourier series from the viewpoint of unitary representations of the torus group T.Since the compactness of the group T is decisive, the general theory of unitary representations of compact groups (Peter-Weyl theory) is given in $3. In & the I,irreducible unitary representations of the group T are determined and the results in $3 are applied to the group T. In $5 and $6, the Fourier series of smooth functions and distributions on T are discussed. In Chapter 11, the theory of Fourier series is developed to the non commutative compact Lie groups SU(2) and SO(3). Using the theory of charactersin $2 and the calculation of the Haar measure in $3, it is proved in §4 that the representations constructed in $1 exhaust the irreducible unitary representations of SU(2). In $5, the notion of Lie algebra is introduced for a linear Lie group. $5 is the preparation to $6 which discusses the Fourier series of smooth functions on SU(2). In $7, the irreducible representations of the rotation group SO(3) and their realizations on the spaces of spherical harmonics are discussed. $8 gives the only exception of the requkment for self-containedness. In this section, E. Cartan's theory of highest weight and Weyl's theory of characters are freely used. The aim in $8 is to show that the results in $6 for SU(2) hold for general compact Lie groups. The reader may wish to skip over $8. In chapter 111, Fourier integral and unitary representations of the group R m are treated. The theory is based on L. Schwartz's theorem (Theorem 1.2) on the Fourier transforms of rapidly decreasing functions. Since R"

PREQACB

xi

is not compact, the notion of the direct integral is needed for irreducible decomposition of representations. Using Bochner's theorem on positive definite functions, an arbitrary unitary representation of Rnis decomposed as the direct integral of irreducible representations in $3. The PaleyWiener-Schwartz theorem characterizing the Fourier transforms of Cmfunctions with compact supports is proved in $4. In $5 the notion of tempered distribution is introduced and their Fourier transform is studied. Chapter IV deals with the Euclidean motion group M(2) as a simple example of non compact and non commutative Lie groups. The group M(2) has a series of S n i t e dimensionalirreducible unitary representations which are good examples of induced representations. Harmonic analysis on M(2) is discussed. In particular, the inversion formula of Fourier transform on M(2) is proved in $3 for rapidly decreasing functions and the Fourier transforms of such functions are characterized in $5. The final and longest Chapter V treats the group SL(2, R), which is a prototype of non compact semisimple Lie groups. The group SL(2,R) appears in many branches of mathematics in different forms. A realization of SL(2, R) as the special unitary group SU(1, 1) of indefinite Hermitian form z1Zl-z212is particularly useful for constructing the representations. The group SL(2, R)z SU(1, 1) has the three different series of irreducible unitary representations; the principal continuous series, the (principal) discrete series and the complementary series. After a short preparation for Iwasawa decomposition in $1, these three series of representations are constructed in $2, $3 and $4 respectively. In $6, the irreducible unitary representations of SL(2, R) are classified by the infinitesimal method. In $7, the characters of irreducible representations are computed explicitly. $8 gives the Fourier inversion formula for the C"-functions with compact supports. In $9 harmonic analysis of two-sided K-invariant functions is discussed. The Notes at the end of the book include certain historical facts and supplementary results together with references €or further study. The bibliography contains a list of papers and books more or less directly related to the subject of this book. The present volume is based on lectures given by the author over the past decade at several universities including Osaka Univ., New Mexico State Univ. and the Univ. of Tokyo. I would like to express my sincere gratitude to Professor W. L. Baily for carefully reading the manuscript and for offering many suggestions to improve the exposition. I would also like to thank Dr. T. Hirai for his advice on the content of Ch. V, and Dr. Y. Shimizu for his help in proofreading. It is also a pleasure to thank the National Science Foundation for its financial support during my stay in the U.S. Finally, I wish to thank

xii

PREFACE

the staff of Kodansha Scientific, whose endurance enabled me finally to complete this book. The book is dedicated to the late Dr. Edward T. Kobayashi who provided me an opportunity to lecture most ofits contents and made the notes for me.

May, 1975

Mitsuo SUGIURA

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface . . .. . . . . . . .. . . , . . . . .. . . . . . .. . . . . . . . . . . .. . . .. . . . . . . . . . . .. i~ Conventions and Notations . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . xv

Fourier series and the torus group T . . . . . . . . . . . . 1

I.

$1. Introduction, 1 $2. Fundamental definitions, 5 $3. Unitary representations of compact groups, 14 &I. Fourier series of square integrable functions, 30 $5. Fourier series of smooth functions and distributions, 34

11.

Representations of SU(2) and SO(3)

. . . . . . . . . . . .47

$1. Construction of irreducible representations of SU(2), 47

$2. Characters of compact groups, 50 $3. Haar measures on SU(2), 54 &I. Enumeration of irreducible representations, 57 $5. Lie algebras and their representations, 59 $6. Fourier series on SU(2), 76 $7. Representations of SO(3) and spherical harmonics, 82 $8. Fourier series on compact Lie groups, 93

111. The Fourier transform and unitary representations of RM . . . . . . . . . . . . . . . . . . . . . . . .lo1 $1. Rapidly decreasing functions, 101 $2. The Plancherel theorem and the decomposition of the regular representation, 115 $3. Positive definite functions and Stone’s theorem, 122 &I. The Paley-Wiener theorem, 142 35. Tempered distributions and their Fourier transforms, 150

IV. The Euclidean motion group . . . . . . . . . . . . . . . . . .155 $1. Construction of irreducible representations, 155 §2. Classification of irreducible unitary representations, 165 93. Fourier transforms of rapidly decreasing functions, 169 &I. The Plancherel theorem, 179 95. Determination of 9 ( G ) and B ( G ) , 187

V. Unitary representation of SL(2, R) . . . . . . . . . . . .205 91. The Iwasawa decomposition, 205 92. Irreducible unitary representations, 213 I. PRINCIPALCONTINUOUS SERIES, 213 93. Irreducible unitary representations, 235 11. F?RINCIPAL DISCRETE SERIES,235 111. T H E LIMIT OF DISCRETE SERIES 246 &I. Irreducible unitary representations, 255 IV. COMPLEMENTARY SERIES, 246 95. K-finite vectors, 265 96. Classification of irreducible unitary representations, 280 97. The characters, 306 98. Inversion formula, 343 99. Harmonic analysis of zonal functions, 362 910. Irreducible unitary representations of SL (2, R),388 I. DISCRETE SERIES, 389 11. COMPLEMENTARY SERIES, 391 111. PRINCIPAL CONTINUOUS SERIES, 393

Appendix ...................................................... 395 Notes .......................................................... 411 Bibliography . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .417 Index . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449

Conventions and Notations

Theorems and Propositions are numbered separately in each section. Therefore both Proposition 3.1 and Theorem 3.1 exist in 43. A Proposition or Theorem is referred simply as (say) Proposition 3.1 in the same chapter. A results in a different chapter is cited as (say) Ch. 11, Proposition 3.1. Notations. 2, N,R, C, denote respectively the set of integers, the set of non negative integers, the set of real numbers and the set of complex numbers with their usual algebraic and topological structures. Re and Im signify the real and imaginary parts of a complex number. A x B denotes the direct product of sets A and B. M n ( C ) denotes the algebra of all matrices of degree n with coefficients in C. Let X be a topological space. Then C ( X ) denotes the vector space of all C-valued continuous functions on X. L ( X ) denotes the subspace of C ( X ) consisting of all functions with compact supports. The uniform norm I(flJm=supIf(x)l is a norm on L(X). Z#X

Let H be a Banach space andf be an H-valued function on I=(a, + m) and g be a real valued function on I. If g(t)-'f(t) is bounded on some subinterval (b, + m) of I, then we write

f(t)= O(g(t))

( t d + m 1.

If lim g(t)-tf(t) = 0, we denote t-+-

f(O=o(g(tN (t-. + m). Similar notations are used for two sequences (fn)neN and ( g n ) n c N . The complexification of a real vector space V is denoted by V C . VC is a complex vector space whose element z is uniquely written as z = x + i y , (x, y E V). If V is a real Lie algebra, then Vc is a complex Lie algebra. The support of a function f is denoted by supp (f).

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CHAPTER I

Fourier series and the torus group T

$1. Introduction A unitary representation is a homomorphism of a given group into a group of unitary operators on a Hilbert space (cf. $2 for the precise definition). The theory of unitary representation is a natural generalization of classical Fourier analysis. Fourier analysis is a method of decomposing “any” oscillation into simple oscillations called harmonic oscillation. So Fourier analysis is also called harmonic analysis. A harmonic oscillation is a motion given by an equation x = a cos n t + b sin nt, where t is the time and x is the coordinate of a moving point. The equation for x can be expressed as follows: x = d cos(nt a) (1.1)

+

where d= .Ja’ +ba, cos a = a/d and sin a =: -b/d. The number d in (1.1) is the amplitude, n, the frequency and a, the initial phase of the oscillation. The phase a is not related to the nature of the oscillation but depends only on the choice of the origin of the time scale. Fourier analysis assures us that any function in a suitable family of functions (such as smooth functions, functions of bounded variations) with the period 2a can be developed in a Fourier series:f can be expressed as w

f(t> =3

(1.2)

2

+ C (a,

+

cos nt b, sinnt)

n-1

where the coefficients a, and b, are determined by (1.3)

an=-

x

s:’

f(t) cos n t dt,

b, =

J:’

f(t) sin nt dt.

The expansion (1.2) means that the oscillation f is a superposition of harmonic oscillations with integral frequencies n=O, 1, 2, .... By Euler’s relation e“ =cos t

+i sin r,

1

2

FOURIER SERIES AND TORUS GROUP

T

the Fourier series (1.2) can be rewritten as

-

f(t)=

(1.4)

C

C,e*nt,

n=--

where the coefficients cn are determined by

(1.5)

1

c --(an n-

2

- ibn)=- f(t)e-tncdt 2n Is’’ 0

c- ---(a,+ib,)=~;; 1 n-

2

r

(n=O, 1, 2, .-).

of(t)e*ncdt.

The function xn(t)= etnc appearing in (1.4) expresses a circular motion with a constant angular velocity n of which the projection to the real (or imaginary) axis is a harmonic oscillation. From our point of view, the “complex” form (1.4) of a Fourier series is preferable to the “real” form (1.2), because the functions xn(t) are strongly connected to the group structure. We shall call the function e‘nt a harmonic oscillation rather than (1.1). The function Xn(f)=efRthas the following four properties: 1) Ixn(t)l= 1 for all t E R. 2) Xn(t)xn(s)= Xn(t +s). 3) x,, is continuous. 4) Xn(2R)= 1 . The properties I), 2), and 3) mean that xn is a continuous homomorphism from the additive group R of real numbers to the one dimensional unitary group U(1) = { z E CI IzI = I}. That is, xn is a unitary representation of the real number n. On the contrary, the group R. This fact hol4s for an’ property (4) holds only when n is an integer. And the function x,, for an integer n is a u n i t a j representation of the torus group T = R/2nZ which is isomorphic to U(1) and to SO(2). (See the last part of this section for the definition of the topological group T). Now we study the Fourier series for a square integrable function. All complex valued functions which are square integrable on the interval [ 0 , 2 ~ form ] a Hilbert space La(T)with the inner product

Another remarkable property of the family of functions ( z J n r Zis its orthogonality : 5) (xn, zn)=6n,m The determination of the Fourier coefficients in (1.5) is explained by the orthogonality relation 5). If a functionfin La(T)can be expanded as

LNTRODUCTION

3

in La(T),then the coefficient cn must be equal to

for all n. So the first problem in the theory of Fourier series for square integrable functions is as follows: Problem A : Can any function f in La(T)be expanded in La(T)as in (1.6)? The answer is affirmative and will be proved in 54. Here we explain the meaning of the expansion in La(T). In a Hilbert space H , a family ( x . ) , , ~of elements in H is called summable with the sum x in H if for any positive number E , there exists a finite subset Bo of A such that for any finite subset B 2 Bo, the inequality Ilx- Ex,l/ < E holds. x is denoted by Ex,. In particular if A = N (the set ..B

urA

of all non-negative integers), this definition implies n

limllx n+co

C xkll= 0, k=O

and if A = 2 (the set of all integers), this implies n

A family ( x . ) . ~of~ elements in a Hilbert space H is called orthonormal if it satisfies

An orthonormal family ( x . ) . ~in~H i s called complete if it satisfies one of the equivalent conditions stated in the following proposition. Proposition 1.1. The following three conditions l), 2) and 3) for an orthonormal family (x,),,~ in a Hilbert space H are mutually equivalent: 1) every element f in H can be expanded in the form

f=

c

(f, xm)xm

9

.CA

2) the Parseval equality

Ilflla

=

I(Lx.4' USA

is valid for allfin H,

4

FOURIER SERIES AND TORUS GROUP

T

3) iff( E H) is orthogonal to xu (that is, (f,x,)=O) for all a in A, then fmust be equal to 0. Proof. 1)02). This equivalence is clear by the equality

c3)is trivial. 3) *l). Put (1.9)

g=

1cf, x&a urA

The right hand side of (1.9) is convergent by the BesseZ’s inequality (1.10)

c I(f,

xa)lls Ilfll’

aeA

which follows from (1.8). So g is a well defined element in H and the equality (1.9) gives the equalities (g, x.)=cf, xu) for all a E A ,

that is, ( g - f , x a ) = O for all a e A . Then the assumption 3) implies g-f=O and f= Ccf, xa)xa. RtA

q.e.d.

Using the concept of completeness of an orthonormal family, problem A can be re-expressed as follows: Problem A’: Is the orthonormal family (x,JnrZ complete? Many proofs for the completeness of the family ( x ~ ) , are , ~ ~known. We shall give a proof from the representation theoretical point of view. Our proof depends on a theorem of Peter and Weyl which asserts that the matricial elements of “all” irreducible unitary representations of a compact group G form a complete orthogonal family. By the Peter-Weyl theorem, the above Problem A’ is reduced to the following Problem B. Problem B : Does there exist an irreducible unitary representation “different” from every x,,? The precise meaning of “different from every” is “not equivalent to any” which is explained in the next section. Now we shall give the precise definition of the torus group T.A set G is called a topological group if it satisfies the following three conditions: 1) G is a group, 2) G is a topological space satisfying Hausdorff separation axiom, 3) the mapping ( x , y ) ~ x y - ’is a continuous mapping from the direct product space G x G into G.

FUNDAMENTAL DEFINITIONS

5

Examples of topological groups. The additive group R of real numbers is a topological group with its usual topology. The condition 3) is satisfied because of the inequality I(x-y)-(x'-y')l4 Ix-x'I +ly-y'l. The complex general linear group GL(n, C ) of order n is a topological group with the topology defined by

( 2 Igrj-hrjla) . Any 111

distance d(g, h) =

subgroup N of a topological

j-1

group G is a topological group with the relative topology of G. So the subgroups of GL(n, C ) , such as SL(n, C), SO(n), U(n), SU(n), are all topological groups. Let H be a closed normal subgroup of a topological group G. Then the set G/H of all cosets gH= {ghlh E H ) of G modulo H forms a group with the product defined by

(XH) 01H)=(XYIH. The group G/H is called a factor group of G. The topology on G/H is given by determining the family &'(G/H) of open sets in G/H. Let p : g w g H be the canonical projection of G onto G/H. Then d ( G / H ) is defined as follows: (1.11) &'(G/H)= {A1 A c G/H and p-l(A) E d(G)). I t is easily verified that the family @(G/H) satisfies the axioms of open sets: any unions and finite intersections of open sets are open. The factor group G/H with the topology defined by ( 1 . 1 1) is a topological group and is called a factor group of the topological group G. For example the group 2 r Z = (2mln E Z } (2 is the group of integers) is a discrete subgroup of R . So the factor group R/2nZ is defined and it is called the 1-dimensional torus group and is denoted by T.The group T is compact because it is the image of a compact interval [0,2r] by the continuous mapping p . The group T is commutative because R is commutative.

$2. Fundamental definitions Definition 1. A unitary representation of a topological group G is a strongly continuous homomorphism U of G into the group U ( H ) of unitary operators on a Hilbert space H. The Hilbert space H i s called the representation space of U and is denoted by H( U ) . The dimension d( U) of H(U) is called the degree of the representation U. More precisely, a mapping U :g H U g of a group G into the group U ( H ) is called a homomorphism if it satisfies Ugh= U,Uh for all g and h in G.

6

FOURIER SERIES AND TORUS GROUP

T

A homomorphism U of a topological group G into U ( H )is called strongly continuous if the mapping gr-.U,,x is a continuous mapping of G into H = H( U)for all x in H. Remark 1. The equality IIUgX-

Unxll =IIUg--IAX-XIl

proves that a homomorphism U of G into U ( H ) is strongly continuous if it is strongly continuous at the identity element e of G . Example 1. Let R be the additive group of real numbers with its usual topology. Each real number s defines a function x , on R by Xs(t)=eft*. Then xI is a unitary representation of R . The representation space H(xs) is a one-dimensional vector space C (the field of complex numbers). The complex number xa(t)is identified with a scalar multiplication m+XS(t)x on C which is a unitary operator on C . Example 2. Let H=M,,(C) be the linear space of all complex square matrices of degree n. H is a Hilbert space with the inner product ( A , B) =Tr(AB*) . A unitary representation A of the unitary group U(n) on H = M , ( C ) is defined by A g ( X )=gXg-1 . The representation A is called the (complex) adjoint representation of

Wn). ExampZe 3. Let H = L a ( R ) be the Hilbert space of all complex-valued square-integrable functions on R . The inner product is defined by ( f , g ) = / -a -I(f)mdt.

A unitary representation R of the group R on L2(R)is defined by

( R t f ) ( s ) = f ( ~ +t), f E L 2 ( R ) . Since the Lebesque measure on R is invariant under the translation S H s + t , the operator R, is a unitary operator on H = L 2 ( R ) . It is easily verified that R is a homomorphism. There remains to prove the strongly continuity of R. By Remark 1, it is sufficient to prove that for any f in H and for E >0, there exists a neighborhood N of 0 in R such that if t belongs to N , then the inequality IlRlf -fll< holds. Since the space L(R) of all continuous functions with compact supports

7

FUNDAMENTAL DEFINITIONS

is dense in H=La(R) (cf. Appendix DS), there exists a function L(R) satisfying

(O

in

(2.1) I V - ~ I I~ € 1 3 . Since Rt is a unitary operator, the inequality (2.2) IIR~~-R < €13 ~~II holds for all r in R. Let U be a fixed compact neighborhood of 0 and C be the support of (o. Then since C is compact, (O is uniformly continuous on R . Hence there exists a neighborhood N = -N of 0 such that N c U and (2.3) llRt~--(ollm satisfying TU,= V,T for all gE G ,

is called an intertwining operator between U and V. Proposition 2.2. Let U and V be two finite-dimensional irreducible representations of G and T be an intertwining operator between U and V. Then either T= 0 or T is a linear isomorphism of H( U)onto H( V). Proof. By the intertwining property of T, the image Im T= T(H(U)) and the kernel Ker T= T-l(O) of T are invariant under V and U respectively. By the irreducibility of U and V, we have ImT= {O) or H(V) and KerT=H(U) or {O}. If ImT= {O) or Ker T=H(U), then we have T=O. Otherwise Im T= H(v> and Ker T= {O}, i.e., T is an isomorphism of H(U) onto H(V). q.e.d.

10

FOURIER SERIES AND TORUS GROUP

T

Remark 1. If U and V are not equivalent, then the only possible intertwining operator T between U and V is 0. This can be seen from Proposition 2.2 and Proposition 2.6 which will be proved later. Proposition 2.3. Let U be a unitary representation of a group G. Then a closed subspace V of H( U ) is invariant under U if and only if the orthogonal projection P on V commutes with U, for all g in G. If V is invariant under U,then the orthogonal complement V l is also invariant under U. Proof. H = H( U ) is decomposed as a direct sum

(2.5)

H = V @ V'

where V' = [ z E H ( ( y ,z ) =0 for all y E V ) is the orthogonal complement of V. If V is invariant under U, then V' is also invariant under U.This fact was proved in the proof of Proposition 2.1. Let x=yi-z,

YE

v,2 6 V'

be the decomposition of an element x in H according to (2.5). Then the orthogonal projection P is defined by Px =y and we have u g x = u , y + u g z , UoyE v, U,Z€ VL,

and PU,x= u,y= UgPX.

We have proved (2.6) PUg=UgP for all g E G . Conversely if (2.6) is valid, then the equalities U,V=U,PH=PU,H=PH= V show that V is invariant under U.

q.e.d.

Let B ( H ) be the algebra of bounded linear operators on a Hilbert space H and M be a subset of B(H). The commutant M' of M is defined by M ' = [ A E B ( H ) I A U = U A forany U E M } . We define M", M"', ..., by M" = (W)',M"'= (M")', ... .

Proposition 2.4. (Schur's Lemma) Let U be a finite-dimensional unitary representation of a group G and M = { UglgE G}. Then U is irreducible if and only if the commutant M' of M is equal to C1. Proof. Let V be a U-invariant subspace of H( U ) and P be the orthogonal projection on V. By Proposition 2.3, P belongs to M'. So if M ' = C I , P = a l , a E C . Since P a = P , a = O or 1, i.e. P=O or 1.

This implies that V = [O} or H ( U ) and U is irreducible. Conversely assume

FUNDAMENTAL DEFINITIONS

11

that U is irreducible. Then Proposition 2.2 proves that M’ consists of 0 and invertible elements. Let A be an element in M’ and a be an eigenvalue of A (the root of the equation, det(x1 -A)=O). Then a1 - A belongs to M’ and is not invertible. So a1 - A must be equal to 0. Since A can be q.e.d. an arbitrary element in M’, we have proved M’=Cl. Proposition 2.4 can be proved without the assumption that U is finitedimensional. Before proving this fact, we give some preliminaries. Let M an N be subsets of B ( H ) . Then the following properties of commutant are obvious from the definition of M’. (2.7) and (2.8)

MC N

3

M ’ 2 N’

=> M”c

N”.

M c M”.

(2.7) and (2.8) imply M ‘ 2 M ” ‘ . On the other hand (2.8) applied to M’ gives M’cM”’. So we have M’ = M”’. (2.9) A subalgebra M of B ( H ) is called a von Neumann algebra (or W*-algebra) if it is equal to M“. It can be proved that a subalgebra M containing 1 is a von Neumann algebra if and only if M is self-adjoint ( M = M * ) and M is weakly closed (cf. Dixmier [l]). Let M be a subset of B ( H ) . Then (2.8) and (2.9) prove that M” is a von Neumann algebra containing M . If A is a von Neumann algebra containing M , then (2.7) proves that M ” c A ” = A . So M“ is the smallest von Neumann algebra containing M . Theorem 2.1. (Schur’s Lemma) Let U be a unitary representation of a group G and M = {U,lg E G } . Then U is irreducible if and only if the commutant M’ is equal to the set C1 of scalar operators. Proof. The proof of the “if part” of the theorem is same as the corresponding part in the proof of Proposition 2.4. We assume that U is irreducible and prove that M ’ = C l . Any operator A E B ( H ) can be written uniquely in the form A=H1+iHa, H,*=Hj ( j = l , 2)

by putting Hl = 2-1(A +A*), H , = (2i)-l(A-A*). The operator A belongs to M’ if and only if Hl and Ha belong to M ’ . So it is sufficient to prove that any self-adjoint operator A in M‘ is a scalar operator. Let (2.10)

A=

1-

ItdE,

-w

12

FOURIER SERIES AND TORUS GROUP

T

be the spectral representation of a bounded self-adjoint operator A belonging to M’. Then we have El E {A}” (2.1 1) (cf. Appendix E.4). Since {A} c M‘, we have {A] ” c &f“’ = M‘ (2.12) by (2.7) and (2.9). (2.11) and (2.12) prove that E,EM’ forall R E R . Therefore by Proposition 2.2, E,H is invariant under U. E,H is equal to [O} or H, because U is irreducible. Since E, is monotone increasing and right continuous, we have 1 ROSA (2.13) E,= 0 I 1, then there exists a closed U-invariant subspace V different from {0} and H(V). For example take any one dimensional subspace. Therefore dimH( U ) must be equal to 1. Remark 2. The proof of Theorem 2.1 depends solely on Proposition 2.3 and the fact that any W*-algebra is generated by the projections. Definition 8. Let U be a unitary representation of a group G. An element x in the representation space H ( U ) is called a cyclic vector if the set [ U,xlg E G} (topologically) spans H(U). That is, the space of all finite linear combinations of the elements in { U,,xlgE G} is dense in H(U). A

unitary representation U which has a cyclic vector in H ( U ) is called a cyclic representation. Proposition 2.5. Every unitary representation U is a direct sum of cyclic representations. Proof. Let r be the set of all families r = (Ha)aeA of mutually orthogonal closed U-invariant subspaces Ha such that each H , has a cyclic vector with respect to the restriction of U to Ha. Then r is an inductively ordered set. The order is given by the inclusion. In fact, if 0 is a totally ordered

FUNDAMENTAL DEFINITIONS

13

r, then TO= u 7 is the upper limit of # in r. By Zorn’s Lemma, there exists a maximal element r = (Ha)aeA in r. Then we have subset of

14.

H=@Ha. .Ted

Because if we assume that HZ @ H a , then there exists a non-zero element . for all x E H( U)and y E H( V ) . Since T satisfies TU,-I = V,-lT for all g E G,

T* satisfies U,T*=T*V,

for all g E G .

The bounded operator T*T on H(U)commutes with U, for all g e G . Let M = { U,lg E G} ;then T*T belongs to M’. Since (T*Tx, x)=IITxll280 and T is injective, T*T is a bounded strictly positive definite self-adjoint operator on H(U). There exists a bounded strictly positive definite selfadjoint operator S such that S’= T*T. S is bijective.

T*T=J AdEl 0

be the spectral representation of T*T. Then

Since S is a weak limit of a linear combinations of Ez’s and El belongs

14

FOURIER SERIES AND TORUS GROUP

T

to {T*T}" (cf. Appendix E), S belongs to [ T * T ) " c M " ' = M ' Let W = TS-1. Then W is an injective, bounded, linear operator from H ( U ) onto H(V). Since W*W=S-lT*TS-l=l, IIWxll2=(W*Wx,x)=[lxl[*.W is an isometry of H(U) onto H(V). W is an intertwining operator between U and V, because WU,= TS-l U,=TU,S-l= V,TS-l = V, W for all g E G.

$3. Unitary representations of compact groups First we define the Haar integral on a locally compact group G . Let' L =L(G, R ) be the vector space of real valued continuous functions on G with compact supports. A Radon integral m on G is a positive linear form on L (cf. Appendix D.3). The linear form m on L can be extended to a linear form on the space L1=L1(G,m) of all complex valued m-integrable functions on G (cf. Appendix D). The extended linear form is also called a Radon integral and is denoted by

A right Haar integral I on a locally compact group G is a Radon integral on G which satisfies

(3.1) Z# 0 and Z ( R , f ) = I ( f ) for all g e G and f E L , (3.2) where the right translation R, is defined by (R,f)(h)=f(hg). A left Haar integral on G is similarly defined by replacing (3.2) by (3.2)'

Z ( L , f ) = I ( f ) for all g E G and f~ L ,

where (L,f)(h)=f(g-lh). A locally compact group G is called unimodular if a right Haar integral of G is also a left Haar integral. In a unimodular group G , the value of a Haar integral I for a function f is denoted by (3.3) and L1(G,dt) is denoted simply by L1(G). All the groups discussed in this book are unimodular. Proposition 3.1. For any locally compact group G , there exists a right Haar integral. Let Il and labe two right (or left) Haar integrals on G. Then there exists a positive constant c > 0 such that Zl = c12.

UNITARY REPRESENTATIONS OF COMPACT GROUPS

I5

We omit the proof of Proposition 3.1. Instead of relying on this general existence theorem, we shall construct explicitly a Haar integral on each Lie group discussed in this book as a Riemann (and Lebesgue) integral with respect to suitable coordinates of the group. If G is a compact group, then the constant 1 belongs to L and Z(I)>O. Dividing Z by the constant Z(1), we obtain a Haar integral satisfying

(3.4) 1(1)= 1. A Haar integral satisfying (3.4) is called normalized. There exists a unique normalized right Haar integral on G. f ( t ) d t is normalized

Example 1.

Haar integral on

T = R/2nZ. Proposition 3.2. The normalized right Haar integral Z on a compact group G is invariant under the right (left) translation R,:i w t s (L, : t ~ s - l t and ) the inversion S : t w t - 1 :

(3.5)

/$t)dt

= / p ) d f = / Q f ( s - l r ) d t = / Q f ( t-')dt

for all S E G and f E L 1 ( G ) . Proof. Since the process of extending the integral from a linear form on L to one on L1(G') isinvariant under these transformations, it is sufficient to prove (3.5) for any function f in L. Then the right invariance is contained in the definition itself. Let s be a fixed element in G and put J ( f )=Z(L,f) for any YE L. Then J is a normalized right Haar integral on G , because J ( R t f )=Z(L,R$) =Z(RtL,f)=Z(L,fl = J ( f ) and J( 1) =I( 1) = 1 . Hence, the uniqueness of Z implies J = Z, namely Z(L, f)=Z ( f ) . To prove the invariance under S, let (Sf)(t)=f(t-l) and

K(f)=&Yf). Since S(Rtf ) =L4S.f).

K is a normalized right Haar integral on G . Therefore, K=Z and Z is invariant under S.

q.e.d.

In the following we always denote the values of the normalized Haar integral on G as in (3.3). Now we return to our original subject, unitary representations of a compact group G . All the results in this section are obtained from Theorem 3.1 on the following page.

16

FOURIER SERIES AND TORUS GROUP

T

Theorem 3.1. Any unitary representation U of a compact group G is a (Hilbert space) direct sum of finitedimensional irreducible unitary representations. In particular, any irreducible unitary representation of a compact group is finitedimensional. Proof. The last part of the theorem is obtained from the first half by assuming that the representation U is irreducible. TO prove the first half of the theorem, we can assume that the representation U is, cyclic by Proposition 2.5. Let z be a cyclic vector in H ( U ) = H with norm llzll= 1. A Hermitian form (x, y ) on H i s defined by <x, Y>

=/ (

~

x,8 z) (2, Ury)h.

The Schwarz inequality implies the inequality

I<x,V > l S llxll llvll Thus ( x , y ) is a bounded Hermitian form on H and there exists a bounded Hermitian operator K on H satisfying (x, y ) =(Kx, y). From the definition of K, we have (3.6)

Jff

(Kx,v)=

(Utx, Z) (z, utY)dt

for all x, y E H . The Hermitian operator K is strictly positive definite, i.e. (3.7) Since

(Kx,x)>O if xfO.

(Kx,x)=

s

j(Usx, z)ladtzO,

K is positive semi-definite. If (Kx, x) =0, then (Ut,x,z ) = (x, Ut-1z)=0 for all t E G, and since z is a cyclic vector, the last equality implies x=O. Thus (3.7) is proved. As a corollary to (3.7), any eigenvalue R of K is positive. That is,

if K x = l x and xfO, By the defining formula (3.6), we have

(3.8)

(KUsx, y)=

then A>O.

(ucrx, z ) (z, uty)dt

=IG

(U'X,Z) (z,Urr-ly)dt

=(Kx, U r l y )=(U*KX,y ) for all x , y E H. We have proved that

17

UNITARY REPRESENTATIONS OF COMPACT GROUPS

13.9)

KU,=U,K Let I be a real number and put

forall s e G .

H(I) = { x E HlKx =2x1 . Then (3.9) proves that 13.10) U,H(I)c H(I) for all s E G. The most important property of the operator K is its compactness. A bounded linear operator K on a Hilbert space H is called compact if any weakly convergent sequence (Xn)nrN in H is mapped by K to a strongly convergent sequence (KXn)nrN (cf. Appendix B). That is, K is compact if the condition lim (X-Xn, y)=O for all y E H implies lim IlKx- Kxn(l=O. n-a,

n--

Since lim(Kx, Kxn) = IIKxlla and n--

+

IIKx-KXnlI’= IIfill’ -(Kx, Kxn)- (KXn, fi) llfinll’ it is sufficient to show that (3.11)

3

lim llKXnll= ilKxli n+-

in order to prove the compactness of K. By Banach-Steinhaus theorem {cf. Appendix A), a weakly convergent sequence (Xn)nrN is b ~ u ~ d e d . There exists a positive number M such that I l x n l lM~ for all n 6 N. The last inequality and the Schwarz inequality prove that I( UtXn, Z) (2, UsXn) (Uit-a, z)lS M’ for all s and t in G. Then by Lebesgue’s bounded convergence theorem (cf. Appendix C.6),

s

lim IIKxnlla=lim(Kxn,Kxn)=lim n--

It--

=lim/DJ n*-

n--

(UtXn,

(UtXn, Z) (z,

UtKxn)dt

D

z) (z, usxn) (Ust-lz, z)hdt

(t

=n Kxll’ . Thus (3.1 1) and the compactness of K are proved. Let E be the set of eigenvalues of K. Then the Hilbert-Schmidt theorem (Appendix B.5) shows that (3.12) H - @H(4, Ar E

18

FOURIER SERIES AND TORUS GROUP

T

and dimH(2)c + m for all 2 E E (remark each R E E is strictly positive: A>0). On the other hand each eigenspace H(2) is invariant under U, by (3.10). If the restriction of U to H(R) is denoted by UA,(3.12) implies

u= @ u.'. It E

Since each U.'is a finite-dimensional unitary representation of G, UA is a direct sum of irreducible (necessarily finite-dimensional) unitary representations (Proposition 2.1). Theorem 3.1 is proved. Another fact basic for the representation theory of compact groups is the orthogonality relations between matricial elements of irreducible representations which were found and effectively used for finite groups by I. Schur. A continuous homomorphism U of a topological group G into the unitary group U(n) of order n is a unitary representation and is called a matricial (unitary) representation. The number n is called the degree of U and is denoted by d ( U ) .

Theorem 3.2. (Orthogonality relations). Let U and V be two irreducible matricial unitary representations of a compact group G and let urr(g)and vrl(g)be the (i,j)-element and (k,&element of the matrices U, and V , respectively. Then the inner products in La(G)between matricial elements are given by

Proof. Let n = d ( U ) and m = d ( V ) and A be a matrix with n rows and m columns. Then the matrix B=/

UtAVt-ldt 0

satisfies U,BV,-l= B i.e. U,B= BV, for all s E G . The matrix B is a linear operator of Cm= H( V ) into Cn=H( U ) and is an intertwining operator (Definition 7 in 92) between V and U. Since both V and U are irreducible, Proposition 2.2 and Remark 1 after it prove that B = 0 if U is not equivalent to V. If U = V , Proposition 2.4 proves that B=bl. The value of the scalar b is obtained by taking the traces of both sides of the last equality:

UNITARY REPRESENTATIONSOF COMPACT GROUPS

19

1 b=-TrB. n If we take Ejl as A, where Ejl is a matrix whose (p, &element is d p j d l q , then the (i, k)-element of the matrix B is equal to n

m

n

on the one hand and on the other hand is equal to 0 if U&V and equal to

if U = V. The theorem is proved. Remark 1. Without the assumption that U and V are matricial, the orthogonality relations can be expressed more generally as if U&V (3.13) (U,x, y ) (Vtz, w)dt= d(U)-l(x, z) (y, w), if U = V.

1

r'

(3.13) is easily obtained from Theorem 3.2, because any unitary representation U is equivalent to a matricial unitary representation U'. The equivalence is realized by taking an orthonormal basis (et) of H( U)and defining U,' as the matrix of the linear transformation U, with respect to the basis ( e t ) . To state the next theorem (Peter-Weyl theorem)' neatly, we introduce the following notation.

Definition 1. The direct sum of m copies of a unitary representation

U is denoted by mU : mU= U @ ... @ U. m times

Definition 2. The set of all equivalence classes of irreducible unitary representations of a topological group G is denoted by 6 and is called the dual of G. Theorem 3.3 (Peter-Weyl). Let G be a compact group and 6 the dual of G. Choose a matricial unitary representation UA =(u$,l) for each class 1 in 8 once for all and denote the degree d(U2) of U Aby d(2). Then the family

20

FOURIER SERIES AND TORUS GROUP

T

9={ q u t j ’ I A E e , 16i,j6d(R)l is a complete orthonormal family in La(@. Let d(l)

H f a = z C u r j Afor

2 ~ 8and i € { 1 , 2,..., d(2)}.

j-1

Then the subspace Hi’ of La(G) is invariant under the right regular representation R of G. The representation induced by R on the subspace Hia is equivalent to U’.The Hilbert space L1(G)and the regular representation R are decomposed as follows: d(l)

L’(G)= @ @Hi’ 2.8

(-1

Proof. The family 9 is orthonormal by the orthogonality relations (Theorem 3.2). Since

z

d(2)

(Rtutj’)(s)=urj’(st)=

utk’(s)uxj’(t)

k-1

the subspace Hi2 is invariant under R and the restriction RtlHf2of the operator Rt to Hii is represented by the matrix Uta. It remains to prove that {Ht212E 6, 1SiSd(2)) spans the Hilbert space La(G). d(l)

Suppose that the closed subspace W = @ @Hf2is not equal to L%(G). ace f-1

Then the orthogonal complement W of W is a non-zero closed subspace of La(@ which is invariant under R. Let U be the unitary representation of G induced on the space WL.Then U is a direct sum of irreducibleunitary representations (Theorem 3.1). There exists a class 2 E 8 and an invariant subspace E+ {0} of W Lon which the regular representation R induces a representation in the class 2. Let n be the degree d(2) of U’.Then there exists an orthonormal basis (xi)lrirnof E such that n

(3.14)

Rtxj=Cutj’(t)xc

-

(1 5j5n)

1-1

The equalities (3.14) imply (3.15)

xjE&Hf’ i-1

( l s j s n ) and E c & H i a c W. f-1

The last inclusion contradicts the fact that (0)# E c WL.Hence (3.15) proves that W=L’(G). The proof of (3.15). The equality (3.14) can be written as

UNITARY REPRESENTATIONS OF COMPACT GROUPS

21

c n

x,(sO = ut,W&).

