Unitary Kloosterman Sums and the Gelfand–Graev Representation of GL2
✍ Scribed by Charles W. Curtis; Ken-ichi Shinoda
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 554 KB
- Volume
- 216
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
In this article, we obtain relations between two types of Kloosterman sums and corresponding ones for finite field extensions, using L-series and Euler product expansions. The methods used were first applied by A. Weil in his proof of the Davenport-Hasse relation for Gauss sums ([8, Chap. 11]), and for other exponential sums in ([13, Appendix V]). Generalized Kloosterman sums have also been considered in the context of l-adic sheaves over certain algebraic varieties by Deligne [6].
The relations proved in Sections 1 and 2 are closely connected with the representation theory over the complex field C of the connected reductive algebraic group G = GL 2 defined over a finite field F = Fq, with Frobenius map o-, so that G '~ = GL2(F). In [5], a norm map A was defined, as a homomorphism of algebras from the Hecke algebra ~m of a Gelfand-Graev representation of G ~m to a corresponding Hecke algebra ~ of G '~. The norm map A is characterized by intertwining relations involving the two classes of ~r-stable maximal tori in G. The identities for Kloosterman *This article was written while the authors were participants in the programme Representation Theory of Algebraic Groups and Related Finite Groups at the Isaac Newton Institute during March 1997. We thank the Institute for its hospitality.