T. Kobayashi, T. Kubo, and M. Pevzner. Conformal Symmetry Breaking Operators for Differential Forms on Spheres, volume 2170 of Lecture Notes in Mathematics. Springer Singapore, 2016. ix+192 pages. DOI: 10.1007/978-981-10-2657-7. arXiv: 1605.09272. Softcover ISBN: 978-981-10-2656-0. eBook ISBN: 978-981-10-2657-7..

We give a complete classification of conformally covariant differential operators between the spaces of i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn-1. Moreover, we find explicit formulæ 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].


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