On the Analytic Structure of Commutative Nilmanifolds

TL;DR

This paper investigates the analytic properties of commutative nilmanifolds, revealing that in exceptional cases, the nilpotent groups have specific semidirect product structures that enable explicit harmonic analysis.

Contribution

It identifies the structure of nilpotent groups in exceptional cases, allowing explicit harmonic analysis and Fourier inversion formulas for commutative nilmanifolds.

Findings

Most nilpotent groups have square-integrable representations modulo their center.

In three exceptional cases, groups are semidirect products involving  and .

Explicit harmonic analysis formulas are derived for these cases.

Abstract

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form $G/K = N \rtimes K/K$ where, in all but three cases, the nilpotent group $N$ has irreducible unitary representations whose coefficients are square integrable modulo the center $Z$ of $N$. Here we show that, in those three "exceptional" cases, the group $N$ is a semidirect product $N_1 \rtimes \mathbb{R}$ or $N_1 \rtimes \mathbb{C}$ where the normal subgroup $N_1$ contains the center $Z$ of $N$ and has irreducible unitary representations whose coefficients are square integrable modulo $Z$. This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.