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 \(\mathfrak {u}\) 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 \(\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}\), 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 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.