Matsuki's double coset decomposition via gradient maps

TL;DR

This paper presents a new proof of Matsuki's double coset decomposition for real-reductive Lie groups using gradient maps, and describes elements in non-closed double cosets.

Contribution

It introduces a novel application of gradient maps to reprove Matsuki's parametrization and characterizes elements in non-closed double cosets.

Findings

New proof of Matsuki's double coset decomposition

Description of elements in non-closed double cosets

Application of gradient maps in Lie group theory

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.