Basic loci of Coxeter type with arbitrary parahoric level

Abstract Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.

1. Introduction 1.1.One way to understand arithmetic information encoded in Shimura varieties is to study the geometry and cohomology of the special fiber of a suitable integral model.This method has been applied with great success for example in the work of Harris and Taylor on the local Langlands correspondence for GL n .The moduli description which is available at least in the case of Shimura varieties of Hodge type, together with the particular structure obtained (ultimately) from the Frobenius morphism, allows to define certain stratifications of the special fiber whose strata can be studied one by one.In particular, there is the Newton stratification which is defined, roughly speaking, by grouping those points into one stratum whose corresponding abelian varieties have isogenous p-divisible groups.Another important stratification is the EKOR stratification defined in [19], a stratification which simultaneously generalizes the Ekedahl-Oort stratification for hyperspecial level and the Kottwitz-Rapoport stratification for Iwahori level.
It has been observed, over the past few decades, that in certain cases the unique closed Newton stratum, the so-called basic locus, has a simple description.More precisely, it has a stratification as a union of Deligne-Lusztig varieties, where the index set of the union and the combinatorics of the closure relations can be described in terms of a certain Bruhat-Tits building attached to the situation at hand.While for the Siegel moduli space of principally polarized g-dimensional abelian varieties this works only when g 2, there are several infinite families where such a description is possible; the cases studied in most detail so far arise from unitary Shimura varieties for unitary groups of signature (1, n − 1).See the paper [44] by Vollaard and Wedhorn for a prototypical example, and Section 6 for a more detailed discussion of individual cases and further references.Results of this type have found applications of different kinds: • Explicit descriptions of the basic locus have been used to compute intersection numbers of special cycles on the special fiber of a Shimura variety, in order to prove results predicted by the Kudla-Rapoport program which relates such intersection numbers to Fourier coefficients of modular forms (in a general sense).As examples, we mention [26], [40], [27].• Similar intersection numbers on Rapoport-Zink spaces play a role in the arithmetic fundamental lemma and in arithmetic transfer conjectures.See for instance [51], [33].• In a different direction, a good understanding of the basic locus has been of high importance for some recent results around the Tate conjecture for the special fiber of certain Shimura varieties.See for example [20], [42], [49].
In this paper, we give a group-theoretic view on this phenomenon, extending previous work (see [8], [9]) in this direction.
1.2.Let us explain the main results of this paper.We fix a connected reductive group G over a non-archimedean local field F and a conjugacy class of cocharacters µ of G over a (fixed) algebraic closure of F .Let τ ∈ B(G, µ) be the unique basic element.Fix a rational level structure K. See Section 2.1 for the notation used here and for more details.
The central object of this paper is the generalized affine Deligne-Lusztig variety X(µ, τ ) K , which can be viewed as a group-theoretic model of the basic locus mentioned above, in those cases where G and µ come from a Shimura datum.
The following definition (which originates from [8]) singles out a class of particularly wellbehaved cases.The idea behind it is to express the condition that X(µ, τ ) K is a union of classical Deligne-Lusztig varieties attached to a twisted Coxeter element (in some finite Weyl group).
We define (cf.Definition 2. 4 for further details and equivalent formulations) Definition 1.1.The datum (G, µ, K) is said to be of Coxeter type if every EKOR stratum that occurs in X(µ, τ ) K is the EKOR stratum of a Weyl group element w that is a twisted Coxeter element in a finite standard parabolic subgroup of the Iwahori-Weyl group W .
The notion of EKOR strata that we use here is the local version of the EKOR strata introduced in [19], an interpolation between Ekedahl-Oort and Kottwitz-Rapoport strata.See Sections 2.3 and 2.4 for further details.
The main novelty in this paper are new characterizations of the cases of Coxeter type, on the one hand by a simple dimension condition, on the other hand, equivalently, by an explicit grouptheoretic condition which involves neither affine Deligne-Lusztig varieties nor the µ-admissible set.As a consequence, we obtain a classification of all Coxeter cases.
We start by establishing the following general lower bound Theorem 1.2.(Theorem 3.5) Let J τ denote the σ-centralizer of τ .Suppose that µ is non-central in every simple factor of the adjoint group G ad over F .We have that dim X(µ, τ ) K rank ss F (J τ ).We can characterize the cases that are of Coxeter type as precisely those cases where equality holds in the previous theorem: Theorem 1.3.(Theorem 4.6) Suppose that µ is non-central in every simple factor of the adjoint group G ad over F .The following conditions are equivalent: (1) The triple (G, µ, K) is of Coxeter type; (2) We have that dim X(µ, τ ) K = rank ss F (J τ ); (3) For any admissible triple (ξ, J, K ′ ) with K ′ ⊇ K, we have that µ, 2ρ ♯{σ-orbits of K ′ ξ } + rank ss F (J τ ).We list Condition (3) in this theorem to indicate that we have a simple group-theoretic characterization of these cases which involves neither the dimension of X(µ, τ ) K , nor the more subtle combinatorics of the admissible set.For the notation used here, we refer to Section 4. This condition allows us to classify all cases of Coxeter type (without using the classification results of [8], [9]).
Theorem 1.4 (Theorem 4.6, Table 1).Assume that G is quasi-simple over F and µ is noncentral.Denote by W a the corresponding affine Weyl group, by µ the image of a dominant representative of µ in the translation lattice of the Iwahori-Weyl group, and by σ the automorphism of W a induced by Frobenius.
The property whether (G, µ, K) is of Coxeter type depends only on the tuple (W a , σ, µ, K).
The quadruples (W a , σ, µ, K) of Coxeter type with K minimal are classified as follows (up to isomorphism, see Section 2.2.1 for the notation): It is easy to see (Remark 2.5) that whenever (W a , σ, µ, K) is of Coxeter type and K ⊆ K ′ , then (W a , σ, µ, K ′ ) is of Coxeter type.From the classification, we also obtain the following, quite surprising result, which does not seem to follow directly from the characterization above: In all cases, there is a unique (up to isomorphism) minimal set K ⊂ S such that (W a , σ, µ, K) is of Coxeter type.Note however that the situation is quite subtle: Starting with a datum (G, µ), the corresponding statement is not true (see the C-BC 2 cases in Table 3) because isomorphisms of the Dynkin diagram might not be automorphisms of the oriented local Dynkin diagram.
In Section 5, we discuss consequences of our results for Rapoport-Zink spaces.For many of the pairs (G, µ) in our list, a stratification of the reduced special fiber by classical Deligne-Lusztig varieties has already been established.In most cases, these results deal with maximal parahoric level structure.The stratification is called the "Bruhat-Tits stratification" because its index set is related to a Bruhat-Tits building.
It is expected -and known in many cases -that one can identify the perfection of the special fiber with a generalized affine Deligne-Lusztig variety of the form X(µ, b) K (see Section 5, in particular Property (♦)).This makes the connection with the group-theoretic results, and the Bruhat-Tits stratification on that side of the story, see Section 2.4.
Proposition 5.8 below allows us to establish a stratification of the special fiber for many nonmaximal level structures in cases of Coxeter type, before passing to the perfection.Note that to prove this result, we need to know a priori that X(µ, τ ) K has a Bruhat-Tits stratification.

