  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.
