Letters in Mathematical Physics

, Volume 95, Issue 1, pp 89–107 | Cite as

Dirac Induction for Loop Groups

  • Hessel Posthuma
Open Access


Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this, we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.

Mathematics Subject Classification (2000)



loop groups Dirac induction 



The author thanks the anonymous referee for valuable comments and pointing out references [15] and [21].

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Alekseev A., Meinrenken E.: The non-commutative Weil algebra. Invent. Math. 139, 135–172 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bott R.: The index theorem for homogeneous differential operators. In: Cairns, S.S. (eds) Differential and Combinatorial Topology, pp. 167–186. Princeton University Press, Princeton (1965)Google Scholar
  3. 3.
    Freed, D.S.: Twisted K-theory and loop groups. In: Proceedings of the ICM, vol. III (Beijing, 2002), pp. 419–430. Higher Ed. Press, Beijing (2002). arXiv:math.AT/0206237Google Scholar
  4. 4.
    Freed, D.S., Hopkins, M.J., Teleman, C.: Twisted K-theory and loop group representations I (2007). arXiv:math.AT/07111906Google Scholar
  5. 5.
    Freed, D.S., Hopkins, M.J., Teleman, C.: Twisted K-theory and loop group representations III (2005). arXiv:math.AT/0312155Google Scholar
  6. 6.
    Gross B., Kostant B., Ramond P., Sternberg S.: The Weyl character formula, the half-spin representations, and equal rank subgroups. Proc. Natl. Acad. Sci. USA 95, 8441–8442 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Kac V.G., Todorov I.T.: Superconformal current algebras and their unitary representations. Commun. Math. Phys. 102, 337–347 (1985)MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Kitchloo N.: Dominant K-theory and integrable highest weight representations of Kac-Moody groups. Adv. Math. 221, 1191–1226 (2009)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kostant B.: Lie algebra cohomology and the generalized Borel–Weil theorem. Ann. Math. 74, 329–387 (1961)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Kostant B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100, 447–501 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Landweber G.: Harmonic spinors on homogeneous spaces. Repr. Theory 4, 466–473 (2000)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Landweber G.: Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups. Duke Math. J. 110, 121–160 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Landweber G.: Twisted representation rings and Dirac induction. J. Pure Appl. Algebra 206, 21–54 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Meinrenken E.: On the quantization of conjugacy classes. Enseign. Math. 55(2), 33–75 (2009)MATHMathSciNetGoogle Scholar
  15. 15.
    Meinrenken, E.: The cubic Dirac operator for infinite-dimensional Lie algebras. Can. J. Math. (to appear)Google Scholar
  16. 16.
    Mickelsson J.: Twisted K theory invariants. Lett. Math. Phys. 71, 109–121 (2005)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Pressley A., Segal G.: Loop Groups. Oxford University Press, Oxford (1988)MATHGoogle Scholar
  18. 18.
    Slebarski S.: The Dirac operator on homogeneous spaces of reductive Lie groups II. Am. J. Math. 109, 499–520 (1987)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Teleman C.: Lie algebra cohomology and the fusion rules, Comm. Math. Phys. 173, 265–311 (1995)MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Teleman C.: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve. Invent. Math. 134, 1–57 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Wassermann, A.: Kac–Moody and Virasoro algebras. Lecture notes (1998). arXiv: 10041287Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations