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 ← L_+ → \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) ← \mathcal N(L_+) → \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\} ≥ \max\{r, s\} , we show that there are open subvariety L'_+ (resp. ( (\mathfrak s^2)' ) of L_+ (resp. \mathfrak s^2 ) and locally closed subvariety (\mathfrak s^1)' of \mathfrak s^1 such that the restrictions of the moment maps \mathcal N((\mathfrak s^1)') ← \mathcal N(L'_+) → \mathcal N((\mathfrak s^2)') give bijections of nilpotent orbits. Furthermore, we show that the bijections preserve the closure relation and the equivalence class of singularities.