Abstract view
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions


Published:20041001
Printed: Oct 2004
Aloysius G. Helminck
Gerald W. Schwarz
Abstract
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.