Let be a compact Lie group acting in a Hamiltonian way on a symplectic manifold which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map is proper so that the reduced space is compact for all . Then, we can define the “formal geometric quantization” of as
The aim of this article is to study the functorial properties of the assignment .
Considérons l’action hamiltonienne d’un groupe de Lie compact sur une variété symplectique préquantifiée par un fibré en droites de Kostant-Souriau. On suppose que l’application moment est propre, ainsi les réductions symplectiques sont compactes pour tout . On peut alors définir la quantification formelle de comme
Le but de ce travail est l’étude de certaines propriétés fonctorielles de l’application .
Accepted:
DOI: 10.5802/aif.2429
Classification: 58F06, 57S15, 19L47, 19L10
Keywords: Geometric quantization, moment map, symplectic reduction, index, transversally elliptic
Author's affiliations:
@article{AIF_2009__59_1_199_0, author = {Paradan, Paul-\'Emile}, title = {Formal geometric quantization}, journal = {Annales de l'Institut Fourier}, pages = {199--238}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {1}, year = {2009}, doi = {10.5802/aif.2429}, zbl = {1163.53056}, mrnumber = {2514864}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2429/} }
TY - JOUR TI - Formal geometric quantization JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 199 EP - 238 VL - 59 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2429/ UR - https://zbmath.org/?q=an%3A1163.53056 UR - https://www.ams.org/mathscinet-getitem?mr=2514864 UR - https://doi.org/10.5802/aif.2429 DO - 10.5802/aif.2429 LA - en ID - AIF_2009__59_1_199_0 ER -
Paradan, Paul-Émile. Formal geometric quantization. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 199-238. doi : 10.5802/aif.2429. https://aif.centre-mersenne.org/articles/10.5802/aif.2429/
[1] Convexity and commuting Hamiltonians, Bull. London Math. Soc., Volume 14 (1982) no. 1, pp. 1-15 | Article | MR: 642416 | Zbl: 0482.58013
[2] The index of elliptic operators. II, Ann. of Math. (2), Volume 87 (1968), pp. 531-545 | Article | MR: 236951 | Zbl: 0164.24201
[3] The index of elliptic operators. I, Ann. of Math. (2), Volume 87 (1968), pp. 484-530 | Article | MR: 236950 | Zbl: 0164.24001
[4] The index of elliptic operators. III, Ann. of Math. (2), Volume 87 (1968), pp. 546-604 | Article | MR: 236952 | Zbl: 0164.24301
[5] The index of elliptic operators. IV, Ann. of Math. (2), Volume 93 (1971), pp. 139-149 | Article | MR: 279834 | Zbl: 0212.28603
[6] Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin, 1974 | MR: 482866 | Zbl: 0297.58009
[7] Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 298, Springer-Verlag, Berlin, 1992 | MR: 1215720 | Zbl: 0744.58001
[8] The Chern character of a transversally elliptic symbol and the equivariant index, Invent. Math., Volume 124 (1996) no. 1-3, pp. 11-49 | Article | MR: 1369410 | Zbl: 0847.46037
[9] L’indice équivariant des opérateurs transversalement elliptiques, Invent. Math., Volume 124 (1996) no. 1-3, pp. 51-101 | Article | MR: 1369411 | Zbl: 0883.58037
[10] Variétés sphériques, Opérations hamiltoniennes et opération de groupes algébriques, Notes de la session de S.M.F., Grenoble, 1997, pp. 1-60
[11] The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv., Volume 73 (1998) no. 1, pp. 137-174 | Article | MR: 1610599 | Zbl: 0935.14029
[12] Décomposition simpliciale d’un réseau, invariante par un groupe fini d’automorphismes, C. R. Acad. Sci. Paris Sér. A-B, Volume 288 (1979) no. 2, p. A137-A139 | MR: 524769 | Zbl: 0406.14022
[13] Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math., Volume 23 (2005) no. 2, pp. 161-170 | MR: 2155008 | Zbl: 1078.14076
