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 Qₚ 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 π₀(Sht(G,b,μ,∞) × Spd Cₚ) with its right G(Qₚ) × Gb (Qp ) × WE -action. As a corollary, we prove new cases of a conjecture of Rapoport and Viehmann.