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

TL;DR

This paper constructs new examples of smooth, absolutely irreducible, non-admissible representations of p-adic GL(n) groups over residue fields, extending previous work to ramified fields using diagram theory.

Contribution

It introduces a method to construct non-admissible irreducible representations of GL(n) over ramified local fields, expanding the understanding of representation theory in characteristic p.

Findings

Constructed non-admissible irreducible representations of GL_2(F) over ramified fields.

Extended the construction to higher dimensions via parabolic induction.

Utilized the theory of diagrams of Breuil and Paskunas.

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 $n>2$.