We study the cohomological equation for discrete
parabolic actions on some higher rank Lie groups via representation theory. Speci cally, 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 G = SL(n, R) are sharp up to nite loss of regularity.
Moreover, we prove that 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.
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