# Averaging functors in Fargues' program for GL_n

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 Kaletha-Weinstein, that it satisfies most of the requirements of Fargues' conjecture for $\mathrm{GL}_n$.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03444489
Contributor : Arthur-César Le Bras Connect in order to contact the contributor
Submitted on : Friday, November 26, 2021 - 11:53:41 AM
Last modification on : Wednesday, December 1, 2021 - 1:33:18 PM

### File

2104.04701.pdf
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### Identifiers

• HAL Id : hal-03444489, version 1
• ARXIV : 2104.04701

### Citation

Johannes Anschütz, Arthur-César Le Bras. Averaging functors in Fargues' program for GL_n. 2021. ⟨hal-03444489⟩

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