1.3.
Comparison with previous results.This article can be seen as a continuation of [8] where a classification of all cases of Coxeter type was obtained under the following additional (and, a priori, quite restrictive) assumption: K = S − {v} is the complement of a single element of S (and, as above, is assumed to be preserved by σ, i.e., v is a fix point of σ).
In this paper we remove this restriction on K, and obtain different more conceptual characterizations of the cases of Coxeter type.
In [9] we introduced and studied the notion of fully Hodge-Newton decomposable pairs (G, µ) (see Section 2.3), and gave a classification of those cases.We will see below (Corollary 4.7) that (G, µ) is fully Hodge-Newton decomposable whenever it is of Coxeter type.Note however that the classification results in the paper at hand do not make use of the classification of fully Hodge-Newton decomposable cases in [9].While in all fully Hodge-Newton decomposable cases, the space X(µ, b) K (for basic b) has a stratification into classical Deligne-Lusztig varieties (called the weak Bruhat-Tits stratification in Section 2.4), this stratification has additional nice properties in the cases of Coxeter type.
1.4.Outline of the paper.We recall some preliminary notions and explain the general setting in Section 2. After recalling the method of Deligne-Lusztig reduction in Section 3.1, we prove the dimension formula characterizing Coxeter type cases in Section 3, and prove the classification in Section 4. In Section 5 we discuss consequences for Rapoport-Zink spaces and relate our results to previous work on the side of Shimura varieties and Rapoport-Zink spaces.See Table 3 in Section 6.1 for a summary.1.5.Acknowledgments.We would like to thank George Pappas for answering questions on parahoric subgroups.

Coxeter type
2.1.Setup.Let F be a non-archimedean local field, fix an algebraic closure F , denote by F the completion of its maximal unramified extension F un ⊂ F , and by σ the Frobenius automorphism of F over F .We usually think of F being of mixed characteristic, i.e., F is a finite extension of Q p .Everything has an equal-characteristic counterpart, however, where F is of the form F q ((t)), the Laurent series field over a finite field F q .In either case, we denote by p the residue characteristic of F .
We fix a connected reductive group G over F .Write G = G( F ). Let Ȋ ⊆ G be a σ-invariant Iwahori subgroup, our standard Iwahori subgroup, and let T be a maximal torus of G such that the alcove a corresponding to Ȋ, the standard alcove in the Bruhat-Tits building of G over F , lies in the apartment attached to T F .Attached to this data, we have the extended affine Weyl group W and the (relative) finite Weyl group W 0 .We fix a special vertex of the base alcove and obtain a splitting W = X * ⋊ W 0 , where X * := X * (T ) Γ0 denotes the coinvariants of the cocharacter lattice of T with respect to Γ 0 = Gal(F /F un ).See [43], [10, §2].
We denote by S the set of simple affine reflections (defined by our base alcove) inside the affine Weyl group W a ⊆ W .The Frobenius σ acts on S (since by assumption the base alcove is fixed by σ).Likewise, if τ ∈ W has length 0, then it fixes the base alcove and thus acts by conjugation on S; we denote this action by Ad(τ ).For an element w = w ′ τ ∈ W a τ , the σ-support supp σ (w) is the smallest subset of S which is Ad(τ ) • σ-stable and contains all simple affine reflections that occur in a reduced expression for w ′ .The final condition can also be rephrased as w ′ ∈ W supp σ (w) , where for a subset K ⊆ S we write W K for the subgroup of W a generated by the elements of K. We denote by K W the set of minimal length representatives of the cosets in W K \ W .
For b ∈ G, we denote by J b the σ-centralizer of b, i.e., If b is understood, we just write J instead of J b .Below we always work with the unique reduced root system Φ underlying the relative root system of G over F (the échelonnage root system).

