Character varieties and Lagrangian submanifolds.

Let \( \Gamma \) be a finitely generated group, and let \( G\) be a reductive affine algebraic group over \(C\).The set of group homomorphisms \(Hom(\Gamma, G) \) together with an affine algebraic structure coming from \(G\) is called \textit{representation variety} of \(\Gamma\) in \(G\). \(G\) acts on \(Hom(\Gamma, G) \) via conjugation. The categorical quotient \(X_G(\Gamma) \) \// G\) is called \textit{character variety}. For the fundamental group \(\pi_1(S)\) of a surface \(S\), the character variety of good represantations, i.e. irreducible representations with closed orbits under the \(G\)-action, is a complex manifold \(X_G^g(\pi_1(S))\) with tangent spaces \(H^1(\pi_1(S), {g}_{Ad, \rho})\). Using a non-degenerate bilinear form \(B: {g} \times {g} \rightarrow \mathbb{C}\), William Goldmam constructed a symplectic form \(\omega^B\) on \(X_G^g(\pi_1(S))\). We will use a mostly algebro-geometric approach in order to understand the construction of this symplectic form.

Afterwards, we will turn to the Lagrangian submanifold theorem as it has been proven by Adam Sikora:

Consider a compact connected 3-manifold \(M\) with boundary \(\partial M = S\). Then the embedding \(S \rightarrow M\)  induces a map \(r* : X_G^g(\pi_1(M)) \rightarrow X_G^g(\pi_1(S)) \) on the character varieties. The non-singular part of the image \(\Y_G(M) = \left[ r*X_G^g(\pi_1(M))\cap X_G^g(\pi_1(S))\right]^{ns}\) is an isotropic submanifolg of \(X_G^g(\pi_1(S))\). We will see under which circumstances this isotropic submanifold is Lagrangian. Another Lagrangian submanifold is the fixed point set \(\mathcal{L}_G\) of a certain anti-symplectic involution on \(X_G^g(\pi_1(S))\). We will demonstrate the construction of a 3-manifold \(M\) by Laura Schaposnik and David Baraglia , for which we have \( Y_G(M)\subset \mathcal{L}_G\) . Finally, we will briefly review the generalzation of the symplectic form \(\omega^B\) to a compact connected Kähler manifold as proven by Yael Karshon. We will conclude by discussing to which extent the Lagrangian submanifold theorem can be generalized in this case.