When is a symplectic quotient an orbifold?

Abstract

Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary ${\mathrm{SU}}_2$-modules yield symplectic quotients that are $\mathbb{Z}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.

14 pages, added extensions of results in version 1 (Theorem 1.3 and Corollary 1.4)

References in corpus (4)

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• Constructing symplectomorphisms between symplectic torus quotients
• When does the zero fiber of the moment map have rational singularities?
• Symplectic quotients and representability: the circle action case
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups
• The symplectic form associated to a singular Poisson algebra
• The partial derivative of ratios of Schur polynomials and applications to symplectic quotients