Bounds for Kloosterman sums on \(\mathrm{GL}(n)\) (Q6624759)

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for \(\operatorname{GL}(n)\) for arbitrary \(n \geqslant 3\), 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 obtain power-saving bounds for all Kloosterman sums on \(\operatorname{GL}(4)\).