Topology of character varieties of Abelian groups

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.

33 pages; version 3: few small changes, one error corrected, one or two additional references; to appear in Topology and its Applications

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Cited by in corpus (34)

• The topology of nilpotent representations in reductive groups and their maximal compact subgroups
• E-polynomial of the SL(3,C)-character variety of free groups
• Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups
• Homotopy Groups of Free Group Character Varieties
• Rank 1 character varieties of finitely presented groups
• Fundamental Groups of Character Varieties: Surfaces and Tori
• Hodge-Deligne polynomials of character varieties of abelian groups
• Covering spaces of character varieties
• Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups
• Decision problems, complexity, traces, and representations
• Poincare series of character varieties for nilpotent groups
• Homological stability for spaces of commuting elements in Lie groups
• Character varieties for real forms
• Commuting elements in reductive groups and Higgs bundles on abelian varieties
• The mapping class group action on SU(3)-character varieties
• Stratified hyperkahler spaces from semisimple Lie algebras
• Invariants of pairs in SL(4,C) and SU(3,1)
• Flawed groups and the topology of character varieties
• Harmonic Higgs Bundles and Coassociative ALE Fibrations
• Character varieties of virtually nilpotent Kähler groups and G-Higgs bundles
• Geometry of $\mathrm{SU}(3)$-character varieties of torus knots
• Poisson maps between character varieties: gluing and capping
• Plethystic exponential calculus and characteristic polynomials of permutations
• Generating series for the $E$-polynomials of $GL(n,\mathbb{C})$-character varieties
• Bad Representations and Homotopy of Character Varieties
• Hodge-Deligne Polynomials of Symmetric Products of Algebraic Groups
• Mixed Hodge structures on character varieties of nilpotent groups
• Character Varieties of Generalized Torus Knot Groups
• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
• Character Varieties of Abelian Groups
• $e$-polynomials of character varieties
• Dynamics on the SU(2,1)-character variety of the one-holed torus
• Homotopy type of free group character varieties
• On GIT quotients and real forms