# On a decomposition of $p$-adic Coxeter orbits

Alexander B. Ivanov  Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn, Germany<EMAIL_ADDRESS>
###### Abstract.

We analyze the geometry of some $p$-adic Deligne–Lusztig spaces $X_{w}(b)$
introduced in [Iva23] attached to an unramified reductive group ${\mathbf{G}}$
over a non-archimedean local field. We prove that when ${\mathbf{G}}$ 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 Frobenius-twisted Steinberg’s cross section.

###### Contents

  1. 1 Introduction
  2. 2 Setup and preliminaries
  3. 3 Rational and stable conjugacy classes of unramified tori
  4. 4 The maps $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$
  5. 5 Variant of Steinberg’s cross section
  6. 6 Decomposition into integral $p$-adic spaces
  7. 7 Estimates via Newton polygons

##  1\. Introduction

Let ${\mathbf{G}}$ be an unramified reductive group over a non-archimedean
local field $k$. In [Iva23] $p$-adic Deligne–Lusztig spaces $X_{w}(b)$
attached to ${\mathbf{G}}$ were constructed. This allows one to apply the rich
variety of methods from Deligne–Lusztig theory [DL76] to the study of
representations of the $p$-adic group ${\mathbf{G}}(k)$. Indeed, many results
in this direction were obtained recently, cf. [CI, CS17, CO, DI20] and
references therein.

The classical Deligne–Lusztig varieties are smooth schemes of finite type over
a finite field. In contrast, $X_{w}(b)$ are sheaves for the arc-topology
[BM21] on perfect algebras over the residue field of $k$. Concerning their
representability the situation is slightly delicate: In many cases it is known
that $X_{w}(b)$ are ind-representable [Iva23, Thm. C]. Also, there are
examples, where $X_{w}(b)$ is known to be an ind-scheme, which is not a scheme
[Iva23, Cor. 10.2]. On the other hand, for ${\mathbf{G}}={\bf GL}_{n}$, $c$
Coxeter and $b$ basic, $X_{c}(b)$ are schemes [CI, Prop. 2.6]. Moreover, the
proof of this last fact has led to a quite explicit description of (a cover
of) $X_{c}(b)$ in this case, eventually allowing one to compute large parts of
its cohomology and relate it to the local Langlands correspondences, which was
the main task of [CI, CI21b].

Motivated by the last sentence, we analyze here the geometry of the spaces
$X_{c}(b)$ and their

natural covers $\dot{X}_{\bar{c}}(b)$ when ${\mathbf{G}}$ is a classical
group111We call an unramified $k$-group _classical_ if each connected
component of its Dynkin diagram is of type
$A_{n-1},B_{m},C_{m},D_{m},{}^{2}A_{n-1}$ or ${}^{2}D_{m}$; see Definition 5.1
for more details., $c$ is a (twisted) Coxeter element and $b$ is basic. Under
these assumptions our main result, Theorem 1.1, shows that $X_{c}(b)$,
$\dot{X}_{\bar{c}}(b)$ admit natural decompositions as disjoint unions of
translates of certain integral level Deligne–Lusztig spaces of Coxeter type.
Those integral level spaces are easier to handle. In particular, they are
affine schemes, and it follows that $X_{c}(b),\dot{X}_{\bar{c}}(b)$ are
schemes, proving [Iva23, Conj. 1.1] in this case. Our decomposition result,
along with the Mackey-type formula shown in [DI20] (as well as techniques from
[Lus76, CO]), can be applied –similarly as in [CI]– to study the cohomology of
$X_{c}(b),\dot{X}_{\bar{c}}(b)$ and smooth ${\mathbf{G}}(k)$-representations
therein. Beside our main result, we prove some related results on rational
conjugacy classes of tori and a version of Steinberg’s cross section, see §1.4
below for a summary. Now, we introduce the relevant notation and state our
main result, Theorem 1.1.

###  1.1. Setup

Let $\breve{k}$ denote the completion of a maximal unramified extension of $k$
and let ${\mathcal{O}}_{k},{\mathcal{O}}_{\breve{k}}$ be the rings of integers
of $k,\breve{k}$. Let $\sigma$ be the Frobenius of $\breve{k}/k$. Let
${\mathbf{T}}\subseteq{\mathbf{B}}\subseteq{\mathbf{G}}$ be a centralizer of a
maximal split $k$-torus and a $k$-rational Borel subgroup of the unramified
group ${\mathbf{G}}$. (Note that ${\mathbf{T}}$ is a $k$-rational maximal
torus of ${\mathbf{G}}$.) Let ${\mathbf{U}}$ resp. ${\mathbf{U}}^{-}$ be the
unipotent radical of ${\mathbf{B}}$ resp. the opposite Borel subgroup. Let $W$
be the Weyl group of ${\mathbf{T}}$, and $S\subseteq W$ the set of simple
reflections determined by ${\mathbf{B}}$. Denote by
${\mathcal{B}}_{{\mathbf{G}},\breve{k}}$ the Bruhat–Tits building of the
adjoint group ${\mathbf{G}}^{\rm ad}$ of ${\mathbf{G}}$ over $\breve{k}$, and
by ${\mathcal{A}}_{{\mathbf{T}},\breve{k}}$ the apartment of ${\mathbf{T}}$
therein. Denote by $\sigma$ the action induced by Frobenius on
$W,S,{\mathcal{B}}_{{\mathbf{G}},\breve{k}},{\mathcal{A}}_{{\mathbf{T}},\breve{k}}$
and ${\mathbf{G}}(\breve{k})$ (as well as its subgroups).

An element $b\in{\mathbf{G}}(\breve{k})$ is called _basic_ , if its Newton
point $\nu_{b}$ factors through the center of ${\mathbf{G}}$, cf. [Kot85].
Twisting $\sigma$ by $\mathop{\rm Ad}(b)$ one gets an inner $k$-form
${\mathbf{G}}_{b}$ of ${\mathbf{G}}$, see §2.1.3. Similarly, for $w\in W$,
twisting the Frobenius on ${\mathbf{T}}(\breve{k})$ by $\mathop{\rm Ad}(w)$
one gets an (outer) $k$-form ${\mathbf{T}}_{w}$ of ${\mathbf{T}}$.

The space $X_{w}(b)$, introduced in [Iva23], is the intersection of
$L{\mathcal{O}}(w)\subseteq L({\mathbf{G}}/{\mathbf{B}})^{2}$ with the graph
of $b\sigma\colon L({\mathbf{G}}/{\mathbf{B}})\rightarrow
L({\mathbf{G}}/{\mathbf{B}})$, where $L$ denotes the loop functor (see
§2.1.1). Similarly, we have its coverings $\dot{X}_{\bar{w}}(b)$, where
$\bar{w}$ lies in $\overline{F}_{w}/\ker\bar{\kappa}_{w}$, a certain discrete
quotient of the set $F_{w}(\breve{k})\subseteq{\mathbf{G}}(\breve{k})$ of all
lifts of $w$, cf. §3.4, §4. For more details on
$X_{w}(b),\dot{X}_{\bar{w}}(b)$, see §2.2 and Definition 4.1. The group
${\mathbf{G}}_{b}(k)$ acts on $X_{w}(b)$ and
${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{w}(k)$ acts on $\dot{X}_{\bar{w}}(b)$.
The natural map $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$ is
${\mathbf{G}}_{b}(k)$-equivariant.

An element $c\in W$ is called _twisted Coxeter_ , or simply _Coxeter_ , if any
of its reduced expressions contains precisely one element from each
$\sigma$-orbit on $S$ [Spr74, §7]. Usually, we denote by $w$ an arbitrary
element of $W$, whereas the letter $c$ is reserved for Coxeter elements.

We refer to §2 for more notation and setup.

###  1.2. Main result

Let $c\in W$ be Coxeter, $\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$ a
lift of $c$ and $b\in{\mathbf{G}}(\breve{k})$ basic. We prove a decomposition
result for $X_{c}(b)$ and $\dot{X}_{\bar{c}}(b)$. 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
$\dot{X}_{\bar{c}}(b)=\varnothing$, or –after an equivariant isomorphism– we
may even assume that $b$ lies over $\bar{c}$. In the rest of §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 $b\sigma$ of
${\mathcal{A}}_{{\mathbf{T}},\breve{k}}$ has precisely one fixed point
${\mathbf{x}}$ (this follows from [Spr74, Lemma 7.4] as $c$ is Coxeter). Let
${\mathcal{G}}_{\mathbf{x}}$ be the corresponding parahoric
${\mathcal{O}}_{\breve{k}}$-model of ${\mathbf{G}}$. It descends to an
${\mathcal{O}}_{k}$-model ${\mathcal{G}}_{{\mathbf{x}},b}$ of
${\mathbf{G}}_{b}$. In particular, the Frobenius
$\sigma_{b}\stackrel{{\scriptstyle\rm def}}{{=}}{\rm Ad}(b)\circ\sigma$ of
$L{\mathbf{G}}$ fixes the subgroup $L^{+}{\mathcal{G}}_{{\mathbf{x}}}$. Here
$L,L^{+}$ denote the functors of (positive) loops, see §2.1.1. We also have
the corresponding parahoric model ${\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}$ of
${\mathbf{G}}^{\rm ad}$. Consider the scheme
$\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$ defined by the cartesian
diagram

$\textstyle{\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{+}({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{+}{\mathcal{G}}_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{+}{\mathcal{G}}_{{\mathbf{x}}}}$
(1.1)

where the lower map is $g\mapsto g^{-1}\sigma_{b}(g)$, and where
$({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})_{{\mathbf{x}}}$ is the closure of
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-}$ in ${\mathcal{G}}_{{\mathbf{x}}}$. We
also consider the quotient for the pro-étale (equivalently, arc-) topology,

$X_{c,b}^{{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}}}}=\dot{X}_{\bar{c},b}^{{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}}}}/{\mathbf{T}}_{c}^{\rm ad}(k),$ (1.2)

where $\dot{X}_{\bar{c},b}^{{\mathcal{G}}^{\rm ad}_{{\mathbf{x}}}}$ is (1.1)
applied to the parahoric model ${\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}$ of the
adjoint group ${\mathbf{G}}^{\rm ad}$. Then $X_{c,b}^{{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}}}}$ and $\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$
are infinite-dimensional affine schemes, cf. Proposition 2.2.

######  Theorem 1.1.

Let ${\mathbf{G}}$ be an unramified group of classical type. Let $c,\bar{c}$
and $b$ be as above. Then there is a ${\mathbf{G}}_{b}(k)$-equivariant
isomorphism

$X_{c}(b)=\coprod_{\gamma\in{\mathbf{G}}^{\rm ad}_{b}(k)/{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}},b}({\mathcal{O}}_{k})}\gamma X_{c,b}^{{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}}}}$ (1.3)

and a ${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{c}(k)$-equivariant isomorphism

$\dot{X}_{\bar{c}}(b)\cong\coprod_{\gamma\in{\mathbf{G}}_{b}(k)/{\mathcal{G}}_{{\mathbf{x}},b}({\mathcal{O}}_{k})}\gamma\dot{X}_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}}.$
(1.4)

After necessary preparations, Theorem 1.1 will be proven in §6 and §7. Let us
outline the proof. Using 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
§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
${\mathbf{G}}$.

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 $\overline{\mathbb{F}}_{q}$, the $p$-adic
Deligne–Lusztig spaces are essentially never of finite type over
$\overline{\mathbb{F}}_{q}$. On the other side, 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 ${\bf GL}_{n}$ is isomorphic to a $p$-adic
Deligne–Lusztig space $\dot{X}_{\bar{c}}(b)$ 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, Conj. 1.1] in the following
cases.

######  Corollary 1.2.

If ${\mathbf{G}}$ is an unramified group of classical type, $c\in W$ Coxeter,
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$ arbitrary and $b$ basic,
then $X_{c}(b)$, $\dot{X}_{\bar{c}}(b)$ are disjoint unions of affine schemes.

###  1.3. Quasi-split case

If $b$ is $\sigma$-conjugate to $1$, Theorem 1.1 admits a clearer
reformulation. Let $c\in W$ be Coxeter and
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$. If
$\dot{X}_{\bar{c}}(1)\neq\varnothing$, then a (any) lift
$\dot{c}\in{\mathbf{G}}(\breve{k})$ of $\bar{c}$ is $\sigma$-conjugate to $1$
(by Corollary 4.6). Moreover, choosing $\dot{c}$ appropriately we may
additionally assume that the (unique and automatically hyperspecial) fixed
point ${\mathbf{x}}={\mathbf{x}}_{\dot{c}}$ of $\dot{c}\sigma$ on
${\mathcal{A}}_{{\mathbf{T}},\breve{k}}$ lies in
${\mathcal{A}}_{{\mathbf{T}},k}={\mathcal{A}}_{{\mathbf{T}},\breve{k}}^{\langle\sigma\rangle}$
(cf. §6.4). In this situation we have the integral level analogues
$X_{c}^{{\mathcal{G}}_{{\mathbf{x}}}}(1)$ and
$\dot{X}_{\dot{c}}^{{\mathcal{G}}_{{\mathbf{x}}}}(1)$ of
$X_{c}(1),\dot{X}_{\bar{c}}(1)$, which were introduced in [DI20, Def. 4.1.1],
see §2.2. In this situation we have the following corollary, proven in §6.4.

######  Corollary 1.3.

With notation as above, there are ${\mathbf{G}}(k)$\- resp.
${\mathbf{G}}(k)\times{\mathbf{T}}_{c}(k)$-equivariant isomorphisms

$X_{c}(1)\cong\coprod_{\gamma\in{\mathbf{G}}^{\rm ad}(k)/{\mathcal{G}}^{\rm
ad}_{{\mathbf{x}},1}({\mathcal{O}}_{k})}\gamma
X_{c}^{{\mathcal{G}}_{\mathbf{x}}}(1)\quad\text{ and
}\quad\dot{X}_{\bar{c}}(1)\cong\coprod_{\gamma\in{\mathbf{G}}(k)/{\mathcal{G}}_{{\mathbf{x}},1}({\mathcal{O}}_{k})}\gamma\dot{X}_{\dot{c}}^{{\mathcal{G}}_{\mathbf{x}}}(1)$

###  1.4. Two related results

In the rest of the introduction we briefly discuss two further themes of this
article. First, we review in §3, following deBacker and Reeder [DeB06, DR09,
Ree11], the parametrization of rational conjugacy classes of unramified
maximal $k$-tori in pure inner forms of ${\mathbf{G}}$. Moreover, in §3.3-3.4
we extend their discussion to all extended pure inner forms, and give a
description of rational conjugacy classes in terms of
$\overline{F}_{w}/\ker\bar{\kappa}_{w}$, the discrete parameter space for the
spaces $\dot{X}_{\bar{w}}(b)$ over $X_{w}(b)$. In §3.5 we in detail
investigate the Coxeter case, where (under some mild assumptions on
${\mathbf{G}}$) it turns out that the set of different covers
$\dot{X}_{\bar{c}}(b)$ 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 ${\mathbf{G}}_{b}$ of ${\mathbf{G}}$, 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 §5, we show a loop version of twisted Steinberg’s cross section.
Steinberg’s original result [Ste65, Prop. 8.9] states roughly that if
$\dot{c}\in F_{c}(\breve{k})$ is any lift of a Coxeter element $c\in W$, any
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}$-orbit on ${}^{c}{\mathbf{U}}$ for the
action given by $g(u)=g^{-1}u\mathop{\rm Ad}(\dot{c})(g)$ contains precisely
one point in ${}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-}$ ($k$ can be replaced by
any field here). In [HL12, 3.14(c)] this is generalized in various respects,
in particular, to the case where $k$ is replaced by a finite field
${\mathbb{F}}_{q}$ and the action is Frobenius-twisted, i.e., given by
$g(u)=g^{-1}u\sigma_{\dot{c}}(g)$, where $\sigma$ is (geometric) Frobenius
over ${\mathbb{F}}_{q}$ and $\sigma_{\dot{c}}=\mathop{\rm
Ad}(\dot{c})\circ\sigma$. This can be used to prove affineness of classical
Deligne–Lusztig varieties attached to Coxeter elements (for example, by
combining [DL76, 1.12] and [HL12, 4.3]). In our setup, with $k$ non-
archimedean local, we need (to perform a reduction step in the proof of
Theorem 1.1) the same result for the action of
$L({}^{c}{\mathbf{U}}\cap{\mathbf{U}})$ on $L({}^{c}{\mathbf{U}})$. A part of
the proof in [HL12] can be adapted, but at some point in [HL12] the
Ax–Grothendieck theorem222It states, for an endomorphism $X\rightarrow X$ of a
scheme of finite type over an algebraically closed field $\Omega$, that if
$X(\Omega)\rightarrow X(\Omega)$ is injective, then it is also surjective. is
used, which is impossible in our setting (as $L{\mathbf{U}}$ is not even a
scheme). Therefore, we carry through a different argument, at least for
classical groups and special Coxeter elements.

### Outline

In §2 we introduce more notation, prove a fact about integral $p$-adic
Deligne–Lusztig spaces, and review the Borovoi fundamental group and
$\sigma$-conjugacy classes. In §3 we review (and extend) some results on
rational/stable conjugacy classes of unramified maximal tori of
${\mathbf{G}}$. In §4 we review and extend some results from [Iva23] about the
maps $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$. In §5 we prove loop (and
integral loop) versions of twisted Steinberg’s cross section. This is used in
§6 to replace $\dot{X}_{\bar{c}}(b)$ (and its integral analogue) by a more
tractable object. The main part of the article consists §6 and §7, where
Theorem 1.1 is proven.

### Acknowledgements

I want to thank Johannes Anschütz, Charlotte Chan, Lucas Mann and Peter
Scholze for several discussions related to the present work. Especially, I
want to thank Charlotte Chan, with whom I initially started to work on
Lusztig’s conjecture, and who, in particular, pointed out the reference [HL12]
to me. I thank Michael Borovoi for pointing out the fact used in the beginning
of §6.3.2 to me, and an anonymous referee for various helpful suggestions. The
author acknowledges support by the DFG via the Leibniz prize of Peter Scholze.

##  2\. Setup and preliminaries

Recall that ${\mathbf{G}}$ is an unramified group over the local field $k$. We
use the notation from §1.1.

###  2.1. More notation

Given a group $G$, $g\in G$ and $H\subseteq G$ any subset, we write
${}^{g}H=gHg^{-1}$. For an abelian group $A$ we write $A_{{\rm tors}}$ for its
torsion subgroup.

Let $p$ be the residue characteristic of $k$. For a ring $R$ of characteristic
$p$, denote by $\operatorname{Perf}_{R}$ the category of perfect algebras over
$R$. Denote by $\varpi$ a uniformizer of $k$ and by ${\mathbb{F}}_{q}$ (resp.
$\overline{{\mathbb{F}}}_{q}$) the residue field of $k$ (resp. $\breve{k}$).

