Mixed Hodge structures on character varieties of nilpotent groups

Abstract

AbstractLet $$\textsf{Hom}^{0}(\Gamma ,G)$$ Hom 0 ( Γ , G ) be the connected component of the identity of the variety of representations of a finitely generated nilpotent group $$\Gamma $$ Γ into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety $$\textsf{Hom}^{0}(\Gamma ,G)$$ Hom 0 ( Γ , G ) and on the character variety $$\textsf{Hom}^{0}(\Gamma ,G)/\!\!/G$$ Hom 0 ( Γ , G ) / / G . We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.