A classification of polyharmonic Maaß forms via quiver representations

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We give a classification of the Harish-Chandra modules generated by the pullback to SL2(R) of polyharmonic Maaß forms for congruence subgroups of SL2(Z) 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.