Differential Geometry
Geometric quantization for proper moment maps
[Quantification géométrique pour les applications moment propres]
Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 389-394.

Nous établissons une formule de quantification géométrique pour les actions hamiltoniennes d'un groupe de Lie compact agissant sur une variété symplectique non-compacte dont l'application moment est propre. En particulier, nous résolvons une conjecture formulée par Michèle Vergne dans son exposé à l'ICM 2006.

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we give a solution to a conjecture of Michèle Vergne.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.003
Ma, Xiaonan 1 ; Zhang, Weiping 2

1 Université Denis-Diderot – Paris 7, UFR de mathématiques, case 7012, site Chevaleret, 75205 Paris cedex 13, France
2 Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China
@article{CRMATH_2009__347_7-8_389_0,
     author = {Ma, Xiaonan and Zhang, Weiping},
     title = {Geometric quantization for proper moment maps},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {389--394},
     publisher = {Elsevier},
     volume = {347},
     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.003},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2009.02.003/}
}
TY  - JOUR
AU  - Ma, Xiaonan
AU  - Zhang, Weiping
TI  - Geometric quantization for proper moment maps
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 389
EP  - 394
VL  - 347
IS  - 7-8
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.crma.2009.02.003/
DO  - 10.1016/j.crma.2009.02.003
LA  - en
ID  - CRMATH_2009__347_7-8_389_0
ER  - 
%0 Journal Article
%A Ma, Xiaonan
%A Zhang, Weiping
%T Geometric quantization for proper moment maps
%J Comptes Rendus. Mathématique
%D 2009
%P 389-394
%V 347
%N 7-8
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.crma.2009.02.003/
%R 10.1016/j.crma.2009.02.003
%G en
%F CRMATH_2009__347_7-8_389_0
Ma, Xiaonan; Zhang, Weiping. Geometric quantization for proper moment maps. Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 389-394. doi : 10.1016/j.crma.2009.02.003. https://www.numdam.org/articles/10.1016/j.crma.2009.02.003/

[1] Atiyah, M.F. Elliptic Operators and Compact Groups, Lecture Notes in Mathematics, vol. 401, Springer-Verlag, Berlin, 1974

[2] Atiyah, M.F.; Patodi, V.K.; Singer, I.M. Spectral asymmetry and Riemannian geometry I, Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 43-69

[3] Bismut, J.-M.; Lebeau, G. Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math., Volume 74 (1991) (1992), ii+298 pp

[4] Braverman, M. Index theorem for equivariant Dirac operators on noncompact manifolds, K-Theory, Volume 27 (2002), pp. 61-101

[5] Dai, X.; Zhang, W. Splitting of the family index, Commun. Math. Phys., Volume 182 (1996), pp. 303-318

[6] Gilkey, P. On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math., Volume 102 (1993), pp. 129-183

[7] Guillemin, V.; Sternberg, S. Geometric quantization and multiplicities of group representations, Invent. Math., Volume 67 (1982), pp. 515-538

[8] Ma, X.; Zhang, W. Geometric quantization for proper moment maps | arXiv

[9] Meinrenken, E.; Sjamaar, R. Singular reduction and quantization, Topology, Volume 38 (1999), pp. 699-762

[10] Paradan, P.-É. Localization of the Riemann–Roch character, J. Funct. Anal., Volume 187 (2001), pp. 442-509

[11] Paradan, P.-É. Spinc-quantization and the K-multiplicities of the discrete series, Ann. Sci. Ecole Norm. Sup. (4), Volume 36 (2003), pp. 805-845

[12] Paradan, P.-É. Multiplicities of the discrete series (38 pp) | arXiv

[13] Tian, Y.; Zhang, W. An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg, Invent. Math., Volume 132 (1998), pp. 229-259

[14] Tian, Y.; Zhang, W. Quantization formula for symplectic manifolds with boundary, Geom. Funct. Anal., Volume 9 (1999), pp. 596-640

[15] Vergne, M. Applications of equivariant cohomology, International Congress of Mathematicians, vol. I, Eur. Math. Soc., Zürich, 2007, pp. 635-664

