A note on the classification of simple $SL_2(\bar{\mathbb{F}}_p)$-modules admitting $\bf T$-stable lines in cross characteristic

TL;DR

This paper classifies irreducible representations of the algebraic group SL_2 over algebraic closures of finite fields that contain stable lines under the diagonal subgroup, enhancing understanding of their module structure.

Contribution

It provides a complete classification of simple modules with T-stable lines for SL_2 over algebraic closures of finite fields, a previously unresolved problem.

Findings

Identified all irreducible modules with T-stable lines

Characterized modules based on their stability properties

Extended classification to cross characteristic cases

Abstract

Let $\bf T$ be the group of diagonal matrices in $SL_2(\bar{\mathbb{F}}_p)$, where $p$ is a prime number. Let $\Bbbk$ be an algebraically closed field of characteristic not equal to $2$ and $p$. We classify all the irreducible $\Bbbk$-representations of $SL_2(\bar{\mathbb{F}}_p)$ that admit $\bf T$-stable lines.