For $R$ in $\operatorname{Perf}_{{\mathbb{F}}_{q}}$, let ${\mathbb{W}}(R)$ be
the unique (up to unique isomorphism) $\varpi$-adically complete and separated
${\mathcal{O}}_{k}$-algebra, in which $\varpi$ is not a zero-divisor and which
satisfies ${\mathbb{W}}(R)/\varpi{\mathbb{W}}(R)=R$. With other words if
$W(R)$ denotes the $p$-typical Witt-vectors of $k$, then
${\mathbb{W}}(R)=W(R)\otimes_{W({\mathbb{F}}_{q})}{\mathcal{O}}_{k}$ if
$\mathop{\rm char}k=0$ and ${\mathbb{W}}(R)=R[\\![\varpi]\\!]$ if $\mathop{\rm
char}k=p$. Denote by $\phi=\phi_{R}$ the Frobenius automorphisms of
${\mathbb{W}}(R)$, ${\mathbb{W}}(R)[1/\varpi]$, induced by $x\mapsto x^{q}$ on
$R$ (this notation is only used in the proof of Proposition 5.3 and §7). We
have ${\mathbb{W}}({\mathbb{F}}_{q})[1/\varpi]=k$ and
${\mathbb{W}}(\overline{{\mathbb{F}}}_{q})[1/\varpi]=\breve{k}$ (so that
$\phi_{\overline{{\mathbb{F}}}_{q}}=\sigma$).

Denote by $k^{\rm alg}$ a fixed algebraic closure of $k$ containing the
maximal algebraic subextension of $\breve{k}/k$. Denote by
$\langle\sigma\rangle\cong\hat{\mathbb{Z}}$ the group of continuous
automorphisms of $\breve{k}/k$.

####  2.1.1. Loop functors

For a $\breve{k}$-scheme $X$, let

$LX\colon\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}\rightarrow{\rm
Sets},\quad R\mapsto X({\mathbb{W}}(R)[1/\varpi^{-1}])$

be the loop space of $X$. If $X$ is affine of finite type over $\breve{k}$,
then $LX$ is representable by an ind-scheme. For an
${\mathcal{O}}_{\breve{k}}$-scheme ${\mathcal{X}}$, let

$L^{+}{\mathcal{X}}\colon\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}\rightarrow{\rm
Sets},\quad R\mapsto{\mathcal{X}}({\mathbb{W}}(R))$

be the space of positive loops of ${\mathcal{X}}$. Moreover, if $n\geq 1$, let

$L^{+}_{n}{\mathcal{X}}\colon\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}\rightarrow{\rm
Sets},\quad R\mapsto{\mathcal{X}}({\mathbb{W}}(R)/\varpi^{n}{\mathbb{W}}(R))$

be the truncated version of $L^{+}{\mathcal{X}}$. By [BGA18, Prop.-Def. 6.2],
$L^{+}_{n}{\mathcal{X}}$ is a scheme (resp. affine scheme) for arbitrary
(resp. affine) ${\mathcal{X}}$, and if ${\mathcal{X}}$ was of finite type,
then $L^{+}{\mathcal{X}}$ is too. Moreover,
$L^{+}{\mathcal{X}}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>along affine transition maps (cf. [BGA18, Prop. 8.5]). Thus, if
${\mathcal{X}}$ is a scheme (resp. affine scheme), then $L^{+}{\mathcal{X}}$
is too.

If $X$ is the base change to $\breve{k}$ of a $k$-scheme $X_{0}$, then the
functor $LX$ carries a natural automorphism, the geometric Frobenius, denoted
by $\sigma_{X_{0}}$ (or simply by $\sigma$, if $X_{0}$ is clear from context).
Similarly, we have the geometric Frobenius $\sigma=\sigma_{{\mathcal{X}}_{0}}$
on $L^{+}{\mathcal{X}}$ if the ${\mathcal{O}}_{\breve{k}}$-scheme
${\mathcal{X}}$ is the base change of an ${\mathcal{O}}_{k}$-scheme
${\mathcal{X}}_{0}$.

####  2.1.2. Groups

We write ${\mathbf{G}}^{\rm ad}$ resp. ${\mathbf{G}}^{\mathop{\rm sc}}$ for
the adjoint group, resp. the simply connected cover of the derived group of
${\mathbf{G}}$. If ${\mathbf{H}}\subseteq{\mathbf{G}}$ is a closed subgroup of
${\mathbf{G}}$, denote by ${\mathbf{H}}^{\rm ad}$ resp.
${\mathbf{H}}^{\mathop{\rm sc}}$ its image in ${\mathbf{G}}^{\mathop{\rm
ad}}$, resp. its preimage in ${\mathbf{G}}^{\mathop{\rm sc}}$. We identify the
Weyl groups of ${\mathbf{G}}^{\rm ad},{\mathbf{G}}^{\mathop{\rm sc}}$ with
$W$. We denote by ${\mathbf{Z}}$ the center of ${\mathbf{G}}$.

Write $X_{\ast}({\mathbf{T}})$ resp. $X^{\ast}({\mathbf{T}})$ for the free
${\mathbb{Z}}$-module of cocharacters resp. characters of ${\mathbf{T}}$.
Denote by $\Phi\subseteq X^{\ast}({\mathbf{T}})$ resp. $\Phi^{\vee}\subseteq
X_{\ast}({\mathbf{T}})$ the set of roots resp. coroots of ${\mathbf{G}}$;
denote by $\Phi^{+}$ resp. $\Delta$ the set of positive resp. simple roots,
determined by ${\mathbf{B}}$; denote by $s_{\alpha}\in W$ the reflection
attached to $\alpha\in\Phi$. For $\xi\in X_{\ast}({\mathbf{T}})$, denote by
$\varpi^{\xi}$ the image in ${\mathbf{T}}(\breve{k})$ of $\varpi$ under $\xi$.

Write ${\mathbf{N}}$ for the normalizer of ${\mathbf{T}}$. We have the
reductive ${\mathcal{O}}_{k}$-model ${\mathcal{T}}$ of ${\mathbf{T}}$, and
${\mathcal{T}}({\mathcal{O}}_{\breve{k}})$ is the maximal compact subgroup of
${\mathbf{T}}(\breve{k})$. The quotient
$\widetilde{W}={\mathbf{N}}(\breve{k})/{\mathcal{T}}({\mathcal{O}}_{\breve{k}})$
is the _extended affine Weyl group_ of ${\mathbf{T}}$ in ${\mathbf{G}}$. It is
naturally an extension $\widetilde{W}=W\ltimes X_{\ast}({\mathbf{T}})$, whose
splittings (resp. $\sigma$-equivariant splittings) $W\rightarrow\widetilde{W}$
correspond to hyperspecial points of ${\mathcal{A}}_{{\mathbf{G}},\breve{k}}$
(resp. of ${\mathcal{A}}_{{\mathbf{G}},\breve{k}}^{\langle\sigma\rangle}$).

####  2.1.3. Twisting Frobenius

For $b\in{\mathbf{G}}(\breve{k})$, we have the $k$-group ${\mathbf{G}}_{b}$,
which is defined as the functorial $\sigma$-centralizer of $b$, cf. [RZ96,
1.12] or [Kot97, 3.3, Appendix A]. Then
${\mathbf{G}}_{b}(k)=\\{g\in{\mathbf{G}}(\breve{k})\colon
g^{-1}b\sigma(g)=b\\}$. If $b$ is basic (cf. §2.4), then ${\mathbf{G}}_{b}$ is
an inner form of ${\mathbf{G}}$, we may identify
${\mathbf{G}}_{b}(\breve{k})={\mathbf{G}}(\breve{k})$ and we denote by
$\sigma_{b}\stackrel{{\scriptstyle\rm def}}{{=}}\mathop{\rm Ad}(b)\circ\sigma$
the twisted Frobenius on ${\mathbf{G}}(\breve{k})$. We also write $\sigma_{b}$
for the corresponding geometric Frobenius on
$L{\mathbf{G}}_{b}=L{\mathbf{G}}$.

Suppose that the basic element $b$ lies in ${\mathbf{N}}(\breve{k})$ and that
${\mathbf{x}}\in{\mathcal{A}}_{{\mathbf{T}},\breve{k}}$ is fixed by the action
of $b\sigma$. Let ${\mathcal{G}}_{{\mathbf{x}}}$ be the corresponding
parahoric model of ${\mathbf{G}}$ over ${\mathcal{O}}_{\breve{k}}$. Then
$\sigma_{b}$ fixes
${\mathcal{G}}_{{\mathbf{x}}}({\mathcal{O}}_{\breve{k}})\subseteq{\mathbf{G}}(\breve{k})$,
and $L^{+}{\mathcal{G}}_{{\mathbf{x}}}\subseteq L{\mathbf{G}}$. Thus
${\mathcal{G}}_{{\mathbf{x}}}$ descends to a ${\mathcal{O}}_{k}$-model of
${\mathbf{G}}_{b}$, which we denote by ${\mathcal{G}}_{{\mathbf{x}},b}$. We
have
$L^{+}{\mathcal{G}}_{{\mathbf{x}},b}\otimes_{{\mathbb{F}}_{q}}\overline{{\mathbb{F}}}_{q}=L^{+}{\mathcal{G}}_{{\mathbf{x}}}$.

Similarly, for $w\in W$ we have the (outer) $\breve{k}/k$-form
${\mathbf{T}}_{w}$ of ${\mathbf{T}}$, satisfying
${\mathbf{T}}_{w}(k)=\\{t\in{\mathbf{T}}(\breve{k})\colon{\rm
Ad}\,w(\sigma(t))=t\\}$. We may identify
${\mathbf{T}}_{w}(\breve{k})={\mathbf{T}}(\breve{k})$ and we denote by
$\sigma_{w}\stackrel{{\scriptstyle\rm def}}{{=}}\mathop{\rm Ad}(w)\circ\sigma$
the twisted Frobenius on this group, as well as on $L{\mathbf{T}}$.

####  2.1.4. Topologies

We work with two topologies on $\operatorname{Perf}_{{\mathbb{F}}_{p}}$: the
pro-étale topology [BS15] and the arc-topology [BM21]. Recall that a map
$S^{\prime}\rightarrow S$ of qcqs schemes is an arc-cover if any immediate
specialization in $S$ lifts to $S^{\prime}$. By [BM21, Thm. 5.16], the arc-
topology on perfect qcqs ${\mathbb{F}}_{p}$-schemes coincides with the
canonical topology. Thus any perfect qcqs scheme is an arc-sheaf. In
particular, for any qcqs scheme ${\mathcal{X}}/{\mathcal{O}}_{k}$,
$L^{+}{\mathcal{X}}$ is an arc-sheaf. For any quasi-projective scheme $X/k$,
$LX$ is an arc-sheaf by [Iva23, Thm. A].

####  2.1.5. Locally profinite groups

As in [Iva23, §4], we regard a locally profinite group $H$ as a
(representable) functor

$S\mapsto C^{0}(|S|,H)$

on $\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$. In particular,
${\mathbf{G}}_{b}(k)$ and ${\mathbf{T}}_{w}(k)$ are such functors. If $H$ is
profinite, this functor is even represented by an affine scheme.

####  2.1.6. Galois cohomology

For definition and standard facts about non-abelian cohomology we refer to
[Ser97]. If $M$ is discrete $\langle\sigma\rangle$-group, where
$\langle\sigma\rangle$ is as in §2.1, we write $H^{1}(\sigma,M)$ for the
corresponding first cohomology set. We identify a continuous $1$-cocylce
$a\colon\langle\sigma\rangle\rightarrow M$ with $a(\sigma)\in M$, and write
$[a(\sigma)]\in H^{1}(\sigma,M)$ for the corresponding cohomology class.

We will write $H^{1}(\sigma,{\mathbf{G}})$ resp.
$H^{1}(\sigma_{w},{\mathbf{T}})$ instead of
$H^{1}(\sigma,{\mathbf{G}}(\breve{k}))$ resp.
$H^{1}(\sigma,{\mathbf{T}}_{w}(\breve{k}))$. By a theorem of Steinberg [Ser97,
Thm. III.1’] resp. its extension [BS68, 8.6] if $\mathop{\rm char}k>0$, we
have $H^{1}(\sigma,{\mathbf{G}})=H^{1}(k,{\mathbf{G}})$ and
$H^{1}(\sigma_{w},{\mathbf{T}})=H^{1}(k,{\mathbf{T}}_{w})$ via inflation along
$\operatorname{Gal}_{k}\twoheadrightarrow\langle\sigma\rangle$. We will always
use the notation $H^{1}(\sigma,{\mathbf{G}})$,
$H^{1}(\sigma_{w},{\mathbf{T}})$ without further comment.

###  2.2. Review of some $p$-adic Deligne–Lusztig spaces

####  2.2.1. Spaces $X_{w}(b)$ and $\dot{X}_{\dot{w}}(b)$

Let $w\in W$, $\dot{w}\in{\mathbf{N}}(\breve{k})$ a lift of $w$ and
$b\in{\mathbf{G}}(\breve{k})$. Denote by
${\mathcal{O}}(w)\subseteq({\mathbf{G}}/{\mathbf{B}})^{2}$ (resp.
${\mathcal{O}}(\dot{w})\subseteq{\mathbf{G}}/{\mathbf{U}}$) the
${\mathbf{G}}$-orbit corresponding to $w$ (resp. $\dot{w}$) under the Bruhat
decomposition. The _$p$ -adic Deligne–Lusztig spaces_ $X_{w}(b)$ and
$\dot{X}_{\dot{w}}(b)$ were defined in [Iva23, Def. 8.3] by the cartesian
diagrams of functors on $\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$,

$\textstyle{X_{w}(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L{\mathcal{O}}(w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(G/B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},b\sigma)}$$\textstyle{L(G/B)\times
L(G/B)}$
and
$\textstyle{\dot{X}_{\dot{w}}(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\dot{\mathcal{O}}(\dot{w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L(G/U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},b\sigma)}$$\textstyle{L(G/U)\times
L(G/U)}$

where $b\sigma\colon
L(G/B)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}L(G/B)$, and $b$ acts by
left multiplication. Then $X_{w}(b),\dot{X}_{\dot{w}}(b)$ are sheaves for the
arc-topology (see [BM21]) on
$\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$ [Iva23, Cor. 8.4]. There
is a natural action of ${\mathbf{G}}_{b}(k)$ on $X_{w}(b)$ and of
${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{w}(k)$ on $\dot{X}_{w}(b)$.

####  2.2.2. Spaces $\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$ and
$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}$

These spaces were introduced in (1.1) and (1.2). Here we prove their
representability.

######  Lemma 2.1.

Let $\\{H_{n}\\}_{n\geq 1}$ be a inverse system of finite groups, acting
compatibly and freely on an inverse system $\\{X_{n}\\}_{n\geq 1}$ of pro-
étale (resp. arc-) sheaves on
$\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$. Write
$H=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>$X=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>The natural map
$X/H\rightarrow\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>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
injectivity, it is enough to show that the map from the pre-sheaf quotient
$R\mapsto X(R)/H(R)$ to
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>is injective. Let $R\in\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$.
Suppose $s_{1},s_{2}\in X(R)$ map to the same element in
$(\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}X_{n}/H_{n})(R)=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}(X_{n}/H_{n})(R)$.
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 $n\geq 1$, $S_{n}=\\{h_{n}\in H_{n}(R)\colon
h_{n}s_{1,n}=s_{2,n}\\}\neq\varnothing$. This sets form an inverse system. By
freeness of the action, we have $\\#S_{n}=1$, and it follows that
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>Then
$h\in\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>H(R)$ satisfies $hs_{1}=s_{2}$. This proves the injectivity. For surjectivity,
let $s_{n}\in X_{n}(R)/H_{n}(R)$ be a compatible collection of elements. For
each $n$, find some pro-étale cover $R\rightarrow R_{n}$ such that
$s_{n}|_{R_{n}}$ lifts to $\tilde{s}_{n}\in X_{n}(R_{n})$. As $R\rightarrow
R^{\prime}=\mathop{\underrightarrow{\lim}}_{n}R_{n}$ is still an pro-étale
cover [BS15, proof of Lm. 4.1.8], we may replace each $R_{n}$ by $R^{\prime}$
and $\tilde{s}_{n}$ by $\tilde{s}_{n}|_{R^{\prime}}$. Now, we construct an
element $\tilde{s}\in X(R^{\prime})$ mapping to $(s_{n})_{n\geq 1}$. Starting
with $\tilde{s}_{1}$ if $n=1$, and working by induction on $n$, suppose that
$(\tilde{s}_{i})_{i=1}^{n}$ is compatible. As the system $(s_{n})_{n\geq 1}$
is compatible, we can find some $h_{n+1}\in H_{n+1}(R^{\prime})$ such that
$h_{n+1}\tilde{s}_{n+1}$ maps to $\tilde{s}_{n}$ under
$X_{n+1}(R^{\prime})\rightarrow X_{n}(R^{\prime})$. Then replace
$\tilde{s}_{n+1}$ by $h_{n+1}\tilde{s}_{n+1}$, so that
$(\tilde{s}_{i})_{i=1}^{n+1}$ becomes compatible. We conclude by induction.
The argument for the arc-topoogy is the same, using [BM21, Cor. 2.18] instead
of [BS15, Lm. 4.1.8] ∎

For $n\geq 1$, let $\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}_{{\mathbf{x}}}}$ be
defined by the diagram (1.1) with $L^{+}$ replaced by $L^{+}_{n}$ everywhere.
Then all $\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}_{{\mathbf{x}}}}$ are affine
schemes and
$\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}_{{\mathbf{x}}}}$.
We also have the truncated version of (1.2): let ${\mathcal{T}}$ be the
closure of ${\mathbf{T}}$ in ${\mathcal{G}}_{{\mathbf{x}}}$ and
${\mathcal{T}}_{c}$ the ${\mathcal{O}}_{k}$-torus, arising by twitsing the
Frobenius on ${\mathcal{T}}$ by $c$. Then put
$X_{c,b,n}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm
ad}}:=\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}/T_{n}$,
where $T_{n}:={\mathcal{T}}_{c}^{\rm ad}({\mathcal{O}}_{k}/\varpi^{n})$.

######  Proposition 2.2.

$\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$ and
$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}$ are affine perfect schemes.
Also, $X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm
ad}}=\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm
ad}}/{\mathbf{T}}_{c}(k)$ remains unchanged if one takes the quotient with
respect to the arc- instead of the pro-étale topology. Moreover,
$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm
ad}}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}X_{c,b,n}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm
ad}}$.

