Stability with respect to actions of real reductive Lie groups

TL;DR

This paper develops a systematic framework for analyzing the stability of actions by real reductive Lie groups on topological spaces, using a generalized Kempf-Ness function to characterize stability conditions.

Contribution

It introduces an abstract setting for non-compact real reductive Lie group actions that allows numerical criteria for stability, semi-stability, and polystability.

Findings

Characterization of stability via maximal weight functions

Application to actions on real compact submanifolds of Kähler manifolds

Extension to actions on measures of submanifolds

Abstract

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this construction is that one can characterize stability, semi-stability and polystability of a point by numerical criteria, that is in terms of a function called maximal weight. We apply this setting to the actions of a real non-compact reductive Lie group G on a real compact submanifold M of a Kaehler manifold Z and to the action of G on measures of M.