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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© Toshiyuki Kobayashi