Heisenberg Idempotents on Unipotent Groups

TL;DR

This paper studies Heisenberg idempotents in the derived category of a unipotent algebraic group over a field of positive characteristic, proving they form the bounded derived category of a modular category, confirming a conjecture by Drinfeld.

Contribution

It establishes that the Hecke subcategory associated with Heisenberg idempotents is equivalent to the bounded derived category of a modular category, confirming Drinfeld's conjecture.

Findings

Hecke subcategory eD_G(G) is the bounded derived category of a modular category.

Confirmation of Drinfeld's conjecture on Heisenberg idempotents.

Structural understanding of idempotents in the derived category of unipotent groups.

Abstract

Let G be an algebraic group over an algebraically closed field of positive characteristic such that its neutral connected component is a unipotent group. We consider a certain class of closed idempotents in the braided monoidal category (under convolution of complexes) D_G(G) known as Heisenberg idempotents. For such an idempotent e, we will prove certain results about the Hecke subcategory eD_G(G) conjectured by V. Drinfeld. In particular, we will see that it is the bounded derived category of a modular category.