The Global Jacquet-Langlands Correspondence via Tensor Products

We prove that the global Jacquet–Langlands correspondence $\mathrm{J}\mathrm{L}$ for $\mathrm{G}\mathrm{L}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of $\mathrm{G}\mathrm{L}(2)$ 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 $G(\mathbb{A})$-$(G(\mathbb{A})\times \mathrm{G}\mathrm{L}(2,\mathbb{A}))$ bimodule such that $L^2(G(F)\mathrm{\backslash }G(\mathbb{A}),\chi ){\otimes }_{\mathcal{H}(G)}{L}^2(D(\mathbb{A})\times {\mathbb{A}}^{\times })\cong {\oplus }_{\pi \in \mathcal{A}(G,{\chi }^{-1})}$ $\pi \otimes \mathrm{J}\mathrm{L}(\pi )\mathrm{.}$ This decomposition into irreducible representations of $G(\mathbb{A})\times \mathrm{G}\mathrm{L}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence.