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My question is as in the title: What advantages do perverse sheaves provide over D-modules? (or vice versa)

As a specific example: could something like the modular generalized Springer correspondence have been proven solely with D-modules? Would there be essential differences between the proof developed by Achar-Henderson-Juteau-Riche, which utilized perverse sheaves, and a proof that solely used D-modules? By essential difference, I mean something that would not be a consequence of passing through the Riemann-Hilbert correspondence.

My basic understanding is that perverse sheaves are better adapted to the algebro-geometric setting, whereas D-modules are more analytic. Nonetheless, a book such as Achar's "Perverse Sheaves and Applications to Representation Theory" still largely emphasizes the analytic topology of complex varieties. Perhaps this is just for the conceptual convenience of the reader (and the nearby cycles stuff), because he also states an étale version and says that most of the results are true in this setting.

Another interesting thing to note is how much more prevalent D-modules are in the literature, especially geometric Langlands and generalizations such as arithmetic D-modules. The prismatic "crew" (Lurie, Bhatt, Scholze, ...) also seems to be interested in (generalizations of) D-modules.

This raises another question: why aren't we using (generalizations of) perverse sheaves?

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    $\begingroup$ Your question surely has many possible answers. Let me give you the first that comes to mind: perverse sheaves are related to regular holonomic D-modules. But we also have a six-functor formalism, nearby / vanishing cycles, etc... for more general holonomic D-modules. (For example, the exponential D-module $(\mathcal{O},d+dx)$ on $\mathbb{A}^1$ is very interesting and appears in a lot of places. But it's irregular and has no perverse sheaf analog.) And that's not saying anything about even more general D-modules. $\endgroup$
    – Gabriel
    Commented May 22 at 11:19
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    $\begingroup$ I think a slightly better way to state the difference is that perverse sheaves are topological in nature, while D-modules more or less live in the (quasi)coherent world. That may partially explain why algebraic constructions are more likely to use the D-module language. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 22 at 13:22
  • $\begingroup$ @R.vanDobbendeBruyn That makes sense. I'm not sure why I had the notion that D-modules were more analytic. Still, given that it is possible (albeit tedious) to define perverse sheaves in the étale setting, and there is a R-H correspondence there, it does still make sense to pursue generalizations of perverse sheaves in other contexts, such as crystalline or prismatic, right? Especially to establish some sort of R-H there. $\endgroup$
    – Andrea B.
    Commented May 23 at 1:51
  • $\begingroup$ Riemann–Hilbert is harder in positive characteristic, because the 'topological' (étale) theory is an $\ell$-adic theory while the 'coherent'/'crystalline' side (I think this is arithmetic D-modules?) is a $p$-adic theory. But each of these objects separately are indeed studied, and maybe people have even thought about connections between them. I am far from an expert, but I do know that going from $p$ to $\ell$ is hard. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 23 at 16:06

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