A classification of harmonic weak Maa{\ss} forms of half-integral weight

TL;DR

This paper classifies certain harmonic weak Maa{ ss} forms of half-integral weight through Harish-Chandra modules, extending previous work and exploring theta lifts and correspondences in representation theory.

Contribution

It extends classification of harmonic weak Maa{ ss} forms to half-integral weights and analyzes their realization via regularized theta lifts, contributing to understanding theta correspondences.

Findings

Classification of Harish-Chandra modules generated by these forms

Realization of modules via regularized theta lifts

Insight into theta correspondence for Harish-Chandra modules

Abstract

We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maa{\ss} 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 Maa{\ss} 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.