Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

Preprint

12 April 1993

preprint
Published by arXiv in arXiv

https://doi.org/10.48550/arXiv.alg-geom/9304004

Abstract

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.

Keywords

SYMPLECTIC REDUCTION
HOLOMORPHIC SLICES
REPRESENTATIONS
MOMENTUM MAP
QUANTIZATION
SINGULAR
ORBITS

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Version 1, 1993-04-12, ArXiv

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