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 Kähler manifold Z. Suppose the action of a compact Lie group with Lie algebra \(\mathfrak {u}\) $\mathfrak{u}$ extends holomorphically to an action of the complexified group \(U^{\mathbb {C}}\) $U^{\mathbb{C}}$ and that the U-action on Z is Hamiltonian. If \(G\subset U^{\mathbb {C}}\) $G\subset U^{\mathbb{C}}$ is compatible, there is a corresponding gradient map \(\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}\) ${\mu }_{\mathfrak{p}}:X\to \mathfrak{p}$ , where \(\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}\) $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ 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 \(\mu _\mathfrak {p}^{-1}(0)\) ${\mu }_{\mathfrak{p}}^{-1}(0)$ in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ( \(\partial _\infty G/K\) ${\partial }_{\infty }G/K$ ) of the symmetric space G/K. We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.