Abstract
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 $$\displaystyle \operatorname {Hom}_{G^{\prime }}(\Pi |{ }_{G^{\prime }}, \pi )\neq \{0\}. $$ In the special case that both Π and π are tempered, this implies the Gross–Prasad conjectures (Gross and Prasad, Canad J Math 44:974–1002, 1992) for tempered representations of SO(n + 1, 1) × SO(n, 1) which are nontrivial on the center.
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