The Global Jacquet-Langlands Correspondence via Tensor Products

TL;DR

This paper demonstrates that the global Jacquet-Langlands correspondence for GL(2) can be explicitly realized through tensor products over Hecke algebras, providing a new structural perspective on the correspondence.

Contribution

It introduces a novel realization of the global Jacquet-Langlands correspondence via tensor products over Hecke algebras, utilizing the similitude theta correspondence.

Findings

Establishes an isomorphism between tensor products of automorphic spaces and direct sums of irreducible representations.

Provides a new structural framework for understanding the Jacquet-Langlands correspondence.

Recovers the full global Jacquet-Langlands correspondence through this tensor product decomposition.

Abstract

We prove that the global Jacquet--Langlands correspondence ${\rm JL}$ for ${\rm GL}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of ${\rm GL}(2)$ over a number field. Using the similitude theta correspondence, the space $L^2(D(\mathbb{A})\times \mathbb{A}^{\times})$ acquires the structure of a $G(\mathbb{A})$-$(G(\mathbb{A})\times {\rm GL}(2,\mathbb{A}))$ bimodule such that $L^2(G(F)\backslash G(\mathbb{A}),\chi)\otimes_{\mathcal{H}(G)}L^2(D(\mathbb{A})\times \mathbb{A}^{\times})~\cong~\oplus_{\pi\in {\mathcal{A}}(G,\chi^{-1})}$ $\pi\otimes{\rm JL}(\pi).$ This decomposition into irreducible representations of $G(\mathbb{A})\times {\rm GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence.