A note on the $L^{2}-$harmonic analysis of the Joint-Eigenspace Fourier transform

TL;DR

This paper investigates the irreducibility of the regular representation of noncompact semisimple Lie groups on the Hilbert space associated with the Joint-Eigenspace Fourier transform, providing a characterization based on spectral simplicity.

Contribution

It offers a new characterization of irreducibility for the regular representation using spectral properties of the Fourier transform on symmetric spaces.

Findings

Irreducibility linked to simplicity of spectral parameters

Complete $L^{2}$-decomposition of the Fourier transform

Characterization in terms of spectral pair simplicity

Abstract

We consider the irreducibility of the regular representation of a noncompact semisimpe Lie group $G$ on the Hilbert space of the image of the Joint-Eigenspace Fourier transform on its corresponding symmetric space $G/K.$ The $L^{2}-$decomposition of the Joint-Eigenspace Fourier transform leads to the complete characterization of the said irreducibility in terms of the simplicity of a pair of members of $\mathfrak{a}^{*}_{\mathbb{C}}.$