A Program for Branching Problems in the Representation Theory of Real Reductive Groups: Classification Problem of Symmetry Breaking Operators. Representation Theory inspired by the Langlands Conjectures (organized by B. Speh and P. Trapa), in connection with the AMS-AWM Noether lecture by Birgit Speh. Denver, USA, 17 January 2020.

A symmetry breaking operator is an intertwining operator that arises from branching problems of representations, thatis, an $H$-intertwining operator from an (irreducible) representation of $G$ to that of the subgroup $H$. It may be an integraloperator, and may be a more singular one such as a differential operator, when representations are realized geometrically. In general, it is a hard problem to classify symmetry breaking operators.

We plan to discuss a criterion for the spaceof symmetry breaking operators to be finite-dimensional, and a classification scheme of symmetry breaking operatorswith some examples for orthogonal groups.

References:

• T. Kobayashi. A program for branching problems in the representation theory of real reductive groups. Progr. Math.312, pp. 277-322, 2015.
• T. Kobayashi, T. Kubo, and M. Pevzner, Conformal symmetry breaking operators for differential forms on spheres, Lecture Notes in Math., 2170, Springer, 2016. viii+192 pages.
• T. Kobayashi, B. Speh. Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of AMS. 238. 2015. vi+118 pages.
• T. Kobayashi, B. Speh, Symmetry Breaking for Representations of Rank One Orthogonal Groups II, Lecture Notes in Math. 2234, Springer, 2018. xv+342 pages.

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© Toshiyuki Kobayashi