The Local Jacquet-Langlands Correspondence Via Fourier Analysis

TL;DR

This paper introduces a new Fourier analysis-based method to construct the local Jacquet-Langlands correspondence, ensuring epsilon factor preservation automatically for representations of division algebras over non-Archimedean fields.

Contribution

It provides a novel Fourier analysis approach to explicitly realize the Jacquet-Langlands correspondence with automatic epsilon factor preservation.

Findings

Constructs the correspondence explicitly using Fourier analysis.

Ensures epsilon factors are preserved automatically.

Offers a new perspective on the representation theory of division algebras.

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 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.