Weyl algebras on Braverman-Kazhdan spaces

Let $G$ be a split, simply connected, almost simple algebraic group over a field of characteristic zero, and let $P$ be a maximal parabolic subgroup of $G$. We study the ring of differential operators on $P^{\mathrm{d}\mathrm{e}\mathrm{r}}\mathrm{\backslash }G$, showing that it shares several key structural properties with classical Weyl algebras. We also develop a corresponding theory of $D$-modules.