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On Riemann-Roch formulas for multiplicities
| Venue: | J. Amer. Math. Soc |
| Citations: | 49 - 2 self |
BibTeX
@ARTICLE{Meinrenken_onriemann-roch,
author = {Eckhard Meinrenken},
title = {On Riemann-Roch formulas for multiplicities},
journal = {J. Amer. Math. Soc},
year = {}
}
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Abstract
A Theorem due to Guillemin and Sternberg [17] 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 symplectically reduced space. 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. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan [20] and Guillemin [14]. Since there are no complex geometry arguments involved, the result also extends to non Kählerian settings. 1
Keyphrases
reduced space riemann-roch formula complex geometry argument compact hler manifold various irreducible component equivariant cohomology explicit formula smooth symplectic manifold compact lie group geometric quantization riemann-roch number hamiltonian action g-invariant subspace localization technique hlerian setting recent work
