Polar orbitopes

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum polytope. There is a tight relation with parabolic subgroups: the set of extreme points of a face is the closed orbit of a parabolic subgroup of G and for any parabolic subgroup the closed orbit is of this form.Comment: 24 pages. To appear on Communications in Analysis and Geometr

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.Comment: 29 pages, minor errors corrected, referee suggestions implemente