  Let $(M,\omega)$ be a $2n$-dimensional smooth compact symplectic manifold
equipped with a Hamiltonian circle action with only isolated fixed points and
let $\mu : M \rightarrow \R$ be a corresponding moment map. Let $\Lambda_{2k}$
be the set of all fixed points of index $2k$. In this paper, we will show that
if $\mu$ is constant on $\Lambda_{2k}$ for each $k$, then $(M,\omega)$
satisfies the hard Lefschetz property. In particular, if $(M,\omega)$ admits a
self-indexing moment map, i.e. $\mu(p) = 2k$ for every $p \in \Lambda_{2k}$ and
$k=0,1,\cdots,n,$ then $(M,\omega)$ satisfies the hard Lefschetz property.
