Birational geometry of symplectic resolutions of nilpotent orbits II

for a parabolic subgroup P ⊂ G. In Part I [Na 2], when g is classical, we have proved that any two symplectic resolutions of Ō are connected by a sequence of Mukai flops of type A or of type D. In this paper (Part II), we shall improve and generalize all arguments in Part I so that the exceptional Lie algebras can be dealt with. We shall replace all arguments of [Na 2] which use flags, by those which use only Dynkin diagrams. In the classical case, we already know which parabolic subgroups P appear as the polarizations of O and when the Springer map μ : YP := T (G/P ) → Ō has degree 1 ([He]); so, in [Na 2], we only had to study the relationship between such polarizations. But, for the exceptional Lie algebras, there seems no complete answer to this question. In this paper, we will start with a nilpotent orbit closure Ō which has a Springer resolution YP0 := T ∗(G/P0) → Ō. Even when g is classical, we will not try to use the classification of polarizations [He] as possible as we can. First we introduce an equivalence relation in the set of parabolic subgroups of G in terms of marked Dynkin diagrams (Definition 1). Our theorem then claims that a parabolic subgroup P ⊂ G always give a Springer resolution of Ō if P is equivalent to P0. Conversely, we conjecture that any symplectic resolution of Ō has this form. In fact,