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

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 ${\mathfrak{a}}_{\mathbb{C}}^{\ast }.$

arXiv commentA continuation of our study of the Joint-Eigenspace Fourier transform on noncompact symmetric spaces. arXiv admin note: substantial text overlap with arXiv:2409.09036