Geodesic
Smoothness of Quotients Associated With a Pair of Commuting Involutions
Canadian journal of mathematics, Tome 56 (2004) no. 5, pp. 945-962

Voir la notice de l'article provenant de la source Cambridge University Press

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 $H\,=K$ .
DOI : 10.4153/CJM-2004-043-7
Mots-clés : 20G15, 20G20, 22E46 
Helminck, Aloysius G.; Schwarz, Gerald W. Smoothness of Quotients Associated With a Pair of Commuting Involutions. Canadian journal of mathematics, Tome 56 (2004) no. 5, pp. 945-962. doi: 10.4153/CJM-2004-043-7
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