Nilpotent Orbits and Complex Dual Pairs

τ(w)(x) = 〈x(w), w〉, w ∈ W, x ∈ g ⊆ End(W ), and similarly for τ ′. Our main theorem describes the behaviour of closures of nilpotent orbits under the action of moment maps. It is easy to see that for a nilpotent coadjoint orbit O ⊆ g∗ the set τ ′(τ−1(O)) is the union of nilpotent coadjoint orbits in g′. It turns out that it is a closure of a single orbit: Theorem 1.1 Let O ⊆ g∗ be a nilpotent coadjoint orbit. There exists a (unique) nilpotent coadjoint orbit O′ ⊆ g′∗ such that τ ′(τ−1(O)) = O′.