Symmetry breaking operators are intertwining operators from a representation of a group $G$ to a representation of a subgroup $G'$, intertwining for the subgroup. For spherical principal series of $G = O(1,n+1)$ and $G' = O(1,n)$, these operators have been classified recently by Kobayashi-Speh in the smooth category. We study symmetry breaking operators in the category of Harish-Chandra modules, recovering the results by Kobayashi-Speh in this setting, and thus providing the $K$-picture of their operators. We further indicate how the same method can be used in the case of $G = U(1,n+1)$ and $G' = U(1,n)$ to classify symmetry breaking operators.

Joint work with Bent Ørsted.