# Affine Deligne–Lusztig varieties with finite Coxeter parts

Xuhua He  The Institute of Mathematical Sciences and Department of
Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR,
China<EMAIL_ADDRESS>,  Sian Nie  Academy of Mathematics and
Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and,
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Chinese Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS>and
Qingchao Yu  Beijing International Center for Mathematical Research, Beijing
University, Haidian, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study affine Deligne–Lusztig varieties $X_{w}(b)$ when the
finite part of the element $w$ in the Iwahori–Weyl group is a partial
$\sigma$-Coxeter element. We show that such $w$ is a cordial element and
$X_{w}(b)\neq\emptyset$ if and only if $b$ satisfies a certain Hodge–Newton
indecomposability condition. The main result of this paper is that for such
$w$ and $b$, $X_{w}(b)$ has a simple geometric structure: the
$\sigma$-centralizer of $b$ acts transitively on the set of irreducible
components of $X_{w}(b)$; and each irreducible component is an iterated
fibration over a classical Deligne–Lusztig variety of Coxeter type, and the
iterated fibers are either $\mathbb{A}^{1}$ or $\mathbb{G}_{m}$.

###### Key words and phrases:

Affine Deligne–Lusztig varieties, Coxeter elements

###### 2010 Mathematics Subject Classification:

11G25, 20G25

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Class polynomials and reduction trees
  4. 4 Cordiality and the set $B(\mathbf{G})_{w}$
  5. 5 Analyzing the reduction paths
  6. 6 Some combinatorial identities
  7. 7 The general case

##  1\. Introduction

###  1.1. Classical/affine Deligne–Lusztig varieties

The classical Deligne–Lusztig varieties were introduced by Deligne and Lusztig
in [3]. They play a crucial role in the representation theory of finite
reductive groups. They are defined for a connected reductive group
$\mathbf{G}$ over a finite field $\mathbb{F}_{q}$. For any element $w$ in the
(finite) Weyl group of $\mathbf{G}(\bar{}\mathbb{F}_{q})$, the corresponding
Deligne–Lusztig variety $X_{w}$ is a certain locally closed subvariety of the
flag variety of $\mathbf{G}(\bar{}\mathbb{F}_{q})$, and it admits a natural
action of the finite reductive group $\mathbf{G}(\mathbb{F}_{q})$. It is known
that

(a) The classical Deligne–Lusztig variety $X_{w}$ is smooth and of dimension
equal to the length of $w$, and the finite reductive group
$\mathbf{G}(\mathbb{F}_{q})$ acts transitively on the set of irreducible
components of $X_{w}$.

Affine Deligne–Lusztig varieties were introduced by Rapoport in [27] as the
affine analog of classical Deligne–Lusztig varieties. They serve as group-
theoretic models for Shimura varieties and shtukas. They are defined for a
connected reductive group $\mathbf{G}$ over a non-archimedean local field $F$.
Let $\breve{F}$ be the completion of the maximal unramified extension of $F$.
For any element $w$ in the Iwahori–Weyl group $\tilde{W}$ of
$\mathbf{G}(\breve{F})$ and any element $b\in\mathbf{G}(\breve{F})$, the
corresponding affine Deligne–Lusztig variety $X_{w}(b)$ is a certain locally
closed subscheme of finite type in the affine flag variety of
$\mathbf{G}(\breve{F})$, and it admits a natural action of the
$\sigma$-centralizer $\mathbf{J}_{b}(F)$ of $b$. Unlike classical
Deligne–Lusztig varieties, which have the nice geometric structure described
in (a), the geometric structures of affine Deligne–Lusztig varieties are very
complicated:

  * •

For many pairs $(w,b)$, $X_{w}(b)$ are empty.

  * •

Even if $X_{w}(b)$ is nonempty, it is not equi-dimensional in general, and it
is very difficult to determine its dimension.

  * •

In general, the group $\mathbf{J}_{b}(F)$ does not acts transitively on the
set of irreducible components of $X_{w}(b)$, and very little is known about
this $\mathbf{J}_{b}(F)$-action.

  * •

The irreducible components of $X_{w}(b)$, in general, have a very complicated
geometric structure.

We refer to the survey article [16] and the paper [17] for recent developments
regarding the nonemptiness pattern and the dimension formula for $X_{w}(b)$.

###  1.2. Main result

In [25], Milićević and Viehmann introduced the notion of cordial elements. The
geometry of the affine Deligne–Lusztig varieties associated with cordial
elements is “well-behaved” in the following sense. If $w$ is a cordial
element, then the elements $b$ with $X_{w}(b)\neq\emptyset$ form a saturated
set in the sense of [25, Theorem 1.1], and for any such $b$, there is a simple
dimension formula for $X_{w}(b)$. Moreover, $X_{w}(b)$ is equi-dimensional.
Schremmer gave a classification of the cordial elements in his recent work
[28].

However, even for a cordial element $w$, very little is known about the
$\mathbf{J}_{b}(F)$-orbits on the set of irreducible components of $X_{w}(b)$
or about the geometric structure of the irreducible components of $X_{w}(b)$.

On the other hand, by [13], there is a family of elements in the Iwahori–Weyl
group whose associated affine Deligne–Lusztig varieties have very simple
geometric structures. We denote by $\sigma$ the Frobenius morphism on
$\mathbf{G}(\breve{F})$ and the induced group automorphism on the Iwahori–Weyl
group $\tilde{W}$. Suppose that $w$ is a minimal length element in its
$\sigma$-conjugacy class of $\tilde{W}$; then there exists a unique
$\sigma$-conjugacy class $[b]$ with $X_{w}(b)\neq\emptyset$. In this case,
there exist a parahoric subgroup $P$ of $\mathbf{J}_{b}(F)$ and a classical
Deligne–Lusztig variety $X$ (associated with the reductive quotient of $P$)
such that $X_{w}(b)\cong\mathbf{J}_{b}(F)\times^{P}X$. Such a simple geometric
structure has been used in the study of certain Shimura varieties with simple
geometric structure (see [6], [8], and [9]). However, these minimal length
elements $w$ form only a tiny fraction of the whole Iwahori–Weyl group, and
such a simple geometric structure only occurs in a few cases.

In this paper, we will focus on another family of elements in $\tilde{W}$. For
any $w\in\tilde{W}$, we define its finite part to be the image of $w$ under
the map $\eta_{\sigma}:\tilde{W}W$ (see §2.5). The main result of this paper
is that if the finite part of $w$ is a $\sigma$-Coxeter element of $W$, then
the associated affine Deligne–Lusztig variety $X_{w}(b)$ for any $b$ has a
simple geometric structure.

######  Theorem 1.1 (See Theorem 2.7).

Let $w\in\tilde{W}$ such that $\eta_{\sigma}(w)$ is a $\sigma$-Coxeter element
of $W$. Then

  1. (1)

$w$ is a cordial element;

  2. (2)

$X_{w}(b)\neq\emptyset$ if and only if $b$ satisfies a certain Hodge–Newton
indecomposability condition;

  3. (3)

For any $b$ with $X_{w}(b)\neq\emptyset$, there exists a parahoric subgroup
$P$ of $\mathbf{J}_{b}(F)$ and a classical Deligne–Lusztig variety $X$ of
Coxeter type, and an iterated fibration $YX$ whose iterated fibers are either
$\mathbb{A}^{1}$ or $\mathbb{G}_{m}$ such that
$X_{w}(b)\cong\mathbf{J}_{b}(F)\times^{P}Y$.

We refer to §2 for the precise statement and the definition of the notions
used here. The special case of Part (3) where $\mathbf{G}=\mathrm{GL}_{n}$,
$b$ is basic and $w$ is certain element with finite Coxeter part was studied
by Shimada [29].

###  1.3. Strategy

One major tool used in the study of affine Deligne–Lusztig varieties is the
Deligne–Lusztig reduction method [13]. Based on the Deligne–Lusztig reduction,
a close relationship between affine Deligne–Lusztig varieties and the class
polynomials of affine Hecke algebras was established in [13]. One remarkable
property of these class polynomials is that they are polynomials in
$(\mathbf{q}-1)$ with non-negative integral coefficients. With each element
$w\in\tilde{W}$, we may associate a reduction tree, which encodes the
information on the reduction steps and determines the class polynomials
associated with $w$. However, obtaining an explicit description of the
reduction trees is quite challenging.

Another key ingredient in this paper is the Chen–Zhu conjecture. This
conjecture predicts the $\mathbf{J}_{b}(F)$-action on the top-dimensional
irreducible components of affine Deligne–Lusztig varieties in the affine
Grassmannian. This conjecture was verified recently in [26], [32], and [21].
Part of the Chen–Zhu conjecture predicts the isotropy group for the
$\mathbf{J}_{b}(F)$-action, which gives some information about the end points
of the reduction trees.

Combining the above two ingredients, in §5, we show that the end points of
each reduction tree for $w$ with finite Coxeter part must be certain
$\sigma$-Coxeter elements, and each path, which corresponds to a
$\sigma$-conjugacy class $[b]$ of $b$, in a reduction tree provides a
$\mathbf{J}_{b}(F)$-orbit of irreducible components of $X_{w}(b)$. It remains
to show that for any $b$, there is at most one path in the reduction tree that
corresponds to $[b]$ (i.e., the “multiplicity one” result). For the (unique)
maximal $\sigma$-conjugacy class $[b]$ with $X_{w}(b)\neq\emptyset$, this
“multiplicity one” result is obvious. For the basic $\sigma$-conjugacy class
$[b]$, one may deduce the “multiplicity one” result by showing that any path
corresponding to $[b]$ is of a unique type. See §5.6.

It is more challenging to determine the numbers of reduction paths for other
$\sigma$-conjugacy classes in a reduction tree. In this paper, we use the
following indirect approach to establish the “multiplicity one” result. We
first interpret the class polynomials as the number of rational points for
certain admissible subsets. We then use the positivity property of the class
polynomials to show that the “multiplicity one” result for all $b$ is
equivalent to the following single combinatorial identity:

$\sum_{[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}(\mathbf{q}-1)^{?}\mathbf{q}^{-??}=1.$

Here $B(\mathbf{G},\mu)_{\mathrm{indec}}$ is the set of all Hodge–Newton
indecomposable $\sigma$-conjugacy classes, and the powers “?” and “??” are
certain non-negative integers determined by $w$ and $b$. We refer to §2.3 and
§5.2 for the precise definitions.

Verifying the combinatorial identity (*) is the most technical part of this
paper and is done in §6. We first establish natural bijections between the
sets $B(\mathbf{G},\mu)_{\mathrm{indec}}$ for various pairs
$(\mathbf{G},\mu)$, which is of independent interest. In combination with
other techniques, we reduce the verification of (*) to the case for simply
laced, $\breve{F}$-simple and split groups and for fundamental coweights.

For classical groups, we may further reduce to the case where $\mu$ is
minuscule. In this case, the “multiplicity one” result follows from the
Chen–Zhu conjecture. For exceptional groups, we use a computer to verify (*).
The most complicated case for the exceptional group is
$(E_{8},\omega^{\vee}_{4})$. In this case, the left-hand side of (*) involves
a summation of $729$ terms. It is also worth mentioning that in the case
$(A_{n-1},\omega^{\vee}_{i})$, we may write the identity (*) explicitly as

$\sum_{\begin{subarray}{c}k\geqslant
1,1>\frac{a_{1}}{b_{1}}>\cdots>\frac{a_{k}}{b_{k}}>0;\\\
a_{i}+\cdots+a_{k}=i,b_{1}+\cdots+b_{k}=n\end{subarray}}(\mathbf{q}-1)^{k-1}\mathbf{q}^{k-1-\frac{\sum_{1\leqslant
l_{1}<l_{2}\leqslant
k}(a_{l_{1}}b_{l_{2}}-a_{l_{2}}b_{l_{1}})+\sum_{1\leqslant l\leqslant
k}\gcd(a_{l},b_{l})}{2}}\\\ =\mathbf{q}^{\frac{i(n-i)-n}{2}}.$

We do not know if there is a purely combinatorial proof of this identity.

Acknowledgments:  XH is partially supported by Hong Kong RGC Grant 14300220,
by funds connected with the Choh-Ming Chair at CUHK, and by the Xplorer prize.

##  2\. Preliminaries

###  2.1. Reductive groups

Let $F$ be a non-archimedean local field with residue field $\mathbb{F}_{q}$
and let $\breve{F}$ be the completion of the maximal unramified extension of
$F$. We write $\Gamma$ for $\operatorname{Gal}(\overline{F}/F)$, and
$\Gamma_{0}$ for the inertia subgroup of $\Gamma$.

Let $\mathbf{G}$ be a quasi-split connected reductive group over $F$. We set
$\breve{G}=\mathbf{G}(\breve{F})$. Let $\sigma$ be the Frobenius morphism of
$\breve{F}$ over $F$. We use the same symbol $\sigma$ for the induced
Frobenius morphism on $\breve{G}$. Let $S$ be a maximal $\breve{F}$-split
torus of $\mathbf{G}$ defined over $F$, which contains a maximal $F$-split
torus. Let $T$ be the centralizer of $S$ in $\mathbf{G}$. Then $T$ is a
maximal torus. We denote by $N$ the normalizer of $T$ in $\mathbf{G}$. Let
$W=N(\breve{F})/T(\breve{F})$ be the relative Weyl group. We fix a
$\sigma$-stable Iwahori subgroup $\breve{}\mathcal{I}$ of $\breve{G}$. Let
$\tilde{W}=N(\breve{F})/T(\breve{F})\cap\breve{}\mathcal{I}$ be the
_Iwahori–Weyl group_. The action $\sigma$ on $\breve{G}$ induces a natural
action on $\tilde{W}$ and $W$, which we still denote by $\sigma$. For any
$w\in\tilde{W}$, we choose a representative $\dot{w}$ in $N(\breve{F})$. We
have the splitting

$\tilde{W}=X_{*}(T)_{\Gamma_{0}}\rtimes W=\\{t^{\lambda}w;\lambda\in
X_{*}(T)_{\Gamma_{0}},w\in W\\}.$

Here $X_{*}(T)_{\Gamma_{0}}$ denotes the $\Gamma_{0}$-coinvariants of
$X_{*}(T)$.

Since $\mathbf{G}$ is quasi-split over $F$, $\sigma$ acts naturally on
$X_{*}(T)_{\Gamma_{0}}$ and on $W$. We denote by $\ell$ the length function on
$\tilde{W}$ and on $W$, and by $\leqslant$ the Bruhat order on $\tilde{W}$ and
on $W$. Let $\tilde{}\mathbb{S}$ be the index set of simple reflections in
$\tilde{W}$ and let $\mathbb{S}\subset\tilde{}\mathbb{S}$ be the index set of
simple reflections in $W$. In other words, the set of simple reflections in
$\tilde{W}$ is $\\{s_{i}\mid i\in\tilde{}\mathbb{S}\\}$.

For any $w\in W$, we denote by $\operatorname{supp}(w)$ the set of $i$ such
that $s_{i}$ occurs in some (or, equivalently, any) reduced expressions of
$w$, and we set
$\operatorname{supp}_{\sigma}(w)=\cup_{l\in\mathbb{N}}\sigma^{l}(\operatorname{supp}(w))$.

An element $c\in W$ is called a (full) $\sigma$-Coxeter element if it is a
product of simple reflections, one from each $\sigma$-orbit of $\mathbb{S}$.
An element $c\in W$ is called a partial $\sigma$-Coxeter element if it is a
product of simple reflections, at most one from each $\sigma$-orbit of
$\mathbb{S}$.

Let $\Phi$ be the reduced root system underlying the relative root system of
$\mathbf{G}$ over $\breve{F}$ (the échelonnage root system). For any
$i\in\mathbb{S}$, we denote by $\alpha_{i}$ and $\alpha_{i}^{\vee}$ the
corresponding (relative) simple root and simple coroot respectively.

###  2.2. The $\sigma$-conjugacy classes of $\breve{G}$

The $\sigma$-conjugation action on $\breve{G}$ is defined by
$g\cdot_{\sigma}g^{\prime}=gg^{\prime}\sigma(g)^{-1}$. For $b\in\breve{G}$, we
denote by $[b]$ the $\sigma$-conjugacy class of $b$. Let $B(\mathbf{G})$ be
the set of $\sigma$-conjugacy classes on $\breve{G}$. The classification of
the $\sigma$-conjugacy classes is due to Kottwitz in [22] and [23]. Any
$\sigma$-conjugacy class $[b]$ is determined by two invariants:

  * •

the element $\kappa([b])\in\pi_{1}(\mathbf{G})_{\sigma}$;

  * •

the Newton point
$\nu_{b}\in\big{(}(X_{*}(T)_{\Gamma_{0},\mathbb{Q}})^{+}\big{)}^{\langle\sigma\rangle}$.

Here $\pi_{1}(\mathbf{G})_{\sigma}$ denotes the $\sigma$-coinvariants of
$\pi_{1}(\mathbf{G})$, and $(X_{*}(T)_{\Gamma_{0},\mathbb{Q}})^{+}$ denotes
the intersection of
$X_{*}(T)_{\Gamma_{0}}\otimes\mathbb{Q}=X_{*}(T)^{\Gamma_{0}}\otimes\mathbb{Q}$
with the set $X_{*}(T)_{\mathbb{Q}}^{+}$ of dominant elements in
$X_{*}(T)_{\mathbb{Q}}$. Denote $V=X_{*}(T)_{\Gamma_{0}}\otimes\mathbb{R}$.
For any $v\in V$, denote $I(v)=\\{i\in\mathbb{S};\langle
v,\alpha_{i}\rangle=0\\}$. Here
$\langle~{},~{}\rangle:V\times\mathbb{R}\Phi\mathbb{R}$ is the natural
pairing. Let $V^{+}$ be the set of dominant vectors $v\in V$, that is,
$\langle v,\alpha_{i}\rangle\geqslant 0$ for $i\in\mathbb{S}$.

The set $B(\mathbf{G})$ is equipped with a natural partial order:
$[b]\leqslant[b^{\prime}]$ if and only if $\kappa([b])=\kappa([b^{\prime}])$
and $\nu_{b}\leqslant\nu_{b^{\prime}}$. Here $\leqslant$ is the dominance
order on the set of dominant elements in $X_{*}(T)_{\mathbb{Q}}$, that is,
$\nu\leqslant\nu^{\prime}$ if $\nu^{\prime}-\nu$ is a non-negative rational
linear combination of positive relative coroots. It is proved in [2, Theorem
7.4] that the poset $B(\mathbf{G})$ is ranked. For any
$[b]\leqslant[b^{\prime}]$ in $B(\mathbf{G})$, we denote by ${\rm
length}([b],[b^{\prime}])$ the length of any maximal chain between $[b]$ and
$[b^{\prime}]$.

Let $\mu$ be a dominant coweight. Let $\mu^{\diamond}$ be the average of the
$\sigma$-orbit of $\mu$. The set of _neutrally acceptable_ $\sigma$-conjugacy
classes is defined by

$B(\mathbf{G},\mu)=\\{[b]\in
B(\mathbf{G});\kappa([b])=\kappa(\mu),\nu_{b}\leqslant\mu^{\diamond}\\}.$

For any $i\in\mathbb{S}$, let $\omega_{i}\in\mathbb{R}\Phi$ be the
corresponding fundamental weight. For any $\sigma$-orbit $\mathcal{O}$ of
$\mathbb{S}$, let $\omega_{\mathcal{O}}=\sum_{i\in\mathcal{O}}\omega_{i}$. The
following length formula is due to Chai (see [2, Theorem 7.4] and [31, Theorem
3.4]).

