Orbits and invariants associated with a pair of commuting involutions

Let sigma, theta be commuting involutions of the connected reductive algebraic group G, where sigma, theta, and G are defined over a (usually algebraically closed) field k, char k not equal 2. We have fixed point groups H := G(sigma) and K := G(theta) and an action (H x K) x G --> G, where ((h, k), g) bar right arrow hgk(-1), h is an element of H, k is an element of K, g is an element of G. Let G parallel to (H x K) denote Spec O(G)(HxK) (the categorical quotient). Let A be maximal among subtori S of G such that theta (s) = sigma (s) = s(-1) for all s is an element of S. There is the associated Weyl group W := W-HxK(A) We show the following. . The inclusion A --> G induces an isomorphism A/W (-->) over bar )over tilde>G parallel to (H x K). In particular; the closed (H x K)-orbits are precisely those which intersect A. . The fibers of G --> G parallel to (H x K) are the same as those occurring in certain associated symmetric varieties. In particular; the fibers consist of finitely many orbits. We investigate . the structure of W and its relation to other naturally occurring Weyl groups and to the action of sigma theta on the A-weighs spaces of g; . the relation of the orbit type stratifications of A/W and G parallel to (H x K). Along the way we simplify some of R. Richardson's proofs for the symmetric case sigma = theta, and at the end we quickly recover results of M. Berger; M. Flensted-Jensen, B. Hoogenboom, and I: Matsuki [Ber], [FJl] [Hoo], [Mat]for the case k = R.