A CONVEXITY THEOREM FOR SEMISIMPLE SYMMETRICAL SPACES

In this paper we prove a convexity theorem for semisimple symmetric sp aces which generalizes Kostant's convexity theorem for Riemannian symm etric spaces. Let tau be an involution on the semisimple connected Lie group G and H = G0t the 1-component of the group of fixed points. We choose a Cartan involution theta of G which commutes with tau and writ e K = G(theta) for the group of fixed points. Then there exists an abe lian subgroup A of G, a subgroup M of K commuting with A , and a nilpo tent subgroup N such that HMAN is an open subset of G and there exists an analytic mapping L: HMAN --> a = L(A) with L(hman) = log a. The se t of all elements in A for which aH subset-or-equal-to HMAN is a close d convex cone. Our main result is the description of the projections L (aH) subset-or-equal-to a for these elements as the sum of the convex hull of the Weyl group orbit of log a and a certain convex cone in a.