(a) For $[b]\in B(\mathbf{G},\mu)$, ${\rm
length}([b],[t^{\mu}])=\sum_{\mathcal{O}\in\mathbb{S}/\langle\sigma\rangle}\lceil\langle\mu-\nu_{b},\omega_{\mathcal{O}}\rangle\rceil.$

###  2.3. Hodge–Newton indecomposable/irreducible set

For any $\sigma$-stable subset $J$ of $\mathbb{S}$, we denote by
$\mathbf{M}_{J}$ the standard Levi subgroup of $\mathbf{G}_{\breve{F}}$
associated with $J$. Let $W_{J}\subseteq W$ be the parabolic subgroup
generated by the simple reflections in $J$. Then $W_{J}$ is the Weyl group of
$\mathbf{M}_{J}$. Let $b\in\breve{G}$. We say that $(\mu,b)$ is Hodge–Newton
decomposable with respect to $\mathbf{M}_{J}$ if $I(\nu_{b})\subseteq J$ and
$\mu^{\diamond}-\nu_{b}\in\sum_{j\in J}\mathbb{R}_{\geqslant
0}\alpha_{j}^{\vee}$. If $(\mu,b)$ is not Hodge–Newton decomposable with
respect to any proper $\sigma$-stable standard Levi subgroup of
$\mathbf{G}_{\breve{F}}$, then we say that $[b]$ is Hodge–Newton
indecomposable. Set

$B(\mathbf{G},\mu)_{\mathrm{indec}}=\\{[b]\in B(\mathbf{G},\mu);[b]\text{ is
Hodge--Newton indecomposable}\\}.$

We say that $(\mu,b)$ is Hodge–Newton $J$-irreducible if
$\mu^{\diamond}-\nu_{b}\in\sum_{j\in J}\mathbb{R}_{>0}\alpha_{j}^{\vee}$. Set

$B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}=\\{[b]\in
B(\mathbf{G},\mu);[b]\text{ is Hodge--Newton $J$-irreducible}\\}.$

We simply write
$B(\mathbf{G},\mu)_{\mathrm{irr}}=B(\mathbf{G},\mu)_{\mathbb{S}\text{-}\mathrm{irr}}$.

We say that $\mu$ is essentially non-central with respect to $\mathbf{M}_{J}$
if it is non-central on every $\sigma$-orbit of connected components of $J$.
It is easy to see that
$B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}\neq\emptyset$ if and only if $\mu$
is essentially non-central with respect to $\mathbf{M}_{J}$. We may simply say
that $\mu$ is essentially non-central if it is essentially non-central with
respect to $\mathbf{G}$. If $\mu$ is essentially non-central, then
$B(\mathbf{G},\mu)_{\mathrm{irr}}=B(\mathbf{G},\mu)_{\mathrm{indec}}$.

Let $\mathfrak{M}_{\mu}$ be the set of $\sigma$-stable subsets
$J\subseteq\mathbb{S}$ such that $\mu$ is essentially non-central in $J$. Then
we have

$B(\mathbf{G},\mu)=\sqcup_{J\in\mathfrak{M}_{\mu}}B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}.$

Let $J=\sigma(J)\subseteq\mathbb{S}$. For $b\in\mathbf{M}_{J}(\breve{F})$, we
denote by $[b]_{\mathbf{M}_{J}}$ the $\sigma$-conjugacy class of $b$ in
$\mathbf{M}_{J}(\breve{F})$, and denote by $\nu_{b}^{\mathbf{M}_{J}}$ its
$\mathbf{M}_{J}$-dominant Newton point. We have the following result.

######  Lemma 2.1.

Let $\mu$ be a dominant coweight and $J$ be a $\sigma$-stable subset of
$\mathbb{S}$. Then

  1. (1)

the map
$\phi_{J}:B(\mathbf{M}_{J},\mu)B(\mathbf{G},\mu),~{}[b]_{\mathbf{M}_{J}}[b]$
is injective;

  2. (2)

the image of $\phi_{J}$ consists of $[b]\in B(\mathbf{G},\mu)$ with
$\mu^{\diamond}-\nu_{b}\in\sum_{i\in J}\mathbb{R}_{\geqslant
0}\alpha_{i}^{\vee}$;

  3. (3)

for $[b]_{\mathbf{M}_{J}}\in B(\mathbf{M}_{J},\mu)$, ${\rm
length}_{\mathbf{G}}([b],[t^{\mu}])={\rm
length}_{\mathbf{M}_{J}}([b]_{\mathbf{M}_{J}},[t^{\mu}]_{\mathbf{M}_{J}})$.

###### Proof.

Let $[b]_{\mathbf{M}_{J}}\in B(\mathbf{M}_{J},\mu)$. Then
$\mu^{\diamond}-\nu_{b}^{\mathbf{M}_{J}}\in\sum_{i\in J}\mathbb{R}_{\geqslant
0}\alpha_{i}^{\vee}$, which implies that $\nu_{b}^{\mathbf{M}_{J}}$ is
dominant with respect to $\mathbf{G}$, and hence
$\nu_{b}^{\mathbf{M}_{J}}=\nu_{b}$. Now the Newton point and the Kottwitz
point of $[b]_{\mathbf{M}_{J}}$ are determined by $[b]$ and $\mu$
respectively. Hence $\phi_{J}$ is injective.

Part (2) follows from [2, §7.1] and [20, Lemma 3.5]. Part (3) follows from
part (2) and Chai’s length formula §2.2(a). ∎

As a consequence, we have the following.

######  Corollary 2.2.

Let $J\in\mathfrak{M}_{\mu}$. Then the map $\phi_{J}$ in Lemma 2.1 induces a
bijection $B(\mathbf{M}_{J},\mu)_{\mathrm{irr}}\cong
B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$.

###  2.4. Affine Deligne–Lusztig varieties

Let $Fl=\breve{G}/\breve{}\mathcal{I}$ be an affine flag variety. For any
$b\in\breve{G}$ and $w\in\tilde{W}$, we define the corresponding affine
Deligne–Lusztig variety in the affine flag variety

$X_{w}(b)=\\{g\breve{}\mathcal{I}\in\breve{G}/\breve{}\mathcal{I};g^{-1}b\sigma(g)\in\breve{}\mathcal{I}\dot{w}\breve{}\mathcal{I}\\}\subset
Fl.$

Let $\mathbf{k}$ be the residue field of $\breve{F}$. In the equal
characteristic setting, the affine Deligne–Lusztig variety $X_{w}(b)$ is the
set of $\mathbf{k}$-valued points of a locally closed subscheme of the affine
flag variety, equipped with the reduced scheme structure. In the mixed
characteristic setting, we consider $X_{w}(b)$ as the $\mathbf{k}$-valued
points of a perfect scheme in the sense of Zhu [33] and Bhatt–Scholze [1], a
locally closed perfect subscheme of the $p$-adic partial flag variety.

We denote by $\Sigma^{\mathrm{top}}(X_{w}(b))$ the set of top-dimensional
irreducible components of $X_{w}(b)$. Let $\mathbf{J}_{b}$ be the
$\sigma$-centralizer of $b$ and let
$\mathbf{J}_{b}(F)=\\{g\in\breve{G};gb\sigma(g)^{-1}=b\\}$ be the group of
$F$-points of $\mathbf{J}_{b}$. The left action of $\mathbf{J}_{b}(F)$ on
$X_{w}(b)$ induces an action of $\mathbf{J}_{b}(F)$ on
$\Sigma^{\mathrm{top}}(X_{w}(b))$. We denote by
$\mathbf{J}_{b}(F)\backslash\Sigma^{\mathrm{top}}(X_{w}(b))$ the set of
$\mathbf{J}_{b}(F)$-orbits on $\Sigma^{\mathrm{top}}(X_{w}(b))$.

Note that if $b$ and $b^{\prime}$ are $\sigma$-conjugate in $\breve{G}$, then
$X_{w}(b)$ and $X_{w}(b^{\prime})$ are isomorphic. Thus the affine
Deligne–Lusztig variety $X_{w}(b)$ (up to isomorphism) depends only on the
element $w$ in the Iwahori–Weyl group $\tilde{W}$ and the $\sigma$-conjugacy
class $[b]$ in $B(\mathbf{G})$. We set

$B(\mathbf{G})_{w}=\\{[b]\in B(\mathbf{G});X_{w}(b)\neq\emptyset\\}.$

There is a unique maximal $\sigma$-conjugacy class in $B(\mathbf{G})_{w}$,
which we denote by $[b_{w}]$. By [25, Lemma 3.2], $\dim
X_{w}(b_{w})=\ell(w)-\langle{\nu}_{b_{w}},2\rho\rangle$. Here $\rho$ is the
half sum of the positive roots in $\Phi$.

###  2.5. Cordial elements

For $b\in\breve{G}$, the _defect_ of $b$ is defined to be

${\mathrm{def}}(b)=\operatorname{rank}_{F}\mathbf{G}-\operatorname{rank}_{F}\mathbf{J}_{b},$

where $\operatorname{rank}_{F}$ denotes the $F$-rank of a reductive group over
$F$.

By [31, Theorem 3.4], we have the following length formula:

(a) For $[b]\in B(\mathbf{G},\mu)$, ${\rm
length}([b],[t^{\mu}])=\langle\mu-\nu_{b},\rho\rangle+\frac{1}{2}{\mathrm{def}}(b).$

Let ${}^{\mathbb{S}}\tilde{W}$ be the set of minimal length representatives
for the cosets in $W\backslash\tilde{W}$. Note that any element
$w\in\tilde{W}$ can be written in a unique way as $w=xt^{\mu}y$ with $\mu$
dominant, $x,y\in W$ such that $t^{\mu}y\in{}^{\mathbb{S}}\tilde{W}$. We have
$\ell(w)=\ell(x)+\langle\mu,2\rho\rangle-\ell(y)$. In this case, we set
$\eta_{\sigma}(w)=\sigma^{-1}(y)x$. The virtual dimension is defined to be

$d_{w}(b)=\tfrac{1}{2}(\ell(w)+\ell(\eta_{\sigma}(w))-{\mathrm{def}}(b)-\langle\nu_{b},2\rho\rangle).$

For $w=t^{\mu}y\in{}^{\mathbb{S}}\tilde{W}$, it is easy to see that
$d_{w}(b)=\langle\mu-\nu_{b},\rho\rangle-\frac{1}{2}{\mathrm{def}}(b)$.

By [13, Theorem 10.3] and [15, Theorem 2.30], we have

(b) For $w\in\tilde{W}$ and $b\in\breve{G}$, $\dim X_{w}(b)\leq d_{w}(b)$.

In the special case $[b]=[b_{w}]$, (b) implies that

$\ell(w)-\ell(\eta_{\sigma}(w))\leq\langle\nu_{b_{w}},2\rho\rangle-{\mathrm{def}}(b_{w}).$

Cordial elements were introduced by Milićević and Viehmann in [25]. By
definition, an element $w\in\tilde{W}$ is cordial if $\dim
X_{w}(b_{w})=d_{w}(b_{w})$. This condition is equivalent to the condition that
$\ell(w)-\ell(\eta_{\sigma}(w))=\langle\nu_{b_{w}},2\rho\rangle-{\mathrm{def}}(b_{w})$.
The following nice properties of the cordial elements are established in [25].

######  Theorem 2.3.

Let $w\in\tilde{W}$ be a cordial element. Then

  1. (1)

$B(\mathbf{G})_{w}$ is saturated, that is, if
$[b_{1}]\leqslant[b_{2}]\leqslant[b_{3}]$ in $B(\mathbf{G})$ and
$[b_{1}],[b_{3}]\in B(\mathbf{G})_{w}$, then $[b_{2}]\in B(\mathbf{G})_{w}$;

  2. (2)

for each $[b]\in B(\mathbf{G})_{w}$, $X_{w}(b)$ is equi-dimensional of
dimension equal to $d_{w}(b)$.

###  2.6. Minimal length elements

We consider the $\sigma$-conjugation action on $\tilde{W}$ defined by
$w\cdot_{\sigma}w^{\prime}=ww^{\prime}\sigma(w)^{-1}$. Let
$B(\tilde{W},\sigma)$ be the set of $\sigma$-conjugacy classes of $\tilde{W}$.
For any $\sigma$-conjugacy class $\mathcal{O}$ of $\tilde{W}$, we let
$\mathcal{O}_{\min}$ be the set of minimal length elements in $\mathcal{O}$,
and we write $\ell(\mathcal{O})=\ell(w)$ for any $w\in\mathcal{O}_{\min}$.

For $w,w^{\prime}\in\tilde{W}$ and $i\in\tilde{}\mathbb{S}$, we write
$w\xrightarrow{s_{i}}_{\sigma}w^{\prime}$ if $w^{\prime}=s_{i}w\sigma(s_{i})$
and $\ell(w^{\prime})\leqslant\ell(w)$. We write $w_{\sigma}w^{\prime}$ if
there is a sequence $w=w_{0},w_{1},\ldots,w_{n}=w^{\prime}$ of elements in
$\tilde{W}$ such that for any $k$, $w_{k-1}\xrightarrow{s_{i}}_{\sigma}w_{k}$
for some $s_{i}\in\tilde{}\mathbb{S}$. We write $w\approx_{\sigma}w^{\prime}$
if $w_{\sigma}w^{\prime}$ and $w^{\prime}_{\sigma}w$.

We call $w,w^{\prime}\in\tilde{W}$ elementarily strongly $\sigma$-conjugate if
$\ell(w)=\ell(w^{\prime})$ and there exists $x\in\tilde{W}$ such that
$w^{\prime}=xw\sigma(x)^{-1}$ and $\ell(xw)=\ell(x)+\ell(w)$ or
$\ell(w\sigma(x)^{-1})=\ell(x)+\ell(w)$. We call $w,w^{\prime}$ strongly
$\sigma$-conjugate if there is a sequence
$w=w_{0},w_{1},\ldots,w_{n}=w^{\prime}$ such that for each $i$, $w_{i-1}$ is
elementarily strongly $\sigma$-conjugate to $w_{i}$. We write
$w\sim_{\sigma}w^{\prime}$ if $w$ and $w^{\prime}$ are strongly
$\sigma$-conjugate.

The following result is proved in [18, Theorem 2.10].

######  Theorem 2.4.

Let $\mathcal{O}$ be a $\sigma$-conjugacy class in $\tilde{W}$. Then the
following hold:

  1. (1)

For each element $w\in\mathcal{O}$, there exists
$w^{\prime}\in\mathcal{O}_{\min}$ such that $w\rightarrow_{\sigma}w^{\prime}$.

  2. (2)

Let $w,w^{\prime}\in\mathcal{O}_{\min}$. Then $w\sim_{\sigma}w^{\prime}$.

###  2.7. Decompositions of affine Deligne–Lusztig varieties

We recall the Deligne–Lusztig reduction method on affine Deligne–Lusztig
varieties.

######  Proposition 2.5.

[21, Proposition 3.3.1] Let $w\in\tilde{W}$, $i\in\tilde{}\mathbb{S}$, and
$b\in\breve{G}$. If $\operatorname{char}(F)>0$, then the following two
statements hold.

  1. (1)

If $\ell(s_{i}w\sigma(s_{i}))=\ell(w)$, then there exists a
$\mathbf{J}_{b}(F)$-equivariant morphism $X_{w}(b)X_{s_{i}w\sigma(s_{i})}(b)$
which is a universal homeomorphism.

  2. (2)

If $\ell(s_{i}w\sigma(s_{i}))=\ell(w)-2$, then $X_{w}(b)=X_{1}\sqcup X_{2}$,
where $X_{1}$ is a $\mathbf{J}_{b}(F)$-stable open subscheme $X$ of $X_{w}(b)$
and $X_{2}$ is its closed complement satisfying the following conditions:

     * •

$X_{1}$ is $\mathbf{J}_{b}(F)$-equivariant universally homeomorphic to a
Zariski-locally trivial $\mathbb{G}_{m}$-bundle over $X_{s_{i}w}(b)$;

     * •

$X_{2}$ is $\mathbf{J}_{b}(F)$-equivariant universally homeomorphic to a
Zariski-locally trivial $\mathbb{A}^{1}$-bundle over
$X_{s_{i}w\sigma(s_{i})}(b)$.

If $\operatorname{char}(F)=0$, then the above two statements still hold, but
with $\mathbb{A}^{1}$ and $\mathbb{G}_{m}$ replaced by the perfections
$\mathbb{A}^{1,\mathrm{perf}}$ and $\mathbb{G}^{\mathrm{perf}}_{m}$,
respectively.

For any $a_{1},a_{2}\in\mathbb{N}$, we say that a scheme $X$ is an iterated
fibration of type $(a_{1},a_{2})$ over a scheme $Y$ if there exist morphisms

$X=Y_{0}Y_{1}\cdots Y_{a_{1}+a_{2}}=Y$

such that for any $i$ with $0\leqslant i<a_{1}+a_{2}$, $Y_{i}$ is a Zariski-
locally trivial $\mathbb{A}^{1,(\mathrm{perf})}$-bundle or
$\mathbb{G}^{(\mathrm{perf})}_{m}$-bundle over $Y_{i+1}$, and there are
exactly $a_{1}$ locally trivial $\mathbb{G}^{(\mathrm{perf})}_{m}$-bundles in
the sequence. In this case, there are exactly $a_{2}$ locally trivial
$\mathbb{A}^{1,(\mathrm{perf})}$-bundles in the sequence.

Now combining Theorem 2.4 with Proposition 2.5, we have the following.

######  Proposition 2.6.

Let $w\in\tilde{W}$ and $b\in\breve{G}$. Then $X_{w}(b)$ admits a
decomposition into finitely many locally closed, $\mathbf{J}_{b}(F)$-stable
subschemes $X_{i}$, and each $X_{i}$ is $\mathbf{J}_{b}(F)$-equivariant
universally homeomorphic to an iterated fibration of type $(a_{1},a_{2})$ over
$X_{w^{\prime}}(b)$ for some $w^{\prime}$ of minimal length in its
$\sigma$-conjugacy class in $\tilde{W}$ and some $a_{1},a_{2}\in\mathbb{N}$.

###  2.8. Classical Deligne–Lusztig varieties

Let $\mathbb{F}_{q}$ be a finite field and let $\bar{}\mathbb{F}_{q}$ an
algebraic closure of $\mathbb{F}_{q}$. Let $\mathbf{H}$ be a connected
reductive group over $\mathbb{F}_{q}$ and
$H=\mathbf{H}(\bar{}\mathbb{F}_{q})$. Let $\sigma_{\mathbf{H}}$ be the
Frobenius morphism on $H$. Let $B$ be a $\sigma_{\mathbf{H}}$-stable Borel
subgroup of $H$. Let $W_{H}$ be the Weyl group of $H$ and let $\mathbb{S}_{H}$
be the set of simple reflections. Then $\sigma_{\mathbf{H}}$ induces a group
automorphism on $W_{H}$ preserving $\mathbb{S}_{H}$. (Classical)
Deligne–Lusztig varieties were introduced by Deligne and Lusztig in [3]. They
are defined as follows. For $x\in W_{H}$, we set

$X^{\mathbf{H}}_{x}=\\{hB\in H/B;h^{-1}\sigma(h)\in B\dot{x}B\\}.$

If $x$ is a $\sigma_{\mathbf{H}}$-Coxeter element of $W_{H}$, then we say that
$X^{\mathbf{H}}_{x}$ is a classical Deligne–Lusztig variety of Coxeter type
for $\mathbf{H}$. It is well known that classical Deligne–Lusztig varieties of
Coxeter type are irreducible.

