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

TL;DR

The article extends the Hilbert-Mumford criterion to polystability for real reductive Lie groups acting on real submanifolds of Kähler manifolds by employing a gradient map $\mu_\mathfrak{p}: X\to \mathfrak{p}$ and a Kempf–Ness framework. It defines a maximal weight function $\lambda_x$ on the boundary at infinity of the symmetric space $M=G/K$ and proves that polystability of a point $x$ is characterized by $\lambda_x\ge0$ with boundary zeros connected along geodesics, without requiring a completeness assumption. The development integrates the structure of compatible subgroups, the symmetric-space geometry, and the parabolic-reduction mechanism to provide a concrete numerical criterion in terms of $\lambda_x$ and the gradient map $\mu_\mathfrak{p}$. This extends noncompact real-group actions in a GIT-like framework, enabling robust quotients and moduli constructions in real-analytic and Kähler settings. The results offer a practical tool for identifying polystable orbits via boundary data and gradient-flow behavior, with potential applications in symplectic and complex geometry where real reductive symmetries arise.

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 $μ_\mathfrak{p}: X\to \mathfrak{p}$, 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 $μ_\mathfrak{p}^{-1}(0)$ 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 $G/K$.

Theorems & Definitions (36)

• Lemma 2.1: LA
• Proposition 2.2
• proof
• Definition 3.1
• Lemma 3.1
• proof
• Definition 4.1
• Proposition 4.1
• Proposition 4.2
• Lemma 4.3