Construction and analysis of symmetry breaking operators for the pair $(\operatorname{GL}(n+1,\mathbb{R}),\operatorname{GL}(n,\mathbb{R}))$

Abstract

The pair of real reductive groups $(G,H)=(\operatorname{GL}(n+1,\mathbb{R}),\operatorname{GL}(n,\mathbb{R}))$ is a strong Gelfand pair, i.e. the multiplicities $\dim\operatorname{Hom}_H(\pi|_H,\tau)$ are either $0$ or $1$ for all irreducible Casselman-Wallach representations $\pi$ of $G$ and $\tau$ of $H$. This paper is concerned with the construction of explicit intertwining operators in $\operatorname{Hom}_H(\pi|_H,\tau)$, so-called symmetry breaking operators, in the case where both $\pi$ and $\tau$ are principal series representations. Such operators come in families that depend meromorphically on the induction parameters, and we show how to normalize them in order to make the parameter dependence holomorphic. This is done by establishing explicit Bernstein-Sato identities for their distribution kernels as well as explicit functional identities for the composition of symmetry breaking operators with standard Knapp-Stein intertwining operators for $G$ and $H$. We also show that the obtained normalization is optimal and identify a subset of parameters for which the family of operators vanishes. Finally, we relate the operators to the local archimedean Rankin-Selberg integrals and use this relation to evaluate them on the spherical vectors.