Topology of character varieties of Abelian groups

TL;DR

This paper studies the topology of character varieties of Abelian groups, showing they are deformation retracts of GIT quotients and providing conditions for irreducibility in complex reductive groups.

Contribution

It establishes a strong deformation retraction between conjugation orbit spaces and GIT quotients for Abelian groups, and characterizes irreducibility conditions for these varieties.

Findings

Hom(A,K)/K is a deformation retract of Hom(A,G)//G

Necessary and sufficient conditions for irreducibility when G is semisimple

Sufficient conditions for irreducibility when G is reductive and A is free abelian

Abstract

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessary and sufficient conditions for the character variety Hom(A,G)//G to be irreducible when G is connected and semisimple. For a general connected reductive G, analogous conditions are found to be sufficient for irreducibility, when A is free abelian.