misc

# Nonemptiness of single affine Deligne-Lusztig varities

Dong Gyu Lim  Evans Hall, University of California at Berkeley, CA, USA
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

Affine Deligne-Lusztig varieties with various level structures show up in the
study of Shimura varieties and moduli spaces of shtukas. Among is the Iwahori
level structure which is the most refined one. We study the nonemptiness
problem of single affine Deligne-Lusztig varieties at Iwahori level in the
basic case. Under a genericity condition (the “shrunken Weyl chambers”
condition), an explicit criterion is known. However, no explicit criterion has
been available without the condition even conjecturally. We conjecture a new
criterion in full generality, and prove it except for finitely many cases. As
an application, the nonemptiness problem for special cases and a new
conjectural dimension formula are discussed.

###### Contents

  1. 1 Introduction
  2. 2 Basics on single affine Deligne-Lusztig varieties
  3. 3 Proof of main theorems
  4. 4 One critical strip
  5. 5 Some computations and applications
  6. 6 Suggestions and future works

##  1\. Introduction

###  1.1. Background

In the study of the special fibers of Shimura varieties, affine Deligne-
Lusztig varieties show up naturally. In his seminal expository article
[Rap05], Rapoport introduced affine Deligne-Lusztig variety over a mixed
characteristic local field as an important piece in the description of
$\overline{\mathbb{F}}_{p}$-points of the special fiber of a certain Shimura
variety with hyperspecial level structure or Iwahori level structure. This was
motivated by the Langlands-Rapoport conjecture in that the $p$-part of the
conjecture is the affine Deligne-Lusztig variety. Since then, affine Deligne-
Lusztig varieties have been exploited in the study of Shimura varieties,
Rapoport-Zink spaces, local Shimura varieties, and moduli spaces of local
shtukas.

Many questions arise naturally including the geometric (or scheme) structure,
the nonemptiness, the dimension formula, and the set of connected components,
etc. These basic questions are not only interesting on their own but useful in
the study of the aforementioned objects. For example, the set of connected
components computed in [CKV15] is used to prove the Langlands-Rapoport
conjecture in [Kis17]. It (resp. the dimension formula) can also be used to
describe the set of connected components (resp. the dimension) of Rapoport-
Zink spaces ([She20, 3], [Zhu17, 3.2]). In this paper, we focus on the
nonemptiness question and the dimension formula.

####  1.1.1. Mazur’s Inequality

Let $G$ be a connected reductive group over $\mathbb{Z}_{p}$ with a maximal
torus $T$ and let $\breve{\mathbb{Q}}_{p}$ be the fraction field of the ring
of Witt vectors
$W(\overline{\mathbb{F}}_{p})\equalscolon\breve{\mathbb{Z}}_{p}$. The
Frobenius morphism on $\overline{\mathbb{F}}_{p}$ lifts uniquely to
$\breve{\mathbb{Z}}_{p}$ by the universal property of the ring of Witt vectors
and then extend to a bijective map (denoted by $\sigma$) on
$\breve{\mathbb{Q}}_{p}$. Now, fix $b\in G(\breve{\mathbb{Q}}_{p})$ and a
dominant cocharacter $\mu\in X_{*}(T)^{+}$. The affine Deligne-Lusztig variety
is defined as

$X_{\mu}(b)\colonequals\\{gG(\breve{\mathbb{Z}}_{p})\in
G(\breve{\mathbb{Q}}_{p})/G(\breve{\mathbb{Z}}_{p}):g^{-1}b\sigma(g)\in
G(\breve{\mathbb{Z}}_{p})p^{\mu}G(\breve{\mathbb{Z}}_{p})\\},$

where $p^{\mu}$ is the image of $p$ under the cocharacter $\mu$.

The very first result on the nonemptiness is due to Rapoport-Richartz ([RR96,
Theorem 4.2]). They showed that, when $G$ is unramified, if $X_{\mu}(b)$ is
nonempty then Mazur’s inequality, that is, $[b]\in B(G,\mu)$ (see definition
2.1) holds. Thanks to [Kot03], [Gas10], and [He14], it is now a theorem that
$X_{\mu}(b)\neq\emptyset$ if and only if $[b]\in B(G,\mu)$ for a general
reductive group $G$ (defined over $\mathbb{Q}_{p}$) and a special maximal
parahoric subgroup $K$ in place of $G(\breve{\mathbb{Z}}_{p})$.

Similarly, using Bruhat-Tits theory, one can consider affine Deligne-Lusztig
varieties with an arbitrary parahoric level structure $K$ as follows.

$\displaystyle X(\mu,b)_{K}\colonequals\\{gK\in
G(\breve{\mathbb{Q}}_{p})/K:g^{-1}b$ $\displaystyle\sigma(g)\in K\dot{x}K$
$\displaystyle\text{ for some }x\in
W_{K}\backslash\operatorname{Adm}(\mu)/W_{K}\\},$

where $W_{K}$ is the group generated by the simple reflections of $K$ and
$\operatorname{Adm}(\mu)$ is the $\mu$-admissible set ([KR00]). Still, Mazur’s
inequality is the necessary and sufficient condition for
$X(\mu,b)_{K}\neq\emptyset$ (see [KR03], [Win05], and [He16]).

Meanwhile, $X(\mu,b)_{K}$ is, from the definition, a disjoint union of several
pieces (where $x$ varies over
$W_{K}\backslash\\!\operatorname{Adm}(\mu)/W_{K}$). These pieces, therefore,
can be thought of as more refined objects or building blocks, which we call as
single111This is the terminology used in the literature occasionally. See
[He14] for example. affine Deligne-Lusztig varieties. One can study their
nonemptiness problem and, to get to the point first, it has a considerably
different flavor.

