## T. Kobayashi and B. Speh.
Distinguished representations of *SO*(*n*+1,1) ×*SO*(*n*,1),
periods and branching laws.
preprint, 32 pages.
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.

[ preprint version(pdf) |
arXiv ]

© Toshiyuki Kobayashi