Formal Geometric Quantization III: Functoriality in the Spin c Setting



In this paper, we prove a functorial aspect of the formal geometric quantization procedure of non-compact spin-c manifolds.


Geometric quantization Dirac operator Equivariant index 

Mathematics Subject Classification (2010)

57S15 53C27 58J20 


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  1. 1.
    Atiyah, M.F.: Elliptic Operators and Compact Groups. Lecture Notes in Mathematics, vol. 401. Springer-Verlag, Berlin (1974)CrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators III. Ann. Math. 87, 546–604 (1968)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Atiyah, M.F., Hirzebruch, F.: Spin Manifold and Group Actions, Essays on Topology and Related Topics (Geneva), vol. 1969. Springer-Verlag, Berlin (1970)Google Scholar
  4. 4.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators, Grundlehren, vol. 298. Springer, Berlin (1991)MATHGoogle Scholar
  5. 5.
    Braverman, M.: Index theorem for equivariant Dirac operators on noncompact manifolds. K-Theory 27, 61–101 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Duistermaat, J.J.: The Heat Equation and the Lefschetz Fixed Point Formula for the Spinc-Dirac operator, Progress in Nonlinear Differential Equation and Their Applications, vol. 18. Birkhauser, Boston (1996)Google Scholar
  7. 7.
    Hattori, A.: spinc-structures and S 1-actions. Invent. Math. 48, 7–31 (1978)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hochs, P., Song, Y.: Equivariant indices of spinc-Dirac operators for proper moment maps. Duke Math. J. 166, 1125–1178 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kawasaki, T.: The index of elliptic operators over V-manifolds. Nagoya Math. J. 84, 135–157 (1981)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ma, X., Zhang, W.: Geometric quantization for proper moment maps: the Vergne conjecture. Acta Math. 212, 11–57 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Meinrenken, E.: Symplectic surgery and the Spinc-Dirac operator. Adv. Math. 134, 240–277 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Meinrenken, E., Sjamaar, R.: Singular reduction and quantization. Topology 38, 699–762 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Paradan, P.-E.: Localization of the Riemann-Roch character. J. Funct. Anal. 187, 442–509 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Paradan, P.-E.: Spinc-quantization and the K-multiplicities of the discrete series. Annales scientifiques de l’E.N.S. 36, 805–845 (2003)MathSciNetMATHGoogle Scholar
  15. 15.
    Paradan, P.-E.: Formal geometric quantization. Ann. Inst. Fourier 59, 199–238 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Paradan, P.-E.: Formal geometric quantization II. Pac. J. Math. 253, 169–211 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Paradan, P.-E., Vergne, M.: Witten non abelian localization for equivariant K-theory and the [Q, R] = 0 Theorem, arXiv: (2015), accepted in Memoirs of the A.M.S.
  18. 18.
    Paradan, P.-E., Vergne, M.: Admissible coadjoint orbits for compact Lie groups. Transformation Groups (2017).
  19. 19.
    Paradan, P.-E., Vergne, M.: Equivariant Dirac operators and differential geometric invariant theory. Acta Math. 218, 137–199 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Tian, Y., Zhang, W.: An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg. Invent. Math. 132, 229–259 (1998)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander GrothendieckUniversité de Montpellier, CNRSMontpellierFrance

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