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arXiv · 1709.06651

On the Kottwitz conjecture for local shtuka spaces

Tasho Kaletha·David Hansen·Jared Weinstein
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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.

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Tasho Kaletha, David Hansen, Jared Weinstein. 2017-09-19. On the Kottwitz conjecture for local shtuka spaces. https://arxiv.org/abs/1709.06651

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