Explicit formulas for mixed Hodge polynomials of character varieties of nilpotent groups

TL;DR

This paper derives explicit formulas for the mixed Hodge polynomials of character varieties associated with finitely generated nilpotent groups into specific classical groups, expanding understanding of their geometric and algebraic structures.

Contribution

It provides explicit partition type formulas for mixed Hodge polynomials of character varieties for G=Sp_{2n} and G=SO_{n}, building on previous formulas by Florentino, Lawton, and Silva.

Findings

Explicit formulas for G=Sp_{2n} character varieties.

Explicit formulas for G=SO_{n} character varieties.

Enhanced understanding of the mixed Hodge structures of these varieties.

Abstract

Let $Hom^0(\Gamma,G)$ be the path-connected component of the identity representation of the variety of representations of a finitely generated nilpotent group $\Gamma$ into a connected reductive complex affine algebraic group $G$. With the formulas given by Florentino, Lawton and Silva, we provide explicit partition type formulas for the mixed Hodge polynomials of character varieties $Hom^0(\Gamma,G)// G$ when $G=Sp_{2n}$ and $G=SO_{n}$.