Canonical Representations and Overgroups for Hyperboloids

For the hyperboloid \(X = G/H\), where G = SO₀(p, q) and H = SO₀(p, q ? 1), we define canonical representations R_λ,ν λ ∈ ?, ν = 0, 1, as the restrictions to G of representations \(\tilde R\lambda ,\nu\), associated with a cone, of the group \(\tilde G = \operatorname{SO} _0 (p + 1,q)\). They act on functions on the direct product Ω of two spheres of dimensions p ? 1 and q ? 1. The manifold Ω contains two copies of \(X\) as open G-orbits. We explicitly describe the interaction of the Lie operators of the group \({\tilde G}\) in \(\tilde R\lambda ,\nu\) with the Poisson and Fourier transforms associated with the canonical representations. These transforms are operators intertwining the representations R_λ,ν with representations of G associated with a cone.