Erratum (October 26, 2012)

Throughout this paper, we denote by $\mathcal{H}_q$ one of the cyclotomic Hecke algebras given by the table in Subsection 1.3. We claim in the beginning of Subsection 2.2 that the algebra $\mathbb{C}(y)\mathcal{H}_q$ is split semisimple based on the fact that $\mathcal{H}_q$ is a free $\mathbb{Z}[\textbf{u},\textbf{u}^{-1}]$-module of rank equal to the order of the group, which was proved for the groups of Table 1.3 by Malle and Michel in [24].

Of course, this is simply conjectured to be true for the cyclotomic Hecke algebras associated with other complex reflection groups. However, the addition of Subsection 2.1 between 1.3 and 2.2 during the corrections phase makes it look as if we claim that the above statement is true for any complex reflection group. This explains also the change of notation for the cyclotomic Hecke algebra between Subsections 2.1 and 2.2.

There has recently been a paper on arXiv by Ivan Marin, stating that the freeness conjecture is not actually proved for any exceptional complex reflection group of rank 2, except for G4 (the same being true about the conjecture of the existence of a canonical symmetrizing form). So we now feel obliged to say that whatever we claim for $\mathcal{H}_q$ in the first two paragraphs of Subsection 2.2 is, for the moment, conjectured, and known only in the case of the group G4.