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

Junbin Dong

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.

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

Abstract

Let be the group of diagonal matrices in , where is a prime number. Let be an algebraically closed field of characteristic not equal to and . We classify all the irreducible -representations of that admit -stable lines.
Paper Structure (5 sections, 10 theorems, 66 equations)

This paper contains 5 sections, 10 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. The trivial character case
  3. A key lemma
  4. The nontrivial character case
  5. Supplementary result

Key Result

Theorem 1.1

Let $\Bbbk$ be an algebraically closed field of characteristic not equal to $2$ and $p$. Then all simple $SL_2(\bar{\mathbb{F}}_p)$-modules over $\Bbbk$ admitting $\bf T$-stable lines are precisely  $\Bbbk_{\operatorname{tr}}$, $\operatorname{St}$, and $\mathbb{M}(\theta)$ for all nontrivial charact

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Lemma \ref{['keylemma']}
  • Lemma 4.1
  • ...and 7 more