Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

Details Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

1993-04-12

arxiv.org/abs/alg-geom/9304004v1

Mathematics

Algebraic Geometry

34 pages, AmS-LaTeX version 1.1

Scientific paper

I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.

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