Unitarity in “Quantization Commutes with Reduction”


Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.

Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M // G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M // G.

This is a preview of subscription content, access via your institution.


  1. ADPW91

    Axelrod S., Della Pietra S. and Witten E. (1991). Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33: 787–902

    MATH  Google Scholar 

  2. BSZ92

    Baez J.C., Segal I.E. and Zhou Z.-F. (1992). Introduction to algebraic and constructive quantum field theory. Princeton Series in Physics. Princeton University Press, Princeton, NJ

    Google Scholar 

  3. BaH04

    Banyaga A. and Hurtubise D.E. (2004). A proof of the Morse–Bott lemma. Exp. Math. 22(4): 365–373

    MATH  Google Scholar 

  4. BlH75

    Bleistein N. and Handelsman R.A. (1975). Asymptotic Expansions of Integrals. Dover Inc., New York

    MATH  Google Scholar 

  5. BPU95

    Borthwick D., Paul T. and Uribe A. (1995). Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122(2): 359–402

    MATH  Article  ADS  Google Scholar 

  6. BU96

    Borthwick D. and Uribe A. (1996). Almost complex strucutres and geometric quantization. Math. Res. Lett. 3(6): 845–861

    MATH  Google Scholar 

  7. BU00

    Borthwick D. and Uribe A. (2000). Nearly Kählerian embeddings of symplectic manifolds. Asian J. Math. 4(3): 599–620

    MATH  Google Scholar 

  8. BdMG81

    Boutetde Monvel L. and Guillemin V. (1981). The Spectral Theory of Toeplitz Operators, Vol. 99 Ann. of Math. Studies. Princeton University Press, Princeton, NJ

    Google Scholar 

  9. Char06a

    Charles L. (2007). Semi-classical properties of geometric quantization with metaplectic correction. Comm. Math. Phys. 270(2): 445–480

    MATH  Article  ADS  Google Scholar 

  10. Char06b

    Charles L. (2006). Toeplitz operators and Hamiltonian torus actions. J. Func. Anal. 236(1): 299–350

    MATH  Article  Google Scholar 

  11. Chav06

    Chavel I. (2006). Riemannian Geometry: A Modern Introduction, Second Edition, Vol. 98 Cambridge studies in advanced mathematics. Cambridge University Press, New York

    Google Scholar 

  12. Czy78

    Czyz J. (1978). On some approach to geometric quantization. Diff. Geom. Methods in Math. Phys. 676: 315–328

    Article  Google Scholar 

  13. Dir64

    Dirac P.A.M. (1964). Lectures on Quantum Mechanics. Yeshiva University, New York

    Google Scholar 

  14. Don04

    Donaldson, S.K.: Remarks on gauge theory, complex geometry and 4-manifold topology. In: S.M. Atiyah, D. Iagolnitzer, editors, Fields Medallists’ Lectures, 2nd Edition, no. 9 in World Scientific Series in 20th Century Mathematics, 2004

  15. DH99

    Driver B.K. and Hall B.C. (1999). Yang–Mills theory and the Segal–Bargmann transform. Commun. Math. Phys. 201: 249–290

    MATH  Article  ADS  Google Scholar 

  16. DH00

    Driver, B.K., Hall, B.C.: The energy representation has no non-zero fixed vectors. In: Stochastic processes, physics and geometry: new interplays, II, number 29 in Conference Proceedings, Providence, RI: Amer. Math. Soc. 2000

  17. DK00

    Duistermaat J. and Kolk J. (2000). Lie Groups. Springer-Verlag, Berlin

    MATH  Google Scholar 

  18. FJ78

    Flensted-Jensen M. (1978). Spherical functions of a real semisimple Lie group. A method of reduction to the complex case. J. Func. Anal. 30(1): 106–146

    MATH  Article  Google Scholar 

  19. FMMN03

    Florentino C., Matias P., Mourão J. and Nunes J.P. (2003). Coherent state transforms and vector bundles on elliptic curves. J. Func. Anal. 204(2): 355–398

    MATH  Article  Google Scholar 

  20. Flu98

    Flude, J.P.M.: Geometric asymptotics of spin. Thesis, U. Nottingham, UK, 1998

  21. GLP99

    Gilkey P.B., Leahy J.V. and Park J. (1999). Spectral geometry, Riemannian submersions, and the Gromov–Lawson conjecture. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL

    Google Scholar 

  22. Got86

    Gotay M.J. (1986). Constraints, reduction and quantization. J. Math. Phys. 27(8): 2051–2066

    MATH  Article  ADS  Google Scholar 

  23. GH78

    Griffiths P. and Harris J. (1978). Principles of Algebraic Geometry. John Wiley & Sons, New York

    MATH  Google Scholar 

  24. GS82

    Guillemin V. and Sternberg S. (1982). Geometric Quantization and Multiplicities of Group Representations. Invent. Math. 67: 515–538

