Normalisateur et groupe dʹArtin de type sphérique

Let (A,S) be an Artin group and X a subset of S; denote by AX the subgroup of A generated by X. When A is of spherical type, we prove that the normalizer and the commensurator of AX in A are equal and are the product of AX by the quasi-centralizer of AX in A. Looking the associated monoids A+ and AX+, we described the quasi-centralizer of AX+ in A+ thanks to results in Coxeter groups. These two results generalize earlier results of Paris [J. Algebra 196 (1997) 369–399]. Finally, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin group and in the Coxeter group.