# On certain Iwahori--Hecke modules of $GL_3$ in characteristic $p$

## TL;DR

This paper demonstrates that a finiteness property of Iwahori--Hecke modules observed in $GL_2$ does not extend to $GL_3$, revealing the existence of infinite codimension submodules in certain invariants.

## Contribution

It shows that the finiteness results for Iwahori--Hecke modules in $GL_2$ do not hold for $GL_3$, highlighting a fundamental difference in module structure.

## Key findings

- Existence of non-zero Iwahori--Hecke submodules of infinite codimension in $GL_3$
- Failure of finiteness results analogous to $GL_2$ case
- Contrasts with known properties of $GL_2$ modules

## Abstract

In this note, we show that the natural analogue of certain finiteness result of Barthel--Livn$\acute{\text{e}}$ on $GL_2$ fails for $GL_3$. More precisely, within the pro-$p$-Iwahori invariants of a maximal compact induction of $GL_3$, we show there exist non-zero Iwahori--Hecke submodules of \emph{infinite} codimension.