A geometric Steinberg formula

TL;DR

This paper establishes a geometric isomorphism for simple perverse sheaves on the affine Grassmannian, linking it to the Steinberg tensor product formula for simple representations in positive characteristic.

Contribution

It provides a geometric proof of the Steinberg tensor product formula via an isomorphism of perverse sheaves on the affine Grassmannian.

Findings

Proves a geometric isomorphism related to the Steinberg tensor product formula

Connects geometric representation theory with classical algebraic results

Supports the Finkelberg-Mirković conjecture in this context

Abstract

We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg-Mirkovi\'c conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic.