T. Kobayashi and B. Speh, Distinguished representations of SO(n+1,1) ×SO(n,1), periods and branching laws, Relative Trace Formulas (W. Müller, S. W. Shin, and N. Templier, eds.), Simons Symposia, Springer, 2021, pp. 291--319, DOI: 10.1007/978-3-030-68506-5_8. arXiv: 1907.05890..

Given irreducible representations Π and π of the rank one special orthogonal groups G=SO(n+1,1) and G'=SO(n,1) with nonsingular integral infinitesimal character, we state in terms of θ-stable parameter necessary and sufficient conditions so that
Hom_G'(Π|_G', π) ≠ {0}.
In the special case that both Π and π are tempered, this implies the Gross-Prasad conjectures for tempered representations of SO(n+1,1) ×SO(n,1) which are nontrivial on the center.

We apply these results to construct nonzero periods and distinguished representations. If both Π and π have the trivial infinitesimal character ρ then we use a theorem that the periods are nonzero on the minimal K-type to obtain a nontrivial bilinear form on the (g,K)-cohomology of the representations.

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