While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in there. Let me describe the construction:
Let $A$ be an almost commutative algebra over an algebraically closed field $k$ (positively filtered whose associated graded is of finite type). Denote by $\sigma: A \to grA$ the multiplicative map sending an element to its corresponding "symbol" in the associated graded.
The following proposition is proved about ore localizations in this context:
Proposition: Let $\tilde{S} \subset grA$ be a multiplicatively closed subset. Then $S = \{a \in A : \sigma(a) \in \tilde{S} \}$ satisfies ore conditions. Meaning there exists a noncommutative localization $S^{-1}A$.
He then introduces a $\mathbb{Z}$-filtration on $S^{-1}A$ naturally as follows:
$$(S^{-1}A)_n = \{s^{-1}a \in S^{-1}A : deg\sigma(a)-deg\sigma(s)=n \}$$
Definition: The completion of $S^{-1}A$ w.r.t. the filtration defined above is called formal microlocalization of $A$ at $S$ and denoted $A_S$.
Further arguments show $U \to A_U$ defines a sheaf of algebras on the affine scheme $Spec(grA)$ And that for any finitely generated $A$-module the assignment $U \to M_U := A_U \otimes_A M$ defines a sheaf of modules over it.
It seems to me that this construction can be generalized to give microlocalization of $\mathcal{D}_X$ for general smooth varieties over $\mathbb{C}$. Let's denote by $\mathcal{E}_X$ the sheaf obtained from the procedure above applied to $\mathcal{D}_X$.
It therefore raises the questions:
How does the sheaf $\mathcal{E}_X$ relate to Sato's sheaf of formal microdifferential operators? (on the analytic cotanget bundle of $X$ of course).
Can $\mathcal{E}_X$ be described explicitly (generators and relations) in simple examples? ($Spec \mathbb{C}[[x]], Spec \mathbb{C}[x], \mathbb{A}^n$).
And finally:
Is this algebraic side of microlocalization developed anywhere in the literature? If so where?
