Abstract
This work is a contribution to the categorification of mathbb{Z}-modular data and deals mainly with ℤ-modular data arising from complex reflection groups, as well as cellular characters for these groups. In his classification of representations of finite groups of Lie type, Lusztig defines a nonabelian Fourier transform, and associate a ℕ-modular datum to each family of unipotent characters. In a generalization of Lusztig's theory to Spetses, Broué, Malle and Michel construct ℤ-modular data associated to some complex reflection groups. We first give a categorical explanation of some of these ℤ-modular data in terms of representation of the Drinfeld double of a finite group. We had to endow the category of representations with a non-spherical structure. The study of slightly degenerate categories shows that they naturally give rise to ℤ-modular data. In order to construct some examples, we consider an extension of the fusion categories associated to Uξ(g), where g is a simple Lie algebra and ξ a root of unity. These categories are constructed as semisimplification of the category of tilting modules of Dξ(g), which is a central extension of Uξ(g). If g={sl}_{n+1}, we show that this category is related to some ℤ-modular data associated to the complex reflection group G(d,1,(n(n+1))/2). Exceptional complex reflection groups are also considered and many different categories appear in the categorification of the associated ℤ-modular data : modules categories over twisted Drinfeld doubles as well as some subcategories of fusion categories of tilting modules over Dξ(g) in type A and B.