Harish-Chandra pairs in the Verlinde category in positive characteristic

TL;DR

This paper establishes an equivalence between affine group schemes and Harish-Chandra pairs within the Verlinde category in positive characteristic, extending to their representation categories and exploring implications for $GL(L)$ representations.

Contribution

It proves a new categorical equivalence in the Verlinde category and extends this to representation theory, providing insights into $GL(L)$ representations in positive characteristic.

Findings

Category of affine group schemes is equivalent to Harish-Chandra pairs in the Verlinde category

Extended the equivalence to representation categories

Analyzed consequences for $GL(L)$ representation theory

Abstract

In this article, we prove that the category of affine group schemes of finite type in the Verlinde category is equivalent to the category of Harish-Chandra pairs in the Verlinde category. Subsequently, we extend this equivalence to an equivalence between corresponding representation categories and then study some consequences of this equivalence to the representation theory of $GL(L)$, with $L$ a simple object in the Verlinde category.