Character varieties

We study properties of irreducible and completely reducible representations of finitely generated groups Γ 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 Γ by conjugation. Let X G ( Γ ) X_G(Γ ) be the G G -character variety of Γ . Γ . We prove that for every completely reducible, scheme smooth ρ : Γ → G ρ :Γ → G \[ T [ ρ ] X G ( Γ ) ≃ T 0 ( H 1 ( Γ , A d ρ ) / /