Canonical Representations and Overgroups for Hyperboloids

For the hyperboloid $\mathcal{X}=G/H$, where $G=\operatorname{SO}_0(p,q)$ and $H=\operatorname{SO}_0(p,q-1)$, we define canonical representations $R_{\lambda,\nu}$, $\lambda\in\mathbb{C}$, $\nu=0,1$, as the restrictions to $G$ of representations $\widetilde{R}_{\lambda,\nu}$, associated with a cone, of the group $\widetilde{G}=\operatorname{SO}_0(p+1,q)$. They act on functions on the direct product $\Omega$ of two spheres of dimensions $p-1$ and $q-1$. The manifold $\Omega$ contains two copies of $\mathcal{X}$ as open $G$-orbits. We explicitly describe the interaction of the Lie operators of the group $\widetilde{G}$ in $\widetilde{R}_{\lambda,\nu}$ with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations $R_{\lambda,\nu}$ with representations of $G$ associated with a cone.