一个弗兰克滤波器的 functorial 细化及其对于自守形式空间的雅各布-朗格兰斯对应关系
A Functorial Refinement of the Franke Filtration and the Jacquet--Langlands Correspondence for Spaces of Automorphic Forms
AI总结：
本文通过扩展弗兰克滤波器的functorial细化，建立了自守形式空间中一般线性群及其内形式之间的雅各布-朗格兰斯对应关系，扩展了Badulescu的对应关系并包含完整functorial提升。
摘要：
全球雅各布-朗格兰斯对应关系是朗格兰斯functoriality的一个实例，即通过L-群的恒等态射将一般线性群的内形式的不可约自守表示提升到分裂形式。通过Badulescu的工作，在离散谱的不可约分量情况下已建立。本文的目的在于将此对应关系扩展到离散谱之外。为此，采用了自守形式空间的弗兰克滤波器的观点。事实上，我们的技术关键成分是一个弗兰克滤波器的functorial细化，这使我们能够建立一般线性群及其内形式上此细化滤波器的连续商之间的雅各布-朗格兰斯对应关系。结果，我们的扩展雅各布-朗格兰斯对应关系正确地扩展了Badulescu的对应关系，并包含了朗格兰斯functoriality所预测的完整functorial提升。
英文摘要：
The global Jacquet--Langlands correspondence is an instance of Langlands functoriality, namely the expected lifting of the irreducible automorphic representations of an inner form of the general linear group to the split form via the identity morphism of $L$-groups. It is established, by the work of Badulescu, in the case of irreducible components of the discrete spectrum. The purpose of this paper is to extend this correspondence beyond the discrete spectrum. To this end, the point of view of the Franke filtration of spaces of automorphic forms is taken. In fact, our technical key ingredient is a functorial refinement of the Franke filtration, which allows us to establish the Jacquet--Langlands correspondence between consecutive quotients of this refined filtration on the general linear group and its inner form. As a result, our extended Jacquet--Langlands correspondence properly extends Badulescu's correspondence and contains the full functorial lift, predicted by Langlands functoriality.
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