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Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of 2 occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras . [ABSTRACT FROM AUTHOR]

For the pair (G;G") = (O(p + 1; q + 1);O(p; + þ 1)), we construct and give a complete classification of intertwining operators (symmetry breaking operators) between most degenerate spherical principal series representations of G and those of the subgroup G0, extending the work initiated by Kobayashi and Speh [Mem. Amer. Math. Soc. 2015] in the real rank one case where q = 0. Functional identities and residue formulæ of the regular symmetry breaking operators are also provided explicitly. The results contribute to Program C of branching problems suggested by the first author [Progr. Math. 2015]. [ABSTRACT FROM AUTHOR]

For a symmetric pair ( G , H ) of reductive groups we construct a family of intertwining operators between spherical principal series representations of G and H that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. In particular, this establishes lower bounds for multiplicities of intertwining operators. We further discuss uniqueness of intertwining operators, and for the rank one cases ( G , H ) = ( SU ( 1 , n ; F ) , S ( U ( 1 , m ; F ) × U ( n − m ; F ) ) ) , with F = R , C , H , O , and for the pair ( G , H ) = ( GL ( 4 n , R ) , GL ( 2 n , C ) ) we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners. [ABSTRACT FROM AUTHOR]