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Pseudo-Riemannian Spectral Triples for $\mathrm{SU}(1,1)$

Jort de Groot

TL;DR

This work constructs both a pseudo-Riemannian spectral triple and an indefinite spectral triple for the noncompact Lie group SU(1,1) using Kostant's cubic Dirac operator. By leveraging the harmonic analysis of SU(1,1) and its Plancherel decomposition, the author obtains a fiberwise, diagonal action on matrix elements across tempered representations, enabling explicit verification of the necessary operator-theoretic properties. A key simplification arises from the cubic Dirac operator for which the second-order obstruction $R_D$ vanishes, and the real/imaginary parts of D anti-commute, facilitating compact resolvent arguments and the doubling construction to obtain even indefinite triples. The results provide a concrete, analysis-driven realization of pseudo-Riemannian and indefinite spectral triples in a noncompact setting and suggest avenues for extending to homogeneous spaces and quantum-group contexts. Overall, the paper advances the understanding of spectral triples beyond the Riemannian compact case by harnessing representation-theoretic techniques.

Abstract

We use the harmonic analysis of $\mathrm{SU}(1,1)$ to show that the triple $(\mathcal{A},\mathcal{H},D)$, with $D$ (the closure of) Kostant's cubic Dirac operator acting on the Hilbert space $\mathcal{H}=L^2(\mathrm{SU}(1,1))\otimes\mathbb{C}^2$, and with $*$-algebra $\mathcal{A}=C^\infty_c(\mathrm{SU}(1,1))\otimes 1$, forms both a pseudo-Riemannian spectral triple in the sense of Van den Dungen, Paschke and Rennie, and an indefinite spectral triple in the sense of Van den Dungen and Rennie.

Pseudo-Riemannian Spectral Triples for $\mathrm{SU}(1,1)$

TL;DR

This work constructs both a pseudo-Riemannian spectral triple and an indefinite spectral triple for the noncompact Lie group SU(1,1) using Kostant's cubic Dirac operator. By leveraging the harmonic analysis of SU(1,1) and its Plancherel decomposition, the author obtains a fiberwise, diagonal action on matrix elements across tempered representations, enabling explicit verification of the necessary operator-theoretic properties. A key simplification arises from the cubic Dirac operator for which the second-order obstruction vanishes, and the real/imaginary parts of D anti-commute, facilitating compact resolvent arguments and the doubling construction to obtain even indefinite triples. The results provide a concrete, analysis-driven realization of pseudo-Riemannian and indefinite spectral triples in a noncompact setting and suggest avenues for extending to homogeneous spaces and quantum-group contexts. Overall, the paper advances the understanding of spectral triples beyond the Riemannian compact case by harnessing representation-theoretic techniques.

Abstract

We use the harmonic analysis of to show that the triple , with (the closure of) Kostant's cubic Dirac operator acting on the Hilbert space , and with -algebra , forms both a pseudo-Riemannian spectral triple in the sense of Van den Dungen, Paschke and Rennie, and an indefinite spectral triple in the sense of Van den Dungen and Rennie.
Paper Structure (12 sections, 14 theorems, 73 equations)

This paper contains 12 sections, 14 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. The cubic Dirac operator on $\mathrm{SU}(1,1)$
  3. Geometric formulation
  4. Spectral decomposition
  5. The Plancherel decomposition
  6. Spectral triples
  7. Pseudo-Riemannian spectral triple
  8. Indefinite spectral triple
  9. Proofs
  10. Pseudo-Riemannian spectral triple
  11. Indefinite spectral triple
  12. The weighted resolvent

Key Result

Theorem 3.1

The Fourier transform $f\rightarrow\hat{f}$ maps $L^1(\mathrm{SU}(1,1))\cap L^2(\mathrm{SU}(1,1))$ into $\int^\oplus\mathcal{H}_\pi\hat{\otimes}\mathcal{H}_{\bar{\pi}}d\mu(\pi)$, and extends to a unitary map from $L^2(\mathrm{SU}(1,1))$ onto $\int^\oplus\mathcal{H}_\pi\hat{\otimes}\mathcal{H}_{\bar{ with $dl$ the counting measure on $\tfrac{1}{2}\mathbb{Z}_{<0}$ and $d\rho$ the Lebesgue measure on

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1: Plancherel theorem for $\mathrm{SU}(1,1)$ harish-chandra_1952
  • Definition 4.1: dungen_paschke_rennie_2013
  • Theorem 4.2
  • Remark 4.3
  • Definition 4.4: dungen_2019
  • Remark 4.5
  • Theorem 4.6
  • ...and 26 more