Restriction of $p$-adic representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ to parahoric subgroups

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$, and we prove that if an irreducible Banach space representation $\Pi$ of $\mathrm{GL}_2(\mathbf{Q}_p)$ has infinite $\mathrm{GL}_2(\mathbf{Z}_p)$-length then a twist of $\Pi$ has locally algebraic vectors. This answers a question of Dospinescu. We make the simplifying assumption that $p > 3$ and that all our representations are generic.