(Enhanced) Tits data and Coxeter data.
To specify the classification results below, two types of data typically arise.On the one hand, we will refer to affine Weyl groups together with an automorphism and a coweight; this kind of data we will call Coxeter data.On the other hand, we will refer to algebraic groups over F together with a conjugacy class of cocharacters; this is what we will call Tits data below (using Tits's translation between isomorphism classes of such data with local Dynkin diagrams).In both cases, we often enhance these data by including a "level structure", i.e., a subset K ⊂ S with W K finite, of the set of simple affine reflections.In the group case, this gives rise to a standard parahoric subgroup.Our level structure will always be assumed to be rational, i.e., K is fixed by the automorphism σ (the automorphism induced by the Frobenius over F in the group case).
We hope that no confusion will arise between the notions of Coxeter datum and of being of Coxeter type, the latter being a property that certain Coxeter data have, and others do notsimilarly as some elements of a Coxeter group are Coxeter elements.Definition 2.1 (cf.[18], [10, §2.6]).(1) A Coxeter datum (over F ) is a tuple ((W a , S), σ, λ) consisting of an affine Coxeter system, a length-preserving automorphism σ and a W 0conjugacy class λ in X * , the coweight lattice.Here W 0 denotes the finite Weyl group of the given affine Coxeter system.An enhanced Coxeter datum is a tuple ((W a , S), σ, λ, K) whose first three entries constitute a Coxeter datum and where K S is a subset with σ(K) = K.Below, we often just write W a instead of (W a , S), or we replace this item by the corresponding affine Dynkin type.
(2) A Tits datum (over F ) is a tuple ( ∆, σ, λ) consisting of an absolute local Dynkin diagram (cf.[43]), a diagram automorphism and a W 0 -conjugacy class λ in the coweight lattice X * .An enhanced Tits datum is a tuple ( ∆, σ, λ, K) whose first three entries constitute a Tits datum and where K is a type of rational parahoric subgroups in the corresponding group.
2.2.1.Notation for automorphisms of Dynkin diagrams.We use the same labeling of the Coxeter graph as in Bourbaki [2, Plate I-X].As in [10], we use the following notation for automorphisms of affine Dynkin diagrams.In case the fundamental coweight ω ∨ i is minuscule, we denote the corresponding length 0 element τ (t ω ∨ i ) by τ i ; conjugation by τ i is a length preserving automorphism of W which we denote by Ad(τ i ).For type A n , the automorphism Ad(τ i ) is the rotation of the affine Dynkin diagram by i steps (i.e., s 0 is mapped to s i , s 1 is mapped to s i+1 , and so on), and we also denote it by ̺ i .We write ς 0 for the automorphism which fixes the vertex 0, and is the unique nontrivial diagram automorphism of the finite Dynkin diagram, if W 0 is of type A n , D n (with n 5) or E 6 .For type D 4 , we also denote by ς 0 the diagram automorphism which interchanges α 3 and α 4 .
For the product Ãn−1 × Ãn−1 , we denote by 1 ς 0 the automorphism which switches the two factors.
2.3.Fully Hodge-Newton decomposable pairs (G, µ).We now fix a conjugacy class µ of cocharacters G m,F → G F over the algebraic closure F of F .We denote by µ + ∈ X * (T ) the dominant representative of this conjugacy class, and by µ the image of µ + in the coweight lattice X * = X * (T ) Γ0 , i.e., the translation lattice of the Iwahori-Weyl group.Cf. [10, §2.2].
We also fix a (representative in G of a) length 0 element τ ∈ W whose σ-conjugacy class is the unique basic element in B(G, µ).
Let π = π K : G/ Ȋ → G/ K denote the projection from the affine flag variety to the partial affine flag variety of level K ( K denotes the standard parahoric subgroup of type K).Recall that Adm(µ) = {w ∈ W ; w t x(µ) for some x ∈ W 0 } denotes the µ-admissible set.By definition, We write K Adm(µ) = Adm(µ)∩ K W for the subset of Adm(µ) consisting of all elements which are of minimal length in their right W K -coset.Below, we sometimes write X K,w (τ ) = π(X w (τ )).
Note that this property, as shown by condition (1), is independent of K, and depends only on the Coxeter datum (W a , σ, µ).
Then we can rewrite condition (2) as Definition 2.3.We say an element w ∈ W a τ is a σ-Coxeter element (or is a twisted Coxeter element ) if from each Ad(τ ) • σ-orbit on S at most one simple reflection appears in some (or equivalently, any) reduced expression of wτ −1 .
We denote by K Cox(µ) ⊂ K Adm(µ) 0 the subset of K Adm(µ) 0 consisting of σ-Coxeter elements.* If K = ∅, then we may simply omit the superscript and write Adm(µ) 0 and Cox(µ).We define Definition 2.4.We say that the triple (G, µ, K) or the corresponding quadruple (W a , µ, σ, K) Note that this is the same definition as given in the introduction.Furthermore, this definition is equivalent to the one in [8, §5.1].In fact, the definition in loc.cit. is easily seen to be equivalent to saying that The equivalence of the two definitions then follows from (2) in Theorem 2.2.
2.4.The Bruhat-Tits stratification.Let (G, µ) be fully Hodge-Newton decomposable.As pointed out above, we have where as before π : G/ Ȋ → G/ K denotes the projection from the full affine flag variety to the partial affine flag variety of type K.
For each w ∈ K Adm(µ) 0 , we have where Y (w) is a classical Deligne-Lusztig variety in a (finite-dimensional) flag variety and P w is an Ad(τ ) • σ-stable parahoric subgroup of G attached to w by an explicit recipe.More precisely, denote by P ♭ w the standard parahoric subgroup generated by supp σ (w), and by K ⊂ G the standard parahoric subgroup of type K.The projection π restricts to an isomorphism from {g Ȋ; g ∈ P ♭ w , g −1 τ σ(g) ∈ Ȋw Ȋ} onto its image Y (w).So Y (w) is isomorphic to the classical Deligne-Lusztig variety attached to wτ −1 in the (finite-dimensional) flag variety P ♭ w / Ȋ, for the Frobenius given by Ad(τ ) • σ.
The parahoric subgroup P w is the standard parahoric generated by supp σ (w) and I(K, w, σ).
Here I(K, w, σ) is the maximal Ad(w) • σ-stable subset of K. Then [9,Prop. 5.7] shows that W supp σ (w)∪I(K,w,σ) is finite.Furthermore, we have the following simple description: Lemma 2.6.Suppose that w is a σ-Coxeter element in W supp σ (w) .Then I(K, w, σ) is the set of all s ∈ S − supp σ (w) with the following two properties: (a) s commutes with every element of supp σ (w), and (b) the Ad(τ ) • σ-orbit of s is contained in K.
In terms of the Dynkin diagram we can express this as saying that I(K, w, σ) is the union of those Ad(τ ) • σ-orbits in K in which no element is connected to any vertex in supp σ (w).
Putting together these stratifications for all the different w, we obtain the decomposition of X(µ, τ ) K as a union of classical Deligne-Lusztig varieties "in a natural way".We call this stratification the weak Bruhat-Tits stratification.
The closure relations between strata are given as follows (cf.[8,Sections 3.3 and 7]).The closure of a stratum jY (w) contains a stratum j ′ Y (w ′ ) if and only if (1) w ′ K,σ w which means by definition that there exists u ∈ W K such that u −1 w ′ σ(u) w, and (2) j ′ (J(F ) ∩ P w ′ ) ∩ j(J(F ) ∩ P w ) = ∅.We can express the second condition in an equivalent way in terms of the building, as follows (see [8,Erratum,Prop. 7.2.2]).We identify the set {h ∈ J(F )/J(F ) ∩ P w ; κ(h) = κ(j)} with the set of simplices of type P w in the rational building of J, and similarly for w ′ .Then (2) above is equivalent to requiring that κ(j ′ ) = κ(j) and that the simplices attached to j and j ′ via the above identification are contained in the closure of some alcove.
If moreover (G, µ, K) is of Coxeter type, then this stratification has further nice properties.
We have the following consequence.
Proof.By the proposition above, w is determined by supp σ (w).So it is enough to show that we can recover supp σ (w) from the union supp σ (w) ∪ I(K, w, σ).Since w ∈ K W , every connected component of supp σ (w) must meet S−K.On the other hand, by definition we have I(K, w, σ) ⊆ K. Therefore the σ-support of w consists exactly of those connected components of the union supp σ (w) ∪ I(K, w, σ) which intersect S − K.
By Corollary 2.8, if (G, µ, K) is of Coxeter type, then the index set w J(F )/(J(F ) ∩ P w ) of the weak BT stratification can be seen as a subset of the set of all simplices in the Bruhat-Tits building of J over F (up to fixing the connected component, i.e., the image under κ).In view of these particularly favorable properties, we call the resulting stratification the Bruhat-Tits stratification of X(µ, τ ) K .

