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).