In a well known construction of finite W-algebras, one first constructs a certain
nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathcal{m}\rightarrow \mathbb{C}$.
Then one defines

$$U(\mathfrak{g},e)=(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

Where $\mathfrak{m}_\chi$ is the set of all $m-\chi(m)$ and
$\mathfrak{m}$ acts on $U(\mathfrak{g})$ by derivations, extending the adjoint action on $\mathfrak{g}$
Is this the same as 

$$U(\mathfrak{g})^{\mathfrak{m}}/(U(\mathfrak{g})\mathfrak{m}_{\chi})^\mathfrak{m}$$

Of course one can reformulate this question and ask if the following cohomology group vanishes:  
$$H^1(\mathfrak{m},U(\mathfrak{g})\mathfrak{m}_{\chi}))=0$$
Maybe this follows from some Lynch style vanishing, but I am not very familiar with these theorems.