Irreducible components of affine Deligne-Lusztig varieties

Abstract

By extending the method of semi-modules developed by de Jong, Oort, Viehmann and Hamacher, we introduce a stratification for the affne Deligne-Lusztig variety (in the affne Grassmannian) attached to attached to a minuscule cocharacter and a basic element. As an application, we complete the proof of a conjecture on the irreducible components of affne Deligne-Lusztig varieties, due to

Geometric Structure of Affine Deligne-Lusztig Varieties for $GL_3$

“…Recently, the parametrization problem of top-dimensional irreducible components of X λ (b) was also solved. See [18] and [22]. Besides the geometric properties as above, it is known that in certain cases, the (closed) affine Deligne-Lusztig variety admits a simple description.…”

Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

“…This change makes Kisin varieties much harder to study compared to affine Deligne-Lusztig varieties. Much is known about the structure of affine Deligne-Lusztig varieties by the study of many people, such as the non-emptyness ( [11], [13], [17], [22], [24], [29]), dimension formula ( [12], [14], [31], [36]), set of connected components ( [4], [5], [6], [18], [25], [32]) and set of irreducible components up to group action ( [15], [26], [34], [35]). One of the powerful tools to study affine Deligne-Lusztig varieties is the semi-module stratification which arises in a group theoretic way.…”

“…For example, de Jong and Oort used this stratification to show that each connected component of superbasic affine Deligne-Lusztig variety for GL n is irreducible ( [7]). The second author used this stratification to give a proof of a conjecture of Xinwen Zhu and the first author about the irreducible components of affine Deligne-Lusztig varieties ( [26], with another proof using twisted orbital integrals given by Zhou and Zhu [35]). In this paper, we want to apply this tool to the study of Kisin varieties.…”