No abstract

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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 G extends holomorphically to an action of the complexified group G C and that with respect to a compatible maximal compact subgroup U of G C the action on Z is Hamiltonian. There is a corresponding gradient map µ p : X → p * where g = k ⊕ p is a Cartan decomposition of g. We obtain a Morse-like function η p := µ p 2 on X. Associated with critical points of η p are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds S β of X which are called pre-strata. In cases where µ p is proper, the pre-strata form a decomposition of X and in cases where 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 where G = U C and X = Z is compact.