ar X iv : 0 90 2 . 25 89 v 1 [ m at h . R T ] 1 6 Fe b 20 09 CHARACTER VARIETIES

Let G be a complex reductive algebraic group and let Γ be a finitely generated group. In this paper we study properties of irreducible and completely reducible representations ρ : Γ → G in the context of the geometric invariant theory of the G-action on Hom(Γ, G) by conjugation. In particular, we prove that ρ is poly-stable (i.e. its orbit is closed) if and only if ρ is completely reducible. We also show that ρ is properly stable (with respect to G/C(G)-action) if and only if ρ is irreducible. The categorical quotient XG(Γ) = Hom(Γ, G)//G is the G-character variety of Γ. We prove that if ρ is scheme smooth and completely reducible then T[ρ] XG(Γ) = T0(H (Γ, Ad ρ)//SG(ρ)) where H1(Γ, Ad ρ) is the 1st cohomology group of Γ with coefficients in the lie algebra g of G twisted by the homomorphism Γ ρ −→ G Ad −→ GL(g) and SG(ρ) is the stabilizer of ρ. Let M be an orientable 3-manifold with a connected boundary F of genus g ≥ 2. Let Xi G(F ) be the subset of the G-character variety of π1(F ) composed of conjugacy classes of irreducible representations. By a theorem of Goldman, X G(F ) is a holomorphic symplectic manifold. We prove that the set of irreducible G-representations of π1(F ) which extend to scheme smooth representations of π1(M) is a complex Lagrangian submanifold of X G(F ).