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arXiv · 2207.02278

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

Claudia Alfes-Neumann·Igor Burban·Martin Raum
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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.

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Claudia Alfes-Neumann, Igor Burban, Martin Raum. 2022-07-05. A classification of polyharmonic Maa\ss{} forms via quiver representations. https://arxiv.org/abs/2207.02278

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