The author’s abstract: “Let \(\Gamma\) be the fundamental group of a hyperbolic surface of genus g; for \(1\leq p\leq q\), PSU(p,q) is the group of isometries of a certain Hermitian symmetric space \(D_{p,q}\) of rank p. There exists a characteristic number c: Hom(\(\Gamma\),PSU(p,q))\(\to {\mathbb{R}}\), which is constant on each connected component and such that \(| c(\rho)| \leq 4p\pi (g-1)\) for every representation \(\rho\). We show that representations with maximal characteristic number (plus some nondegeneracy condition if \(p>2)\) leave invariant a totally geodesic subspace of \(D_{p,q}\) isometric to \(D_{p,p}\).”