# On certain Iwahori representations of unramified $U(2, 1)$ in characteristic $p$

## TL;DR

This paper studies certain smooth representations of the unramified unitary group U(2,1) over a local field of characteristic p, focusing on Iwahori subrepresentations to understand their finite presentation properties.

## Contribution

It provides explicit descriptions of Iwahori subrepresentations for specific classes of representations and establishes conditions for non-finite presentation.

## Key findings

- Identifies a sufficient condition for non-finite presentation of representations.
- Explicitly determines Iwahori subrepresentations for spherical universal Hecke modules.
- Analyzes irreducible principal series representations in this context.

## Abstract

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$, and $K$ be a maximal compact open subgroup of $G$. For an $\overline{\mathbf{F}}_p$-smooth representation $\pi$ of $G$ containing a weight $\sigma$ of $K$, we follow the work of Hu (\cite{Hu12}) to attach $\pi$ a certain $I_K$-subrepresentation, where $I_K$ is the Iwahori subgroup in $K$. In terms of such an $I_K$-subrepresentation, we prove a sufficient condition for $\pi$ to be non-finitely presented. We determine such an $I_K$-subrepresentation explicitly, when $\pi$ is either a spherical universal Hecke module or an irreducible principal series.