Introduced by Deligne, mixed Hodge structures provide a generalization of the Kähler decomposition of cohomology. They were motivated by Deligne's attempt to assign “virtual” Hodge numbers to non-smooth varieties, a problem in turn related to the Weil conjectures.

I will give a short introduction to mixed Hodge structures, and calculate them in some simple illustrative examples. Afterwards, I will present a sketch of how to obtain these structures for varieties given by quotients of diagonal actions. Finally, I will finish by applying these results to the free abelian character varieties of $\operatorname{GL}(n,\mathbb{C})$ and $\operatorname{SL}(n,\mathbb{C})$.