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

Published 3 Oct 2024 in math.FA | (2410.09075v1)

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