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AVERAGING FUNCTORS IN FARGUES’ PROGRAM FOR $\mathrm {GL}_n$

Part of: Algebraic number theory: local and $p$-adic fields (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  24 September 2025

Johannes Anschütz Open the ORCID record for Johannes Anschütz [Opens in a new window]  and
Arthur-César Le Bras Open the ORCID record for Arthur-César Le Bras [Opens in a new window]
Johannes Anschütz
Affiliation:
Université Paris-Saclay , Laboratoire de Mathématiques d’Orsay, Orsay, France (janschuetz@math.cnrs.fr)
Arthur-César Le Bras*
Affiliation:
Université de Strasbourg UFR de Mathématique et d’Informatique , France
*
lebras@math.uni-bonn.de
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Abstract

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

MSC classification

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory 14F20: Étale and other Grothendieck topologies and (co)homologies 14G22: Rigid analytic geometry

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 40
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Bernstein, JN, Deligne, P, et al. (1984) Le centre de Bernstein. In Representations of Reductive Groups over a Local Field. Paris: Hermann, 1–32.Google Scholar
Bushnell, CJ and Henniart, G (2003) Generalized Whittaker models and the Bernstein center. Amer. J. Math. 125(3), 513–547.10.1353/ajm.2003.0015CrossRefGoogle Scholar
Dat, J-F (2012) Théorie de lubin–tate non abélienne l-entière. Duke Math. J. 161(6), 951–1010 (15 April 2012). DOI: 10.1215/00127094-1548425 10.1215/00127094-1548425CrossRefGoogle Scholar
Dat, J-F, Helm, D, Kurinczuk, R and Moss, G (2025) Moduli of langlands parameters. J. Eur. Math. Soc. (JEMS) 27(5), 1827–1927.10.4171/jems/1599CrossRefGoogle Scholar
Drevon, B and Sécherre, V (2021) Block decomposition of the category of $\ell$ -modular smooth representations of finite length of $gl\left(m,d\right)$ . arXiv preprint 2102.05898.Google Scholar
Drinfeld, VG (1983) Two-dimensional $\ell$ -adic representations of the fundamental group of a curve over a finite field and automorphic forms on $GL(2)$ . Amer. J. Math. 105(1), 85–114.10.2307/2374382CrossRefGoogle Scholar
Fargues, L (2016) Geometrization of the local Langlands correspondence: an overview. arXiv preprint arXiv:1602.00999.Google Scholar
Fargues, L (2020) G-torseurs en théorie de hodge p-adique. Available at https://webusers.imj-prg.fr/~laurent.fargues/Prepublications.html.Google Scholar
Fargues, L (2006) Dualité de Poincaré et involution de Zelevinsky dans la cohomologie étale équivariante des espaces analytiques rigides. https://webusers.imj-prg.fr/~laurent.fargues/Prepublications.html.Google Scholar
Fargues, L (2017) Simple connexité des fibres d’une application d’Abel-Jacobi et corps de classe local. arXiv preprint 1705.01526.Google Scholar
Fargues, L and Scholze, P (2021) Geometrization of the local Langlands correspondence. arXiv preprint arXiv:2102.13459.Google Scholar
Frenkel, E, Gaitsgory, D and Vilonen, K (2002) On the geometric Langlands conjecture. J. Amer. Math. Soc. 15(2), 367–417.10.1090/S0894-0347-01-00388-5CrossRefGoogle Scholar
Gaisin, I and Imai, N (2016) Non-semi-stable loci in hecke stacks and fargues’ conjecture. arXiv preprint 1608.07446.Google Scholar
Gaitsgory, D A generalized vanishing conjecture. Available at https://people.mpim-bonn.mpg.de/gaitsgde/GL/GenVan.pdf.Google Scholar
Gaitsgory, D (2004) On a vanishing conjecture appearing in the geometric Langlands correspondence. Ann. of Math. 160(2), 617–682.10.4007/annals.2004.160.617CrossRefGoogle Scholar
Hamannn, L (2022) Geometric eisenstein series, intertwining operators, and shin’s averaging formula. arXiv preprint arXiv:2209.08175.Google Scholar
Hamann, L and Imai, N (2025) Dualizing complexes on the moduli of parabolic bundles. J. Reine Angew. Math. (0).Google Scholar
Hansen, D On the supercuspidal cohomology of basic local Shimura varieties. http://www.davidrenshawhansen.net/.Google Scholar
Hansen, D, Kaletha, T and Weinstein, J (2022) On the Kottwitz conjecture for local shtuka spaces. In Forum of Mathematics, Pi, vol. 10. Cambridge: Cambridge University Press, e13. doi: 10.1017/fmp.2022.7.Google Scholar
Henniart, G and Vignéras, M-F (2020) Representations of a reductive $p$ -adic group in characteristic distinct from $p$ . arXiv preprint:2010.06462.Google Scholar
Gel’fand, I M Kazhdan, D (1972) On the representation of the group $GL(n,K)$ where $K$ is a local field. Funct Anal Its Appl. 6, 315–317. https://doi.org/10.1007/BF01077652.CrossRefGoogle Scholar
Laumon, G (1987) Correspondance de Langlands géométrique pour les corps de fonctions. Duke Math. J. 54(2), 309–359.10.1215/S0012-7094-87-05418-4CrossRefGoogle Scholar
Sécherre, V, Minguez, A and Stevens, S (2014) Types modulo $\ell$ pour les formes intérieures de $G{L}_n$ sur un corps local non archimédien. Proc. Lond. Math. Soc. 109(4), 823–891.Google Scholar
Scholze, P (2017) Étale cohomology of diamonds. Available at the author’s webpage: https://people.mpim-bonn.mpg.de/scholze/papers.html#Papers.Google Scholar
Scholze, P and Weinstein, J (2014) $p$ -adic geometry. notes from lecture course at UC Berkeley. Available at http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf.Google Scholar
Vignéras, M-F (1989) Représentations modulaires de $GL\left(2,F\right)$ en caractéristique $\ell$ , $F$ corps $p$ -adique, $p\ne \ell$ . Compos. Math. 72(1), 33–66.Google Scholar
Zhu, X (2020) Coherent sheaves on the stack of Langlands parameters. arXiv preprint 2008.02998.Google Scholar