Character D-modules via Drinfeld center of Harish-Chandra bimodules

TL;DR

This paper connects character D-modules to the Drinfeld center of Harish-Chandra bimodules, providing new insights into their structure, classification, and relations to convolution and cohomology in representation theory.

Contribution

It introduces a novel approach to character D-modules via the Drinfeld center and offers a new classification method for irreducible character sheaves over complex numbers.

Findings

Realization of character D-modules as Drinfeld center of Harish-Chandra bimodules

Classification of irreducible character sheaves using this new framework

Description of top cohomology of convolution of character sheaves

Abstract

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg. Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category arXiv:math/0605628v3 this allows us to derive classification of irreducible character sheaves over $\mathbb C$ obtained by Lusztig by a different method. We also give a simple description for the top cohomology of convolution of character sheaves over $\mathbb C$ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini-Procesi compactification.