Jantzen coefficients and simplicity of parabolic Verma modules

Abstract

We systematically study the Jantzen coefficients. They share some properties with the leading coefficients of Kazhdan-Lusztig polynomials (the μ-functions) and are relatively easy to calculate. We develop a reduction process which reduces the computation of the Jantzen coefficients to the case of basic parabolic Verma modules. Through a classification of such modules, all the Jantzen coefficients for classical Lie algebras are obtained up to a sign. As an application of our results, we give refinements of Jantzen's simplicity criteria.