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SUMMARY:Arthur-César Le Bras (University of Paris 13)
DTSTART:20211021T160000Z
DTEND:20211021T172000Z
DTSTAMP:20260423T035753Z
UID:RAMpAGe/55
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/RAMpAGe/55/"
 >Bun_G minicourse:  Construction of the eigensheaf</a>\nby Arthur-César L
 e Bras (University of Paris 13) as part of Recent Advances in Modern p-Adi
 c Geometry (RAMpAGe)\n\n\nAbstract\nThis talk is the sixth part of a six-p
 art series "$\\mathrm{Bun}_G$\, Shtukas\, and the Local Langlands Program"
 \, held Tuesdays and Thursdays between 5 and 21 October\, 2021.\n\nRecordi
 ngs and slides will appear here:  https://sites.google.com/view/rampagesem
 inar/home\n\nSeries abstract:  The recent manuscript of Fargues-Scholze ai
 ms to "geometrize" the Langlands program for a p-adic group $G$\, by relat
 ing 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 $G$.\n\
 nThe 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 loc
 al shtuka spaces by Hansen-Kaletha-Weinstein\,  and the construction of ei
 gensheaves on $\\mathrm{Bun}_G$ when $G=\\mathrm{GL}_n$ by Anschütz-le Br
 as. \n\nTalk abstract: In these last two talks\, the Galois group finally 
 enters the picture. Let $E$ be a local field and a reductive group $G$ ove
 r $E$. Following Dat-Helm-Kurinczuk-Moss\, Zhu and Fargues-Scholze\, we wi
 ll first explain how to construct the \\textit{stack of $L$-parameters}\, 
 which is an ind-Artin-stack parametrizing $\\hat{G}$-valued continuous rep
 resentations of the Weil group of $E$ (for simplicity\, we will restrict o
 ur attention to characteristic zero coefficients). Then we will explain ho
 w to construct an action (called the \\textit{spectral action}) of the cat
 egory of perfect complexes on the stack of $L$-parameters on the derived c
 ategory of $\\ell$-adic sheaves on $\\mathrm{Bun}_G$. This is the main res
 ult 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.\nThe existence of the spect
 ral action allows one to go from the « automorphic side » to the « Galo
 is side »\, and conversely. In one direction\, we will see that it implie
 s quite directly the construction of $L$-parameters attached to smooth irr
 educible representations of $G(E)$. In the other direction\, Fargues formu
 lated in 2014 a striking conjecture predicting that one can attach to a di
 screte $L$-parameter an \\textit{Hecke eigensheaf} on $\\mathrm{Bun}_G$ wi
 th 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 irreduc
 ible\, by using the spectral action together with the results of the previ
 ous talks.\n
LOCATION:https://researchseminars.org/talk/RAMpAGe/55/
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