Resumen de Multivariable (\varphi ,\Gamma )-modules and smooth o-torsion representations
Let G be a Qp -split reductive group with connected centre and Borel subgroup B=TN . We construct a right exact functor D∨Δ from the category of smooth modulo pn representations of B to the category of projective limits of finitely generated étale (φ,Γ) -modules over a multivariable (indexed by the set of simple roots) commutative Laurent series ring. These correspond to representations of a direct power of Gal(Qp¯¯¯¯¯¯/Qp) via an equivalence of categories. Parabolic induction from a subgroup P=LPNP gives rise to a basechange from a Laurent series ring in those variables with corresponding simple roots contained in the Levi component LP . D∨Δ is exact and yields finitely generated objects on the category SPA of finite length representations with subquotients of principal series as Jordan–Hölder factors. Lifting the functor D∨Δ to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf Yπ,Δ on G / B and a G-equivariant continuous map from the Pontryagin dual π∨ of a smooth representation π of G to the global sections Yπ,Δ(G/B) . We deduce that D∨Δ is fully faithful on the full subcategory of SPA with Jordan–Hölder factors isomorphic to irreducible principal series.
