A classification of harmonic Maass forms

TL;DR

This paper classifies the Harish-Chandra modules generated by harmonic Maass forms for congruence subgroups, revealing nine possible types and providing explicit examples for each.

Contribution

It provides a complete classification of modules generated by harmonic Maass forms, including explicit examples for all nine types, and discusses non-harmonic cases.

Findings

Nine possible Harish-Chandra modules identified

Explicit examples constructed for each module type

Discussion of non-harmonic forms and complex modules

Abstract

We give a classification of the Harish-Chandra modules generated by the pullback to $\text{SL}_2(\mathbb R)$ of harmonic Maass forms for congruence subgroups of $\text{SL}_2(\mathbb Z)$ with exponential growth allowed at the cusps. We assume that the weight is integral but include vector-valued forms. Due to the weak growth condition, these modules need not be irreducible. Elementary Lie algebra considerations imply that there are 9 possibilities, and we show, by giving explicit examples, that all of them arise from harmonic Maass forms. Finally, we briefly discuss the case of forms that are not harmonic but rather are annihilated by a power of the Laplacian, where much more complicated Harish-Chandra modules can arise.