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For a smooth complex variety, one can consider the category of say holonomic $\mathcal D$ modules on it. It is equipped with the deRham functor, which turns a $\cal D$-module into a constructible sheaf.

My question is, is there an analogue of constructible sheaves and the deRham functor for rings which behave like $\cal D$? A typical example I would be interested in is rings of twisted differential operators.

Edit: For example I wonder whether one could complete the following scheme:

Principal block of category $\cal O$ $\rightsquigarrow$ ${\cal D}$-modules on $G/B$ $\rightsquigarrow$ Perverse sheaves on $G/B$

SIngular block of category $\cal O$ $\rightsquigarrow$ twisted $\cal D$-modules on $G/B$ $\rightsquigarrow$ ?? on $G/B$

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  • $\begingroup$ There is a substantial tradition, going back at least to work of Beilinson and Bernstein around 1980, involving twisted differential operators in the framework of D-modules. This is discussed for example in the expanded English version of lectures by Hotta-Tanisaki: D-Modules, Perverse Sheaves, and Representation Theory, Birkhauser, 1995 (see 11.2). Can you clarify how your question fits into the existing framework? (Also, a tag ag.algebraic-geometry might be useful.) $\endgroup$
    – Jim Humphreys
    Commented Feb 2, 2013 at 17:54
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    $\begingroup$ P.S. I agree with Ardakov that what you are asking for is oversimplified. I'd especially recommend that you study Remark 12.2.8 in the expanded English version of the Japanese notes by Hotta-Tanisaki (with help from K. Takeuchi) published by Birkhauser in 2008. They give detailed references on further work for arbitrary weights, but for integral weights the simplest approach seems to rely on Jantzen's translation functors. From there it gets less direct, but perverse sheaves on flag varieties still play a leading role. $\endgroup$
    – Jim Humphreys
    Commented Feb 3, 2013 at 14:35

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I'm not sure what you mean by "rings which look like D" but here's one point of view: the de Rham functor is just derived Hom from the structure sheaf $O$, i.e. you're testing all D-modules against your favorite one. One can imagine an analog in any context where you have a favorite module. For D-modules with an integral twist, you can just use the corresponding line bundle instead of O, and again get a Riemann-Hilbert correspondence with constructible sheaves (this is a little silly though since all of these categories with integral twists are canonically equivalent).

More generally if your twist is a complex linear combination of line bundles (eg always in the analytic topology) you can think of twisted D-modules as monodromic D-modules: these are D-modules on the total space of a principal torus bundle, which are weakly torus equivariant, and have some fixed monodromy along the fibers. For example in the case of the flag variety all sheaves of twisted differential operators can be viewed this way using the "basic affine space" $G/N$. So then you can apply the ordinary de Rham functor upstairs. This gives a Riemann-Hilbert correspondence with monodromic constructible sheaves on the total space, or if you prefer, with constructible sheaves on a gerbe over the base (I think this is addressed in another MO answer about complex powers of line bundles, I learned it from a paper of Kashiwara). The class of the gerbe in $H^2(X, C^\times)$ is just the exponential of the Chern class of the TDO, considered as a class in $H^2(X,C)$. Is that the kind of answer you're looking for?

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  • $\begingroup$ Thanks for your answer! However it is not yet quite what I am looking for! What I would prefer is some category of constructible sheaf like objects on $G/B$ and not some slightly different space. Is this possible? $\endgroup$
    – Jan Weidner
    Commented Feb 3, 2013 at 6:33
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    $\begingroup$ Perhaps part of the problem is that your squiggly arrow from Singular block of category $\mathcal{O}$ to twisted $\mathcal{D}$-modules on $G/B$ does not even induce a derived equivalence? If you don't want to pass to $G/N$, will you consider partial flag varieties $G/P$? I have this paper here (arxiv.org/abs/1011.0896) in mind. $\endgroup$
    – user91132
    Commented Feb 3, 2013 at 11:28
  • $\begingroup$ Yes I know, that the functor from twisted D-modules to singular category O is not an equivalence. However I would also not expect the thing on the "constructible sheaves" side to be equivalent to category O. Also my question is not really representation theory, I only added this after Jim Humphreys question. So I think the flag manifold being affine for some TDOs is not part of the problem. $\endgroup$
    – Jan Weidner
    Commented Feb 3, 2013 at 15:34
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    $\begingroup$ Jan- It is exactly what you are looking for. You are not going to do better than constructible sheaves over a gerbe. $\endgroup$
    – Ben Webster ♦
    Commented Feb 3, 2013 at 20:53
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    $\begingroup$ @ChrisKuo Representation Theory and D-modules on flag manifolds, Asterisque 173-174 (1989) 55-109. There are several sections describing twisted sheaves and twisted D-modules and their equivalence. See also: Equivariant derived category and representation of real semisimple Lie groups. Representation theory and complex analysis, 137-234, Lecture Notes in Math., 1931, Springer, Berlin, 2008. $\endgroup$
    – David Ben-Zvi
    Commented Apr 5, 2023 at 0:29

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