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

TL;DR

This paper classifies all differential symmetry breaking operators between principal series representations of the de Sitter and Lorentz groups, proving their localness and sporadic nature.

Contribution

It constructs and classifies all such operators, establishing their differential nature and demonstrating they are sporadic, not derived from residue formulas.

Findings

All symmetry breaking operators are differential operators.

Operators are necessarily local (differential).

Operators are sporadic, not obtained via residue formulas.

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.