Conformal Symmetry Breaking Operators for Differential Forms on Spheres

Abstract

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 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 matrixvalued 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].

Bernstein-Sato identities and conformal symmetry breaking operators

“…In the final section we highlight different origins of Bernstein-Sato operators. Furthermore, we discuss several applications related to conformal symmetry breaking differential operators [KØSS15,FJS16,KKP16]. As a consequence we shall observe that the Bernstein-Sato operators recover conformal symmetry breaking differential operators for functions, spinors and differential forms by partially new formulas.…”

“…Our convention for the distribution kernels are as duals to what they are considered as in the standard literature mentioned above. This choice is justified by the use of Fourier transform, which is applied to these distribution kernels directly without further dualization and leads to generalizations of singular vectors studied in [KØSS15,FJS16,KKP16]. The proposal to find Bernstein-Sato operators of interest in the context of conformal symmetry breaking operators was initiated in [PS15], where certain shift operators for Gegenbauer polynomials regarded as the residues of Fourier transformed K ± λ,ν (x ′ , x n ) are studied.…”

Symmetry breaking operators for real reductive groups of rank one

“…This approach is essentially the F-method proposed by Kobayashi [Kob13] which was previously applied in several situations where the nilpotent radical is abelian (see e.g. [KØSS15, KKP16,KP16]).…”

An extension problem related to the fractional Branson–Gover operators

“…For 1 2 < s < n 2 − p one of these representations is the corresponding complementary series representation of O(1, n) onḢ s− 1 2 ,p (R n−1 ), and the restriction mapḢ s,p (R n ) →Ḣ s− 1 2 ,p (R n−1 ) projects onto this component. This observation makes it possible to use the machinery of symmetry breaking operators whose study was recently initiated by Kobayashi [6] (see also [4,7,8,9]). In this language the differential operator ∆ a,p (a = 2(1 − s)) corresponds to the action of the Casimir element of O(1, n) inḢ s,p (R n ) and the fractional Branson-Gover operators are the standard Knapp-Stein intertwining operators between principal series representations of the group O(1, n).…”