Generalized Poincar\'e series for $\mathrm{S}\mathrm{U}(2,1)$

We define and study ‘non-abelian’ Poincar'e series for the group $G=\mathrm{S}\mathrm{U}(2,1)$, i.e. Poincar'e series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincar'e series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained in arXiv:1912.01334. We compute the inner product of truncations of these series and those associated to unitary characters of $N$ with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincar'e series for $\mathrm{S}\mathrm{L}(2,\mathbb{R})$.