On conjugation orbits of semisimple pairs in rank one

Abstract

Abstract We consider the Lie groups SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} and Sp ⁢ ( n , 1 ) {\mathrm{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 ) {\mathrm{Sp}(n,1)} and SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in Hom ⁢ ( F 2 , G ) / G {{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both ρ ⁢ ( x ) {\rho(x)} and ρ ⁢ ( y ) {\rho(y)} are semisimple elements in G, where F 2 = 〈 x , y 〉 {{\mathrm{F}}_{2}=\langle x,y\rangle} , G = Sp ⁢ ( n , 1 ) {G=\mathrm{Sp}(n,1)} or SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} . We use the PSp ⁢ ( n , 1 ) {{\mathrm{PSp}}(n,1)} -configuration space M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} of ordered m-tuples of distinct points in 𝐇 ℍ n ¯ {\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 M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} .