On a decomposition of p -adic Coxeter orbits

. We analyze the geometry of some p -adic Deligne–Lusztig spaces X w ( b ) , introduced by the author in a paper in Crelle, attached to an unramiﬁed reductive group G over a non-archimedean local ﬁeld. We prove that when G is classical, b basic and w Coxeter, X w ( b ) decomposes as a disjoint union of translates of a certain integral p -adic Deligne–Lusztig space. Along the way we extend some observations of DeBacker and Reeder on rational conjugacy classes of unramiﬁed tori to the case of extended pure inner forms and prove a loop version of the Frobenius-twisted Steinberg cross section.


Setup
Let k denote the completion of a maximal unramified extension of k, and let O k , O k be the rings of integers of k, k.Let σ be the Frobenius of k/k.Let T ⊆ B ⊆ G be a centralizer of a maximal split k-torus and a k-rational Borel subgroup of the unramified group G. (Note that T is a k-rational maximal torus of G.) Let U, resp.U − , be the unipotent radical of B, resp. the opposite Borel subgroup.Let W be the Weyl group of T and S ⊆ W the set of simple reflections determined by B. Denote by B G, k the Bruhat-Tits building of the adjoint group G ad of G over k and by A T, k the apartment of T therein.Denote by σ the action induced by Frobenius on W , S, B G, k , A T, k and G( k) (as well as its subgroups).
An element b ∈ G( k) is called basic if its Newton point ν b factors through the center of G; cf.[Kot85].Twisting σ by Ad(b), one gets an inner k-form G b of G, see Section 2.1.3.Similarly, for w ∈ W , twisting the Frobenius on T( k) by Ad(w), one gets an (outer) k-form T w of T.
The space X w (b), introduced in [Iva23], is the intersection of LO(w) ⊆ L(G/B) 2 with the graph of bσ : L(G/B) → L(G/B), where L denotes the loop functor (see Section 2.1.1).Similarly, we have its coverings Ẋ w(b), where w lies in F w / ker κw , a certain discrete quotient of the set F w ( k) ⊆ G( k) of all lifts of w; cf.Sections 3.4 and 4. For more details on X w (b) and Ẋ w(b), see Section 2.2 and Definition 4.1.The group G b (k) acts on X w (b), and G b (k) × T w (k) acts on Ẋ w(b).The natural map Ẋ w(b) → X w (b) is G b (k)-equivariant.
An element c ∈ W is called twisted Coxeter, or simply Coxeter, if any of its reduced expressions contains precisely one element from each σ -orbit on S; cf.[Spr74,Section 7].Usually, we denote by w an arbitrary element of W , whereas the letter c is reserved for Coxeter elements.
We refer to Section 2 for more notation and setup.

Main result
Let c ∈ W be Coxeter, c ∈ F c / ker κc a lift of c and b ∈ G( k) basic.We prove a decomposition result for X c (b) and Ẋc (b).The space X c (b) is always non-empty, and-after an equivariant isomorphism-one can assume that c is a special Coxeter element (see Definition 5.2) and b is a lift of c.Moreover, in Proposition 6.1, we will see that either Ẋc (b) = ∅, or-after an equivariant isomorphism-we may even assume that b lies over c.In the rest of Section 1.2 we work under these simplifying assumptions (which do not result in any loss of generality).
As b lies over c, the affine transformation bσ of A T, k has precisely one fixed point x (this follows from [Spr74,Lemma 7.4] where the lower map is g → g −1 σ b (g) and where ( c U ∩ U − ) x is the closure of c U ∩ U − in G x .We also consider the quotient for the pro-étale (equivalently, arc-)topology, (1.2) X After necessary preparations, Theorem 1.1 will be proven in Sections 6 and 7. Let us outline the proof.By a v-descent argument, (1.3) follows from a particular case of (1.4) for absolutely almost simple adjoint groups.Showing (1.4) in this case presents the main technical difficulty, which is solved in Section 7 by using a special property of the Newton polygons of isocrystals and analyzing each irreducible Dynkin type separately (here the assumption that c is Coxeter becomes crucial).Finally, (1.4) can be deduced in general from (1.3).We expect Theorem 1.1 to hold for all unramified reductive groups G.
At this point, let us stress that p-adic Deligne-Lusztig spaces are quite different in nature from affine Deligne-Lusztig varieties (say, at Iwahori level).The latter were introduced by Rapoport in [Rap05].They are important because their unions appear as special fibers of Rapoport-Zink spaces.Note, for example, that whereas affine Deligne-Lusztig varieties are (perfect) schemes locally of finite type over F q , the padic Deligne-Lusztig spaces are essentially never of finite type over F q .On the other hand, there is, remarkably, some relation between these two types of objects.In [CI21a] it was shown that the inverse limit of a certain system of affine Deligne-Lusztig varieties (of increasing level) for the group GL n is isomorphic to a p-adic Deligne-Lusztig space Ẋc (b) for the same group and a Coxeter element c.We expect this result to generalize to other groups, especially in light of the similarity of the decomposition of Coxeter-type spaces X c (b) provided by Theorem 1.1 and a very similar-looking decomposition result of He-Nie-Yu [HNY22, Theorem 1.1(3)] for affine Deligne-Lusztig varieties attached to elements of the affine Weyl group with finite Coxeter part.
Let us also note that Theorem 1.1 implies [Iva23, Conjecture 1.1] in the following cases.

Quasi-split case
If b is σ -conjugate to 1, Theorem 1.1 admits a clearer reformulation.Let c ∈ W be Coxeter and c ∈ F c / ker κc .If Ẋc (1) ∅, then a (i.e., any) lift ċ ∈ G( k) of c is σ -conjugate to 1 (by Corollary 4.6).Moreover, choosing ċ appropriately, we may additionally assume that the (unique and automatically hyperspecial) fixed point x = x ċ of ċσ on A T, k lies in A T,k = A ⟨σ ⟩ T, k (cf.Section 6.4).In this situation we have the integral level analogues X G x c (1) and ẊG x ċ (1) of X c (1) and Ẋc (1), which were introduced in [DI20, Definition 4.1.1];see Section 2.2.In this situation we have the following corollary, proven in Section 6.4.

Two related results
In the rest of the introduction we briefly discuss two further themes of this article.First, we review in Section 3, following DeBacker and Reeder [DeB06,DR09,Ree11], the parametrization of rational conjugacy classes of unramified maximal k-tori in pure inner forms of G.Moreover, in Sections 3.3-3.4we extend their discussion to all extended pure inner forms and give a description of rational conjugacy classes in terms of F w / ker κw , the discrete parameter space for the spaces Ẋ w(b) over X w (b).In Section 3.5 we investigate in detail the Coxeter case , where (under some mild assumptions on G) it turns out that the set of different covers Ẋc (b) of X c (b) for b basic is in canonical bijection with the set of rational conjugacy classes of unramified Coxeter tori in the inner form G b of G; see Corollary 4.7.Thus the p-adic Deligne-Lusztig space X c (b) "can see" the difference between the various rational conjugacy classes of tori whose stable class is given by c.
Secondly, in Section 5 we show a loop version of the twisted Steinberg cross section.Steinberg's original result [Ste65,Proposition 8.9] states roughly that if ċ ∈ F c ( k) is any lift of a Coxeter element c ∈ W , any c U ∩ U-orbit on c U for the action given by g(u) = g −1 u Ad( ċ)(g) contains precisely one point in c U ∩ U − (k can be replaced by any field here).In [HL12, Section 3.14(c)] this is generalized in various respects, in particular to the case where k is replaced by a finite field F q and the action is Frobenius-twisted, i.e., given by g(u) = g −1 uσ ċ(g), where σ is the (geometric) Frobenius over F q and σ ċ = Ad( ċ) • σ .This can be used to prove affineness of classical Deligne-Lusztig varieties attached to Coxeter elements (for example, by combining [DL76, Corollary 1.12] and [HL12, Section 4.3]).In our setup, with k non-archimedean local, (to perform a reduction step in the proof of Theorem 1.1) we need the same result for the action of L( c U ∩ U) on L( c U).A part of the proof in [HL12] can be adapted, but at some point in [HL12] the Ax-Grothendieck theorem (2) is used, which is impossible in our setting (as LU is not even a scheme).Therefore, we carry through a different argument, at least for classical groups and special Coxeter elements.

Outline
In Section 2 we introduce more notation, prove a fact about integral p-adic Deligne-Lusztig spaces and review the Borovoi fundamental group and σ -conjugacy classes.In Section 3 we review (and extend) some results on rational/stable conjugacy classes of unramified maximal tori of G.In Section 4 we review and extend some results from [Iva23] about the maps Ẋ w(b) → X w (b).In Section 5 we prove loop (and integral loop) versions of the twisted Steinberg cross section.This is used in Section 6 to replace Ẋc (b) (and its integral analogue) by a more tractable object.The main part of the article consists of Sections 6 and 7, where Theorem 1.1 is proven.

A. B. Ivanov 6
A. B. Ivanov Let p be the residue characteristic of k.For a ring R of characteristic p, denote by Perf R the category of perfect algebras over R. Denote by ϖ a uniformizer of k and by F q (resp.F q ) the residue field of k (resp.k).
For R in Perf F q , let W(R) be the unique (up to unique isomorphism) ϖ-adically complete and separated O k -algebra in which ϖ is not a zero-divisor and which satisfies W(R) /ϖW(R) = R.In other words, if W (R) denotes the p-typical Witt-vectors of k, then W(R) = W (R) ⊗ W (F q ) O k if char k = 0 and W(R) = R [[ϖ]] if char k = p.Denote by φ = φ R the Frobenius automorphisms of W(R), W(R)[1/ϖ], induced by x → x q on R (this notation is only used in the proof of Proposition 5.3 and Section 7).We have Denote by k alg a fixed algebraic closure of k containing the maximal algebraic subextension of k/k.Denote by ⟨σ ⟩ Ẑ the group of continuous automorphisms of k/k.

