The Global Jacquet-Langlands Correspondence via Tensor Products

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.