On Conjugation orbits of semisimple pairs in rank one

Abstract: We consider Lie groups ${\rm SU}(n,1)$ and ${\rm Sp}(n,1)$ that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in ${\rm Sp}(n,1)$ and ${\rm SU}(n,1)$ up to conjugacy. This gives local parametrization of the representations $\rho$ in $Hom(F_2, G)/G$ such that both $\rho(x)$ and $\rho(y)$ are hyperbolics, where $F_2=\langle x, y\rangle$, $G={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We use the ${\rm PSp}(n,1)$-configuration space $M(n,i,m-i)$ of ordered $m$-tuples of points on $\overline{{\bf H}_{\mathbb H}^n}$, where first $i$ points in an $m$-tuple are null points, to classify the semisimple pairs. Further, we also classify points on $M(n,i,m-i)$.