Selecta Mathematica

, Volume 24, Issue 2, pp 935–995 | Cite as

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

  • Gergely Zábrádi


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.

Mathematics Subject Classification

Primary 11S37 Secondary 11S20 20G05 20G25 



My debt to the works of Christophe Breuil [8], Pierre Colmez [12, 13], Peter Schneider, and Marie-France Vignéras [25, 26] will be obvious to the reader. I would also like to thank Márton Erdélyi, Jan Kohlhaase, Vytautas Paškūnas, Peter Schneider, and Tamás Szamuely for discussions on the topic. I am grateful to the referee for the careful reading of the manuscript and for their various comments


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.BudapestHungary

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