2025/01/30 by Gergely Jakovác, Jakovác, Gergely, Gergely Zábrádi +1
Mathematics · #11F70 #11F80 #11F85 #11S37 #22E50 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2501.18396
openalex publication_date 2025/01/30 · openalex created_date 2025/02/01 · openalex updated_date 2026/07/31
The second named author previously constructed a functor \mathbbV^\vee∘ D^\veeΔ from the category of smooth p-power torsion representations of GLn(ℚp) to the category of inductive limits of continuous representations on finite p-primary abelian groups of the direct product Gℚp,Δ× ℚp^× of (n-1) copies of the absolute Galois group of ℚp and one copy of the multiplicative group ℚp^×. In the present work we show that this functor attaches finite dimensional representations on the Galois side to smooth p-power torsion representations of finite length on the automorphic side. This has some implications on the finiteness properties of Breuil's functor, too. Moreover, \mathbbV^\vee∘ D^\veeΔ produces irreducible representations of Gℚp,Δ× ℚp^× when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of D^\veeΔ when restricted to successive extensions of subquotients of principal series. We use this to characterize representations that are parabolically induced from the product of a torus and GL2(ℚp). Finally, we formulate a conjecture and prove partial results on the essential image.