Some dimension formulas
We first prove (in Section 3.2) the following inequality on the dimension of affine Deligne-Lusztig varieties.As before, τ is a fixed representative of a length 0 element in W whose σ-conjugacy class is the basic element in B(G, µ).
Let x, x ′ ∈ W and s ∈ S. We write We write x → σ x ′ if there exists a sequence x 0 , x 1 , . . ., x r in W and a sequence s 1 , s 2 , . . ., s r in S such that [17] For each x ∈ W there exists an element y ∈ W which is of minimal length inside its σ-conjugacy class such that x → σ y; The following theorem, which is referred to as the reduction á la Deligne and Lusztig, is proved in [4, proof of Theorem 1.6] (parts (i) and (ii)) and [15,Theorem 4.8], see also [7,Corollary 2.5.3].
, where νb denotes the Newton vector of b.

3.2.
Proof of Proposition 3.1.We argue by induction on the length of w.If w is of minimal length in its σ-conjugacy class, by Theorem 3.3, dim X w (τ ) = ℓ(w) and the statement follows.Otherwise, by Theorem 3.2, there exist u ∈ W and s ∈ S such that w ≈ σ u and suσ(s) < u.Thus, supp σ (u) = supp σ (w) and Moreover, dim X w (τ ) = dim X u (τ ) and either X suσ(s) (τ ) = ∅ or X su (τ ) = ∅.Let us assume that the former case occurs; the proof in the other case is basically the same.By induction hypothesis,
Now we prove the main result of this section.
Theorem 3.5.Suppose that µ is non-central in every simple factor of the adjoint group G ad over F .Then dim X(µ, τ ) K rank ss F (J τ ).If moreover the equality holds, then (G, µ) is fully Hodge-Newton decomposable.
Proof.We may reduce to the case where G is quasi-simple over F and that it is semisimple of adjoint type.Under this assumption, we have where each G i is a simple reductive group over F , and σ(G j ) = G j+1 for all j.Here we set . ., µ r ) with respect to the decomposition above.For J ⊆ S we set J j = J ∩ Sj .We may write ρ as ρ = ρ 1 + . . .+ ρ l , where ρ i is the half sum of positive roots corresponding to the root system associated to Si .We also have that Ω = Ω 1 × . . .× Ω l .We write τ = (τ 1 , . . ., τ l ), where Now we assume that dim X(µ, τ . By Corollary 3.4, we have supp σ (w) S and hence By Section 2.3, (G, µ) is fully Hodge-Newton decomposable.

