Heisenberg harmonic weak Maass-Jacobi forms

The space of harmonic weak Maass-Jacobi forms has originally been
defined by Bringmann and Richter. We consider a subspace, that we call
the subspace of Heisenberg harmonic Maass-Jacobi forms. This space, as
opposed to the space of all Maass-Jacobi forms, allows for a detailed
analysis. We will decompose the space with respect to an analog of the
xi-operator for harmonic weak Maass forms, and we will demonstrate how
a theta-like decomposition comes up in the theory. We also discuss
singularities of Heisenberg harmonic Maass-Jacobi forms, which play a
role akin to the singularities at the cusps that harmonic weak Maass
forms have. In particular, the singularities of the modular completion
of Zwegers's mu-function are shed light on in the context of thisnew
space.

The talk is based on research performed jointly with K. Bringmann and
O. Richter.