Categorical local Langlands and torsion classes of some Shimura varieties

TL;DR

This paper advances the understanding of the categorical local Langlands correspondence for $GL_n$ by explicitly describing spectral actions and proving conjectures related to Shimura varieties and torsion coefficients.

Contribution

It provides an explicit spectral description for local Shimura varieties and proves a strongly generic case of the categorical local Langlands conjecture with torsion coefficients.

Findings

Explicit description of spectral action for certain local Shimura varieties.

Proof of a strongly generic case of the categorical local Langlands conjecture.

New vanishing results for cohomology of Shimura varieties of type A.

Abstract

We study the cohomology of various local Shimura varieties for $GL_n$. This provides an explicit description of the spectral action constructed by Fargues-Scholze in certain cases and allows us to prove some strongly generic part of the categorical local Langlands conjecture for $GL_n$ with torsion coefficients. As applications, we are able to prove an analogue of the Harris-Viehmann conjecture and deduce new vanishing results for the cohomology of Shimura varieties of type A in the torsion coefficient setting.