Koszul duality and modular representations of semi-simple Lie algebras

TL;DR

This paper demonstrates that blocks of the restricted enveloping algebra of a semi-simple Lie algebra over a large characteristic field can be given Koszul gradings, extending previous results and analyzing their dual rings.

Contribution

It proves the existence of Koszul gradings on all blocks of the restricted enveloping algebra for semi-simple Lie algebras in large characteristic, expanding prior work.

Findings

All blocks of (Ug)_0 admit Koszul gradings.

Provides information on Koszul dual rings.

Utilizes localization theory in positive characteristic.

Abstract

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin.