A new Fourier transform

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In order to define a geometric Fourier transform, one usually works with either ℓ-adic sheaves in characteristic p>0 or with D-modules in characteristic 0. If one considers ℓ-adic sheaves on the stack quotient of a vector bundle V by the homothety action of Gₘ, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on [V/Gₘ] 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 D-modules (when the latter are defined).