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Construction and classification of differential symmetry breaking operators for principal series representations of the pair $(SO_0(4,1), SO_0(3,1))$ for special parameters

Víctor Pérez-Valdés

TL;DR

This work classifies differential symmetry breaking operators between principal series representations of $G=SO_0(4,1)$ and $G'=SO_0(3,1)$ for the special parameter regime $|m|=N$. Using the F-method, the authors reduce the problem to solving a polynomial/PDE system; they prove a sharp criterion: a nonzero operator exists iff $\nu-\lambda\in\mathbb{N}$ and the space is 1-dimensional. For the pivotal case $|m|=N$, they construct an explicit generator $\mathbb{D}_{\lambda,\nu}^{N,\pm N}$ in terms of renormalized Gegenbauer polynomials, with distinct expressions for $m=N$ and $m=-N$, and establish a duality between the two. The approach hinges on decomposing the problem into Step 1 (finding $L'$-equivariant polynomial generators) and Step 2 (solving the resulting differential equations), and then solving associated ODE systems to obtain complete classifications. A duality between $m$ and $-m$ reduces the analysis to the $m\ge N$ case, while the appendix collects the Gegenbauer toolkit essential to the explicit operator formulas. These results advance explicit branching tools in conformal and representation-theoretic settings, enabling precise construction of symmetry breaking operators in this Lorentzian geometry context.

Abstract

We construct and give a complete classification of all the differential symmetry breaking operators $\mathbb{D}_{λ, ν}^{N,m}: C^\infty(S^3, \mathcal{V}_λ^{2N+1}) \rightarrow C^\infty(S^2, \mathcal{L}_{m, ν})$, between the spaces of smooth sections of a vector bundle of rank $2N+1$ over the $3$-sphere $\mathcal{V}_λ^{2N+1} \rightarrow S^3$, and a line bundle over the $2$-sphere $\mathcal{L}_{m, ν} \rightarrow S^2$ in the special case $|m| = N$.

Construction and classification of differential symmetry breaking operators for principal series representations of the pair $(SO_0(4,1), SO_0(3,1))$ for special parameters

TL;DR

This work classifies differential symmetry breaking operators between principal series representations of and for the special parameter regime . Using the F-method, the authors reduce the problem to solving a polynomial/PDE system; they prove a sharp criterion: a nonzero operator exists iff and the space is 1-dimensional. For the pivotal case , they construct an explicit generator in terms of renormalized Gegenbauer polynomials, with distinct expressions for and , and establish a duality between the two. The approach hinges on decomposing the problem into Step 1 (finding -equivariant polynomial generators) and Step 2 (solving the resulting differential equations), and then solving associated ODE systems to obtain complete classifications. A duality between and reduces the analysis to the case, while the appendix collects the Gegenbauer toolkit essential to the explicit operator formulas. These results advance explicit branching tools in conformal and representation-theoretic settings, enabling precise construction of symmetry breaking operators in this Lorentzian geometry context.

Abstract

We construct and give a complete classification of all the differential symmetry breaking operators , between the spaces of smooth sections of a vector bundle of rank over the -sphere , and a line bundle over the -sphere in the special case .

Paper Structure

This paper contains 24 sections, 29 theorems, 224 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Main results
  4. Review of the F-method
  5. Principal Series Representations of $G = SO_0(4,1)$ and $G^\prime = SO_0(3,1)$
  6. Step 1: Generators of $\text{\normalfont{Hom}}_{L^\prime}\left(V_\lambda^{2N+1}, {\mathbb C}_{m,\nu} \otimes \text{\normalfont{Pol}}(\mathfrak{n}_+)\right)$
  7. Finite-Dimensional Representation Theory of SO(3)
  8. Characterization of $\text{\normalfont{Hom}}_{L^\prime}\left(V_\lambda^{2N+1}, {\mathbb C}_{m,\nu} \otimes \text{\normalfont{Pol}}(\mathfrak{n}_+)\right)$
  9. Step 2: Solving the Differential Equation (\ref{['F-system-2-concrete']}), case $m \geq N$
  10. Description of $\text{\normalfont{Sol}}(\mathfrak{n}_+; \sigma_\lambda^{2N+1}, \tau_{N, \nu})$ and proof of Theorems \ref{['mainthm1']} and \ref{['mainthm2']} for $m = N$
  11. Reduction Theorems
  12. Proof of Theorem \ref{['Thm-findingequations']} (Finding Equations)
  13. Decomposition of the equation $\left(\widehat{d\pi_\mu}(C_1^+)\otimes \text{\normalfont{id}}_{{\mathbb C}_{m,\nu}}\right)\psi=0$
  14. Vector coefficients of $\left(\widehat{d\pi_{(\sigma^{2N+1}, \lambda)^*}}(C_1^+)\otimes \text{\normalfont{id}}_{{\mathbb C}_{m,\nu}}\right)\psi$
  15. Explicit formulæ for $M_s(\psi)$
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $\lambda, \nu \in {\mathbb C}$, $N \in {\mathbb N}$ and $m = \pm N$. Then, the following three conditions on the quadruple $(\lambda, \nu, N, m)$ are equivalent:

Figures (2)

  • Figure 7.1: Hierarchy for $N = 1$
  • Figure 7.2: Hierarchy for $N = 2$

Theorems & Definitions (58)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1: kob-pev1
  • Theorem 2.2: kob-pev1
  • Lemma 4.1
  • proof
  • Lemma 4.2: kkp
  • Proposition 4.3
  • proof
  • ...and 48 more