The Radon transform on ${\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$

Let $G$ be $SL(2,{\mathbf{R}})$. $G$ acts on the upper half-plane $\mathcal{H}$ by the Möbius transformation, providing $\mathcal{H}$ with the Riemannian metric structure along with the Laplacian, $\Delta$. We study the integral transform along each geodesic. $G$ acts on $\mathcal{P}$, the space of all geodesics, in a natural way, providing $\mathcal{P}$ with its invariant measure and its own Laplacian. ( $\mathcal{P}$ actually is a coset space of $G$.) Therefore the above transform can be viewed as a map from a suitable function space on $\mathcal{H}$ to a suitable function space on $\mathcal{P}$. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.