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A characteristic variety for the primitive spectrum of a semisimple lie algebra

  • A. Joseph
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 587)

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.

Keywords

Weyl Group Minimal Element Characteristic Zero Verma Module Nilpotent Orbit 
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References

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    W. Borho and J.C. Jantzen, Über primitive ideale in der Einhüllenden einer halbeinfacher Lie-algebra, preprint, Bonn, 1976.Google Scholar
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    W. Borho and H. Kraft, Über die Gelfand-Kirillov-Dimension; Math. Annalen, 22, 1976, pp. 1–24.MathSciNetCrossRefzbMATHGoogle Scholar
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    R.W. Carter, Simple groups of Lie type, Monographs in pure and applied mathematics, XXVIII, John Wiley, London, 1972.Google Scholar
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    J. Dixmier, Algèbres enveloppantes, cahiers scientifiques, XXXVII, Gauthier-Villars, Paris, 1974.Google Scholar
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    M. Duflo, Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple, preprint, Paris, 1976.Google Scholar
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    J.C. Jantzen, Kontravariante Formen auf induzierte Darstellungen, halbeinfacher Lie-algebren, preprint, Bonn, 1975.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • A. Joseph
    • 1
  1. 1.Department of MathematicsTel-Aviv UniversityRamat-AvivIsrael

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