Smoothness of Quotients Associated with a Pair of Commuting Involutions

Let σ , θ be commuting involutions of the connected semisimple algebraic group G where σ , θ and G are defined over an algebraically closed field k, char k = 0. Let H := Gσ and K := Gθ be the fixed point groups. We have an action (H × K )×G → G, where ((h, k), g) 7→ hgk−1, h ∈ H , k ∈ K , g ∈ G. Let G/ (H × K ) denote the categorical quotient SpecO(G)H×K . We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where σ = θ and H = K .