Analysis and Geometry on Complex Homogeneous Domains

528 Pages·2000·21.835 MB·English

by  Jacques Faraut, Soji Kaneyuki, Adam Korányi, Qi-keng Lu, Guy Roos (auth.)

Progress in Mathematics
Volume 185
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
J acques Faraut
Soji Kaneyuki
Adam Koranyi
Qi-keng Lu
GuyRoos
Analysis and Geometry on
Complex Homogeneous
Domains
Springer Science+Business Media, LLC
Jacques Faraut Soji Kaneyuki
Institut de Mathematiques Department of Mathematics
Universite Pierre et Marie Curie Sophia University
F-75252 Paris Chiyoda-ku, Tokyo 102-8554
France Japan
Adam Korănyi Qi-keng Lu
Dept. Mathematics & Computer Science Institute of Mathematics
H.H. Lehman College Academia Sinicia
Bronx, NY 10468-1589 Beijing 100080
USA China
Guy Roos
Departement de Matbematiques
Universite de Poitiers
F-86022 Poitiers Cedex
France
Ubrary of Congress Cataloging-in-Publication Data
Analysis and geometry on complex homogeneous domains / Jacques Faraut ... [et aL].
p. cm. - (Progress in mathematics ; v. 185)
Includes bibliographical references and index.
ISBN 978-1-4612-7115-4 ISBN 978-1-4612-1366-6 (eBook)
DOI 10.1007/978-1-4612-1366-6
1. Functions of several complex variables. 2. Mathematical analysis. 3. Geometry. 1.
Faraut, Jacques, 1940- II. Progress in mathematics (Boston, Mass.); voI. 185.
QA331.7.A5 2000 99-055697
515'.94-<1c21 CIP
AMS Subject Classifications: Primary-32MlO, 32M15, 32A37, 22E46, 43A85
Secondary-32-02,22-01,22-02
Printed on acid-free paper.
©2000 Springer Science+Business Media New York
Originally published by Birkhliuser Boston in 2000
Softcover reprint of the hardcover 1s t edition 2000
Chinese Edition, ©Science Press Beijing
AII rights reserved. This work may not be translated or copied in whole or in part without the written
permissionofthepublisher Springer Science+Business Media, LLC,
except forbrief excerpts in connection with reviews or scholarlyanalysis.
Use in connection with any form of information storage and retrieval, electronic adaptat ion, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by the
Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7115-4 SPIN 10728121
Typeset by Guy Roos in Jt\TEX,
9 8 765 4 3 2 1
Contents
Preface xi
Part I Function Spaces on Complex Semi-groups
by Jacques Faraut 1
Introduction 3
I Hilbert Spaces of Holomorphic Functions 5
Ll Reproducing kernels . . . . . . . . . . . . . . . .. 5
1.2 Invariant Hilbert spaces of holomorphic functions. 15
II Invariant Cones and Complex Semi-groups 19
ILl Complex semi-groups. . . . . . . . . . . . 19
11.2 Invariant cones in a representation space. 21
11.3 Invariant cones in a simple Lie algebra 26
III Positive Unitary Representations 33
III.1 Self-adjoint operators ...... 33
III.2 Unitary representations ..... 38
III.3 Positive unitary representations . 41
vi Contents
IV Hilbert Function Spaces on Complex Semi-groups 45
IV.1 Schur orthogonality relations . . . . . . . . . . . 45
IV.2 The Hardy space of a complex semi-group. . . . 53
IV.3 The Cauchy-Szego kernel and the Poisson kernel 59
IV.4 Spectral decomposition of the Hardy space 62
q
V Hilbert Function Spaces on SL(2, 65
q
V.l Complex Olshanski semi-group in SL(2, 65
V.2 Irreducible positive unitary representations 67
V.3 Characters and formal dimensions of the represen-
tations 7rm . . . . . . . • . . . . . . . . . . . . . • •. 73
V.4 Bi-invariant Hilbert spaces of holomorphic functions 76
V.5 The Hardy space . . 78
V.6 The Bergman space .................. 79
VI Hilbert Function Spaces on a Complex Semi-simple
Lie Group 83
Vl.l Bounded symmetric domains . . . . . . . . 83
VI.2 Irreducible positive unitary representations 88
VI.3 Characters and formal dimensions ..... 96
VI.4 Bi-invariant Hilbert spaces of holomorphic functions 98
References 99
Part II Graded Lie Algebras and Pseudo-hermitian
Symmetric Spaces
by Soji Kaneyuki 103
Introduction 105
I Semisimple Graded Lie Algebras 107
1.1 Root theory of real semisimple Lie algebras .107
1.2 Semi simple graded Lie algebras .111
1.3 Example. .116
1.4 Tables ....... .119
II Symmetric R-Spaces 127
11.1 Symmetric R-spaces and their noncom pact duals .127
