On Duflo's Theorem That Minimal Primitive Ideals Are Centrally Generated

Abstract

Let g be a complex semisimple Lie algebra. Duflo's theorem states that each minimal primitive ideal of the enveloping algebra U(g) is generated by its intersection with the center. We prove the following generalization. For the “relative enveloping algebra” A=U(g)/I relative to a parabolic subalgebra q of g, where I denotes the annihilator of the induced module U(g)⊗U([q,q])C, let Z denote the center of A, let Z̃ denote its normalization, and let Ã=AZ̃ be the slight extension of A obtained by normalizing the center. We present a theorem that states that under certain conditions, which are always satisfied if g=sln, we have that each minimal primitive ideal of à is generated by its intersection with the center Z̃. Duflo's theorem is the special case where q is a Borel subalgebra (then I=0, Z̃=Z, and Ã=A=U(g)). Note that the normalization of the center is really necessary for the theorem: The corresponding statement for A and Z instead of à and Z̃ fails for g=sl3.