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

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

  1. W. Borho and J.C. Jantzen, Über primitive ideale in der Einhüllenden einer halbeinfacher Lie-algebra, preprint, Bonn, 1976.

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  2. W. Borho and H. Kraft, Über die Gelfand-Kirillov-Dimension; Math. Annalen, 22, 1976, pp. 1–24.

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  4. J. Dixmier, Algèbres enveloppantes, cahiers scientifiques, XXXVII, Gauthier-Villars, Paris, 1974.

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  5. M. Duflo, Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple, preprint, Paris, 1976.

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  6. J.C. Jantzen, Kontravariante Formen auf induzierte Darstellungen, halbeinfacher Lie-algebren, preprint, Bonn, 1975.

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© 1977 Springer-Verlag

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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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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08245-3

  • Online ISBN: 978-3-540-37365-0

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