  Let $\mathbb{G}$ denote a higher rank $\mathbb R$-split simple Lie group of
the following type: $SL(n,\mathbb R)$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and
$E_{8(8)}$, where $m\geq 4$ and $n \geq 3$. We study the cohomological equation
for discrete parabolic actions on $\mathbb G$ via representation theory.
Specifically, we characterize the obstructions to solving the cohomological
equation and construct smooth solutions with Sobolev estimates. We prove that
global estimates of the solution are generally not tame, and our non-tame
estimates in the case $\mathbb G=SL(n,\mathbb R)$ are sharp up to finite loss
of regularity. Moreover, we prove that for general $\mathbb G$ the estimates
are tame in all but one direction, and as an application, we obtain tame
estimates for the common solution of the cocycle equations. We also give a
sufficient condition for which the first cohomology with coefficients in smooth
vector fields is trivial. In the case that $\mathbb G=SL(n,\mathbb R)$, we show
this condition is also necessary. A new method is developed to prove tame
directions involving computations within maximal unipotent subgroups of the
unitary duals of $SL(2,\mathbb R)\ltimes\mathbb R^2$ and $SL(2,\mathbb
R)\ltimes\mathbb R^4$. A new technique is also developed to prove non-tameness
for solutions of the cohomological equation.