We remark that, among parahoric level structures, the Iwahori level structure
contains the finest information via the natural projection map from the affine
flag variety (Iwahori level) to the affine partial flag variety (parahoric
level) or the affine Grassmannian (hyperspecial level). From now on, we
restrict ourselves to the Iwahori level.

###  1.2. Single affine Deligne-Lusztig variety at Iwahori level

Along with the notations from section 1.1.1, let $I$ be a $\sigma$-stable
Iwahori subgroup of $G(\breve{\mathbb{Q}}_{p})$ stabilizing a base alcove,
$\widetilde{W}$ be the Iwahori-Weyl group, and $W_{0}$ be the relative Weyl
group (see section 2.1). For $x\in\widetilde{W}$ and $b\in
G(\breve{\mathbb{Q}}_{p})$, the single affine Deligne-Lusztig variety (at
Iwahori level) is defined by

$X_{x}(b)\colonequals\\{gI\in G(\breve{\mathbb{Q}}_{p})/I:g^{-1}b\sigma(g)\in
I\dot{x}I\\}$

where $\dot{x}$ is an element in a subgroup of $G(\breve{\mathbb{Q}}_{p})$
which is a lift of $x\in\widetilde{W}$.

The question on the nonemptiness criterion (and the dimension formula) for
$X_{x}(b)$ first appeared in [Rap00, Problem 4.5] and some cases when
$G=\operatorname{GL}_{2}$ were studied in [Rap05, Example 4.3]. For a more
general setting, the first partial conjecture was posed by Reuman ([Reu04,
Conjecture 7.1]) where he detected an element in $W_{0}$, now called
$\eta_{\sigma}(x)$ (see definition 2.9), that gives a good amount of
information on the nonemptiness (and even on the dimension formula). The
conjecture was reformulated and proved partially in [GHKR10] and then
completely by [GHN15, Theorem B] in the basic case. For convenience, let us
name the theorem GHN B.

We note first that there is an obvious obstruction for the nonemptiness.
Recall the Kottwitz map (cf. Definition 2.1) defined in [Kot97, 4.5]. Then,

if $X_{x}(b)\neq\emptyset$ then $\kappa_{G}(x)=\kappa_{G}(b)$,

because the Kottwitz map applied to the condition
$g^{-1}b\sigma(g)\\!\in\\!I\dot{x}I$ results in
$\kappa_{G}(b)\\!=\\!\kappa_{G}(\dot{x})\\!=\\!\kappa_{G}(x)$. Geometrically,
this means simply ‘$x$ and $b$ are in the same connected component of the loop
group $G(\breve{\mathbb{Q}}_{p})$’.

We may and will reduce to the case where $G$ is simple, quasi-split, and
adjoint (Section 2.2). Then, GHN B tells us that, under the condition called
Shrunken Weyl chambers (Definition 2.4), there is only one interesting
obstruction ($\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}$)
other than the obvious one ($\kappa_{G}(x)=\kappa_{G}(b)$).

Figure 1. An apartment of the Bruhat-Tits building of $\operatorname{PGL}_{3}$

###### GHN B.

Let $b$ be basic. If $x$ lies in the shrunken Weyl chambers then

$X_{x}(b)\neq\emptyset\text{ if and only if }\kappa_{G}(x)=\kappa_{G}(b)\text{
and }\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}.$

Here, $\mathbb{S}$ is the set of simple reflections of $W_{0}$ and
$\operatorname{supp}_{\sigma}$ is the map sending $w\in W_{0}$ to the minimal
$\sigma$-stable subset of $\mathbb{S}$ containing all simple reflections from
any reduced expression of $w$ (Definition 2.10). Shrunken Weyl chambers are,
intuitively, the complement of the red strips (called critical strips) in fig.
1 and the critical strips are defined to be the strips passing through the
base alcove (the black triangle).

####  1.2.1. A side remark on explicit criteria

Here, we clarify the term ‘explicit criterion’ mentioned in [GHN15, 1.2] and
will be used in this paper at times. As this is not related to the rest of the
paper, one may skip this remark.

In loc.cit., we have the following theorem (name it GHN A).

###### GHN A.

Let $b$ be basic. Then, $X_{x}(b)\neq\emptyset$ if and only if, for all pairs
$(J,w)$ such that $x$ is a $(J,w,\delta)$-alcove, the following holds:

$\kappa_{M_{J}}(w^{-1}x\delta(w))\in\kappa_{M_{J}}\left([b]\cap
M_{J}(\breve{\mathbb{Q}}_{p})\right).$

The Levi subgroup of $G$ corresponding to $J\subset\mathbb{S}$ is denoted by
$M_{J}$ and $\kappa_{M_{J}}$ is its Kottwitz map. We refer to definition 2.8
for the term $(J,w,\delta)$-alcove.

Practically, GHN B is used often in applications222See, for example, [He21,
Theorem 1.1] and [MV20, Remark 3.18]. Using our new explicit criterion, we
will give more applications. while GHN A is not. However, they solve the same
problem and, actually, GHN B is more restrictive. Taking that into account, we
can see that GHN B is more applicable and explicit already. Let us now see an
example explaining this more clearly.

