On the Poles of p-adic Complex Powers and the b-Functions of Prehomogeneous Vector Spaces

Abstract

Introduction. In the real case, it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b-functions (cf. [B]). Many examples suggest that a similar relation between them would exist also in the p-adic case. Recently the case of polynomials over a p-adic field in two variables has been closely examined (cf. F. Loeser [L], W. Veys [V]). In this paper we shall deal with the case of relative invariants of prehomogeneous vector spaces over a p-adic field. Let us explain our problem more precisely. Let K be a p-adic field, i.e., a finite algebraic extension of Qp. We denote by ?)K the maximal compact subring of K, by rrT)K the ideal of nonunits of )K, and put q = #(DKI/'XC)K). We denote by | IKthe absolute value on K normalized as ITWK = q-. Let fQ(KX) be the set of quasi-characters of KX. Then the identity component fl(KX)O of fl(KX) consists of w,(t) = Itls for all s in C and fl(Kx)o CX under ws q-S. Let (G, p, V) be a prehomogeneous vector space (abbrev. P.V.) defined over K, where we assume that