On categorical local langlands program for $GL_n$

TL;DR

This paper advances the categorical local Langlands program for $GL_n$ by explicitly describing the spectral action, constructing Hecke eigensheaves, and proving new cases of related conjectures using moduli spaces of local Shtukas.

Contribution

It provides an explicit description of the spectral action in the local Shtukas setting, enabling new constructions and proofs within the $GL_n$ local Langlands framework.

Findings

Explicit description of the spectral action for $GL_n$

Construction of Hecke eigensheaves for certain Weil representations

Proof of new cases of the Harris-Viehmann conjecture

Abstract

We study various moduli spaces of local Shtukas in the setting of Fargues' program for $GL_n$. In certain cases, this gives us an explicit description of the spectral action which was recently introduced by Fargues and Scholze. This description sheds light to the categorical local Langlands program for $GL_n$ and allows us to construct Hecke eigensheaves associated to certain $\ell$-adic Weil representations of rank $n$ and to prove some parts of Fargues' conjecture. Moreover, by using this description, we can prove new cases of the Harris-Viehmann conjecture for non-basic Rapoport-Zink spaces and compute some parts of the cohomology of the Igusa varieties associated to $GL_n$.