$S{L}_4$-Kloosterman sum via the Bruhat decomposition

Abstract

We define the Kloosterman sum for $S{L}_4$ over the Kloosterman set via the Bruhat decomposition and stratify the Kloosterman set using the reduced word decomposition of the Weyl group element. The Kloosterman sum for an $S{L}_4$-long word is decomposed into finer parts (called the fine Kloosterman sum), and can be written as a finite sum of a product of two classical Kloosterman sums.