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.

Abstract. We consider actions of real Lie subgroups G of complex reductive Lie groups on Kählerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.

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18 pagesNous étudions l'action d'un groupe réel-réductif $G=K\exp(\lie{p})$ sur une sous-variété réel-analytique $X$ d'une variété kählérienne $Z$. Nous supposons que l'action de $G$ peut être prolongée à une action holomorphe du groupe complexifié $G^\mbb{C}$ telle que l'action d'un sous-groupe maximal compact de $G^\mbb{C}$ soit hamiltonienne. L'application moment $\mu$ induit une application gradient $\mu_\lie{p}\colon X\to\lie{p}$. Nous montrons que $\mu_\lie{p}$ separe les orbites de $K$ si et seulement si un sous-groupe minimal parabolique de $G$ possède une orbite ouverte dans $X$. Ce résultat généralise la charactérisation de Brion des variétés kählériennes sphériques qui admettent une application moment

Abstract. We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X/ /G for the G-action, we introduce a stratification which is defined with respect to orbit types of closed orbits. Our main result is a description of the quotient X/ /G in terms of quotients by normalizer subgroups associated to the stratification.