Finiteness properties of generalized Montréal functors with applications to mod $p$ representations of ${\mathrm{GL}}_n({\mathbb{Q}}_p)$

Abstract

The second named author previously constructed a functor ${\mathbb{V}}^{\vee }\circ D_{\Delta }^{\vee }$ from the category of smooth $p$-power torsion representations of ${\mathrm{GL}}_n({\mathbb{Q}}_p)$ to the category of inductive limits of continuous representations on finite $p$-primary abelian groups of the direct product $G_{{\mathbb{Q}}_p,\Delta }\times {\mathbb{Q}}_p^{\times }$ of $(n-1)$ copies of the absolute Galois group of ${\mathbb{Q}}_p$ and one copy of the multiplicative group ${\mathbb{Q}}_p^{\times }$. 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, ${\mathbb{V}}^{\vee }\circ D_{\Delta }^{\vee }$ produces irreducible representations of $G_{{\mathbb{Q}}_p,\Delta }\times {\mathbb{Q}}_p^{\times }$ when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of $D_{\Delta }^{\vee }$ 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 ${\mathrm{GL}}_2({\mathbb{Q}}_p)$. Finally, we formulate a conjecture and prove partial results on the essential image.

v2: 45 pages, sharpened results in section 3.2, new section on GL3 in the introduction, submitted version, comments welcome