Fourier coefficients of Jacobi forms of real weights (Q6594434)

Harmonic Maaß-Jacobi forms generalize the notion of harmonic Maaß forms introduced by \textit{J. H. Bruinier} and \textit{J. Funke} [Duke Math. J. 125, No. 1, 45--90 (2004; Zbl 1088.11030)] to the setting of Jacobi form. Jacobi forms in the present paper depend on two scalar variables, \(\tau\) on the Poincaré upper half plane and \(z\) on the complex plane. Among the various notions of Maaß-Jacobi forms the most favorable one to work with is the one of Maaß-Jacobi forms that are holomorphic in \(z\). The reason is that the theta decomposition for Jacobi forms extends, thus allowing to effectively work with usual harmonic Maaß forms for Weil representations, while stating results for Maaß-Jacobi forms.\N\NPoincaré series for a group \(\Gamma \subset \mathrm{SL}_2(\mathbb{Z})\) that contains the group of upper triangular matrices \(\Gamma_\infty\) can be simplified to a sum over contributions of double cosets \(\Gamma_\infty \backslash \Gamma \slash \Gamma_\infty\) by evaluating a suitable integral at the infinite place. This type of calculation has been performed in, for instance, Theorem 1.4 of [\textit{J. H. Bruinier}, Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors. Berlin: Springer (2002; Zbl 1004.11021)].\N\NThe authors combine these two techniques to obtain formulas for the Fourier coefficients of Maaß-Jacobi forms that can be written as a linear combination of Poincaré series corresponding to their principle parts. They include arbitrary negative real weights and multiplier systems in their theorems.\N\NThe authors also employ Serre duality as manifested, for instance, in Theorem 1.17 of [\textit{J. H. Bruinier}, Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors. Berlin: Springer (2002; Zbl 1004.11021)] or Theorem 1.1 of [\textit{J. H. Bruinier} and \textit{J. Funke}, Duke Math. J. 125, No. 1, 45--90 (2004; Zbl 1088.11030)] to obtain a characterization of Maaß-Jacobi forms that are weakly holomorphic. They restrict to Maaß-Jacobi forms whose principle part is concentrated at infinity. In their proof, this restriction is required to simplify a sum over cusps. An alternative approach, which emphasizes the connection to harmonic Maaß forms as before, can be based on the induction-restriction of Weil representations. In this language, the condition imposed by the authors simplifies the duality pairing for the relevant representation.