  Generalizing Lusztig's work, Malle has associated to some imprimitive complex
reflection group $W$ a set of "unipotent characters", which are in bijection of
the usual unipotent characters of the associated finite reductive group if $W$
is a Weyl group. He also obtained a partition of these characters into families
and associated to each family a $\mathbb{Z}$-modular datum. We construct a
categorification of some of these data, by studying the category of tilting
modules of the Drinfeld double of the quantum enveloping algebra of the Borel
of a simple complex Lie algebra.
