Normalizers of Sylow subgroups in finite reflection groups

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Let W be a finite reflection group, either real or complex, and S l a Sylow l -subgroup of W. We prove the existence of a semidirect product decomposition of N W ( S l ) in terms of the unique parabolic subgroup of W minimally containing S l and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow l -subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow l -subgroup.