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.

Review

"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

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