  Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed
$(1,1)$-form $\omega$. Then we have a natural double complex
$\overline{\partial}+\overline{\partial}^\Lambda$, where
$\overline{\partial}^\Lambda$ denotes the symplectic adjoint of the
$\overline{\partial}$-operator. We study the Hard Lefschetz Condition on the
Dolbeault cohomology groups of $X$ with respect to the symplectic form
$\omega$. In \cite{TW}, we proved that such a condition is equivalent to a
certain symplectic analogous of the $\partial\overline{\partial}$-Lemma, namely
the $\overline{\partial}\, \overline{\partial}^\Lambda$-Lemma, which can be
characterized in terms of Bott--Chern and Aeppli cohomologies associated to the
above double complex. We obtain Nomizu type theorems for the Bott--Chern and
Aeppli cohomologies and we show that the $\overline{\partial}\,
\overline{\partial}^\Lambda$-Lemma is stable under small deformations of
$\omega$, but not stable under small deformations of the complex structure.
However, if we further assume that $X$ satisfies the
$\partial\overline{\partial}$-Lemma then the $\overline{\partial}\,
\overline{\partial}^\Lambda$-Lemma is stable.
