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Classical and adelic automorphic forms from SL(n) to GL(n) over number fields

GL}_2^+(\mathbb{R}) \times \text{GL}_n(\hat{\mathcal{O}})) = \text{GL}^+_2(\mathcal{O}) = Z(\mathcal{O}^\times) \text{SL}_2(\mathcal{O}), $$ which also appears in Gelbart's book, Automorphic forms on adeles