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The essentially tame local Langlands correspondence, I

Authors: Colin J. Bushnell and Guy Henniart
Journal: J. Amer. Math. Soc. 18 (2005), 685-710
MSC (2000): Primary 22E50
Published electronically: April 25, 2005
MathSciNet review: 2138141
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Abstract: Let $F$ be a non-Archimedean local field (of characteristic $0$ or $p$) with finite residue field of characteristic $p$. An irreducible smooth representation of the Weil group of $F$ is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension $n$ is denoted $\mathcal {G}^\mathrm {et}_n(F)$. The Langlands correspondence induces a bijection of $\mathcal {G}^\mathrm {et}_n(F)$ with a certain set $\mathcal {A}^\mathrm {et}_n(F)$ of irreducible supercuspidal representations of $\mathrm {GL}_n(F)$. We consider the set $P_n(F)$ of isomorphism classes of certain pairs $(E/F,\xi )$, called “admissible”, consisting of a tamely ramified field extension $E/F$ of degree $n$ and a quasicharacter $\xi$ of $E^\times$. There is an obvious bijection of $P_n(F)$ with $\mathcal {G}^\mathrm {et}_n(F)$. Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of $P_n(F)$ with $\mathcal {A}^\mathrm {et}_n(F)$, generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of $\mathcal {G}^\mathrm {et}_n(F)$ with $\mathcal {A}^\mathrm {et}_n(F)$. We show that one obtains the Langlands correspondence by composing the map $P_n(F) \to \mathcal {A}^\mathrm {et}_n(F)$ with a permutation of $P_n(F)$ of the form $(E/F,\xi )\mapsto (E/F,\mu _\xi \xi )$, where $\mu _\xi$ is a tamely ramified character of $E^\times$ depending on $\xi$. This answers a question of Moy (1986). We calculate the character $\mu _\xi$ in the case where $E/F$ is totally ramified of odd degree.

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Additional Information

Colin J. Bushnell
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
MR Author ID: 43795

Guy Henniart
Affiliation: Département de Mathématiques & UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France
MR Author ID: 84385

Keywords: Explicit local Langlands correspondence, base change, automorphic induction, tame lifting, admissible pair
Received by editor(s): March 29, 2004
Published electronically: April 25, 2005
Additional Notes: Much of the work in this programme was carried out while the first-named author was visiting, and partly supported by, l’Université de Paris-Sud. Both authors were also partially supported by the EU network “Arithmetical Algebraic Geometry”.
Dedicated: To the memory of Albrecht Fröhlich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.