Nilpotent Orbits of ℤ₄-Graded Lie Algebra and Geometry of Moment Maps Associated to the Dual Pair (<em>U</em>(<em>p</em>, <em>q</em>), <em>U</em>(<em>r</em>, <em>s</em>))

Abstract

Let s1 ← L+ → s2 be the _K_ℂ-versions of the moment maps associated to the dual pair (U(p, q), U(r, s)) and N(s1) ← N(L+) → N(s2) 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} ≥ max{r, s}, we show that there are open subvariety L'+ (resp. (s2)') of L+ (resp. s2) and locally closed subvariety (s1)' of s1 such that the restrictions of the moment maps N((s1)') ← N(L'+) → N((s2)') give bijections of nilpotent orbits. Furthermore, we show that the bijections preserve the closure relation and the equivalence class of singularities.