  We show that for a parabolic R^d-action on a compact quotient of PSL(2,R)^d,
the cohomologies in degrees 1 through d-1 trivialize, and we give the
obstructions to solving the degree-d coboundary equation, along with bounds on
Sobolev norms of primitives. In previous papers we have established these
results for certain Anosov systems. The present work extends the methods of
those papers to systems that are not Anosov. The main new idea is in Section 4,
where we define special elements of representation spaces that allow us to
modify the arguments from the previous papers. In Section 7 we discuss how one
may generalize this strategy to R^d-systems coming from a product of Lie
groups, like in the systems we have here.
