Gauss sums for general and special linear groups over a finite field

Abstract.

For a nontrivial additive character $\lambda$ of the finite field with q elements and each positive integer r, the 'Gauss' sums $\sum \lambda ({(\rm tr}\, g)^r)$ over $g \in {\rm GL}\,(n,q)$ and over $g \in {\rm SL}\,(n,q)$ are considered. We show that both of them can be expressed as polynomials in q with coefficients involving certain exponential sums. As applications, we derive from these expressions the formulas for the number of elements g in GL (n,q) and SL (n,q) with ${\rm tr}\, g = \beta$, for each $\beta$ in the finite field with q elements.