Artin–Tits groups with image Deligne complex

Let (A,S) be an Artin–Tits system, and AX be the standard parabolic subgroup of A generated by a subset X of S. Under the hypothesis that the Deligne complex has a image geometric realization, we prove that the normalizer and the commensurator of AX in A are equal. Furthermore, if AX is of spherical type, these subgroups are the product of AX with the quasi-centralizer of AX in A. For two-dimensional Artin–Tits groups, the result still holds without any sphericality hypothesis on X. We explicitly describe the elements of this quasi-centralizer.