Some results about the geometric Whittaker model
Abstract
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 Gvariety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of ladic sheaves on $X$ with respect to a generic character of U commutes with Verdier duality. Namely, in the first example we take X to be an arbitrary Gvariety and we prove the above property for all $\oU$equivariant sheaves on X where $\oU$ is the unipotent radical of an opposite parabolic subgroup; in the second example we take X=G and we prove the corresponding result for sheaves which are equivariant under the adjoint action (the latter result was conjectured by B.C.Ngo who proved it for G=GL(n)). Analogous results hold also for Dmodules when k is replaced by the field of complex numbers. As an application of the proof of the first statement we reprove a theorem of N.Katz and G.Laumon about local acyclicity of the kernel of the FourierDeligne transform.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2002
 arXiv:
 arXiv:math/0210250
 Bibcode:
 2002math.....10250B
 Keywords:

 Algebraic Geometry;
 Representation Theory
 EPrint:
 8 pages