# Mixed Hodge Polynomial for Algebraic Stacks

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

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