A convexity theorem for semisimple symmetric spaces

Abstract

In this paper we prove a generalization of the convexity theorem in [3] for a symmetric space G/H. Here G is a connected semisimple Lie group with finite center, σ an involution of G and H an open subgroup of the group G of fixed points for σ. The generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type of parabolic subgroup considered in [3]. From now on we assume more generally that G is a real reductive group of the HarishChandra class; this will allow an inductive argument relative to the real rank of G. Let θ : G → G be a Cartan involution of G that commutes with σ; for its existence, see [20, Thm. 6.16]. The associated group K := G of fixed points is a maximal compact subgroup of G. For the infinitesimal involutions determined by σ and θ we use the same symbols: θ, σ : g → g; here g denotes the Lie algebra of G. With respect to the infinitesimal involutions, g decomposes as