Affine Deligne-Lusztig varieties and the action of 𝐽

Chen, Miaofen; Viehmann, Eva

doi:10.1090/jag/693

Cited by 18 publications

(12 citation statements)

References 21 publications

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“…Note that varying b within [b] only changes X µ (b) by an isomorphism. For suitably chosen b ∈ [b], the connected components of C μ are precisely the intersections of X µ (b) with some Iwahori-orbit on Gr G (see [4,Section 3]). Since the latter form a stratification on Gr G , we can apply the localisation long exact sequence to calculate the cohomology of X µ (b).…”

Section: Conjecture 13 (Chen Zhumentioning

confidence: 99%

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: Conjecture 13 (Chen Zhumentioning

confidence: 99%

Irreducible components of minuscule affine Deligne–Lusztig varieties

Hamacher

Viehmann

2018

Alg. Number Th.

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“…This will probably require generalizing, via Bruhat–Tits theory, the algebra of lattices in quadratic spaces we use in this paper. In another direction, it would also be interesting to understand our results from the point of view of the stratifications introduced by Chen and Viehmann in [CV15].…”

Section: Introductionmentioning

confidence: 99%

Rapoport–Zink spaces for spinor groups

Howard¹,

Pappas²

2017

Compositio Math.

155

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After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport-Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport-Zink space is determined explicitly.

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“…Proof. This follows almost immediately from the theory of affine Deligne-Lusztig varieties (see, for example, [5]) since we are comparing the geometric points of RZ-spaces for the isomorphic groups GL 2 (E) and GSp 2 (E).…”

Section: Theorem 52 (1)mentioning

confidence: 97%