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Kostant's problem and goldie rank

Part of the Lecture Notes in Mathematics book series (LNM,volume 880)

Abstract

Although Kostant's problem was answered negatively by Conze-Berline and Duflo ([5], 6.5) its further investigation is still of considerable interest. This is because the fact that it holds for certain classes of modules and the determination of the precise extent to which it fails for others gives an enormous amount of information on primitive ideals and in particular on the Goldie ranks of primitive quotients. This paper does not review this development in any detail; but concentrates on four examples which form part of the ongoing investigation. Two of these were discussed at the meeting, the third which involves only Goldie rang was developed shortly afterwards following discussions with R.Howe. The fourth example is an application of the Enright functor. The first example shows that for Whittaker modules, Kostant's problem has a negative answer even in type AI, whereas for highest weight modules a negative answer first arises in type B2 (or type C2) as reported by Conze-Berline and Duflo. The second example makes precise this failure in type G2 (and in type B2) and shows that it is an inevitable consequence of a constraint imposed by an additivity principle for Goldie rank. This result perceived in ([15], Sect, 9) has since undergone a considerable extension [18]. The third example shows that in type G2 there is at least one completely prime primitive ideal whose associated variety is the Zariski closure of the unique eight dimensional orbit (in the dual of the Lie algebra). This is part of a general program [18] to determine the scale factors in the Goldie rank polynomials ([17], 1.4). It may also be considered to be part of a program to parametrize the completely prime primitive ideals through the orbit space. In the example the Goldie ranks of all the primitive ideals associated with this eight dimensional orbit can be computed and one finds that there are exactly two which are completely prime. This provides the first unequivocal counterexample to the existence of a natural bijection between these two sets.

Keywords

  • Verma Module
  • Poisson Algebra
  • Zariski Closure
  • Primitive Ideal
  • High Weight Module

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. I.N. Bernstein and S.I. Gelfand: Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compos. Math. 41 (1980).

    Google Scholar 

  2. W. Borho: Über Schichten halbeinfacher Lie-Algebren, Preprint, Wuppertal, 1979.

    Google Scholar 

  3. W. Borho and J.C. Jantzen: Über primitive Ideals in der Einhüllenden einer halbeinfacher Lie-Algebren, Invent. Math. 39 (1977), 1–53.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. N. Conze: Algèbres d'opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France, 102 (1974), 379–415.

    MathSciNet  MATH  Google Scholar 

  5. N. Conze-Berline and M. Duflo: Sur les représentations induites des groupes semi-simples complexes, Compos. Math., 34 (1977), 307–336.

    MathSciNet  MATH  Google Scholar 

  6. V. V. Deodhar: On a construction of representations and a problem of Enright, Invent. Math., 57 (1980), 101–118.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. J. Dixmier: Algèbres enveloppantes, Cahiers Scientifiques, XXXVII, Gauthier-Villars, Paris, 1974.

    MATH  Google Scholar 

  8. M. Duflo: Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple, Ann. Math., 105, (1977), 107–130.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. T.J. Enright: On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. Math., 110, (1979), 1–82.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. O. Gabber and A. Joseph: On the Bernstein-Gelfand-Gelfand resolution and the Duflo sum formula, Compos. Math. (In the press).

    Google Scholar 

  11. J.C. Jantzen: Moduln mit einen höchsten Gewicht, L.N. 750, Springer-Verlag, Berlin/Heidelberg/New-York, 1979.

    MATH  Google Scholar 

  12. A. Joseph: The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Ec. Norm. Sup., 9 (1976), 1–30.

    MathSciNet  MATH  Google Scholar 

  13. A. Joseph: A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Alg., 48, (1977), 241–289.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. A. Joseph: On the annihilators of simple subquotients of the principal series, Ann. Ec. Norm. Sup., 10 (1977), 419–440.

    MathSciNet  MATH  Google Scholar 

  15. A. Joseph: Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math., 56 (1980), 191–213.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. A. Joseph: Goldie rank in the enveloping algebra of a semisimple Lie algebra, I, J. Alg. 66 (1980).

    Google Scholar 

  17. A. Joseph: Goldie rank in the enveloping algebra of a semisimple Lie algebra, II, J. Alg. 66 (1980)

    Google Scholar 

  18. A. Joseph: Goldie rank in the enveloping algebra of a semisimple Lie algebra, III, to appear.

    Google Scholar 

  19. A. Joseph: Dixmier's problem for Verma and principal series submodules, J. Lond. Math. Soc., 20 (1979), 193–204.

    CrossRef  MATH  Google Scholar 

  20. B. Kostant: On Whittaker vectors and representation theory, Invent. Math., 48 (1978), 101–184.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. J.P. Serre: Algèbres de Lie semi-simples complexes, W.A. Benjamin, Amsterdam/New-York, 1966.

    MATH  Google Scholar 

  22. P. Tauvel: Sur les representations des algèbres de Lie nilpotentes, C.R. Acad. Sci., A278 (1974), 977–979.

    MathSciNet  MATH  Google Scholar 

  23. D. Vogan: A generalized τ-invariant for the primitive spectrum of a semisimple Lie algebra, Math. Ann., 242 (1979), 209–224.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1981 Springer-Verlag

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Joseph, A. (1981). Kostant's problem and goldie rank. In: Carmona, J., Vergne, M. (eds) Non Commutative Harmonic Analysis and Lie Groups. Lecture Notes in Mathematics, vol 880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090412

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  • DOI: https://doi.org/10.1007/BFb0090412

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