Classification
Note that the fully Hodge-Newton decomposable cases are classified in [9].By further studying these cases via a case-by-case analysis, one may get a classification of the Coxeter types.However, there is a more direct approach (without using the classification of the Hodge-Newton decomposable cases).This approach classifies the cases where dim X(µ, τ ) K = rank ss F (J τ ), and in particular, by analyzing all these cases, we show that this equality implies that (G, µ, K) is of Coxeter type and thus we obtain a classification of the Coxeter types.This is what we will do in this section.
We may assume that G is quasi-simple over F , and that it is semisimple of adjoint type (cf.[9, Section 3.3]).Under this assumption, we have where each G i is a simple reductive group over F , and σ(G j ) = G j+1 for all j.Here we set G r+1 = G 1 .Write S = S1 ⊔ • • • ⊔ Sr and µ = (µ 1 , . . ., µ r ) with respect to the decomposition above.For J ⊆ S we set J j = J ∩ Sj .We assume further each factor µ j is non-central.
Recall that Φ denotes the unique reduced root system underlying the relative root system of G over F (the échelonnage root system).
4.1.Admissible triples.Let s ∈ S. If s ∈ W 0 , set α s to be the simple root corresponding to s.Otherwise, set α s = −θ, where θ ∈ Φ + is the highest root of the j-component of the decomposition (4.1),where s ∈ Sj ; then s = t θ ∨ s θ .Let J ⊆ S such that W J is finite.We denote by Φ J the root system spanned by α s for s ∈ J.
Lemma 4.1.For λ ∈ X * we have t λ ∈ K W if and only if λ, α s 0 for all s ∈ K.In particular, there exists ξ ∈ W 0 • µ such that t ξ ∈ K W .
Proof.The first statement follows immediately from the definitions.For the second one, notice that {α s ; s ∈ K} is the set of simple roots for Φ K whose Weyl group is p(W K ) ⊆ W 0 .Thus, each p(W K )-orbit in W 0 • µ contains a unique cocharacter ξ such that ξ, α s 0 for s ∈ K, that is, t ξ ∈ K W as desired.
Definition 4.2.We say the triple (ξ, J, K) with where C ranges over the connected components C of K on which the p(σ)-average ξ ⋄ is nonzero.In other words, K ξ is the minimal σ-stable subset of K such that ξ ⋄ J ∈ RΦ ∨ K ξ .
Lemma 4.3.Let (ξ, J, K) be an admissible triple.Then there exists some σ-Coxeter element c ∈ W K ξ such that In particular, t ξ c ∈ K W ∩ Adm(λ) and the Newton point of t ξ c is central.
Proof.The existence of c such that ℓ(t ξ c) = ℓ(t ξ ) − ℓ(c) and hence t ξ c ∈ K W ∩ Adm(λ) follows exactly along the same lines as [9, Lemma 6.4, Proposition 6.7].It remains to show that the Newton point of t ξ c is central.By the proof of [9, Lemma 6.4], it suffices to show that the p(cσ)-average ν of ξ J is zero.Write Proof.Let c be as in Lemma 4.3.Then we need to show that ℓ(t ξ c) rank ss F (J τ ).But our assumption implies, by Theorem 3.5, that (G, µ) is fully Hodge-Newton decomposable, and hence we have ℓ(t ξ c) = dim X t ξ c (τ ) (note that this is an easy consequence of (3.2) and does not require the use of classification results).
Altogether we obtain Given a σ-stable subset K ⊆ S with W K finite, it follows from Lemma 4.1 that there always exists some admissible triple (ξ, J, K).
In particular, µ, 2ρ 2 rank ss F (G). (c) We can now state the following equivalent characterizations of being of Coxeter type.Note that there is an obvious notion of product of Coxeter data.We call a Coxeter datum irreducible, if it cannot be decomposed as a product in a non-trivial way.

Strategy.
In Table 1, we list the minimal irreducible enhanced Coxeter data (up to isomorphism) satisfying condition (3) of Theorem 4.6 together with the set K Adm(µ) 0 .It is easy to see that in all these cases, K Adm(µ) 0 = K Cox(µ).Therefore, we also have Note that the condition (3), although a bit technical, is the most elementary one among the three conditions and only involves the root system.In the rest of the section, we will analyze the condition (3), and give a classification of the irreducible enhanced Coxeter data that satisfy this condition, and finally show that those cases are of Coxeter type.This finishes the direction (3) ⇒ (1).
Our strategy is as follows.
In step (I), we show that the condition (3) implies that the Coxeter datum (W a , σ, µ) is one of those listed in Table 2.This is done by the inequalities (b) and (c) in Corollary 4.5 in most of the cases.The only exception is in Type D, where in some cases we have to use the full strength of condition (3).
Note that the cases in this table are the fully Hodge-Newton decomposable cases.We will then further analyze these cases and give a complete classification.
In step (II), we show that the condition (3) implies that for the Coxeter datum in the Table above, the parahoric subgroup K must be as specified in Theorem 4.6/Table 1.
In Table 2 we list the remaining cases together with the σ-orbits and the Ad(τ ) • σ-orbits on S.This information will be used in the case-by-case analysis in the remainder of this section.
Then the inequality (a) fails for the admissible triple (ξ, J, K ′ ).
. The semisimple rank of J τ is zero and the inequality (b) fails.

4.5.
Step (III): Final verification.To finish the proof of Theorem 4.6, it only remains to check in each case that the set K Adm(µ) 0 is the set given in Table 1.From the explicit description, we see that these cases are of Coxeter type.We omit the explicit computations.

