Serge lang SL2(R) (1985) 978 1 4612 5142 2
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Graduate Texts in Mathematics
105
Editorial Board
S. Axler F.W. Gehring
Springer
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Springer Books on Elementary Mathematics by Serge Lang
MATH! Encounters with High School Students
1985, ISBN 96129-1
The Beauty of Doing Mathematics
1985, ISBN 96149-6
Geometry. A High School Course (with G. Murrow)
1991, ISBN 96654-4
Basic Mathematics
1988, ISBN 96787-7
A First Course in Calculus, Fifth Edition
1998, ISBN 96201-8
Calculus of Several Variables
1987, ISBN 96405-3
Introduction to Linear Algebra
1988, ISBN 96205-0
Linear Algebra
1989, ISBN 96412-6
Undergraduate Algebra, Second Edition
1990, ISBN 97279-X
Undergraduate Analysis, Second Edition
1997, ISBN 94841-4
Complex Analysis
1993, ISBN 97886-0
Real and Functional Analysis
1993, ISBN 94001-4
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Serge Lang
With 33 Figures
i
Springer
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Serge Lang
Department of Mathematics
Yale University
New Haven, Connecticut 06520
U.S.A.
Editorial Board
S. Axler
F. W. Gehring
K.A. Ribet
Department of
Mathematics
San Francisco State
University
San Francisco, CA 94132
U.S.A.
Department of
Mathematics
University of Michigan
Ann Arbor, MI 48109
U.S.A.
Department of
Mathematics
University of California
at Berkeley
Berkeley, CA 94720
U.S.A.
AMS Subject Classification: 22E46
Library of Congress Cataloging in Publication Data
Lang, Serge
SL 2 (R).
(Graduate texts in mathematics; 105)
Originally published: Reading, Mass.:
Addison-Wesley, 1975.
Bibliography: p.
Includes index.
I. Lie groups. 2. Representations of groups.
I. Title. II. Series.
QA387.L35 1985
512'.55
85-14802
This book was originally published in 1975 © Addison-Wesley Publishing Company,
Inc., Reading, Massachusetts.
© 1985 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1985
All rights reserved. No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New
York 10010, U.S.A.
9 8 7 6 5 4 3 2 (Corrected second printing, 1998)
ISBN-13: 978-1-4612-9581-5
DOl: 10.1007/978-1-4612-5142-2
e-ISBN-13: 978-1-4612-5142-2
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Foreword
Starting with Bargmann's paper on the infinite dimensional representations of
SLiR), the theory of representations of semisimple Lie groups has evolved to
a rather extensive production. Some of the main contributors have been:
Gelfand-Naimark and Harish-Chandra, who considered the Lorentz group in
the late forties; Gelfand-Naimark, who dealt with the classical complex
groups, while Harish-Chandra worked out the general real case, especially
through the derived representation of the Lie algebra, establishing the
Plancherel formula (Gelfand-Graev also contributed to the real case); Cartan, Gelfand-Naimark, Godement, Harish-Chandra, who developed the
theory of spherical functions (Godement gave several Bourbaki seminar
reports giving proofs for a number of spectral results not accessible otherwise); Selberg, who took the group modulo a discrete subgroup and obtained
the trace formula; Gelfand, Fomin, Pjateckii-Shapiro, and Harish-Chandra,
who established connections with automorphic forms; lacquet-Langlands,
who pushed through the connection with L-series and Hecke theory. This
history is so involved and so extensive that I am incompetent to give a really
good account, and I refer the reader to bibliographies in the books by
Warner, Gelfand-Graev-Pjateckii-Shapiro, and Helgason for further information. A few more historical comments will be made in the appropriate
places in the book.
It is not easy to get into representation theory, especially for someone
interested in number theory, for a number of reasons. First, the general
theorems on higher dimensional groups require massive doses of Lie theory.
Second, one needs a good background in standard and not so standard
analysis on a fairly broad scale. Third, the experts have been writing for each
other for so long that the literature is somewhat labyrinthine.
I got interested because of the obvious connections with number theory,
principally through Langlands' conjecture relating representation theory to
elliptic curves [La 2]. This is a global conjecture, in the adelic theory. I
v
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vi
FOREWORD
realized soon enough that it was best to acquire a good understanding of the
real theory before getting everything on the adeles. I think most people who
have worked in representations have looked at SL2(R) first, and I know this is
the case for both Harish and Langlands.
