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On sporadic symmetry breaking operators for principal series representations of the de Sitter and Lorentz groups

Víctor Pérez-Valdés

Abstract

In this paper, we construct and classify all differential symmetry breaking operators between certain principal series representations of the pair $SO_0(4,1) \supset SO_0(3,1)$. In this case, we also prove a localness theorem, namely, all symmetry breaking operators between the principal series representations in concern are necessarily differential operators. In addition, we show that all these symmetry breaking operators are sporadic in the sense of T. Kobayashi, that is, they cannot be obtained by residue formulas of meromorphic families of symmetry breaking operators.

On sporadic symmetry breaking operators for principal series representations of the de Sitter and Lorentz groups

Abstract

In this paper, we construct and classify all differential symmetry breaking operators between certain principal series representations of the pair . In this case, we also prove a localness theorem, namely, all symmetry breaking operators between the principal series representations in concern are necessarily differential operators. In addition, we show that all these symmetry breaking operators are sporadic in the sense of T. Kobayashi, that is, they cannot be obtained by residue formulas of meromorphic families of symmetry breaking operators.

Paper Structure

This paper contains 21 sections, 37 theorems, 250 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Main results
  4. Review of Principal Series Representations of the de Sitter and Lorentz Groups
  5. Applying the F-method
  6. Description of $\text{\normalfont{Sol}}(\mathfrak{n}_+; \sigma_\lambda^{2N+1}, \tau_{m, \nu})$ for $m > N$
  7. Proof of Theorems \ref{['thm_main_class']} and \ref{['thm_main_const']} for $m > N$
  8. Proof of Corollary \ref{['corol_main']}
  9. Reduction Theorems
  10. Proof of Theorem \ref{['Thm-solvingequations']}: Solving the System $\Xi(\lambda, a, N, m)$
  11. Phase 1: Obtaining $f_{\pm N}$ up to constant–Necessary condition on $\lambda$
  12. Phase 2: Obtaining all $f_{\pm j}$ up to constant
  13. Phase 3: Proving $f_{-0} = f_{+0}$
  14. Proof of Theorem \ref{['Thm-solvingequations']}
  15. The case $m < -N$
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $\lambda, \nu \in {\mathbb C}$, $N \in {\mathbb N}$ and $m \in {\mathbb Z}$, and suppose that $|m| > N$. Then, the following three conditions on the quadruple $(\lambda, \nu, N, m)$ are equivalent:

Figures (5)

  • Figure 1.1: Distribution of the parameters $(\lambda, \nu, N, m)$ satisfying (iii) of Theorem \ref{['thm_main_class']}
  • Figure 4.2: Hierarchy for $N = 1$
  • Figure 4.3: Hierarchy for $N = 2$
  • Figure 4.4: Hierarchy for $N = 3$
  • Figure 4.5: Hierarchy for $N = 4$

Theorems & Definitions (73)

  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6: Localness Theorem
  • Definition 3.1: KP16a
  • Theorem 3.2: F-method, KP16a
  • Proposition 3.3: Per-Val24
  • Theorem 3.4
  • proof : Proof of Theorem \ref{['thm_main_class']} for $m > N$
  • proof : Proof of Theorem \ref{['thm_main_const']} for $m>N$
  • ...and 63 more