On conjugation orbits of semisimple pairs in rank one

Abstract

AbstractWe consider the Lie groups {\mathrm{SU}(n,1)} and {\mathrm{Sp}(n,1)} that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in {\mathrm{Sp}(n,1)} and {\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in {{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both {\rho(x)} and {\rho(y)} are semisimple elements in G, where {{\mathrm{F}}_{2}=\langle x,y\rangle}, {G=\mathrm{Sp}(n,1)} or {\mathrm{SU}(n,1)}. We use the {{\mathrm{PSp}}(n,1)}-configuration space {{\mathrm{M}}(n,i,m-i)} of ordered m-tuples of distinct points in {\overline{{\mathbf{H}}_{{\mathbb{H}}}^{n}}}, where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on {{\mathrm{M}}(n,i,m-i)}.