Automorphisms of complex reflection groups
arXiv:math/0701266
Abstract
Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when is irreducible, apart from the exception $G=\Sgot_6$, as well as for a large class of non-irreducible groups, any automorphism of 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 $N_{\GL(\BC^r)}(G)$ and of a "Galois" automorphism: we show that $\Gal(K/\BQ)$, where is the field of definition of , injects into the group of outer automorphisms of , and that this injection can be chosen such that it induces the usual Galois action on characters of , apart from a few exceptional characters; further, replacing if needed by an extension of degree 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 can be chosen rational.