A new Fourier transform

2014/02/22 by Wang, Jonathan
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1402.5555

Abstract

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 \mathbbG_m, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on [V/\mathbbG_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).

Related

• Transformation de Fourier homogene
• The Fourier Transform and Characteristic Cycles of Monodromic ℓ-adic Sheaves
• The derived homogeneous Fourier transform
• Calculation of l-adic local Fourier transformations
• Sheaves with connection on abelian varieties