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 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 µp: X → p ∗ where g = k⊕p is a Cartan decomposition of g. We obtain a Morse like function ηp: = 1 2 ‖µp‖2 on X. Associated to critical points of ηp are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called prestrata. In case that µp 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 G = U C and X = Z is compact.