Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties

We study properties of the restriction of discrete series representations of $G=U(p,q)$ to $G^{\mathrm{\prime }}=U(p-1,q)$ and the corresponding symmetry breaking operators in ${\mathrm{Hom}}_{G^{\mathrm{\prime }}}(\pi {\mathrm{\mid }}_{G^{\mathrm{\prime }}},{\pi }^{\mathrm{\prime }})$. This leads to the introduction of elementary and coherent pairs of discrete series representations and their classification. Translations of symmetry breaking operators are defined via tensor products with finite-dimensional representations, which leads to the study of the coherent cohomology of discrete series representations under restriction and translations. This is applied to the study of cup products of coherent cohomology of associated Shimura varieties, and to the arithmetic of central values of certain Rankin–Selberg $L$-functions of $GL(n+1)\times GL(n)$.