Stabilizers of irreducible components of affine Deligne--Lusztig varieties

We study the $J_b(F)$-action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of $J_b(F)$ of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.

1. Introduction 1.1.Affine Deligne-Lusztig varieties and their irreducible components.Affine Deligne-Lusztig varieties were introduced by Rapoport in [Rap05].In the equal characteristic setting, affine Deligne-Lusztig varieties are related to the moduli space of local shtukas.In the mixed characteristic setting, they are related to the geometry of Rapoport-Zink spaces and hence to the geometry of certain distinguished loci in the special fiber of Shimura varieties via the p-adic uniformization.Therefore studying the geometry of affine Deligne-Lusztig varieties can give useful information on the geometry of special cycles on Shimura varieties.
This paper is concerned with studying the set of top-dimensional irreducible components of affine Deligne-Lusztig varieties.To state our main results we fix some notation.Let F be a local field with ring of integers O F , and let F be the completion of the maximal unramified extension of F .Let G be a reductive group over F , which we assume is unramified in the introduction for simplicity.For b ∈ G( F ) and µ a cocharacter of G, we have the affine Deligne-Lusztig variety X µ (b) which is a locally closed subscheme of the affine Grassmannian.We refer to §2.4.1 for the precise definition.
If F is of equal characteristic, X µ (b) is locally of finite type.If F is of mixed characteristic, X µ (b) is a perfect scheme and is locally of perfectly finite type.In either case, it is known that X µ (b) is finite dimensional.We write Σ top (X µ (b)) for the set of top-dimensional irreducible components of X µ (b).
The scheme X µ (b) is equipped with an action of J b (F ), the F -rational points of a certain reductive group J b over F (the Frobenius-centralizer of b).This induces an action of J b (F ) on Σ top (X µ (b)).The goal of this paper is to understand the J b (F )-set Σ top (X µ (b)).This amounts to considering the following two problems.
(ii) For each Z ∈ Σ top (X µ (b)), determine the stabilizer of Z in J b (F ).
For (i), M. Chen and X. Zhu conjectured (see [HV18,Conjecture 1.3]) that the set of the J b (F )-orbits in Σ top (X µ (b)) should be in natural bijection with the Mirkovic-Vilonen basis MV µ (λ b ) for a certain weight space of a representation of the dual group G. (See §4.1 for the definition of MV µ (λ b ).) 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.
For (ii), Xiao-Zhu [XZ17, Theorem 4.4.14]showed that if the element b ∈ G( F ) is unramified, then the stabilizer of every Z ∈ Σ top (X µ (b)) is a hyperspecial subgroup of J b (F ) (see also [ZZ20, Theorem 6.2.2]).For general b, it was conjectured by X. Zhu1 that every stabilizer should be a parahoric subgroup of J b (F ) of maximal volume. 2 Our first main result confirms this conjecture.
Theorem A (See Theorem 4.1.2and Corollary 4.1.4).For each Z ∈ Σ top (X µ (b)), the stabilizer of Z in J b (F ) is a very special parahoric subgroup of J b (F ).In particular, there is an isomorphism of J b (F )-sets where J a ⊂ J b (F ) is a very special parahoric subgroup.
We refer to §2.2.1 for the definition of very special parahoric subgroups, and Proposition 2.2.5 for the equivalence of this condition with that of having maximal volume.After this result was announced, S. Nie informed us that he could also prove this result using a different method.
For a reductive group over F with no factors of type C-BC n , the condition that a parahoric is very special determines the parahoric up to conjugation in the adjoint group.Thus when J b has no factors of type C-BC n , Theorem A determines the stabilizers up to conjugation by J ad b (F ).It is an interesting problem to determine the stabilizers up to J b (F )-conjugacy.However Theorem A is already enough for some important applications explained below.
1.2.Application to Shimura varieties.Let (G, X) be a Shimura datum, and let K ⊂ G(A f ) be a sufficiently small compact open subgroup.Then we have the associated Shimura variety Sh K (G, X) which is an algebraic variety defined over a number field E. Let p > 2 be a prime.We assume that (G, X) is of Hodge type, and that K = K p K p where K p is a compact open subgroup of G(A p f ) and K p is a hyperspecial subgroup of G(Q p ). Then by work of Kisin [Kis10], for any prime v|p of E, there is a smooth canonical integral model S K (G, X) of Sh K (G, X) over O E (v) .We write Sh K for its special fiber.
Write G for G = G Qp .There is a stratification of Sh K indexed by the Kottwitz set B(G, µ) (cf.§2.4.4).We let [b] bas denote the unique basic element of B(G, µ), and we write Sh K,bas for the stratum corresponding to [b] bas .This is known as the basic locus, and is a generalization of the supersingular locus in the special fiber of a modular curve.The Rapoport-Zink uniformization (see e.g.[XZ17, Corollary 7.2.16])implies that there is an isomorphism of perfect schemes ( †) Sh pfn K,bas Here I is a certain reductive group over Q with I⊗ Q A p f ∼ = G⊗ Q A p f and I⊗ Q Q p ∼ = J b , and the left hand side denotes the perfection of Sh K,bas .The following theorem then follows immediately from Theorem A and the above isomorphism.

Theorem B (See Corollary 5.2.3). There exists a bijection between the set of topdimensional irreducible components of
where I p ∼ = K p and I a p is a very special parahoric subgroup of J b (Q p ). Moreover the bijection is equivariant for prime-to-p Hecke operators.
In fact, Sh K,bas is equidimensional by [HV18,Theorem 3.4], so we have obtained a description of the set of all irreducible components in this case.We also remark that for Theorem A, the assumption that G is unramified over F is not necessary, and we in fact obtain results for general quasi-split G over F .This allows us to obtain a generalization of Theorem B. The key input for this is a generalization of ( †) for the integral models constructed by Kisin-Pappas [KP18], which we prove in §5.
Theorem B and its generalization reflect the general philosophy going back to Serre and Deuring that components of the basic locus are parameterized by class sets for an inner form of the structure group.We refer to [VW11], [HP17], and [LT20] for some special cases of this result.
The main contribution of this paper is the information that the compact open subgroups I a p ⊂ I(Q p ) are very special.For many applications this is a crucial piece of information.For example, in [LT20], the authors used the description of irreducible components in the supersingular locus of quaternionic Shimura varieties to prove an arithmetic level raising result on the way to proving cases of the Beilinson-Bloch-Kato conjecture.For this, they used the interpretation of functions on Σ top (Sh K,bas ) as automorphic forms for I. Thus the knowledge of I a p is needed to determine the level of these automorphic forms.In [LMPT19], the authors used a formula for the number of irreducible components in the supersingular locus of unitary Shimura varieties to prove results on the image of the Torelli map.This requires information on the volume of I a p .
1.3.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 .After a series of reduction steps, we can assume that G is an unramified adjoint group over F , that F has characteristic 0, and that b ∈ B(G, µ) is basic.It is known that the stabilizer of every Z ∈ Σ top (X µ (b)) is a parahoric subgroup of J b (F ), so the question is to prove that such a parahoric subgroup must have maximal volume.
The proof proceeds in two steps.
(1) Show that there exists Z ∈ Σ top (X µ (b)) whose stabilizer is a parahoric subgroup of J b (F ) of maximal volume.
(2) Show that all the stabilizers have the maximal volume.
The Deligne-Lusztig reduction method in [He14] works for the affine Deligne-Lusztig varieties in the affine flag variety.It keeps track of geometric information such as the dimension and the number of irreducible components of top dimension.To keep track of the stabilizers of top-dimensional irreducible components under the action of J b (F ), we introduce a refined reduction method in the context of motivic counting.Then we use the explicit dimension formula for X µ (b) and a certain affine Deligne-Lusztig variety X w0t µ (b) in the affine flag variety to obtain a J b (F )-equivariant bijection Σ top (X w0t µ (b)) ∼ − → Σ top (X µ (b)).We combine the explicit reduction path constructed in [He14] with a refinement of the argument in [HY12] to obtain an element of Σ top (X w0t µ (b)) whose stabilizer in J b (F ) has maximal volume.This finishes step (1).
For step (2), consider the quantity where the sum is over a set of representatives for the J b (F )-orbits in Σ top (X µ (b)).
The results of [ZZ20] imply that Q(µ, b) depends only on b, not on µ.Moreover, for the given b there exists µ 1 ∈ X * (T ) + such that |J b (F )\Σ top (X µ1 (b))| = 1.By step (1) applied to (µ 1 , b), we know that Q(µ 1 , b) is equal to the inverse of the maximal volume attained by parahoric subgroups of ) must be a parahoric subgroup of maximal volume for each Z.This finishes step (2).For step (2), the assumption that F has characteristic 0 is crucial.This is due to the fact that the results we use from [ZZ20] rely on the Base Change Fundamental Lemma, a result only known for characteristic 0 local fields in general.
1.4.Outline of the paper.In §2 we introduce notations and some preliminary group theoretic results.In §2.2, we define very special parahoric subgroups and prove the equivalence of this condition with that of having maximal volume and maximal log volume.We then introduce affine Deligne-Lusztig varieties and establish the relation between components of X µ (b) and X w0t µ (b) in §2.4.In §3, we give a reinterpretation of the Deligne-Lusztig reduction method in terms of motivic counting.We apply this in §3.4 to show the existence of a component in X w0t µ (b) whose stabilizer is a very special parahoric.In §4 , we prove Theorem A. In §4.2 and §4.3, we reduce the proof to the case where char(F ) = 0, G is adjoint, unramified over F , and F -simple, and b is basic.The proof then proceeds in §4.5 and §4.6 as outlined above, with some extra work needed to handle the case of type A and E 6 , which is the content of §4.4.Finally in §5, we apply our results to study the basic locus of Shimura varieties and prove Theorem B. As mentioned, the key input is an analogue of the p-adic uniformization for the integral models of Shimura varieties constructed by Kisin-Pappas, which we prove following the method in [XZ17, §7] using results of [Zho20].
Acknowledgments: We would like to thank S. Nie, G. Prasad, M. Rapoport, and X. Zhu for useful conversations regarding this project.This work was partially inspired by the comments of M.
2.1.1.Let F be a non-archimedean local field with valuation ring O F and residue field k F = F q .We fix an algebraic closure F of F .Let F ur be the maximal unramified extension of F inside F , and let F be the completion of F ur .We denote by O F the valuation ring of F , and denote by k the residue field of F , which is an algebraic closure of k F .Fix an algebraic closure F of F , and fix an F uralgebra embedding F → F .We write Γ for Gal(F /F ) and write Γ 0 for the inertia subgroup of Γ, which is identified with Gal( F / F ).We let σ ∈ Aut( F /F ) denote the q-Frobenius.
Let G be a connected reductive group over F .We fix a maximal F ur -split torus S in G defined over F , which exists by [BT84,Corollaire 5.1.12].By [Rou77, Proposition 2.3.9],S is also maximal F -split.Let T be the centralizer of S in G.By Steinberg's theorem G is quasi-split over F , so T is a maximal torus in G. Let N be the normalizer of T in G, and let In other words W0 is the relative Weyl group of G F .
The Iwahori-Weyl group is defined to be where T ( F ) 1 is the kernel of the Kottwitz homomorphism T ( F ) → X * (T ) Γ0 .We have a natural short exact sequence For each λ ∈ X * (T ) Γ0 , we write t λ for the corresponding element of W .Such elements of W are called translation elements.
2.1.2.Let Ȃ be the apartment of G F corresponding to S F .Thus Ȃ is an affine R-space under X * (T ) Γ0 ⊗ Z R. The Frobenius σ and the Iwahori-Weyl group W act on Ȃ via affine transformations.Since Ȃ is naturally identified with the apartment of G F ur corresponding to S F ur , there exists a σ-stable alcove in Ȃ by [Tit79, §1.10.3] as the residue field of F is finite.We fix such a σ-stable alcove ȃ.Let Ȋ ⊂ G( F ) be the Iwahori subgroup corresponding to ȃ.Then Ȋ is σ-stable and we write I for the corresponding Iwahori subgroup Ȋσ of G(F ).
As explained in [HR08], the choice of ȃ gives rise to a subgroup Wa of W called the affine Weyl group.This is by definition the subgroup generated by the set S of simple reflections in the walls of ȃ.The pair ( Wa , S) is a Coxeter group.
Let Ω be the stabilizer of ȃ in W . Then by [HR08, Lemma 14], we have and Ω is (canonically) isomorphic to π 1 (G) Γ0 .The length function on the Coxeter group ( Wa , S) extends to a function with respect to which Ω is the set of length-zero elements of W .The Frobenius σ naturally acts on W , stabilizing the subset S ⊂ W (as ȃ is σ-stable).In particular, σ induces an automorphism of the Coxeter group ( Wa , S).By [HR08,p. 195], there exists a reduced root system Σ such that where Q ∨ (Σ) and W (Σ) denote the coroot lattice and Weyl group of Σ respectively.The roots of Σ are proportional to the roots of the relative root system for G F .However the root systems themselves may not be isomorphic.
2.1.3.Let K be a subset of S. We write W K ⊂ W for the subgroup generated by K. We let W K (resp.K W ) denote the set of minimal length representatives for the cosets in W / W K (resp.W K \ W ).
For each w ∈ W , we choose a lift ẇ ∈ N ( F ) of w.We assume furthermore that σ( ẇ) = ẇ if σ(w) = w.Indeed, to see that this can always be arranged, it suffices to see that the Lang map The desired surjectivity follows from Greenberg's theorem [Gre63, Proposition 3] (whose proof holds regardless of the characteristic of F ) applied to T 0 .
Let K be a subset of S such that W K is finite.In this case K corresponds to a standard parahoric subgroup of G( F ) containing Ȋ, which we denote by K.By the Bruhat decomposition, the map w → ẇ induces a bijection If furthermore K is σ-stable, then so is K, and we write K = Kσ for the corresponding parahoric subgroup of G(F ).In what follows we will often abuse notation and write K (resp.K) for the parahoric group scheme over O F (resp.O F ) when there is no risk of confusion.The same is applied to the notations Ȋ and I.

