Splitting fields for extended complex reflection groups and Hecke algebras

Abstract

Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.