Abstract:

The two (equivariant) moment maps for commuting symplectically orthogonal actions form a (Weinstein) dual pair.
There is a general principle that symplectic reduction for one Lie Lie group provides coadjoint orbits of the other group.
I will give examples with matrix groups and diffeomorphism groups, mentioning some coadjoint orbits of diffeomorphism groups one gets via symplectic reduction.