Symplectic quotients have symplectic singularities

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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 zero 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 implies in particular that the real symplectic quotient is graded Gorenstein. In case$K$is a torus or$\operatorname{SU}_{2}$, we show that these results hold without the hypothesis that$(V,G)$is$3$-large.