Tori Invariant under an Involutorial Automorphism I

Let G be a connected reductive linear algebraic group defined over an field k of characteristic not 2, θ ∈ Aut(G) an involutional k-automorphism of G and K = Gθ = {g ∈ G | θ(g) = g} the set of fixed points of θ. Denote the set of k-rational points of G by Gk. In this paper we shall classify the K-conjugacy classes of θ-stable maximal tori of G. This is shown to be independent of the characteristic of k and can be applied to describe all the orbits of affine symmetric spaces under the action of a minimal parabolic subgroup in the case that the affine symmetric space is of type (G,K) (see Helminck [8]). This paper is part of a series of papers leading towards a classification of all orbits of affine symmetric spaces under the action of a minimal parabolic subgroup (see also [10]). The results and techniques in this paper will be used for the classification in the remaining cases. The K-conjugacy classes of θ-stable maximal tori of G occur (in a slightly different form) in the representation theory of real semisimple Lie groups. Namely if θ is a Cartan involution of G as in [10, 11.7], then the K-conjugacy classes of θ-stable maximal tori of G correspond one to one with the Gk-conjugacy classes of maximal k-tori of G. So for k = R this leads to a one to one correspondence with the conjugacy classes of Cartan subalgebras of a real simisimple Lie algebra. We give two characterizations of these conjugacy classes. The first characterization relates the K-conjugacy classes with conjugacy classes of involutions ∗Partially supported by N.S.F. grant number DMS-8600037