Springer theory in braid groups and the Birman-Ko-Lee monoid, preprint

We state a conjecture about centralizers of certain roots of central elements in braid groups, and check it for braid groups of type A, B, G(d, 1, r) and a couple of other cases. Our proof makes use of results from Birman-Ko-Lee, of which we give a new intrinsic account. Notations. If G is a group acting on a set X, we denote by XG the subset of X of elements fixed by all elements of G. If (X,x) is a pointed topolog-ical space, we denote by Ω(X,x) the corresponding loop space, by ∼ the homotopy relation on Ω(X,x) and by pi1(X,x) the fundamental group. For all n ∈ N, we denote by µn the set of n-th roots of unity in C. 0. Introduction. Springer theory of regular elements (introduced in [Sp]) explains how certain complex reflection groups naturally arise as centralizers in other complex reflection groups of particular elements, the regular elements. The construc-tion relies on invariant theory and elementary algebraic geometry, and give