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

Abstract

AbstractWe 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 $$\mathfrak {u}$$ u extends holomorphically to an action of the complexified group $$U^{\mathbb {C}}$$ U C and that the U-action on Z is Hamiltonian. If $$G\subset U^{\mathbb {C}}$$ G ⊂ U C is compatible, there is a corresponding gradient map $$\mu _\mathfrak {p}: X\rightarrow \mathfrak {p}$$ μ p : X → p , where $$\mathfrak {g}= \mathfrak {k}\oplus \mathfrak {p}$$ 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 $$\mu _\mathfrak {p}^{-1}(0)$$ 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 ($$\partial _\infty G/K$$ ∂ ∞ G / K ) of the symmetric space G/K. We also establish the Hilbert–Mumford criterion for polystability of the action of G on measures.