Rigidity of conformal measure preserving representations via self-joinings

I will discuss a rigidity problem for representations of discrete Zariski dense subrgoups, 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 has found a new application to Anosv representations, in particular to Hitchin representations. (Based on joint work with Dongryul Kim)