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.

We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$. Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$. If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$-modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$.