Let
A denote the maximal F -split subtorus of S, which is also a maximal F -split torus in G.We write Z A and N A for the centralizer and normalizer of A in G respectively.Since Z A is anisotropic modulo center over F , there is a unique parahoric subgroup Z A of Z A (F ).The relative Iwahori-Weyl group is defined to be It admits a natural map to the relative Weyl group We write D for the relative local Dynkin diagram of (G, A, F ), and write ∆ for the set of vertices of D. Let A be the apartment associated to A, and let a be the base alcove in A determined by the Iwahori subgroup I of G(F ).For each v ∈ ∆, let α v be the corresponding non-divisible simple affine root on A. As explained in [Tit79, 1.11], ∆ is naturally identified with the set of σ-orbits C in S such that WC is finite.For v ∈ ∆, we write C v ⊂ S for the corresponding σ-orbit, and write s v ∈ W for the reflection in A along α v .By [Ric16, Lemma 1.6], there is a natural isomorphism W ∼ = W σ induced by the inclusion map N A (F ) → N ( F ).By [Lus03, A.8], s v corresponds to the longest element of WCv under this isomorphism.We set We also note that if w ∈ W , then the lifting ẇ in N ( F ) chosen in §2.1.3 is contained in N A (F ), which follows from our assumption that ẇ is σ-invariant.

Parahoric subgroups of maximal volume.
We keep the notations of §2.1.In this subsection we give a description of the parahoric subgroups of G(F ) that have the maximal volume .
2.2.1.For a vertex v ∈ ∆, we define d(v) := l(s v ).When G is simply connected and absolutely almost simple, this coincides with the integer attached to v in [Tit79, 1.8], cf.[Ric16, Remark 1.13 (ii)].We say that a special vertex v ∈ ∆ is very special if d(v) is minimal among all special vertices v ′ lying in the connected component of D containing v.
Let x ∈ A be a point lying in the closure a of a.We associate to x a set of vertices There is also a notion of a very special parahoric subgroup defined in [Zhu17, Definition 6.1].When G is quasi-split, it can be shown that these two notions are equivalent.However, they differ for non-quasi-split G (cf. [Zhu17, Lemma 6.1]).
2.2.4.We now fix a choice of Haar measure on G(F ) such that all Iwahori subgroups of G(F ) have volume 1.Let K be a parahoric subgroup of G(F ) and K the associated parahoric subgroup of G( F ).We define the log-volume of K by (2.2.4.1) log vol(K) := dim K/I, where K (resp.I) denotes the reductive quotient of the special fiber of K (resp.the image of the special fiber of Ȋ in K).If K is a standard parahoric corresponding to a σ-stable subset K ⊂ S, then we have where w K is the longest element of W K .We have the Bruhat decompositions w) .Proposition 2.2.5.Let K be a parahoric subgroup of G(F ).Then the following are equivalent: (1) K is a very special parahoric; (2) K is of maximal volume among all the parahoric subgroups of G(F ); (3) K has maximal log-volume.Remark 2.2.6.When G is simply connected and absolutely almost simple, the equivalence between (1) and ( 2) is [BP89, Proposition A.5].The equivalence between (3) and the other two conditions will be used in the proof of Corollary 4.2.4 below, especially when we alter the local field.
2.2.7.To prove Proposition 2.2.5 we follow the method in [BP89,A.4].We begin with some preparation.Assume that G is almost simple over F and let Φ be the relative root system Φ(G, A).We let Φ nd denote the system of non-divisible roots in Φ and we write W for the Weyl group of Φ nd , which is identified with the relative Weyl group W 0 of G.
For an element v ∈ ∆, we define K(v) := S \ {s v } ⊂ S. We let W K(v) denote the subgroup of W ∼ = W σ generated by K(v).Then the natural map Aff(A) → GL(X * (A)⊗R) (i.e., taking the linear part) induces an identification between W K(v) and a subgroup of W, which we denote by W v .We denote the inverse isomorphism by where we consider W K(v) as a subgroup of W .For each v ′ ∈ ∆ \ {v}, we write α v ′ for the unique proportion of the vector part of α v ′ that lies in Φ nd .We let Φ v denote the sub-root system of Φ nd generated by α v ′ with v ′ ∈ ∆ \ {v}.
We define an ordering on Φ v by specifying the positive simple roots to be given by α v ′ with v ′ ∈ ∆ \ {v}, and we write Φ + v (resp.Φ − v ) for the subset of positive (resp.negative) roots.Note that the ordering on Φ v depends on v; it is possible that there exist v 1 , v 2 ∈ ∆ such that Φ v1 = Φ v2 but Φ + v1 = Φ + v2 .For α ∈ Φ v , we define an integer d(α, v) as follows.If α = α v ′ for some v ′ ∈ ∆ \ {v}, then we define d(α, v) = d(v ′ ).In general, we define d(α, v) by specifying that its dependence on α is W v -invariant.This is well-defined since if v 1 , v 2 , ∈ ∆ \ {v} are such that α v1 and α v2 are W v -conjugate, then d(v 1 ) = d(v 2 ); cf.[BP89,A.4 . By [Ric16, Sublemma 1.12] and induction, we have and the result follows.
Proof.Since d(α, v) and d(α, v 0 ) only depend on the W v -orbit of α, it suffices to prove this in the case that α ∈ Φ + v is a simple root, i.e. α = α v ′ with v ′ ∈ ∆\{v}.If v ′ = v 0 , then α is also a simple root for Φ + v0 and we have by inspection of Tits' table [Tit79,§4], we find that Proof of Proposition 2.2.5.It suffices to prove the result for K a standard parahoric.We first consider the case where G is adjoint and simple over F .Let K0 , K ⊂ S be σ-stable subsets with corresponding parahoric subgroups K 0 and K of G(F ), and corresponding subsets K 0 , K 0 ⊂ S. Assume that K 0 is a very special parahoric.Then we need to show that and that strict inequality holds in each case if K is not very special.
Since K 0 is very special, we have K 0 = K(v 0 ) for v 0 ∈ ∆ a very special vertex.Moreover, since K is contained inside a parahoric corresponding to some v ∈ ∆, we may assume K = K(v).
Since v 0 is a very special vertex, W v0 = W and we have where the first inequality follows from Lemma 2.2.9.Thus If K is not special, then the second inequality is strict.If K is special but not very special, then the first inequality is strict.We thus obtain the equivalence (1) ⇔ (2).
Similarly, if we let w v ∈ W v (resp.w v0 ∈ W v0 ) denote the image of w K (resp.w K0 ), then we have If K is not very special, then the inequality is strict.Thus we obtain (1) ⇔ (3).
The case with general G is reduced to the above special case by considering the direct product decomposition of G ad into F -simple factors.In fact, by (2.2.4.1) (resp.(2.2.4.3)), we know that the log-volume (resp.volume) of a parahoric subgroup of G(F ) is equal to the product of the log-volumes (resp.volumes) of corresponding parahoric subgroups of the F -simple factors of G ad .

