The local Jacquet-Langlands correspondence via Fourier analysis
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 483-512.

Let F be a locally compact non-Archimedean field, and let B/F be a division algebra of dimension 4. The Jacquet-Langlands correspondence provides a bijection between smooth irreducible representations π of B × of dimension >1 and irreducible cuspidal representations of GL 2 (F). We present a new construction of this bijection in which the preservation of epsilon factors is automatic. This is done by constructing a family of pairs (,ρ), where M 2 (F)×B is an order and ρ is a finite-dimensional representation of a certain subgroup of GL 2 (F)×B × containing × . Let ππ be an irreducible representation of GL 2 (F)×B × ; we show that ππ contains such a ρ if and only if π is cuspidal and corresponds to π ˇ under Jacquet-Langlands, and also that every π and π arises this way. The agreement of epsilon factors is reduced to a Fourier-analytic calculation on a finite ring quotient of .

Soit F un corps local non archimédien et localement compact, et soit B/F un corps de quaternions. La correspondance de Jacquet-Langlands fournit une bijection entre les représentations lisses et irréductibles de B × de dimension >1 et les représentations cuspidales et irréductibles de GL 2 (F). Nous présentons une nouvelle construction de cette bijection pour laquelle la préservation des facteurs epsilon est automatique. Nous construisons une famille de paires (,ρ), ou M 2 (F)×B est un ordre et ρ est une représentation d’une certaine sous-groupe de GL 2 (F)×B × qui contient × . Soit ππ une représentation irréductible de GL 2 (F)×B ×  ; nous prouvons que ππ contient une telle ρ si et seulement si π est cuspidale et correspond à π ˇ sous la correspondence de Jacquet-Langlands. On y voit tous les π et les π . L’égalité des facteurs epsilon est reduite à un calcul Fourier-analytique sur un anneau quotient de .

DOI: 10.5802/jtnb.728
Jared Weinstein 1

1 UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA
     author = {Jared Weinstein},
     title = {The local {Jacquet-Langlands} correspondence via {Fourier} analysis},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {483--512},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {2},
     year = {2010},
     doi = {10.5802/jtnb.728},
     mrnumber = {2769075},
     zbl = {1223.11066},
     language = {en},
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Jared Weinstein. The local Jacquet-Langlands correspondence via Fourier analysis. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 483-512. doi : 10.5802/jtnb.728.

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