Matsuki's double coset decomposition via gradient maps

Abstract

Let $G$ be a real-reductive Lie group and let $G_1$ and $G_2$ be two subgroups given by involutions. We show how the technique of gradient maps can be used in order to obtain a new proof of Matsuki's parametrization of the closed double cosets $G_1\backslash G/G_2$ by Cartan subsets. We also describe the elements sitting in non-closed double cosets.