## 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.

[ DOI |
arXiv |
preprint version(pdf) |
full text(pdf) ]

© Toshiyuki Kobayashi