Generalized character sums associated to regular prehomogeneous vector spaces

The purpose of this note is to give a short derivation of the finite field analogue of Sate's functional equation for the zeta function associated wit h a prehomogeneous vector space (see [S]). We restrict ourselves to the cas e of a regular prehomogeneous vector space, however, we allow twisting of o ur character sums by local systems associated to arbitrary representations of the component group of the stabilizer of a generic point. The main idea of our approach is to use the Picard-Lefschetz formula in l-adic cohomology instead of using a lift of a prehomogeneous space to the characteristic ze ro las is done in [DeG]). Also we deduce another functional equation associ ated with a regular prehomogeneous vector space (Theorem 1.4).