\(L\) functions of some exponential sums of finite classical groups (Q1407667)

The authors consider exponential sums \(E_n(V,f)\) and the \(L\) function \(L(t,V,f)\) associated to a variety \(V\), a regular function \(f\) on it and a fixed nontrivial additive character \(\lambda\) of \({\mathbb F}_q\). They give a formula for the exponential sum \(\sum_{g\in G} \lambda(\text{tr}\, g)\) as the alternating sum of traces of the Frobenius map acting on a Weyl group invariant for certain cohomology groups of \((G/T)\times T\), where \(T\) is a maximal torus of \(G\). Explicit computations are given for the classical groups \(\text{GL}_n\), \(\text{SL}_n\) and \(\text{Sp}_{2n}\). In these cases the \(L\) functions are explicitly described. The the exponential sum \(\sum_{g\in \text{GL}_n({\mathbb F}_q)} \,\lambda(\text{tr}\,g + \text{tr}\, g^{-1}) \) is also computed.