## *Symmetry Breaking Operators for Rank One Orthogonal Groups*.
Analysis, Geometry and
Representations on Lie Groups and Homogeneous Spaces (Conference in honor of
Prof. Takeshi Kawazoe and Prof. Ahmed Intissar). Marrakech, Morocco, 8-12
December 2014.

Branching problems ask how irreducible representations ƒÎ of groups *G* ''decompose” when restricted to subgroups *G*'.
For real reductive groups, branching problems include various
important
special cases, however, it is notorious that ''infinite
multiplicites”
may well happen in general even if (*G*,*G*') are natural pairs such as
symmetric pairs.
By using analysis on (real) spherical varieties, we explain a
general
project on branching pborlems (K- 2014),
a necessary and sufficient condition on the pair of reductie groups
for
the multiplicities to be always finite (and also to be of uniformly
bounded) (K-T.Oshima, 2013)), and its classfication results
(K-Matsuki, 2014)).
I give a classification of all symmetry breaing operators that
intertwines two spherical principal series representations of two
groups
*O*(*n*+1,1) to *O*(*n*,1). The last part is a joint work with B. Speh (to
appear in Memoirs of AMS).

© Toshiyuki Kobayashi