Drinfeld double of quantum groups, tilting modules, and \(\mathbb{Z}\)-modular data associated to complex reflection groups (Q2210590)

Summary: 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. As an application, we obtain a proof of a conjecture by Cuntz at the decategorified level.