Periodic Reporting for period 1 - MotLang (Motives and the Langlands program)
Période du rapport: 2022-04-01 au 2024-09-30
The project uses algebraic cycles in the form of motivic sheaves to address fundamental questions about the Langlands program for function fields. This program conjectures a decomposition of the space of automorphic forms by motivic Langlands parameters. A breakthrough of V. Lafforgue for function fields with generalizations by Xue, Zhu, Drinfeld and Gaitsgory led to tremendous progress in the construction of l-adic Langlands parameters. This should be viewed as the l-adic realization of the conjectured motivic parametrization. Independently, recent advances for motivic sheaves, as envisioned by Grothendieck and realized by Voevodsky, Ayoub, Cisinski--Déglise, provide powerful techniques to handle algebraic cycles. Despite the immense progress in both areas, hardly any advances have been made specifying the motivic nature of Langlands original prediction. The project aims to formulate and implement motivic Langlands parameterizations. It is divided into three subprojects: (1) Satake equivalence, (2) Drinfeld's lemma and (3) Langlands parametrizations. Subprojects (1) and (2) aim to lay the necessary foundations. Their results are combined in Subprojects (3) towards the overall goal of the project.
In the reporting period the activities were focused in reaching the objectives of Subprojects (1) and (2). In Subproject (1), achievements of team members include extensions of the Satake equivalence for motivic sheaves to include constant term functors, twisted groups and versions for the Beilinson--Drinfeld Grassmannian, a classification of normal Schubert varieties in the affine Grassmannian and the Arkhipov-Bezrukavnikov equivalence for twisted groups. In Subproject (2), achievements include the development of the theory of constructible sheaves with condensed coefficients and building on this the categorical Drinfeld's lemma for constructible Weil sheaves as well as Frobenius rigidity for cellular objects in the stable homotopy category providing first instances of a motivic version of Drinfeld's lemma.
The results are expected to be useful in addressing the fundamental problem of constructing motivic Langlands parametrization for function fields and beyond. This would specify and possibly address a long-standing open problem in the areas still lacking a precise formulation.