Advertisement

Functional Analysis and Its Applications

, Volume 42, Issue 4, pp 308–316 | Cite as

Family algebras and generalized exponents for polyvector representations of simple Lie algebras of type B n

  • A. A. Kirillov
Article

Abstract

We give an explicit formula for the exterior powers ∧ k π 1 of the defining representation π 1 of the simple Lie algebra ςο(2n + 1, ℂ). We use the technique of family algebras. All representations in question are children of the spinor representation σ of g2ο(2n + 1, ℂ). We also give a survey of main results on family algebras.

Key words

family algebra generalized exponent representation of Lie algebra spinor representation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. B. Dynkin, “Some properties of the weight systems of linear representations of semisimple Lie groups,” Dokl. Akad. Nauk SSSR, 71:2 (1950), 221–224.MATHMathSciNetGoogle Scholar
  2. [2]
    R. K. Brylinski (Gupta), “Limits of weight spaces, Lusztig’s q-analogs, and fibering of adjoint orbits,” J. Amer. Math. Soc., 2:3 (1989), 517–533.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. K. Gupta, “Generalized exponents via Hall-Littlewood symmetric functions,” Bull. Amer. Math. Soc., 16:2 (1987), 287–291.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    W. Hesselink, “Characters of nullcone,” Math. Ann., 252:3 (1980), 179–182.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. A. Kirillov, “Family algebras,” Electron. Res. Announc. Amer. Math. Soc., 6:1 (2000), 7–20 (electronic).MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    A. A. Kirillov, “Introduction to family algebras,” Moscow Math. J., 1:1 (2001), 49–63.MATHMathSciNetGoogle Scholar
  7. [7]
    B. Kostant, “Lie group representations on polynomial rings,” Amer. J. Math., 85 (1963), 327–404.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Lusztig, “Singularities, character formulas, and a q-analogue of Kostant’s weight multiplicity formula,” in: Analyse et Topologie sur les Espaces Singuliers (II–III), Asterisque, vol. 101–102, Soc. Math. France, Paris, 1983, 208–227.Google Scholar
  9. [9]
    D. Panyushev, “Weight multiplicity free representations, g-endomorphism algebras and Dynkin polynomials,” J. London Math. Soc., 69:2 (2004), 273–290.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations