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If we have filtration on D-module compatible with some good filtration on differential operators sheaf then adjoint graded module is module over functions on cotangent bundle. So morally D-module is quantization of something lying on cotangent bundle. There is some rigid statement around this? Some concrete expectations. From here Derived (non-commutative) geometry, geometric constructions in explicit form we could get some dg-algebras for derived categories of D-modules and coherent sheaves on cotangent bundle resp., for their suitable models (~quasi-isomorphism) can we observe first thing as "quantization" of second precisely on the level of differential? Second. Peoples in geometric representation theory often work with coherent sheaves on cotangent bundle of flag varieties, but by the pointed view their constructions must admit quantization, does such thoughts been exploited? It is can be funny if geometric representation construction was connected with some quantum group, because D-modules on flag variety is almost representations of lie algebra

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  • $\begingroup$ look also at quantization of lagrangian submanifolds $\endgroup$
    – Alexander Chervov
    Commented Mar 1, 2013 at 19:46
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    $\begingroup$ I'm not really sure what question you're asking (perhaps the issue is alluded to in your handle). Geometric representation theorists have certainly noticed that D-modules are quantized coherent sheaves on the cotangent bundle and exploited this fact. So what about it? $\endgroup$
    – Ben Webster ♦
    Commented Mar 1, 2013 at 23:37
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    $\begingroup$ D-modules on flag varieties have been extensively studied, e.g., in Beilinson-Bernstein. There is also work connected to quantum groups using a quantum flag variety, in work by Rosenberg and later Backland-Kremnitzer. In other words, such thoughts have been exploited in the past. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 2, 2013 at 0:22
  • $\begingroup$ I know nothing about geometric representation theory and was interested in that from Khovanov homology side. There is construction by Joel Kamnitzer in which cotangent bundle to non-complete (generalized) flag variety plays a role. So 'there is' some quantization of construction by move from sheaves on cotangent bundle to d-modules. So I want example of quantization of some construction in sheaves to d-modules where output is deformed first one. $\endgroup$
    – Bad English
    Commented Mar 11, 2013 at 19:12
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    $\begingroup$ If I assume the question is asking something along the lines of what information on the cotangent bundle recovers the D-module then this is going to be very hard. For example, to make a statement precise one will need to dive into the depths of the proof of the codimension three conjecture of Kashiwara and Vilonen: front.math.ucdavis.edu/1209.5124. To make things even harder this conjecture, well theorem, only applies to regular holonomic D-modules. I have not heard of any proposed conjecture for non-holonomic D-modules... $\endgroup$
    – Jeremy Pecharich
    Commented Mar 13, 2013 at 18:42

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