On classification of typical representations for ${\rm GL}_3(F)$

TL;DR

This paper classifies typical representations for Bernstein components of ${ m GL}_3(F)$ over non-Archimedean local fields, completing the classification for residue field size greater than 2.

Contribution

It provides a complete classification of typical representations for ${ m GL}_3(F)$, covering inertial classes and principal series components for certain residue field sizes.

Findings

Classification of typical representations for inertial classes with $q_F>2$.

Classification of typical representations for principal series with $q_F>3$.

Complete classification of ${ m GL}_3(F)$ typical representations for $q_F>2$.

Abstract

Let $F$ be any non-Archimedean local field with residue field of cardinality $q_F$. In this article, we obtain a classification of typical representations for the Bernstein components associated to the inertial classes of the form $[{\rm GL}_n(F)\times F^\times, \sigma\otimes\chi]$ with $q_F>2$, and for the principal series components with $q_F>3$. With this we complete the classification of typical representations for ${\rm GL}_3(F)$, for $q_F>2$.