  We study the cohomological equation for discrete horocycle maps on $SL(2,
\mathbb{R})$ and $SL(2,\mathbb R)\times SL(2, \mathbb{R})$ via representation
theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions
of the cohomological equation of horocycle maps in representations of
$SL(2,\mathbb R)$. Our estimates improve on previous results and are sharp up
to a fixed, finite loss of regularity. Moreover, they are tame on a
co-dimension one subspace of $sl(2, \mathbb R)$, and we prove tame cocycle
rigidity for some two-parameter discrete actions, improving on a previous
result. Our estimates on the cohomological equation of horocycle maps overcome
difficulties in previous papers by working in a more suitable model for $SL(2,
\mathbb R)$ in which all cases of irreducible, unitary representations of
$SL(2, \mathbb R)$ can be studied simultaneously.
  Finally, our results combine with those of a very recent paper by the authors
to give cohomology results for discrete parabolic actions in regular
representations of some general classes of simple Lie groups, providing a
fundamental step toward proving differential local rigidity of parabolic
actions in this general setting.