σ-conjugacy classes.
We keep the setting of §2.1, and assume in addition that G is quasi-split over F .2.3.1.Under the assumption that G is quasi-split over F , we can fix a σ-stable special point s lying in the closure of ȃ (cf.[Zhu15, Lemma 6.1]).For an abelian group X and a Z-algebra R, we write X R for X ⊗ Z R. The choice of s gives rise to a σ-equivariant isomorphism which sends 0 to s.We let S0 ⊂ S denote the subset of simple reflections fixing s.Then S0 is preserved by the action of σ.The identification (2.3.1.1)determines a chamber X * (T ) Γ0,R + in X * (T ) Γ0,R ∼ = X * (S) R (with respect to the relative roots of (G F , S F )), namely the one whose image under (2.3.1.1)contains the alcove ȃ.We let X * (T ) Γ0 + (resp. Note that X * (T ) Γ0,R + gives rise to an ordering of the relative roots of (G F , S F ).
Since G is quasi-split over F , this uniquely determines an ordering of the absolute roots in X * (T ), and determines a Borel subgroup of We shall sometimes write [b] G if we want to specify G. Let B(G) be the set of σ-conjugacy classes in G( F ).
The elements of B(G) have been classified by Kottwitz in [Kot97].For b ∈ G( F ), we write ν b ∈ (X * (T ) Γ0,Q + ) σ for its dominant Newton point.(Note that We let κ : G( F ) → π 1 (G) Γ0 denote the Kottwitz homomorphism and we write for the composition of κ with the natural projection π 1 (G) Γ0 → π 1 (G) Γ .This factors through a map B(G) → π 1 (G) Γ , which we still denote by κ.By [Kot97, §4.13], the map is injective.We sometimes write ν G and κ G for ν and κ if we want to specify G.
for any F -algebra R. Let M be the centralizer of ν b , where we consider ν b as an element of (X * (T ) Q + ) Γ ⊂ (X * (T ) Γ ) Q as explained in §2.3.2.Then M is a Levi subgroup of G defined over F and J b is an inner form of M over F .2.3.4.The maps ν and κ on B(G) can be described in a more explicit way as follows.Let B( W , σ) be the set of σ-conjugacy classes in W .The map W → G( F ), w → ẇ defined in §2.1 induces a well-defined map For each w ∈ W , there exists a positive integer n such that σ n acts trivially on W and such that wσ(w) • • • σ n−1 (w) = t λ for some λ ∈ X * (T ) Γ0 .We set ν w := λ n ∈ X * (T ) Γ0,Q and we let ν w denote the unique W0 -conjugate of ν w that lies in X * (T ) Γ0,Q + .Then ν w is necessarily fixed by σ.We let κ(w) ∈ π 1 (G) Γ0 denote the image of w under the quotient map W → W / Wa ∼ = π 1 (G) Γ0 , and we let κ(w) be the image of κ(w) in π 1 (G) Γ .By [He14], we have a commutative diagram: 2.4.Affine Deligne-Lusztig varieties.We keep the setting and notation of §2.3.We assume in addition that G splits over a tamely ramified extension of F and that char(F ) is either zero or coprime to the order of π 1 (G ad ).
is the set of k-points of a locally closed sub-scheme X K,w (b) of the partial affine flag variety Gr K.In this case X K,w (b) is locally of finite type over k (cf.[PR08]).If char(F ) = 0, then X K,w (b)(k) is the set of kpoints of a locally closed sub-scheme X K,w (b) of the Witt vector partial affine flag variety Gr K constructed by X. Zhu [Zhu17] and Bhatt-Scholze [BS17].In this case X K,w (b) is locally of perfectly finite type over k (see [HV20, Theorem 1.1]).In both cases, we call X K,w (b) the affine Deligne-Lusztig variety associated to b, w, and K.
The group J b (F ) (see §2.3.3)acts on X K,w (b) via k-scheme automorphisms.By [HV20, Theorem 1.1], the induced J b (F )-action on the set of irreducible components of X K,w (b) has finitely many orbits.The results in [HV20] also have the following easy consequence.

Lemma 2.4.2. Every irreducible component of X
Proof.Let Z be an irreducible component of X K,w (b).By [HV20, Proposition 5.4], there is a dense open subset U ⊂ Z which is contained in a finite union i S i of Schubert varieties in Gr K. Since the Schubert varieties are closed in Gr K, we have the Schubert varieties are of finite type over k when char(F ) > 0 and of perfectly finite type over k when char(F ) = 0 (cf.[HV20, §4]), so the underlying topological space of i S i is Noetherian.It follows that Z is quasi-compact.2.4.3.We are mainly interested in X K,w (b) in the following two cases: • (Iwahori level.)We have K = ∅, i.e., K = Ȋ.
• (Maximal special level.)We have K = S0 , i.e., K is the maximal special parahoric subgroup corresponding to the special point s.
When K = ∅, we simply write X w (b) for X ∅,w (b).When K = S0 , the restriction of the natural map W → W0 to W K ⊂ W induces an isomorphism W K ∼ − → W0 .In other words, our choice of s determines a splitting of the exact sequence (2.1.1.1).In this case we shall identify W0 with W K , viewed as a subgroup of W .We have natural bijections where the second map is induced by the inclusion We sometimes write λ is a nonnegative rational linear combination of the positive coroots in X * (S) (with respect to (G F , S F ) and the ordering defined in §2.3.1).
For µ ∈ X * (T ) Γ0 + , we define Here µ ♮ is the image of µ in π 1 (G) Γ , and µ ⋄ ∈ X * (T ) Γ0,Q + denotes the average of the σ-orbit of the image of µ in X * (T ) Γ0,Q + .The set B(G, µ) has a unique basic element, which is also the unique minimal element with respect to the natural partial order on B(G, µ) (see [HR17,§2]).
The following criterion for the non-emptiness of X µ (b), originally conjectured by Kottwitz and Rapoport, was proved by Gashi [Gas10] for unramified groups and by the first-named author [He14, Theorem 7.1] in general.

Now we recall the dimension formula for
We let ρ denote the half sum of positive roots in the root system Σ (see §2.1).The following theorem was proved by Görtz-Haines-Kottwitz-Reumann [GHKR06] and [Vie06] for split G, and by Hamacher [Ham15] and X. Zhu [Zhu17] independently for unramified groups.The result in general was proved by the first-named author [He16, Theorem 2.29].
Definition 2.4.8.For a scheme X of finite Krull dimension and each non-negative integer d, we write Σ d (X) for the set of irreducible components of X of dimension d (which is allowed to be empty).We write Σ top (X) for Σ dim(X) (X).We write Σ(X) for the set of all irreducible components of X.
2.4.9.The main object of interest in this paper is the set Σ top (X µ (b)).To study this set it will be useful to relate X µ (b) to a certain affine Deligne-Lusztig variety with Iwahori level.
We have a natural projection map between the partial affine flag varieties, which exhibits Gr Ȋ as an étale fibration over Gr K with fibers isomorphic to K/I when char(F ) > 0 (resp.the perfection of K/I when char(F ) = 0).See §2.2.4 for K/I.As in §2.4.3, we identify W0 with the subgroup WS 0 of W .For µ ∈ X * (T ) Γ0 + , the map π induces a J b (F )-equivariant map (2.4.9.1) In fact, the left hand side is equal to π −1 (X µ (b)).

Deligne-Lusztig reduction method and motivic counting
3.1.The Grothendieck-Deligne-Lusztig monoid.Recall that k is a fixed algebraic closure of F q .Let H be an abstract group.We retain the notations introduced in Definition 2.4.8.Definition 3.1.1.Let S H be the category of perfect k-schemes V that are equipped with an H-action and satisfy the following conditions: (1) The scheme V is locally of perfectly finite type over k.
(2) Each irreducible component of V is quasi-compact.
(3) The H-action on Σ(V ) has finitely many orbits.We define morphisms in S H to be the k-scheme morphisms that are H-equivariant.
3.1.2.It is a simple exercise to check that the category S H is essentially small.Thus the isomorphism classes in S H form a set.Let N[S H ] be the free commutative monoid generated by this set.For any object V in S H , we denote by [V ] the element of N[S H ] given by the isomorphism class of V .
For any k-scheme Q, we write Q pfn for the perfection of Q, which is a perfect kscheme.We write A 1 for A 1 k , and write equipped with the product H-action is also in S H .We thus define an endomorphism for any object V in S H . Lemma 3.1.3.Let V be an object in S H , and let U be an H-stable open subscheme of V .Then U equipped with the induced H-action is an object in S H . Proof.Clearly U satisfies condition (1) in Definition 3.1.1.We verify the other two conditions.For each Hence we have a bijection The right hand side is an H-stable subset of Σ(V ), and the bijection is H-equivariant.Since V satisfies condition (3), so does U .
Since V satisfies conditions (1) and (2), each 3.1.5.We recall the general notion of a quotient monoid.Let (M, +) be a commutative monoid.An equivalence relation x i and y = n i=1 y i for some x i , y i ∈ M such that x i ∼ y i for each i.Definition 3.1.6.Let ≡ be congruence on N[S H ] associated to ∼, and let GDL H be the quotient monoid N[S H ]/≡. We call GDL H the Grothendieck-Deligne-Lusztig monoid.For any object V in S H , we denote the image of 3.1.7.One easily checks that the endomorphism T of N[S H ] descends to an endomorphism of GDL H , which we still denote by T. We write L for T + 1 ∈ End(GDL H ).

Calculus of top irreducible components.
3.2.1.Let H be an abstract group as before.One can formally calculate "topdimensional irreducible components" of elements of GDL H .To this end we first introduce a commutative monoid TIC H which is much simpler than GDL H and serves to record information about top-dimensional irreducible components.Let Set H f be the category of H-sets which contain only finitely many H-orbits.This is an essentially small category.We let TIC H be the set of pairs (Σ, d), where Σ is an isomorphism class in Set H f , and d ∈ Z ≥0 .Given two elements (Σ 1 , d 1 ), (Σ 2 , d 2 ) ∈ TIC H , we define their sum to be This makes TIC H a commutative monoid.In the above definition of the sum, if d 1 ≥ d 2 , then we say that (Σ 1 , d 1 ) makes non-trivial contribution to the sum.
Define an endomorphism T of TIC H by We write L for T + 1 ∈ End(TIC H ); it is easy to see that in fact L = T in End(TIC H ).
Note that every object V in S H has finite Krull dimension.The sets Σ(V ) and Σ d (V ) for all d ∈ Z ≥0 (Definition 2.4.8)equipped with the natural H-actions are all objects in Set H f .Definition 3.2.2.For any which is an object in Set H f .The pair consisting of the isomorphism class of Σ top (X) and the integer dim X is thus an element of TIC H , which we denote by C(X) ∈ TIC H . Lemma 3.2.3.The map C : N[S H ] → TIC H is a monoid homomorphism, and descends to a monoid homomorphism C : GDL H → TIC H .Moreover, C is equivariant with respect to the endomorphisms T on GDL H and T on TIC H (see §3.1.7 and §3.2.1).
Proof.It follows from the definitions that C is a monoid homomorphism.To show that C descends to GDL H , it suffices to check that any X, X ′ ∈ N[S H ] with X ∼ X ′ satisfies C(X) = C(X ′ ).For this, we only need to analyze the three situations in Definition 3.1.4.Namely, we may assume that X and X ′ are the two sides of ∼ in those situations.
In situation (1), we have For the same reason, we also have One treats situation (2) similarly, noting that Observe that for each Z ∈ Σ(V ), precisely one of the following two statements holds: It follows that C is equivariant with respect to T on the two sides.Since C is induced by C, it is also equivariant with respect to T on the two sides.

Class polynomials and motivic counting.
We assume that G is as in §2.4,i.e., G is quasi-split, tamely ramified, and char(F ) ∤ |π 1 (G ad )| if char(F ) > 0. Then we have the affine Deligne-Lusztig variety X w (b) associated to w ∈ W and b ∈ G( F ).The motivation behind the definition of the Grothendieck-Deligne-Lusztig monoid is that it gives a natural setting to apply the Deligne-Lusztig reduction method for X w (b).We recall the reduction method in the proposition below.(1) If l(swσ(s)) = l(w), then there exists a

the following conditions:
• There exist a k-scheme Y 1 with a J b (F )-action, and stable.There exist a k-scheme Y 2 with a J b (F )-action, and , where f 2 is a Zariski-locally trivial G m -bundle and g 2 is a universal homeomorphism.If char(F ) = 0, then the above two statements still hold, but with "A 1 -bundle" and "G m -bundle" replaced by "A 1,pfn -bundle" and "G pfn m -bundle" respectively.Proof.The equal characteristic case is proved in [GH10, §2.5].The mixed characteristic case follows from the same proof.

