Gauss sums for symplectic groups over a finite field (Q1264305)

The Gauss sum \(\sum_{g\in G} \chi(\det g) \lambda (\operatorname {tr}g)\) is evaluated in the cases where \(G= \text{Sp} (2n,q)\) is the symplectic group and \(G= \text{GSp} (2n,q)\) is the symplectic similitude group over \(\mathbb{F}_q\), in terms of the usual Kloosterman sums. Here \(\lambda\) is a nontrivial additive character, \(\chi\) denotes a multiplicative character of a finite field \(\mathbb{F}_q\), and \(G\) is a subgroup of the general linear group \(\text{GL} (2n, \mathbb{F}_q)\).