I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the symplectic Hodge operator $\ast_{s}:\Omega^{k}(M)\longrightarrow \Omega^{2n-k}(M)$given by
$\eta\wedge \ast_{s} \xi= \left(\omega^{-1} \right) \left(\eta,\xi\right)d$Vol $=\dfrac{1}{k!}\left(\omega^{-1} \right)^{i_{1}j_{1}}\ldots\left(\omega^{-1} \right)^{i_{k}j_{k}}\eta_{i_{1}i_{2}\ldots i_{k}}\xi_{j_{1}j_{2}\ldots j_{k}}\dfrac{\omega^n}{n!}$
it is the symplectic analogous of the Hodge start, but instead of the metric $g$ we use the symplectic form $\omega$
The symplectic adjoint of the standard exterior derivative acting on $k$-forms is given by
$d^{\Lambda}=\left(-1 \right)^{k+1} \ast_{s}d\ast_{s} : \Omega^{k}(M)\longrightarrow \Omega^{k-1}(M)$
Using this they defined some complexes and their cohomologies, one of them is the following
$\Omega^{k}\overset{d^{\Lambda}} \longrightarrow \Omega^{k-1}$
and the cohomology group is
$H^{k}_{d^{\Lambda}}(M)=\dfrac{\mathtt{ker}\left( d^{\Lambda}\right) \cap \Omega^{k}(M)}{\mathtt{Im}d^{\Lambda}\cap \Omega^{k}(M)}$
After that they introduce the Laplacian associated with this cohomology in the usual way
$\Delta_{d^{\Lambda}}=d^{\Lambda}d^{\Lambda\ast}+ d^{\Lambda \ast}d^{\Lambda}$
and they prove a Hodge decomposition type theorem for this Laplacian. No problem up to this point.
After that they consider another complex
$\Omega^{k}\overset{dd^{\Lambda}} \longrightarrow \Omega^{k}\overset{d+d^{\Lambda}} \longrightarrow \Omega^{k+1} \oplus \Omega^{k-1} $
and the cohomology group is
$H^{k}_{d+d^{\Lambda}}(M)=\dfrac{\mathtt{ker}\left(d+ d^{\Lambda}\right) \cap \Omega^{k}(M)}{\mathtt{Im}dd^{\Lambda}\cap \Omega^{k}(M)}$
and they introduce what they called the Laplacian associated with this cohomology to prove a Hodge decomposition type theorem
$\Delta_{d+d^{\Lambda}}=dd^{\Lambda}\left(dd^{\Lambda}\right)^{\ast}+\lambda \left( d^{\ast}d+d^{\Lambda \ast}d^{\Lambda}\right)$
where $\ast$ is the usual Riemannian Hodge start and they mention "where we have inserted an undetermined real constant $\lambda>0$ that gives the relative weight between the terms"
Here is where my problem starts:
i)It is not clear for me what is the role of the constant $\lambda$?
ii) why do we need that constant there?
iii) and what is "the relative weight between the terms"?
It seems that they put that weight to compensate something that does not appear in the definition of the standard Laplacian on forms, but I could not figure out exactly what,(even their definition of $\Delta_{d+d^{\Lambda}}$ is not still clear at all for me) it is the first time that I deal with modified Laplacians
Many Thanks for your help
