Automorphisms of complex reflection groups

Ivan Marin (1) , Jean Michel (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\Sgot_6$, 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 $N_{\GL(\BC^r)}(G)$ and of a ``Galois'' automorphism: we show that $\Gal(K/\BQ)$, 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 if needed $K$ 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 $G$ can be chosen rational.

Domaines: Théorie des représentations [math.RT]; Théorie des groupes [math.GR]

Ivan Marin, Jean Michel. Automorphisms of complex reflection groups. 2009. ⟨hal-00123495v3⟩