It is proved in [13, Theorem 4.8] that if $w\in\tilde{W}$ is a minimal length
element in its $\sigma$-conjugacy class, then $X_{w}(b)\neq\emptyset$ if and
only if $b$ and $\dot{w}$ are in the same $\sigma$-conjugacy class of
$\breve{G}$. In this case,

$X_{w}(b)\cong\mathbf{J}_{b}(F)\times^{P}X.$

Here $\cong$ means a $\mathbf{J}_{b}(F)$-equivariant universal homeomorphism,
$P$ is a parahoric subgroup of $\mathbf{J}_{b}(F)$, and $X$ is (the perfection
of) a classical Deligne–Lusztig variety for some connected reductive group
$\mathbf{H}$ over $\mathbb{F}_{q}$ with $\mathbf{H}(\mathbb{F}_{q})$
isomorphic to the reductive quotient of $P$. The group $P$ acts on
$\mathbf{J}_{b}(F)\times X$ by $p\cdot(g,z)=(gp^{-1},p\cdot z)$, and
$\mathbf{J}_{b}(F)\times^{P}X$ is the quotient space.

###  2.9. Very special parahoric subgroups

We follow [21, §2] for the definition of very special parahoric subgroups.

Let $\mathbf{G}_{1}$ be a (not necessarily quasi-split) connected reductive
group over $F$ and let $\sigma_{1}$ be its Frobenius morphism. Let
$\tilde{W}_{1}$ be the Iwahori–Weyl group $\mathbf{G}_{1}$. We still denote by
$\sigma_{1}$ the action on $\tilde{W}_{1}$ induced from the Frobenius morphism
on $\mathbf{G}_{1}$. Let $\tilde{}\mathbb{S}_{1}$ be the set of simple
reflections in $\tilde{W}_{1}$.

A parahoric subgroup $P$ of $\mathbf{G}_{1}(F)$ is called very special if it
is of maximal volume among all the parahoric subgroups of $\mathbf{G}_{1}(F)$.
A $\sigma_{1}$-stable subset $\breve{K}\subseteq\tilde{}\mathbb{S}_{1}$ is
called very special with respect to $\sigma_{1}$ if the parabolic subgroup
$W_{\breve{K}}$ generated by $\breve{K}$ is finite and the longest element of
$W_{\breve{K}}$ is of maximal length among all such $\sigma_{1}$-stable
subsets of $\tilde{}\mathbb{S}_{1}$. By [21, Proposition 2.2.5],

(a) A $\sigma_{1}$-stable subset $\breve{K}$ of $\tilde{}\mathbb{S}_{1}$ is
very special if and only if $\breve{P}_{\breve{K}}\cap\mathbf{G}_{1}(F)$ is a
very special parahoric parahoric subgroup of $\mathbf{G}_{1}(F)$, where
$\breve{P}_{\breve{K}}$ is the parahoric subgroup of
$\mathbf{G}_{1}(\breve{F})$ corresponding to $\breve{K}$.

###  2.10. Statement of the main result

Let $w\in\tilde{W}$. We say that $w$ has finite $\sigma$-Coxeter part if
$\eta_{\sigma}(w)$ is a partial $\sigma$-Coxeter element. In this case, we set
$J(w)=\operatorname{supp}_{\sigma}(\eta_{\sigma}(w))$. For any $\sigma$-stable
subset $J$ of $J(w)$, denote by $J_{\mu}^{\prime}$ (resp.
$J_{\mu}^{\prime\prime}$) the union of all $\sigma$-orbits of connected
components of $J$ in which $\mu$ is non-central (resp. central). Then $\mu$ is
essentially non-central in $J_{\mu}^{\prime}$. By §2.3,
$B(\mathbf{G},\mu)_{J^{\prime}_{\mu}\text{-}\mathrm{irr}}\neq\emptyset$, and
we have a natural bijection
$B(\mathbf{M}_{J_{\mu}^{\prime}},\mu)_{\mathrm{irr}}\cong
B(\mathbf{G},\mu)_{J^{\prime}_{\mu}\text{-}\mathrm{irr}}$.

Let $J_{0}(w)$ be the subset of $\mathbb{S}$ with
$\mu-\nu_{b_{w}}\in\sum_{i\in J_{0}(w)}\mathbb{R}_{>0}\alpha_{i}^{\vee}$. Then
$J_{0}(w)=J_{0}(w)^{\prime}_{\mu}$. By definition,
$[\dot{w}]\leqslant[b_{w}]$. Thus $\nu_{\dot{w}}\leqslant\nu_{b_{w}}$ and
$J_{0}(w)\subseteq J(w)$.

For any $\sigma$-stable $J_{1},J_{2}\subseteq\mathbb{S}$, denote by
$[J_{1},J_{2}]$ the set of $\sigma$-stable subsets $J\subseteq\mathbb{S}$ such
that $J_{1}\subseteq J\subseteq J_{2}$ and denote by $[J_{1},J_{2}]_{\mu}$ the
set of $J\in[J_{1},J_{2}]$ such that $\mu$ is essentially non-central in $J$
(or equivalently, $J=J_{\mu}^{\prime}$).

Now we state the main result of this paper.

######  Theorem 2.7.

Let $w\in Wt^{\mu}W$ such that $\eta_{\sigma}(w)$ is a partial
$\sigma$-Coxeter element. Then

  1. (1)

$w$ is a cordial element;

  2. (2)

$B(\mathbf{G})_{w}=\sqcup_{J\in[J_{0}(w),J(w)]_{\mu}}B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$;

  3. (3)

for any $[b]\in B(\mathbf{G})_{w}$, we have

$X_{w}(b)\cong\mathbf{J}_{b}(F)\times^{P}Y,$

where $P$ is a parahoric subgroup of $\mathbf{J}_{b}(F)$, and $Y$ is an
iterated fibration over (the perfection of) a classical Deligne–Lusztig
variety of Coxeter type for some connected reductive group $\mathbf{H}$ over
$\mathbb{F}_{q}$ with $\mathbf{H}(\mathbb{F}_{q})$ isomorphic to the reductive
quotient of $P$.

######  Remark 2.8.

  1. (1)

In particular, $\mathbf{J}_{b}(F)$ acts transitively on the set of irreducible
components of $X_{w}(b)$.

  2. (2)

We also have an explicit description of the parahoric subgroup $P$ of
$\mathbf{J}_{b}(F)$ in Theorem 2.7(3) (see §7). In particular, if
$w\in{}^{\mathbb{S}}\tilde{W}$ and $\eta_{\sigma}(w)$ is a (full)
$\sigma$-Coxeter element, then the parahoric subgroup $P$ is very special.

Parts (1) and (2) will be proved in §4. Part (3) is the most difficult part of
this paper and will be proved in §§6 and 7. The proof is based on a deep
analysis of the reduction tree of $w$, which will be introduced in §3.

##  3\. Class polynomials and reduction trees

In this section, we recall the class polynomials of Hecke algebras and the
connection with affine Deligne–Lusztig varieties discovered in [13]. We then
introduce the reduction tree, which encodes more information than the class
polynomials.

###  3.1. Hecke algebras and their cocenters

Let $\mathbf{q}$ be an indeterminate. Let $H$ be the Hecke algebra associated
with $\tilde{W}$, that is, it is the $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-algebra
generated by $T_{w}$ for $w\in\tilde{W}$ subject to the relations

  * •

$T_{w}T_{w^{\prime}}=T_{ww^{\prime}}$ if
$\ell(ww^{\prime})=\ell(w)+\ell(w^{\prime})$;

  * •

$(T_{s_{i}}+1)(T_{s_{i}}-\mathbf{q})=0$ for $i\in\tilde{}\mathbb{S}$.

The action of $\sigma$ on $\tilde{W}$ induces an action on $H$, which we still
denote by $\sigma$. The $\sigma$-commutator $[H,H]_{\sigma}$ is the
$\mathbb{Z}[\mathbf{q}^{\pm 1}]$-submodule generated by
$hh^{\prime}-h^{\prime}\sigma(h)$ for $h,h^{\prime}\in H$. The
$\sigma$-cocenter of $H$ is defined to be

$\bar{H}_{\sigma}=H/[H,H]_{\sigma}.$

By Theorem 2.4(2), for any $\sigma$-conjugacy class $\mathcal{O}$ of
$\tilde{W}$ and $w,w^{\prime}\in\mathcal{O}_{\min}$, we have
$T_{w}+[H,H]_{\sigma}=T_{w^{\prime}}+[H,H]_{\sigma}$. We write
$T_{\mathcal{O}}=T_{w}+[H,H]_{\sigma}\in\bar{H}_{\sigma}$ for any
$w\in\mathcal{O}_{\min}$. It is proved in [18, Theorem 6.7] that

######  Theorem 3.1.

$\bar{H}_{\sigma}$ is a free $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-module with
basis $\\{T_{\mathcal{O}}\\}_{\mathcal{O}\in B(\tilde{W},\sigma)}$.

Moreover, by [13, §2.3] and [15, §2.8.2], for any $w\in\tilde{W}$ and
$\mathcal{O}\in B(\tilde{W},\sigma)$, there exists a unique polynomial
$F_{w,\mathcal{O}}\in\mathbb{N}[q-1]$ such that

$T_{w}+[H,H]_{\sigma}=\sum_{\mathcal{O}\in
B(\tilde{W},\sigma)}F_{w,\mathcal{O}}T_{\mathcal{O}}\in\bar{H}_{\sigma}.$

The polynomials $F_{w,\mathcal{O}}$ are called class polynomials111The
polynomials we use here coincide with those in [15] and differ from the
polynomials used in [13] by a certain monomial. See [15, footnote on p. 106]..

###  3.2. Class polynomials and affine Deligne–Lusztig varieties

The class polynomials encode a lot of information about affine Deligne–Lusztig
varieties.

Let $\tilde{W}B(\mathbf{G})$ be the map sending $w\in\tilde{W}$ to the
$\sigma$-conjugacy class $[\dot{w}]$ of $\breve{G}$. It is known that this map
is independent of the choice of the representative $\dot{w}$ of $w$, and it
induces a map

$\Psi:B(\tilde{W},\sigma)B(\mathbf{G}).$

By [13, Theorem 3.7], the map $\Psi$ is surjective.

Let $w\in\tilde{W}$ and $[b]\in B(\mathbf{G})$. We set

$F_{w,[b]}=\sum_{\mathcal{O}\in
B(\tilde{W},\sigma);\Psi(\mathcal{O})=[b]}\mathbf{q}^{\ell(\mathcal{O})}F_{w,\mathcal{O}}\in\mathbb{N}[\mathbf{q}-1].$

Here $\ell(\mathcal{O})=\ell(x)$ for any $x\in\mathcal{O}_{\min}$.

The following “dimension = degree” theorem is established in [13, Theorem
6.1].

######  Theorem 3.2.

Let $w\in\tilde{W}$ and $b\in\breve{G}$. Then $\dim X_{w}(b)=\deg
F_{w,[b]}-\langle\nu_{b},2\rho\rangle$.

######  Remark 3.3.

Here, by convention, $\dim\emptyset=\deg 0=-\infty$.

Moreover, we have the following “leading coefficients = irreducible
components” theorem. This is established in [15, Theorem 2.19]. See also [21,
Theorem 3.3.9 and Corollary 3.3.11].

######  Theorem 3.4.

Let $w\in\tilde{W}$ and $b\in\breve{G}$. Then the cardinality of
$\mathbf{J}_{b}(F)\backslash\Sigma^{\mathrm{top}}(X_{w}(b))$ equals the
leading coefficient of $F_{w,[b]}$.

Although not needed in this paper, it is also worth mentioning that in the
superbasic case, the class polynomial gives the number of rational points of
an affine Deligne–Lusztig variety. This is established in [13, Proposition
8.3].

######  Proposition 3.5.

Suppose that the residue field of $F$ is $\mathbb{F}_{q}$. Assume that
$\mathbf{G}=PGL_{n}$ split over $F$ and $b\in\mathbf{G}(F)$ is a superbasic
element in $\breve{G}$. Then

$\sharp X_{w}(b)^{\sigma}=nF_{w,[b]}|_{\mathbf{q}=q}.$

###  3.3. An identity on the class polynomials

We have the following identity on the class polynomials. We first give a proof
using representations of Hecke algebras. Then we provide a geometric
interpretation of this identity.

######  Proposition 3.6.

Let $w\in\tilde{W}$. Then

$\mathbf{q}^{\ell(w)}=\sum_{\mathcal{O}\in
B(\tilde{W},\sigma)}\mathbf{q}^{\ell(\mathcal{O})}F_{w,\mathcal{O}}=\sum_{[b]\in
B(\mathbf{G})}F_{w,[b]}.$

###### Proof.

We prove the first equality. The second follows from the definition.

Let $\pi:H\mathbb{Z}[\mathbf{q}^{\pm 1}]$ be the homomorphism of
$\mathbb{Z}[\mathbf{q}^{\pm 1}]$-algebras sending $T_{s_{i}}$ to $\mathbf{q}$
for any $i\in\tilde{}\mathbb{S}$. As $\pi\circ\sigma=\pi$,
$\pi([H,H]_{\sigma})=0$ and thus $\pi$ induces a homomorphism of the
$\mathbb{Z}[\mathbf{q}^{\pm 1}]$-modules
$\bar{H}_{\sigma}\mathbb{Z}[\mathbf{q}^{\pm 1}]$, which we still denote by
$\pi$.

We have $T_{w}+[H,H]_{\sigma}=\sum_{\mathcal{O}\in
B(\tilde{W},\sigma)}F_{w,\mathcal{O}}T_{\mathcal{O}}$. Applying $\pi$ to both
sides, we obtain $\mathbf{q}^{\ell(w)}=\sum_{\mathcal{O}\in
B(\tilde{W},\sigma)}\mathbf{q}^{\ell(\mathcal{O})}F_{w,\mathcal{O}}$. ∎

In the rest of this subsection, we assume furthermore that
$F=\mathbb{F}_{q}((\epsilon))$ and that $\mathbf{G}$ is split over $F$. We
give a geometric interpretation of the above identity.

For any $n\in\mathbb{N}$, let $\breve{I}_{n}$ be the $n$th congruence subgroup
of $\breve{I}$. Following [4, §2.10], we call a subset $X$ of $\breve{G}$
admissible if for any $w\in\tilde{W}$, there exists $n\in\mathbb{N}$ such that
$X\cap\breve{I}\dot{w}\breve{I}$ is stable under right multiplication of
$\breve{I}_{n}$. In this case, the action of $\breve{I}_{n}$ on
$X\cap\breve{I}\dot{w}\breve{I}$ is free, and
$\frac{\sharp\bigl{(}(X\cap\breve{I}\dot{w}\breve{I})/\breve{I}_{n}\bigr{)}^{\sigma}}{\sharp\breve{I}_{n}^{\sigma}}$
is independent of the choice of such $n$. We set

$\breve{\sharp}(X\cap\breve{I}\dot{w}\breve{I})=\frac{\sharp\bigl{(}(X\cap\breve{I}\dot{w}\breve{I})/\breve{I}_{n}\bigr{)}^{\sigma}}{\sharp\breve{I}_{n}^{\sigma}}.$

An admissible subset $X$ is called bounded if
$X\cap\breve{I}\dot{w}\breve{I}=\emptyset$ for all but finitely many
$w\in\tilde{W}$. For any bounded admissible subset $X$, we set

$\breve{\sharp}X=\sum_{w\in\tilde{W}}\breve{\sharp}(X\cap\breve{I}\dot{w}\breve{I})$

and call it the normalized cardinality of the rational points of $X$. By [14,
Theorem A.1], each $\sigma$-conjugacy class of $\breve{G}$ is admissible. We
have the following geometric interpretation of the class polynomials.

######  Proposition 3.7.

Let $w\in\tilde{W}$ and $[b]\in B(\mathbf{G})$. Then

$\breve{\sharp}([b]\cap\breve{I}\dot{w}\breve{I})=F_{w,[b]}|_{\mathbf{q}=q}.$

###### Proof.

We argue by induction on $\ell(w)$.

If $w$ is a minimal length element in its $\sigma$-conjugacy class in
$\tilde{W}$, then

$[b]\cap\breve{I}\dot{w}\breve{I}=\begin{cases}\breve{I}\dot{w}\breve{I}&\text{if
}[b]=[\dot{w}],\\\ \emptyset&\text{otherwise}.\end{cases}$

On the other hand, by [15, §2.8.2],

$F_{w,[b]}=\begin{cases}\mathbf{q}^{\ell(w)}&\text{if }[b]=[\dot{w}],\\\
0&\text{otherwise}.\end{cases}$

Thus the proposition holds in this case.

Now we assume that $w$ is not a minimal length element in its
$\sigma$-conjugacy class. By Theorem 2.4(1), there exist
$w^{\prime}\in\tilde{W}$ and $i\in\tilde{}\mathbb{S}$ such that
$w\approx_{\sigma}w^{\prime}$ and $s_{i}w^{\prime}\sigma(s_{i})<w^{\prime}$.
Set $w_{1}=s_{i}w^{\prime}$ and $w_{2}=s_{i}w^{\prime}\sigma(s_{i})$. Then
$\ell(w_{1}),\ell(w_{2})<\ell(w)$. By the proof of [15, Theorem 2.16],

$\breve{\sharp}([b]\cap\breve{I}\dot{w}\breve{I})=\breve{\sharp}([b]\cap\breve{I}\dot{w}^{\prime}\breve{I})=(q-1)\breve{\sharp}([b]\cap\breve{I}\dot{w}_{1}\breve{I})+q\breve{\sharp}([b]\cap\breve{I}\dot{w}_{2}\breve{I}).$

On the other hand, by [15, §2.8.2],
$F_{w,[b]}=(\mathbf{q}-1)F_{w_{1},[b]}+\mathbf{q}F_{w_{2},[b]}$. Now the
statement for $w$ follows from the inductive hypothesis on $w_{1}$ and
$w_{2}$. ∎

We have the decomposition $\breve{I}\dot{w}\breve{I}=\sqcup_{[b]\in
B(\mathbf{G})}[b]\cap\breve{I}\dot{w}\breve{I}$. Thus

$\mathbf{q}^{\ell(w)}=\breve{\sharp}(\breve{I}\dot{w}\breve{I})=\sum_{[b]\in
B(\mathbf{G})}\breve{\sharp}([b]\cap\breve{I}\dot{w}\breve{I})=\sum_{[b]\in
B(\mathbf{G})}F_{w,[b]}.$

This gives an alternative proof of Proposition 3.6 and a geometric
interpretation of Proposition 3.6 in the case where the $\sigma$-action on
$\tilde{W}$ is trivial via counting (the normalized cardinality of) the
rational points of $\breve{I}\dot{w}\breve{I}$.

###  3.4. Reduction trees

Let $w\in\tilde{W}$. We construct the reduction tree for $w$, which encodes
the Deligne–Lusztig reduction for the affine Deligne–Lusztig varieties
associated with $w$ (and with all $b\in\breve{G}$).

The vertices of the graphs are the elements of $\tilde{W}$, and the (oriented)
edges are of the form $x\rightharpoonup y$, where $x,y\in\tilde{W}$, and there
exists $x^{\prime}\in\tilde{W}$ and $i\in\tilde{}\mathbb{S}$ with
$x\approx_{\sigma}x^{\prime}$, $s_{i}x^{\prime}\sigma(s_{i})<x^{\prime}$ and
$y\in\\{s_{i}x^{\prime},s_{i}x^{\prime}\sigma(s_{i})\\}$. Some elements of
$\tilde{W}$ may occur more than once in a reduction tree.

The reduction trees are constructed inductively.

Suppose that $w$ is of minimal length in its $\sigma$-conjugacy class of
$\tilde{W}$. Then the reduction tree of $w$ consists of a single vertex $w$
and no edges.

Suppose that $w$ is not of minimal length in its $\sigma$-conjugacy class of
$\tilde{W}$ and that a reduction tree is given for any $z\in\tilde{W}$ with
$\ell(z)<\ell(w)$. By Theorem 2.4(1), there exist $w^{\prime}\in\tilde{W}$ and
$i\in\tilde{}\mathbb{S}$ with $w\approx_{\sigma}w^{\prime}$ and
$s_{i}w^{\prime}\sigma(s_{i})<w^{\prime}$. The reduction tree of $w$ is the
graph containing the given reduction tree for $s_{i}w^{\prime}$ and the
reduction tree for $s_{i}w^{\prime}\sigma(s_{i})$, and the edges
$w\rightharpoonup s_{i}w^{\prime}$ and $w\rightharpoonup
s_{i}w^{\prime}\sigma(s_{i})$.

