  In this article we propose a vanishing conjecture for a certain class of
$\ell$-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 $\ell$-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 $G=\GL_n$.
