Root data in character varieties

TL;DR

This paper introduces a new stratification of $G$-character varieties based on conjugacy classes of parabolic subgroups, facilitating motivic computations and detailed formulas for classical types.

Contribution

It provides a novel stratification approach for character varieties using root data, enabling explicit motivic formulas for classical groups.

Findings

Stratification of character varieties via parabolic conjugacy classes.

Construction of pseudo-quotients capturing topological invariants.

Explicit motivic formulas for types A, B, C, D.

Abstract

Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $\Gamma$ a finitely generated group, we provide a stratification of the $G$-character variety of $\Gamma$ in terms of conjugacy classes of parabolic subgroups of $G$. Each stratum has the structure of a pseudo-quotient, which is a relaxed GIT notion capturing the topology of the quotient and, therefore, behaving well for motivic computations of invariants of the character varieties. These stratifications are constructed by analyzing the root datum of $G$ to encode parabolic classes. Finally, detailed and explicit motivic formulae are provided for cases with Dynkin diagram of types $A$, $B$, $C$ and $D$.