Abstract
Suppose that one of the real vector spaces V and W is symplectic and the other is quadratic. Let g 1 and g 2 denote the Lie algebras of the groups of isometries of the two spaces, and let τ i : V ⊗ R{double-struck} W → g i be their respective moment maps for i = 1, 2. Suppose that O and L, are nilpotent orbits in g 1 and g 2 , respectively. We prove that τ 2 (τ 1 -1 (O)) and τ 1 (τ 2 -1 (L))) are each the union of at most two closures of nilpotent orbits in g 1 and g 2 , respectively (where P denotes the closure of a nilpotent orbit P). © 2010 by Pacific Journal of Mathematics.