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

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 $\mathbf{V}_{\Delta}(\Pi)\simeq \overline{L}^{\boxtimes}(\rho)$.