Hyperbolic spaces, principal series and ${\rm O}(2,\infty)$

Abstract

We prove that there exists no irreducible representation of the identity component of the isometry group ${\rm PO}(1,n)$ of the real hyperbolic space of dimension $n$ into the group ${\rm O}(2,\infty)$, if $n\geq 3$. This is motivated by the existence of irreducible representations (arising from the spherical principal series) of ${\rm PO}(1,n)^{\circ}$ into the groups ${\rm O}(p,\infty)$ for other values of $p$.