Kloosterman sets in reductive groups

We find an explicit formula for the generating function for the sizes of Kl oosterman sets (or equivalently, the local Kloosterman zeta function for tr ivial unipotent characters) in the context of a split reductive connected a lgebraic group over an nonarchimedean local field K. 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 connec ted Chevalley groups over Q, the global Kloosterman zeta function correspon ding to the pair of trivial unipotent characters is a product of Riemann ze ta functions. (C) 1998 Academic Press.