Consequences for RZ spaces
In this section, we explain consequences of our results for Rapoport-Zink spaces.Also compare the paper [47] by H. Wang for applications to Shimura varieties.5.1.Definitions.We consider the situation where F = Q p , and where the pair (G, µ) corresponds to a Rapoport-Zink space.As before, we consider the basic case, i.e., we denote by b the basic σ-conjugacy class in B(G, µ).
Since there are different constructions of RZ spaces in the PEL case and the more general case of Hodge type, we will axiomatize the properties that we require, rather than fixing one of the constructions.
In all cases, the RZ space M(G, µ, b) K is a formal scheme over Ȏ, the ring of integers of F .We denote by k the residue class field of F , and by a subscript − k indicate the base change to k.
Note that the results in the previous sections are group-theoretic in nature and hence concern parahoric level structures, but the known constructions of RZ spaces work for stabilizers of facets in the Bruhat-Tits building.See the discussion in Section 5.5.To take this possible difference into account right from the beginning, we change notation as follows: We denote by P the stabilizers of facets of the base alcove, and by P • the corresponding parahoric.
The setup of the theory entails that P and P • are always defined over F , i.e., fixed by σ.
We denote by Gr P • the partial affine flag variety for P • (i.e., with k-valued points G/P • ), and similarly by Gr P the "partial affine flag variety" for P with k-valued points G/P .Let π : Gr P • → Gr P denote the projection.
Since P • ⊆ P is a normal subgroup, the (finite) quotient group P/P • acts on Gr P • on the right by gP • • p = gpP • , p ∈ P/P • , and Gr P is the quotient by this action.Let κ : G → π 0 (Gr P • ) = π 1 (G) Γ0 be the Kottwitz homomorphism.Since P • is the kernel of κ|P , and correspondingly, the parahoric group scheme corresponding to P • is the connected component of the "stabilizer group scheme" corresponding to P , the restriction of π to any connected component of Gr P • is an isomorphism onto a connected component of Gr P ; cf.[29, Thm.1.4, App.Prop.3].In other words, π identifies those connected components which are mapped to each other by P .
In this way, we obtain a perfect scheme Gr P with k-valued points G/P and such that the projection π : Gr P • → Gr P is an isomorphism when restricted to a connected component of Gr P • .
We write X P := X(µ, b) P = {g ∈ Gr P ; g −1 bσ(g) ∈ P Adm(µ)P }, which also inherits the structure of a perfect scheme.Similarly as in [9], we consider the following condition: (♦) For facet stabilizers P ⊂ P ′ , we have a projection M(G, µ, b) P → M(G, µ, b) P ′ , and there are isomorphisms of perfect schemes, compatible with the projections for inclusions P ⊂ P ′ .
X GL (µ ′ , b) P ′ under the embedding G/P ⊆ G ′ /P ′ .This follows once we can prove that Adm(µ) P embeds into the admissible set Adm(µ ′ ) P ′ ⊂ P ′ \G ′ /P ′ .This compatibility of admissible sets follows from the inclusions , where M GL P ′ denotes the local model for the general linear group, using once more the assumption A(µ) P = Adm(µ) P and the fact that for the general linear group the corresponding equality is known, as well.
It is clear that everything above is compatible with the projections arising from forgetting some of the lattices in our chain.
Remark 5.2.The condition A(µ) P = Adm(µ) P is known to hold in many cases.Note that almost the same condition is posed as Axiom 3.2 in [19]; the only difference is that in our setting we can (and need to) be a little bit more precise as to how this identification arises.
(1) Assume that p is odd, G/Q p is connected and splits over a tamely ramified extension and that the stabilizer P of our lattice chain is a parahoric subgroup.Then by the work of Pappas and Zhu [31, Thm.1.1, Thm.1.2], in many individual cases, its predecessors), the special fiber of M flat P is the union of Schubert varieties (in an equal characteristic affine Grassmannian) indexed by the admissible set Adm(µ) P .See also loc.cit., Section 8.2.
(2) The condition has been checked in many individual cases, including cases where the stabilizer P is not parahoric.Specifically, see Smithling's papers [36], [38] for ramified unitary groups, and [37] for split even orthogonal groups.
Proposition 5.3.For RZ spaces M P of PEL type the projections in condition (♦) exist and condition (♣) is satisfied.
Proof.It is clear that for P ⊆ P ′ there is a projection morphism between the corresponding RZ spaces.Now, for the "naive" versions the definition in terms of chains of p-divisible groups shows immediately that the morphism in question is a monomorphism.Since the irreducible components of the source are proper, cf.[35, Prop.2.32], and the source is locally of finite type over k ( [35,Thm. 3.25]), it follows that the morphism is a closed immersion.The "flat" RZ spaces are closed formal subschemes of the naive RZ spaces, so the above property continues to hold.

RZ spaces of Hodge type.
In the work of Kim [24] and Hamacher and Kim [12] where RZ spaces for data of Hodge type are constructed, the bijection M(G, µ, b) P (k) ∼ = X(µ, b) P (k) on k-valued points is an essential feature of the construction, see [12,Prop. 4.3.5].Zhu ([52,Prop. 3.11]) proved that this set-theoretical equality implies the above isomorphism of perfect schemes, using results of Gabber and Lau on Dieudonné theory over perfect rings to handle the case G = GL n , and then embedding the general situation into a suitable GL n -situation.While in [52] it was assumed that K is hyperspecial, the only reason for this assumption is that at the time of writing RZ spaces of Hodge type had been constructed only in this special situation; the paper [12] appeared only later.Note however that the previous paragraph concerns only the situation for a fixed level P .Because of the way RZ spaces are defined in the Hodge type situation, it is not clear that these bijections are compatible with the projection maps attached to a pair P ⊂ P ′ of parahoric subgroups.In fact, the definition relies on embedding the situation into an RZ space of Siegel type (i.e., associated with a group of symplectic similitudes and hyperspecial level structure).However, it is not evident whether the result is independent of the choice of embedding, and it does not seem clear whether such embeddings can be chosen in a compatible way given P ⊂ P ′ .5.4.The weak Bruhat-Tits stratification.Next we discuss the question of defining a (weak) Bruhat-Tits stratification on Rapoport-Zink spaces.Hence, we now restrict to the fully Hodge-Newton decomposable case.Then we have the weak Bruhat-Tits stratification on X(µ, b) P • (Section 2.4).
Since property (♦) involves P instead of P • , we first need to discuss the question of defining a weak BT stratification on X(µ, b) P .To this end, we impose the following additional assumption: (♥) The projection X P • → X P is surjective.
(with π ′ : Gr P ′ • → Gr P ′ the projection) where the vertical morphisms are isomorphisms.Since the upper horizontal map maps each BT stratum in X c isomorphically onto a BT stratum in X c ′ , the same holds true for the lower horizontal map.We also record the following easy lemma: Lemma 5.6.Let f , f i , i ∈ I, be facets of the base alcove (viewed as simplices, i.e., as subsets of the set of vertices of the base alcove).Denote by P , P i the facet stabilizers, and by P • , P • i the corresponding parahoric subgroups.The following are equivalent: (1) 1) and ( 2) are equivalent.Furthermore, (3) and ( 1) are equivalent since a parahoric subgroup is determined by the set of affine simple reflections which it contains.Now we can invoke property (♦) and obtain that in the fully Hodge-Newton decomposable case and under the above assumptions, the spaces M(G, µ, b) p −∞ P,k carry a weak BT stratification which is compatible with the weak BT stratification on X P via (♦).For an inclusion P ⊂ P ′ , the corresponding map of RZ spaces maps each BT stratum in the source isomorphically (in the sense of perfect schemes) onto a BT stratum in the target.
Since the morphism is a homeomorphism, we likewise get Corollary 5.7.In the fully Hodge-Newton decomposable case and under the above assumptions, the spaces M(G, µ, b) P,k carry a weak Bruhat-Tits stratification into locally closed reduced subschemes which is compatible with the weak BT stratification on X P via passing to perfections and the isomorphism (♦).For an inclusion P ⊂ P ′ , the corresponding map of RZ spaces maps each BT stratum in the source homeomorphically onto a BT stratum in the target.
The perfection of each stratum is isomorphic to the perfection of a classical Deligne-Lusztig variety.
The goal of this section is to investigate the following property (under the assumption of full Hodge-Newton decomposability): Property BT P : In the (weak) Bruhat-Tits stratification of M(G, µ, b) red P,k , each stratum is isomorphic to the corresponding classical Deligne-Lusztig variety, without passing to the perfection.
Our result will be that if (BT Pi ) holds for all i, then (BT P ) also holds for P = ∩ i P i .Note that we need to use in the proof that we already know the result after perfection for level P .
For most maximal parahoric level structures P , the property (BT P ) has been established by now, and this result will allow us to deduce the same property for many non-maximal level structures, and in particular for most level structures of Coxeter type.See Section 6 for a discussion of the individual cases.
Proposition 5.8.Let (G, µ) be fully Hodge-Newton decomposable.Assume that properties (♦), (♣) and (♥) hold.As above, let P i be facet stabilizers (of facets of the base alcove which are fixed by σ) such that BT Pi holds for all i.Then BT P holds for P := i P i .Proof.To shorten the notation, we write M P for M(G, µ, b) red P,k .The assumption implies that M P has a Bruhat-Tits stratification as explained above.Consider a BT stratum S ⊂ M p −∞ P (i.e., S is isomorphic to the perfection of a classical Deligne-Lusztig variety X).We denote by Y the reduced locally closed subscheme inside M P corresponding to S.
For each i, S maps isomorphically onto a BT stratum S i ⊂ M p −∞