(3.16)

i-1

Putting s= e (the identity element of G), we get

This “proof” is incomplete because (3.14) is a system of equations in La(G)and the two sides of (3.16) are equal only when s does not belong to a null set N(t) (of measure 0) for all f E G. The identity element e may be contained in the exceptional set N(t). The exact proof is as follows. Let

[

M(s)= t E GI x&)

ud(t)xt(s)}

# t-1

and

and let xar be the defining function of the set M and m the normalized Haar measure on G. Then by Fubini’s theorem

=SslS.

1

x ~ ( st)ds , dt =

SG

so

m(N(t))dt=0

.

m(M(s))=0 for almost all s E G There exists thus a null set N such that m(M(s))=O for all S E G-N. Since m(G)= 1>0, there exists an element s in G such that m(M(s))=0. For this element s, the equality (3.16) is valid for almost every t in G. Fix such an element s and put st=a. Then (3.16) can be written as

c n

XAU) =

(3.17)

ut,2(s-lu)xi(s)

(-1 n

=

u*l’(a)Ck k=1

9

22

FOURIER SERIES AND TORUS GROUP

T

where c k = ~ U ~ ~ ~ ( Sis- a~ constant ) X ~ ( S(s)is a fixed element). The equality i=1

(3.17) is valid for almost all a in G. So we have proved x j E &Ti2. (-1

Theorem

3.3 is proved. By the definition of completenessof an orthonormal family (Proposition 1.1), the content of Theorem 3.3, 1) can be expressed as follows. COROLLARY 1 to Theorem 3.3. panded in the Fourier series:

Any function f in Lz(G) can be ex-

(3.18) where the series converges in the norm of L2(G). The Parseval equality (3.19) holds for any f in L2(G). Now we shall rewrite (3.18) and (3.19) in another way. Definition 3. The Fourier trandormf of a function f in L1(G)is the matrix-valued function on the dual 6 of the compact group G defined by

(3.20)

sn

f(n) =

f ( t ) u i , - l d t , for R E 6 .

The value f(;O off at R E d is a matrix of degree d(2). In the following, we denote the Hilbert-Schmidt norm of a matrix A=(au) of degree n by IlAll: n

llAlla=Tr(AA*)=

~aij~a.

6, j = 1

COROLLARY 2 to Theorem 3.3. Any functionf in L2(G)can be expanded as (3.21) where the series converges to the function f in the norm of L2(G).The Parseval equality for the function f E L2(G)can be written as (3.22)

UNITARY REPRESENTATIONS OF COMPACT GROUPS

23

Proof. By the definition of the Fourier transformf, the (i,j)-element

f(& of the matrix {(I) is given by

Therefore (3.21) and (3.22) are easily obtained from (3.18) and (3.19) respectiGely. In fact, we have

c c

d(l)

T r ( f ( 4 VQ2> =

f(&*utm

i, j = l d(l)

=

(f,uiJ2)utj2(g)

f, j=l

and q.e.d. In order to see clearly the meaning of the Fourier transform on L2(G), we introduce on the dual 6 of a compact group G a function space ~ “ 6given ) as follows: Definition 4. The space L2(6)is the set of all fucctions p on the dual

6

of a compact group G with values in the set 6Mn(C)of matrices n=l satisfying 1) p(l) E M ~ ( ~ ) ( cfor ) all I E c3 2) Cd(~)llp(J)ll2 -'

+ R&),

dC. By Fig.1 and the law of sine, we have

where R,(z) =

+ C l = IsinOI- Isin41 '1 2 rlsine1. In particular, we get 11 + clz 1 if 181 4 2 . Hence we have 11

The above inequalities show that the remainder term R, (z) tends to zero when n tends to +m. Therefore the equality (5.17) holds for every point z on the unit circle U except at the point z = - 1. If z belongs to U then

44

FOURIER SERIES AND TORUS GROUP

T

r = 1 and the angle 4 in Fig.1 is equal to 0/2, and so the equality (5.19) becomes (5.20)

When 0 = fa, the right side of (5.20) is equal to 0. Put (5.21)

Then g(0) is a periodic function with the period 2a and (5.22)

g (e) = g (e

+ 2nn) =

I?

--Ic(e,

( ~ , ( ~ ) s isinpdedpdqj. n~e

Proof. Since the Riemannian metric of S8 is induced by the Euclidean metric of R4,its line element is given by

+sinzed$+ sina@sin'p

=dea

d#'. Therefore the volume element dx= (det(g exp ;u~(Y)) t (uexp?ueXp+ ) =lim exp n

>") n

n--(

=exp t(U'(X)+ U'(Y)) for all t E R. Hence we have U'(X+ Y) = V'(X)+ U'( Y) Similarly by (5.28), we have U'([X, Y]) = [ U'(X), U'( Y)]

for all X and Y in g. for all X and Y in g.

Definition 6. The representation U' of g is called the direrentialrepresentation of U and is denoted by the same symbol U if there exists no possibility of confusion. Example 4. Let g be an element of a linear Lie group G. Then the linear transformation Adg on the Lie algebra g of G is defined by (Adg)X=gXg-' for X E g.

Since exp t(gXg-'>=g(exp tX)g-l for all t E R, gXg-' belongs to g and Adg is actually a linear transformation on g. And the mapping Ad: g H Adg is a continuous real representation of G on g. The representation is called the adjoint representation of G. The differential representation ad of Ad is defined by (exp t x ) Y(exp( - tm)] t = O =X Y - YX= [x, Y]. The representation ad of a Lie algebra g is called the adjoint representation of g. Now we come back to our original subject, i.e., the representations of SU(2). First we give a basis of the Lie algebra Su(2) of SU(2). Proposition 5.10. Put

Then they form a basis of the Lie algebra Bu(2). Their Lie products are given by [Xi, Xa ]=x3. (5.31) [X,, X3 ]=XI, [X3, XI ]=x,,

68

REPRESENTATIONS OF

Proof.

sU(2) AND s0(3)

Since en(2)=

{(:+cyb+2) - I

a, b, C E R)

by Proposition 5.7., ( X + ) l s i s 8is a basis of Bn(2). The Lie products are easily calculated. Proposition 5.11. Let U" be the irreducible representation of SU(2) constructed in $1. and (X,)lst., be the basis of Su(2) given in Proposition 5.10. Then the differential representation of U" is given by i

(k +( n-k )yt+l

U"(X1)pt =1

(ot-l

(5.32) we have by Proposition 5.8

1

sin-

2

sin - cos 2

cos-

-

and exp tX3= Then we have by (5.32) and (5.33) c u n ~ i ) (21, p ~za)=

i =

jk

t-0

y

-

({;[ 2

-)

+

t t z1cos +iza sin 2t t ( i z1 sin z) 2 (n-W z1 za"-t-l 2

~ (n+ -k ) (or+1] (21, za)

C U ~ ( X p, )t ] (z1,zs) =[${(zl cos

and t -+ 2

za sin

t

2

2

LIE ALGEBRAS A N D THEIR REPRESENTATIONS

69

Similarly we have

q.e.d. Remark. The eigenvalues o f the operator iU"(X8) (or iUn(Xl) or iU"(X2)) are called the weights of the representation Un. It should be remarked that the weights determine the irreducible representations of SU(2). In fact, the class of U" is the only class of irreduciblerepresentations of SU(2) whose maximal (or highest) weight is equal to n/2. With the inner product defined in Proposition 1.2., the representation space Vn of Un is a Hilbert space and U" is a unitary representation. The Hilbert-Schmidt norm \\All of a linear transformation A of Vn is defined in terms of this inner product. More explicitly, since (5.34)

is an orthonormal base of V,, the Hilbert-Schmidt norm l]A]lis defined by

c n

llAlla=

11Aji11'.

k=O

Proposition 5.12. Let X be any element in Bu(2), and U" be an irreducible unitary representation of SU(2). Then the Hilbert-Schmidt norm IIUn(X)ll of U"(X) satisfies the following equality.

-

IIU" (X)llSn dn+l IIXllS IIXII. Proof. For any element X E lu(2),there exists an element g E SU(2) and

70

REPRESENTATIONS OF

=2-1na(n

su(2)AND s0(3)

+ 1) 11 XIIagna(n+ 1) 11 Xl12~ ( +n1)‘ )I

q.e.d.

An associative algebra A over a field K has the structure of Lie algebra if the Lie product is defined by [x, YI =X Y -Y& where the products on the right hand side are the original products in A. This Lie algebra is denoted by L(A). In particular, the associative algebra p( V ) of all linear transformations on a vector space Y has the structure of a Lie algebra. The Lie algebra L ( y (V)) is denoted by g1(V). The Lie algebra g of a linear Lie group G is a real Lie subalgebra of an associative algebra Mn(C). Now for each Lie algebra g, we construct an associative algebra U(g) in which g is imbedded.

Definition 7. Let g be a Lie algebra over a field K, and T be the tensor algebra of g. Then T, as a vector space, is the direct sum of the spaces Bmg and the product in T is defined from the tensor product. Let I be the two-sided ideal of T generated by the set

[x@ Y -

Y @ X x - [ X , Y ] I x,YE g}. Then the factor algebra U(g) = T/Iis called the universal enveloping algebra of g. Let (i, be the canonical projection of T onto U(g) and (O be the restriction of $ to g. Now is a linear mapping of g into U(g) satisfying

d[X Yl) =lo@-) 94y>- $4Y ) Cp(J-7 =[&n cP(Y)l. is called the canonical mapping of g into U(g). The universality of U(g) is described in the following Proposition.

‘p

Proposition 5.13. Let g be a Lie algebra over a field K, A be an associative algebra over K and u be a homomorphism of g into L(A). Then there exists a homomorphism U’ of U(g) into A such that I

u yo=u.

LIE ALGEBRAS AND THEIR REPRESENTATIONS

71

Proof. The linear mapping u of g into A is extended to a homomorphism uo of T into A by defining U O ( X I @ ... @X,) =u(X1) ... u(X,). The extended homomorphism u0 sends the ideal I to 0 because

u o ( X @ Y - Y @ X - [ X , y1) = u ( X ) . ( Y ) - o ( Y ) o ( X ) - o ( [ X , Y])=O.

Therefore, the image U O ( ? ) of an element ? E T depends only on the coset t + Z and hence there exists a homomorphism U' of U(g) into A satisfying U

/

oy=o.

COROLLARY to Proposition 5.13. Let g be the Lie algebra of a linear Lie group G . Then the canonical mapping y of g into U(g) is injective. Proof. Let G be a closed subgroup of GL(n, C). Proposition 5.4 shows that g is a Lie subalgebra of gl(n, C ) = L ( M ,( C ) ) .Let u be the inclusion mapping of g into gI(n, C ) : o ( X )=X. Then there exists a homomorphism 7 of U(g) into M,,(C) satisfying ~ O J D = U . Since u is injective, (O must be injective. q.e.d.

By this Corollary, we shall always identify XE g with y ( X ) E U(g) and g is regarded as a Lie subalgebra of L (U(g)). Any representation u of g on Y may be uniquely extended to a representation U' of U(g) by Proposition 5.12. For the sake of simplicity, u' is also denoted by U . Remark. It can be proved that any finite-dimensional Lie algebra g has an injective representation (Theorem of Ado-Iwasawa cf. Bourbaki [4] Ch. I $7. THEOREM 3 and Stminaire Sophus Lie [ l ] ,expost 8). Thus y is injective for any Lie algebra g.

Definition 8. Let g be a Lie algebra and X be an element of g. Then a linear transformation ad X on g is defined by (ad X ) Y = [X, Y ] The mapping ad : Xwad X is a representation of g on g (i.e. a homomorphism of g into gl(g)). This is easily verified by the Jacobi identity. The bilinear form B on g x g defined by B(X, Y )=Tr(ad X ad Y )

is called the Killing form of g. B satisfies the following identity (5.35)

B([X, Y ] ,Z ) = B ( X , [ Y, 21) for any X , Y, Z E g, because Tr([a, b] c ) =Tr(abc)-Tr(bac)=Tr(abc)-Tr(acb)=Tr(a[b, c]).

72

REPRESENTATIONS OF

SU(2) AND SO(3)

A finite-dimensional Lie algebra g over a field of characteristic 0 is called semisimple if its Killing form is non-degenerate. A connected linear Lie group is called semisimple if its Lie algebra is semisimple. Example 5 . The Lie algebra lu(2) is semisimple. By (5.31), we have

Therefore we have (5.36)

B(X#,X,)=

-at, (lSi,jS3),

and det(B(Xc,Xi))zO.

Definition 9. Let (Xt)la/rn be a basis of a semisimple Lie algebra and gt, =B ( X , X’) and

W’)=(gt,)-’. Then the element

c n

Q=

gO, f is proved to be a C"-function on G. COROLLARY to Theorem 6.2. Let f be an integrable function on G = SU(2). Then the Fourier transformf is rapidly decreasing if and only if f coincides with a C"-function almost everywhere.

Definition 2. is defined by

For any element D E U(Bu(2)), a seminorm p D on C"(G)

pD(f )= !@fIl = where D is regarded as a differential operator on G=SU(2). The family of seminorms {pDID E U(Bu(2))I gives the structure of a complete locally convex topological vector space to C"(G). The topological vector space so obtained is denoted by d ( G ) . For any s 2 0, a seminorm q, on 9 ( d )is defined by 4 r ( d = SUP Ilngdn)ll ncN

~ ( 6is )regarded as the topological vector space defined by the family {q,lszO} of seminorms.

FOURIER SERIES ON SU(2)

*

81

Theorem 6.3. Let G=SU(2). Then the Fourier transform :fwfis a topological isomorphism of g ( G ) onto ~ ( 6 ) . Proof. 9is surjective by Theorem 6.2. Assume that two functionsj and A, have the same Fourier transform f=fl, then by the uniqueness property of the Fourier transform (Ch. I. Theorem 3.6) f coincides with f1 almost everywhere. Sincef andS, are continuous, they coincide everyis injective. where. So By the Corollary to Proposition 6.1, we get

*

(+))*llf(n)Il

=If

= ll(QW(~)Il

(QY)k)U,"_l&ll s/ol(QY)(dl llU;-lll&

a

s JnflIIJaXfll-. So we have n2k--"111f(n)lls8*llQYll-, i.e., qw-i/a(f) 4 8'

since q&) d qt(p) if s5 t, 1 if s 5 2k --. 2 Since k can be taken arbitrarily large, the last inequality proves the continuity of The inverse Fourier transform S - l is also continuous. In fact, by Proposition 5.12., Theorem 6.2. and (6.10), we get q , ( f )s 8* p & f )

(6.1 1)

*.

Il&

-

Xlf 11-5 C ( n + 1)~'~llf(n)ll IIvn(mll-..IIUn(xl)ll n-0

Since the series C (n+ l)ak+s~2/na converges if s > 2k + 512, the above in-

82

REPRESENTATIONS OF

SU(2) AND SO(3)

equality proves the continuity of X-l.(Note that every element D of U(Q) is a linear combination of “monomials” Xr ... X I . )

COROLLARY to Theorem 6.3. the family of seminorms,

The Topology of g ( G ) is defined by

(6.12) (Pdlk E N l . Proof. Let g o ( G ) be the topological vector space whose underlying vector space is C”(G) and whose topology is given by the seminorms (6.12). The identity mapping is a continuous mapping of S(G‘) onto s o ( G )because Qn is an element of U(g). Conversely the Fourier transform .9- is a continuous mapping of g o ( G ) onto 9(&) by (6.11) and X-l is a continuous mapping of g ( 6 ) onto g ( G ) . Therefore the identity mapping Id=..F-lo..F is q.e.d. a continuous mapping of go(@ onto g ( G ) .

$7. Representation of SO(3) and spherical harmonics The representation theory of the three-dimensional rotation group SO(3) is derived from that of SU(2) described in the preceeding sections, because SU(2) is a covering group of SO(3). The irreducible unitary representations of SO(3) are realized on the space of harmonic polynomials or spherical harmonics. First we give the relation between SU(2) and SO(3). Proposition 7.1. Let (et)lStg3 be an orthonormal base of three dimensional Euclidean space R3, and Hi be the subgroup of SO(3) consisting of the rotations around the axis e,(l Si53). Then the group SO(3) is generated by two of the three subgroups Hi(l S i 5 3 ) , e.g., by H, and Hs. More precisely any element g of SU(3) can be written as g=rst, r, t E H3, s E Ha Proof. We can rotate the vector ge3 into the (el, es)-plane by a rotation rl in H3. Then we can send rlge3 to e3 by a rotation slE Ha : slrlg e3= e3. Finally slrlg = t belongs Hs and g =rst where r =rl-l E H3 and s =sl-lE H z. Let (Xt)l~i63 be the basis of the Lie algebra 2 4 2 ) defined in Proposition 5.1 O., i.e., Xl=i(p X2=1(1 1 0 -1o) and

-9>,

i),

and put 1

(X,Y ) = --B(X, 2

Y)

REPRESENTATIONSOF

SO(3) AND

83

SPHERICAL HARMONICS

where B is the Killing form of Su(2). Then by (5.36), we have

Therefore the symmetric bilinear form (X, Y) is positive definite and

(Xi)lstss is an orthonormal base of Su(2)with respect,to this inner product. Hereafter we identify the vector space Su(2) with RSby the mapping tfX,-(rl,

fa,

tJ).

f-1

Proposition 7.2. The adjoint representation Ad of SU(2) (cf. 55, Example 4) is a continuous homomorphism of SU(2) onto SO(3). The kernel of the adjoint representation is [ & 1) and SU(2)/ { 2 1) G SO(3). (7.1) Proof. Since ad(gXg-l) = Adg adXo Adg-l, we have (AdgX, AdgY) =(gXg-l, gYg-') 0

= -I T , ( Adgo adXo Adg-10 Adgo ad Yo Adg-I)

2

1

= --Tr(adXadY)=(X,

Y) 2 for any g in SU(2) and any X, Y in Su(2). Therefore Ad is a continuous homomorphism of SU(2) into O(3). Since SU(2),which is homeomorphic to Ss, is connected, det Ad g = 1 for all g e SU(2); hence, Ad S U ( 2 ) c SO(3). By (5.32) and (5.33), we have 1 cos t/2 -sin "2) cos t/2 sin t/2 ( ~ exp d tXa)X1 = T(sin t/2 cos t/2 -sin t / 2 cos t/2

(p

); (

1 -isint icost icost i s i n t

=T(

=(cos t)Xl

- (sin t)&.

Similarly, we have and (Ad exp tXa)Xa =Xa (Ad exp tXs)Xs=(sin t)Xl +(cos t)Xs. Therefore cost 0 sint (7.2) 0 cost -sin r

)

84

REPRESENTATIONS OF

SU(2) AND SO(3)

and [Ad exp tX,lt E R} =H2(cf. Proposition 7.1). Similarly, -sin t (7.3) Adexp t X 8 = E : cost 0 1

"0)

and {Ad exp tX81tE R}=Ha. Since the subgroups H2and H8 generate S0(3), we have proved that Ad SU(2)=SO(3). Now we determine the kernel N of Ad. Let

be any element in N:Ad g = 1. Since (Ad g)X8=Xs,we have gX8 =Xa g, i.e.,

-ib -id)=(-: This imples b=c=O and g=

;(

-3). Moreover we have

(Ad g)X2=Xr, gX, =Xr g, i.e.,

-;)=(: SO

a9 = 1,

-f>.

a= & 1 and g = -t- 1.

q.e.d.

Theorem 7.1. For any non negative integer 1, there exists an irreducible unitary representation Do1of SO(3) which is given by (7.4) Do' Ad= U a l , ( U a l is the representation of SU(2) defined in Proposition 1.1.) Any irreducible unitary representation D of SO(3) is equivalent to Dozfor some non negative integer 1. If l#l', then Do1is not equivalent to Do1'. Proof. Let (ox (0sk g n ) be the basis of the representation space V,, of Undefined by (1.5). Then we have Q

(Un(- 1)pr) (21,z~)=(pk(-~i,

-4

=(- 1)" (or(z1,zd

and (7.5.)

U"(- 1) =(- 1)" If n =21 ( I E N),then the kernel of U" contains N = { f1) and there exists a well-defined homomorphism Do' of SO(3) into U(Vn)satisfying (7.4). DO'is an irreducible unitary representation of SO(3) as is U a L .

REPRESENTATIONS OF

SO(3) AND SPHERICAL HARMONICS

85

Let D be an irreducible unitary representation of SO(3). Then U=Dn Ad is an irreducible unitary representation of SU(2), and so is equivalent to Unfor some n E N (Theorem 4.1). Since Un(- 1 ) =D(Ad( - 1 ) ) =D( 1) = 1, n must be even and equal to 21 for some I € N by (7.5). Therefore D is equivalent to Do1. If I#I', then deg Do'f-deg Do1'and DoLis not equivalent to Do1'. q.e.d. We construct a realization of the representation Do1which has a close connection with the nature of SO(3). If we pursue the analogy to the representation Un of SU(2), we may take the representation T1on the space WI of homomogeneous polynomials of degree I in three variables x, y, z defined by

(TPtf)(4=f (%I, where x = ( x , y , z). However it turns out that T' is not irreducible if 122. (7.6)

+

For example, W2contains the invariant subspace spanned by xa y a+z'. So we must take an irreducible subspace of Wl in order to obtain an irreducible representation. Such a subspace may be obtained as an eigenspace of a differential operator, the Laplacian. Definitionl. Let

be the Laplacian on Ra and let Hi= { f W ~ Ld~f = O } . An element of H Lis called a harmonic polynomial or a solid harmonic of degree 1.

Proposition 7.3. dim W z =(z+1)(1+2) and dim HL=21+1. 2 Proof. The monomials xpyqz' ; p , q, rE N , p + q + r = l form a basis of the space Wl of homogeneous polynomials of x, y , z of degree 1. Since the dimension of the space of homogeneous polynomials of x and y with degree I-r is I-r+ 1, dim Wl is equal to I

(1+2) ( I + 1 )

dim w , = C +-0( I - r + l ) =

2

Any polynomial F i n WLcan be written uniquely as

86

REPRESENTATIONS OF

SU(2) AND SO(3)

(7.7)

Where Fn(y, z ) is a homogeneous polynomial of y and z of degree I-n. Since

We have (7.8)

So an element F in Hl is uniquely determined by giving FOand Fl and it is denoted by F(Fo,F l ) ; moreover, Fo and Fl can be taken arbitrarily. Let Wl' be the space of homogeneous polynomials in y and z of degree 1. Then the mappings (DO :FOwF(Fo,0) and p1 :B w F ( 0 , Fl) are linear Since F=O if and only if F, = O mappings from W,' and Wl'-l into Hi. ( O s n ~ l )y,o and pl are injective. The vector space Hi is the direct sum of yo( Wl') and yl( Wt-l'), because F(Fo,Fl)=F(F0,O) +F ( O , 4 ) . Since dimWl'=Z+l, we have dim Hi=(Z+1)+Z=2Z+1. q.e.6.

Proposition 7.4. Let 3-be the space of complex valued C"-functions on R 3 . Then a representation T of SO(3) on ..Fis defined by ( T , f ) (4=f (xg) for g E S0(3),f E .Fand x E R3. The Laplacian

Commutes with Tofor any g E SO(3): doT,,=T,,od.

Proof. Since SO(3) acts on R 3 from the right as a group of linear transformations, Tgf belongs to .-Fwith f and T is a homomorphism of SO(3) into G L ( x ) . For any function f in S,put Tpf=cp and y=xg and write x=(xl, xS, xs), Y = (yl, y,,ys). Then we have y j =Ex' gtj and i

Thus we get

REPRESENTATIONS OF

SO(3) A N D SPHERICAL

HARMONICS

= [(Ta0 4fl (x).

87

q.e.d.

Theorem 7.2. We preserve the notation in Propositionb7.4. 1) The space HL of harmonic polynomials of degree 1 is invariant under the representation T. Put TgIHl=Dzg.Then D1 :gHD', is a continuous, irreducible unitary representation of SO(3). 2) D' is equivalent to Doz. 3) Any irreducible unitary representation D of SO(3) is equivalent to D1 for some 1E N . Proof. 1) Iff belongs to H E ,then T,f belongs to Hc for all g E SO(3), because AT,f = TgAf =0 by Proposition 7.4. Since the matrix elements of DL(g)are polynomials in those of g , D1 is continuous. Dz(g) is a unitary operator on a Hilbert space HZ whose inner product is given by

where dx is the normalized measure on the unit sphere Ss.(In polar coordinates, dx = (4~)-1sin 8 d8dp) 2) Let Dz Ad=Bz. Then Bz is a unitary representation of SU(2). We recall that an irreducible representation of SU(2) is determined by its highest weight (cf. Remark after Proposition 5.11). Let 0

fr(x, y , z) =( x

+iyy.

Thenfz belongs to H z . From the relation (7.3), we have (B'(exp iXslfi) (x, y, z) =( x cos t +y sin t -ix sin t iy cos t)'

+

=(xe-ft

and (7.9)

+ jye-L')Z

=e-Lzt

h(X7

Y7

4

88

REPRESENTATIONS OF

SU(2) AND SO(3)

The finite-dimensional unitary representation BL is decomposed into a direct sum of irreducible representations. Let (7.10) be the decomposition into such a sum. By (7.9), one of the irreducible components, say Unci)has the weight 1. Then by Proposition 5.11, 15 n(i)/2 namely. (7.1 1) 21+ 1s n ( i ) + 1. On the other hand, by (7.10) we have (7.12) 21+ 1=deg Bz2 deg Un(O=n(i)+ 1. From (7.11) and (7.12), we see that 21+1=n(i)+l and that B L = U n ( { ) is irreducible. Since D' Ad = UsL=DLo Ad, DL is equivalent to DLo. 3) follows from 2) and Theorem 7.1. q.e.d. 0

0

Now we express the elements of Ht more explicitly in terms of known functions. Proposition 7.5. Put 0 0 0

Then ( Z , ) l s t s 3is a basis of the Lie algebra 4 3 ) of SO(3).The Lie products between them are given by

[za,Zs]=Zi, [Za, Zi]=Za, [ZI,Za]=Za. Proof. By Proposition 5.7, the Lie algebra ~ ( 3 of ) SO(3) consists of all real skew-symmetric matrics of degree 3. It is clear that (Zi)lscra is a basis of o(3). The Lie products [Z', Z,] can be calculated easily. q.e.d. Remark 1. By comparison between Proposition 5.10 and Proposition 7.5, the two Lie algebras h ( 2 ) and 4 3 ) are mutually isomorphic. This fact also follows from Proposition 7.2. The isomorphism is given by the adjoint representation ad of m(2) which is the differential representation of the adjoint representation Ad of the group SU(2). More precisely, we have seen in Example 5 in $5 that

ad(X,)=Zr

(1 riS3).

We have defined the differential representation for a finite-dimensional

REPRESENTATIONS OF

SO(3) AND

SPHERICAL HARMONICS

89

representation of a linear Lie group. Although the representation T defined by (7.6) is not finite-dimensional, the differential representation T' of o(3) is defined similarly. proposition 7.6. Let X be an element of o(3). A linear transformation T ' ( X ) is defined on the space S of C-fun0 tions on Ra by

Then T' is a representation of the Lie algebra o(3) on F. In particular we have T'(Z1)=z-

a - y-, a az

ay

a

a

T'(Z2)=x- -zaz ax

and

a

a

T'(Z*)=y--x-. ax ay

Proof. Let Z=(zr,) be an element of o(3) and f E fl be a C"-function on RS.Then

So we have for two elements Z = ( z t j ) , and Y=&)

in 0(3),

T'(aX)=aT'(X),

=T'(Y+Z)

and

= T'([Y,Z l )

Therefore T' is a representation of the Lie algebra o(3) on pression for T'(Z$)is easily obtained using (7.13).

F. The ex-

90

REPRESENTATIONS OF

s u ( 2 ) AND s0(3)

Proposition 7.7. In the polar coordinates (r, 8, p), the vector fields T'(Zi)(1 5 is 3) are represented as follows: T'(Z1)= sin p-

a

ae

ap

a q-+cot ae

T'(Z)=-cos T'(Z3)=

e cos p- a

+cot

e sin p- a

ap

--.aPa

Proof. Since x = r sin e cos p, y =r sin e sin p, z =r cos 8, we have

/a ar

ax

i a --

r

=

ae

i a ~-

a

cosecosp

cosesinp

-sine

-

- sin p

cos (0

0

a -

aY

/

\

iazf

Since the coefficient matrix A in the above equality is orthogonal, we have A-l= ' A and sinecosp

cosecosp

-sinp

sine sin p

cos e sin p

cos p

cose

- sin e

0

la

I

\

ar l a -r ae l a -, r s i n e ap,

So we have T'(Z1)=z-

a -y- a az

ay

a

cos e sin p

= r cos B (sin e sin r+ ar

a

sin e r

-r sin e sin p (cos e----)

ar

a ae

=sin p-+cot

a -+ae

cos a ) r sine -ap

a ae

e cos p-. a

ap

The expressions for TI(&) and T'(Zs)are obtained similarly.

Proposition 7.8.

Put

H = iT'(Zs),

E=2-'(iT'(Z1)

- T'(Z2))

REPRESENTATIONS OF

s0(3) AND

SPHERICAL m O N I C S

91

and F= iT'(Z1) + T'(Z2). Then in the polar coordinates, these have the expressions

(7.14) F=e-tP

(-6:

+ i cot 6 ) .a aP

Their Lie products are given by (7.15) [H, E ] = E , [H, F]=-F, [E, F]=H. Proof. The expressions in polar coordinates are easily obtained using Proposition 7.5. The Lie products of H, E and Fare obtained from those of Zl, Za and Zs and from the fact that T' is a Lie algebra homomorphism (Propositions 7.3 and 7.4). q.e.d. Since any element f in Hi is a homogeneous polynomial of degree I, it satisfies (7.16) f(rx, ry, rz) =rff(x, Y, 4 for any r 5 0 . Let D c f ) be the restriction offto Saand Hi' = { D ( f ) l f ~Hi] . Then d, is a linear isomorphism of the vector space Hi onto Hi'. HI is recovered from Hi' by (7.16). The elements of Hi' are called spherical harmonics of degree 1.

Theorem 7.3. Let Hi' be the space of spherical harmonics of degree 1. Then Hi' has a basis (Yim)-lsmL1 defined by Yim(B,v) =e*mrPi-m(cos6), where PIm(x)is the associated Legendre function defined by

Proof. As was shown in the proof of Theorem 7.2, ( ~ + i y belongs )~ to H l . Hence (7.17) Yi1(6,y)=cetl~sini8 (c=const.) belongs to Hi'. Let YLm =F1-m YIi, -15m $1. (7.18) Then YIm satisfies FYim= YLn'-l and (7.19)

92

REPRESENTATIONS OF

(7.20)

HYcm=m YZm

su(2)AND sO(3) - I s m 41.

(7.19) is obvious. (7.20) is proved by downward induction on m starting from 1. (7.20) is valid for m = l by (7.'17) and (7.14). If (7.20) is valid for m, then by (7.15) and (7.19) we get HY2m-l=HFYzm=(FH-F)YZm= (m- 1)YZm-l. Since H = -ia/ap, (7.20) implies that YZmhas the form Ycm(8,p) =etm(Pf,(0). (7.21) By (7.19), the functionsf m and fm-l satisfy

Putting s=COse andfa(@)=pm(s), we get, using (7.21), (7.22) Again putting (7.23) pm(s)=(l -s')-m'aum(s)9 (7.22) can be written as (7.24)

*

durn --&-l.

and d2-m

pm(s)=c(l- - s y - - - - - ( l - - s ' ) 2 . ds2-m Putting c = (- 1)z/l!2c,we get pm(s)=P2-m(s), fm@) =P I - m (e)~ and ~~

Ycm(e,y)=etmPPz-m(cos8). These 21+ 1 functions Yzm(- 1 S m s l ) are mutually orthogonal in Ls(Sa), because (etmp)m.Eare orthogonal in La([O,24). Therefore ( Y t n ) - z j m Lisz a linearly independent family in the (21+1)-dimensional space Hz', i.e., q.e.d. it is a basis of Hz'. Remark 2. Since the identity (7.25)

(1-m)! Pz-"(s) =(- l)"-----P2"(s) (l+m)!

FOURIER SERIES ON COMPACT LIE GROUPS

93

holds, the functions

(-1Sm51) form a basis of Hc'.The identity (7.25) can be proved group theoretically. Let UgCbe the representation of SU(2) constructed in $1. US1is regarded as a matricial representation using the orthonormal basis z ~ ~ + ~ z ~ ~ - ~ / J(l+k)! (l-k)!, (-ZSkSl). Put U g f C = ( ~ i(rg )n) -cc s k , r n + r and dk,,'(Adg)= uk,,'(g). Then the function do.rn'( - Z s m S l ) is invariant under the left translation of the elements in the subgroup K = {exp fZ& E R)of G = SO(3). Hence do,,( is a function on the sphere Sa=K\G. It can be proved that if the coset Kg corresponds to a point on S%with the spherical coordinates (8, p), then eim~Plm(cos 8)

The proof of (7.26) is similar to that of Theorem 7.3. Using (7.26) and the relations exp( - 8Za)expxZI=exp&exp( -BZ~),

we get (7.25).

$8. Fourier series on compact Lie groups The theory of Fourier series on SU(2)developed in $6 can be generalized to any connected compact Lie group G. In this section we give an outline of the theory by using freely the results of representation theory of compact Lie groups (cf. J. P. Serre [l], M. Sugiura 121). For the details of the proof, see Sugiura [4]. Throughout this section we use the following notation. G : a compact connected Lie group; Go:the commutator subgroup of G; T: a maximal toral subgroup of G ; l=dim T=the rank of G ; p : the rank of Go;n=1+2m=the dimension of G (where m is the number of positive roots of Go);g: the Lie algebra of G; 8": the complexification of g; t: the Lie algebra of T; dg: the normalized Haar measure of G,La(G)=LB(G, dg); Ck(G):the set of all k-times continuously differentiable complexvalued functions on G ; ~~A~~=Tr(AA*)l'a: the Hilbert-Schmidt norm of a matrix A; 6 :the dual of G = the set of all equivalence classes of irreducible unitary representation of G. In this section, a finite-dimensional,matricial, unitary representation of G is called, simply, a representation of G.

94

REPRESENTATIONS OF

SU(2) AND SO(3)

The difl'erential representation U' of a representation U of G is a representation of the Lie algebra g. Instead of writing U ' ( X ) , we write U(X). Since U(t) is a commutative set of skew-Hermitian matrices, there exist a unitary matrix Q and pure imaginary-valued linear forms 21, ..., r7r on t such that

RdH)

QUW)P=[

-,.,.*

0

1

-...._____

RdH) for all H in t. The linear forms R1, ..., I r on t are called the weights of U. In particular, a non-zero weight of the adjoint representation Ad of G is called a root of G with respect to T. The set of roots with respect to T is denoted by R:We fix once for all a positive definite inner product (X, Y) on g which is invariant under AdG. Put 1 x1=(X, X)lfa.A pure imaginaryvalued linear form (in particular a weight of a representation)I is identified with an element h, in t which satisfies R(H)=i(h2, H) for all H in t. We put R(H)=i(R, H) and (1, p)=(h2, hJ. Let r(G)be the kernel of the homomorphism expa: t+T. Then r(G) is a discrete subgroup of t of rank I (i.e. r(G)zZ l ) . Let Z be the set defined by Z= [ R E t l ( R , H ) E 2nZ for all H E r(G)}. Then an element R in Z is called a G-integral form on t. The set Z of all G-integral forms coincides with the set of all weights of the representations of G. We choose once for all a lexicographic order @ in t with respect to a basis. Let P be the set of all positive roots with respect to @. The elements of D = {A E ZI(2, a)LO for all a E P) are called dominant G-integral forms. The maximal element with respect to @ in the set of all weights of a representation U is called the highest weight of U. A linear form on t is a highest weight of a representation of G if and only if R is a dominant G-integral form, i.e., R E D. Since two irreducible representations of G are equivalent if and only if their highest weights coincide (cf. Serre [l] Ch. VII Th. 1, Sugiura 121 Th. 2), there exists a natural bijection from D onto the dual 6 of G which maps a dominant G-integral form R E D to a class of irreducible representations (U2) having highest weight R . We identify 6 with D by this bijection. If G is a direct product G1 x Ga of two compact Lie groups G1 and Gs, then the dual G of G is identified with 6, x 6,. The identification is realized as follows. If (A, p) is an element of x 6 , and U ' and Up are representations in the classes R and p respec-

el

FOURIER SERIES ON COMPACT LIE GROUPS

95

tively, then the representation UQU.: (gl,ga)wUg,'@Ug2pis an irreducible representation of G. Conversely, every irreducible representation U of G = GIx Ga is equivalent to a certain representation of the form U2@Up(RE P E 6%). By the mapping ( A , p ) w ( U L @ U . ) , 6, x 6, is identified with 6. In particular, if to is the center of g and To be the connected Lie subgroup of G with the Lie algebra to, then Gox To is afinite covering of G and G is isomorphic to (Gox To)/F,where F is a finite normal subgroup of Gox To,8 is the subset of 8, x foconsisting of representation classes whose kernels contain the subgroup F. Since To is commutative, any irreducible representation of TOis one dimensional (Ch. I. Corollary to Theorem 2.1.). So the degree of the ,a E f0) is equal to deg irreducible representation UA+r= UL@Up ( A B 60, UA =d(n). Hence, the degree of the irreducible representation U2(2E 6) is given by Weyl's dimension formula,

el,

where 6 =2-lC a (cf. Serre [13, Sugiura [2n. .€P

Since d(2) is a polynomial of degree m, there exists a positive constant C such that (8.2) d(~)sCl,p for a n y ~ ~ D ~ = D - { 0 ) . The Casimir element L? of the universal enveloping algebra U(g) is defined by n

(8.3)

-sa=

1g''X#X, 6.'-1

is a basis of g and (Xc,X,) = g t j , (g") =(g',)-l. where (X')l If g is simple (i.e. if the only ideals of g are {0)and g), then the operator L? defined by (8.3) differs from the Casimir element in Definition 9 of $5 by a scalar factor. The Casimir element Q belongs to the center of U(g). (Proposition 5.14). Let 2 be an element of 6. Since the representation U' and its differential representation are irreducible, V ( Qis) equal to a scalar operator cl by Schur's Lemma (Ch. I. Proposition 2.4). (Remark Schur's Lemma for finite-dimensional representations is valid not only for groups but also for Lie algebras (or for any set of operations on a vector space).) The value of the scalar c can be calculated easily. The result is as follows. Proposition 8.1.

1) For any compact group G, we have ($2) =w(R)1

96

REPRESENTATIONS OF

SU(2) AND SO(3)

where w ( R ) = ( R ,n+26). 2) The matricial element uAl* xrRn Remark. The symbol Lmis often used for the space of all essentially bounded functions. Our space La is a proper subspace of the latter space. Definition 2. The convolution f * g of two functions f and g in L1 is defined by

De6nition 3. The Fourier transform f = T of f a function f in L1 is defined as (1.1)

f(c) =

1

f(x)e-'(=*t) dm(x), (E E R").

Rn

The conjugate (or inverse) Fourier transform N * f of f is defined as

103

RAPIDLY DECREASING FUNCTIONS

=f(-X) . Proposition 1.1. 1) The Fourier transform{ of a functionf in L1 is a bounded continuous function on R". It satisfies

Ilf 11- 4Ilf 111* (1.3) 2) Iffand g belongs to L1,then we have

( f *g>- =AProof. (1-4) we have

Since If(x)e-s(t.Ol= If(x)l

If (€)I5 1R ,If (x)ldm(x)=llflll for any E E Rn.For any Eo E Rn,we have l i d ( € )= f ( ~ o ) ('(0

because we can apply Lebesgue's dominated convergence theorem by (1.4). 2) By Fubini's theorem we have

Definition 4. For any multi-indexm =(ml, ...,m,) E N",and x E R", put 1m1=lm1l + ... Im,l, x"=x1"1 ... Xn"" and D"=(--)"l l a ... l a i axl i ax, *

+

(--)""

A complex valued Cm-functionf on Rn is called rapidly decreasing if

(1.5)

pN,m

cf)=suP (1 + Ixla)>"l(D"f)

+

O0

104

FOURIER TRANSFORM A N D UNITARY REPRESENTATIONS OF

R"

for all N E N and m E Nn.The set of all rapidly decreasing functions on R" is denoted by 9= 9 ( R " ) . Example 1. Any C"-function with compact support belongs to 9.For a concrete example of such a function, see Lemma 1 below. Example 2. The function u(x)=e-lsil belongs to 9,because D*u(x)= (2i)j"j xmu(x)and Jim xmu(x)=O for any m E N". Irl*+-

+

Proposition 1.2. 1) 9c Lp, for any p E [l, m]. 2) 9 is a metrizable, complete, locally convex, topological vector space (FrCchet space). €'roo$ 1) Since

+

If(x)l $ P N , O C f ) (1 Ixls)-N for any x E R", f is bounded and belongs to L". If 1 j p < Q) the above inequality implies (1.6)

+

Ilf IlP SPN,O(f) 1/-.(1+ Ixl2)-PNdm(x)1l ' P
n/2p). 2) 9 is a vector space by its definition. Since 9 is topologized by the I N E N , m E N")of seminorms, 9is metrizable. countable family { p N P m Then (pN,m(fn))n.N is a Cauchy Let (fn)n.N be a Cauchy sequence in 9. D ~ ~ converges .) unisequence of real numbers and ((1 I X ~ ~ ) ~ ((X))nrN formly to a function. Thus, for any 1, m E N", the sequence (Xz(Dmfn) ( x ) ) ~ , N converges uniformly on R" to a function fr,m(x)=x'(D*f) (x). q.e.d. Hence the sequence (fn),,eN converges to f in 9. The most important and interesting property of the Fourier transform lies in the fact that 5 transforms the operation of differentiation into the operation of multiplication of coordinates and vice versa. More precisely we have the following proposition.

+

Proposition 1.3. I f f is a rapidly decreasing function, then we have (1.7) (Dmf)"(()=Emf(€) for any m e N" and E E Rn. Moreoverf is a C"-function and satisfies

(W)(€I=(- 1)'"'(Xrnf(X))"(€). Proof. From integration by parts, we have

(1.8)

RAPIDLY DECREASING FUNCTIONS

105

because lim f(x)=O. By repeated differentiation, we get (1.7). s -is

Since xmf ( x ) belongs to 9 for any m E Nn, Ixkf(x)e-i("*oI= Ixkf(x)I belongs to L1(Rn).Hence we can differentiatef(E) under the integral sign and we obtain

k " We have proved (1.8)when m=(O, ..., 0, 1, 0, ..., 0).Repeating the operation of differentiation, we get (1.8) for every m E N m . q.e.d.

Proposition 1.3 implies that S and T *are continuous linear m a p 'into 9'. (See Theorem 1.2 below.) To prove that fl is surjecpings of 9 tive, we use the elementary solution uc of the heat equation au/at=Au.

Proposition 1.4.

1) Let u(x)=e-ltlalt. Then we have u E 9 and

(1 *9)

2) Let u@)

6=u. =(2t)-n"e-I="l4' for

t > 0. Then

rir (t)=e-glcla, y * f l u , = u t .

3 ) Iffis bounded and continuous, then we have

lim (ut*f)(x)=f(x) t*+O

for all x E R". Proof. 1) Since u(x)=

fi e-*k21aand e-"*.c' = fi e-(=ktk,it is sufficient

k-1

2) Since uc(x)=(2t)-n%(x/4/), we have

k-1

106

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Rn

Sincef is bounded, the integrand is 5 (1 f Jl,e-I"l' which belongs to 9 ' c L'. So we can apply Lebesgue's theorem and get lim ( u t * f ) (x)=H-"~'

e-'*la f ( x ) d z = f ( x )

t-to

by the continuity off.

q.e.d.

Theorem 1.2 (L. Schwartz). Iff belongs to 9, then the Fourier invertion formula (1.19)

=I

f (x)

f(E)etCa,6)dm(0

RR

holds for any x. Namely we have (1.11) F * F f = f and F F * f = f for any f in 9. The Fourier transform fl is a topological isomorphism of 9 onto 9 and the conjugate Fourier transform fl*is the inverse mapping of fl. Proof. First we prove that

(1.12) F9c9, x*9c9. Let f E 9 and 1, m E Nn.Put g(x) = (- l)lml xmf(x). Then g E 9, and by (1.8) we have S=Dmf and Ez(Dmf)(E)=(Dcg)*([).Since D l g E 9 c L ' .

RAPIDLY DECREASING FUNCTIONS

107

(Dlg)*is bounded by Proposition 1.1. Hence f belongs to 9'.Since ( S * f )(()=f(-E), S*fbelongs to 9 '. We have proved (1.12). Proposition 1.4. 2) and Fubini's theorem prove that

=IRnGt(f)f(t)et(**C)dm(E).

Since f~ 9 c L 1 and lim Gt(E)=lim t*+O c++o (1.13) tends to S,nf(E)ef(**o

ME)

e-'Itla=

1, the right hand side of

=(.F*T~)

(XI

as t (>0) tends to zero. On the other hand, the left hand side of (1.13) tends tof(x) by Proposition 1.4, 3). We have proved (l.lO), i.e. S*flf=$ Since ( S *(€)= f) f(-E), we have

s

(flfl*f) (x) = f(-E)e-'(*.E) d m ( ~ ) = (fl*flf) (XI

=fO.

We have proved (1.11) and that flis a linear isomorphism of 9 'onto 9. Now we shall prove the continuity of R.Suppose a sequence (fn)nsN converges to f in 9'. The inequality (1.6) proves that (fn) converges to f i n L'. Then ( f n ) n . N converges uniformly tof by (1.3). Replacingf(x) by Hence 5 : fwfis, Dlxmf(x), we see that (f,JnCN converges to f in 9'. continuous. Since ( X * f )( E ) = ( N f ) ( - E ) = ( S X f ) (E) and S : ft+f (f(x) =f(-x)) is continuous in 9, fl*=f l - 1 is also continuous. Therefore fl is a homeomorphism of 9 'onto 9'. q.e.d. Theorem 1.2 is the basis of harmonic analysis on Rn. All the main results in this chapter are obtained from Theorem 1.2. As the first application of Theorem 1.2, we prove the following theorem.

Theorem 1.3. For any f in 9 'and g in L1,we have

-

(1.14)

/ R n g ( x ~ ) d m ~=x/Rn~(~)f(c) ) dm(E).

In particular, the Parseval equality

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

108

IIfIIa=

(1.15)

IIPII,

holds for any f E 9'. Proof. By the inversion formula (1.10) and Fubini's theorem, we have

=/ s ) / g ( x ) e - i c z J )

dm(x)dm(t)

q.e.d. Putting g= f i n this equation, we get (1.15). Now we shall prove that .9' is dense in the Banach space L p for all p (1 s p s + a). We need some lemmas.

LEMMA 1. For any 6 >0, put

m)=[ e ~ p ( ~ ) i f , x E u a = { x €I ~ InO if xa+ 1 . Since g E 9 ' and f is bounded, lim +(x-y)l f(y)l =O for any y E K. Ill+-"

On the other hand +(x-y)l f ( y ) I , being a continuous function of y, is bounded on the compact set K. So we can apply Lebesgue's theorem to (1.17) and prove that lim Ixl'l(Dmh)(x)l =O Ill-+-

for any 1 E N and m E Nn.The last equality proves that h belongs to 9. q.e.d.

110

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

LEMMA 3. Iffbelongs to L, then we have lim Ilhd*f-fllp=O for everyp (1 r p $ d4O

+ a).

Proof. l ) p = + w . Sincef is uniformly continuous on R",for any E >0 there exists a positive number 6 ( ~ such ) that if Ix-yl O, and f E Lp, there exists a function g in L such that

+

Ilf-gllp O and X E Rn). ut(~)=(2t)-"'~

Then 1) ut* I(, =ut+, for any r, s>0, and 2) for any positive definite function p on Rn, we have JRnp(x)ur(x)

for any t > O .

Proof: 1) By Proposition 1.1,2) and Proposition 1.4, (uc* u , ) A ( ~ ) = l i c ( ~ ) l i , ( ~ ) = e - ( t + ' ) l " a =( U t + , )

YE).

Since ut E 9c L1, we get ut * u, =ut+, by Theorem 1.2 and Theorem 1.7. 2) Using the result in l), we get

The positive measure p defined by dp(x)=u&)dm(x) p(Rn)=(2xt)-n/a

e-ltlaWx=z-n/a

is bounded by

e-l~lady=1. So we can apply

JBn

Proposition 3.4 and it follows that the right hand side of the above equality is positive. q.e.d. Proposition 3.8. Let (p,,JnzfNbe a sequence of positive Radon measures on Rn with total measure 1 and let

be the Fourier transform of pm.Assume that the sequence (p,,Jnr~converges to a function p uniformly on every compact set. Then p is positive definite, and there exists a positive Radon measure p with total measure 1 such that p(x) =JUn e*(=* dp (€1 for any X E Rn. Proof: The function p is continuous because it is the limit of a sequence of continuous functions which converges uniformly on compact sets. Since pmis positive definite, it satisfies the inequalities

128

FOURIERTRANSFORM AND UNITARY REPRESENTATIONS OF

Letting m tend to

R"

+ a,we get

c n

. , f

(0 ( X j - x o

t' €/LO.

f-1

Hence p is positive definite. Let T be the linear form on 9defined by

T(f)=/ f ( x ) ( 0 ( 4 d m W .

(3.7)

R"

Then T is a bounded linear form on 9. In fact, iff belongs to 9, by Fubini's theorem and Theorem 1.2, we get

~ ( f=Jim )

m*+-/Rn

f ( x ) (on(x>dm(x)

=m-fimIRnY(E)d p m (E).

Since

ISf(E)dpm(BI

5 pm (Rn)llfll- = Ilfll-,

T(f)satisfies

IT(f)lS llfllfor every f in 9. Since 9 is dense in L" (the space of continuous functions f on Rn which vanish at infinity) by Theorem 1.4, the linear form T

on 9 can be uniquely extended to a bounded linear form on L" which is also denoted by T. In particular T is a bounded linear form on L. So T is a Radon integral on Rn. Since pmis a positive measure, T is positive o n 9 (fZO+T(f)LO). If an elementfin L is positive (fZO), then there exists a sequence (fn),,.~of positive elements in 9(e.g., takefn=hlln*f) converging to f i n the uniform norm 11 I]_. Hence T is positive on L. So there exists a positive Radon measure p on Rn such that

for any f E L" (cf. Appendix D, 4). Using the inversion formula (Theorem 1.2), (3.7), and (3.8), we get

POSITIVE DEFINITE FUNCTIONS AND STONE'S THEOREM

129

s

for a n y f ~9.Since p(x) and e*(=,e)dp([)=4(x) are bounded and belong

to the dual space of L1,and since 9 =9 is dense in L1,the last equality implies that erCr*f)dp ( E )

(o(x)=/

(3.9)

R"

for almost every X E R". Since both sides of (3.9) are continuous, (3.9) is valid for all X E R". Puttingx=Oin (3.9), we get p(R")=p(O)=lim ym(0)= m-lim pm(Rn)= 1. q.e.d. m--

Theorem 3.1 (Bochner). If (O is a positive definite function on R", then there exists a unique positive Radon measure p with total measure p(R")= p(0) such that

for all x E R P . Proof. Existence of p. If p(0) =0, then y =0 by Proposition 3.1. In this case, we can take p=O. So we assume that p(O)>O. By replacing y with q(O)-'p, we can assume that ~(0) = 1. Put ut(x) =(2t)-n~ae-i"isr4t and Vm(x) = ( ~ ~ ) ' " ' U ~ ( X ) ( O ( X )

for any m B N and dp,(E) = @,(€)dm((). Then pmis an integrable function 'and cp is bounded, and p,,, is a bounded Radon because urnbelongs to 9 measure on R". If g E .si' is positive, then the function ~(XX =)/Rned(i.6 )

g(E) d m ( ~ )

-

is positive definite by Proposition 3.6. So the functionf(x) =g(x) is also positive definite. By Theorem 1.3, we get P

P

130

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

s

=(2m)R12

2 0.

-

umW (DWf(x) dm(x)

-

The last inequality is valid by Proposition 3.7 because v(x)f(x) is positive definite (Proposition 3.5). The inequality (3.10) implies for all E E Rn.

@,,,(E)TO

(3.11)

Suppose @,,,(to) = SUP

9

6-1

where the supremum is taken over all decompositions of A into a disjoint union of a finite number of sets in b.The total variation 1p1 of p is a finite positive measure on 123. Let p1 and p, be the real and imaginary parts of p. Then the positive and negative parts p; of p k are defined by pk+(A)= SUP P k ( B ) BcA

and

p;(A) = -inf pk(B). BcA

p; and pi are finite positive measures on b and pk = p l - p ; (k= 1,2). A complex-valued measurable function f on X is called p-integrable if I f 1 is Ipl-integrable. Since Ogp:(A)g [pI(A) for any A E 8 (k= 1,2), f is then

s

&integrable. The integral f d p is defined by

134

FOURIER TRANSFORM A N D UNITARY REPRESENTATIONS OF

R"

and satisfies (3.20)

Proposition 3.10. Iff and g are square-integrable functions on X with respect to the measures pu and purespectively, then the product f(x)g(x) is integrable with respect to pU,,,and satisfies

Proof.

Because of the general inequality (3.20), it is sufficient to prove

So we can assume fro and g2O by replacing f and g with respectively.

If I

and lgl

n

If A = u At (disjoint union), we have 1-1 n

n

= I I ~ ( 4 u l lIIE(A)vll by the Schwarz inequality in Hand in Rn and by Proposition 3.9,2). Thus we have proved

(3.23) Ipu,ul(A)~pU(A)*~u(A)* for any A E 8 Now we prove (3.22) for simple functions n

n

i=1

J-1

.

and (Bj)lsjsn are two disjoint families in where A r 0. Since X A X B= x A B , we have by the inequality (3.23)

B and ar20,

POSITIVE D ~ M T FUNCTIONS E AND STONE'S THEOREM

n

m

135

n n l

Iff 20 and g L 0 are square-integrable with respect to p, and p,, then there exist two monotone increasing sequences (fn ) and (g,) of positive simple functions which converge to f and g respectively. Then we have

Since I fnl d f and lgnl $ g, we can apply Lebesgue's theorem on both sides of the last inequality. Letting n tend to infinity, we get (3.22). q.e.d. The following proposition will be used frequently. Proposition 3.11. 1) Let B be a fixed element in b.Then the complex measure p:," defined by p t , , ( A ) = (E(A)E(B)u,v ) satisfies p:,,(A) = p.,,(A n B)= (E(B)E(A)u,v). For any p:,,-integrable functionf,

,.

c

(3.25)

is a complex measure. Iff is p-integrable, then we have c

c

(3.26)

Proof. 1) Since E(A)E(B)=E(An B) (Proposition 3.9), we have p:,, (A) =pu.,(A nI?)=(E(B)E(A)u,v) and (3.25). 2) Since p is an indefinite integral with respect to the measure gp,.,, (3.26) holds. (cf. Halmos [l] p. 134, Theorem B). Proposition 3.12. Let E be a spectral measure on b in H, f and g be

136

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Rn

complex-valued measurable functions on X. 1) Then the set D( f) of all vectors u in H satisfying (3.27)

Jxlf(x)l~dllE(x)ull' < +

is a subspace of Hand there exists a linear operator T(f):D ( f ) + H such that (3.28)

for all u E D ( f ) and v E H. 2) D ( f ) = D ( f )and for any elements u, v in D(f), ( T ( f ) u ,v> = (4T(P)V). 3) For any set A in 123 and u E D ( f ) , E(A)u belongs to D(f) and we have T ( f ) E ( A ) u= E ( A ) T ( f ) u . 4) For u E D(f)and v E D(g),

f ( x ) r n ( E ( x ) u ,9 .

( T ( f ) u , T(g)v)= J x

In particular (3.29)

llT(f)ulI'=/

X

If(x)12dllm)ull'

5 ) For any a E C,D ( f ) c D ( u f ) . For any u E D(f),T(uf)u=aT(f)u. 6) D(f)n D(g) c W+d.For any u E N f )n D(gh

(3.30) T(f+g)u = + TWu. 7) Let u E D ( f ) . Then T ( f ) u belongs to D(g) if and only if u E D ( g f ) . If this condition is satisfied, then T ( g ) T ( f ) u= T(gf)u. 8) Iff is a bounded measurable function then D ( f ) = H and T(f)is a bounded linear operator on H. Proof. 1) If u, v belong to D(f),then Proposition 3.10 shows thatfis square-integrable with respect to p,,, and /I,,,,, because 1 is ,u,-integrable. Hence the equality

+

1I-W)(u+ v)l12 = IIE(A)ul12+(E(A)u, v) +(E(&, u) IIE(A)vlla shows that f is square /I,+,-integrable, namely, u + v belongs to Wf). Define the mapping B : D(f)x H+C by

N u',) V

f(x)d(E(x)u,v). /X

Then UHB(U,v) is a linear mapping and VHB(U,v) is an anti-linear mapping. Moreover, we have by Proposition 3.10,

POSITIVE DEFINITE FUNCnONS AND STONE’S THEOREM

137

So by Riesz’s theorem, there exists a linear operator T ( f ) : D( f )+H satisfying B(u, v) = ( T ( f ) u ,v) for any u E D(f)and v E H. 2) By the definition of D(f),D ( f ) = D ( f ) . If u and v belong to D(f), we have

P

because E(B)E(A)=E(A)E(B). 4) By 2), 3) and Proposition 3.11.2), we have

5 ) is trivial.

6) Since I f + g l a 9 (If12+lgla),D(f)n D ( g ) c D ( f + g ) . We have (3.30) from (3.28). 7) For any u E D ( f ) , we have by Proposition 3.1 1,

138

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

Hence T(f )u E D(g) if and only if u E D(gf ). If this condition is satisfied we have

8) Iff is a bounded, measurable function, then f is square p,--integrable for any u E H, because pu is a finite positive measure. That is D ( f )=H. The equality (3.29) proves that IIT(f)ull S Ilfllmllull, where 11f 11- is the supremum of If I. So T(f)is bounded.

Definition 4. For the linear operator Tcf> :D O - H defined by (3.28) we write

=I

T(f)

f(x)dE(x).

X

Theorem 3.2 (Stone). If U is a unitary representation of Rnon H, then there exists a spectral measure E on b (the set of all Bore1 sets in R") in H such that U, =[Rnei[z.el dE(E) for every x E Rn . A bounded linear operator T on H commutes with every U, (XE Rn) if and only if T commutes with every E ( A ) (A E b). In particular U,E(A) = E(A)Ux. Proof.. For any vector u E H, the function v,(x) = (Uxu, u) is positive definite (Example 1). So there exists a unique bounded positive Radon measure pu satisfying

(3.31) and

(3.32) pu(R") = llul12 by Bochner's theorem (Theorem 3.1). For any two vectors u and v in H, put pu. 0

(A)=4-' Ip*+u ( A )-pu-u ( A )

+

ipu+to

(4- ipu-iv ( 4 1 .

POSITIVE DEFINITEFUNCTIONS AND STONE'S THEOREM

139

Then pU,,,is the unique complex measure on R n satisfying

(3.33)

(U,u, v) =

1

Rn

ef("*a) dpu.0 (8

for every x E R". For any fixed Bore1 set A, the mapping (u, v ) n p u . , ( A ) is a positive Hermitian form on H x H. It is bounded, because by (3.32) Ipu,v(A>I 5 pu(~)"2pu(A)"'

5 llull llvll -

Hence there exists a unique, bounded, Hermitian operator E(A) such that

(3.34) pu,.(A)=(E(A)u, v) for all u, v E H. The mapping E : A n E ( A ) is a spectral measure on b in H. In fact, by (3.32) and (3.34), we get

(B(R")u,V ) = p u , u ( R n ) = ( u , v); hence, E(Rn)= 1. If B= u B, is a countable disjoint union, then n

(E( U Bn)u, v)=pu.u(

B.)=Cpu.u(Bn)=C(E(Bn)u, V) n

n

because pu,uis a complex measure. Hence

It remains to prove that E(A) is a projection. Since E(A) is Hermitian, it suffices to prove E(A) is idempotent. More generally, we prove (3.35)

E(A)E(B)= E(A

We have

n B) for any A, BE b.

SRnf?-

%f(E(c)u,u, v) = (U,U,u, v)

(3.36)

et("+u.E)d(E(€)u, v)

S n

where p(A)=

ef(gsE)d(E(c)u, v) for any A E 9.For any fixed A E b,put

A

F(B)=E(A

B). Then we can write p(A) as p(A) =/R,et(u. E ) d(F(c)u,v).

By the uniqueness of the measure pUyu,,, the equality (3.36) yields

140

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF R"

(3.37)

J

(E(A)U,,u,v)=p(A) = Rne*(Y*t)d(F(e)u, v)

for any A E 23. On the other hand, we have

(3.38)

SI.

(E(A)U,u, v) =(Up,E(A)v)=

et(v*e)d(E(E)u,E(A)v).

Again by the uniqueness of p,,,E(A),,, (3.37) and (3.38) yield (E(A n B)u, V ) =(F(B)u, V ) =(E(A)E(B)u,V ) for any B E b. Since A , u, and v are arbitrary, we get (3.35). The uniqueness of E is clear from the uniqueness of pu,u. Let T be a bounded linear operator on H. Then we have for any u, V E H and X E Rn, (3.39)

(TU,u, v)=

e((=." d(E(c)u, T v)-

*

I-R"

and

(3.40)

e*(=.W(TE(e)u,v)

-/Rn

)-IRn

( U,Tu, v -

e*("*W(E(€)Tu, v).

Hence if TE(A)=E(A)T for every A E 23, then we have TU, = U,T for every x E Rn. Conversely if TU,= U,T for every X E Rn, then by the uniqueness of pu,T.u,we get TE(A)=E(A)T for every A E b .

q.e.d.

Remark. We can easily derive Bochner's theorem from Stone's theorem. Let 9 be a positive definite function on R". Then there exists a unitary representation U of R" and a vector u in H = H ( U ) such that cp(x)=(U,u, u) (Proposition 3.3). By Stone's theorem U can be written as

s

Us= e"**"dE(E). So &c)= finite Borel measure.

s

e*("."dp,(E)where pu : Aw(E(A)u, u ) is a

Theorem 3.3. If U is a cyclic unitary representation of Rn on H, then there exists a finite Borel measure p on Rn such that

.

where ~ ( ( x=)e*(=.b ) Proof. By Stone's theorem (Theorem 3.2), there exists a spectral measure E on b in H such that

POSITIVE DEFINITE FUNCTIONS A N D STONE'S THEOREM

U,=/Rnet(s.

141

dE(€).

Let u be a cyclic vector in H and define a finite Bore1 measure p by

.

p(A) = IIE(A)ulls=(E(A)u,u) for any A E b Let f be an element in L'(R", p). Now u belongs to o(f) in the notation of Proposition 3.12, and the linear operator

T ( f ) = / R n f ( €dE(€) ) :W ) - + H

is defined. Then the linear mapping 0 : L' (R", p)+H defined by 0 ( f )= T ( f ) u is an isometry by (3.29). The image H' of 0 is invariant under U because we have u,u=Jlnet(s*~)d ~ ( € ) u = T(x.)uE H'

for any X E R". Since u is a cyclic vector, H' is dense in H. O n the other hand, H' is closed because it is isometric to the complete space La(R",p). So we have proved that H ' = H and 0 is an isometry of L'(Rn, p ) onto H . Since

L1(R", p)=J

R"

Cl3 Cedp(€1,

where Ct = C for all 6E R", a unitary representation U oon La(R",p ) is defined by

U'=/

R"

a3 xedp (8*

Then we have (3.41)

U p 0 =00Uj for every x E R"

.

In fact, for any f E La(Rn,p ) and v E H, we have by Proposition 3.1 1,

u:>f,v) =s R p"f(€)d(E(OU, . v)

((00

=IRn

f ( € ) d e (R/" e'(=*?)d/(E(7?)~, E(€)V))

=Sf(g)d,(U=u, E ( W =

s

f(€)d(E(E)UdJ,v )

142

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

Thus we obtain (3.41).

R" q.e.d.

Remark. We can drop the assumption that U be cyclic in Theorem 3.3.

In the general case, an irreducible representation xr can be contained in U with some multiplicity. Let U be an arbitrary unitary representation of Rn. Then U is a direct sum of cyclic representations Ua(aE A):

u=

(3.42)

@

u a

atA

(Ch. I. Proposition 2.5). Each cyclic representation U a is decomposed in the form

ua=[R"

(3.43)

@ xdpa (E)

by Theorem 3.3. Let J= A x Rn. Define a measure p on J by p(a x B)= p a ( B ) and put 7j)=(a,6) and x,,=xr. Then (3.42) and (3.43) yield

$4. The Paley-Wiener theorem

In this section we give a characterization by means of the Fourier transform of the infinitely differentiable functions on Rn with compact supports. This result due to Paley-Wiener, reveals a remarkable relation between a certain class of holomorphic functions and harmonic analysis of functions with compact support. First we give fundamental definitions and results on holomorphic functions of several complex variables. Let Cn be the space of all vectors z =(zl,..., z,) with n complex conponents. The norm IzI of z is defined by IZI =(Izlp+ ...+ IZnla)l'a. The canonical inner product in Rn is extended to a bilinear form (z, w)= n

c.?ewk

on C" x c".If zk =

+ivh and &,

vk

E R,then the real and imaginary

k=l

parts of z are defined by Re z=(ul, ..., u,) and Im z=(vl, ..., vn). For any multi-index m E N",we put m ! = m l ! ... m,!, We introduce an order in N" by defining p hq if and only if pk2 qk for all k e {1,2, ..., n}. Z ~ = Z ~ ~ . , . Zand ~ "

143

PALEY-WIENER THEOREM

Definition 1. A complex-valued continuous functionf on a connected open set A in Cn is called holomorphic if the function f, : cwf(a1, .-.,ak-1, c, aX+l, .-*,an) is a holomorphic function of the complex variable C on an open neighbork of a&in C for every u=(al, ..., a,,) E A and k E {I, 2, ..., n}. hood u We recall that a functionf, of a complex variable C is holomorphic on a connected open set UX,if (4.1)

exists for every

To e UX.The limit in (4.2) is called the derivative off, at

To and is denoted byf,'(Co) or by

(Co).&'(aX)is the partial derivative of

f a t a with respect to the k-th coordinate and is denoted by (ayaz,) (a). Let f=u+iv and z=x+iy be the decompositions off and z into real and imaginary parts. Iff is holomorphic on A, then the functionf, defined in (4.1) is a holomorphic function of a complex variable. Hence the Cauchy-Riemann equations a v =-- au and (4.3) axX

hold on A for any k E { 1,2, bY

auk

...,n} . The partial derivatives off

are given

(4.4)

Let rr be a piecewise smooth, simple, closed curve in the complex plane C and let Dk be the domain interior to r r(1 5 k 5 n). Iff is holomorphic on an open set A containing 6, x ... x fin,then from the Cauchy integral formula for one complex variable we obtain

for any z=(zl, ..., z,) E D1 x ... x D,. (4.5) is the Cauchy integral formula for a holomorphic function of several complex variables. Put r =rlx ... x rn, D=D1 x ... x D,, and r;=(Cl, ..., then (4.5) can be written in the form

m);

f(z) =( 2 ~ i ) - * / ~ -dC

for z E D.

If r k is the circle lzX-ukl =rk and f is holomorphic on an open set A containing the polydisc 6,then f ( z ) can be expanded into a power series with the center a = (al, ..., a,). In fact, put

1 4

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

f (C) (C -a)-'"+" dc

a, =(27r9-n

(4.7)

R"

s r

..., m,+1)

foranymENn, wherem+l=(ml+l, and

C a,(z

gN(z)=f(z)-

-a)" for N E N".

NL,>O

Then we get

If c is a real number in the interval (0,1) and z belongs to the polydisc 0,= { zI I zk -akI 5 crk (1 5 k i n) }, then we have a constant M such that IgjV(z)I

5 M {1 - (1 - &+1)

...(1 - &+')}.

Hence when Nk+ + 00 (1 $k$n), gN converges to 0 uniformly on D,. We have proved the following theorem. Taylor's Theorem. Iff is holomorphic on an open set containing the polydisc D with the center a, then f can be expanded in a power series: (4.9)

f(z)=

c

a,

(Z-u)m

.

mrNn

The coefficients a, in (4.9) are given by (4.7). Since the power series converging in a polydisc D can be differentiated arbitrarily many times, a holomorphic function can be differentiated arbitrarily many times. Differentiating(4.9), we get the following Corollary. COROLLARY to Taylor's Theorem. The coefficient a,,, in (4.9) is given by (4.10) where D:=(=)"'a

... (-)"" a azn

Hence the power seriesexpansion of a holomorphicfunctionf is uniquely determined by f and a.

Definition 2. A function f which is holomorphic on the whole space C" is called entire. LEMMA1. If an entire function f on C" vanishes on Rn,then f=O on Cn. Proof. All the coefficients a, in the Taylor expansion (4.9) off with the center a=O are equal to 0, because by (4.4) and (4.10) we have

145

PALEY-WIENER THEOREM

q.e.d.

Definition 3. Let r 2 0. Then an entire function ~pon Cnis said to be of exponential type 5 r if there exists a constant C> 0 such that lq(z)ld CerlIrnz' for all z E Cn . We denote by S r = S r ( C n )the space of all entire functions f on C" such that zmf (z) is of exponential type r for every m E Nn.The vector space p,is topologized by the family of seminorms:

The topology of g f , just defined coincides with the topology defined by the family of seminorms:

(4.12)

sm ((.)=sup {e-,lIrn*I Izmp(z)l], rrCn

(mE N") .

n

In fact, since 1zm15(1+(z(')N if I m ( l = ~ r n r 5 2 Nsm(v)dqN(q) , if ImllS k-1

2N. Conversely, since (1 + lzla)" is a linear combination of the monomials

the inequality qN((o)d

1

CP S%P (9)

IpllSN

holds for any (o E s

r ,

where the cp's are certain constantsz0.

Definition 4. The space of all complex valued C"-functions on R" with compact supports is denoted by g = g ( R n ) . The subspace of 9 consisting of those functions whose supports are contained in B, = { x E RnI 1x1 $ r ) is denoted by 9,.9,is topologized by the family of seminorms:

(4.13)

I(omf) (XI, (m€Nn) pm(f>=SUP zdn

9

where Dm is the partial derivation defined in Difinition 4 in $1.

Proposition 4.1. The spaces S,and g rare Frechet spaces (i.e. complete, metrizable, locally convex vector spaces) if r >0. Proof. They are metrizable, locally convex spaces, because their topologies are defined by countable families of seminorms. Now we prove their completeness. Let ( f & ~be a Cauchy sequence in the space Br.Then (f,JnrN and (Dmfn),,.N converge uniformly. So f = lim fi is a C"-function

146

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

whose support is contained in B,, i.e. f E 9 , . Let ( q & . ~ be a Cauchy Then for any m E Nn and E >0, there exists N, E N such sequence in Sr. that

.

then S, ( y p- ~ , J < E (4.14) if p , q 2 N,,, Since the inequality

(4.15) 12" ( p p (z)-pq(z))I S sm (pp-pJer'Irna' holds for all z E C", the sequence ( Z ~ ~ ~ ( Z ) ) , , ~ Nconverges to a function g , and uniformly so on every compact set. Hence the limit function g, is entire and g,(z)=zmgo(z) for every m E N" and z e Cn. Since Is,(pp)sm(pq)lSs,(pP-pq), the real sequence ( S , ( ~ ~ ) ) , , ~has N a limit c,zO. Taking the limit when p-+ + to in the inequality

lzmpp(z)le-,

Irnlrl

ds, (pp),

we get lzmgo(z)le-'Ixrn 11 I:C ,

for any z E Cn. Hence gobelongs to Sr and go =lim pn in the topology n-+-

of ZI.

q.e.d.

Definition 5. The Fourier-Laplace transform Sc f of an integrable functionf on Rn with compact support is defined by (4.16) flc f

Hc((f>

f ( x ) e -t(*,*)h(x).

(z)=/ R"

is a function on Cn.

Theorem 4.1 (Paley-Wiener). The Fourier-Laplace transform X cis a topological linear isomorphism of 9, onto s r for any r >0. The inverse mapping of X cis given by the conjugate Fourier transform on Rn. and p =S C f b ethe Fourier-Laplace Proof. Let f be a function in gr transform off. Then p is given by

j'

(4.17)

p(z) =

f(x)e-'(s*a) h ( x ) ,

Br

where Br= { x e RnI Ixl$r}. Since le-*(z.a)l=e(*Jrna)<erlXm for any x E B, and since a continuous function f ( x ) is integrable on the compact set B,, the integral (4.17) converges for every z E C" and p is a continuous function of z by Lebesgue's theorem. Since e-'(***' is an entire function of z and

-I a

azk

(f(x)e-'(*S*))l=I-ixk f(x)e-'(".')I 5 Ixkf(x)lerlxm'l

147

PALEY-WIENER THEOREM

is integrable on B,, p(z) is holomorphic at every point with respect to the k-th coordinate z k for any k. Hence p is holomorphic on Cn and is an entire function. Moreover, by repeating integration by parts as in the proof of Proposition 1.3, we get

(DY)(x)e-6('".a ) dm(x)

(4.18)

foranyzECnandkENn. Hence we have (4.19) Izk'p(z)l5 IIDy:flller1Im *I 5 rn(Br)llD"fll&lm 'I for any z E C" and k E N n . (4.19) proves that 'p belongs to g?? and that RC is a continuous mapping of sT into z,. Sc is injective on 9,.In fact, if flc f=O for a function f E .gr, then the restriction flf=f of F c f to Rn is equal to 0. Hencef =0 by the uniqueness of Fourier transform on 9 2 3 (Theorem 1.2). It remains to prove that S c 3=,p,and ( f l c ) - l . is continuous. Let Q be a function in Z1 and put (4.20)

f ( x )=

SpRn

p(t) e$(=* E) d m ~ .

Since Ip(€)ISqN(y)(1 +I€I')-" for every (E Rn and N E N, p is integrable on Rn and f can be differentiated arbitrarily many times under the integral sign. Hence f is a C"-function on Rn. Let z k = t k + i v k , ( k , ?lk E R ; then the integral (4.21)

Im'p(z)e'('"*

d&

is independent of 77r.Let r be the rectangular path in the &-plane given in Fig. 4. Since p E Zrsatisfies

l'p(z)I ICl (1 + Izl2)-l e' the vertical integral tends to 0 when b-+ have

Fig. 4

, and

IIrn 00

c+-co.

