A classification of polyharmonic Maa\ss{} forms via quiver representations

TL;DR

This paper classifies polyharmonic Maass forms using quiver representations, extending previous harmonic case results, and introduces a new case for weights greater than one, supported by a computational framework.

Contribution

It introduces a novel classification of polyharmonic Maass forms via quiver representations, including a new case for weights greater than one, and develops a computational approach.

Findings

Classification includes ten cases, expanding beyond the harmonic case.

Polyharmonic Maass forms correspond to cyclic, indecomposable quiver representations.

Develops a computational framework for explicit examples.

Abstract

We give a classification of the Harish-Chandra modules generated by the pullback to~$\SL{2}(\RR)$ of \emph{poly}harmonic Maa\ss{} forms for congruence subgroups of~$\SL{2}(\ZZ)$ with exponential growth allowed at the cusps. This extends results of Bringmann--Kudla in the harmonic case. While in the harmonic setting there are nine cases, our classification comprises ten; A new case arises in weights $k > 1$. To obtain the classification we introduce quiver representations into the topic and show that those associated with polyharmonic Maa\ss{} forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maa\ss{} forms. To realize all possible cases of Harish-Chandra modules we develop a theory of weight shifts for Taylor coefficients of vector-valued spectral families. We provide a comprehensive computer implementation of this theory, which allows us to provide explicit examples.