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arXiv · 2106.14200

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

Roelof W. Bruggeman·Roberto J. Miatello
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Abstract

We define and study 'non-abelian' Poincar\'e series for the group $G=\mathrm{SU} (2,1)$, 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 $\mathrm{SL}(2,\mathbb{R})$.

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Roelof W. Bruggeman, Roberto J. Miatello. 2021-06-27. Generalized Poincar\'e series for $\mathrm{SU}(2,1)$. https://arxiv.org/abs/2106.14200

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