Divisorial inertia and central elements in braid groups

Given a complex reflection group W, we will show how the generators of the centers of the parabolic subgroups of the pure braid group P(W) can be represented by loops around irreducible divisors of the corresponding minimal De Concini-Procesi model XW. We will also show that a more subtle construction gives representations of the generators of the centers of the parabolic subgroups of the braid group B(W) as loops in the (not smooth) quotient variety XW/W.