# Nilpotent Orbits of ℤ<sub>4</sub>-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>))

### Takuya Ohta

Tokyo Denki University, Japan

## Abstract

Let **s**1 ← *L*+ → **s**2 be the _K_ℂ-versions of the moment maps associated to the dual pair (*U*(*p*, *q*), *U*(*r*, *s*)) and *N*(**s**1) ← *N*(*L*+) → *N*(**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*} ≥ max{*r*, *s*}, we show that there are open subvariety *L'*+ (resp. (**s**2)') of *L*+ (resp. **s**2) and locally closed subvariety (**s**1)' of **s**1 such that the restrictions of the moment maps *N*((**s**1)') ← *N*(*L'*+) → *N*((**s**2)') give bijections of nilpotent orbits. Furthermore, we show that the bijections preserve the closure relation and the equivalence class of singularities.

## Cite this article

Takuya Ohta, Nilpotent Orbits of ℤ<sub>4</sub>-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>)). Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, pp. 723–756

DOI 10.2977/PRIMS/1145475228