Generalized Poincaré series for SU(2,1)

Abstract

We introduce and study ‘non-abelian’ Poincaré series for the group G = 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 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 SU(2, 1) that, in particular, generalize those valid for the classical holomorphic (and antiholomorphic) Poincaré series for SL(2, R).