Generalized Poincaré series for SU(2,1)

Bruggeman, Roelof; Miatello, Roberto

doi:10.1090/ert/674

Generalized Poincaré series for SU(2,1)
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by Roelof W. Bruggeman and Roberto J. Miatello;

Represent. Theory 28 (2024), 381-433

DOI: https://doi.org/10.1090/ert/674

Published electronically: September 17, 2024

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Abstract:

We introduce and study ‘non-abelian’ Poincaré series for the group $G=\operatorname {SU}(2,1)$, that is, Poincaré series attached to a Stone-Von Neumann representation of the unipotent subgroup $N$ of $G$. Such Poincaré series have in general exponential growth. In this study we use results on abelian and non-abelian Fourier term modules obtained by Bruggeman and Miatello [Representations of $\mathrm {SU}(2,1)$ in Fourier term modules, Springer, Cham, 2023] to compute the inner product of truncations of these series with square integrable automorphic forms, in connection with their Fourier expansions. As a consequence, we obtain general completeness results for $\operatorname {SU}(2,1)$ that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincaré series for $\operatorname {SL}(2,\mathbb {R})$.

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