The local Jacquet-Langlands correspondence via Fourier analysis

Jared Weinstein

doi:10.5802/jtnb.728

Jared Weinstein ¹

¹ UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 483-512.

Abstract (VO)
Résumé

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 $B^{\times}$ 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 $(ℒ, ρ)$ , where $ℒ \subset M_{2} (F) \times B$ is an order and $ρ$ is a finite-dimensional representation of a certain subgroup of ${GL}_{2} (F) \times B^{\times}$ containing $ℒ^{\times}$ . Let $π \otimes π^{'}$ be an irreducible representation of ${GL}_{2} (F) \times B^{\times}$ ; we show that $π \otimes π^{'}$ contains such a $ρ$ if and only if $π$ is cuspidal and corresponds to ${\overset{ˇ}{π}}^{'}$ 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 $ℒ$ .

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 $B^{\times}$ 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 $(ℒ, ρ)$ , ou $ℒ \subset M_{2} (F) \times B$ est un ordre et $ρ$ est une représentation d’une certaine sous-groupe de ${GL}_{2} (F) \times B^{\times}$ qui contient $ℒ^{\times}$ . Soit $π \otimes π^{'}$ une représentation irréductible de ${GL}_{2} (F) \times B^{\times}$ ; nous prouvons que $π \otimes π^{'}$ contient une telle $ρ$ si et seulement si $π$ est cuspidale et correspond à ${\overset{ˇ}{π}}^{'}$ 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 $ℒ$ .

MR Zbl

DOI : 10.5802/jtnb.728

Affiliations des auteurs :

Jared Weinstein ¹

¹ UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA

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Jared Weinstein. The local Jacquet-Langlands correspondence via Fourier analysis. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 483-512. doi : 10.5802/jtnb.728. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.728/

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