Conformal symmetry breaking operators for differential forms on spheres

TL;DR

This paper classifies all conformally covariant differential operators between forms on spheres and provides explicit formulas and factorization identities, advancing understanding of symmetry breaking operators in conformal geometry.

Contribution

It offers a complete classification and explicit formulas for conformally covariant operators on differential forms, introducing new matrix-valued operators and factorization identities.

Findings

Explicit formulas for conformally covariant operators on forms

Complete classification of symmetry breaking operators

Matrix-valued factorization identities established

Abstract

We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formul{\ae} for these new matrix-valued operators in the flat coordinates in terms of basic operators in differential geometry and classical orthogonal polynomials. We also establish matrix-valued factorization identities among all possible combinations of conformally covariant differential operators. The main machinery of the proof is the "F-method" based on the "algebraic Fourier transform of Verma modules" (Kobayashi-Pevzner [Selecta Math. 2016]) and its extension to matrix-valued case developed here. A short summary of the main results was announced in [C. R. Acad. Sci. Paris, 2016].