Nilpotent Orbits of $Z_4$-Graded Lie Algebra and Geometry of Moment Maps Associated to the Dual Pair $(U(p,q),U(r,s))$

Abstract

Let ${\mathfrak{s}}^1\leftarrow L_{+}\to {\mathfrak{s}}^2$ be the $K_C$-versions of the moment maps associated to the dual pair $(U(p,q),U(r,s))$ and $\mathcal{N}({\mathfrak{s}}^1)\leftarrow \mathcal{N}(L_{+})\to \mathcal{N}({\mathfrak{s}}^2)$ their restrictions to the nilpotent varieties. In this paper, we first describe the nilpotent orbit correspondence via the moment maps explicitly. Second, under the condition $\min \{p,q\}\ge \max \{r,s\}$, we show that there are open subvariety $L_{+}^{\prime }$ (resp. ($({\mathfrak{s}}^2{)}^{\prime }$) of $L_{+}$ (resp. ${\mathfrak{s}}^2$) and locally closed subvariety $({\mathfrak{s}}^1{)}^{\prime }$ of ${\mathfrak{s}}^1$ such that the restrictions of the moment maps $\mathcal{N}(({\mathfrak{s}}^1{)}^{\prime })\leftarrow \mathcal{N}(L_{+}^{\prime })\to \mathcal{N}(({\mathfrak{s}}^2{)}^{\prime })$ give bijections of nilpotent orbits. Furthermore, we show that the bijections preserve the closure relation and the equivalence class of singularities.