"Galois" automorphisms of complex reflection groups

Ivan Marin (1) , Jean Michel (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let $G$ be a complex reflection group and $K$ its field of definition (the subfield of $\BC$ generated by the reflection character). We show that $\Gal(K/\BQ)$ injects into the group $A$ of outer automorphisms of $G$ which preserve reflections, 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 representations affords a model which is equivariant with respect to this lifting. Along the way we give the structure of the group $A$ and 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. "Galois" automorphisms of complex reflection groups. 2007. ⟨hal-00123495v1⟩