Sheets and Topology of Primitive Spectra for Semisimple Lie Algebras,

Let be a semisimple Lie algebra. Consider the set of primitive ideals of the enveloping algebra U( ), given the Jacobson topology. A basic open problem is to describe as a countable union of algebraic varieties with strata as large as possible. Towards this aim the notion of a sheet in is introduced here in analogy with the notion of sheet in the adjoint orbit space /G. The sheets in 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 with maximal Goldie-rank n. For n = 1, which corresponds to completely prime ideals, an extension of the Dixmier-map (from some adjoint orbits to primitive ideals) is introduced with the aim of showing that the sheets with n = 1 are homeomorphic to sheets in /G 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 is also required, and for this a new approach is given, which is both simpler and more general.