Induced nilpotent orbits and birational geometry

Let G be a complex simple algebraic group and let g be its Lie algebra. A nilpotent orbit O in g is an orbit of a nilpotent element of g by the adjoint action of G on g. Then O admits a natural symplectic 2-form ω and the nilpotent orbit closure Ō has symplectic singularities in the sense of [Be] and [Na3] (cf. [Pa], [Hi]). In [Ri], Richardson introduced the notion of so-called the Richadson orbit. A nilpotent orbit O is called Richardson if there is a parabolic subgroup Q of G such that O ∩ n(q) is an open dense subset of n(q), where n(q) is the nil-radical of q. Later, Lusztig and Spaltenstein [L-S] generalized this notion to the induced orbit. A nilpotent orbit O is an induced orbit if there are a parabolic subgroup Q of G and a nilpotent orbit O in the Levi subalgebra l(q) of q := Lie(Q) such that O meets n(q)+O in an open dense subset. If O is an induced orbit, one has a natural map (cf. (1.2)) ν : G× (n(q) +O) → Ō.