Unitarization and Inversion Formulae for the Radon Transform Between Dual Pairs

We consider the Radon transform associated to dual pairs (X,Ξ) in the sense of Helgason, with X = G/K and Ξ = G/H, where G is a locally compact group and K and H are closed subgroups of G. Under some technical assumptions, we prove that if the quasi-regular representations of G acting on L2(X) and L2(Ξ) are irreducible, then the Radon transform admits a unitarization intertwining the two representations. If, in addition, the representations are square- integrable, we provide an inversion formula for the Radon transform based on the voice transform associated to these representations.