## T. Kobayashi and B. Speh.
*Symmetry Breaking for Representations of Rank One Orthogonal
Groups II*, volume 2234 of *Lecture Notes in Mathematics*.
Springer, 2018.
xv+342 pages, ISBN: 978-981-13-2900-5,
eBook: 978-981-13-2901-2.
arXiv: 1801.00158.
DOI:
10.1007/978-981-13-2901-2..

For a pair (*G*,*G*')=(*O*(*n*+1,1), *O*(*n*,1))
of reductive groups,
we investigate intertwining operators (*symmetry breaking operators*) between principal series representations
*I*_{δ}(*V*,λ) of *G*,
and *J*_{ε}(*W*,ν) of the subgroup *G*'.
The representations are parametrized by finite-dimensional representations *V*, *W* of *O*(*n*) respectively of *O*(*n*-1),
characters δ, ε of *O*(1),
and λ, νinC.
Denote by [*V*:*W*] the multiplicity of *W*
occurring in the restriction *V*|_{O(n-1)},
which is either 0 or 1.
If [*V*:*W*] =/=0
then
we construct a holomorphic family of symmetry breaking operators
and prove that
_{C} *Hom*_{G'}(*I*_{δ}(*V*, λ)|_{G'}, *J*_{ε}(*W*, ν))
is nonzero
for all the parameters λ, ν and δ, ε,
whereas if [*V*:*W*] = 0 there may exist *sporadic* differential symmetry breaking operators.
We propose a *classification scheme* to find all matrix-valued symmetry breaking operators explicitly,
and carry out this program completely in the case
(*V*,*W*)=(^{i}(*C*^{n}), ^{j}(*C*^{n-1})).
In conformal geometry, our results yield the complete classification
of conformal covariant operators from differential forms on a Riemannian manifold *X*
to those on a submanifold *Y* in the model space (*X*, *Y*) = (*S*^{n}, *S*^{n-1}).

We use this information to determine the space of symmetry breaking operators
for any pair of irreducible representations
of *G* and the subgroup *G*' with trivial infinitesimal character.
Furthermore we prove the multiplicity conjecture by B. Gross and D. Prasad for tempered principal series representations of (*SO*(*n*+1,1), *SO*(*n*,1))
and also for 3 tempered representations Π, π, ϖ of
*SO*(2*m*+2,1), *SO*(2*m*+1,1) and *SO*(2*m*,1)
with trivial infinitesimal character.
In connection to automorphic form theory,
we apply our main results to find *periods*
of irreducible representations of the Lorentz group
having nonzero (*g*, *K*)-cohomologies.

This book is an extension of the recent work in the two research monographs:
Kobayashi-Speh [Memoirs Amer. Math. Soc., 2015]
for *spherical* principal series representations
and
Kobayashi-Kubo-Pevzner [Lecture Notes in Math., 2016]
for conformally covariant *differential* symmetry breaking operators.

[ DOI | arXiv |
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© Toshiyuki Kobayashi