Classification of differential symmetry breaking operators for differential forms

TL;DR

This paper classifies all conformally covariant differential operators between differential forms on spheres and hyperspheres, providing explicit formulas and factorization identities, advancing understanding of symmetry breaking in geometric analysis.

Contribution

It offers a complete classification and explicit formulas for symmetry breaking operators between differential forms on spheres, a novel contribution in conformal geometry.

Findings

Complete classification of conformally covariant operators

Explicit formulas for matrix-valued operators

Factorization identities for these operators

Abstract

We give a complete classification of conformally covariant differential operators between the spaces of differential $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the restriction of principal series representations of the Lie group $O(n+1,1)$. Further, we provide explicit formul\ae{} for these matrix-valued operators in the flat coordinates and find factorization identities for them.