Character varieties

Abstract:

We study properties of irreducible and completely reducible representations of finitely generated groups $\Gamma$ into reductive algebraic groups $G.$ In particular, we study the geometric invariant theory of the $G$ action on the space of $G$-representations of $\Gamma$ by conjugation.

Let $X_G(\Gamma )$ be the $G$-character variety of $\Gamma .$ We prove that for every completely reducible, scheme smooth $\rho :\Gamma \to G$ \[ T_{[\rho ]} X_G(\Gamma )\simeq T_0 \big (H^1(\Gamma ,Ad \rho )//S_\Gamma \big ),\] where $H^1(\Gamma ,Ad \rho )$ is the first cohomology group of $\Gamma$ with coefficients in the Lie algebra $\mathfrak {g}$ of $G$ twisted by $\Gamma \stackrel {\rho }{\longrightarrow } G\stackrel {Ad}{\longrightarrow } GL(\mathfrak {g})$ and $S_\Gamma$ is the centralizer of $\rho (\Gamma )$ in $G.$ The condition of $\rho$ being scheme smooth is very important as there are groups $\Gamma$ such that \[ dim T_{[\rho ]} X_G(\Gamma )< T_0 H^1(\Gamma ,Ad \rho ),\] for a Zariski open subset of points in $X_G(\Gamma ).$ 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 $g\geq 2.$ Let $X_G^g(F)$ be the subset of the $G$-character variety of $\pi _1(F)$ composed of conjugacy classes of good representations $\rho : \Gamma \to G,$ i.e., irreducible representations such that the centralizer of $\rho (\Gamma )$ 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 $\pi _1(F)$ which extend to representations of $\pi _1(M)$ 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.