Skip to main content
SpringerLink
Log in

A direct proof of a generalized harish-chandra isomorphism

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

A direct proof of a Harish-Chandra isomorphism recently established by Khoroshkin, Nazarov and Vinberg [10] involving Zhelobenko operators, is given. A key point is the computation of certain determinants analogous to those of Parthasarathy, Ranga Rao, Varadaragan [14]. This analysis avoids the passage to graded objects and the subsequent geometric arguments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Balagovic, Chevalley restriction theorem for vector-valued functions on a quantum group, arXiv:math/1004.0371.

  2. W. Borho, R. Rentschler, Oresche Teilmengen in Einhüllenden Algebren, Math. Ann. 217 (1975), 201–210.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Joseph, Enveloping algebras: problems old and new. in: Lie Theory and Geometry, Progress in Mathematics, Vol. 123, Birkhäuser, Boston, MA, 1994, pp. 385–413.

  4. A. Joseph, Sur l’annulateur d’un module de Verma, with an outline of the annihilation theorem by M. Gorelik and E. Lanzmann, in: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, (Montreal, PQ, 1997), Kluwer, Dordrecht, 1998, pp. 237–300.

  5. A. Joseph, An algebraic slice in the coadjoint action of the Borel and the Coxeter element, Adv. Math. 227 (2011), 522–585.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Joseph, G. Letzter, Evaluation of the qunatum affine PRV determinants, Math. Research Lett. 9 (2002), 307–322.

    MathSciNet  MATH  Google Scholar 

  7. S. Khoroshkin, M. Nazarov, Yangians and Mickelsson algebras I, Transform. Groups 11 (2006), no. 4, 625–658.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Khoroshkin, M. Nazarov, Mickelsson algebras and representations of Yangians, arXiv:math/0912.1101.

  9. S. Khoroshkin, O. Ogievetsky, Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113–2165.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Khoroshkin, M. Nazarov, E. Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), 1168–1180.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Moeglin, Idéaux bilatères des algèbres enveloppantes, Bull. Soc. Math. France 108 (1980), no. 2, 143–186.

    MathSciNet  MATH  Google Scholar 

  13. S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Mathematics, Vol. 818, Springer, Berlin, 1980.

    Google Scholar 

  14. K. P. Parthasarathy, R. Ranga Rao, V. S. Varadarajan, Representations of complex semisimple Lie groups and Lie algebras, Ann. Math. 85 (1967), 383–429.

    Article  MathSciNet  MATH  Google Scholar 

  15. Д. П. Желобенко, Экстремальные коциклы на группах Вейля., Функц. анализ и его прил. 21 (1987), no. 3, 11–21. Engl. transl.: D. Zhelobenko, Extremal cocycles on Weyl groups, Funct. Anal. Appl. 21 (1987), 183–192.

Download references

Author information

Authors and Affiliations

  1. Donald Frey Professional Chair, Department of Mathematics, Weizmann Institute of Science, 2 Herzl Street, Rehovot, 76100, Israel

    ANTHONY JOSEPH

Authors
  1. ANTHONY JOSEPH
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ANTHONY JOSEPH.

Additional information

Work supported in part by Israel Science Foundation Grant, no. 710724.

Rights and permissions

Reprints and permissions

About this article

Cite this article

JOSEPH, A. A direct proof of a generalized harish-chandra isomorphism. Transformation Groups 17, 513–521 (2012). https://doi.org/10.1007/s00031-012-9173-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-012-9173-x

Keywords

Search

Navigation