Abstract :
Let be a parabolic subalgebra in a semisimple Lie algebra . We study the moduleM (λ) induced from a one-dimensional -module of weight λ. LetU = U( )/Ibe the quotient of the enveloping algebraU( ) by the annihilatorIof the generic module induced from . Let denote the integral closure of the centerZofU, and = U. ThenM (λ) is not only aU-module, but even a -module [[1]], so its central characterZ → extends to a central character → . We prove that this extended central character is useful: It is unique if and only if Dixmierʹs map is injective. Moreover, the “modified Dixmier-map” of the authorʹs work (1998,Abh. Math. Sem. Univ. Hamburg68, 25–44) from the sheetS/Gdetermined by into the space of minimal primitive ideals of is a homeomorphismS/G → given by Ann M (λ). As an application, we obtain the following result related to Dufloʹs theorem that minimal primitive ideals are induced and centrally generated: Each minimal primitive idealJof is induced and “almost” generated by its intersection with the center, in the sense that . As further application of the main theorem we get results on the well-definedness of Dixmierʹs map.
