The Global Jacquet-Langlands Correspondence via Tensor Products
Jun Yang
TL;DR
The paper shows that the global Jacquet--Langlands correspondence for ${\rm GL}(2)$ can be realized through a tensor-product construction governed by Howe duality for a quaternionic similitude dual pair. By formulating both local and global similitude theta lifts as modules over appropriate Hecke algebras and the global induced Weil representation ${\Omega}_{\mathbb A}$, it proves a canonical isomorphism ${\Theta}(\pi) \cong {\pi}^{\vee} \otimes_{ {\mathcal H}(G) } {\Omega}_{\mathbb A}$ and derives the global decomposition ${L^2}(G(F)\backslash G({\mathbb A}),\chi) \otimes_{ {\mathcal H}(G) } L^2(D({\mathbb A}) \times {\mathbb A}^{\times}) \cong \bigoplus_{\pi} {\pi} \otimes {\rm JL}( {\pi} )$, where ${\rm JL}( {\pi} )$ is the Jacquet--Langlands transfer of $\pi$. This perspective treats archimedean and non-archimedean places uniformly and avoids trace formulas, highlighting Howe duality as a structural mechanism for theta-functorial transfers and suggesting a general tensor-product strategy for other theta-induced transfers.
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 $L^2(G(F)\backslash G(\mathbb{A}),χ)\otimes_{\mathcal{H}(G)}L^2(D(\mathbb{A})\times \mathbb{A}^{\times})~\cong~\oplus_{π\in {\mathcal{A}}(G,χ^{-1})}$ $π\otimes{\rm JL}(π).$ This decomposition into irreducible representations of $G(\mathbb{A})\times {\rm GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence.