On the Conjugacy of Cartan Subspaces

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 -adic symmetric k-varieties the space L(Gk/Hk ) of square integrable functions decomposes into several series, one for each Hk-conjugacy class of Cartan subspaces of Gk/Hk. In this paper we give a characterization of the Hk-conjugacy classes of these Cartan subspaces in the case that there exists a splitting extension of order 2 and (G, σ) satisfies the additional condition that all tori which are maximal σ-split and k-split are Hk-conjugate. This condition is satisfied for k the the real numbers and many of the symmetric k-varieties with a splitting extension of order 2. For k = we prove a number of additional results as well.