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 µp : 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 term of the gradient map µp.
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