Realization of unitary representations of the Lorentz group on de Sitter space

Frahm J, Neeb KH, Ólafsson G (2025)

Publication Type: Journal article

Publication year: 2025

Journal

Indagationes Mathematicae Elsevier

Book Volume: 36

Pages Range: 61-113

Journal Issue: 1

DOI: 10.1016/j.indag.2024.04.002

Abstract

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups G acting on a non-compactly causal symmetric space M=G/H, every irreducible unitary representation of G can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over M. In the present paper we discuss this procedure for the connected Lorentz group G=SO1,d(R)e acting on de Sitter space M=dSd. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990’s, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain Gℂcut in the complexified group Gℂ=SO1,d(ℂ), which for d=1 specializes to the complex cut plane ℂ∖(−∞,0]. A number of special situations is discussed specifically: (a) The case d=1, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for d>2 is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case d=3, where we obtain rather explicit formulas for the matrix-valued spherical functions.

Authors with CRIS profile

Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

Involved external institutions

Aarhus University DK Denmark (DK) Louisiana State University US United States (USA) (US)

How to cite

APA:

Frahm, J., Neeb, K.H., & Ólafsson, G. (2025). Realization of unitary representations of the Lorentz group on de Sitter space. Indagationes Mathematicae, 36(1), 61-113. https://doi.org/10.1016/j.indag.2024.04.002

MLA:

Frahm, Jan, Karl Hermann Neeb, and Gestur Ólafsson. "Realization of unitary representations of the Lorentz group on de Sitter space." Indagationes Mathematicae 36.1 (2025): 61-113.

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