Orbits and invariants associated with a pair of commuting involutions

Abstract

Let σ,θ be commuting involutions of the connected reductive algebraic group G, where σ,θ, and G are defined over a (usually algebraically closed) field k, char k≠2. We have fixed point groups H:≠G^σ and K:≠G^θ and an action (H×K)×G→G, where ((h, k), g)↦hgk⁻¹, h∈H, k∈K, g∈G. Let G//(H×K) denote Spec $\mathscr{O}$(G)^H×K (the categorical quotient).

Let A be maximal among subtori S of G such that θ(s)=σ(s)=s⁻¹ for all s∈S. There is the associated Weyl group W:=W_H×K(A). We show the following.

· The inclusion A→G induces an isomorphism A/W$\widetilde{\to}$G//(H×K). In particular, the closed (H×K)-orbits are precisely those which intersect A.

· The fibers of G→G//(H×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 σθ on the A-weight spaces of $\mathfrak {g}$;

· the relation of the orbit type stratifications of A/W and G//(H×K).

Along the way we simplify some of R. Richardson's proofs for the symmetric case σ=θ, and at the end we quickly recover results of M. Berger, M. Flensted-Jensen, B. Hoogenboom, and T. Matsuki [Ber], [FJ1], [Hoo], [Mat] for the case k=ℝ.