Orbits, invariants, and representations associated to involutions of reductive groups

Abstract

By an involution of an algebraic group or Lie algebra, we mean an automorphism 0 such that 0 2 is the identity automorphism. Let 0 be an involution'of a complex reductive Lie algebra .q and let p denote the 1 eigenspace of 0 on ,q. get G be the adjoint group of.q and let K={g~GlOg=gO}. Kostant and Rallis [10] have made a detailed study of the action of K on p. In particular they obtain very precise information concerning the orbits of K on p and the representation of K on C[p], the algebra of polynomial functions on p. In this paper we consider a "global" analogue of the problem studied by Kostant and Rallis. Let G be a reductive affine algebraic group over an algebraically closed field k of characteristic +2. Let 0 be an involution of G and let K={geGI0(g ) =g}. If G is semisimple and the base field is C, then G/K is the "complexification" of a Riemannian symmetric space. We study the action of K (and of groups closely related to K) by left translation on the coset space G/K. For technical reasons it is convenient to study a slightly different, but equivalent, problem. Let r: G-+G be defined by r(g)=g0(g) i. Then z (G)=P is a closed subvariety of G. Furthermore z is constant on left cosets of G modulo K and induces an isomorphism o: G/K--+P. We have o ( k g K ) = k o ( g ) k ~ for k e K and geG. Thus P is stable under conjugation by K and a is a K-isomorphism of affine K-varieties. We consider the action of K on P by conjugation. Let x e G have Jordan decomposition x=x,x , , . Then we show that x e P if and only if both x~ and x,, are in P. Thus we have a Jordan decomposition for elements of P. This will play a key role in our results. A torus S of G is O-anisotropic if O(s)=s -1 for every seS. It is known that any two maximal 0-anisotropic tori of G are K~ Let A be a maximal 0-anisotropic torus of G. The following result characterizes semisimple elements of P: