Multivariable $(\phi ,\mathrm{\Gamma })$-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 $D_{\mathrm{\Delta }}^{\vee }$ from the category of smooth modulo $p^n$ representations of $B$ to the category of projective limits of finitely generated 'etale $(\phi ,\mathrm{\Gamma })$-modules over a multivariable (indexed by the set of simple roots) commutative Laurent-series ring. These correspond to representations of a direct power of $\mathrm{G}\mathrm{a}\mathrm{l}(\overline{{\mathbb{Q}}_p}\mathrm{/}{\mathbb{Q}}_p)$ via an equivalence of categories. Parabolic induction from a subgroup $P=L_P{N}_P$ corresponds to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component $L_P$. $D_{\mathrm{\Delta }}^{\vee }$ is exact and yields finitely generated objects on the category $S{P}_A$ of finite length representations with subquotients of principal series as Jordan-H"older factors. Lifting the functor $D_{\mathrm{\Delta }}^{\vee }$ to all (noncommuting) variables indexed by the positive roots allows us to construct a $G$-equivariant sheaf ${\mathfrak{Y}}_{\pi ,\mathrm{\Delta }}$ on $G\mathrm{/}B$ and a $G$-equivariant continuous map from the Pontryagin dual ${\pi }^{\vee }$ of a smooth representation $\pi$ of $G$ to the global sections ${\mathfrak{Y}}_{\pi ,\mathrm{\Delta }}(G\mathrm{/}B)$. We deduce that $D_{\mathrm{\Delta }}^{\vee }$ is fully faithful on the full subcategory of $S{P}_A$ with Jordan-H"older factors isomorphic to irreducible principal series.