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

Jan Frahm, Karl-Hermann Neeb, Gestur Olafsson

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 $G = SO_{1,d}(R)_e$ acting on de Sitter space $M = dS^d$. 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 $G_C^{cut}$ in the complexified group $G_C = \SO_{1,d}(C)$, which for $d = 1$ specializes to the complex cut plane $C \setminus (-\infinity, 0]$. 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.

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

Abstract

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups acting on a non-compactly causal symmetric space , every irreducible unitary representation of can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over . In the present paper we discuss this procedure for the connected Lorentz group acting on de Sitter space . 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 in the complexified group , which for specializes to the complex cut plane . A number of special situations is discussed specifically: (a) The case , which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for 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 , where we obtain rather explicit formulas for the matrix-valued spherical functions.
Paper Structure (35 sections, 44 theorems, 335 equations)

This paper contains 35 sections, 44 theorems, 335 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Notation
  4. The modular flow
  5. Lie group data
  6. Crown domains and tubes
  7. SL$_2$-notation
  8. Distribution vectors by analytic continuation
  9. Realization in vector bundles
  10. The cut domain in de Sitter space
  11. Analytic continuation of the generalized spherical function
  12. Existence of the analytic continuation
  13. Boundary behavior of the analytic continuation
  14. Locality of the net
  15. Spacelike pairs in de Sitter space
  16. ...and 20 more sections

Key Result

Lemma 3.1

$A_{\mathbb C}^{\rm{cut}}.i{\bf{e}}_0 = \exp(\mathcal{S}_{\pm \pi}h).i{\bf{e}}_0 = \{ z \in A_{\mathbb C}.i{\bf{e}}_0 \colon -i z_0 \not\in (-\infty, -1]\}.$

Theorems & Definitions (95)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • ...and 85 more