Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps

TL;DR

This paper investigates the properties of semistable points and convexity phenomena related to gradient maps in the context of real reductive group actions on Kahler manifolds, extending classical convexity results.

Contribution

It establishes openness and connectedness of semistable point sets and proves convexity theorems for specific group actions on Kahler manifolds.

Findings

Openness and connectedness of semistable point sets.

Convexity theorem for G-action on G-invariant Lagrangian submanifolds.

Convexity result for two-orbit varieties.

Abstract

We study the action of a real reductive group $G$ on a Kahler manifold $Z$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ We assume that the action of $U$, a maximal compact connected subgroup of $U^\mathbb{C}$ on $Z$ is Hamiltonian. If $G\subset U^\mathbb{C}$ is compatible, there is a corresponding gradient map $\mu_\mathfrak{p}: Z\to \mathfrak{p}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ is a Cartan decomposition of the Lie algebra of $G$. Our main results are the openness and connectedness of the set of semistable points associated with $G$-action on $Z$, a convexity theorem for the $G$-action on a $G$-invariant compact Lagrangian submanifold of $Z$, and a convexity result for two-orbit variety.