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

摘要

Let G be a -split reductive group with connected centre and Borel subgroup . We construct a right exact functor from the category of smooth modulo representations of B to the category of projective limits of finitely generated ,tale -modules over a multivariable (indexed by the set of simple roots) commutative Laurent series ring. These correspond to representations of a direct power of via an equivalence of categories. Parabolic induction from a subgroup gives rise to a basechange from a Laurent series ring in those variables with corresponding simple roots contained in the Levi component . is exact and yields finitely generated objects on the category of finite length representations with subquotients of principal series as Jordan-Holder factors. Lifting the functor to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf on G / B and a G-equivariant continuous map from the Pontryagin dual of a smooth representation of G to the global sections . We deduce that is fully faithful on the full subcategory of with Jordan-Holder factors isomorphic to irreducible principal series.