Real double coset spaces and their invariants

TL;DR

This paper studies the structure of double coset spaces in real reductive groups using moment map and invariant theory, providing a stratification of the quotient where closed double cosets form trivial bundles.

Contribution

It introduces a method to compute double coset spaces for real forms of complex reductive groups, revealing a stratification and fiber structure of closed double cosets.

Findings

Stratification of the quotient space by double cosets

Closed double cosets form trivial bundles over a torus quotient

Application of moment map and invariant theory techniques

Abstract

Let G be a real form of a complex reductive group. Suppose that we are given involutions \sigma and \theta of G. Let H=G^\sigma denote the fixed group of \sigma and let K=G^\theta denote the fixed group of \theta. We are interested in calculating the double coset space H\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.