For the sake of simplicity, let $G$ be split or residually split for a moment.
Let $x$ be a translation element $t^{\mu}$. Then, GHN B directly implies that,
if $t^{\mu}$ lies in the shrunken Weyl chambers then
$X_{t^{\mu}}(b)=\emptyset$ always.333This is because $\eta_{\sigma}(x)$ is the
identity element $\operatorname{id}_{W_{0}}$. See definition 2.9 for
$\eta_{\sigma}(x)$. Moreover, our new (explicit) criterion will show that
$X_{t^{\mu}}(b)\neq\emptyset$ if and only if $[t^{\mu}]=[b]\in B(G)$. This
recovers [GHKR10, Corollary 9.2.1] for $b$ basic.

On the other hand, it is not easy to find all possible pairs $(J,w)$ such that
$x$ is a $(J,w,\delta)$-alcove (especially definition 2.8 (2) is not easily
manageable) even when $x=t^{\mu}$, which is a necessary step to apply GHN A.
In addition, it is not an easy take to compute the values in
$\kappa_{M_{J}}([b]\cap M_{J}(\breve{\mathbb{Q}}_{p}))$ afterwards.

###  1.3. Main Conjecture

Our goal is to remove the shrunken Weyl chambers condition. We will suggest a
general conjecture on an explicit nonemptiness criterion in the basic case and
prove it for all but finitely many $x$’s and specify some classes of elements
satisfying the conjecture. Let $b$ be basic.

There is a new assumption in our conjecture that the following example can
justify. Let $x=\operatorname{id}_{\widetilde{W}}$, the identity element in
$\widetilde{W}$, and $b=1$. Then, directly from the definition,
$X_{x}(b)\neq\emptyset$. However,
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\emptyset$ as
$\eta_{\sigma}(x)=\operatorname{id}_{W_{0}}$. Hence, there are some
exceptional $x$’s not having the $\operatorname{supp}_{\sigma}$-obstruction.

In the following, we denote by $\widetilde{\operatorname{supp}_{\sigma}}$ the
$\sigma$-support function on $\widetilde{W}$.444We found that it may cause
confusion to use the same notations for two $\sigma$-support functions each
defined on $W_{0}$ and $\widetilde{W}$ and decided to use
$\widetilde{\operatorname{supp}_{\sigma}}$ instead of
$\operatorname{supp}_{\sigma}:\widetilde{W}\to 2^{\widetilde{\mathbb{S}}}$.

######  Lemma 1.1.

Assume that $\kappa_{G}(x)=\kappa_{G}(b)$. If
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$ then
$X_{x}(b)\neq\emptyset$.

###### Proof.

Let $\nu_{x}$ be the image of $x$ under the Newton map. The condition
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$
implies that $\nu_{x}$ is central because the group
$W_{\widetilde{\operatorname{supp}_{\sigma}}(x)}$ generated by the elements of
$\widetilde{\operatorname{supp}_{\sigma}}(x)$ is finite in such a case.
However, if $\nu_{x}$ is central, for a representative $\dot{x}$, we have
$\kappa(\dot{x})=\kappa(b)$ and $\bar{\nu}_{\dot{x}}=0=\bar{\nu}_{b}$ so that
$[\dot{x}]=[b]\in B(G)$. Hence, $X_{x}(b)\neq\emptyset$. ∎

Lemma 1.1 shows that we only need to consider the case
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$. Our main
conjecture is the following: (we follow the notations from section 2)

######  Conjecture 1.

Let $G$ be a simple and quasi-split reductive group of adjoint type. Let $x$
be an element of Iwahori-Weyl group $\widetilde{W}$ and $b\in\breve{G}$ be
basic. Assume that $\kappa_{G}(b)=\kappa_{G}(x)$ and
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$. Then,

$X_{x}(b)\neq\emptyset\text{ if and only if
}\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})=\mathbb{S}\text{
for all }r\in W_{x},$

where $W_{x}$ is the subset of $W_{0}$ defined in definition 3.5.

Conjecture 1 claims that, except for the lemma 1.1 cases, the critical
obstruction for the nonemptiness is whether or not the $\sigma$-supports of
certain $\sigma$-conjugates of $\eta_{\sigma}(x)$ are all full. For
convenience, we label by $\Leftarrow$ (resp. $\Rightarrow$) the ‘if’ (resp.
‘only if’) direction. Our main theorem is the following.

######  Theorem 1.2.

Conjecture 1 holds for all but finitely many $x$. More precisely, $\Leftarrow$
holds for all $x$ and $\Rightarrow$ holds for all but finitely many cases. In
addition, $\Rightarrow$ holds for

  1. (1)

$x$ lying in exactly one critical strip (see definition 2.4),

  2. (2)

a translation element or $vt^{\mu}$-form element (for $\mu$ dominant), and

  3. (3)

$x$ in type $A_{n}$ under the condition of proposition 5.2.

Note that GHN B takes care of infinitely many $x$’s, but it does not apply to
infinitely many cases as well. We prove $\Leftarrow$ in corollary 3.8 in full
generality and prove $\Rightarrow$ in proposition 3.10 for $x$ such that
$\ell(x)\gg 0$. An efficetive bound for the length can be computed and we do
so for type $A_{n}$ in proposition 5.2.

Let us discuss Theorem 1.2 (3) in more detail. For simplicity, let
$G=\operatorname{PGL}_{n}$. There are two assumptions in proposition 5.2. The
first assumption is that $\kappa_{G}(x)=0\in\mathbb{Z}/n\simeq
X^{*}(Z(\hat{G})^{\Gamma})$. It means that $x$ can be written as
$t^{\lambda}w$ where
$\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{Z}^{n}_{\text{sum}=0}$ and
$w\in S_{n}$ (symmetric group). And then the second assumption is that
‘$\max_{i}{\lambda_{i}}>1$ and $\min_{i}{\lambda_{i}}<-1$’.

