A Hilbert-Mumford Criterion for polystability for actions of real reductive Lie groups

TL;DR

This paper establishes a criterion for polystability of real reductive Lie group actions on submanifolds of Kahler manifolds, extending geometric invariant theory concepts to a broader setting involving gradient maps and boundary at infinity analysis.

Contribution

It introduces a Hilbert-Mumford criterion for polystability in the context of real reductive Lie group actions, linking orbit intersection properties with boundary behavior of maximal weight functions.

Findings

Characterizes G-orbits intersecting the zero level set of the gradient map.

Relates orbit stability to boundary maps on the symmetric space.

Extends stability criteria to real reductive group actions on Kahler submanifolds.

Abstract

We presented a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra $\mathfrak{u}$ extends holomorphically to an action of the complexified group $U^\mathbb{C}$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $\mu_\mathfrak{p}: X\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Under some mild restrictions on the $G$-action on $X,$ we characterize which $G$-orbits in $X$ intersect $\mu_\mathfrak{p}^{-1}(0)$ in terms of the maximal weight function, which we viewed as a collection of maps defined on the boundary at infinity ($\partial_\infty G/K$) of the symmetric space $G/K$.