A Hilbert–Mumford criterion for polystability for actions of real reductive Lie groups

We study a Hilbert-Mumford criterion for polystablility associated with an action of a real reductive Lie group $G$ on a real submanifold $X$ of a \Keler manifold $Z$. Suppose the action of a compact Lie group with Lie algebra $\liu$ extends holomorphically to an action of the complexified group $U^\C$ and that the $U$-action on $Z$ is Hamiltonian. If $G\subset U^\C$ is compatible, there is a corresponding gradient map $\mu_\mathfrak{p}: X\to \mathfrak{p}$, where $\lieg = \liek \oplus \liep$ 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 $\mu_\liep^{-1}(0)$ in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ($\partial_\infty G/K$) of the symmetric space $G/K$. We also establish the Hilbert-Mumford criterion for polystability of the action of $G$ on measures.