Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -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.