Zelevinsky Duality on Basic Local Shimura Varieties

TL;DR

This paper provides a straightforward proof of how Zelevinsky involution acts on the cohomology of basic local Shimura varieties, extending previous results for specific groups using Fargues-Scholze machinery.

Contribution

It offers a simplified proof of Zelevinsky involution action and generalizes prior results to broader classes of local Shimura varieties.

Findings

Zelevinsky involution action described on cohomology

Generalization of earlier results for $GL_{n}$ and $GSp_{4}$

Utilization of Fargues-Scholze machinery

Abstract

We give a simple proof of a general result describing the action of the Zelevinsky involution on the cohomology of certain basic local Shimura varieties, using the machinery of Fargues-Scholze. As an application, we generalize earlier results of Fargues and Mieda on the action of the Zelevinsky involution on the cohomology of $GL_{n}$ and $GSp_{4}$ type basic local Shimura varieties, respectively.