  We determine the action of the Torelli group on the equivariant cohomology of
the space of flat SL(2,C) connections on a closed Riemann surface. We show that
the trivial part of the action contains the equivariant cohomology of the even
component of the space of flat PSL(2,C) connections. The non-trivial part
consists of the even alternating products of degree two Prym representations,
so that the kernel of the action is precisely the Prym-Torelli group. We
compute the Betti numbers of the ordinary cohomology of the moduli space of
flat SL(2,C) connections. Using results of Cappell-Lee-Miller we show that the
Prym-Torelli group, which acts trivially on equivariant cohomology, acts
non-trivially on ordinary cohomology.
