On the classification of k-involutions

Let G be a connected reductive algebraic group defined over a held k of cha racteristic not 2, theta an involution of G defined over k, H a k-open subg roup of the fixed point group of theta, and G(k) (resp. H-k) the set of k-r ational points of G (resp. H). The variety G(k)/H-k is called a symmetric k -variety. These varieties occur in many problems in representation theory, geometry, and singularity theory. Over the last few decades the representat ion theory of these varieties has been extensively studied for k = R and C. As most of the work in these two cases was completed, the study of the rep resentation theory over other fields, like local fields and finite fields, began. The representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space, like the restricted root syst ems with Weyl groups, etc. Thus it is essential to study first this structu re and the related geometry. In this paper we give a characterization of th e isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fin e structure for the real numbers, p-adic numbers, finite fields and number fields. (C) 2000 Academic Press.