Algebraic Groups with a Commuting Pair of Involutions and Semisimple Symmetric Spaces

Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describe all semisimple locally symmetric spaces together with their line structure. Involutorial automorphisms of g occur in several places in the literature. Cartan has already shown that for F= C, the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple Lie algebras, which correspond in their turn to the isomorphism classes of Riemannian symmetric spaces (see Helgason [ 111). If one lifts this involution to the group G, then the present work gives a characteristic free description of these isomorphism classes. In a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g. Namely let (go, a) be a semisimple locally symmetric pair; i.e., go is a real semisimple Lie algebra and rr E Aut(g,) an involution. Then by a result of Berger [2], there exists a Cartan involution 8 of go, such that 00 = ea. If we denote the complexilication of go by g, then o and 8 induce a pair of commuting involutions of g. Conversely, if c, 8 E Aut(g) are commuting involutions, then c and 8 determine two locally semisimple symmetric pairs. For if u is a Oand O-stable compact real form with conjugation r, then (ger, (~1 ge,) and (g,,, f3 Iger) are semisimple locally symmetric pairs where