    MATH  Article  ADS  Google Scholar 

  25. Hal01

    Hall B.C. (2001). Coherent states and the quantization of (1 + 1)-dimensional Yang–Mills theory. Rev. Math. Phys. 13(10): 1281–1305

    MATH  Article  Google Scholar 

  26. Hal02

    Hall B.C. (2002). Geometric quantization and the generalized segal–bargmann transform for lie groups of compact type. Commun. Math. Phys. 226: 233–268

    MATH  Article  ADS  Google Scholar 

  27. HHL94

    Heinzner P., Huckleberry A. and Loose F. (1994). Kählerian extensions of the symplectic reduction. J. reine angew. Math. 455: 123–140

    MATH  Google Scholar 

  28. Hue06

    Huebschmann J. (2006). Kähler quantization and reduction. J. reine angew. Math. 591: 75–109

    MATH  Google Scholar 

  29. JK97

    Jeffrey L.C. and Kirwan F.C. (1997). Localization and the quantization conjecture. Topology 36(3): 647–693

    MATH  Article  Google Scholar 

  30. KN79

    Kempf, G., Ness, L.: The length of vectors in representation spaces. In: Algebraic Geometry (Proceedings of the Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), no. 732 Lecture Notes in Math., Berlin: Springer, 1979, pp 233–243

  31. Kna02

    Knapp A. (2002). Lie Groups: Beyond an Introduction, 2nd Edition, Vol. 140 Progress in Mathematics. Basel-Boston, Birkhäuser

    Google Scholar 

  32. Lan95

    Landsman N. (1995). Rieffel induction as generalized quantum Marsden–Weinstein reduction. J. Geom. Phys. 15(4): 285–319

    MATH  Article  Google Scholar 

  33. LW97

    Landsman N. and Wren K. (1997). Constrained quantization and θ-angles. Nuc. Phys. B 502(3): 537–560

    MATH  Article  ADS  Google Scholar 

  34. MM04

    Ma X. and Marinescu G. (2004). Generalized Bergman kernels on symplectic manifolds. C. R. Acad. Sci. Paris, Ser. I 339(7): 493–498, Full Version: http://arxiv.org/list/math.DG/0411559, 2004

    MATH  Google Scholar 

  35. MZ05

    Ma X. and Zhang W. (2005). Bergman kernels and symplectic reduction. C. R. Acad. Sci. Paris, Ser. I 341: 297–302

    MATH  Google Scholar 

  36. MZ06

    Ma, X., Zhang, W.: Bergman kernels and symplectic reduction. http://arxiv.org/list/math.DG/0607605, 2006

  37. MW74

    Marsden J. and Weinstein A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1): 121–130

    MATH  Article  Google Scholar 

  38. Mei98

    Meinrenken E. (1998). Symplectic Surgery and the Spinc-Dirac Operator. Adv. Math. 134(2): 240–277

    MATH  Article  Google Scholar 

  39. MFK94

    Mumford D., Fogarty J. and Kirwan F. (1994). Geometric Invariant Theory, Third Edition, Volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin

    Google Scholar 

  40. Pao05

    Paoletti R. (2005). The Szëgo Kernel of a symplectic quotient. Adv. Math. 197(2): 523–553

    MATH  Article  Google Scholar 

  41. Sja95

    Sjamaar R. (1995). Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. (2) 141(1): 87–129

    MATH  Article  Google Scholar 

  42. Sja96

    Sjamaar R. (1996). Symplectic reduction and Riemann–Roch formulas for multiplicities. Bull. Amer. Math. Soc. (N.S.) 33(3): 327–388

    MATH  Article  Google Scholar 

  43. Ste99

    Stenzel M. (1999). The Segal–Bargmann transform on a symmetric space of compact type. J. Func. Anal. 165: 44–58

    MATH  Article  Google Scholar 

  44. TZ98

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

    MATH  Article  Google Scholar 

  45. Woo91

    Woodhouse N.M.J. (1991). Geometric Quantization. Oxford University Press, Inc., New York

    Google Scholar 

  46. Wre98

    Wren K. (1998). Constrained quantization and θ-angles. II. Nuc. Phys. B 521(3): 471–502

    MATH  Article  ADS  Google Scholar 

  47. Zhe00

    Zheng, F.: Complex Differential Geometry, Volume 18 of Studies in Advanced Mathematics. Providence, RI: Amer. Math. Soc./ Int. Press, 2000

Download references

Author information



Corresponding author

Correspondence to Brian C. Hall.

Additional information

Supported in part by NSF Grant DMS-02000649.

Communicated by J.Z. Imbrie

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hall, B.C., Kirwin, W.D. Unitarity in “Quantization Commutes with Reduction”. Commun. Math. Phys. 275, 401–442 (2007). https://doi.org/10.1007/s00220-007-0303-6

Download citation


  • Line Bundle
  • Toeplitz Operator
  • Holomorphic Section
  • Bergman Kernel
  • Geometric Quantization