PUB - Publikationen an der Universität Bielefeld

We give a classification of the Harish-Chandra modules generated by the pullback to SL2(R) of polyharmonic Maa ss 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 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).