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

TL;DR

This paper constructs a categorification of $Z$-modular data linked to complex reflection groups by studying tilting modules of the Drinfeld double of quantum groups, extending Lusztig and Malle's work.

Contribution

It introduces a new categorification framework for $Z$-modular data associated with complex reflection groups using tilting modules of quantum doubles.

Findings

Established a connection between tilting modules and $Z$-modular data.

Extended Lusztig's and Malle's work to a categorified setting.

Provided new insights into the structure of complex reflection groups.

Abstract

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.