###### Proof.

The claim for $\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$ follows
from the definition and the affineness of $L^{+}{\mathcal{X}}$ for an affine
scheme ${\mathcal{X}}/{\mathcal{O}}_{\breve{k}}$, cf. §2.1.1. Suppose that
${\mathbf{G}}$ is adjoint. For $n\geq 1$ let $T_{n}={\mathcal{T}}_{c}^{\rm
ad}({\mathcal{O}}_{k}/\varpi^{n})$ be as above. Note that the finite group
$T_{n}$ acts freely on the affine scheme
$\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}$ and we have
$X_{c,b}^{{\mathcal{G}}}=(\dot{X}_{\bar{c},b}^{{\mathcal{G}}}/T)_{\rm
proet}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}(\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}/T_{n})_{\rm
proet}$ by Lemma 2.1. By [Sta14, Tag 07S7], the fppf-quotient
$(\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}/T_{n})_{\rm fppf}$ is a scheme, which
is even affine (since $\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}$ is) and
$\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}\rightarrow(\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}/T_{n})_{\rm
fppf}$ is an fppf-torsor. As $T_{n}$ is finite and acts freely on
$\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}$, this is even an étale torsor. From
this, it follows that $(\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}/T_{n})_{\rm
fppf}=(\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}}/T_{n})_{\rm et}$, 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 $X_{c,b}^{{\mathcal{G}}}$, being the inverse limit of
affine perfect schemes, is itself one. ∎

######  Remark 2.3.

The map $g\mapsto g^{-1}\sigma_{b}(g)\colon L^{+}{\mathcal{G}}_{\bf
x}\rightarrow L^{+}{\mathcal{G}}_{\bf x}$ is pro-étale and –by Lang’s theorem
applied to all truncations– also surjective. Thus
$\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$,
$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}$ and their truncations are
always non-empty. Also it follows that
$\dot{X}_{\bar{c},b}^{{\mathcal{G}}_{{\mathbf{x}}}}$,
$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}$ are infinite-dimensional.
The truncations are perfections of finite-dimensional schemes. More precisely,
as the Lang map is étale, we have

$\dim\dot{X}_{\bar{c},b,n}^{{\mathcal{G}}_{{\mathbf{x}}}}=\dim
X_{c,b,n}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}=\dim
L^{+}_{n}({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})_{\bf x}=\ell(c)n,$

where $\ell(c)$ denotes the length of the Coxeter element $c\in W$.

######  Conjecture 2.4.

$X_{c,b}^{{\mathcal{G}}_{{\mathbf{x}}}^{\rm ad}}$ is an
${\mathbb{A}}^{\infty}$-bundle over a classical Deligne–Luszig variety
attached to the reductive quotient of the special fiber of
${\mathcal{G}}_{{\mathbf{x}}}$.

This is easily verified by hand for ${\mathbf{G}}={\bf GL}_{n}$, using the
explicit description in [CI21a].

####  2.2.3. Spaces $X_{w}^{{\mathcal{G}}}(1)$,
$\dot{X}_{\dot{w}}^{{\mathcal{G}}}(1)$

These spaces only appear in §1.3 and §6.4. Suppose
${\mathbf{x}}\in{\mathcal{A}}_{{\mathbf{T}},\breve{k}}^{\langle\sigma\rangle}$
is a hyperspecial point, and let ${\mathcal{G}}={\mathcal{G}}_{{\mathbf{x}}}$
be the corresponding reductive ${\mathcal{O}}_{\breve{k}}$-model of
${\mathbf{G}}$. The reductive ${\mathcal{O}}_{k}$-group
${\mathcal{G}}_{{\mathbf{x}},1}$ is quasi-split, and the closure
${\mathbf{B}}_{{\mathbf{x}}}$ of ${\mathbf{B}}$ in ${\mathcal{G}}$ is a
${\mathcal{O}}_{k}$-rational Borel subgroup. Moreover, ${\mathcal{G}}$ has a
Bruhat decomposition, in the sense that
$({\mathcal{G}}/{\mathbf{B}}_{{\mathbf{x}}})^{2}\otimes_{{\mathcal{O}}_{k}}{\mathcal{O}}_{\breve{k}}$
contains locally closed subschemes ${\mathcal{O}}(w)$ ($w\in W$), flat over
${\mathcal{O}}_{\breve{k}}$, which fiberwise induce the usual Bruhat
decomposition. If ${\mathbf{N}}_{{\mathbf{x}}},{\mathbf{U}}_{{\mathbf{x}}}$
are the closures of ${\mathbf{N}},{\mathbf{U}}$ in ${\mathcal{G}}$, one
similarly has
${\mathcal{O}}(\dot{w})\subseteq({\mathcal{G}}/{\mathbf{U}}_{{\mathbf{x}}})^{2}\otimes_{{\mathcal{O}}_{k}}{\mathcal{O}}_{\breve{k}}$
($\dot{w}\in{\mathbf{N}}_{{\mathbf{x}}}({\mathcal{O}}_{\breve{k}})$). Taking
$b=1$, $\dot{w}\in{\mathbf{N}}_{{\mathbf{x}}}({\mathcal{O}}_{\breve{k}})$ and
replacing ${\mathbf{G}},{\mathbf{B}},{\mathbf{U}}$ by
${\mathcal{G}},{\mathbf{B}}_{{\mathbf{x}}},{\mathbf{U}}_{{\mathbf{x}}}$ and
$L$ by $L^{+}$ everywhere in the diagrams in §2.2.1, we obtain integral
$p$-adic Deligne–Lusztig spaces $X_{w}^{{\mathcal{G}}}(1)$,
$\dot{X}_{\dot{w}}^{{\mathcal{G}}}(1)$, as introduced in [DI20, Def. 4.1.1].
They are acted on by ${\mathcal{G}}_{{\mathbf{x}},1}({\mathcal{O}}_{k})$ and
${\mathcal{G}}_{{\mathbf{x}},1}({\mathcal{O}}_{k})\times{\mathcal{T}}({\mathcal{O}}_{k})$
respectively.

###  2.3. Fundamental group and the Kottwitz map

The natural map ${\mathbf{T}}^{\mathop{\rm sc}}\rightarrow{\mathbf{T}}$
induces an injection $X_{\ast}({\mathbf{T}}^{\mathop{\rm sc}})\rightarrow
X_{\ast}({\mathbf{T}})$, and we identify $X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})$ with its image. The fundamental group of ${\mathbf{G}}$ in the sense of
Borovoi [Bor98] is the group

$\pi_{1}({\mathbf{G}})=X_{\ast}({\mathbf{T}})/X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}}),$

It is independent of the choice of ${\mathbf{T}}$, functorial in
${\mathbf{G}}$, the induced $W$-action on it is trivial, and it admits a
Frobenius action, which we denote by $\sigma$. Kottwitz constructed a
homomorphism
$\tilde{\kappa}_{\mathbf{G}}\colon{\mathbf{G}}(\breve{k})\rightarrow\pi_{1}({\mathbf{G}})$,
see [Kot97, §7] and [PR08, 2.a.2]. Reformulated, a classical result of
Kottwitz [Kot84, §6] states that $\tilde{\kappa}_{\mathbf{G}}$ induces an
isomorphism
$H^{1}(\sigma,{\mathbf{G}})\cong\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle,{\rm
tors}}$. For tori we have $\pi_{1}(-)=X_{\ast}(-)$. For any $w\in W$, any
$k$-rational embedding ${\mathbf{T}}_{w}\rightarrow{\mathbf{G}}$ induces the
map

$X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle,{\rm
tors}}=H^{1}(\sigma_{w},{\mathbf{T}})\rightarrow
H^{1}(\sigma,{\mathbf{G}})=\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle,{\rm
tors}}.$

which is obtained from
$X_{\ast}({\mathbf{T}})\rightarrow\pi_{1}({\mathbf{G}})$ by taking
$\sigma_{w}$\- resp. $\sigma$-invariants and passing to the torsion subgroup.
From the definition of $\pi_{1}({\mathbf{G}})$ we deduce an exact sequence

$X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})_{\langle\sigma_{w}\rangle}\stackrel{{\scriptstyle\beta_{w}}}{{\rightarrow}}X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}\rightarrow
0.$ (2.1)

In particular,

$\ker\left(H^{1}(\sigma_{w},{\mathbf{T}})\rightarrow
H^{1}(\sigma,{\mathbf{G}})\right)=\operatorname{im}\left(\beta_{w}\right)_{\rm
tors}$ (2.2)

######  Remark 2.5.

If ${\mathbf{G}}$ is adjoint and $w$ Coxeter, then $\beta_{w}=0$ and
(equivalently)
$X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}\cong\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$.
For split groups this follows from [Spr74, Cor. 8.3]. In the non-split case,
note that there is some $n\geq 1$ such that $\sigma_{w}^{n}=\sigma_{w_{\rm
spl}}$ as automorphisms of $X_{\ast}({\mathbf{T}})$ and
$X_{\ast}({\mathbf{T}}^{\rm sc})$, where $w_{\rm spl}$ is some (non-twisted)
Coxeter element in $W$ and $\sigma_{w_{\rm spl}}:={\rm Ad}(w_{\rm
spl})\circ\sigma^{n}$ is formed with respect to the base change of
${\mathbf{G}}$ to the unramified extension of $k$ of degree $n$. Then, as
automorphisms of $X_{\ast}({\mathbf{T}})$ and of $X_{\ast}({\mathbf{T}}^{\rm
sc})$, $\sigma_{w_{\rm spl}}-1$ factors through $\sigma_{w}-1$, so that we
have a surjection
$X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}\twoheadrightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w_{\rm spl}}\rangle}$ and a similar
surjection for $X_{\ast}({\mathbf{T}}^{\rm sc})$. These surjections form a
commutative square with the maps $\beta_{w}$ and $\beta_{w_{\rm spl}}$ (as
both are induced by the inclusion $X_{\ast}({\mathbf{T}}^{\rm sc})\rightarrow
X_{\ast}({\mathbf{T}})$). As $\beta_{w_{\rm spl}}=0$ by the split case, it
follows that also $\beta_{w}=0$.

By [PR08, §5] we may reinterpret $\tilde{\kappa}_{\mathbf{G}}$ as a map of
ind-schemes $\tilde{\kappa}_{\mathbf{G}}\colon
L{\mathbf{G}}\rightarrow\pi_{1}({\mathbf{G}})$, where we regard
$\pi_{1}({\mathbf{G}})$ as discrete perfect scheme via §2.1.5. Passing to
$\sigma$-coinvariants gives a surjective map

$\kappa_{\mathbf{G}}\colon
L{\mathbf{G}}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}.$ (2.3)

Finally, let us note that for $w\in W$ we have the exact sequence

$1\rightarrow{\mathbf{T}}_{w}(k)\rightarrow
L{\mathbf{T}}\stackrel{{\scriptstyle\sigma_{w}-1}}{{\longrightarrow}}L{\mathbf{T}}\stackrel{{\scriptstyle\kappa_{w}}}{{\longrightarrow}}X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}\rightarrow
1$ (2.4)

where we write $\kappa_{w}$ instead of $\kappa_{{\mathbf{T}}_{w}}$. Moreover,
$\kappa_{w}$ factors through the natural projection $\bar{\kappa}_{w}\colon
X_{\ast}({\mathbf{T}})\rightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$.

###  2.4. Newton map and $\sigma$-conjugacy classes

Let ${\mathfrak{f}}\in\operatorname{Perf}_{{\mathbb{F}}_{q}}$ be an
algebraically closed field, so that $L={\mathbb{W}}({\mathfrak{f}})[1/\varpi]$
is an extension of $\breve{k}$. Then $L/k$ is equipped with the Frobenius
$\sigma$ lifting that of ${\mathfrak{f}}/{\mathbb{F}}_{q}$ and $L^{\sigma}=k$.
In [Kot85] the set

$B({\mathbf{G}})=H^{1}(\sigma^{\mathbb{Z}},{\mathbf{G}}(L))\cong{\mathbf{G}}(L)/\mathop{\rm
Ad}\nolimits_{\sigma}{\mathbf{G}}(L)$

of $\sigma$-conjugacy classes of ${\mathbf{G}}(L)$ in considered, where $x,y$
are $\sigma$-conjugate if there is some $g\in{\mathbf{G}}(L)$ with
$g^{-1}x\sigma(g)=y$. Denote the $\sigma$-conjugacy class of
$b\in{\mathbf{G}}(L)$ by $[b]$ (or $[b]_{\mathbf{G}}$ if an ambiguity is
possible). The set $B({\mathbf{G}})$ is independent of the choice of
${\mathfrak{f}}$. To a $\sigma$-conjugacy class $[b]$ one can attach the
_Newton point_ $\nu_{b}\in(W\backslash
X_{\ast}({\mathbf{T}})_{{\mathbb{Q}}})^{\langle\sigma\rangle}$. Moreover, the
Kottwitz map $\kappa_{\mathbf{G}}({\mathfrak{f}})$ from (2.3) factors through
a map $\kappa_{\mathbf{G}}\colon
B({\mathbf{G}})\twoheadrightarrow\pi({\mathbf{G}})_{\langle\sigma\rangle}$.
Then the map

$(\nu,\kappa_{\mathbf{G}})\colon B({\mathbf{G}})\rightarrow(W\backslash
X_{\ast}({\mathbf{T}})_{{\mathbb{Q}}})^{\langle\sigma\rangle}\times\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
(2.5)

is injective. An element $b\in{\mathbf{G}}(L)$ is called _basic_ if $\nu_{b}$
is central in ${\mathbf{G}}$. We denote by $B({\mathbf{G}})_{\rm bas}\subseteq
B({\mathbf{G}})$ the subset of basic classes. There is a canonical injection
$H^{1}(\sigma,{\mathbf{G}})\hookrightarrow B({\mathbf{G}})_{\rm bas}$. In
[Kot85], the isomorphism
$H^{1}(\sigma,{\mathbf{G}})\cong\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle,{\rm
tors}}$ was extended to the natural commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 27.0pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}(\sigma,{\mathbf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
73.51384pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 92.48636pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.74318pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.74318pt\raise-32.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
92.48636pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle,{\rm
tors}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
117.72786pt\raise-8.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
117.72786pt\raise-32.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
28.82726pt\raise-43.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{B({\mathbf{G}})_{\rm
bas}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
77.01329pt\raise-38.78482pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
97.95413pt\raise-43.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
97.95413pt\raise-43.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
(2.6)

where the lower map is the restriction of $\kappa_{{\mathbf{G}}}$ to
$B({\mathbf{G}})_{\rm bas}$.

We recall how to compute the Newton point $\nu_{b}$ of $b\in{\mathbf{N}}(L)$,
see e.g. [RV14, Rem. 2.3(iii)]. Let $\tilde{w}\in\widetilde{W}$ be the image
of $b$ under ${\mathbf{N}}(L)\twoheadrightarrow\widetilde{W}$. There is an
integer $d>0$, such that $\sigma^{d}$ acts trivially on $\widetilde{W}$ and
such that $\mu=\tilde{w}\sigma(\tilde{w})\dots\sigma^{d-1}(\tilde{w})\in
X_{\ast}({\mathbf{T}})$. Then $\nu_{b}$ is the image of $d^{-1}\mu$ in
$W\backslash X_{\ast}({\mathbf{T}})_{\mathbb{Q}}$. In particular, $\nu_{b}$
only depends on $\tilde{w}$ and it makes sense to define
$\nu_{\tilde{w}}=\nu_{b}$. Moreover, we call $\tilde{w}$ basic if
$\nu_{\tilde{w}}$ is central.

Finally, we recall that any $\sigma$-conjugacy class has a representative in
${\mathbf{N}}(\breve{k})$, and that the induced surjection
${\mathbf{N}}(\breve{k})\twoheadrightarrow B({\mathbf{G}})$ factors through
$\widetilde{W}\twoheadrightarrow B({\mathbf{G}})$, cf. [GHKR10, Corollary
7.2.2] for split groups and [He14, Theorem 3.7] in general. In particular, the
Kottwitz homomorphism induces a map
$\bar{\kappa}_{\mathbf{G}}\colon\widetilde{W}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$.

##  3\. Rational and stable conjugacy classes of unramified tori

Recall that two maximal $k$-tori ${\mathbf{T}}_{1},{\mathbf{T}}_{2}$ in
${\mathbf{G}}$ are _stably conjugate_ if there exists $g\in{\mathbf{G}}(k^{\rm
alg})$ such that ${}^{g}({\mathbf{T}}_{1}(k))={\mathbf{T}}_{2}(k)$. 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 ${\mathbf{G}}$. Then we investigate the
case of _extended_ pure inner forms of ${\mathbf{G}}$. 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 $\dot{X}_{\bar{w}}(b)$.

###  3.1. Stable conjugacy classes

For $b\in{\mathbf{N}}(\breve{k})$, the twisted Frobenius $\sigma_{b}={\rm
Ad}\,b\circ\sigma$ of ${\mathbf{G}}(\breve{k})$ induces one on $W$. Suppose
now $b\in{\mathbf{N}}(\breve{k})$ 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 ${\mathbf{G}}_{b}$ into $H^{1}(\sigma_{b},W)$, which is
even a bijection when ${\mathbf{G}}_{b}$ is quasi-split (e.g., $[b]=[1]$).
Identifying
$H^{1}(\sigma_{b},W)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(\sigma,W)$
via the natural twisting map $x\mapsto xb$ on cocycles (using [Ser97, Prop.
I.35 bis]), we get

$\\{\text{stable conjugacy classes of $k$-tori in
${\mathbf{G}}_{b}$}\\}\hookrightarrow H^{1}(\sigma,W),$ (3.1)

mapping the class of ${\mathbf{T}}(\subseteq{\mathbf{G}}_{b})$ to the image of
$b$ in $H^{1}(\sigma,W)$.

######  Definition 3.1.

For $b\in{\mathbf{N}}(\breve{k})$ basic and $w\in W$, denote by
${\mathfrak{T}}({\mathbf{G}}_{b},w)$ the _stable conjugacy class_ of all
unramified maximal $k$-tori in ${\mathbf{G}}_{b}$, which corresponds to
$[w]\in H^{1}(\sigma,W)$ under (3.1).

The elements of ${\mathfrak{T}}({\mathbf{G}}_{b},w)$ are precisely the images
of $k$-embeddings ${\mathbf{T}}_{w}\rightarrow{\mathbf{G}}_{b}$.

