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Pré-Publication, Document De Travail Année : 2021

Averaging functors in Fargues' program for GL_n

Résumé

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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Dates et versions

hal-03444489 , version 1 (26-11-2021)

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