Unitarization of the Horocyclic Radon Transform on Symmetric Spaces

Abstract

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