Extended Central Characters and Dixmier's Map

✍ Scribed by Walter Borho

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 107 KB
• Volume: 213
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7666

✦ Synopsis

Let ᒍ be a parabolic subalgebra in a semisimple Lie algebra ᒄ. We study the Ž . module M induced from a one-dimensional ᒍ-module of weight . Let U s ᒍ Ž . Ž . U ᒄ rI be the quotient of the enveloping algebra U ᒄ by the annihilator I of the generic module induced from ᒍ. Let Z denote the integral closure of the center Z ˜˜Ž . w x of U, and U s ZU. Then M is not only a U-module, but even a U-module 1 , ᒍ so its central character Z ª ‫ރ‬ extends to a central character Z ª ‫.ރ‬ 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 Ž . 1998, Abh. Math. Sem. Uni¨. Hamburg 68, 25᎐44 from the sheet SrG determined by ᒍ into the space ᑲ of minimal primitive ideals of U is a homeomor-Ũ Ž . phism SrG ª ᑲ given by O O ¬ Ann M . As an application, we obtain the Ũ ᒍ following result related to Duflo's theorem that minimal primitive ideals are ĩnduced and centrally generated: Each minimal primitive ideal J of U is induced and ''almost'' generated by its intersection ᒊ with the center, in the sense that ' J s ᒊU . As further application of the main theorem we get results on the well-definedness of Dixmier's map. ᒍ UŽ ᒍ .