Kloosterman sums for Chevalley groups

Abstract:

A generalization of Kloosterman sums to a simply connected Chevalley group $G$ is discussed. These sums are parameterized by pairs $(w,t)$ where $w$ is an element of the Weyl group of $G$ and $t$ is an element of a ${\mathbf {Q}}$-split torus in $G$. The $SL(2,{\mathbf {Q}})$-Kloosterman sums coincide with the classical Kloosterman sums and $SL(r,{\mathbf {Q}})$-Kloosterman sums, $r \geq 3$, coincide with the sums introduced in [B-F-G,F,S]. Algebraic properties of the sums are proved by root system methods. In particular an explicit decomposition of a general Kloosterman sum over ${\mathbf {Q}}$ into the product of local $p$-adic factors is obtained. Using this factorization one can show that the Kloosterman sums corresponding to a toral element, which acts trivially on the highest weight space of a fundamental irreducible representation, splits into a product of Kloosterman sums for Chevalley groups of lower rank.