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

TL;DR

The article develops a comprehensive theory for the generalized Montréall functor in the p-adic Langlands context for GL_n(Q_p), proving finiteness properties that attach finite-dimensional Galois representations to finite-length automorphic representations and establishing irreducibility-detection and kernel-characterization results. It builds a robust framework using multivariable (φ_Δ,Γ_Δ)-modules, noncommutative lattices, and G/B-equivariant sheaves to control size and structure, and applies these tools to the study of higher extensions of principal series, parabolic induction, and the interplay with Galois-side extensions. A central achievement is showing that D^∨_Δ is fully faithful on the quotient SP_h and providing precise criteria for when automorphic representations arise from parabolic induction, as well as formulating a conjectural description of the essential image of D^∨_Δ on SP_h with supporting results. The work also connects to Breuil’s and Colmez’s functors, offering potential new methods for computing extension groups and suggesting a unified, higher-rank p-adic Langlands program guided by these generalized Montréal-type functors.

Abstract

The second named author previously constructed a functor $\mathbb{V}^\vee\circ D^\vee_Δ$ 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 $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ 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, $\mathbb{V}^\vee\circ D^\vee_Δ$ produces irreducible representations of $G_{\mathbb{Q}_p,Δ}\times \mathbb{Q}_p^\times$ when applied to irreducible objects on the automorphic side and detects isomorphisms unless it vanishes. Further, we determine the kernel of $D^\vee_Δ$ 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.

Theorems & Definitions (137)

• Theorem A
• Theorem B
• Theorem C
• Theorem D: Theorem \ref{['fullyfaithful']}
• Theorem E: Theorem \ref{['lookslikeparindisparind']}
• Theorem F: Corollary \ref{['gl3supercuspidal']}
• Proposition 2.1.1
• proof
• Proposition 2.1.2
• proof