Representation Theory and Noncommutative Harmonic Analysis II: Homogeneous Spaces, Representations and Special Functions

273 Pages·1995·6.56 MB·English

by  V. F. Molchanov (auth.), A. A. Kirillov (eds.)

Encyclopaedia of
Mathematical Sciences
Volume 59
Editor- in-Chief: R. V. Gamkrelidze
Springer-Verlag Berlin Heidelberg GmbH
A. A. Kirillov (Ed.)
Representation Theory
and Noncommutative
Harmonic Analysis II
Homogeneous Spaces, Representations
and Special Functions
With 2 Figures
Springer
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko,
N.M. Ostianu, V.P. Sakharova, A.B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 59, Teoriya predstavlenij i nekommutativnyj
garmonicheskij analiz 2
Publisher VINITI, Moscow 1990
Mathematics Subject Classification (1991):
22E15, 22E45, 22E46, 33-02, 33C80, 43A85, 43A90, 44A05, 53C15
ISBN 978-3-642-08126-2
Ubrary of Congress Cataloging-in-Publication Data
Teoriia predstavlenil i nekommutativnyl garmonicheskil analiz II.
English. Representation theory and noncommutative harmonic analysis II I A. A. Kirillov (ed.).
p. cm. - (Encyclopaedia of mathematical sciences; v. 59)
Includes bibliographical references and indexes.
ISBN 978-3-642-08126-2 ISBN 978-3-662-09756-4 (eBook)
DOI 10.1007/978-3-662-09756-4
1. Representations of groups. 2. Kac-Moody algebras .. 3. Harmonic anaIysis.
1. KiriIlov, A. A. (Aleksandr Aleksandrovich), 1936-. II. litle. III. Series.
QAI76.T46213 1995 512'.55-dc20 95-1411 CIP
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed,
specificaIly the rights of translation, reprinting, reuse of iIIustrations, recitation, broadcasting, reproduction on
microfilm or in any other way, and storage in data banks. Duplication ofthis publication or parts thereofis permitted
only under the provisions ofthe German Copyright Law of September 9, 1965. in its current version, and permission
for use must always be obtained from Springer-VerIag Berlin Heidelberg GmbH.
Violations are Iiable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1995
OriginaIly pubIished by Springer-VerIag Berlin Heidelberg New York in 1995
Typesetting: Camera-ready copy produced from the authors'
and translators' input files using a Springer TEX macro package.
SPIN 10031273 41/3140 - 5 4321 0- Printed on acid-free paper
List of Editors, Authors and Translators
Editor-in-Chief
R. v. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul.
Vavilova 42, 117966 Moscow, Institute for Scientific Infonnation (VINITI), ul.
Usievicha 20a, 125219 Moscow, Russia; e-mail:
Contents
I. Harmonic Analysis on Homogeneous Spaces
V. F. Molchanov
1
ll. Representations of Lie Groups and Special Functions
A. U. Klimyk, N. Va. Vilenkin
137
Author Index
261
Subject Index
264
I. Harmonic Analysis on
Homogeneous Spaces
V. F. Molchanov
Translated from the Russian
by G. van Dijk and V. F. Molchanov
Contents
§o. Introduction ............................................... 3
Chapter 1. Harmonic Analysis on Homogeneous Spaces
of Lie Groups ................................................. 11
§1 . Manifolds ................................................. 11
§2. Lie Groups and Lie Algebras ................................ 12
§3. Homogeneous Spaces of Lie Groups ........................... 16
§4. Symmetric Spaces .......................................... 18
§5. Semisimple Symmetric Spaces ............................... 21
§6. Riemannian Symmetric Spaces ............................... 26
§7 . Invariant Differential Operators .............................. 28
§8. Unitary Representations of Class One ......................... 33
§9. Relative Discrete Series ..................................... 39
§10. Gelfand Pairs .............................................. 41
§1 1. Harmonic Analysis on Symmetric Spaces
of the Non-Compact Type ................................... 45
§12. Harmonic Analysis on Symmetric Spaces
of the Compact Type ....................................... 52
§13. A Duality Principle ........................................ 56
§14. Harmonic Analysis on Semisimple Symmetric Spaces ........... 60
2 V. F. Molchanov
Chapter 2. Harmonic Analysis on Semisimple Symmetric Spaces
of Rank One .................................................. 62
§15. Semisimple Symmetric Lie Algebras of Rank One .............. 63
§16. The Tangent Representation ................................. 68
§17. Orbits of the Group H on X ................................ 75
§18. The Averaging Map ........................................ 78
§1 9. The Manifolds Y and S ..................................... 83
§20. The Laplace-Beltrami Operator .............................. 85
§21. The Group V .............................................. 87
§22. Orbits of the Group H on the Manifold S ..................... 89
§23. Representations of the Non-Unitary Principal Series ............ 91
§24. Intertwining Operators ............. . . . . . . . . . . . . . . . . . . . . . . . . 92
§25. Intertwining Operators on Simplest K-types ................... 97
§26. Invariants of the Group H ........ . . . . . . . . . . . . . . . . . . . . . . . . . . 99
§27. The Fourier and Poisson Transform. Spherical Functions ........ 101
§28. Eigenfunctions of the Radial Part of the
Laplace-Beltrami Operator .................................. 104
§29. Explicit Expressions for the Spherical Functions ................ 108
§30. The Plancherel Formula ..................................... 110
§31. The Case r+ = 1 ........................................... 120
References .................................................... 127
I. Harmonic Analysis on Homogeneous Spaces 3
§O. Introduction
Harmonic analysis on homogeneous spaces is a far-reaching generalization
of the classical theory of Fourier series and Fourier integrals. It is a branch of
functional analysis which is vigorously developing now. The principal contents
is closely connected with group representation theory in infinite-dimensional
spaces. On the other hand, it interacts with such diverse fields as algebra,
algebraic geometry, spectral theory of operators, number theory, Hamiltonian
mechanics, quantum mechanics.
The main problem of harmonic analysis can be stated as follows. Let G
be a Lie group, H a closed subgroup, X the homogeneous space G / H. As a
rule we shall realize homogeneous spaces as right coset spaces, so G acts on
X from the right: x -+ xg. Let L be a G-invariant space of functions on X,
then a representation U of G acts on L by translations:
U(g)f(x) = f(xg). (0.1)
The problem is to determine the structure of U, namely to find out whether
U is irreducible and if not what the composition series of U is; in particular
if U can be decomposed into a direct sum of irreducible representations then
one wants to know which representations enter into the decomposition and
what their multiplicities are. If U is unitary one would like to write down a
Plancherel formula and an inversion formula.
A unitary U appears for example when there exists a positive measure dx
on X which is invariant under G. For L we then take the space L2(X) with
respect to dx provided with the inner product
(h, h) = Ix h(x)h(x)dx.
The operators (0.1) preserve this inner product, so the representation U of G
by translations on L2(X) is unitary. It is called the quasiregular representation
of G on X. The most important problem of harmonic analysis is to decom-
pose (as explicitly as possible) the quasiregular representation into irreducible
unitary ones.
Very often there are differential operators on X which commute with the
action of G and which are essentially self-adjoint on L2(X). In that case G
preserves their spectral decomposition, and the decomposition of L2(X) in
terms of eigenfunctions of these operators is often an important step in the
decomposition of U.
Let us therefore consider the algebra D(X) of all G-invariant differential
operators on X. Let X be a homomorphism D(X) -+ C. The subspace Ex(X)
of COO(X) which consists of functions f satisfying
Df = x(D)f, DE D(X),
4 V. F. Molchanov
is called a joint eigenspace for D(X). This subspace is invariant under the
translations (0.1). Take it as L. Denote by Tx the corresponding representa-
tion of G on Ex. by translations. In order to give the general problem for-
mulated above a more concrete form, let us pose the following problems (cf.
Helgason 1984): a) describe Ex; b) determine for which X the representation
Tx. is irreducible; c) expand an "arbitrary" function on X in terms of joint
eigenfunctions of D(X).
At present harmonic analysis is most advanced for semisimple symmetric
spaces. The last decades have shown a growing interest in this field. Really
brilliant and deep results have been obtained here. These spaces (for their def-
inition, see §4) form a very important and wide class of homogeneous spaces,
containing for example:
(a) Riemannian semisimple symmetric spaces, see Helgason 1962. The clas-
sification of such spaces from a local point of view was given in 1926-27 by E.
Cartan, the father of the theory of symmetric spaces.
(b) Semisimple Lie groups. A semisimple Lie group G1 can be viewed as a
space G / H where G = G 1 X Gland H is the diagonal, that is H = {(g, g) I
9 E Gd· The local classification was given by E. Cartan in 1914.
(c) Hyperbolic spaces, see Wolf 1972. These are divided into three classes:
SOo(p,q) /SOo(p,q -1),
SU(p,q)/S(U(p,q -1) x U(I)).
Sp(p,q)/Sp(p,q -1) x Sp(l)
the real hyperbolic spaces (hyperboloids), the complex hyperbolic spaces and
the quaternion hyperbolic spaces respectively.
The classification of semisimple symmetric spaces from a local point of view
was given in Berger 1957, see also Koh 1965. For the list, see Table 7; it is
sufficient to assume that G is simple.
Let us briefly touch upon the history of the problem.
For compact (hence Riemannian) symmetric spaces X = G/H (here G is
compact) harmonic analysis was developed by E. Cartan in 1929 in his classical
work Cartan 1929. He showed that the quasiregular representation of G on X
is decomposed into a multiplicity-free direct sum of irreducible representations
of class one with respect to H (that is representations having a non-zero H-
invariant vector). In every irreducible constituent V C L2(X) there is just
one (appropriately normalized) spherical function, that is a function invariant
under H; the linear combinations of its translates are dense in V. Moreover
he showed that if X is a sphere then the V are precisely the eigenspaces of
the Laplace operator on X.
For arbitrary Riemannian symmetric spaces (H compact) spherical func-
tions were introduced in the fundamental work of LM. Gel'fand (cf. Gel'fand
1950). He recognized their importance and stated their basic properties. Some
of his ideas were further developed in the paper Berezin-Gel'fand 1956. For
Riemannian symmetric spaces of the non-compact type, a decomposition of