The adjointness of the maps between the space of Jacobi forms and the space of modular forms (Q1326959)

\textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms (Prog. Math. 55) (1985; Zbl 0554.10018)] constructed a map from a space of Jacobi forms to a space of elliptic modular forms. On the other hand, \textit{T. Satoh} [Math. Ann. 285, 463-480 (1989; Zbl 0678.10021)] constructed a map from a space of cusp forms to a space of Jacobi cusp forms. In this paper, we prove a conjecture of \textit{N. P. Skoruppa} [J. Reine Angew. Math. 411, 66-95 (1990; Zbl 0702.11028)] to the effect that these maps are, up to constant, adjoint with respect to the Petersson products.