Loop functors.
For a k-scheme X, let be the loop space of X.If X is affine of finite type over k, then LX is representable by an ind-scheme.For an O k -scheme X , let be the space of positive loops of X .Moreover, if n ≥ 1, let n X is a scheme (resp.affine scheme) for arbitrary (resp.affine) X , and if X is of finite type, then L + X is too.Moreover, L + X = lim ← − −n L + n X , along affine transition maps (cf.[BGA18, Proposition 8.5]).Thus, if X is a scheme (resp.affine scheme), then L + X is too.
If X is the base change to k of a k-scheme X 0 , then the functor LX carries a natural automorphism, the geometric Frobenius, denoted by σ X 0 (or simply by σ if X 0 is clear from context).Similarly, we have the geometric Frobenius σ = σ X 0 on L + X if the O k -scheme X is the base change of an O k -scheme X 0 .

Groups.
We write G ad , resp.G sc , for the adjoint group, resp.the simply connected cover of the derived group, of G.If H ⊆ G is a closed subgroup of G, denote by H ad , resp.H sc , its image in G ad , resp.its preimage in G sc .We identify the Weyl groups of G ad and G sc with W .We denote by Z the center of G.
Write N for the normalizer of T. We have the reductive O k -model T of T, and

Twisting Frobenius.
For b ∈ G( k), we have the k-group G b , which is defined as the functorial σ -centralizer of b; cf.[RZ96, 1.12] or [Kot97,3.3Suppose that the basic element b lies in N( k) and that x ∈ A T, k is fixed by the action of bσ .Let G x be the corresponding parahoric model of Similarly, for w ∈ W , we have the (outer) k/k-form T w of T, satisfying T w (k) = {t ∈ T( k) : Ad w(σ (t)) = t}.We may identify T w ( k) = T( k), and we denote by σ w := Ad(w) • σ the twisted Frobenius on this group, as well as on LT.

Topologies.
We work with two topologies on Perf F p : the pro-étale topology, cf.[BS15], and the arc-topology, cf.[BM21].Recall that a map S ′ → S of qcqs schemes is an arc-cover if any immediate specialization in S lifts to S ′ .By [BM21, Theorem 5.16], the arc-topology on perfect qcqs F p -schemes coincides with the canonical topology.Thus any perfect qcqs scheme is an arc-sheaf.In particular, for any qcqs scheme X /O k , L + X is an arc-sheaf.For any quasi-projective scheme X/k, LX is an arc-sheaf by [Iva23, Theorem A].

Locally profinite groups.
As in [Iva23, Section 4], we regard a locally profinite group H as a (representable) functor S −→ C 0 (|S|, H) on Perf F q .In particular, G b (k) and T w (k) are such functors.If H is profinite, this functor is even represented by an affine scheme.

Galois cohomology.
For the definition and standard facts about non-abelian cohomology, we refer to [Ser94].If M is a discrete ⟨σ ⟩-group, where ⟨σ ⟩ is as in Section 2.1, we write H 1 (σ , M) for the corresponding first cohomology set.We identify a continuous 1-cocycle a : ⟨σ ⟩ → M with a(σ ) ∈ M and write [a(σ )] ∈ H 1 (σ , M) for the corresponding cohomology class.

Spaces
Lemma 2.1.Let {H n } n≥1 be an inverse system of finite groups, acting compatibly and freely on an inverse system {X n } n≥1 of pro-étale (resp.arc-)sheaves on is an isomorphism, where the quotients are formed for the pro-étale (resp.arc-)topology.
Proof.Consider the case of pro-étale topology.The existence of the map is easy.For the injectivity, it is enough to show that the map from the presheaf quotient Then for each n, the images s 1,n , s 2,n of s 1 , s 2 in X n (R)/H n (R) coincide (as any quotient presheaf is automatically separated).Thus for each n ≥ 1, S n = {h n ∈ H n (R) : h n s 1,n = s 2,n } ∅.This set forms an inverse system.By the freeness of the action, we have #S n = 1, and it follows that lim This proves the injectivity.For the surjectivity, let s n ∈ X n (R)/H n (R) be a compatible collection of elements.For each n, find some pro-étale cover we may replace each R n with R ′ and sn with sn | R ′ .Now, we construct an element s ∈ X(R ′ ) mapping to (s n ) n≥1 .Starting with s1 if n = 1, and working by induction on n, suppose that (s i ) n i=1 is compatible.As the system (s n ) n≥1 is compatible, we can find some h n+1 ∈ H n+1 (R ′ ) such that h n+1 sn+1 maps to sn under X n+1 (R ′ ) → X n (R ′ ).Then replace sn+1 by h n+1 sn+1 , so that (s i ) n+1 i=1 becomes compatible.We conclude by induction.The argument for the arc-topology is the same, using Proof.The claim for ẊG x c,b follows from the definition and the affineness of L + X for an affine scheme X /O k ; cf.Section 2.1.1.Suppose that G is adjoint.For n ≥ 1, let T n = T ad c (O k /ϖ n ) be as above.Note that the finite group T n acts freely on the affine scheme ẊG c,b,n and we have As T n is finite and acts freely on ẊG c,b,n , this is even an étale torsor.From this, it follows that ( ẊG c,b,n /T n ) fppf = ( ẊG c,b,n /T n ) ét , the quotient for the étale topology.In particular, the latter is an affine perfect scheme, hence already a pro-étale and even an arc-sheaf.This implies that the sheafification of the quotient for the arc-and pro-étale topologies agree, as well as the fact that X G c,b , being the inverse limit of affine perfect schemes, is itself one.This is easily verified by hand for G = GL n , using the explicit description in [CI21a].

Fundamental group and the Kottwitz map
The natural map T sc → T induces an injection X * (T sc ) → X * (T), and we identify X * (T sc ) with its image.The fundamental group of G in the sense of Borovoi [Bor98] is the group It is independent of the choice of T and functorial in G, the induced W -action on it is trivial, and it admits a Frobenius action, which we denote by σ .Kottwitz constructed a homomorphism κG : G( k) → π 1 (G); see [Kot97, Section 7] and [PR08, Section 2.a.2].Reformulated, a classical result of Kottwitz [Kot84, Section 6] states that κG induces an isomorphism H 1 (σ , G) π 1 (G) ⟨σ ⟩,tors .For tori, we have π 1 (−) = X * (−).For any w ∈ W , any k-rational embedding T w → G induces the map which is obtained from X * (T) → π 1 (G) by taking σ w -(resp.σ -)invariants and passing to the torsion subgroup.From the definition of π 1 (G) we deduce an exact sequence (2.1) In particular, Remark 2.5.If G is adjoint and w Coxeter, then β w = 0 and (equivalently) X * (T) ⟨σ w ⟩ π 1 (G) ⟨σ ⟩ .For split groups, this follows from [Spr74, Corollary 8.3].In the non-split case, note that there is some n ≥ 1 such that σ n w = σ w spl as automorphisms of X * (T) and X * (T sc ), where w spl is some (non-twisted) Coxeter element in W and σ w spl := Ad(w spl ) • σ n is formed with respect to the base change of G to the unramified extension of k of degree n.Then, as automorphisms of X * (T) and of X * (T sc ), σ w spl − 1 factors through σ w − 1, so that we have a surjection X * (T) ⟨σ w ⟩ ↠ X * (T) ⟨σ w spl ⟩ and a similar surjection for X * (T sc ).These surjections form a commutative square with the maps β w and β w spl (as both are induced by the inclusion X * (T sc ) → X * (T)).As β w spl = 0 by the split case, it follows that also β w = 0.
By [PR08, Section 5] we may reinterpret κG as a map of ind-schemes κG : LG → π 1 (G), where we regard π 1 (G) as a discrete perfect scheme via Section 2.1.5.Passing to σ -coinvariants gives a surjective map Finally, let us note that for w ∈ W , we have the exact sequence where we write κ w instead of κ T w .Moreover, κ w factors through the natural projection κw : X * (T) → X * (T) ⟨σ w ⟩ .

Newton map and σ -conjugacy classes
Let f ∈ Perf F q be an algebraically closed field, so that L = W(f)[1/ϖ] is an extension of k.Then L/k is equipped with the Frobenius σ lifting that of f/F q and L σ = k.In [Kot85] the set where the lower map is the restriction of κ G to B(G) bas .We recall how to compute the Newton point ν b of b ∈ N(L); see e.g.[RV14, Remark 2.3(iii)].Let w ∈ W be the image of b under N(L) ↠ W .There is an integer d > 0 such that σ d acts trivially on W and such that µ = wσ ( w) . . .σ d−1 ( w) ∈ X * (T).Then ν b is the image of d −1 µ in W \X * (T) Q .In particular, ν b only depends on w, and it makes sense to define ν w = ν b .Moreover, we call w basic if ν w is central.

Rational and stable conjugacy classes of unramified tori
Recall that two maximal k-tori T 1 , T 2 in G are stably conjugate if there exists a g ∈ G(k alg ) such that g (T 1 (k)) = T 2 (k).If two maximal k-tori are rationally conjugate, they are also stably conjugate.Thus any stable conjugacy class breaks up into a (finite) union of rational conjugacy classes.We review from [DeB06,DR09,Ree11] how this happens for unramified tori of pure inner forms of G. Then we investigate the case of extended pure inner forms of G.We wish to point out that the description from [DeB06] in terms of the Bruhat-Tits building covers this case, but we are interested in a slightly different description, which is closely related to the spaces Ẋ w(b).

Stable conjugacy classes
Replacing g by g v for any lift v ∈ N( k) of some v ∈ W has the effect of replacing w by v −1 wσ (v).

