Goldie rank in the enveloping algebra of a semisimple Lie algebra, II

Abstract

Let g be a complex semisimple Lie algebra, U (g) the enveloping algebra of g and Prim U (g) the set of primitive ideals of U (g). This is the second of a two-part series in which the Goldie ranks of the primitive quotients { U (g)/ J : J ∈ Prim U (g)} are computed. Lethbe a Cartan subalgebra forgandh * the dual of h. In the first part it was shown that these ranks can be described by a family of polynomial functions onh * . Here a formula is obtained for these polynomials in terms of the multiplicities of the simple factors of the Verma modules. In particular this gives an affirmative answer to conjectures (i), (ii) of [14, 7.4] except for the choice ofΩ λ .