Normalisers of parabolic subgroups in finite unitary reflection groups

TL;DR

This paper extends the understanding of normalisers of parabolic subgroups from finite Coxeter groups to all finite unitary reflection groups, showing they form semidirect products and exploring their actions as reflection groups.

Contribution

It generalizes known results about normalisers in Coxeter groups to the broader class of finite unitary reflection groups, including the structure of complements.

Findings

Normalisers are semidirect products in all cases.

In many cases, complements are stabilisers of root sets.

Some complements act as reflection groups on fixed point spaces.

Abstract

It is well known that the normaliser of a parabolic subgroup of a finite Coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots. We show that a similar result holds for all finite unitary reflection groups: namely, the normaliser of a parabolic subgroup is a semidirect product and in many, but not all, cases the complement can be obtained as the stabiliser of a set of roots. In addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup.