Abstract: I begin with some general results on anaylsis on homogeneous spaces and restricting representations of reductive groups to their subgroups. Then I will focus on a concrete geometric question arising from conformal geometry, giving the complete classification of conformally covariant “symmetry breaking operators” for differential forms on the model space for codimension one submanifolds. Some of the symmetry breaking operators are given as differential operators, whereas some others include integral operators and its meromorphic continuation.

If time permits, I would like to discuss some applications and related questions including a conjecture of Gross and Prasad.

References:

_{ [1] T. Kobayashi. A program for branching problems in the representation theory of real reductive groups. Progr. Math. 312, pp. 277-322, 2015.}

_{[2] T. Kobayashi, T. Kubo, and M. Pevzner, Conformal symmetry breaking operators for differential forms on spheres, viii+192 pages. Lecture Notes in Mathematics, vol. 2170, 2016.}

_{ [3] T. Kobayashi and B. Speh. Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of Amer. Math. Soc. 238. 2015. 118 pages.}

_{ [4] T. Kobayashi and B. Speh, – II, xv+342 pages, Lecture Notes in Math. 2234, Springer-Nature, 2018.}