Cohomological equation and cocycle rigidity of parabolic actions in $SL(n,\RR)$

Abstract

For any unitary representation $(\pi,\mathcal{H})$ of $G=SL(n,\RR)$, $n\geq 3$ without non-trivial $G$-invariant vectors, we study smooth solutions of the cohomological equation $\mathfrak{u}f=g$ where $\mathfrak{u}$ is a vector in the root space of $\mathfrak{sl}(n,\RR)$ and $g$ is a given vector in $\mathcal{H}$. We characterize the obstructions to solving the cohomological equation, construct smooth solutions of the cohomological equation and obtain tame Sobolev estimates for $f$. We also study common solutions to (the infinitesimal version of) the cocycle equation $\mathfrak{u}h=\mathfrak{v}g$, where $\mathfrak{u}$ and $\mathfrak{v}$ are commutative vectors in different root spaces of $\mathfrak{sl}(n,\RR)$ and $g$ and $h$ are given vectors in $\mathcal{H}$. We give precisely the condition under which the cocycle equation has common solutions: $(*)$ if $\mathfrak{u}$ and $\mathfrak{v}$ embed in $\mathfrak{sl}(2,\RR)\times \RR$, then the common solution exists. Otherwise, we show counter examples in each $SL(n,\RR)$, $n\geq 3$. As an application, we obtain smooth cocycle rigidity for higher rank parabolic actions over $SL(n,\RR)/\Gamma$, $n\geq 4$ if the Lie algebra of the acting parabolic subgroup contains a pair $\mathfrak{u}$ and $\mathfrak{v}$ satisfying property $(*)$ and prove that the cocycle rigidity fails otherwise. Especially, the cocycle rigidity always fails for $SL(3,\RR)$. The main new ingredient in the proof is making use of unitary duals of various subgroup in $SL(n,\RR)$ isomorphic to $SL(2,\RR)\ltimes\RR^2$ or $(SL(2,\RR)\ltimes\RR^2)\ltimes\RR^3$ obtained by Mackey theory.