Fourier-Jacobi expansion and Ikeda lifting

We consider the following map $\Psi^{(2n-1)} $ from elliptic modular forms to Siegel modular forms of half-integral weight, which is the decomposition of the following three maps: $\Psi^{(2n-1)} $ : $S_{2k}(SL_{2}(\mathbb{Z}))arrow S_{k+n}(\Gamma_{2n})arrow J_{k+n,1}^{\mathrm{c}usp}(\Gamma_{2n-1}^{J})arrow S_{k+n-\frac{1}{2}}^{+}(\Gamma_{0}^{(2n-1)}(4)) $