A classification of harmonic Maass forms

Abstract

We give a classification of the Harish-Chandra modules generated by the pullback to $$\mathrm{SL}_2(\mathbb {R})$$ of harmonic Maass forms for congruence subgroups of $$\mathrm{SL}_2(\mathbb {Z})$$ 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 do not need to be irreducible. Elementary Lie algebra considerations imply that there are nine 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. We hope that our classification will prove useful in understanding harmonic Maass forms from a representation theoretic perspective and that it will illustrate in the simplest case the phenomenon of extensions occurring in the space of automorphic forms.