Unitarization of the Horocyclic Radon Transform on Symmetric Spaces

We consider the Radon transform for a dual pair $(X,\Xi)$, where $X=G/K$ is noncompact symmetric space and $\Xi$ of horocycles $X$. address unitarization problem that was considered (and solved in some cases) by Helgason, namely determination pseudo-differential operator such pre-composition with extends to unitary $\mathcal{Q}\colon L^2(X)\to L_\flat^2(\Xi)$, $L_\flat^2(\Xi)$ closed subspace $L^2(\Xi)$ which accounts Weyl symmetries. Furthermore, we show extension intertwines quasi-regular representations $G$ on $L^2(X)$ $L_\flat^2(\Xi)$.