Automorphisms of complex reflection groups

I. MARIN AND J. MICHEL Abstract. Let G ⊂ GL(Cr) be an irreducible finite complex reflection group. We show that (apart from the exception G = S6) 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 NGL(Cr)(G) and of what we call a “Galois” automorphism: we show that Gal(K/Q), where K is the field of definition of G (the subfield of C 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.