Advertisement

Mathematische Zeitschrift

, Volume 246, Issue 1–2, pp 125–154 | Cite as

Capelli Identities for the dual pair (O M ,Sp N )

  • Minoru ItohEmail author
Article

Abstract.

Analogues of the Capelli identity are given from the viewpoint of dual pair theory. The Capelli identity is a famous formula in the invariant theory, and can be regarded as an explicit description of a correspondence of invariant differential operators associated to the dual pair (GL r ,GL s ). The main results are Capelli type identities in this sense for the dual pair (O M ,Sp N ).

Keywords

Differential Operator Invariant Theory Type Identity Dual Pair Explicit Description 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Capelli, A.: Über die Zurückführung der Cayley’schen Operation Ω auf gewöhnliche Polar-Operationen. Math. Ann. 29, 331–338 (1887)Google Scholar
  2. 2.
    Capelli, A.: Sur les opérations dans la théorie des formes algébriques. Math. Ann. 37, 1–37 (1890)zbMATHGoogle Scholar
  3. 3.
    Gould, M.D.: Characteristic identities for semisimple Lie algebras. J. Aust. Math. Soc. Ser. B 26, 257–283 (1985)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Howe, R.: θ-series and invariant theory. In: Automorphic Forms, Representations, and L-Functions, Proc. Symp. Pure Math. Am. Math. Soc., Corvallis/ Oregon 1977, Proc. Symp. Pure Math. 33, 1, 1979, pp. 275–285Google Scholar
  5. 5.
    Howe, R.: Remarks on classical invariant theory. Trans. Am. Math. Soc. 313, 539–570 (1989); Erratum. Trans. Am. Math. Soc. 318, 823 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Howe, R.: Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In: the Schur Lectures (1992), Piatetski-Shapiro, Ilya, et al. (eds.), Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 8, 1995Google Scholar
  7. 7.
    Howe, R., Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions. Math. Ann. 290, 565–619 (1991)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Itoh, M.: Explicit Newton’s formulas for . J. Algebra 208, 687–697 (1998)Google Scholar
  9. 9.
    Itoh, M.: Capelli elements for the orthogonal Lie algebras. J. Lie Theory 10, 463–489 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Itoh, M.: A Cayley-Hamilton theorem for the skew Capelli elements. J. Algebra 242, 740–761 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Itoh, M.: Correspondences of the Gelfand invariants in the reductive dual pairs. To appear in J. Aust. Math. Soc.Google Scholar
  12. 12.
    Itoh, M., Umeda,T.: On central elements in the universal enveloping algebras of the orthogonal Lie algebras. Compos. Math. 127 333–359 (2001)Google Scholar
  13. 13.
    Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representation and harmonic polynomial. Invent. Math. 44, 1–47 (1978)zbMATHGoogle Scholar
  14. 14.
    Mœ glin, C., Vignéras, M.-F., Waldspurger, J.-L.: Correspondances de Howe sur un corps p-adique. Lecture Notes in Mathematics, 1291, Springer-Verlag, 1987Google Scholar
  15. 15.
    Molev, A.: Sklyanin determinant, Laplace operators, and characteristic identities for classical Lie algebras. J. Math. Phys. 36, 923–943 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Molev, A., Nazarov, M.: Capelli identities for classical Lie algebras. Math. Ann. 313, 315–357 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Nazarov, M.: Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys. 21, 123–131 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Noumi, M., Umeda, T., Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculus on GL q(n). Duke Math. J. 76, 567–594 (1994)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ochiai, G.: A Capelli type identity associated with the dual pair . Master’s thesis at Kyoto University, 1996 Feb.Google Scholar
  20. 20.
    Umeda, T.: The Capelli identities, a century after. Am. Math. Soc, Transl., 2. Ser. 183, 51–78 (1998); translation from Sugaku 46(3), 206–227 (1994)Google Scholar
  21. 21.
    Umeda, T.: On the proof of the Capelli identities. Preprint, 1997Google Scholar
  22. 22.
    Umeda, T.: Newton’s formula for . Proc. Am. Math. Soc. 126, 3169–3175 (1998)Google Scholar
  23. 23.
    Umeda, T.: On Turnbull identity for skew symmetric matrices. Proc. Edinb. Math. Soc., II. Ser. 43, 379–393 (2000)Google Scholar
  24. 24.
    Želobenko, D.P.: Compact Lie Groups and their Representations. Translations of Mathematical Monographs. Vol. 40. Providence, R.I.: American Mathematical Society, 1973Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

Personalised recommendations