TITLE: Mixed Hodge Polynomial for Algebraic Stacks
QUESTION [5 upvotes]: Let $X$ be a complex algebraic variety. The numerical invariants associated with the Mixed Hodge Structure of $X$ can be encoded in a polynomial in three variables called the mixed Hodge polynomial $H(X, x,y,t)$. There is also a version for compactly supported cohomology $H_c(X, x,y,t)$. The polynomial $E(X,x,y)=H_c(X,x,y,-1)$ is called the $E$-polynomial of $X$. See, e.g., $\S 2$ of this paper for a review. 
Question: Are there analogs of these polynomials for a complex algebraic stack $Y$? 
As a first step, we can consider a quotient stack $X/G$. Thus, I'm asking whether the constructions of mixed Hodge structures and associated polynomials work for equivariant cohomology. 
When $G$ is a finite group then the answer is yes, see, e.g., this post. I'm interested in the case that $G$ is not finite; e.g., $G=\mathrm{GL}_n(\mathbb{C})$. One hopes that in favorable situations $E(Y)=E(X)/E(G)$.

REPLY [3 votes]: It has been shown by Toën https://arxiv.org/abs/math/0509098 and Joyce https://arxiv.org/abs/math/0509722 that the Grothendieck group $K_0$ of Artin stacks of finite type with affine stabilizers (over $\mathbb{C}$) is isomorphic to the group obtained from $K_0$
of varieties after inverting the class $\mathbb{L}$ of the affine line, and the classes
$\mathbb{L}^i-1$ for $i\geq 1$. In particular, all additive invariants of varieties which are invertible on these particular classes (such as the E-polynomial viewed as a rational function) extend to algebraic stacks with affine stabilizers.