Connectivity and Kirwan surjectivity for isoparametric submanifolds

Abstract

Atiyah's formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope, and (b) all preimages of the moment map are connected. Part (a) was generalized by Terng to the wider context of isoparametric submanifolds in euclidean space. In this paper we prove a generalization of part (b) for a certain class of isoparametric submanifolds (more precisely, for those with all multiplicities strictly greater than 1). For generalized real flag manifolds, which are an important class of isoparametric submanifolds, we give a surjectivity criterium of a certain Kirwan map (involving equivariant cohomology with rational coefficients) which arises naturally in this context. Examples are also discussed.