Let $X$ be an affine smooth irreducible variety over an algebraically closed field of zero characteristics, $\mathcal{V}$ the Lie algebra of vector fields on $X$.
We have a natural map $\phi: U(\mathcal{V}) \rightarrow \mathcal{D}(X)$ of the envelopping algebra of $\mathcal{V}$ to the ring of differential operators on $X$, as $\mathcal{V}$ is a Lie subalgebra of $\mathcal{D}(X)^{(-)}$.
- Are there conditions where we can say something about the kernel and image of $\phi$?