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Consider the Hamiltonian action of a compact Lie group on a symplectic manifold which has the strong Lefschetz property. We first establish an equivariant version of the MerkulovGuillemin d?lemma, and an improved version of the KirwanGinzburg equivariant formality theorem, which says that every cohomology class has a canonical equivariant extension. We then proceed to extend the equivariant d?lemma to equivariant differential forms with generalized coefficients. Finally we investigate the subtle differences between an equivariant Kaehler manifold and a Hamiltonian symplectic manifold with the strong Lefscehtz property. Among other things, we construct sixdimensional compact nonKaehler Hamiltonian circle manifolds which each satisfy the Hard Lefschetz property, but nevertheless each have a symplectic quotient which does not satisfy the strong Lefschetz property. As an aside we prove that the strong Lefschetz property, unlike that of equivariant Kaehler condition, does not guarantee the DuistermaatHeckman function to be logconcave.
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