Compact orbits of Parabolic subgroups

TL;DR

This paper investigates the structure of compact orbits of parabolic subgroups within real reductive groups acting on submanifolds of Kahler manifolds, utilizing gradient maps derived from Hamiltonian actions.

Contribution

It provides a description of compact orbits of parabolic subgroups using gradient maps in the context of Hamiltonian actions of real reductive groups.

Findings

Characterization of compact orbits via gradient maps

Extension of Hamiltonian actions to complexified groups

Description of orbit structure in terms of gradient maps

Abstract

We study the action of a real reductive group $G$ on a real submanifold $X$ of a Kahler manifold $Z$. We suppose that the action of a compact connected Lie group $U$ 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 exists a gradient map $\mu_{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$. In this paper we describe compact orbits of parabolic subgroups of $G$ in term of the gradient map $\mu_\mathfrak p$.