Mixed Hodge structures on character varieties of nilpotent groups

TL;DR

This paper explicitly determines the mixed Hodge structures and polynomials of representation and character varieties of finitely generated nilpotent groups into complex reductive groups, providing formulas for their invariants.

Contribution

It introduces explicit formulas for the mixed Hodge structures and polynomials of these varieties, advancing understanding of their geometric and topological properties.

Findings

Explicit mixed Hodge polynomial formulas derived

Closed and recursive formulae provided

Enhanced understanding of nilpotent group representation varieties

Abstract

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety R and on the character variety R//G. We obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomial of these representation and character varieties.