Differential Geometry
Geometric quantization for proper moment maps
Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 389-394.

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.

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.

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Accepted:
Published online:
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
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Ma, Xiaonan; Zhang, Weiping. Geometric quantization for proper moment maps. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 389-394. doi : 10.1016/j.crma.2009.02.003. http://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

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