# Understanding an involution of the category of perverse sheaves on $\mathbb{C}$

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?

• 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). Apr 14, 2022 at 2:24