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.
