Kloosterman Sets in Reductive Groups Original Research Article

We find an explicit formula for the generating function for the sizes of Kloosterman sets (or equivalently, the local Kloosterman zeta function for trivial unipotent characters) in the context of a split reductive connected algebraic group over an nonarchimedean local fieldK. We provide two proofs of the formula: One is based on a representation theoretical interpretation of the generating function, the other uses an explicit parametrization of the Kloosterman sets. The formula implies that in the case of simply connected Chevalley groups over image, the global Kloosterman zeta function corresponding to the pair of trivial unipotent characters is a product of Riemann zeta functions.