Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

On Riemann-Roch Formulas for Multiplicities


Author: Eckhard Meinrenken
Journal: J. Amer. Math. Soc. 9 (1996), 373-389
MSC (1991): Primary 53C15, 58F05, 58G07
DOI: https://doi.org/10.1090/S0894-0347-96-00197-X
MathSciNet review: 1325798
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A theorem of Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kähler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectic quotient. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. The result extends to non-Kählerian settings, if one defines the representation by the equivariant index of the $\text{Spin}^c$-Dirac operator associated to the quantizing line bundle.


References [Enhancements On Off] (What's this?)

  • 1. M. F. Atiyah: Elliptic operators and compact groups, Lecture Notes in Mathematics 401, Springer-Verlag (1974). MR 58:2910
  • 2. M. F. Atiyah, R. Bott: The moment map and equivariant cohomology, Topology 23, 1--28 (1984). MR 85e:58041
  • 3. M. F. Atiyah, G. Segal: The index of elliptic operators II. Ann. Math. 87, 531--545 (1968). MR 38:5244
  • 4. M. F. Atiyah, I. Singer: The index of elliptic operators III. Ann. Math. 87, 546--604 (1968). MR 38:5245
  • 5. N. Berline, M. Vergne: Classes charactéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295, 539--541 (1982). MR 83m:58002
  • 6. N. Berline, M. Vergne: The equivariant index and Kirillov's character formula, Am. J. Math. 107, 1159--1190 (1985). MR 87a:58143
  • 7. N. Berline. E. Getzler, M. Vergne: Heat kernels and Dirac operators, Springer-Verlag 1992. MR 94e:58130
  • 8. J. J. Duistermaat: Equivariant cohomology and stationary phase, Symplectic Geometry and Quantization (Sanda and Yokohama, 1993), Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 45--62. CMP 95:09
  • 9. J. J. Duistermaat, G. Heckman: On the variation in the cohomology of the symplectic form of the reduced phase space, Inv. Math. 69, 259--268 (1982). Addendum, 72, 153--158 (1983). MR 84h:58051a,b
  • 10. M. Duflo, M. Vergne: Cohomologie équivariante et descente, Asterisque 215 (1993). MR 95f:22018
  • 11. E. Ehrhart: Polynômes arithméthiques et méthode des polyèdres en combinatoire, Birkhäuser Verlag, 1977. MR 55:5544
  • 12. C. Farsi: K-theoretical Index theorems for orbifolds, Quart. J. Math. Oxford 43 (2), 183--200 (1992). MR 93f:58231
  • 13. V. Guillemin: Reduction and Riemann-Roch, In: Lie groups and geometry in honour of B. Kostant. Progr. Math. Birkhäuser Boston, 1994.
  • 14. V. Guillemin, E. Lerman, S. Sternberg: On the Kostant multiplicity formula J. Geom. Phys. 5, 721--750 (1988). MR 92f:58058
  • 15. V. Guillemin, E. Prato: Heckman, Kostant and Steinberg formulas for symplectic manifolds, Adv. Math. 82, 160--179 (1990). MR 91h:58041
  • 16. V. Guillemin, S. Sternberg: Geometric quantization and multiplicities of group representations, Invent. Math. 67, 515--538 (1982). MR 83m:58040
  • 17. V. Guillemin, S. Sternberg:Symplectic techniques in physics, Cambridge University Press 1984. MR 86f:58054
  • 18. L. Hörmander: The analysis of linear partial differential operators I. Springer 1990. MR 91m:35001a
  • 19. L. Jeffrey, F. Kirwan: Localization for nonabelian group actions, Topology 34, 291--327 (1995). CMP 95:08
  • 20. T. Kawasaki: The Riemann-Roch theorem for complex V-manifolds, Osaka J. Math. 16, 151--157 (1979). MR 80f:58042
  • 21. I. Satake: The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan, 9, 464--492 (1957). MR 20:2022
  • 22. R. Sjamaar: Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. Math. 141, 87--129 (1995). CMP 95:07
  • 23. M. Vergne: Quantification géométrique et multiplicités, C. R. Acad. Sci., 319, 327--332 (1994). CMP 94:16
  • 24. M. Vergne: A note on Jeffrey-Kirwan-Witten's localization formula, Preprint DMI, École Norm. Sup., Paris (1994).
  • 25. E. Witten: Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303--368. MR 93m:58017

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 53C15, 58F05, 58G07

Retrieve articles in all journals with MSC (1991): 53C15, 58F05, 58G07


Additional Information

Eckhard Meinrenken
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S0894-0347-96-00197-X
Received by editor(s): June 16, 1994
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society