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- Hodge-deligne polynomials of character varieties of free abelian groupsPublication . Florentino; Silva, JaimeLet F be a finite group and X be a complex quasi-projective F-variety. For r is an element of N, we consider the mixed Hodge-Deligne polynomials of quotients X-r/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G = GL(n, C) and SL(n, C), we get even more concrete expressions for these polynomials, using the combinatorics of partitions.
- Mixed Hodge structures on character varieties of nilpotent groupsPublication . A. A. Florentino, Carlos; Lawton, Sean; Silva, JaimeLet Hom0(Ƭ, G) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety Hom0(Ƭ, G) and on the character variety Hom0(Ƭ, G)//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.