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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]. §1. IntroductionIn this paper we deal with the b-functions of the invariants on the flag manifolds G/P . In the case where P is a Borel subgroup, Kashiwara [3] determined the b-functions by using the universal Verma modules. For general parabolic subgroups P we show that b-functions are regarded as generators of ideals defined by universal generalized Verma modules. When the unipotent radical of P is commutative, we determine the generator.Let g be a simple Lie algebra over the complex number field C, and let G be a connected simply-connected simple algebraic group with Lie algebra g. Fix a parabolic subalgebra p of g. We denote the reductive part of p and the nilpotent part of p by l and n respectively. Let L be the subgroup of G

We show that there exist natural q-analogues of the b-functions for the prehomogeneous vector spaces of commutative parabolic type and calculate them explicitly in each case. Our method of calculating the b-functions seems to be new even for the original case q = 1. 