[14] Complete symmetric varieties, Invariant theory (Montecatini, 1982) (Lecture Notes in Math.) Volume 996, Springer, Berlin, 1983, pp. 1-44 | Zbl: 0581.14041
[15] Complete symmetric varieties. II. Intersection theory, Algebraic groups and related topics (Kyoto/Nagoya, 1983) (Adv. Stud. Pure Math.) Volume 6, North-Holland, Amsterdam, 1985, pp. 481-513 | Zbl: 0596.14041
[16] The heat kernel Lefschetz fixed point formula for the spin- Dirac operator, Progress in Nonlinear Differential Equations and their Applications, 18, Birkhäuser Boston Inc., Boston, MA, 1996 | MR: 1365745 | Zbl: 0858.58045
[17] Geometric quantization and multiplicities of group representations, Invent. Math., Volume 67 (1982) no. 3, pp. 515-538 | Article | MR: 664118 | Zbl: 0503.58018
[18] Localization and the quantization conjecture, Topology, Volume 36 (1997) no. 3, pp. 647-693 | Article | MR: 1422429 | Zbl: 0876.55007
[19] The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) (Lecture Notes in Math.) Volume 732, Springer, Berlin, 1979, pp. 233-243 | MR: 555701 | Zbl: 0407.22012
[20] Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, Volume 31, Princeton University Press, Princeton, NJ, 1984 | MR: 766741 | Zbl: 0553.14020
[21] Quantization and unitary representations. I. Prequantization, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, p. 87-208. Lecture Notes in Math., Vol. 170 | MR: 294568 | Zbl: 0223.53028
[22] Symplectic cuts, Math. Res. Lett., Volume 2 (1995) no. 3, pp. 247-258 | MR: 1338784 | Zbl: 0835.53034
[23] Nonabelian convexity by symplectic cuts, Topology, Volume 37 (1998) no. 2, pp. 245-259 | Article | MR: 1489203 | Zbl: 0913.58023
[24] On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc., Volume 9 (1996) no. 2, pp. 373-389 | Article | MR: 1325798 | Zbl: 0851.53020
[25] Symplectic surgery and the -Dirac operator, Adv. Math., Volume 134 (1998) no. 2, pp. 240-277 | Article | MR: 1617809 | Zbl: 0929.53045
[26] Singular reduction and quantization, Topology, Volume 38 (1999) no. 4, pp. 699-762 | Article | MR: 1679797 | Zbl: 0928.37013
[27] Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Volume 34, Springer-Verlag, Berlin, 1994 | MR: 1304906 | Zbl: 0797.14004
[28] Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 15, Springer-Verlag, Berlin, 1988 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR: 922894 | Zbl: 0628.52002
[29] Localization of the Riemann-Roch character, J. Funct. Anal., Volume 187 (2001) no. 2, pp. 442-509 | Article | MR: 1875155 | Zbl: 1001.53062
[30] Symplectic reduction and Riemann-Roch formulas for multiplicities, Bull. Amer. Math. Soc. (N.S.), Volume 33 (1996) no. 3, pp. 327-338 | Article | MR: 1364017 | Zbl: 0857.58021
[31] An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math., Volume 132 (1998) no. 2, pp. 229-259 | Article | MR: 1621428 | Zbl: 0944.53047
[32] Multiplicities formula for geometric quantization. I, II, Duke Math. J., Volume 82 (1996) no. 1, p. 143-179, 181–194 | Article | MR: 1387225 | Zbl: 0855.58034
[33] Quantification géométrique et réduction symplectique, Astérisque (2002) no. 282, pp. 249-278 (Séminaire Bourbaki, Vol. 2000/2001, Exp. No. 888, viii) | Numdam | MR: 1975181 | Zbl: 1037.53062
[34] Non-abelian symplectic cuts and the geometric quantization of noncompact manifolds, Lett. Math. Phys., Volume 56 (2001) no. 1, pp. 31-40 (EuroConférence Moshé Flato 2000, Part I (Dijon)) | Article | MR: 1848164 | Zbl: 1024.53051
[35] Two-dimensional gauge theories revisited, J. Geom. Phys., Volume 9 (1992) no. 4, pp. 303-368 | Article | MR: 1185834 | Zbl: 0768.53042
[36] The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom., Volume 14 (1996) no. 1, pp. 3-42 | Article | MR: 1375064 | Zbl: 0877.58022
Cited by Sources: