Averaging functors in Fargues' program for GL_n

TL;DR

This paper explores averaging functors within Fargues' geometric Langlands framework for GL_n, providing new insights into spectral actions and verifying properties of Fargues-Scholze parameters without relying on the local Langlands correspondence.

Contribution

It explicitly constructs averaging functors in Fargues' program, verifies irreducibility of parameters for supercuspidal representations, and attaches Hecke eigensheaves to Weil representations, advancing understanding of Fargues' conjecture.

Findings

Fargues-Scholze parameters for supercuspidal GL_2 are irreducible.

Constructed Hecke eigensheaves for irreducible Weil representations.

Validated key aspects of Fargues' conjecture 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 $\mathrm{GL}_n$.