Symplectic reduction and Riemann-Roch formulas for multiplicities
Author:
Reyer Sjamaar
Journal:
Bull. Amer. Math. Soc. 33 (1996), 327-338
MSC (1991):
Primary 58F06; Secondary 14L30, 19L10
DOI:
https://doi.org/10.1090/S0273-0979-96-00661-1
MathSciNet review:
1364017
Full-text PDF
References | Similar Articles | Additional Information
- 1. S. Axelrod, S. Della Pietra, and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991), 787--902. MR 92i:58064
- 2. A. Canas da Silva, Y. Karshon, and S. Tolman, Quantization of presymplectic manifolds and circle actions, preprint, Massachusetts Institute of Technology, in preparation.
- 3. J. J. Duistermaat, V. Guillemin, E. Meinrenken, and S. Wu, Symplectic reduction and Riemann-Roch for circle actions, Math. Res. Letters 2 (1995), 259--266. MR 1:338 785
- 4. C. Duval, J. Elhadad, and G. M. Tuynman, The BRS method and geometric quantization: Some examples, Comm. Math. Phys. 126 (1990), 535--557. MR 91b:81054
- 5. M. J. Gotay, Constraints, reduction, and quantization, J. Math. Phys. 27 (1986), 2051--2066. MR 87k:58103
- 6. M. Grossberg and Y. Karshon, Equivariant index and the moment map for completely integrable torus actions, Adv. in Math., to appear.
- 7. V. Guillemin, Reduced phase spaces and Riemann-Roch, Lie Groups and Geometry in Honor of B. Kostant (Massachusetts Institute of Technology, 1994) (R. Brylinski et al., eds.), Progress in Mathematics, vol. 123, Birkhäuser, Boston, 1995, pp. 305--334. MR 1:327 539
- 8. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515--538. MR 83m:58040
- 9. ------, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344--380. MR 84d:58034
- 10. L. C. Jeffrey and F. C. Kirwan, Localization and the quantization conjecture, preprint, 1995.
- 11. ------, Localization for nonabelian group actions, Topology 34 (1995), 291--327. MR 1:318 878
- 12. ------, On localization and Riemann-Roch numbers for symplectic quotients, preprint, 1995.
- 13. J. Kalkman, Cohomology rings of symplectic quotients, J. Reine Angew. Math. 458 (1995), 37--52. MR 96a:55014
- 14. F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton Univ. Press, Princeton, NJ, 1984. MR 96i:58050
- 15. B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Phys. 176 (1987), 49--113. MR 88m:58057
- 16. E. Lerman, Symplectic cuts, Math. Res. Letters 2 (1995), 247--258. MR 1:338 784
- 17. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121--130. MR 53:6633
- 18. E. Meinrenken, On Riemann-Roch formulas for multiplicities, J. Amer. Math. Soc., to appear.
- 19.
------, Symplectic surgery and the Spin
-Dirac operator, Adv. in Math., to appear.
- 20. ------, Vielfachheitsformeln für die Quantisierung von Phasenräumen, Ph.D. thesis, Universität Freiburg, 1994.
- 21. K. Meyer, Symmetries and integrals in mathematics, Dynamical Systems (Univ. of Bahia, 1971) (M. M. Peixoto, ed.), Academic Press, New York, 1973. MR 48:9760
- 22. R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. of Math. (2) 141 (1995), 87--129. MR 96a:58098
- 23. M. Vergne, Multiplicity formula for geometric quantization I, II, Duke Math. J., to appear.
- 24. ------, A note on Jeffrey-Kirwan-Witten's localisation formula, Topology, to appear.
- 25. ------, Equivariant index formula for orbifolds, preprint, École Normale Supérieure, Paris, 1994.
- 26. ------, Quantification géométrique et multiplicités, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 327--332. MR 1:289 306
- 27. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303--368. MR 93m:58017
- 28. N. M. J. Woodhouse, Geometric quantization, second ed., Oxford Univ. Press, Oxford, 1992. MR 94a:58082
- 29. S. Wu, An integration formula for the square of moment maps of circle actions, Lett. Math. Phys. 29 (1993), 311--328. MR 95b:58070
Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 58F06, 14L30, 19L10
Retrieve articles in all journals with MSC (1991): 58F06, 14L30, 19L10
Additional Information
Reyer Sjamaar
Affiliation:
Cornell University, Ithaca, New York 14853-7901
Email:
sjamaar@math.cornell.edu
DOI:
https://doi.org/10.1090/S0273-0979-96-00661-1
Keywords:
Momentum mappings,
geometric quantization,
equivariant index theorem
Received by editor(s):
September 15, 1995
Received by editor(s) in revised form:
December 24, 1995
Additional Notes:
I gratefully acknowledge support from the Ruhr-Universität Bochum. I wish to thank L. Jeffrey, Y. Karshon, and E. Meinrenken for helpful comments.
Article copyright:
© Copyright 1996
American Mathematical Society