P
. So S i is the perfection of the same DL variety X.By the assumption BT Pi , the stratum Y i ⊂ M Pi corresponding to S i is isomorphic to X. Now consider the closed embedding M P → i M Pi given by Property (♣).Restricting to Y , we obtain a closed embedding Y → i Y i .Passing to perfections we obtain an embedding S → i S i .
Then the reduced closed subscheme of i Y i with underlying space S is isomorphic to the Deligne-Lusztig variety X: Both are reduced locally closed subschemes, so it is enough to check that they have the same underlying topological space.But this can be checked on the perfections, where we know the result from the group-theoretic situation.
This implies that the reduced subscheme Y of M P with same topological space as S is isomorphic to X, as desired.5.5.Stabilizers versus parahoric subgroups.It remains to discuss in more detail in which cases the assumption (♥) is satisfied, i.e., when the map X P • → X P is surjective.Let us briefly recall the setup.
Let P ⊂ G be the stabilizer of a (poly-)simplex in the Bruhat-Tits building.The intersection P • of P with the kernel of the Kottwitz homomorphism κ : G → π 0 (L G) = π 1 (G) Γ0 is a parahoric subgroup (and all parahoric subgroups arise in this way), but in general, P • ⊂ P is a proper normal subgroup.
Remark 5.9.This phenomenon occurs in several cases: (i) The case of ramified unitary groups has been discussed in detail by Pappas  If G is semi-simple and simply connected, or more generally if π 0 (L G) is torsion-free, then the finite subgroup P/P • ⊆ π 0 (L G) must be trivial, so that we have P • = P .
In the sequel, we assume that P • is a standard parahoric subgroup, or in other words, that P is the stabilizer of a face of the base alcove.As mentioned above, the group-theoretic results obtained in this paper concern a priori the "parahoric setting", but RZ spaces have been defined, so far, in the "stabilizer setting".See also Remark 5.4 in the paper [12] by Hamacher and Kim.
(iii) We have P Adm(µ)P = P • Adm(µ)P .Proof.To prove part (1), say that P is the stabilizer of the face f of the base alcove.Under the assumption that vω stabilizes f , we need to show that the same is true for ω.Assume that ωf = f .Since ω preserves the base alcove, this means that ωf is a face of a type different from the type of f .Because the action of W a preserves the type of faces, it is then impossible that vωf = f .For (2), note that conjugation preserves the orbit W 0 (µ) and conjugation by length 0 elements preserves the Bruhat order.Now we prove part (3).By part (1), we can write P = p P • p, where p runs through a system of representatives of P/P • given by (representatives of) length 0 elements in W . Since P • ⊂ P is normal, the claimed statement follows from (2).Now for gP • ∈ Gr P • , we have The action of σ on π 0 (Gr P • ) is trivial.(For instance, this holds if G splits over a totally ramified extension of F .) Proposition 5.13.If the equivalent conditions of the previous lemma are satisfied, then the diagram In particular, in this situation assumption (♥) is satisfied.

Known results, new results and open cases
As in the previous section, we assume that (G, µ) comes from a Rapoport-Zink space.
Table 3 lists the cases that come from Shimura varieties and where µ is non-central in each Ffactor of G.As we stated before, it has been checked in many cases that the strata of the Bruhat-Tits stratification are classical Deligne-Lusztig varieties (also before passing to perfections).The strategy in the papers cited below consists of the following steps: Define a set-theoretic bijection in terms of Dieudonné theory, extend it to a morphism of schemes using Zink's display theory, and check that one obtains an identification of schemes using Zariski's main theorem and the normality of Deligne-Lusztig varieties.In all cases, the stratification is given by viewing the Bruhat-Tits building of the group J in terms of lattices ("vertex lattices"), and defining the strata by considering relative positions of Dieudonné modules and such vertex lattices.This means that Dieudonné theory gives a suitable way to set up the identifications (♦), and that then the BT stratification as defined in this paper coincides with the stratifications defined in the papers mentioned below.
The meaning of the symbols used in the table is as follows: a known case, ?an open case, N a case newly settled by this paper (Prop.5.8), N * a case newly settled by this paper (Prop.5.8), up to the verification of the assumptions (♦), (♣), (♥) for the involved RZ spaces.
Below we give further remarks on some of the cases.on the affine Dynkin diagram.We have dim X(µ, τ ) K = 0 for all K S, so the set-theoretic bijection defining the Bruhat-Tits stratification is automatically an isomorphism before passing to the perfection.This case arises from unitary Shimura varieties attached to groups that split over Q p .Cf. the book [13] by Harris and Taylor.

