Let $lambda$ be a nontrivial additive character of the finite field $F_q to C^ imes and let lambda$ be a multiplicative charater of $F^lambda_q to C^ imes$. For a reductive group (or a finite group of Lie type) G defined over $F_q$ (see [1] and [2]) and its finite dimensional linear representation $phi$ over $F_q$, we define the Gauss sum $G(G, phi, lambda, X)$ as follows; $$ G(G, phi, lambda, X) = sum_{g in G} X(det(phi(g)) cdot lambda(tr(phi(g)). $$