J. C. Taylor

Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian

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The Karpelevič Compactification

This compactification was introduced by Karpelevic in [K3]. The original inductive definition of \({\overline X ^K}\) is recalled in § 5.3. By examining the closure of a flat A · o in \({\overline X ^K}\), a non-inductive characterization of the closure \({\overline {{A^ + } \cdot o} ^K}\) of \({\overline {A \cdot o} ^K}\) is obtained (see Theorem 5.6). The nature of the Karpelevic topology restricted to the flat is clarified by the introduction of the class of K- fundament al sequences. Using this concept, one shows that (mathtype) is isomorphic to a compactification of a determined by its polyhedral structure. This compactification of a, referred to as the Karpelevic compactification of a, is used to give a new proof that the Karpelevic topology is compact. Lemma 5.26, Proposition 5.27, and Corollary 5.28 explain the relations between the Karpelevic compactification and the conic and dual cell compactifications. Finally, in Remark 5.32 a new way to define the Karpelevic compactification is presented. It consists of fitting together the Karpelevic compactifications of the flats kA · o, k ⊀ K, in exactly the same way that the dual cell compactification is obtained from the polyhedral compactifications of the flats kA · o.

Harnack Inequality, Martin’s Method and The Positive Spectrum for Random Walks

The study of positive eigenfunctions of the Laplace operator L is closely related to the study of convolution equations defined by probability measures p. With applications to other non-semisimple Lie groups in mind, several results for general convolution equations on a locally compact metrizable group H are established in this chapter.

Random Walks and Ground State Properties

The main questions previously examined can also be considered in the general framework of random walks. If one takes into account the results in Chapters IX and X, this leads to new proofs and new formulations of many of the results discussed earlier.

Subalgebras and Parabolic Subgroups

The key to understanding the geometrical structure of the compactifications of symmetric spaces of non-compact type is given by the family of parabolic subgroups of G. In this chapter the relation between these subgroups and sets of simple roots is discussed. Additional details for matters treated in this chapter may be found in Helgason [H2] or Warner [W1]. This chapter begins by introducing the two basic decompositions of G.

The Satake-Furstenberg Compactifications

The compactifications of symmetric spaces defined by Satake [S1], referred to nowadays as Satake compactifications, were motivated by the theory of automorphic forms and of representations. Furstenberg [F3] considered boundary value problems at infinity for the Laplacian on symmetric spaces and was led to isomorphic compactifications, as was shown by Moore [M8]. While these two families of compactifications are isomorphic, they are defined by quite different methods.

Compactification via the Ground State

The K-invariant probability m on F = G/P represents, by means of the square root of the Poisson kernel, a unique solution of the equation Lu + λ 0u = 0 with u(o) = 1. It is the spherical function Φ defined by \(\Phi \left( {g \cdot o} \right) = \int {_K{e^{ - \rho \left( {H\left( {{g^{ - 1}}k} \right)} \right)}}} dk = \int {_\mathcal{F}{P^{1/2}}\left( {x,b} \right)dm\left( b \right),} \) where x = g · o ∈ X, b = kP, and \(P\left( {x,b} \right) = {e^{ - 2\rho \left( {H\left( {{g^{ - 1}}k} \right)} \right)}}\) is the Poisson kernel on X (see §7.21 and § 8.27). It plays a basic role in harmonic analysis on semisimple groups, for example it dominates all the spherical functions associated with the unitary principal series, and is called the Harish-Chandra spherical function (see [G1]). It arose earlier in Chapter VII when determining the limit functions for the Martin compactification at the bottom of the positive spectrum.

The Furstenberg Boundary and Bounded Harmonic Functions

Let L denote the Laplace—Beltrami operator on X = G/K. The main purpose of this chapter is to give another, elementary, and self-contained proof of the so-called Poisson formula (see Theorem 12.10) for the integral representation of the bounded harmonic functions, i.e., solutions of the equation Lf = 0 [F3]. This was proved earlier (see Corollary 8.29), using the Martin boundary of X for λ = 0. The key to the proof, presented here, is the fact that (G, K) is a Gelfand pair. As a result it follows, see Corollary 12.9, that a bounded C2-function is harmonic if and only if it satisfies the mean-value property. This is not so easily proved as in Euclidean space because, if the rank of X is greater than one, K is not transitive on the geodesic spheres centered at o.

An Intrinsic Approach To The Boundaries of X

It is well-known that any symmetric space X of non-compact type can be realized as the space So of maximal compact subgroups by associating with g · o its isotropy subgroup gKg-1. The space S of closed subgroups of G is compact in the topology of Hausdorff convergence on the compact subsets of G. As a result, the closure \(\overline {{S_0}} \) is a compactification of X.

Geometrical Constructions of Compactifications

In this chapter several geometrical compactifications are described that are relevant to the rest of this book. The first one is the conic compactification (see §3.1). When X is identified with p, it amounts to adjoining a sphere of codimension 1 at infinity to a Euclidean space in the usual way. It turns out that this sphere X(∞) at infinity may be given the structure of a simplicial complex Δ(X) (see Theorem 3.15 and Proposition 3.18) with respect to which it is a spherical Tits building (Definition 3.14). This is accomplished by identifying the sets of points in X(∞) stabilized by the various parabolic subgroups (see Proposition 3.9).