Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping class to the induced action on $H^1(\Sigma,\mathbf Z)$. Composing $t$ with any representation of the symplectic group produces a large number of linear representations of $\Gamma_g$.

These are only a small fraction of all representations of the mapping class groups. Others can for instance be obtained from 3D TQFTs or by from different constructions involving lower central series. My question is however whether the symplectic representations are the only ones that can be defined "algebro-geometrically".

Let me ask a more concrete question. A representation of $\Gamma_g$ is the same as a local system on the moduli space of curves of genus $g$, $M_g$. For a representation which factors through $\mathrm{Sp}(2g,\mathbf Z)$ this local system underlies a polarized variation of Hodge structure, since it is pulled back from a PVHS on the Shimura variety parametrizing principally polarized abelian varieties of genus $g$. Is the converse true - if a local system (say with $\mathbf Q$ coefficients) on $M_g$ underlies a PVHS, is it isomorphic to one of the symplectic local systems?