Geodesic
On the Kottwitz conjecture for local shtuka spaces
Forum of Mathematics, Pi, Tome 10 (2022)

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Kottwitz’s conjecture describes the contribution of a supercuspidal representation 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 $\operatorname {\mathrm {GL}}_n$, the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands. 
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David Hansen; Tasho Kaletha; Jared Weinstein. On the Kottwitz conjecture for local shtuka spaces. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2022.7

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