From Groups to Symmetric Spaces

This paper is based on a talk given at the conference ”Representation theory of real reductive groups”, Salt Lake City, July 2009. We fix an algebraically closed field k of characteristic exponent p. (We assume, except in §18, that either p = 1 or p ≫ 0.) We also fix a symmetric space that is a triple (G, θ,K) where G is a connected reductive algebraic group over k, θ : G −→ G is an involution and K is the identity component of the fixed point set of θ (K is a connected reductive algebraic group). We shall often write (G,K) instead of (G, θ,K). Let g = Lie (G), k = Lie (K), p = g/p. Note that K acts naturally on p by the adjoint action. If H is a connected reductive algebraic group over k then H gives rise to a symmetric space (H×H,H) where H is imbedded in H×H as the diagonal; here θ(a, b) = (b, a). (Such a symmetric space is said to be diagonal.) In this paper we examine various properties/constructions which are known for groups (or for diagonal symmetric spaces) and we do some experiments to see to what extent they generalize to non-diagonal symmetric spaces.