BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Johannes Anschütz (University of Bonn) DTSTART:20211019T160000Z DTEND:20211019T172000Z DTSTAMP:20260423T021303Z UID:RAMpAGe/54 DESCRIPTION:Title: Bun_G minicourse: The spectral action\nby Johannes Anschütz (Univer sity of Bonn) as part of Recent Advances in Modern p-Adic Geometry (RAMpAG e)\n\n\nAbstract\nThis talk is the fifth part of a six-part series "$\\mat hrm{Bun}_G$\, Shtukas\, and the Local Langlands Program"\, held Tuesdays a nd Thursdays between 5 and 21 October\, 2021.\n\nRecordings and slides wil l appear here: https://sites.google.com/view/rampageseminar/home\n\nSerie s 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. La fforgue\, the main result of [FS] is the construction of an L-parameter at tached to a smooth irreducible representation of $G$.\n\nThe goal of this series is to review the main ideas of this work\, and to discuss two relat ed results: progress on the Kottwitz conjecture for local shtuka spaces b y Hansen-Kaletha-Weinstein\, and the construction of eigensheaves on $\\m athrm{Bun}_G$ when $G=\\mathrm{GL}_n$ by Anschütz-le Bras. \n\nTalk abstr act: 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 D at-Helm-Kurinczuk-Moss\, Zhu and Fargues-Scholze\, we will first explain h ow to construct the \\textit{stack of $L$-parameters}\, which is an ind-Ar tin-stack parametrizing $\\hat{G}$-valued continuous representations of th e Weil group of $E$ (for simplicity\, we will restrict our attention to ch aracteristic zero coefficients). Then we will explain how to construct an action (called the \\textit{spectral action}) of the category of perfect c omplexes 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-Sch olze and is obtained by combining the general version of the geometric Sat ake equivalence with a presentation of this category of perfect complexes by generators and relations.\nThe existence of the spectral action allows one to go from the « automorphic side » to the « Galois side »\, and c onversely. In one direction\, we will see that it implies quite directly t he construction of $L$-parameters attached to smooth irreducible represent ations of $G(E)$. In the other direction\, Fargues formulated in 2014 a st riking conjecture predicting that one can attach to a discrete $L$-paramet er 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 th e spectral action together with the results of the previous talks.\n LOCATION:https://researchseminars.org/talk/RAMpAGe/54/ END:VEVENT END:VCALENDAR