A vanishing conjecture: the GLn case

In this article we propose a vanishing conjecture for a certain class of l-adic complexes on a reductive group G which can be regarded as a generalization of the acyclicity of the Artin-Schreier sheaf. We show that the vanishing conjecture contains, as a special case, a conjecture of Braverman and Kazhdan on the acyclicity of rho-Bessel sheaves (Braverman and Kazhdan in Geom Funct Anal I:237-278, 2002). Along the way, we introduce a certain class of Weyl group equivariant l-adic complexes on a maximal torus called central complexes and relate the category of central complexes to the Whittaker category on G. We prove the vanishing conjecture in the case when G = GL(n).