2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78723In this paper we consider the analogue of the Sato's functional equation for the prehomogeneous vector spaces over finite fields. The corresponding character sums depend on a relative invariant on such a space and an irreducible representation of the group of components of the stabilizer of a generic point. The proof is based on the Picard-Lefschetz formula in $l$-adique cohomology.AMSLatex, 16 pages, new theorem addedAlgebraic Geometrytext