Let w ∈ W and b ∈ G( F )
. By the discussion in §2.4.1 and Lemma 2.4.2, we know that the perfection Using the formalism in §3.1, we can reformulate Proposition 3.3.1 in the following proposition (which is weaker, but more convenient for applications).(1) If l(swσ(s)) = l(x), then Proof.This follows from Proposition 3.3.1 and the following three observations.Firstly, if a morphism of k-schemes is universally homeomorphic, then the perfection of this morphism is an isomorphism, by [BS17, Lemma 3.8].Secondly, if a morphism of k-schemes is a Zariski-locally trivial A 1 -bundle (resp.G m -bundle), then the perfection of this morphism is a Zariski-locally trivial A 1,pfn -bundle (resp.G pfn mbundle).Thirdly, the perfections of the k-schemes X 1 , X 2 , Y 1 , Y 2 in Proposition 3.3.1 (2), equipped with the natural J b (F )-actions, are all objects in S J b (F ) .Indeed, the assertion for X 2 follows from the fact that X w (b) pfn is in S J b (F ) and Lemma 3.1.3.The assertion for Y 2 follows from the fact that X sx (b) pfn is in S J b (F ) , and the fact that the perfection of g 2 is a J b (F )-equivariant isomorphism.The assertion for Y 1 follows from the fact that X sxσ(x) (b) pfn is in S J b (F ) , and the fact that the perfection of g 1 is a J b (F )-equivariant isomorphism.The assertion for X 1 follows from the assertion for Y 1 , the fact that the perfection of f 1 is locally of perfectly finite type, and the fact that pulling back along the perfection of 3.3.4.In order to effectively use Proposition 3.3.3 to study the J b (F )-action on Σ top (X w (b)), we need a refined version of the class polynomials for affine Hecke algebras.We first recall the definition of the usual class polynomials.Here we use the convention of [He16, §2.8.2], which differs from that in [He14].
Let q be an indeterminate, and let Z[q ±1 ] be the Laurent polynomial ring.Let H be the affine Hecke algebra over Z[q ±1 ] attached to W . Thus H is the associative Z[q ±1 ]-algebra generated by symbols {T w | w ∈ W } subject to the following relations: The action of σ on W induces an automorphism σ of H characterized by σ(T w ) = T σ(w) for all w ∈ W . Define [H, H] σ to be the Z[q ±1 ]-submodule of H generated by hσ(h ′ ) − h ′ σ(h), where h and h ′ run over elements of H. Define the σ-cocenter (or simply cocenter) to be the quotient module Hσ := H/[H, H] σ .
For any O ∈ B( W , σ), let O min be the set of minimal length elements of O.By [HN14, Theorem 5.3, Theorem 6.7], the cocenter Hσ is a free Z[q ±1 ]-module with a basis given by {T O | O ∈ B( W , σ)}.Here T O is the image of T w in Hσ for some (or equivalently, any) w ∈ O min .Moreover, for any w ∈ W , we have where F w,O ∈ Z[q] is the class polynomial, uniquely determined by the above identity.
3.3.5.As indicated above, we need a refinement of the polynomials F w,O where (w, O) ∈ W × B( W , σ).The refined polynomials will be indexed by pairs (w, C) ∈ W ×C ( W ), where C ( W ) is a set more refined than B( W , σ).We now define C ( W ).
For w, w ′ ∈ W and s ∈ S, we write We now construct the refined polynomials in the following theorem.Let N[q − 1] denote the set of polynomials in the variable q−1 with positive integral coefficients.The second statement in the theorem can be viewed as a "motivic counting" result.
Theorem 3.3.9.Fix w ∈ W .There exists a map satisfying the following conditions.
(2) For each b ∈ G( F ), we have Proof.We prove the statement by induction on ℓ(w).
If w ∈ Wσ,min , then by [He16, §2.8.2], for any O ∈ B( W , σ), we have On the other hand, for C ∈ C ( W ), we set In this case, the map C → F w,C satisfies conditions (1) and (2).Now assume that w / ∈ Wσ,min .Then by Theorem 3.3.6,there exists w ′ ∈ W and s ∈ S such that w ≈σ w ′ and sw ′ σ(s) < w ′ .By [He16, §2.8.2], for any O ∈ B( W , σ), we have For C ∈ C ( W ), we set where F sw ′ ,C (q − 1) and F sw ′ σ(s),C (q − 1) are defined by the induction hypothesis.Since condition (1) holds for sw ′ and sw ′ σ(s), it also holds for w.By Proposition 3.3.3(2), for any b ∈ G( F ) we have By the induction hypothesis, we have the following identities in GDL J b (F ) : Then Thus (2) holds for w.
Remark 3.3.10.(1) The polynomials F w,C are not uniquely characterized by condition (1) in Theorem 3.3.9.This is because the cocenter of the affine Hecke algebra over Z[q] has a torsion part, cf.[He15, §5.2].(In contrast, as we have mentioned above, the cocenter of the affine Hecke algebra over Z[q ±1 ] is free.) (2) Fix b ∈ G( F ), and let K J b (F ) 0 be the Grothendieck group of the monoid GDL J b (F ) .The endomorphism L of GDL J b (F ) gives rise to a Z[q]-module structure on K necessarily torsion-free as a Z[q]-module.It would be interesting to compare the torsion phenomenon here with the cocenter of the affine Hecke algebra over Z[q].
(3) As we have seen in the proof of Theorem 3.3.9, the construction of F w,C depends on G only via the triple ( W , l : W → Z ≥0 , σ ∈ Aut( W )).This will allow us to reduce the study of general G to unramified groups.
) is given by the first (resp.second) coordinate of the element Proof.Note that the isomorphism class of the J b (F )-set Σ top (X w (b)) and the integer dim X w (b) do not change if we replace X w (b) by its perfection.The corollary then follows from applying the T-equivariant homomorphism C in Lemma 3.2.3 to the two sides of (3.3.9.1).
Remark 3.3.12.Fix b ∈ G( F ). 3.4.Stabilizer of one irreducible component.We keep the setting and notation of §3.3.In this subsection we assume in addition that G is F -simple and adjoint.We will apply the results in §3.3 to study the stabilizers for the J b (F )action on Σ top (X w0t µ (b)).
3.4.1.Recall that for δ an automorphism of ( Wa , S) and K ⊂ S a δ-stable subset, a δ-twisted Coxeter element of W K is an element which can be written as where s 1 , . . ., s n ∈ W K are distinct and form a set of representatives of the δ-orbits in K.For w ∈ Wa we write supp δ (w) for the smallest δ-stable subset K of S such that w ∈ W K .As explained in §2.4.3, we identify W0 with the subgroup WS 0 of W .Note that every w ∈ W can be written in a unique way as w = xt µ y, where µ ∈ X * (T ) Γ0 + , x, y ∈ W0 , and t µ y ∈ S0 W .Moreover, l(w) = l(x) + l(t µ ) − l(y).The following result gives a refinement of [He14,Proposition 11.6].
Then there exists a σ-twisted Coxeter element c of W0 with t µ c ∈ S0 W such that for each b ∈ G( F ), we have for some P ∈ GDL J b (F ) .
We proceed by induction on | K|.The case | K| = 0 is clear, as we can take c = y.We thus assume that the result is true for all K′ K. We may also assume that the result is true for all x ′ ∈ W K with supp σ (x ′ ) = K and l(x ′ ) < l(x).We set Then as in [He14, Proposition 11.6], K1 is a proper subset of K, and every s ∈ K1 commutes with y and with t µ y.
3.4.3.For an element τ ∈ Ω, the Iwahori-Weyl group and affine Weyl group of J τ are isomorphic to W and Wa respectively, and the Frobenius actions are both given by Ad(τ ) • σ.We need the following result which is proved in [HY12].Set V := X * (T ) Γ0 ⊗ Z R. Remark 3.4.5.In Proposition 3.4.4,the unique K is explicitly computed in each case in [HY12].The "moreover" part of the proposition immediately follows from the explicit description.
The main result of this subsection is the following proposition.Proof.Since µ is dominant, t µ ∈ S0 W .If µ = 0, then we may take b = 1.In this case, J b (F ) = G(F ) and X µ (b) = G(F )/K is discrete; here K ⊂ G(F ) is the parahoric subgroup corresponding to S0 which is very special (cf.Remark 2.2.3).
For any Z ∈ X µ (b), the stabilizer Stab Z (J b (F )) is conjugate to K and thus is a very special parahoric subgroup of G(F ).Now the statement on X w0t µ (b) follows from Proposition 2.4.10.Now assume that µ = 0.By Proposition 3.4.2applied to K = S0 and w = w 0 t µ , there exists a σ-twisted Coxeter element c of W0 such that Let τ ∈ Ω be the unique element such that κ(τ ) = µ ♮ ∈ π 1 (G) Γ0 .Upon replacing b by another representative in [b], we may assume b = τ .By Proposition 3.4.4and Theorem 3.3.6,there exists an Ad(τ ) • σ-stable subset K ⊂ S and an Ad(τ ) • σtwisted Coxeter element c ′ of W K such that the associated parahoric J of J τ (F ) is very special, c ′ τ is of minimal length in its σ-conjugacy class, and t µ c → σ c ′ τ .By Proposition 3.3.3,we have for some P ∈ GDL J τ (F ) .By Lemma 3.2.3, the above equality implies that for some H ∈ TIC J τ (F ) .Here on the right side, the addition is in the monoid TIC J τ (F ) .
By [Kot06, §1.9], l(c) − l(c ′ ) = def G ( τ ).By Theorem 2.4.7 and [He14, Theorem 10.1], where the fourth equality follows from the fact that t µ c ∈ S0 W .By the above computation, the first term in the sum makes a non-trivial contribution to the sum in the sense of §3.2.1.Thus we have a , where X K c ′ τ ( τ ) is a classical Deligne-Lusztig variety (resp.perfection of a classical Deligne-Lusztig variety) if char(F ) > 0 (resp.char(F ) = 0) defined by as J τ (F )-sets and the stabilizer of the elements are isomorphic to J .

Component stabilizers for X µ (b)
4.1.The main theorem and some consequences.
4.1.1.We keep the notation and assumptions of §2.4.In particular, G is a quasisplit tamely ramified reductive group over F , and char(F ) We now state our main theorem, which confirms conjectures made by X. Zhu and Rapoport.

Corollary 4.1.4.
There is an identification of J b (F )-sets where J i ⊂ J b (F ) is a very special parahoric subgroup for each i.
4.1.5.When G is unramified, an explicit formula for N (µ, b) was conjectured by M. Chen and X. Zhu, and was proved independently by the second and third named authors in [ZZ20] and by S. Nie in [Nie18a].In the appendix of [ZZ20], a generalization of this formula for ramified G is given.We now recall this formula when G is unramified, as this will be needed in §4.3 below.Consider the dual group G of G over C. We fix a pinning ( B, T , X + ) of G, and fix an isomorphism between the based root datum of ( G, B, T ) and the dual of the based root datum of (G, B, T ).(See §2.3.1 for B.) We then have a unique Γ-action on G via automorphisms preserving ( B, T , X + ) such that the induced Γ-action on the based root datum of ( G, B, T ) is compatible with the natural Γ-action on the based root datum of (G, B, T ), see for instance [ZZ20, §5.1].Now Γ 0 acts trivially on X * (T ), so the element µ ∈ X * (T ) + Γ0 can be viewed as a B-dominant character of T .Let V µ be the irreducible representation of G of highest weight µ.Let S be the identity component of the Γ-fixed points of T .Then X * ( S) is identified with the maximal torsion-free quotient of X * (T ) Γ = X * (T ) σ .As in [ZZ20, Definition 2.6.4],b determines an element λ b ∈ X * ( S).We omit the explicit definition of λ b here.Let V µ (λ b ) be the weight space in the S-representation V µ of weight λ b .The geometric Satake provides us with a canonical basis MV µ (λ b ) of V µ (λ b ).
In the theorem below, the numerical identity is proved independently by the second and third named authors [ZZ20, Theorem A] and Nie [Nie18a, Theorem 0.5].The second statement is due to Nie [Nie18a, Theorem 0.5].
Theorem 4.1.6.Keep the assumptions in §2.4, and assume that G is unramified over F .We have Moreover, there is a natural bijection between J b (F )\Σ top (X µ (b)) and MV µ (λ b ).

