Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications
Year: 2026
Type: article
Abstract: We realize all irreducible unitary representations of the group SO(n+1,1)_0 on explicit Hilbert spaces of vector-valued L^2-functions on R^n\{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 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.
Source: Advances in Mathematics
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
OpenAlex ID: W7130361562