TL;DR
This paper extends parity sheaf theory to twisted equivariant settings, constructing a modular monodromic Hecke category and establishing new categorifications and equivalences in representation theory.
Contribution
It introduces a formalism for twisted equivariant parity sheaves and constructs a modular monodromic Hecke category, providing new categorifications and equivalences.
Findings
Constructed a modular incarnation of Lusztig and Yun's monodromic Hecke category.
Established a modular categorification of the monodromic Hecke algebra.
Proved a monoidal equivalence between monodromic and ordinary Hecke categories on endoscopic groups.
Abstract
Generalizing the theory of parity sheaves on complex algebraic stacks due to Juteau-Mautner-Williamson, we develop a theory of twisted equivariant parity sheaves. We use this formalism to construct a modular incarnation of Lusztig and Yun's monodromic Hecke category. We then give two applications: (1) a modular categorification of the monodromic Hecke algebra, and (2) a monoidal equivalence between the monodromic Hecke category of parity sheaves and the ordinary Hecke category of parity sheaves on the endoscopic group.
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