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# Smoothness of Quotients Associated \\With a Pair of Commuting Involutions

Published:2004-10-01
Printed: Oct 2004
• Aloysius G. Helminck
• Gerald W. Schwarz
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## 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 $H=K$.
 MSC Classifications: 20G15 - Linear algebraic groups over arbitrary fields 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 22E15 - General properties and structure of real Lie groups 22E46 - Semisimple Lie groups and their representations