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

Normalisers of parabolic subgroups in finite unitary reflection groups