Therefore, as I learned the theory myself it seemed a good idea to write
up SL 2(R). The topics are as follows:
I. We first show how a representation decomposes over the maximal
compact subgroup K consisting of all matrices
COS (J
sin (J ),
cos (J
and see that an irreducible representation decomposes in such a way that
each character of K (indexed by an integer) occurs at most once.
2. We describe the Iwasawa decomposition G = ANK, from which most
of the structure and theorems on G follow. In particular, we obtain representations of G induced by characters of A.
3. We discuss in detail the case when the trivial representation of K
occurs. This is the theory of spherical functions. We need only Haar measure
for this, thereby making it much more accessible than in other presentations
using Lie theory, structure theory, and differential equations.
4. We describe a continuous series of representations, the induced ones,
some of which are unitary.
5. We discuss the derived representation on the Lie algebra, getting into
the infinitesimal theory, and proving the uniqueness of any possible unitarization. We also characterize the cases when a unitarization is possible, thereby
obtaining the classification of Bargmann. Although not needed for the
Plancherel formula, it is satisfying to know that any unitary irreducible
representation is infinitesimally isomorphic to a subrepresentation of an
induced one from a quasicharacter of the diagonal group. The derived
representation of the Lie algebra on the algebraic space of K-finite vectors
plays a crucial role, essentially algebraicizing the situation.
6. The various representations are related by the Plancherel inversion
formula by Harish-Chandra's method of integrating over conjugacy classes.
7. We give a method of Harish-Chandra to unitarize the "discrete
series," i.e. those representations admitting a highest and lowest weight vector
in the space of K-finite vectors.
8. We discuss the structure of the algebra of differential operators, with
special cases of Harish-Chandra's results on SL 2(R) giving the center of the
universal enveloping algebra and the commutator of K. At this point, we have
enough information on differential equations to get the one fact about
spherical functions which we could not prove before, namely that there are no
other examples besides those exhibited in Chapter IV.
(
- sin
(J
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FOREWORD
va
The above topics in a sense conclude a first part of the book. The second
part deals with the case when we take the group modulo a discrete subgroup.
The classical case is SL 2(Z). This leads to inversion formulas and spectral
decomposition theorems on L2(f\ G), which constitute the remaining chapters.
I had originally intended to include the Selberg trace formula over the
reals, but in the case of non-compact quotient this addition would have been
sizable, and the book was already getting big. I therefore decided to omit it,
hoping to return to the matter at a later date.
A good portion of the first part of the book depends only on playing with
Haar measure and the Iwasawa decomposition, without infinitesimal considerations. Even when we use these, we are able to carry out the Plancherel
formula and the discussion of the various representations without caring
whether we have "all" irreducible unitary representations, or "an" spherical
functions (although we prove incidentally that we do). A separate chapter
deals with those theorems directly involving partial differential equations via
the Casimir operator, and analytical considerations using the regularity
theorem for elliptic differential equations. The organization of the book is
therefore designed for maximal flexibility and minimal a priori knowledge.
The methods used and the notation are carefully chosen to suggest the
approach which works in the higher dimensional case.
Since I address this book to those who, like me before I wrote it, don't
know anything, I have made considerable efforts to keep it self-contained. I
reproduce the proofs of a lot of facts from advanced calculus, and also
several appendices on various parts of analysis (spectral theorem for bounded
and unbounded hermitian operators, elliptic differential equations, etc.) for
the convenience of the reader. These and my Real Analysis form a sufficient
background.
The Faddeev paper on the spectral decomposition of the Laplace operator on the upper half-plane is an exceedingly good introduction to analysis,
placing the latter in a nice geometric framework. Any good senior undergraduate or first year graduate student should be able to read most of it, and
I have reproduced it (with the addition of many details left out to more expert
readers by Faddeev) as Chapter XIV. Faddeev's method comes from perturbation theory and scattering theory, and as such is interesting for its own
sake, as well as to analysts who may know the analytic part and may want to
see how it applies in the group theoretic context. Kubota's recent book on
Eisenstein series (which appeared while the present book was in production)
uses a different method (Selberg-Langlands), and assumes most of the details
of functional analysis as known. Therefore, neither Kubota's book nor mine
makes the other unnecessary.
