Abstract
M. Duflo [5] has recently shown that the primitive spectrum of a split semisimple Lie algebra over a field of characteristic zero is just the set of annihilators of simple quotients of Verma modules. Following this a characteristic variety is defined for two-sided ideals in the enveloping algebra and used to give a new and elementary proof of Duflo's ordering principle on the fibre of primitive ideals with the same central character. The main new result of this paper (Theorem 15) exhibits a decomposition of the Weyl group into disjoint subsets (cells) so that each point in a given cell defines the same ideal (via Duflo's theorem). It is conjectured that different cells correspond to different ideals, a result which would classify the primitive spectrum.
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References
W. Borho and J.C. Jantzen, Über primitive ideale in der Einhüllenden einer halbeinfacher Lie-algebra, preprint, Bonn, 1976.
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Joseph, A. (1977). A characteristic variety for the primitive spectrum of a semisimple lie algebra. In: Carmona, J., Vergne, M. (eds) Non-Commutative Harmonic Analysis. Lecture Notes in Mathematics, vol 587. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087917
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DOI: https://doi.org/10.1007/BFb0087917
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