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SUMMARY:Linus Hamann (Princeton)
DTSTART:20201029T160000Z
DTEND:20201029T172000Z
DTSTAMP:20260423T052758Z
UID:RAMpAGe/23
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/RAMpAGe/23/"
 >Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands Corre
 spondences</a>\nby Linus Hamann (Princeton) as part of Recent Advances in 
 Modern p-Adic Geometry (RAMpAGe)\n\n\nAbstract\nIn upcoming work\, Fargues
  and Scholze construct a candidate for a general local Langlands correspon
 dence\,  \nassociating to a smooth irreducible representation of a connect
 ed reductive group $G/\\mathbf{Q}_{p}$ a continuous semisimple Weil parame
 ter\, using the action of excursion operators\non the moduli space of $G$-
 bundles on the Fargues-Fontaine curve. It is a natural question to ask whe
 ther this correspondence is compatible with known instances of the local L
 anglands correspondence after semi-simplification. For $G = \\mathrm{GL}_{
 n}$\, this compatibility is deduced from the fact that correspondence of H
 arris-Taylor is realized in the cohomology of the Lubin-Tate tower at infi
 nite level\, via its interpretation as a moduli space of mixed characteris
 tic shtukas. For $G = \\mathrm{GSp}_{4}$ or its inner form $\\mathrm{GU}_{
 2}(D)$\, there is a local Langlands correspondence constructed by Gan-Take
 da and Gan-Tantono\, respectively. We will discuss upcoming work related t
 o proving compatibility in this case. Similar to the case of $\\mathrm{GL}
 _{n}$\, this involves realizing this local Langlands correspondence in the
  cohomology of the local Shimura varieties at infinite level associated wi
 th these groups. We do this by applying basic uniformization of these loca
 l Shimura varieties due to Shen\, as well as results on Galois representat
 ions in the cohomology of the relevant global Shimura varieties due to Sor
 ensen and Kret-Shin. After proving this compatibility\, we employ various 
 new ideas from the geometry of the Fargues Scholze correspondence to obtai
 n a complete description of the $\\rho$-isotypic part of the cohomology of
  this local Shimura variety at infinite level\, where $\\rho$ is a represe
 ntation of $G$ with supercuspidal Gan-Takeda or Gan-Tantono parameter\, th
 ereby verifying the strongest form of the Kottwitz conjecture for these sp
 ecific representations.\n
LOCATION:https://researchseminars.org/talk/RAMpAGe/23/
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