Local and Global Geometric Langlands Correspondence

这个项目的主旨是几何朗兰兹对应理论的发展。朗兰兹现象的核心思想是数学中自然产生的一些基本对称定律是成对出现的，被称为朗兰兹对偶。从经验上观察到，服从一组对称定律的数学结构等同于对偶对称定律的根本不同结构。这个项目的目标是在几何朗兰兹对应的情况下，形式化并使这种对偶性的基本结构在数学上变得严谨。更详细地说，为了理解几何朗兰兹对应，人们应该进行量子变形，在这种情况下，群及其朗兰兹对偶的角色变得更加对称，也就是说，不再有“自同构与伽罗瓦”，而是“扭曲自同构与扭曲自同构”。在进行量子变形之后，我们看到在几何朗兰兹对应的原点应该是局部范畴的某种等价，这是几年前J. Lurie和PI提出的：这是Whittaker范畴（对于G群）与Kazhdan-Luztig范畴（对于G的朗兰兹对偶）的等价。本项目将建立这种等效性并发展其后果。

The thrust of this project is the development of the theory of the geometric Langlands correspondence. The core idea of the Langlands phenomenon is that some basic symmetry laws that arise naturally in mathematics come in pairs, known as Langlands dual pairs. It has been observed empirically that mathematical constructions that obey one set of symmetry laws are equivalent to fundamentally different constructions for the dual symmetry laws. The goals of this project are to formalize and make mathematically rigorous the underlying structures responsible for this duality in the case of the geometric Langlands correspondence.In more detail, it has become clear that in order to understand the geometric Langlands correspondence, one should perform a quantum deformation, in which case the roles of the group and its Langlands dual become more symmetric, that is, there is no more "automorphic vs. Galois", but rather "twisted automorphic vs. twisted automorphic." After performing the quantum deformation, one sees that at the origin of the geometric Langlands correspondence should be a certain equivalence of local categories, proposed by J. Lurie and the PI several years ago: this is the equivalence of the Whittaker category (for a group G) vs the Kazhdan-Luztig category (for the Langlands dual of G). This project will establish this equivalence and develop its consequences.