Convolution morphisms and Kottwitz conjecture

TL;DR

This paper advances the understanding of the Kottwitz conjecture by defining etale cohomology for moduli spaces of mixed characteristic local shtukas, establishing new cases of the conjecture, and exploring its limitations.

Contribution

It introduces a new framework for etale cohomology of local shtukas, proves the Kottwitz conjecture in several new cases, and analyzes its failure in non-minuscule and non-cuspidal scenarios.

Findings

Kottwitz conjecture holds for all inner forms of GL_3 with minuscule cocharacters.

The conjecture is valid for any inner form of GL_2 with cuspidal Langlands parameters.

The conjecture fails in non-minuscule cases with non-cuspidal Langlands parameters.

Abstract

We define etale cohomology of the moduli spaces of mixed characteristic local shtukas so that it gives smooth representations including the case where the relevant elements of the Kottwitz set are both non-basic. Then we relate the etale cohomology of different moduli spaces of mixed characteristic local shtukas using convolution morphisms, duality morphisms and twist morphisms. As an application, we show the Kottwitz conjecture in some new cases including the cases for all inner forms of $\mathrm{GL}_3$ and minuscule cocharacters. We study also some non-minuscule cases and show that the Kottwitz conjecture is true for any inner form of $\mathrm{GL}_2$ and any cocharacter if the Langlands parameter is cuspidal. On the other hand, we show that the Kottwitz conjecture does not hold as it is in non-minuscule cases if the Langlands parameter is not cuspidal. Further, we show that a generalization of the Harris--Viehmann conjecture for the moduli spaces of mixed characteristic local shtukas does not hold in Hodge--Newton irreducible cases.