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 Sato's functional equation for the zeta function associated with a prehomogeneous vector space (see [S]). We restrict ourselves to the case of a regular prehomogeneous vector space, however, we allow twisting of our 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 zero (as is done in [DeG]). Also we deduce another functional equation associated with a regular prehomogeneous vector space (Theorem 1.4).