Artin–Tits groups with CAT(0) Deligne complex

Let (A,S)(A,S) be an Artin–Tits system, and AXAX be the standard parabolic subgroup of AA generated by a subset XX of SS. Under the hypothesis that the Deligne complex has a CAT(0) geometric realization, we prove that the normalizer and the commensurator of AXAX in AA are equal. Furthermore, if AXAX is of spherical type, these subgroups are the product of AXAX with the quasi-centralizer of AXAX in AA. For two-dimensional Artin–Tits groups, the result still holds without any sphericality hypothesis on XX. We explicitly describe the elements of this quasi-centralizer.