Reduction to adjoint unramified F -simple groups in characteristic 0.
In this subsection, we show that to prove Theorem 4.1.2,it suffices to prove it in the case where char(F ) = 0, and G is an adjoint F -simple unramified group over F .

Let w ∈ W and [b] ∈ B(G).
We first construct some combinatorial data involving only the affine Weyl group Wa together with the length function l and the action of σ on Wa , but not the reductive group G.This allows us to connect different reductive groups over different local fields.
Let Aut 0 ( Wa ) be the group of length-preserving automorphisms of Wa .We may regard σ as an element of Aut 0 ( Wa ).Let W a = Wa ⋊ Aut 0 ( Wa ).We have a natural group homomorphism Moreover, the map i is compatible with the actions of σ. (Here the action of σ on W a is given by (w, We say that the triples (G, b, w) and (G ′ , b ′ , w ′ ) are associated if the following conditions are satisfied: • There exists a length-preserving isomorphism f : Wa is the natural homomorphism analogous to i.
In this case, f induces an isomorphism from the affine Weyl group of J b to the affine Weyl group of J b ′ .We thus obtain a bijection between the standard parahoric subgroups of J b (F ) and those of J b ′ (F ), cf.[ZZ20, Lemma 3.2.2].Let J ⊂ J b (F ) and J ′ ⊂ J b ′ (F ′ ) be parahoric subgroups.We say that J and J ′ are associated with respect to f , if there exist j ∈ J b (F ) and j ′ ∈ J b ′ (F ′ ) such that jJ j −1 and j ′ J ′ j ′−1 are standard parahoric subgroups which correspond to each other under the above-mentioned bijection.
satisfying the following condition: Here C ′ runs over C ( W ′ ), the set of ≈σ ′ -equivalence classes in W ′ σ ′ ,min .Note that in the proof of Theorem 3.3.9, the construction of the polynomials F w,C only involves σ-conjugation of Wa on W , and the construction remains the same if we replace W by W a .The identification f : Wa Proof.We first assume Theorem 4.1.2is true for unramified adjoint groups over local fields of characteristic 0. Let G be an arbitrary (i.e., quasi-split, tamely ramified, reductive) group over an arbitrary local field F .By Proposition 2.4.10, it suffices to show that the stabilizer in J b (F ) of every element of Σ top (X w0t µ (b)) is a very special parahoric of J b (F ).
Since X µ (b) = ∅, we have X w0t µ (b) = ∅ and thus κ G (w 0 t µ ) = κ G (b).By [He14, Theorem 3.7], there exists w ∈ W such that [b] = [ ẇ].In particular, we have κ G (w 0 t µ ) = κ G (w).By replacing w by a suitable element in the σ-orbit of w, we may assume furthermore that w 0 t µ Wa = w Wa .We choose F ′ a local field of characteristic 0 and G ′ an adjoint unramified group over F ′ such that there is a length-preserving isomorphism . By the equivalence (1) ⇔ (3) in Proposition 2.2.5 and by the formula (2.2.4.2) for the log-volume, we know that Stab Z (J b (F )) is a very special parahoric subgroup of J b (F ).
Now the reduction from the adjoint unramified case to the adjoint unramified F -simple case follows from the fact that any adjoint unramified group over F is a direct product of adjoint unramified F -simple groups.
4.3.Reduction to the basic case.We assume that char(F ) = 0 and that G is an adjoint F -simple unramified group over F .By Corollary 4.2.4,we can reduce the proof of Theorem 4.1.2to this case.In this subsection we show that we can further reduce the proof to the case where b is basic.We follow the strategy of [HV18,§5].4.3.1.Let K = S0 , and let K and K be the corresponding parahoric subgroups of G(F ) and G( F ) respectively, as in §2.4.3.In our current setting, K is in fact a hyperspecial subgroup of G(F ).
Let M ⊂ G denote the standard Levi subgroup of G given by the centralizer of ν G b .We view Ȃ as an apartment for M and let ȃM ⊂ Ȃ be the (unique) alcove with respect to M such that ȃ ⊂ ȃM .We denote by WM the Iwahori-Weyl group for M and denote by Ω M the subgroup of length zero elements determined by ȃM Let P be the standard parabolic subgroup of G with Levi subgroup M .Let N be the unipotent radical of P .Let KM (resp.KP ) denote the intersection M ( F )∩ K (resp P ( F ) ∩ K).These arise from group schemes K M and K P defined over O F , and K M (O F ) is a hyperspecial subgroup of M (F ).As in [HV18, §5], we define These can be identified with the sets of k-points of perfect subschemes X M⊂G µ (b) and X P ⊂G µ (b) of Gr KM and Gr KP respectively.The natural maps M ← P → G induce maps which are easily seen to be J b (F )-equivariant.The same argument as [Ham15, Lemma 2.2] and the paragraph preceding it shows that the map q is a decomposition of X µ (b) into locally closed subschemes (cf.[HV18, Lemma 5.2]) and hence we obtain a J b (F )-equivariant bijection 4.3.2.Let X and X be smooth finite-type affine group schemes over F and O F respectively.The loop group LX and the positive loop group LX are defined to be the functors on perfect k-algebras R given by Then LX is representable by an ind-perfect ind-group-scheme, and L + X is representable by the perfection of an affine group scheme over k.We also define the n th jet-group L n X to be the functor on perfect k-algebras R given by where π is a uniformizer in F .Then L n X is representable by the perfection of an algebraic group over k.
Lemma 4.3.3.The map Proof.Recall we have assumed b = τ for τ ∈ Ω M .Choose s sufficiently divisible such that τ σ(τ ) . . .σ s−1 (τ ) = t λs where λ s := sν b ∈ X * (T ) + .(Note that since we have assumed G is unramified, Γ 0 acts trivially on X * (T ).) Then we have b s ∈ ṫλs T ( F ) 1 and it suffices to show that the map For r ≥ 0, we define N r := N (F ) ∩ I r where I r is the r th -subgroup in the Moy-Prasad filtration of I. Then N r = N r (O F ) for an O F -group scheme N r and we have Since λ s ∈ X * (T ) + , we have ṫλs σ s (N r ) ṫ−λs ⊂ N r for all r.It follows that f s b induces a morphism f s b,r : L r N 0 −→ L r N 0 for each r.In fact f s b,r is naturally defined before taking perfections and is an étale morphism since it induces multiplication by −1 on tangent spaces.It follows that f s b,r is an étale covering.Let F s be the degree s unramified extension of F and let J Now fix an element χ ∈ X * (T ) +,σ ∩X * (Z M ), where Z M is the center of M .Using the fact that Ad ṫχ ṫχ is an isomorphism.Taking an inductive limit over χ, we find that f s b : LN → LN is an isomorphism.4.3.4.We identify Gr K with the fpqc quotient LG/L + K.For λ ∈ X * (T ), recall the semi-infinite orbit We let Gr K,µ denote the Schubert cell L + K ṫµ L + K/L + K and Gr K, µ the corresponding Schubert variety which is defined to be the closure of Gr K,µ inside Gr K.
Let M ⊂ G denote the Levi subgroup determined by M and the fixed pinning from §4.1.5.For an M -dominant element λ ∈ X * (T ), we may consider λ as element of X * ( T ) which is M -dominant with respect to the ordering determined by B ∩ M .We write V M λ for the irreducible representation of M of highest weight λ.We let a λ,µ denote the multiplicity of V M λ appearing in the M -representation V µ | M , and we write ρ M (resp.ρ N ) for the half sum of positive roots in M (resp.roots in N ).The same argument as [GHKR06, Proposition 5.4.2]shows that and that we have It follows that multiplication by k M induces an automorphism of S N,λ with inverse given by multiplication by k −1 M , and hence an automorphism of S N,λ ∩ Gr K,µ .The group ṫ−λ KM ṫλ ∩ KM arises as the O F -points of a smooth connected O Fscheme Kλ .Then as above, left multiplication induces a map . The image of this map is an irreducible subscheme of S N,λ ∩ Gr K,µ containing Z, hence is equal to Z.It follows that k M (Z) = Z.4.3.6.We define the sets Then there is a decomposition (4.3.6.1) where each Proof.(1) By [Ham15, Lemma 2.8, Proposition 2.9 (3)], which also holds in the mixed characteristic setting, we have The first and third equalities follow Theorem 2.4.7and the second equality follows from the identities def G (b) = def M (b) and ν b , ρ N = λ, ρ N .By [GHKR06, Proposition 5.4.2], which again holds in mixed characteristic, the first inequality is an equality if and only if a λ,µ = 0.
(2) By [Ham15, Proposition 2.9 (2)] and a similar calculation as in (1), for any . It follows that these quantities are equal and we have a λ,µ = 0. We write p −1 (Y ′ ) for the fiber product As in [HV18, Proposition 5.6], we set Then the natural map γ : Φ → p −1 (Y ′ ) is a KN -torsor.Moreover upon shrinking Y and replacing ι if necessary (cf.[HV18, Proof of Proposition 5.6]) we may assume m −1 bσ(m) ∈ ṫλ KM for any m ∈ ι(Y ′ ).We then define We write Ad M : LM × LN → LM × LN for the map (m, n) → (m, mnm −1 ).This is easily seen to be an isomorphism with inverse given by Ad . By Lemma 4.3.3,fb is an isomorphism.The restriction of fb to Φ gives an isomorphism fb : Φ → E and we have a Cartesian diagram: We consider the projection We write pr : E → S N,λ ∩ Gr K,µ for the composition of projection onto the second component pr 2 : Since the maps γ, fb , and pr all induce bijections on irreducible components, it follows that θ is a bijection.
Let U ∈ Σ top (p −1 (Z)) and let U 1 , U 2 ⊂ Φ denote the preimages of U and jU respectively.
Then we have z ∈ U 2 (k), and one computes that Then by Lemma 4.3.5, we have pr• fb (x) ∈ Z ′ 2 .Since this is true for a dense set of x in U 1 , it follows that pr • fb (U 1 ) ⊂ Z ′ 2 , and hence Corollary 4.3.9.
Proof.Let U P ∈ Σ top (X P ⊂G µ (b)) be the component corresponding to U and let By the J b (F )-equivariance of p, we have Since The statement is proved.
Proposition 4.3.10.In order to prove Theorem 4.1.2,it suffices to prove it when char(F ) = 0, G is F -simple, adjoint, and unramified over F , and b is basic.
Proof.This follows from Corollary 4.2.4 and Corollary 4.3.9.
4.4.The special case of a sum of dominant minuscule cocharacters.We assume that char(F ) = 0, that G is F -simple, adjoint, and unramified over F , and that b is basic.Our goal in this subsection is to prove a partial result towards Theorem 4.1.2when µ is a sum of minuscule dominant cocharacters.We use the idea of X. Zhu (see [Zhu17, §3.1.3])that one can "separate" the summands of µ by constructing a convolution map from the affine Deligne-Lusztig variety of a Weil-restriction group to the original affine Deligne-Lusztig variety.This idea was originally used in loc.cit. to establish the dimension formula, and it was S. Nie who first applied this idea to the study of irreducible components (see [Nie18b] and [Nie18a]).
4.4.1.Let F r denote the unramified extension of F of degree r inside F .Let H be an unramified reductive group over F r and let G ′ := Res Fr/F H.We canonically identify F with Fr .For b ∈ H( F ) and µ a geometric cocharacter of H, we have the affine Deligne-Lusztig variety write X H µ (b) as in §2.4.3.In this subsection we denote this by X H µ (bσ r ) to emphasize that H is a group over F r and the Frobenius relative to F r is σ r .We also write J (r) b for J b (defined with respect to H over F r ), and write B (r) (H) for the set of σ r -conjugacy classes in H( F ).
