Organised by Pol van Hoften . This quarter we are studying moduli spaces of (local) Langlands parameters, following the paper of Dat-Helm-Kurinczuk-Moss, in order to understand the statement of a conjectural local Langlands correspondence "in families". We will also try to understand some of the results of Fargues-Scholze and perhaps briefly discuss their categorical version of the local Langlands correspondence, which implies a correspondence in families.

Seminar Schedule

• (April 5th 2022): Introduction, overview and basics on Weil-Deligne representations (Pol). Notes .
• (April 12th 2022): Moduli spaces of tame Langlands parameters (Vaughan).
• (April 19th 2022): Reduction to tame parameters up to the proof of Theorem 3.4 of Dat-Helm-Kurinczuk-Moss (Ben). Notes .
• (April 26th 2022): The proof of Theorem 3.4 of Dat-Helm-Kurinczuk-Moss (Sean).
• (May 3rd 2022): Section 4 of Dat-Helm-Kurinczuk-Moss, with an emphasis on Theorem 4.13 and its proof (Connor)
• (May 10th 2022): Connected components of moduli spaces of Langlands-parameters (Mark)
• (May 17th 2022): Smooth representations of p-adic reductive groups I: Basics, parabolic induction, examples in depth zero (Thomas)
• (May 24th 2022): Smooth representations of p-adic reductive groups II: Supercuspidal supports, blocks, and the Bernstein centre (Konstantin).
• (May 31st 2022): Overview of the results of Fargues-Scholze and application to finiteness of Hecke algebras (Lie).

References

• Moduli of Langlands Parameters by Dat-Helm-Kurinczuk-Moss.
• Finiteness for Hecke algebras of p-adic groups by Dat-Helm-Kurinczuk-Moss
• Geometrization of the local Langlands correspondence by Fargues-Scholze. Sections 1,2 and 3 of chapter VIII give a different approach to the construction of the moduli space of Langlands-parameters. Chapter IX.7 discusses their main results in terms of Bernstein centers.
• Notes on smooth representations of p-adic reductive groups by Florian Herzig.
• Representations des groupes reductifs p-adiques by Renard-Schwartz.
• Coherent sheaves on the stack of Langlands parameters by Xinwen Zhu. This gives yet another construction of moduli stacks of Langlands parameters, and also discusses moduli spaces of global Langlands parameters over function fields.
• Coherent Springer theory and the categorical Deligne-Langlands correspondence by Zvi-Chen-Helm-Nadler.
• On the derived category of the Iwahori-Hecke algebra by Hellmann.