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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 $E(Y)=E(X)/E(G)$.

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  • $\begingroup$ Did you already look at Kai Behrend's PhD thesis? $\endgroup$ – Jason Starr Feb 5 at 18:56

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