### A COMPACT EMBEDDING OF SEMISIMPLE SYMMETRIC SPACES

#### Abstract

Let G be a connected real semisimple Lie group with finite center

and ? be an involutive automorphism of G. Suppose that H is a closed

subgroup of G with G^?_e \subset H \subset G^?, where G^? is the fixed points group of

? and G^?_e denotes its identity component. The coset space X=G/H is

then a semisimple symmetric space. Let ? be a Cartan involution which

commutes with ? and K be the set of all fixed points of ?. Then K is

a ?-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.

and ? be an involutive automorphism of G. Suppose that H is a closed

subgroup of G with G^?_e \subset H \subset G^?, where G^? is the fixed points group of

? and G^?_e denotes its identity component. The coset space X=G/H is

then a semisimple symmetric space. Let ? be a Cartan involution which

commutes with ? and K be the set of all fixed points of ?. Then K is

a ?-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.

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