Can anyone explain to me what the Hodge decomposition form of a symplectic form in a special symplectic manifold looks like?

Using the additional information that the OP provided in the comments to Yael Fregier's answer, I can elaborate as follows: I still don't know what "special complex manifold" means, but in any case, I will assume the following. If $(M, J, \nabla)$ is a complex manifold with a connection $\nabla$ coming from a metric $g$, (that is, $\nabla$ is the LeviCivita connection of $g$), then we get an associated symplectic form $\omega(X,Y) = g(JX, Y)$, and $\omega$ is parallel with respect to $\nabla$ if and only if $J$ is parallel with respect to $\nabla$, if and only if $J$ is integrable and $\omega$ is closed. That is, $(M, g, J, \omega)$ is Kaehler. In this case, $\omega$ is harmonic, so its Hodge decomposition is $\omega = \omega \in \Delta_2$, where $\Delta_2$ is the space of harmonic $2$forms on $M$, using the notation of the OP. If $\nabla$ does not come from a metric, you still need some metric to define the coderivative $d^* = \delta$ of $d$, and to define the Laplacian $\Delta$. One can indeed do this with a different connection $\nabla$, as long as you have a metric. But in this case it is not clear to me what the symplectic form $\omega$ is, and how it is related to $J$ and $\nabla$. Added later: I think I just realized that the OP is not asking about the Hodge decomposition of the form $\omega$ in particular, just the "Hodge decomposition" for a "special symplectic manifold." There is a version of "symplectic Hodge theory." See, for example, these notes by Victor Guillemin: http://wwwmath.mit.edu/~vwg/shlomonotes.pdf  I don't know if this is the same thing mentioned in Yael Fregier's answer. Otherwise, I remain confused by the question. 


I agree with Spiro Karigiannis that the question would need some clarifications. I slightly modify the question by replacing the first "form" by "for". In this case I also agree with Spiro about the fact that a metric would be needed to speak of Hodge decomposition... Unless you refer to the HodgeLepage decomposition which has a meaning for a symplectic manifold without metric : Given a symplectic vector space $(V,\omega)$, one can associate to it two operators $\omega^+$ and $\omega^$ which act on the exterior algebra on $V^*$ by respectively left multiplying a given form $\alpha$ by $\omega$ (i.e. $\omega^+(\alpha)=\omega\wedge\alpha$) or by contracting $\alpha$ by the Poisson bivector $\pi$ associated to $\omega$ ($\omega^(\alpha)=i_\pi(\alpha)$). These two operators satisfy the relations of the Lie algebra $sl(2)$ and they cut out the space of differential forms into irreducible $sl(2)$modules which are also modules over the Lie algebra of symplectomorphisms since the operators $\omega^+$ and $\omega^$ are invariant under the action of this Lie algebra. This decomposition is called the HodgeLepage decomposition, and the highest weight vectors are called effective forms. One can find all the details, and some explicit formulas in Darboux coordinates in chapter 5 of the book "Contact Geometry and Nonlinear Differential Equations" by Kushner, Lychagin and Roubtsov (encyclopedia of Mathematics and its Applications (No. 101)) at Cambridge University Press. 

