Bun_G minicourse: The spectral action

Abstract: This talk is the fifth part of a six-part series "$\mathrm{Bun}_G$, Shtukas, and the Local Langlands Program", held Tuesdays and Thursdays between 5 and 21 October, 2021.

Recordings and slides will appear here: sites.google.com/view/rampageseminar/home

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 $\mathrm{Bun}_G$. 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 $\mathrm{Bun}_G$ when $G=\mathrm{GL}_n$ by Anschütz-le Bras.

Talk 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.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

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Recent Advances in Modern p-Adic Geometry (RAMpAGe)

Series comments: The Zoom Meeting ID is: 995 3670 1681. The password for the series is: *The first three-digit prime*. Please visit the external homepage for notes and videos from past talks.

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