2002/10/16 by Roman Bezrukavnikov, Bezrukavnikov, Roman, Alexander Braverman +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.math/0210250
openalex publication_date 2002/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup P in G and denote by U its unipotent radical. Let X be a G-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on X with respect to a generic character of U commutes with Verdier duality. In the first example we take χ to be an arbitrary G-variety and we prove the above property for all P-equivariant sheaves on X where P is an opposite parabolic subgroup assuming χ satisfies a strong nondegeneracy condition (such a χ exists for some but not all choices of P). In the case when P is a Borel subgroup it is enough to require that the sheaf in question is U equivariant where U is the unipotent radical of P. In the second example we take X = G where G acts by left translations and we prove the corresponding result when P is a Borel subgroup for sheaves equivariant under the adjoint action of G (the latter result was conjectured by B. C. Ngo who proved it for G = GL(n)). As an application we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier-Deligne transform.