# Typical representations for level zero Bernstein components of ${\rm GL}_n(F)$

## TL;DR

This paper classifies typical representations of ${
m GL}_n( ext{integers }F)$ for level zero Bernstein components, extending prior results to all characteristics and clarifying their role in the structure of smooth irreducible representations.

## Contribution

It provides a classification of typical representations for level zero Bernstein components of ${
m GL}_n(F)$, generalizing previous work to all characteristics.

## Key findings

- Classified typical representations for level zero Bernstein components.
- Extended results of Henniart and Paskunas to all characteristics.
- Clarified the role of typical representations in the cuspidal support.

## Abstract

Let $F$ be a non-discrete non-Archimedean locally compact field. In this article for a level zero Bernstein component $s$, we classify those irreducible smooth representations of ${\rm GL}_n{\integers{F}}$ (called typical representations) whose appearance in a smooth irreducible representation $\pi$ of ${\rm GL}_n{F}$ implies that the cuspidal support of $\pi$ is $s$. These results extend, for level zero representations, the results of Henniart and Pa\v{s}k\={u}nas on cuspidal representations. The results are independent of the characteristic of the base field.