Non-admissible irreducible representations of $p$-adic $\mathrm{GL}_{n}$ in characteristic $p$

TL;DR

This work constructs irreducible non-admissible smooth representations of $p$-adic groups in characteristic $p$, focusing on $\mathrm{GL}_n(F)$. It develops a novel spliced module and an infinite-dimensional diagram built from Breuil-Paskunas diagrams, enabling irreducible non-admissible $\mathrm{GL}_2(F)$ representations and, via parabolic induction, irreducible non-admissible $\mathrm{GL}_n(F)$ representations for all $n\ge 2$. For $n=2$, the representation is realized inside an injective envelope to produce an irreducible non-admissible $\mathrm{GL}_2(F)$-module with a nontrivial endomorphism algebra; for higher $n$, irreducibility is secured by $\mathrm{M}$-regularity of $\mathrm{GL}_n(\mathcal{O}_F)$-weights and Hecke-algebra arguments. The results extend earlier unramified cases and connect to conjectures about functors from mod $p$ representations to moduli stacks of Langlands parameters, highlighting implications for the mod $p$ Langlands program.

Abstract

Let $p>3$ and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. We construct smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_2(F)$ defined over the residue field of $F$ extending the earlier results of the authors for $F$ unramified over $\mathbb{Q}_{p}$. This construction uses the theory of diagrams of Breuil and Paskunas. By parabolic induction, we obtain smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_n(F)$ for $n>2$.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.6
• Lemma 2.7