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

Abstract

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.

Accepted version. Some corrections, main results unchanged

References in corpus (1)

• On some $p$-adic and mod $p$ representations of quaternion algebra over ${\mathbb{Q}}_p$