Multivariable $(φ,Γ)$-modules and smooth $o$-torsion representations

Let $G$ be a $\mathbb{Q}_p$-split reductive group with connected centre and Borel subgroup $B=TN$. We construct a right exact functor $D^\vee_Δ$ from the category of smooth modulo $p^n$ representations of $B$ to the category of projective limits of finitely generated étale $(φ,Γ)$-modules over a multivariable (indexed by the set of simple roots) commutative Laurent-series ring. These correspond to representations of a direct power of $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ via an equivalence of categories. Parabolic induction from a subgroup $P=L_PN_P$ corresponds to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component $L_P$. $D^\vee_Δ$ is exact and yields finitely generated objects on the category $SP_A$ of finite length representations with subquotients of principal series as Jordan-Hölder factors. Lifting the functor $D^\vee_Δ$ to all (noncommuting) variables indexed by the positive roots allows us to construct a $G$-equivariant sheaf $\mathfrak{Y}_{π,Δ}$ on $G/B$ and a $G$-equivariant continuous map from the Pontryagin dual $π^\vee$ of a smooth representation $π$ of $G$ to the global sections $\mathfrak{Y}_{π,Δ}(G/B)$. We deduce that $D^\vee_Δ$ is fully faithful on the full subcategory of $SP_A$ with Jordan-Hölder factors isomorphic to irreducible principal series.

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