The essentially tame local Langlands correspondence, I |
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Authors: | Colin J. Bushnell Guy Henniart |
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Affiliation: | Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom ; Département de Mathématiques & UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France |
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Abstract: | Let be a non-Archimedean local field (of characteristic or ) with finite residue field of characteristic . An irreducible smooth representation of the Weil group of 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 is denoted . The Langlands correspondence induces a bijection of with a certain set of irreducible supercuspidal representations of . We consider the set of isomorphism classes of certain pairs , called ``admissible', consisting of a tamely ramified field extension of degree and a quasicharacter of . There is an obvious bijection of with . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of with , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of with . We show that one obtains the Langlands correspondence by composing the map with a permutation of of the form , where is a tamely ramified character of depending on . This answers a question of Moy (1986). We calculate the character in the case where is totally ramified of odd degree. |
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Keywords: | Explicit local Langlands correspondence base change automorphic induction tame lifting admissible pair |
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