Harmonic analysis on Riemannian semi simple symmetric spaces and on special types of non-Riemannian semi simple symmetric spaces are well-established theories. This book presents a systematic treatment of the basic problems on semi simple symmetric spaces and a discussion of some of the more important recent developments in the field. The author's primary contribution has been his idea of how to construct the discrete series for such a space.In this book a fundamental role is played by the ideas behind that construction, namely the duality principle, the orbit picture related to it, and the definition of representations by means of distributions on the orbits. Intended as a text at the upper graduate level, the book assumes a basic knowledge of Fourier analysis, differential geometry, and functional analysis. In particular, the reader should have a good knowledge of the general theory of real and complex Lie algebras and Lie groups and of the root and weight theories for semi simple Lie algebras and Lie groups.
Structure and classification of symmetric spaces Harmonic analysis on semisimple symmetric spaces The noncompact Riemannian form $X^r$ of a semisimple symmetric space The Poisson transform on a symmetric space of noncompact type The $H^d$-orbits on the boundary and the corresponding representation of $G$ Representations related to the closed $H^d$-orbits The discrete series for a semisimple symmetric space.