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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 Z denote its normalization, and let A = AZ 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  =  s l n, we have that each minimal primitive ideal of A is generated by its intersection with the center Z. Duflo's theorem is the special case where q is a Borel subalgebra (then I = 0, Z = Z, and A = 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 A and Z fails for g  =  s l 3.