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 ${\mathrm{GL}}_n$.

References in corpus (3)

• Geometrization of the local Langlands correspondence
• Representations of a reductive $p$-adic group in characteristic distinct from $p$
• Block decomposition of the category of l-modular smooth representations of finite length of GL(m,D)