Note that the reduction trees of $w$ are not unique. They depend on the
choices of $w^{\prime}$ and $s_{i}$ in the construction. We will see in the
rest of this section that the reduction trees encode more information than the
class polynomials.

###  3.5. Reduction path

Let $\mathcal{T}$ be a reduction tree of $w$. An end point of the tree
$\mathcal{T}$ is a vertex $x$ of $\mathcal{T}$ such that there is no edge of
the form $x\rightharpoonup x^{\prime}$ in $\mathcal{T}$. By Theorem 2.4, each
end point is of minimal length in its $\sigma$-conjugacy class. A reduction
path in $\mathcal{T}$ is a path $\underline{p}:w\rightharpoonup
w_{1}\rightharpoonup\cdots\rightharpoonup w_{n}$, where $w_{n}$ is an end
point of $\mathcal{T}$. The length $\ell(\underline{p})$ of the reduction path
$\underline{p}$ is the number of edges in $\underline{p}$. We also write
$\operatorname{end}(\underline{p})=w_{n}$ and
$[b]_{\underline{p}}=\Psi(\operatorname{end}(\underline{p}))\in
B(\mathbf{G})$.

Note that if $x\rightharpoonup y$, then $\ell(x)-\ell(y)\in\\{1,2\\}$. We say
that the edge $x\rightharpoonup y$ is of type I if $\ell(x)-\ell(y)=1$ and of
type II if $\ell(x)-\ell(y)=2$. For any reduction path $\underline{p}$, we
denote by $\ell_{I}(\underline{p})$ the number of type I edges in
$\underline{p}$ and by $\ell_{II}(\underline{p})$ the number of type II edges
in $\underline{p}$. Then
$\ell(\underline{p})=\ell_{I}(\underline{p})+\ell_{II}(\underline{p})$.

The following relation between class polynomials and reduction trees follows
easily from the inductive construction, and we omit the details of its proof.

######  Lemma 3.8.

Let $w\in\tilde{W}$ and let $\mathcal{T}$ be a reduction tree of $w$. Then,
for any $\sigma$-conjugacy class $\mathcal{O}$ of $\tilde{W}$, we have

$F_{w,\mathcal{O}}=\sum_{\underline{p}}(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})},$

where $\underline{p}$ runs over all the reduction paths in $\mathcal{T}$ with
$\Psi(\operatorname{end}(\underline{p}))=\mathcal{O}$.

Combining Proposition 2.5 with the construction of the reduction trees, we
obtain the following result.

######  Proposition 3.9.

Let $w\in\tilde{W}$ and $\mathcal{T}$ be a reduction tree of $w$. Then, for
any $b\in\breve{G}$, there exists a decomposition

$X_{w}(b)=\bigsqcup_{\underline{p}\mathrm{\;is\;a\;reduction\;path\;of\;}\mathcal{T};[b]_{\underline{p}}=[b]}X_{\underline{p}},$

where $X_{\underline{p}}$ is a locally closed subscheme of $X_{w}(b)$, and is
$\mathbf{J}_{b}(F)$-equivariant universally homeomorphic to an iterated
fibration of type $(\ell_{I}(\underline{p}),\ell_{II}(\underline{p}))$ over
$X_{\operatorname{end}(\underline{p})}(b)$.

######  Remark 3.10.

Since $\operatorname{end}(\underline{p})$ is a minimal length element in its
$\sigma$-conjugacy class, by §2.8(a) we have
$X_{\operatorname{end}(\underline{p})}(b)\cong\mathbf{J}_{b}(F)\times^{P}X$,
where $P$ is a parahoric subgroup of $\mathbf{J}_{b}(F)$ and $X$ is (the
perfection of) an irreducible component of a classical Deligne–Lusztig
variety. Thus each irreducible component $Y$ of $X_{\underline{p}}$ is
universally homeomorphic to an iterated fibration of type
$(\ell_{I}(\underline{p}),\ell_{II}(\underline{p}))$ over $X$. We have a
natural action of $\mathbf{J}_{b}(F)$ on $X_{w}(b)$, and $X_{\underline{p}}$
is stable under this action. In this case,
$X_{\underline{p}}\cong\mathbf{J}_{b}(F)\times^{P}Y$.

##  4\. Cordiality and the set $B(\mathbf{G})_{w}$

###  4.1. Maximal Hodge–Newton irreducible elements

Recall that $[b_{t^{\mu}}]$ is the unique maximal element of
$B(\mathbf{G},\mu)$. We have the following result on
$B(\mathbf{G},\mu)_{\mathrm{indec}}$.

######  Proposition 4.1.

The set $B(\mathbf{G},\mu)_{\mathrm{indec}}$ contains a unique maximal
element.

######  Remark 4.2.

We denote this element by $[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$.

###### Proof.

Following [2], for any subset $E\subseteq(V^{+})^{\sigma}$, we set

$C_{\geqslant E}=\\{v\in(V^{+})^{\sigma};v\geqslant v^{\prime},\text{ for any
}v^{\prime}\in E\\}.$

For any $i\in\mathbb{S}$, denote by $\mathcal{O}_{i}$ the $\sigma$-orbit of
$i$. Set $\omega_{\mathcal{O}_{i}}=\sum_{j\in\mathcal{O}_{i}}\omega_{j}$.
Define $e_{i}\in\mathbb{Q}\omega_{i}^{\vee}$ with

$\langle
e_{i},\omega_{i}\rangle=\frac{1}{\sharp\mathcal{O}_{i}}\max\\{0,\langle\mu,\omega_{\mathcal{O}_{i}}\rangle-1\\}.$

Set $E=\\{e_{i};i\in\mathbb{S}\\}$. By [2, Theorem 6.5], there exists a unique
$\sigma$-conjugacy class $[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]\in
B(\mathbf{G},\mu)$ such that
$\nu_{b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}}=\min C_{\geqslant E}$ and
$\langle\mu-\nu_{b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}},\omega_{\mathcal{O}_{k}}\rangle=1$
for any $k\in\mathbb{S}-I(\nu)$. Set
$\nu=\nu_{b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}}$.

Suppose that $(\mu,[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}])$ is
Hodge–Newton decomposable with respect to some standard Levi subgroup
$\mathbf{M}_{J}$ with $J=\sigma(J)\subsetneqq\mathbb{S}$. Let
$j\in\mathbb{S}-J$. By definition,
$\langle\mu-\nu,\omega_{\mathcal{O}_{j}}\rangle=0$. On the other hand, as
$I(\nu)\subseteq J$ we have $\langle\nu,\alpha_{j}\rangle\neq 0$. Thus
$\langle\nu,\omega_{\mathcal{O}_{j}}\rangle=1$, which is a contradiction.
Therefore $[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}$.

On the other hand, let $[b^{\prime}]\in B(\mathbf{G},\mu)_{\mathrm{indec}}$.
Set $\nu^{\prime}=\nu_{b^{\prime}}$. For any $i\in\mathbb{S}-I(v^{\prime})$,
we denote by $pr_{(i)}:V=\mathbb{R}\omega^{\vee}_{i}\oplus\sum_{j\neq
i}\mathbb{R}\alpha_{j}^{\vee}\mathbb{R}\omega_{i}^{\vee}$ the natural
projection. Set
$e^{\prime}_{i}=pr_{(i)}(v^{\prime})\in\mathbb{R}\omega_{i}^{\vee}$. Let
$E^{\prime}=\\{e^{\prime}_{i};i\in\mathbb{S}-I(v^{\prime})\\}$. Again by [2,
Theorem 6.5], we have $v^{\prime}=\min C_{\geqslant E^{\prime}}$. By §2.2(a),
we have
$\langle\mu-\nu^{\prime},\omega_{\mathcal{O}_{i}}\rangle\in\mathbb{Z}_{\geqslant
1}$ for $i\in\mathbb{S}-I(v^{\prime})$. This means that
$e_{i}^{\prime}\leqslant e_{i}$ for $i\in\mathbb{S}-I(v^{\prime})$. So $v\in
C_{\geqslant E^{\prime}}$ and $v\geqslant\min C_{\geqslant
E^{\prime}}=v^{\prime}$. Hence $[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$ is
the unique maximal element of $B(\mathbf{G},\mu)_{\mathrm{indec}}$. ∎

Combining the description of $[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$ in
the proof above with §2.2(a), we have the following result.

######  Corollary 4.3.

Suppose that $\mu$ is essentially non-central. Then

${{\rm
length}}([b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}],[t^{\mu}])=\sharp(\mathbb{S}/\langle\sigma\rangle).$

###  4.2. Proof of Theorem 2.7(1) and (2) for $w=t^{\mu}c$

Now we prove Theorem 2.7(1) and (2) in the case
$w=t^{\mu}c\in{}^{\mathbb{S}}\tilde{W}$, where $c$ is a partial
$\sigma$-Coxeter element. In this case, it is easy to see that $\mu$ is
essentially non-central in $\operatorname{supp}_{\sigma}(c)$. Set
$J=\operatorname{supp}_{\sigma}(c)$. We have
$J(w)=J(w)_{\mu}^{\prime}=J_{0}(w)=J$. We need to show that $w$ is cordial and
$B(\mathbf{G})_{w}=B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$.

By §2.3, we have a natural bijection
$B(\mathbf{M}_{J},\mu)_{\mathrm{irr}}\cong
B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$. By [7, Theorem 3.3.1], we have
$B(\mathbf{G})_{w}=B(\mathbf{M}_{J})_{w}$. Note that
$\mu^{\diamond}-\nu_{b_{w}}\in\sum_{j\in J}\mathbb{R}\alpha_{j}^{\vee}$. Hence
$\langle\mu-\nu_{b_{w}},\rho\rangle=\langle\mu-\nu_{b_{w}},\rho_{J}\rangle$,
where $\rho_{J}$ is the half sum of positive roots of $\mathbf{M}_{J}$. By
definition, $w$ is cordial in $\mathbf{G}$ if and only if it is cordial in
$\mathbf{M}_{J}$. Hence we may assume that $J=\mathbb{S}$ and that $c$ is a
$\sigma$-Coxeter element.

We first show that

(a) $B(\mathbf{G})_{w}\subseteq B(\mathbf{G},\mu)_{\mathrm{indec}}$.

Suppose that $X_{w}(b)\neq\emptyset$ and $(\mu,b)$ is Hodge–Newton
decomposable with respect to some proper standard Levi subgroup $\mathbf{M}$.
By [8, Theorem 1.11], there is some $u\in W$ such that $u^{-1}w\sigma(u)$ lies
in $\tilde{W}_{\mathbf{M}}$, which contradicts the fact that $c$ is
$\sigma-$Coxeter. Thus (a) is proved.

Next we show that

(b) $[b_{w}]=[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$ and $w$ is a cordial
element.

By §2.5(b), we have

$\langle\mu,2\rho\rangle-2\ell(c)=\ell(w)-\ell(\eta_{\sigma}(w))\leqslant\langle\nu_{b_{w}},2\rho\rangle-{\mathrm{def}}(b_{w}).$

Combined with §2.5 (a), we get ${\rm
length}([b_{w}],[t^{\mu}])\leqslant\ell(c)=\sharp(\mathbb{S}/\langle\sigma\rangle)$.
However, by Corollary 4.3, we have ${\rm
length}([b_{w}],[b_{t^{\mu}}])\geqslant{\rm
length}([b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}],[b_{t^{\mu}}])=\sharp(\mathbb{S}/\langle\sigma\rangle)$.
Thus we must have $[b_{w}]=[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$ and
$\dim X_{w}(b_{w})=d_{w}(b_{w})$. Thus (b) is proved.

Finally, we show that

(c)
$B(\mathbf{G})_{w}=B(\mathbf{G},\mu)_{\mathrm{indec}}=B(\mathbf{G},\mu)_{\mathrm{irr}}$.

Let $[b_{\min}]$ be the unique basic $\sigma$-conjugacy class in
$B(\mathbf{G})$ with $\kappa(b_{\min})=\kappa(\mu)$. Then $[b_{\min}]$ is the
unique minimal element in $B(\mathbf{G},\mu)_{\mathrm{indec}}$. Note that $c$
is a $\sigma$-Coxeter element of $W$. Thus $\nu_{\dot{w}}$ is central and
$[\dot{w}]=[b_{\min}]$. In particular, $[b_{\min}]\in B(\mathbf{G})_{w}$. By
Theorem 2.3, $B(\mathbf{G})_{w}$ is saturated and hence must be equal to
$B(\mathbf{G},\mu)_{\mathrm{indec}}$.

This completes the proof of the $t^{\mu}c$ case.

###  4.3. Partial conjugation

To handle the general case, we use the partial conjugation method introduced
in [12].

By partial conjugation, we mean conjugating by elements in the finite Weyl
group $W$. For any $x\in{}^{\mathbb{S}}\tilde{W},$ set

$I(x)=\max\\{J\subseteq\mathbb{S};{\mathrm{Ad}}(x)\sigma(J)=J\\}.$

This is well-defined. Indeed, if ${\mathrm{Ad}}(x)\sigma(J_{i})=J_{i}$ for
$i=1,2$, then ${\mathrm{Ad}}(x)\sigma(J_{1}\cup J_{2})=J_{1}\cup J_{2}$. Let
$W_{I(x)}$ be the subgroup of $W$ generated by the simple reflections in
$I(x)$. Then ${\mathrm{Ad}}(x)\circ\sigma$ gives a length-preserving group
automorphism on $W_{I(x)}$. By [12, Proposition 2.4], we have

$\tilde{W}=\bigsqcup_{x\in{}^{\mathbb{S}}\tilde{W}}W\cdot_{\sigma}(W_{I(x)}x)=\bigsqcup_{x\in{}^{\mathbb{S}}\tilde{W}}W\cdot_{\sigma}(xW_{\sigma(I(x))}).$

Moreover, by [12, Proposition 3.4], we have the following:

(a) For any $w\in\tilde{W}$, there exists $x\in{}^{\mathbb{S}}\tilde{W}$ and
$u\in W_{I(x)}$ such that $w_{\sigma}ux$ and all the simple reflections
involved in the conjugations are in $\mathbb{S}$.

By [13, Proposition 4.9], we have the following:

(b) Let $x\in{}^{\mathbb{S}}\tilde{W}$ and $u\in W_{I(x)}$. Then $B(G)_{ux}$ =
$B(G)_{x}$ and $\dim X_{ux}(b)=\dim X_{x}(b)+\ell(u)$ for any $[b]\in
B(\mathbf{G})_{x}$.

Similar to §3.4, we may consider partial reductions. By partial reduction, we
mean reduction $w\rightharpoonup s_{i}w$ or $w\rightharpoonup
s_{i}w\sigma(s_{i})$ with $i\in\mathbb{S}$. We shows that partial reduction
preserves elements with finite Coxeter parts.

######  Lemma 4.4.

Let $w\in\tilde{W}$ with $\eta_{\sigma}(w)$ a partial $\sigma$-Coxeter element
of $W$. Let $i\in\mathbb{S}$ with $s_{i}w<w$. Then

  1. (1)

$\eta_{\sigma}(s_{i}w\sigma(s_{i}))$ is a partial $\sigma$-Coxeter element of
$W$ and
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(s_{i}w\sigma(s_{i})))=\operatorname{supp}_{\sigma}(\eta_{\sigma}(w))$.

  2. (2)

If, moreover, $s_{i}w\sigma(s_{i})<w$, then $\eta_{\sigma}(s_{i}w)$ is a
partial $\sigma$-Coxeter element of $W$ and
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(s_{i}w))=\operatorname{supp}_{\sigma}(\eta_{\sigma}(w))-\\{\sigma^{l}(i^{\prime});l\in\mathbb{Z}\\}$
for some $i^{\prime}\in\operatorname{supp}_{\sigma}(\eta_{\sigma}(w))$.

###### Proof.

We prove part (1). The proof of part (2) is similar, and we skip the details.

Write $w=xt^{\mu}y$ with $t^{\mu}y\in{}^{\mathbb{S}}\tilde{W}$. Set
$c=\eta_{\sigma}(w)=\sigma^{-1}(y)x$. If
$y\sigma(s_{i})\in{}^{\mathbb{S}}\tilde{W}$, then
$\eta_{\sigma}(s_{i}w\sigma(s_{i}))=\eta_{\sigma}(w)$, and the statement is
obvious. Now assume that $y\sigma(s_{i})=s_{i^{\prime}}y$ for some
$i^{\prime}\in I(\mu)$. Then we have $x^{-1}(\alpha_{i})<0$ and
$\sigma^{-1}(y)(\alpha_{i})>0$. Thus
$(\sigma^{-1}(y)x)(-x^{-1}(\alpha_{i}))<0$. It follows that
$\sigma^{-1}(y)s_{i}x=(\sigma^{-1}(y)x)(x^{-1}s_{i}x)<\sigma^{-1}(y)x$. By the
cancellation property of Coxeter groups, we conclude that
$\sigma^{-1}(y)s_{i}x$ is a partial $\sigma$-Coxeter element and
$\ell(\sigma^{-1}(y)s_{i}x)=\ell(\sigma^{-1}(y)x)-1$. Write
$c^{\prime}=\sigma^{-1}(y)s_{i}x$. Notice that
$\sigma^{-1}(s_{i^{\prime}})c=\sigma^{-1}(s_{i^{\prime}}y)x=\sigma^{-1}(y)s_{i}x=c^{\prime}$.
It follows that $\sigma^{-1}(i^{\prime})$ is not in the $\sigma$-support of
$c^{\prime}$. Hence
$\eta_{\sigma}(s_{i}w\sigma(s_{i}))=\sigma^{-1}(y)s_{i}xs_{i^{\prime}}=c^{\prime}s_{i^{\prime}}$
is a partial $\sigma$-Coxeter element, and

$\operatorname{supp}_{\sigma}(\eta_{\sigma}(s_{i}w\sigma(s_{i})))=\operatorname{supp}_{\sigma}(\eta_{\sigma}(w)).$

The statement is proved. ∎

###  4.4. Proof of Theorem 2.7(1) and (2): general case

Let $w=xt^{\mu}y\in\tilde{W}$ with $t^{\mu}y\in{}^{\mathbb{S}}\tilde{W}$. We
assume that $\eta_{\sigma}(w)$ a partial $\sigma$-Coxeter element. We prove
Theorem 2.7(1) and (2) by induction on $\ell(x)$.

The case $\ell(x)=0$ has already been proved in §4.2. Assume that $\ell(x)>0$.
Let $i\in\mathbb{S}$ such that $s_{i}x<x$. There are three different cases.

Case (1): $\ell(s_{i}w\sigma(s_{i}))<\ell(w)$. Write $w_{1}=s_{i}w$ and
$w_{2}=s_{i}w\sigma(s_{i})$. By Lemma 4.4, $\eta_{\sigma}(w_{1})$ and
$\eta_{\sigma}(w_{2})$ are both partial $\sigma$-Coxeter elements. The
inductive hypothesis applies for $w_{1}$ and $w_{2}$. Since $w_{1}>w_{2}$, we
have $[b_{w}]=\max([b_{w_{1}}],[b_{w_{2}}])=[b_{w_{1}}]$. Then
$d_{w}(b_{w})=d_{w_{1}}(b_{w})+1$ and $\dim X_{w_{1}}(b_{w})+1=\dim
X_{w}(b_{w})$. Thus $w$ is cordial.