It would have been incoherent to expand the present book to a global
context with adeles. I hope nevertheless that the reader will be well prepared
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VIII
FOREWORD
to move in that direction after having gotten acquainted with SL 2(R). The
book by Gelfand-Graev-Pjateckii -Shapiro is quite useful in that respect.
I have profited from discussions with many people during the last two
years, some of them at the Williamstown conference on representation theory
in 1972. Among them I wish to thank specifically Godement, HarishChandra, Helgason, Labesse, Lachaud, Langlands, C. Moore, Sally, Wilfried
Schmid, Stein. Peter Lax and Ralph Phillips were of great help in teaching me
some POE. I also thank those who went through the class at Yale and made
helpful contributions during the time this book was evolving. I am especially
grateful to R. Bruggeman for his careful reading of the manuscript. I also
want to thank Joe Repka for helping me with the proofreading.
Serge Lang
New Haven, Connecticut
September 1974
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Contents
Notation.
. .
. xv
Chapter I General Results
1
2
3
4
The representation on Cc(G)
A criterion for complete reducibility
L 2 kernels and operators
Plancherel measures
1
9
12
15
Chapter II Compact Groups
I Decomposition over K for SLiR)
2 Compact groups in general
19
26
Chapter HI Induced Representations
Integration on coset spaces
2 Induced representations
3 Associated spherical functions
4 The kernel defining the induced representation .
37
43
45
47
Chapter IV Spherical Functions
2
3
4
5
Bi-invariance
Irreducibility
The spherical property
Connection with unitary representations
Positive definite functions
51
53
55
61
62
Chapter V The Spherical Transform
. 67
1 Integral formulas .
ix
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x
CONTENTS
2
3
4
5
The Harish transform
The Mellin transform
The spherical transform
Explicit formulas and asymptotic expansions
69
74
78
83
Chapter VI The Derived Representation on the Lie Algebra
2
3
4
5
6
7
The derived representation
The derived representation decomposed over K
Unitarization of a representation
The Lie derivatives on G
Irreducible components of the induced representations
Classification of all unitary irreducible representations
Separation by the trace
. 89
. 100
. 108
. 113
. 116
. 121
. 124
Chapter VII Traces
I
2
3
4
5
Operators of trace class .
Integral formulas .
The trace in the induced representation
The trace in the discrete series
Relation between the Harish transforms on A and K .
Appendix. General facts about traces
.
.
.
.
127
134
147
150
. 153
. 155
Chapter VIII The Plancherel Formula
Calculus lemma
2 The Harish transforms discontinuities
3 Some lemmas .
4 The Plancherel formula .
· 164
· 166
· 169
· 172
Chapter IX Discrete Series
I
2
3
4
5
Discrete series in L 2( G) .
Representation in the upper half plane
Representation on the disc.
The lifting of weight m .
The holomorphic property .
.179
· 181
· 185
· 187
· 189
Chapter X Partial Differential Operators
The universal enveloping algebra
2 Analytic vectors
3 Eigenfunctions of C£: (f) .
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· 191
· 198
· 199
XI
CONTENTS
Chapter XI The Wei! Representation
1 Some convolutions
2 Generators and relations for SL 2
3 The Wei! representation
.205
.209
.211
°
Chapter XII Representation on L 2(f \ G)
Cusps on the group . . .
2 Cusp forms. . . . . .
3 A criterion for compact operators
4 Complete reducibility of °L 2(f\ G)
Chapter
xm
1
2
3
4
5
6
7
8
The Continuous Part of L 2(f \ G)
An orthogonality relation . . .
The Eisenstein series. . . . .
Analytic continuation and functional equation
Mellin and zeta transforms
Some group theoretic lemmas. . . . . .
An expression for TOTrp
. . . . . . .
Analytic continuation of the zeta transform of TOTrp
The spectral decomposition . . . . . . . .
Chapter XIV Spectral Decomposition of the Laplace Operator on
I
2
3
4
5
6
7
8
9
10
11
12
13
14
.219
.227
.232
.234
.239
.243
.245
.248
.251
.253
.255
.259
r\~
Geometry and differential operators on ~ . . . .
A solution of Irp = s(l - s)cp. . . . . . . . .
The resolvant of the Laplace operator on ~ for C1 > 1
Symmetry of the Laplace operator on r\~ .
The Laplace operator on r\~ . . . . . . . .
Green's functions and the Whittaker equation . . .
