Stability, Analytic Stability for Real Reductive Lie Groups

Abstract

We presented a systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group G on a real submanifold X of a Kähler manifold Z. More precisely, we suppose that the action of a compact Lie group U 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. The concept of energy complete action of G on X is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a G-equivariant function called maximal weight. We also prove the classical Hilbert–Mumford criteria for semistability and polystability conditions.