Sheets and Topology of Primitive Spectra for Semisimple Lie Algebras
✍ Scribed by Walter Borho; Anthony Joseph
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 579 KB
- Volume
- 244
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Let ᒄ be a semisimple Lie algebra. Consider the set X X of primitive ideals of the Ž . enveloping algebra U ᒄ , given the Jacobson topology. A basic open problem is to describe X X as a countable union of algebraic varieties V V with strata V V as large as possible. Towards this aim the notion of a sheet in X X is introduced here in analogy with the notion of sheet in the adjoint orbit space ᒄrG. The sheets in X X are classified and given a purely topological description. Moreover Goldie-rank is constant and maximal on a dense open subset of a sheet. For every positive integer n it is shown that there are only finitely many sheets in X X with maximal Goldie-rank n. For n s 1, which corresponds to completely prime ideals, an Ž . extension of the Dixmier-map from some adjoint orbits to primitive ideals is introduced with the aim of showing that the sheets with n s 1 are homeomorphic to sheets in ᒄrG and hence explicitly described as algebraic varieties. This goal is partly though not fully achieved. The proofs of the above results require a detailed knowledge of Goldie-rank polynomials and even a new difficult result concerning the support of their restrictions to the walls. An extension of a theorem of Soergel on the topology of X X is also required, and for this a new approach is given, which is both simpler and more general.