In fact, we

148

FOURIER TRANSFORM AND UNITARY REPRESENTATIONSOF

Similarly lim C*--

(4.22)

s'

R"

=O. So we have proved

C+%

+

f ( x ) = I R n y(t ir/)e(( 2 , t+w WE)

satisfies for any x and '1 in R".Since (o E S+ lp(z)ls Cn (1 + IzIl)-" erlxmrl 5 Cn (1 I€l')-" erlrl for any z = € + iv E C", (4.22) yields

+

(4.23) 5 Coerlql - ( 2 . d for any x, 7 E Rn , where Co= C n h l iI ~ l a ) - ~ d m>(0~is) a constant. For any x E R" sati sfy ing 1x1>r, put '1=rx/]xl, where r is an arbitrary real number >O. Then (4.23) gives

If(x)lS Coet(+-121) for any r>O. Making t- + co in the last inequality, we have proved that f ( x )=O if Since g+c.Y,we have jxl > r and f belongs to g+. (4.24) T*csf=f by theorem 1.2. On the other hand, let 4 be the restriction of Then the definition (4.20) off can be written as

(o

to R".

9-4 =J: (4.25) (4.24), (4.25) and the uniqueness property of the Fourier transform (Theorem 1.5) implies that ( T f - 4 ) (x)=O for every x E R" because f - 4 is continuous. Hence the entire function sc f - p vanishes on Rn. So s c f - p = O on Cn by Lemma 1. We have proved that X cis a mapping of sr onto sr and the inverse mapping of T c is given by the conjugate Fourier transform T *on Rn. Now we prove the continuity of ( S C ) - ' . Let k E Nn be a multi-index. Replacing cp(z) in (4.20) by z*p(z), we get (4.26)

(DY) (x) =

J

R"

P ~ Ge') ( 2 , o h ~ .

149

PALEY-WIENER THEOREM

If 2N> Max (lkl, n), then (o E py satisfies

Izip(z)le-r I*m 5 -(1+ Izl')Nlp(z)le-"lm'f

(4.27)

81

5 (1+ 1Z1')-"qlN

($0)

for any z E Cn.(4.26) and (4.27) prove that (4.28)

IIDyfl-$Aq2N (p)

J+

where A = (1 lCls)-Ndm(€)is a positive constant. (4.28) proves that (Fc)-l

:pwfis continuous.

q.e.d.

Now we extend the result in Theorem 4.1 to the space 9 = u gr. It r>O

is necessary to introduce a topology in g which is related to the topology in grin a reasonable way. Although the family (4.13) of seminorms defines a topology on 9,this topology has the disadvantage of not being complete. We shall give a complete locally convex topology on 9 which induces the original topology on 9 7 . This topology is also important for defining a distribution which will be discussed in the next section. The is likewise to be equipped with a topology in a way space p =u Sr r>O

similar to that for 9.So we shall give a general way of introducing a locally convex topology on a vector space starting from the topologies of certain subspaces. A subset B of a complex vector space E is called disced if ax E B whenever xEBand l a l s l .

Proposition 4.2. Let E be a complex vector space and let (E&N be a family of subspaces of E satisfying the following conditions: (a) Enc (b) u En=E, (c) En has a locally convex topology T,, (d) the n topology induced by Tn+lon En coincides with T,. Then we have: 1) There exists a unique strongest locally convex topology T on E which :En+E continuous for all n E N. makes the injection in 2) A linear mappingfof E into a locally convex topological vector space F is continuous (in T) if and only if fn =fo in :E,+F is continuous for every n E N. Proof. 1) Let N, be the system of neighborhoods of 0 in En, and for any element ( Vn)nrN of 17 N,, let U((V,)) be the smallest convex, disced nrN

set containing u Vn. Then the set B= { U((V,)) I

(Vn)nrNE

17n.N N,} is a

ntN

fundamental system of neighborhoods of 0 in a topology on the vector space E with respect to which E is locally convex. Since V, c U((V,)), the injection inis continuous for every n E N. Let T' be a locally convex topology on E in which every in is continuous and let V be a closed.

150

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

convex, disced neighborhood of 0 in T'. Since in is continuous, i;'(V)= V n En belongs to N,, for any n E N. Therefore U((V n En)) = V belongs to B. Hence T is stronger than T'. 2) If f is continuous, then fn =f .in is continuous. Conversely, if f n is continuous for every n E N , then for any convex, disced neighborhood V of 0 in F, f ;l( V )= i;l( f - l ( V ) )= f l (V )n En belongs to N,. Hence U = U((f-'(V) nEn)) belongs to B. Since f - l ( V ) is convex and disced, U = f - l ( Y ) andf-l(V) belong to B. Thusfis continuous. q.e.d. Definition 6. The locally convex topological vector space E with the topology T defined in Proposition 4.1 is called the strict inductive limit of En and we write E=lim En. -----,

Remark. It can be proved that the topology on En induced by T coincides with T,, and that E is complete if all En are complete. (cf. N. Bourbaki [2] Ch. 11, !$I.)

Definition 7. The vector space 9 = 9 ( R n ) is topologized as the strict inductive limit of d m where 9, is the subspace of 9 consisting of those functions in 9 whose supports are contained in B, = {x E RnI 1x1

-

5 m} . The topology of the space S = u S ,= u g?, of all entire funcT>O m-1 tions f on Cn for which z*f(z) is of exponential type for every m E N n is defined as that of the strict inductive limit of g?,. With these definitions, the first result in Theorem 4.1 can be stated as follows. Theorem 4.2. The Fourier-Laplace transform isomorphism of .g=g ( R " ) onto p= u Sr.

P is a topological

7>0

Proof. Since 9 =

6 sm and Z = 6 g?,,and ,, X cinduces a linear

m-1

m-1

onto Z,,,by Theorem 4.1, SC is a linear isomorisomorphism of sm be the restriction of N C to B,,,, regarded phism of 9 onto S . Let 5; as a mapping of 9, onto Zm, and let imandj, be the injections gm-+ 9 and S,-+Srespectively. Then S; is continuous by Theorem 4.1. So j , , , ~ ~ ~ = ~is ccontinuous. o i m Hence is continuous by Proposition 4.2. Similarly ( . F c ) - I is continuous.

$5. Tempered distributions and their Fourier transforms In this section we define distributions on Rn and discuss their Fourier

TEMPERED DISTRIBUTIONS A N D THEIR FOURIER TRANSFORMS

15 1

transforms. Since the mechanism of Fourier transform does not work smoothly on the space 9' of distributions, we introduce the subspace 9' of tempered distributions. Definition 1. Let 9 = 9 ( R n ) be the space of complex-valued C"functions on Rn with compact supports, 9 is equipped with the topology Then, a of the strict inductive limit of g n = 9 ( B n ) introduced in continuous linear form T o n 9 is called a distribution on Rn. By Proposition 4.2, a linear form T on 9 is continuous if and only if the restriction T, of T to s m = d ( B m )is continuous for every m>O. The space of all distributions on Rn is denoted by 9 ' = 9 ' ( R n ) . The canonical bilinear form on 9 x 9' is defined by

w.

=T(p)foryE9and T E ~ ' .

Example 1. The Dirac measure aa at a is defined by

=CpW. It can be easily verified that 6, is a distribution. Example 2. Any Radon measure p on Rn may be viewed as a distribution if we take

For, the inequality

I I 5 I l ~ l l - l(Bmh ~l which holds for p in gm, proves that p is continuous on 9". Example 3. A measurable function f on Rn is called locally summable iff is integrable with respect to the Lebesgue measure dm(x) on every compact set. Then f defines a distribution Tf by =

f ( x ) y(x) dm(x),

because the measure p j defined by dp (x)=f(x)dm(x) is a Radon measure (Example 2). Since the mappingfi+Tf is an injection of the space of locally summable functions into g',f is identified with T,.

Definition 2. If T is a distribution on R" and a ... then the linear form i axl i ax,, p+(- 1)'"' is a continuous linear form on 3, i.e., a distribution on Rn. By definition this distribution is DmT:

(- -)"'

D m= i

(IL)",

152

FOURIER TRANSFORM AND UNITARY REPRESENTA~ONS OF R"



Example 4.

=(- 1)Irn1(Drn(p)(a). Proposition 5.1. Let 9 = 9 ( R n ) be the space of rapidly decreasing functions (cf. Definition 1 in $1). Then we have: 1) The injection i : d + y is continuous. 2) d is dense in 9'. 3) If T is a continuous linear form on .9' (i.e. T E.Si"), then TwToi is an injection of 9'into g'. = g ( B , ) , then Proof. 1) If p belongs to drn PL,N(V)=SUP I(1-k lXI')N(DXp) s ( 1 +m')NIIDkv!lfor any k E N" and N E N. So the injection i, : gm+9 is continuous for every m. Hence i is continuous by Proposition 4.2. 2) Choose a function (0 in g such that $(x) = 1 if 1x1 5 1. For any function f i n 9, put &(x) =f(x)$(rx) for r > 0 and x E R" Thenf; belongs to 9.Now we prove that fi converges to f in 9 when r-+ +0. If P is a polynomial and m E N" is a multi-index, then by Leibnitz' law of differentiation we see that (5.1)

P(x)Dm( f - f i ) ( x ) = P ( x )

,C8(Dm-kf)(x)r'"Dk(l-$ (rx)).

esm

1 By the choice of (0, Dm(l-$(rx)) =0 if 1x1 5 - for every rn E N".Sincef r belongs to 9, P(x) (Dm-Lf)( x ) is bounded on R" and the right hand side of (5.1) converges to 0 uniformly on R" when r++O. Hence we have proved that fr converges to f in 9. 3) By l), Toi is a continuous linear form on 9.Therefore the mapping I: THToi is a linear mapping of 9' into 9'.I is injective by 2).

Definition 3. If T Ey',then the distribution Toi defined in Proposition 5.1 is called a tempered distribution. Since I : T w Toi is an injection of 9'' into g',we identify T E9' with Toi a 9'. Example 5 . The Dirac measure 6, and its derivatives D6, are tempered distributions. Example 6. A positive Radon measure on R" is called slowly increasing if

TEMPERED DISTRIBUTIONS AND T H ~ R FOURIER TRANSFORMS

153

for some N E N . A slowly increasing measure p is a tempered distribution. In fact, we see that

I IS

Po, N ( d J R n ( + l Ixl2>-"4 (4.

Definition 4. The Fourier transform f ' = T T of a tempered distribution T is the tempered distribution defined by (5.2) cp, f'> = for all p E 9. Since the Fourier transform N : C ~ H is Q a topological isomorphism of 9 onto 9 '(Theorem 1.2), the linear form f defined by (5.2) is continuous and is actually a tempered distribution. Similarly the conjugate (or inverse) Fourier transform N * T of T is defined by = for all p E 9.

Example 7. Iff E L', the space of summable functions, then we have f,=T;. Namely, the Fourier transform off regarded as a distribution (Ex. 3) is equal to the Fourier transform of the functionf. In fact, by Fubini's theorem we get

Example 8. Let 6=a0. Then 6 = 1 and I = S . In fact,

=

and =

=@(O)=

s

s

y(x)dm(x)= c v , I >

= @(x)dm(x)=(O(O)=.

Theorem 5.1. The Fourier transform N is a topological automorphism of 9 '' equipped with the weak*-topology (Appendix F). The inverse of fl is the conjugate Fourier transform fl*. Proof. 5 and N*are linear mappings of 9' into 9".Moreover for any T E9" and p E 9, we have = < X * 5 p , T> =

154

FOURIER TRANSFORM AND UNITARY REPRESENTATIONS OF

R"

T * X T > = < T T * p , T> = by theorem 1.2. Hence 9 * T T = T and 9 9 * T = T for every T E9'.So fl is a linear automorphism of 9' and 9-l=S Take *. a finite number of elements pl,..., pmin 9 and a positive number E , and and put , 4) r(a)t(z)r(a)-l=t(r(a)z), 5) h(z, a)-'=h(-r(-a)z, -a), 6 ) h(z, a)h(w, B)=h(z+r(a)w, a+@. Proof. These relations are easily proved by direct calculation of products of matrices. For example, 6) is proved by the relation

1) and 3) are particular cases of 6). 2) follows from 1). As transformations on C, these elements satisfy the relations

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

157

+

r(a)t(z)r(a)-lw=r(a)(r(a)-'w+z ) = w r(a)z=t(r(a)z)w for any w E C. So we have 4). 5 ) is proved by using 2), 3) and 4).

q.e.d.

Remark. Proposition 1.1, 5 ) and 6) show that G=M(2) is actually a subgroup of GL(2, C ) . Proposition 1.2. Let T = {t(z)JzE C) and K = {r(a)laE R} Then 1) T and K are subgroups of G= M(2) and we have T=Ra, KrU(1)zT. 2) G=TK, T n K = { I ) . 3) T is a normal subgroup of G. 4) G is the semidirect product of T and K. G is homeomorphic to each of the product spaces T x K and Rax T. Proof. Proposition 1.2 is a direct consequence of Proposition 1.1.

.

To construct the irreducible unitary representations of G= M(2), we use the method of induced representations. This technique was discovered and effectively used by Frobenius for finite groups. This is a method of constructing a representation of a group from a representation of some subgroup of it. The theory of induced unitary representations of locally compact groups was initiated by G. W. Mackey [3]. We shall give a brief account of induced representations of topological groups in the last part of this section. Here we construct directly the irreducible unitary representations in the form most convenient for later use. The representation to be constructed is induced by an irreducible unitary representation za :t(z)I+ efCXsa) of the subgroup T. This fact will be proved later. Since K G U(1)z T =R/2xZ, (1 -2) dr =da/2n is the normalized Haar measure of the compact group K. In this section, the Hilbart space La(K)=L3([0,2x],da/2x) is denoted by H.

Theorem 1.1. Let a be any element in Ra. Then there exists a unitary representationU" of G=M(2) on H = La(K)defined by (U,"F)(s) =ef(*.'"'F(r(a)-'s), (1.3) where g=f(z)r(a)=f(z)rand F E H. Proof. Since

158 ( 1.4)

EUCLIDEAN MOTION GROUP

(Ug"F,UguF')=

F(r-ls)F'(r-'s)ds

SR S K

=

~ ( s ) ~ m=d(F,s F')

for any F, F' E H , UgUis a unitary operator for every g E G. If gt = firi, fr=t(zt)ET, r t E K ( i = l , 2), then glgs=t(zl +r1z2)r1ra. Thus we have (uVl92"F)(s) =ei(si+riza, s@)F(ra-lrl-ls) -eKX1, IU),+a. ~ l - l ~ u ) F ( r ~ - l ( r ~ - l ~ ) )

UgaUF) (r1-ls) = [uol'(ug,uF) I($> and =ei(zls'u)(

Uglgaa= UglaUgza* It remains to prove that the mapping U" :g n Uguis strongly continuous. It is sufficient to show that for any I; in H and any E>O, there exists a neighborhood N of 1 in G such that

(1.5)

(1.6) IIU,"F-F\I<E for any g E N . If F=O, (1.6) is trivial. We assume that FzO. There exists a continuous function p E C ( K ) satisfying

IIF-v11a(

ne2

Cnxn)

(0) =

1c n e - ~ n ~ x n (.~ ) nci

Hence F= CC,,~,, E H satisfies UraF= F for every r e K if and only if F= cox0 =cO.

160

EUCLIDEAN MOTION GROUP

2) By (1.13), we get

=L/ax etap eosede

277 0 =Jo(ap). The last equality is the definition of the Bessel function Jo. (Bessel's integral formula). Theorem 1.3. Let a, b B R1.Then Ua is equivalent to U bif and only if la1 = lbl. Proof. By Theorem 1.2, it is sufficient to prove that if U a s U e for a, b z O , then a=b. If U az Ub, there exists a unitary operator T on H such that Tuga= /. UgBTfor every g E G. Therefore UTbT1= TUral = T1 for any r E K, hence T1= c by Proposition 1.4. Since T is unitary, IcI = 1. So we have pa(g) = (Ug'l,

1)=(Tuga 1, T1) =(UgbT1,T1)=lcla(Ugbl,1) = q ~ b ( g ) for any g E G.

Thus we have (1.14) Since

Jo(ap)=Jo(bp) for every

oER

.

we have (1.15) Let fa(x)=Jo(ax).Then we have

.

(1.16) fa"(0)=aaJo"(0) Therefore (1.14) implies that aaJo"(0)=baJo"(0),hence, aa=bs by (1.15). So we have proved that if U a sU band aLO, b z O , then a=b. q.e.d.

Theorem 1.4. If la1 >0, the unitary representation U a is irreducible. Proof. We can assume that a>O by Theorem 1.3. It is sufEcient to prove that if a projection operator P on H=L'(K) satisfies

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

161

Puga = UgaP for any g E G , then either P=O or P = 1 (Ch. I. Proposition 2.3). For any integer n, put Xn(e)= eine and Px,=f n . Then we have Sn(8-a) = (Ur(m)aPXn) (8) = (PUr(a)aXn)(8) -P(efn(e-u)) =e-tnufn(e) for any a

.

Hence we have fn(a) =cnetnu

i.e. (1.17)

for almost all a ,

PI,,=cnzn for every n E Z .

(Ch. I. (3.15)). If x E R, then

P(etascoseetne) =(PUt(.)'Xn)(B)=(Llt(=)"PXn)(8) =Cn(Ut(rlaXn)

(0) = Cneiaf

coseeins

.

Hence

Since

by Taylor's theorem, we have

(1.20) =cniacosBetne,so that (1.18)and (1.20)prove that P(iu cos8etn@) P(cos8 ein8)=cn cose efne; (1.21) considering t(xi) instead of t(x), we get similarly P(sin8 erne)=cn sin&P . (1.22) (1.21) and (1.22)prove (1.23) Pxn+l =CnXn+l for every n E Z (1.17) and (1.23) prove that Cn+l=Cn and hence Cn =CO for every n E 2. Since (x,,),,,~ is an orthonormal base of H, we have

162

EUCLIDEAN MOTION GROUP

P=col.

(1.24)

Since P a=P,coa= co and co= O or 1. We have proved that P=O or 1.

q.e.d.

Definition 2. The set of representations P= {U"la>O} is called the principal series of irreducible unitary representations of M(2).

Induced representations There exists a canonical method of constructing a representation T of a group G from a representation r of a subgroup H of G. The representation T is said to be induced by T. In its essence, the induced representation T is the natural representation of G on the space r ( E ) of sections of the homogeneous vector bundle E associated with T. Let V be the representation space of r. Then E is constructed as follows. An equivalence relation R is defined on the product space G x V defining (g, x)R(g', x') if and only if there exists an element h in H such that g'=gh and x'=r(h)-'x. Let E= (G x V ) / R be the quotient space of G x V by R . If H is a closed subgroup of a Lie group G and r is a continuous representation of H , then E is a vector bundle over the homogeneous space G/H associated with the principal bundle G(G/H, H ) . But here we are giving the description of the induced representation in an abstract setting and we make no such additional assumptions. Let ( g , x ) be the equivalence class containing (g, x) E G x V under the relation R and p : (g, x) W g H be the canonical projection of E onto G/H. G acts on E from the left by the rule

mo,

g(g0, x>= x> * A section f of E is, by definition, a mapping f:G/H+E satisfying paf = Ido,x. Let r ( E ) be the set of all sections of E. Then r ( E ) has the structure of a vector space. If fi,& E r ( E ) and fi(gH) = ( g , xi> (i= 1,2), then fl+fa and ufl (a is a scalar) are defined by

(fi+fa)

= ( g , xi

+xa>,

(ah)( g H )= -

G acts on r ( E ) canonically from the left and r ( E ) is a G-module in virtue of the representation T of G on r ( E ) defined by ( T d )(goH)=g *f(g-lgoH). The representation T of G is said to be induced by the representation t of H. The induced representation can be realized in another way. Let W be the space of all mappings F: G-, V satisfying F(gh)=~(h)-lF(g) for any g E G and any h E H . Then a representation S of G on W is defined by

setting

CONSTRUCTION OF IRREDUCIBLE REPRESENTATIONS

163

-

( S l a (go)=F(g-lg0) The representation S is equivalent to the induced representation T. Let F E W and r be the mapping of G into G x V defined by 7 ( g )=( g , F(g)). Then r(gh)= (gh, r(h)-lF(g)) is equivalent to r(g) for any g E G and h E H. Hence we get a mapping f : G/H+E by passing to the quotient:

G - G xI V

4-

Ip

G/H

f

E

-

f k H ) =( g , F O ) The mapping 0 : F H f is a linear isomorphism of W onto r ( E ) . In fact it is clear that 0 is injective. Let f be an element in r ( E ) and put f ( g H ) = ( g , x ) . Then an element F in W is defined by putting F ( g ) = x and it satisfies 0 ( F ) = f . Hence 0 is surjective. The linear isomorphism 0 is an intertwining mapping between T and S, i.e. 0 satisfies O O S , = T , O O forevery

gEG.

In fact, we see that "0 SJFI(goH)= (go9 (&F) (go)) =g(g-lgo, Fk-'go)> =&W) 1(g-'goH) = U l 7 @IF1 (gem. O

O

Detailed studies of the induced representations have been made under the assumption of additional conditions on G , H and r by taking a suitable subspace ro of r ( E ) . For the case where G is a finite group, see, e.g., Curtis-Reiner [11. For the case where G is a compact group, see Weil [l]. For the case where G is a complex Lie group and rois the space of holomorphic sections, see Bott [l] and Kostant [2]. For the case where G is a Lie group, see Bruhat [l]. For the case where G is a semisimple Lie group and rois the space of square-integrable sections, see Okamoto-Ozeki [11, Narashimann-Okamot0 [l], and Hotta [l]. Here we give the definition of unitary induced representation of a locally compact group due to G. W. Mackey [3]. Definition 3. Let G be a locally compact group, H a closed subgroup of G and assume that GIH is o-compact and that there exists a positive Ginvariant Radon measure AL, # O on G/H. Let r be a unitary representation of H on a separable Hilbert space V

164

EUCLIDEAN MOTION GROUP

and let H be the space of all functionsf on G with values in V satisfying the following three conditions: 1) gt+(f(g), v) is measurable for every v E V, 2) f(gh)=r(h)-'f(g) for every g E G and h E H,

where g=gH. Then W is a Hilbert space with the inner product

(fi,fa)= ~ G , H ( x ( g ) . $ d ( g ) ) d P ( 9 ) *

A unitary representation T of G on H is defined by setting (1.25) ( T d )(go) =f(g-'go). The unitary representation defined by the last equality is called the representation induced by r and denoted by T=Ind t. HtQ

Proposition 1.5. The principal series representation U aof the motion group M(2) is equivalent to the representation Ta induced by the irreducible unitary representation

x a : t(z)i+ef(s.a) of the translation subgroup T. Proof: Define the linear mapping A of La(K)=H(Ua)into H = H(Ta) by setting (AF)(t(z)r)=xa(t(-r-'z))F(r) for F E L s ( K ) .

Since (AF)(t(z)r t(zo))=(AF) (t(z+rzo)r) =xa(t( -r-'z-zo))F(r)= Xa(Zo)-'(AF)(t(z)r) A F actually belongs to H. Moreover, since dr is the G-invariant measure on K = G / T , we see that

Thus A is an isometry of La(K)into H. A is surjectivebecause A C ( K ) is dense in H. A is an intertwining operator between U aand T a ,i.e. A satisfies A 0 U g a = T g a o A forany g e G . In fact, if g = t(w)s and s E K, then

CLASSIFICATIONS OF IRREDUCIBLE UNITARY REPRESENTATIONS

165

In the above definition of unitary induced representation, we assume that-there exists a positive G-invariant measure p # O on G/H. However no such a measure p exists in many important cases. In general case, a quasi-invariant measure p plays the same role as the invariant measure in the above discussion. A positive Radon measure p on G / H is called quasi-invariant if the measure gp ( ( g p ) (A) = p(g-lA)) is absolutely continuous with respect to p for any g E G. It can be proved that there always exists a quasi-invariant measure # 0 on G/H where G is a locally compact group and H is a closed subgroup (cf. Bourbaki [3] Ch. VII. $1. no 9). The measure gp has the density (Radon-Nykodimderivative) d(gp)/dp with respect to p. Using the quasi-invariant measure p, we can define the induced representation T o n the space H in Definition 3 by setting

(Tuf)ko)=(dp(g-'b)/~p(80) )'f(g-lgo) .

52. Classification of irreducible unitary representations There exist irreducible unitary representations other than the principa) series representations constructed in $1. Let p :G H K be the projection of G= T K onto K : p(t(z)r)=r. Then p is a homomorphism of G onto K whose kernel is T. Since G I T z K , any irreducible unitary representation x of K defines an irreducible unitary representation x op of G. We know that 12- ( ~ ~ : r ( c u ) ~ e ' " - I n ~ ~ ) is the set of all irreducible unitary representations of K (more precisely l? should be called a complete set of representatives of the set of classes of irreducible unitary representations).

Theorem 2.1. (It0 [I]-Mackey [3]). Any irreducible unitary representation U of the motion group G=M(2) is equivalent to one of the elements in the set

e=(U"la>O) u (XnoplnE21. No two elements in

6 are equivalent to each other.

166

EUCLIDEAN MOTION GROUP

Proof. Since Ua is infinite-dimensional and zn op is one-dimensional, Ua is not equivalent to zn op. Hence the last part of Theorem 2.1 is clear

from Theorem 1.3 and Ch. I, Theorem 4.2. Let U be an irreducible unitary representation of G=M(2) on a Hilbert space H and put Urtr)=V, and

Urta)=W,

.

Then V is a unitary representation of R a g T.Hence by Stone's theorem (Ch. I11 Theorem 3.2), there exists a unique spectral measure E on the family B of Borel sets in R1such that

For any Borel set A E: b and any x E Ra, we have (2.2) V,E(A) =E(A)v, (cf. Ch. 111. Theorem 3.2). Since r(a)t(x)r(a)-l=t(r(a)x), we get

s

e"". W (W,E(y)W,-I) = W, V , W,-'

= Vr(a)s = = /e'(*.

s

e"".r(a,-'ff)&(y)

u)dE(r(a)y).

Hence for any Borel set A E 23 and a E R,we have WaE(A)Wm-'=E(r(a)A) (2.3) by the uniqueness of E. In particular, we see that (2.4) if r ( a ) A = A , then W,E(A)=E(A)W,. By (2.2), (2.4) and the irreducibility of U, we see that (2.5) if r(a)A= A for every a E R, then E(A)=O or 1. In particular an open disc Br= { X E R21 1x1 < r } of radius r>O is invariant under every rotation, and we get (2.6) E(B,)=O or 1 for any rZO . Since u Bn=Ra and E(Ra)= 1, E(B,)= 1 for sufficiently large c by the n

strong continuity of E(Ch. I11 Proposition 3.9.). Put a = sup [r IE(B,) =01 . Then we have O$a< + a. Since u Ba-l,n=Ba, we have n

CLASSIFICATIONSOF IRREDUCIBLE UNITARY REPRESENTATIONS

(2.7)

167

E(B,)=O.

Put Dr= { x e Ral I x l s r } . Then E(Dr)=l if r > a . Since n Da+l,n=D,, we have n

(2.8) E(D,) = 1 by Ch. 111, Proposition 3.9. (2.7) and (2.8) show that the spectral measure E is concentrated on the circle S,= { x e RaI Ixl=a). There exist two possibilities: a=O and a>O. We examine the two cases separately.

Case I, a =0. In this case, we get ”

for every X E R a . Hence the kernel of U contains the subgroup T and U can be regarded as an irreducible unitary representation of K. Therefore U is equivalent to xn o p for some n E Z by Ch. I. Theorem 4.2. Case 11, a>O Since W is a unitary representation of the compact group T E K, W is a direct sum of irreducible unitary representations (Ch. I. Theorem 3.1). In particular there exists an integer n and the representation space H of W contains a unit vector u satisfying (2.9) Wau= einau for any a E R . Let 9 be the mapping r(a)nae*”.Then 9 is a homeomorphism of the group K onto the circle S,. Put F(B)=E($(B)) for B E b(K), where b(K) is the set of all Bore1 sets in K. Then F is a spectral measure on B(K). (2.1) can be re-written as

Let p be the measure on b(K) defined by p(B)= IIF(B)ull’ for B E b(K). Then by Ch. I11 Proposition 3.12, for each function f E La(K,p ) , there exists a linear operator

defined on the space

168

EUCLIDEAN MOTION GROUP

The mapping CJ : f ~ H ( fis) an isometry of La(K,p ) onto a subspace M = M ( u ) of H {Ch. 111. Proposition 3.12). M is isometric to L'(K, p) and is complete, hence is closed. Now we prove that the closed subspace M is invariant under the representation U.By (2.3) and (2.9), we see that ( ~ u ~ ( f )v)u=,r>(n)dA(waF(l)u, v) =

J)(~)Mw +a) ~ . u v) ,

=/ ; e t n q f ( l

-a)d(F(R)u, V) .

Putting

f=W=etm-f(A-a) , we get (2.11) W , H ( f ) u = H ( f " ) u for any a E R and f E La(K,p ) . We recall here that the integer n in (2.11) depends on the choice of u in (2.9). (2.10)

If we put ~ ~ ( r ) = e ~ ( * .then ' " ) ,we may write V,=H(x.). Hence if we put g,(r) = ~.(r)f(r) = e((".ra)f(r then by Ch. 111, Proposition 3.12, 7), we get 1

9

.

V,H(f)u=H(g,)u for any z E C and f E L'(K, p ) (2.12) (2.11) and (2.12) prove that M = { H ( f ) u l f L2(K, ~ p)) is invariant under U. Since U is irreducible, M must be 0 or H . Since M contains u= H(l)u+O, M must be the whole space H. Put e(l)=e6n2;then since emn(;c)= efnaeinCa-a' =&), (2.1 1) shows that uo=H(e)u satisfies (2.13) Wuuo=uo for all a E R . Hence we can take uo instead of u in (2.9). Starting from uo, the above argument shows that the isometry CJ is the desired intertwining operator between U and U".In fact, the measure po(B)=IIF(B)uoll' is invariant under the translation r ~ r o rin the group K by (2.3), (2.13) and satisfies p o ( K ) = 1. Hence dpo(l)=dR/2n. Furthermore, the isometry CJ : f ~ H ( fis) an isometry of L'(K) =La([O,2x1, d2/2n) onto M(u0)=H. Moreover, the function fqo(l) in (2.10) for uo is given by fa0(A) =f(l -a) =(U,c.)"f)(A). Hence (2.11) and (2.12) can be re-written as KO-1

urCu, WI (8) = f ( e ) = (urcU)af) (0) a

[(CJ-l o

Utca)o @)f](0) =g.(8) = =

e'("9

r(s)a)

(utca)af) (0) .

f(e)

d

FOURIER TRANSFORMS OF RAPIDLY DECREASINGFUNCTIONS

So we have proved that O-l.UvaO=Uga forevery g E G , and U is equivalent to Ua.

169

q.e.d.

$3. Fourier transforms of rapidly decreasing functions In the following three sections, we discuss the Fourier transform on the group M(2) which is always denoted by G. In this section, we introduce the space Y(G) of rapidly decreasing functions on G=M(2). It turns out that any rapidly decreasing functionf has an expansion with respect to the principal series representations Ua(a> 0). It is a remarkable fact that the one-dimensional irreducible representations x,, ap of G play no part in the expansion. It is also noteworthy that there exists a close relation between the Fourier transform on G=M(2) and the ordinary Fourier transform on R'. For a function f on G =M(2) we write f ( g ) = f ka )= f kr )

if g = t(z)r(a)= t(z)r. Proposition 3.1. Let g = t(z)r(a)= t(x+yi)r(a). Then dg =dzdr =dxdyda is a left invariant Haar measure on G =M(2). It is also right invariant and G is a unimodular group. Proof. Since t(zo)rot(z)r= t(zo+r0z)rOr and the Lebesgue measure dz on Ra is invariant under the rigid motion zHroz+zo,and since dr is the Haar measure of the group K, we have d(gog)=dg. So dg is a left invariant Haar measure on G =M(2). Similarly we have d(ggo)=dgo and G is unimodular. q.e.d. Definition 1. The measure on G given by dg=dzda =dm(z)dr (2d' is called the normalized Haar measure on G. In the following dg always denotes the normalized Haar measure given by (3.1). Definition 2. The Fourier transformf of a functionf E L 1 ( Q is a function on R*+=(O, + a) with values in the Banach space B(L'(K)) of aU bounded linear operators on L1(K)defined by

170

EUCLIDEAN MOTION GROUP

{(a) =

(3.2)

1

f(g)Ua,-,dg for a> 0 ,

Q

where U ais one of the principal series unitary representations of G =M(2) constructed in $1. Proposition 3.2.

Iff and h are integrable functions on G, then we have

1) IIf(~)lldllflllfor any a>O,

2)

and

(f*W=v,

3) (I*)* (a)=tf(a))*, where f * ( g ) = f j . Proof. 1) For any vector u E H, we have

Ilf(a)ull.~ ff If(s)l II~"0-144?

5 llflll llull * 2) By Fubini's theorem, we have

= h(a)f(a). 3) For any two vectors u, v E H,

Definition 3. A complex-valued C"-function f on G = M ( 2 ) is called rapidly decreasing if for any N E N and m E N 3 we have

(3.3)

where

m,m(f)=

SUP I(l+lZl')"(~"f)(Z,

arR, a c C

all < +

9

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

171

The vector space of all rapidly decreasing functions on G is denoted by

9=.y(G). 9 is a Frdchet space in the topology given by the family of seminorsms { p N , , I N ~N,m E N 3 }

.

Proposition 3.3. 9(G)cLp(G) for any p 2 1. Proof. Since f E 9 ( G ) satisfies

Ifk a>lIpiv.o(f)(1 + I z l Y for any N E N, Proposition 3.3 is a direct consequence of Proposition 3.1. q.e.d. Proposition 3.4.

Iffis a rapidly decreasing function on G,then we have

(3.4)

for any u>O and F E La(K), where

.

(3.5)

f ( z , rs-')e-*(".ra)dm(z)

Proof. If F and F' belong to La(K)and g = t(z)r, then g-l= r l t ( -z ) = t( -r-lz)r-l and n

n

,

.

n

n

Since F' is arbitrary element in La(K),we have proved that (3.4) is valid for almost every s E K . In the above proof of Proposition 3.4, we actually prove the following corollary.

COROLLARY 1 to Proposition 3.4. If YE Ll(G), then (3.4) is vaIid for almost all s E K. Proposition 3.4 shows that i f f € g ( G ) , thenf(a) is an integral operator on La(K)whose kernel k, is given by (3.5). COROLLARY 2 to Proposition 3.4. We denote the ordinary Fourier transform off(z, r ) , regarded as a function of z E Ra, by f ( f , r ) :

172

EUCLIDEAN MOTION GROUP

(3.6) Then the kernel k&; s, r ) is given by (3.7)

kf(a;s, r ) =f(ra7rs-l) .

Definition 4. A bounded linear operator A on a separable Hilbert space H is said to be of truce cZms if for any orthonormal basis ( p n ) n rof ~ H, the series

-

(3.8)

C

(On)

n-0

converges to a finite sum which is independent of the choice of (lon). The sum of (3.8) is called the truce of A and is denoted by TrA. If A is of trace class, the series (3.8) converges absolutely, because the sum is invariant with respect to any change of ordering among the pn's.

LEMMA1. Let H be a separable Hilbert space. If a bounded linear operator A on H satisfies (3.9)

for a fixed orthonormal base then A is of trace class. Moreover, if U and V are two bounded operators on H, then UAV, AVU and VUA are of trace class and have the same trace. Proof. Let urnn=(A~rn,pn), urnn=(Uprn, pn)7 and Vrnn=(Vprn, pn)Then we have

Hence we see that (3.10)

UA Vprn =

~(rnn&kvkl(bL n. k. 1

The Schwarz inequality in 1 9 proves that

-

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

by the assumption (3.9). Hence the series C

&nankVLm

173

converges abso-

m.n.k

lutely and we can change the order of terms without affecting the sum. So we have by the Parseval equality

Let (+,,),,.N be another orthonormal basis of H. Then the linear operator W on N defined by Wp, =$,, is a unitary operator. By (3.12) we have

Hence UAV is of trace class. In particular, putting U = V = 1, we see that A is of trace class. Similarly A VU and VUA are of trace class. (3.12) shows that Tr(UAV)=Tr(AVU)=Tr(VUA). LEMMA 2. If k(0, p) is a C2-functionon T', then the Fourier series m

(3.13)

C

um,nei(m8+n+

m.n---

converges absolutely and uniformly. Proof. Let

of k

174

EUCLIDEAN MOTION GROUP

Then the Fourier coefficient's bm,, of h satisfy bm,n =(Ak,e(m,n)) = (k, Ae(m,n)) = -(ma+n3)am,,.

Since h =Ak E C ( T a )c La(T),Bessel's inequality shows that

-

1 Ibm, s llhllsa< + n12

m. n---

Hence by Lemma 1 in Ch. 11, $8 we have

q.e.d. Proposition 3.5. If k(e, v) is a Ca-functionon K a=K x K s T a, then the integral operator L on La(K)defined by

satisfies the assumption of Lemma 1 for an orthonormal basis (x,,)~.z. Hence L is of trace class and has the trace

Proof. Let Xn(B)=eine.Then ( p J n r zis an orthonormal base of La(K). Lemma 2 proves that the Fourier series (3.13) of k converges absolutely and uniformly. Since

-am,

-n

it follows that

C

I(Lxm, xn)I < + 00, i.e.9

m ,n=--

L satisfies the assumption of Lemma 1 for (&.Z. Hence Lemma 1 proves that L is of trace class. Since the series k(B,O)=

C m . n=--

am,

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

175

converges uniformly on T,we can integrate it term by term and get

=Tr L by (3.14). Theorem 3.1. (Inversion formula) Any function f in p ( G ) may be recovered from its Fourier transform f,by the formula f ( g ) = r m T r (Ugaf(a))adu.

