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

Abstract

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

8 pages