Some perverse equivalences of $SL(2,q)$ in its defining characteristic

TL;DR

This paper explores novel autoequivalences in the derived category of modular representations of SL(2,q), revealing new symmetries and structures in the representation theory of finite groups in characteristic p.

Contribution

It introduces a new type of autoequivalence for the derived category of SL(2,q) representations, utilizing perverse equivalences and Brauer correspondence techniques.

Findings

Discovered a new autoequivalence with unique features

Connected representation theory with derived category symmetries

Provided a new perspective on modular representations of SL(2,q)

Abstract

In this article, we study the modular representations of the special linear group of degree two over a finite field in defining characteristic. In particular, we study the automorphisms of derived category of representations. We have been able to obtain a new type of autoequivalence. This autoequivalence has some uncommon features. It is more conveniently conceived and proved using the representation theory of its Brauer correspondence but at the same time it can be very neatly described, using a type of derived equivalence called perverse equivalence, in global settings.