######  Remark 3.2.

We recall the construction from the proof of [DeB06, Lemma 4.3.1]. Any
unramified maximal $k$-torus of ${\mathbf{G}}_{b}$ is of the form
${}^{g}{\mathbf{T}}$ for some $g\in{\mathbf{G}}_{b}(\breve{k})$. Moreover,
${}^{g}{\mathbf{T}}$ is $k$-rational if and only if
$\sigma_{b}({}^{g}{\mathbf{T}})={}^{g}{\mathbf{T}}$, which is equivalent to
$g^{-1}b\sigma(g)\in{\mathbf{N}}(\breve{k})$. In that case, if $w\in W$ is the
image of $g^{-1}b\sigma(g)$, then
${}^{g}{\mathbf{T}}\in{\mathfrak{T}}({\mathbf{G}}_{b},w)$. Replacing $g$ by
$g\dot{v}$ for any lift $\dot{v}\in{\mathbf{N}}(\breve{k})$ of some $v\in W$
has the effect of replacing $w$ by $v^{-1}w\sigma(v)$.

###  3.2. Rational conjugacy classes for pure inner forms

It is easy to see that the surjection
${\mathbf{N}}(\breve{k})\twoheadrightarrow\widetilde{W}$ induces an
isomorphism333Indeed, one can use [Ser97, Cor. 2 to Prop. I.39; Cor. to Prop.
I.41] along with pro-version of Lang’s theorem for the group
${\mathcal{T}}({\mathcal{O}}_{\breve{k}})=\ker({\mathbf{N}}(\breve{k})\twoheadrightarrow\widetilde{W})$
with varying rational structures given by $\sigma_{b}$
($b\in{\mathbf{N}}(\breve{k})$). $H^{1}(\sigma,{\mathbf{N}}(\breve{k}))\cong
H^{1}(\sigma,\widetilde{W})$.

The following is a consequence of [Ree11, Prop. 6.1]444In fact, [Ree11] works
with $\operatorname{Gal}_{k}$-cohomology and _all_ maximal $k$-tori. To deduce
our statements, one can run literally the same arguments with
$\langle\sigma\rangle$ replacing $\operatorname{Gal}_{k}$ and unramified
$k$-tori replacing all $k$-tori., see also [DR09, §2] and [DeB06, §4.3].
Suppose that $[b]\in B({\mathbf{G}})_{\rm bas}$ comes from a class $\xi\in
H^{1}(\sigma,{\mathbf{G}})$ (cf. (2.6)), i.e., ${\mathbf{G}}_{b}$ is a pure
inner form of ${\mathbf{G}}$. Then rational conjugacy classes of unramified
maximal $k$-tori in ${\mathbf{G}}_{b}$ are in bijection with preimages of
$\xi$ under the natural map
$H^{1}(\sigma,\widetilde{W})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}H^{1}(\sigma,{\mathbf{N}}(\breve{k}))\rightarrow
H^{1}(\sigma,{\mathbf{G}})$, cf. [Ree11, Lemma 6.2] (and use twisting). There
are maps

$H^{1}(\sigma,W)\stackrel{{\scriptstyle
q_{0}}}{{\leftarrow}}H^{1}(\sigma,\widetilde{W})\stackrel{{\scriptstyle\pi_{0}}}{{\rightarrow}}H^{1}(\sigma,{\mathbf{G}}).$
(3.2)

By §3.1, given $[w]\in H^{1}(\sigma,W)$, represented by $w\in W$, and $\xi\in
H^{1}(\sigma,{\mathbf{G}})$, represented by $b\in{\mathbf{N}}(\breve{k})$, the
set ${\mathfrak{T}}({\mathbf{G}}_{b},w)/\mathop{\rm Ad}{\mathbf{G}}_{b}(k)$ of
rational conjugacy classes in ${\mathfrak{T}}({\mathbf{G}}_{b},w)$ is in
bijection with $\pi_{0}^{-1}(\xi)\cap q_{0}^{-1}([w])$. In particular, $[w]$
is in the image of (3.1) if and only if $\pi_{0}^{-1}(\xi)\cap
q_{0}^{-1}([w])\neq\varnothing$.

###  3.3. Rational conjugacy classes for extended pure inner forms

We wish to extend (3.2) to all extended pure inner forms. We have
$X_{\ast}({\mathbf{Z}})\subseteq
X_{\ast}({\mathbf{T}})\subseteq\widetilde{W}$. Define

$\widetilde{Z}^{1}(\sigma,\widetilde{W})=\left\\{\tilde{w}\in\widetilde{W}\colon\nu_{\tilde{w}}\in
X_{\ast}({\mathbf{Z}})_{{\mathbb{Q}}}\right\\},$

where $\nu_{\tilde{w}}$ is as in §2.4. With other words,
$\widetilde{Z}^{1}(\sigma,\widetilde{W})$ is precisely the set of all basic
elements in $\widetilde{W}$ in the sense of §2.4. Define an equivalence
relation on $\widetilde{Z}^{1}(\sigma,\widetilde{W})$ by $x\sim y$ if and only
if $\exists g\in\widetilde{W}$ such that $y=g^{-1}x\sigma(g)$. Put

$\widetilde{H}^{1}(\sigma,\widetilde{W})=\widetilde{Z}^{1}(\sigma,\widetilde{W})/\sim$

Note that we have
$H^{1}(\sigma,\widetilde{W})\subseteq\widetilde{H}^{1}(\sigma,\widetilde{W})$
and
$Z^{1}(\sigma,\widetilde{W})\subseteq\widetilde{Z}^{1}(\sigma,\widetilde{W})$
is precisely the set of those $\tilde{w}$ with $\nu_{\tilde{w}}=0$. Recalling
from (2.6) that $H^{1}(\sigma,{\mathbf{G}})\subseteq B({\mathbf{G}})_{\rm
bas}$ we have the following lemma.

######  Lemma 3.3.

There is a natural surjection
$\pi\colon\widetilde{H}^{1}(\sigma,\widetilde{W})\rightarrow
B({\mathbf{G}})_{\rm bas}$ extending the surjection $\pi_{0}\colon
H^{1}(\sigma,\widetilde{W})\rightarrow H^{1}(\sigma,{\mathbf{G}})$.

###### Proof.

Suppose
$\tilde{w}_{1}\sim\tilde{w}_{2}\in\widetilde{Z}^{1}(\sigma,\widetilde{W})$,
i.e., $\tilde{w}_{2}=g^{-1}\tilde{w}_{1}\sigma(g)$. Then obviously
$\bar{\kappa}_{{\mathbf{G}}}(\tilde{w}_{1})=\bar{\kappa}_{{\mathbf{G}}}(\tilde{w}_{2})\in\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
and an easy computation shows that $\nu_{\tilde{w}_{1}}=\nu_{\tilde{w}_{2}}$.
Thus some (any) lifts to ${\mathbf{G}}(\breve{k})$ of $\tilde{w}_{1}$,
$\tilde{w}_{2}$ are $\sigma$-conjugate. This gives a map
$\widetilde{H}^{1}(\sigma,\widetilde{W})\rightarrow B({\mathbf{G}})$, whose
image is contained in $B({\mathbf{G}})_{\rm bas}$ as each
$\tilde{w}\in\widetilde{Z}^{1}(\sigma,\widetilde{W})$ is basic. The
surjectivity is clear. ∎

The projection $\widetilde{W}\twoheadrightarrow W$ induces a map
$q\colon\widetilde{H}^{1}(\sigma,\widetilde{W})\rightarrow H^{1}(\sigma,W)$.
Altogether, we have the diagram

$H^{1}(\sigma,W)\stackrel{{\scriptstyle
q}}{{\leftarrow}}\widetilde{H}^{1}(\sigma,\widetilde{W})\stackrel{{\scriptstyle\pi}}{{\rightarrow}}B({\mathbf{G}})_{\rm
bas}.$ (3.3)

extending (3.2). DeBacker and Reeder’s parametrization from §3.2 admits the
following completely analogous generalization to all extended pure inner
forms.

######  Proposition 3.4.

Given $[w]\in H^{1}(\sigma,W)$ and $[b]\in B({\mathbf{G}})_{\rm bas}$, we have
a natural bijection ${\mathfrak{T}}({\mathbf{G}}_{b},w)/\mathop{\rm
Ad}{\mathbf{G}}_{b}(k)\cong\pi^{-1}([b])\cap q^{-1}([w])$. In particular,
$[w]$ is in the image of (3.1) if and only if $\pi^{-1}([b])\cap
q^{-1}([w])\neq\varnothing$.

###### Proof.

Any unramified maximal $k$-torus in ${\mathbf{G}}_{b}$ is of the form
${}^{g}{\mathbf{T}}$ with $g^{-1}b\sigma(g)\in{\mathbf{N}}(\breve{k})$, see
Remark 3.2. Let $\tilde{w}_{g}\in\widetilde{W}$ be the image of
$g^{-1}b\sigma(g)$. Then $\nu_{\tilde{w}_{g}}=\nu_{b}$, and hence
$\tilde{w}_{g}\in\widetilde{Z}^{1}(\sigma,\widetilde{W})$ and
$\pi([\tilde{w}_{g}])=[b]$. Let ${}^{g_{i}}{\mathbf{T}}$ ($i=1,2$) be two such
tori with corresponding elements
$\tilde{w}_{i}\in\widetilde{Z}^{1}(\sigma,\widetilde{W})$. Suppose that
${}^{g_{i}}{\mathbf{T}}$ are rationally conjugate in ${\mathbf{G}}_{b}$, i.e.,
there is some $h\in{\mathbf{G}}_{b}(k)$ with
${}^{g_{1}}{\mathbf{T}}={}^{hg_{2}}{\mathbf{T}}$. Then
$g_{1}^{-1}hg_{2}\in{\mathbf{N}}(\breve{k})$ and letting
$\tilde{v}\in\widetilde{W}$ be its image, we compute

$\tilde{v}\tilde{w}_{2}\sigma(\tilde{v})^{-1}=g_{1}^{-1}hg_{2}g_{2}^{-1}b\sigma(g_{2})\sigma(g_{2}^{-1}h^{-1}g_{1})=g_{1}^{-1}hb\sigma(h^{-1})\sigma(g_{1})=g_{1}^{-1}b\sigma(g_{1})=\tilde{w}_{1},$
(3.4)

i.e., $\tilde{w}_{1}$ and $\tilde{w}_{2}$ define the same class in
$\widetilde{H}^{1}(\sigma,\widetilde{W})$. Conversely, given
$[\tilde{w}]\in\tilde{\pi}^{-1}([b])$, we can lift $\tilde{w}$ to some
$\dot{w}\in{\mathbf{N}}(\breve{k})$, which necessarily belongs to $[b]$. If
$\dot{w}=g^{-1}b\sigma(g)$, one immediately checks that
$\tilde{w}\mapsto{}^{g}{\mathbf{T}}$ induces an inverse to the above map. This
shows that rational conjugacy classes of unramified tori in ${\mathbf{G}}_{b}$
are in bijection with $\pi^{-1}([b])$, establishing the analogue of [Ree11,
Lemma 6.2]. The rest of the proof (the analogue of [Ree11, Lemma 6.4]) is done
analogously.∎

###  3.4. A description of $\pi^{-1}([b])\cap q^{-1}([w])$

Fix $w\in W$ and $[b]\in B({\mathbf{G}})_{\rm bas}$. Put

$F_{w}\stackrel{{\scriptstyle\rm def}}{{=}}\text{fiber of
${\mathbf{N}}\rightarrow W$ over $w$}\quad\text{ and
}\quad\overline{F}_{w}\stackrel{{\scriptstyle\rm def}}{{=}}\text{fiber of
$\widetilde{W}\rightarrow W$ over $w$}$

We regard $F_{w}$ as a $\breve{k}$-scheme and $\overline{F}_{w}$ as a discrete
$\overline{{\mathbb{F}}}_{q}$-scheme (or, sometimes, as a discrete set). Then
$F_{w}$ is a (trivial) ${\mathbf{T}}_{\breve{k}}$-torsor and
$\overline{F}_{w}$ is a (trivial) $X_{\ast}({\mathbf{T}})$-torsor. The action
of ${\mathbf{T}}_{\breve{k}}$ on $F_{w}$ is given by $t,\tilde{w}\mapsto
t\tilde{w}$ and the action of $X_{\ast}({\mathbf{T}})$ on $\overline{F}_{w}$
is induced by this. Recall the maps $\kappa_{w}\colon L{\mathbf{T}}\rightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$ and $\bar{\kappa}_{w}\colon
X_{\ast}({\mathbf{T}})\rightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$ from §2.3.

######  Lemma 3.5.

The natural surjection $F_{w}(\breve{k})\twoheadrightarrow\overline{F}_{w}$
induces an isomorphism
$LF_{w}/\ker\kappa_{w}\cong\overline{F}_{w}/\ker\overline{\kappa}_{w}$. This
discrete set is a (trivial)
$X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$-torsor.

###### Proof.

As $LF_{w}$ (resp. $\overline{F}_{w}$) is a trivial $L{\mathbf{T}}$-torsor
(resp. $X_{\ast}({\mathbf{T}})$-torsor), everything follows from
$L{\mathbf{T}}/\ker\kappa_{w}=X_{\ast}({\mathbf{T}})/\ker\bar{\kappa}_{w}=X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$.
∎

Observe that for $\tilde{w}\in\overline{F}_{w}$ the values of
$\nu_{\tilde{w}}$ and $\bar{\kappa}_{\mathbf{G}}(\tilde{w})$ remain constant
if $\tilde{w}$ is multiplied by an element in $\ker\bar{\kappa}_{w}$. Indeed,
for $\bar{\kappa}_{w}$ this is clear and for $\nu$ this follows from the
description in §2.4. In particular,
$\bar{\kappa}_{{\mathbf{G}}}|_{\overline{F}_{w}}$ induces a map

$\bar{\kappa}^{w}\colon\overline{F}_{w}/\ker\bar{\kappa}_{w}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
(3.5)

Moreover, from §2.4 we immediately deduce the following lemma.

######  Lemma 3.6.

(2.5) factors through an injection
$(\nu,\bar{\kappa}^{w})\colon\overline{F}_{w}/\ker\bar{\kappa}_{w}\rightarrow(W\backslash
X_{\ast}({\mathbf{T}})_{{\mathbb{Q}}})^{\langle\sigma\rangle}\times\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$.

Fix $w\in W$ and $[b]\in B({\mathbf{G}})_{\rm bas}$. Put

$[b]\cap\overline{F}_{w}\stackrel{{\scriptstyle\rm def}}{{=}}{\rm
im}\left([b]\cap F_{w}(\breve{k})\hookrightarrow
F_{w}(\breve{k})\twoheadrightarrow\overline{F}_{w}\right).$

Then
$[b]\cap\overline{F}_{w}\subseteq[b]\cap\widetilde{Z}^{1}(\sigma,\widetilde{W})$,
inducing a map
$[b]\cap\overline{F}_{w}\rightarrow\pi^{-1}([b])\subseteq\widetilde{H}^{1}(\sigma,\widetilde{W})$.
This obviously factors through a map
$[b]\cap\overline{F}_{w}\rightarrow\pi^{-1}([b])\cap q^{-1}([w])$. By Lemma
3.6, $[b]\cap\overline{F}_{w}$ is stable under the action of
$\ker\bar{\kappa}_{w}$, and we have the (finite) quotient set
$[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$, consisting of all
$\bar{w}\in\overline{F}_{w}/\ker\bar{\kappa}_{w}$ satisfying
$\nu_{\bar{w}}=\nu_{b}$ and
$\bar{\kappa}^{w}(\bar{w})=\kappa_{\mathbf{G}}(b)$.

######  Proposition 3.7.

  * (i)

$[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$ is either empty or a
$\ker(H^{1}(\sigma_{w},{\mathbf{T}})\rightarrow
H^{1}(\sigma,{\mathbf{G}}))=\operatorname{im}(\beta_{w})_{\rm tors}$-torsor
(cf. (2.2)). It is empty if and only if
${\mathfrak{T}}({\mathbf{G}}_{b},w)=\varnothing$.

  * (ii)

The centralizer $C_{W}(w\sigma)\subseteq W$ of $w\sigma\in
W\rtimes\langle\sigma\rangle$ acts on
$[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$ in a natural way and there are
canonical isomorphisms

$\left([b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}\right)/C_{W}(w\sigma)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\pi^{-1}([b])\cap
q^{-1}([w])\cong{\mathfrak{T}}({\mathbf{G}}_{b},w)/\mathop{\rm
Ad}{\mathbf{G}}_{b}(k).$

###### Proof.

We have
${\mathfrak{T}}({\mathbf{G}}_{b},w)\neq\varnothing\,\Leftrightarrow\,\exists
g\in{\mathbf{G}}(\breve{k})\text{ such that }g^{-1}b\sigma(g)\in
F_{w}(\breve{k})\,\Leftrightarrow\,[b]\cap\overline{F}_{w}\neq\varnothing$,
where the first equivalence follows from Remark 3.2. To show the first claim
of (i) we may assume that there is an element
$\tilde{w}_{1}\in[b]\cap\overline{F}_{w}\neq\varnothing$. Recall that
$\overline{F}_{w}$ is a $X_{\ast}({\mathbf{T}})$-torsor.

######  Lemma 3.8.

For $\tau\in X_{\ast}({\mathbf{T}})$ we have
$\tau\tilde{w}_{1}\in[b]\cap\overline{F}_{w}\Leftrightarrow\bar{\kappa}_{w}(\tau)\in\ker(H^{1}(\sigma_{w},{\mathbf{T}})\rightarrow
H^{1}(\sigma,{\mathbf{G}}))$.

