The local Jacquet-Langlands correspondence via Fourier analysis

Jared Weinstein[1]

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

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 2, page 483-512
  • ISSN: 1246-7405

Abstract

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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 ( ℒ , ρ ) , where ℒ ⊂ M 2 ( F ) × B is an order and ρ is a finite-dimensional representation of a certain subgroup of GL 2 ( F ) × B × containing ℒ × . Let π ⊗ π ′ be an irreducible representation of GL 2 ( F ) × B × ; we show that π ⊗ π ′ 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 ℒ .

How to cite

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Weinstein, Jared. "The local Jacquet-Langlands correspondence via Fourier analysis." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 483-512. <http://eudml.org/doc/116416>.

@article{Weinstein2010,
abstract = {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 $&gt;1$ and irreducible cuspidal representations of $\operatorname\{GL\}_2(F)$. 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 $(\mathcal\{L\},\rho )$, where $\mathcal\{L\}\subset M_2(F)\times B$ is an order and $\rho $ is a finite-dimensional representation of a certain subgroup of $\operatorname\{GL\}_2(F)\times B^\times $ containing $\mathcal\{L\}^\times $. Let $\pi \otimes \pi ^\{\prime\}$ be an irreducible representation of $\operatorname\{GL\}_2(F)\times B^\{\times \}$; 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 $\mathcal\{L\}$.},
affiliation = {UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA},
author = {Weinstein, Jared},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {locally compact non-Archimedean field; division algebra of dimension 4},
language = {eng},
number = {2},
pages = {483-512},
publisher = {Université Bordeaux 1},
title = {The local Jacquet-Langlands correspondence via Fourier analysis},
url = {http://eudml.org/doc/116416},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Weinstein, Jared
TI - The local Jacquet-Langlands correspondence via Fourier analysis
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 483
EP - 512
AB - 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 $&gt;1$ and irreducible cuspidal representations of $\operatorname{GL}_2(F)$. 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 $(\mathcal{L},\rho )$, where $\mathcal{L}\subset M_2(F)\times B$ is an order and $\rho $ is a finite-dimensional representation of a certain subgroup of $\operatorname{GL}_2(F)\times B^\times $ containing $\mathcal{L}^\times $. Let $\pi \otimes \pi ^{\prime}$ be an irreducible representation of $\operatorname{GL}_2(F)\times B^{\times }$; 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 $\mathcal{L}$.
LA - eng
KW - locally compact non-Archimedean field; division algebra of dimension 4
UR - http://eudml.org/doc/116416
ER -

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