COMPACT ORBITS OF PARABOLIC SUBGROUPS

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U 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 exists a gradient map μ:X⟶p where g=k⊕p is a Cartan decomposition of g . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map μp