Analysis and Geometry on Complex Homogeneous Domains

1. Front Matter

   Pages i-xvii

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2. Function Spaces on Complex Semi-groups

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction

      • Jacques Faraut

      Pages 3-4

   3. Hilbert Spaces of Holomorphic Functions

      • Jacques Faraut

      Pages 5-18

   4. Invariant Cones in a Lie Algebra and Complex Semi-groups

      • Jacques Faraut

      Pages 19-32

   5. Positive Unitary Representations

      • Jacques Faraut

      Pages 33-43

   6. Hilbert Function Spaces on Complex Semi-groups

      • Jacques Faraut

      Pages 45-63

   7. Hilbert Function Spaces on a Complex Olshanski Semi-group in SL (2, ℂ)

      • Jacques Faraut

      Pages 65-81

   8. Hilbert Function Spaces on a Complex Olshanski Semi-group in a Complex Semi-simple Lie Group

      • Jacques Faraut

      Pages 83-102

3. Graded Lie Algebras, Related Geometric Structures and Pseudo-hermitian Symmetric Spaces

   1. Front Matter

      Pages 103-103

      PDF

   2. Introduction

      • Soji Kaneyuki

      Pages 105-106

   3. Semisimple Graded Lie Algebras

      • Soji Kaneyuki

      Pages 107-126

   4. Symmetric R-Spaces

      • Soji Kaneyuki

      Pages 127-150

   5. Pseudo-Hermitian Symmetric Spaces

      • Soji Kaneyuki

      Pages 151-182

4. Function Spaces on Bounded Symmetric Domains

   1. Front Matter

      Pages 183-183

      PDF

   2. Introduction

      • Adam Korányi

      Pages 185-186

   3. Bergman Kernel and Bergman Metric

      • Adam Korányi

      Pages 187-191

   4. Symmetric Domains and Symmetric Spaces

      • Adam Korányi

      Pages 193-202

   5. Construction of the Hermitian Symmetric Spaces

      • Adam Korányi

      Pages 203-210

   6. Structure of Symmetric Domains

      • Adam Korányi

      Pages 211-213

A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

• Complex analysis
• Lie group
• Matrix
• algebra
• differential geometry
• function space
• group theory
• manifold
• topological groups/Lie groups

"The book is clearly an important contribution to the literature on the subject. A number of results presented here is not accessible in book form elsewhere. Both research students and professional mathematicians will find this valuable volume an extremely useful guide and reference work."

--Publicationes Mathematicae

"This book, which is a useful text for advanced graduate students and researchers…gives a comprehensive account of the field of homogeneous complex domains…The exposition is rapidly paced and efficient, without compromising proofs. Moreover, plenty of examples are given, enabling the reader to understand the essential ideas behind the notions and the theorems."

--ZAA

"This book has been written by five outstanding experts with the intention of surveying the most important goals and viewpoints of the field…It is a pleasant reading, details of proofs are supplied or omitted in a very well chosen manner…Many new top results are described and it contains the most important references after each part. I recommend it first of all since, using this book, one can reach the research level with considerably less effort than by the aid of other means of the recent literature."

--ASM

• Institut de Mathématiques, Université Pierre et Marie Curie, Paris, France

  Jacques Faraut

• Department of Mathematics, Sophia University, Tokyo, Japan

  Soji Kaneyuki

• Dept. Mathematics & Computer Science, H.H. Lehman College, Bronx, USA

  Adam Korányi

• Institute of Mathematics, Academia Sinicia, Beijing, China

  Qi-keng Lu

• Département de Mathématiques, Université de Poitiers, Poitiers Cedex, France

  Guy Roos