[Quantification géométrique pour les applications moment propres]
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.
Accepté le :
Publié le :
@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 = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1016/j.crma.2009.02.003/
[1] Elliptic Operators and Compact Groups, Lecture Notes in Mathematics, vol. 401, Springer-Verlag, Berlin, 1974
[2] Spectral asymmetry and Riemannian geometry I, Proc. Camb. Philos. Soc., Volume 77 (1975), pp. 43-69
[3] Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math., Volume 74 (1991) (1992), ii+298 pp
[4] Index theorem for equivariant Dirac operators on noncompact manifolds, K-Theory, Volume 27 (2002), pp. 61-101
[5] Splitting of the family index, Commun. Math. Phys., Volume 182 (1996), pp. 303-318
[6] On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math., Volume 102 (1993), pp. 129-183
[7] Geometric quantization and multiplicities of group representations, Invent. Math., Volume 67 (1982), pp. 515-538
[8] Geometric quantization for proper moment maps | arXiv
[9] Singular reduction and quantization, Topology, Volume 38 (1999), pp. 699-762
[10] Localization of the Riemann–Roch character, J. Funct. Anal., Volume 187 (2001), pp. 442-509
[11] -quantization and the K-multiplicities of the discrete series, Ann. Sci. Ecole Norm. Sup. (4), Volume 36 (2003), pp. 805-845
[12] Multiplicities of the discrete series (38 pp) | arXiv
[13] An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg, Invent. Math., Volume 132 (1998), pp. 229-259
[14] Quantization formula for symplectic manifolds with boundary, Geom. Funct. Anal., Volume 9 (1999), pp. 596-640
[15] Applications of equivariant cohomology, International Congress of Mathematicians, vol. I, Eur. Math. Soc., Zürich, 2007, pp. 635-664
Cité par Sources :
