Stability, Analytic Stability for Real Reductive Lie Groups

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 Kählermanifold Z .More precisely, we suppose that the action of a compact Lie group with Lie algebra u extends holomorphically to an action of the complexified group U^C and that the U -action on Z is Hamiltonian. If G ⊂ U C is compatible, there is a corresponding gradient map μp : X → p, where g = k ⊕ 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