###### Proof.

$\tau\tilde{w}_{1}\in[b]\cap\overline{F}_{w}$ if and only if
$\bar{\kappa}_{\mathbf{G}}(\tau)=0$ (as $\kappa_{\mathbf{G}}$ is a group
homomorphism) and $\nu_{\tau\tilde{w}_{1}}=\nu_{\tilde{w}_{1}}$. Now,
$\bar{\kappa}_{\mathbf{G}}(\tau)=0$ means that the image
$\bar{\kappa}_{w}(\tau)\in X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$
maps to $0\in\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$, which is
equivalent to $\bar{\kappa}_{w}(\tau)\in\operatorname{im}(\beta_{w})$, with
$\beta_{w}$ as in (2.1). Now we use the algorithm to compute $\nu$ from §2.4.
Note that the integer $d$ can be chosen the same for $\tilde{w}_{1}$,
$\tau\tilde{w}_{1}$. Let then $\mu_{\tilde{w}_{1}}$, $\mu_{\tau\tilde{w}_{1}}$
be as described in §2.4. Then $\nu_{\tau\tilde{w}_{1}}=\nu_{\tilde{w}_{1}}$ is
equivalent to the existence of some $v\in W$ such that the equality

$\sum_{i=0}^{d-1}(w\sigma)^{i}(\tau)=v(\mu_{\tilde{w}_{1}})-\mu_{\tilde{w}_{1}}$

holds in $X_{\ast}({\mathbf{T}})$. But $\tilde{w}_{1}$ is basic, so
$\mu_{\tilde{w}_{1}}$ is central, i.e., the right hand side is zero. Thus
$\tau\tilde{w}_{1}\in[b]\cap\overline{F}_{w}$ if and only if
$\bar{\kappa}_{w}(\tau)\in\operatorname{im}(\beta_{w})$ and
$\sum_{i=0}^{d-1}(w\sigma)^{i}(\tau)=0$. By [DR09, Lemma 2.4.1], the latter
condition is equivalent to
$\tau\in\bar{\kappa}_{w}^{-1}(X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle,{\rm
tors}})=\bar{\kappa}_{w}^{-1}(H^{1}(\sigma_{w},{\mathbf{T}}))$. ∎

By Lemma 3.8, $[b]\cap\overline{F}_{w}$ (being non-empty) is a torsor under
the preimage in $X_{\ast}({\mathbf{T}})$ of
$\ker(H^{1}(\sigma,{\mathbf{T}}_{w})\rightarrow
H^{1}(\sigma,{\mathbf{G}}))\subseteq
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$. This preimage contains
$\ker\bar{\kappa}_{w}$ 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
$[b]\cap\overline{F}_{w}\longrightarrow\pi^{-1}([b])\cap q^{-1}([w])$, and one
checks that it factors through $[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$.
Moreover, surjectivity of this map amounts to the tautological fact that any
element in $[w]\subseteq W$ is $\sigma$-conjugate to $w$. Next, let us
describe the natural action of $C_{W}(w\sigma)$ on
$[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$. Let
$\bar{w}\in[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$ with preimage
$\tilde{w}\in[b]\cap\overline{F}_{w}$. Let $v\in C_{W}(w\sigma)$ and let
$\tilde{v}\in\widetilde{W}$ be any preimage. Then the action of $v$ sends
$\bar{w}$ to the image of $\tilde{v}^{-1}\tilde{w}\sigma(\tilde{v})$ in
$[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$. This is independent of the
choice of $\tilde{v}$, as if $\tilde{v}^{\prime}$ is another choice, putting
$\tau=\tilde{v}^{-1}\tilde{v}^{\prime}\in X_{\ast}({\mathbf{T}})$ gives the
element
$\tilde{v}^{\prime-1}\tilde{w}\sigma(\tilde{v}^{\prime})=\tau^{-1}\tilde{v}^{-1}\tilde{w}\sigma(\tilde{v})\sigma(\tau)$,
which has the same image in $[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$.
Similarly, one shows that it is independent of the choice of $\tilde{w}$. Now,
$\bar{w}^{\prime}\in[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$ has the same
image as $\bar{w}$ in $\pi^{-1}([b])\cap q^{-1}([w])$ if and only if there is
some $\tilde{v}\in\widetilde{W}$ with
$\tilde{v}^{-1}\tilde{w}\sigma(\tilde{v})=\tilde{w}^{\prime}$ for some lift
$\tilde{w}^{\prime}\in\overline{F}_{w}$ of $\bar{w}^{\prime}$. By definition
of the $C_{W}(w\sigma)$-action this is equivalent to the fact that
$\bar{w},\bar{w}^{\prime}$ lie in the same $C_{W}(w\sigma)$-orbit. Thus
$C_{W}(w\sigma)$-orbits in $[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$
coincide with the fibers of the surjection onto $\pi^{-1}([b])\cap
q^{-1}([w])$, and we are done.∎

######  Remark 3.9.

Let us explicate the correspondence in Proposition 3.7(ii). If
$\bar{w}\in[b]\cap\overline{F}_{w}/\ker\bar{\kappa}_{w}$ represents a class on
the left side, $\dot{w}\in[b]\cap F_{w}(\breve{k})$ is a (any) lift of
$\bar{w}$, and $g\in{\mathbf{G}}(\breve{k})$ satisfies
$g^{-1}b\sigma(g)=\dot{w}$, then the map ${\mathop{\rm
Ad}(g)}\circ\iota\colon{\mathbf{T}}_{w}\rightarrow{\mathbf{G}}_{b}$ is
$k$-rational, its image ${}^{g}{\mathbf{T}}\subseteq{\mathbf{G}}_{b}$ lies in
${\mathfrak{T}}({\mathbf{G}}_{b},w)$, and the ${\mathbf{G}}_{b}(k)$-conjugacy
class of ${}^{g}{\mathbf{T}}$ is the image of $\bar{w}$ under the above map.

###  3.5. Case of Coxeter tori

We keep the setup of §3.3,3.4. Furthermore, throughout §3.5 we assume that
$w=c\in W$ _is a Coxeter element_. Recall $\beta_{c}\colon
X_{\ast}({\mathbf{T}}^{\mathop{\rm sc}})_{\langle\sigma_{c}\rangle}\rightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{c}\rangle}$ from (2.1).

######  Proposition 3.10.

Let $[b]\in B({\mathbf{G}})_{\rm bas}$ and $c\in W$ Coxeter. Then the
following hold.

  * (i)

${\rm im}(\beta_{c})$ is finite.

  * (ii)

We have

$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}=(\bar{\kappa}^{c})^{-1}(\kappa_{\mathbf{G}}(b)).$

This is a ${\rm im}(\beta_{c})$-torsor. In particular,
$[b]\cap\overline{F}_{c}\neq\varnothing$.

###### Proof.

(i): As $c$ Coxeter, $X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})^{\langle\sigma_{c}\rangle}=0$, hence $X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})_{\langle\sigma_{c}\rangle}$ is finite. Hence ${\rm im}(\beta_{c})$ is
finite. For (ii) we first record the following easy and well-known lemma.

######  Lemma 3.11.

If $c$ is Coxeter, all elements in $\overline{F}_{c}$ are basic.

###### Proof.

Let $\tilde{c}\in\overline{F}_{c}$. If $\mu$ is as in the algorithm to compute
$\nu_{b}$ from §2.4, we see that $\mu\in
X_{\ast}({\mathbf{T}})^{\langle\sigma_{c}\rangle}_{{\mathbb{Q}}}\subseteq
X_{\ast}({\mathbf{Z}})_{{\mathbb{Q}}}$ the last inclusion being true as $c$ is
Coxeter. ∎

Now note that
$\bar{\kappa}_{\mathbf{G}}|_{\overline{F}_{c}}\colon\overline{F}_{c}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
is surjective. This holds as $\overline{F}_{c}$ is a
$X_{\ast}({\mathbf{T}})$-torsor, $\kappa_{{\mathbf{G}}}$ is a group
homomorphism and
$X_{\ast}({\mathbf{T}})\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
is surjective. Taking any preimage $b^{\prime}$ of $\kappa_{\mathbf{G}}(b)$ in
$\overline{F}_{c}$, Lemma 3.11 shows that $b^{\prime}$ is basic. Moreover, we
know that $\kappa_{\mathbf{G}}$ induces a bijection $B({\mathbf{G}})_{\rm
bas}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$. It follows that
$b^{\prime}\in[b]$, i.e., $[b]\cap\overline{F}_{c}\neq\varnothing$. Moreover,
as this holds for any $b^{\prime}$ in the preimage, we even get the equality
claimed in (ii). Now, $(\bar{\kappa}^{c})^{-1}(\bar{\kappa}_{\mathbf{G}}(b))$
is a
$\ker(X_{\ast}({\mathbf{T}}_{\langle\sigma_{c}\rangle}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle})=\operatorname{im}(\beta_{c})$-torsor.
∎

######  Remark 3.12.

It follows from Proposition 3.10(ii) that for any $[b]\in B({\mathbf{G}})_{\rm
bas}$ and any Coxeter element $c$, there exists a representative of $[b]$ in
$F_{c}(\breve{k})$. This is a converse to Lemma 3.11.

######  Proposition 3.13.

Let $c\in W$ be Coxeter. Suppose one of the following holds:

  * (a)

$[b]=1$, or

  * (b)

${\mathbf{G}}=\prod_{i=1}^{s}{\mathbf{G}}_{i}$, where for each $i$,
${\mathbf{G}}_{i}={\rm Res}_{k_{i}/k}{\mathbf{G}}_{i}^{\prime}$ for a finite
unramified extension $k_{i}/k$ and an absolutely almost simple $k_{i}$-group
${\mathbf{G}}^{\prime}_{i}$.

Then the $C_{W}(c\sigma)$-action on
$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}$ is trivial and there is a
natural bijection

$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathfrak{T}}({\mathbf{G}}_{b},c)/\mathop{\rm
Ad}{\mathbf{G}}_{b}(k).$

###### Proof.

Choose some $\sigma$-equivariant splitting $\iota\colon
W\rightarrow\widetilde{W}$ (which exists, as ${\mathbf{G}}$ is unramified) and
denote by $\dot{W}$ its image. Let $\dot{c}\in\overline{F}_{c}\cap\dot{W}$.
Let $\tau\in X_{\ast}({\mathbf{T}})$, such that
$\tau\dot{c}\in[b]\cap\overline{F}_{c}$. Let $v\in C_{W}(c\sigma)$ and let
$\dot{v}=\iota(v)$. The action of $v$ sends the image of $\tau\dot{c}$ in
$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}$ to the image in
$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}$ of the element
$\dot{v}^{-1}\tau\dot{c}\sigma(\dot{v})$. As $\iota$ is $\sigma$-equivariant,
$\dot{v}\in C_{\dot{W}}(\dot{c}\sigma)$, so
$\dot{v}^{-1}\tau\dot{c}\sigma(\dot{v})=\dot{v}^{-1}\tau\dot{v}\dot{c}={\rm
Ad}(v)^{-1}(\tau)\dot{c}$, i.e., we must show (replacing $v$ by $v^{-1}$) that
$({\rm Ad}\,v-\operatorname{id})(\tau)=\mathop{\rm
Ad}\,v(\tau)-\tau\in\ker\bar{\kappa}_{c}$.

In case (a) we have
$[1]\cap\overline{F}_{c}=X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})\cdot\dot{c}$. Thus, as $\ker\bar{\kappa}_{c}^{\mathop{\rm
sc}}\subseteq\ker\bar{\kappa}_{c}$, it suffices to show that $({\rm
Ad}\,v-\operatorname{id})(X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}}))\subseteq\ker\bar{\kappa}_{c}^{\mathop{\rm sc}}$. With other words, we
are reduced to the case that ${\mathbf{G}}$ is simply connected, which is
covered by (b).

In case (b), one is immediately reduced to the case of
${\mathbf{G}}=\mathop{\rm Res}_{k^{\prime}/k}{\mathbf{G}}^{\prime}$ with
$k^{\prime}/k$ a finite subextension of $\breve{k}/k$ and
${\mathbf{G}}^{\prime}$ an absolutely almost simple $k^{\prime}$-group. Let
$d=[k^{\prime}\colon k]$. Write
$X_{\ast}({\mathbf{T}})=\bigoplus_{i=1}^{d}X_{\ast}({\mathbf{T}}^{\prime})$
and $W=\prod_{i=1}^{d}W^{\prime}$, with $\sigma$-action given by
$(v_{1},\dots,v_{d})\mapsto(\sigma^{\prime}(v_{d}),v_{1},\dots,v_{d-1})$ on
both objects, where $\sigma^{\prime}$ is the $\breve{k}/k^{\prime}$-Frobenius
for ${\mathbf{G}}^{\prime}$. If $c,c_{1}\in W$ are $\sigma$-conjugate, the
statement claimed in the proposition holds for $c$ if and only if it holds for
$c_{1}$. Thus we may assume that $c=(c^{\prime},1,\dots,1)$ where
$c^{\prime}\in W^{\prime}$ is a Coxeter element in $W^{\prime}$. Suppose we
know the claim for ${\mathbf{G}}^{\prime},c^{\prime}$, i.e., for any
$v^{\prime}\in C_{W^{\prime}}(c^{\prime}\sigma^{\prime})$, $(\mathop{\rm
Ad}v^{\prime}-\operatorname{id})(X_{\ast}({\mathbf{T}}^{\prime}))\subseteq(\sigma^{\prime}_{c^{\prime}}-\operatorname{id})(X_{\ast}({\mathbf{T}}^{\prime}))$.
We have

$\displaystyle\ker\bar{\kappa}_{c}$
$\displaystyle=(\sigma_{c}-\operatorname{id})(X_{\ast}({\mathbf{T}}))=\left\\{\sigma^{\prime}_{c^{\prime}}(\alpha_{d})-\alpha_{1},\alpha_{1}-\alpha_{2},\dots,\alpha_{r-1}-\alpha_{d}\colon\alpha_{i}\in
X_{\ast}({\mathbf{T}}^{\prime})\text{ for }1\leq i\leq d\right\\}$
$\displaystyle=\left\\{\left(\sigma^{\prime}_{c^{\prime}}(\alpha_{r})-\alpha_{d}-\sum_{i=1}^{d-1}\beta_{i},\beta_{1},\dots,\beta_{d-1}\right)\colon\beta_{i},\alpha_{d}\in
X_{\ast}({\mathbf{T}}^{\prime})\text{ for }1\leq i\leq d-1\right\\}$

Inclusion $W^{\prime}\rightarrow W$ into first factor induces an isomorphism
$C_{W^{\prime}}(c^{\prime}\sigma^{\prime})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}C_{W}(c\sigma)$,
$v^{\prime}\mapsto(v^{\prime},\dots,v^{\prime})=v$. For $(e_{i})_{i=1}^{d}\in
X_{\ast}({\mathbf{T}})$, we have $(\mathop{\rm
Ad}v-\operatorname{id})((e_{i})_{i})=((\mathop{\rm
Ad}v^{\prime}-\operatorname{id})(e_{i}))_{i=1}^{d}$. This lies in
$\ker\bar{\kappa}_{c}$, as, setting $\beta_{i}:=(\mathop{\rm
Ad}v^{\prime}-\operatorname{id})(e_{i+1})$ for $1\leq i\leq d-1$, it suffices
to find an $\alpha_{d}\in X_{\ast}({\mathbf{T}}^{\prime})$ such that
$\sigma^{\prime}_{c^{\prime}}(\alpha_{d})-\alpha_{d}=\mathop{\rm
Ad}v^{\prime}(\sum_{i=1}^{d}e_{i})-\sum_{i=1}^{d}e_{i}$, which works by
assumption for ${\mathbf{G}}^{\prime}$.

It remains to check the claim for an absolutely almost simple group
${\mathbf{G}}$ over $k$. Let $d>0$ be the smallest positive integer, such that
$\sigma^{d}$ acts trivially on the Dynkin diagram of ${\mathbf{G}}$. By
[Spr74, Thm. 7.6(v)], $C_{W}(c\sigma)$ is the cyclic subgroup of $W$ generated
by $c^{\rm spl}=c\sigma(c)\dots\sigma^{d-1}(c)$. Thus, it suffices to check
that $(\mathop{\rm Ad}c^{\rm
spl}-\operatorname{id})(X_{\ast}({\mathbf{T}}))\subseteq\ker\bar{\kappa}_{c}=\operatorname{im}(\sigma_{c}-\operatorname{id}\colon
X_{\ast}({\mathbf{T}})\rightarrow X_{\ast}({\mathbf{T}}))$. Observe that
$\sigma_{c}(\ker\bar{\kappa}_{c})=\ker\bar{\kappa}_{c}$ (as $\sigma_{c}$ is an
automorphism and for any $v\in X_{\ast}({\mathbf{T}})$,
$\sigma_{c}(\sigma_{c}(v)-v)=\sigma_{c}(\sigma_{c}(v))-\sigma_{c}(v)\in\ker\bar{\kappa}_{c}$).
Now $\sigma^{d}$ acts trivially on $X_{\ast}({\mathbf{T}})$, so it suffices to
check that for all $\tau\in X_{\ast}({\mathbf{T}})$, $c^{\rm
spl}\sigma^{d}(\tau)-\tau\in\ker\bar{\kappa}_{c}$, or equivalently,
$\sigma_{c}^{d}(v)-v\in\ker\bar{\kappa}_{c}$. We now prove that
$\sigma_{c}^{i}(v)-v\in\ker\bar{\kappa}_{c}$ for all $i>0$. For $i=1$, this
holds as $\ker\bar{\kappa}_{c}=\operatorname{im}(\sigma_{c}-1)$. For $i>1$, we
have
$\sigma_{c}^{i}(v)-v=\sigma_{c}^{i}(v)-\sigma_{c}^{i-1}(v)+\sigma_{c}^{i-1}(v)-v=\sigma_{c}^{i-1}(\sigma_{c}(v)-v)+\sigma_{c}^{i-1}(v)-v$.
Now we have $\sigma_{c}^{i-1}(v)-v\in\ker\bar{\kappa}_{c}$ by induction on
$i$, and
$\sigma_{c}^{i-1}(\sigma_{c}(v)-v)\in\sigma_{c}^{i-1}(\ker\bar{\kappa}_{c})=\ker\bar{\kappa}_{c}$.
∎

Proposition 3.13 becomes false without any assumptions on ${\mathbf{G}}$, $b$,
as the next example shows.

######  Example 3.14.

Suppose $\mathop{\rm char}k\neq 2$. Let ${\mathbf{G}}^{\prime}={\bf
SL}_{2}\times{\bf SL}_{2}$ and ${\mathbf{G}}={\mathbf{G}}^{\prime}/\mu_{2}$,
embedded diagonally. Let ${\mathbf{T}}^{\prime}\subseteq{\mathbf{G}}^{\prime}$
be the product of diagonal tori in ${\bf SL}_{2}$ and ${\mathbf{T}}$ its image
in ${\mathbf{G}}$. Let $\varepsilon_{1}=((1,-1),(0,0))$,
$\varepsilon_{2}=((0,0),(1,-1))$ be a ${\mathbb{Z}}$-basis of
$X_{\ast}({\mathbf{T}}^{\prime})$. We may identify
$X_{\ast}({\mathbf{T}})=\\{\frac{a_{1}\varepsilon_{1}+a_{2}\varepsilon_{2}}{2}\in
X_{\ast}({\mathbf{T}}^{\prime})_{\mathbb{Q}}\colon
a_{1},a_{2}\in{\mathbb{Z}},a_{1}\equiv a_{2}\mod 2\\}$. Let $s$ be the image
of $\left(\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right)$ in the Weyl
group of the diagonal torus in ${\bf SL}_{2}$, so that $c=(s,s)$ represents a
Coxeter element in $W$. Let $\dot{c}_{0}=\left(\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)\right)\in F_{c}(\breve{k})$ be a hyperspecial
lift, and let $\bar{c}_{0}$ be its image in $\overline{F}_{c}$. We have
$C_{W}(c)=W\cong{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$.
Note that
$\tau_{1}=(\varpi^{\frac{1}{2}},-\varpi^{-\frac{1}{2}}),(\varpi^{\frac{1}{2}},-\varpi^{-\frac{1}{2}})\in{\mathbf{G}}(k)$
by Galois descent. The action of Frobenius $\sigma$ on all objects below is
trivial, so we ignore it. We have
$\pi_{1}({\mathbf{G}})=X_{\ast}({\mathbf{T}})/X_{\ast}({\mathbf{T}}^{\prime})\cong{\mathbb{Z}}/2{\mathbb{Z}}$,
the non-trivial element being the class of
$\frac{\varepsilon_{1}+\varepsilon_{2}}{2}$. It is represented by the basic
element $b_{1}=\dot{c}_{0}\tau_{1}$. We have
$\ker\bar{\kappa}_{c}={\mathbb{Z}}\cdot
2\varepsilon_{1}+{\mathbb{Z}}\cdot(\varepsilon_{2}-\varepsilon_{1})\subseteq
X_{\ast}({\mathbf{T}})$.

