Character varieties

We study properties of irreducible and completely reducible representations of finitely generated groups Γ into reductive algebraic groups G. In particular, we study the geometric invariant theory of the G action on the space of G-representations of Γ by conjugation. Let be the G-character variety of Γ. We prove that for every completely reducible, scheme smooth: G where H ¹ is the first cohomology group of Γ with coefficients in the Lie algebra g of G twisted by and SΓ is the centralizer of in G. The condition of ρ being scheme smooth is very important as there are groups Γ such that dim for a Zariski open subset of points in XG(Γ). We prove, however, that all irreducible representations of surface groups are scheme smooth. Let M be an orientable 3-manifold with a connected boundary F of genus g ≥ 2. Let X ^g _G(F) be the subset of the G-character variety of π1(F) composed of conjugacy classes of good representations ρ: Γ → G, i.e., irreducible representations such that the centralizer of is the center of G. By a theorem of Goldman, X ^g _G(F) is a holomorphic symplectic manifold. We prove that the set of good G-representations of π1(F) which extend to representations of π1(M) is a complex isotropic subspace of X ^g _G(F). It is Lagrangian, if these representations correspond to reduced points of the G-character variety of M. It is an open problem whether it is always the case. 2012 American Mathematical Society.