Martin Boundaries for Real Semisimple Lie Groups

We construct the Martin boundaries for symmetric spaces of noncompact type X = G/K, where G is a real semisimple Lie group with a finite center. This work is a generalization of a result of Dynkin who constructed them in the SL(n, C) case. The main result is an explicit form of the Martin kernel. It is obtained by means of investigation of asymptotics of Green′s functions and zonal spherical functions on G. Comparison of the Martin compactification and the Satake and Karpelevich compactifications is presented. Finally we give an explicit form of our construction for the Cartan domain of the type 1. In conclusion we discuss an application of our approach to the scattering problem of the quantum Calogero-Moser system and a generalization to the quantum and p-adic groups.