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 H0 for the normalizer N W (W 0 ), one first considers a natural subquotient B0 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 H0 . We also give classes of both split and non-split examples in the non-Coxeter case. Contents 1. Introduction 2. Definitions and preliminaries 3. Reflection subgroups of Coxeter groups 3.1. Reflection subgroups and normalizers 3.2. Reducing non-reduced expressions 3.3. Properties of positive lifts 3.4. The finite Coxeter case 3.5. Hecke algebras 4. Groupoid descriptions of normalizers 4.1. Preliminaries on groupoids 4.2. Groupoids of reflection subgroups 4.3. Artin groupoids 5. An example of splitting for the group G(d, 1, n) 5.1. Preliminaries 5.2. A direct product decomposition 6. Counter-examples in the general case 6.1. Non-parabolic reflection subgroups with no complement in their normalizer 6.2. Central elements of braid groups References