On conjugation orbits of semisimple pairs in rank one

Abstract

We consider the Lie groups SU(n,1) and Sp(n,1) that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in Sp(n,1) and SU(n,1) up to conjugacy. This gives local parametrization of the representations ρ in Hom(F2,G)/G such that both ρ(x) and ρ(y) are semisimple elements in G, where F2=⟨x,y⟩, G=Sp(n,1) or SU(n,1). We use the PSp(n,1)-configuration space M(n,i,m−i) of ordered m-tuples of distinct points in ¯¯¯¯¯¯HnH, where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M(n,i,m−i).