_Case $b=1$_. We have
$[1]\cap\overline{F}_{c}=X_{\ast}({\mathbf{T}}^{\prime})\cdot\bar{c}_{0}$. As
$X_{\ast}({\mathbf{T}}^{\prime})/\ker\bar{\kappa}_{c}=\\{\bar{0},\bar{\varepsilon}_{1}\\}\cong{\mathbb{Z}}/2{\mathbb{Z}}$,
we have
$[1]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}=(X_{\ast}({\mathbf{T}}^{\prime})/\ker\bar{\kappa}_{c})\cdot\bar{c}_{0}=\\{\bar{0},\bar{\varepsilon}_{1}\\}\cdot\bar{c}_{0}$.
Now it is immediate to check that the action of $C_{W}(c)$ from Proposition
3.7 on this set is trivial. For example $\mathop{\rm Ad}(s,1)$ send
$\varepsilon_{i}\in X_{\ast}({\mathbf{T}}^{\prime})$ to
$(-1)^{i}\varepsilon_{i}$ ($i\in 1,2$), so $\mathop{\rm
Ad}(s,1)(\varepsilon_{i})-\varepsilon_{i}\in\ker\bar{\kappa}_{c}$. This is in
agreement with Proposition 3.13 and also with the Bruhat–Tits picture:
${\mathbf{G}}^{\rm ad}\cong{\bf PGL}_{2}\times{\bf PGL}_{2}$, so an alcove of
${\mathcal{B}}_{{\mathbf{G}},k}$ is a square, whose vertices are hyperspecial
of type $(\alpha,\beta)$ with $\alpha,\beta\in{\mathbb{Z}}/2{\mathbb{Z}}$
being the parity of the valuation of determinant of the lattice representing a
vertex of ${\mathcal{B}}_{{\bf PGL}_{2}}$. The action of
$\tau_{1}\in{\mathbf{G}}(k)$ identifies the vertices
$(0,0)\leftrightarrow(1,1)$ and $(0,1)\leftrightarrow(1,0)$, so that there are
two ${\mathbf{G}}(k)$-orbits in hyperspecial vertices of a base alcove, and
hence (by [DeB06]), two rational conjugacy classes of Coxeter tori.

_Case $b=b_{1}$_. We have
$[b_{1}]\cap\overline{F}_{c}=(X_{\ast}({\mathbf{T}}){\,\smallsetminus\,}X_{\ast}({\mathbf{T}}^{\prime}))\cdot\bar{c}_{0}$.
As
$(X_{\ast}({\mathbf{T}}){\,\smallsetminus\,}X_{\ast}({\mathbf{T}}^{\prime}))/\ker\bar{\kappa}_{c}=\\{\frac{\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}}{2},\frac{\bar{\varepsilon}_{1}+3\bar{\varepsilon}_{2}}{2}\\}$,
we have
$[b_{1}]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}=\\{\frac{\bar{\varepsilon}_{1}+\bar{\varepsilon}_{2}}{2},\frac{\bar{\varepsilon}_{1}+3\bar{\varepsilon}_{2}}{2}\\}\cdot\dot{c}_{0}$.
The action of $(s,1)\in C_{W}(c)$ identifies the two elements in
$[b_{1}]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}$, as
$\frac{\varepsilon_{1}+3\varepsilon_{2}}{2}-\mathop{\rm
Ad}(s,1)(\frac{\varepsilon_{1}+\varepsilon_{2}}{2})=\frac{\varepsilon_{1}+3\varepsilon_{2}}{2}-\frac{-\varepsilon_{1}+\varepsilon_{2}}{2}=\varepsilon_{1}+\varepsilon_{2}\in\ker\bar{\kappa}_{c}$,
i.e., the $C_{W}(c)$-action is non-trivial. This agrees with Bruhat–Tits
theory, as the adjoint building of ${\mathbf{G}}_{b_{1}}$ over $k$ is a point
(=the barycenter of an alcove of ${\mathcal{B}}_{{\mathbf{G}},k}$), so that
there is just one ${\mathbf{G}}_{b_{1}}(k)$-conjugacy class of Coxeter tori.

##  4\. The maps $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$

Here we review and extend some results from [Iva23, §11]. By [Iva23, Lemma
11.5] the space $\dot{X}_{\dot{w}}(b)$ depends – up to an
${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{w}(k)$-equivariant isomorphism – only
on the image of $\dot{w}$ in the discrete set $LF_{w}/\ker\kappa_{w}$ and not
on $\dot{w}$. As the concrete choice of $\dot{w}$ never plays an important
role, we make the following simplifying definition. Recall from Lemma 3.5 that
$LF_{w}/\ker\kappa_{w}=\overline{F}_{w}/\ker\bar{\kappa}_{w}$.

######  Definition 4.1.

For $b\in{\mathbf{G}}(\breve{k})$ and
$\bar{w}\in\overline{F}_{w}/\ker\bar{\kappa}_{w}$ we denote by
$\dot{X}_{\bar{w}}(b)$ the space $\dot{X}_{\dot{w}}(b)$ for an arbitrary lift
$\dot{w}\in F_{w}(\breve{k})$ of $\bar{w}$.

######  Proposition 4.2 ([Iva23], Prop. 11.4 and 11.9).

Let $w\in W$ and $b\in{\mathbf{G}}(\breve{k})$. There is a canonical map of
arc-sheaves

$\alpha_{w,b}\colon
X_{w}(b)\rightarrow\overline{F}_{w}/\ker\overline{\kappa}_{w},$

defining a clopen decomposition
$X_{w}(b)=\coprod_{\bar{w}\in\overline{F}_{w}/\ker\overline{\kappa}_{w}}X_{w}(b)_{\bar{w}}$,
where $X_{w}(b)_{\bar{w}}\stackrel{{\scriptstyle\rm
def}}{{=}}\alpha_{w,b}^{-1}(\\{\bar{w}\\})$. For
$\bar{w}\in\overline{F}_{w}/\ker\bar{\kappa}_{w}$,
$\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)_{\bar{w}}$ is a pro-étale
${\mathbf{T}}_{w}(k)$-torsor. In particular,
$\dot{X}_{\bar{w}}(b)\neq\varnothing$ if and only if
$\bar{w}\in\operatorname{im}(\alpha_{w,b})$.

###### Proof.

Combine [Iva23, Prop. 11.4 and 11.9] with Lemma 3.5. ∎

We have the following emptyness criterion for $\dot{X}_{\bar{w}}(b)$, i.e., a
constraint on the image of $\alpha_{w,b}$. Recall the map
$\bar{\kappa}^{w}\colon\overline{F}_{w}/\ker\bar{\kappa}_{w}\rightarrow\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}$
from (3.5).

######  Proposition 4.3.

Let $b\in{\mathbf{G}}(\breve{k})$,
$\bar{w}\in\overline{F}_{w}/\ker\bar{\kappa}_{w}$. If
$\dot{X}_{\bar{w}}(b)\neq\varnothing$, then
$\kappa_{{\mathbf{G}}}(b)=\bar{\kappa}^{w}(\bar{w})$. In particular,
$\alpha_{w,b}$ factors through a map $\alpha_{w,b}\colon
X_{w}(b)\rightarrow(\bar{\kappa}^{w})^{-1}(\kappa_{\mathbf{G}}(b))$.

###### Proof.

Let $\dot{w}\in F_{w}(\breve{k})$ be a lift of $\bar{w}$. As
$\dot{X}_{\dot{w}}(b)\neq\varnothing$, there is some algebraically closed
field ${\mathfrak{f}}\in\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$,
such that $\dot{X}_{\dot{w}}(b)({\mathfrak{f}})\neq\varnothing$. Let
$\bar{g}\in\dot{X}_{\dot{w}}(b)({\mathfrak{f}})$, and let
$g\in{\mathbf{G}}(L)$ be a lift of $\bar{g}$. Then
$g^{-1}b\sigma(g)\in{\mathbf{U}}(L)\dot{w}{\mathbf{U}}(L)$. We clearly have
$\kappa_{\mathbf{G}}(g^{-1}b\sigma(g))=\kappa_{\mathbf{G}}(b)$. Furthermore,
any element of ${\mathbf{U}}(L)$ can be conjugated to some Iwahori subgroup
and the Kottwitz map is trivial on Iwahori subgroups. Thus
$\kappa_{\mathbf{G}}({\mathbf{U}})=1$. Thus
$\kappa_{\mathbf{G}}({\mathbf{U}}(L)\dot{w}{\mathbf{U}}(L))=\kappa_{\mathbf{G}}(\dot{w})$,
and the claim follows. The last claim follows from the first and Proposition
4.2. ∎

Recall the map $\beta_{w}\colon X_{\ast}({\mathbf{T}}^{\mathop{\rm
sc}})_{\langle\sigma_{w}\rangle}\rightarrow
X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}$ from (2.1). We have the
following diagram with compatible simply transitive actions:

$\textstyle{(\bar{\kappa}^{w})^{-1}(\kappa_{{\mathbf{G}}}(b))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{F}_{w}/\ker\bar{\kappa}_{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\kappa}^{w}}$$\textstyle{\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}}$$\textstyle{\operatorname{im}(\beta_{w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circlearrowleft}$$\textstyle{X_{\ast}({\mathbf{T}})_{\langle\sigma_{w}\rangle}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circlearrowleft}$$\textstyle{\pi_{1}({\mathbf{G}})_{\langle\sigma\rangle}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circlearrowleft}$
(4.1)

where the actions are induced by left multiplication action of
$X_{\ast}({\mathbf{T}})$ on $\overline{F}_{w}$.

######  Corollary 4.4.

If $\beta_{w}=0$, then $\alpha_{w,b}$ factors through a point, i.e., there is
at most one $\bar{w}\in\overline{F}_{w}/\ker\bar{\kappa}_{w}$ such that
$\dot{X}_{\bar{w}}(b)\neq\varnothing$. In this case,
$X_{w}(b)\neq\varnothing\Leftrightarrow\dot{X}_{\bar{w}}(b)\neq\varnothing$
and $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$ is a pro-étale
${\mathbf{T}}_{w}(k)$-torsor.

######  Corollary 4.5.

If ${\mathbf{G}}$ is of adjoint type and $w$ is Coxeter, the conclusions of
Corollary 4.4 hold. Moreover, in this case the map
$\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$ is quasi-compact.

###### Proof.

The first claim follows from Corollary 4.4 and Remark 2.5. The last claim
follows as $\dot{X}_{\bar{w}}(b)\rightarrow X_{w}(b)$ is a torsor under the
profinite group ${\mathbf{T}}_{w}(k)$. ∎

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
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$ such that
$X_{\bar{c}}(b)\neq\varnothing$, there exists $b_{1}\in[b]\cap
F_{c}(\breve{k})$ lying over $\bar{c}$ and there is a
${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{c}(k)$-equivariant isomorphism
$\dot{X}_{\bar{c}}(b)\cong\dot{X}_{\bar{c}}(b_{1})$.

###### Proof.

By Proposition 4.3, $\dot{X}_{\bar{c}}(b)\neq\varnothing$ forces
$\bar{c}\in(\bar{\kappa}^{c})^{-1}(\bar{\kappa}_{{\mathbf{G}}}(b))=[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}$,
where the equality holds by Proposition 3.10. This implies the existence of
$b_{1}$. For any $g\in{\mathbf{G}}(\breve{k})$ such that
$g^{-1}b\sigma(g)=b_{1}$, $x\mapsto gx$ induces an equivariant isomorphism
$\dot{X}_{\bar{c}}(b)\cong\dot{X}_{\bar{c}}(b_{1})$ by [Iva23, Rem. 8.7]. ∎

Let us also record the following consequence.

######  Corollary 4.7.

Suppose $b$ is basic and $c$ is Coxeter. Then $\alpha_{c,b}$ factors through a
surjection

$\alpha_{c,b}\colon
X_{c}(b)\rightarrow[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}.$

There is a canonical surjection from the set of different non-empty pieces
$X_{c}(b)_{\bar{c}}\subseteq X_{c}(b)$ to the set
${\mathfrak{T}}({\mathbf{G}}_{b},c)/\mathop{\rm Ad}{\mathbf{G}}_{b}(k)$ of
rational conjugacy classes of unramified Coxeter tori in ${\mathbf{G}}_{b}$.

In particular, if ${\mathbf{G}},b$ satisfy the assumptions of Proposition
3.13, then this surjection is a $1$-to-$1$ correspondence.

###### Proof.

The surjectivity of the map $\alpha_{c,b}$ 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.

######  Remark 4.8.

If $[b]$ is non-basic and $c$ Coxeter, then
$[b]\cap\overline{F}_{c}/\ker\bar{\kappa}_{c}=\varnothing$ by Lemma 3.11.
Hence –although we still have $\operatorname{im}(\beta_{c})_{\rm
tors}=\operatorname{im}(\beta_{c})$ just as in Proposition 3.10– we cannot
conclude that
$[b]\cap\overline{F}_{c}/\ker\kappa_{c}=(\bar{\kappa}^{c})^{-1}(\kappa_{\mathbf{G}}(b))$.
In particular, our arguments do not imply that $X_{c}(b)$ (equivalently, all
of its covering spaces $\dot{X}_{\bar{c}}(b)$ with
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$) is empty. In agreement with
this, $X_{c}(b)$ indeed can be non-empty [Iva23, Thm. 13.2].

##  5\. Variant of Steinberg’s cross section

By the _Dynkin diagram_ ${\rm Dyn}({\mathbf{G}})$ of the unramified group
${\mathbf{G}}$ we mean the Dynkin diagram of the split group
${\mathbf{G}}_{\breve{k}}$ endowed with the action induced by Frobenius
$\sigma$. There is a unique decomposition into $\sigma$-stable subdiagrams
${\rm Dyn}({\mathbf{G}})=\coprod_{i}\Gamma_{i}$, such that $\sigma$ cyclically
permutes $\pi_{0}(\Gamma_{i})$ for each $i$. For each $i$, let $\Gamma_{i,0}$
be a connected component of $\Gamma_{i}$, so that
$\Gamma_{i}=\coprod_{j=1}^{d_{i}}\sigma^{j}(\Gamma_{i,0})$. Then
$(\Gamma_{i,0},\sigma^{d_{i}})$ 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 ${\mathbf{G}}$ is _of classical type_ , if
each connected component $(\Gamma_{i,0},\sigma^{d_{i}})$ as above of ${\rm
Dyn}({\mathbf{G}})$ is of one of the following types: $A_{n-1}$ ($n\geq 2$),
$B_{m}$ ($m\geq 2$), $C_{m}$ ($m\geq 2$), $D_{m}$ ($m\geq 4$), ${}^{2}A_{n-1}$
($n\geq 3$) or ${}^{2}D_{m}$ ($m\geq 4$).

Although Theorem 1.1 and its corollaries essentially hold for all Coxeter
elements (cf. the discussion preceeding Theorem 1.1), it will be convenient to
work with particular ones. Therefore we introduce the following auxiliary
notion.

######  Definition 5.2.

Let ${\mathbf{G}}$ be of classical type. Let $c\in W$ be a Coxeter element. We
call $c$ _special_ in the following cases:

  * •

If ${\mathbf{G}}$ is absolutely almost simple, $c$ is special if it is of the
form555The explicit shape of these elements only gets important when
everything boils down to explicit calculations in §7. For the sake of better
readability we postpone the explicit description, until the relevant notation
is introduced. as in (7.5) (resp. (7.7), (7.9), (7.13), (7.18), (7.26)) if
${\mathbf{G}}$ is of type $A_{n-1}$ (resp. $C_{m}$, $B_{m}$, $D_{m}$,
${}^{2}A_{n-1}$, ${}^{2}D_{m}$)

  * •

If ${\mathbf{G}}=\mathop{\rm Res}_{k^{\prime}/k}{\mathbf{G}}^{\prime}$ for a
finite unramified extension $k^{\prime}/k$ and an absolutely almost simple
$k^{\prime}$-group ${\mathbf{G}}^{\prime}$: we may identify the Weyl group $W$
of ${\mathbf{G}}$ with $\prod_{i=1}^{[k^{\prime}:k]}W^{\prime}$, where
$W^{\prime}$ is the Weyl group of ${\mathbf{G}}^{\prime}$ and $\sigma$ acts
cyclically. Then $c$ is special if it is of the form $(c^{\prime},1,\dots,1)$,
where $c^{\prime}$ is a special Coxeter element of $W^{\prime}$

  * •

If ${\mathbf{G}}$ is isogeneous to a product $\prod_{i=1}^{m}{\mathbf{G}}_{i}$
of almost simple $k$-groups ${\mathbf{G}}_{i}$: we may identify
$W=\prod_{i=1}^{m}W_{i}$, where $W_{i}$ is the Weyl group of
${\mathbf{G}}_{i}$. Then $c=(c_{i})_{i=1}^{m}$ is special if all $c_{i}$ are
special.

###  5.1. Loop version of Steinberg’s cross section

For classical groups and special Coxeter elements we now prove the existence
of certain Steinberg cross secions in our setup. This is a Frobenius-twisted
loop version of [Ste65, Prop. 8.9] and [HL12, Thm. 3.6 and §3.14], which will
be an important ingredient in the proof of Theorem 1.1.

