Beurling-Fourier algebras on Lie groups and their spectra

TL;DR

This paper studies Beurling-Fourier algebras on various Lie groups, analyzing their spectra and how different weights affect their properties, with a focus on specific examples like $SU(n)$ and the Heisenberg group.

Contribution

It introduces a refined definition of weights on duals of locally compact groups and determines spectra for Beurling-Fourier algebras on key Lie groups, linking them to complexifications.

Findings

Spectra are explicitly determined for several Lie groups.

Polynomially growing weights do not alter the spectra.

Regularity of Beurling-Fourier algebras is established.

Abstract

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras.