Isomorphisms of Symplectic Torus Quotients

Abstract: We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $\mu\colon V\oplus V^\to\mathfrak{g}^$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/!!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}N N\mathrm{sing}\geq 4$. If $\operatorname{codim}NN\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}{N'}(N')\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/!!/ G'\xrightarrow{\sim} N/!!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.