On the other hand, any element $x=t^{\lambda}w$ satisfying
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$ (cf.
Lemma 1.1) has the property ‘$\max_{i}\lambda_{i}=1$ and
$\min_{i}\lambda_{i}=-1$’. Hence, there is only a small gap left in the
conjecture under the assumption that $\kappa_{G}(x)=0$. This way, we can view
Theorem 1.2 (3) as an evidence for Conjecture 1 in the sense that elements in
lemma 1.1 are the only exceptions.

######  Remark 1.

It is worth pointing out that GHN B and Theorem 1.2 for $x$ lying in the
shrunken Weyl chambers are almost the same, but there is a subtle difference.
For example, when $x=w_{0}$ (the longest element of $W_{0}$) as an element of
$\widetilde{W}$, we apply GHN B to get $X_{x}(1)\neq\emptyset$. However, $x$
does not fit into Theorem 1.2 as
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$, so we
should apply lemma 1.1 to get $X_{x}(1)\neq\emptyset$ here.

######  Remark 2.

One might wonder what the condition
‘$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$’ means.
Surprisingly, a simple concrete picture exists for this condition in the sense
that “$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$
if and only if the action of $x\circ\sigma$ fixes a point in the closure of
the base alcove”. Moreover, under the obvious obstruction (that is,
$\kappa_{G}(x)=\kappa_{G}(b)$) when $b$ is basic, it is equivalent to that
$I\dot{x}I\subset[b]$. ([GHN19, Proposition 5.6 and Lemma 5.8])

Surprisingly, our result says that the abstract criterion (GHN A) can be made
much stronger as follows. This was observed by Sian Nie.

######  Theorem 1.3 (Stronger GHN A).

Let $b$ be basic and suppose that
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$ with
$\ell(x)\gg 0$. Then, $X_{x}(b)\neq\emptyset$ if and only if $x$ is not a
$(J,w)_{\sigma}$-alcove for any proper $J\subsetneq\mathbb{S}$.

We note that the condition ‘$\ell(x)\gg 0$’ can be removed once Conjecture 1
is fully proved. In short, for $x$ not intersecting with the base alcove,
$X_{x}(b)\neq\emptyset$ if and only if $x$ is not “coming from a proper Levi
subgroup”.

###  1.4. New Ideas and Sketch of Proof

The novelty of our work lies in the introduction of the set $W_{x}$ which
arises naturally in the following sense.

The idea is that a critical strip behaves as if it belongs to the shrunken
Weyl chambers adjacent to the strip. Hence, if $x$ lies in a critical strip,
we can “embed” $x$ into each shrunken Weyl chamber adjacent to the strip.
Noting that, when $x$ lies in a shrunken Weyl chamber, the rule for
$X_{x}(b)\neq\emptyset$ is
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}$, we can presuppose
that, in general, the rule for $X_{x}(b)\neq\emptyset$ would be
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x^{\prime}))=\mathbb{S}$ for each
embedding $x^{\prime}$ of $x$ into each shrunken Weyl chamber adjacent to the
strips.

The set $W_{x}$ is defined to be the set of elements in $W_{0}$ that,
intuitively, embed $x$ into those shrunken Weyl chambers. Practically, for $x$
in the shrunken Weyl chambers, we get $W_{x}=\\{\operatorname{id}_{W_{0}}\\}$
and, for $x$ in one critical strip (cf. Theorem 1.2 (1)), we have
$W_{x}=\\{\operatorname{id}_{W_{0}},s\\}$ where $s$ is the simple reflection
related to the critical strip containing $x$.

In order to define $W_{x}$, we make a new observation on a certain structure
of the set of critical strips containing $x$ (Proposition 3.4). More
precisely, let $\Phi$ be the set of (relative) roots of $G$ and, for
simplicity, $x$ lie in the dominant Weyl chamber. Next, denote by $\Phi_{x}$
the set of positive roots whose critical strip contains $x$. Then, $\Phi_{x}$
is “anti-closed” in the sense that if $\alpha+\beta\in\Phi_{x}$ for two
positive roots $\alpha$ and $\beta$ then $\alpha\in\Phi_{x}$ or
$\beta\in\Phi_{x}$. Alternatively, it is equivalent to that the complement of
$\Phi_{x}$ in the set of positive roots is closed.

Using this, we show that the set $W_{x}$ is well-defined (Proposition 3.3) and
that $W_{x}$ contains exactly the elements needed for the ‘$\sigma$-support
test’ (cf. Lemma 3.7). After that, we follow the strategy of [GHN15] as
explained in section 2.4 with more careful study on the $W_{x}$-action on
dominant cocharacters. We also use the positivity of the coroot-coefficients
of dominant cocharacters (see Lemma 2.18) and work with the exact (positive)
coefficients in the proof of Theorem 1.2 (1) and (3).

######  Remark 3.

There has been an interesting coincidence happening while we have been working
on this problem. Felix Schremmer ([Sch22]) recently defined a set called
‘length-positive’ in his work on the generic Newton point. Our work and
Schremmer’s work do not overlap, that is, the results are rather
complementary. However, the length-positive set of Schremmer turns out to be a
$W_{0}$-conjugate of the set $W_{x}$ of ours.

###  1.5. Applications and future works

####  1.5.1. The set $B(G)_{x}$ and cordial elements

The question of whether $X_{x}(b)$ is nonempty is equivalent to the question
of whether $I\dot{x}I\cap[b]$ is nonempty. In this perspective, we can denote
the set of $[b]\in B(G)$ such that $I\dot{x}I\cap[b]\neq\emptyset$ by
$B(G)_{x}$ and ask to describe it. This approach first appeared in [Bea09].

