CONFORMAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS ON SPHERES

Toshiyuki Kobayashi (1) , Toshihisa Kubo , Michael Pevzner (2)

1 Graduate School of Mathematics
2 LMR - Laboratoire de Mathématiques de Reims

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 formulae 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 co-variant 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].

Mots clés: Symmetry breaking operators; branching law; F-method; conformal geometry; Verma module; Lorentz group

Domaines: Théorie des représentations [math.RT]

Toshiyuki Kobayashi, Toshihisa Kubo, Michael Pevzner. CONFORMAL SYMMETRY BREAKING OPERATORS FOR DIFFERENTIAL FORMS ON SPHERES. 2016. ⟨hal-01907092⟩