Implications of Breuil-Herzig-Hu-Morra-Schraen's conjectures on Zábrádi's functor
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 $\mathbf{V}_Δ(Π)\simeq \overline{L}^{\boxtimes}(ρ)$.