Local \(b\)-functions of prehomogeneous Lagrangians (Q1321789)

A \(b\)-function \(b(s)\) is a polynomial defined for the complex power \(f(x)^s\) of a polynomial \(f(x)\). But in the microlocal point of view, the \(b\)-function should be considered on each of the irreducible simple Lagrangian subvarieties of the support of the holonomic system to \(f(x)^s\). This fact was pointed out in the hand-written manuscript on the microlocal analysis of prehomogeneous vector spaces by \textit{M. Kashiwara} et al. but it has not been published. The author of this paper proves this result by his formulation and in a different way.