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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, 20G20, 22E15, 22E46 show english descriptions Linear algebraic groups over arbitrary fields
Linear algebraic groups over the reals, the complexes, the quaternions
General properties and structure of real Lie groups
Semisimple Lie groups and their representations
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
 

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