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 G k (resp. H k ) the set of k-rational points of G (resp. H). The variety G k /H k is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space L 2 (G k /H k ) of square integrable functions decomposes into several series, one for each H k -conjugacy class of Cartan subspaces of G k /H k .In this paper we give a characterization of the H k -conjugacy classes of these Cartan subspaces in the case that there exists a splitting extension of order 2 and (G, σ) is (σ, k)-split conjugate (see Subsection 3.8). This condition is satisfied for k the real numbers and several other fields for which the symmetric k-variety has a splitting extension of order 2. For k = R we prove a number of additional results as well.