We compute the connected components of arbitrary parahoric level affine Deligne–Lusztig varieties and local Shimura varieties, thus resolving a folklore conjecture raised in (He in Some results on affine Deligne–Lusztig varieties. YouTube video, 2018; Zhou in Duke Math. J. 169(15):2937–3031, 2020) in full generality (even for non-quasisplit groups). We achieve this by relating them to the connected components of infinite level moduli spaces of $p$ -adic shtukas, where we use v-sheaf-theoretic techniques such as the specialization map of kimberlites. Along the way, we give a $p$ -adic Hodge-theoretic characterization of HN-irreducibility. As applications, we obtain many results on the geometry of integral models of Shimura varieties of Hodge type at arbitrary stabilizer-parahoric levels. In particular, we deduce new CM lifting results on integral models of Shimura varieties for quasisplit groups at parahoric levels that arise as stabilizer Bruhat–Tits group schemes.