Observe that

$B(\mathbf{G})_{w}=B(\mathbf{G})_{w_{1}}\cup
B(\mathbf{G})_{w_{2}}=\sqcup_{J\in[J_{0}(w_{1}),J(w_{1})]_{\mu}\cup[J_{0}(w_{2}),J(w_{2})]_{\mu}}B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}.$

Note that $J(w_{2})=J(w)$, $J_{0}(w_{1})=J_{0}(w)$ and $J(w_{1})\subset J(w)$.
By §2.5(b), $B(\mathbf{G})_{w}$ is saturated. Hence we must have

$[J_{0}(w_{1}),J(w_{1})]_{\mu}\cup[J_{0}(w_{2}),J(w_{2})]_{\mu}=[J_{0}(w),J(w)]_{\mu}.$

This proves part (2) of Theorem 2.7.

Case (2): $y\sigma(s_{i})<y$. Write
$w^{\prime}=s_{i}w\sigma(s_{i})=s_{i}xt^{\mu}y\sigma(s_{i})$. The inductive
hypothesis applies for $w^{\prime}$. Note that $w\approx_{\sigma}w^{\prime}$,
in particular, $B(\mathbf{G})_{w}=B(\mathbf{G})_{w^{\prime}}$. Note also that
$J(w)=J(w^{\prime})$. Hence the statements hold for $w$.

Case (3): $y\sigma(s_{i})=s_{i^{\prime}}y$ for some $i^{\prime}\in I(\mu)$ and
$\ell(s_{i}xs_{i^{\prime}})=\ell(x)$. Write
$w^{\prime}=s_{i}w\sigma(s_{i})=s_{i}xs_{i^{\prime}}t^{\mu}y$. Then
$w\approx_{\sigma}w^{\prime}$. By Lemma 4.4,
$\eta_{\sigma}(w^{\prime})=\sigma^{-1}(y)s_{i}xs_{i^{\prime}}$ is a partial
$\sigma$-Coxeter element with length equal to $\ell(\sigma^{-1}(y)x)$. Hence
the statements hold for $w$ if and only if they hold for $w^{\prime}$. We
continue the procedure until case (1) or (2) happens. If case (3) happens all
the time and the procedure does not end, then $x\in W_{I(t^{\mu}y)}$, and both
$x$ and $y$ are partial $\sigma$-Coxeter elements. Then the statements follow
from §4.3(b).

##  5\. Analyzing the reduction paths

###  5.1. $\sigma$-Conjugacy classes of $\tilde{W}$

We first recall the definition of elliptic conjugacy classes. Let $W_{1}$ be a
Coxeter group and let $S_{1}$ be the index set of simple reflections in
$W_{1}$. Let $\delta$ be a length-preserving group automorphism on $W_{1}$. A
$\delta$-conjugacy class $C$ of $W_{1}$ is called elliptic if it contains no
elements in any proper $\delta$-stable standard parabolic subgroup of $W_{1}$.

Let $x\in\tilde{W}$. We regard $x\sigma$ as an element in
$\tilde{W}\rtimes\langle\sigma\rangle$. There exists a positive integer such
that $(x\sigma)^{n}=t^{\lambda}$ for some $\lambda\in X_{*}(T)_{\Gamma_{0}}$.
Then we set $\nu_{x}=\lambda/n\in V$. It is easy to see that $\nu_{x}$ is
independent of the choice of $n$. Moreover, the unique dominant element in the
$W$-orbit of $\nu_{x}$ equals the (dominant) Newton point $\nu_{\dot{x}}$ for
$\dot{x}\in\breve{G}$.

We follow [15, §1.8.3]. Let $J\subseteq\mathbb{S}$. Let
$\tilde{W}_{J}=X_{*}(T)_{\Gamma_{0}}\rtimes W_{J}$ be the Iwahori–Weyl group
of the standard Levi subgroup $\mathbf{M}_{J}$ of $\mathbf{G}$. Let
${}^{J}\tilde{W}$ be the set of minimal length representatives for cosets in
$W_{J}\backslash\tilde{W}$. Let $\tilde{J}\supseteq J$ be the set of simple
reflections for the Iwahori–Weyl group $\tilde{W}_{J}$.

We say that $(J,x,\breve{K},C)$ is a standard quadruple if

  1. (1)

$J\subseteq\mathbb{S}$ with $\sigma(J)=J$;

  2. (2)

$x\in{}^{J}\tilde{W}$ such that $\nu_{x}$ is dominant, $J=I(\nu_{x})$, and
${\mathrm{Ad}}(x)\circ\sigma$ preserves $\tilde{J}$;

  3. (3)

$\breve{K}\subseteq\tilde{J}$ with $W_{\breve{K}}$ finite and
${\mathrm{Ad}}(x)\delta(\breve{K})=\breve{K}$;

  4. (4)

$C$ is an elliptic $({\mathrm{Ad}}(x)\circ\sigma)$-conjugacy class of
$W_{\breve{K}}$.

We say that the standard quadruples $(J,x,\breve{K},C)$ and
$(J^{\prime},x^{\prime},\breve{K}^{\prime},C^{\prime})$ are equivalent in
$\tilde{W}$ if $J=J^{\prime}$, there exists a length-zero element $\tau$ of
$\tilde{W}_{J}$ with $x^{\prime}=\tau x\sigma(\tau)^{-1}$, and there exists
$w\in\tilde{W}_{J}$ with $x^{\prime}\sigma(w)(x^{\prime})^{-1}=w$ and
$C^{\prime}=w\tau C(w\tau)^{-1}$.

By [15, Theorem 1.19], we have the following:

(a) The map $(J,x,\breve{K},C)\tilde{W}\cdot_{\sigma}Cx$ induces a bijection
between the equivalence classes of standard quadruples and the set of
$\sigma$-conjugacy classes of $\tilde{W}$.

Let $\mathcal{O}\in B(\tilde{W},\sigma)$ and let $(J,x,\breve{K},C)$ be a
standard quadruple associated with $\mathcal{O}$. We say that $\mathcal{O}$ is
Coxeter (resp. elliptic) associated with $[b]\in B(\mathbf{G})$ if
$\Psi(\mathcal{O})=[b]$, $\breve{K}\subset\tilde{J}$ is very special with
respect to ${\mathrm{Ad}}(\dot{x})\circ\sigma$, and $C$ is an
$({\mathrm{Ad}}(x)\circ\sigma)$-Coxeter (resp. elliptic) conjugacy class of
$W_{\breve{K}}$. We say that $w\in\tilde{W}$ is a $\sigma$-Coxeter element
associated with $[b]$ if it is a minimal length element in a Coxeter
$\sigma$-conjugacy class associated with $[b]$.

###  5.2. Description of reduction trees

For any $[b]\in B(\mathbf{G},\mu)$, set

$\displaystyle\ell_{I}(\mu,[b])$
$\displaystyle=\sharp(\mathbb{S}/\langle\sigma\rangle)-\sharp(I(\nu_{b})/\langle\sigma\rangle),$
$\displaystyle\ell_{II}(\mu,[b])$ $\displaystyle={\rm
length}([b],[t^{\mu}])-\sharp(\mathbb{S}/\langle\sigma\rangle).$

We have the following description of reduction trees.

######  Theorem 5.1.

Let $c$ be a $\sigma$-Coxeter element of $W$ such that
$t^{\mu}c\in{}^{\mathbb{S}}\tilde{W}$. Let $\mathcal{T}$ be a reduction tree
of $t^{\mu}c$. Then, for any reduction path $\underline{p}$ in $\mathcal{T}$,
we have

  1. (1)

$\ell_{I}(\underline{p})=\ell_{I}(\mu,[b]_{\underline{p}})$ and
$\ell_{II}(\underline{p})=\ell_{II}(\mu,[b]_{\underline{p}})$;

  2. (2)

$\operatorname{end}(\underline{p})$ is a $\sigma$-Coxeter element associated
with $[b]_{\underline{p}}$.

Moreover, for any $[b]\in B(\mathbf{G},\mu)_{\mathrm{indec}}$, there exists a
unique reduction path $\underline{p}$ in $\mathcal{T}$ with
$[b]_{\underline{p}}=[b]$.

Combining Theorem 5.1 with Proposition 3.9 and Remark 3.10, we obtain part (3)
of Theorem 2.7 for $w=t^{\mu}c$. We will describe the reduction trees of the
elements with finite partial $\sigma$-Coxeter part in §7 and deduce Theorem
2.7(3) for such elements.

In the rest of this section, we shall prove parts (1) and (2) of Theorem 5.1.
The “moreover” part (i.e., the multiplicity-one result) is the most difficult
part and will be proved in §6.

###  5.3. Estimate $\ell_{I}$

Let $\operatorname{Aff}(V)$ be the group of affine transformations on $V$. For
any $g\in\operatorname{Aff}(V)$, denote $V^{g}=\\{v\in V\mid g(v)=v\\}$. We
have a natural projection map
$p:\tilde{W}\rtimes\langle\sigma\rangle\operatorname{Aff}(V)\mathrm{GL}(V)$.
For any $w\in\tilde{W}$, define $V_{w}=\\{v\in V;w\sigma(v)=v+\nu_{w}\\}$. We
have $\dim V_{w}=\dim V^{p(w\sigma)}$.

It is easy to see that for any $g\in\mathrm{GL}(V)$ and any reflection
$r\in\mathrm{GL}(V)$, we have

$|\dim V^{rg}-\dim V^{g}|\leqslant 1.$

In particular, for any $w\in\tilde{W}$ and $i\in\tilde{}\mathbb{S}$, we have

$|\dim V_{s_{i}w}-\dim V_{w}|\leqslant 1.$

Now let $w=t^{\mu}c$ be as in Theorem 5.1. Let $\mathcal{T}$ be a reduction
tree of $w$ and let $\underline{p}$ be a reduction path in $\mathcal{T}$. Set
$e=\operatorname{end}(\underline{p})$ and $[b]=\Psi(e)\in B(\mathbf{G})$. Let
$(J,x,\breve{K},C)$ be a standard quadruple associated with the
$\sigma$-conjugacy class of $e$.

Consider the variation of $\dim V_{-}$ along the reduction path
$\underline{p}$. Note that type II edges do not change $\dim V_{-}$, and type
I edges change $\dim V_{-}$ by at most 1. Therefore
$\ell_{I}(\underline{p})\geqslant|\dim V_{e}-\dim V_{w}|$. Since $p(w)=c$ is a
$\sigma$-Coxeter element, $\dim V^{p(w\sigma)}=0$. Since $C$ is
${\mathrm{Ad}}(x)\circ\sigma$-elliptic in $\breve{K}$, we have $\dim
V_{e}=\dim V_{x}-\sharp(\breve{K}/\langle{\mathrm{Ad}}(x)\circ\sigma\rangle)$.
Hence

$\ell_{I}(\underline{p})\geqslant\dim
V_{x}-\sharp(\breve{K}/\langle{\mathrm{Ad}}(x)\circ\sigma\rangle).$

By [24, §1.9], ${\mathrm{def}}(b)=\sharp(\mathbb{S}/\langle\sigma\rangle)-\dim
V_{x}$. Note that
$\ell_{I}(\underline{p})+2\ell_{II}(\underline{p})=\ell(t^{\mu}c)-\ell(e)$.
Moreover,

$\ell(e)\geqslant\langle\nu_{b},2\rho\rangle+\sharp(\breve{K}/\langle{\mathrm{Ad}}(x)\circ\sigma\rangle),$

with equality holding if and only if $C$ is an
$({\mathrm{Ad}}(x)\circ\sigma)$-Coxeter conjugacy class in $\breve{K}$. We
have

$\displaystyle\dim X_{\underline{p}}$
$\displaystyle=\ell_{I}(\underline{p})+\ell_{II}(\underline{p})+\ell(e)-\langle\nu_{b},2\rho\rangle$
$\displaystyle=\tfrac{1}{2}(\ell_{I}(\underline{p})+\langle\mu,2\rho\rangle-\sharp(\mathbb{S}/\langle\sigma\rangle)+\ell(e))-\langle\nu_{b},2\rho\rangle$
$\displaystyle\geqslant\langle\mu-\nu_{b},\rho\rangle+\tfrac{1}{2}(\dim
V_{x}-\sharp(\mathbb{S}/\langle\sigma\rangle))$
$\displaystyle=\langle\mu-\nu_{b},\rho\rangle-\tfrac{1}{2}{\mathrm{def}}(b)=d_{w}(b).$

By §2.5(b), we have $\dim X_{\underline{p}}\leqslant\dim X_{w}(b)\leqslant\dim
d_{w}(b)$. Thus the inequalities in (a) and (b) are equalities, and $\dim
X_{\underline{p}}=\dim X_{w}(b)$. In particular, $C$ is an
$({\mathrm{Ad}}(x)\circ\sigma)$-Coxeter conjugacy class in $\breve{K}$.

###  5.4. Affine Deligne–Lusztig varieties in the affine Grassmannian

It remains to show that $\breve{K}$ occurring in §5.3 is very special. To do
this, we need some information on affine Deligne–Lusztig varieties in the
affine Grassmannian.

Let $\breve{P}\subseteq\breve{G}$ be a special parahoric subgroup containing
$\breve{I}$. The affine Deligne–Lusztig variety in the affine Grassmannian
$\breve{G}/\breve{P}$ is defined by

$X_{\mu}(b)=\\{g\in\breve{G}/\breve{P};g^{-1}b\sigma(g)\in\breve{P}t^{\mu}\breve{P}\\}.$

The following dimension formula is proved in [4] and [30] for split groups,
[10] and [33] for unramified groups, and [15, Theorem 2.29] in general.

######  Theorem 5.2.

Suppose that $[b]\in B(\mathbf{G},\mu)$. Then $\dim
X_{\mu}(b)=\langle\mu-\nu_{b},\rho\rangle-\frac{1}{2}{\mathrm{def}}(b)$.

Let $\Sigma^{\mathrm{top}}(X_{\mu}(b))$ be the set of top-dimensional
irreducible components of $X_{\mu}(b)$.

Let $\widehat{\mathbf{G}}$ be the Langlands dual of $\mathbf{G}$ over the
complex number field $\mathbb{C}$. Let $\widehat{T}$ be the maximal torus dual
to $T$. Then $\sigma$ acts on $\widehat{T}$ in a natural way, and we denote by
$\widehat{T}^{\sigma}$ the $\sigma$-fixed points of $\widehat{T}$. Let
$\lambda_{b}\in X^{*}(\widehat{T}^{\sigma})$ be the “best integral
approximation” of the Newton point of $b$ in the sense of [11, Definition
2.1]. Let $V_{\mu}$ be the irreducible representation of
$\widehat{\mathbf{G}}$ with highest weight $\mu$. Write $V_{\mu}(\lambda_{b})$
for the corresponding $\lambda_{b}$-weight subspace of $\widehat{T}^{\sigma}$.
The following result was conjectured by M. Chen and X. Zhu, and is proved in
[32], [21], and [26].

######  Theorem 5.3.

The number of $\mathbf{J}_{b}(F)$-orbits on
$\Sigma^{\mathrm{top}}(X_{\mu}(b))$ equals $\dim V_{\mu}(\lambda_{b})$.
Moreover, the stabilizer of each element in
$\Sigma^{\mathrm{top}}(X_{\mu}(b))$ is a very special parahoric subgroup of
$\mathbf{J}_{b}(F)$.

We also need the following result that connects affine Deligne–Lusztig
varieties in the affine flag and in the affine Grassmannian.

######  Lemma 5.4.

The $\mathbf{J}_{b}(F)$-equivariant projection map $X_{t^{\mu}c}(b)X_{\mu}(b)$
is injective.

###### Proof.

Let $g\breve{I},g^{\prime}\breve{I}\in\breve{G}/\breve{I}$ be in the same
fiber of the natural projection map $X_{t^{\mu}c}(b)X_{\mu}(b)$. Then
$g^{\prime-1}g\in\breve{P}$. We have
$g^{\prime-1}g\in\breve{I}\dot{x}\breve{I}$ for some $x\in W$.

Since
$(g^{\prime-1}g)(g^{-1}b\sigma(g))=(g^{\prime-1}b\sigma(g^{\prime}))\sigma(g^{\prime-1}g)$,
then
$(\breve{I}\dot{x}\breve{I})(\breve{I}t^{\mu}\dot{c}\breve{I})\cap(\breve{I}t^{\mu}\dot{c}\breve{I})\sigma(\breve{I}\dot{x}\breve{I})\neq\emptyset$.
Since $t^{\mu}c\in{}^{\mathbb{S}}\tilde{W}$,
$(\breve{I}\dot{x}\breve{I})(\breve{I}t^{\mu}\dot{c}\breve{I})=\breve{I}\dot{x}t^{\mu}\dot{c}\breve{I}$.
Thus we have $xt^{\mu}c=t^{\mu}c\sigma(x)$ and
$\operatorname{supp}_{\sigma}(x)\subset I(t^{\mu}c)$. As $c$ is
$\sigma$-elliptic, we conclude that
$\operatorname{supp}_{\sigma}(x)=\emptyset$ and hence
$g^{\prime}g^{-1}\in\breve{I}$ as desired. ∎

###  5.5. Proof of Theorem 5.1(1) and (2)

We continue our analysis of reduction paths. All the notation is the same as
in §5.3.

Note that the equalities in §5.3(a) and (b) hold. Note also that
${\mathrm{def}}(b)=\sharp(I(\nu_{b})/\langle\sigma\rangle)-\sharp(\breve{K}/\langle{\mathrm{Ad}}(x)\circ\sigma\rangle)$.
It follows that
$\ell_{I}(\underline{p})=\sharp(\mathbb{S}/\langle\sigma\rangle)-\sharp(I(\nu_{b})/\langle\sigma\rangle)=\ell_{I}(\mu,[b])$.
Using §2.5(a) and the simple fact that
$\ell_{I}(\underline{p})+2\ell_{II}(\underline{p})=\ell(w)-\ell(e)$, one can
prove that $\ell_{II}(\underline{p})=\ell_{II}(\mu,[b])$. This proves part (1)
of Theorem 5.1.

By §2.8(a), the stabilizer in $\mathbf{J}_{b}(F)$ of any irreducible component
of $X_{\underline{p}}$ is isomorphic to the parahoric subgroup
$\breve{P}_{\breve{K}}\cap\mathbf{J}_{b}(F)\subseteq\mathbf{J}_{b}(F)$. By
§5.3, we have $\dim X_{\underline{p}}=d_{w}(b)=\dim X_{\mu}(b)$. By Lemma 5.4,
the image of each irreducible component $Z$ of $X_{\underline{p}}$ in
$X_{\mu}(b)$ is an open dense subset of some top-dimensional irreducible
component $Y$ of $X_{\mu}(b)$. Thus the stabilizers of $Z$ and $Y$ coincide.
By Theorem 5.3, $\breve{P}_{\breve{K}}\cap\mathbf{J}_{b}(F)$ is a very special
parahoric subgroup of $\mathbf{J}_{b}(F)$. Hence, by §2.9(a),
$\breve{K}\subset\tilde{J}$ is very special with respect to
${\mathrm{Ad}}(\dot{x})\circ\sigma$. By 5.3, $C$ is an
$({\mathrm{Ad}}(x)\circ\sigma)$-Coxeter conjugacy class of $W_{\breve{K}}$.
This proves part (2) of Theorem 5.1.

###  5.6. The extreme cases

Let $\mathcal{T}$ be a reduction tree of $t^{\mu}c$. Let $[b]\in
B(\mathbf{G})_{t^{\mu}c}$ and let $\underline{p}$ be a path in $\mathcal{T}$
such that $[b]_{\underline{p}}=[b]$.

If $[b]=[b_{\mu,\mathbf{G}\text{-}\mathrm{indec}}]$, then
$\ell_{II}(\underline{p})=\ell_{II}(\mu,[b])=0$. Therefore $\underline{p}$
consists only of type I edges and is unique.

If $[b]$ is basic. Then $I(\nu_{b})=\mathbb{S}$ and
$\ell_{I}(\underline{p})=\ell_{I}(\mu,[b])=0$. Therefore $\underline{p}$
consists only of type II edges and is also unique.

