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I am interested in studying the Fourier transform for perverse sheaves on "nice" spaces, say affine complex space stratified by the action of an algebraic group into finitely many orbits.

In particular, there is a beautiful construction of the Fourier transform for $D$-modules, where, in the affine case, if $A_n = \mathbb{C}[x_1, \dots, x_n] \langle \delta_1, \dots, \delta_n\rangle$ is the $n$-th Weyl alebra, then the Fourier transform is given by pushforward along the automorphism $f:A_n \rightarrow A_n$ provided by $x_i \mapsto \delta_i$, $\delta_i \mapsto - x_i$ (or equivalent).

Is there any intrinsic way to construct the functor for Perverse Sheaves on the same space, so that it commutes with the Fourier transform for $D$-modules across the Riemann-Hilbert correspondence? (i.e., so that the Fourier transform commutes with the De-Rham functor).

Thank you!

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    $\begingroup$ In the setting of constructible sheaves in the analytic topology it is often called the Fourier-Sato transform. See Kashiwara-Schapira "Sheaves on Manifolds" section 3.7 for the general construction. Then I think it is shown somewhere in Chapter 10 that it preserves perverse sheaves (with appropriate shift?). $\endgroup$
    – Sam Gunningham
    Commented Jun 15 at 23:52
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    $\begingroup$ I am not sure of a reference for the fact that the Fourier-Sato transform corresponds to the usual Fourier transform for algebraic D-modules. Note that the D-module transform does not preserve regularity (at $\infty$) so one should restrict to $\mathbb G_m$-monodromic objects to get a well-defined statement (this is also the generality in which the Fourier-Sato transform is defined). There is some discussion here: mathoverflow.net/questions/3139/… $\endgroup$
    – Sam Gunningham
    Commented Jun 15 at 23:55
  • $\begingroup$ Wonderful! Thank you very much, I will look into this book. $\endgroup$
    – James Steele
    Commented Jun 19 at 20:43

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