Kloosterman sets in reductive groups. (English) Zbl 0919.11055
Authors’ abstract: 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 a 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 connected Chevalley groups over \(\mathbb{Q}\), the global Kloosterman zeta function corresponding to the pair of trivial unipotent characters is a product of Riemann zeta functions.
Reviewer: A.Deitmar (Heidelberg)
MSC:
| 11L05 | Gauss and Kloosterman sums; generalizations |
| 11M41 | Other Dirichlet series and zeta functions |
| 22E35 | Analysis on \(p\)-adic Lie groups |
Keywords:
reductive groups; generating function; sizes of Kloosterman sets; local Kloosterman zeta function for trivial unipotent characters; Chevalley groupsReferences:
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