On the mapping class group action on the cohomology of the representation space of a surface

The mapping class group of a $d$-pointed Riemann surface has a natural $C^{\infty }$ action on any moduli space of parabolic bundles with the marked points as the parabolic points. We prove that under some numerical conditions on the parabolic data, the induced action of the mapping class group on the cohomology algebra of the moduli space of parabolic bundles factors through the natural epimorphism of the mapping class group onto the symplectic group.