A GENERAL NONLINEAR CONVEXITY THEOREM

If G is a connected semisimple real Lie group with Iwasawa decompositi on G = KAN, then Kostant's non-linear convexity theorem describes the image of a K-coset aK under the mapping L : KAN --> a, gan bar arrow p ointing right log a as the convex hull of a Weyl group orbit of log a. In this paper we give a direct geometric proof of a generalization of the inclusion part of this convexity theorem (the other part is false in general) to more general symmetric pairs which are not necessarily reductive. We also explain how it can be applied to study semigroup a ctions on holomorphic vector bundles und the algebraic and geometric s tructure of certain subsemigroups of complex groups.