For some work I'm doing, I need a version of the Hodge to de Rham spectral sequence for stacks. I am not at all an expert on stacks, so please excuse me if I make minor technical mistakes in stating it.

I only need to deal with quotient stacks. Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$ and let $G$ be a finite group acting on $X$. I am trying to understand the cohomology of the quotient stack $X/G$. Sheaves on $X/G$ should be the same thing as $G$-equivariant sheaves on $X$ and sheaf cohomology should be the derived functors of the $G$-equivariant global sections functor. What I think should be true is that there should be a Hodge to de Rham spectral sequence converging to $H^{\ast}(X/G;\mathbb{C})$ with

$$E_1^{pq} = H^p(X/G;\Omega^q).$$

Here $\Omega^q$ is the sheaf of $G$-equivariant holomorphic $q$-forms on $X$.

This should be able to be proved following the usual proof of the Hodge to de Rham spectral sequence: one first proves that the constant sheaf $\mathbb{C}$ on $X/G$ is quasi-isomorphic to the de Rham complex $\Omega^{\ast}$, and the one looks at the hypercohomology spectral sequence. The only place where I can see any issue is proving the exactness of the de Rham complex, which would require an equivariant version of the Poincare lemma. However, this should be able to be derived from the usual Poincare lemma by averaging.

**Question**: Am I correct that this spectral sequence exists, and if so can anyone give me a reference for it?

I can find plenty of papers that investigate situations where a version of the Hodge to de Rham spectral sequence for stacks degenerates, but all of them are working in much more generality than I am and it is hard for me to verify that the spectral sequence they deal with is the same one I (tried) to state above. I don't need the spectral sequence to degenerate, though it presumably does in the situation I am working in.