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.

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.