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.