Bounds for Kloosterman sums on GL(n)

Abstract

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n), as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak's density conjecture for the principal congruence subgroup of prime level. We also bound all Kloosterman sums for GL(4).