On the Kottwitz conjecture for local shtuka spaces

TL;DR

This paper proves an extension of Kottwitz's conjecture for local shtuka spaces using a new trace formula, confirming the compatibility of Fargues-Scholze's L-parameters with classical local Langlands parameters.

Contribution

The paper introduces a new Lefschetz-Verdier trace formula for v-stacks and proves the extended Kottwitz conjecture, linking local shtuka cohomology with the local Langlands correspondence.

Findings

Proved the extended Kottwitz conjecture for local shtukas.

Confirmed the compatibility of Fargues-Scholze L-parameters with classical parameters.

Developed a new trace formula for v-stacks.

Abstract

Kottwitz's conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze's more general spaces of local shtukas. Using a new Lefschetz-Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that for an irreducible smooth representation of an inner form of $\mathrm{GL}_n$, the $L$-parameter constructed by Fargues-Scholze agrees with the usual semisimplified parameter arising from local Langlands.