Let $c\in W$ be a Coxeter element. If $b\in{\mathbf{G}}(\breve{k})$ is any
lift of $c$, then $b\sigma$ has a unique fixed point
${\mathbf{x}}={\mathbf{x}}_{b}$ on ${\mathcal{A}}_{{\mathbf{T}},\breve{k}}$.
Let ${\mathcal{G}}_{{\mathbf{x}}}$ denote the corresponding parahoric
${\mathcal{O}}_{\breve{k}}$-model of ${\mathbf{G}}$. We have the twisted
Frobenius $\sigma_{b}=\mathop{\rm Ad}(b)\circ\sigma$ on
$L{\mathbf{G}},L^{+}{\mathcal{G}}_{\mathbf{x}}$. Let
$({}^{c}{\mathbf{U}}\cap{\mathbf{U}})_{{\mathbf{x}}}$ be the closure of
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}$ in ${\mathcal{G}}_{{\mathbf{x}}}$.
Similarly, we have the closures
$({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})_{{\mathbf{x}}}$ and
$({}^{c}{\mathbf{U}})_{{\mathbf{x}}}$ of
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-}$ and ${}^{c}{\mathbf{U}}$ in
${\mathcal{G}}_{{\mathbf{x}}}$.

######  Proposition 5.3.

Let ${\mathbf{G}}$ be of classical type and $c\in W$ a special Coxeter
element. Then

$\alpha_{b}\colon L({}^{c}{\mathbf{U}}\cap{\mathbf{U}})\times
L({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})\rightarrow
L({}^{c}{\mathbf{U}}),\quad(x,y)\mapsto x^{-1}y\sigma_{b}(x)$

is an isomorphism. The injectivity of $\alpha_{b}$ holds for arbitrary element
$c\in W$, which is of minimal length in an elliptic $\sigma$-conjugacy class
(cf. [HL12]).

Furthermore, $\alpha_{b}$ restricts to an isomorphism

$\alpha_{b,{\mathbf{x}}}\colon
L^{+}({}^{c}{\mathbf{U}}\cap{\mathbf{U}})_{{\mathbf{x}}}\times
L^{+}({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})_{{\mathbf{x}}}\rightarrow
L^{+}({}^{c}{\mathbf{U}})_{{\mathbf{x}}}.$

######  Remark 5.4.

It looks very plausible that Proposition 5.3 holds for arbitrary Coxeter
element $c$. It should be possible to prove this by adapting [HL12, 3.5, 3.6
and 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, Thm. 3.6] says that
${\mathbf{U}}\times({\mathbf{U}}\cap{}^{b}{\mathbf{U}}^{-})\rightarrow{\mathbf{U}}b{\mathbf{U}}$,
$(u,z)\mapsto uz\sigma(u)^{-1}$ is an isomorphism if ${\mathbf{G}}$ is
reductive over a finite field with Frobenius morphism $\sigma$ (and, in fact,
in various other setups). As an easy corollary of this, [HL12, §3.14] states
that
$\alpha_{b}^{\prime}\colon({\mathbf{U}}\cap{}^{\sigma^{-1}(c)^{-1}}{\mathbf{U}})\times({\mathbf{U}}\cap{}^{c}{\mathbf{U}}^{-})\rightarrow{\mathbf{U}}$,
$(x,y)\mapsto xy\sigma_{b}(x)^{-1}$ is an isomorphism in the same setup. Now
the finite field version (without the loop functors) of the map $\alpha_{b}$
of Proposition 5.3 is obtained from $\alpha_{b}^{\prime}$ by a series of
changes of variables (cf. the first paragraph of the proof of Proposition
5.3), so that $\alpha_{b}$ is an isomorphism if and only if
$\alpha_{b}^{\prime}$ is.

###### Proof of Proposition 5.3.

We work with pro-étale topology on
$\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$, and regard $\alpha_{b}$
as a map of pro-étale sheaves. Injectivity of $\alpha_{b}$ may be derived from
[HL12]. First, change the variable by setting $y=\sigma_{b}(y_{1})$. It
remains to show that $L({\mathbf{U}}\cap{}^{c}{\mathbf{U}})\times
L({\mathbf{U}}\cap{}^{\sigma^{-1}(c^{-1})}{\mathbf{U}}^{-})\rightarrow
L({}^{c}{\mathbf{U}})$, $x,y_{1}\mapsto x^{-1}\sigma_{b}(y_{1}x)$ is
injective. Compose this map with the isomorphism
$(\cdot)^{-1}\circ\sigma^{-1}\circ{\rm Ad}(b)\colon
L({}^{c}{\mathbf{U}})\rightarrow L{\mathbf{U}}$. It remains to show that this
composition, given by $x,y_{1}\mapsto x^{-1}y_{1}^{-1}\sigma^{-1}(b^{-1}xb)$,
is injective. Finally, change the variables by putting $x=x_{2}^{-1}$ and
$y_{1}=y_{2}^{-1}$, and set $\tilde{b}:=\sigma^{-1}(b)^{-1}$ with image
$\tilde{c}$ in $W$. We are reduced to showing that
$L({\mathbf{U}}\cap{}^{\sigma(\tilde{c})^{-1}}{\mathbf{U}})\times
L({\mathbf{U}}\cap{}^{\tilde{c}}{\mathbf{U}}^{-})\rightarrow L{\mathbf{U}}$,
$x_{2},y_{2}\mapsto x_{2}y_{2}\tilde{b}\sigma^{-1}(x_{2})^{-1}\tilde{b}^{-1}$
is injective. This follows from [HL12, 3.14(b)] applied to
$A={\mathbb{W}}(R)[1/\varpi]$ (for any
$R\in\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$) and
$\chi=\phi_{R}^{-1}$.

We now show the surjectivity of $\alpha_{b}$. We may assume that
${\mathbf{G}}$ is of adjoint type. Then
${\mathbf{G}}=\prod_{i=1}^{r}{\mathbf{G}}_{i}$ with ${\mathbf{G}}_{i}$ almost
simple $k$-group, and we are immediately reduced to the case that
${\mathbf{G}}$ is almost simple over $k$.

######  Lemma 5.5.

Let $k_{d}/k$ be an unramified extension of degree $d\geq 1$, and suppose that
${\mathbf{G}}={\rm Res}_{k_{d}/k}{\mathbf{G}}^{\prime}$ for a $k_{d}$-group
${\mathbf{G}}^{\prime}$. If the surjectivity claim for $\alpha_{b}$ of
Proposition 5.3 holds for ${\mathbf{G}}^{\prime}$, then it holds for
${\mathbf{G}}$.

###### Proof.

Identify
${\mathbf{G}}_{\breve{k}}=\prod_{i=1}^{d}{\mathbf{G}}^{\prime}_{\breve{k}}$
and similarly for the Weyl groups $W$, $W^{\prime}$ and unipotent radicals
${\mathbf{U}},{\mathbf{U}}^{\prime}$ of Borels of
${\mathbf{G}},{\mathbf{G}}^{\prime}$, so that the Frobenius $\sigma$ permutes
cyclically the components. As $c$ is special, $c=(c^{\prime},1,\dots,1)$ for a
special Coxeter element $c^{\prime}\in W^{\prime}$. We then have
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}=({}^{c^{\prime}}{\mathbf{U}}^{\prime}\cap{\mathbf{U}}^{\prime})\times{\mathbf{U}}^{\prime}\times\dots\times{\mathbf{U}}^{\prime}$
and
${}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-}=({}^{c^{\prime}}{\mathbf{U}}^{\prime}\cap{\mathbf{U}}^{\prime-})\times
1\times\dots\times 1$. The lift $b$ of $c$ is then given as
$(b^{\prime},\tau_{2},\dots,\tau_{d})$ for a lift $b^{\prime}$ of $c^{\prime}$
to ${\mathbf{G}}^{\prime}(\breve{k})$ and some
$\tau_{i}\in{\mathbf{T}}^{\prime}(\breve{k})$. Then the map $\alpha_{b}$ from
Proposition 5.3 is given by

$(x_{i})_{i=1}^{d},y_{1}\mapsto(x_{1}^{-1}y_{1}\mathop{\rm
Ad}(b^{\prime})(\sigma^{\prime}(x_{d})),x_{2}^{-1}{\rm
Ad}(\tau_{2})(x_{1}),\dots x_{d}^{-1}{\rm Ad}(\tau_{2})(x_{d-1})).$

From this, surjectivity of $\alpha_{b}$ for ${\mathbf{G}}$ immediately reduces
to surjectivity of $\alpha_{b^{\prime}}$ for ${\mathbf{G}}^{\prime}$. ∎

######  Lemma 5.6.

If $\widetilde{\alpha}_{b}\colon
L({\mathbf{U}}\,\cap\,{}^{c^{-1}}{\mathbf{U}})\times
L({\mathbf{U}}\,\cap\,{}^{c^{-1}}{\mathbf{U}}^{-})\rightarrow L{\mathbf{U}}$
denotes the map $\widetilde{\alpha}_{b}(x,y)=x^{-1}y\sigma(bxb^{-1})$, then
$\mathop{\rm Ad}(b^{-1})(\alpha_{b}(x,y))=\widetilde{\alpha}_{b}(\mathop{\rm
Ad}(b^{-1})(x),\mathop{\rm Ad}(b^{-1})(y))$.

###### Proof.

Immediate computation. ∎

As ${\mathbf{G}}$ splits over $\breve{k}$, Lemma 5.5 reduces us to the case
that ${\mathbf{G}}$ is absolutely almost simple over $k$. It suffices to show
that the map $\widetilde{\alpha}_{b}$ from Lemma 5.6 is surjective. Write
$A=L({\mathbf{U}}\cap{}^{c^{-1}}{\mathbf{U}})$,
$B=L({\mathbf{U}}\cap{}^{c^{-1}}{\mathbf{U}}^{-})$, $C=L{\mathbf{U}}$. For
$\star\in\\{A,B,C\\}$, write $\Phi_{\star}=\\{\alpha\in\Phi\colon
L{\mathbf{U}}_{\alpha}\subseteq\star\\}$, where
${\mathbf{U}}_{\alpha}\subseteq{\mathbf{G}}$ denotes the root subgroup
corresponding to $\alpha$. We call a subset $\Psi\subseteq\Phi$ _closed under
addition_ , if for any $\alpha,\beta\in\Psi$ with $\alpha+\beta\in\Phi$ we
have $\alpha+\beta\in\Psi$.

######  Lemma 5.7.

There exists a positive integer $r$ and subsets

$\Phi_{C}=\Psi_{1}\supseteq\Psi_{2}\supseteq\dots\supseteq\Psi_{r}=\Phi_{B},$

satisfying the following conditions:

  * (1)

For all $1\leq i\leq r$, $\Psi_{i}$ and $\Psi_{i}{\,\smallsetminus\,}\Psi_{r}$
are closed under addition.

  * (2)

For all $1\leq i<r$ and all $\alpha,\beta\in\Psi_{i}$ the implication
$\alpha+\beta\in\Phi\Rightarrow\alpha+\beta\in\Psi_{i+1}$ holds.

  * (3)

For all $1\leq i\leq r$,
$\sigma(c(\Psi_{i}{\,\smallsetminus\,}\Psi_{r}))\subseteq\Psi_{i}$.

###### Proof.

We give the desired partitions in §7.1.4 (type $A_{n-1}$), §7.2.4 (types
$B_{m},C_{m}$), §7.4.9 (type $D_{m}$),§7.5.7 (type ${}^{2}A_{n-1}$), §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 and we omit this checking. ∎

######  Lemma 5.8.

Let $I$ be a finite set, not containing $0$. Let $\lambda\colon I\rightarrow
I\cup\\{0\\}$ be a map, such that $\lambda^{-1}(i)$ has at most one element if
$i\in I$. For any $i\in I\cap\lambda(I)$, let
$\varepsilon_{i}\in\breve{k}^{\times}$. Consider the map
$\delta\colon\prod_{i\in I}L{\mathbb{G}}_{a}\rightarrow\prod_{i\in
I}L{\mathbb{G}}_{a}$, given by $\delta((a_{i})_{i\in I})=(b_{i})_{i\in I}$,
where $b_{i}=0$ if $\lambda^{-1}(i)=\varnothing$ and
$b_{i}=\varepsilon_{i}\phi(a_{\lambda^{-1}(i)})$ otherwise. Then the map
$x\mapsto x-\delta(x)$ is surjective for the pro-étale topology.

###### Proof.

There are some integers $s\geq r\geq 0$ and a disjoint decomposition
$I=\coprod_{i=1}^{r}I_{i}\dot{\cup}\coprod_{i=r+1}^{s}I_{i}$, such that for
$1\leq i\leq r$, $\lambda$ restricts to a map $I_{i}\rightarrow I_{i}$ which
is a full cycle of length $\\#I_{i}$ (first type), and for $r+1\leq i\leq s$,
$\lambda$ restricts to a map $I_{i}\rightarrow I_{i}\dot{\cup}\\{0\\}$, which
is, after choosing an isomorphism $I_{i}\cong\\{1,\dots,m\\}$, given by
$\lambda(j)=j-1$ for all $j\in I_{i}$ (second type). It suffices to show that
the restriction of $\delta$ to $\prod_{j\in I_{i}}L{\mathbb{G}}_{a}$ is
surjective. Thus we may assume that $I$ is one of the $I_{i}$’s. If
$I=\\{1,\dots,m\\}$ is of second type, one has the filtration $G^{\bullet}$ of
$G=\prod_{i=1}^{m}L{\mathbb{G}}_{a}$ by closed sub-(ind-group schemes)
$G_{j}=\prod_{i=1}^{j}L{\mathbb{G}}_{a}$, stable by $\delta$, such that
$x\mapsto x-\delta(x)$ induces the identity on associated graded object ${\rm
gr}\,G^{\bullet}$. From this, the result easily follows. If $I$ is of the
first type, we may assume $I={\mathbb{Z}}/m{\mathbb{Z}}$, and
$\lambda(i)=i+1$. Replacing $\phi$ by $\phi^{m}$, one is then immediately
reduced to the case that $m=1$ and $\delta\colon L{\mathbb{G}}_{a}\rightarrow
L{\mathbb{G}}_{a}$, $\delta(x)=\varepsilon\phi(x)$ for some
$\varepsilon\in\breve{k}^{\times}$. Then Lemma 5.9 finishes the proof. ∎

######  Lemma 5.9.

Let $\varepsilon\in\breve{k}$. Then the map
$\operatorname{id}-\varepsilon\phi\colon L{\mathbb{G}}_{a}\rightarrow
L{\mathbb{G}}_{a}$, $x\mapsto x-\varepsilon\phi(x)$ is surjective for the pro-
étale topology.

###### Proof.

If $\varepsilon\not\in{\mathcal{O}}_{\breve{k}}^{\times}$, then
$\operatorname{id}-\varepsilon\phi$ is an isomorphism. Indeed, if $\mathop{\rm
ord}_{\varpi}(\varepsilon)>0$, $\sum_{i=0}^{\infty}(\varepsilon\phi)^{i}$ is a
meaningful endomorphism of $L{\mathbb{G}}_{a}$ (as ${\mathbb{W}}(R)$ is
$\varpi$-adically complete for each $R\in\operatorname{Perf}$), and the
inverse of $\operatorname{id}-\varepsilon\phi$. If $\mathop{\rm
ord}_{\varpi}(\varepsilon)<0$, a similar argument goes through using
$\operatorname{id}-\varepsilon\phi=-\varepsilon\phi(\operatorname{id}-(\varepsilon\phi)^{-1})$.
Now assume $\varepsilon\in{\mathcal{O}}_{\breve{k}}^{\times}$. We have
$L{\mathbb{G}}_{a}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>and it suffices to show that $\operatorname{id}-\varepsilon\phi$ is surjective
on
$L^{+}{\mathbb{G}}_{a}=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\leftarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
lim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>Here it follows from a pro-version of Lang’s theorem. ∎

We now proof the surjectivity of $\alpha_{b}$. For a subset
$\Psi\subseteq\Phi^{+}$ closed under addition, let ${\mathbf{U}}_{\Psi}$
denote the unique closed subgroup of ${\mathbf{U}}$, whose $\breve{k}$-points
are generated by $\\{{\mathbf{U}}_{\alpha}(\breve{k})\\}_{\alpha\in\Psi}$. As
a $\breve{k}$-scheme, ${\mathbf{U}}_{\Psi}$ is isomorphic to
$\prod_{\alpha\in\Psi}{\mathbf{U}}_{\alpha}$. Write
$U_{\Psi}=L{\mathbf{U}}_{\Psi}$. Let $\\{\Psi_{i}\\}_{i=1}^{r}$ be a
filtration as in Lemma 5.7. For $1\leq i\leq r$, we have the closed sub-(ind-
group schemes) $C_{i}:=U_{\Psi_{i}}$ of $L{\mathbf{U}}$, with $C_{1}=C$,
$C_{r}=B$. Moreover, condition (2) from Lemma 5.7 guarantees that
$C_{i+1}\subseteq C_{i}$ is a closed normal subgroup, with abelian quotient.
As $H^{1}(k,{\mathbf{H}})=0$ for a split unipotent $k$-group ${\mathbf{H}}$,
we have
$C_{i}/C_{i+1}\cong\prod_{\alpha\in\Psi_{i}{\,\smallsetminus\,}\Psi_{i+1}}L{\mathbf{U}}_{\alpha}$.
Moreover, by condition (1), for $1\leq i\leq r-1$, $C_{i}$ has the closed
subgroup $A_{i}=U_{\Psi_{i}{\,\smallsetminus\,}\Psi_{r}}$, and condition (2)
implies that $A_{i+1}$ is normal in $A_{i}$, and that the inclusion
$A_{i}\hookrightarrow C_{i}$ induces an isomorphism $\gamma_{i}\colon
A_{i}/A_{i+1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}C_{i}/C_{i+1}$.
Moreover, by condition (3), the map $\sigma\circ\mathop{\rm Ad}(b)\colon
A_{i}\rightarrow C_{i}$ induces a map $\delta_{i}\colon
A_{i}/A_{i+1}\rightarrow C_{i}/C_{i+1}$. The composition
$\gamma_{i}^{-1}\delta_{i}$ is an endomorphism of
$A_{i}/A_{i+1}\cong\prod_{\alpha\in\Psi_{i}{\,\smallsetminus\,}\Psi_{i+1}}L{\mathbf{U}}_{\alpha}$,
which is of the form as in Lemma 5.8. From Lemma 5.8 it follows that $a\mapsto
a\cdot(\gamma_{i}^{-1}\delta_{i})(a)^{-1}$ is a surjective endomorphism of
$A_{i}/A_{i+1}$.