######  Remark 4.

For clarification, we remark that the main difference between the approach
using $B(G)_{x}$ and ours is which variable is fixed. In our approach, we fix
$b$ and study the nonemptiness, but the study of $B(G)_{x}$ fixes $x$. As we
will see in a moment, these two approaches are complementary.

When $x$ is cordial (see definition 5.3), the set $B(G)_{x}$ has the property
called saturated (see lemma 5.4), which makes the complete description of
$B(G)_{x}$ easier. Combining this with Theorem 1.2 (3), we obtain a full
description of $B(G)_{x}$ in some special cases as follows.

######  Theorem 1.4.

Let $x=vt^{\mu}$ for a dominant non-central555If $\mu$ is central, it is
obvious that $B(G)_{x}=\\{[t^{\mu}]\\}=B(G,\mu)$ via lemma 1.1. $\mu$ and
$v\in W_{0}$, and let $W_{0}(\mu)$ be the stabilizer subgroup of $W_{0}$
fixing $\mu$. Then,

$W_{x}=\\{r\in W_{0}{(\mu)}:\ell(vr^{-1})=\ell(v)+\ell(r)\\}.$

Now, if
$\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})=\mathbb{S}$
for all $r\in W_{x}$ then $B(G)_{x}=B(G,\mu)$.

####  1.5.2. Future works on new dimension formulas

The following dimension formula ([He14]) is known for $x$ lying in the
shrunken Weyl chambers:

$\dim
X_{x}(b)=\frac{1}{2}\left(\ell(x)+\ell(\eta_{\sigma}(x))-\operatorname{def}_{G}(b)\right).$

On the contrary, outside of the shrunken Weyl chambers, even a conjectural
formula for $\dim X_{x}(b)$ is still mysterious. However, if more nonemptiness
results are found, we can approach this problem with the following recursive
formula ([He14, Proposition 4.2]): For $s\in\mathbb{S}$ satisfying
$\ell(sx\sigma(s))=\ell(x)-2$,

$\dim X_{x}(b)=\max\\{\dim X_{sx}(b),\dim X_{sx\sigma(s)}(b)\\}+1.$

Typically, this is a bottom-up formula for the length reason (that is,
$\ell(x)=\ell(sx)+1=\ell(sx\sigma(s))+2$). However, for example, if
$X_{sx\sigma(s)}(b)$ is empty then we can also compute $\dim X_{sx}(b)$ from
$\dim X_{x}(b)$, which is top-down.

In future work, using Theorem 1.2 (2), we will show the following dimension
formula which is new even in the rank two case:

######  Theorem 1.5.

Let $G$ be residually split with $\operatorname{rk}\\!{G_{\text{sc}}}=2$ and
$b$ be basic. For $x\in\widetilde{W}$ lying in only one critical strip
(associated to $v\alpha$), if $X_{x}(b)\neq\emptyset$ then

$\dim
X_{x}(b)=\frac{1}{2}\left(\ell(x)+\min\\{\ell(\eta_{\sigma}(x)),\ell(\sigma^{-1}(s_{\alpha})\eta_{\sigma}(x)s_{\alpha})\\}-\operatorname{def}_{G}(b)\right)-\epsilon,$

where $\epsilon=1$ if $\eta_{\sigma}(x)=w_{0}$ and $\epsilon=0$ otherwise.

###  1.6. Organization

We describe the organization here.

§2 is a preliminary section. We recall the setup of [GHN15] and note down some
computations related to critical strips. Following loc.cit., we summarize the
proof of GHN B and collect the lemmas subject to be generalized.

In §3, we introduce the set $W_{x}$. Then, we discuss some properties of
closed subsets of a root system ([ĐCH94]). Then we prove Theorem 1.2 and
Theorem 1.2 (2).

In §4, we prove Theorem 1.2 (1) by handling some possibly exceptional cases.
We study in detail using the exact positive coefficients from [OV90].

In §5, we wrap up some tedious computations postponed in §3 and §4 proving
Theorem 1.2 (3). In the application part, we recall some facts from [MV20] and
[He21] and then prove Theorem 1.4.

### Acknowledgements

This is a part of the author’s thesis. The author thanks to his advisor Sug
Woo Shin for his introduction to the Deligne-Lusztig world. The author also
feels gratitude to Ulrich Görtz and Xuhua He for their kind replies and
encouragement, and to Sian Nie and Felix Schremmer for their careful readings
and comments on the first draft. Lastly, Michael Rapoport’s comments on the
history of affine Deligne-Lusztig varieties which the author was not aware of,
are appreciated.

This project started when the pandemic covered the world and many ideas came
to the author during the stay in Korea Institute for Advanced Study (KIAS). We
would like to thank the institution for the support and hospitality.

##  2\. Basics on single affine Deligne-Lusztig varieties

Let $F$ be a nonarchimedean local field with a uniformizer $t$ and $G$ be a
connected reductive group over $F$. Denote by
$\breve{F}\colonequals\widehat{F^{\operatorname{nr}}}$ the completion of
maximal unramified extension of $F$ and by $\breve{G}$ the $\breve{F}$-points
of $G$. The Frobenius map on the residue field of $\breve{F}$ lifts to that of
$\breve{F}$ which we denote by $\sigma$ again. The induced map on $\breve{G}$
will be denoted the same. Finally, we denote the set of $\sigma$-conjugacy
classes of $\breve{G}$ by $B(G)$.

