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.

Inhalt

I Function Spaces on Complex Semi-groups by Jacques Faraut.- I Hilbert Spaces of Holomorphic Functions.- I.1 Reproducing kernels.- I.2 Invariant Hilbert spaces of holomorphic functions..- II Invariant Cones and Complex Semi-groups.- II.1 Complex semi-groups.- 1I.2 Invariant cones in a representation space.- II.3 Invariant cones in a simple Lie algebra.- III Positive Unitary Representations.- III.1 Self-adjoint operators.- III.2 Unitary representations.- III.3 Positive unitary representations.- IV Hilbert Function Spaces on Complex Semi-groups.- IV.1 Schur orthogonality relations.- IV.2 The Hardy space of a complex semi-group.- IV.3 The Cauchy-Szegö kernel and the Poisson kernel.- IV.4 Spectral decomposition of the Hardy space.- V Hilbert Function Spaces on SL(2,?).- V.1 Complex Olshanski semi-group in SL(2,?).- V.2 Irreducible positive unitary representations.- V.3 Characters and formal dimensions of the representations ?m.- V.4 Bi-invariant Hilbert spaces of holomorphic functions.- V.5 The Hardy space.- V.6 The Bergman space.- VI Hilbert Function Spaces on a Complex Semi-simple Lie Group.- VI.1 Bounded symmetric domains.- VI.2 Irreducible positive unitary representations.- VI.3 Characters and formal dimensions.- VI.4 Bi-invariant Hilbert spaces of holomorphic functions.- References.- II Graded Lie Algebras and Pseudo-hermitian Symmetric Spaces by Soji Kaneyuki.- I Semisimple Graded Lie Algebras.- I.1 Root theory of real semisimple Lie algebras.- I.2 Semisimple graded Lie algebras.- I.3 Example.- I.4 Tables.- II Symmetric R-Spaces.- II.1 Symmetric R-spaces and their noncompact duals.- II.2 Sylvester's law of inertia in simple GLA's.- II.3 Generalized conformal structures and causal structures.- III Pseudo-Hermitian Symmetric Spaces.- III.1 Pseudo-Hermitian spaces and nonconvex Siegel domains.- III.2 Simple reducible pseudo-Hermitian symmetric spaces.- References.- III Function Spaces on Bounded Symmetric Domains by Adam Kordnyi.- I Bergman Kernel and Bergman Metric.- I.1 Domains in Cr".- 1.2 Bergman kernel, reproducing kernels.- I.3 The Bergman metric.- II Symmetric Domains and Symmetric Spaces.- II.1 Basic facts, definitions.- II.2 Riemannian symmetric spaces.- II.3 Theory of oiLa's.- II.4 OiLa's of bounded symmetric domains.- II.5 Cartan subalgebras.- III Construction of the Hermitian Symmetric Spaces.- III.1 The Borel imbedding theorem.- III.2 The Harish-Chandra realization.- III.3 Remarks on classification.- IV Structure of Symmetric Domains.- IV.1 Restricted root system, boundary orbits.- IV.2 Decomposition under the Cayley transform.- V The Weighted Bergman Spaces.- V.1 Analysis on symmetric domains.- V.2 Decomposition under K.- V.3 Spaces of holomorphic functions.- VI Differential Operators.- VI.1 Properties of ?s.- VI.2 Invariant differential operators on ?.- VI.3 Further results on $$ \mathbb{D}$$(?).- VI.4 Extending D? to p+.- VII Function Spaces.- VII.1 The holomorphic discrete series.- VII.2 Analytic continuation of the holomorphic discrete series.- VII.3 Explicit formulas for the inner products.- VII.4 L9-spaces and Bergman type projections.- VII.5 Some questions of duality.- VII.6 Further results.- References.- 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.- References.- V Jordan Triple Systems by Guy Roos.- I Polynomial Identities.- I.1 Definition of Jordan triple systems.- I.2 Identities of minimal degree.- 1.3 Jordan representations and duality.- 1.4 The fundamental identity of degree 7.- 1.5 The Bergman operator.- II Jordan Algebras.- II.1 Jordan algebras arising from a JTS.- II.2 Identities in a Jordan algebra.- II.3 The JTS associated to a Jordan algebra.- III The Quasi-inverse.- III.1 Quasi-invertibility in a Jordan algebra.- 111.2 Quasi-invertibility in a JTS.- 11I.3 Identities for the quasi-inverse.- 1II.4 Differential equations.- 1I1.5 Addition formulas.- IV The Generic Minimal Polynomial.- IV.1 Unital Jordan algebras.- IV.2 General Jordan algebras.- IV.3 Jordan triple systems.- V Tripotents and Peirce Decomposition.- V.1 Tripotent elements.- V.2 Peirce decomposition.- V.3 Orthogonality of tripotents.- V.4 Simultaneous Peirce decomposition.- VI Hermitian Positive JTS.- VI.1 Positivity.- VI.2 Spectral decomposition.- VI.3 Automorphisms.- VI.4 The spectral norm.- VI.5 Classification of Hermitian positive JTS.- VII Further Results and Open Problems.- VII.1 Schmid decomposition.- VII.2 Compactification of an hermitian positive JTS.- VII.3 Projective imbedding.- VII.4 Volume computations.- VII.5 Some open problems.- References.