For an irreducible representation π of a group G and a subgroup G', the restriction π|G' may not be well under control as a representation of G'. I proposed in  to study branching problems of reductive groups into thrree stages:
Stage A: Abstract feature of the restriction π|G'
Stage B: Branching laws.
Stage C: Construction of symmetry breaking operators (SBO).
One of the results in Stage A is:
Theorem 1 (geometric criteria for finite/bounded multiplicities,)
(1) The space of SBOs are finite dimensional for any irreps Leftrightarrow G×G'/diag(G') is real spherical.
(2) The space of SBOs are of uniformly bounded dimension Leftrightarrow G×G'/diag(G') is spherical.
A classification of (G,G') satisfying the geometric criteria was accomplished in . Then, a priori estimate in Stage A predicts potentially interesting settings for a detailed study of branching problems (Stages B and C). I illustrate this program by the first complete solution to Stage C.
Another motivation also comes from conformal geometry:
Question 2. Given a Riemannian manifold X and its hypersurface Y, find conformally covariant SBOs from differential i-forms on Xj-forms on Y.
The model space with largest symmetries satisfies the criterion in Theorem 1, and Question 2 is regarded as Stages B and C of branching problems. I explain a complete solution [2,5,6]. Some of the methods developed in the proof are applicable in a more general setting that Stage A (Theorem 1) suggests.
Finally, some applications of these results include: periods of irreducible unitary representations with nonzero cohomologies, an evidence of Gross-Prasad conjecture  for O(n,1), a construction of discrete spectrum of the branching laws of complementary series .
T.Kobayashi, A program for branching problems, Progr Math. 312 (2015), pp.277-322 (Vogan volume)
T.Kobayashi-K.Kubo, M.Pevzner, Lecture Notes in Math. vol. 2170 (2016)
T.Kobayashi-T.Matsuki, Transformation Groups (2014) (Dynkin volume)
T.Kobayashi-T.Oshima, Adv Math 2013
T.Kobayashi-B.Speh, Memoirs of AMS, vol.1126 (2015).
© Toshiyuki Kobayashi