The local Jacquet-Langlands correspondence via Fourier analysis

Weinstein, Jared

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 ${\pi }^{\prime }$ of $B^{\times }$ of dimension $>1$ and irreducible cuspidal representations of ${GL}_2\left(F\right)$. 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 $(ℒ,\rho )$, where $ℒ\subset M_2\left(F\right)\times B$ is an order and $\rho$ is a finite-dimensional representation of a certain subgroup of ${GL}_2\left(F\right)\times B^{\times }$ containing ${ℒ}^{\times }$. Let $\pi \otimes {\pi }^{\prime }$ be an irreducible representation of ${GL}_2\left(F\right)\times B^{\times }$; we show that $\pi \otimes {\pi }^{\prime }$ contains such a $\rho$ if and only if $\pi$ is cuspidal and corresponds to ${\check{\pi }}^{\prime }$ under Jacquet-Langlands, and also that every $\pi$ and ${\pi }^{\prime }$ 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 ${GL}_2\left(F\right)$. Nous présentons une nouvelle construction de cette bijection pour laquelle la préservation des facteurs epsilon est automatique. Nous construisons une famille de paires $(ℒ,\rho )$, ou $ℒ\subset M_2\left(F\right)\times B$ est un ordre et $\rho$ est une représentation d’une certaine sous-groupe de ${GL}_2\left(F\right)\times B^{\times }$ qui contient ${ℒ}^{\times }$. Soit $\pi \otimes {\pi }^{\prime }$ une représentation irréductible de ${GL}_2\left(F\right)\times B^{\times }$ ; nous prouvons que $\pi \otimes {\pi }^{\prime }$ contient une telle $\rho$ si et seulement si $\pi$ est cuspidale et correspond à ${\check{\pi }}^{\prime }$ sous la correspondence de Jacquet-Langlands. On y voit tous les $\pi$ et les ${\pi }^{\prime }$. L’égalité des facteurs epsilon est reduite à un calcul Fourier-analytique sur un anneau quotient de $ℒ$.

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