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We give a complete classification of intertwining operators (symmetry breaking operators) between spherical principal series representations of G = O(n+1, 1) and G ′ = O(n, 1). We construct three meromorphic families of the symmetry breaking operators, and find their distribution kernels and their residues at all poles explicitly. Symmetry breaking operators at exceptional discrete parameters are thoroughly studied.We obtain closed formulae for the functional equations which the composition of the the symmetry breaking operators with the KnappStein intertwining operators of G and G ′ satisfy, and use them to determine the symmetry breaking operators between irreducible composition factors of the spherical principal series representations of G and G ′ . Some applications are included.

No abstract

We find upper and lower bounds of the multiplicities of irreducible admissible representations $\pi$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^G\tau$ from irreducible representations $\tau$ of a closed subgroup $H$. As corollaries, we establish geometric criteria for finiteness of the dimension of $Hom_G(\pi,Ind_H^G \tau)$ (induction) and of $Hom_H(\pi|_H,\tau)$ (restriction) by means of the real flag variety $G/P$, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.Comment: to appear in Advances in Mathematic