manuscripta mathematica

, Volume 130, Issue 3, pp 295–310 | Cite as

Symplectic Dirac operators on Hermitian symmetric spaces

  • Steffen Brasch
  • Katharina Habermann
  • Lutz Habermann
Open Access


We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.

Mathematics Subject Classification (2000)

32M15 53C35 53D05 58J50 


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.
    Bär, C.: Das Spektrum von Dirac-Operatoren. Dissertation, Universität Bonn (1990)Google Scholar
  2. 2.
    Bieliavsky P.: Semisimple symplectic symmetric spaces. Geom. Dedicata 73, 245–273 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Mathematical Surveys and Monographs 67. Providence: AMS (2000)Google Scholar
  4. 4.
    Cahen M., Gutt S.: Spin structures on compact simply connected Riemannian symmetric spaces. Simon Stevin 62, 209–242 (1988)MATHMathSciNetGoogle Scholar
  5. 5.
    Cahen M., Gutt S., Rawnsley J.: Symmetric symplectic spaces with Ricci-type curvature. Conférence Moshé Flato Quantization, deformations, and symmetries (Dijon, 1999). Math. Phys. Stud. 22, 81–91 (2000)MathSciNetGoogle Scholar
  6. 6.
    Fricke J., Habermann L.: On the geometry of moduli spaces of symplectic structures. Manuscripta Math. 109, 405–417 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fricke J., Habermann K., Habermann L.: On the existence of pseudo-Riemannian metrics on the moduli space of symplectic structures. Differ. Geom. Appl. 23, 17–25 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Habermann K.: The Dirac operator on symplectic spinors. Ann. Glob. Anal. Geom. 13, 155–168 (1995)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Habermann K.: Basic properties of symplectic Dirac operators. Commun. Math. Phys. 184, 629–652 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Habermann K.: Harmonic symplectic spinors on Riemann surfaces. Manuscr. Math. 94, 465–484 (1997)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Habermann, K., Klein, A.: A Fourier transform for symplectic spinors and applications. Preprint, Universität Greifswald (2002)Google Scholar
  12. 12.
    Habermann K., Habermann L.: Introduction to Symplectic Dirac Operators. Lecture Notes in Mathematics 1887. Springer, Berlin (2006)Google Scholar
  13. 13.
    Habermann K., Habermann L., Rosenthal P.: Symplectic Yang-Mills theory, Ricci tensor, and connections. Calc. Var. Partial Differ. Equ. 30, 137–152 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kadlčáková, L.: Dirac operator in contact symplectic parabolic geometry. Proceedings of the 21st Winter School “Geometry and Physics” (Srní, 2001). Rend. Circ. Mat. Palermo (2) Suppl. 69, 97–107 (2002)Google Scholar
  15. 15.
    Kobayashi S., Nomizu K.: Foundations of differential geometry II. Interscience Publishers, New York (1969)MATHGoogle Scholar
  16. 16.
    Kostant, B.: Symplectic Spinors. Symp. math. 14, Geom. simplett., Fis. mat., Teor. geom. Integr. Var. minim., Convegni 1973, 139–152 (1974)Google Scholar
  17. 17.
    Krýsl, S.: Classification of 1st order symplectic spinor operators over contact projective geometries. arXiv:0710.1425v1 [math.DG]Google Scholar
  18. 18.
    Loos O.: Symmetric spaces I. W.A. Benjamin, Inc., New York (1969)MATHGoogle Scholar
  19. 19.
    Rudnick S.: Symplektische Dirac-Operatoren auf symmetrischen Räumen. Universität Greifswald, Diplomarbeit (2005)Google Scholar
  20. 20.
    Sommen, F.: An extension of Clifford analysis towards super-symmetry. Papers of the 5th International Conference Clifford Algebras and their Applications in Mathematical Physics (Ixtapa-Zihuatanejo, 1999). Prog. Phys. 19, 199–224 (2000)Google Scholar
  21. 21.
    Tondeur B.: Affine Zusammenhänge auf Mannigfaltigkeiten mit fastsymplektischer Struktur. Comment. Math. Helv. 13, 234–244 (1961)MathSciNetGoogle Scholar
  22. 22.
    Wallach, N.R.: Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics 19. Marcel Dekker, Inc., New York (1973)Google Scholar
  23. 23.
    Wilson P.M.H.: Some remarks on moduli of symplectic structures. Manuscripta Math. 116, 93–98 (2005)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wyss C.: Symplektische Diracoperatoren auf dem komplexen projektiven Raum. Universität Bremen, Diplomarbeit (2003)Google Scholar
  25. 25.
    Zelobenko, P.D.: Compact Lie groups and their representations. Translations of Mathematical Monographs 40. American Mathematical Society, Providence (1973)Google Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Steffen Brasch
    • 1
  • Katharina Habermann
    • 2
  • Lutz Habermann
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of GreifswaldGreifswaldGermany
  2. 2.SUB GöttingenGöttingenGermany
  3. 3.Institute of Differential GeometryUniversity of HannoverHannoverGermany

Personalised recommendations