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

2024/06/17 by Christian Arends, Arends, Christian, Frederik Juul Bang-Jensen +3
Computer Science · #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2406.11349

Abstract

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.

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