Rational conjugacy classes for extended pure inner forms
We wish to extend (3.2) to all extended pure inner forms.We have where ν w is as in Section 2.4.In other words, Z 1 (σ , W ) is precisely the set of all basic elements in W in the sense of Section 2.4.Define an equivalence relation on Z 1 (σ , W ) by x ∼ y if and only if there exists a g ∈ W such that y = g −1 xσ (g).Put H 1 (σ , W ) = Z 1 (σ , W )/ ∼ .Note that we have H 1 (σ , W ) ⊆ H 1 (σ , W ) and Z 1 (σ , W ) ⊆ Z 1 (σ , W ) is precisely the set of those w with ν w = 0. Recalling from (2.6) that H 1 (σ , G) ⊆ B(G) bas , we have the following lemma. (4)In fact, [Ree11] works with Gal k -cohomology and all maximal k-tori.To deduce our statements, one can run literally the same arguments with ⟨σ ⟩ replacing Gal k and unramified k-tori replacing all k-tori.
, and letting ṽ ∈ W be its image, we compute

A description of π
We regard F w as a k-scheme and F w as a discrete F q -scheme (or, sometimes, as a discrete set).Then F w is a (trivial) T k -torsor and F w is a (trivial) X * (T)-torsor.The action of T k on F w is given by t, w → t w, and the action of X * (T) on F w is induced by this.Recall the maps κ w : LT → X * (T) ⟨σ w ⟩ and κw : Proof.As LF w (resp.F w ) is a trivial LT-torsor (resp.X * (T)-torsor), everything follows from LT/ ker κ w = X * (T) / ker κw = X * (T) ⟨σ w ⟩ .□ Observe that for w ∈ F w , the values of ν w and κG ( w) remain constant if w is multiplied by an element in ker κw .Indeed, for κw this is clear, and for ν this follows from the description in Section 2.4.In particular, κG | F w induces a map Moreover, from Section 2.4 we immediately deduce the following lemma.
ker κw in a natural way, and there are canonical isomorphisms where the first equivalence follows from Remark 3.2.To show the first claim of (i), we may assume that there is an element , with β w as in (2.1).Now we use the algorithm to compute ν from Section 2.4.Note that the integer d can be chosen the same for w1 and τ w1 .Then let µ w1 and µ τ w1 be as described in Section 2.4.Then the equality ν τ w1 = ν w1 is equivalent to the existence of some v ∈ W such that the equality holds in X * (T).But w1 is basic, so µ w1 is central; i.e., the right-hand side is zero.
This preimage contains ker κw , and factoring it out, we arrive at the claim.This finishes the proof of (i).
In (ii), it is clear that we have a map ), and one checks that it factors through [b] ∩ F w / ker κw .Moreover, surjectivity of this map amounts to the tautological fact that any element in , and let ṽ ∈ W be any preimage.Then the action of v sends w to the image of ṽ−1 wσ ( ṽ) in [b] ∩ F w / ker κw .This is independent of the choice of ṽ, as if ṽ′ is another choice, putting τ = ṽ−1 ṽ′ ∈ X * (T) gives the element ṽ′−1 wσ ( ṽ′ ) = τ −1 ṽ−1 wσ ( ṽ)σ (τ), which has the same image in [b] ∩ F w / ker κw .Similarly, one shows that it is independent of the choice of w.Now, w′ ∈ [b] ∩ F w / ker κw has the same image as w in π −1 ([b]) ∩ q −1 ([w]) if and only if there is some ṽ ∈ W with ṽ−1 wσ ( ṽ) = w′ for some lift w′ ∈ F w of w′ .By the definition of the C W (wσ )-action, this is equivalent to the fact that w and w′ lie in the same C W (wσ )-orbit.Thus C W (wσ )-orbits in [b] ∩ F w / ker κw coincide with fibers of the surjection onto ) and the G b (k)-conjugacy class of g T is the image of w under the above map.

The case of Coxeter tori
We keep the setup of Sections 3.3 and 3.4.Furthermore, throughout Section 3.5 we assume that bas and c ∈ W be Coxeter.Then the following hold: ) is finite.For (ii) we first record the following easy and well-known lemma.
Moreover, as this holds for any b ′ in the preimage, we even get the equality claimed in (ii).Now, ( κc ker κc is trivial, and there is a natural bijection Proof.Choose some σ -equivariant splitting ι : W → W (which exists, as G is unramified), and denote its image by In case (i), we have Thus, as ker κsc c ⊆ ker κc , it suffices to show that we have In other words, we are reduced to the case that G is simply connected, which is covered by (ii).
In case (ii), one is immediately reduced to the case that This lies in ker κc as, setting Observe that σ c (ker κc ) = ker κc (as σ c is an automorphism and for any v Proposition 3.13 becomes false without any assumptions on G, b, as the next example shows. Example 3.14.Suppose char k 2. Let G ′ = SL 2 × SL 2 and G = G ′ /µ 2 , embedded diagonally.Let T ′ ⊆ G ′ be the product of diagonal tori in SL 2 and T its image in G. Let ε 1 = ((1, −1), (0, 0)), ε 2 = ((0, 0), (1, −1)) be a Z-basis of X * (T ′ ).We may identify Let s be the image of 0 1 −1 0 in the Weyl group of the diagonal torus in SL 2 , so that c = (s, s) represents a Coxeter element in W .Let ċ0 = 0 1 −1 0 , 0 1 −1 0 ∈ F c ( k) be a hyperspecial lift, and let c0 be its image in F c .We have by Galois descent.The action of the Frobenius σ on all objects below is trivial, so we ignore it.We have π 1 (G) = X * (T) /X * (T ′ ) Z/2Z, the non-trivial element being the class of ε 1 +ε 2 2 .It is represented by the basic element b 1 = ċ0 τ 1 .We have This is in agreement with Proposition 3.13 and also with the Bruhat-Tits picture: G ad PGL 2 × PGL 2 , so an alcove of B G,k is a square whose vertices are hyperspecial of type (α, β) with α, β ∈ Z/2Z the parity of the valuation of the determinant of the lattice representing a vertex of B PGL 2 .The action of τ 1 ∈ G(k) identifies the vertices (0, 0) ↔ (1, 1) and (0, 1) ↔ (1, 0), so that there are two G(k)-orbits in hyperspecial vertices of a base alcove and hence (by [DeB06]) two rational conjugacy classes of Coxeter tori.
, the C W (c)-action is non-trivial.This agrees with Bruhat-Tits theory as the adjoint building of G b 1 over k is a point (it is the barycenter of an alcove of B G,k ), so that there is just one G b 1 (k)-conjugacy class of Coxeter tori.

The maps Ẋ w(b) → X w (b)
Here we review and extend some results from [Iva23,Section 11].By [Iva23,Lemma 11.5] the space Ẋ ẇ(b) depends-up to a G b (k) × T w (k)-equivariant isomorphism-only on the image of ẇ in the discrete set LF w / ker κ w and not on ẇ.As the concrete choice of ẇ never plays an important role, we make the following simplifying definition.Recall from Lemma 3.5 that LF w / ker κ w = F w / ker κw .
Proof.Combine [Iva23, Propositions 11.4 and 11.9] with Lemma 3.5.□ We have the following emptyness criterion for Ẋ w(b), i.e., a constraint on the image of α w,b .Recall the map κw : , and let g ∈ G(L) be a lift of ḡ.Then g −1 bσ (g) ∈ U(L) ẇU(L).We clearly have κ G (g −1 bσ (g)) = κ G (b). Furthermore, any element of U(L) can be conjugated to some Iwahori subgroup, and the Kottwitz map is trivial on Iwahori subgroups.Thus κ G (U) = 1.Thus κ G (U(L) ẇU(L)) = κ G ( ẇ), and the first claim follows.The last claim follows from the first and Proposition 4.2.□ Recall the map β w : X * (T sc ) ⟨σ w ⟩ → X * (T) ⟨σ w ⟩ from (2.1).We have the following diagram with compatible simply transitive actions: where the actions are induced by the left multiplication action of X * (T) on F w .There is a canonical surjection from the set of different non-empty pieces In particular, if G, b satisfy the assumptions of Proposition 3.13, then this surjection is a 1-to-1 correspondence.
Proof.The surjectivity of the map α c,b in the corollary follows (for example) from Theorem 1.1.The rest immediately follows from Propositions 3.7, 3.10 and 3.13 □ We finish this section with the following remark on the non-basic case.

A variant of Steinberg's cross section
By the Dynkin diagram Dyn(G) of the unramified group G, we mean the Dynkin diagram of the split group G k endowed with the action induced by the Frobenius σ .There is a unique decomposition into σ -stable subdiagrams Dyn(G) = i Γ i such that σ cyclically permutes π 0 (Γ i ) for each i.For each i, let Γ i,0 be a connected component of Γ i , so that Γ i = d i j=1 σ j (Γ i,0 ).Then (Γ i,0 , σ d i ) is a connected Dynkin diagram of a group over k d i , an unramified extension of k of degree d i .
Definition 5.1.We say that the unramified group G is of classical type if each connected component (Γ i,0 , σ d i ) as above of Dyn(G) is of one of the following types: Although Theorem 1.1 and its corollaries essentially hold for all Coxeter elements (cf. the discussion preceding Theorem 1.1), it will be convenient to work with particular ones.Therefore, we introduce the following auxiliary notion.Definition 5.2.Let G be of classical type.Let c ∈ W be a Coxeter element.We call c special in the following cases: (5) given in (7.5) (resp.(7.7), (7.9), (7.13), (7.18), (7.26) we may identify the Weyl group W of G with i=1 is special if all c i are special.
(5) The explicit shape of these elements only becomes important when everything boils down to explicit calculations in Section 7.For the sake of better readability, we postpone the explicit description until the relevant notation is introduced.

