Explicit Matching of Objects in the Categorical Local Langlands Conjectures

par Joshua Lam (IHES)

mercredi 24 nov. 2021, 16:30 → 17:30 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Séminaire de géométrie arithmétique

Recently, it was realized that the local Langlands correspondence has a geometric and categorical formulation, with variants of such a formulation 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; some of these coherent sheaves are conjectured by Zhu to give 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 of categories. 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.

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IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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