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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Kloosterman sums for Chevalley groups

Author: Romuald Dąbrowski
Journal: Trans. Amer. Math. Soc. 337 (1993), 757-769
MSC: Primary 11L05; Secondary 20G05
MathSciNet review: 1102221
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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.

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