This proves the “moreover” part of Theorem 2.7 for these two extreme cases.

##  6\. Some combinatorial identities

###  6.1. Reduction to combinatorial identities

In this section, we assume that $\mu$ is essentially non-central. Let $c$ be a
$\sigma$-Coxeter element of $W$ such that
$t^{\mu}c\in{}^{\mathbb{S}}\tilde{W}$. Let $\mathcal{T}$ be a reduction tree
of $t^{\mu}c$. For any $[b]\in B(\mathbf{G},\mu)_{\mathrm{indec}}$, let
$n_{[b]}$ be the number of reduction paths $\underline{p}$ in $\mathcal{T}$
with $[b]_{\underline{p}}=[b]$. By Theorem 2.7(2), $n_{[b]}\geqslant 1$ for
all $[b]\in B(\mathbf{G},\mu)_{\mathrm{indec}}$. By §5.6, $n_{[b]}=1$ if $[b]$
is either the minimal or the maximal element in
$B(\mathbf{G},\mu)_{\mathrm{indec}}$.

Combining Theorem 5.1(1) and (2) with Proposition 3.6 and Lemma 3.8, we have

$\mathbf{q}^{\langle\mu,2\rho\rangle-\sharp(\mathbb{S}/\langle\sigma\rangle)}=\sum_{[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}n_{[b]}(\mathbf{q}-1)^{\ell_{I}(\mu,[b])}\mathbf{q}^{\ell_{II}(\mu,[b])+\ell_{[b]}},$

where
$\ell_{[b]}=\langle\nu_{b},2\rho\rangle+\sharp(I(\nu_{b})/\langle\sigma\rangle)-{\mathrm{def}}(b)$
and equals $\ell(\mathcal{O})$ for any $\sigma$-Coxeter class $\mathcal{O}$
associated with $[b]$.

Note that
$(\mathbf{q}-1)^{a}\mathbf{q}^{a^{\prime}}\in\mathbb{N}[\mathbf{q}-1]$ for all
$a,a^{\prime}\in\mathbb{N}$. Thus, to show that $n_{[b]}=1$ for all $[b]$, it
suffices to show that

$\sum_{[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}(\mathbf{q}-1)^{\ell_{I}(\mu,[b])}\mathbf{q}^{\ell_{II}(\mu,[b])+\ell_{[b]}}=\mathbf{q}^{\langle\mu,2\rho\rangle-\sharp(\mathbb{S}/\langle\sigma\rangle)}.$

(In fact, it is enough to prove the inequality $\geqslant$.)

Using §2.5(a), one computes that

$\ell_{II}(\mu,[b])+\ell_{[b]}-(\langle\mu,2\rho\rangle-\sharp(\mathbb{S}/\langle\sigma\rangle))=\sharp(I(\nu_{b})/\langle\sigma\rangle)-{\rm
length}([b],[t^{\mu}]).$

Thus, the equality $({\spadesuit})$ is equivalent to

$\sum_{[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}(\mathbf{q}-1)^{\sharp(\mathbb{S}/\langle\sigma\rangle)-\sharp(I(\nu_{b})/\langle\sigma\rangle)}\mathbf{q}^{\sharp(I(\nu_{b})/\langle\sigma\rangle)-\mathrm{length}([b],[t^{\mu}])}=1.$

###  6.2. Reduction to unramified adjoint groups

Let $\mathbf{G}_{{\mathrm{ad}}}$ be the adjoint group of $\mathbf{G}$ and let
$T_{{\mathrm{ad}}}$ be the image of $T$ in $\mathbf{G}_{{\mathrm{ad}}}$. We
denote by $\mu_{{\mathrm{ad}}}$ the image of $\mu$ in
$X_{*}(T_{{\mathrm{ad}}})_{\Gamma_{0}}$. For any $b\in\breve{G}$, we denote by
$b_{{\mathrm{ad}}}$ its image in $\breve{G}_{{\mathrm{ad}}}$. By [23,
Proposition 4.10], the map $\mathbf{G}\mathbf{G}_{{\mathrm{ad}}}$ identifies
the reduced root system of $\mathbf{G}_{{\mathrm{ad}}}$ with that of
$\mathbf{G}$ and induces an isomorphism of posets

$B(\mathbf{G},\mu)_{\mathrm{indec}}\cong
B(\mathbf{G}_{{\mathrm{ad}}},\mu_{{\mathrm{ad}}})_{\mathrm{indec}},\qquad[b][b_{{\mathrm{ad}}}].$

Therefore, the equality (${\spadesuit}^{\prime}$) for $\mathbf{G}$ is
equivalent to that for $\mathbf{G}_{{\mathrm{ad}}}$. We can therefore assume
that $\mathbf{G}$ is adjoint. In this case, it is convenient to work with the
reduced root system $\Phi$ of $\mathbf{G}$. Define

$B(\Phi,\sigma,\mu)=\\{\nu\in(V^{+})^{\sigma};\langle\mu^{\diamond}-\nu,\omega_{\mathcal{O}_{i}}\rangle\in\mathbb{Z}_{\geqslant
0}\text{ for any }i\in\mathbb{S}-I(v)\\},$

where $\mathcal{O}_{i}$ denotes the $\sigma$-orbit of $i$. By Lemma [20, Lemma
3.5], the map $[b]\nu_{b}$ identifies $B(\mathbf{G},\mu)$ with
$B(\Phi,\sigma,\mu)$ as posets. For any $\nu\in B(\Phi,\sigma,\mu)$, we set
${\rm
length}(\nu,\mu^{\diamond})=\sum_{\mathcal{O}\in\mathbb{S}/\langle\sigma\rangle}\lceil\langle\mu^{\diamond}-\nu,\omega_{\mathcal{O}}\rangle\rceil$.
Then ${\rm length}([b],[t^{\mu}])={\rm length}(\nu_{b},\mu^{\diamond})$ for
any $[b]\in B(\mathbf{G},\mu)$.

We set

$f_{\Phi,\sigma,\mu}(\nu)=(\mathbf{q}-1)^{\sharp(\mathbb{S}/\langle\sigma\rangle)-\sharp(I(\nu)/\langle\sigma\rangle)}\mathbf{q}^{\sharp(I(\nu)/\langle\sigma\rangle)-{\rm
length}(\nu,\mu^{\diamond})}.$

Now the equality (${\spadesuit}^{\prime}$) can be reformulated in a purely
combinatorial way as

$\sum_{\nu\in B(\Phi,\sigma,\mu)_{\mathrm{indec}}}f_{\Phi,\sigma,\mu}(\nu)=1.$

As any triple $(\Phi,\sigma,\mu)$ arises from an unramified group, it suffices
to prove (${\spadesuit}^{\prime\prime}$) for the triples $(\Phi,\sigma,\mu)$
arising from unramified adjoint groups. In the rest of this section, we assume
that $\mathbf{G}$ is an unramified adjoint group.

###  6.3. Reduction to $F$-simple groups

Write $\mathbf{G}=\mathbf{G}_{1}\times\cdots\times\mathbf{G}_{l}$, where
$\mathbf{G}_{i}$ are $F$-simple adjoint groups. Write
$\mu=(\mu_{1},\ldots,\mu_{l})$, where $\mu_{i}$ is a dominant coweight of
$\mathbf{G}_{i}$. Also we have $\Phi=\Phi_{1}\sqcup\cdots\sqcup\Phi_{l}$,
where $\Phi_{i}$ is the root system of $\mathbf{G}_{i}$. It is easy to see
that

$B(\Phi,\sigma,\mu)_{\mathrm{indec}}=B(\Phi_{1},\sigma_{1},\mu_{1})_{\mathrm{indec}}\times\cdots\times
B(\Phi_{l},\sigma_{l},\mu_{l})_{\mathrm{indec}},$

where $\sigma_{i}$ is the restriction of $\sigma$ on $\Phi_{i}$. Moreover, it
is easy to see that
$f_{\Phi,\sigma,\mu}=\prod_{i=1}^{l}f_{\Phi_{i},\sigma_{i},\mu_{i}}$. In the
rest of this section, we assume that $\mathbf{G}$ is an $F$-simple unramified
adjoint group.

###  6.4. Reduction to $\breve{F}$-simple groups

We have $\Phi=\Phi_{1}\sqcup\cdots\sqcup\Phi_{l}$, where
$\Phi_{1}\cong\cdots\cong\Phi_{l}$ are irreducible root systems and $\sigma$
induces an isomorphism from $\Phi_{i}$ to $\Phi_{i+1}$. Here, by convention,
we set $\Phi_{l+1}=\Phi_{1}$. Then the map

$\nu=(\nu_{1},\dots,\nu_{l})|\nu|=\nu_{l}+\sigma(\nu_{l-1})+\cdots+\sigma^{l-1}(\nu_{1})$

induces an isomorphism of posets

$B(\Phi,\sigma,\mu)_{\mathrm{indec}}\overset{\sim}{}B(\Phi_{l},\sigma^{l},|\mu|)_{\mathrm{indec}}.$

Moreover, there is a natural bijection
$\mathbb{S}/\langle\sigma\rangle\cong\mathbb{S}_{l}/\langle\sigma^{l}\rangle$,
where $\mathbb{S}_{l}$ is the set of simple reflections for $\Phi_{l}$. Thus
$f_{\Phi,\sigma,\mu}=f_{\Phi_{l},\sigma^{l},|\mu|}$. In the rest of this
section, we assume that $\mathbf{G}$ is an $\breve{F}$-simple unramified
adjoint group.

###  6.5. Reduction to split groups

Let $\mathcal{O}$ be a $\sigma$-orbit of $\mathbb{S}$. If all the simple roots
in $\mathcal{O}$ commute with each other, we define
$\alpha_{\mathcal{O}}^{\prime}=\sum_{i\in\mathcal{O}}\alpha_{i}$. If
$\mathcal{O}=\\{i_{0},j_{0}\\}$ with
$\langle\alpha_{j_{0}}^{\vee},\alpha_{i_{0}}\rangle=\langle\alpha_{i_{0}}^{\vee},\alpha_{j_{0}}\rangle=-1$,
we define $\alpha_{\mathcal{O}}^{\prime}=2(\alpha_{i_{0}}+\alpha_{j_{0}})$.
Let $\mathbb{S}^{\prime}=\mathbb{S}/\langle\sigma\rangle$ and let
$\Phi^{\prime}$ be the root system generated by
$\alpha_{\mathcal{O}}^{\prime}$ for all $\mathcal{O}\in\mathbb{S}^{\prime}$.
The coroot corresponding to $\mathcal{O}$ is given by
$\alpha_{\mathcal{O}}^{\prime\vee}=\frac{1}{\sharp\mathcal{O}}\sum_{i\in\mathcal{O}}\alpha_{i}^{\vee}$,
and the fundamental weight corresponding to $\mathcal{O}$ is given by
$\omega_{\mathcal{O}}^{\prime}=\sum_{i\in\mathcal{O}}\omega_{i}$. For any
$\nu\in(V^{+})^{\sigma}$, $\nu\in B(\Phi,\sigma,\mu)$ if and only if
$\langle\mu^{\diamond}-\nu,\omega_{\mathcal{O}}^{\prime}\rangle\in\mathbb{Z}_{\geqslant
0}$ for any $\mathcal{O}\in\mathbb{S}/\langle\sigma\rangle$ such that
$\langle\nu,\alpha_{\mathcal{O}}^{\prime}\rangle\neq 0$, which is also
equivalent to $\nu\in B(\Phi^{\prime},\mathrm{id},\mu^{\diamond})$. Hence we
have the following:

(a) The natural identification
$(\mathbb{R}\Phi^{\vee})^{\sigma}=\mathbb{R}{\Phi^{\prime}}^{\vee}$ induces an
bijection of posets

$B(\Phi,\sigma,\mu)_{\mathrm{indec}}\overset{\sim}{}B(\Phi^{\prime},\mathrm{id},\mu^{\diamond})_{\mathrm{indec}}.$

It follows from (a) that
$f_{\Phi,\sigma,\mu}=f_{\Phi^{\prime},\mathrm{id},\mu^{\diamond}}$. Therefore,
it suffices to prove (${\spadesuit}^{\prime\prime}$) for the triples
$(\Phi,\sigma,\mu)$ arising from split groups. In the rest of this section, we
assume that $\mathbf{G}$ is a split $\breve{F}$-simple adjoint group. We
identify $B(\mathbf{G},\mu)$ with $B(\Phi,\sigma,\mu)$ and write
$f_{\mathbf{G},\mu}(\nu)=f_{\Phi,\sigma,\mu}(\nu)$.

###  6.6. Reduction to simply laced groups

By §6, the identity (${\spadesuit}$) is equivalent to the condition that in
some (or, equivalently, any) reduction tree of $t^{\mu}c$, there exists only
one reduction path whose end point is associated with a given $[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}$.

There exists an irreducible, simply laced, extended affine Weyl group
$(\tilde{W}^{\prime},\tilde{}\mathbb{S}^{\prime})$ of adjoint type and a
length-preserving automorphism $\delta$ on $\tilde{W}^{\prime}$ such that
$\tilde{W}=(\tilde{W}^{\prime})^{\delta}$. We have a natural bijection between
$\tilde{}\mathbb{S}$ and $\tilde{\mathbb{S}}^{\prime}/\langle\delta\rangle$.
Moreover, we may assume that the simple reflections in each $\delta$-orbit in
$\tilde{}\mathbb{S}^{\prime}$ commute. More explicitly,

  * •

if $\tilde{W}$ is of type $\tilde{B}_{n}$, then we take $\tilde{W}^{\prime}$
to be of type $\tilde{D}_{n+1}$ and $\delta$ is of order $2$;

  * •

if $\tilde{W}$ is of type $\tilde{C}_{n}$, then we take $\tilde{W}^{\prime}$
to be of type $\tilde{A}_{2n-1}$ and $\delta$ is of order $2$;

  * •

if $\tilde{W}$ is of type $\tilde{F}_{4}$, then we take $\tilde{W}^{\prime}$
to be of type $\tilde{E}_{6}$ and $\delta$ is of order $2$;

  * •

if $\tilde{W}$ is of type $\tilde{G}_{2}$, then we take $\tilde{W}^{\prime}$
to be of type $D_{4}$ and $\delta$ is of order $3$.

Let $\iota:\tilde{W}\tilde{W}^{\prime}$ be the natural embedding. For each
$i\in\tilde{}\mathbb{S}$, we have $\iota(s_{i})=s_{i_{1}^{\prime}}\cdots
s_{i_{k}^{\prime}}$, where $i_{1}^{\prime},\ldots,i_{k}^{\prime}$ are the
$\delta$-orbits of $i$ in $\tilde{}\mathbb{S}^{\prime}$. Let $w\in\tilde{W}$
and $w\rightharpoonup s_{i}w$ be a type I reduction edge (see §3.4). Then one
can construct a $k$-step reduction path

$\iota(w)\rightharpoonup s_{i_{k}^{\prime}}\iota(w)\rightharpoonup
s_{i_{k-1}^{\prime}}s_{i_{k}^{\prime}}\iota(w)\rightharpoonup\cdots\rightharpoonup
s_{i_{1}^{\prime}}\cdots s_{i_{k}^{\prime}}\iota(w)=\iota(s_{i}w)$

in $\tilde{W}^{\prime}$. Similarly, a type II reduction edge $w\rightharpoonup
s_{i}ws_{i}$ corresponds to a $k$-step reduction path from $\iota(w)$ to
$\iota(sws)$, whose edges are all of type II. Now considering a reduction tree
$\Gamma$ of $w$, we can construct a reduction tree $\mathcal{T}^{\prime}$ of
$\iota(w)$ such that $\mathcal{T}$ can be viewed as a subtree of
$\mathcal{T}^{\prime}$ in the above way. Hence the multiplicity-one result of
$\iota(t^{\mu}c)\in\tilde{W}^{\prime}$ implies the multiplicity-one result of
$t^{\mu}c\in\tilde{W}$.

In the rest of this section, we assume that $\mathbf{G}$ is a split
$\breve{F}$-simple simply laced adjoint group. We will then reduce to the case
where $\mu$ is a fundamental coweight. We first need a combinatorial identity
on finite graphs.

###  6.7. A combinatorial identity on graphs

Let $X$ be a finite graph and $Y\subseteq X$. Denote by $\mathcal{A}(Y,X)$ the
set of subsets $J\subseteq X$ such that none of the connected components of
$X-J$ lies in $Y$. Define

$f_{Y,X}=\sum_{J\in\mathcal{A}(Y,X)}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\mathbf{q}^{\sharp(Y\cap J^{\circ})}\in\mathbb{Z}[\mathbf{q}],$

where $J^{\circ}=\\{i\in J;\text{$i$ has no neighbors in $X-J$}\\}$ is the
interior of $J$.

######  Lemma 6.1.

We have $f_{Y,X}=\mathbf{q}^{\sharp X}$ for any $Y\subseteq X$.

###### Proof.

Define

$\alpha:\\{(J,K);J\in\mathcal{A}(Y,X),K\subseteq Y\cap
J^{\circ}\\}\\{\text{subsets of $X$}\\},~{}(J,K)J-K.$

We construct the inverse map $\beta$ of $\alpha$ as follows. Let $H\subseteq
X$. Let $C$ be the union of connected components of $X-H$ that are contained
in $Y$. Then we define $\beta(H)=(H\sqcup C,C)$. By definition,
$\alpha\circ\beta=\mathrm{id}$. On the other hand, for any
$J\in\mathcal{A}(Y,X)$ and $K\subseteq Y\cap J^{\circ}$, $\alpha((J,K))=J-K$.
Moreover, $X-(J-K)=(X-J)\sqcup K$. As $K\subseteq Y\cap J^{\circ}$, $K$ and
$X-J$ are not connected with each other. Hence $K$ is the union of connected
components of $X-(J-K)$ contained in $Y$. Therefore
$\beta\circ\alpha((J,K))=\beta(J-K)=((J-K)\sqcup K,K)=(J,K)$ and hence
$\beta\circ\alpha=\mathrm{id}$. Therefore $\alpha$ is a bijection. Using the
binomial expansion, we get

$\displaystyle f_{Y,X}$
$\displaystyle=\sum_{J\in\mathcal{A}(Y,X)}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\mathbf{q}^{\sharp(Y\cap J^{\circ})}$
$\displaystyle=\sum_{J\in\mathcal{A}(Y,X)}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\sum_{K\subseteq Y\cap J^{\circ}}(q-1)^{\sharp(Y\cap J^{\circ}-K)}$
$\displaystyle=\sum_{J\in\mathcal{A}(X,Y)}\sum_{K\subseteq Y\cap
J^{\circ}}(\mathbf{q}-1)^{\sharp(J-K)}$ $\displaystyle=\sum_{H\subseteq
X}(\mathbf{q}-1)^{\sharp H}=\mathbf{q}^{\sharp X}.$

The lemma is proved. ∎

###  6.8. Reduction to fundamental coweights

Assume that $\mu$ is not a fundamental coweight. Then there exist
$i,j\in\mathbb{S}$ (here $i$ and $j$ are not necessarily distinct) such that
$\mu-\omega_{i}^{\vee}-\omega_{j}^{\vee}$ is also dominant. Let $X$ be the
(unique) path in the Dynkin diagram of $\mathbb{S}$ with end points $i,j$.

Let $\lambda=\mu-\sum_{k\in X}\alpha_{k}^{\vee}$ and let $\mathcal{A}$ be the
set of subsets $J\subseteq X$ such that $\lambda$ is non-central on each
connected component of $\mathbb{S}-J$ (equivalently,
$B(\mathbf{M}_{\mathbb{S}-J},\lambda)_{\mathrm{irr}}\neq\emptyset$). Note that
$\lambda$ is non-central on each connected component of $\mathbb{S}-X$ and
that the connected components of $\mathbb{S}-J$ that do not intersect
$\mathbb{S}-X$ are the connected components of $\\{i\in X-J;i\text{ has no
neighbors in }\mathbb{S}-X\\}$.

