For an irreducible representation π of a groupGand a subgroupG', the restriction π|_{G}' may not be well under control as a representation ofG'. I proposed in [1] 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,[4])

(1) The space of SBOs are finite dimensional for any irreps LeftrightarrowG×G'/diag(G') is real spherical.

(2) The space of SBOs are of uniformly bounded dimension LeftrightarrowG×G'/diag(G') is spherical.A classification of (

G,G') satisfying the geometric criteria was accomplished in [3]. 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

Xand its hypersurfaceY, find conformally covariant SBOs from differentiali-forms onXj-forms onY.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 [6] for

O(n,1), a construction of discrete spectrum of the branching laws of complementary series [5].References:

[1]T.Kobayashi, A program for branching problems, Progr Math. 312 (2015), pp.277-322 (Vogan volume)

[2]T.Kobayashi-K.Kubo, M.Pevzner, Lecture Notes in Math. vol. 2170 (2016)

[3]T.Kobayashi-T.Matsuki, Transformation Groups (2014) (Dynkin volume)

[4]T.Kobayashi-T.Oshima, Adv Math 2013

[5]T.Kobayashi-B.Speh, Memoirs of AMS, vol.1126 (2015).

[6]T.Kobayashi-B.Speh, arXiv:1702.00263

© Toshiyuki Kobayashi