A classification of polyharmonic Maaß forms via quiver representations

We give a classification of the Harish-Chandra modules generated by the pullback to~$\text{SL}2(\text{RR})$ of \emph{poly}harmonic Maaß forms for congruence subgroups of~$\text{SL}2(\text{ZZ})$ 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 Maaß forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maaß 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.