(3.15)

0

In particular, Uguf(a)andf(a) are of trace class. Proof. Since f ( u ) is an integral operator with C"-kernel (Proposition 3.4), it satisfies the assumption of Lemma 1 by Proposition 3.5. Hence Ugaf(u)and f ( a ) are of trace class. Let g=t(z)u, u E K. Then Ugaf(a) is an integral operator with kernel m f ,#(a;s, r ) =e"Z8 aa)f(ru, rs-1 u ) . In fact, we have (u,~~(u)F) (s) =e"'".a a ) ( f ( u ) ~ (u-1s) )

=J

e*('.Ja)f(ru,rs-1u)~(r)dr

by Proposition 3.4 and Corollary 2 to it. Hence by Proposition 3.5, Tr(u.mf(a))=J

X

mf.g(f.2;Y r r)dr

=j-ne".'"'f(ra, u)dr . If r is fixed, then the functionf, :zl-.f(z, r ) is a rapidly decreasing function on Ra. Since

f ( u , r )=

j-

f ( z , r)e-f(z+)dm(z)

Ra

is the Fourier transform of A E 9 ( R a ) on Ra, the inversion formula for

f7 E 9 ( R s )gives

176

EUCLIDEAN MOTION GROUP

=fm

Proposition 3.6. Iff and h belong to 9 ( G ) ,thenf * h and f * (g) belong to 9 ( G ) . Proof. Since(r(z)r)-l=t(-r-lz)r-l,f*(z,a)=f(-r(-a)z, -a). Hence f * E 9 iff E 9.Since f E 9is bounded and h E 9is integrable (Proposition 3.3), f(ggl)h(g1-') is integrable with respect to gl. Hence

(f* h) k ) = J

fkgl)h(gl-l)&l (I

is defined and has a finite value for all g E G. If g=t(z)r and gl =t(zl)rl, then ggl =t(z rzl)rrl Hence for any m E N 8and N E N,we have

+

.

l(1 +Izl')NDc,,r,mf(z+rZI, rrl)h( -r1-lzIy r1-91 JPN.m(f)Ih(-rl-'zl, rl)l * Thus we can differentiatef * h under the integral sign and have

(3.15)

(1 + Izl')"D"(f*

=so

4(g)

(1+ IZl")"(a"f)(ggl)hkl)&

*

Since

(1 + IZl')"l(D"f )(ggl)t s ( 1 + IZI')-'PN+%m(f) we see that (3.15) tends to zero when lzl++ m by Lebesgue's theorem. Hence the left hand side of (3.15) is bounded andf*h belongs to 9 ( G ) . q.e.d. 9

Definition 5. Let Hand H' be two separable Hilbert spaces. The set of bounded linear operators from H into H' is denoted by B(H, H'). Let (pn).,.,v be an orthonormal basis of H. For any element A of B(H, H'). Put

-

(3.16)

IIAII~~= 1IIApnIIf n=O

-

FOURIER TRANSFORMS OF RAPIDLY DECREASING FUNCTIONS

177

The value of (3.16) is independent of the choice of (+). In fact, if ( @ J n r N is an orthonormal basis of H', then, by the Parseval equality, we have

Hence the value of (3.16) is independent of equality shows that

Moreover, the last

(cp,).

(3.18) llAll2= IIA*llz An operator A E B(H, H') is called a Hilbert-Schmidt operator (abr. H-S operator) if IIAlla< 03. Let

+

+

(3.19) Ba(H, H')= { A E B(H, H')I llAlla< 00) . be the set of H-S operators from H into H'. Then Ba(H, H') is a subspace of B(H, H') and llAlla is a norm on Ba(H, H') which is called the Hilbert-Schmidt norm (abr. H-S norm). Moreover if A and B belong to Ba(H, H'), then the inner product (3.20) is defined. With the inner product (3.20), Ba(H, H') is a Hilbert space. A bounded operator A E B(H, H') is an H-S operator if and only if A*A is an operator of trace class on H. If A and B are H-S operators in Ba(H, H'), then B*A is an operator of trace class and we have (A, B) =Tr(B*A) .

(3.21)

Theorem 3.2. (Parseval equality) Iff belongs to 9 ( G ) , thenf(a) is a Hilbert-Schmidt operator on H = La(K)and it satisfies (3.22) Proof. Put h= f *f *. Then h belongs to 9 ( G ) by Proposition 3.6. Since &a) is of trace class (Theorem 3.1 1) and Tr &a) =Tr ((f*>"(a)f(a)) =

Tr(f(a) *f(a))= IIf(a)l12' by Proposition 3.1 and (3.21),f(a) is a HilbertSchmidt operator. Moreover, Theorem 3.1 proves that

:1

h( 1) =

Tr (&(a))a& =J:-Il

f(a)llAzda

.

On the other hand, by the definition of the convolution product, we have

h(l)=j-@fk)E)&=J We have proved (3.22).

Q

Ifk)l@r. q.e.d.

178

EUCLIDEAN MOTION GROUP

The image 9= { f If€ g ( G ) } will be determined in $5. As an application of properties of the Fourier transform on G we calculate the character of U a . As we have seen in Chapter 11, the characters determine the unitary representations of compact groups. The characters play the same role for the non-compact group G =M(2). However, here is a difficulty in defining the character of U",because an infinite-dimensional unitary operator U," does not belong to the trace class. In fact, the series "

(3.23) does not converge. This difficulty is overcome by considering (3.23) as a distribution on G. Definition 6. Let g ( G ) be the space of all complex-valuedC"-functions on G with compact supports. For any 00, put B5= {t(z)rE GI lzlsa} and

a u = g ( B u ) = { f ~ s ( G ) l f ( z ,a)=O if lzl>a} . a(&) is a Frkchet space with respect to the family of seminorms:

(3.24) IPm(f)= IP"f Il-lm w Then g ( G ) = 6 s ( B n ) is topologized as the strict inductive limit of n=1

9 ( B n )(cf. Ch. 111. Definition 6). A continuous linear form on the topological vector space a(G) is called a distribution on G. The space of all distributions on G is denoted by m G ) . Example 1. Iff is a continuous function on Raand p is a Radon measure on K E T,then the linear form f@,u on =(G) defined by

P H (lo, f @ p )

f (z)lo(z, a)dm(z)dl*(a)

=

(lo

Es

( G ))

S R J K

is a distribution on G. Now we consider (3.23) as a distribution x5 on G. This means that the character xu is the linear form on s ( G ) defined by

C J f(g)(Ugaxn, Xn)dg= C

Xu :fH

s---~l

where

Q

n--*

(U/axn,

Xn)=TrU/"s

PJANCHEREL THEOREM

179

Theorem 3.3. (Vilenkin) For any fixed 0-0, the linear form xu :fe Tr Ufais a distribution on G. In fact, xa is equal to JO(alzl)@(a) (cf. Example l), where JO is the Bessel function of order 0 and 6 is the Dirac measure at 0 on T z K. Proof. By Proposition 3.4 and 3.5, Ufais oftrace class and 1

Tr UfU=&J

Par

kh(a;8, e)de 0

q.e.d. Remark. From Theorem 3.3, we obtain another proof of Theorem 1.3. In fact, if U u z U b then we have U I u t z and X l o l = X l b l by Theorem 1.2. Hence we have la1 = lbl by Theorem 3.3.

$4. The Plancherel theorem In this section, we discuss the Fourier transform of the square-integrable functions on the group G=M(2). We shall prove an analogue of the Plancherel theorem which asserts that the Fourier transform fl ;f-f on G can be extended uniquely to an isometry of La(G)onto LaB,(R*+,ada) =/ : @ H ,

a&, where H , = Ba(La(K))is the Hilbert space of H-S opera-

tors on La(K)(Theorem 4.2). The analogue of the Plancherel theorem can be interpreted as the decomposition of the regular representation into irreducible representations (Theorem 4.3).

LEMMA1. Let H=La(X,p ) , H'=L'(Y, v ) , and let @ be the mapping of H"=L3(Xx Y, p x V ) into the space Ba(H', H) of H-S operators (cf. $3, Definition 5) which maps k E H" =L2(Xx Y, p x V ) into the integral operator K with the kernel k. Then @ is an isometry of H I onto Ba(H', H) = Ba. Proof. Let (p,,) be an orthonormal basis of H'. Then applying the Parseval equality to k(x, y), regarded as a function of y, we get

180

EUCLIDEAN MOTION GROUP

= CI(Ki2 (x)l'

*

I

Integrating the last equality over X,we get llkli2'=/

X

1

~ ~ ~ & b ' = ~ ~ ~ ~ ~

Ik(x, Y)l'dp(X)dp(Y)= f

Y

(G)

because is an orthonormal base of H'. The last equality proves that 0 is an isometry of H" into Br. Now we prove that Q, is surjective. Let K be an element of B, and define an operator K,, for each n E N by putting

Then K n is an H-S operator and ~~K-Knllzl= IlKv#+O

(4.1)

if n--r

+ . Q)

p>n

The operator Kn is an integral operator with the kernel n

kn (x, Y )=

C

*

(KPP)

P-0

in fact, since any f in H' can be written as

f= C ( f ,$Dk)$Dk ,

we see that n

c n

=

(f,v k ) ( K n p k ) ( x )

k-0

=(Kaf ) (4*

Since (4.2)

Ilkn-knrll,"~~Kn-Knrlln'=

IIKvkll'

if n > m ,

k-m+l

(K,) and (k,) are Cauchy sequences in Br and in H" respectively. Let k =lim k,, be the limit of (k,,) in H" and KO=@(k).Then since @[kn)= n-oD

K,, we have

181

PLANCHEREL THEOREM

limIIK~-KnII,=limIlk-knll,=O.

(4.2)

n--

(1-01

(4.1) and (4.2) prove that K = KO=Q(k).

q.e.d.

Definition 1. Let X and Y be two sets and T ( X ) , S ( Y ) , and ~ ( X Y)Xbe respectively the vector spaces of all complex-valued functions on X,Y, and X x Y. For any two functionf E f l ( X ) and g E .F(Y), Put ( f o g )(x, Y )=fW g ( Y ) * Then f o g E S ( X x Y). Since the mapping cf,g)Hf o g is a bilinear into ~ ( X Y), X there exists a linear mapmapping of x ( X ) x x(Y) ping @ : f l ( X ) @ T ( Y ) 4 f l ( X x Y)such that Q(f@d=fOg. Q is injective. In fact, let Q(h)=O. Then h can be written as

h=

Camnfm@gn

5

m. n

where

and (g,Jn are two linearly independent families in x ( X ) and

X(Y).Since

Cam.nfm(x)gn(Y)=o

for all x E x,Y E Y,

m. n

we have Cam,n fm=O for all n, by the linear independence of (gn).Hence U ~ , ~ = for O all m,n and h=O.

LEMMA 2. Let H=L2(X, p ) , H’=L’(Y, v), and H”=La(Xx Y, p x v ) . Then the mapping Q in Definition 1 can be extended uniquely to an isometry 0‘ of the Hilbert space tensor product H@H’ (cf. Ch. I11 53 Definition 2) onto HI‘. Proof. IfS, k E H and g , h E H’, then by Fubini’s theorem, we have

(fOg, k O h )=JX JYf(x)g(Y)koh(y)dp(x)dv(Y) = (f,k ) ( 8 9 h)

=(f@g, k0.h). Hence, if ((0,) and (+,) are orthonormal bases of H and of H’ respectively, (cpn@$m)n,m is an orthonormal system in H”. Since ( ~ n @ + m ) n . m is an orthonormal basis of H B H ’ , any element h in H@H’ can be developed in a series, (4.3)

h=

1(h, yn @ n. m

+m)pn

@ (In,

and

182

EUCLIDEAN MOTION GROUP

IW=

(4.4)

C~ ( h ,

-

pn@+m)Ia

n. m

The mapping 0,originally defined on the algebraic tensor product of H and H‘ can be extended to a linear mapping Of of H@H’ into H” by setting

~ ’ ( h=)

C(h,

pn 8 +m)vn

o

+m

n, m

for any h in H@H‘ (cf. (4.3)). The we have

z

IIO’(~)II~~= ~ ( hv,n B + m ) P

(4.5)

-

n, m

(4.4) and (4.5) prove that Of is an isometry of H@H‘ into H”. It remains is to prove that 0’ is surjective. It is sufficient to prove that (pn @,,,,q,m,) complete. To show this, it suffices to prove that if k E H” satisfies (k, (on @ +,J =O (4.6) then k=O. If k satisfies (4.6), then

for all n, m,

(k,f @g)=O for all f E H and g E H , because f = 2 anpn and g = C bm+m . Let K be the integral operator mapping H’ into H having kernel k. Then by Lemma 1 and (4.7), we see that llkllaa= llKIIaa=

1IIKG~II~= C IW+~,

vn)~2

n. m

7n

=

1

R.

I(k~n@+m)l’=O

m

q.e.d.

and thus k=O in H”.

By the isometry @’ in Lemma 2, we identify f @ g with f o g . So we write

(f@ g ) (x, v)= f (XI

*

With this notation, the following Corollary has been proved in the course of the proof of Lemma 2. COLLORARY to Lemma 2.

1) Iff, k E H and g, h E H ’ , then we have

(f@g,k@hh)=(f,k)(g,h), (4.7) 14.8) IIf@gIIa= llflla Ilglla 2) If ((on)n and (+m)m are orthonormal bases of H and of H’ respectively, then (cpn@gm),,, is an orthonormal basis of HI‘.

-

PLANCHEREL THEOREM

183

Theorem 4.1. Let G = M(2). Then 9 ( G ) is dense in La(@. Proof. By Lemma 2, La(Ra)@La(K)=La(G); hence, it is sufficient to show that for any f E La(Ra),g E La(K)and E >0, there exists an element h in p ( G ) such that (4.9)

Ilf@g-hIla<E

-

We can assume that f # 0 and g # 0. Since the set of trigonometric polynomials R(T) is dense in La(T)(cf. Ch. I. Theorem 4.3), there exists an element glE Cm(K)such that (4.10)

I I ~ - ~ i l l a < ~ / O l l f l l a ).

We can assume that ~/(211f11a)< Ilglla. Then we have gl #O. Moreover, since 9 ( R a )is dense in La(Ra)(Ch. 111. Theorem 1.4), there exists an element fi E . y ( R a ) such that

-

Ilf-fib <E/(2Ilgilld Let h =fl@gl. Then h is a C”-function on G = Ra x K and belongs to 9 ( G ) . By (4.8), we see that (4.1 1)

IIf@ g-hlli 5 If@ g-f@giIIa + IIf@ gi -fi @ gills = Ilf Ila Ik-gilla + IIf-fiII2 llgilla <E/2+€/2=E.

q.e.d.

Theorem 4.2. (Plancherel Theorem for M(2)). Let Ba = Ba(La(K))be the Hilbert space of all Hilbert-Schmidt operators on La(K). Put Ha=Ba for all a>O and H =

s:p

@Haadz. Then the Fourier transform

fl :fwfon .9’(G‘) (cf. 3.2)) can be extended uniquely to an isometry F of La(G)onto H. Proof. Theorem 3.2 proves that fl is an isometry of .9’(G) into H . Since g ( G ) is dense in La(@ (Theorem 4.1), 9can be extended uniquely to an isometry I; of La(G)into H . It remains to prove that F is surjective. Let L be an element in H . Then L is a function on R+* = (0,+ a) with values in B,. Since the value L(a) of L at a is an H-S operator on La(K), L(a) is an integral operator with kernel ka E La(Kx K) by Lemma 1. Put ka(s,r ) =k(a; s, r). Then we have

= / : 7 n J ~ / c ( u ;s, r)l’hdruh. Hence 0 :L w k ( a ; s, r ) is an isometry of H onto La(R+*x K x K). We identify H with La(R+*x K x K ) by the mapping 0.

184

EUCLIDEAN MOTION GROUP

Let (P :R+* x K+RP be the transformation (u,r)k+ru. Then the mapping gt+gov is an isometry of Ls(Rs)onto L2(R+*x K ) (the transformation to polar coordinates). We identify L’(R’) with L2(R+*x K ) and LS(R1 x K ) with Ls(R+*x K x K ) = H. Since F is an isometry, the image Im F is closed in H.Hence to prove the surjectivity of F,it is sufficient to show that Im 9is dense in If.Moreover since Im fl is contained in Im F, it suBces to prove that for any k E H =L’(R1 x K ) and E >0, there exists an elementf in P(G) such that

Ilk-k/Ila 0. Then u E La(Ra).Since y ( R a ) is dense in La(RS)(Ch. 111. Theorem 1.4), there exists an element v in 9 ( R a )such that

I ~ u-Vila <E . Let y * v = w be the inverse Fourier transform of v on Rs. Then w E 9 ( R a ) (Ch. 111. Theorem 1.2). Hence f=w@X-n belongs to Y ( G ) . This function f = w @ ~ - , satisfies , (4.12) for k=g&,,. In fact, since k/(a;S, r ) = ( X w ) (ra)X-n(rS-l) =v(ra) X-n(r)Xn(s)

(3.7)

9

we have

Ilg@Xn-k/lIx=II(X-nU)@ =IIX-n(U-~)llr

Xn-(X-nv)@ ~nll, IIXnlls=II~-~lll<E.

q.e.d.

Definition 2. The isometry F in Theorem 4.2 is called the FourierPluncherel transform on G =M(2). The Plancherel theorem for M(2) gives the irreducible decomposition of the regular representation in the same way as the original Plancherel Theorem gives the irreducible decomposition of the regular representation of Rn (cf. Ch. 111. Theorem 2.2). First we give a general proposition.

185

PLANCHEREL THEOREM

Proposition 4.1. Let U be a unitary representation of a topological group G on a separable Hilbert space H. Let Ba= B,(H) be the Hilbert space of all Hilbert-Schmidt operators on Hand define a unitary representation V of G on Bs by setting (4.15) V,(A)=U,A for A E B ~and g E G . Then V can be decomposed as the direct sum of countable copies of U. More precisely, let (pn),,*~be an orthonormal basis of H, P, be the projection on Cp,,, and B P = { AE BsIAP,=A). Then we have

-

B I = @ Bsn and

(4.16) (4.17) Proof.

n-0

V I B l n z U forall n e N . V, is a unitary operator on B,, because

-

c

(VAA), v~(B))= (UJpn,

uJ+n)

n-0

=

f (Ah, BPR)

n-0

= ( A , B) for any A, BE Ba. It is obvious that VPlrt= V,,V,, for any gl and g3 in G. Now we prove the strong continuity of V. It is sufficient to prove that for any A E Bl and E >0,there exists a neighborhood N of the identity e in G such that (4.18) IIV,(A)-AIIsO. (4.22) nrZ

ProoJ Let V5 be the unitary representation of G on Ha=B,(L'(K)) defined by Vg5(A)=Ug5A for A E H 5 and g E G . Since (x,,),,.z is an orthonormal basis of La(K), V" is decomposed as in (4.22) by Proposition 4.1. Let F be the Fourier-Plancherel transform on G. Then we have (4.23)

a [ (

Vgaudu) F for every g E G .

Fo R , =

0

0

Since both sides of (4.23) are isometries of L'(G) onto H =

1:'

$H5ada

(Theorem 4.2) and y ( G ) is dense in LYG) (Theorem 4.1), it is sufficientto show that (4.24)

( F oRa)f =

[(/+-a Vgaadu) F ] f 0

for anyf E 9 ( G ) .

Hence (4.24) is proved.

q.e.d.

$5. Determination of 9 ( G ) and &(G)

In this section, we determine the images 9 ( G )= {fIf E .9(G)] of (fIf E .g(G)) of . g ( G ) under the Fourier transform

9 ( G ) and &(G)=

188

EUCLIDEAN MOTION GROUP

fl on G=M(2). We shall see that the analogues of the Theorems of Schwartz (Ch. 111. Theorem 1.2) and of Payley-Wiener(Ch. 111. Theorem 4.1, 4.2) are valid for the non-compact, non-commutative solvable Lie group G=M(2). In $3, we have defined the Fourier transform{ off E L1(G)as a function on R+*=(O, + a). In this section, we regardf as a function on R'. Definition 1. The Fourier transformf of a functionf in L1(G)is defined for any C E R2 as

f(C)=/ f(g)Ucq-l&, C E R a 9

(5.1)

(i

where UC is the principal series representation corresponding to the parameter value c (i.e. (UCt(.)$)(s)=et(x.*c)F(r-ls)). If C=u>O, then.f(c) = f ( u ) coincides with the value given under the original definition (3.2). The extended Fourier transform f(C) satisfies the following functional equation. Proposition 5.1. Let f be an integrable function on G. Thenf satisfies R, o f ( C ) RV-I=f(rC) for any r E K and C E R9 . (5.2) Proof. Since U,C satisfies 0

R, U,' R,-I = UqrC for any g E G and r E K by Theorem 1.2, we get (5.2). 0

0

q.e.d.

Proposition 5.2. Let f E 9 ( G ) and E R*. 1) Then f ( C ) is an integral operator with the kernel k / ( s , r)=k,(C :s, r ) given by r

=f(rc, rs-9 Y

where f ( C , r ) is the ordinary Fourier transform of the function ZH f ( z , r ) on R f . k , is a C"'-function on R' x K x K. 2) The inversion formula (5.4)

fk)= J B f T r ( u q m1M

C )

holds. 3) (5.5.)

kf(uC;s.r ) = k,(C; su, ru) for any r, s, U E Kand CER'.

DETERMINATION OF

P ( G ) AND

&(a

189

Proof. 1 ) is Proposition 3.4. The proof of Proposition 3.4 is valid for any U E Ra. The expression (5.3) shows that k, is a C"-function on R a x K x K. 2) Put =ra, a 2 0, r E K. Then we have Tr( U,cf(C))=Tr(R,UgaRr-' R,f(a)R,-l) =Tr( U g a f ( a ) )

by Proposition 5.1. Hence by Theorem 3.1, we get f ( g )=["Tr(

Ugaf(a))ada= /:-SKTr( Ugraf(ra))adadr

0

-

= S R a T r ( U g m)dm(C)

=(f(c)R,-lF) (su) =

1

k,(C ;su, ru)F(r)dr.

kf(C;su, r)F(ru-l)dr=

K

S K

for any F in H = La(K).Hence we get 3). 4) Putting r = 1 in (5.3) and using the Fourier inversion formula on

R2

(Ch. 111. Theorem 1.2.), we have f(z, s-l) =

1

k,(C ;s, I)ei('.C'dm(C)

.

' R

q.e.d. In the sequel we always use the following notation. For function F on

K,we write F(r)=F(r(a))=F(a). Let Do be the differential operator Do=--I d i da' As an operator on 9'(G), Do should be interpreted by the formula Do= i-l a/aa. Then Do satisfies (DO& Fa)= (FI,DoF2) for any Fl, Fa E C 1 ( K ) . (A) Moreover, put Xp(a)=efpn for any p E 2. Then we have (B)

DOXP "PXP

-

190

EUCLIDEAN MOTION GROUP

Proposition 5.3. Let L be an integral operator on the Hilbert space H = La(K) with a continuous kernel k. Then the following five conditions 1)-5) are mutually equivalent. 1) The kernel k is a C"-function on K x K. 2) For any m and n in N, there exists an integral operator L,,,,, with a Cm-kernelk,,, such that

(Dorn L Don)F=Lm,,F for any F E Cm(K). 3) For any m,nE N, there exists a constant Cm,,2 0 satisfying II(Dom L Don)xpII5 Cm,n for any p E 2. 0

0

0

0

4) For any m,nE N, m

C I((DomoLoDo")~p,x31< +

-

PA=-"

5 ) For any m,nE N, the linear operator DornoLoDon on Cm(K)can be extended to a bounded linear operator L,,, on H = L2(K). Proof. Since the implications 2)==-5)*3) are obvious, it is sufficient to prove that the conditions l), 2), 3) and 4) are mutually equivalent. 1)*2). If k@, a) is a C"-function on K x K a T x T,then for any m,n E N, (5.7)

is well-defined. Let Lm,,,be the integral operator with the kernel k,,,. By differentiation under the integral sign and integration by parts, we easily see that

(Dom L Do")F= L m,nF for any F E C m ( K ) . 2)*3). It is clear that 0

0

II(Dom L Do")xpll= IILm.nxpllS IILm.nll= Cm,n for any p E 2. 3)+4). If L satisfies 3), then for any m,nE N, there exists a constant am,,,2 0 such that 0

0

~~(l+Doa)Dom~L~Don(l+Doa)~p~~Ium,n for any P E 2. (We can take Urn,, = Cm,n Hence we have

+ Cm+a.n + Cm.n+a +

Cm+2.n+a

-1

m

1 I(DomLDonX*,X J l P.P--" m

=

1 I(DomLDo"(l+Doa)xp,(1+Do')xq)I/(1+~')(1+q') 9,P-m

DETERMINATION OF

Q(G) AND &(G)

191

because

f p.q---

= (1

+ 43-'=(

(1 + P T ' (1

+ n2/3y < +

1 p---Ol

+2

P-1 2

09.

4) =-1). If L satisfies 4), then for any m,n E N, we have lim

Ipl--,lqi

SO

--I(DomLDon~-p,

Xq)l

=o

that for some positive b m , n I(Dom~Dony-p, ~31 S b m , n for any p , q E 2.

Since l(~omL~onX--9, X J I =IP"4"l KLX-p, X P ) l = IP"4"l I(k, X P 8 X q ) l 9

the last inequality implies that the Fourier coefficient LCp, q)= (k, xP@xq)of k is rapidly decreasing. Then k may be expanded in a uniformly convergent Fourier series

c ca

k=

(k,XP€3Xxp)XP€3XXo

p.q=--

and the series can be differentiated term by term arbitrarily many times. Hence k is a C"-function on K x K E T x T. q.e.d.

Remark 1. Since C ( K ) ,is dense in La(K),the bounded linear operator

Lm,,, in the condition 5) is uniquely determined by L and is equal to the with integral operator Lm,,,with the kernel (5.7). We identify DOmoLoDon Lm,,,.With this convention, the condition 5) can be written as follows: For any m,nE N,we have (5.8) 1 p o m L Don\( < + ca . Remark 2. By the equalities (A) and (B) in p. 183, the condition 4) in Proposition 5.3, can be written as follows: 0

(5.9)

c-

0

For any m,nE N, IP"4"(LXP, X 3 1
O. Hence 9" is a topological isomorphism of 9 =s ( G ) onto Proof. Let f be a function in Saand k,(. 5

Hence N E R by the isomorphism &me. We have

Hence A z R by the isomorphism ?Hat. Since Xoa= - 1,

cos el2 sin 812 -sin 812 cos 812

n-0

n-0

=

Hence, by the isomorphism eHUe, K G R/4nZ T. Remark. K is a maximal compact subgroup of SL(2,R).

Proposition 1.3. Any element g E SL(2, R ) is uniquely decomposable in the form (1.4) g =U d Z t n ( . If g=

(:

i)

E SL(2, R), then the triple (8, t, E ) in (1.4) is given by the

relations

Proof. Taking the product of matrices we have

e

e

Hence we get a= ella cos- c = -ella sin-, from which we derive a - ic = 2’ 2

207

IWASAWA DECOMPOSITION

e(tlS)+(‘@I2).

Hence la- icl

and we get

a-ic = --=== Jaa

+

and et =a’

CS

+ca.

On the other hand, we also have ab +cd= etE from the matrix calculation above, and this gives us [=

ab + cd

~

aa+ca

*

We have proved that every element g in SL(2, R ) can be written uniquely as in (1.4) where (0, t, E ) is given by (1.5). Proposition 1.4. The mappingf:(u, a, n ) ~ u a is n an analytic diffeomorphism of K x A x N onto SL(2, R ) . In particular, SL(2, R ) is connected. Proof. f is a bijection of K x A x N onto SL(2, R ) by Proposition 1.3. The mapping f is real analytic because the operation of multiplication is a polynomial mapping. The inverse mapping off is also real analytic as is seen by (1.5). (Note that if g belongs to SL(2, R), then (a, c) # 0 and aa+ ca>0.) Since K, A , and N are connected, SL(2, R ) is connected. Proposition 1.5. For an element g in SL(2, R ) and 0 E R,let (1.6)

W e =Ug.eat(p.

e ) k g , el

be the Iwasawa decompositicjn of gue. Then the following holds: 1 ) For g, g‘ in SL(2, R ) we have (1.7)

(gg’).O=g.(g‘.O) mod 477,

(1.8) (1.9)

t(gg’, 0) = t(g, g’-e>+ t(g’, 8) g . ( e + 2 K ) = g . e + 2 ~mod 477,

2) If g =

I;]E SL(2, R), then we have e

(a- ic)cos(1.10)

2

e((~.e)12 =

1-8=0,

t(g, e + 2 ~ ) = t ( g 0). ,

+(- b +id)sin-62

e

e s

I(a-ic)cos-+(-b+id)sin--l 2

e

2

e ,

(1.11)

et(QSe)=I(a-ic)cos-+(-b+id)sin-~a

(1.12)

E(g, e) =e-‘(p.

2

2

1 {(ab+ cd)coso +z(u3

-b2+c2-ds)sin8}.

K

N (because N is normal in AN).

A

By the uniqueness of the Iwasawa decomposition, we have (gg') 8 = 8" = g - 8' =g.(g'. 8) mod 411,

-

t(gg', 8) =t"

+t'= t(g, g'4) +t(g', 8).

To get the second half of l), we observe that uB+lr=-uI, and hence that guo+s.= -gu.9=

-Ug-eat(g,~)nt(g,e)

=U g . s + a P t c g , a+il)nc(g, e+ad,

so that again by the uniqueness of the Iwasawa decomposition, we have g . ( 8 + 2 7 ~ ) = g . 8 + 2 ~mod 411 and t(g, 8 + 2 a ) = t ( g , 8). 2 ) The assertion 2 ) is obtained by a straightforward application of Proposition 1.3. to

3) From 2) we obtain the relation ef(g.@)lPet(& @)is

" +(-b +id)sin-.2

= (a- jc)cos-

0

2

Differentiating the both sides of this equality with respect to 8, we obtain

By taking the imaginery parts of the both sides of this relation equal, we obtain

IWASAWA DECOMPOSITION

209

To construct a realization of SL(2, R), we consider the group SL(2, C) operating on the Riemann sphere C = C u [ 00 ] (the one point compactification of the complex plane C). As is well known, an element g =

[:: "d]

of SL(2, C ) acts on C as a linear fractional transformation (D(g),defined bY az+b +(g) : ZHgZ=( zE C ) . cz+d It is easy to see that, for g,g' E SL(2, C ) and z E C , we have {(gg') z=g(g'z), lz=z.

The map $ is a homomorphism from SL(2, C) to the group A ( C ) of automorphism of C,where, by an automorphism of C, we mean a bijective holomorphic transformation of C whose inverse is also holomorphic. One can prove, that 4 is surjective (cf. H. Cartan, Elementary theory of analytic functions of one or several complex variables, Hermann-Addison-Wesley, 1963, pp. 180-181). Hence, we have SL(2, C ) / { + l ]EA(C). If we let C+ denote the upper half-plane, { z=x + iy E C Iy >01, then, for g E SL(2, C ) , we have g c + = c + s g E SL(2, R ) and sL(2, R)/ { 11 EA(C+) (ibid. pp. 182-183). Let D = { z E CI IzI < l} be the unit disc. The Cayley transformation z-i c :Z H C Z = z+i transforms C+ onto D. [Observe that Z E C + o l z - i l < l z + i l . ] We have an isomorphism from A(C+)onto A ( D ) given by gncgc-'. Let C be the 2 x 2 unitary matrix given by c=-[1 1 E U(2). a1 We now have the following proposition.

*

-3

Proposition 1.6. 1) Let G=C-SL(2,R).C-l. Then G =

dl

If C[C a b C-l=["

B

"1, @

then

I["

"]I B @

la12-1p12=1).

210

SL(2,R)

UNITARY REPRESENTATION OF

a=

1

+

{(a+d) i(b-c)}

1

p = [(a- d )- i(b +c)] 2) We have G=SU(l, l), where SU(1, I ) = { g E M 2 ( C )I g * e g = e l , and

[:

det g= 1) and el=

-3

If H is a Hermitian form on C’, given by H(z)=zli-l-zzlzs,

.=“‘I

za

E C’,

then we can write G=SU(l, 1)= { g E M 2 ( C )I H(gz)=H(z) for all Z E C’,det g=l}. 3) By the isomorphism h :gnCgC-’ from SY2, R ) to G, we have

Proof. 1) Let g=@

[:ii, -it].

:] A

ne=[:

E

i]

E M2(R).Then we have

=-[ 1 (a+ d ) + i(b -

c)

2 (a-d)+i(b+c)

Hence CgC-1 is of the form lolalz-

i],

3

E SL(2, C), we have

+

(a- d ) - i(b c)

(a+d)-i(b-c)

1 ’

where a, B E C. We have det g =

1 ~ 1 ’ = 1. The assertion 1) follows.

2) For g =

1-iy

21 1

IWASAWA DECOMPOSITION

1;

p

-l 6 6 1 = L

-/I a1

Using this, we can easily deduce (g*elg=el and det g = 1) o (a=& ,8=p and

Ia12 -I,811=1).

Hence we have the first assertion in 2). The condition g*elg = el is nothing but the matrix form of the condition H ( g z ) = H ( z ) for all Z E Ca, so the second assertion in 2) follows. The assertion 3) follows directly from the truth of assertion 1). q.e.d. Proposition 1.7. Identifying corresponding object by the isomorphism h : SL(2, R)+SU(l, l), we write h(ue)=ue, h(a,)=ac, h(ne)=ne and

.

h ( g 4 = u, eat(,, 8) nt(,.e).

If g = [ "

1'

B a

E SU(1, l), then we have

(1.13) (1.14)

etQ*

= lac

+

a = Id

+pCla, where

= ete.

Proof. The proposition can be obtained from Proposition 1.5,2) via the conversion formula of Proposition 1.6, 1). It can also be proved by observing that from Proposition 1.6, 1) we get a - ic = a ,9; so Proposition 1.3 now reads

+

(1.15)

Now we calculate and obtain (in SU(1, l), using Proposition 1.6, 3))

Then we apply (1.15) to a'=aereIa, $=,!?e-telato obtain (1.13) and (1.14).

Definition 3. The 3-dimensional Lorentz group G(2)= O(2, 1) is the subgroup of GL(3, R) defined by G(2)= ( g e GL(3, R)l'gBg=B} where B=

(- 0 la ). The connected component G+(2)of G(2) containing

identity 1 is called the 3-dimensional proper Lorentz group.

212

UNITARY REPRESENTATION OF

Proposition 1.8.

SL(2,R)

Let

and eo=X(l,O,O), el =X(0,l ,O), e, =X(O,O,1). For any g E SL(2, R ) let s(g) be the matrix of the linear transformation x w g 2 g in W with respect to the basis (e,). Then s is a continuous homomorphism of SL(2, R) onto G+(2) with the kernel Z = { -+ 1). Proof. The vector space W is the subspace of M,(R) consisting of all symmetric matrices. Therefore, s(g)x=gxtg belongs to W for any x E W. Hence x-gx’g is a linear transformation on W. Since - det X(xo,xl, x,) = -xoa + xlS+xsa is invariant under s(g), s is a continuous homomorphism of SL(2, R) into G(2). Since SL(2, R) is connected, the image s(SL(2, R)) is connected, and contained in G+(2). To find ker s, we let g =

(z ”d>

E ker s.

Then the equations eo =s(g)eo=gcg =

(.I +” ac+ bd

c2+ d a

and

show that a’+bz=l, a’-.bS=1 ca+dS=l, cS-d’= - 1, from which we derive b=c=O, a = k 1, d= k 1. Since det g=1, we conclude g = k 1. Since s(- 1)= 1, ker s is actually equal to { k l}. It is easily seen that the kernel ker s’ of the differential representation s’ (Ch. 11. 55) is equal to the Lie algebra of ker s= { k 1). Hence ker s‘= {0} and s’ is a linear injection of Bl(2, R) into the Lie algebra g(2) of G+(2). Since dim g(2) = 3 =dimBK(2, R),s‘ is a linear isomorphism of GK(2, R) onto g(2). Since expg(2) is a neighborhood of 1 in G+(2) and the connected group G+(2) is generated by every neighborhood of the identity, every element h in G+(2) can be written in the form h =exp tl Yl ......exp tr Yk where YrE g(2) and tr E R. Since s’ is surjective, there exist k elements Xl, ..., Xk in Gl(2, R) such that s’(Xt)= Yt (1 S i S k ) . Put g=exp tlXl ... exptXXk.Then g belongs to SL(2, R ) and s(g)=h, because s(exp t X ) = exp rs’(X) for every XEBI(~,R), by Ch. 11, Proposition 5.9. We have proved that s is a continuous mapping of SL(2, R) onto G+(2),with kernel q.e.d. equal to Z = { k l}.

PRINCIPAL CONTINUOUS SERIES

213

Remark. Let g,,(O 5 i, j g 2) be the (i, j)-element of g E G(2). Then the relation lgBg=B yields goo'=l +g,O2+g~o2~1 and det g= & 1. Hence either gooZ1 or good - 1. It can be proved easily that G+(2) consists of all elements g in G(2) satisfying (1.16)

det g= 1 and goal 1.

