Symplectic quotients have symplectic singularities

Abstract

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V,G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or ${\mathrm{SU}}_2$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.

35 pages. v2: Fixed gap in first version (the proof of Corollary 3.14 in v1 only applies if the slice is taken at a real point). Hypothesis for the main results changed from 2-large to 3-large, added Sections 3.3 and 3.5, and improved exposition. v3: Added Lemma 3.17, improved exposition. Final preprint version accepted by Compositio Mathematica (not the published version)

References in corpus (1)

• Towards a symplectic version of the Chevalley restriction theorem

Cited by in corpus (7)

• Hilbert series associated to symplectic quotients by ${\mathrm{SU}}_2$
• Higher Koszul brackets on the cotangent complex
• Hilbert series of symplectic quotients by the 2-torus
• When does the zero fiber of the moment map have rational singularities?
• The partial derivative of ratios of Schur polynomials and applications to symplectic quotients
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups
• The symplectic form associated to a singular Poisson algebra