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 $\mu :V\oplus V^{\ast }\to {\mathfrak{g}}^{\ast }$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N//G$. We show that $M$ determines $V\oplus V^{\ast }$ and $G$, up to isomorphism, if ${\mathrm{codim}}_N{N}_{\mathrm{sing}}\ge 4$. If ${\mathrm{codim}}_N{N}_{\mathrm{sing}}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G^{\prime }\subset G$ and a faithful $1$-modular $G^{\prime }$-submodule $V^{\prime }\subset V$ with shell $N^{\prime }$ such that ${\mathrm{codim}}_{N^{\prime }}(N^{\prime }{)}_{\mathrm{sing}}\ge 4$. Moreover, there is a $G^{\prime }$-equivariant morphism $N^{\prime }\to N$ inducing an isomorphism $N^{\prime }//G^{\prime }\stackrel{\sim }{\to }N//G$. Thus, up to isomorphism, $M$ determines $V^{\prime }\oplus (V^{\prime }{)}^{\ast }$ and $G^{\prime }$, hence also $N^{\prime }$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.

24 pages

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