Restriction of p-adic representations of GL₂(Q_p) to parahoric subgroups

2021/11/24 by Andrea Dotto, Dotto, Andrea
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2111.12827

openalex publication_date 2021/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Without using the p-adic Langlands correspondence, we prove that for many finite length smooth representations of GL₂(Q_p) on p-torsion modules the GL₂(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 GL₂(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 Gal(Q_p/Q_p), and we prove that if an irreducible Banach space representation Π of GL₂(Q_p) has infinite GL₂(Z_p)-length then a twist of Π 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.

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