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Modular Koszul duality

Part of: Linear algebraic groups and related topics Rings and algebras arising under various constructions Special varieties

Published online by Cambridge University Press:  13 December 2013

Simon Riche ,
Wolfgang Soergel  and
Geordie Williamson
Simon Riche
Affiliation:
Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448, F-63000 Clermont-Ferrand, France email simon.riche@math.univ-bpclermont.fr CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubière, France email simon.riche@math.univ-bpclermont.fr
Wolfgang Soergel
Affiliation:
Mathematisches Institut, Universität Freiburg, Eckerstraße 1, D-79104 Freiburg, Germany email Wolfgang.Soergel@math.uni-freiburg.de
Geordie Williamson
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany email geordie@mpim-bonn.mpg.de
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Abstract

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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 $2$ .

Keywords

Koszul dualitycategory $ \mathcal{O} $formalityparity sheavesperverse sheaves

MSC classification

Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 20G05: Representation theory 16S37: Quadratic and Koszul algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 2 , February 2014 , pp. 273 - 332
Copyright
© The Author(s) 2013 

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