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

TL;DR

This paper studies a functor linking smooth p-power torsion representations of GL_n(Q_p) to Galois representations, showing finiteness, irreducibility detection, and characterizing induced representations, with implications for p-adic representation theory.

Contribution

It demonstrates that the functor attaches finite dimensional Galois representations to finite length automorphic representations and characterizes its kernel and image properties.

Findings

Functor attaches finite dimensional Galois representations to automorphic representations.

Functor produces irreducible Galois representations from irreducible automorphic objects.

Characterizes representations induced from a torus and GL_2(Q_p).

Abstract

The second named author previously constructed a functor $\mathbb{V}^\vee\circ D^\vee_\Delta$ 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, $\mathbb{V}^\vee\circ D^\vee_\Delta$ 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 $D^\vee_\Delta$ 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.