Background 1) Knop's theorem
----------------------------


In [fundamental paper in Annals 1994](http://www.algeo.math.uni-erlangen.de/fileadmin/algeo/users/knop/papers/HC.html) 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"


<http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf>


See [Capelli determinant = Duflo ( determinant) - was it known ?](https://mathoverflow.net/questions/92348/capelli-determinant-duflo-determinant-was-it-known)  , 
[Is the Duflo map for Lie algs. unique ?](https://mathoverflow.net/questions/80025/is-the-duflo-map-for-lie-algs-unique)
for some info on Duflo map.


Question
--------


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.



Can Duflo type map be defined for invariant differential operators ? (In a way compatible with "Harish-Chandra" isomorphism defined by F. Knop)
Let $X=T^\*M$, and apply Dolgushev's equivariant formality theorem (see <http://arxiv.org/abs/math/0307212>) to get a $G$-equivariant $L\_\infty$-quasi-isomorphism
$$
\mathcal U:T\_{poly}X \longrightarrow D\_{poly}X
$$


Let $\pi$ be the Poisson structure on $X$ which comes from the canonical symplectic form on $T^\*M$. Then one gets a $G$-equiavriant quasi-isomorphism of complexes
$\mathcal U\_\pi^1$ from the Poisson cochain complex of $(X,\pi)$ to the Hochschild cochain complex of $(\mathcal O\_X,\star\_\pi)$, where $\star\_\pi$ is the star-product the quantizes $\pi$ through $\mathcal U$.


Moreover, one has a homotopy betwen the cup-product one both sides, and it is $G$-equivariant (compatibility with cup-products is sketched in the original paper of Kontsevich <http://arxiv.org/abs/q-alg/9709040>, is rigorously proved in <http://arxiv.org/abs/math/0106205> by Manchon and Torossian, and the globalization is adressed in <http://arxiv.org/abs/0805.3444> - in this last reference you will easily see that if you start with a $G$-invariant connection to perform globalization then your homotopy will be $G$-invariant too).


**Conclusion**. taking $G$-invariants and restricting yourself to the degree zero part of cohomology (Poisson and Hochschild, respectively), you'll find that the Poisson center of $\mathcal O\_X^G$ is isomorphic, *as an algebra*, to the center of the subalgebra of $G$-invariants elements in the quantization.


**Things I have been hiding under the carpet**. 1. issues involving $\hbar$. In this case you have to prove that the quantization is actually convergent and can be specialized to $\hbar=1$. 2. You have to prove that the quantization really gives back differential operators... this is probably the more tricky part.


