Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups

TL;DR

This paper investigates the cohomological equation and cocycle rigidity for discrete parabolic actions on higher rank Lie groups, providing new insights into obstructions, solution estimates, and conditions for trivial first cohomology.

Contribution

It introduces novel methods for analyzing tame and non-tame estimates, characterizes obstructions, and establishes conditions for trivial first cohomology in higher rank Lie groups.

Findings

Characterization of obstructions to solving the cohomological equation.

Construction of smooth solutions with Sobolev estimates.

Identification of sharp non-tame estimates in the case of SL(n, R).

Abstract

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.