H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series

Abstract

It was shown in previous work that the one-variable $\hat{\mu }$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r+1,1)$ are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of "product type" for non-degenerate lattices of signature $(r+s,s)$. We thus obtain a clean generalization of $\hat{\mu }$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic Maaß-Jacobi forms, we obtain harmonic weak Maaß forms of higher depth in the sense of Zagier and Zwegers. In particular, we explain the modular completion of some, so-called degenerate indefinite theta series in the context of higher depth mixed mock modular forms. The structure theory for Heisenberg harmonic Maaß-Jacobi forms developed in this paper also explains a curious splitting of Zwegers's two-variable $\hat{\mu }$-function into the sum of a meromorphic Jacobi form and a one-variable Maaß-Jacobi form.

30 pages, accepted for Res. Math. Sci

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• Mock Theta Functions
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Cited by in corpus (3)

• Indefinite theta series and generalized error functions
• On the Fourier coefficients of negative index meromorphic Jacobi forms
• Jacobi Forms of Indefinite Lattice Index