On the geometric connected components of moduli spaces of $p$-adic shtukas and local Shimura varieties

We study topological properties of moduli spaces of p-adic shtukas and local Shimura varieties. On one hand, we construct and study the specialization map for moduli spaces of p-adic shtukas at parahoric level whose target is an affine Deligne-Lusztig variety. On the other hand, given a p-adic shtuka datum $(G, b, μ)$, with $G$ unramified over $\mathbb{Q}_p$ and such that $(b, μ)$ is HN-irreducible, we determine the set of geometric connected components of infinite level moduli spaces of p-adic shtukas. In other words, we understand $π_0(\mathrm{Sht}_{(G,b,μ,\infty)} \times \mathrm{Spd} \mathbb{C}_p)$ with its right $G(\mathbb{Q}_p) \times G_b (\mathbb{Q}p ) \times W_E$ -action. As a corollary, we prove new cases of a conjecture of Rapoport and Viehmann.