Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let ${\varphi }_{Iₙ\left(h\right),1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and ${\sigma }_{n-1}\left({\varphi }_{Iₙ\left(h\right),1}\right)$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to ${\varphi }_{Iₙ\left(h\right),1}$ under the...