The modular group $\Gamma\_{g}$ of isotopy classes of diffeomorphisms of a genus $g$ surface $S$ acts on $H^1(S,\mathbb{Q})$ (or $H^1(S,\mathbb{Z})$) respecting the intersection pairing. This gives a homomorphism $\Gamma\_{g}\to Sp(2g)$. We also know that this homomorphism is surjective. Therefore, to understand $H^1(S,\mathbb{Q})$ as a representation of the modular group it suffices to view it as a representation of $Sp(2g)$.


Has this representation been studied somewhere? E.g. is the decomposition in irreducible representations known? I could not find this anywhere, so I apologize if this is all well-known stuff, but it is rather far from what I usually deal with. I would really appreciate a reference for these things.


Also, is it known if the map $\Gamma\_{g}\to Sp(2g)$ is surjective when we restrict it to the hyperelliptic modular group? If yes, can we at least in that case understand the representation? 



Symplectic representation of modular group
For your last question: the map from the hyperelliptic modular group to $\operatorname{Sp}(2g,\mathbb{Z}) $ is *not* surjective as soon as $g\geq 3$. This was proved by V. Arnold,
*A remark on the branching of hyperelliptic integrals as functions of the parameters*,
Functional Anal. Appl. 2 (1968), 187–189.


