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

TL;DR

This work characterizes the full submodule structure of Fourier term modules for ${\mathrm{SU}}(2,1)$, encompassing both abelian and non-abelian Fourier terms, and accommodates exponential growth in automorphic contexts. By formulating a detailed ($\mathfrak{g},K$)-module framework and employing explicit shift operators, the authors classify how Fourier terms decompose into abelian characters, Stone–von Neumann realizations, and theta-function data, for both generic and integral Weyl parametrizations. They prove that, depending on the parametrization, large families of modules (including principal series, large discrete series, and Langlands-type representations) occur as submodules or quotients, and they provide a systematic description of kernels of shift operators which governs submodule lattices. The analysis connects central action via characters of $ZU(\mathfrak g)$ to eigenfunction equations and boundary behavior, yielding precise multiplicity and sector-structure results for $K$-types. These results have direct implications for the Fourier–Jacobi expansions of automorphic forms on $\Gamma\backslash G$, and inform Poincaré series and their analytic properties with exponential growth. Overall, the paper delivers a comprehensive, computable atlas of Fourier-term representations for the smallest non-abelian unipotent case, with broad consequences for automorphic forms on ${\mathrm{SU}}(2,1)$.

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. Poincaré 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.

Figures (45)

• Figure 1: Square with factors for downward shift operators
• Figure 2: Squares with factors for downward shift operators.
• Figure 3: Propagation of the kernel of $S^{3}_{-1}.$
• Figure 4: A connected collection of zero squares.
• Figure 5: $K$-types $\tau^h_p$ depicted in the $(h/3,p)$-plane. The arrows indicate the change of $K$-type given by the four shift operators. Repeated application of shift operators leaves invariant the set of thick points, which satisfy $\frac{h}{3}\equiv p\bmod 2$.

Theorems & Definitions (169)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Lemma I.2.1
• proof
• Definition I.4.1
• Proposition I.4.2
• Lemma I.4.3
• proof