Normalisateurs et groupes d'Artin-Tits de type sphérique

Abstract

Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the associated monoids $A_S^{+}$ and $A_X^{+}$, we describe the quasi-centralizer of $A_X^{+}$ in $A_S^{+}$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits group and in the Coxeter group.

14 pages

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