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 $\mathrm{GL}_n$.