$2. Irreducible unitary representations

I. PRMCIPALCONTINUOUS SERIES Let G= KAN be the Iwasawa decomposition of G = S U ( l , 1) (Proposition 1.3). We can identify K with the torus T = R/4 7~ Z by ue= btela e-,@:] we (mod 4 ~ ) .As we have seen in Proposition 1.5, G x K-tK, defined ~, (gg').e =g.(g'-O) and 1.8 = 0, hence, by identifying by (g, U ~ ) H U ~ . satisfies K with T, we can say that G acts on K from the left. On the other hand, we can map K onto U = {C E C I = 1) by (I, :ubw e(@.The map (I, is a homomorphism from K onto U with ker (I,= { t l } = {uo,ua.}, or we can say that (I, is a homomorphism from T onto U with ker Q = (0,27~(mod 47~)).If we deiine the action of G on U so that the following diagram: GxT+T

. 1 1 G X U-+U

is commutative, then we have

In fact, if C=efo, then using Proposition 1.5, 2) we have

We define a function t ; G x U+R by t(g, t;) = t(g, 81, where I: =(I,(us). This function is well-defined, because t(g, e +2n) = t(g, 19)(Proposition 1.5, 1)). By Proposition 1.7, we have e"g*C)= IbC+al'. Let the function u : G x U-+U be defined by

2 14

UNITARY REPRESENTATION OF

and for s E C and j = O ,

SL(2,R)

1 -, set 2 vf. 8 ( g ,

c) =u(g, @Ve-st(g*C).

Let p be the normalized Haar measure of the compact group U,i.e., 1 dp(C)=-dO 2z

We usually denote the integral

(C=ete).

1

f(C)

dp(C) by

1

f(C) dC.

U

U

Then we have the following proposition.

Proof. By Proposition 1.7, we have

Taking the complex conjugate of this relation, we get

from which we derive u(g, O, there exists an M > 0 such that

;1

%C,

s)l4

M for any c E U and s E [t -a, t +a].

Since IF&, t, h)l6 Mahaif Ihl $ a and lim F&, t, h)=O L-0

for any (C, t),

we can apply Lebesgue's theorem and get

s

limllFn(C,f, h)llz=lim h-0

h*O

27

IF&, t, h)lzdC=O.

Hence the function t-?FPLis differentiable with respect to tk and the derivative is equal to the function &4?F(C, t)/af,. In particular (2.19) is proved. Similarly we can prove aa?Ft/atcat,exists and is equal to c-aa?F(c, t)/at,at, and so on. We see that V t has derivatives of any order and is an 8valued C"-function on U. q.e.d.

where

3) If # E 6-belongs to a closed subspace 8' of 8 which is invariant under V.',then dVx'.*# belongs to 8' for all X E g. 4) We can extend the definition of ~ V ~ . ' ( g) X to E X E gc formally by

PRINCIPAL CONTINUOUS SERIES

dVd.' =dV&'+idVd.', Then we have

223

where Z = A + i B e go, A, B E 8.

dJ'Z'. ' f p =(S + p +j)fp+l, dVz'*'f,= ( s - p -j)fp-iy for Z = X l + i Y and z = X l - i Y . Proof. 1) Let ar=exp tXl, then we have

(2.20)

(Vex,tg,l*J#) (C) =e-Jr(5~-'~c)u(at-l, C)2j#(at-1C)

and Ch-

t

ShT

at =exp tXl =[ s h i

chi]'

We want to differentiate (2.20) with respect to t and set t=O. We first observe that the product of the first two factors of the right member cf (2.20) is equal to &J+j ) r ( 5t - 1,C)

(-

C sh- t

+ch-)Xj. t

2

2

(cf. (1.14) and the definition of u). We have

'

To deal with the third factor of the right

member of (2.20), we first calculate

If we set at-'. C = etB(')and write #(ar-l-C)=f(O(t)), then we have d

1

x#(ar-l-~)

We note, thatf'(e(0))

= (D#) (C).

t-0

=f(e(o))e'(o).

We obtain e'(0) from

d ( ~ ~ ' c ) l r - O = e gie'(O)=i@'(O), @(o) which, combined with the result

224

UNITARY REPRESENTATION OF

SL(2,R)

above, gives us 8 ' ( 0 ) =2i3 . Putting these together, we have

(~v,,,~.B)(c)=o.+s~+(c)+2i(-~)+(c)-~~-c-') c+rl

(09)(0,

giving the first formula in 1). If we replace the at above by

and make the corresponding changes all the way through, then we arrive at the second formula of 1). 2) To obtain the results in 2), we apply 1) to =fp and use

Cfm

+

=fp-l(C),

m P t r )

c-YP(c) =fP+l(C), =

- @fP(C).

3) If # E 8- n @', then t-l(Vexp t X j s * # - #) belongs to 8'. Therefore, the strong limit lim t-l( Vex,t x j * - +)=dVxjS8# t.0

+

belongs to 8'. 4) Obvious. LEMMA 1. Every element g in G = S U ( l , 1) or Go=SL(2, R) can be written as follows: g= ucatu+. Proof. Since G is isomorphic to Go, it suffices to prove the lemma for the group Go. Let g be an element in Go, and P be the set of all positive definite real symmetric matrices of order 2. Then Igg belongs to P.So there exists an element p in P such that 'gg=pa. Let u=gp-'. Then we have $uu=p-ltggp-'= 1, i.e., u is an orthogonal matrix. Since detg= 1, we have detp = detu= 1, i.e., p, u E Go= SL(2, R). Since p E P n Go, there exists a v E SO(2) =K such that p = v-latv for some t. Since the elements uv-l and v in K can be written as uv-l = u+ and v = u+, we have g= uv-latv= u+atu+. q.e.d. Remark. Concerning a direct proof for the group G and the uniqueness of the expression for g, see Proposition 5.2.

Theorem 2.1 1 1 1 1) If Re s =- and (j,s) # (- -), then Vjl is an irreducible unitary repre2 27 2 sentation of G.

225

PRINCIPAL CONTINUOUS SERIES

2) Let $3' (resp. 8-) be the closed subspace of 8 spanned by { fp I p r o ) (resp. { fp I p0 and let t =d/2>0. Let

B'= ICE CJdist(B,C)Sc}. Then B' is a compact subset of D . For any z E B, D.(z) c B'. Let

PRINCIPAL DISCRETE SERIES

24 1

Then

3) The third assertion is clear from the second. m-om be a sequence in $n converging to f in g n . Then by 2), [fm} ,,, is convergent to some function #(on D) uniformly on every compact subset B in D. Since # is a uniform limit of a sequence of holomorphic functions, q5 is holomorphic on D. We have 4) Let [fm}

# ( I ) =lim fm(C), m-lim Ilfm-fll=O. m-m

Some subsequence {fm,} of Ifm} converges to f almost everywhere. (This fact is contained in the usual proof of the completeness of Ls(D,mn)= gn.) So f and # are equal almost everywhere, and +=fin gn. Hence # E -pn and f=# E Qn. Therefore, Qn is closed in 9%.

Theorem 3.1. LetnE*Z,n>l.Then 1) Q n + 1 EQn 2) Every closed subspace 4 ' 2 {0} of Qn which is invariant under T", contains 1. 3) Let U;= T," IQn be the restriction of Tonto 8.. Then Unis an irreducible unitary representation of G. ProoJ 1) We have seen that 11 1II = 1< Q) ,n 2 1 (Proposition 3.3, 1)). (Remark: If n < l , then 11111'= a.) Hence 1 E $n and Qn # {O}. 2) LetfE 8.. We can write

-

242

UNITARY REPRESENTATION OF

SL(2,R)

in D. We have

Hence we have (3.9)

+-/re'ne(

If f belongs to

then #=-

uZef)(c)cio =a0 =f(o). eine(U&f)d8 also belongs to 4'. In

's"

2K

0

tnek

8k+l-@k , and 2n k since by the invariance of 8' we have U; f E 8' for each k, each such ek sum is also in 8'. Since 8' is a closed subspace of 8, we conclude that the limit of the sum is also in 8'. But $(C;)=f(O)for all f E D, as was shown in (3.9), hence q5 is a constant and # =f(O) E 8'. If 8' # {O], then there is an f E Q', f#O, and there is some point co E D such that f(Co) it 0. Since G acts transitively on D (Proposition 3.1), there is a g E G such that g-l.O=c0. Hence we have fact, # is the limit of the Riemann sumz e

(Uiekf)

( U , " f )(0)=a-znf(g-1.0)=a-2nf(To) #O, where g-l=

CBa :I*

By the invariance of 8' we have U ; f E Q ' ; by what was said above, ( U ; f ) (0) E 8'; and since (U;f)(0) # 0, we conclude that 1 E 8'. 3) Let 8'2 {0} be a closed invariant subspace of 8%.If @'#$jn7 then the orthogonal complement 8" of 8' is not {O} and is a closed invariant subspace of 8".Hence, by 2), 1 E 8' n @'l= {0], a contradiction. q.e.d. 1 Let n E 32, n2 1 and let Qn (resp. 8-n)be the subspace of g,,of holomorphic (resp. antiholomorphic)functions in 9" Let. u be the map from p,, onto itself given by u :~ ITheJmap u is an isometric antilinear Define the operator T;" on 9% by mapping and we have o(Qn)=&. Tpn=u T ; u and the operator U;" on 8 - n by U - ; = u U;(I=T-;18-,,. 1

Theorem 3.2. Let n E -2, n 2 1. Then .Q-" is a closed subspace of sn, 2 which is invariant by T-". U-"is an irreducible unitary representation of G on Q-,,.

PRINCIPAL DISCRETE SERIES

243

Proof. Via the isometry u : Qn-&,, the first assertion follows from Proposition 3.4, 4), and the second assertion follows from Theorem 3.1,

3). 1 2, Definition.1. The set of irreducible unitary representations { UnI n E 7 In1 2 1) is called the discrete series of G.

Then {O; I p E N } (resp. {@ 1 p E N})is a complete orthonormal system n in 8 n (resp. 8 - n ) >

k

2)

i.

(ue)O;

U&@;=X-n-p

( n 2 1, p

CJi:e = X n + p

€

N)

(~g)?

Proof. Let n 2 1. We have

Letting r s = t this becomes (2n- 1) ,&,,q/l(l

-t)2n-s fPdi

0

=6,,,(2n-

1) B(2n- 1, p + 1)

Hence {O; I p E N } is an orthonormal system. The system is complete. In fact, if f ( C )

amCm E Qn is orthogonal to Opn for all p E N , then

= m=O

am=O for

all m E N , and hence f=O. Replacing C P by Q', we can use the same argument to show that p E N } is a complete orthonormal system in &n. 2) We have (Uugn O p n )(c) =e-"e Opn(e-(@C) = e-t(n+p)@ Opn(C) =X-n-p(ue)

(U,,g-n@

@pn(C)

{el

9

(c) =U(e+@ OPn(e-*@C)) =er(n+p)@ (c) =Xn+p(ue)

@(C)

-

1 1) If n, m E -2 and In[,Imlr 1 and n#m, then Unand 2 Urnare not equivalent. Theorem 3.3.

244

UNITARY REPRESENTATION OF SL(2,R)

2) Un is not equivalent to V'.'. 3) YO.' is not equivalent to V1IrJ. Proof.

1

1) Let I E 72, 12 1. Then the sets of weight functions on K

associated to the representations U', U-'of G are respectively A'= { z - i , A-'= { X I ,

x-i-1,

X-1-a,

~ i + i X, I + S ,

...I , ...I .

1 if n, m a - 2 and In!, lml2 1, then U n z U" implies An=A"; but the 2 description of A', A-' above shows that if An =A", then n =m. 2) From Proposition 2.2,4), we know that the set Aj*' of weight functions on K associated to the representation Vf.' of G is

A'*'= { X p + j I p a 21 . Since Aj-*# A', A-l, we conclude that Un is not equivalent to Vj.*. 3) Finally, Ao.*= { x P I ~ 2 1 E is not equal to A 1 f a - c ={ X p + l l l I P E 21, so Yo*'is not equivalent to V1la.t. q.e.d. The representation Tn can be regarded as an induced representation. We I

shall explain this fact. For n E -2,n2 1, 2 pn (ue)=efne gives an irreducible unitary representation of K. By the isomorphism of

,,7-

K onto U ( l ) = { a ~ Clal=l}, l defined by us= Eel8 ef81a, we identify K with U(1). Then pn is the unitary representation of U(1) defined by pn(u)=aXn . pn can be extended to a holomorphic (rational) representation the complexification U(l)c= C* = { z E C I z#O] by putting pnC(z) =ZZ"

pnC

of

.

Let V" be the unitary representation of G induced by the representation pn of K. The representation space g,, of ' Vn is the set of all functions f satisfying the following three conditions. 1) f : G+C is measurable. 2) f ( g k ) = p 4 k ) f ( g ) , for all g E G, k E K. 2n- 1 3 ) 1 1 f 1 1 2 = y / G , KIf(g>l' dR@)< Q),where dR(C)=(l -Kll)-adm(O is the

245

PRINCIPAL DISCRETE SERIES

invariant measure on G / K = D . Then g',, is a Hilbert space with norm given by 3). The representation Vn is given by (VL7o"f 1 ( g ) =f(go-lg),

fE P ' n

Propsition 3.6. The representations V n and Tn are equivalent: Vn E Tn. Proof. 1) Let A :pn-+g',, be the operator defined by (SO)

(A$) (g)=pnc(J(g, O))-l)#

Note that since J(g, C)= jC+ a, for g =

($ E -Yn)*

[- i], we have J ( g , O ) = n E

C*. We want to show that A# =f thus defined is indeed in PI,. Of the three conditions for elements of 1) is clearly satisfied f o r 5 To check 2) we calculate as follows:

f k k ) = p n c ( J ( g k , O))-l $(gkO)

(by Proposition 3.2) = PnC((J(g, WJ(kO))-') #(go) = P*(W PnC(Jk,0))-' #(go) = P.(k)f(t?)

and find that 2) is satisfied. For 3) we calculate again: 2n-1

l l f l l % = ~ [

Ipnc(Jk, fflK

o>-l)l~ I#W)I' dA(g) (gO=E=C E D )

=lI$lP< *. So condition 3) is verified. Furthermore, the above shows that A : g,-. 9'is . an isometry. We shall now show that A g , , = g t n . Letfs g',, and let

fi(d=pnc(J(g, 0 ) ) f k ) . Thenf l is constant on each coset of G modulo K. In fact, f l k k ) = pnC(J(gk, O)lf(gk)= pnc(J(g, kO))pnC(J(k,O))pn(klf(g). Since M) =0, and since J(ue,0) = e-tbla gives pn( J(k, 0)) = p,,(k)-l, we get fi(gk)=fi(g). Hence fi defines a function # on G / K z D by #(go)=

246

UNITARY REPRESENTATION OF s u 2 , R )

Propsition 3.7. The representation U n(I n I 2 1) defines a representation of the Lorentz group G+(2) if and only if n E Z . Proof. U"defines a representation of G+(2) if and only if Un(-1) = 1. We have - 1 = u2* and for n 2 1 Uu2=QPn = e - i ( n + p ) 2 n U&nq

@ n

P

= e-2nnI

@ n

P

= &("+P)Zn p - = elnut P

@:

Hence U n ( - 1) = 1 e n n ~ Z (for In1 2 1).

111. THE LIMITOF DISCRETE SERIES As we saw in 52, the representation V3v3 is the direct sum of two irreducible representations. In this subsection we shall construct these two irreducible representations on the Hardy space H 2 (0). For each holomorphic function f on the unit disc Definition 2. D and real numbers p > 0 and r (0 S r < l ) , put (3.10) Let Hp = H p (0)be the set of all holomorphic functions f: D fying (3.1 1)

MP

-

C satis-

( r , f > IM

for some constant M and for all r E [0, 1) = { r E RIO 5 r < 1 }. Hp is called a Hardy space. We need only H2. H 2 is a Hilbert space as is shown in the following proposition. Proposition 3.8. The following two conditions (a) and (b) for a holomorphic function m

(3.12)

f ( z )=

1anzn n-0

247

LIMIT OF DISCRETE SERIES OD

are mutually equivalent: (a) f = H 2 , (b)

I a, I < + 00. H 2 is a Hilbert

n-0

space with the norm (3.13) n-0

forfas given in (3.12). Let B be the mapping which maps the functionf OD

in (3.12) to the “boundary function”

4 (ele)= 1andhe.Then B is an n-0

isometry from H 2on to the closed subspace $0 of the Hilbert space $ = L2 (U, dp) spanned by {f-p (&) = dPeIp 2 0). Proof. The orthogonality of the family {elnejneNin $ leads to the equality

n-0

Hence M 2(r,f) is a monotone increasing function of r. We have DD

(3.15)

lim M~(r,f12 = rtl

1 Ia, I ’. n-0

Therefore (a) implies (b). We shall prove the converse. Let A be the mapping which sends the function f in (3.12) to the sequence (a,). Then A is a mapping from H 2 into f2 (N). If we define the norm in HZby (3.13), then A is a norm preserving mapping. cu

Conversely let (a,) be a sequence in f2 (N)and put f(z) =

1a j n . n-0

Then, by Schwarz inequality, we have

Thus the power series UJ” converges on D and the convergence is uniform on any disc lzl S p with p < 1. Hencef(z) is holomorphic on D

248

UNITARY REPRESENTATION OF

sL(2,R)

and belongs to H 2 by (3.15). Therefore (b) implies (a). We have proved that (a) is equivalent to (b). Moreover A is an isometric mapping from H 2 onto the Hilbert space l2 (N). So H 2 itself is a Hilbert space with the norm (3.13). Let C be the mapping which maps (a,) in 1’ (N)to

- a, f-. in $

=

“-0

L2 (U)where I.(&B) = e-me. Then C is an isometry from l2 (N) onto the closed subspace &j0-= 0 Cf-.. Hence the mapping B = C 0 A is an isonZN

metry from H 2onto $;. q.e.d. .and @. for each Previously we have defined two Hilbert space 9 1 1 - n 2, ~ n 2 1. But 9, and 8. are defined for each real number n > 2 2. and 4, with a real parameter n > 1 Propositions 3.3 and 3.4 hold for 9 and no alteration of the earlier proof is needed. The following Proposition shows that the Hilbert space HZis the “limit” of the space $r as t tends to 1 2‘ 1y Proposition 3.9. For each real number t > T let $r be the space of all holomorphic functionsf o n D which belong to the Hilbert space Yr= L2 (D, n - l ( 2 t - 1) (1 - IzI 2 ) 2 r - 1 dm(z)). Then $r is a Hilbert space and contains H2. Moreover we have

llfll zr?g llfllr

(3.16)

forf€H2,

llJlll is the norm in H 2 consists of all elements f in fl 8, for t>l/Z which the limit lim l l f l l r exists and is finite. rll/2 Proof. Let PP( z ) = (r(2t + p ) / T (2t) T ( p + l))l?zP forp E Nand t >

where

1 Then { PPI p E N }is a complete orthonormal system in $r (Proposition -

2‘ 3.5.1)). We remark that

(3.17)

r(P

+ 1)r(2t) -

+ p)

P 2t+p-l

-

P-1 2t+p-2

1 . . . -< 2r

Hence a holomorphic function f (z) = c a n t ” in $r satisfies n-0

1.

249

LIMIT OF DISCRETJ2 SERIES 0)

1

I

luR12=

~lfllz

for all t > 1/2.

R-0

Therefore we have

H Zc at

for all t

> 1/2.

Moreover, iffbelongs to HZ, then the series in the right side of (3.18) is convergent. Hence the series in the left side converges uniformly on the range t 2 1/2. Conversely let f b e a function in n $t for which lim l l f l l r exists and is t>1/2

r11/2

finite. Then the inequality

implies

llfllf 2

(3.19)

-

1

1u,12rzp

= M~( r , n z

for r = ( 2 t ) - l / z .

R-0

llfllt exists and is finite thenfbelongs to a2.We have proved that H 2 consists of all elementsfin r,l/f% for which lim Ilfllt exists and r1112 Hence if lim

r1112

is finite.

q.e.d.

Delinition 3. Let C = e'e be a fixed point on the unit circle U and a be a non negative real number less than 7r/2 (0 5 a < 4 2 ) . Then the angular domain A , (0= {re', I 1 arg (1 I < a} is the domain indicated in Fig. 5. Note that /3 = arg (1 - re'(#-e))=

Fig. 5

250

arg

UNITARY REPRESENTATION OF

SL(2,R)

(F)L

Ocz, where z = rer+'. If a function f(z) has a finite

=

limit c when z approaches to a point c on U along a curve in an angular domain A, (0,then c is called the non-tangential limit off at c. Proposition 3.10. 1 ) Let f be a function in H2and 4 = Bf be the ''boundary function" off(cf. Proposition 3.8.). Then f is represented as the Cauchy integral of 4:

(3.20) 2) f is also represented as the Poisson integral of

for any z

= reie E

4:

D ; and

3) f has a non-tangential limit 4 (&r) at almost every point ( = erron the unit circle U. m

C OD

4 (0= anpfor c = e". Since n-0 n-0 4 belongs to L2(U)(Proposition 3.9), 4 belongs to L1(U).For any fixed Proof. 1) Let f ( z ) =

ad". Then

m

Z E D ,(( - z)-l

= zr"C-'"+"

converges uniformly on U.Hence we have

n-0

The integral in the right side of the above equality is the inner product of 4 and f-,, (e'? = elnrin L2( U ) . Hence the expression on the right side of the last equality is equal to the following:

= n-0

wherefn (0= C-" = e-lnr 2) Similarly the series

u#Jn

=f (z),

LIMIT OF DISCRETE SERIES

25 1

converges uniformly on U for a fixed Z E D . We have

Substract (3.22) from (3.20) we get

for z = refe, C = e" and d( = icdt. 3) Sincefis represented by the Poisson integral (3.21),f(rde) has a nontangential limit 4 ( & I ) for almost every t by a well known theorem of Fatou on the Poisson integral. See Appendix G for the proof of Fatou's theorem. q.e.d. Proposition 3.11. 1) Let j = -T, SEC and f E $ = L2 (U).Put

for g-'

=

6:)

E G = SU(1,I).

Then VJo"is a bounded representaion

of G. In particular, if Re s = 1/2, then Vj.' is a unitary representation of G. 2) Put (3.23)

(m(0= Cf(0

Then T is a unitary operator on (3.24)

8.

f o r f a T satisfies

T OV;' = V-l;. o T

for all gEG.

Hence we have (3.24)'

V-1IZ.r - VlIZ,.

for all SEC.

Proof: 1) The proof of 1) is same as that of Proposition 2.2. 2) Since I 61 = 1, T is a unitary operator on $ = L2 (U).We have the equalities

252

UNITARY REF'RESENTATION OF SL(2,R)

= [(V:/z,' o 2")f ] We have proved (3.24) and (3.24)'.

for all g E G .

(0

Remark. The explicit expression of the operator V;1/2,112 is as follows:

P I W ~ W ~3.~ 12.~ O 1) ILet I f~ H Z ,g-' =

:)

E G = SU(1,l) and put

(3.27) Then U:I2f belongs to H 2 and U1t zis a unitary representation of G. 2) Let B be the mapping which sends f E H 2 to its "boundary function" 4 = Bf (cf. Proposition 3.8). Then we have (3.28)

B o U:I2 = V;1/21/2 o B

for a l l g ~ G .

Put S = T 0 B where T is defined by (3.23). Then we have (3.29)

S

o

UiIz = V:/2.1/z 0 S

for all g E G .

The closed subspace 8; of 8 = L2(V)spanned by { f-,(c) = P ' ( p 2 0} is invariant under V-1/2~1/2. The closed subspace 8- spanned by If-,! p > 0 ) is invariant under V1/z*1/2.We have (3.30)

u1/2

v-1/2.1/2

Proof. 1) Put w = g-l

18;;

y1/2.1/2

.z. Then we have

1 - Iw12 = I/Jz

h ( w )=

1/92

I @-.

+ aI-2 (1 - lzl') + h(z)

and

4 - 4

by Proposition 3.2. Since g E G induces a complex analytic automorphism of the unit disc D, U:/2f is holomorphic on D. Moreover U:/2f belongs to n ,fjt along withf. So we have r>ln

LIMIT OF DISCREIE SERIES

(3.31) lim IIU:/2ffl = lim I 11/2

Ill/2

= lim

111/2

Ilfll:

=

253

n

Ilfl12 < +

03.

When t decreases monotonicaly, then 11 U:12fllIis monotone increasing. Hence lirn 11 U:/2fll,exists and is finite. Therefore U:l2f belongs to H2 by I

I1/2

Proposition 3.9. Moreover we have (3.32)

IIU:/2fII 5 llfll for allfEH2 andgEG.

It is easily verified that (3.33)

U:/2 Uij2 = U:i2 for all g and h in G.

(cf. Proposition 3.3). By (3.32) and (3.33) we have and

llfll = lu:!3 u:/2fll5 IIU:/2fll 5 llfll IIU:/2fll = l l f l l for allf€H2 and gEG.

Hence U:I2 is a unitary operator on H2.The continuity of the mapping U:/2f is proved similarly as in Proposition 3.3. So U1I2is a unitary representation of G. 2) Since the linear fractional transformation

g-

(3.34) induced by g-' EG is a conformal mapping which transforms a circle to a circle and leaves the circle U invariant, the transformation (3.34) maps a diameter of U passing e'e to a circle orthogonal to U. Hence the limit

is a non-tangential limit. So, by Proposition 3.10,3), the non-tangential limit lirnf(g-' rt 1

exists. We have,

re'e) = (Bf)(g-l

do)

254

UNITARY REPRESENTATION OF

SL(2,R)

[(B o U:l2)fl ( 8 6 ) = lim (U:I2f) (refe) r11

= [(V;1/2~1/zo B ) f ]

(89.

(3.27) is proved. Put S = To B. Then S is an isometry from H Z into

8 = Lz (U). By Proposition 3.8, we have (3.35) BH2 = 8; and SH2 = @-. Hence U'I2

V--1/2.1/21@0 -2: V 1 / 2 * 1 / 2 ~ ~ -

by (3.28) and (3.29). Proposition 3.13.

q.e.d. 1) Let u be the complex conjugation on functions

(uf)(z)=fin

Put uH2 = H2 and Q U : / ~Q = U;1/2. Then U-'I2 is a unitary representation of G on F . 2) Let 8' be the closed subspace of 8 = Lz (U) spanned by { f. (89 - .-id? In h 0). Then the mapping B = uBu is an isometry from HZ onto 8'. We have

B o U;1/2 = V:/2,1/2 o B for all g E G

(3.36) (3.37)

u-1'2

v1/2,1/2

and

18'.

Proof. 1) If we introduce an inner product in H2 by (a-ag) = (f,g) for any f and g in H 2 , then ZP is a Hilbert space, and U;'l2 is a unitary operator on F , and U - 1 / 2is a unitary representation of G on H2. 2) First we have the equality (3.38)

for all gEG,

=

O V ; ~ / Z * ~ / ~ UV1/z,1/2

because

(0= (st + a1-1 f(g-1 0 = (/?t+ 6)186 + zI -"(g-'

(u~;1/2.1/2~f)

r ) = ( v y 2 f ) (0.

Multiplying the equality (3.28) by u on both the left and the right, we get (3.36). Since of-,,=I. we have

B (H2) = uBH' Thus B is an isometry from (3.36).

= 08; = @

Cf-. = 8'.

.SO

P onto 8'. Hence (3.37) follows from

255

COMPLEMENTARY SERIES

Remark. H 2 consists of all antiholomorphic functionsf on D for which

M z(r,f) is bounded on the range 0 S r < 1. The operator U;'Iz is expressed as follows :

Theorem 3.3. U1l2 and U - l l 2 are irreducible unitary representations of G on H 2 and F respectively. They are equivalent to two irreducible components of the unique reducible representation V1/2*1/z in the principal continuous series: u1/2E V1/2.1/21$- and lJ-112 E Vl/z.l/z 1 8 ' 9 (3.40) where

6-= Q Cfp, @+ = @ Cf,and& (0=

C-P

= e-rpe.

PSO

P and

and put

257

COMPLEMENTARY SERIES

(4.2)

and by Schwarz's inequality this is S~11#11 lI+lf

0

(1-cosaY-l da

(by Proposition 4.1) = llill 11+11
--.2

We shall now show that ($, +), is positive definite for 1/2=e-""uu-l'

~ u C~ ) 0

,

C)

=$(et

Theorem 4.1. For any u satisfying Q 1). (cf. Theorems 6.4 and 6.5). Remark. V' is not equivalent to Vo*++*u Proposition 4.8.

P(3 < u c 1) defines a representation of G+(2). proof. This is clear from the fact that VU2; = 1.

Proposition 4.9. If O= 1, then the Hermitian form (#, +)1 is positive semi-definite. The completion of 8 with respect to (4, +)1 is isomorphic to @//8(0),where @(O)= {#I(#. #)l=O). Moreover dim&= 1. The representation V1 induced by VOJ on Q1 is the (trivial) identity representation I of G. Proof. By Proposition 4.3, Ro( 1) = 1 and In 1) =0 ( for n # 0. Hence we have

-

($9

+)I=

C

( # , f n ) W = ( # , f ~ )

n--O

from which the first statement follows. Since

(+,fa>,

K-FINITE VECTORS

265

(VgO*'$,V17°*14)1=($,$91, + E Q, g e G, holds (cf. proof of Proposition 4.6), Q(0) is invariant under VgO*l,g E G. Let $o =f& Q(0)E Q/Q(O). We have $9

Vvl$o = (Vv0*'h,h)$o. Hence, in order to prove that V1 is the identity representation of G, it suffices to prove that ( V g ' J ~ l ~ , f 0 )for = l ,allgEG.

By $2, Lemma 1, any g e G can be written in the form g=u,atuo, and since V,,O*lfo=fo ,

we have

(v,O*lfo,fo)=(vat OJvqO.'foo,

vu+-1O J f O ) = ( v a , O * ' f ~ o , f O ) (Proposition 4.5)

$5. &finite vectors The present section is in preparation for the classification of irreducible unitary representations. In this section we shall prove that each irreducible unitary representation U of SL(2, R) contains each irreducible unitary representation xm(mE 2-lZ) of its maximal compact subgroup K at most once. Then the representation space 8 of U is decomposed as the direct sum of one-dimensionalsubspaces am(mE M) :a=@ am,where UIQm= mriY x,,,. The elements of the algebraic sum Q K = C Qm of the subspaces Qm m t I

are called K-finite vectors for U.It will be shown that QK is contained in the space 8.. of C"-vectors and is invariant under the differential representation U' of U.The restriction U,(x> of U ' ( X ) to Q K for X E 11(2, R) defines a representation UK of the Lie algebra 11(2, R) on The classification of irreducible unitary representations U of SL(2, R) can be reduced to the classification of Ug.This process is carried out in the next section.

ax.

266

UNITARY REPRESENTATION OF

SL (2,R)

First we calculate the Haar measure on G=SU(l, 1).

Proposition 5.1. The left-invariant Haar integration on G=SU(l, 1) (or Go= SL(2, R)) is given by

for any continuous functionf with compact support, where g =uoatne is the Iwasawa decomposition of g (cf. Proposition 1.3). Proof. Put dg =e’dedtdt. We shall prove that d(gog)=dg for any goE G. By Lemma 1 in $2 (or the following Proposition 5.2), every element go in G can be written as go=u,a,u,. Hence it is sufficient to prove d(g0g) =dg for go= u, and ar. u,ueatnt = u,+oat nt, we get d(u,g) =etd(p e)dtdc=etdedtdE=dg. By a simple calculation we get

Since

+

atneat-’=nett. Put t’=t(a,, 0 ) and €’=€(a,, 0 ) in the notation of Proposition 1.5. Then we have

a,ueatnc=u,, .eat+zclatnt - ua,.eatlatat-lnetatne = U,,.&#+tne-tt.+t

Hence we have by Proposition 2.1, 3), d(a,g)=et+t’d(a,-B)d(t+t’)d(E+e-tE’) -et+t‘e-t’dedtdE=dg.

q.e.d.

In the following, the Haar measure dg of SU(1, 1) (or of SL(2, R))is fixed by the normalization given by (5.1). Proposition 5.2. Any element g in G=SU(l, 1) (or in Go=SL(2, R)) can be written as (5.2)

g=u,atu#, O s p < 4 n 7 Ost, 05$O, there exists a C"-function f E g ( G ) such that IIU,a-allAx for any x E HK and X E g by Proposition 6.4. Conversely, assume that there exists an isometry A of H onto L satisfying AHg=& and (6.12). Then again by Proposition 6.3, we get

+

A UexpgxX= VexptxAX for any x E Hg, X E g and t E R. The last equality implies that AU,= U,A for any g E G in the same way as in the proof of Proposition 6.4, because Hg is dense in H. Hence U is equivalent to V. q.e.d.

Proposition 6.5 is valid for any connected semisimple Lie group (Note that Theorem 5.2 is valid for any such group). Moreover, the following proposition was proved by Harish-Chandra [3], I.

Proposition. Let U and V be two irreducible unitary representations of a connected semisimple Lie group G and K be a maximal compact subgroup of G. Then U is equivalent to V if and only if Ug is equivalent to VK* Proof. Since we do not use this proposition later, we give an outline of proof. Let H and L be the representation spaces of U and V respectively. The “only if” part of Theorem can be proved similarly as in the proof of Proposition 6.5. Suppose that there exists a linear isomorphism A of HK onto Lg such that AUg(X)v= Vg(X)Av for all v E HK and X E g. Then for each R E g, we have A H K ( I )=LK(I).Since the representations of K induced on HK(R) and Lg(R) are unitarily equivalent each other, there exists an isometry B, of Hg(R) onto Lg(R) such that B , U k X = vkB,x for every X E Hg(R) and k E K. Since H&) (resp. L&)) is orthogonal to H&) (resp. LK(p))if # p , and since Hg (resp. Lg) is dense in H (resp. L), there exists an isometry 3 of H onto L whose restriction to H&) is equal to B, for each E I?. Let S be a linear isomorphism of HK onto itself defined by Sx=B-IAx (XE Hg). Then S and S-I leave Hg(R) and &(A) invariant respectively. Since dimHg(R) c + co , there exists a linear transformation S* of Hg into itself satisfying (S*x,y)=(x,Sy) for all x and y in Hg.Furthermore S*-1 exists and S* and S*-’ leave Hg(R) invariant. Put T= S*S. Then we claim that TUg(X)T-lx= Ug(X)x for all X E g and x E HK (*) First we notice that if X E g and x , y E Hg, then we have ~

(uK(X)X,Y)= -(x,uK(X)y) .

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

287

Similar equality holds for VK.SinceBSVK(X)S-' =A UK(X)A-'B= Vg(X)B, we have for all x , y E HK and X E g (TUg(X)T-'x, y ) = (S*SuK(X)S-'S*-'x, y) =(SUJ?(X)S-1S*-'x, Sy) =(BSUg(X)S-'S*-'x, BSy) = (VK(X)BS*-lx, Ay) = -(BS*-'x, Vg(X)Ay)= - (BS*-'x, A UK(X)Y) = - (S*-'x, B-'A UK(X)Y)= - (S*-'x, SU,(X)y) = -(x, UK(X)Y)= (UK(X>X,Y ) . We have proved (*). Now choose xo E H&o) such that xo# 0 and TXO= cxo for some C E C . Since cIIxolla=(S*Sxo,xo)=IISx~l12>0,c is positive. Let U be the universal enveloping algebra of 9". Then it can be proved that the closure FV of W = UK(lI)xois invariant under U, for any g E G (cf. Prop. 6.9). Hence, by irreducibility, we get P = H . Since dim HK(R) c + co ,we have wn HK(R)=H&) for any R E 2 and W =HK. For each x E HK, there exists an element b E U such that x = U K ( ~ ) X Hence O. we have Tx=TUK(b)xo=U ~ ( b ) T x ~ = by c x (*) and IIAxll'=IISxll'=(Tx, x ) = cllxllp for all x E HK. Hence A is bounded on HK and can be extended to a bounded operator A on H. A satisfies llAxlla=cllxlla for all x E H . Hence c-lIaA= R is an isometry from H onto L. Moreover R satisfies RUK(X)x= VK(X)Rxfor all X E HK and X E g. Now we can prove that U is equivalent to V in a similar way as in the proof of Proposition 6.5. q.e.d.

Definition 4. Let U be a Banach representation of a Lie group G on a Banach space H. Then a vector x in H is called an anaIytic vector for U if the function ,t : gt-+U,x is an analytic function on G. Since U is strongly continuous, a vector x is analytic for U if the function g w c U,x, 'p > is analytic for each 'p in H* (Ch. IV, $5, LEMMA 1). The set of analytic vectors for U is denoted by H,. H , is a linear subspace of H contained in the space H, of C"-vectors for U. Proposition 6.6. The space H , of analytic vectors is invariant under U and U'. Proof. Let go be an element of G and x be an analytic vector. Then since the right translation Roo:g-rggo is analytic, the mapping RgOoR: gwU,,,x is analytic. Hence U,,x belongs to H,. Since the adjoint representation Ad is analytic on G, the mapping gwU,U'(X)x= U'(AdgX) U,x is analytic and U'(X)x belongs to H . for any X E g. q.e.d. Definition 5. For any XE g, we denote the restriction of U ' ( X ) to H . by U-(X). Then U, is a representation of the Lie algebra g on H.. U. can be extended to a representation of the universal enveloping algebra of g. U. is called the analytic diferential representation of U.

