Certain L-series of Rankin-Selberg type associated to Siegel modular forms of degree \(g\) (Q911627)

Let \(F\) and \(G\) be Siegel modular forms of weight \(k\) and \(\ell\), resp. on \(\Gamma_ g=SP_{2g}({\mathbb{Z}})\). Assume that \(F\) is cuspidal, \(k>\ell\), \(k\equiv \ell (\bmod 2)\) and let \(f\) be a non-zero Siegel-Hecke eigenform which is acusp form of weight \(k-\ell\) on \(\Gamma_{g-j}\) where \(j\) is a fixed integer with \(1\leq j\leq g\). We define \[ D_{F,G;j}(s;f)=\sum_{\{m>0\}/\sim}(1/\epsilon (m))<\phi_ m,f\psi_ m>(\det m)^{-s}, \] where the summation is over a set of representatives for the usual right-action of \(GL_ j({\mathbb{Z}})\) on the set of positive definite half-integral (j,j)-matrices, \(\epsilon(m)\) is the number of \(GL_ j({\mathbb{Z}})\)-units of m, \(\phi_ m\) resp. \(\psi_ m\) denotes the \(m\)-th Fourier-Jacobi coefficients of F resp. G and \(<, >\) is the Petersson scalar product on the space of Jacobi forms of weight \(k\) and index \(m\). This series is similar to that studied for \(g=2\) by \textit{N.- P. Skoruppa} and the author [Invent. Math. 95, No.3, 541-558 (1989; Zbl 0665.10019)] and by \textit{T. Yamazaki} [preprint] for arbitrary \(g\). In the present paper we prove that \(D_{F,G;j}(s;f)\) has a meromorphic continuation to \({\mathbb{C}}\) and satisfies a functional equation. We also study the action of Aut(\({\mathbb{C}}/{\mathbb{Q}})\) on the special value \(D_{F,G;j}(s_ 0;f)\) for a certain integer \(s_ 0\) in case F is a Hecke eigenform.