Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Z\'abr\'adi's functor

TL;DR

This paper investigates the relationship between certain admissible representations of fffffff(fffffff) and Galois representations, revealing limitations in recovering Galois representations via Zfabrfadi's functor.

Contribution

The paper provides new insights into how Zfabrfadi's functor behaves with compatible representations, especially in reducible cases for higher dimensions.

Findings

ffffffff(fffffff) cannot recover fffffff( ho) in certain cases.

Compatibility conditions are insufficient for reconstructing Galois representations in higher dimensions.

Limitations are especially pronounced when ffffffff is reducible and dimension f f f f.

Abstract

Let $\rho$ be an $n$-dimensional representation of $\mathcal{G}_{\mathbb{Q}_p}$ over $\overline{\mathbb{F}_p}$. When $\rho$ 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 $\rho$ 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 $\overline{L}^{\boxtimes}(\rho)$ and constructed from $\rho$, from some $\Pi$ compatible with $\rho$ by using Z\'abr\'adi's functor $\mathbf{V}_{\Delta}$. We give a range of results, for an arbitrary $\Pi$ verifying some "weak" compatibilities with $\rho$, about how badly $\mathbf{V}_{\Delta}(\Pi)$ behaves. In particular, when $\rho$ is reducible and $n\geq 3$, no $\Pi$ compatible with $P_{\rho}$ can verify $\mathbf{V}_{\Delta}(\Pi)\simeq \overline{L}^{\boxtimes}(\rho)$.