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).Comment: 20 page