A CLASS OF PARABOLIC k-SUBGROUPS ASSOCIATED WITH SYMMETRIC k-VARIETIES

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, σ an involution of G defined over k, H a k-open subgroup of the fixed point group of σ, Gk (resp. Hk) the set of k-rational points of G (resp. H) and Gk/Hk the corresponding symmetric k-variety. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L(Gk/Hk ) if and only if the parabolic k-subgroup is σ-split. So for a study of these induced representations a detailed description of the Hk-conjucagy classes of these σ-split parabolic k-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric kvarieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and -adic symmetric k-varieties.