Modular perverse sheaves on flag varieties I: tilting and parity sheaves

TL;DR

This paper establishes a deep connection between parity complexes and tilting perverse sheaves on flag varieties, revealing a graded equivalence that impacts the understanding of modular categories and decomposition numbers.

Contribution

It proves that the category of parity complexes is a graded version of tilting perverse sheaves on dual flag varieties, extending the understanding of these categories in good characteristic.

Findings

Parity complexes form a graded version of tilting perverse sheaves.

Implications for Soergel's modular category O.

Results on multiplicities and decomposition numbers.

Abstract

In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group, for any field of coefficients whose characteristic is good. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.