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

Abstract

In this paper, we study the $\mathbb J$-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter $μ$. 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 $μ$. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.