  We consider parabolic flows on 3-dimensional manifolds which are renormalized
by circle extensions of Anosov diffeormorphisms. This class of flows includes
nilflows on the Heisenberg nilmanifold which are renormalized by partially
hyperbolic automorphisms. The transfer operators associated to the
renormalization maps, acting on anisotropic Sobolev spaces, are known to have
good spectral properties (this relies on ideas which have some resemblance to
representation theory but also apply to non-algebraic systems). The spectral
information is used to describe the deviation of ergodic averages and solutions
of the cohomological equation for the parabolic flow.
