Regular theory in complex braid groups

Abstract

In his seminal paper [2], Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection between regular elements in the reflection group, and roots of the “full twist” element of the pure braid group.He then suggests that it would be possible to extend the conclusion of this theorem to centralizers of regular elements in well-generated groups. In this paper we give a positive answer to this question and we show moreover that these results hold for an arbitrary reflection group. As a byproduct, we get a generalization of a theorem from Shvartsman regarding the torsion of the quotient of an irreducible braid group by its center.