In this paper we prove a
convexity theorem for semisimple symmetric spaces which generalizes Kostant’s
convexity theorem for Riemannian symmetric spaces. Let τ be an involution on the
semisimple connected Lie group G and H = G0τ the 1-component of the group of
fixed points. We choose a Cartan involution 𝜃 of G which commutes with τ and write
K = G𝜃 for the group of fixed points. Then there exists an abelian subgroup A of G,
a subgroup M of K commuting with A, and a nilpotent 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) =loga. The set of all elements in A
for which aH ⊆ HMAN is a closed convex cone. Our main result is the
description of the projections L(aH) ⊆ a for these elements as the sum of the
convex hull of the Weyl group orbit of loga and a certain convex cone in
a.
