On $\mathbb{J}$-strata with Parahoric Stabilizers in Affine Deligne-Lusztig Varieties

In this paper, we study the $\mathbb{J}$-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter $\mu$. This stratification was introduced by Chen-Viehmann and has been expected to serve as an interesting tool for studying basic loci in Shimura varieties. We parametrize the $\mathbb{J}$-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. We further prove that the cardinality of these sets is equal to that of a certain subset of the Weyl group orbit of $\mu$. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.