$SL_4$-Kloosterman sum via the Bruhat decomposition

TL;DR

The paper develops a higher-rank generalization of Kloosterman sums via Bruhat decomposition, focusing on the SL_4 long-word case and its fine-cell decomposition. It provides a detailed parametrization of the fine Kloosterman cells, showing that the SL_4 long-word sum factors into a finite sum of products of two classical Kloosterman sums, with explicit divisibility and congruence conditions. A Weil-type bound is established for the SL_4 long-word sum, and the work outlines a parallel SL_5 construction with corresponding fine-cell parametrization. The results extend the group-theoretic framework for Kloosterman sums to higher rank, enabling refined analytic tools for automorphic forms and trace formulas.

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.

Theorems & Definitions (15)

• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• Proposition 3.5
• Lemma 3.6
• proof
• Theorem 3.7