On harmonic weak Maass forms of half integral weight (Q2839348)

\textit{M. Eichler} and \textit{D. Zagier}'s book [The theory of Jacobi forms. Boston etc.: Birkhäuser (1985; Zbl 0554.10018)] gives isomorphisms between the spaces of holomorphic Jacobi forms, certain half-integral weight modular forms, and vector-valued modular forms. The authors extend these results to a real-analytic setting by establishing isomorphisms between spaces of harmonic Maaß-Jacobi forms and scalar-valued and vector-valued harmonic weak Maaß forms. The proof of the isomorphism between harmonic Maaß-Jacobi forms and scalar-valued harmonic weak Maaß forms follows exactly as in the holomorphic case, while the isomorphism between the spaces of scalar-valued and vector-valued harmonic weak Maaß forms relies on properties of the Weil representation.