  In his seminal paper on complex reflection arrangements, 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.