Let $R\in\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$. Let $x_{i}\in
C_{i}(R)$. Let $\bar{x}_{i}$ be its image in $(C_{i}/C_{i+1})(R)$. By the
above, after replacing $R$ by a pro-étale cover, we may find some $a_{i}\in
A_{i}(R)$ with image $\bar{a}_{i}\in(A_{i}/A_{i+1})(R)$, such that
$\bar{a}_{i}\cdot(\gamma_{i}^{-1}\delta_{i})(\bar{a}_{i}^{-1})=\gamma_{i}^{-1}(\bar{x}_{i})$.
Applying $\gamma_{i}$ we deduce the equation
$\gamma_{i}(\bar{a}_{i})^{-1}\bar{x}_{i}\delta_{i}(\bar{a}_{i})=1$ in
$(C_{i}/C_{i+1})(R)$. The left hand side is precisely the image of
$\widetilde{\alpha}_{b}(a_{i},x_{i})\in C_{i}(R)$ in $(C_{i}/C_{i+1})(R)$.
With other words, for each $x_{i}\in C_{i}(R)$, we may find (after possibly
enlaring $R$) some $a_{i}\in A_{i}$, such that
$\widetilde{\alpha}_{b}(a_{i},x_{i})\in C_{i+1}$. Starting with an arbitrary
$x=x_{1}\in C(R)$, and applying this procedure consequently for
$i=1,2,\dots,r-1$, we deduce that after possibly enlarging $R$, there are some
$a_{1},\dots,a_{r-1}\in A(R)$, such that for $a=a_{r-1}a_{r-2}\dots a_{1}$ we
have $\widetilde{\alpha}_{b}(a,x)\in C_{r}(R)=B(R)$. (Here we use that
$\widetilde{\alpha}_{b}(b,\widetilde{\alpha}_{b}(a,x))=\widetilde{\alpha}_{b}(ab,x)$.
This finishes the proof that $\alpha_{b}$ is an isomorphism.

Next we turn to $\alpha_{b,{\mathbf{x}}}$. Note that it is well-defined, as
for $x\in L^{+}({}^{c}{\mathbf{U}}\cap{\mathbf{U}})_{{\mathbf{x}}}$,
$\sigma_{b}(x)\in L({}^{c}{\mathbf{U}})\cap
L^{+}{\mathcal{G}}_{{\mathbf{x}}}=L^{+}({}^{c}{\mathbf{U}}_{{\mathbf{x}}})$,
the intersection taken in $L{\mathbf{G}}$. Being the restriction of
$\alpha_{b}$, it is injective, as $\alpha_{b}$ is, and it remains to prove its
surjectivity. Applying ${\rm Ad}(b^{-1})$, which conjugates
${\mathcal{G}}_{{\mathbf{x}}}$ to ${\mathcal{G}}_{{\mathbf{y}}}$, where
${\mathbf{y}}=b^{-1}({\mathbf{x}})$ (and the analogue of Lemma 5.6) it
suffices to show that (with the obvious notation) the map
$\widetilde{\alpha}_{b,{\mathbf{x}}}\colon
L^{+}({\mathbf{U}}\cap{}^{c^{-1}}{\mathbf{U}})_{{\mathbf{y}}}\times
L^{+}({\mathbf{U}}\cap^{c^{-1}}{\mathbf{U}}^{-})_{{\mathbf{y}}}\rightarrow
L^{+}{\mathbf{U}}_{{\mathbf{y}}}$,
$\widetilde{\alpha}_{b}(x,y)=x^{-1}y\sigma(bxb^{-1})$ is surjective. By the
same reasoning as in Lemma 5.5 (and the paragraph preceding it) we are reduced
to the case that ${\mathbf{G}}$ is absolutely almost simple over $k$. For
$\Psi\subseteq\Phi^{+}$, let ${\mathbf{U}}_{\Psi,{\mathbf{y}}}$ denote the
closure of ${\mathbf{U}}_{\Psi}$ in ${\mathcal{G}}_{{\mathbf{y}}}$. We have
${\mathbf{U}}_{\Psi,{\mathbf{y}}}\cong\prod_{\alpha\in\Psi}{\mathbf{U}}_{\alpha,{\mathbf{y}}}$.
Replacing $C_{i},B,A_{i}$ above by
$C_{i}^{+}=L^{+}{\mathbf{U}}_{\Psi_{i},{\mathbf{y}}}$,
$B^{+}=L^{+}({\mathbf{U}}\cap{}^{c^{-1}}{\mathbf{U}}^{-})$,
$A_{i}^{+}=L^{+}{\mathbf{U}}_{\Psi_{i}{\,\smallsetminus\,}\Psi_{r},{\mathbf{y}}}$
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
$\alpha_{b,{\mathbf{x}}}$. ∎

###  5.2. A variant of a Deligne–Lusztig space

Throughout this section we work with the arc-topology on
$\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$. Steinberg’s cross section
allows us to give a different presentation of Deligne–Lusztig spaces
$X_{c}(b)$, $\dot{X}_{\bar{c}}(b)$. We have the following general definition.

######  Definition 5.10.

Let $w\in W$, $\dot{w}\in F_{w}(\breve{k})$ and $b\in{\mathbf{G}}(\breve{k})$.
Define $\dot{X}_{\dot{w},b}$ and $\tilde{X}_{w,b}$ by the following cartesian
diagrams:

$\textstyle{\dot{X}_{\dot{w},b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L({}^{w}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot\dot{w}b^{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L{\mathbf{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\mapsto
g^{-1}\sigma_{b}(g)}$$\textstyle{L{\mathbf{G}}}$
and
$\textstyle{\tilde{X}_{w,b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L({}^{w}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot
L{\mathbf{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L{\mathbf{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\mapsto
g^{-1}\sigma_{b}(g)}$$\textstyle{L{\mathbf{G}}}$

and let $X_{w,b}\stackrel{{\scriptstyle\rm
def}}{{=}}\tilde{X}_{w,b}/L{\mathbf{T}}$.

######  Remark 5.11.

The map $g\mapsto g^{-1}\sigma_{b}(g)\colon L{\mathbf{G}}\rightarrow
L{\mathbf{G}}$ factors through
$\kappa_{\mathbf{G}}^{-1}(\kappa_{\mathbf{G}}(b))\subseteq L{\mathbf{G}}$.
Thus $\tilde{X}_{c,b}\rightarrow({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot
L{\mathbf{T}}$ factors through
$({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot\kappa_{{\mathbf{G}}}|_{L{\mathbf{T}}}^{-1}(\kappa_{{\mathbf{G}}}(b))$.
Thus, in the upper right entry of the right diagram of Definition 5.10, we
could have written
$({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot\kappa_{{\mathbf{G}}}|_{L{\mathbf{T}}}^{-1}(\kappa_{{\mathbf{G}}}(b))$,
without changing the definition.

The group ${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{w}(k)$ acts on
$\dot{X}_{\dot{w},b}$ by $g,t\colon x\mapsto gxt$ and ${\mathbf{G}}_{b}(k)$
acts on $X_{w,b}$ by $g\colon x\mapsto gx$. We only use these spaces in the
case that $w=c$ is a (special) Coxeter element.

######  Proposition 5.12.

Let $b\in{\mathbf{G}}(\breve{k})$. Let $c\in W$ be a Coxeter element and
$\dot{c}\in F_{c}(\breve{k})$. Whenever the conclusion of Proposition 5.3
holds (e.g., ${\mathbf{G}}$ classical, $c$ special Coxeter), the following
hold.

  * (i)

There is a ${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{c}(k)$-equivariant
isomorphism $\dot{X}_{\dot{c}}(b)\cong\dot{X}_{\dot{c},b}$.

  * (ii)

If $b\in F_{c}(\breve{k})$, then there is a ${\mathbf{G}}_{b}(k)$-equivariant
isomorphism $X_{c}(b)\cong X_{c,b}$.

###### Proof.

(i): Consider the sheafification $\dot{X}^{\prime}$ of the functor on
$\operatorname{Perf}_{\overline{{\mathbb{F}}}_{q}}$,

$X_{1}\colon R\mapsto\left\\{g\in L{\mathbf{G}}(R)\colon
g^{-1}\sigma_{b}(g)\in
L({}^{c}{\mathbf{U}})(R)\dot{c}b^{-1}=\dot{c}L{\mathbf{U}}(R)b^{-1}\right\\}/L({\mathbf{U}}\cap{}^{c}{\mathbf{U}})(R).$

By [Iva23, Prop. 12.1] $X_{\dot{c}}(b)$ and $\dot{X}^{\prime}$ are
equivariantly isomorphic. It remains to show that
$\dot{X}^{\prime}\cong\dot{X}_{\dot{c},b}$ equivariantly. Note that
$\dot{X}_{\dot{c},b}$ is the sheafification of

$X_{2}\colon R\mapsto\left\\{g\in L{\mathbf{G}}(R)\colon
g^{-1}\sigma_{b}(g)\in
L({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})(R)\cdot\dot{c}b^{-1}\right\\}.$

There is an obvious equivariant map $X_{2}\rightarrow X_{1}$, of which we
claim that it is an isomorphism. This follows, once we check that

$L({}^{c}{\mathbf{U}}\cap{\mathbf{U}})\times
L({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})\cdot\dot{c}b^{-1}\rightarrow\dot{c}L{\mathbf{U}}b^{-1}=L({}^{c}{\mathbf{U}})\cdot\dot{c}b^{-1},\quad
x,y^{\prime}\mapsto x^{-1}y^{\prime}\sigma_{b}(x)$

is an isomorphism. Replacing $y^{\prime}=y\dot{c}b^{-1}$, with $y$ ranging in
$L({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})$, this map gets
$(x,y)\mapsto\left(x^{-1}y\dot{c}\sigma(x)\dot{c}^{-1}\right)\cdot\dot{c}b^{-1}$,
which is an isomorphism by Proposition 5.3. The proof of (ii) is completely
analogous. ∎

######  Proposition 5.13.

Suppose ${\mathbf{G}}$ is of adjoint type. Let $c\in W$ be as in Proposition
5.12 and $b\in F_{c}(\breve{k})$. For any $\dot{c}\in F_{c}(\breve{k})$
satisfying $\kappa_{\mathbf{G}}(b)=\kappa_{\mathbf{G}}(\dot{c})$ there are
${\mathbf{G}}_{b}(k)$-equivariant isomorphisms $X_{c}(b)\cong
X_{c,b}\cong\dot{X}_{\dot{c},b}/{\mathbf{T}}_{c}(k)$.

###### Proof.

By Proposition 5.12, it suffices to show that
$\dot{X}_{\dot{c},b}/{\mathbf{T}}_{c}(k)\cong X_{c,b}$. Writing
$\tau:=\dot{c}^{-1}b\in(L{\mathbf{T}}\cap\kappa_{\mathbf{G}}^{-1}(\kappa_{\mathbf{G}}(b)))(\overline{{\mathbb{F}}}_{q})$,
these sheaves are the sheafifications of the quotient presheaves

$\displaystyle Y_{1}\colon R$ $\displaystyle\mapsto\\{g\in
L{\mathbf{G}}(R)\colon g^{-1}\sigma_{b}(g)\in
L({\mathbf{U}}^{-}\cap{}^{c}{\mathbf{U}})(R)\cdot\tau\\}/{\mathbf{T}}_{c}(k)(R)\qquad\quad\text{and}$
$\displaystyle Y_{2}\colon R$ $\displaystyle\mapsto\\{g\in
L{\mathbf{G}}(R)\colon g^{-1}\sigma_{b}(g)\in
L({\mathbf{U}}^{-}\cap{}^{c}{\mathbf{U}})(R)\cdot
L{\mathbf{T}}(R)\\}/L{\mathbf{T}}(R),$

respectively. We have an obvious map $Y_{1}\rightarrow Y_{2}$, 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 $g,h\in Y_{1}(R)$. Then
$\sigma_{b}(g)=gu_{g}\tau$, $\sigma_{b}(h)=hu_{h}\tau$ for some
$u_{g},u_{h}\in L({\mathbf{U}}^{-}\cap{}^{c}{\mathbf{U}})(R)$. Suppose their
images in $Y_{2}(R)$ agree, i.e., there is some $t\in L{\mathbf{T}}(R)$, such
that $gt=h$. Then $\sigma_{b}(g)\sigma_{b}(t)=\sigma_{b}(h)$ and hence
$gu_{g}\tau\sigma_{b}(t)=hu_{h}\tau$, i.e.,

$\sigma_{c}(t)=\sigma_{b}(t)=\tau^{-1}u_{g}^{-1}tu_{h}\tau$

(first equation holds as $b\in F_{c}(\breve{k})$). Thus, $\sigma_{c}(t)=tu$
for some $u\in L({\mathbf{U}}^{-}\cap{}^{c}{\mathbf{U}})(R)$, which implies
$u=1$, and hence $t\in(L{\mathbf{T}})^{\sigma_{c}}(R)={\mathbf{T}}_{c}(k)(R)$.

Secondly, let $g\in L{\mathbf{G}}(R)$ satisfy
$\sigma_{b}(g)=gu_{g}\tau^{\prime}$ for some
$u\in({}^{c}{\mathbf{U}}\cap{\mathbf{U}}^{-})(R)$ and $\tau^{\prime}\in
L{\mathbf{T}}(R)$. By Remark 5.11,
$\kappa_{\mathbf{G}}(\tau^{\prime})=\kappa_{\mathbf{G}}(b)$. Changing $g$ to
$gt$ for some $t\in L{\mathbf{T}}(R)$ has no effect on the image of $g$ in
$Y_{2}(R)$, but amounts to replacing $\tau^{\prime}$ to
$\tau^{\prime}t^{-1}\sigma(t)$. Thus, to show that $Y_{1}\rightarrow Y_{2}$ is
surjective when passing to sheafification, it remains to check that the
$\sigma_{c}$-twisted action morphism

$L{\mathbf{T}}\times(L{\mathbf{T}}\cap\kappa_{\mathbf{G}}^{-1}(\kappa_{\mathbf{G}}(b)))\rightarrow
L{\mathbf{T}}\cap\kappa_{\mathbf{G}}^{-1}(\kappa_{\mathbf{G}}(b)),\qquad
t,\tau^{\prime}\mapsto t^{-1}\tau^{\prime}\sigma_{c}(t)$

is surjective for the arc-topology. In fact it is even surjective for the pro-
etale topology. Indeed, the space
$L{\mathbf{T}}\cap\kappa_{\mathbf{G}}^{-1}(\kappa_{\mathbf{G}}(b))$ is fibered
over
$(\bar{\kappa}_{\mathbf{G}}|_{X_{\ast}({\mathbf{T}})})^{-1}(\kappa_{\mathbf{G}}(b))$,
all fibers being isomorphic to $L^{+}{\mathcal{T}}$. Using pro-Lang for the
$\sigma_{c}$-stable connected subscheme $L^{+}{\mathcal{T}}\subseteq
L{\mathbf{T}}$, we are reduced to show that the map of discrete abelian groups
$X_{\ast}({\mathbf{T}})\times(\bar{\kappa}_{\mathbf{G}}|_{X_{\ast}({\mathbf{T}})})^{-1}(\kappa_{\mathbf{G}}(b))\rightarrow(\bar{\kappa}_{\mathbf{G}}|_{X_{\ast}({\mathbf{T}})})^{-1}(\bar{\kappa}_{\mathbf{G}}(b))$
given $\mu,\chi\mapsto\chi+\sigma_{c}(\mu)-\mu$ is surjective, i.e., that the
fibers of $\bar{\kappa}_{\mathbf{G}}|_{X_{\ast}({\mathbf{T}})}$ are torsors
under $\operatorname{im}(\sigma_{c}-\operatorname{id}\colon
X_{\ast}({\mathbf{T}})\rightarrow
X_{\ast}({\mathbf{T}}))=\ker\bar{\kappa}_{c}$. This holds by Remark 2.5 as
${\mathbf{G}}$ is adjoint and $c$ Coxeter. ∎

##  6\. 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
$X_{c}(b)$, $\dot{X}_{\bar{c}}(b)$ for an unramified classical group
${\mathbf{G}}$, $c$ Coxeter and $b$ basic.

######  Proposition 6.1.

Let $b\in{\mathbf{G}}(\breve{k})$ basic, $c\in W$ Coxeter, and
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$. Suppose
$\dot{X}_{\bar{c}}(b)\neq\varnothing$. Then there exist

  * •

a special Coxeter element $c_{1}\in W$ (cf. Definition 5.2),

  * •

a lift $\bar{c}_{1}\in\overline{F}_{c_{1}}/\ker\bar{\kappa}_{c_{1}}$, and

  * •

an element $b_{1}\in[b]\cap F_{c}(\breve{k})$, which lies over $\bar{c}_{1}$,

such that $X_{c}(b)\cong X_{c_{1}}(b_{1})$ and
$\dot{X}_{\bar{c}}(b)\cong\dot{X}_{\bar{c}_{1}}(b_{1})$,
${\mathbf{G}}_{b}(k)$\- resp.
${\mathbf{G}}_{b}(k)\times{\mathbf{T}}_{c}(k)$-equivariantly.

###### Proof.

By [Iva23, Cor. 8.18] there exists a special Coxeter element $c_{1}$, such
that $X_{c}(b)\cong X_{c_{1}}(b)$ equivariantly. Moreover, we may find some
$\bar{c}_{1}$ over $c_{1}$ such that $\dot{X}_{\bar{c}}(b)$ is equivariantly
isomorphic to $\dot{X}_{\bar{c}_{1}}(b)$, cf. [Iva23, Lm. 8.16]. By Corollary
4.6 we may pick $b_{1}\in[b]\cap F_{c_{1}}(\breve{k})$ lying over
$\bar{c}_{1}$. ∎

The rest of the article is devoted to the proof of Theorem 1.1. Thus we start
with

  * •

a special Coxeter element $c\in W$ and
$\bar{c}\in\overline{F}_{c}/\ker\bar{\kappa}_{c}$,

  * •

a basic element $b$, which lies in $F_{c}(\breve{k})$ and lifts $\bar{c}$.

In §6.1 we reduce (1.3) to the case that ${\mathbf{G}}$ is absolutely almost
simple and adjoint. In §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 §6.3. Finally, in §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.

###  6.1. 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 ${\mathbf{G}}^{\rm ad}$ (for the left hand side, use that
${\mathbf{G}}/{\mathbf{B}}={\mathbf{G}}^{\rm ad}/{\mathbf{B}}^{\rm ad}$).
Hence we may assume that ${\mathbf{G}}$ is of adjoint type. Then
${\mathbf{G}}$ 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 ${\mathbf{G}}$ is of adjoint type and almost simple.

If ${\mathbf{G}}$ is adjoint and almost simple, then ${\mathbf{G}}=\mathop{\rm
Res}_{k^{\prime}/k}{\mathbf{G}}^{\prime}$ for some finite unramified extension
$k^{\prime}/k$ and some absolutely almost simple $k^{\prime}$-group
${\mathbf{G}}^{\prime}$. As $B({\mathbf{G}})=B({\mathbf{G}}^{\prime})$ [Kot85,
1.10] (which allows to change $b$ appropriately inside its $\sigma$-conjugacy
class), Lemma 6.2 reduces us to the case that ${\mathbf{G}}$ is adjoint
absolutely almost simple.