Irreducible components of minuscule affine Deligne–Lusztig varieties

Abstract

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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