A COMPACT EMBEDDING OF SEMISIMPLE SYMMETRIC SPACES

Tran Dao Dong, Tran Vui

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.

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