## T. Kobayashi and A. Leontiev.
Image of conformally covariant, symmetry breaking operators for
*R*^{p,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*
*A*_{λ, 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 *R*^{p,q}
*R*^{p-1,q}.
In this article,
we determine explicitly
the image of the renormalized operator
*A*_{λ, ν}
for all (λ, ν) in*C*^{2}.
In particular,
the complex parameters (λ, ν)
for which the image of *A*_{λ, 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