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.

Kottwitz 的猜想描述了超尖表示对当地 Shimura 变体的上同调的贡献，即当地 Langlands 对应。这一猜想的自然延伸涉及 Scholze 更一般的局部 shtukas 空间。使用 v 堆栈的新 Lefschetz-Verdier 迹公式，我们证明了扩展猜想，忽略 Weil 群的作用，并取模一个虚拟表示，其特征在椭圆元素的轨迹上消失。作为一个应用程序，我们表明，对于 $\operatorname {\mathrm {GL}}_n$ 的内部形式的不可约平滑表示，由 Fargues–Scholze 构造的L参数与通常由局部 Langlands 产生的半简化参数一致.