Multivariable $(\phi ,\Gamma )$-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 $D_{\Delta }^{\vee }$ from the category of smooth modulo $p^n$ representations of $B$ to the category of projective limits of finitely generated étale $(\phi ,\Gamma )$-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 $P=L_P{N}_P$ corresponds to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component $L_P$. $D_{\Delta }^{\vee }$ is exact and yields finitely generated objects on the category $S{P}_A$ of finite length representations with subquotients of principal series as Jordan-Hölder factors. Lifting the functor $D_{\Delta }^{\vee }$ 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 $D_{\Delta }^{\vee }$ is fully faithful on the full subcategory of $S{P}_A$ with Jordan-Hölder factors isomorphic to irreducible principal series.

55 pages, revised, to appear in Selecta Mathematica

References in corpus (4)

• Multivariable $(\phi ,\Gamma )$-modules and locally analytic vectors
• Multivariable $(\phi ,\Gamma )$-modules and smooth $o$-torsion representations
• Frobenius modules over multivariate Robba rings
• Complements sur les extensions entre series principales p-adiques et modulo p de G(F)

Cited by in corpus (3)

• Multivariable $(\phi ,\Gamma )$-modules and smooth $o$-torsion representations
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• A note on multivariable $(\phi ,\Gamma )$-modules