A vanishing conjecture: the GL_n case

In this article we propose a vanishing conjecture for a certain class of $\mathrm{\ell }$-adic complexes on a reductive group $G$, which can be regraded 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 \cite{BK1}. Along the way, we introduce a certain class of Weyl group equivariant $\mathrm{\ell }$-adic complexes on a maximal torus called \emph{central complexes} and relate the category of central complexes to the Whittaker category on $G$. We prove the vanishing conjecture in the case when $G={\text{GL}}_n$.