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


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 


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  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

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