Skip to main content
SpringerLink
Log in

On the Vergne conjecture

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Braverman, Index theorem for equivariant Dirac operators on noncompact manifolds, \(K\)-Theory 23 (2002), 61–101.

  3. M. Braverman, The index theory on non-compact manifolds with proper group action, J. Geom. Phys. 98 (2015), 275–284.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Guillemin and S. Sternberg, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344–380.

  5. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515–538.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Guillemin, Reduced phase spaces and Riemann-Roch, In: Lie theory and geometry, Progr. Math., 123, pages 305–334, Birkhäuser Boston, Boston, MA, 1994.

  7. P. Hochs and V. Mathai, Geometric quantization and families of inner products, Adv. Math. 282 (2015), 362–426.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Hochs and Y. Song, An equivariant index for proper actions I, J. Funct. Anal., to appear, 2016. (doi:10.1016/j.jfa.2016.08.024, 44 pp.).

  9. P. Hochs and Y. Song, An equivariant index for proper actions II: properties and applications, 2016. (arXiv:1602.02836, 39 pp.).

  10. P. Hochs and Y. Song, An equivariant index for proper actions III: the invariant and discrete series indices, Differential Geom. Appl. 49 (2016), 1–22.

  11. P. Hochs and Y. Song, Equivariant indices of Spin\(^c\)-dirac operators for proper moment maps, Duke Math. J., to appear, 2016. (arXiv:1503.00801, 60 pp.).

  12. T. Kawasaki, The index of elliptic operators, over \(V\)-manifolds, Nagoya Math. J. 84 (1981), 135–157.

    Article  MathSciNet  MATH  Google Scholar 

  13. F. C. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. Math. (2) 122 (1985), 41–85.

  14. E. Meinrenken, Symplectic surgery and the Spin\(^c\)-Dirac operator, Adv. Math. 134 (1998), 240–277.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Meinrenken and R. Sjamaar, Singular reduction and quantization, Topology 38 (1999), 699–762.

  16. X. Ma and W. Zhang, Geometric quantization for proper moment maps, C. R. Math. Acad. Sci. Paris 347 (2009), 389–394.

    Article  MathSciNet  MATH  Google Scholar 

  17. X. Ma and W. Zhang, Geometric quantization for proper moment maps: the Vergne conjecture, Acta Math. 212 (2014), 11–57.

    Article  MathSciNet  MATH  Google Scholar 

  18. P.-É. Paradan, Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), 442–509.

    Article  MathSciNet  MATH  Google Scholar 

  19. P.-É. Paradan, \({\rm Spin}^c\)-quantization and the \(K\)-multiplicities of the discrete series, Ann. Sci. École Norm. Sup. (4) 36 (2003), 805–845.

  20. P.-É. Paradan, Formal geometric quantization II, Pacific J. Math. 253 (2011), 169–211.

    Article  MathSciNet  MATH  Google Scholar 

  21. P.-É. Paradan and M. Vergne, Witten non abelian localization for equivariant \(K\)-theory, and the \([Q,R]=0\) theorem, 2015. (arXiv:1504.07502, 90 pp.)

  22. Y. Tian and W. Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), 229–259.

  23. Y. Tian and W. Zhang, Quantization formula for symplectic manifolds with boundary, Geom. Funct. Anal. 9 (1999), 596–640.

  24. M. Vergne, Applications of equivariant cohomology, In: International Congress of Mathematicians Vol. I, pages 635–664, Eur. Math. Soc., Zürich, 2007.

  25. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303–368.

Download references

Acknowledgements

The authors would like to thank Maxim Braverman, Nigel Higson, Paul-Émile Paradan, Eckhard Meinrenken, and Michèle Vergne for many useful discussions. Special thanks go to Xiaonan Ma and Weiping Zhang for proposing this topic and kind help. Peter Hochs was supported by the European Union, through Marie Curie fellowship PIOF-GA-2011-299300.

Author information

Authors and Affiliations

  1. University of Adelaide, Adelaide, Australia

    Peter Hochs

  2. Dartmouth College, Hanover, NH, USA

    Yanli Song

Authors
  1. Peter Hochs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanli Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Hochs.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hochs, P., Song, Y. On the Vergne conjecture. Arch. Math. 108, 99–112 (2017). https://doi.org/10.1007/s00013-016-0997-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-016-0997-9

Mathematics Subject Classification

Keywords

Search

Navigation