Compactifications of Symmetric Spaces

The concept of symmetric space is of central importance in many branches of mathematics. Compactifications of these spaces have been studied from the points of view of representation theory, geometry, and random walks. This work is devoted to the study of the interrelationships among these various compactifications and, in particular, focuses on the martin compactifications. It is the first exposition to treat compactifications of symmetric spaces systematically and to uniformized the various points of view.

I. Introduction.- Statement of the main new results.- Characterizations of the compactification $${bar X^{SF}}$$.- The Karpelevi? compactification $${bar X^K}$$.- Fibers of maps between the compactifications.- Application to Brownian motion.- Eigenfunctions and Martin's method.- Methods of proof.- Open problems.- Conventions.- Study guide.- II. Subalgebras and parabolic subgroups.- The Iwasawa and Cartan decompositions.- Parabolic subgroups.- Subsets of ? and Lie subalgebras.- The Langlands decomposition of PI and the symmetric space XI.- Bruhat decompositions.- III. Geometrical constructions of compactifications.- The conic compactification $${bar X^c}$$.- The conical decomposition of a and the Weyl group.- Parabolic subgroups and stabilizers of the points in X(?).- Flats through the base point and Proposition 3.8.- The Tits building ?(G) of G and its geometrical realization ?(X).- The polyhedral compactification of a flat.- The dual cell complex ?*(X).- The dual cell compactification X ? ?*(X).- IV. The Satake-Furstenberg compactifications.- Finite dimensional representations.- Weights and highest weights.- Representation and parabolic subgroups.- Satake compactifications.- Furstenberg compactifications.- V. The Karpelevi? compactification.- The Karpelevi? compactification.- Convergence in the Karpelevi? topology restricted to a flat.- The Karpelevi? compactification of a.- The Karpelevi? topology is compact.- The relation between the Karpelevi? compactification, conical and dual cell compactifications.- A characterization of the Karpelevi? compactification.- VI. Martin compactifications.- The Martin compactification.- Convergence of Brownian motion.- Extension of the group action to the Martin compactification.- The Martin compactification fora random walk.- VII. The Martin compactification X ? ?X(?0).- The Laplacian in horocyclic coordinates.- Generalized horocyclic coordinates and the Laplacian.- Computation of the limit functions: reduction.- The limit of a CI-canonical sequence.- Classification of limit functions and the topology of X ? ?X(?0).- VIII. The Martin compactification X ? ?X(?).- The case of X = SL(n, ?)/SU(n) for ?