Hecke duality of Ikeda lifts (Q462830)

Let \(S_k\) be the space of cusp forms of weight \(k\) for the full modular group and let \(S^n_k\) stand for the space of Siegel cusp forms of weight \(k\) and degree \(n\). Let \(\{k,n\}\subseteq\mathbb{N}\) and \(k= n\pmod 2\). The authors study the subspace \(I^{(2n)}_k\) of \(S^{2n}_{k+n}\) generated by the Ikeda lifts of the primitive Hecke newforms in \(S_{2k}\) and prove, in particular, that if \(F\in K^{(2n)}_k\), then \(F\) possesses the Hecke duality property. They go on to prove the following two converse theorems for the Saito-Kurokawa lifting. Let \(k\) be an even positive integer, \(f\in S^{2k-2}\), and \(F\in S^2_k\); suppose that \(F\) is a Hecke eigenform in the sense of \textit{T. Ikeda} [Ann. Math. (2) 154, No. 3, 641--681 (2001; Zbl 0998.11023)], that \(F\) is an eigenform of Hecke operators with respect to the Hecke pair \((\Gamma_2,\text{Sp}_2(\mathbb{Q}))\). Denote by \(L(f,s)\) and \(L(f,s)\) the standard \(L\)-functions attached to \(F\) and \(f\), respectively. If NEWLINE\[NEWLINEL(f,s)= \zeta(s) L(s+ k-1,f) L(s+k-2, f),NEWLINE\]NEWLINE then \(F\) is the Saito-Kurokawa lift of \(f\) (Theorem 1). Moreover, let \(\pi= \bigotimes_v\pi_v\) be the automorphic representation attached to \(F\). If \(\pi_v\) is a degenerate principal series representation for almost all finite places \(v\), then \(F\) is the Saito-Kurokawa lift of a cuspform in \(S_{2k-2}\) (Theorem 2).