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

TL;DR

This paper introduces and analyzes non-abelian Poincaré series for the group SU(2,1), extending classical results and providing new completeness theorems for automorphic forms using advanced Fourier analysis techniques.

Contribution

It defines non-abelian Poincaré series for SU(2,1) and establishes their properties, generalizing classical Poincaré series results for SL(2,R).

Findings

Computed inner products of truncated series and automorphic forms.

Established completeness results for non-abelian Poincaré series.

Extended classical Poincaré series theory to SU(2,1).

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 $\mathrm{SL}(2,\mathbb{R})$.