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

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 ${\mu }_{\mathfrak{p}}:X\to \mathfrak{p}$, 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 ${\mu }_{\mathfrak{p}}^{-1}(0)$ in terms of the maximal weight function, which we viewed as a collection of maps defined on the boundary at infinity (${\mathrm{\partial }}_{\mathrm{\infty }}G\mathrm{/}K$) of the symmetric space $G\mathrm{/}K$.