Modular Koszul duality

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.

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References in corpus (3)

• Parity Sheaves
• Koszul duality and modular representations of semi-simple Lie algebras
• Decomposition numbers for perverse sheaves

Cited by in corpus (11)

• Parity Sheaves
• Cohomology of cluster varieties. I. Locally acyclic case
• Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture
• Modular perverse sheaves on flag varieties II: Koszul duality and formality
• Derived categories of graded gentle one-cycle algebras
• Mixed Motives and Geometric Representation Theory in Equal Characteristic
• Grassmannians and Koszul duality
• Modular equivariant formality
• Notes on exotic and perverse-coherent sheaves
• Extensions of Verma modules
• Unicity of graded covers of the Category $\mathcal{O}$ of Bernstein-Gelfand-Gelfand