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

TL;DR

This paper constructs explicit Hilbert space models for all irreducible unitary representations of the rank one orthogonal group SO_0(n+1,1), using Fourier analysis to facilitate applications like Whittaker models and decomposition results.

Contribution

It provides explicit realizations of all irreducible unitary representations of SO_0(n+1,1) on vector-valued L^2-spaces, with a novel use of Fourier transforms for intertwining operators.

Findings

Explicit Hilbert space models for all irreducible unitary representations.

Explicit formulas for Knapp-Stein intertwining operators.

Simplified proof of decomposition into parabolic subgroup representations.

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 $\mathbb{R}^n\setminus\{0\}$. 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 $\mathbb{R}^n$. 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.