A classification of harmonic weak Maaß forms of half-integral weight

Abstract

We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by Schulze-Pillot and parallels recent work by Bringmann–Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. Harish-Chandra modules in both integral and half-integral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to an initial understanding of a theta correspondence for extensions of Harish-Chandra modules.