Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications

We realize all irreducible unitary representations of the group SO₀ (n+1, 1) on explicit Hilbert spaces of vector-valued L²-functions on Rⁿ\0\. The key ingredient in our construction is an explicit expression for the standard Knapp-Stein intertwining operators between arbitrary principal series representations in terms of the Euclidean Fourier transform on a maximal unipotent subgroup isomorphic to Rⁿ. As an application, we describe the space of Whittaker vectors on all irreducible Casselman-Wallach representations. Moreover, the new realizations of the irreducible unitary representations immediately reveal their decomposition into irreducible representations of a parabolic subgroup, thus providing a simple proof of a recent result of Liu-Oshima-Yu.