288

UNITARY REPRESENTATION OF

SL (2,R)

LEMMA 1. Let D be an elliptic differential operator with analytic coefficients on a real analytic manifold M and g be an analytic function on M. Iff is a C"-solution of the differential equation ( D- a)f= g, (6.13) for a constant a,then f is analytic on M . Moreover iff is a C"-solution of (6.14) (D ='")a 0 for some n E N,then f is analytic on M. Proof. By a classical theorem of S. Bernstein, a C"-solution of (6.13) is analytic. (For the proof of Bernstein's theorem, see F. John [I] p. 144.) The analyticity of a C"-solution of (6.14) is proved by induction on n. If n = 0, then f = 0 is analytic. Assume that n 2 1 and any C"-solution of (6.14) for n - 1 (in place of n) is analytic. Put g = ( D - a)J Then the function g satisfies ( D - a)"-'g = 0. Hence g is analytic by assumption. The original function f satisfies (6.13) and is analytic on A4 by the first q.e.d. half of the lemma.

2. Let G be a Lie group, K be a compact subgroup of G, and LEMMA

U be the universal enveloping algebra of g regarded as the algebra of left invariant differential operators on G. Then there exists an element D of U such that D is an elliptic differential operator on G and (Adk)D=D for every k E K. Proof. Since K i s a compact subgroup of G, there exists an inner product (X, Y) on the Lie algebra g of G which is invariant under AdK. Let X I , ..., X , be an orthonormal basis of g with respect to this inner product and put D=Xi'+ ...+X n 2 . Then (Adk)D= D for every k E K and D is an elliptic differential operator on G.

Proposition 6.7. Let U be an irreducible Banach representation of a Lie group G on a Banach space H, and K be a compact subgroup of G. If there exists an element 20 in (the set of equivalence classes of irreducible unitary representations of K ) satisfying Ocdim H(R0)c+ 00, the space H. of analytic vectors for U is dense in H. Proof. Since H. is invariant under U (Proposition 6.6), the closure I?, is also invariant under U because of the boundedness.of U,.Since U is irreducible, it follows that if H, f {0), then p-=H. Hence it remains to prove that H . # (0}. By Proposition 5.13, we have H, n H(lo)=H(Ao) and H , ~ H ( R ~Let ) . D be a differential operator on G as in Lemma 2.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

289

Since U'(D) commutes with Ukfor any k in K, the subspace H ( h ) is invariant under U'(D). Since dimH(Ro)c+ 00, there exists a basis XI, ..., xn of H(l0) such that for each i there exists at E C and an integer n > O satisfying (U'(D)-at)"xt=O ( l $ i $ n ) (Jordan normal form theorem). For any element q in the dual space H* of H, put A(g) =

'

Then since xrE H(Ao)c H,, we have

for any X in the Lie algebra g of G and

[(D- at)$] ( g )= < Ug( U'(D)- a$%,

q > =O

.

Hence the functionfi is analytic by Lemma 1. So the vector xi is a weakly analytic vector and hence an analytic vector by Ch. IV, $5, Lemma 1. q.e.d. Remark. It can be proved that for any Banach representation U of a Lie group G, H. is dense in H (Nelson's theorem). Cf. Nelson [l], GArding [2] and Warner [l]. Proposition 6.8. (Taylor's formula). If v is an analytic vector for U,then there exists an open neighborhood N of 0 in the Lie algebra g of G such that (6.15)

for any X E N . Proof. If we regard an element X of g as a left-invariant vector field on G, then for any H-valued analytic function f on G, we have

for any m E N. Hence the Taylor expansion of the function ti+f(gexp t X ) around t = O is given by

Sincef is analytic at e (the identity element of G), there exists a star-shaped of g and open neighborhood N of 0 in g with respect to a basis (Xr)lstrn

290

UNITARY REPRESENTATION OF

SL (2,R)

a power series P(xl,...,xn) convergent on N such that n

f(exp

(1XZJ)

...

=P(x~, ,xn)

I=1

for each X = c xtX, in N. Then we have f(exp tx)=P(txl,..., txn)=

- 1 1 -amt m!

m-0

for any t E [0, I]. Since dm

am=[&(exP

tx)lt-0=(Xmf) (e),

we get (6.16)

f(exp X ) =

1- Z1 ( X (e) ~J for)any x

E N.

me0

Now let v be an analytic vector for U.Then applying the formula (6.16) to the function 5 :gHUgv, we get (6.15), because

q.e.d. Proposition 6.9. Let G be a connected Lie group, U, the universal enveloping algebra of the complexification Q“ of the Lie algebra g of G, and let U be a Banach representation of G on a Banach space H. 1) If x is an analytic vector for U,then the closure U,(U)x of the orbit of x is invariant under U. 2) If L is a subspace of H , invariant under U,,then the closure I? of L is invariant under U. ProoJ 1) Let be an element in the dual space H* of H satisfying

< U,(z)x, (D> =O for any z E U. Let a be any element in U,(U)x; then a belongs to H,(Proposition 6.6) and there exists a symmetric open neighborhood N of 0 in g such that

uer, a =

- 1 1 -U,(x)ma m!

m-0

for any XE N.

CLASSIFICATION OF IRREDUCIBLE UNITARY REPRESENTATIONS

29 1

for any XEN. We can assume that the exponential mapping is a homeomorphism of N onto an open set Win G by taking a smaller neighborhood if necessary. Since the real analytic function f(g) = ( Uga,p) on G is equal to 0 on a non-empty open set W, f is identically zero on the connected group G. Hence we have U,a E -for all g E G by the Hahn-Banach theorem. 2) Let x be an element of L and g be in G . To prove that UgxE t,it is sufficient to show that for any E > O there exists an element y in L satisfying IIU,x-yll<s. Put M=IIU,II. Since U, is invertible, we see that M>O. Since x belongs to E, there exists an element z in L such that -=4(2M).

IIX-ZII

By l), we have

-~

u,z E U,(U)ZC

U,(U)LC

L.

Hence there exists an element y in L such that

I I u,z -A I f . In this case there exists a unique real number Y >0 such that q= 4 +v2. Put s = + + i v . Then M(U)=M(V+**)and q(U)=q(V+.*). Hence U is equivalent to V+s0by Theorem 6.3. iii) M =M,+, q =n( 1 -n) (n E 2-lZ, n >0). We divide this case iii) into two subscases iii.a) and iii.b) according to whether n = 8 or n > &. iii.a) M=M++,4 = f . In this case M(U)=M(V*.*&j+)and q(U)=q(V+**IQ+). Hence Uis equiva-

+

306

UNITARY REPRESENTATION OF

SL (2,R)

lent to V*.*1$+by Theorem 6.3. iii.b) M=Mn+, q=n(l -n) ( n ~ 2 - ' 2 ,n>)). In this case, M(U)=M(U-n) and q(U)=q(U-^) (cf. Proposition 6.10 and Corollary to Theorem 6.1). Hence U is equivalent to U-". iv.a) M = M , - , q = ) . In this case U is equivalent to V**+ I$-. iv.b) M=M-,-,

-7.

q=n(l -n) (.E 2-'2, n < y

In this case U is equivalent to Un. v) M = {O}, q=o. In this case, U is the identity representation as was proved in Theorem 6.2. Since the pair (M(U), q(U)) takes different values for the different representations enumerated in Theorem 6.4, no two of them are equivalent to q.e.d. each other by Theorem 6.3.

$7 Thecharacters The character of an irreducible unitary representation of the Euclidean motion group M(2) was defined as a distribution on the group. (Ch. IV, $3.) Similarly, the character of an irreducible unitary representation U of G = SU(1, 1) is defined as a distribution on G.Namely, for any complexvalued C"-function f on G with compact support, the operator n

(7.1) belongs to the trace class, and the mapping fHTr U, is a distribution on G. We shall calculate the characters of the representations of G which belong to the principal continuous series, the complementary series, and the discrete series. It turns out that the characters of these representations are functions on G. Proposition 7.1. Let 9 ( G ) be the space of all complex-valued C"functions with compact supports on a Lie group G, U be a Banach representation of G on a Banach space H and V be the uniform closure of the subspace ( UfI f € a ( G ) ) in B= B(H) (the Banach algebra of bounded operators on H). 1) Put f,(A)= U,A and r,(A) = A U,-l

for g B G and A E V. Then I and r are Banach representations of G on V.

307

THE CHARACTERS

The operator U, (f E 9 ( G ) ) is a C"-vector for the representations I and r. Let L and R be the left and right regular representations of G and I', r', L', R' be the differential representations of I, r, L,R respectively. Then we have

l'(Ddr'(D4 U,= U L * C D ~ ) R , C D ~ > ,

(7.2)

for every f in d ( G ) and for all D1and Da in the universal enveloping algebra U(g) of the Lie algebra g of G. 2) Let K be a compact subgroup of G, let A be an element of the dual R of K, and let d(A) and x1 be the degree and the character of 1. Let E(R) be the projection defined by

sg

E(R)=d(A)

(7.3)

xA(k-l)U&

and let Et(A), &(A) be the similarly defined projections for the representations I and r. Then we have B ( I ) ( A )=E(A)A and Er(R)(A)=AE(A'),

(7.4)

where 1' is the class contragredient to A. Proof. 1)The operators I, and r, leave the subspace Vinvariant because W

(7.5)

1

f =

f(g1)U,,,dg1=

ULgf

and

1 0

r a ( u f ) = / f(gl)Ug1o-'dgl= u R g / . Since ~ ~ l , ( A ) l 1 5 ]IIAll, / U g ~I,~ is a bounded operator on V. It is clear that I and r are representations of G on V. Now we prove that I is strongly continuous. Take an element A in V and a positive number E . Then there exists a functionfin S ( G ) such that

(7.6) IlU,-4IO

for all g c W.

There exists a functionfin S1(G) which vanishes outside of Wand satisfies

fro

and

1.

f(g)dg=l.

Hence we have

Iv(go)l=I

s.

(p(g0)-

(p(g))f(g)dglS E. Since c can be taken arbitrarily small, we get p(go)=O. q.e.d.

Now we proceed to the calculation of the character of the representation U" in the discrete series. We use Abel's method of summation for

337

THE CHARACTERS

the divergent series C(UgnYpn,V p n ) and prove that the sum taken by P Abelian summation is equal to the character, i.e., the distributional sum of the series. Before stating the theorem, we recall results on the representation of the conjugate classes in G'. Every element g in GI)= u goH'g0-' may be written gocG

&?=go(-l)S'acgo-'

for a certain goE G, t E R, and j~ {0, &}. The values of j and It1 are uniquely determined by g (Proposition 7.9). Every element g in Gal= u goK'go-lmay be written VO'G

g =goUego-'

for a certain go E G and 19E [0, 4n). I9 is uniquely determined by g.

Theorem 7.4. Let U" (n E 2-'2, In[2 1) be a representation of G = SU(1, 1) in the discrete series (cf. $3). Then the character of Unis equal to the function e-(lnl-t)ltl lecrs-e-tia

I

(-l)tn9

if g = g o ( - l ) s ~ a c g ~GI', -l~

if g E G-G'

,

where j = 0 or 4 and e(n) =sign n. Namely, we have

s.

Tr U p =

(7.88)

f(g)@"(g)dg for all f E S ( G ) .

Proof. We introduce the convergence factor r (0 c r < 1) in the summa-

tion of

i

(

~

~

n

K,"). ~ ~ nPut ,

P-0

@,"(g)=

-

CrP(Ug"Kp*,vPn). P-0

Then the series converges uniformly on G for each fixed r E (0, l), because lupn(g)l5 1 for all g E G. Hence the function 8," is continuous on G. Since up-"(g)= up"(g),we have (7.89)

-

@r-"(g)=@rn(g)

Hence it is sufficient to calculate 8," for n2 1. Assume that n~ 1. Then by Proposition 7.17 and Lemma 7, we have

338

UNITARY REPRESENTATION OF

+ +

SL (2,R)

t

-In

=e-tncp+*~~-1(2-1(~ w I) )l-an P - j- )

where w=re-t(p++)and R=[1-2

,

t ( I-2th2~)w+wa]'.

Hence a finite limit exists for lim @,"(g) for every g=u,atu+ E G-K. r-1-0

Since R =R(w) takes real values on the positive real axis, we see that lim R ( r ) = ( 4 t h aty ) * =21th$/ and 7-14

By (7.89), we have

and for all n E 2-'Z, In)2 1. On the other hand, on the subgroup K,we have

and us+ f1. Hence

exists and is finite. By (7.89), we have

Hence lim @,"(g)=Oln(g)

r-1-0

exists and is finite for every g E G'. Moreover, we have

.

&"(g)=@(g) if g E H' u K' To prove eln=8" on G', it is sufficient to show that elnis invariant under inner automorphisms. This fact will be demonstrated at the end

(7.90)

339

THE CHARACTERS

of the proof. Since the operator Ufnfor eachf in 9 ( G ) is of trace class, the series

converges absolutely. Therefore, by Abel's theorem on power series, we get m

(7.91)

Tr U p = P-0

m

(UlnVPn,lPpn)= r-1-0 lirn

rP(UfnFpn, VPn) P-0

for every f s g(G'). Since

c OD

s r"llfll Ilu,"ll s [4n(214-l)-'I'

llfll(1

+

00

P-0

by Proposition 7.18, the right hand side of (7.91) is equal to n

n

Thus we have proved that r

(7.92)

Tr U," = lim

r-1-0

J f(g)@r"(g)dg Q

for every f E g ( G ) . Put R = R(r, 1 ) = (1 -2( 1-2 th$)r+

ra)+and R1= lim R(r, t) r-1-0

Then we have 0 6Ri- R= (Ri'-Ra) (RI R)-' (7.93)

+

I ;6(1 -r)Rl-'

Since R1-1=2-1(1 +e-t)ll-e-cl-l R*=R-{O}, we see that

. is bounded on every compact set in

lim R=Rl r-1-0

340

UNITARY REPRESENTATION OF

SL (2,R)

uniformly on every compact set in R - {O} . Hence 8," converges uniformly to eln on every compact set in G -K. On the other hand eraconverges to elnuniformly on every compact subset of K' because 1-r IQ~"(~o)-@i"(~a)l = I -re*tel 11 -psi * Therefore, 9," converges to elRuniformly on every compact set in G . Hence Qln is a continuous function on G . Moreover, by (7.92), we have

Tr U3"=S6'g)Q1"(g)&

(7.94)

for every f in g l ( G ) = { f ~ g ( G ) I s u p p f c G ' } . (7.94) proves that the distribution fwTr U," coincides with the function Elln on the open set G' of regular elements. For each x E G, put f.C(g)=f(x-lgx). Then we have j-/(g)@l"(xgx-')&= =Tr U;= =Tr U3"=

f"(g)@l"(g)&

/ff

1

f(g)Ol"(g)dg.

ff

By Lemma 8, we get (7.95) Ql"(xgx-l)=Ol"(g) for all g E G' and x E G

.

By (7.90) and (7.95), we have Qln(g)=@n(g) for all g E G' . Since Il-e**B111--re*~o~-1~4if O < r < l , we see that

(sin~~j@,"(u,)-e.(u,)I 64(1-r) and (7.96)

lim (sin~~(Q,"(ua)-Bn(u.))= 0

r-1-0

uniformly on K. Let ro be a fixed real number in the interval (0, 1). Then R(r0, t ) = 2 l-ro for all t E R. Hence for every r in the inter-

Val [ro,l), we have by (7.93)

THE CHARACTERS

341

We see that

uniformly on every compact set oft. Therefore, (7.97)

lim r-1-0

(sh

fa,)-@( k a t ) )=0

uniformly on every compact subset of the Cartan subgroup H = M A . By the relations (7.96) and (7.97) and the integral formula (7.52) (Proposition 7.13), we have (7.98)

lim /Gf(g)Qrn(g)dg=

r-1-0

1

f(g)@"(g)dg

for allfin d ( G ) . (7.92) and (7.98) prove the desired relation (7.88). q.e.d. Theorems 7.2, 7.3 and 7.4 show that the characters of representations in the principal continuous, discrete, and complementary series are in fact functions which are real analytic on each of the connected components Gl' and G2'of G' and are integrable on every compact set. This is a special case of theorem due to Harish-Chandra [6], [lo] stating that a central eigendistribution (in particular a character of an irreducible unitary representation) on a connected semisimple Lie group is a locally summable function which is analytic on each component of G . At the end of this section, we give a remarkable relation between the character Of*"+'of the non-unitary representation VJ,"+jin the continuous series (92) and the character 8" of the discrete series representation. In this relation, the character of a finite-dimensional representation appears.

Proposition 7.19. Let Vnbe the vector space of all homogeneous polynomials of degree n in two variables zl, z2. Let F" be the representation of G=SU(l, 1) on V, defined by (7.99)

(F"(g)9)(4=(OM*

(cf. Ch. XI, $1). Then the character En of F" is given as follows. If A and A-' are the eigenvalues of a regular element g E G , then An+l--R-(n+l)

En&)=

2-2-1

-

ProoJ Actually (7.99) defines a representation Fn of the complex Lie group GC=SL(2, C). If g is regular, then 2 # P , and there exists an element in G' such that

342

UNITARY REPRESENTATION OF

SL (2,R)

Hence we have €"lg)=TrF'(g)=TrFn(() R

O

- (In+I-R-(n+l) 1t (1-2-9 as was shown in the proof of Ch. 11, Proposition 2.5.

Theorem 7.5. For each TI E N,n r 1, and j we have (7.100) Proof.

+

E

q.e.d.

{O,g) and for all g E G ,

+

@j.n+j(g)= (a(n+J-l) (g) @"+j(g) @-"-j(g)

.

By Theorems 7.2 and 7.4, we have

and

Hence we have (7.100) for all g E G1'. On the other component GI'of G , we have OJ*"+'(goUago-') =0 ,

and (@+j

(n+j-# +@-n-')(gousgo-') = -sinsin (0/2)

0

Hence (7.100) is valid for all g E G,'.

'

q.e.d.

Remark. Theorem 7.5 reflects the following relations among the representations, VJsn+j,Facn+j-')and Un+J@U-"-j. By Proposition 2.6,4), we see that dVzf*n+jf-,-a,=dVjj.n+jf*=0

and the subspace Q(j, n+j) of Q=La(U) spanned by IfPlpzn or pS-n-2jl is invariant under V f ~ " +The j . subspace Q ( j ,n + j ) has codimension 2(n + j)-1 and the representation induced by Vfvn+j on the quotient space

343

INVERSION FORMULA

Q/@(j,n+j) is equivalent to composed thus :

The subspace Q(j,n+j) is de-

FZcn+j-l).

W, n+j) =Q+Ci, n + A @ @-(j, n + A , where Q+(j,n+j) and &(j, n+j) are spanned by {fplp>,n}and {f,lpS -n-2j} respectively. Roughly speaking, the representations induced by Vjtn+fon Q+(j,n + j ) and on &(j, n+j) are equivalent to U-"-j and to Un+jrespectively. More precisely, we get Ur("+J) after the completion of a dense subspace with respect to a suitable inner product. These circumstances are reflected in the relation (7.100). Another explanation is also possible. By Theorem 7.2, we see that (7.101) @.J=@jJ-J. (if s = + + i l , (7.101) is another expression of the result of Theorem 2.2: V j J z VJ-'). By (7.101)7(7.100) can be rewritten as a j . 1 - n - f - z(n+j-l) + @n+j + 0 - n - j (7.102) -t This relation can be interpreted as follows. The representation VjJ-"-j leaves invariant the (2n +2j- 1)-dimensional subspace Q(j, n j ) l of 8 and induces a representation equivalent to F2(n+j-1) on Q(j, n + j ) l . The representation induced by VjJ-"-j on the quotient space Q/@(j,n + j ) l is equivalent to Un+J@U-"-jafter extension to a suitable completion of a dense subspace.

+

$8. Inversion formula As for the groups SU(2), R" and M(2) treated in the preceding chapters, we can develop the theory of the Fourier transform on the group G = SU(1, 1). Let 6 be the dual of G. This means that 6 is the set of all equivalence classes of irreducible unitary representations of G . We can choose a representation U Afor each class 2 in 6. Then the Fourier transformf of a function f in L1(G)is defined by the formula n

Thus f is an operator-valued function on 6. We can expect that the original functionf may be recovered from its Fourier transform f,at least for a function f in s ( G ) . Similar inversion formulae for the groups SU(2), Rn and M(2) suggest that there exists a measure p on 6 such that

344

UNITARY REPRESENTATION OF

SL (2,R)

In fact such a measure exists and is given as follows. Put fcj, s )=

1

f ( g )V , - i W g = V f

a

and

f ( n ) = / f(g)Un,-ldg= Ufv n , where f"(g)=f(g-l). Then we have +f(g)= Tr (f(0,i (8.3)

&Io

+1

+iv) Vgo*++ru)~ th

K V ~ V

+

(2n -1) {Tr( f ( n )U,") Tr (f(-n) U,-")} nta-12 nS1-

for every f e g ( G ) . Hence the support of the Plancherel measure p is equal to the union of the principal continuous series and the discrete series. The complementary series do not contribute to the inversion formula for the functionsf in s ( G ) or L2(G).Putting g = 1 in (8.3), we get +(8.4) f(l)=&I0 Oo.*+fu(f)v thxvdv

where W'(f) = Tr( Vf

=

f(g-')W'(g)dg

and

/a

-s.

W f )=T W f 1v

"

-

f(g-l)@n(g)&

The functions W 8and 8" are the characters of W aand U n respectively (cf. $7). The general formula (8.3) is obtained from the special case (8.4) by replacingfwith L,-lf. Hence we study mainly the formula (8.4). In the study of harmonic analysis on the group G = S U ( l , l), the integral formula in Proposition 7.13 plays an essential role. This integral formula gives a decomposition of the Haar integration on the group G into two stages; one is the integral (the mean) over each conjugacy class consisting of semisimple ( =diagonalizable) elements; the other is Haar

345

INVERSION FORMULA

integration over each of the Cartan subgroups H and K with the weight functions 2(sht/2)' and 4(sin 6/2)* respectively. In virtue of this integral formula, the Fourier transform {W'cf), P c f ) } off E g ( G ) is obtained by first taking the means Ff and Ff on the hyperbolic and elliptic conjugacy classes, and then carrying out the Fourier transforms on the abelian Lie groups Hand K.We shall see later that the derivative F f ' ( 6 ) of the mean FfKgives the value f(1) off at the identity element 1, and that consideration of the jump of Ff=(6) at 6 =O gives a relation between Ff and FfK.Then these two facts and harmonic analysis on A E R and K s T give the inversion formula on G. The results and the proofs of this section are due to Harish-Chandra [2], [8]. First we give two Lemmas which will be decisive in the proofs of the above stated results.

f(6e', 6e-t)(eC-e-')dt,

g(6)=6

(820).

J O

Then g is a C-function on R - {0} with compact support and satisfies lim-(O)= dg 0-0 de

-2 f(0,O).

In particular dgld6 can be extended to a continuous function on R. Proof. Since f belongs to g ( R a ) , g is a C"-function on R - (0), and we have

ss

f (Be', 6e-')(ec-e-')dr

g'(e)=

{e%+e-'fv} ( W ,ee-')(e*-e-')dt

+\:B

=J +J

e-tf dt

( e ' f + eeslf,)dr-j 0

0

(6fv-6fs-6e-2t

fv)df for any 6 2 0,

0

lo rn

where fs = aflax and fy =af/ay. Since 1f(x, y)]g ]Ifl]-

and

we have

(8.6)

limj e-C f(Bet, Be-')dt =f(O, 0) 0-0

0

by Lebesgue's dominated convergence theorem. Similarly

e-Vr = 1,

346

UNITARY REPRESENTATION OF

SL (2,R)

Since the support off is compact, there exists a T>O such thatf(x, y)=O unless Max(lx1, Iyl)s T. Then f,(Be', Be-')=O ( t 2 0 ) unless lelet$ T. Hence

Therefore we have

J

f,(eet, ee-')dt=O,

lim 8 0-0

0

because lim 0 log 1191=0. Similarly 0-0

rm

lim 0 8-0

J

f=(Bec,Be-')dt = O . 0

Put x=Be' and y=Be-'. Then we have

and

Since +/;$(xf)dt=

B-l[(xf)(Se',Be-')];= -f(B,

0) ,

we have

Summing up these results we get Lemma 1 .

LEMMA2. The function g in Lemma 1 satisfies

Proof. Let O<e$T and Bef=x. Then

q.e.d.

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Since If(x, Px-')lS Ilfll-,

(8.8)

347

we get

lim 01; etfdt =/:~(x, 0)dx =/"f(x,

R*+O

o)&.

By (8.6) and (8.8), we get the first equality in (8.7). If 8(x, 0

o)&

~x-')dx-/->(x, 0

as 8- -0.

q.e.d.

Delinition 1. We denote by go(@ the set of all functionsf in g(G) satisfyingf(x) =f(x-') =f(uxu-') for every x E G and for every u E K.

Proposition 8.1. For each functionfin g o ( G ) ,put (E

= f1)

,

where dg and dg* are G-invariant measures on GIH and G / K (cf. Proposition 7.6 and 7.12). Then the function F f K is a C"-function on R-2aZ and its graph has jumps at e =0 and 8 =28. The jumps are given by (8.9) F f K ( + O ) - F f K ( - O ) = ~ F f ( l )and (8.10) FfK(2r+ 0) -FfK(2z-0) = -A F ~-(1 ) . ProoJ Since K is an Abelian subgroup and a null set in G, we have

by Lemma 1 in $7.By Proposition 5.3, the last integral is equal to (8.11)

.

2n/~f(a,u,~~-~)shtdf

Since

and f E g o ( G ) has the compact support, there exists a constant M>O such that f(atueat-l)=O if Itl>M and e # 2 d .

348

UNITARY REPRESBNTATION OF

SL (2,R)

Hence the integral (8.11) is convergent and FIK(R) is a Cm-functionon If 0 = 2 n s (n E Z), then the integral (8.11) R - ~ ~ Tfor Z eachf in go(@. is divergent unlessf((- 1)")=0. By (8.12) and Lemma 2, we have

Ffg(+0) = lim sin (e/2)H(6) =2-1 lim 0 H(0) e-+o

8-+O

Similarly we have

FlK(e+ 2 ~ ) =-F,lK(e)

(8.13)

wherefi(x) =f(-x), we have (8.10).

Proposition 8.2.

For each f E .go(G), put d G f ( 6 ) = ~ F 7 ( 0 ) , 0 E R-2nZ.

Then Gf(R) is a C--function on R -2 x 2 and can be extended to a continuous function on R. If the extended function is also denoted by GI, we have Gf(0) = -rf( 1) and G,(27~)=sf( -1) . Proof: By (8.11), (8.12) and Lemma 1, we have d -2nf(l)=lim-(~H(~)) e-o de

d d8

=2 lim-(sin (@/2)H(@)) =2 lim GI(@ e-o

.

By (8.13) and the above result at 8=0, we have limG,(O)=zf(-l). 3-9. Hence lirn GI(@ exists and is finite for each n E 2, and G, can be extended e+'.mt

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to a continuous function on R.

349

q.e.d.

The derivative dG f / d e=daFfy e ) / d e 2is discontinuous at e = 2nx (n E Z ) , but it has finite limits when e tends to 2 n x 4 0 . Hence GI is a piecewise C1-function. To prove this fact, we need the notion of "radial part" of the Casimir operator. Although this notion may be defined in a general setting, we restrict ourselves to give the necessary minimum. (For the general definition and properties of the radial part of a differential operator, see Helgason [6].) Definition 2. Let w=XOa-Xl'- Y' be the Casimir operator on G = SU(1, 1). For each function 4 in g ( K ' ) , put f(gusg-')=@(ue).Then f belongs to C"(G,'). Let (of)' be the restriction of wf to K'. Then the is a linear transformation in C"(K') and mapping d ( w ) : $-(wf)supp (d(w)$) c supp (+). Hence d(w) is a differential operator on K' which is called the radial part of o on K'. For each f in g ( G , ' ) , put (8.14)

f(gusg-')dg*.

F(B)=/

GlR

Then F belongs to s ( K ' ) . Since 0 is invariant under the left and right translations La and Rg, we have i3lR

Proposition 8.3. Put a(@)= 2sin(B/2) and L= da/dea.Then we have d ( w ) =6-' L 6-4-', (8.16) where 6 in (8.16) implies the operator f~ 6f , and 0

(8.17)

0

F . f K ( ~ ) = ( ~ + 4 - 1 ) ~ f ~ ( ~ )for all f e g o ( @ and 6

E 2aR.

The second derivative daFf(@)/dB1 has finite limits as B-+2nxfOfor each n E 2.In particular, Gf(B)=(Ffg)'(B)is a piecewise C1-function on R. Proof. Since o =Xol X12 Y', A(@) is a second order differential oper-

-

-

ator on K' whose highest term is equal to Xoa=da/de'=L,i.e. (8.18)

d ( w ) =L

+lower order terms .

350

UNITARY REPRESENTATION OF

SL (2,R)

Let d ~ ( G 2 'be ) the set of all real-valued C"-functions on Gt' with compact supports. Then we have (8.19) (of,&) =(f,4) for everyf andji in d ~ ( G s ' )where , the inner product is given by (fi,A)=

lGs,fi(g)A(g)dg. ~f fi satisfies fi(gu,gg-l)=fi(u,) for every g E G and

U ~ E K 'we , have by (8.14), (8.15)and Proposition 7.13

inner automorphisms, we and products are defined by ($1,

$l(us)$a(us)6(B)2d8. The mapping fH F is a surjective

+a)=('Ir)-'J 0

mapping from d ~ ( G s 'onto ) d a ( K ' ) . (The proof is entirely similar to that of Lemma 1,l) in 97.) Hence&) is a symmetric operator. Since 6-10L06is also a symmetric operator whose highest term is equal to L, d(o)-6-10 Lo6 is a symmetric operator of order S 1. Since there exists no symmetric differential operator of order 1, d(0)-6-~oL06 must be a function c. Operating with d(0)-6-~oLo6 on the constant 1, we get c=d(o)1-6-'L(6)=(01)-+4-'=4-'

.

we have Hence we have d(0)=6-~0L06+4-'. For each f in go(@,

LK

Fe,K(e) =2 - 1 ~ ) (of)(gusg-l)dg*

+

=2-1s(e) ( A ( ~ ) F (e)) =2-1(~6 4 - 9 ) ~ ( e )

By Proposition 8.1, both F,g(B) and F-,"(e) have finite limits when e-+ 2 n r f 0 (n E Z), hence the second derivative (Ffg)"has the same property by (8.17). Thus (Fig)"is a piecewise continuous function and G, =(Fjg)' is a piecewise C1-function on R. Definition 3. For each function f in d ( G ) =

cf : G-CI f is a C"-func-

351

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tion with compact support},put

As we shall see later, the different parts of the inversion formula correspond to f + and f - . Hence it is convenient to study f + and f - separately. These two parts are analogous to Fourier cosine and sine series which correspond to even and odd functions respectively.

+

Put s =t i;l and

Proposition 8.4.

p*(t) =~,,(a) =e ~ / ~ / ~ ~ f * ( u t n . ) ~ .

Then for every f E go(@, we have (8.20) W'( f - ) =@'**(f+) =o , Do**(f + )=

(8.21) (8.22)

G***(f-) =

1:. S

p+(t)e*Wt and (p-(r)e"tdt

.

-0I

Proof.

By Proposition 7.5, we have

{l m

o j - s ( f )=2-1

e"tF,(ut)dt+ (- I)s~[

-m

e"CF,(--a,)dt)

-_

.

for every f s go(G), where j E {0,+} and s=++i17. Since f*(-ut)= ff(ut) and F l ( f u t ) = fF,(ut), we see that (8.20), (8.21) and (8.22) hold. Definition 4.

For each n E 2-'2 and n r 1 , and for each f E S ( G ) , put D"( f

Proposition 8.5.

(8.23) (8.24)

)=en( f ) +e-yf ) .

For each f E s o ( G ) , P ( f - ) = O if n~ 2 and Dn(f+)=O if n s f . 2 - 2 .

Put

F*(O)=

1

QlK

f*(gusg-')dg*

.

Then for every f s SO(@ and n E 2,n 2 1, we have

352

UNITARY REPRBSBNTATION OF

-2n-l

SL (2,R)

1:.

sin (n-+) e sin+c?F+(B)d@

and (8.26)

On++($) =2

e-n5p-(t)dt J

O

s:.

-27~-1

sin no sin +8F-(B)dB .

By Theorem 7.4, the distribution D" is equal to a function e-(n-+)ltllsh+tl-l(- l ) 4 n j if g=go(- l)afutgo-lE GI' , D"(g)= -sin(n-+)O(sin+O)-' if g=goueg0-l E Ga',

Proof.

lo

if g E G - G .

Hence we have by Proposition 7.13 D"(f>=2fme-(n-+)tsh +t

+ 2(-1)sn[e-(n-h)t

Hf(gatg-l)dg

sh+rdt/

f(g(-at)g-1)dg

QfR

By Propositions 7.6 and 7.7 and by the assumption that f(uxu-l)=f(x), we have

We treat the case 1 ) n E Z and the case 2) n E + Z - Z = {nlnE $2 and n 4 Z } separately. 1) n E Z . Then (- l)sn= 1 and f,(g(-a)g-l) = &f,(gatg-l). Hence for Dncf+) (or Dn(S-)), the first integral in (8.27) is equal to (or cancels) the second one.

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353

+

Since sin (n-3)e sin +e =+[ -cos no cos (n- 1)8] has the period 27~and f*(gUb+lxg-l)=f*(-gusg-l) = ff*(gubg-’) , the third integral in (8.27) is equal to

(8.28)

Hence we have proved (8.23) and (8.25). 2) n E 3 2 - 2 . In this case (-l)2n=-l and a(e)=sin(n.-$)ex Hence we can prove (8.24) and sin38 satisfies a(8+2n)=--a(B). (8.26) similarly.

Proposition 8.6. Put G+=GI, for each f E 9 ( G ) and

1:.

G+(O)cos(n+3)8dB,

c~=zc-’

g , = .-1[

-.

G-(8) cos n8 d8 for each n E N,

and put

Then we have

(8.29) and m

(8.30)

G-(O)=

C

6ngn

.

n=O

Proof. By (8.28), G-has the period 2n and G+has the period 4n. Hence A ( @= ) G+(B)cos38 has the period 27~.Since both Ga and A are continuous and piecewise C1-functions (Proposition 8.3), they can be expanded in Fourier series which are everywhere convergent (Ch. I. Theorem 5.1). Since G- is continuous at 8 =0 and is an even function, we have G-(0) = m

C &gn. Similarly we have

n-0

(8.31) where

354

UNITARY REPRESENTATION OF

SL (2,R)

an=x-l[G+(@) cos 319cos no do .

Since B(0) = G+(B)sin+e is an odd function with the period 28, we have m

0 =B(0)=

(8.32)

1bn ,

where

n-1

bn=r - 1 1 G+(e) sin 30 sin no de . -x

Since

an--bn=cn

-

and an+bn=cn-1 for all n2 1, we get

C

G+(~)=+co+

n=l

m

Cn=+co+

C

Cn

9

n=O

by taking the difference and sum (8.31)+(8.32). Since the series Ccn m

ca

is convergent, we conclude that co=0 and G+(O)= C cn = C cn . n-1

n-0

Proposition 8.7. For each functionfin ~ o ( G )we , have

Proof. Put Ht=F,&:lag.Then Propositions 8.2, 8.6 and 8.1 and integration by parts show that

Similarly we get (8.34) by integrating (8.30) by parts.

q.e.d.

355

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LEMMA 3. For any 1)

/:@+)-'sin

3)

/::thysin

A E R,we have

At dt =2n th xA ,

Atdt= 2r cthA r ,

t

t d t cthye+ao+a1z+ .... Q

Hence the integral on the semicircle around 0 is equal to a/: id8= -da= -xiRes f ( z ) = -2wi. r-0

Similarly, the integral on the semicircle around 2xi is equal to -ni Resf(z) =2 ~ i e - ~ ~ l . .-Zd

Summing up the above results and using Cauchy's theorem,

we get

+

i(1 e-~r~)l=2ni(l-e-3*~) and t=2nthwA.

"(

2) Since a1 (sh+)-'sin

I f ) = t (sh+)-'

cos At and t

over R,we can differentiate 1) under the integral sign and get

1:-

t(sh t/2)-l cos At dt = 27cd(th nA)/dA.

Since t(sh t/2)-' is an even function, we get 2).

1:-

3) Put J=

erlccth t/2 dt and g(z)=e*llcth (42).

The poles of g(z) are same as those of f(z) in 1). We integrate g(z) along the same contour C as in 1). In this case g(z 2 4 =e-l=lg(z), Res g(z) = 2 , Res g(2)=2e-ax2

+

S-0

a-lrt

.

Hence we have (l-e-an2)J=2ai(l +e-z*2) and J=2x cth nR . 4) We can differentiate 3) under the integral sign and get 4). Proposition 8.8.

For any f in g o ( G ) , we have

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INVERSION FORMULA

I::+*.

(8.36) ProoJ

f)R cth X R d~

++(I(

Put and @ L 2 ( f ) = @ i * + + t 2 ( f - ) .

@+l(f)=@O~++~~(f+)

Then by Proposition 8.4, we have

1-

~ f= )

pa(t)e(a'dt

.

-01

Since (p+(t)=F,,(at) belongs to s ( R ) , we get S_p.'(t)e(lldt=

-iAQal(f)

by the integration by parts. Put

+,(A)

=

1-

t-lp*'(t)e$Jtdt.

