A new Fourier transform

In order to define a geometric Fourier transform, one usually works with either $\ell$-adic sheaves in characteristic $p \gt 0$ or with $\mathcal{D}$-modules in characteristic $0$. If one considers $\ell$-adic sheaves on the stack quotient of a vector bundle $V$ by the homothety action of $\mathbb{G}_m$, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on $[V / \mathbb{G}_m]$ is closely related to the category of (unipotently) monodromic sheaves on $V$. In this article, we introduce a new functor, which is defined on all sheaves on $V$ in any characteristic, and we show that it restricts to an equivalence on monodromic sheaves. We also discuss the relation between this new functor and Laumon’s homogeneous transform, the Fourier–Deligne transform, and the usual Fourier transform on $\mathcal{D}$-modules (when the latter are defined).

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Accepted 2 April 2015

Published 13 April 2016