Background 1) Knop's theorem
In fundamental paper in Annals 1994 F. Knop proved the following theorem.
Let G be a connected reductive group and X a smooth G-variety.
Theorem: Assume that X is either spherical or affine. Then the center Z(X) of the ring of G-invariant differential operators on X is a polynomial ring. More precisely, Z(X) is isomorphic to the ring of invariants of a finite reflection group.
Background 2) Duflo's map
M. Duflo defined fundamental construction: for any Lie algebra $g$ a map: $DufloMap: S(g) \to U(g) $, such that it is
1) Restricted to the $S(g)^g$ it will give isomorphism of commutative algebras $S(g)^g \to Z(U(G))$
2) It is isomorphism of graded vector spaces, moreover it is identity map on "principal symbolds"
3) Isomorphism of $g$ modules
D. Calaque, C. Rossi "Lectures on Duflo isomorphisms in Lie algebras and complex geometry"
See Capelli determinant = Duflo ( determinant) - was it known ? , Is the Duflo map for Lie algs. unique ? for some info on Duflo map.
Let $G$ be a Lie group (reductive may be necessary). $M$ is manifold (may be affine).
Question Is it possible to find such a map from Functions(T*M) to Differential operators on $M$, such that it satisfies requirements similar to the Duflo map, where the role of $S(g)$ is played by $Fun(T^*M)$ and of $U(g)$ is played by $Dif(M)$ ?
Moreover we should require is is compatible with Knop-Harish-Chandra isomorphism.
If center of $Dif(M)^G$ is trivial, then there is no point to ask the question, but if center is non-trivial the requirements seems quite non-trivial.