-OD

Since p+ is an even function, p*' is an odd function and t - I p+'(t) belongs to g ( R ) . Hence belongs to 9 ( R ) (Ch. 111, Theorem 1.2) and can be differentiated under the integral sign. We have

+,

#*'(A)

=i

1:-

p*'(t)eWt .

Since t - l ~ + ' ( t belongs ) to y ( R ) and t(sh$)-l

belongs to L1(R),we

can apply the Parseval equality for the Fourier transform on R (Ch. 111, Theorem 1.3), and get d t-'p+'(t) t(sh t/2)-'dt= ++(A)z(thnR)dR

/Im

jm -_

=-

1:-

j:-

++'(A)thddA = -

R th ZR ~ D + ~ ( f ) d 2 .

Similarly, we get t-'p-'(t)t cth t/2 dt = -

Proposition 8.9. For any f E go(@, we have

and

358

UNITARY REPRESENTATION OF

277f-(1)=

(8.38) Proof.

fb

SL (2,R)

m

@-'(f)IcthxRdR+ CnD"+'(f-). n-1

By Proposition 8.5 and integration by parts, we get

r-

m+l

+

1:

v+'(t)(sh t/2)-ldt

=2lrf,(l)-+f-

-m

@ + ~ ( Jth) dR d . ~

The last equality follows from Propositions 8.7 and 8.8 and the fact that q+'(t)(sh t/2)-' is an even function. Since V j .*+%a z Vj* (Theorem 2.2), we have O*^(f)=cD*-^(f). Hence the last expression in the above sequence of equalities is equal to 277J+(l)-/m@+2(f)R 0 thxRdI. We have proved (8.37).

s:

Put D * ( f - ) = 2

p-(t)dt. Then we have

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{

m

=2 lim (m++)(o+(O)-2n-'E m-+"

359

n r sinnOH-(e)de} n-1

-r

+jLp-'(t)cth-dtt 2

Dewtion 5. For each functionf in L1(G),the Fourier transform f of f is defined as the operator-valued function on = {(j,3 + iR) I R >0, j E {O, &I1 u {nln E 2-'Z, In1 2 1) given by

e0

f ( j , &+in)=

f(g)V,-if.*+*Adg=vjvj,++*2 So

and

f(n)=/

f(g)U,-lndg= UIvn,

1 where the Haar measure dg is normalized as dg=-eecdedtdf. 47r

Theorem 8.1. (The inversion formula for d ( G ) ) . For each function f i n g ( G ) , we have (8.39)

++a)Vgo,++*l)l thddr2

f ( g ) = z1/ " Tr(f(0,

+

Tr(f(+,+ 0

1 +-4nnr%-le E (2n-1)

+ il) V,*.*+*')A

cth n l dR

{Tr(f(n) V,")+Tr(f(-n)

V,-")} .

n2l

Proof. First we assume thatfbelongs to go(@, i.e.,fsatisfiesf(um-1) = f ( x ) =f(x-l) for all x E G and for all u E K. Then by Proposition 8.4, we have Tr(f(O,+

+ 9 ) ) =O0*+ + c l ( f=) O0* (f+) *+*l

=@+^(f).

Similarly

+

Tr(f(&, & iR)) =@ - l ( f ) By Proposition 8.5, we have Tr(f(n))+Tr(f(-n))=

.

K(f+) D

if n E Z ,

(f-) if nE3.Z-Z

360

UNITARY REPRESENTATION OF

SL (2,R)

Hence by adding (8.37) to (8.38), we get

(8.40)

f(1) =

$1:

Tr(f(O,3 +il))R th nR d2

1

1

+& n r t e . n p i

(2-

1)(Tr(f(4)+Tr(f(--n))

for all f e ~ o ( G ) . Now we turn to the general case and put f"(g)=J

K f(kgk-')clk.

Then we have Vh-lj.JV,f.*Vhj.*&.

V,OhJ= /K

Since V,j**is of trace class (Theorem 7.1),

-

(V,j**V&*fP,

=Tr

Vlj.Jfp)

V,jTJ

p--m

for every k E K and the series is absolutely convergent. Hence we have

=IK

Tr(VfVj.*)dk=Tr(Vjvji~)=Tr(f(j,s)).

Similarly, we have Now put f"(g)=f(g-l).

Trf(n)=Trf(n) . Since W(*a+) =P( +a,) ,

by Theorem 7.2, we have W(g-') =W*(g) and

By Theorem 7.4, we see that

36 1

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Hence we have Tr((f")W) =Tr(f(-n))

*

Put

h =.)(P +P") Then by the above results, we have Tr h(j, s) =Trfu, s) (8.41)

.

and Tr h(n)+Tr h( -n) =Trf(n) +Trf( -n)

(8.42)

.

Since h belongs to so(G), (8.40) is valid for h. Hence by (8.41) and (8.42), (8.40) is also valid for$ We have now proved (8.39) for every f in 3(@. Put (L,f)(x)=f(g-lx). Then we have

(L, f )"(j, s) = f(j,s) Vg-lf*'and (L, f )^(n) =f ( n ) V,-l . Therefore, we obtain (8.39) by replacingfin (8.40) by L,-$ q.e.d.

Tbeorem 8.2. (Parseval equality). For every f in s ( G ) , we have

where I] Proof.

[I, means the Hilbert-Schmidt norm. Put a= f * f *

where

f*k) =f(g-').

for each f ES(G)

Then u(1)=JG I f(g)lVg ,

Tr 40,s) =Tr(f(j, s ) f ( j ,s)*) = IIfG, s)ll2 for s = 2 4 + i 2

and Trd(n) =Tr(f(n)f(n)*) = Ilf(n)llla .

Hence equation (8.40) applied to u gives us Theorem 8.2.

q.e.d.

362

UNITARY REPRESENTATION OF

SL (2,R)

99 Harmonic analysis of zonal functions Definition 1. A function f on G=SU(l, 1) is said to be two-sided Kinvariant or a zonal function, if it satisfies f(kgk') =f ( g ) for each g E G and k, k' E K. The set of all complex valued zonal functions on G is denoted by A. In this section, we study Fourier transforms of zonal functions. In particular the images of the spaces of C"-functions with compact supports and of rapidly decreasing functions are determined. Let n be a half-integer (i.e. n E $2)and put xn(ue)=elne

The set of all C-valued continuous functions on G with compact supports is denoted by YG). Proposition 9.1. Let U be a Banach representation of G=SU(l, 1) on a Hilbert space H such that u k is a unitary operator for each k E K and fn andf, be two vectors in H satisfying (9.1) Utft=xi(kYi for each k E K (i=n, m)

.

Then the operator

Uf =/Gf(g)uGd.g for a functionf in L (G) n A satisfies

.

(Uff n ,f,) =0 unless (n,m)=(0,O)

Proof.

Sincef is two sided K-invariant, we have (uffn,fin) =

1

f(k-lgK-1) (u a f n fm)dg 9

Proposition 9.2. If the restriction to K of a unitary representation U of G does not contain the identity representation xo, then U,=0 for every function f in L(G) n A. In particular, f(+,s)=O for every S E C and f(n)=O for every nE+Z,InlBl. Proof: The representation space H of U has an orthonormal basis

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

363

Cfr)rcr satisfying (9.1). Hence the first part of Proposition 9.2 is a direct consequence of Proposition 9.1. The last half is clear from the first half and Theorem 6.2. Proposition 9.3.

Put

fn(c)=cn

for ~ E (lCl=l). Z Then we have

( V , o ~ ~ ~=,0f unless ~ ) (n,m)= (0,O)for every f E L(G) nA and s E %. Proof. Clear from Proposition 9.1. By Proposition 9.2, the support of the Fourier transform f of a zonal functionfis contained in the set { Vo.++321120}. By Proposition 9.3, the operator valued Fourier transform f ( 0 , s) of f E L(G) n A is reduced to a C-valued function ( Vlo~*fo,fo)which will be denoted by {(s). The inversion formula for a function f in g ( G ) n A can be written in terms of the scalar valued Fourier transformf(s). In fact we have the following proposition.

el=

Proposition 9.4.

Iff belongs to s ( G )n A, then we have

+f ( g ) = k j f(++il)#(g, ++il)AthxldR for each gEG

Proposition 9.5. The function #(g, s) defined by (9.2) satisfies (9.4) #(kgk', s)=#(g, s) for every k,k' E K and g E G, P

364

UNITARY REPRESENTATION OF

SL (2,R)

Delinition 2. The function $(g, s ) is called a zonal spherical function. Proof. Sincefo satisfies (9.1) for i=O, 4 satisfies (9.4). By the definition of Vg0.*,we have

(vgO,* f)(C)=e-*t(o-'.C)f(g-'.C) * Hence +(g, s)=

e-*t(g-l.c)dC

(~>ssfo,fo)=/ l7

q.e.d. Proposition 9.6. I f f belongs to L1(G)n A, then the Fourier transform f(s) is given by (9.6)

f ( s )=

[Ff(t)e(*-'%ft for Re

s=

1 3

-m

in terms of the Radon transform Ff ( t )=et"

L-

f(atne)& .

Proof. By Proposition 9.4, we have

n

n

q.e.d. Dewtion 3. The transformation :f H f l defined by (9.3) is called the spherical transform on G. By Proposition 9.6, the spherical transform f-f on the integrable zonal functions is decomposed into the Radon transform f - F f and the

365

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

ordinary Fourier transform on the real line (PA). This decomposition enables us to determine the precise image of g ( G ) nA.

LEMMA1. 1) If v E C ( R ) is an odd function, then there exists an even function E C"(R) such that

+

.

v(t)=t+(t) for every t E R 2) Iff E C"(R) is an even function, then the function,

+

g(x)= f(&) is a C"-function on [0, a). 3) The function k(x) = [log(x J k T ) ] * belongs to C"([1, 4) Iff E C"(R) is an even function, then the function h(x)=f(log ( x + J m ) ) belongs to C"([l, a)). Proof. 1) It is sufficient to put

+

+

03)).

+

+

2) It is clear that g is a C"-function on the open interval (0, a). Hence it is sufl6cient to show that g is differentiable infinitely many times at x=O from the right. Put x = P . Then we have (-$>.,(x)

=

zyf(f),

(&

for every k E N and x >0.

Sincef is an even COD-function, the right hand side of the above equation is also an even C"-function. Hence by l), the limit

exists and is finite. Hence g is differentiable k-times at x=O from the right. Since k is an arbitrary integer 20,g belongs to C"([O. 03)).

+

t

3) Put f(t)=xf. Then f is an analytic function on a neighborhood of t = O . Put

Then I(x)=f(t) and

Hence I belongs to @"([1, + a)) by an argument similar to that in 2). Hence k ( x ) = [ l o g ( x + , / ~ ) ] 2 = ( x 2 - 1 ) I ( x ) fbelongs to C"([1, a)).

+

4) Since h(x)=f(t)

=g(t ') = (g k)(x), 0

366

UNITARY REPRESENTATION OF

h belongs to C"([l,

+ a)).

SL (2,R) q.e.d.

Definition 4. Let C+"(R) be the set of complex-valued, even C"-functions on R and 9 + ( R )be the subset of C+"(R) consisting of functionswith compact supports. For each function f in C+"(R), we put f [ x ]=f(cht) for x =cht . By Lemma 1, the mapping f(t)t+f[x] is a bijection from C+"(R) onto C"([l, w)) and maps 9 + ( R ) onto 9+([1, a)).

+

+

Proposition 9.7. If the functionf belongs to C"(G)n A, then the function d t )=f(at) belongs to C+"(R). The mapping f ~p is a bijection from C"(G) n A onto C+"(R) and maps g ( G ) n A onto g + ( R ) . Proof. Since the Iwasawa decompositiongnusatncis a diffeomorphism (Proposition 1.4), (@,t, c) can be take as coordinates on G. Hence p(t)= f(at)is a C"-function on R. p is an even function by Proposition 2.11, 2). Since every element g in G can be written as g=u,at,u, (proposition 5.2), a function f in A is uniquely determined by its restrictionflA to A. Thus the mappingf-p is injective. Let p be a function in C+"(R). Put (9.7)

Then f is a well defined function in C"(G) n A. In fact, the expression g= u,atu, (OO. Then by the Payley-Wiener theorem (Ch. 111, Theorem 4.1) there exists a function F in g , , ( R ) such that l?(l)=cp(++i;O for all 1 E R. Since cp

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

369

satisfies the functional equation (9.13), F is an even function of t. Hence by Proposition 9.8 and its corollary, there exists a function f ~ g ( G ) nA satisfying (9.14) and Ff=F. Thus we have proved that f= 9. q.e.d. The next aim of this section is to establish the uniqueness of the inverse spherical transform, which will be done in Theorem 9.2. For this purpose we give some elementary properties of the zonal sphericalfunction $(g, s). Most of these properties are valid for any non-compact semisimple Lie group G and a maximal compact subgroup K of G. Proposition 9.9. The zonal spherical function $(g, s) =(Vgo.yo, yo)has the following properties: 1) For each fixed g E G, $(g, s) is an entire function of s.

2) $(g, s) =$(g, 1 -s )

for every s E C and g E G,

3) $ k , s) = $ k 9 3)

for every s E C and g E G.

4) $(g-l, s)= $(g, s )

for Re s= 1/2 and g EG.

Proof. 1) From the proof of Proposition 9.5, we see that (9.15)

$(g, ~ ) = / ~ e - ~ ~ ( dk. g-',~)

Hence $(g, s) is an entire function of s. 2) If Re s= 112, then there exists a unitary operator A on .fj=L4(U,dc) such that A V;J-' = V$'A for every g E G. (Theorem 2.2) Since V$'Afo=AV$l-#fo=Ah

for every k E K, Afo must be a scalar multiple of fo.Namely there exists a complex number c such that Afo=cJb, Icl=l.

Hence we have #(g, 1- ~ ) = ( V ~ ' - ' f o , f o ) = ( V ~Ah, ' Ah)

= ( v ; ~ y o , f o ) = ~ (s) g,

for Re s= 1/2. Since $(g,s) and $(g,l -s) are entire functions of s. $(g, 1 --s)=$(g, s ) for all s E C .

370

UNITARY REPRESENTATION OF

SL (2,R)

3) The expression (9.15) of #(g, s) gives 3) immediately. 4) Since Yo*'is a unitary representation when Res- 112, we see that

fW1,s)=(vv-~O.Jfo,fo)=(fo, V v O

~ ' h ) = ~ .

Proposition 9.10. Let s=o+i2 be a complex number in D = Is E C104 Re s$ I}. Then Proof.

ldg, s>ld ldg, dl d 1 for every g E G By (9.15) we have

1

I#(g, s>ls

.

e-*t(g-l.k)dk=+(g, a)

K

If 1/2 c a < 1, then there exists the complementary series representation V'. By Propositions 4.3 and 4.4, we have m

( Vg"fO,fO).=

1

=

~ n ( u ) (Vgo.*fo,fn)Cfn,fO)

(vo~*"f~,f~) = @(g,

0)

n=-m

Since V' is a unitary representation, +(g,0 ) is positive definite and satisfies (9.16) l $ ( g , ~ ) l s d 1 , ~ ) = 1forevery g e G (Ch. 111, Proposition 3.1). By the functional equation $(g, 1-s) = #(g, s), (9.16) is valid for OO, there exists a function f E 9 ( G ) satisfying (9.17). Put (9.18)

f"(x)=

11 K

f(kxk')dkdk'.

K

Then we have f" E 9 ( G )n A and

=Ilf-gll1(t)(a-las-)$(u,, 0

(9.28)

S)dm(t).

Moreover (dm/dsn)f(s) is bounded on D. By (9.26) and Proposition 9.16, we have

1-

(~"f)^(4 = bnf)(t)$(ut, s) dm@) 0

4)dm(t)

= /d(t)(o.g(al, =sn( 1-s)nf(s)

.

Replacingfin (9.28) by d f and using (9.27), we get (9.29) and (dm/ds*)(sn(l-s>.f(s)) is bounded on D. Hence f belongs to 9. The above formula (9.29) proves the continuity of y.In fact if

Isht(l+~~+~)(o"f)(t)l5&/2n for all t 2 0 , then we have

I(d"/ds")(s"(l-s)"f(s))l

$E

(1 +tn+2)-'tmdt=Ce. q.e.d.

for all s E D.

Proposition 9.21. Iff belongs to 9,then for each integer m 2 0, there exists a constant C , such that Ix logn xf"x]l$ C,,, for all x E [ 1+ a)

.

380

UNlTARY REPRESENTATION OF

f”x] belongs to L1([l,

SL (2,R)

+ m), dx).

Pro05 Iff belongs to 9and m E N,then

s:

ZJ(t) =

Umshuf (u)du

is a bounded function of t. In fact, we have IZmf(t)I 51‘ l(1 +us)umshuf(u)l(l +u2)-1du

r-

The function Jmf(t)=

1:

umshUf’(u)du

is also bounded. For by integration by parts, we get Jmf(t)= tmshtf(t)-

and cht-sht (Itl++

a). Since

s:

(mum-lshu+u”chu)f (u)du

-sht(wf)(t)=(shtf‘(t))’, we have (u”shuf’(u))’ =murn-lshuf’(u)+ umshu(wf )(u) . Since wf belong to 9for eachf in 9, tmshtf‘(t)=mJm-l’(t)+Irnm’(t) is a bounded function oft. Then we see that tmf’(t) is bounded on R. Therefore

is bounded. Since log ( x + JxB--l) -log x, Jthat x l o g 9 * f”x]

-x(x-,

+ a), we see

+

is bounded on [l, m), for each m E N. Since (x log’ x)-1 is integrable on [2, + m), f”x] is integrable on [2, + a).Sincef”x1 is a C--function on [l, + m ) (Lemma l ) , f ” x ] is integrable on [l, 21. Hencef”~]is integrable on [l, a).

+

Theorem 9.3. (The inversion formula for 9). For every f in 9,we have fl*flf=f,

is the spherical transform f-f where the inverse spherical transform defined by

defined by (9.3) and

Y* is

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

9-* ;F Hfk)=

(9.30)

381

F(s)dg, s)dpo(s)*

J L

Proof. I f f belongs to 9,then t%ht f(t) is bounded on R. Thus log*x f [ x ] is bounded on [l, + a ) and f [ x ] is integrable on [l, +a). Hence the Radon transform -m

is defined. The above infinite integral converges uniformly with respect to x in a compact set. Similarly, by Proposition 9.21,f"xl is integrable on [l, a). Therefore the infinite integral

+

/:_f'[x+t?'ld7jl convergesuniformly with respect to x contained in each compact set. Hence F,[x] is a differentiable function of x and we have

Then we have

in exactly same way as in the proof of Proposition 9.8. Put f=2sh(t/2). Then we have 1+ p = c h t , df=ch (t/2) dt, hence (9.31)

F,'[cht]chTdt t

f(l)=G[ -m

Since there exists a constant M>O such that Ixf[x]lsM for all {l, a),we have

+

Hence F,(t) is integrable on R. By Proposition 9.20 Therefore

X E

f belongs to 9.

is a bounded function of A for all m E N and f ( + + i A ) is an integrable function of A. By Proposition 9.6, {(++in) is the Fourier transform

382

UNITARY REPRESENTATION OF

SL (2,R)

of F,(t) multiplied by 4%. By Ch. 111, Theorem 1.8, the Fourier inversion formula holds for F,(t) and we have

1'

+-

=

*

o

f(+ +i

~COSA ) tdA

.

Differentiating both sides of above equality with respect to x=cht, we get

By (9.31), (9.32) and Fubini's theorem, we get

='I 2*

+-

f(++iA)AthlsAdA .

0

The last equality follows from Lemma 3 in $8. By Proposition 9.14, 2), we have

(9.34)

(9.

Remark. The proof of Theorem 9.3 does not depend on the results of 58 except for Lemma 3 of that section which gives the Fourier transform

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

383

of (shr/2)-' on R and is independent of the other parts of $8. Hence Theorem 9.3 affords an another proof of Proposition 9.4.

Theorem 9.4. (Ehrenpreis-Mautner) The spherical transform 9 is a ' onto 9. The inverse mapping 9 - - l topological isomorphism of 9 of 9coincides with the inverse spherical transform 9-*given in (9.30). Proof. By Proposition 9.20, 9is a continuous mapping of 9 'into 9.And Theorem 9.3 proves that Y*Y f=f for every f E 9'. Let F be an arbitrary element in 9, and put f = Y * F . Since

+

+

+

sn( 1 -s)nF(s)$(g, s) = (1/4 Ra)"F(+ iR)$(g, 3 2)

is a bounded function of R for every n E N, and g , F(++iR)#(g, 3+iR) is integrable with respect to the spherical Plancherel measure p o . Hence f= r * F is defined. Now we shall prove the following two facts. (A) Y * F =f belongs to 9'. (B) 9-*is a continuous mapping from 9 into 9. If (A) is established, then Theorem 9.3 proves that 9-* Y f = f = T * F . we have Since 9-fand F belongs to 9, (C) Y f = F i.e. 99-*F=F by the uniqueness of the inverse spherical transform 9-*(Theorem 9.2). The assertions (B) and (C) together with Proposition 9.20 and Theorem 9.3 constitute the content of Theorem 9.4. Therefore it is sufficient to prove the assertions (A) and (B). Let F be an element of 9and put f = Y * F . Since Idm$(at,s)/atml$MmlP(s)l for all 220, m E N and s E D by Propositions 9.10 and 9.19, the infinite integral

1

lII+i"

Ira

F(s)(a "$/a t "1 (at, s)dpo(s)

converges uniformly on any compact subset of 0 4t < + a. Hence f is a C"-function and satisfies

dm

F f(0=

113

Therefore we have

By Propositions 7.16, 9.5 and 9.16, we have

384

UNITARY REPRESENTATION OF

SL (2,R)

Put t

y=sha2

and #(ar,s)=#(y, s).

Then we have #(Y, 4 = ( 1 +y)-*F(s,s, 1; Y/(l +Y))

-

Put z=y/(l +Y)

*

Then by Kunmmer's formula F(u, b, c; z)=(l-z)-aF(u, b, c ; z/(z-1)) #(y, s)=F(s, l-s, 1: -y). Now we put p ( y , s)=y-ar(1-2s)r(l

-s)-V(s,

(cf. ErdClyi [l] p. 64),

s,2s; -J+)

and p y y , s)=y*-T(2s- l)I-(s)-'F(l-s,

1-s, 2-2s; -y-l).

Then we have

.

(9.35) p y y , s)=#"'(y, 1-s) By Gauss' formula r ( ~ ) - ' I ; ( a , b , ~ : z ) = r ( b - ~ ) [ r ( b ) r ( ~ - ~ ) ] - l ( - ~l)--~"+Fa( ,a1,-~+u;z-') +F(U-b) [r(u)r(c-b)]-'(-z)-bF(b, 1-c+ b, 1--a+ b :z-l) (cf. ErdClyi [I] p. 63), we have (9.36) d Y , 8) = 5 w Y , s) +P Y Y , s) * By Euler's formula F(u, b,

C:

s:

z)=Z"(~)[I'(b)I'(c-b)]-~~*-l(l-r)e-a-l(l-r~)dr

(cf. ErdClyi [l], p. 59), we get

s:

=(2~)-'y-~tanns r*-l(l-r)'-l(l

+~y-l)-~dr

using r ( s ) r(l-s)=~/sinns. Since the integrand is an entire function of s, qP(y, s) is a meromorphic function of s in the half-plane Re s>O with simple poles at s= 1/2, 3/2, 5/2,... for y>O. Hence (y, s)= $(l) (y, 1-s) is a meromorphic function in the half-plane Re s< 1 with simple poles at s= 1/2, -1/2, -3/2,.- for y>O. Moreover

385

HARMONIC ANALYSIS OF ZONAL FUNCTIONS

Res $(a)(y,s) = -Res $(l'(y, s) a-l/a

a-111

.

Therefore q P ( y , s)+qP(y, s) is regular analytic in the interior D for y >0. Put W ) ( t ,s)=2-'i(+-s) tan x(+-s)#"(at,

b

of

s)

for j = 1,2 and

@"'(Y,

4=y-'+(y, 8 )

Then by (9.37), we have

Hence +(y, s) is regular analytic in the half-plane u =Re s >0 for y >0. If O0 such that

(9.38)

Put

386

UNITARY REPRESENTATION OF

1

SL (2,R)

iia+t-

f ( j ) ( y )=

F(s)@(j)(y, s)ds

(j= 1,2).

iia-to,

Then using (9.35) and the relation F(s)=F(l-s), we have (9.40) f‘a’(Y) =f‘”(v) * Hence by (9.36) and (9.443, we get

f(v)=2 - l ( f C W+f‘”(Y))

=f(’’(y)

9

namely

J

iia+i-

(9.41)

f(t)=

F(s)@“’(at,s)&.

iia-t-

For each integer n >0, we have

Since 0y,=[s(l-s)]~9J and [s(l-s)]”F(s) belongs to (9.41) for the function s”(l-s)”F(s) and get

J

9,we can apply

iia+&-

(9.43)

(o”f)(t)=

s”(l-s)”F(s)cD‘”(ae, s)&

1

iia-i-

For each y>O, F(s)O(l)(y,s) is analytic in 1/2sRe s< 1 and continuous on D s = ( s ~ C ( 1 / 2 S R e s S 1 }By(9.37)and . thefact that F e Q , wehave lim I F(s)W )(y ,s)l =0 IIm J I *-

and the convergence is uniform on D1. Hence we can shift the line of integration in (9.41) and get (9.44)

f ( t )= S1+*-F(s)@“’(u,, l-is)dr

1

1+i-

=

F(s)y-J+(at,s)ds.

l-i-

Since we have proved thatfis a C”-function, the proof of the assertion (A) at the beginning of the proof is reduced to the proof of the following assertion (A). (A) pm,n(f) O such that if

for all s E D, then IylogmYf(Y)lS BBrn.0 for all y > 0. For the purpose of proving this Lemma, we prove the more general Proposition T,,,,,,. For an m,n E N , and each function F i n 9

is bounded in y>O. Moreover we can find a constant B,,,>O the conditions IF(S)lSe(l+lSl)-',

...,

$E(~+ISI)-~ for all

such that

S ED

imply lylogLySlifmF(s)y-g,,+(y, an

5eBrn.n.

1-t-

Our Lemma 3 is the conjunction of Propositions Tm.ofor all m e N. Let W , be the conjunction of Propositions T,,,, for all n E N. Then we shall prove Proposition W , by the induction on m. Proof of Wo. For each integer n10,we have

for all s in D1={sll/4sRes g 1) and all y>O by (9.38). If F E 9 satisfies IF(s)l S e ( I + IS^)-^ for all s E D,then we have

388

UNITARY REPRESENTATION OF SL42,R)

where go,,= xC,. Proof of W,. Let m >0 and assume W , for all p <m. If F belongs to 9,then the inequality (9.45) holds for all y>O. Since d m F / h mbelongs . for every m, the repeated use of integration by parts shows that to Q

where A, and Ak are some constants. Since d m F / h mand F belong to

9,the induction hypothesis and (9.46) prove Proposition W,. q.e.d. COROLLARY to Theorem 9.4. The space 9is complete and hence a Frtchet space. Proof. 9 'is topologically isomorphic to a FrCchet space 9.

§lo. Irreducible unitary representations of SL(2,R) In sections 2,3 and 4, we have constructed irreducible unitary representations of the group SU(1,l). We have chosen SU(1,l) there instead of SL(2, R) because the behaviour of a maximal compact subgroup K in irreducible representations is easier to describ for SU(1,l). Since the group Go = SL(2, R) is isomorphic to G = SU(1,l) by the isomorphism (10.1)

y:Go

- G,

g' =

(g) = CgC-',

j ,(;

C =-

-ij)

each unitary representation V of G gives a unitary representation V' of Go. V' is defined by V' (g) = V(g'). V' is irreducible if and only if V is so. We put (10.2)

REPRESENTATION OF SL(2, a)

389

However this realization of representations for SL(2,R) is unnatural because the irreducible representations of SU(1,l) are constructed on spaces of functions on the unit disc D or the unit circle U. The group SL(2, R)/{f 1 ) is the automorphism group of the upper half-plane C+ while the group SU(l,l)/{ f1 } is the automorphism group of the unit disc. Hence it is natural to construct the representations of SL(2, R) on the space of functions on the upper half-plane C+ or its boundary, the real line R plus 00. In this section, we construct such representations for the group SL (2, R).

I. DISCRETE SERIES Let c be the Cayley transformation from the upper half-plane C+ = {z = x i y ) y > 0) onto the unit disc D = {CEC~ < 1):

+

z-i C =c(z) =z + i'

(10.3)

The inverse transformation c-l is given by (10.4)

The Cayley transformation c is a complex analytic diffeomorphism of C+ onto D. Let n E -12 and n > 112, and let 8, be the representation space of the 2

discrete series representation U" of SU (1, 1) (cf. $3). Namely sists of all holomorphic functions # on D with a finite norm

where dm (0= dudv for bY

C = u + iv. The representation

8. con-

Un is defined

(10.6)

Let H, be the Hilbert space of all holomorphic functionsfon C+with a finite norm (10.7)

where z = x

Ilfll2= (2n - l$Jc+

If ( z )I 2YZn-2dxdr

+ iy.

Proposition 10.1. 1) Let W,be the mapping from 8, into H , which maps # to f = W,# given by

UNITARY REPRESENTATIONOF SU2,R)

390

+

f ( z ) = ~ ( z i1-h I

(10.8)

z-i (m),

where A = JZ2-(*-I). Then W. is an isometry from @, onto H.. 2) Let D" be the irreducible unitary representation of SL(2, R) defined bY

0;

(10.9)

5:

Then the operator

w, u; 0

0

w;1.

0; is given by

(10.10)

for the element g-' in (10.2).

+

Proof. 1) Since the Cayley transformation C = c (z) = u iv is holomorphic, f = WnI is holomorphic on C+. Moreover we have

by the Cauchy-Riemann equations. A direct calculation gives the equality (10.11)

1

4Y

- IC12=

lz+i12'

Hence we have

X IZ

+ i14 dxdy = (2n -

If ( z ) I zyh-2 dxdy = ~

~ f ~ ~ z .

Hence W. is an isometry of @, into H.. Since (10.12)

z+i=-

2i 1 -C'

for z E C+ and

C E U,

the inverse mapping W,,-I of W. is given by (10.13)

4 (0 = (W,,-l f) (0 = A-I (2i)2n(1 - O-'"f(i l1 -+CC

).

For each function f in H,, the function 4 given by (10.13) belongs to @,, and satisfies W,, 4 =f.Hence W, is an isometry of onto H.. 2) The relations between the matrix entries of g'-I and g-' given in Proposition 1.6,l) imply that

a,,

REPRESENTATIONS OF s y 2 ,

391

R)

(10.14) Since g-' (10.15)

- C),

z = c-l ((g'-') az cz

we have

+b +d

The equalities (10.6), (10.13), (10.14) and (10.15) give

Corollary to Proposition 10.1. Let n E T1Z , n

1 and > -, 2

@-,

be the

Hilbert space on which the discrete series representation U-"is realized (cf. Theorem 3.2) and put W-, = u W,a,

where uf = f i s the complex conjugation. Then W-. is an isometry of &, onto UH,= H-,. H - , is the space of all antiholomorphic functions on C+ with the norm (10.7) finite. Thus also 0;" = W-,U>" W-,-'

(10.16)

is an irreducible unitary representation of SL (2, I?). The representation operator is given by (10.17)

11. Let 1 < u < 1 and

COMPLEMENTARY SERIES

$o

be the Hilbert space of all complex valued

measurable functions on the unit circle U with a finite norm

UNITARY REPRESENTATION OF Sa2,R)

392

where d( = (24-l d8 and & = (279-l d y for C = e'e and q = e ' y . @, is the representation space of the complementary series representation V" (cf. $4). The operator V;, for the element g' in (10.2) is given by (10.19)

Let H, be the Hilbert space of all complex valued measurable functions

f on the real line R with the norm (10.20)

Ilfll'

I X - ~ l - " " - ' " f ( ~ ) f Q d x d< y + 00.

=

PropositiOn 10.2. 1) Let S, be the mapping which maps a function to the function f = So/defined by

4 in @,

(10.21)

where A, = 2('-1)'2n-1.Then S, is an isometry of $, onto H,. 2) Put

c; = s, v;,

(10.22)

0

0

S;].

Then C" is an irreducible unitary representation of SL (2, R) on the Hilbert space H,. The representation operator C ; is given by (10.23)

for an element g - I

=

t f;).

Proof. 1) Put C = c ( x ) and q = c ( y ) (x, y Iy+ il = Iy - i l . Then it is easily verified that (10.24) (10.25)

1 - Re(Cq) =

dC = (2~)-'d8 =

E R)

21x - Y I 2

Ix

+ i I 2 Iy + il' dx

n(1

+ x')

-

and

dx nix

and note that

+ iI2'

By the equalities (10.21), (10.24) and (10.25) we have

393

onto H,. Hence S, is an isometry of 2) Since the inverse transformation S;’ : f-

4

is given by

(10.26)

we can prove (10.23) in the same way as (10.10). In fact we have

( C V )(XI

= [(SU

= Icx

v:, C’)fl(4

+ dl

-2,f

(n). m+b

111. PRINCIPAL CONTINUOUS SERIES The principal continuous series representation VJJ( j E { 0, 1/2}, s E C, Re s = 1/2) is realized on the Hilbert space @ = L2 (U,d o . The representation operator V i a is defined by

The corresponding representation PJ.’ of SL (2, R) is realized on the Hilbeit space $ = L2(R,dx). Proposition 10.3 Let j E { 0, 1/2} and Re s = 1/2. 1) Then the transformation TI,#: 4 f defined by ++

is isometry of $ = L2(U)onto H = L2 (R);and 2) if we put

394

UNITARYREPRESENTATIONOF SL(2,R)

P i J = T/.I VJ9ST-1 (10.29) r' J,r then PJUI is an irreducible unitary representation of SL (2, R). The representation operator p';* for g-' in (10.2) is given by 9

(10.30) (Pi'f) (x) = [sign (cx

+ d)I2JIcx+ dl+f(-).ax + b

Proof: 1) By (10.25) and (10.28), we have

=J- I1(x)I2dx

=

11f1I2

-0.

if Re s = 1/2. Hence Tj,, is an isometry of 8 = L2 (U)onto H = L2 (R). 2) The inverse transformation Ti,.-' is given by

By the equalities (10.14), (10.15), (10.26), (10.28) and (10.31) we have

('j'f)

= [(T,.8

'i;

T,,J-l).f]

Appendix

A. Banach-Steinhaus theorem 1. A subset B of a topological space X i s said to be nowhere dense if the closure has no inner points. A subset M of X is called meager or of the first category if M is the union of a countable family of nowhere dense subsets. If A is not of the first category, A is called of the second category. 2. A toplogical space X is called a Baire space if every non empty open subset of X is of the second category. Theorem (The Baire category theorem). A complete metric space and a locally compact Hausdorff space are Baire spaces. cf. Bourbaki [l] Ch. 9 $5. In particular a Banach space is a Baire space and of the second category. 3. Theorem (The uniform boundedness theorem). Let X be a Banach space and Y be a normed space. Let (Ta)asA be a family of continuous (=bounded) linear mappings of X into Y. If the set {TaxlaE A } is bounded for each x E X , then we have the strong limit lim T,x=O 0*0

uniformly in a E A . In particular {llTaIlla E A } is bounded. cf. Yoshida [l], p. 68, 69. Let X be a Banach space and X * be the dual space of X. X * is the Banach space of all bounded-linear forms on X with- the norm llfll= suplf(x)l. iisnsl

A Banach space Xis regarded as a subspace of the dual space X** of X * . Then Theorem A.3 implies the following theorem. Theorem (Banach-Steinhaus theorem). If a sequence ( x , ) , . ~ in a Banach space is weakly convergent, then the sequence (IIx,ll),.~ is bounded.

4.

B. Hilbert-Schmidt theorem 1. A linear mapping T of a Banach space X into a Banach space Y is called compact (or completely continuous) if every bounded subset B of 395

3 96

APPENDIX

Xis mapped onto a totally bounded subset T(B)of Y (that is, the closure T(B) is compact). Namely T is compact if the image (Tx,),,Nof each bounded sequence (x,),,*N contains a strongly convergent subsequence. 2. If H is a Hilbert space, a linear operator T on H is compact if and only if every weakly convergent sequence (X,,),,~N is mapped to a strongly convergent sequence ( T x ~ ) , , * N . The “if” part is easily proved. In fact if (x,),0, there exists an integer n such that 5.

Let xtl be the d e h i n g function of Un and put fn(x) = xn(x)f(X). Then we have (1)

Ilf-fnllp

=

f(x)dE(x).

3. The adjoint T* of a linear operator-T with dense domain D(T)is defined in Ch. V. $6, Definition 1. T is called self-adjoint if T = T*. THEOREM (von Neumann). If T is self-adjoint operator in a Hilbert space H, then there exists a unique spectral measure E on b =b(R) in H such. that

.=Iw

RdE,.

(1)

-ca

(1) is called the spectral representation of T. If T is bounded, the integral is taken over a finite interval. 4. THEOREM. Let A be a bounded self-adjoint operator on a Hilbert

space H, and A =

SI.

;tdEi be the spectral representation of A . If M is a

von Neumann algebra (cf. Ch. I, $2) containing A, then EA belongs to., M for all R E R. In particular EAE { A }

".

404

APPENDIX

Proof. If A 2 c (or A 5 -c), then En = 1 (or El = 0) belongs to M. Assume that -c < A. < c. Let E > 0 and pa (x) be a polynomical satisfying 1--e