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

TL;DR

This paper investigates the solutions and obstructions to cohomological and cocycle equations in unitary representations of $SL(n,\RR)$, establishing conditions for rigidity and constructing smooth solutions with Sobolev estimates.

Contribution

It characterizes obstructions and solutions to cohomological equations in $SL(n,\RR)$ representations and determines when cocycle rigidity holds for higher rank parabolic actions.

Findings

Characterizes obstructions to solving the cohomological equation.

Constructs smooth solutions with tame Sobolev estimates.

Establishes conditions for cocycle rigidity in higher rank parabolic actions.

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.