###  2.1. Iwahori-Weyl group and $B(G)$

Let $S$ be a maximal $\breve{F}$-split torus of $G$ defined over $F$ and $T$
be the centralizer of $S$. Note that $T$ is a maximal torus of $G$ as $G$
becomes quasi-split over $\breve{F}$ by Steinberg’s Theorem.

####  2.1.1. Iwahori-Weyl group $\widetilde{W}$

The Iwahori-Weyl group associated to $S$ is defined as

$\widetilde{W}\colonequals N_{S}(G)(\breve{F})/T(\breve{F})_{1}$ (1)

where $T(\breve{F})_{1}$ is the unique parahoric subgroup of $T(\breve{F})$.
We can fit $\widetilde{W}$ into the following short exact sequence of groups
([HR08] Definition 7)

$0\to X_{*}(T)_{\Gamma_{0}}\to\widetilde{W}\to W_{0}\to 1$

where ${\Gamma_{0}}$ is the absolute Galois group of $\breve{F}$ and $W_{0}$
is the relative Weyl group

$W_{0}\colonequals N_{S}(G)(\breve{F})/T(\breve{F})$ (2)

Now, we fix a $\sigma$-invariant base alcove $\mathbf{a}$ in the apartment of
$S$ and let $I$ be the Iwahori subgroup of $G$ corresponding to $\mathbf{a}$.
By fixing a special vertex in the apartment, we get a section
$W_{0}\to\widetilde{W}$ (not necessarily $\sigma$-equivariant) which allows us
to identify $\widetilde{W}$ with $X_{*}(T)_{\Gamma_{0}}\rtimes W_{0}$.

The Newton map $\nu:\widetilde{W}\to
X_{*}(T)_{\Gamma_{0},\mathbb{Q}}^{\sigma}$ is defined as follows. The action
of $\sigma$ on $\widetilde{W}$ is of finite order so that, given
$x\in\widetilde{W}$, there exists a positive integer $N$ such that
$\prod_{i=0}^{N-1}\sigma^{i}(x)$ belongs to $X_{*}(T)_{\Gamma_{0}}^{\sigma}$,
say $\mu$. The Newton map is defined to send $x$ to $\nu_{x}=\frac{\mu}{N}$.
This does not depend on the choice of $N$.

Think: Wanna mention things about extended affine Weyl group and Iwahori-Weyl
group? Torsion-related things can be found in Haines-He Vertexwise paper 8.1
and Prasad-Kaletha book and find ‘induced’. It can also be found in BT2 4.4.16
with the induced Galois module thing.

Let $G_{\text{sc}}$ be the simply connected cover of the derived subgroup of
$G$ and $T_{\text{sc}}$ the inverse image of $T$ via $G_{\text{sc}}\to
G_{\text{der}}\to G$. The Iwahori-Weyl group of $G_{\text{sc}}$ is the affine
Weyl group $W_{a}$ and gives rise to the following short exact sequence
([HR08] Lemma 14):

$1\to
W_{a}\to\widetilde{W}\overset{\tilde{\kappa}_{G}}{\longrightarrow}X^{*}(Z(\hat{G})^{\Gamma_{0}})\to
1.$ (3)

Denoting by $\Omega\subset\widetilde{W}$ the stabilizer of the base alcove, we
have an isomorphism $X^{*}(Z(\hat{G})^{\Gamma_{0}})\simeq\Omega$ which gives a
section of $\tilde{\kappa}_{G}$. This presents $\widetilde{W}$ as
$W_{a}\rtimes X^{*}(Z(\hat{G})^{\Gamma_{0}})$. Now, the Bruhat order on
$W_{a}$ extends onto $\widetilde{W}$ by making two elements be comparable when
their projections to $X^{*}(Z(\hat{G})^{\Gamma_{0}})$ agree.

Note that $W_{a}$ is the affine Weyl group generated by orthogonal reflections
with respect to the hyperplanes in
$X_{*}(T_{\text{sc}})_{\Gamma_{0}}\otimes\mathbb{R}$. Hence, by [Bou81, Ch.VI,
§2.5. Proposition 8], there exists a reduced root system $\Sigma$ whose affine
Weyl group is canonically isomorphic to $W_{a}$. We denote by $Q^{\vee}$ the
coroot lattice of $\Sigma$ and $P^{\vee}$ its coweight lattice. Lastly, let
$\mathbb{S}$ be the set of simple reflections of the finite Weyl group of
$\Sigma$ (that is, $W_{0}$) and $\widetilde{\mathbb{S}}$ the set of affine
simple reflections.

The map $\sigma$ on $\breve{G}$ induces an action on $\mathbb{S}$ which we
will denote by $\sigma$ again. We call $J\subset\mathbb{S}$ a $\sigma$-stable
subset if $\sigma(J)=J$. For any $\sigma$-stable subset $J$, we denote
$X_{*}(T)_{\Gamma_{0}}\rtimes W_{J}$ by $\widetilde{W}_{J}$ where $W_{J}$ is
the subgroup of $W_{0}$ generated by the simple reflections of $J$.

A comment on the notation: we will use the notation $v\cdot\mu$ when
considering $W_{0}$-action on $X_{*}(T)_{\Gamma_{0}}$. So, for example,
$vt^{\mu}\in\widetilde{W}$ can also be written as $t^{v\cdot\mu}v$.

