Stratifications with respect to actions of real reductive groups

We study the action of a real reductive group G on a real submanifold X of a K"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata. In case that the gradient map is proper, the pre-strata form a decomposition of X and in case that X is compact they are the strata of a Morse type stratification of X. Our results are generalizations of results of Kirwan obtained in the case that X=Z is compact and the group itself is complex reductive.

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