Connected components of affine Deligne–Lusztig varieties in mixed characteristic

Chen, Miaofen; Kisin, Mark; Viehmann, Eva

doi:10.1112/s0010437x15007253

Cited by 37 publications

(76 citation statements)

References 33 publications

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“…Let G ad be the adjoint group of G. We denote by a subscript "ad" the image of an element of G(L), X * (T ) or π 1 (G) in G ad (L), X * (T ad ) or π 1 (G ad ), respectively. By [5,Cor. 2.4.2], the homeomorphism of Proposition 3.1 induces a universal homeomorphism…”

Section: Equidimensionalitymentioning

confidence: 97%

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Irreducible components of minuscule affine Deligne–Lusztig varieties

Hamacher

Viehmann

2018

Alg. Number Th.

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We examine the set of J b (F )-orbits in the set of irreducible components of affine Deligne-Lusztig varieties for a hyperspecial subgroup and minuscule coweight µ. Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with µ of the dual group.The authors were partially supported by ERC starting grant 277889 "Moduli spaces of local G-shtukas".The following basic assertion seems to be well-known, but we could not find a reference in the literature.Lemma 1.1. The scheme X µ (b) is locally of finite type in the equal characteristic case and locally of perfectly finite type in the case of unequal characteristic.Proof. The proof of this is the same as the corresponding part of the analogous statement for moduli spaces of local G-shtukas, compare the proof of Theorem 6.3 in [12] (where only the first half of p. 113 of loc. cit. is needed). In that proof, the case of equal characteristic and split G is considered. However, the general statement follows from the same proof.Notice that in general X µ (b) is not quasi-compact since it may have infinitely many irreducible components. It is conjectured to be equidimensional, but this has not been proven in full generality yet. In Section 3 we give an overview about the cases where equidimensionality has been proven. In the case of µ minuscule, which we are primarily interested in here, there are only a few exceptional cases where this is not yet known.Definition 1.2. For a finite-dimensional k-scheme X we denote by Σ(X) the set of irreducible components of X and by Σ top (X) ⊂ Σ(X) the subset of those irreducible components which are top-dimensional.The affine Deligne-Lusztig varieties X µ (b) and X µ (b) carry a natural action (by left multiplication) by the groupThis action induces an action of J b (F ) on the set of irreducible components.A complete description of the set of orbits was previously only known for the groups GL n and GSp 2n and minuscule µ where the action is transitive ([24],[25]), and for some other particular cases, see for example [28] for a particular family of unitary groups and minuscule µ.To describe the (conjectured) number of orbits, denote byĜ the dual group of G in the sense of Deligne and Lusztig. That is,Ĝ is the reductive group scheme over O F that contains a Borel subgroupB with maximal torusT and maximal split torusŜ such that there exists an Galois equivariant isomorphism X * (T ) ∼ = X * (T ) identifying simple coroots ofT with simple roots of T . For any µ ∈ X * (T ) dom = X * (T ) dom we denote by V µ the associated Weyl module ofĜ OL .In the following we use an element λ G (b) ∈ X * (T Γ ) that we define in Section 2. Its restriction λ toŜ can be seen as a 'best integral approximation' of the Newton point ν b of [b], while its precise value in X * (T Γ ) will depend on the Kottwitz point κ G (b). We choose a liftλ ∈ X * (T ). Conjecture 1.3 (Chen, Zhu). There exists a canonical bijection between J b (F )\Σ(X µ (b)) and the basis of V µ (λ G ...

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Section: Equidimensionalitymentioning

confidence: 97%

“…In the function field case, let R = R Proof. This is [5,Lemma 3.4.4], except for the fact that in loc. cit., R is assumed to be smooth, and only the case of mixed characteristic is considered.…”

Section: Reduction To the Superbasic Casementioning

confidence: 99%

Irreducible components of minuscule affine Deligne–Lusztig varieties

Hamacher

Viehmann

2018

Alg. Number Th.

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“…Then π 0 (X λ (b)) ∼ = π 1 (G) σ Γ 0 . This was first proved by Viehmann for split groups, and then by Chen, Kisin and Viehmann [5] for quasi-split unramified groups and for λ minuscule. The description of π 0 (X λ (b)) for G quasi-split unramified, and λ non-minuscule, was conjectured in [5] and was established by Nie [58].…”

Section: Connected Componentsmentioning

confidence: 82%

Some Results on Affine Deligne–lusztig Varieties

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.2010 Mathematics Subject Classification. 14L05, 20G25.Affine Deligne-Lusztig varieties are schemes locally of finite type over F q . Also the varieties are isomorphic if the element b is replaced by another element b ′ in the same σ-conjugacy class.A major difference between affine Deligne-Lusztig varieties and classical Deligne-Lusztig varieties is that affine Deligne-Lusztig varieties have the second parameter:

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“…Proof. By Lemma 2.3.6 of [7], the map ω G is compatible with the J b (Q p )-actions on both sides. By construction, J b (Q p ) acts on π 1 (G) Γ by left multiplication via the map…”

Section: 2mentioning

confidence: 90%

On Some Generalized Rapoport–Zink Spaces

Shen¹

2019

Can. J. Math.-J. Can. Math.

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We enlarge the class of Rapoport-Zink spaces of Hodge type by modifying the centers of the associated p-adic reductive groups. These such-obtained Rapoport-Zink spaces are called of abelian type. The class of Rapoport-Zink spaces of abelian type is strictly larger than the class of Rapoport-Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport-Zink spaces of abelian type can be viewed as moduli spaces of local G-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport-Zink spaces of abelian type. We construct and study the Ekedahl-Oort stratifications for the special fibers of Rapoport-Zink spaces of abelian type. As an application, we deduce a Rapoport-Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

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Connected components of affine Deligne–Lusztig varieties in mixed characteristic

Cited by 37 publications

References 33 publications

Irreducible components of minuscule affine Deligne–Lusztig varieties

Irreducible components of minuscule affine Deligne–Lusztig varieties

Some Results on Affine Deligne–lusztig Varieties

On Some Generalized Rapoport–Zink Spaces

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