Generating operators of symmetry breaking -- from discrete to continuous

TL;DR

This paper introduces a method to construct a variety of fundamental symmetry breaking operators with continuous parameters, expanding the toolkit for analyzing representations and transforms in mathematical physics.

Contribution

It develops a systematic approach to generate continuous-parameter operators from differential symmetry breaking operators using the generating operator framework.

Findings

Constructed invariant trilinear forms on infinite-dimensional representations.

Derived Fourier and Poisson transforms on anti-de Sitter space.

Produced integral symmetry breaking operators for fusion rules.

Abstract

Based on the "generating operator" of the Rankin--Cohen brackets introduced in Kobayashi-Pevzner [arXiv:2306.16800], we present a method to construct various fundamental operators with continuous parameters such as invariant trilinear forms on infinite-dimensional representations, the Fourier and the Poisson transforms on the anti-de Sitter space, and integral symmetry breaking operators for the fusion rules, among others, out of a countable set of differential symmetry breaking operators.