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
$\underset{\scriptscriptstyle-}{k},$
char
$\underline{k}$
=0. Let
$H:={{G}^{\sigma }}$
and
$K:={{G}^{\theta }}$
be the fixed point groups. We have an action
$\left( H\,\times \,K \right)\,\times \,G\,\to \,G$
, where
$\left( \left( h,\,k \right),\,g \right)\,\mapsto \,hg{{k}^{-1}},\,h\,\in \,H$
,
$k\,\in \,K,g\,\in \,G$
. Let
$G\,//\,\left( H\,\times \,K \right)$
denote the categorical quotient Spec
$\mathcal{O}{{(G)}^{H\times K}}$
. 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
$\sigma \,=\,\theta $
and
$H\,=K$
.