Mini-courses

Dima Arinkin: The Hitchin fibration and the classical limit of the Langlands conjecture

Let G be a reductive group and X a compact Riemann surface. The global geometric Langlands conjecture (which is now a theorem) is, roughly speaking, an equivalence between the category of D-modules on the moduli of G-bundles and the category of quasicoherent sheaves on the moduli of G'-local systems. Here G' is the Langlands dual group of G. Interestingly, although the relation between G and G' is symmetric, the formulation of the conjecture is not. We may recover the symmetry by taking the `classical limit' of the conjecture, when the two categories degenerate into the categories of quasicoherent sheaves on the Hitchin fibration for G and G'. The classical limit of the geometric Langlands conjecture refers to the conjectural Fourier-Mukai transform between these two categories. The conjecture is interesting on its own (independently of the Langlands conjecture) and invites a new set of tools and ideas. While there was a significant progress on this statement for G=GL(n), much less is known about the case of general G. I will start my talks in this `global' setting, and discuss the relation between the geometric Langlands and its classical limit. I will then focus on the classical statement and its unique advantages and challenges. In particular, I would like to discuss the relation between local and global objects in the classical limit.

Linus Hamann: The Local Langlands Correspondence and its Geometrization

Let $G/\mathbb{Q}_{p}$ be a connected reductive group over the p-adic numbers. The local Langlands correspondence is a bridge between the analytic world (smooth irreducible representations of $G$) and the arithmetic world (representations of Weil group of $\mathbb{Q}_{p}$). Recently, Fargues and Scholze recast this correspondence in the framework of the geometric Langlands correspondence, replacing smooth representations with certain l-adic sheaves on $\mathrm{Bun}_{G}$, the moduli stack of $G$-bundles on the Fargues-Fontaine curve, and Weil group representations with coherent sheaves on the moduli stack of such representations. On the one hand, this directly generalizes the usual local Langlands correspondence and its various refinements, by embedding the derived category of smooth representations of $G$ and its inner forms as the category of sheaves on the Harder-Narasimhan strata of $\mathrm{Bun}_{G}$. On the other hand, it also gives insight into other important phenomenon in arithmetic geometry. For example, both sides of this correspondence are equipped with a compatible Hecke action that encodes information about how local Langlands is realized in the cohomology of global Shimura varieties, which is important in various parts of number theory. The goal of this course will be to explain some of these exciting new developments, highlighting the connection to classical local Langlands as well as its applications to the cohomology of Shimura varieties through various examples.

Abstracts

Rızacan Çiloğlu: On two geometric realizations of an affine Hecke algebra for twisted groups

For a split reductive group $G$ over an equal characteristic local field, Roman Bezrukavnikov has proved an equivalence of categories between $\ell$-adic sheaves on the associated Iwahori–Hecke stack and coherent sheaves on the Steinberg variety of the Langlands dual group $\check G$. This equivalence has had broad applications, ranging from the representation theory of quantum groups to, more recently, Zhu’s proof of the tame categorical local Langlands conjecture. In this talk, I will discuss extending Bezrukavnikov’s equivalence to tamely ramified quasi-split groups. The main technical difference from the split setting is the absence of a suitable analogue of Gaitsgory’s central functor. I will also outline how to address this using a coherent description of Iwahori–Whittaker sheaves on the affine flag variety, generalizing the work of Arkhipov–Bezrukavnikov from the split case to tamely ramified quasi-split groups.

Duong Dinh: Classical limit of the geometric Langlands correspondence for SL(2, C)

Over a compact Riemann surface C, we consider the moduli of pairs (E, L) where E is a rank-2 holomorphic bundle and L a sub-line bundle. Equipping this pair with a Higgs field, we can define an effective divisor on the corresponding spectral curve. Via this construction, we define a rational local symplectomorphism from the cotangent of pairs (E, L) to a symmetric product of the cotangent of C. This map can be interpreted as the classical limit of V. Drinfeld's construction of Hecke eigensheaves in the geometric Langlands correspondence for rank-2 case.

Robert Hanson: Twistors and hyperholomorphic branes in geometric Langlands

Work of Kapustin and Witten in supersymmetric gauge theory suggests that hyperholomorphic boundary conditions, known as BBB-branes, should play some role within the geometric Langlands program. The purpose of this talk is to propose a spectral category of nilpotent and ind-coherent BBB-branes, constructed using the twistor theory of local systems founded by Deligne and Simpson. We shall explain how our constructions pass certain “Eisenstein tests” with respect to a twistor variant of geometric Eisenstein series functor. This is based on forthcoming work and a previous collaboration with Emilio Franco (arXiv:2311.10032).