  • Loizides, Yiannis Index Formula for Hamiltonian Loop Group Spaces, Communications in Mathematical Physics, Volume 405 (2024) no. 8 | DOI:10.1007/s00220-024-05089-1
  • Lin, Yi; Loizides, Yiannis; Sjamaar, Reyer; Song, Yanli Riemannian foliations and geometric quantization, Journal of Geometry and Physics, Volume 198 (2024), p. 105133 | DOI:10.1016/j.geomphys.2024.105133
  • Hsiao, Chin-Yu; Huang, Rung-Tzung; Li, Xiaoshan; Shao, Guokuan G-invariant Bergman kernel and geometric quantization on complex manifolds with boundary, Mathematische Annalen, Volume 390 (2024) no. 4, p. 4889 | DOI:10.1007/s00208-024-02865-1
  • Galasso, Andrea; Hsiao, Chin-Yu Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds, Mathematische Zeitschrift, Volume 308 (2024) no. 1 | DOI:10.1007/s00209-024-03561-1
  • Hsiao, Chin-Yu; Ma, Xiaonan; Marinescu, George Geometric quantization on CR manifolds, Communications in Contemporary Mathematics, Volume 25 (2023) no. 10 | DOI:10.1142/s0219199722500742
  • Ma, Xiaonan Quantization Commutes with Reduction, a Survey, Acta Mathematica Scientia, Volume 41 (2021) no. 6, p. 1859 | DOI:10.1007/s10473-021-0604-4
  • Loizides, Yiannis; Song, Yanli Norm-square localization and the quantization of Hamiltonian loop group spaces, Journal of Functional Analysis, Volume 278 (2020) no. 9, p. 108445 | DOI:10.1016/j.jfa.2019.108445
  • Hochs, Peter; Song, Yanli; Yu, Shilin A geometric realisation of tempered representations restricted to maximal compact subgroups, Mathematische Annalen, Volume 378 (2020) no. 1-2, p. 97 | DOI:10.1007/s00208-020-02006-4
  • Hochs, Peter; Song, Yanli On the Vergne conjecture, Archiv der Mathematik, Volume 108 (2017) no. 1, p. 99 | DOI:10.1007/s00013-016-0997-9
  • Song, Yanli A K-homological approach to the quantization commutes with reduction problem, Journal of Geometry and Physics, Volume 112 (2017), p. 29 | DOI:10.1016/j.geomphys.2016.08.017
  • Hochs, Peter; Mathai, Varghese Formal geometric quantisation for proper actions, Journal of Homotopy and Related Structures, Volume 11 (2016) no. 3, p. 409 | DOI:10.1007/s40062-015-0109-8
  • Hochs, Peter; Mathai, Varghese Geometric quantization and families of inner products, Advances in Mathematics, Volume 282 (2015), p. 362 | DOI:10.1016/j.aim.2015.07.004
  • Ma, Xiaonan; Zhang, Weiping Geometric quantization for proper moment maps: the Vergne conjecture, Acta Mathematica, Volume 212 (2014) no. 1, p. 11 | DOI:10.1007/s11511-014-0108-3
  • Fujita, Hajime; Furuta, Mikio; Yoshida, Takahiko Torus Fibrations and Localization of Index II, Communications in Mathematical Physics, Volume 326 (2014) no. 3, p. 585 | DOI:10.1007/s00220-014-1890-7
  • Fujita, Hajime; Furuta, Mikio; Yoshida, Takahiko Torus Fibrations and Localization of Index III, Communications in Mathematical Physics, Volume 327 (2014) no. 3, p. 665 | DOI:10.1007/s00220-014-2039-4
  • Ma, Xiaonan; Zhang, Weiping Transversal Index and L2-index for Manifolds with Boundary, Metric and Differential Geometry, Volume 297 (2012), p. 299 | DOI:10.1007/978-3-0348-0257-4_10
  • Paradan, Paul-Émile Formal geometric quantization II, Pacific Journal of Mathematics, Volume 253 (2011) no. 1, p. 169 | DOI:10.2140/pjm.2011.253.169

Cité par 17 documents. Sources : Crossref