Holomorphic Slices, Symplectic Reduction and Multiplicities of Representations

Abstract

Abstract. I prove the existence of slices for an action of a reductive complex Lie group on a Kähler 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.

Homogeneous Lagrangian submanifolds

“…Indeed Ω is always contained in M ss , moreover (see e.g., [30]) M ss is the smallest G-invariant subset of M that contains μ −1 (0), therefore M ss = Ω. In the Kähler case it is easy to see that the stratum associated to the minimum critical set of μ 2 (see [20, p. 80] for the precise definition) coincides with the set of semistable points.…”

Properties of the residual circle action on a hypertoric variety

“…Then there is a unique hyperkähler structure on the hyperkähler quotient M = N / / / / ξ G := µ If ξ ∈ g * ⊕ g * C is fixed by the coadjoint action of G, the inverse image µ −1 C (ξ C ) is preserved by G, and is a (singular) Kähler subvariety with respect to ω R . Then by [HL] (see also [Na,3.2] and [Sj,2.5…”

Toric symplectic singular spaces I: isolated singularities

“…Therefore, by [24, Theorem 2.8], every toric symplectic singular space is a complex analytic space. Furthermore by [24,Lemma 2.16] it is a Kähler space in the sense of Grauert. The Kähler structures on these spaces are described elsewhere [7].…”