Local and Global Geometric Langlands Correspondence

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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.