Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms (Q1959459)

After reviewing harmonic Maass forms and skew-holomorphic Jacobi forms, the authors define harmonic Maass-Jacobi forms of weight \(k\) and index \(m\). These are real-analytic functions \(\phi : \mathbb{H} \times \mathbb{C} \to \mathbb{C}\) which are invariant under the usual slash operator, annihilated by a certain differential operator, and which satisfy certain growth conditions. The classical Jacobi forms of Eichler-Zagier and the real-analytic Jacobi forms of Zwegers are notable examples. Let \(\widehat{\mathbb{J}}_{k,m}\) denote the space of harmonic Maass-Jacobi forms of weight \(k\) and index \(m\) which are also holomorphic in the Jacobi variable \(z \in \mathbb{C}\). Motivated by the operator \(\xi_k\), which plays an important role in the theory of harmonic Maass forms, the authors introduce a differential operator \(\xi_{k,m}\) and show that it maps \(\widehat{\mathbb{J}}_{k,m}\) to \(J_{3-k,m}^{sk!}\), the space of weak skew-holomorphic Jacobi forms of weight \(k\) and index \(m\). Next they present Maass-Jacobi-Poincaré series \(\mathcal{P}_{k,m}^{(n,r)} \in \widehat{\mathbb{J}}_{k,m}\). They compute their Fourier expansions, establish ``Zagier-type'' dualities, and show that \(\xi_{k,m}\left(\mathcal{P}_{k,m}^{(n,r)}\right) = J_{3-k,m}^{(n,r)sk}\), the skew-holomorphic Jacobi-Poincaré series of weight \(3-k\) and index \(m\). Finally, they prove the commutativity of a diagram involving \(\xi_{k-1/2}\), \(\xi_{k,1}\), and two lifts given in terms of theta functions