Analysis and Geometry on Complex Homogeneous Domains, Softcover reprint of the original 1st ed. 2000 Progress in Mathematics Series, Vol. 185

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.

I Function Spaces on Complex Semi-groups by Jacques Faraut.- I Hilbert Spaces of Holomorphic Functions.- II Invariant Cones and Complex Semi-groups.- III Positive Unitary Representations.- IV Hilbert Function Spaces on Complex Semi-groups.- V Hilbert Function Spaces on SL(2,?).- VI Hilbert Function Spaces on a Complex Semi-simple Lie Group.- II Graded Lie Algebras and Pseudo-hermitian Symmetric Spaces by Soji Kaneyuki.- I Semisimple Graded Lie Algebras.- II Symmetric R-Spaces.- III Pseudo-Hermitian Symmetric Spaces.- III Function Spaces on Bounded Symmetric Domains by Adam Kordnyi.- I Bergman Kernel and Bergman Metric.- II Symmetric Domains and Symmetric Spaces.- III Construction of the Hermitian Symmetric Spaces.- IV Structure of Symmetric Domains.- V The Weighted Bergman Spaces.- VI Differential Operators.- VII Function Spaces.- IV The Heat Kernels of Non Compact Symmetric Spaces by Qi-keng Lu.- I Introduction.- II The Laplace-Beltrami Operator in Various Coordinates.- III The Integral Transformations.- IV The Heat Kernel of the Hyperball R?(m, n).- V The Harmonic Forms on the Complex Grassmann Manifold.- VI The Horo-hypercircle Coordinate of a Complex Hyperball.- VII The Heat Kernel of RII(m).- VIII The Matrix Representation of NIRGSS.- V Jordan Triple Systems by Guy Roos.- I Polynomial Identities.- II Jordan Algebras.- III The Quasi-inverse.- IV The Generic Minimal Polynomial.- V Tripotents and Peirce Decomposition.- VI Hermitian Positive JTS.- VII Further Results and Open Problems.