Decomposition of the resolvant on r\~ for C1 > 3/2 .
s(l - s)
The equation -1.);"(y) =
2
1.);(y) on [a, 00)
y
Eigenfunctions of the Laplacian in L 2(r\~) = H .
The resolvant equations for 0 < C1 < 2
The kernel of the resolvant for 0 < C1 < 2
The Eisenstein operator and Eisenstein functions
The continuous part of the spectrum
Several cusps . . . . . . . . . . . .
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. 266
. 272
. 275
.280
.284
. 287
.294
.309
. 314
.321
. 328
. 338
. 346
. 349
xii
CONTENTS
Appendix 1 Bounded Hermitian Operators and Schur's Lemma
Continuous functions of operators
2 Projection functions of operators
.355
.363
Appendix 2 Unbounded Operators
I Self-adjoint operators
2 The spectral measure
3 The resolvant formula
.369
.377
.379
Appendix 3 Meromorphic Families of Operators
Compact operators
2 Bounded operators
.383
.387
Appendix 4 Elliptic PDE
I
2
3
4
5
Sobolev spaces.
Ordinary estimates
Elliptic estimates .
Compactness and regularity on the torus
Regularity in Euclidean space
.389
.395
.400
.404
.407
Appendix 5 Weak and Strong Analyticity
I Complex theorem .
2 Real theorem
.411
.415
Bibliography
.419
Symbols Frequently Used
.423
Index .
.427
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Notation
To denote the fact that a function is bounded, we write f = 0(1). If f, g are
two functions on a space X and g ;;;. 0, we write f = O( g) if there exists a
constant C such that If(x)1 ,;;; Cg(x) for all x E X. If X = R is the real line,
say, the above relation may hold for x sufficiently large, say x ;;;. x o, and then
we express this by writing x ~ 00. Instead of f = O(g), we also use the
Vinogradov notation,
f«
g.
On a topological space X, C(X) is the space of continuous functions. If X
is a Coo manifold (nothing worse than open subsets of euclidean space, or
something like SL 2(R), with obvious coordinates, will occur), we let COO(X)
be the space of Coo functions. We put a lower index c to indicate compact
support. Hence Cc(X) and Cc""(X) are the spaces of continuous and Coo
functions with compact support, respectively.
By the way, SLz
category. An automorphism is an isomorphism of an object with itself. For
instance, a continuous linear automorphism of a normed vector space H is a
continuous linear map A: H ~ H for which there exists a continuous linear
map B: H ~ H such that AB = BA = I. A Coo isomorphism is a Coo
mapping having a CO() inverse.
If H is a Banach space, we let En(H) denote the Banach space of
continuous linear maps of H into itself. If H is a Hilbert space, we let Aut(H)
be the group of unitary automorphisms of H. We let GL(H) be the group of
continuous linear automorphisms of H with itself.
If G' is a subgroup of a group G we let
G'\G
XIII
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XIV
NOTATION
be the space of right cosets of G'. If I' operates on a set
~,
we let
be the space of I'-orbits. Certain right wingers put their discrete subgroup I'
on the right. Gelfand-Graev-Pjateckii-Shapiro and Langlands put it on the
left. I agree with the latter, and hope to turn the right wingers into left
wmgcrs.
For the convenience of the reader we also include a summary of objects
used frequently throughout the book, with a very brief indication of their
respective definitions at the end of the book for quick reference.
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I
General Results
§1. THE REPRESENTATION ON CC
Banach space (which in most of our applications will be a Hilbert space). A
representation of G in H is a homomorphism
'IT: G ~ GL{H)
of G into the group of continuous linear automorphisms of H, such that for
each vector v E H the map of G into H given by
x ~ 'IT(x)v
is continuous. One may say that the homomorphism is strongly continuous,
the strong topology being the norm topology on the Banach space. [We recall
here that the weak topology on H is that topology having the smallest family
of open sets for which all functiohals on H are continuous.]
A representation is called bounded if there exists a number C > 0 such
that 1'IT(x)1 " C for all x E G. If H is a Hilbert space and 'IT(x) is unitary for
all x E G, i.e. preserves the norm, then the representation 'IT is caned unitary,
and is obviously bounded by 1.
For a representation, it suffices to verify the continuity condition above
on a dense subset of vectors; in other words:
Let 'IT: G ~ GL(H) be a homomorphism and assume that for a dense set
of v E H the map x ~ 'IT(x)v is continuous. Assume that the image of some
neighborhood of the unit element e in G under 'IT is bounded in GL(H). Then
'IT is a representation.
