On generalized Cartan subspaces

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. For real and p-adic symmetric k-varieties the space L2(Gk/Hk) of square integrable functions decomposes into several series, one for each Hk-conjugacy class of Cartan subspaces of Gk/Hk.