Twisted orbital integrals and irreducible components of affine Deligne–Lusztig varieties

Abstract

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and qanalogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirković-Vilonen basis of a certain weight space of a representation of the Langlands dual group.Understanding of basic geometric properties of affine Deligne-Lusztig varieties has been fruitful for arithmetic applications. For instance an understanding of the connected components [CKV15] was applied to the proof of a version of Langlands-Rapoport conjecture by Kisin [Kis17]. The geometry of the supersingular locus of Hilbert modular varieties, which is also a question closely related to affine Deligne-Lusztig varieties via p-adic uniformizatoin, was applied to arithmetic level raising in the recent work of Liu-Tian [LT17].In this paper, we concern the problem of parameterizing the irreducible components of affine Deligne-Lusztig varieties. This problem was initiated in the work of Xiao-Zhu [XZ17]. These authors studied this problem in some special cases, as an essential ingredient in their proof of a version of Tate's conjecture for special fibers of Shimura varieties. After that, Miaofen Chen and Xinwen Zhu formulated a general conjecture, relating the set of top dimensional irreducible components of general affine Deligne-Lusztig varieties to the Mirković-Vilonen cycles in the affine Grassmannian, and thus to the representation theory of the Langlands dual group via the geometric Satake. Partial results towards this conjecture have been obtained by Xiao-Zhu [XZ17], Hamacher-Viehmann [HV17], and Nie [Nie18a], based on a common idea of reduction to the superbasic case (which goes back to [GHKR06]).In this paper we present a new method and prove:Theorem. The Chen-Zhu Conjecture (see Conjecture 1.2.1) holds in full generality.Our proof is based on an approach completely different from the previous works. The problem is to compute the number of top dimensional irreducible components of an affine Deligne-Lusztig variety (modulo a certain symmetry group). We use the Lang-Weil estimate to relate this number to the asymptotic behavior of the number of points on the affine Deligne-Lusztig variety over a finite field, as the finite field grows. We show that the number of points over a finite field is computed by a twisted orbital integral, and thus we reduce the problem to the asymptotic behavior of twisted orbital integrals. We study the latter using explicit methods from local harmonic analysis and representation theory, including the Base Change Fundamental Lemma and the Kato-Lusztig formula.An interesting point in our proof is that we apply the Base Change Fundamental Lemma, which is only available in general for mixed characteristic local fields as the current known proofs of it r...

Stabilizers of irreducible components of affine Deligne--Lusztig varieties

“…The proof of Theorem A. Our proof of Theorem A makes use of techniques from p-adic harmonic analysis developed in [ZZ20], and the Deligne-Lusztig reduction method for affine Deligne-Lusztig varieties developed in [He14]. For simplicity in the introduction, we assume that G has no factors of type A or E 6 .…”

“…Special cases of this conjecture was proved by Xiao-Zhu [XZ17], Hamacher-Viehmann [HV18] and Nie [Nie18b]. The conjecture was finally proved by Nie [Nie18a], and by the second and third authors [ZZ20] using different methods.…”

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.…”