A loop version of Steinberg's cross section
For classical groups and special Coxeter elements, we now prove the existence of certain Steinberg cross sections in our setup.This is a Frobenius-twisted loop version of [Ste65, Proposition 8.9] and [HL12, Theorem 3.6 and Section 3.14], which will be an important ingredient in the proof of Theorem 1.
is an isomorphism.The injectivity of α b holds for arbitrary elements c ∈ W which are of minimal length in an elliptic σ -conjugacy class (cf.[HL12] ).
Furthermore, α b restricts to an isomorphism Finally, change the variables by putting x = x −1 2 and y 1 = y −1 2 , and set b := σ −1 (b) −1 with image c in W .We are reduced to showing that L(U∩ σ ( c R .We now show the surjectivity of α b .We may assume that G is of adjoint type.Then G = r i=1 G i with G i almost simple k-groups, and we are immediately reduced to the case that G is almost simple over k.Proof.Identify G k = d i=1 G ′ k and similarly for the Weyl groups W , W ′ and unipotent radicals U, U ′ of Borels of G, G ′ , so that the Frobenius σ permutes cyclically the components.As c is special, c = (c ′ , 1, . . ., 1) for a special Coxeter element c ′ ∈ W ′ .We then have Then the map α b from Proposition 5.3 is given by Proof.This follows immediately from a computation.□ As G splits over k, Lemma 5.5 reduces us to the case that G is absolutely almost simple over k.It suffices to show that the map α b from Lemma 5.6 is surjective.
Lemma 5.7.There exist a positive integer r and subsets satisfying the following conditions: Proof.We give the desired partitions in Section 7.1.4(type A n−1 ), Section 7.2.4 (types B m , C m ), Section 7.4.9(type D m ), Section 7.5.7 (type 2 A n−1 ) and Section 7.6.4(type 2 D m ) after the relevant notation is introduced.
In each case, it is straightforward to check that the conditions of the lemma hold for the given partition; we omit this checking.□ Lemma 5.8.Let I be a finite set not containing 0. Let λ : is surjective for the pro-étale topology.
Proof.There are some integers s ≥ r ≥ 0 and a disjoint decomposition I = r i=1 I i ∪ s i=r+1 I i such that for 1 ≤ i ≤ r, λ restricts to a map I i → I i which is a full cycle of length #I i (first type), and for r + 1 ≤ i ≤ s, λ restricts to a map I i → I i ∪{0}, which is, after choosing an isomorphism I i {1, . . ., m}, given by λ(j) = j − 1 for all j ∈ I i (second type).It suffices to show that the restriction of δ to j∈I i LG a is surjective.Thus we may assume that I is one of the I i .If I = {1, . . ., m} is of the second type, one has the filtration LG a by closed subschemes (ind-group schemes) LG a , stable by δ, such that x → x − δ(x) induces the identity on the associated graded object gr G • .From this, the result easily follows.If I is of the first type, we may assume I = Z/mZ and λ(i) = i + 1. Replacing φ with φ m , one is then immediately reduced to the case that m = 1 and δ : LG a → LG a , δ(x) = εφ(x) for some ε ∈ k× .Then Lemma 5.9 finishes the proof.□ Lemma 5.9.Let ε ∈ k.Then the map id −εφ : LG a → LG a , x → x − εφ(x) is surjective for the pro-étale topology. Proof.
LG a (as W(R) is ϖ-adically complete for each R ∈ Perf) and the inverse of id −εφ.If ord ϖ (ε) < 0, a similar argument goes through using id −εφ = −εφ(id −(εφ) −1 ).Now assume ε ∈ O × k .We have LG a = lim − − →m→−∞ ϖ m L + G a , and it suffices to show that id −εφ is surjective on This follows from a pro-version of Lang's theorem. □ We now prove the surjectivity of α b .For a subset Ψ ⊆ Φ + closed under addition, let U Ψ denote the unique closed subgroup of U whose k-points are generated by {U α ( k)} α∈Ψ .As a k-scheme, U Ψ is isomorphic to α∈Ψ U α .Write U Ψ = LU Ψ .Let {Ψ i } r i=1 be a filtration as in Lemma 5.7.For 1 ≤ i ≤ r, we have the closed subschemes (ind-group schemes)

and condition (ii) implies that A i+1 is normal in A i and that the inclusion
, which is of the form given in Lemma 5.8.From Lemma 5.8 it follows that a → a , and let xi be its image in (C i /C i+1 )(R).By the above, after replacing R with a pro-étale cover, we may find some In other words, for each x i ∈ C i (R), we may find (after possibly enlarging R) some a i ∈ A i such that α b (a i , x i ) ∈ C i+1 .Starting with an arbitrary x = x 1 ∈ C(R) and applying this procedure consequently for i = 1, 2, . . ., r − 1, we deduce that after possibly enlarging R, there are some a 1 , . . ., a r−1 ∈ A(R) such that for a = a r−1 a r−2 . . .a 1 , we have α b (a, x) ∈ C r (R) = B(R).(Here we use that α b (b, α b (a, x)) = α b (ab, x).)This finishes the proof that α b is an isomorphism.
Next we turn to α b,x .Note that it is well defined as for ), the intersection taken in LG.Being the restriction of α b , it is injective, as α b is, and it remains to prove its surjectivity.Applying Ad(b −1 ), which conjugates G x to G y , where y = b −1 (x), and the analogue of Lemma 5.6, it suffices to show that (with the obvious notation) the map α b,x : ) is surjective.By the same reasoning as in Lemma 5.5 (and the paragraph preceding it), we are reduced to the case that G is absolutely almost simple over k.For Ψ ⊆ Φ + , let U Ψ ,y denote the closure of U Ψ in G y .We have U Ψ ,y α∈Ψ U α,y .Replacing C i , B, A i above with ,y and using the integral version of Lemma 5.8 (with essentially the same proof), we can run the above inductive argument for the quotients A + i /A + i+1 , deducing the surjectivity of α b,x .□

A variant of a Deligne-Lusztig space
Throughout this section we work with the arc-topology on Perf LG and Xw,b LG, and let X w,b := Xw,b /LT.
Remark 5.11.The map g → g −1 σ b (g) : Proof.(i) Consider the sheafification Ẋ′ of the functor on Perf F q , By [Iva23, Proposition 12.1] X ċ(b) and Ẋ′ are equivariantly isomorphic.It remains to show that Ẋ′ Ẋ ċ,b equivariantly.Note that Ẋ ċ,b is the sheafification of There is an obvious equivariant map X 2 → X 1 , of which we claim that it is an isomorphism.This follows once we check that is an isomorphism.Substituting y ′ = y ċb −1 , with y ranging in L( c U ∩ U − ), this map becomes (x, y) → x −1 y ċσ (x) ċ−1 • ċb −1 , which is an isomorphism by Proposition 5.3.The proof of (ii) is completely analogous.□ Proposition 5.13.Suppose G is of adjoint type.Let c ∈ W be as in Proposition 5.12 and b ∈ F c ( k).For any Proof.By Proposition 5.12, it suffices to show that Ẋ ċ,b /T c (k) X c,b .Writing ), these sheaves are the sheafifications of the quotient presheaves and respectively.We have an obvious map Y 1 → Y 2 , and we claim its sheafification is an isomorphism.First, the map between presheaves itself is injective (which already implies that its sheafification is injective, as quotient presheaves are separated).Indeed, let g, h ∈ Y 1 (R).Then σ b (g) = gu g τ, σ b (h) = hu h τ for some u g , u h ∈ L(U − ∩ c U)(R).Suppose their images in Y 2 (R) agree, i.e., there is some t ∈ LT(R) such that gt = h.Then σ b (g)σ b (t) = σ b (h), and hence gu g τσ b (t) = hu h τ; i.e., Changing g to gt for some t ∈ LT(R) has no effect on the image of g in Y 2 (R) but amounts to replacing τ ′ with τ ′ t −1 σ (t).Thus, to show that Y 1 → Y 2 is surjective when passing to sheafification, it remains to check that the σ c -twisted action morphism is surjective for the arc-topology.In fact, it is even surjective for the pro-étale topology.Indeed, the space ), all fibers being isomorphic to L + T .Using pro-Lang for the σ c -stable connected subscheme L + T ⊆ LT, we are reduced to showing that the map of discrete abelian groups X * (T) × ( κG , that the fibers of κG | X * (T) are torsors under im(σ c − id : X * (T) → X * (T)) = ker κc .This holds by Remark 2.5 as G is adjoint and c Coxeter. □

Decomposition into integral p-adic spaces
In this section we begin with the proof of Theorem 1.1.Before that, we show that this theorem is indeed sufficiently general to describe all spaces X c (b) and Ẋc (b) for an unramified classical group G, c Coxeter and b basic.• a basic element b which lies in F c ( k) and lifts c.In Section 6.1 we reduce (1.3) to the case that G is absolutely almost simple and adjoint.In Section 6.2 we then prove (1.3) by a v-descent argument, assuming the technical Proposition 6.3 (which essentially is (1.4) for absolutely almost simple adjoint groups, cf.Corollary 6.4).Then it remains to deduce (1.4) from (1.3).This is done in Section 6.3.Finally, in Section 7 we prove the remaining Proposition 6.3, which is the most technical part of the proof.We also point out that specialness of c is only used in the proof of Proposition 6.3 and nowhere else in the proof of Theorem 1.1.

Reduction of (1.3) to the adjoint absolutely almost simple case
First, note that both sides of the equality claimed in the theorem only depend on G ad (for the left-hand side, use that G/B = G ad /B ad ).Hence we may assume that G is of adjoint type.Then G decomposes as a direct product of almost simple k-groups, and both sides of the claimed isomorphism (1.3) do so accordingly.Thus we are reduced to the case that G is of adjoint type and almost simple.
If G is adjoint and almost simple, then G = Res k ′ /k G ′ for some finite unramified extension k ′ /k and some absolutely almost simple k ′ -group G ′ .As B(G) = B(G ′ ), cf.[Kot85, 1.10], (which allows us to change b appropriately inside its σ -conjugacy class), Lemma 6.2 reduces us to the case that G is adjoint absolutely almost simple.
We state and prove Lemma 6.2 in bigger generality than just for adjoint absolutely almost simple groups and Coxeter elements.Let us fix notation first.Let k ′ /k be a finite unramified extension of degree d, let H ′ be an unramified reductive group over k ′ , with maximal quasi-split torus T ′ .Let H = Res k ′ /k H ′ .Let W ′ and W d i=1 W ′ be the Weyl groups of (H ′ , T ′ ) and (H, Res k ′ /k T).Identifying the buildings of H over k and of H ′ over k ′ , any point x ∈ A T, k defines a parahoric model H ′ x of H ′ , and we have Note that for definition (1.1) to work, we only need that the affine transformation bσ fixes x (assumptions on c to be Coxeter and on b to be basic are redundant), so part (ii) of the following lemma makes sense.

These natural isomorphisms are compatible with the natural identifications
Proof.(i) Let B ′ be a k ′ -rational Borel subgroup of H ′ which contains T ′ , and let B = Res k ′ /k B ′ be the corresponding Borel subgroup of H.As k-groups, we have H d i=1 H ′ , with Frobenius permuting the components cyclically.As L(•) commutes with products, we have L(H/B) = d i=1 L(H ′ /B ′ ).Let g = (g 1 , . . ., g d ) ∈ L(G/B).Then g ∈ X w (b) if and only if the relative position of (g 1 , . . ., g d ) and (σ ′ (g d ), g 1 , . . ., g d− 1) is (w ′ , 1, . . ., 1), where σ ′ is the geometric Frobenius on L(G ′ /B ′ ).This hold if and only if (ii) Just as above, we have The same holds after applying which is the case if and only if g 1 ∈ ẊH ′ x w,b and the g i (i = 2, . . ., d) are determined by appropriate formulas in terms of g 1 .Thus g → g 1 defines the required isomorphism.The last isomorphism follows from the first.□ Altogether, in the proof of (1.3) we may assume G to be absolutely almost simple and adjoint.
We then have the following result, whose proof occupies Section 7.
Proof.Consider the composed map where the second map is the natural projection.By Proposition 6.3, for any R ∈ Perf F q and any g ∈ Ẋb,b (R), LG is a σ b -stable subsheaf; thus σ b acts on LG/L + G x .It follows that (6.1) factors through We now prove (1.3) for G as in Corollary 6.4 (this suffices by Section 6.1).We need to descend the disjoint union decomposition of Corollary 6.4 to X c,b .Recall that by [Ryd10, Corollary 2.9], a quasi-compact morphism of schemes is universally subtrusive if and only if it satisfies the condition in [BS17, Definition 2.1], i.e., is a v-cover.We need the following version of [Ryd10, Theorem 4.1].Lemma 6.5.Let S ′ → S be a quasi-compact universally subtrusive map between schemes.Let E ⊆ S be a subset, Proof.It suffices to prove the claim for closed sets.The "only if" part is clear.For the other direction, it suffices by [Ryd10, Corollary 1.5] to show that E is closed in the constructible topology and stable under specializations if the same holds for E ′ .As f is quasi-compact, f is closed in the constructible topology; cf.[Ryd10, Proposition 1.2].Hence E = f (E ′ ) is closed in the constructible topology.Let x 0 ⇝ x 1 be a specialization relation in S with x 0 ∈ E. As f is universally subtrusive, x 0 ⇝ x 1 lifts to a specialization relation y 0 ⇝ y 1 .Then y 0 ∈ E ′ and as E ′ is closed, also

Proof of (1.4)
Using Proposition 5.12 we may replace the left-hand side in (1.4) by Ẋb,b (as b lies over c).If G is of adjoint type, (1.4) follows from Corollary 6.4 and the discussion in Section 6.1.We now proceed in two steps.

Special case:
The center Z of G is an unramified induced torus.Lemma 6.6.The natural map p : ) is surjective, and its fibers are X * (Z) ⟨σ ⟩torsors.
Proof.As Z is an induced torus, Factoring this sequence out from the first one, and using Z(k)/Z(O k ) X * (Z) ⟨σ ⟩ (which again follows from H 1 (σ , Z(O k )) = 0), we obtain the lemma.□

Let π :
LG → LG ad be the natural map.It is surjective as , and the quotient is the discrete scheme γ τY , where where we have σ b ( τ) = σ ( τ) as τ is central.This cannot lie in ), the disjoint decomposition for Ẋad b,b (cf.beginning of Section 6.3) implies, together with (6.2) and Lemmas 6.6 and 6.7, that Ẋb,b is the disjoint union of all γ ẊG x b,b , where γ varies through G b (k)/G x,b (O k ), which is precisely (1.4).

General case.
An arbitrary G admits a derived embedding G → G such that G is unramified reductive and the center of G is an unramified induced torus.(This was pointed out to us by M. Borovoi; cf.[Bor21].)Now the result follows from Section 6.3.1 and the next proposition.Proposition 6.8.Let G → G be a derived embedding.If (1.4) holds for G, then it holds for G.
Proof.We identify G with a subgroup of G.
Lemma 6.9.Let f ∈ Perf F q be an algebraically closed field.Let where Fix(x) denotes the stabilizer of the vertex x in the respective group (we identify the adjoint groups of G and G).
we have ker κ G ∩ G(L) = ker κG .As the actions on the adjoint building are compatible with the inclusion G → G, we have Putting all these together, we deduce Taking σ b -invariants in Lemma 6.9, we deduce that Proof.By assumption, the sheaf γ Ẋ G x c,b ∩ LG admits a geometric point.Let f ∈ Perf F q be algebraically closed, such that there exists some Lemma 6.11.Let X be an ind-(reduced scheme) over F q and let Y , Z be two closed subfunctors, which are themselves ind-(reduced schemes).If Y (f) = Z(f) for any algebraically closed field f/F q , then Y = Z.
Proof.Write X = lim − − →i X i for qcqs reduced schemes X i and an index set I. Then Y ∩ X i , Z ∩ X i are closed reduced subschemes of X i , which agree on geometric points.Thus Y ∩ X i = Z ∩ X i for each i, and hence also Y = Z as the X n exhaust X. □ By combining Lemma 6.11 with Lemma 6.9 we see that LG ∩ L + G x = L + G x (note that any ind-(perfect scheme) is automatically an ind-(reduced scheme)).As where the second equality is by the assumption of the proposition, the third is by Lemma 6.10 and the fifth holds by (6.3).This proves the proposition.□

Quasi-split case
We work in the setup of Section 1.3.In particular, we have the Coxeter element c, and we have As c is Coxeter, the affine transformation ċσ of A T, k admits a unique fixed point, which necessarily equals x.Comparing the two parametrizations of T(G, c) / Ad G(k) above, we see that the image of ċ equals the image of c in ([1] ∩ F c / ker κc ) /C W (cσ ).Replacing ċ with γ ċ for an appropriate γ ∈ C W (cσ ), we have that ċ lifts c.Now we prove Corollary 1.3.We only prove the first isomorphism (the second is completely analogous).We may assume that G is adjoint.By the pro-version of Lang's theorem for G x , we may find an (1).We compute (quotients are taken in the category of arc-sheaves) (1), where the third isomorphism is g → hg and in the fourth isomorphism, we exploit the last statement of Proposition 5.3.

Estimates via Newton polygons
In this section we prove Proposition 6.3 for an absolutely almost simple classical group G ad of adjoint type (so, compared to Section 6.2, we change our notation, and the group which was denoted by G there is G ad here).It will be more convenient to work with a certain central extension p : G → G ad , whose kernel Z = ker(p) is an unramified torus.(Which central extension we take is specified in each particular case of our case-by-case study below; see Sections 7.1.1,7.2.1, 7.3.1,7.4.1,7.5.1,7.6.1.)Then we have the exact sequence (7.1) Remark 7.1.In our case-by-case study below, the left map will be injective, except in type 2 A n−1 with n even.
(This is clear in the split cases A n−1 , B m , C m , D m ; see Sections 7.5.2 and 7.6.2for the cases 2 A n−1 (n odd) and 2 D m .)In type 2 A n−1 with n even, the left map will be the zero map Z/2Z → Z/2Z (see Section 7.5.2).
We let b ∈ F ad c ( k) ⊆ G ad ( k) be a basic element lying over a Coxeter element c ∈ W , so that we have to show that X ad b, b = X ad b, b,O within LG ad .It suffices to show that the Lang map g → g −1 σb(g) : As all rings in Perf F q are reduced, it suffices to do so on geometric points.Let f denote a fixed algebraically closed field in Perf F q and put L : where the objects on the left (resp.right) are contained in LG (resp. LG ad ), and where Lemma 7.2.We have The exactness of (2.4) implies that there exists some τ ∈ Z(L) such that z = τ −1 σ (τ)ϖ ζ .Replacing g with gτ −1 and recalling that To show that the right arrow in (7.2) is surjective, it suffices (using Lemma 7.2) to show that the left vertical arrow is surjective on f-points for all ζ.Thus, finally, we are reduced to showing that the inclusion we will do this in the following subsections case by case.As mentioned in Remark 7.1, we have # ker X * (Z) ⟨σ ⟩ → π 1 (G) ⟨σ ⟩ = 1, and it will suffice to prove only the surjectivity of Ẋb,b,O (f) → Ẋb,b (f), with the only exception (treated in Section 7.5.6) in the case 2 A n−1 with n even, when # ker X * (Z) ⟨σ ⟩ → π 1 (G) ⟨σ ⟩ = 2.We will use the following well-known property of isocrystals.
Lemma 7.3.Let (V , ϕ) be an isocrystal over f which is isoclinic of slope λ and dimension n ≥ 1.Let v be a cyclic vector of (V , ϕ), that is, an element v ∈ V such that the set Proof.The Newton polygon of (V , ϕ) is the straight line segment connecting the points (0, 0) and (n, λn) in the plane.It coincides with the Newton polygon of the polynomial e., we have (7.4) bσ (g) = gyb.
We will now proceed case by case.Below we will denote the fixed point x of bσ by x b (to make the notation non-ambiguous).Also recall the notation from Section 2, especially Section 2.1.2.

