Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the PBW isomorphism, inverse to that natural inclusion of the symmetric algebra.

Let $C^{Hoch}_{*} (U\mathfrak{g})$ be the graded vector space underlying the standard complex computing Hochschild homology of $U\mathfrak{g}$, ie forget the differential for now. Composition of $\iota$ with the HKR morphism induces a map of graded vector spaces, $C^{Hoch}_{*} (U\mathfrak{g})\rightarrow \Omega^{*} (S\mathfrak{g})$. This respects the natural filtrations on both sides.

What I'd like to know is whether there's a nice differential on the right hand side, such that our map intertwines the Hochschild differential on the LHS with this new differential. The leading order term (wrt the filtration) for this differential should be the lie derivative wrt the Poisson-Kirillov bracket on $S(\mathfrak{g})$.