The Lagrangian Radon Transform and the Weil representation

We consider the operator $\mathcal R$, which sends a function on $\mathbb R^{2n}$ to its integrals over all affine Lagrangian subspaces in $\mathbb R^{2n}$. We discuss properties of the operator $\mathcal R$ and of the representation of the affine symplectic group in the space of functions on $\mathbb R^{2n}$.