Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}\_{flat}(\Sigma)$ and $\overline {\mathcal A}\_{flat}(Y)$ denote the gauge equivalence classes of flat connections on the trivial bundles $\Sigma \times G$ and $Y \times G$ respectively. The space $\overline {\mathcal A}\_{flat}(\Sigma)$ is a finite dimensional symplectic manifold with singularitites. I am trying to understand why the image of the restriction map $\overline {\mathcal A}\_{flat}(Y) \to \overline {\mathcal A}\_{flat}(\Sigma)$ is a Lagrangian submanifold.
I know why the image is an isotropic subspace, but I am having a problem with the dimensions. In particular, the space
$$\overline {\mathcal A}\_{flat}(\Sigma)\simeq \operatorname{Hom}(\pi\_1(\Sigma),G)/G$$
has dimension $(2\operatorname{genus}(\Sigma)-2)\dim(G)$, but I can not see why the space $\operatorname{Hom}(\pi\_1(Y),G)/G$ has half that dimension.


For completeness, I include a proof of why the image of $\overline {\mathcal A}\_{flat}(Y)$ in $\overline {\mathcal A}\_{flat}(\Sigma)$ under the restriction map is isotropic.
Let $\mathcal A(Y)$ be the space of connections on $Y \times G$. There is a one-form $\mathcal F$ on $\mathcal A(Y)$ defined as 
$${\mathcal F}\_A(\alpha):=\int\_Y \langle \alpha \wedge F\_A\rangle,$$
where $A \in \mathcal A(Y)$ is a connection, $F\_A \in \Omega^2(Y,\mathfrak g)$ is its curvature and $\alpha \in T\_A\mathcal A(Y)$. We can see by some calculations that
$$d\mathcal F\_A(\alpha,\beta)=\int\_\Sigma \langle \alpha \wedge \beta \rangle,$$
where $\alpha, \beta \in T\_A\mathcal A(Y)$, which coincides with the symplectic form on $\mathcal A(\Sigma)$ the space of connections on $\Sigma \times G$. When restricted to $\mathcal A\_{flat}(Y)$, $\mathcal F$ vansihes, and so does $d\mathcal F$. Therefore the restriction map takes $\mathcal A\_{flat}(Y)$ to an isotropic submanifold of $\mathcal A\_{flat}(\Sigma)$. By gauge invariance, the restriction map takes $\overline {\mathcal A}\_{flat}(Y)$ to an isotropic submanifold of $\overline {\mathcal A}\_{flat}(\Sigma)$.



Flat connections on 3-manifold with boundary
Another good reference is Chris Herald's paper. Legendrian cobordism and Chern-Simons theory on 3-manifolds with boundary. Comm. Anal. Geom. 2 (1994), no. 3, 337–413. 


It is an easy exercise with Poincare duality to see that the image of the map in cohomology (so at the level of "Zaraski" tangent spaces)
$$
H^1(Y;ad\_\rho) \to H^1(\Sigma;ad\_\rho)
$$
is half dimensional. Indeed this map appears in the long exact sequence 
of the pair $(Y,\Sigma)$ 
$$
H^1(Y;ad\_\rho) \to H^1(\Sigma;ad\_\rho)\to H^2(Y,
\Sigma;ad\_\rho)
$$
and the two arrows are dual by PD so have the same rank. The exactness tells you
the sum of these ranks is $dim(H^1(\Sigma;ad\_\rho))$.


Here $\rho$ is a representation of $\pi\_1(Y)\to G$ corresponding to the given flat connection and $ad\_\rho$ is the local system with fiber $\mathfrak{g}$ corresponding to the adjoint action of $\rho$ on $\mathfrak{g}$.


