Multivariable $(\varphi ,\Gamma )$-modules and smooth o-torsion representations

Abstract

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