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

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 $G\mathrm{/}K\mathrm{.}$ 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 ${\mathfrak{a}}_{\mathbb{C}}^{\ast }\mathrm{.}$