Conformal measure rigidity for representations via self-joinings

Abstract

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\rho:\Gamma\to G_2$ and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations. More precisely, for a given representation $\rho$ with a boundary map $f$ defined on the limit set $\Lambda$, we ask whether the extendability of $\rho$ to $G$ can be detected by the property that $f$ pushes forward some $\Gamma$-conformal measure class $[\nu_\Gamma]$ to a $\rho(\Gamma)$-conformal measure class $[\nu_{\rho(\Gamma)}]$. When $\Gamma$ is of divergence type in a rank one group or when $\rho$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $\Gamma_\rho=(\text{id} \times \rho)(\Gamma)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f)_*\nu_\Gamma$ and $(f^{-1}\times \text{id})_*\nu_{\rho(\Gamma)}$, which are higher rank $\Gamma_\rho$-conformal measures, cannot be in the same measure class.