Minimal polynomials and annihilators of generalized Verma modules of the scalar type

Abstract

Let g be a complex reductive Lie algebra and U(g) the universal enveloping algebra of g. Associated to a faithful irreducible finite dimensional representation of g, a square matrix F with entries in U(g) naturally arises and if we consider the entries of F are elements in End(M) of a given U(g)-module M, the minimal polynomial of F is defined as the usual one for an associative algebra over the complex field. Suppose M is a generalized Verma module induced from a character of a parabolic subalgebra of g. In this paper a polynomial q(x) with the parameter of the character is constructed, which equals the minimal polynomial for the generic parameter. Then the two-sided ideal of U(g) generated by the entries of q(F) is studied. We give a sufficient condition for the parameter such that the ideal describes the difference of two left ideals related to M and the corresponding Verma module. The result has many applications. For example we can explicitly give a generator system of the annihilator of M for the generic parameter. This paper also deals with many concrete examples.