On Riemann-Roch Formulas for Multiplicities

Meinrenken, Eckhard

doi:10.1090/S0894-0347-96-00197-X

On Riemann-Roch Formulas for Multiplicities
HTML articles powered by AMS MathViewer

by Eckhard Meinrenken PDF

J. Amer. Math. Soc. 9 (1996), 373-389 Request permission

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

Michael Francis Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics, Vol. 401, Springer-Verlag, Berlin-New York, 1974. MR 0482866, DOI 10.1007/BFb0057821
M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. MR 721448, DOI 10.1016/0040-9383(84)90021-1
M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531–545. MR 236951, DOI 10.2307/1970716
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Nicole Berline and Michèle Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541 (French, with English summary). MR 685019
Nicole Berline and Michèle Vergne, The equivariant index and Kirillov’s character formula, Amer. J. Math. 107 (1985), no. 5, 1159–1190. MR 805808, DOI 10.2307/2374350
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat kernels and Dirac operators, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720, DOI 10.1007/978-3-642-58088-8
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.
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Michel Duflo and Michèle Vergne, Cohomologie équivariante et descente, Astérisque 215 (1993), 5–108 (French). Sur la cohomologie équivariante des variétés différentiables. MR 1247060
E. Ehrhart, Polynômes arithmétiques et méthode des polyèdres en combinatoire, International Series of Numerical Mathematics, Vol. 35, Birkhäuser Verlag, Basel-Stuttgart, 1977. MR 0432556
Carla Farsi, $K$-theoretical index theorems for orbifolds, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 170, 183–200. MR 1164622, DOI 10.1093/qmath/43.2.183
V. Guillemin: Reduction and Riemann-Roch, In: Lie groups and geometry in honour of B. Kostant. Progr. Math. Birkhäuser Boston, 1994.
V. Guillemin, E. Lerman, and S. Sternberg, On the Kostant multiplicity formula, J. Geom. Phys. 5 (1988), no. 4, 721–750 (1989). MR 1075729, DOI 10.1016/0393-0440(88)90026-5
Victor Guillemin and Elisa Prato, Heckman, Kostant, and Steinberg formulas for symplectic manifolds, Adv. Math. 82 (1990), no. 2, 160–179. MR 1063956, DOI 10.1016/0001-8708(90)90087-4
V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), no. 3, 515–538. MR 664118, DOI 10.1007/BF01398934
Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065993, DOI 10.1007/978-3-642-61497-2
L. Jeffrey, F. Kirwan: Localization for nonabelian group actions, Topology 34, 291–327 (1995).
Tetsuro Kawasaki, The Riemann-Roch theorem for complex $V$-manifolds, Osaka Math. J. 16 (1979), no. 1, 151–159. MR 527023
Ichirô Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 95520, DOI 10.2969/jmsj/00940464
R. Sjamaar: Holomorphic slices, symplectic reduction and multiplicities of representations, Ann. Math. 141, 87–129 (1995).
M. Vergne: Quantification géométrique et multiplicités, C. R. Acad. Sci., 319, 327–332 (1994).
M. Vergne: A note on Jeffrey-Kirwan-Witten’s localization formula, Preprint DMI, École Norm. Sup., Paris (1994).
Edward Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303–368. MR 1185834, DOI 10.1016/0393-0440(92)90034-X