Type A n−1
This case is treated in [CI21a,CI23].We include it for the sake of completeness.

7.1.1.
Let n ≥ 2, and let V 0 be an n-dimensional k-vector space with an (ordered) basis {e 1 , . . ., e n }.Whenever convenient, identify V 0 with k n via this basis.Let G = GL n (V 0 ).

7.1.2.
Let T ⊆ B ⊆ G be the diagonal torus and the upper triangular Borel subgroup.Then X * (T) Then π 1 (G ad ) = X * (T)/ZΦ ∨ + X * (Z) Z/nZ, the isomorphism given by sending 1 to the class βi of β i for any i.
We have For 0 ≤ κ < n, consider the lift ċκ of c to G( k) given by e i → e i+1 for 1 ≤ i < n and e n → ϖ κ e 1 .Then ċκ maps to κ β1 ∈ π 1 (G ad ), and G ċκ run through all inner forms of G.
We identify A T ad , k with X * (T ad ) R by sending 0 ∈ X * (T) R to the σ -stable hyperspecial point corresponding to . Let g ∈ Ẋb,b (f).Then (7.4) holds with y as described in Section 7.1.2.We have to show that y ∈ G x b (O L ), or equivalently (see e.g.[MP94, Section 2.5]) Then σ (g)(e i ) = σ (g i ) for all i.Thus, (7.4) implies ϕ(g i ) = gyb(e i ).Using this equation for all i n and writing v := g 1 , we deduce gyb(e i ) for i = n and inserting the values for g i , we arrive at the equation This gives (7.6), and we are done.
The action of c on Φ + is as follows:

7.2.1.
Let m ≥ 2. Let V 0 be a 2m-dimensional k-vector space with an (ordered) basis {e 1 , . . ., e m , e −m , . . .e −1 }.Whenever convenient, we will identify V 0 with k 2m via this basis.Let ⟨•, •⟩ be the alternating bilinear form on V 0 determined by ⟨e ±i , e ±j ⟩ = 0 if i j and ⟨e i , e where λ(g) ∈ R × is a unit only depending on g.The group G is the group of symplectic similitudes, and the character λ : G → G m is called the similitude character.

7.2.2.
Let T ⊆ B ⊆ G be the diagonal torus and the Borel subgroup consisting of upper triangular matrices in G.
We identify A T ad , k with X * (T ad ) R by sending 0 ∈ X * (T) R to the σ -stable hyperspecial point ("the origin") corresponding to the self-dual lattice given by e 1 → e 1 + m i=2 a i−1 e i + a m e −m ; e i → e i for 2 ≤ i ≤ m and i = −1; e −i → e −i − a i−1 e −1 for all 2 ≤ i ≤ m.

7.2.3.
Let b = ċκ with κ ∈ {0, 1}.We compute that x b = 0 if κ = 0 and that x b = m i=1 λ i ε i with f).Then (7.4) holds with y as described in Section 7.2.2, depending on some a 1 , . . ., a m ∈ L. We have to show that y ∈ G x b (O L ), which is (as in Section 7.1.3)equivalent to Consider the φ-linear isocrystal (V , ϕ) = (V 0 ⊗ k L, bσ ) over f.It is isoclinic of slope κ 2 .For i ∈ {±1, . . ., ±m}, let g i := g(e i ).Then σ (g)(e i ) = σ (g i ) for all i.Hence (7.4) implies ϕ(g i ) = gyb(e i ).Using this equation for all i m + 1 and writing v := g 1 , we deduce that In particular, {ϕ i (v)} 2m−1 i=0 is a basis of V ; i.e., v is a cyclic vector for V .Finally, applying ϕ(g i ) = gyb(e i ) for i = m + 1 and inserting the above formulas for the g i , we deduce The action of c on Φ + is as follows: for We set r = m+1; m and i 0 < j}.Replacing all roots with the corresponding coroots, we obtain by duality the desired partition for type B m (under the induced isomorphism of Weyl groups, the special Coxeter element for type C m maps to the one in Section 7.3.2).

7.3.1.
Let m ≥ 2. Let V 0 be a (2m + 1)-dimensional k-vector space with an (ordered) basis {e 1 , . . ., e m , e m+1 , e −m , . . .e −1 }.Whenever convenient, identify V 0 with k 2m+1 via this basis.Let Q : V 0 → k be the split quadratic form given by Q m i=1 (a i e i + a −i e i ) + a m+1 e m+1 = m i=1 a i a −i + a 2 m+1 .Let G = SO(V 0 , Q) be the split odd orthogonal group attached to (V 0 , Q); i.e., It is of adjoint type.(The computations below work independently of the characteristic and the residue characteristic of k.)

7.3.2.
Let T ⊆ B ⊆ G be the diagonal torus and the Borel subgroup consisting of upper triangular matrices in G. Then T = {(t 1 , . . ., t m , 1, t −1 m , . . ., t −1 1 ) : generated by the class of ε i for any i.
We identify A T, k with X * (T) R by sending 0 ∈ X * (T) R to the σ -stable hyperspecial point ("the origin") corresponding to the self-dual lattice The group c U ∩ U − is generated by root subgroups attached to α i−1 (2 ≤ i ≤ m) and α −1 .Explicitly, any element y ∈ ( c U ∩ U − )(R) is an R-linear map in GL(V 0 )(R) given by e 1 → e 1 + m i=2 a i−1 e i + a m e m+1 − a 2 m e −1 ; e i → e i for 2 ≤ i ≤ m and i = −1; e m+1 → e m+1 − 2a m e −1 ; e −i → e −i − a i−1 e −1 for 2 ≤ i ≤ m.
Now suppose a m 0. Then a m ∈ L × , and λ := a m φ(a m ) ∈ L × satisfies ord ϖ (λ) = 0. Apply λϕ(•) to (7.12), and add the result to (7.12); then use (7.11) to eliminate u: where the * denote some unspecified elements of L.Moreover, v is a cyclic vector for V .Indeed, {ϕ i (v)} 2m−1 i=0 ∪ {u} is a basis of V , and hence it follows from (7.12) and a m 0, that {ϕ i (v)} 2m i=0 also is.Using the cyclicity of v, the last equation, the fact that ord ϖ (λ) = 0 and Lemma 7.3, we deduce by induction on i that ord ϖ (a i ) ≥ 0 for i = 1, . . ., m and that, moreover, ord ϖ (a m ) ≥ 1 2 if κ = 1.This agrees with what we had to show in (7.10).

7.4.1.
Let m ≥ 4. Let V 0 be a 2m-dimensional k-vector space with an (ordered) basis {e 1 , . . ., e m , e −m , . . .e −1 }.Whenever convenient, we use this basis to identify V 0 with k 2m .Consider the quadratic form Q m i=1 (a i e i + a −i e −i ) = m i=1 a i a −i on V 0 .We have the orthogonal group O(V 0 , Q) ⊆ GL(V 0 ).Via the Clifford algebra attached to Q, we have the surjective Dickson morphism Its kernel is the special orthogonal group SO(V 0 , Q) of Q.Finally, let G = GSO(V 0 , Q) be the closed subgroup of GL(V 0 ) of elements which up to a scalar lie in SO(V 0 , Q). (6)
We identify A T ad , k with X * (T ad ) R by sending 0 ∈ X * (T) R to the σ -stable hyperspecial point ("the origin") corresponding to the self-dual lattice given by e 1 → e 1 + m i=2 a i−1 e i + a m e −m − a m−1 a m e −1 ; e i → e i for 2 ≤ i ≤ m − 1 and i = −1; e m → e m − a m e −1 ; e −i → e −i − a i−1 e −1 for all 2 ≤ i ≤ m.

7.4.3.
Let b = ċκ with κ ∈ {0, 1, 2}.We compute . Then (7.4) holds, with n as described in Section 7.4.2,depending on some a 1 , . . ., a m ∈ L. We have to show that y ∈ G x b (O L ), which (as in Section 7.1.3)is equivalent to For i ∈ {±1, . . ., ±m}, put g i := ϕ(e i ).Then (7.4) implies that ϕ(g i ) = gyb(e i ) for all i.Using this equation for all i m + 1, m + 2 and writing v := g 1 and u := g m , we compute that If char k 2, then the Dickson morphism is not necessary to describe G.For any k-algebra R, we have where λ(g) ∈ R × is a unit only depending on g.
where A From the formulas for the g ±i above, it follows that the vectors are linearly independent, hence form a basis by dimension reasons.□ Now, evaluating the equation ϕ(g i ) = gyb(e i ) for i = m + 1 and i = m + 2 gives the formulas We have five cases, which we consider separately in the following subsections.
7.4.4.Case a m 0, a m−1 0, µ 0. Suppose a m a m−1 µ 0. Apply a m φ(a m−1 ) ϕ(•) to (7.15), and subtract (7.15) from the resulting equation.This gives an expression of a m φ(a m−1 ) ϕ 2m−1 (v) as a linear combination of {ϕ i (v)} 2m−2 i=0 , ϕ 2 (u) and u.Substitute (7.16) into this expression to eliminate ϕ 2 (u).The coefficient of u in the resulting equation is µ 0, and it gives an expression of u as a linear combination of {ϕ i (v)} 2m−1 i=0 : where the * ∈ L denote unspecified coefficients; all E, E i , As a m 0, the coefficient of ϕ 2m−1 (v) in (7.17) is non-zero.Using this observation, as well as the version of it after applying ϕ(•), shows, together with Lemma 7.4, that v is a cyclic vector for V .Now insert (7.17) into (7.15) to eliminate u there.This gives (after multiplication with an element in L × ) an expression . First assume κ = 0.If ord ϖ (a m−1 ) > ord ϖ (a m ), then , where the fraction is just γ from case κ = 0 for a m−1 and a ′ m , and we are done by applying the above.Finally, suppose κ = 2. Then µ = ϖa m φ(a m ) − a m−1 φ(a m−1 ), and the claim is immediately checked in both cases, ord ϖ (a m ) < ord ϖ (a m−1 ) and the opposite.□ Now we proceed case by case.
We have As the two summands have different valuations by assumption, both valuations must be non-negative, which implies ord ϖ (a m−1 ), ord ϖ (a m ) ≥ 0. We are done by (7.14).
The slope is again λ = 0, so that β i ≥ 0 for all i.We have As in the case b = ċ0 , we deduce ord ϖ (a i ) ≥ 0 for 1 ≤ i ≤ m − 2. We have β m−1 = a m−1 a m + ϖφ(a m−2 )γ.Using Lemma 7.5, we deduce that if ord ϖ (a m−1 ) = ord ϖ (a m ) − 1, then this number must be non-negative, and we are done according to (7.14).Assume ord ϖ (a m−1 ) ord ϖ (a m )−1.We compute The slope is λ = 1 2 , so that for any i, As in the previous two cases, we deduce that (7.14) holds.
In the following cases, we do not give all details as the computations are similar to (and easier than) those in Section 7.4.4.7.4.5.Case a m−1 0, a m = 0.

Case a
This case is completely analogous to the one in Section 7.4.5 (one has ϕ m−1 (v) = 1 a m−1 (u − ϕ 2 (u)), from which the cyclicity of u and an explicit formula ϕ 2m (u) = 2m−1 i=0 β i ϕ i (u) follow).

Case a
By Lemma 7.4 v is a cyclic vector for the (2m−2)-dimensional subisocrystal with L-basis {ϕ i (v)} 2m−3 i=0 .The result follows easily from Lemma 7.3 by looking at the coefficients of (7.15).
7.4.8.Case a m−1 0, a m 0, µ = 0. (Note that κ 2 in this case.)As a m−1 0, (7.15) along with Lemma 7.4 show that the vectors {ϕ i (v)} 2m−2 i=0 are linearly independent.Further, following the same steps as in the first lines of Section 7.4.4 and exploiting µ = 0, we arrive at an expression a m φ(a m−1 ) ϕ 2m−1 (v) = 2m−2 i=0 β i ϕ i (v), thus showing that v is a cyclic vector for the (2m − 1)-dimensional subisocrystal generated by it.Proceeding as in the previous cases, one verifies (7.14).This finishes the proof in type D m .7.4.9.Proof of Lemma 5.6 for type D m .

7.5.1.
Let n ≥ 2. Let m = ⌊ n 2 ⌋, so that n = 2m + 1 if n odd and n = 2m if n is even.Let V 0 , e i be as in Section 7.1.1.Let G be the unitary group attached to the standard hermitian form on V 0 ⊗ k k 2 , relative to the unramified extension k 2 /k of degree 2. We may identify G ⊗ k k 2 = GL n (V 0 ⊗ k k 2 ), so that for any F q -algebra R 0 with R = W(R 0 ⊗ F q F q )[1/ϖ], the Frobenius σ on G( R) = GL n (V 0 )( R) is given by g −→ σ (g) = Jσ 0 (g) −1,T J −1 , where σ 0 is the Frobenius on GL(V 0 )( R) corresponding to the natural k-structure on GL(V 0 ), (•) T is transposition and J ∈ GL n (V 0 )(k) is given by e i → e n+1−i (1 ≤ i ≤ n).

7.5.2.
Let T, B, β i , α i−j be as in Section 7.1.2.Note that T, B are both k-rational (with respect to σ ).Moreover, the action of σ on π 1 (G ad ) is by multiplication with −1.Thus, π 1 (G ad ) ⟨σ ⟩ = 1 if n is odd, and π 1 (G ad ) ⟨σ ⟩ Z/2Z, with the non-trivial element represented by βi for any i, when n is even.We have an isomorphism Z ∼ − → π 1 (G), 1 → class of β 1 , under which X * (Z) ⊆ π 1 (G) corresponds to nZ.The action of σ corresponds to multiplication by −1.Justifying Remark 7.1 in this case, we deduce from this that the induced map X * (Z) ⟨σ ⟩ Z/2Z → Z/2Z π 1 (G) ⟨σ ⟩ is an isomorphism when n odd and is 0 when n even.
We identify A T ad , k with X * (T ad ) R by sending 0 to the σ -stable hyperspecial point ("the origin") corresponding to the standard lattice n i=1 O k e i .The group c U ∩ U − is generated by the root subgroups attached to α i−1 (2 ≤ i ≤ m + 1).For a k-algebra R, any element y ∈ ( c U ∩ U − )(R) is an element of GL(V 0 )(R), determined by e 1 → e 1 + m λ=1 a λ e λ+1 , e i → e i (2 ≤ i ≤ n) for some a 1 , . . ., a m ∈ R.

(
as c is Coxeter).Let G x be the corresponding parahoricO k -model of G.It descends to an O k -model G x,b of G b .In particular, the Frobenius σ b := Ad(b) • σ of LG fixes the subgroup L + G x .Here L and L + denote the functors of (positive) loops; see Section 2.1.1.We also have the corresponding parahoric model G adx of G ad .Consider the scheme ẊG x c,b defined by the cartesian diagram /T ad c (k), where ẊG ad x c,b is (1.1) applied to the parahoric model G ad x of the adjoint group G ad .Then X G ad x c,b and ẊG x c,b are infinite-dimensional affine schemes; cf.Proposition 2.2.Theorem 1.1.Let G be an unramified group of classical type.Let c, c and b be as above.Then there exist a G b (k)-equivariant isomorphism (1.3) X c (b) = γ∈G ad b (k)/G ad x,b (O k ) γX G ad x c,b and a G b (k) × T c (k)-equivariant isomorphism (1.4) Ẋc (b) γ∈G b (k)/G x,b (O k ) γ ẊG x c,b .

Corollary 1. 2 .
If G is an unramified group of classical type, c ∈ W Coxeter, c ∈ F c / ker κc arbitrary and b basic, then X c (b) and Ẋc (b) are disjoint unions of affine schemes.

Corollary 1. 3 .
With notation as above, there areG(k)-(resp.G(k) × T c (k)-)equivariant isomorphisms X c (1) γ∈G ad (k)/G ad x,1 (O k ) then G b is an inner form of G, we may identify G b ( k) = G( k) and we denote by σ b := Ad(b) • σ the twisted Frobenius on G( k).We also write σ b for the corresponding geometric Frobenius on LG b = LG.
Deligne-Lusztig spaces 2.2.1.Spaces X w (b) and Ẋ ẇ(b).Let w ∈ W , ẇ ∈ N( k) be a lift of w and b ∈ G( k).Denote by O(w) ⊆ (G/B) 2 (resp.O( ẇ) ⊆ G/U) the G-orbit corresponding to w (resp.ẇ) under the Bruhat decomposition.The p-adic Deligne-Lusztig spaces X w (b) and Ẋ ẇ(b) were defined in [Iva23, Definition 8.3] by the cartesian diagrams of functors on Perf F q X w (b) where bσ : L(G/B) ∼ − → L(G/B) and b acts by left multiplication.Then X w (b) and Ẋ ẇ(b) are sheaves for the arc-topology (see [BM21]) on Perf F q ; cf.[Iva23, Corollary 8.4].There is a natural action of G b (k) on X w (b) and of G b (k) × T w (k) on Ẋw (b).
ẊG x c,b and X G ad x c,b .
[BM21, Corollary 2.18] instead of [BS15, Lemma 4.1.8]□ For n ≥ 1, let ẊG x c,b,n be defined by the diagram (1.1) with L + replaced by L + n everywhere.Then all ẊG x c,b,n are affine schemes and ẊG x c,b = lim ← − −n ẊG x c,b,n .We also have the truncated version of (1.2): let T be the closure of T in G x and T c the O k -torus, arising by twisting the Frobenius on T by c.Then put X G ad x c,b,n := ẊG ad x c,b,n /T n , where T n := T ad c (O k /ϖ n ).Proposition 2.2.The schemes ẊG x c,b and X G ad x c,b are affine and perfect.Also, X /T c (k) remains unchanged if one takes the quotient with respect to the arc-topology instead of the pro-étale topology.Moreover, X pro-étale and-by Lang's theorem applied to all truncations-also surjective.Thus ẊG x c,b and X G ad x c,b and their truncations are always non-empty.Also, it follows that ẊG x c,b and X G ad x c,b are infinite-dimensional.The truncations are perfections of finite-dimensional schemes.More precisely, as the Lang map is étale, we have dim ẊG x c,b,n = dim X G ad x c,b,n = dim L + n ( c U ∩ U − ) x = ℓ(c)n, where ℓ(c) denotes the length of the Coxeter element c ∈ W . Conjecture 2.4.The scheme X G ad x c,b is an A ∞ -bundle over a classical Deligne-Lusztig variety attached to the reductive quotient of the special fiber of G x .

