On some nonadmissible smooth irreducible representations of $\mathrm{GL}_2$

TL;DR

This paper provides examples of smooth, absolutely irreducible representations of () over _{p^3} that are not admissible, challenging assumptions about the structure of such representations.

Contribution

It constructs explicit examples of nonadmissible smooth irreducible representations of (), expanding understanding of representation theory over local fields.

Findings

Examples of nonadmissible irreducible representations are constructed.

The representations are smooth and absolutely irreducible.

These examples demonstrate exceptions to typical admissibility expectations.

Abstract

Let $p>2$ be a prime. We give examples of smooth absolutely irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_{p^3})$ over $\mathbb{F}_{p^3}$ which are not admissible.