Toledo invariant of lattices in SU(2,1) via symmetric square

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existence of holomorphic horizontal lift to various period domains of $SU(n,2)$.