Normalizers of Sylow subgroups in finite reflection groups

Abstract

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