# Symplectic quotients have symplectic singularities

## TL;DR

This paper proves that under a generic condition, symplectic quotients arising from certain group actions have symplectic singularities and are graded Gorenstein, with some cases allowing relaxation of the conditions.

## Contribution

It establishes that symplectic quotients have symplectic singularities and are graded Gorenstein when the pair is 3-large, extending to specific groups without this assumption.

## Key findings

- Complex symplectic quotients have symplectic singularities.
- Real symplectic quotients are graded Gorenstein.
- Results hold for certain groups without the 3-large condition.

## 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 $\operatorname{SU}_2$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.