A restriction problem for mod-$p$ representations of $\mathrm{SL}_2(F)$

TL;DR

This paper investigates how smooth irreducible mod-$p$ representations of $ ext{SL}_2(F)$ behave when restricted to a Borel subgroup, revealing that the larger group's action is governed by the subgroup, extending prior work by Paskunas.

Contribution

It provides the $ ext{SL}_2$-analogue of Paskunas' restriction results, clarifying the relationship between $ ext{SL}_2(F)$ representations and their Borel subgroup restrictions.

Findings

The action of $ ext{SL}_2(F)$ on irreducible representations is controlled by the Borel subgroup.

Established an analogue of Paskunas' restriction theorem for $ ext{SL}_2$.

Enhanced understanding of mod-$p$ representation restrictions for non-archimedean local fields.

Abstract

Let $p$ be a prime and $F$ a non-archimedean local field of residue characteristic $p$. In this paper, we study the restriction of smooth irreducible $\bar{\mathbb{F}}_p$-representations of $\mathrm{SL}_2(F)$ to its Borel subgroup. In essence, we show that the action of $\mathrm{SL}_2(F)$ on its irreducibles is controlled by the action of the Borel subgroup. The results of this paper constitute the $\mathrm{SL}_2$-analogue of a work of Pa\v{s}k\=unas\cite{PaskunasRestriction}.