This is trivially proved by three epsilons. Indeed, it suffices to verify the
continuity at the unit element. Let v E H and select VI close to v such that
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2
[I, § 1]
GENERAL RESULTS
x H
'IT{x)vJ is continuous. We then use the triangle inequality
to prove our assertion.
A representation 'IT: G ~ GL(H) is locally bounded, i.e. given a compact
subset K of G, the set 'IT(K) is bounded in GL(H).
Proof Let K be a compact subset of G. For each v E H the set 'IT(K)v is
compact, whence bounded. By the uniform boundedness theorem (Real
Analysis, VIII, §3) it foHows that 'IT(K) is bounded in GL(H).
For the convenience of the reader, we recall briefly the uniform boundedness theorem.
Let {T;} iEI be a family of bounded operators in a Banach space E, and
assume that for each vEE the set {Tjv LEI is bounded. Then the family
{ T; LEI is bounded, as a subset of End( E).
Proof Let Cn be the set of elements vEE such that
all iE/.
Then Cn is closed, and E is the union of the sets Cn' It follows by Baire's
theorem that some C n contains an open ball. Translating this open baH to the
origin yields an open baH B such that the union of the sets T;(B), i E I, is
bounded, whence the family {T;};El is bounded, as desired.
We let Cc(G) denote the space of continuous functions on G with
compact support. It is an algebra under convolution, i.e. the product is
defined by
where dy is a Haar measure on G. We shall assume throughout that G is
unimodular, meaning that left Haar measure is equal to right Haar measure.
For any functionf on G we denote by f- the functionj- (x) = j(x-'). Then
f j(x) dx f f(x=
I)
dx
=
J
f- (x) dx.
Remark. When G is not unimodular, then by uniqueness of Haar measure, there is a modular function ~: G ~ R+ which is a continuous
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[I, § l]
THE REPRESENTATION ON Cc(G)
3
homomorphism into the positive reals, such that
fGf(xa) dx
= ~(a) Lf(x) dx.
One then has
by an obvious argument. It follows that ~(x) dx is right Haar measure. The
typical non-unimodular group which will concern us, but not until Chapter
III, is the group of triangular matrices
or
For this chapter, you can forget about the non-unimodular case,
The modular function occurs in a slightly more general context than
above. Let '1": G ~ G be either an automorphism (group and topological) of
G, or an anti-automorphism, meaning
(xy)'" = y'"x'",
We write either x,. or 1'x for the effect of 'I" on an element x E G, By the
invariance of Haar measure, there exists a positive number ~('I") such that
Lf(x1') dx =
~('I") Lf(x) dx,
because the expression on the left is a non-trivial invariant positive functional
on Cc ( G). We have the obvious composition rule
~('I"(J)
=
~('I") ~«J).
In many applications, we have '1"2 = !d, and therefore ~('I") = 1, i.e. 'I" is
unimodular. This occurs in the context of matrices, when for instance 'I" is the
transpose.
The basic example of a unimodular group is the group of matrices
G
= GL,,(R).
The change of variables formula shows that Haar measure on G is equal to
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4
GENERAL RESULTS
[I, § I J
where d + x is Lebesgue measure on the additive space of n x n matrices. The
above measure on GLn(R) is therefore both right and left Haar measure.
Since
where GL2+(R) is the group of 2 X 2 matrices with positive determinant, and
R + is the group of positive reals, it follows that left Haar measure on SLiR)
is also right invariant, i.e. SLz(R) is unimodular. A better proof is to observe
that left and right Haar measures differ by a continuous homomorphism of
the group into the positive reals, and that SL 2(R) has no such non-trivial
homomorphism. (By looking at conjugacy classes of elements and using
various decompositions of SL 2(R) given later in the book, you should be able
to work this out as an exercise.) Later we shan give explicit descriptions of the
Haar measure on SL 2(R) in terms of various choices of coordinates, and
hence we do not stop here for a more thorough discussion.
We return to an arbitrary locally compact group G. Let 17 be a representation of G in H, and let cp E Cc(G). We define what will be an algebra
homomorphism
by letting
The integral is defined because x f--') cp(X)17(X)V is a continuous map with
compact support from G into H. [If one develops ordinary integration theory
in a natural way over the real or complex numbers, one sees that positivity is
not needed, only linearity and completeness in the space of values of the
functions to be integrated. Cf. my Real Analysis, for instance. Thus the
integral is the ordinary integral, with values in H.]
