Katz's middle convolution for automorphic sheaves

• Applicant: Dr. Konstantin Jakob
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 566801746

In the 1990s, Nick Katz studied rigid local systems, such as the solution sheaf of the Gaussian hypergeometric equation. Rigid local systems are by definition completely determined by the isomorphism classes of their local monodromy. His primary tool for this was his middle convolution with Kummer sheaves, an operation that can reduce the rank of a local system on an open subset of the projective line. If it is possible to reduce the rank to 1 through convolution, the local system is rigid. On the other hand, the convolution is invertible and can therefore be used to construct rigid local systems. Such systems have applications in, for example, inverse Galois theory and have been used to realize exceptional groups as motivic Galois groups.In the non-rigid case, the convolution produces isomorphisms between moduli spaces of local systems with prescribed local monodromy. These spaces are also called character varieties (or character stacks). The geometric Langlands program relates these moduli spaces to automorphic sheaves—perverse sheaves on the moduli stack of parabolic vector bundles. Roughly speaking, each local system should correspond to a Hecke eigensheaf, whose eigenvalue under geometric Hecke operators recovers the local system. Categorically, one expects an equivalence between the category of ind-coherent sheaves on the moduli space of local systems and the category of D-modules on the moduli stack Bun_n of parabolic vector bundles. This equivalence is characterized by its spectral origin: it arises from the action of QCoh(LocSys), via Hecke operators, on the category of D-modules on Bun_n. This equivalence was recently proven by Arinkin, Gaitsgory, Raskin, and many others in the unramified case, i.e., without parabolic structure.The geometric Langlands correspondence thus predicts that Katz's middle convolution induces an operation on automorphic sheaves. The aim of this project is to define and study this operation on the automorphic side. In particular, this operation is intended to be used to explicitly describe the tame geometric Langlands correspondence when the character variety is two-dimensional. It is expected that, in this case, the correspondence can essentially be reduced to three primitive cases via convolution. These cases correspond to the star-shaped affine Dynkin diagrams of type ADE, which encode the parabolic structure of the vector bundles.

• DFG Programme: Research Grants