Periodic Reporting for period 5 - NewtonStrat (Newton strata - geometry and representations)
Reporting period: 2023-08-01 to 2024-06-30
The Langlands programme is a far-reaching web of conjectural or proven correspondences joining the fields of representation theory and of number theory. It is one of the centerpieces of arithmetic geometry, and has in the past decades produced many spectacular breakthroughs, for example the proof of Fermat’s Last Theorem by Taylor and Wiles.
The most successful approach to prove instances of Langlands’ conjectures is via algebraic geometry, by studying suitable moduli (or parameter) spaces. Their cohomology carries actions both of a linear algebraic group (such as GL_n) and a Galois group associated with the number field one is studying. A central tool in the study of the arithmetic properties of these moduli spaces is the Newton stratification, a natural decomposition based on the moduli description of the space. Recently the theory of Newton strata has seen two major new developments: Representation-theoretic methods and results have been successfully established to describe their geometry and cohomology. Furthermore, an adic version of the Newton stratification has been defined and is already of prime importance in new approaches within the Langlands programme.
This project aims at uniting these two novel developments to obtain new results in both contexts with direct applications to the Langlands programme, as well as a close relationship and dictionary between the classical and the adic stratifications. We investigate the geometry of Newton strata in loop groups and Shimura varieties, and representations in their cohomology. Secondly, we study corresponding geometric and cohomological properties of adic Newton strata. Finally, we establish closer ties between the two contexts.
The most successful approach to prove instances of Langlands’ conjectures is via algebraic geometry, by studying suitable moduli (or parameter) spaces. Their cohomology carries actions both of a linear algebraic group (such as GL_n) and a Galois group associated with the number field one is studying. A central tool in the study of the arithmetic properties of these moduli spaces is the Newton stratification, a natural decomposition based on the moduli description of the space. Recently the theory of Newton strata has seen two major new developments: Representation-theoretic methods and results have been successfully established to describe their geometry and cohomology. Furthermore, an adic version of the Newton stratification has been defined and is already of prime importance in new approaches within the Langlands programme.
This project aims at uniting these two novel developments to obtain new results in both contexts with direct applications to the Langlands programme, as well as a close relationship and dictionary between the classical and the adic stratifications. We investigate the geometry of Newton strata in loop groups and Shimura varieties, and representations in their cohomology. Secondly, we study corresponding geometric and cohomological properties of adic Newton strata. Finally, we establish closer ties between the two contexts.
For the first part of the project we studied affine Deligne-Lusztig varieties. These arise naturally as a description in terms of linear algebra of Newton strata in the moduli spaces that arise in the Langlands programme. We found new ways to describe which of these affine Deligne-Lusztig varieties are non-empty, to compute their dimensions and the closure relations between the associated Newton strata, and various other geometric properties such as a new proof of normality of Schubert varieties. We also gave a complete description of affine Deligne-Lusztig varieties for certain unitary groups exceeding the previously studied fully Hodge-Newton decomposable cases. Further, we applied geometric results to show that Langlands correspondences are realized in the cohomology of certain unitary Shimura varieties.
For the second part of the project, we used new methods introduced by Fargues and Fontaine in p-adic Hodge theory to study the recently defined adic Newton strata. We determined which strata lie in the closure of a given one, and considered intersections with the so-called weakly admissible locus. Furthermore, we developed a new Harder-Narasimhan formalism generalizing the weakly admissible locus and studied its properties. We also extended the foundations and cohomological methods, among others by proving descent for solid quasi-coherent sheaves on perfectoid spaces and by developing a 6-functor formalism for pro-étale Q_p-cohomology on Scholze's diamonds and v-stacks.
Our findings on affine Deligne-Lusztig varieties beyond the fully Hodge-Newton decomposable cases have parallel results for a similar class of cases for the generic fiber. Concering cohomological questions related to both the generic and the special fiber and applications to the Langlands program, we construct the central functor and the Arkhipov--Bezrukavnikov functor for p-adic groups, and proved the Arkhipov--Bezrukavnikov equivalence for GL_n. We also proved the geometric Satake equivalence with mod l coefficients for ramified groups over Laurent series fields.
All of our results are published on the arxiv preprint server as soon as the preprints are available and submitted to peer-reviewed journals, and we presented them in several talks in international conferences in America, Asia, and Europe. By now, several of our papers have inspired new results by colleagues all over the world.
For the second part of the project, we used new methods introduced by Fargues and Fontaine in p-adic Hodge theory to study the recently defined adic Newton strata. We determined which strata lie in the closure of a given one, and considered intersections with the so-called weakly admissible locus. Furthermore, we developed a new Harder-Narasimhan formalism generalizing the weakly admissible locus and studied its properties. We also extended the foundations and cohomological methods, among others by proving descent for solid quasi-coherent sheaves on perfectoid spaces and by developing a 6-functor formalism for pro-étale Q_p-cohomology on Scholze's diamonds and v-stacks.
Our findings on affine Deligne-Lusztig varieties beyond the fully Hodge-Newton decomposable cases have parallel results for a similar class of cases for the generic fiber. Concering cohomological questions related to both the generic and the special fiber and applications to the Langlands program, we construct the central functor and the Arkhipov--Bezrukavnikov functor for p-adic groups, and proved the Arkhipov--Bezrukavnikov equivalence for GL_n. We also proved the geometric Satake equivalence with mod l coefficients for ramified groups over Laurent series fields.
All of our results are published on the arxiv preprint server as soon as the preprints are available and submitted to peer-reviewed journals, and we presented them in several talks in international conferences in America, Asia, and Europe. By now, several of our papers have inspired new results by colleagues all over the world.