Let aEG and define in this section 'TaCP(x) = cp(a-Ix). Then the left
invariance of Haar measure immediately yields
(I)
Furthermore one also sees that
product, i.e.
17 1
is a homomorphism for the convolution
(2)
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[I, § 1]
5
Cc(G)
THE REPRESENTATION ON
Indeed,
'1T'(!p *~) = L(!p * ~)(X)'1T(X) dx
=
LL !p(Xy-'}~(y)'1T(X)
Reversing the order of integration and letting x
=
~
LL !p(x)~(Y)'1T(x)'1T(y)
dydx.
xy, this is
dxdy
= '1T'(!p)'1T'(~).
In the above proof, for simplicity, we omitted placing a vector v to the right
of '1T1(!p * ~), and to the right of every expression inside the integral signs. The
integrals are meant in this sense.
Since !p has compact support and '1T is locally bounded, it follows that
'1T1(!p) is a bounded operator, i.e. '1T1(!p)EEnd(H).
If '1T is a bounded representation, then instead of using functions
!pECc(G), we could have taken functionsfEe'(G) and formulas (1), (2)
remain valid. In other words, '1T1 extends to el(G), and furthermore we have
the inequality
(3)
Thus '1T I is a continuous linear homomorphism (representation) of e I( G) into
End( H), as Banach algebras.
If H is a Hilbert space, and'1T is unitary, then we also have the formula
(4)
where !p* is the function such that !p*(x) = !p(x I). This follows at once
from the definition of the symbols involved.
One can recover the values '1T(a) for aE G by knowing the values '1T1(!p)
for !p E Cc ( G), as follows. By a Dirac sequence on G we mean a sequence of
functions {!Pn}' real valued, in Cc(G), satisfying the following properties:
DIR 1. We have !Pn ;.. 0 for all n.
DIR 2. For all n, we have
L
!Pn(x) dx = 1.
DIR 3. Given a neighborhood V of e in G, the support of!p" is contained in
V for all n sufficiently large.
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6
GENERAL RESULTS
[I, § I]
The third condition shows that for large n, the area under CPn is concentrated
near the origin. A Dirac sequence looks like Fig. 1.
Figure 1
I t is obvious that Dirac sequences exist. If G has a Coo structure, like SLiR),
one can even take the functions CPn to be Coo. It is frequently convenient to
use a slightly weaker condition than DIR 3, namely
DIR 3'. Given a neighborhcod V of e in G, and its complement Z, and to, we
have
h
CPnex) dx
<
E:
for all n sufficiently large.
In other words, instead of assuming that the supports of the functions CPn
shrink to e, we merely assume the corresponding L 1 condition. It is slightly
more intuitive to work with the stronger condition which suffices for almost
all applications. When the need arises for the condition DIR 3', we shall
assume that the reader can verify for himself the needed convergence statements valid with the same proof as for the other case.
As will be mentioned later when we discuss analytic vectors, the condition DIR 3' becomes essential if we want the function CPn to be analytic
functions (they cannot have compact support).
At the beginning of this book, and for several chapters, we are principally
interested in the measure theoretic aspects, or the COO aspects, of representations. Consequently we don't need any more about Dirac sequences than
their definitions. It may nevertheless be helpful to realize explicitly that some
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[I, § 1]
THE REPRESENTATION ON Cc(G)
7
convolutions arising in the classical literature are taken with Dirac sequences.
We have the fan owing examples on R.
i) Let cp be a Coo function on R which is positive, has compact support,
and is such that
L
OO
oo
cp(t) dt = 1.
Then the sequence cp,,(t) = ncp(nt) is a Dirac sequence.
ii) Let cp(t) = 'IT- 1/ 2e- I ". Let cp" be defined by the same formula as in (i).
Then {cp,,} is a Dirac sequence.
iii) Let
Then {cp.} is a Dirac family for
Dirac sequence take t: = 1/ n).
E
~
0 (in an obvious sense, to get the
In cases Oi) and (iii) the factor involving 'IT is there to insure that the
integral is equal to 1. The verification that the above are Dirac sequences is at
the level of freshman calculus. Note that the examples (ii) and (iii) do not
have compact support. Example (ii) is the one which is useful in the discussion of analytic vectors. For a use of Example (iii), see Appendix 2, §3. The
Fejer and Poisson kernels in the theory of Fourier series also provide
examples of Dirac sequences. The explicit formulas are irrelevant for the
basic properties, and we now return to the general properties of Dirac
sequences, even reproducing some basic approximation results from Real
Analysis.
