The $b$-functions for Prehomogeneous Vector Spaces of Commutative Parabolic Type and Universal Generalized Verma Modules

Abstract

We shall give a new elementary proof of the uniform expression for the b -functions of prehomogeneous vector spaces of commutative parabolic type obtained by Muller, Rubenthaler and Schiffmann [5] by using micro-local analysis. Our method is similar to Kashiwara’s approach using the universal Verma modules. We shall also give a new proof for the criterion of the irreducibility of the generalized Verma module in terms of b -functions due to Suga [10], Gyoja [1], Wachi [13].