( Ãn
The automorphism τ is the same as in Section 6.1.1,so the composition Ad(τ ) • σ is the identity.The only rational level structure is K = ∅.We have dim X(µ, τ ) = n − 1, and in this case, the set B(G, µ) has only one element: The basic locus equals the whole Shimura variety.The description given by Drinfeld ( [5], see also [35,Thm. 3.72] and the subsequent discussion) of the RZ space as a formal scheme (that can be constructed by gluing pieces indexed by simplices in the Bruhat-Tits building of the corresponding group J (split of type A n−1 )) shows in particular that property BT ∅ holds.
The maximal elements in Adm(µ) are The automorphism Ad(τ ) • σ is the identity map on S, and in particular the σ-support of an element is simply the support of wτ −1 .From this description one sees that each element w ∈ Adm(µ) is determined by its σ-support supp σ (w), and that for all K the order K,σ (see Section 2.4) coincides with the Bruhat order on K W .The individual strata of the BT stratification are isomorphic to classical Deligne-Lusztig varieties in products of general linear groups.In fact, for each w the ambient group is the reductive group with Dynkin diagram given by supp σ (w) (i.e., all other vertices and all edges that involve one of those are discarded) with Frobenius action given by Ad(τ ) • σ = id.
For each w, the index set for the strata of type w is given by J(F )/(J(F ) ∩ P w ), where P w is the standard parahoric subgroup generated by supp σ (w)⊔I(K, w, σ), and where I(K, w, σ) is the subset of K comprising all those vertices which are not connected to supp σ (w), cf.Lemma 2.6.6.1.3.( Ã2m , ς 0 , ω ∨ 1 , S − {s 0 }) -odd unramified unitary group case.The automorphism τ is the same as in Section 6.1.1,so the composition Ad(τ ) • σ acts as the reflection 1 ↔ 0, 2 ↔ 2m, . . .The only level structure of Coxeter type, K = S − {s 0 }, is hyperspecial, and it was shown in [44] that property BT K holds.6.1.4.( Ã2m+1 , ς 0 , ω ∨ 1 , S − {s 0 , s m }) -even unramified unitary group case.The automorphism τ is the same as in Section 6.1.1,so the composition Ad(τ ) • σ acts as the reflection 1 ↔ 0, 2 ↔ 2m + 1, . . .In this case, σ fixes two vertices in the affine Dynkin diagram, namely 0 and m.Both level structures S − {s 0 } and S − {s m } are hyperspecial, and we can again apply the results of [44] to see that property BT K holds for K hyperspecial.It then follows from Proposition 5.8 that the same is true for K = S − {s 0 , s m }.Note that Propositions 5.1 and 5.3 ensure that (♦) and (♣) hold in this case.The sets A(µ) P and Adm(µ) P consist only of the double coset of t µ .Since P is a parahoric subgroup, (♥) is satisfied trivially.
Note that the condition i + j m − 2 ensures that this is a proper subset of S. We see that each element is determined by its σ-support.We write the σ-support as a disjoint union of two intervals (possibly empty, depending on the choice of i and j) which are disconnected in the Dynkin diagram.The individual strata of the BT stratification are classical Deligne-Lusztig varieties in the group (over the finite residue class field of F ) specified by the Dynkin diagram supp σ (w), i.e., a product of two unitary groups (or just one if one of the intervals is empty).
For each w, the index set for the strata of type w is given by J(F )/(J(F ) ∩ P w ), where P w is the standard parahoric subgroup generated by supp σ (w) ⊔ I(K, w, σ), and where I(K, w, σ) can be described as in Lemma 2.6.Remark 6.1.Cho [3] studied the unramified unitary group case for non-hyperspecial maximal parahoric level structure.This case is fully HN decomposable, but not Coxeter type.6.1.5.( Ã3 , id, ω ∨ 2 , S − {s 0 , s 3 }).This case corresponds to a "split U (2, 2)".Note that for each i ∈ Z/4Z, ( Ã3 , id, ω ∨ 2 , S − {s i , s i+1 }) is isomorphic to the above datum.For every i, the level structure S−{s i } is hyperspecial and for those cases it was shown by Fox [6] that the Bruhat-Tits strata are isomorphic to Deligne-Lusztig varieties before perfection.
Note that Propositions 5.1 and 5.3 ensure that (♦) and (♣) hold in this case.It is well-known that the sets A(µ) P and Adm(µ) P coincide in this case.Since P is a parahoric subgroup, (♥) is satisfied trivially.Hence Proposition 5.8 can be applied.6.1.6.(B-C n , ω ∨ 1 , S−{s 0 , s n }) -even ramified unitary group case.To apply Proposition 5.8, we need to check that the assumptions (♦), (♣) and (♥) are satisfied.For the first two, we can use Propositions 5.1 and 5.3.In fact, it follows from Smithling's paper [38] that A(µ) P = Adm(µ) P .It follows from Proposition 5.11 that (♥) is satisfied.
For each w, the index set for the strata of type w is given by J(F )/(J(F ) ∩ P w ), where P w is the standard parahoric subgroup generated by supp σ (w), since in this case, I(K, w, σ) = ∅ for all w ∈ K Adm(µ) 0 (cf.Lemma 2.6).Remark 6.2.In [46], Wang studied the 2 C 2 case with Iwahori level structure.6.1.9.(D n , ω ∨ 1 , S − {s 0 , s n }).Since this case is not of PEL type, it is less clear that the RZ spaces for the different level structures (as defined by Kim [24], Howard and Pappas [22] in the hyperspecial case, and by Hamacher and Kim [12] in general) are compatible in the sense of assumption (♣) above.Once this has been established, the result of Howard and Pappas together with Proposition 5.8 implies the result for level structure S − {s 0 , s n }.
The irreducible enhanced Coxeter data of Coxeter type (with the minimal level structure), up to isomorphism.Notation: In type Ãn−1 , by convention we set s n = s 0 .We use the labeling of the affine Dynkin diagram as in [2].Set s [a,b] = s a s a−1 • • • s b if a b, and s [a,b] = 1 otherwise.Now we come to the classical groups.
Applying restriction of scalars along an unramified extension to the above examples, one can construct further examples, and in particular obtains examples where the action of σ on P/P • is not trivial.