Canonical Representations and Overgroups for Hyperboloids

Abstract

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 ∈ ℂ, 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.