Set $Y=\\{i\in X\cap I(\lambda);i\text{ has no neighbors in }\mathbb{S}-X\\}$.
Then it is easy to see that $\mathcal{A}=\mathcal{A}(Y,X)$ defined in §6.7.

######  Lemma 6.2.

We have the following:

  1. (1)

$\lambda$ is dominant;

  2. (2)

$B(\mathbf{G},\mu)_{\mathrm{indec}}=\sqcup_{J\in\mathcal{A}}B(\mathbf{G},\lambda)_{(\mathbb{S}-J)\text{-}\mathrm{irr}}$.

###### Proof.

Let $l\in\mathbb{S}$. If $l\in\mathbb{S}-X$; then
$\langle\lambda,\alpha_{l}\rangle\geqslant-\langle\sum_{k\in
X}\alpha_{k}^{\vee},\alpha_{l}\rangle\geqslant 0$. If $l\in X-\\{i,j\\}$, then
$\langle\lambda,\alpha_{l}\rangle=-\langle\sum_{k\in
X}\alpha_{k}^{\vee},\alpha_{l}\rangle=-\langle\alpha_{k}^{\vee}+\alpha_{l}^{\vee}+\alpha_{k^{\prime}}^{\vee},\alpha_{l}\rangle\geqslant-(-1+2-1)=0$,
where $k$ and $k^{\prime}$ are the neighbors of $l$ in $X$. If $l\in\\{i,j\\}$
and $i\neq j$, then
$\langle\lambda,\alpha_{l}\rangle=\langle\mu,\alpha_{l}\rangle-\langle\sum_{k\in
X}\alpha_{k}^{\vee},\alpha_{l}\rangle\geqslant 1-1=0$. If $l=i=j$, then
$\langle\lambda,\alpha_{l}\rangle=\langle\mu,\alpha_{l}\rangle-\langle\sum_{k\in
X}\alpha_{k}^{\vee},\alpha_{l}\rangle\geqslant 2-2=0$. This proves part (1).

By definition,
$B(\mathbf{G},\lambda)_{(\mathbb{S}-J)\text{-}\mathrm{irr}}\subseteq
B(\mathbf{G},\mu)_{\mathrm{indec}}$ for any $J\subseteq X$. Now we prove the
other direction. Let $E=\\{e_{i};i\in\mathbb{S}\\}$ be as in the proof of
Proposition 4.1. Then, for each $i\in\mathbb{S}$, by the definition of
$\lambda$, we have

$\langle\lambda,\omega_{i}\rangle\geqslant\max\\{0,\langle\mu,\omega_{i}\rangle-1\\}=\langle
e_{i},\omega_{i}\rangle.$

By [2, Theorem 6.5], this implies that $\lambda\in C_{\geqslant E}$ and hence
$\lambda\geqslant\min C_{\geqslant E}=[b]_{\mathbf{G}\text{-}\mathrm{indec}}.$

For any $\nu\in B(\mathbf{G},\mu)_{\mathrm{indec}}$, there exists a unique
subset $J\subseteq X$ such that
$\lambda-\nu\in\sum_{l\in\mathbb{S}-J}\mathbb{R}_{>0}\alpha_{l}^{\vee}$. By
Lemma 2.1, $\nu\in
B(\mathbf{G},\lambda)_{(\mathbb{S}-J)\text{-}\mathrm{irr}}$. Part (2) is
proved. ∎

######  Proposition 6.3.

Suppose that the equality (${\spadesuit}^{\prime\prime}$) holds for all
fundamental coweights. Then it holds for all dominant coweights.

###### Proof.

We argue by induction on the semisimple rank of $\mathbf{G}$ and the number
$\langle\rho,\mu\rangle$.

Suppose $\mu$ is not fundamental. Let the notation be as in §6.8. By Corollary
2.2, we identify $B(\mathbf{G},\lambda)_{(\mathbb{S}-J)\text{-}\mathrm{irr}}$
with $B(\mathbf{M}_{\mathbb{S}-J},\mu)_{\mathrm{irr}}$ for $J\in\mathcal{A}$.
We show that

(a) For any $J\in\mathcal{A}$ and $\nu\in
B(\mathbf{M}_{\mathbb{S}-J},\mu)_{\mathrm{irr}}$, we have

$f_{\mathbf{G},\mu}(\nu)=\mathbf{q}^{-\sharp X}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\mathbf{q}^{\sharp Y\cap
J^{\circ}}f_{\mathbf{M}_{\mathbb{S}-J},\lambda}(\nu).$

By definition, $f_{\mathbf{M}_{K},\mu}(\nu)=(\mathbf{q}-1)^{\sharp(K-K\cap
I(\nu))}\mathbf{q}^{\sharp K\cap I(\nu)-{\rm
length}_{\mathbf{M}_{K}}(\mu,\nu)}$ for any $K\subseteq\mathbb{S}$.

Note that ${\rm
length}_{\mathbf{G}}(\lambda,\mu)=\langle\mu-\lambda,\rho\rangle=\sharp X$,
and hence

${\rm length}_{\mathbf{G}}(\nu,\mu)={\rm
length}_{\mathbf{G}}(\nu,\lambda)+{\rm length}_{\mathbf{G}}(\lambda,\mu)={\rm
length}_{\mathbf{G}}(\nu,\lambda)+\sharp X.$

By Lemma 2.1, we have ${\rm length}_{\mathbf{G}}(\nu,\mu)={\rm
length}_{\mathbf{M}}(\nu,\mu)$. To show (a), it remains to show that
$I(\nu)-(\mathbb{S}-J)\cap I(\nu)=J\cap I(\nu)=Y\cap J^{\circ}$. Let $l\in J$.
Write $\nu=\lambda-\delta$ for some
$\delta\in\sum_{k\in\mathbb{S}-J}\mathbb{R}_{>0}\alpha_{k}^{\vee}$. Then $l\in
J\cap I(\nu)$ if and only if
$\langle\nu,\alpha\rangle=\langle\lambda,\alpha_{l}\rangle-\langle\delta,\alpha_{l}\rangle=0$,
which is also equivalent to
$\langle\lambda,\alpha_{l}\rangle=\langle\delta,\alpha_{l}\rangle=0$, that is,
$l\in Y\cap J^{\circ}$ as desired. (a) is proved.

Now, by Lemma 6.1 and the inductive hypothesis on $\mathbf{M}_{\mathbb{S}-J}$
and $\lambda$, we have

$\displaystyle\sum_{\nu\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}f_{\mathbf{G},\mu}(\nu)=\sum_{J\in\mathcal{A}}\sum_{\nu\in
B(\mathbf{M}_{\mathbb{S}-J},\lambda)_{\mathrm{irr}}}f_{\mathbf{G},\mu}(\nu)$
$\displaystyle\hskip 56.0pt=\mathbf{q}^{-\sharp
X}\sum_{J\in\mathcal{A}}\sum_{\nu\in
B(\mathbf{M}_{\mathbb{S}-J},\lambda)_{\mathrm{irr}}}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\mathbf{q}^{\sharp Y\cap
J^{\circ}}f_{\mathbf{M}_{\mathbb{S}-J},\lambda}(\nu)$ $\displaystyle\hskip
56.0pt=\mathbf{q}^{-\sharp
X}\sum_{J\in\mathcal{A}}(\mathbf{q}-1)^{\sharp(J-Y\cap
J^{\circ})}\mathbf{q}^{\sharp Y\cap J^{\circ}}=1.$

The proof is finished. ∎

Now it is sufficient to deal with the fundamental coweights of $\mathbf{G}$.

###  6.9. Proof for the minuscule coweights

Let $\mu$ a be (non-zero) minuscule coweight. Then $\dim
V_{\mu}(\lambda_{b})=1$ for any $[b]\in B(\mathbf{G},\mu)$. By Theorem 5.3, we
conclude that
$\sharp(\mathbf{J}_{b}(F)\backslash\Sigma^{\mathrm{top}}(X_{\mu}(b)))=1$ for
any $[b]\in B(\mathbf{G},\mu)$. As in §6.1, we have

$\mathbf{q}^{\langle\mu,2\rho\rangle-\sharp(\mathbb{S}/\langle\sigma\rangle)}=\sum_{[b]\in
B(\mathbf{G},\mu)_{\mathrm{indec}}}n_{[b]}(\mathbf{q}-1)^{\ell_{I}(\mu,[b])}\mathbf{q}^{\ell_{II}(\mu,[b])+\ell_{[b]}},$

where $n_{[b]}$ is the number of reduction paths $\underline{p}$ in a given
reduction tree $\mathcal{T}$ of $t^{\mu}c$ with $[b]_{\underline{p}}=[b]$.
Note also that at the end of §5.3, we showed that $\dim X_{\underline{p}}=\dim
X_{t^{\mu}c}([b]_{\underline{p}})=\dim X_{\mu}([b]_{\underline{p}})$ for any
reduction path $\underline{p}$. Using Lemma 5.4, we conclude that all
$n_{[b]}=1$. This implies the combinatorial identity $({\spadesuit})$, and
then $({\spadesuit}^{\prime\prime})$ follows.

In particular, the combinatorial identity (${\spadesuit}^{\prime\prime}$)
holds for type $A$, since all the fundamental coweights are minuscule. For
$(A_{n-1},\omega^{\vee}_{i})$, we may write the identity $({\spadesuit})$
explicitly as

$\sum_{\begin{subarray}{c}k\geqslant
1,1>\frac{a_{1}}{b_{1}}>\cdots>\frac{a_{k}}{b_{k}}>0;\\\
a_{i}+\cdots+a_{k}=i,b_{1}+\cdots+b_{k}=n\end{subarray}}(\mathbf{q}-1)^{k-1}\mathbf{q}^{k-1-\frac{\sum_{1\leqslant
l_{1}<l_{2}\leqslant
k}(a_{l_{1}}b_{l_{2}}-a_{l_{2}}b_{l_{1}})+\sum_{1\leqslant l\leqslant
k}\gcd(a_{l},b_{l})}{2}}\\\ =\mathbf{q}^{\frac{i(n-i)-n}{2}}.$

We do not know if there is a purely combinatorial proof of this identity.

###  6.10. Type $D_{n}$ case

In this subsection, we assume that $\mathbf{G}$ is of type $D_{n}$
$(n\geqslant 4)$. Note that the fundamental coweights
$\omega^{\vee}_{1},\omega^{\vee}_{n-1},\omega^{\vee}_{n}$ are minuscule and
have already been dealt with in §6.9. Here we deal only with
$\omega^{\vee}_{i}$ for $2\leq i\leq n-2$.

For any integer $k\geqslant 2$, denote by $[1,k]$ the set
$\\{1,2,\ldots,k\\}$. We have a natural embedding
$B(\mathbf{G},\omega_{i-2}^{\vee})\rightarrow
B(\mathbf{G},\omega_{i}^{\vee})$. Set

$\displaystyle B_{I}$ $\displaystyle=\bigsqcup_{i-2\leqslant k\leqslant
n-3}B(\mathbf{G},\omega^{\vee}_{i-2})_{[1,k]\text{-}\mathrm{irr}}\cong\bigsqcup_{i-2\leqslant
k\leqslant n-3}B(\mathbf{M}_{[1,k]},\omega^{\vee}_{i-2})_{\mathrm{irr}},$
$\displaystyle B_{II}$
$\displaystyle=\bigsqcup_{J\subseteq\\{n-1,n\\}}B(\mathbf{G},\omega^{\vee}_{i-2})_{(\mathbb{S}-J)\text{-}\mathrm{irr}}\cong\bigsqcup_{J\subseteq\\{n-1,n\\}}B(\mathbf{M}_{\mathbb{S}-J},\omega^{\vee}_{i-2})_{\mathrm{irr}}.$

For $i\leqslant k\leqslant n-2$, the adjoint group of
$\mathbf{M}_{\mathbb{S}-\\{k\\}}$ is of type $A_{k-1}\times D_{n-k}$. Here, by
convention, type $D_{3}$ is the same as type $A_{3}$, and type $D_{2}$ is the
same as type $A_{1}\times A_{1}$. Set
$\mu_{k}=(1^{i-1},0^{k-i+1},1,0^{n-k-1})$. Then $\mu$ and $\omega^{\vee}_{i}$
are in the same $W$-orbit. The restriction of $\mu_{k}$ to
$\mathbf{M}_{\mathbb{S}-\\{k\\}}$ is (dominant) minuscule, and its projection
to the adjoint group of $\mathbf{M}_{\mathbb{S}-\\{k\\}}$ is the coweight
$(\omega^{\vee}_{i-1},\omega^{\vee}_{1})$ if $k<n-2$ and
$(\omega^{\vee}_{i-1},\omega^{\vee}_{1},\omega_{1}^{\vee})$ if $k=n-2$. As in
§2.3, we identify $B(\mathbf{M}_{\mathbb{S}-\\{k\\}},\mu_{k})$ with its
natural image in $B(\mathbf{G},\omega_{i})$. Set

$B_{III}=\bigsqcup_{i\leqslant k\leqslant
n-2}B(\mathbf{G},\mu_{k})_{(\mathbb{S}-\\{k\\})\text{-}\mathrm{irr}}\cong\bigsqcup_{i\leqslant
k\leqslant n-2}B(\mathbf{M}_{\mathbb{S}-\\{k\\}},\mu_{k})_{\mathrm{irr}}.$

By direct computation, $B_{I}$, $B_{II}$, and $B_{III}$ are disjoint in
$B(\mathbf{G},\omega^{\vee}_{i})$. In the rest of this section, we will show
that

$\sum_{\nu\in B_{I}\sqcup B_{II}\sqcup
B_{III}}f_{\mathbf{G},\omega^{\vee}_{i}}(\nu)=1.$

In §6.1, we have already shown that the left-hand side of (**) is less than or
equal to 1. Hence, if we prove (**), the equality $B_{I}\sqcup B_{I}\sqcup
B_{III}=B(\mathbf{G},\omega_{i}^{\vee})_{\mathrm{indec}}$ also follows.

It can be checked directly that (**) holds for $D_{4}$. Using induction, we
may assume that (**) holds for groups of type $D$ with semisimple rank less
than $n$. Note also that (${\spadesuit}^{\prime\prime}$) for type $A$ has
already been proved in §6.9. Therefore we have

(a) $\sum_{\nu\in
B_{II}}f_{\mathbf{G},\omega^{\vee}_{i}}(\nu)=\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle}\cdot(1+2(\mathbf{q}-1)^{2}+(\mathbf{q}-1)^{2})=\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle}\cdot\mathbf{q}^{2}$.

Next we handle $B_{I}$ and $B_{III}$. Let

$\alpha^{\vee}=\omega^{\vee}_{i}-\omega^{\vee}_{i-2}=\alpha_{i-1}^{\vee}+2\alpha_{i}^{\vee}+\dots+2\alpha_{n-2}^{\vee}+\alpha_{n-1}^{\vee}+\alpha_{n}^{\vee}.$

Note that $\langle\alpha^{\vee},\rho\rangle=2n-2i+1$. We claim that

(b) For $k\in[i-2,n-3]$ and $\nu\in
B(\mathbf{M}_{[1,k]},\omega^{\vee}_{i-2})_{\mathrm{irr}}$, we have

$f_{\mathbf{G},\omega^{\vee}_{i}}(\nu)=(\mathbf{q}-1)\cdot\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle+n-k-1}\cdot
f_{\mathbf{G},\omega^{\vee}_{i-2}}(\nu).$

(c) For $k\in[i,n-2]$ and $\nu\in
B(\mathbf{M}_{\mathbb{S}-\\{k\\}},\mu_{k})_{\mathrm{irr}}$, we have

$f_{\mathbf{G},\omega^{\vee}_{i}}(\nu)=(\mathbf{q}-1)\cdot\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle+2n-k-i}\cdot
f_{\mathbf{M}_{\mathbb{S}-\\{k\\}},\mu_{k}}(\nu).$

We prove (c) here. The proof of (b) is similar.

By definition, we have

$f_{\mathbf{G},\omega^{\vee}_{i}}(\nu)=(\mathbf{q}-1)\cdot\mathbf{q}^{n-k-
n_{0}}\cdot f_{\mathbf{M}_{\mathbb{S}-\\{k\\}},\mu_{k}}(\nu),$

where $n_{0}={\rm length}_{\mathbf{G}}(\nu,\omega^{\vee}_{i})-{\rm
length}_{\mathbf{M}_{\mathbf{G}-\\{k\\}}}(\nu,\mu_{k})$. It can be checked
directly that

$\langle\mu_{k}-\nu,\rho_{\mathbf{M}_{\mathbb{S}-\\{k\\}}}\rangle=\langle\omega_{i-1}^{\vee}-\nu,\rho\rangle+\langle(0^{k},1,0^{n-k-1}),\rho_{\mathbf{M}_{[k+1,n]}}\rangle=\langle\omega_{i-1}^{\vee}-\nu,\rho\rangle+1.$

Note also that
${\mathrm{def}}_{\mathbf{G}}(\nu)={\mathrm{def}}_{\mathbf{M}_{\mathbb{S}-\\{k\\}}}(\nu)$.
Therefore, by the length formula, we have

$\displaystyle n_{0}$
$\displaystyle=\langle\omega^{\vee}_{i}-\nu,\rho\rangle-\langle\mu_{k}-\nu,\rho_{\mathbf{M}_{\mathbb{S}-\\{k\\}}}\rangle+\frac{1}{2}\left({\mathrm{def}}_{D_{n}}(\nu)-{\mathrm{def}}_{\mathbf{M}_{\mathbb{S}-\\{k\\}}}(\nu)\right)$
$\displaystyle=\langle\omega^{\vee}_{i}-\omega_{i-1}^{\vee},\rho\rangle+1=n-i+1=\langle\alpha^{\vee},\rho\rangle-n+i.$

The statement (c) follows.

Combining (a), (b), and (c) with the combinatorial identity $({\spadesuit})$
for type $A$ and for type $D_{l}$ with $l<n$, we have

$\displaystyle\sum_{\nu\in B_{I}\sqcup B_{II}\sqcup
B_{III}}f_{D_{n},\omega^{\vee}_{i}}(\nu)$ $\displaystyle\hskip
64.0pt=\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle}\cdot\left(\mathbf{q}^{2}+(\mathbf{q}-1)\\!\left(\sum_{k=i-2}^{n-3}\mathbf{q}^{n-k-1}+\sum_{k=i}^{n-2}\mathbf{q}^{2n-k-i}\right)\right)$
$\displaystyle\hskip
64.0pt=\mathbf{q}^{-\langle\alpha^{\vee},\rho\rangle}\cdot\mathbf{q}^{2n-2i+1}=1.$

This completes the proof of (**).

###  6.11. Type $E$ case

In this subsection, we assume that $\mathbf{G}$ is of type $E_{n}$. We verify
the combinatorial identity $({\spadesuit}^{\prime\prime})$ by computer. Recall
that a vector $v\in(V^{+})^{\sigma}$ lies in $B(\mathbf{G},\mu)$ if and only
if $\langle\mu-v,\omega_{i}\rangle\in\mathbb{Z}_{\geqslant 0}$ for any
$i\in\mathbb{S}-I(v)$. As a consequence, we have the following
characterization of $B(\mathbf{G},\mu)_{\mathrm{indec}}$:

(a) The set $B(\mathbf{G},\mu)_{\mathrm{indec}}$ equals the set of dominant
vectors of the form $\nu=pr_{I}(\mu-\sum_{i\in I}c_{i}\alpha^{\vee}_{i})$,
where $I$ is a $\sigma$-stable subset of $\mathbb{S}$, $c_{i}\in\mathbb{Z}$,
$1\leq c_{i}\leq\langle\mu,\omega_{i}\rangle$, and
$pr_{I}:V\rightarrow\bigoplus_{i\in I}\mathbb{R}\omega^{\vee}_{i}$ is the
natural orthogonal projection.

