## Duke Mathematical Journal

### Koszul duality and modular representations of semisimple Lie algebras

Simon Riche

#### Abstract

In this article we prove that if $G$ is a connected, simply connected, semisimple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra $(\mathcal{U} \mathfrak{g})_0$ of the Lie algebra $\mathfrak{g}$ 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. In the case of the block associated to a regular character $\lambda$ of the Harish-Chandra center, the dual ring is related to modules over the specialized algebra $(\mathcal{U} \mathfrak{g})^{\lambda}$ with generalized trivial Frobenius character. Our main tool is the localization theory developed by Bezrukavnikov, Mirković, and Rumynin

#### Article information

Source
Duke Math. J., Volume 154, Number 1 (2010), 31-134.

Dates
First available in Project Euclid: 14 July 2010

https://projecteuclid.org/euclid.dmj/1279140506

Digital Object Identifier
doi:10.1215/00127094-2010-034

Mathematical Reviews number (MathSciNet)
MR2668554

Zentralblatt MATH identifier
1264.17005

#### Citation

Riche, Simon. Koszul duality and modular representations of semisimple Lie algebras. Duke Math. J. 154 (2010), no. 1, 31--134. doi:10.1215/00127094-2010-034. https://projecteuclid.org/euclid.dmj/1279140506