Let τ 0 : F r ֒→ F be the inclusion and write τ i for σ i (τ 0 ) for i = 1, . . ., r − 1.Thus {τ 0 , . . ., τ r−1 } is the set of F -algebra embeddings F r → F .There is a canonical identification Let T H be the centralizer of a fixed maximal F r -split torus in H. Let T ′ = Res Fr /F T H , which we view as an F -subgroup of G ′ .Then T ′ is the centralizer of a maximal F -split torus in G ′ .A cocharacter of T ′ is the same as a sequence µ ′ = (µ 0 , . . ., µ r−1 ), where µ i ∈ X * (T H ). Fix a Borel subgroup of H containing T H and use it to define the dominant cocharacters X * (T H ) + .This also defines a Borel subgroup of G ′ containing T ′ and defines X * (T ′ ) + .We fix a hyperspecial subgroup of H(F r ) that is compatible with our choice of the maximal Fr -split F r -rational torus of H.This also determines a hyperspecial subgroup of G ′ (F ).We use these hyperspecial subgroups to define affine Deligne-Lusztig varieties at hyperspecial level for H and G ′ .For b ′ = (b 0 , . . ., b r ) ∈ G ′ ( F ), we define Nm(b ′ ) (F r )).Proof.The morphism θ is given by the isomorphism in [Zhu17, Lemma 3.5] and the left vertical map in the diagram on p. 459 of [Zhu17].The . We claim that J := Stab U (J Nm(b ′ ) (F r )) acts trivially on Σ top (θ −1 (U )).In fact, by the diagram on p. 459 of [Zhu17], there exists m ∈ N and an L m H-torsor U ′ over U equipped with a J -action such that U ′ → U is J -equivariant and such that there exists a J -equivariant , where J acts trivially on F .Our claim follows.By the claim, we have . The lemma follows.Let Z ∈ Σ top (X µ (b)) and we let J ⊂ J b (F ) denote the stabilizer of Z.Let P be the standard parabolic subgroup of G with Levi factor M .By [ZZ20, Theorem 3.1.1],J is a parahoric subgroup of J b (F ).By Theorem 4.1.6and the "only if" part of [HV18, Theorem 5.12], J b (F ) ∩ P ( F ) acts transitively on each J b (F )-orbit in Σ top (X µ (b)).Hence we have Thus by [BT72, Proposition 4.4.2], the equality (4.4.3.1)implies that J is contained in a special parahoric subgroup J 1 of J b (F ).Note that by (4.4.3.1),there exists j ∈ Q(F ) such that jJ j −1 is associated with a facet in the standard apartment A. Thus up to replacing Z by jZ, we may assume that both J and J 1 are associated with facets in A. Then from (4.4.3.1)we get Let J 1 denote the reductive quotient of the special fiber of J 1 .Then the images of J 1 ∩ Q(F ) and J in J 1 are (k F -points of) a Borel subgroup B and a parabolic subgroup P respectively, and B ∩ P contains a maximal torus in J 1 .By (4.4.3.2) we have J 1 = BP, and by the Bruhat decomposition this is possible only when P = J 1 , or equivalently J = J 1 .We have thus proved that J is a special parahoric subgroup of J b (F ).
We now consider the case when µ is a sum of r dominant minuscule cocharacters.Let H be the pinned unramified reductive group over F r such that its based root datum with the σ r -action is identified with the based root datum of (G, B, T ) with the σ-action.Let T H be the maximal torus in the pinning of H. Then we have a canonical identification X * (T ) + ∼ = X * (T H ) + , and the image of µ in X * (T H ) + , denoted by µ H , is also a sum of r dominant minuscule cocharacters.We have canonical identifications G( F ) ∼ = H( F ) and ( W , σ) ∼ = ( WH , σ r ).Let b H ∈ H( F ) correspond to b ∈ G( F ), and let w 0,H denote the longest element of WH .Then (G, b, w 0 t µ ) and (H, b H , w 0,H t µH ) are associated as in §4.2.By Proposition 4.2.3,Proposition 2.4.10, and the fact that association of parahoric subgroups preserves being very special (see the proof of Corollary 4.2.4), it suffices to prove the result for X H µH (b H σ r ).We decompose µ H as r−1 i=0 σ i (µ i ), and define ), and by the previous part of the proof, we know that Stab Z (J b ′ (F )) is a special parahoric.The desired result for X H µH (b H σ r ) follows by noting that the natural map bH (F r )-equivariant bijection between the sets of top-dimensional irreducible components.4.5.Numerical relations.Another key ingredient in our proof of Theorem 4.1.2is a set of numerical relations deduced from results in [ZZ20], which we discuss here.4.5.1.We assume that char(F ) = 0, that G is F -simple, adjoint, and unramified over F , and that b is basic.We also assume that [b] is not unramified, i.e., we assume that def (1) Assume that G is not a Weil restriction of the split adjoint group of type E 6 .Then there exists a minuscule µ 1 ∈ X * (T ) + such that N (µ 1 , b) = 1, and such that for all µ ∈ X * (T ) + we have (2) Assume that G is a Weil restriction of the split adjoint group of type E 6 .(The Weil restriction is necessarily along an unramified extension of F since G is unramified).Then there exist µ 1 , µ 2 ∈ X * (T ) + , where µ 1 is minuscule and µ 2 is a sum of dominant minuscule cocharacters, such that N (µ 1 , b) = 1 and such that for all µ ∈ X * (T ) + we have for some C(µ) ∈ Q.
Proof.The proposition follows from the main result of [ZZ20] (i.e., the Chen-Zhu Conjecture), and the proof of [ZZ20, Theorem 6. Remark 4.5.3.In Proposition 4.5.2, the conclusion in case (2) is weaker than that in case (1), and this originates from the dichotomy in [ZZ20, Proposition 6.3.2].It turns out that in case (2), there is extra difficulty in trying to establish the key estimate [ZZ20, (6.3.1)], and in fact only the weaker statement [ZZ20, Proposition 6.3.2 (2)] is proved.If G is a Weil restriction of PGL n , there seems to be even more serious difficulty in trying to establish [ZZ20,(6.3.1)].As a result the type A case is not considered in [ZZ20, Proposition 6.3.2].After the publishing of [ZZ20], the authors have realized that one can actually prove [ZZ20, (6.3.1)] when G is a Weil restriction of an adjoint unramified unitary group.We will not need this for the purposes of the current paper.4.6.Proof of Theorem 4.1.2.By Proposition 4.3.10,we may assume without loss of generality that char(F ) = 0, that G is F -simple, adjoint, and unramified over F , and that b is basic.If [b] is unramified, then Theorem 4.1.2is already proved in [XZ17, Theorem 4.4.14 (1)], cf.[ZZ20, Theorem 6.2.2].We hence assume that [b] is not unramified.Thus we are in the same setting as §4.5.1.
Let vol max be the volume of a very special parahoric subgroup of J b (F ), where the Haar measure on J b (F ) is as in §4.5.1.We know that every stabilizer for the J b (F )-action on Σ top (X µ (b)) is a parahoric subgroup of J b (F ), see Remark 3.3.12and [ZZ20, Proposition 3.1.4].As a result, the volume of such a stabilizer will be at most vol max , and equality holds if and only if the stabilizer is very special.Since the quantity Q(µ, b) defined in (4.5.1.1)is the average of the volumes of these stabilizers, we see that Theorem 4.1.2for (µ, b) is equivalent to the relation Since G is F -simple, the simple factors of G F are isomorphic to each other.If they are of type A, then µ is necessarily a sum of dominant minuscule cocharacters in X * (T ).In this case, Theorem 4.1.2follows from Proposition 4.4.3 if we know that every special parahoric subgroup of J b (F ) is automatically very special.Since J b is an inner form of G and hence also of type A, it is indeed the case that special parahoric subgroups of J b (F ) are automatically very special, by inspecting the tables in [Tit79,§4].
We are left with the case where G is a Weil restriction of the split adjoint group of type E 6 .In this case, let µ 1 and µ 2 be as in Proposition 4.5.2(2).Since J b is also of type E 6 , by inspecting the tables in [Tit79, §4] we see that every special parahoric subgroup of J b (F ) is automatically very special.Thus by Proposition 4.4.3we know that Theorem 4.1.2holds for (µ 1 , b) and (µ 2 , b).It follows that max .Substituting this back to (4.5.2.1), we obtain (4.6.0.1) for (µ, b), and this implies that Theorem 4.1.2holds for (µ, b).
The proof of Theorem 4.1.2is complete.
5.1.1.We use the previous section to describe the irreducible components in the basic locus of certain Hodge type Shimura varieties constructed in [KP18].Let G be a connected reductive group over Q and X a conjugacy class of homomorphisms , where c is the complex conjugation.For h ∈ X we let µ h denote the cocharacter of G C given by where R is an arbitrary C-algebra and the first map is z → (z, 1).The conjugacy class of µ −1 h is defined over a number field E := E(G, X) ⊂ C and we write {µ} for the corresponding geometric conjugacy class of cocharacters over E.
Let p be an odd prime and we write G := G Qp for the base change of G to Q p .We let A f denote the ring of finite adeles and A p f the finite adeles with trivial component at p.
arises as the complex points of an algebraic variety Sh K (G, X) defined over E. 5.1.2.From now on, we will assume the datum (G, X) is of Hodge type.This means that there exists an embedding of Shimura data ρ : (G, X) −→ (GSp(V, ψ), S ± ) where (V, ψ) is a symplectic space over Q and (GSp(V, ψ), S ± ) is the standard Siegel Shimura datum.We will also make the following assumptions.
( †) The group G := G Qp is quasi-split and splits over a tamely ramified extension of Q p .Moreover p ∤ |π 1 (G der )|, and K p is a connected very special parahoric subgroup of G(Q p ).
Here we say a parahoric K p is connected if it is the same as the stabilizer of a facet in the building for G.When G is unramified, every parahoric which is contained in a hyperspecial parahoric is connected.In the sequel we let G be the group scheme over Z p corresponding to the parahoric K p .
Let v be a prime of E lying above p with residue field k v = F q .We write O for the ring of integers of E and O (v) for the localization of O at v.Under the assumptions above, Kisin-Pappas [KP18] have constructed an integral model S K (G, X) for Sh K (G, X) over O (v) .We briefly recall the construction below.
By the discussion in [KP18, §2.3.15],upon replacing ρ with a different Hodge embedding, we may assume that there exists a Z p -lattice V Zp ⊂ V Qp such that ρ induces a closed immersion G → GL(V Zp ).From now on we fix ρ such that this condition is satisfied.We let sufficiently small compact open subgroup.By [Kis10, Lemma 2.1.2],up to shrinking K p we may choose a sufficiently small K ′p such that the Hodge embedding ρ defines a closed immersion ).The choice of V Z (p) gives rise to an interpretation of Sh K ′ (GSp(V ), S ± ) as a moduli space of abelian varieties and hence to an integral model S K ′ (GSp(V ), S ± ) over Z (p) ; see [KP18,§4] and [Zho20,§6].The integral model S K (G, X) is defined to be the normalization of the closure of Sh K (G, X) in S K ′ (GSp(V ), S ± ) ⊗ Z (p) O (v) .We will write A for the pullback of the universal abelian scheme on S K ′ (GSp(V ), S ± ) ⊗ Z (p) O (v) to S K (G, X).

