SYMMETRIC k - VARIETIES

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. For k = R and C the representation theory of these varieties has been studied extensively. To study the representation theory over other fields, like local fields and finite fields, more needs to be known about their structure and geometry. In this paper we discuss a number of recent results about symmetric k-varieties over fields, other than R or C. This includes a description of the orbits of minimal parabolic k-subgroups on Gk/Hk and also a partial classification of the k-involutions, together with all the fine structure of restricted root systems with Weyl groups and multiplicities.