Let {cp,,} be a Dirac sequence. Then for each vEH, the sequence {'lT1CCP,,}v}
converges to v.
Proof We have
L
cp,,(x)'IT(x)v dx - v =
=
L
[CPn(x)'IT(x) - cp,,(x)]v dx
h cp,,(x)['IT(x)v So
vJ dx,
where SrI is the support of CPn' From the continuity condition on a representation, it is clear that this last integral tends to 0 as n ~ 00.
Let a E G. If {cp,,} is a Dirac sequence, then {7"aCP,,} is a Dirac sequence at
a (in the obvious sense). It is clear from (1) that
'lT1(7"affJn)v ~ 'IT(a)v
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8
[I, § I]
GENERAL RESULTS
as n ~ 00. The value 'IT(a)v is therefore obtained as a limit of values 'lT1(qJ)V
for suitable functions qJ E Cc ( G).
Let W be a subspace of H. (By subspace we shall always mean closed
subspace unless otherwise specified, in which case we sayan algebraic
subspace.) By a dense subspace we mean a dense algebraic subspace. We say
that W is G-invariant if 'IT(x) W c W for aU x E G. We make a similar
definition for Cc (G)-invariant.
Quite generally, let S be a family of operators on H. We say that W is
S-invariant if AWe W for every A E S. Let Wo be a dense algebraic subspace
of W. If Wo is S-invariant, then it is clear that W is also S-invariant.
From the limiting property obtained above, we conclude:
A subspace W of H is G-invariant if and only if W is Cc(G)-invariant.
Let (f be a dense subspace of el(G) and assume that 'IT is bounded. A
subspace W of H is G-invariant if and only if W is also (f -invariant.
For the convenience of the reader, we also recall convergence properties
G).
of Dirac convolutions in
e'(
Let fE el(G) and let Z be a compact set on whichf is continuous. Let {qJn}
be a Dirac sequence. Then qJn * f converges to f uniformly on Z.
Proof We have
qJn *f(x)
f(x}
f
=f
=
qJn(xy-l)f(y) dy =
f
qJn(y)f(y-'x) dy
qJn{y)f(x) dy.
Hence
There exists a neighborhood U of e in G such that if y E U, then for all x E Z,
we have
For n large, the support of qJn is contained in U, whence our integral is
concentrated in U, and is obviously estimated by £. This proves our assertion.
The support of qJn * f is contained in (supp qJn)(supp j), because in the
integral for the convolution, we can limit the integral to xy - I E supp qJ and
yEsuppj. Hence:
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[1, §2]
9
A CRITERION FOR COMPLETE REDUCIBILITY
If f is continuous with compact support, then {
Since Cc(G) is LI-dense in e\G), we obtain also:
Let f E
el ( G). Then
Proof First find
{
1I
fll! <
t:.
Then
Since II g . . hili';;; II glllllhih for two functions g, hE el(G), and since 11
The same argument applies to LP instead of L 1, 1 .;;; p
purposes, the most we would want it for is L2.
<
00.
For our
§2. A CRITERION FOR COMPLETE REDUCIBILITY
Let
'TT:
G ~ GL(H)
and
'TT':
G
~
GL(H')
be representations. A morphism of 'TT into 'TT' is a continuous linear map
~ H' such that for every x E G the following diagram is commutative.
A: H
H
A
~
".(x) ~
H
H'
~ 'If'(x}
~
A
H'
(In the literature, a morphism is sometimes caned an intertwining operator.)
We say that A is an embedding if A is a topological linear isomorphism of H
onto a subspace of H'. We say that A is an isomorphism if there is a
morphism B of 'TT' into 'TT such that AB and BA are the identities of H' and H
respectively. An isomorphism is also called an equivalence. When H, H' are
Hilbert spaces, and 'TT, 'TT' are unitary, then we may deal exclusively with
unitary maps, i.e. require that A be unitary. The context will always make it
clear whether this additional restriction is intended. We say that 'TT occurs in 'TT'
if there exists an embedding of 'TT in 'TT'.
