Bounds for Kloosterman Sums for ${\mathrm{GL}}_n$

Abstract

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for ${\mathrm{GL}}_n$ for $n>2$ are proven, improving the trivial bound by DÄ browski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.

V2: Added a uniform bound. Added a subsection on character dependency. Added further examples and explanations of the main ideas. Some minor changes