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 $(\phi ,\Gamma )$-module $D_{\xi }^{\vee }(\pi )$ attached to a smooth $o$-torsion representation $\pi$ of $B=\mathbf{B}({\mathbb{Q}}_p)$ is isomorphic to the pseudocompact completion of the basechange ${\mathcal{O}}_{\mathcal{E}}{\otimes }_{\Lambda (N_0),\ell }\widetilde{D_{SV}}(\pi )$ to Fontaine's ring (via a Whittaker functional $\ell :N_0=\mathbf{N}({\mathbb{Z}}_p)\to {\mathbb{Z}}_p$) of the étale hull $\widetilde{D_{SV}}(\pi )$ of $D_{SV}(\pi )$ defined by Schneider and Vigneras. Moreover, we construct a $G$-equivariant map from the Pontryagin dual ${\pi }^{\vee }$ to the global sections $\mathfrak{Y}(G/B)$ of the $G$-equivariant sheaf $\mathfrak{Y}$ on $G/B$ attached to a noncommutative multivariable version $D_{\xi ,\ell ,\infty }^{\vee }(\pi )$ of Breuil's $D_{\xi }^{\vee }(\pi )$ whenever $\pi$ comes as the restriction to $B$ of a smooth, admissible representation of $G$ of finite length.

50 pp, revised

References in corpus (1)

• Exactness of the reduction on étale modules