Canonical representations on two-sheeted hyperboloids

The two-sheeted hyperboloid $\mathcal L$ in $\mathbb R^n$ can be identified with the unit sphere $\Omega$ in $\mathbb R^n$ without the equator. Canonical representations of the group $G=\mathrm{SO}_0(n-1,1)$ on $\mathcal L$ are defined as the restrictions to $G$ of the representations of the overgroup $\widetilde G=\mathrm{SO}_0(n,1)$ associated with a cone. They act on functions and distributions on the sphere $\Omega$. We decompose these canonical representations into irreducible constituents and decompose the Berezin form.