Inverse image of D-modules and quasi-b-functions

Abstract

The usual b-function of a holonomic -module is associated to the Euler vector field but the elementary case of a ramification map shows that this Euler vector field is not preserved under inverse image. We define quasi-b-functions, that is b-functions associated to a quasi-homogeneity and use them to state an inverse image theorem for b-functions of holonomic -modules. We apply this result to an explicit calculation of the usual b-function of the Kashiwara-Hotta module on the Grothendieck's simultaneous resolution of a semi-simple Lie algebra. © 2008 Springer Japan.