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].
In this paper we construct conformally invariant systems of first order and second order differential operators associated to a homogeneous line bundle L s → G 0 /Q 0 with Q 0 a maximal parabolic subgroup of quasi-Heisenberg type. This generalizes the results by Barchini, Kable, and Zierau. To do so we use techniques different from ones used by them.
In this paper we give Peter-Weyl type formulas for the space of K-finite solutions to intertwining differential operators between degenerate principal series representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of SL(3, R) attached to the minimal nilpotent orbit.2010 Mathematics Subject Classification. 22E46, 17B10.
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