On the Classification of k-Involutions

Abstract

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, θ an involution of G defined over k, H a k-open subgroup of the fixed point group of θ, and Gk (resp. Hk) the set of k-rational points of G (resp. H). The variety Gk/Hk 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 representation 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 representation 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 systems with Weyl groups, etc. Thus it is essential to study first this structure and the related geometry. In this paper we give a characterization of the isomorphy classes of these symmetric k-varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers, p-adic numbers, finite fields and number fields.