Kac–Wakimoto characters and almost holomorphic modular forms

In joint work with Amanda Folsom,  we resolve a question of Kac, and explain the automorphic properties of  certain characters due to Kac and Wakimoto.  We prove that they   are essentially holomorphic parts of certain generalizations of  harmonic weak Maass forms which we call ``almost harmonic Maass forms".   Loosely speaking,  such functions may be viewed as sums of harmonic weak Maass forms under iterates of the raising operator (themselves therefore non-harmonic weak Maass forms), multiplied by almost holomorphic modular forms.