On Dufloʹs Theorem That Minimal Primitive Ideals Are Centrally Generated

Let be a complex semisimple Lie algebra. Dufloʹs theorem states that each minimal primitive ideal of the enveloping algebra U( ) is generated by its intersection with the center. We prove the following generalization. For the “relative enveloping algebra” A = U( )/I relative to a parabolic subalgebra of , where I denotes the annihilator of the induced module U( ) U([ , ]) , let Z denote the center of A, let denote its normalization, and let à = A 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 = n, we have that each minimal primitive ideal of à is generated by its intersection with the center . Dufloʹs theorem is the special case where is a Borel subalgebra (then I = 0, = Z, and à = A = U( )). Note that the normalization of the center is really necessary for the theorem: The corresponding statement for A and Z instead of à and fails for = 3.