Filtrations of tilting modules and costalks of parity sheaves

TL;DR

This paper generalizes the geometric and algebraic filtrations of tilting modules and their interpretation via the geometric Satake correspondence from characteristic zero to positive characteristic fields.

Contribution

It extends the known filtrations and geometric interpretations of tilting modules and parity sheaves to positive characteristic settings.

Findings

Established a filtration of tilting modules in positive characteristic.

Connected filtrations to geometric Satake correspondence in new settings.

Provided insights into the structure of parity sheaves and costalks.

Abstract

Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski constructed a filtration of weight spaces of a G module, using the action of a principal nilpotent element of the Lie algebra, and proved that this filtration corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg discovered that this filtration has an interesting geometric interpretation via the geometric Satake correspondence. The goal of this article is to generalize these results to positive characteristics.