Symmetry breaking for real reductive groups and applications to automorphic forms

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Let G be a Lie group and H a closed subgroup. For an irreducible representation of G we consider its restriction to H and are interested in finding its irreducible constituents. In the category of smooth representations of real reductive groups this involves the study of intertwining operators between an irreducible representation of G and an irreducible representation of H, intertwining for H. Such operators are also called symmetry breaking operators, and in this project we intend to construct and study families of symmetry breaking operators between principal series representations which depend meromorphically on the induction parameters. For G and H groups of split rank one, and for G and H general linear groups, the first main objective is to completely describe all symmetry breaking operators between (spherical) principal series in terms of such meromorphic families and their residues.Furthermore, symmetry breaking operators turn out to have interesting applications in number theory, where they can be used to find asymptotic estimates for period integrals of automorphic forms on locally symmetric spaces. We plan to use the above families of symmetry breaking operators to study period integrals for rank one locally symmetric spaces, and for locally symmetric spaces for general linear groups.