####  2.1.2. $B(G)$ with Newton map and Kottwitz map

Recall the Newton map and the Kottwitz map from [Kot97, 4.5] that give an
injective homomorphism $(\bar{\nu},\kappa_{G}):B(G)\to
X_{*}(T)_{\mathbb{Q}}^{\Gamma,+}\times X^{*}(Z(\hat{G})^{\Gamma})$ where
$\Gamma$ is the absolute Galois group of $F$.

######  Definition 2.1 ($B(G,\mu)$).

Let $\mu\in X_{*}(T)^{+}$ be a dominant cocharacter of $G$. We define
$B(G,\mu)$ as the subset of $B(G)$ consisting of $[b]\in B(G)$ such that

$\bar{\nu}_{b}\leq\mu^{\diamond}\text{ and
}\kappa_{G}([b])=\kappa_{G}([t^{\mu}])$

where $\mu^{\diamond}$ is the $\Gamma$-average666To be precise, this is true
only when $G$ is quasi-split. In general, it is the average of the dominant
representatives of Galois orbits. For a more detailed explanation, see [HN18,
2.4]. of $\mu$ and $t^{\mu}$ is the image of $t$ under $\mu:\mathbb{G}_{m}\to
T$. Mazur’s inequality mentioned in section 1.1.1 refers to
$\bar{\nu}_{b}\leq\mu^{\diamond}$.

We note that the map $\tilde{\kappa}_{G}$ in eq. 3 followed by the projection
$X^{*}(Z(\hat{G})^{\Gamma_{0}})\to X^{*}(Z(\hat{G})^{\Gamma})$ gives
$\kappa_{G}:\widetilde{W}\to X^{*}(Z(\hat{G})^{\Gamma})$ and it is compatible
with $\kappa_{G}$ on $B(G)$ via the lifting from $\widetilde{W}$ to
$N_{S}(G)(\breve{F})\subset\breve{G}$.

###  2.2. Single affine Deligne-Lusztig variety

The Iwahori-Bruhat decomposition says

$\breve{G}=\bigsqcup_{x\in\widetilde{W}}I\dot{x}I$

where $\dot{x}\in N_{S}(G)(\breve{F})$ is a representative of
$x\in\widetilde{W}$.

######  Definition 2.2 (affine Deligne-Lusztig “variety”).

For $x\in\widetilde{W}$ and $b\in B(G)$, the single affine Deligne-Lusztig
variety associated to $x$ and $b$ is

$X_{x}(b)\colonequals\\{gI\in\breve{G}/I:g^{-1}b\sigma(g)\in I\dot{x}I\\}.$

A priori, it is an affine Deligne-Lusztig set and we do not have a natural
scheme structure on it. In fact, in the case of an equal characteristic local
field $F$, it is not difficult to identify $X_{x}(b)$ as the
$\overline{\mathbb{F}}_{q}$-points of a (locally of finite type) locally
closed subscheme in the affine flag variety over $\overline{\mathbb{F}}_{q}$.
It is the mixed characteristic case where we need distinguished works of
[Zhu17] and [BS17] to give a scheme structure on $X_{x}(b)$.

######  Remark 5.

We want to make some comments about the case of classical Deligne-Lusztig
varieties: In [DL76], Deligne and Lusztig defined a system of varieties that
naturally generates representations of finite groups of Lie type. More
precisely, for a connected reductive group $\mathbb{G}$ defined over
$\mathbb{F}_{q}$ and $G=\mathbb{G}(\overline{\mathbb{F}}_{q})$ with a Borel
subgroup $B\subset G$, they considered the following varieties

$X_{w}\colonequals\\{gB\in G/B:g^{-1}\sigma(g)\in BwB\\}$

for elements $w$ of the Weyl group $W$.777It is $\widetilde{X_{w}}=\\{gU\in
G/U:g^{-1}\sigma(g)\in IwI\\}$, a $T(\mathbb{F}_{q})$-torsor over $X_{w}$,
which plays an important role in the study of finite Lie group representation
in [DL76], but this is irrelevant to the content of this paper. In the case of
finite fields, by Lang’s theorem, $g^{-1}\sigma(g):G\to G$ is surjective so
that there is only a single $\sigma$-conjugacy class in $G$ whence no choice
of $b$ (a $\sigma$-conjugacy class) is necessary. Moreover, by Lang’s theorem,
one can always find $g$ such that $g^{-1}\sigma(g)$ belongs to $BwB$ so that
$X_{w}$ is always nonempty. Hence, no interesting questions about emptiness
arise there. Also, it is an easy exercise to find the dimension formula ($\dim
X_{w}=\ell(w)$) of classical Deligne-Lusztig variety.

In order to study the nonemptiness pattern, it is more convenient to do some
reductions. By [HZ20, Corollary 4.4 and Section 4.3], we can reduce the
nonemptiness problem to the case when $G$ is the quasi-split inner form of an
adjoint group. Finally, if $G=G_{1}\times G_{2}$ then $B(G)=B(G_{1})\times
B(G_{2})$ and
$\widetilde{W}_{G}=\widetilde{W}_{G_{1}}\times\widetilde{W}_{G_{2}}$. Hence,
$X_{x}(b)=X_{x_{1}}(b_{1})\times X_{x_{2}}(b_{2})$ where $x_{i}$’s are the
projections of $x$ onto $\widetilde{W}_{G_{i}}$ and $b_{i}$’s are that of $b$
onto $B(G_{i})$. Now, we may assume that $G$ is a simple quasi-split reductive
group of adjoint type.

######  Remark 6.

In loc.cit., it is assumed that $G$ is tamely ramified over $F$ and
$p\nmid\\!\pi_{1}(G^{\text{ad}})$ in the equal characteristic case. However,
this assumption is not necessary for the nonemptiness results as a universal
homeomorphism preserves the nonemptiness.

