Conformal measure rigidity for representations via self-joinings

Let Γ 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 ρ with a boundary map f defined on the limit set Λ, we ask whether the extendability of ρ to $G_1$ can be detected by the property that f pushes forward some Γ-conformal measure class $\left[{\nu }_{\Gamma }\right]$ to a $\rho \left(\Gamma \right)$-conformal measure class $\left[{\nu }_{\rho \left(\Gamma \right)}\right]$. When Γ is of divergence type in a rank one group or when ρ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining ${\Gamma }_{\rho }=(\text{id}\times \rho )\left(\Gamma \right)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures ${(\text{id}\times f)}_{⁎}{\nu }_{\Gamma }$ and ${(f^{-1}\times \text{id})}_{⁎}{\nu }_{\rho \left(\Gamma \right)}$, which are higher rank ${\Gamma }_{\rho }$-conformal measures, cannot be in the same measure class.