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

Abel Lacabanne (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

Generalizing Lusztig's work, Malle as associated to any imprimitive complex reflection $W$ group 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.

Domaines: Algèbres quantiques [math.QA]; Théorie des représentations [math.RT]

Abel Lacabanne. Drinfeld double of quantum groups, tilting modules and $\mathbb{Z}$-modular data associated to complex reflection groups. 2018. ⟨hal-01827849v2⟩