Analysis and Geometry on Complex Homogeneous

This book, authored by five experts in their respective fields, covers a number of important areas in complex analysis and geometry. Unfolding the subjects from the basics to the more complex, the book gives a rapid paced and efficient exposition, without compromising proofs and examples. Useful as a self-study resource, it will guide readers through the subject considerably faster and more efficiently than competing books.

Preface

xi

Part I Function Spaces on Complex Semi-groups

1

(102)

Jacques Faraut

Introduction

3

(2)

Hilbert Spaces of Holomorphic Functions

5

(14)

Reproducing kernels

5

(10)

Invariant Hilbert spaces of holomorphic functions

15

(4)

Invariant Cones and Complex Semi-groups

19

(14)

Complex semi-groups

19

(2)

Invariant cones in a representation space

21

(5)

Invariant cones in a simple Lie algebra

26

(7)

Positive Unitary Representations

33

(12)

Self-adjoint operators

33

(5)

Unitary representations

38

(3)

Positive unitary representations

41

(4)

Hilbert Function Spaces on Complex Semi-groups

45

(20)

Schur orthogonality relations

45

(8)

The Hardy space of a complex semi-group

53

(6)

The Cauchy-Szego kernel and the Poisson kernel

59

(3)

Spectral decomposition of the Hardy space

62

(3)

Hilbert Function Spaces on SL(2, C)

65

(18)

Complex Olshanski semi-group in SL(2, C)

65

(2)

Irreducible positive unitary representations

67

(6)

Characters and formal dimensions of the representations πm

73

(3)

Bi-invariant Hilbert spaces of holomorphic functions

76

(2)

The Hardy space

78

(1)

The Bergman space

79

(4)

Hilbert Function Spaces on a Complex Semi-simple Lie Group

83

(20)

Bounded symmetric domains

83

(5)

Irreducible positive unitary representations

88

(8)

Characters and formal dimensions

96

(2)

Bi-invariant Hilbert spaces of holomorphic functions

98

(5)

References

99

(4)

Part II Graded Lie Algebras and Pseudo-hermitian Symmetric Spaces

103

(80)

Soji Kaneyuki

Introduction

105

(2)

Semisimple Graded Lie Algebras

107

(20)

Root theory of real semisimple Lie algebras

107

(4)

Semisimple graded Lie algebras

111

(5)

Example

116

(3)

Tables

119

(8)

Symmetric R-Spaces

127

(24)

Symmetric R-spaces and their noncompact duals

127

(6)

Sylvester's law of inertia in simple GLA's

133

(8)

Generalized conformal structures and causal structures

141

(10)

Pseudo-Hermitian Symmetric Spaces

151

(32)

Pseudo-Hermitian spaces and nonconvex Siegel domains

151

(15)

Simple reducible pseudo-Hermitian symmetric spaces

166

(17)

References

179

(4)

Part III Function Spaces on Bounded Symmetric Domains

183

(100)

Adam Koranyi

Introduction

185

(2)

Bergman Kernel and Bergman Metric

187

(6)

Domains in Cn

187

(1)

Bergman kernel, reproducing kernels

188

(2)

The Bergman metric

190

(3)

Symmetric Domains and Symmetric Spaces

193

(10)

Basic facts, definitions

193

(1)

Riemannian symmetric spaces

194

(2)

Theory of oiLa's

196

(1)

OiLa's of bounded symmetric domains

197

(3)

Cartan subalgebras

200

(3)

Construction of the Hermitian Symmetric Spaces

203

(8)

The Borel imbedding theorem

203

(2)

The Harish-Chandra realization

205

(4)

Remarks on classification

209

(2)

Structure of Symmetric Domains

211

(14)

Restricted root system, boundary orbits

211

(6)

Decomposition under the Cayley transform

217

(8)

The Weighted Bergman Spaces

225

(18)

Analysis on symmetric domains

225

(5)

Decomposition under K

230

(4)

Spaces of holomorphic functions

234

(9)

Differential Operators

243

(14)

Properties of δs

243

(3)

Invariant differential operators on Ω

246

(2)

Further results on D(Ω)

248

(3)

Extending Dα to p+

251

(6)

Function Spaces

257

(26)

The holomorphic discrete series

257

(2)

Analytic continuation of the holomorphic discrete series

259

(5)

Explicit formulas for the inner products

264

(3)

Lq-spaces and Bergman type projections

267

(3)

Some questions of duality

270

(4)

Further results

274

(9)

References

277

(6)

Part IV The Heat Kernels of Non Compact Symmetric Spaces

283

(142)

Qi-keng Lu

Introduction

285

(18)

The Laplace-Beltrami Operator in Various Coordinates

303

(18)

The Integral Transformations

321

(16)

The Heat Kernel of the Hyperball RR (m, n)

337

(14)

The Harmonic Forms on the Complex Grassmann Manifold

351

(14)

The Horo-hypercircle Coordinate of a Complex Hyperball

365

(16)

The Heat Kernel of RII(m)

381

(12)

The Matrix Representation of NIRGSS

393

(32)

References

423

(2)

Part V Jordan Triple Systems

425

(110)

Guy Roos

Introduction

427

(2)

Polynomial Identities

429

(22)

Definition of Jordan triple systems

429

(2)

Identities of minimal degree

431

(4)

Jordan representations and duality

435

(5)

The fundamental identity of degree 7

440

(1)

The Bergman operator

441

(10)

Jordan Algebras

451

(10)

Jordan algebras arising from a JTS

451

(1)

Identities in a Jordan algebra

452

(6)

The JTS associated to a Jordan algebra

458

(3)

The Quasi-inverse

461

(14)

Quasi-invertibility in a Jordan algebra

461

(5)

Quasi-invertibility in a JTS

466

(3)

Identities for the quasi-inverse

469

(1)

Differential equations

470

(2)

Addition formulas

472

(3)

The Generic Minimal Polynomial

475

(22)

Unital Jordan algebras

475

(11)

General Jordan algebras

486

(2)

Jordan triple systems

488

(9)

Tripotents and Peirce Decomposition

497

(10)

Tripotent elements

497

(1)

Peirce decomposition

498

(3)

Orthogonality of tripotents

501

(2)

Simultaneous Peirce decomposition

503

(4)

Hermitian Positive JTS

507

(22)

Positivity

507

(3)

Spectral decomposition

510

(8)

Automorphisms

518

(5)

The spectral norm

523

(2)

Classification of Hermitian positive JTS

525

(4)

Further Results and Open Problems

529

(6)

Schmid decomposition

529

(1)

Compactification of an hermitian positive JTS

530

(1)

Projective imbedding

531

(1)

Volume computations

531

(3)

Some open problems

534

(1)

References

535

(2)

Index

537

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Analysis and Geometry on Complex Homogeneous Domains

Summary

Table of Contents

Supplemental Materials