• DOI:10.4310/MRL.2015.V22.N5.A12
• Corpus ID: 119676562

A new Fourier transform

@article{Wang2014ANF,
  title={A new Fourier transform},
  author={Jonathan Wang},
  journal={arXiv: Algebraic Geometry},
  year={2014}
}

• Jonathan Wang
• Published 22 February 2014
• Mathematics
• arXiv: Algebraic Geometry

In order to define a geometric Fourier transform, one usually works with either $\ell$-adic sheaves in characteristic $p>0$ or with $D$-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 $D$-modules (when the latter are defined).