Note on a certain category of mod $p$ representations

Abstract

Let $p>3$ be a prime number, $f\geq1$ an integer. We consider a certain full subcategory $\mathcal C$ of the category of smooth admissible mod $p$ representations of either $\text{GL}_2\mathbf Q_{p^f}$ or of the group of units of the quaternion algebra over $\mathbf Q_{p^f}$. This category was introduced in the context of the mod $p$ Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the $\text{GL}_2$-case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod $p$ representation $\pi$ (with central character) belongs to $\mathcal C$ is completely determined by the restriction of $\pi$ to an arbitrarily small open subgroup.