Compact orbits of Parabolic subgroups

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 ${\mu }_{\mathfrak{p}}:X⟶\mathfrak{p}$ 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 ${\mu }_{\mathfrak{p}}$.

11 pages