Links between generalized Montréal-functors

Abstract

Let $o$ be the ring of integers in a finite extension $K/\mathbb{Q}_p$ and $G=\mathbf{G}(\mathbb{Q}_p)$ be the $\mathbb{Q}_p$-points of a $\mathbb{Q}_p$-split reductive group $\mathbf{G}$ defined over $\mathbb{Z}_p$ with connected centre and split Borel $\mathbf{B}=\mathbf{TN}$. We show that Breuil's pseudocompact $(φ,Γ)$-module $D^\vee_ξ(π)$ attached to a smooth $o$-torsion representation $π$ of $B=\mathbf{B}(\mathbb{Q}_p)$ is isomorphic to the pseudocompact completion of the basechange $\mathcal{O_E}\otimes_{Λ(N_0),\ell}\widetilde{D_{SV}}(π)$ to Fontaine's ring (via a Whittaker functional $\ell\colon N_0=\mathbf{N}(\mathbb{Z}_p)\to \mathbb{Z}_p$) of the étale hull $\widetilde{D_{SV}}(π)$ of $D_{SV}(π)$ defined by Schneider and Vigneras. Moreover, we construct a $G$-equivariant map from the Pontryagin dual $π^\vee$ to the global sections $\mathfrak{Y}(G/B)$ of the $G$-equivariant sheaf $\mathfrak{Y}$ on $G/B$ attached to a noncommutative multivariable version $D^\vee_{ξ,\ell,\infty}(π)$ of Breuil's $D^\vee_ξ(π)$ whenever $π$ comes as the restriction to $B$ of a smooth, admissible representation of $G$ of finite length.