Automorphisms of complex reflection groups

Abstract

Let G ⊂ G L ( C r ) G\subset \mathrm {GL}(\mathbb {C}^r) be a finite complex reflection group. We show that when G G is irreducible, apart from the exception G = S 6 G=\mathfrak {S}_6 , as well as for a large class of non-irreducible groups, any automorphism of G 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 G L ( C r ) ( G ) N_{\mathrm {GL}(\mathbb {C}^r)}(G) and of a “Galois” automorphism: we show that G a l ( K / Q ) \mathrm {Gal}(K/\mathbb {Q}) , where K K is the field of definition of G G , injects into the group of outer automorphisms of G G , and that this injection can be chosen such that it induces the usual Galois action on characters of G G , apart from a few exceptional characters; further, replacing K K if needed by an extension of degree 2 2 , the injection can be lifted to A u t ( G ) \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 G can be chosen rational.