Differential operators on M/G are factor of invariant  differential operators on M ? (Quantization commute with reduction ) What is state of art ? Naively we may expect: Diff(M/G) = Diff(M) // G, where Diff(M) // G = Diff(M)^G /I, where "I" is two-sided ideal in  Diff(M)^G of operators which act by zero on invariant functions.
This is example of quantization commute with reduction ideology, since Diff(N) - quantization of T*N. 
My questions is what is the state of art ? 
I would expect it is known to be  true if action of M on G is free, M - smooth oriented manifold. If yes what is the reference ?

There is paper by B. Fedosov:
Non-Abelian Reduction in Deformation Quantization, Lett. Math. Phys. 1998 Volume 43, Number 2, 137-154, 
http://www.springerlink.com/content/rv230884l0617558/
As far as I understand his result should imply the positive answer for the compact group G.
But I am not sure about the details, may be he assume some compactness, or some other assumptions...
If any one would be so kind to send me this paper, it would be very kind of him. 
 A: In the algebraic situation one could do it as follows: First of all, you have a canonical map $Diff(M) // G\rightarrow Diff(M/G)$. Indeed if $f$ is a function on the quotient, $D\in Diff(M)^G$, then $D(\pi^* f)$ is $G$-invariant, so it descents to the quotient. 
Now lets assume that the action on $M$ is free and proper. Then $\pi:M\rightarrow M/G$ is a principal $G$-bundle. So for checking whether our canonical map is an isomorphism we can restrict to the product situation $G\times X\rightarrow X$. In this case we compute: $$Diff(M) // G = Diff(G\times X)^G /I=(U(\mathfrak g)\otimes Diff(X))/(U(\mathfrak g)\otimes 1) =Diff(X)=Diff(M/G)$$
So here we used $Diff(X\times Y)=Diff(X)\otimes Diff(Y)$, in the algebraic situation this is literally true. In the analytic situation I would guess it is true if one interpretes the $\otimes$ in the right way... Maybe someone more experienced with these things can tell whether the above computation can be translated into the smooth manifold situation?