Rapoport-Zink Uniformization.
5.2.1.We fix a maximal Qp -split Q p -rational torus S in G (cf. §2.1.1)such that K p corresponds to a σ-stable special point s in the apartment corresponding to S. We let T denote the centralizer of S and we fix B a Borel subgroup of G containing T (which exists as we have assumed that G is quasi-split).We let µ ∈ X * (T ) + Γ0 denote the image of a dominant representative µ ∈ X * (T ) + of {µ}.(Here Γ 0 is as in §2.1.1 with respect to F = Q p .)Then for b ∈ B(G, µ) we have the associated affine Deligne-Lusztig variety X µ (b) as in §2.4 corresponding to the very special parahoric K p .
To ease notation we write Sh K for the geometric special fiber of S K (G, X).By [Zho20, §8], there exists a map which induces the Newton stratification on Sh K .We let [b] basic ∈ B(G, µ) denote the unique basic σ-conjugacy class in B(G, µ) and we write Sh K,bas for the preimage of [b] basic under N .By [RR96, Theorem 3.6] this is a closed subscheme of Sh K , which is known as the basic locus.
Our goal is to understand the set Σ top (Sh K,bas ) of top-dimensional irreducible components of Sh K,bas .This will follow from our study of X µ (b) above and the following result, which is the analogue in our context of the Rapoport-Zink uniformization.

Proposition 5.2.2. Let b ∈ [b]
basic .There exists an isomorphism of perfect schemes where I is a certain inner form of G with Moreover this isomorphism is equivariant for prime-to-p Hecke operators.

Corollary 5.2.3. There exists an identification
f .Moreover the following statements hold.
(1) The identification is compatible with prime-to-p Hecke operators.
(2) If G is unramified, we may replace the indexing set with MV µ (λ b ).
Proof.This follows from Proposition 5.2.2, Corollary 4.1.4,and the fact that the topology of a scheme is invariant under taking perfection.
The rest of the section will be devoted to the proof of Proposition 5.2.2.The case when G is an unramified group is proved in [XZ17, Corollary 7.2.6], a key input being the existence of a natural map and we let I ⊂ Aut(A x ⊗ k F p ) denote the subgroup which preserve the tensors s α,0,x and s α,ℓ,x for all ℓ = p.We have the following facts about these groups for points x in the basic locus.
commutes up to inner automorphism for any prime ℓ. 5.2.7.For (γ 0 , (γ ℓ ) ℓ =p , δ) a Kottwitz triple of level r, (γ m 0 , (γ m ℓ ) ℓ =p , δ) is a Kottwitz triple of level rm.We consider the smallest equivalence relation on the set of all Kottwitz triples of all levels under which (γ 0 , (γ ℓ ) ℓ =p , δ) is equivalent to (γ m 0 , (γ m ℓ ) ℓ =p , δ) for all m ≥ 1.An equivalence class under this relation is called a Kottwitz triple.For x ∈ Sh K,bas (F p ), we know that x is defined over some k = F p r , and the associated Kottwitz triple (γ 0 , (γ ℓ ) ℓ =p , δ) of level r defines a Kottwitz triple which is independent of the choice of F p r .
Recall the following notion of isogeny classes introduced in [Zho20].
Definition 5.2.8.Let x, x ′ ∈ Sh K (F p ).We say x and x ′ are isogenous if there exists a quasi-isogeny A x → A x ′ which takes s α,ℓ,x to s α,ℓ,x ′ and s α,0,x to s α,0,x ′ .Clearly this defines an equivalence relation on Sh K (F p ), and the equivalence classes will be called isogeny classes.
Proof.We fix a sufficiently large finite field k = F p r such that x and x ′ are both defined over k and we fix representatives (γ 0 , (γ ℓ ) ℓ =p , δ) and (γ ′ 0 , (γ ′ ℓ ) l =p , δ ′ ) of level r for t and t ′ respectively.Write I and I ′ for the Q-groups associated to x and x ′ as above.We first claim that there exists n ≥ 1 such that γ n 0 and γ ′n 0 are central.Indeed this follows verbatim from the argument in [XZ17, Lemma 7.2.14] which works without the assumption that G is unramified.Therefore upon extending k we may assume γ 0 and γ ′ 0 are both central.Let Z • denote the connected component of the center of G. Upon enlarging k, we may assume t := γ −1 0 γ ′ 0 ∈ Z • (Q).We claim that the image of t in Z • (A f ) lies in a compact open subgroup H.For ℓ = p, we have γ 0 = γ ℓ , hence γ 0 lies in a compact subgroup of Z • (Q ℓ ) since γ ℓ is the Frobenius automorphism of the ℓ-adic Tate module.The same argument works for γ ′ 0 and hence t lies in a compact open subgroup of G(A p f ).For ℓ = p, we have that γ 0 and γ ′ 0 both have the same image in π 1 (G) Γ since δ and δ ′ are both basic.Since the kernel of the map X * (Z • ) σ Γ0 → π 1 (G) Γ is torsion, it follows that upon further extending k, we may assume that γ and γ ′ 0 have the same image under the Kottwitz map κ : Z • (Q p ) → X * (Z • ) σ Γ0 .Thus t lies in the kernel of κ which is a compact open subgroup of Z • (Q p ).
Since G and I are inner forms (recall γ 0 is central), we may naturally consider Z • as a subgroup of I which contains the scalars G m .Then the compactness of Hence there exists m such that γ m 0 = γ ′m 0 .Upon extending k, we may assume γ 0 = γ ′ 0 .This implies γ ℓ = γ ′ ℓ .Now since x, x ′ ∈ Sh K,bas (k), there exists g ∈ G( Qp ) such that g −1 δσ(g) = δ ′ .Taking norms, we obtain g −1 γ 0 σ r (g) = γ ′ 0 = γ 0 and hence g −1 σ r (g) = 1 since γ 0 is central.This implies g ∈ G(Q p r ) and hence δ and δ ′ are σ-conjugate in G(Q p r ).It follows that t ∼ t ′ .Proposition 5.2.6 and Proposition 5.2.10 together with the Hasse principle for adjoint groups imply the following corollary.
Corollary 5.2.11.Let x, x ′ ∈ Sh K,bas (F p ). Then the groups I and I ′ are isomorphic as inner forms of G. Proposition 5.2.12.Let x, x ′ ∈ Sh K,bas (F p ). Then x and x ′ lie in the same isogeny class.
Proof.Let k = F p r be a sufficiently large finite field such that x and x ′ are both defined over k.We let I and I ′ be the groups associated to x and x ′ respectively.We let Isog(A x , A x ′ ) be the scheme of quasi-isogenies between A x ′ and A x ′ .We define P sα (x, x ′ ) ⊂ Isog(A x , A x ′ ) to be the subscheme which takes (s α,ℓ,x ) l =p (resp.s α,0,x ) to (s α,ℓ,x ′ ) ℓ =p (resp.s α,0,x ′ ).It suffices to show that P sα (x, x ′ ) is a trivial I-torsor.We first show P sα (x, x ′ ) is an I-torsor.By Corollary 5.2.11, we may fix an isomorphism I ∼ = I ′ .Let T ⊂ I ∼ = I ′ be a maximal torus.The proof of [Zho20, Theorem 9.4] shows that upon modifying x and x ′ in its isogeny class, we may assume that x and x ′ admit lifts x and x′ to Sh K (G, X)(Q) satisfying the conditions: (1) T ⊂ Aut(A x ) and T ⊂ Aut(A x ′ ) lift to T ⊂ Aut(A x) and T ⊂ Aut(A x′ ).respectively.Here Sh(T, h T ) is the Shimura variety for (T, h T ) and E T is its reflex field.We let P ⊂ Isog(A x, A x′ ) be the scheme of isogenies which respect the Hodge cycles and the action of T. We claim that P is a T-torsor; for this it suffices to show that P is non-empty.
By Proposition 5.2.6, the map is conjugate to the natural inclusion, and a similar statement holds for the map It follows that there exists g ∈ G(Q) such that gig −1 = i ′ .Since i(T) is its own centralizer in G, we have c τ = g −1 τ (g) ∈ i(T)(Q) for any τ ∈ Gal(Q/Q).Let By [Kis17, Lemma 4.4.5]applied to the inclusion T R → K ∞ , the image of c in H 1 (R, T) is trivial, and hence the image of c in H 1 (Q, I) lies in ker(H 1 (Q, I) → H 1 (R, I)).Since the image of c in H 1 (Q, G) is trivial, we have that c is trivial in H 1 (Q, I).It follows that the I-torsor P sα (x, x ′ ) is trivial.
Proof of Proposition 5.2.2.Let x ∈ Sh K,bas (F p ).We first define a natural map X µ (δ) → Sh pfn K,bas .The key input for this is the existence of such a map on F ppoints which was constructed in [Zho20, Proposition 7.7].We may then argue as in [XZ17, Lemma 7.2.12];we sketch the argument emphasizing the points which do not directly carry over to the ramified case.
As in [XZ17, 7.2.6],we may construct an abelian variety A over X µ (b) equipped with a p-power quasi isogeny A → A x × X µ (b).Moreover this quasi-isogeny equips A with tensors s ′ α,0 ∈ D(A[p ∞ ]) ⊗ , as well as a weak polarization and a prime-to-p level structure.Hence we obtain a map We claim ι ′ lifts to a unique map ι : X µ (b) −→ Sh pfn K such that for each closed point y ∈ X µ (b), we have s α,0,y = s ′ α,0,y .The uniqueness follows from [Zho20, Corollary 6.3] and the fact that two maps between perfect schemes coincide if and only if they coincide on the level of closed points.Thus it suffices to prove the lifting locally.
Let y be a closed point of X µ (b) and U ⊂ X µ (b) an affine open neighborhood containing y which is perfectly of finite presentation.We may assume U is the perfection of a reduced affine scheme U 0 ∼ = Spec R and that the quasi-isogeny A| U → A x × U comes from pullback from a quasi-isogeny A 0 → A x × U 0 over U 0 .We thus obtain a map ι ′ 0 : U 0 → S K ′ (GSp(V ), S ± ) ⊗ Z (p) F p and it suffices to show ι ′ 0 can be lifted to ι : U 0 → Sh K .We form the pullback diagram Then Y is equipped with a polarized abelian variety (A Y , λ Y ) and tensors where the s ′ α,0,Y are obtained from pullback of s ′ α,0 along Y → Spec R, and the s α,0,Y are obtained from pullback of s α,0 along Y → Sh K .We let Y • denote the union of connected components which contain an F p -point y such that s α,0,y = s ′ α,0,y .By [MP16, Lemma 5.10], s α,0,Y • = s ′ α,0,Y • .By [Zho20, Proposition 6.5 (i)], the map Y • → Spec R is bijective on F p -points and by [KP18, Proposition 4.2.2], the map Y • → Spec R is finite and is a closed immersion when completed at every point of the domain.In addition R is reduced; it follows that Y • → Spec R is an isomorphism.
The map ι induces a finite map which is bijective on closed points by [Zho20, Proposition 9.1] and Proposition 5.2.12, and is a closed immersion when completed at every closed point of the domain.It follows that ι isog is an isomorphism.

