Certain complex representations of $SL_2(\bar{\mathbb{F}}_q)$

TL;DR

This paper develops a new category of representations for connected reductive algebraic groups over finite fields, demonstrating its favorable properties and classifying simple objects specifically for SL_2 over algebraic closures.

Contribution

It introduces the category al({f G}) for such groups, proves it is abelian and highest weight, and classifies simple objects for SL_2(ar{\u211d}_q).

Findings

The category al({f G}) is abelian.

al({f G}) is a highest weight category.

Simple objects are classified for SL_2(ar{\u211d}_q).

Abstract

We introduce the representation category $\mathscr{C}({\bf G})$ for a connected reductive algebraic group ${\bf G}$ which is defined over a finite field $\mathbb{F}_q$ of $q$ elements. We show that this category has many good properties for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$. In particular, it is an abelian category and a highest weight category. Moreover, we classify the simple objects in $\mathscr{C}({\bf G})$ for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$.