Conformal symmetry breaking on differential forms and some applications

TL;DR

This paper classifies symmetry breaking operators for conformal representations on differential forms, extending previous scalar and local cases, with applications to automorphic forms and conformal geometry.

Contribution

It provides a new classification scheme for symmetry breaking operators on differential forms, generalizing prior scalar and local operator results.

Findings

Classification scheme for symmetry breaking operators on differential forms

Extension of scalar case to differential forms

Applications to automorphic forms and conformal geometry

Abstract

Rapid progress has been made recently on symmetry breaking operators for real reductive groups. Based on Program A-C for branching problems (T.Kobayashi [Progr.Math.2015]), we illustrate a scheme of the classification of (local and nonlocal) symmetry breaking operators by an example of conformal representations on differential forms on the model space $(X,Y)=(S^n, S^{n-1})$, which generalizes the scalar case (Kobayashi--Speh [Memoirs of Amer.Math.Soc. 2015]) and the case of local operators (Kobayashi--Kubo--Pevzner [Lecture Notes in Math. 2016]). Some applications to automorphic form theory, motivations from conformal geometry, and the methods of proof are also discussed.