Extension to Semisimple Algebraic Groups Defined Over a Local Field

Abstract

In this chapter it is indicated how to extend the main results about random walks from Chapters IX, XII, XIII and XIV to the case of a semisimple algebraic group defined over a non-archimedean local field F. More explicitly, the bounded harmonic functions are described, the minimal eigenfunctions are determined, and the Martin compactification at the bottom of the positive spectrum is described. As the general outline of the proofs of the relevant results have already been given (see Chapters IX, XII, XIII, and XIV), the additional indications will be very brief. The aim here is to explain and state the results. Also, answers are given to some of the questions raised by Cartier in [C1]. This chapter should be considered as a first step toward the formulation and understanding of the general theory of Martin compactifications for random walks on semisimple groups defined over a local field. The results are very similar to those in the case of the real field. However, to carry out the proofs used in the real case, it is necessary to have structural information about the group of rational points over the fieldF. Most of this information can be found in [M4, pp. 8–56].