Automorphisms of complex reflection groups

Abstract:

Let $G\subset \mathrm {GL}(\mathbb {C}^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\mathfrak {S}_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of $N_{\mathrm {GL}(\mathbb {C}^r)}(G)$ and of a “Galois” automorphism: we show that $\mathrm {Gal}(K/\mathbb {Q})$, where $K$ is the field of definition of $G$, injects into the group of outer automorphisms of $G$, and that this injection can be chosen such that it induces the usual Galois action on characters of $G$, apart from a few exceptional characters; further, replacing $K$ if needed by an extension of degree $2$, the injection can be lifted to $\mathrm {Aut}(G)$, 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.