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On automorphic forms on the unitary symplectic group \(\mathrm{Sp}(n)\) and \(\mathrm{SL}_ 2(\mathbb R)\). (English) Zbl 0634.10024
In standard notation the authors consider the two natural embeddings \[ (\mathrm{Sp}(1)\times\mathrm{Sp}(n))/\pm (1,1)\hookrightarrow \mathrm{SO}(4n), \tag{i} \]
\[ \mathrm{SL}(2,\mathbb R)\times \mathrm{SO}(4n)\hookrightarrow \mathrm{Sp}(4n,\mathbb R).\tag{ii} \] They study a correspondence between automorphic forms \(F\) on \(\mathrm{Sp}(1)\times \mathrm{Sp}(n)\) and \(\theta_ F\) on \(\mathrm{SL}(2,\mathbb R)\). This correspondence arises from (i), (ii) by the use of theta series and is suggested by the theory of Weil representations and Howe’s theory of dual reductive pairs. For \(n\geq 2\) the correspondence is explicitly given in terms of the associated \(L\)-functions, where the case \(n=2\) was already dealt with by the second author [J. Math. Soc. Japan 16, 214–225 (1964; Zbl 0126.07101)].
Let \(F\) be a simultaneous eigenform under all Hecke operators. Theorem 1 describes \(L(s,F)\) as a finite product involving \(L(s,\theta_ F)\) and another Dirichlet series of elliptic modular type associated with \(F\). Theorem 2 expresses the eigenvalues of \(F\) as a finite sum of special values of \(F\). Theorem 1 is derived from Theorem 2 by the application of the commutation relation between Hecke operators and the Siegel \(\phi\)-operator.

11F27 Theta series; Weil representation; theta correspondences
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F25 Hecke-Petersson operators, differential operators (one variable)
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