11.2 Sylvester's law of inertia in simple GLA's .... .133
Contents vii
II.3 Generalized conformal structures and causal
structures . . . . . . . . . . . . . . . . . 141
III Pseudo-Hermitian Symmetric Spaces 151
III.1 Pseudo-Hermitian spaces and nonconvex Siegel
domains . . . . . . . . . . . . . . . . . . . . .. . 151
III.2 Simple reducible pseudo-Hermitian symmetric
spaces . . . . . . . . . . . . . . . . . . . . . . . . 166
References 179
Part III Function Spaces on Bounded
Symmetric Domains
by Adam K oranyi 183
Introduction 185
I Bergman Kernel and Bergman Metric 187
en . . . . . . . . . . . .
1.1 Domains in .187
1.2 Bergman kernel, reproducing kernels .188
1.3 The Bergman metric . . . . . . . . . .190
II Symmetric Domains
and Symmetric Spaces 193
11.1 Basic facts, definitions ..... .193
11.2 Riemannian symmetric spaces . .194
II.3 Theory of oiLa's . . . . . . . . .196
11.4 OiLa's of bounded symmetric domains .197
11.5 Cartan subalgebras . . . . . . . . . . . .200
III Construction of the Hermitian Symmetric Spaces 203
III. 1 The Borel imbedding theorem . . . 203
111.2 The Harish-Chandra realization . . 205
111.3 Remarks on classification .... . 209
IV Structure of Symmetric Domains 211
IV.1 Restricted root system, boundary orbits . 211
IV.2 Decomposition under the Cayley transform . 217
viii Contents
V The Weighted Bergman Spaces 225
V.l Analysis on symmetric domains . .225
V.2 Decomposition under K ..... .230
V.3 Spaces of holomorphic functions .234
VI Differential Operators 243
VLl Properties of b.s . . . . . . . . . . . .243
VL2 Invariant differential operators on 0 .246
VL3 Further results on JI)(O) .248
VI.4 Extending Do to p+ .251
VII Function Spaces 257
VILI The holomorphicdiscrete series . . . . . . . . . . . . 257
VII.2 Analytic continuation of the holomorphic discrete
series. . . . . . . . . . . . . . . . . . . . . . 259
VIL3 Explicit formulas for the inner products . . 264
VII.4 Lq-spaces and Bergman type projections . 267
VII.5 Some questions of duality . 270
VII.6 Further results . . . . . . . 274
References 277
Part IV The Heat Kernels of Non Compact
Symmetric Spaces
by Qi-keng Lu 283
I Introduction 285
II The Laplace-Beltrami Operator in
Various Coordinates 303
III The Integral Transformations 321
IV The Heat Kernel of the Hyperball RIR(m, n) 337
V The Harmonic Forms on the Complex
Grassmann Manifold 351
VI The Horo-hypercirde Coordinate of a Complex
Hyperball 365
Contents ix
VII The Heat Kernel of'R]](m) 381
VIII The Matrix Representation of NIRGSS 393
References 423
Part V Jordan Triple Systems
by Guy Roos 425
Introduction 427
I Polynomial Identities 429
I.1 Definition of Jordan triple systems .429
L2 Identities of minimal degree . . . . .431
L3 Jordan representations and duality .435
104 The fundamental identity of degree 7 . .440
L5 The Bergman operator . . . . . . . . . .441
II Jordan Algebras 451
11.1 Jordan algebras arising from a JTS . .451
11.2 Identities in a Jordan algebra .... .452
11.3 The JTS associated to a Jordan algebra .458
III The Quasi-inverse 461
111.1 Quasi-invertibility in a Jordan algebra .461
I1L2 Quasi-invertibility in a JTS . . .466
I1L3 Identities for the quasi-inverse . .469
IlIA Differential equations . .470
I1L5 Addition formulas ...... . .472
IV The Generic Minimal Polynomial 475
IV.1 Unital Jordan algebras . .475
IV.2 General Jordan algebras .486
IV.3 Jordan triple systems .. .488
V Tripotents and Peirce Decomposition 497
V.1 Tripotent elements .... . .497
V.2 Peirce decomposition ........ . .498
V.3 Orthogonality of tripotents .... . .501
VA Simultaneous Peirce decomposition. .503
x Contents
VI Hermitian Positive JTS 507
VI.1 Positivity ...... . .507
VI.2 Spectral decomposition. .510
VI.3 Automorphisms . . . . . .518
VIA The spectral norm ... .523
VI.5 Classification of Hermitian positive JTS .525
VII Further Results and Open Problems 529
VII.1 Schmid decomposition . . . . . . . . . . . . .. . 529
VII.2 Compactification of an hermitian positive JTS . 530
VII.3 Projective imbedding. . 531
VilA Volume computations . 531
VII.5 Some open problems .534
References 535
Index 537