Intertwining operators and the restriction of representations of rank-one orthogonal groups

We give a complete classification of intertwining operators (breaking symmetry operators  ) between spherical principal series representations of O(n+1,1)$O(n+1,1)$ and O(n,1)$O(n,1)$ together with explicit formulae of the distribution kernels. Further we use this to determine the breaking symmetry operators between their irreducible composition factors.