Extended Central Characters and Dixmier's Map

Abstract

Let p be a parabolic subalgebra in a semisimple Lie algebra g. We study the moduleMp(λ) induced from a one-dimensional p-module of weight λ. LetU=U(g)/Ibe the quotient of the enveloping algebraU(g) by the annihilatorIof the generic module induced from p. LetZ̃denote the integral closure of the centerZofU, andŨ=Z̃U. ThenMp(λ) is not only aU-module, but even aŨ-module [1], so its central characterZ→C extends to a central characterZ̃→C. We prove that this extended central character is useful: It is unique if and only if Dixmier's map is injective. Moreover, the “modified Dixmier-map” of the author's work (1998,Abh. Math. Sem. Univ. Hamburg68, 25–44) from the sheetS/Gdetermined by p into the space XŨof minimal primitive ideals ofŨis a homeomorphismS/G→XŨgiven by O↦AnnMp(λ). As an application, we obtain the following result related to Duflo's theorem that minimal primitive ideals are induced and centrally generated: Each minimal primitive idealJofŨis induced and “almost” generated by its intersection m with the center, in the sense thatJ=mŨ. As further application of the main theorem we get results on the well-definedness of Dixmier's map.