Representations of the unitary group SU(2,1) in Fourier term modules

TL;DR

This paper analyzes Fourier term modules of automorphic forms on SU(2,1), describing their submodule structures and growth behaviors, including both abelian and non-abelian cases, with implications for automorphic Fourier expansions.

Contribution

It provides a complete classification of Fourier term modules for SU(2,1), including their submodule structures and growth properties, extending understanding of automorphic forms.

Findings

Complete description of submodule structures

Inclusion of exponential growth in Fourier modules

Analysis of abelian and non-abelian Fourier terms

Abstract

We study Fourier term modules on $\mathrm{SU}(2,1)$, which are the modules arising in Fourier expansions of automorphic forms. Maximal unipotent subgroups $N$ of $\mathrm{SU}(2,1)$ are non-abelian, and we consider the ``abelian'' Fourier term modules connected to characters of $N$, and also the ``non-abelian'' modules described with theta functions. Poincar\'e series for $\mathrm{SU}(2,1)$ have in general exponential growth. To deal with such generalized automorphic forms we allow exponential growth for the functions in Fourier term modules. We give a complete description of the submodule structure of all Fourier term modules, and discuss the consequences for Fourier expansions of automorphic forms.