Series abstract: The recent manuscript of Fargues-Scholze aims to "geometrize" the Langlands program for a p-adic group G, by relating the players in that story to the stack Bun_G. Following a strategy of V. Lafforgue, the main result of [FS] is the construction of an L-parameter attached to a smooth irreducible representation of G.
The goal of this series is to review the main ideas of this work, and to discuss two related results: progress on the Kottwitz conjecture for local shtuka spaces by Hansen-Kaletha-Weinstein, and the construction of eigensheaves on Bun_G when G = GL_n.
Speaker: Jared Weinstein
Abstract: We will give a historically motivated introduction to the story, reviewing moduli spaces of p-divisible groups, the Fargues-Fontaine curve, and the stack Bun_G of G-bundles on it. We will then define the moduli spaces of local shtukas, and state our result on their cohomology.
Slides: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/IntroToBunG.pdf
Speaker: Tasho Kaletha
Abstract: We will review some representation-theoretic inputs to [HKW]. We’ll begin with reviewing the statements of the basic and refined local Langlands correspondence and the status of their proofs. We will then define the relative position of two members of a compound L-packet, which is an input to the Kottwitz conjecture, and the relative position of two regular semi-simple elements in inner forms. Based on the latter, we will define a Hecke transfer operator that transfers conjugation-invariant functions between inner forms, and discuss its effect on characters of supercuspidal representations.
Slides: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/Kaletha.pdf
Speaker: Jared Weinstein
Abstract: In this talk we will discuss a very general form of the Lefschetz-Verdier trace formula which applies to stacks (both of schemes and of diamonds). As an application, we will show that if a locally pro-p group G acts on a proper diamond X, and if A is a G-equivariant l-adic sheaf on X which is "dualizable" (= universally locally acyclic), then the cohomology R\Gamma(X,A) is an admissible representation of G, whose Harish-Chandra distribution can be computed in terms of local terms living on the fixed-point locus of G on X.
Slides: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/LefschetzForDiamonds.pdf
Speaker: David Hansen
Abstract: In this lecture, we will give a detailed sketch of the proof of the main theorem of [HKW], building on the material in the first three lectures. The idea that the Kottwitz conjecture should follow from some form of the Lefschetz trace formula goes back to Harris in the '90s. We will try to emphasize the new ingredients which allow us to implement this idea in full generality.
Slides: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/Hansen.pdf
Speaker: Johannes Anschütz
Abstract: In these last two talks, the Galois group finally enters the picture. Let $E$ be a local field and a reductive group $G$ over $E$. Following Dat-Helm-Kurinczuk-Moss, Zhu and Fargues-Scholze, we will first explain how to construct the \textit{stack of $L$-parameters}, which is an ind-Artin-stack parametrizing $\hat{G}$-valued continuous representations of the Weil group of $E$ (for simplicity, we will restrict our attention to characteristic zero coefficients). Then we will explain how to construct an action (called the \textit{spectral action}) of the category of perfect complexes on the stack of $L$-parameters on the derived category of $\ell$-adic sheaves on $\mathrm{Bun}_G$. This is the main result of Fargues-Scholze and is obtained by combining the general version of the geometric Satake equivalence with a presentation of this category of perfect complexes by generators and relations.
The existence of the spectral action allows one to go from the « automorphic side » to the « Galois side », and conversely. In one direction, we will see that it implies quite directly the construction of $L$-parameters attached to smooth irreducible representations of $G(E)$. In the other direction, Fargues formulated in 2014 a striking conjecture predicting that one can attach to a discrete $L$-parameter an \textit{Hecke eigensheaf} on $\mathrm{Bun}_G$ with nice properties. We will recall what this conjecture says when $G=GL_n$, and explain how to prove it when the parameter is assumed to be irreducible, by using the spectral action together with the results of the previous talks.
Notes: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/Anschuetz.pdf
Speaker: Arthur-César le Bras
Abstract: In these last two talks, the Galois group finally enters the picture. Let $E$ be a local field and a reductive group $G$ over $E$. Following Dat-Helm-Kurinczuk-Moss, Zhu and Fargues-Scholze, we will first explain how to construct the \textit{stack of $L$-parameters}, which is an ind-Artin-stack parametrizing $\hat{G}$-valued continuous representations of the Weil group of $E$ (for simplicity, we will restrict our attention to characteristic zero coefficients). Then we will explain how to construct an action (called the \textit{spectral action}) of the category of perfect complexes on the stack of $L$-parameters on the derived category of $\ell$-adic sheaves on $\mathrm{Bun}_G$. This is the main result of Fargues-Scholze and is obtained by combining the general version of the geometric Satake equivalence with a presentation of this category of perfect complexes by generators and relations.
The existence of the spectral action allows one to go from the « automorphic side » to the « Galois side », and conversely. In one direction, we will see that it implies quite directly the construction of $L$-parameters attached to smooth irreducible representations of $G(E)$. In the other direction, Fargues formulated in 2014 a striking conjecture predicting that one can attach to a discrete $L$-parameter an \textit{Hecke eigensheaf} on $\mathrm{Bun}_G$ with nice properties. We will recall what this conjecture says when $G=GL_n$, and explain how to prove it when the parameter is assumed to be irreducible, by using the spectral action together with the results of the previous talks.
Notes: https://math.bu.edu/people/jsweinst/rampage/BunGCourse/LeBras.pdf