Character Varieties of Abelian Groups

TL;DR

This paper establishes a natural normalization map for character varieties of abelian groups related to reductive groups, proving isomorphisms for classical groups and analyzing tangent spaces and symplectic structures.

Contribution

It introduces a normalization map from T^N/W to character varieties, proves it is an isomorphism for classical groups, and studies the geometric and symplectic properties of certain components.

Findings

Normalization map chi is an isomorphism for classical groups.

Tangent spaces match H^1(Z^N, Ad rho) despite no irreducible representations.

The irreducible component admits a Goldman symplectic form for N=2.

Abstract

We prove that for every reductive group G with a maximal torus T and the Weyl group W there is a natural normalization map chi from T^N/W to an irreducible component of the G-character variety of Z^N. We prove that chi is an isomorphism for all classical groups. Additionally, we prove that even though there are no irreducible representations in the above mentioned irreducible component of the character variety for non-abelian G, the tangent spaces to it coincide with H^1(Z^N, Ad rho). Consequently, this irreducible component has the "Goldman" symplectic form for N=2, for which the combinatorial formulas for Goldman bracket hold.