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

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₎, m ≥ 4. For any unitary representation ${(\pi,\mathcal{H})}$ of ${\mathbb{G}}$ without non-trivial ${\mathbb{G}}$-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 ${\mathfrak{G}}$, the Lie algebra of ${\mathbb{G}}$ 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 ${\mathfrak{G}}$ and g and h are given vectors in ${\mathcal{H}}$. We give the sufficient condition under which the cocycle equation has a common solution: (∗) if ${\mathfrak{u}}$ and ${\mathfrak{v}}$ embed in ${\mathfrak{sl}(2,\mathbb{R})\times \mathbb{R}}$, then the common solution exists. In fact, condition (∗) is also necessary when ${\mathbb{G}=SL(n,\mathbb{R})}$. We show counter examples in each ${SL(n,\mathbb{R})}$, n ≥ 3 if condition (∗) is not satisfied. Especially, the cocycle rigidity always fails for ${SL(3,\mathbb{R})}$. As an application, we obtain smooth cocycle rigidity for higher rank parabolic actions over ${\mathbb{G}/\Gamma}$ if the Lie algebra of the acting parabolic subgroup contains a pair ${\mathfrak{u}}$ and ${\mathfrak{v}}$ satisfying property (∗). The main new ingredient in the proof is making use of unitary duals of ${SL(2,\mathbb{R})\ltimes\mathbb{R}^2}$ and ${(SL(2,\mathbb{R})\ltimes\mathbb{R}^2)\ltimes\mathbb{R}^3}$ obtained by Mackey theory.