Mixed modular perverse sheaves on affine flag varieties and Koszul duality

TL;DR

This paper establishes a multiplicity formula for indecomposable tilting perverse sheaves on affine flag varieties in characteristic p, linking them to p-Kazhdan--Lusztig polynomials, and constructs a functor connecting mixed modular and ordinary perverse sheaves.

Contribution

It introduces a new multiplicity formula and a degrading functor under certain conditions, advancing the understanding of modular perverse sheaves on affine flag varieties.

Findings

Proves a multiplicity formula for tilting perverse sheaves in terms of p-Kazhdan--Lusztig polynomials.

Constructs a degrading functor relating mixed modular and ordinary perverse sheaves.

Provides insights into the structure of perverse sheaves in positive characteristic.

Abstract

Under some technical assumptions, and building on joint work with Bezrukavnikov, we prove a multiplicity formula for indecomposable tilting perverse sheaves on affine flag varieties, with coefficients in a field of characteristic $p$, in terms of $p$-Kazhdan--Lusztig polynomials. Under the same assumptions, we also explain the construction of a "degrading functor" relating mixed modular perverse sheaves (as defined in joint work with Achar) on such varieties to ordinary perverse sheaves.