Some results on Igusa local zeta functions associated with simple prehomogeneous vector spaces (Q1389881)

The purpose of this paper is to give an extension of Igusa's result on the Fourier transforms of relatively invariant functions on prehomogeneous vector spaces defined over the \(p\)-adic field \(K\). Igusa proved that, under a suitable condition, a \(p\)-adic zeta function \(Z_K(s)\) on a ``reduced irreducible'' regular prehomogeneous vector space \(G,\rho,V)\) has a functional equation of the form \(\widehat{Z_K(s)}=\Gamma_K(s)Z^*(s^*)\). Here, \(Z^*(s^*)\) is the \(p\)-adic zeta function on the dual prehomogeneous vector space \((G, \rho^*, V^*)\) and \(\Gamma_K(s)\) is a function determined from the \(b\)-function of the relative invariant of \((G,\rho,V)\). The author of this paper gives a similar theorem in a little extended form. He proves that the above formula of the Fourier transform is valid for some of the ``simple'' regular prehomogeneous vector spaces. In such cases, \(p\)-adic zeta functions may be multi-variable functions. Some explicit calculations are also given.