The Surjectivity Theorem, Characteristic Polynomials and Induced Ideals

Abstract

1.1. Let 9 be a complex semisimple Lie algebra and U(g) its enveloping algebra. One has now a very explicit description of th~ set Prim U (g) of primitive ideals of U(g) and one could hope to say how the subs;t Prime U(g) of completely prime, primitive ideals lies in Prim U(g). When 9 is of type An (or even just has An factors) a theorem of C-:-Moeglin [26] asserts that if I E Prime U(g) then I is induced from a character of a parabolic subalgebra of 9.: (The converse assertion which holds for all 9 is an easier and older result due to N. Conze [8], 3.1). Although this is very satisfactory it does not merge well with the Goldie rank picture. Thus one cannot say which coherent families of primitive ideals have a completely prime member equivalently which Goldie rank polynomials take the value 1. Conversely in type An the Goldie ranks are completely known at least implicitly; but this does not allow us to extract Moeglin's theorem.