Explicit Geometric Langlands Correspondence for Rigid Local Systems
Final Report Abstract
The goal of this project was to build new frameworks for the construction of rigid local systems through the geometric Langlands correspondence and to explicitly understand new instances of the correspondence in wildly ramified situations. Together with my host Z. Yun at the Massachusetts Institute of Technology we developed the notion of euphotic automorphic data realizing precisely such a framework. Roughly speaking we impose local conditions of a special kind on the space of automorphic forms such that this space is small in a suitable sense. The functions that satisfy the given local conditions should turn out to be Hecke eigenfunctions. Via Grothendieck’s sheaf-to-function dictionary we upgrade this idea to geometry and consider categories of perverse sheaves on suitable moduli stacks of G-bundles equipped with extra structures (called level structures). The goal becomes to construct so-called Hecke eigensheaves whose eigenvalues under geometric versions of Hecke operators are now local systems. These are the local systems we want to construct. A crucial difference to previous work on rigid local systems and construction of corresponding Hecke eigensheaves is the following. In previously known cases, the categories of perverse sheaves equipped with certain equivariant structures decomposed into pieces with one simple object that are stable under geometric Hecke operators. These simple objects turned out to be Hecke eigensheaves. In the euphotic situation this is no longer true in general. Our solution is to extend previous work of my host Z. Yun to extract eigen local systems from a Hecke eigencategory rather than from a Hecke eigensheaf. In this way we construct new classes of conjecturally rigid local systems and give a list of potential examples for which it remains to be checked that the corresponding space of automorphic forms is small. Our framework generalizes the construction of (generalised) Kloosterman sheaves of J. Heinloth, B.-C. Ngô and Z. Yun and our potential examples are expected to cover hypergeometric local systems for which the geometric Langlands correspondence has been established by M. Kamgarpour and L. Yi. The work on euphotic automorphic data led to collaboration with M. Kamgarpour and L. Yi. In our joint work we constructed another class of rigid local systems called Airy sheaves, generalizing the classical Airy equation. The method of construction follows the same strategy, but in these cases we can construct Hecke eigensheaves and not only Hecke eigencategories. Furthermore this construction does not fit the euphotic framework since Airy sheaves are expected to have deeper ramification than local systems coming from euphotic automorphic data. The Airy equation is particularly famous for being studied in the context of the Stokes phenomenon related to the asymptotics of solutions to irregular differential equations. Together with A. Hohl we therefore studied and computed Stokes matrices for certain Airy equations for G = GLn which are special cases of my construction with M. Kamgarpour and L. Yi. In particular we could prove that the Stokes multipliers of generalized Airy equations are regular unipotent.
Publications
- „Euphotic representations and rigid automorphic data“, 2020
Konstantin Jakob, Zhiwei Yun
(See online at https://doi.org/10.1007/s00029-022-00789-9) - „Airy sheaves for reductive groups“, 2021
Konstantin Jakob, Masoud Kamgarpour and Lingfei Yi
(See online at https://doi.org/10.48550/arXiv.2111.02256) - „Irregular Hodge numbers for rigid G2 -connections“, International Mathematics Research Notices, 2021
Konstantin Jakob, Stefan Reiter
(See online at https://doi.org/10.1093/imrn/rnab168) - „Stokes matrices for Airy equations“, Tohoku Mathematical Journal, 2021
Konstantin Jakob, Andreas Hohl
(See online at https://doi.org/10.2748/tmj.20210506)
