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We study properties of irreducible and completely reducible representations of finitely generated groups Γ into reductive algebraic groups G. G. In particular, we study the geometric invariant theory of the G G action on the space of G G -representations of Γ by conjugation. Let X G (Γ) XG () be the G G -character variety of Γ. . We prove that for every completely reducible, scheme smooth ρ: Γ → G: G \ T [ ρ X G (Γ) ≃ T 0 (H 1 (Γ, A d ρ) / /
Adam S. Sikora (Tue,) studied this question.