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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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The essentially tame local Langlands correspondence, I
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by Colin J. Bushnell and Guy Henniart
J. Amer. Math. Soc. 18 (2005), 685-710
Published electronically: April 25, 2005


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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Bibliographic Information
  • Colin J. Bushnell
  • Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
  • MR Author ID: 43795
  • Email:
  • Guy Henniart
  • Affiliation: Département de Mathématiques & UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France
  • MR Author ID: 84385
  • Email:
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 18 (2005), 685-710
  • MSC (2000): Primary 22E50
  • DOI:
  • MathSciNet review: 2138141