A representation p: G ~ GL(E) is called irreducible if E has no invariant subspace other than {OJ and E itself. Let S be a set of operators on E.
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10
GENERAL RESULTS
[I, §2]
We say that E is S-irreducible if E has no S-invariant subspace other than
{O} and E itself.
Let H be a Hilbert space. If there exist irreducible subspaces E 1, ••• ,Em
of H which are all G-isomorphic (under 'IT) to (p, E), and such that H can be
expressed as a direct sum
and F contains no subspace 'IT ( G)-isomorphic to the Ei' then we say that E
occurs with multiplicity m in H. It is easy to see that if this is the case, then in
any expression of H as a direct sum,
H = E; E9 E2 E9 ... E9 E; E9 P,
where the E/ are 'IT ( G)-isomorphic to the E i , and (p, E) does not occur in P,
then r = m. For the needed technique to reduce the proof to standard
algebraic arguments of semi-simplicity, see Real Analysis, Chapter VII, Exercise 19. We call m the multiplicity of p in 'IT (or of E in H).
Let H be a Hilbert space and 'IT a representation of G in H. We say that
H is completely reducible for 'IT, or that 'IT is completely reducible, if H is the
orthogonal direct sum of irreducible subspaces. We write such a direct sum as
"
H=EBH,.,
iEI
where {i} ranges over a set of indices I, the Hi are subspaces invariant under
G, mutually orthogonal, and H is the closure of the algebraic space generated
by the Hi' This closure is indicated by the roof over the direct sum sign,
which signifies algebraic direct sum. We also say that the family {Hi} is an
orthogonal decomposition of H.
Let A; H ---? H be an operator (continuous linear map). We recall that A
is called compact if A maps bounded sets into relatively compact sets (sets
whose closure is compact). Alternatively, we could say that if {vn } is a
bounded sequence, then {Avn} has a convergent subsequence. A vector v E H
is called an eigenvector for A if Av = AV for some complex number A. Given
AEC, the set of elements v E H such that Av = AV, together with 0, is a
subspace H", called the A-eigenspace of A.
Spectral theorem for compact operators. Let A be a compact hermitian
operator on the Hilbert space E. Then the family of eigenspaces {E,,}, where
A ranges over all eigenvalues (including 0), is an orthogonal decomposition
of E.
Proof Let F be the closure of the subspace generated by all E". Let H
be the orthogonal complement of F. Then H is A -invariant, and A induces a
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[I, §2]
II
A CRITERION FOR COMPLETE REDUCIBILITY
compact hermitian operator on H, which has no eigenvalue. We must show
that H = {O}. This will follow from the next lemma.
Lemmo. Let A be a compact hermitian operator on the Hilbert space
H -=1= {O}. Let c = IAI. Then cor - c is an eigenvalue for A.
Proof There exists a sequence {xn } in H such that
Ixnl
=
1 and
Selecting a subsequence if necessary, we may assume that
<Axn, x n> ~ a
for some number a, and a = ± IA I. Then
o,
IAxn - ax,,12 = <Ax" - ax". Ax" - ax,,>
The right-hand side approaches 0 as n tends to infinity. Since A is compact,
after selecting a subsequence, we may assume that {Ax,,} converges to some
vector y, and then {ax,,} must converge to y also. If a = 0, then IA I = 0 and
A = 0, so we are done. If a -=1= 0, then {xn } itself must converge to some
vector x, and then Ax = ax so that a is the desired eigenvalue for A, thus
proving our lemma, and the theorem.
We observe that each EA has a Hilbert basis consisting of eigenvectors,
namely any Hilbert basis of EA because all non-zero elements of EA are
eigenvectors. Hence E itself has a Hilbert basis consisting of eigenvectors.
Thus we recover precisely the analog of the theorem in the finite dimensional
case. Furthermore, we have some additional information, which follows
trivially:
Each EA is finite dimensional if A -=1= 0, otherwise a denumerable subset
from a Hilbert basis would provide a sequence contradicting the compactness
of A. For a similar reason, given r > 0, there is only a finite number of
eigenvalues A such that IAI ~ r. Thus 0 is a limit of the sequence of eigenvalues if E is infinite dimensional. If H is a Hilbert space and A a compact
operator on H, we may therefore write
00
"
A
H
= ffi
HA
A
=
ffiI H A •,
j=
where the eigenvalues Aj are so ordered that IAi+ Ii
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' IAil,
and lim A; =
o.
