On the Kottwitz conjecture for local shtuka spaces

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