On freeness of compactly induced mod-$p$ representations of $\rm{SL}_{2}(F)$

TL;DR

This work establishes that compactly induced mod-$p$ representations of $\mathrm{SL}_{2}(F)$ from a weight $\sigma_{\vec{r}}$ of $K_0={\rm SL}_{2}(\mathcal{O}_{F})$ form a free left module of infinite rank over the spherical Hecke algebra $\mathcal{H}(G_S,K_0,\sigma_{\vec{r}})$. The authors isolate a general linear-algebraic framework for graded spaces with commuting operators and verify it in the $\tau$-action on the pro-$p$-Iwahori invariants, plus they prove the non-vanishing of $\tau$ on the central Bruhat--Tits tree vertex. Their approach clarifies the mechanism behind freeness and separates combinatorial linear-algebra from group-specific computations, potentially enabling similar results for other split reductive groups. The results extend known freeness phenomena from $\mathrm{GL}_2$ and unramified unitary groups to $\mathrm{SL}_2$, contributing to the mod-$p$ Langlands program by detailing the Hecke-module structure of compact inductions in the rank-one case.

Abstract

Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers $\mathcal{O}_{F}$. Set $G_{S}:={\rm SL}_{2}(F)$and $K_{0}:={\rm SL}_{2}(\mathcal{O}_{F})$ . For a smooth irreducible $\bar{\mathbb{F}}_{p}$-representation $σ$ of $K_{0}$, we study the structure of the compact induction ${\rm ind}_{K_{0}}^{G_{S}}(σ)$ as a left module over the standard spherical Hecke algebra ${\rm End}_{G_{S}}\left({\rm ind}_{K_{0}}^{G_{S}}(σ)\right)$. We prove that it is free and of infinite rank.

Theorems & Definitions (20)

• Theorem 1.1: Theorem \ref{['thm:freeness theorem']}
• Proposition 2.1
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Theorem 4.3