Drinfeld double of quantum groups, tilting modules, and $\mathbb Z$-modular data associated to 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. As an application, we obtain a proof of a conjecture by Cuntz at the decategorified level.

A Soergel-like category for complex reflection groups of rank one

“…One can cite for instance the 'Spetses' program [4], which provides a sort of generalization of unipotent characters of reductive groups to non-existing reductive groups attached to complex reflection groups. In this framework, some categorification results were obtained for cyclic groups in [1] (building up on [6,7]) and later extended in [14,15]. Note that the ring considered in [1,Theorem 5.5] is related to the ring A W studied in the present paper, as observed in Remark 5.4.…”

Slightly Degenerate Categories and ℤ-Modular Data

“…As an application, we will reinterpret the example of Bonnafé and Rouquier in this setting of slightly degenerate categories. This approach will be generalized in [Lac18].…”

Fourier matrices for $G(d,1,n)$ from quantum general linear groups

“…In [14], the author explained how to reinterpret the category of Bonnafé and Rouquier into the framework of slightly degenerate categories. This framework turned out to be well adapted for the problem of categorifying modular data: the modular data of some families of the complex reflection group G(d, 1, n) arise from the representation of the Drinfeld double of the quantum enveloping algebra of the standard Borel of sl m [15].…”