Journal of the
Korean Mathematical Society JKMS

For a lifted nontrivial additive character $\lambda'$ and a multiplicative character $\chi$ of the finite field with $q^2$ elements, the `Gauss' sums $\sum \lambda' (\text{tr} \, w)$ over $ w \in SU(2n+1,q^2)$ and $\sum \chi (\text{det} \, w) \lambda' (\text{tr} \, w)$ over $w \in U(2n+1,q^2)$ are considered. We show that the first sum is a polynomial in $q$ with coefficients involving certain new exponential sums and that the second one is a polynomial in $q$ with coefficients involving powers of the usual twisted Kloosterman sums and the average (over all multiplicative characters of order dividing $q-1$) of the usual Gauss sums. As a consequence we can determine certain `generalized Kloosterman sum over nonsingular Hermitian matrices' which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

Keywords: Gauss sum, multiplicative character, additive character, unitary group, twisted Kloosterman sum, Bruhat decomposition, maximal parabolic subgroup

MSC numbers: Primary 11T23, 11T24 ; Secondary 20G40, 20H30