A geometric model for blocks of Frobenius kernels

TL;DR

This paper introduces a geometric framework using perverse sheaves on the affine Grassmannian to model blocks of Frobenius kernel representations of algebraic groups in positive characteristic.

Contribution

It defines new categories of perverse sheaves that serve as geometric models for Frobenius kernel blocks, establishing their structure as highest weight categories.

Findings

Categories have enough projective and injective objects.

Related to tilting perverse sheaves.

They form highest weight categories in a generalized sense.

Abstract

Building on a geometric counterpart of Steinberg's tensor product formula for simple representations of a connected reductive algebraic group $G$ over a field of positive characteristic, and following an idea of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirkovi\'c, we define and initiate the study of some categories of perverse sheaves on the affine Grassmannian of the Langlands dual group to $G$ that should provide geometric models for blocks of representations of the Frobenius kernel $G_1$ of $G$. In particular, we show that these categories admit enough projective and injective objects, which are closely related to some tilting perverse sheaves, and that they are highest weight categories in an appropriate generalized sense.