For
b ∈ N( k), the twisted Frobenius σ b = Ad b • σ of G( k) induces one on W . Now suppose b ∈ N( k) is basic.By [DeB06, Lemma 4.3.1],there is a natural injection from the set of stable conjugacy classes of unramified maximal k-tori in G b into H 1 (σ b , W ), which is even a bijection when G b is quasi-split (e.g., [b] = [1]).Identifying H 1 (σ b , W ) ∼ − → H 1 (σ , W ) via the natural twisting map x → xb on cocycles (using [Ser94, Proposition I.35 bis]), we get (3.1){stable conjugacy classes of k-tori in G b } −→ H 1 (σ , W ), mapping the class of T(⊆ G b ) to the image of b in H 1 (σ , W ). Definition 3.1.For b ∈ N( k) basic and w ∈ W , denote by T(G b , w) the stable conjugacy class of all unramified maximal k-tori in G b , which corresponds to [w] ∈ H 1 (σ , W ) under (3.1).The elements of T(G b , w) are precisely the images of k-embeddings T w → G b .Remark 3.2.We recall the construction from the proof of [DeB06, Lemma 4.3.1].Any unramified maximal k

( 3 )
Indeed, one can use [Ser94, Corollary 2 to Proposition I.39 and Corollary to Proposition I.41] along with the pro-version of Lang's theorem for the group T (O k ) = ker(N( k) ↠ W ) with varying rational structures given by σ b (b ∈ N( k)).
w1 and w2 define the same class in H 1 (σ , W ). Conversely, given [ w] ∈ π−1 ([b]), we can lift w to some ẇ ∈ N( k), which necessarily belongs to [b].If ẇ = g −1 bσ (g), one immediately checks that w → g T induces an inverse to the above map.This shows that rational conjugacy classes of unramified tori in G b are in bijection with π −1 ([b]), establishing the analogue of [Ree11, Lemma 6.2].The rest of the proof (the analogue of [Ree11, Lemma 6.4]) is done analogously.□ stable under the action of ker κw , and we have the (finite) quotient set [b] ∩ F w / ker κw , consisting of all w ∈ F w / ker κw satisfying ν w = ν b and κw ( w) = κ G (b).
and we are done.□ Remark 3.9.Let us explicate the correspondence in Proposition 3.7(ii).If w ∈ [b] ∩ F w / ker κw represents a class on the left side, ẇ ∈ [b] ∩ F w ( k) is a (i.e., any) lift of w, and g torsor.□ Remark 3.12.It follows from Proposition 3.10(ii) that for any [b] ∈ B(G) bas and any Coxeter element c, there exists a representative of [b] in F c ( k).This is a converse to Lemma 3.11.Proposition 3.13.Let c ∈ W be Coxeter.Suppose one of the following holds: which works by assumption for G ′ .It remains to check the claim for an absolutely almost simple group G over k.Let d > 0 be the smallest positive integer such that σ d acts trivially on the Dynkin diagram of G.By [Spr74, Theorem 7.6

Definition 4. 1 .
For b ∈ G( k) and w ∈ F w / ker κw , we denote by Ẋ w(b) the space Ẋ ẇ(b) for an arbitrary lift ẇ ∈ F w ( k) of w.Proposition 4.2 (cf.[Iva23, Propositions 11.4 and 11.9]).Let w ∈ W and b ∈ G( k).There is a canonical map of arc-sheaves α w,b : X w (b) −→ F w / ker κ w defining a clopen decomposition X w (b) = w∈F w / ker κ w X w (b) w, where X w (b) w

Corollary 4. 4 .
If β w = 0, then α w,b factors through a point, i.e., there is at most one w ∈ F w / ker κw such that Ẋ w(b) ∅.In this case, X w (b) ∅ if and only if Ẋ w(b) ∅ and Ẋ w(b) → X w (b) is a pro-étale T w (k)-torsor.Corollary 4.5.If G is of adjoint type and w is Coxeter, the conclusions of Corollary 4.4 hold.Moreover, in this case, the map Ẋ w(b) → X w (b) is quasi-compact.Proof.The first claim follows from Corollary 4.4 and Remark 2.5.The last claim follows as Ẋ w(b) → X w (b) is a torsor under the profinite group T w (k).□ In the case that b is basic and w = c is Coxeter, the situation simplifies as follows.Corollary 4.6.Suppose b is basic and c is Coxeter.For any c ∈ F c / ker κc such that X c(b) ∅, there exists a b 1 ∈ [b] ∩ F c ( k) lying over c, and there is a G b (k) × T c (k)-equivariant isomorphism Ẋc (b) Ẋc (b 1 ).Proof.By Proposition 4.3, Ẋc (b) ∅ forces c ∈ ( κc ) −1 ( κG (b)) = [b] ∩ F c / ker κc , where the equality holds by Proposition 3.10.This implies the existence of b 1 .For any g ∈ G( k) such that g −1 bσ (g) = b 1 , x → gx induces an equivariant isomorphism Ẋc (b) Ẋc (b 1 ) by [Iva23, Remark 8.7].□ Let us also record the following consequence.Corollary 4.7.Suppose b is basic and c is Coxeter.Then α c,b factors through a surjection α c,b : X c (b) −→ [b] ∩ F c / ker κc .
1. Let c ∈ W be a Coxeter element.If b ∈ G( k) is any lift of c, then bσ has a unique fixed point x = x b on A T, k .Let G x denote the corresponding parahoric O k -model of G.We have the twisted Frobenius σ b = Ad(b) • σ on LG, L + G x .Let ( c U ∩ U) x be the closure of c U ∩ U in G x .Similarly, we have the closures ( c U ∩ U − ) x and ( c U) x of c U ∩ U − and c U in G x .Proposition 5.3.Let G be of classical type and c ∈ W a special Coxeter element.Then Remark 5.4.It looks very plausible that Proposition 5.3 holds for arbitrary Coxeter elements c.It should be possible to prove this by adapting [HL12, Section 3.5, Theorem 3.6 and Section 3.14] to the present setting.Let us make the relation of Proposition 5.3 with [HL12] more clear.With obvious notation (in particular, c is Coxeter and b any lift of it), [HL12, Theorem 3.6] says that U× (U ∩ b U − ) → UbU, (u, z) → uzσ (u) −1 is an isomorphism if G isreductive over a finite field with Frobenius morphism σ (and, in fact, in various other setups).As an easy corollary of this, [HL12, Section 3.14] states that α ′ b an isomorphism in the same setup.Now the finite field version (without the loop functors) of the map α b of Proposition 5.3 is obtained from α ′ b by a series of changes of variables (cf. the first paragraph of the proof of Proposition 5.3), so that α b is an isomorphism if and only if α ′ b is.Proof of Proposition 5.3.We work with the pro-étale topology on Perf F q and regard α b as a map of pro-étale sheaves.The injectivity of α b may be derived from[HL12].First, change the variable by setting y = σ b (y 1 ).It remains to show that L

Lemma 5. 5 .
Let k d /k be an unramified extension of degree d ≥ 1, and suppose that G = Res k d /k G ′ for a k d -group G ′ .If the surjectivity claim for α b of Proposition 5.3 holds for G ′ , then it holds for G.
F q .Steinberg's cross section allows us to give a different presentation of the Deligne-Lusztig spaces X c (b), Ẋc (b).We have the following general definition.Definition 5.10.Let w ∈ W , ẇ ∈ F w ( k) and b ∈ G( k).Define Ẋ ẇ,b and Xw,b by the following cartesian diagrams: in the upper right entry of the right diagram of Definition 5.10, we could have written ( c U ∩ U − ) • κ G | −1 LT (κ G (b)) without changing the definition.The group G b (k) × T w (k) acts on Ẋ ẇ,b by g, t : x → gxt, and G b (k) acts on X w,b by g : x → gx.We only use these spaces in the case that w = c is a (special) Coxeter element.Proposition 5.12.Let b ∈ G( k).Let c ∈ W be a Coxeter element and ċ ∈ F c ( k).Whenever the conclusion of Proposition 5.3 holds (e.g., G classical, c special Coxeter), the following hold:

Proposition 6. 3 .
Suppose G is absolutely almost simple and adjoint.We have Ẋb,b = Ẋb,b,O .Corollary 6.4.Suppose G is absolutely almost simple and adjoint.We have a G b ,b is an ind-scheme by Proposition 5.12 and [Iva23, Theorem C].Thus for any map T → X c,b with T a quasi-compact scheme, the fiber product T × X c,b Ẋb,b with the scheme Ẋb,b is again a scheme.Moreover, the map T × X c,b Ẋb,b → T is quasi-compact and a v-cover by Corollary 4.5.Hence this map is universally subtrusive, cf.[Ryd10, Corollary 2.9], and by Lemma 6.5 and Corollary 6.4, T admits a disjoint union decomposition T = γ∈G b (k)/G x,b (O k ) T γ which is functorial in T .Doing this for all maps T → X c,b from a quasi-compact scheme T into X c,b , we deduce a disjoint union decomposition of X c,b indexed by the same set.The subscheme (ind-scheme) corresponding to γ = 1 is the quotient of ẊG x c,b by T c (k), which equals X G x c,b by definition.This proves (1.3).
be the class corresponding to c under the surjection [1] ∩ F c / ker κc ↠ T(G, c) / Ad G(k) from Proposition 3.7.Then letx be any hyperspecial vertex in B G,k corresponding to [T c] under the above bijection.As x is hyperspecial, c admits a lift ċ ∈ F c ( k) ∩ G x (O k ).
i−m) (a 2m−1−i )ϕ i (v) + a m u. (7.24) ,b (f), we have g −1 σ b (g) ∈ ( c U ∩ U − ) x b (O L )ϖ ζ .As in (7.4), we can write (7.25) bσ (g) = gyϖ ζ b = G x be the corresponding reductive O k -model of G.The reductive O k -group G x,1 is quasi-split, and the closure B x of B in G is an O k -rational Borel subgroup.Moreover, G has a Bruhat decomposition, in the sense that (G/B x ) 2 ⊗ O k O k contains locally closed subschemes O(w) (w ∈ W ), flat over O k , which fiberwise induce the usual Bruhat decomposition.If N x , U x are the closures of N, U in G, one similarly has O and replacing G, B, U with G, B x , U x and L with L + everywhere in the diagrams in Section 2.2.1, we obtain integral p-adic(O k ) and G x,1 (O k ) × T (O k ), respectively.
conjugate, the statement claimed in the proposition holds for c if and only if it holds for c 1 .Thus we may assume that c = (c ′ , 1, . . ., 1), where c ′ ∈ W ′ is a Coxeter element in W ′ .Suppose we know the claim for G ′ , c ′ ; i.e., for any v This shows (7.8), and we are done.