Generalized character sums associated to regular prehomogeneous vector spaces (Q5930041)

In the theory of prehomogeneous vector spaces, the fundamental theorem is the functional equation of the zeta function associated to a relative invariant of a prehomogeneous vector space. This is due to M. Sato in the real and complex cases, and to J. Igusa in the \(p\)-adic case. The character sum is nothing but an analogue of such a zeta function in the case of a finite field. Its functional equation, which corresponds to the above `fundamental theorem', is investigated by \textit{J. Denef} and \textit{A. Gyoja} [Compos. Math. 113, 273-346 (1998; Zbl 0919.11086)] using a lift of a prehomogeneous space to the characteristic zero. On the other hand the authors of this paper give this functional equation by using the Picard-Lefshetz formula in \(l\)-adic cohomology.