Representations of the unitary group SU(2,1) in Fourier term modules

We study Fourier term modules on $\mathrm{SU}(2,1)$, which are the modules arising in Fourier expansions of automorphic forms. Maximal unipotent subgroups $N$ of $\mathrm{SU}(2,1)$ are non-abelian, and we consider the ``abelian'' Fourier term modules connected to characters of $N$, and also the ``non-abelian'' modules described with theta functions. Poincar\'e series for $\mathrm{SU}(2,1)$ have in general exponential growth. To deal with such generalized automorphic forms we allow exponential growth for the functions in Fourier term modules. We give a complete description of the submodule structure of all Fourier term modules, and discuss the consequences for Fourier expansions of automorphic forms.