Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor

TL;DR

The paper investigates whether Zábrádi's functor $\mathbf{V}_{\Delta}$ can recover the $(n-1)$-fold Galois data $\overline{L}^{\boxtimes}(\rho)$ from a GL$_n(\mathbb{Q}_p)$-representation $\Pi$ that is compatible with a given $\rho$ in the Breuil–Herzig–Hu–Morra–Schraen framework. It develops a GL$_3$ toy case to expose structural obstructions: even under weak compatibility with $\widetilde{P}_{\rho}$, the image $\mathbf{V}_{\Delta}(\Pi)$ cannot equal $\overline{L}^{\boxtimes}(\rho)$ when $\rho$ is reducible and $n\ge 3$, due to mismatches in Jordan–Hölder data and finiteness properties of $\mathbf{V}_{\Delta}$. The work also establishes dimension bounds for $\mathbf{V}_{\Delta}(\Pi)$ in the toy case and proves that for irreducible $\rho$ there are infinitely many dimensions where no supersingular $\Pi$ yields the desired equality, highlighting fundamental obstructions to reconstructing $\overline{L}^{\boxtimes}(\rho)$ from $\Pi$ via $\mathbf{V}_{\Delta}$. Collectively, these results reveal pervasive limitations of Breuil–Herzig–Hu–Morra–Schraen-type conjectures when translated through Zábrádi’s functor, even in restricted or toy settings.

Abstract

Let $ρ$ be an $n$-dimensional representation of $\mathcal{G}_{\mathbb{Q}_p}$ over $\overline{\mathbb{F}_p}$. When $ρ$ is generic and a good conjugate, the article "Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$", by Breuil-Herzig-Hu-Morra-Schraen, introduces the notion of compatibility with $ρ$ for an admissible representation of $\mathrm{GL}_n(\mathbb{Q}_p)$. In loc. cit., the five authors also question whether one could recover a representation of $\mathcal{G}_{\mathbb{Q}_p}^{n-1}$, called $\overline{L}^{\boxtimes}(ρ)$ and constructed from $ρ$, from some $Π$ compatible with $ρ$ by using Zábrádi's functor $\mathbf{V}_Δ$. We give a range of results, for an arbitrary $Π$ verifying some "weak" compatibilities with $ρ$, about how badly $\mathbf{V}_Δ(Π)$ behaves. In particular, when $ρ$ is reducible and $n\geq 3$, no $Π$ compatible with $P_ρ$ can verify $\mathbf{V}_Δ(Π)\simeq \overline{L}^{\boxtimes}(ρ)$.

Theorems & Definitions (56)

• Definition 2.1
• Definition 2.2: See marquis_formalisme
• Theorem 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Corollary 2.6
• proof
• Definition 2.7
• Definition 2.8