Multiplicity-free Hamiltonian actions need not be Kähler

Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible Kähler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T³. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible Kähler structure.