Zagier-type dualities and lifting maps for harmonic Maass–Jacobi forms

The real-analytic Jacobi forms of Zwegers' PhD thesis play an important role in the study of mock theta functions and related topics, but have not been part of a rigorous theory yet. In this paper, we introduce harmonic Maass–Jacobi forms, which include the classical Jacobi forms as well as Zwegers' functions as examples. Maass–Jacobi–Poincaré series also provide prime examples. We compute their Fourier expansions, which yield Zagier-type dualities and also yield a lift to skew-holomorphic Jacobi–Poincaré series. Finally, we link harmonic Maass–Jacobi forms to different kinds of automorphic forms via a commutative diagram.