Let $M = text{Spec}(R)$ be a symplectic affine variety of dimension $n$ with the symplectic form $omega$. There is the symplectic Hodge star $star: Omega^k_{M} to Omega^{2n-k}_{M}$ given by the condition that for any $alpha, beta in Omega^k$ we should have $beta wedge alpha = Lambda^k G(beta, alpha) cdot v_n$ where $G(-, -): TM^* times TM^* to R$ is a bivector field corresponding to $omega$ and $v_n = omega^n/n!$. I want to understand the first nontrivial (for me) case: $M = T^* mathbb{A}^2$. In this case $omega = dx_1 dx_2 + dx_3 dx_4$ and $v_2 = dx_1 dx_2 dx_3 dx_4$. Also, $G = frac{partial}{partial_{x_1}} frac{partial}{partial_{x_2}} + frac{partial}{partial_{x_3}} frac{partial}{partial_{x_4}}.$ It seems to me that we have the following

$star(f) = f dx_1 dx_2 dx_3 dx_4$

$star(alpha) = alpha wedge omega$ where $alpha in Omega^1_{M}$

$star(zeta) = -zeta$ for $zeta in Omega^{2}_{M}$

$star(eta) = i_{G} eta$ where $eta in Omega^{3}_{M}$ and $i_{G}$ is the contraction with $G$.

OK, then let us see how the Koszul differential works and it is the place where I met a contradiction in math (of course, the problem is with me but I cannot figure it out).

$textbf{Setup.}$ The Koszul differential $B: Omega^{k}_{M} to Omega^{k-1}_{M}$ has two interpretations, the first one is $B = (i_G, d)$. The second is $B = (-1)^{k+1} star d star.$

$textbf{My misunderstanding.}$ Let $x in Omega^3_{M}$. The first definition gives $Bx = i_G dx – d i_{G}x$. The second definition gives $Bx = star d star x = star d i_{G} x = -d i_{G} x$. As we see, we should have $i_{G} dx=0$ which is not the case, for example, for $x_1 dx_2 dx_3 dx_4.$

$textbf{My quesion.}$ Where is the mistake? Or, can we write down the star isomorphism in the general case? I think it is related to the multiplication by $omega^{n-k}$, but mostly I’m interested how to write down the formula for $star: Omega^{k}_{M} to Omega^{2n-k}_{M}$ for $k>n$. Is there any reference?