Author | Guivarc'h, Yves. author |
---|---|
Title | Compactification of Symmetric Spaces [electronic resource] / by Yves Guivarc'h, Lizhen Ji, J. C. Taylor |
Imprint | Boston, MA : Birkhรคuser Boston, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2452-5 |
Descript | XIV, 284 p. online resource |
I. Introduction -- Statement of the main new results -- Characterizations of the compactification $${\bar X̂{SF}}$$ -- The Karpelevi? compactification $${\bar X̂K}$$ -- 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 $${\bar Xĉ}$$ -- 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 SatakeโFurstenberg 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 for a 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 the minimal 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