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 u u extends holomorphically to an action of the complexified group UC U C and that the U -action on Z is Hamiltonian. If G⊂ UC G ⊂ U C is compatible, there is a corresponding gradient map μ ₚ: X→ p μ p : X → p , where g= k⊕ p g = k ⊕ 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 ₚ⁻¹(0) μ p - 1 ( 0 ) in terms of the maximal weight functions, which we viewed as a collection of maps defined on the boundary at infinity ( ∂ _∞ G/K ∂ ∞ G / K ) of the symmetric space G / K . We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.