Advertisement

Annals of Global Analysis and Geometry

, Volume 17, Issue 4, pp 385–396 | Cite as

The Spectrum of the Dirac Operator on G2/SO(4)

  • L. Seeger
Article

Abstract

The spectrum of the Dirac operator on the compact symmetric space G2/SO(4) is calculated using Parthasarathy's formula and classical harmonic analysis. Furthermore, the neccessary branching rules for G2 and its subgroup SO(4) are given.

branching rules Dirac operator exceptional Lie algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bär, C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces, Arch. Math. 59 (1992), 65–79.Google Scholar
  2. 2.
    Cahen, M. and Gutt, S.: Spin structures on compact simply connected Riemannian symmetric spaces, Simon Stevin Quart. J. Pure Appl. Math. 62(3–4) (1988), 209–242.Google Scholar
  3. 3.
    Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin, 1987.Google Scholar
  4. 4.
    Kramer, W., Semmelmann, U. and Weingart, G.: Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds, SFB 256, Preprint 507, Bonn, 1997.Google Scholar
  5. 5.
    Kramer, W., Semmelmann, U. and Weingart, G.: The first eigenvalue of the Dirac operator on quaternionic Kähler manifolds, Preprint 1997(90), Max-Planck-Institut für Mathematik, Bonn, 1997.Google Scholar
  6. 6.
    Lawson, H. B. and Michelsohn, M.-L.: Spin Geometry, Princeton University Press, Princeton, NJ, 1989.Google Scholar
  7. 7.
    McKay, W. G., Patera, I. and Rand, D. W.: Tables of Representations of Simple Lie Algebras, Vol. 1: Exceptional Simple Lie Algebras, Publication CRM, Université de Montréal, Canada, 1990.Google Scholar
  8. 8.
    Milhorat, J.-L.: Spectre de l'opérateur de Dirac sur les espaces projectifs quaternioniens, C.R. Acad. Sci. Paris 314 (1992), 69–72.Google Scholar
  9. 9.
    Milhorat, J.-L.: Spectrum of the Dirac operator on Gr2(ℂm+2), J. Math. Phys. 39 (1998), 594–609.Google Scholar
  10. 10.
    Parthasarathy, R.: Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1–30.Google Scholar
  11. 11.
    Seifarth, S. and Semmelmann, U.: The spectrum of the Dirac operator on the odd dimensional complex projective space ℂP 2n−1, SFB 288 Preprint No. 95, Berlin, 1993.Google Scholar
  12. 12.
    Strese, H.: Über den Dirac Operator auf Grassmann-Mannigfaltigkeiten, Math. Nachr. 98 (1980), 53–59.Google Scholar
  13. 13.
    Strese, H.: Spektren symmetrischer Räume, Math. Nachr. 98 (1980), 75–82.Google Scholar
  14. 14.
    Wallach, N.: Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, New York, 1973.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • L. Seeger
    • 1
  1. 1.Mathematisches Institut der Universität FreiburgFreiburgGermany

Personalised recommendations