Berkeley-Caltech-Stanford Joint Number Theory Seminar
In the last several years, it has been realized that the local Langlands correspondence has a categorical formulation, with variants given independently by Emerton and Helm, Fargues and Scholze, Hellmann, and Zhu. All of these approaches essentially postulates an equivalence of categories, with one side being of automorphic flavor and the other side Galois. One advantage of such an approach is that it makes sense to consider more general representations on the automorphic side, which on the Galois side corresponds to coherent sheaves more general than skyscraper sheaves; a remarkable conjecture of Zhu's also relates the cohomologies of some of these sheaves to cohomologies of Rapoport-Zink spaces in the local case and Shimura varieties in the global setup. In this talk I will formulate several conjectures attempting to match up explicitly objects under this equivalence. First, I will give conjectural duals to the so-called degenerate Whittaker representations studied by Moeglin-Waldspurger: this conjecture is inspired by analogous conjectures of Gaiotto-Witten on S-duality of boundary conditions in supersymmetric Yang-Mills, and should be closely related to work of Ben-Zvi, Sakellaridis and Venkatesh. Secondly, starting on the Galois side, I will give conjectural duals to the coherent sheaves coming from the ramification filtration. I will then indicate how to prove various special cases of these conjectures.