T. Kobayashi and A. Leontiev. Image of conformally covariant, symmetry breaking operators for Rp,q. In V. Dobrev, editor, Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics. Volume 1. LT-XII/QTS-X 2017, volume 263 of Springer Proceedings in Mathematics & Statistics, pages 3-31, 2018. DOI: 10.1007/978-981-13-2715-5_1..

We consider the meromorphic continuation of an integral transform that gives rise to a conformally covariant, symmetry breaking operator Aλ, ν between the natural family of representations I(λ) and J(ν) of the indefinite orthogonal group G=O(p+1,q+1) and its subgroup G'=O(p,q+1), respectively, realized in function spaces on the conformal compactifications of flat pseudo-Riemannian manifolds Rp,q Rp-1,q. In this article, we determine explicitly the image of the renormalized operator Aλ, ν for all (λ, ν) inC2. In particular, the complex parameters (λ, ν) for which the image of Aλ, ν coincides with {0}, C, finite-dimensional representations, the minimal representation, or discrete series representations for pseudo-Riemannian space forms are explicitly classified. A graphic description of the K-types of the image is also provided. Our results extend a part of the prior results of Kobayashi and Speh [Memoirs of Amer. Math. Soc. 2015] in the Riemannian case where q=0.

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