$SL_4$-Kloosterman sum via the Bruhat decomposition

TL;DR

This paper introduces a new approach to defining and decomposing $SL_4$ Kloosterman sums using Bruhat decomposition and Weyl group stratification, leading to a finer analysis of these sums.

Contribution

It presents a novel method to decompose $SL_4$ Kloosterman sums into finer components and expresses them as finite sums of products of classical sums.

Findings

Decomposition of $SL_4$ Kloosterman sums into finer parts

Representation of these sums as finite sums of classical Kloosterman sums

Stratification of the Kloosterman set via Weyl group decomposition

Abstract

We define the Kloosterman sum for $SL_4$ 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 $SL_4$-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.