Compactifications of Symmetric Spaces [Paperback]

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 Martins 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 SatakeFurstenberg 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 als0