  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.
