Free-monodromic mixed tilting sheaves on flag varieties

TL;DR

This paper constructs a monoidal category of free-monodromic tilting sheaves on Kac-Moody flag varieties within the mixed modular derived category framework, advancing modular Koszul duality theory.

Contribution

It introduces a new construction of free-monodromic tilting sheaves in the modular setting, enabling a modular Koszul duality for flag varieties.

Findings

Construction of a monoidal category of tilting sheaves in the modular setting

Establishment of properties analogous to characteristic zero case

Foundation for modular Koszul duality on flag varieties

Abstract

In this paper we propose a construction of a monoidal category of "free-monodromic" tilting perverse sheaves on (Kac-Moody) flag varieties in the setting of the "mixed modular derived category" introduced by the first and third authors. This category shares most of the properties of their counterpart in characteristic 0, defined by Bezrukavnikov-Yun using certain pro-objects in triangulated categories. This construction is the main new ingredient in the construction of a "modular Koszul duality" equivalence for constructible sheaves on flag varieties, see [P. Achar, S. Makisumi, S. Riche, and G. Williamson, "Koszul duality for Kac-Moody groups and characters of tilting modules", J. Amer. Math. Soc. 32 (2019)].