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

Abstract 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 U^ {C} U C and that the U -action on Z is Hamiltonian. If G U^ {C} G ⊂ U C is compatible, there is a corresponding gradient map _ 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 _ 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.