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

Abstract

Let g be a complex semisimple Lie algebra,U (g) the enveloping algebra ofg, and PrimU (g) the set of primitive ideals ofU (g). In this two-part series the Goldie ranks of the primitive quotients {U(g)/J :J ∈ PrimU(g)} are computed. Lethbe a Cartan subalgebra for g and h* the dual of h. After Duflo [4] there exists a surjection λ ↦ J(λ) of h* onto PrimU(g). In the first part it is shown thath* can be written as a disjoint union of infinite subsets with the following property. On each subset Λ there exists a polynomialpΛ such that for allλ ∈ Λ, pΛ(λ) is the Goldie rank ofU(g). In Part II the computation of thepΛ is reduced to a knowledge of the multiplicities of the simple factors of the Verma modules.