A COMPACT EMBEDDING OF SEMISIMPLE SYMMETRIC SPACES

Abstract

Let G be a connected real semisimple Lie group with finite center and σ be an involutive automorphism of G. Suppose that H is a closed subgroup of G with Gσ e ⊂H ⊂ Gσ, where Gσ is the fixed points group ofσ and Gσ e denotes its identity component. The coset space X = G/H is then a semisimple symmetric space. Let θ be a Cartan involution which commutes with σ and K be the set of all fixed points of θ. Then K is a σ-stable maximal compact subgroup of G and the coset space G/K becomes a Riemannian symmetric space of noncompact type. By using the action of the Weyl group, we have constructed a compact real analytic manifold in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it. The purpose of this note is to apply the above construction to the case of semisimple symmetric spaces X = G/H. Our construction is similar to those of Schlichtkrull, Lizhen Ji, Oshima for Riemannian symmetric spaces and similar to those of Kosters, Sekiguchi, Oshima for semisimple symmetric spaces.