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

Abstract

We realize all irreducible unitary representations of the group ${\mathrm{SO}}_0(n+1,1)$ on explicit Hilbert spaces of vector-valued $L^2$-functions on $R^n\setminus \left\{0\right\}$. 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 $R^n$.

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.