2 3 A ug 2 00 9 CHARACTER VARIETIES

Let G be a complex reductive algebraic group and let Γ be a finitely generated group. 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 the space of G-representations of Γ by conjugation. Let XG(Γ) be the G-character variety of Γ. We prove that if ρ : Γ → G is completely reducible and it represents a reduced point of the character variety then T[ρ] XG(Γ) = T0(H (Γ, Ad ρ)//CG(ρ(Γ))) 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 CG(ρ(Γ)) is the centralizer of ρ(Γ) in G. Let M be an orientable 3-manifold with a connected boundary F of genus g ≥ 2. Let X G(F ) be the subset of the G-character variety of π1(F ) composed of conjugacy classes of good representations, that is irreducible representations ρ : Γ → G such that the centralizer of ρ(Γ) is the center of G. By a theorem of Goldman, X G(F ) is a holomorphic symplectic manifold. The main goal of this paper is to prove that the set of good G-representations of π1(F ) which extend to representations of π1(M) is a complex isotropic submanifold of X g G(F ). Furthermore, if these representations correspond to reduced points of the Gcharacter variety of M , then this submanifold is Lagrangian. This result has important applications to Chern-Simons theory and quantum topology.