Automorphisms of complex reflection groups
Résumé
Let $G\subset\GL(\BC^r)$ be an irreducible finite complex reflection group. We show that (apart from the exception $G=\Sgot_6$) any automorphism of $G$ is the product of an automorphism induced by tensoring by a linear character, of an automorphism induced by an element of $N_{\GL(\BC^r)}(G)$ and of what we call a ``Galois'' automorphism: we show that $\Gal(K/\BQ)$, where $K$ is the field of definition of $G$ (the subfield of $\BC$ generated by the reflection character), injects into the group of outer automorphisms of $G$, and that this injection with a few exceptions can be chosen such that it commutes with the Galois action on characters of $G$; further, replacing if needed $K$ by an extension of order $2$, the injection can be lifted to $\Aut(G)$, and every irreducible representation admits a model which is equivariant with respect to this lifting. Along the way we show that the fundamental invariants of $G$ can be chosen rational.
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