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It is well-known (for example: chapter 2 in [GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble, 35 (1985), 1–48) that a perverse sheaf on $X=\mathbb{C}$ with stratification $X=\mathbb{C}^\times\sqcup \{0\}$ is equivalent to the datum of $(E,F,u,v)$, where $E$, $F$ are finite dimensional vector spaces, $u:E\to F$, $v:F\to E$ are linear maps such that $1+uv$ is invertible.

There is an obvious involution on the category of such data, namely $$(E,F,u,v)\mapsto (F,E,v,u).$$

My question is: what is the corresponding operation on perverse sheaves?

If I understand the correspondence correctly, the case $(E,F,u,v)=(\mathbb{C},0,0,0)$ corresponds to the constant sheaf $\underline{\mathbb {C}}_X$, and the case $(E,F,u,v)=(0,\mathbb{C},0,0)$ corresponds to the shifted sheaf $\underline{\mathbb{C}}_0[-1]$ supported at the origin. (I am using the convention of [GGM], so in particular the constant sheaf $\underline{\mathbb {C}}_X$ without shift is perverse.)

Is there any interesting interpretation of this involution?

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    $\begingroup$ I believe it's the Fourier transform. (Say, pass from perverse sheaves to $D$-modules, then do the $D$-module Fourier transform, then go back). $\endgroup$
    – Will Sawin
    Apr 14, 2022 at 2:24

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It is the Fourier–Sato transform. You can find a detailed discussion in e.g. section 4D of this article: Bezrukavnikov and Kapranov - Microlocal sheaves and quiver varieties.

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