Symmetry breaking for real reductive groups and applications to automorphic forms

设G是李群，H是闭子群.对于G的不可约表示，我们考虑它对H的限制，并有兴趣找到它的不可约成分。在类别的光滑表示的真实的约化群这涉及到研究的交织运营商之间的一个不可约表示的G和一个不可约表示的H，交织为H。这样的运营商也被称为对称破缺运营商，在这个项目中，我们打算构建和研究家庭的对称破缺运营商之间的主系列表示，亚纯依赖于诱导参数。对于分裂秩为1的群G和H，以及一般线性群G和H，第一个主要目标是完全描述所有对称破缺算子，（球面）主级数的亚纯族及其留数。此外，对称破缺算子在数论中有着有趣的应用，在那里，它们可以用来找到局部对称空间上自守形式的周期积分的渐近估计。我们计划利用上述对称破缺算子族来研究秩为一的局部对称空间的周期积分，以及一般线性群的局部对称空间的周期积分。

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.