A vanishing conjecture: the GL_n case

TL;DR

This paper introduces a vanishing conjecture for $ ext{ell}$-adic complexes on reductive groups, generalizing known acyclicity results, and proves it for the case of $GL_n$, connecting it to the Whittaker category.

Contribution

It formulates a new vanishing conjecture for $ ext{ell}$-adic complexes, relates it to existing conjectures, and proves it for $GL_n$, advancing understanding of these complexes on reductive groups.

Findings

Proposes a vanishing conjecture generalizing acyclicity of Artin-Schreier sheaves.

Establishes a connection between central complexes and the Whittaker category.

Proves the conjecture specifically for the group $GL_n$.

Abstract

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 $G=\GL_n$.