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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

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

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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On Riemann-Roch Formulas for Multiplicities
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by Eckhard Meinrenken PDF
J. Amer. Math. Soc. 9 (1996), 373-389 Request permission


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.
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Additional Information
  • Eckhard Meinrenken
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Received by editor(s): June 16, 1994
  • © Copyright 1996 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 9 (1996), 373-389
  • MSC (1991): Primary 53C15, 58F05, 58G07
  • DOI:
  • MathSciNet review: 1325798