Restriction of <i>p</i>-adic representations of GL₂(Q_<i>p</i>) 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 $p &gt; 3$ and that all our representations are generic.