Gauss sums for symplectic groups over a finite field

Abstract

For a nontrivial additive character λ and a multiplicative character χ of the finite field withq elements, the ‘Gauss’ sums Σλ(trg) overg∈Sp(2n,q) and Σχ(detg)λ(trg) overg∈GSp(2n, q) are considered. We show that it can be expressed as a polynomial inq with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain ‘generalized Kloosterman sums over nonsingular matrices’ and ‘generalized Kloosterman sums over nonsingular alternating matrices’, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.