A Duflo Star Product for Poisson Groups

Let $G$ be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of $G$ provided by an Etingof–Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions on $G$. This recovers and generalizes Duflo's theorem which gives an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra $\mathfrak{a}$ and the subalgebra of ad-invariant in the symmetric algebra of $\mathfrak{a}$. As our proof relies on Etingof–Kazhdan construction it ultimately depends on the existence of Drinfeld associators, but otherwise it is a fairly simple application of graphical calculus. This shed some lights on Alekseev–Torossian proof of the Kashiwara–Vergne conjecture, and on the relation observed by Bar-Natan–Le–Thurston between the Duflo isomorphism and the Kontsevich integral of the unknot.