What question did this study set out to answer?

This note aims to construct irreducible smooth representations of GL_n(F) without a central character.

January 18, 2026

Irreducible smooth representations in defining characteristic without central character

Key Points

This note aims to construct irreducible smooth representations of GL_n(F) without a central character.
Utilized methods from prior works by Le (2019) and Ghate, Le, and Sheth (2023).
Examined representations over a non-archimedean local field F.
Explored constructions both without and with a central character and nonscalar endomorphisms.
Successfully constructed irreducible smooth representations of GL_n(F) without a central character.
Created representations with simultaneous central characters and nonscalar endomorphisms when n > 3.
Demonstrated that representations can exist without a Hecke eigenvalue under certain conditions.

Abstract

Let p > 3 p>3, n > 1 n>1 an integer, and F F be a non-archimedean local field with residue field a proper finite extension of F p Fₚ. Let E E be an algebraically closed countable field extension of the residue field of F F. In this short note, we explain how the methods from Le Math. Res. Lett. 26 (2019), 1747–1758 and Ghate, Le, and Sheth Represent. Theory 27 (2023), 1088–1101 can be used to construct irreducible smooth representations of GL n ⁡ (F) GLₙ (F) over E E without a central character. We also construct irreducible smooth representations of GL n ⁡ (F) GLₙ (F) over E E with simultaneously a central character, nonscalar endomorphisms, and if n > 3 n>3, without a Hecke eigenvalue.

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Cite This Study

Daniel Le (Fri,) studied this question.

synapsesocial.com/papers/696c7817eb60fb80d13964f4 https://doi.org/https://doi.org/10.1090/proc/17439

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