Braid groups of normalizers of reflection subgroups

Abstract

Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra ${\tilde{H}}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient ${\tilde{B}}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra ${\tilde{H}}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.

22 pages. To appear in Annales de l'Institut Fourier