On the basis of (a), we can use a computer program to list all the elements in
$B(\mathbf{G},\mu)_{\mathrm{indec}}$ and then verify the combinatorial
identity $({\spadesuit}^{\prime\prime})$ directly.

In the following tables, we provide the numbers of elements in
$B(\mathbf{G},\mu)_{\mathrm{indec}}$ for all the fundamental, non-minuscule
coweights in type $E$. The most complicated case is $E_{8}$, and
$\mu=\omega^{\vee}_{4}$, in which $B(\mathbf{G},\mu)_{\mathrm{indec}}$
contains $729$ elements.

Table 1. Type $E_{6}$

$\mu$ | $\omega_{2}^{\vee}$ | $\omega_{3}^{\vee}$ | $\omega_{4}^{\vee}$ | $\omega_{5}^{\vee}$
---|---|---|---|---
$\sharp B(\mathbf{G},\mu)_{\mathrm{indec}}$ | $7$ | $15$ | $30$ | $15$

Table 2. Type $E_{7}$

$\mu$ | $\omega_{1}^{\vee}$ | $\omega_{2}^{\vee}$ | $\omega_{3}^{\vee}$ | $\omega_{4}^{\vee}$ | $\omega_{5}^{\vee}$ | $\omega_{6}^{\vee}$
---|---|---|---|---|---|---
$\sharp B(\mathbf{G},\mu)_{\mathrm{indec}}$ | $13$ | $26$ | $50$ | $125$ | $69$ | $32$

Table 3. Type $E_{8}$

$\mu$ | $\omega_{1}^{\vee}$ | $\omega_{2}^{\vee}$ | $\omega_{3}^{\vee}$ | $\omega_{4}^{\vee}$ | $\omega_{5}^{\vee}$ | $\omega_{6}^{\vee}$ | $\omega_{7}^{\vee}$ | $\omega_{8}^{\vee}$
---|---|---|---|---|---|---|---|---
$\sharp B(\mathbf{G},\mu)_{\mathrm{indec}}$ | $56$ | $126$ | $254$ | $729$ | $424$ | $220$ | $94$ | $27$

##  7\. The general case

###  7.1. Description of the reduction trees

Let $w\in Wt^{\mu}W$ with finite partial $\sigma$-Coxeter part, that is,
$\eta_{\sigma}(w)$ is a partial $\sigma$-Coxeter element. For any
$J\in[J_{0}(w),J(w)]_{\mu}$ and $[b]\in
B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$, we set

$\displaystyle J^{\flat,w}$ $\displaystyle=\\{i\in
I(\mu^{\diamond})\cap(J(w)-J)\mid i\text{ commutes with }J\\},$
$\displaystyle\ell_{I}(w,[b],J)$
$\displaystyle=\sharp(J(w)/\langle\sigma\rangle)-\sharp(J^{\flat,w}/\langle\sigma\rangle)-\sharp(I(\nu_{b})\cap
J/\langle\sigma\rangle),$ $\displaystyle\ell_{II}(w,[b],J)$
$\displaystyle={\rm
length}([b],[t^{\mu}])-\sharp(J_{0}(w)/\langle\sigma\rangle).$

By Lemma 2.1, there exists a unique $\sigma$-conjugacy class
$[b]_{\mathbf{M}_{J}}\in B(\mathbf{M}_{J},\mu)$ such that
$[b]_{\mathbf{M}_{J}}\subseteq[b]$. We similarly define
$[b]_{\mathbf{M}_{J^{\flat,w}\cup J}}\in B(\mathbf{M}_{J^{\flat,w}\cup
J},\mu)$. In this case, a $\sigma$-Coxeter element associated with
$[b]_{\mathbf{M}_{J^{\flat,w}\cup J}}$ is equal to the product of a
$\sigma$-Coxeter element of $W_{J^{\flat,w}}$ and a $\sigma$-Coxeter element
associated with $[b]_{\mathbf{M}_{J}}$.

The main result of this section is the following description of the reduction
tree of $w$.

######  Theorem 7.1.

Let $w\in Wt^{\mu}W$ with $\eta_{\sigma}(w)$ a partial $\sigma$-Coxeter
element. Let $\mathcal{T}$ be a reduction tree of $w$. Then, for any
$J\in[J_{0}(w),J(w)]_{\mu}$ and $[b]\in
B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$, there exists a unique reduction
path $\underline{p}$ in $\mathcal{T}$ with $[b]_{\underline{p}}=[b]$.
Moreover,

  1. (1)

$\ell_{I}(\underline{p})=\ell_{I}(w,[b],J)$ and
$\ell_{II}(\underline{p})=\ell_{II}(w,[b],J)$;

  2. (2)

$\operatorname{end}(\underline{p})$ is a $\sigma$-Coxeter element associated
with $[b]_{\mathbf{M}_{J^{\flat,w}\cup J}}$.

Combining Theorem 7.1 with Proposition 3.9 and Remark 3.10, we obtain Theorem
2.7(3) for $w$.

###  7.2. Strategy

The strategy for proving Theorem 7.1 is very different from that adopted for
the proof of Theorem 5.1. In the latter case, we used the Chen–Zhu conjecture
and the dimension formula to determine the end points of the reduction trees
of $t^{\mu}c$. However, such a method, when applied to general $w$, cannot
determine the end points.

The approach we use here is as follows. We first apply the partial reduction
method and the class polynomials for $t^{\mu}c$ to calculate the class
polynomials for $w$. As we mentioned earlier, class polynomials, in general,
contains less information than reduction trees. Fortunately, for the elements
$w$ we consider here, by combining the information on the class polynomials
and the estimates on the type I-edges, we obtain the required information for
any reduction tree.

The information about the class polynomials we need is contained in the
following equality on the $\sigma$-cocenter of the Iwahori–Hecke algebra $H$:

$T_{w}+[H,H]_{\sigma}=\sum_{\begin{subarray}{c}J\in[J_{0}(w),J(w)]_{\mu},\\\
[b]\in
B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}\end{subarray}}(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}T_{\mathcal{O}_{w,[b]}}+[H,H]_{\sigma},$

where ${\mathcal{O}_{w,[b]}}$ is the $\sigma$-conjugacy class containing a
$\sigma-$Coxeter element associated with $[b]_{\mathbf{M}_{J^{\flat,w}\cup
J}}$.

###  7.3. Proof of the identity $(\diamondsuit)$

#### 7.3.1.

We consider the case where $w=t^{\mu}c\in{}^{\mathbb{S}}\tilde{W}$ for some
partial $\sigma$-Coxeter element $c$ of $W$. Let
$J=\operatorname{supp}_{\sigma}(c)$. In this case, $J_{0}(w)=J(w)=J$.

If $J=\mathbb{S}$, that is, $c$ is a (full) $\sigma$-Coxeter element, then
$(\diamondsuit)$ follows from Lemma 3.8 and the description of the reduction
tree in Theorem 5.1. In other words, the class polynomial $F_{w,\mathcal{O}}$
is given by

$F_{w,\mathcal{O}}=\begin{cases}(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}&\text{if
}\mathcal{O}=\mathcal{O}_{w,[b]}\text{ for some }[b]\in
B(\mathbf{G},\mu)_{\mathrm{irr}},\\\ 0&\text{otherwise.}\end{cases}$

Assume $J\subsetneq\mathbb{S}$. It follows from [19, Theorem 7.3] that
$F_{w,\mathcal{O}}=\sum_{\mathcal{O}^{J}\subseteq\mathcal{O}}F_{w,\mathcal{O}^{J}}^{J}$,
where $\mathcal{O}^{J}$ denotes a $\sigma$-conjugacy class of $\tilde{W}_{J}$
and $F_{w,\mathcal{O}^{J}}^{J}$ denotes the corresponding class polynomial for
$\mathbf{M}_{J}$. For any $[b]\in B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$,
we have $\mathcal{O}_{w,[b]}\supset\mathcal{O}_{w,[b]_{\mathbf{M}_{J}}}$.
Hence
$F_{w,\mathcal{O}_{w,[b]}}=F^{J}_{w,\mathcal{O}_{w,[b]_{\mathbf{M}_{J}}}}=(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}$.
And $F_{w,\mathcal{O}}=0$ if $\mathcal{O}\neq\mathcal{O}_{w,[b]}$ for any
$[b]\in B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$. This proves
$(\diamondsuit)$.

#### 7.3.2.

We consider the case where $w=c_{1}t^{\mu}c_{2}$ for some partial
$\sigma$-Coxeter elements $c_{1},c_{2}$ of $W$ such that
$t^{\mu}c_{2}\in{}^{\mathbb{S}}\tilde{W}$ and $c_{1}\in W_{I(t^{\mu}c_{2})}$.

Set $J_{1}=\operatorname{supp}_{\sigma}(c_{1})$ and
$J_{2}=\operatorname{supp}_{\sigma}(c_{2})$. One can construct a reduction
tree $\mathcal{T}$ of $t^{\mu}c_{2}$ in $\tilde{W}_{J_{2}}$ such that $c_{1}$
commutes with all the vertices in $\mathcal{T}$. In particular, if
$w_{1}\rightharpoonup w_{2}$ is one edge in $\mathcal{T}$, then we have
$c_{1}w_{1}\rightharpoonup c_{1}w_{2}$ in $\tilde{W}_{J}$. Note that $c_{1}$
is a minimal length element in its $\sigma$-conjugacy class in
$\tilde{W}_{J_{1}}$, and the simple reflections in $\tilde{W}_{J_{1}}$ commute
with the simple reflections in $\tilde{W}_{J_{2}}$. It is easy to see that if
$w_{1}$ is an end point in $\mathcal{T}$ (and hence is a minimal length
element in its $\sigma$-conjugacy class in $\tilde{W}_{J_{2}}$ and commutes
with $c_{1}$), then $c_{1}w_{1}$ is a minimal length element in its
$\sigma$-conjugacy class in $\tilde{W}_{J}$.

Let $\mathcal{T}^{\prime}$ be the tree with vertices $c_{1}w_{1}$ for all
vertices $w_{1}\in\mathcal{T}$ and the edges $c_{1}w_{1}\rightharpoonup
c_{1}w_{2}$ for all edges $w_{1}\rightharpoonup w_{2}$ in $\mathcal{T}$. Then,
from the above discussion, $\mathcal{T}^{\prime}$ is a reduction tree of
$c_{1}t^{\mu}c_{2}$ in $\tilde{W}_{J}$. Note that for $J,[b]$ as in Theorem
7.1, we have $\ell_{I}(w,[b],J)=\ell_{I}(t^{\mu}c_{2},[b],J)$ and
$\ell_{II}(w,[b])=\ell_{II}(t^{\mu}c_{2},[b])$. Hence the identity
$(\diamondsuit)$ for $w$ follows from the identity for $t^{\mu}c_{2}$
established in §7.3.1.

#### 7.3.3.

We consider the case where $w=c_{1}t^{\mu}c_{2}$ for partial $\sigma$-Coxeter
elements $c_{1}$ and $c_{2}$ in $W$ such that
$t^{\mu}c_{2}\in{}^{\mathbb{S}}\tilde{W}$ and
$\operatorname{supp}_{\sigma}(c_{1})\cap\operatorname{supp}_{\sigma}(c_{2})=\emptyset$.

We prove the identity $(\diamondsuit)$ by induction on $\ell(c_{1})$. The case
$\ell(c_{1})=0$ is proved in §7.3.1.

Assume that $\ell(c_{1})>0$. Let $i\in\mathbb{S}$ such that
$s_{i}c_{1}<c_{1}$. There are two cases as follows.

Case (1): $c_{2}\sigma(s_{i})\in{}^{I(\mu)}W$. Write $w_{1}=s_{i}w$ and
$w_{2}=s_{i}w\sigma(s_{i})=s_{i}c_{1}t^{\mu}c_{2}\sigma(s_{i})$. Then
$T_{w}+[H,H]_{\sigma}=(\mathbf{q}-1)T_{w_{1}}+\mathbf{q}T_{w_{2}}+[H,H]_{\sigma}$.
Note that $J(w_{2})=J(w)$, $J_{0}(w_{1})=J_{0}(w)$,
$J(w_{1})=J(w)-\\{\sigma^{\ell}(s_{i})\mid\ell\in\mathbb{Z}\\}$, and
$J_{0}(w_{2})=J_{0}(w)\sqcup\\{\sigma^{\ell}(s_{i})\mid\ell\in\mathbb{Z}\\}$.
Then

$[J_{0}(w),J(w)]_{\mu}=[J_{0}(w_{1}),J(w_{1})]_{\mu}\sqcup[J_{0}(w_{2}),J(w_{2})]_{\mu}.$

By the induction hypothesis, it suffices to prove that

  1. (a)

if $J\in[J_{0}(w_{1}),J(w_{1})]_{\mu}$, then $J^{\flat,w}=J^{\flat,w_{1}}$;
and

  2. (b)

if $J\in[J_{0}(w_{2}),J(w_{2})]_{\mu}$, then $J^{\flat,w}=J^{\flat,w_{2}}$.

Statement (b) is obvious since $J(w_{2})=J(w)$. Let us prove (a). Suppose
$J^{\flat,w}\neq J^{\flat,w_{1}}$; then $i\in I(\mu^{\diamond})$ and $s_{i}$
commutes with $J$. Since $J\supseteq J_{0}(w)$, $s_{i}$ also commutes with
$J_{0}(w)$. Then $\mu$ is not essentially non-central over
$J_{0}(w_{2})=J_{0}(w)\sqcup\\{\sigma^{l}(i)\mid l\in\mathbb{Z}\\}$, which is
a contradiction. This completes the proof.

Case (2): $\sigma(i)\in I(\mu)$, and $\sigma(s_{i})$ commutes with $c_{2}$.
Then the equality $(\diamondsuit)$ holds for $w$ if and only if it holds for
$s_{i}w\sigma(s_{i})$. We continue with the procedure until Case (1) happens.
If Case (2) happens all the time and the procedure does not stop, then
$c_{1}\in W_{I(t^{\mu}c_{2})}$, and the equality $(\diamondsuit)$ follows from
§7.3.2.

###  7.4. Proof of Theorem 7.1

#### 7.4.1.

We consider the case $w=ct^{\mu}$, where $c$ is a partial $\sigma$-Coxeter
element of $W$. Let $\mathcal{T}$ be a reduction tree of $w$. By Lemma 3.8 and
the identity $(\diamondsuit)$ for $w$, we have that for any $\sigma$-conjugacy
class $\mathcal{O}$ of $\tilde{W}$,

$\sum_{\underline{p};\operatorname{end}(\underline{p})\in\mathcal{O}}(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})}\\\
=\begin{cases}(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}&\text{if
}\mathcal{O}=\mathcal{O}_{w,[b]}\text{ for some }[b],\\\
0&\text{otherwise}.\end{cases}$

Let $\underline{p}$ be a path in $\mathcal{T}$. Set
$e=\operatorname{end}(\underline{p})$ and $[b]=[b]_{\underline{p}}$. Assume
that $[b]\in B(\mathbf{G},\mu)_{J\text{-}\mathrm{irr}}$ for some $J$. As
$(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})}\in\mathbb{N}[\mathbf{q}-1]$,
there is no cancellation involved in the left-hand side of the above equality.
Therefore, $e$ must be contained in $\mathcal{O}_{w,[b]}$.

As in §5.3, we have

$\displaystyle\ell_{I}(\underline{p})$ $\displaystyle\geqslant\dim V_{e}-\dim
V_{w}=\sharp(J(w)/\langle\sigma\rangle)-\sharp(J^{\flat,w}/\langle\sigma\rangle)-\sharp(I(\nu_{b})\cap
J/\langle\sigma\rangle)$ $\displaystyle=\ell_{I}(w,[b],J).$

Note that
$\ell_{I}(\underline{p})+2\ell_{II}(\underline{p})=\ell_{I}(w,[b],J)+2\ell_{II}(w,[b],J)$.
Thus
$\deg_{\mathbf{q}}(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})}\geqslant\deg_{\mathbf{q}}(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}$,
with equality holding if and only if
$\ell_{I}(\underline{p})=\ell_{I}(w,[b],J)$. Again, since there is no
cancellation involved in
$\sum_{\underline{p};\operatorname{end}(\underline{p})\in\mathcal{O}}(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})}$,
we must have $\ell_{I}(\underline{p})=\ell_{I}(w,[b],J)$. In this case,
$(\mathbf{q}-1)^{\ell_{I}(\underline{p})}\mathbf{q}^{\ell_{II}(\underline{p})}=(\mathbf{q}-1)^{\ell_{I}(w,[b],J)}\mathbf{q}^{\ell_{II}(w,[b],J)}$.
This also shows that for each $\mathcal{O}=\mathcal{O}_{w,[b]}$, there is a
unique reduction path $\underline{p}$ with
$\operatorname{end}(\underline{p})\in\mathcal{O}$.

This completes the proof of Theorem 7.1 for $w=ct^{\mu}$.

#### 7.4.2.

Now we consider the general case. Let $w=xt^{\mu}y$ with
$t^{\mu}y\in{}^{\mathbb{S}}\tilde{W}$. Set $c=\sigma^{-1}(y)x$ and
$w^{\prime}=ct^{\mu}$. We relate $w$ and $w^{\prime}$ as in the proof of [13,
Theorem 10.3]. Let $y=s_{1}s_{2}\cdots s_{r}$ be a reduced expression. Let
$w^{(0)}=w^{\prime}$, $w^{(1)}=s_{1}w^{(0)}\sigma(s_{1})$,
$w^{(2)}=s_{2}w^{(1)}\sigma(s_{2}),\ldots,w^{(r)}=w$. We have
$w^{(0)}\rightarrow_{\sigma}w^{(1)}\rightarrow_{\sigma}\cdots\rightarrow_{\sigma}w^{(r)}$.
This give a path $w^{\prime}\rightharpoonup w$, consisting of
$\frac{1}{2}(\ell(w^{\prime})-\ell(w))$ type II edges. We denote this path by
$\underline{p}_{0}$.

Let $\mathcal{T}$ be a reduction tree of $w$. One may construct a reduction
tree $\mathcal{T}^{\prime}$ of $w^{\prime}$ containing the concatenation
$\underline{p}_{0}\circ\mathcal{T}$ as a subgraph. In particular, for any
reduction path $\underline{p}$ in $\mathcal{T}$, the concatenation
$\underline{p}^{\prime}:=\underline{p}_{0}\circ\underline{p}$ is a reduction
path in $\mathcal{T}^{\prime}$. By definition,
$\ell_{I}(\underline{p})=\ell_{I}(\underline{p}^{\prime})$ and
$\ell_{II}(\underline{p})+\frac{1}{2}(\ell(w^{\prime})-\ell(w))=\ell_{II}(\underline{p}^{\prime})$.
It is obvious that $J(w)=J(w^{\prime})$ and $J_{0}(w^{\prime})=\emptyset$.
Hence $\ell_{I}(w,[b],J)=\ell_{I}(w^{\prime},[b],J)$. On the other hand, we
have $\sharp(J_{0}(w)/\langle\sigma\rangle)={\rm
length}([b_{w}],[t^{\mu}])=\frac{1}{2}(\ell(\eta_{\sigma}(w)+\ell(y)-\ell(x)))=\frac{1}{2}(\ell(w^{\prime})-\ell(w))$,
where the second equality follows from the definition of cordial elements.
Hence
$\ell_{II}(w,[b],J)+\frac{1}{2}(\ell(w^{\prime})-\ell(w))=\ell_{II}(w^{\prime},[b],J)$.
The statements for $\mathcal{T}$ can now be deduced from the statements for
$\mathcal{T}^{\prime}$.

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