Smoothness of Quotients Associated With a Pair of Commuting Involutions

[摘要] Let $sigma$, $heta$ be commuting involutions of the connected semisimplealgebraic group $G$ where $sigma$, $heta$ and $G$ are defined overan algebraically closed field $k$, $Char k=0$. Let $H:=G^sigma$and $K:=G^heta$ be the fixed point groups. We have an action$(Himes K)imes Go G$, where $((h,k),g)mapsto hgkinv$, $hinH$, $kin K$, $gin G$. Let $quot G{(Himes K)}$ denote thecategorical quotient $Spec O(G)^{Himes K}$. We determine when thisquotient is smooth. Our results are a generalization of those ofSteinberg cite{Steinberg75}, Pittie cite{Pittie72} and Richardsoncite{Rich82b} in the symmetric case where $sigma=heta$ and $H=K$.

[学科分类] 数学（综合）