Modular Koszul duality

TL;DR

This paper establishes a form of Koszul duality for the category  of a reductive group in positive characteristic, using formality of dg-algebras of extensions of parity sheaves, despite the absence of Koszul rings.

Contribution

It introduces a modular analogue of Koszul duality for category  in positive characteristic, expanding the theoretical framework without relying on Koszul rings.

Findings

Proves formality of dg-algebras of extensions of parity sheaves in certain characteristics.

Establishes an analogue of Koszul duality in positive characteristic setting.

Highlights the absence of Koszul rings and Kazhdan--Lusztig conjecture analogues.

Abstract

We prove an analogue of Koszul duality for category $\mathcal{O}$ of a reductive group $G$ in positive characteristic $\ell$ larger than 1 plus the number of roots of $G$. However there are no Koszul rings, and we do not prove an analogue of the Kazhdan--Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus 2.