  For even dimensional conformal manifolds several new conformally invariant
objects were found recently: invariant differential complexes related to, but
distinct from, the de Rham complex (these are elliptic in the case of
Riemannian signature); the cohomology spaces of these; conformally stable form
spaces that we may view as spaces of conformal harmonics; operators that
generalise Branson's Q-curvature; global pairings between differential form
bundles that descend to cohomology pairings. Here we show that these operators,
spaces, and the theory underlying them, simplify significantly on conformally
Einstein manifolds. We give explicit formulae for all the operators concerned.
The null spaces for these, the conformal harmonics, and the cohomology spaces
are expressed explicitly in terms of direct sums of subspaces of eigenspaces of
the form Laplacian. For the case of non-Ricci flat spaces this applies in all
signatures and without topological restrictions. In the case of Riemannian
signature and compact manifolds, this leads to new results on the global
invariant pairings, including for the integral of Q-curvature against the null
space of the dimensional order conformal Laplacian of Graham et al..
