Compactifications of Symmetric Spaces

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 ? < ?0.- Computation of the limit functions for a general semisimple group.- Determination of the Martin compactification.- Bounded harmonic functions on X.- An application to convergence of Brownian motion.- IX. An intrinsic approach to the boundaries of X.- The space of closed subgroups.- Limit groups.- Limits of group spheres.- Parabolic subgroups and boundary theory.- The maximal Furstenberg compactification.- X. Compactification via the ground state.- The twisted action.- Compactification of X via the ground state.- XI. Harnack inequality, Martin’s method and the positive spectrum for random walks.- Basic notations.- Cones with compact bases and the Harnack inequality.- Martin’s method for a random walk.- The positive spectrum of a random walk.- The fixed line property.- Formulas for r(p),r0(p).- Outline of the following chapters.- XII. The Furstenberg boundary and bounded harmonic functions.- Basic notations.- The mean-value property.- Harmonic functions and the mean-value property.- Convergence theorems for harmonic functions.- The Poisson formula for random walks.- XIII. Integral representation of positive eigenfunctions of convolution operators.- The main result of this chapter.- An extension of the main result.- Analytic determination of the minimal eigenfunctions of the Laplacian.- The Busemann cocycle and a geometrical determination of theminimal eigenfunctions of the Laplacian.- Minimal eigenfunctions for random walks.- XIV. Random walks and ground state properties.- Basic definitions and properties.- Convolution.- Spherical functions and minimal eigenfunctions.- Ground state properties.- Random walks, eigenfunctions of the Laplacian and X ? ?X(?0).- The Martin compactification of X determined by a random walk.- An application to parabolic subgroups.- XV. Extension to semisimple algebraic groups defined over a local field.- Some notations and fundamental properties.- Extension of the main results of Chapters XII, XIII, XIV.- Appendix A.- Appendix B.- List of symbols.