Tori Invariant under an Involutorial Automorphism II

Abstract

The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p-adic numbers, finite fields, and number fields.