Erratum for [ZZ20]
In [ZZ20, Definition 5.2.7], the two appearances of C[Y * ] should be replaced by C. In [ZZ20, Proposition 5.5.1], the identity is between two elements of C[q −1 ].
Sh K,bas and the set a∈MVµ(λ b ) Definition 3.1.4.Let ∼ be the minimal equivalence relation on N[S H ] generated by the following rules.
. The following theorem is proved in [HN14, Theorem 2.9].Theorem 3.3.6.Let O be a σ-conjugacy class in W . Then for each w ∈ O, there exists w ′ ∈ O min such that w → σ w ′ .Definition 3.3.7.Let Wσ,min be the set of w ∈ W such that w has minimal length in its own σ-conjugacy class.We write C ( W ) for the set Wσ,min / ≈σ , and we view each element of C ( W ) as a subset of W .We denote by π the natural map C ( W ) → B( W , σ) sending C ∈ C ( W ) to the unique σ-conjugacy class in W containing C. We denote the composition of the map (2.3.4.1) with π by Ψ : C ( W ) → B(G).3.3.8.For any C ∈ C ( W ) and b ∈ G( F ), we write [[X C (b)]] for [[X w (b)]] ∈ GDL J b (F ) for arbitrary w ∈ C. By Proposition 3.3.3(1), the definition of [[X C (b)]] is independent of the choice of w.
For x ∈ Wσ,min , by [He14, Theorem 4.8] we know that X x (b) = ∅ if and only if Ψ(x) = [b], that X x (b) is equidimensional, and that the J b (F )-action on Σ top (X x (b)) is transitive.Moreover, when X x (b) = ∅, we have an explicit formula for dim X x (b) (see [He14, Theorem 4.8]), and we know that the stabilizer of each irreducible component of X x (b) in J b (F ) is a parahoric subgroup of J b (F ) with an explicit description (see the proof of [ZZ20, Proposition 3.1.4]).The upshot is that we explicitly understand the elements (Σ top (X wC (b)), dim X wC (b)) ∈ TIC J b (F ) for all C ∈ C ( W ) and w C ∈ C. Thus by Corollary 3.3.11,the determination of the J b (F )-set Σ top (X w (b)) and dim X w (b) for general w ∈ W reduces to the computation of the polynomials F w,C .It also follows that for general w, the stabilizer of each element of Σ top (X w (b)) in J b (F ) is a parahoric subgroup, cf.[ZZ20, Proposition 3.1.4].

Proposition 3.4. 6 .
Assume that G is F -simple and adjoint.Let [b] ∈ B(G, µ) be the unique basic element.Then there exists Z ∈ Σ top (X w0t µ (b)) such that Stab Z (J b (F )) is a very special parahoric subgroup of J b (F ).

Theorem 4.1. 2 .
Let µ ∈ X * (T ) + Γ0 and [b] ∈ B(G, µ).Each stabilizer for the J b (F )-action on Σ top (X µ (b)) is a very special parahoric subgroup of J b (F ).By [ZZ20, Proposition 3.1.4],we already know that each stabilizer for the J b (F )action on Σ top (X µ (b)) is a parahoric subgroup of J b (F ).In the proof of Theorem 4.1.2below we shall freely use this fact.We now deduce an immediate consequence of Theorem 4.1.2.Definition 4.1.3.Fix µ and b as in Theorem 4.1.2.We write N (µ, b) for the number of J b (F )-orbits in Σ top (X µ (b)).

Proposition 4.2. 3 .
Suppose that (G, b, w) and (G ′ , b ′ , w ′ ) are associated, and fix f : Wa ∼ − → W ′ a as in §4.2.2.Then there is a bijection the parahoric subgroups Stab Z (J b (F )) and Stab Z ′ (J b ′ (F ′ )) are associated with respect to f .Proof.By Corollary 3.3.11,the isomorphism class of the J b (F )-set Σ top (X w (b)) depends only on F w,C and the J b (F )-sets Σ top (X u (b)) for u ∈ Wσ,min .Similarly, the isomorphism class of the J b It follows that it suffices to prove the proposition for w ∈ Wσ,min .Assume that w ∈ Wσ,min .Since the maps i : W → W a and i ′ : W ′ → W ′ a are compatible with the conjugation actions and preserves the length functions, we havew ′ ∈ W ′ σ ′ ,min .By [He14, Theorem 4.8], J b (F ) (resp.J b ′ (F ′ )) acts transitively on Σ top (X w (b)) (resp.Σ top (X w ′ (b ′ )))and hence we obtain the desired bijection Θ.The "moreover" part follows from the explicit description of X w (b) in terms of finite Deligne-Lusztig varieties given in the proof [He14, Theorem 4.8], cf. the proof of [ZZ20, Proposition 3.1.4].Corollary 4.2.4.To prove Theorem 4.1.2,it suffices to prove it when char(F ) = 0 and G is an adjoint F -simple unramified group over F .
. Upon replacing b by an element of its σ-conjugacy class in G( F ), we may assume that b ∈ M ( F ) and that ν M b = ν G b (see e.g.[CKV15, Lemma 2.5.1]).Then b is basic in M .Upon further replacing b by an element of its σ-conjugacy class in M ( F ), we may assume that b = τ for some τ ∈ Ω M .
s ) denote the σ-centralizer group J (s) bs (F s ) := {g ∈ G( F ) : g −1 b s σ s (g) = b s }; then J (s) bs (F s ) ⊂ M ( F ).Since the fibers of f s b : LN → LN are torsors for J (s) bs (F s ) ∩ N ( F ) = {1}, it follows that the "pro-étale" covering f s b | L + N0 : L + N 0 → L + N 0 obtained by taking the inverse limit of the f s b,r is trivial and hence f s b | L + N0 is an isomorphism (cf.[XZ17, Lemma 4.3.4]).

Proposition 4.3. 8 .
Let λ ∈ I µ,b,M with a λ,µ = 0 and let Z ∈ Σ top (X M λ (b)).Then Stab Z (J b (F )) acts trivially on Σ top (p −1 (Z)).Proof.Let Y be an open subscheme of Z and let Y ′ be an étale cover of Y such that the inclusion map Y → X M λ (b) lifts to a map ι : Y ′ → LM .The existence of Y ′ follows from the same argument as [PR08, Theorem 1.4].Upon replacing Y ′ with an irreducible component, we may assume that Y ′ is also irreducible.

Proposition 4.4. 3 .
Assume that µ is a sum of dominant minuscule cocharacters and[b] ∈ B(G, µ) is basic.Then for any Z ∈ Σ top (X µ (b)), Stab Z (J b (F )) is a special parahoric subgroup of J b (F ).Proof.We first consider the case where µ is minuscule.Let M ⊂ G be a standard Levi subgroup such that there exists b ∈ [b] ∩ M ( F ) which is superbasic in M .We use the same notations as in §4.3.1 with respect to M .We choose b ∈ [b] ∩ M ( F ) that is superbasic in M , and upon σ-conjugating b in M ( F ) we may assume that b = τ for some τ ∈ Ω M .

Proposition 5.2. 6 .
Let k = F p r a finite extension of k v and x ∈ Sh K,bas (F p r ).(1)There existsγ 0 ∈ G(Q) which is elliptic in G(R) such that (γ 0 , (γ ℓ ) ℓ =p , δ) forms a Kottwitz triple of level r in the sense of [Kis17, §4.3.1].In particular, γ 0 is G(Q ℓ )-conjugate to γ ℓ for all ℓ (including ℓ = p).(2) For any prime ℓ (including ℓ = p), the natural map I ⊗ Q Q ℓ → I ℓ isan isomorphism, and the group (I/G m )(R) is compact.Here G m ⊂ I arises from the image of the weight homomorphism of the Shimura datum (G, X). (3) We write I 0 ⊂ G for the centralizer of γ n 0 for sufficiently divisible n such that the centralizers stabilize.Then there exists an inner twisting I ⊗ Q Q ∼ − → I 0 ⊗ Q Q which makes I an inner form of I 0 and such that the diagram

( 2 )
The Hodge filtrations on H 1 dR (A x) and H 1 dR (A x′ ) are induced by the same Tvalued cocharacter µ T .(3)If i, i ′ : T → G are the inclusions obtained by regarding T as a subgroup of the Mumford-Tate groups of A x and A x′ (these are well-defined up to G(Q)conjugacy), then x and x′ are in the images of the mapsi : Sh(T, h T ) → Sh K (G, X) E T i ′ : Sh(T, h T ) → Sh K (G, X) E T K ∞ denote the centralizer of i • h T .Then by the same argument as in [Kis17, Proposition 4.4.13], the image of c in H 1 (R, K ∞ ) is trivial.This defines a T-torsor P′ which is isomorphic to P by [Kis17, Proposition 4.2.6].Indeed the proposition in loc.cit.shows that A x′ is isomorphic to the twisted abelian variety A P′ x as in [Kis17, §4.1] equipped with its collection of Hodge cycles and action of T induced from A x.It then follows by the construction of A P′ x that P ∼ = P′ .It follows that P sα is the I-torsor obtained by pushout from the T-torsor P. By [Kis17, Lemma 4.4.3],there is an isomorphism ker(H 1 (Q, I) → H 1 (R, I)) ∼ = ker(H 1 (Q, G) → H 1 (R, G)).
Rapoport after the third-named author's talk at University of Maryland in Fall 2018.X.H. is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by Hong Kong RGC grant 14300220.R.Z. was supported by NSF grant DMS-1638352 through membership of the Institute for Advanced Study, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 804176).Y.Z. is partially supported by NSF grant DMS-1802292, and by a startup grant at University of Maryland.
[BP89,easy to see that ∆ x has non-empty intersection with each connected component of D. A point x lying in the closure a of a is said to be very special if ∆ x contains exactly one very special vertex in each connected component of D. A parahoric subgroup of G(F ) is said to be very special if it is G(F )-conjugate to a standard parahoric subgroup associated to a very special x ∈ a.Remark 2.2.3.When G is simply connected and absolutely almost simple, our definition of a very special parahoric subgroup is the same as that in[BP89, A.4].
]. Let s 1 • • • s n be a reduced word decomposition for w ∈ W v , where s i is the simple reflection corresponding to α vi for v i ∈ ∆ \ {v}.For i = 1, . . ., n, set ′ be a connected reductive group over a (possibly different) local field F ′ , let b ′ ∈ G ′ ( F ′ ), and let w ′ be an element of the Iwahori-Weyl group W ′ of G F ′ .Note that any length-preserving isomorphism of Wa to W ′ a extends in a unique way to a group isomorphism W a → W ′ a .Write σ ′ for the Frobenius in Aut( F ′ /F ), and write [b ′ .2.2.By [He14, Theorem 3.7], the set B(G) is in natural bijection with a certain subset of σ-conjugacy classes in W .By composing with the map i, we may associate to any[b] ∈ B(G) a σ-conjugacy class C [b] in W a .Let G