###  2.3. Terminologies on positions of alcoves

From now on, we assume that $G$ is a simple quasi-split reductive group of
adjoint type. For simplicity, we abusively use $x$ to denote the alcove
$x\mathbf{a}$. For example, $\operatorname{id}_{\widetilde{W}}$ denotes the
base alcove.

Let $\Phi$ be the set $\Phi(G,S)$ of relative roots and $\Phi^{+}$ (resp.
$\Phi^{-}$) be the subset of positive (resp. negative) roots. Let
$V\colonequals X_{*}(T)_{\Gamma_{0}}\otimes\mathbb{R}$ which is isomorphic to
$X_{*}(T_{\text{sc}})_{\Gamma_{0}}\otimes\mathbb{R}$ as $G$ is semisimple. For
$\alpha\in\Phi$, the hyperplane $H_{\alpha}$ in $V$ is defined by
$\\{\mathbf{v}\in V:\langle\alpha,\mathbf{v}\rangle=0\\}$ and, more generally,
for any integer $k$, we define $H_{\alpha}(k)\colonequals\\{\mathbf{v}\in
V:\langle\alpha,\mathbf{v}\rangle=k\\}$. Note that
$H_{\alpha}(k)=H_{-\alpha}(-k)$.

######  Definition 2.3 ([GHKR10, Section 2.1], $k$-value of an alcove with
respect to a root).

For $\alpha\in\Phi$ and $x\in\widetilde{W}$, let us denote by $k(\alpha,x)$
the integer $k$ such that $x$ is located in between the hyperplanes
$H_{\alpha}(k)$ and $H_{\alpha}(k+1)$.

######  Definition 2.4 (The critical strips and shrunken Weyl chambers).

For each positive root $\alpha$, we call the set of alcoves between
$H_{\alpha}(0)$ and $H_{\alpha}(1)$ the critical strip associated to $\alpha$
and denote by $C_{\alpha}$.888One can define the critical strip of a negative
root $\alpha$ to be the set of alcoves between $H_{\alpha}(0)$ and
$H_{\alpha}(-1)$ and also denote by $C_{\alpha}$. This convention will be
secretly used. The set of alcoves which do not lie in any critical strip is
called the shrunken Weyl chambers.

The following computes the $k$-values explicitly. Here, the expression
$t^{\mu}w$ is an alcove in the dominant Weyl chamber always and $v$ is an
element of $W_{0}$. Both $a$ and $\alpha$ mean roots, but the latter will
denote a positive root mostly.

######  Lemma 2.5.

For any root $a\in\Phi$,

$k(a,t^{\mu}w)=\left\\{\begin{array}[]{ll}\langle a,\mu\rangle&\text{if
}w^{-1}a>0,\\\ \langle a,\mu\rangle-1&\text{otherwise.}\end{array}\right.$

###### Proof.

We only need to consider the case where the alcove is represented by a finite
Weyl group element and, in this case, the value $\delta_{w^{-1}\alpha}$
decides whether $w$ is in the $a$-direction or $-a$-direction.∎

In general, $k(a,vt^{\mu}w)=\langle a,v\mu\rangle+\delta_{w^{-1}v^{-1}a}$
where $\delta_{\alpha}=0$ if $\alpha\in\Phi^{+}$ or $=-1$ otherwise.

######  Corollary 2.6.

Let $a$ be a root in (1) and $a$, $b$, and $a+b$ are roots in (2).

  1. (1)

$k(a,t^{\mu}w)+k(-a,t^{\mu}w)=-1$.

  2. (2)

$k(a+b,t^{\mu}w)=k(a,t^{\mu}w)+k(b,t^{\mu}w)$ or
$k(a,t^{\mu}w)+k(b,t^{\mu}w)+1$.

###### Proof.

(1): $w^{-1}a>0$ implies $w^{-1}(-a)<0$ and vice versa. (2): Apply lemma 2.5
noting that the cases where both $w^{-1}a$ and $w^{-1}b$ are positive (resp.
negative) but $w^{-1}(a+b)$ is negative (resp. positive) are not possible. ∎

From lemma 2.5, $vt^{\mu}w\in C_{v\alpha}$ for some $\alpha\in\Phi^{+}$ if and
only if $\langle\alpha,\mu\rangle+\delta_{w^{-1}\alpha}-\delta_{v\alpha}=0$.
Noting that $k(\alpha,t^{\mu}w)\geq 0$ when $\alpha\in\Phi^{+}$, we have the
following corollary easily:

######  Corollary 2.7.

Let $\alpha\in\Phi^{+}$ and suppose $vt^{\mu}w\in C_{v\alpha}$. Then
$v\alpha\in\Phi^{+}$ and one of the following holds:

  1. (1)

$\langle\alpha,\mu\rangle=0$ and $w^{-1}\alpha\in\Phi^{+}$,

  2. (2)

$\langle\alpha,\mu\rangle=1$ and $w^{-1}\alpha\in\Phi^{-}$.

When $vt^{\mu}w\not\in C_{v\alpha}$, if $\langle\alpha,\mu\rangle=0$ then
$v\alpha\in\Phi^{-}$ and $w^{-1}\alpha\in\Phi^{+}$.

Now, we recall the definition of $(J,w)_{\sigma}$-alcove999In [GHN15], this is
denoted by $(J,w,\delta)$-alcove. We choose to use the notation
‘$(J,w)_{\sigma}$-alcove’ in this paper not just for the sake of simplicity.
In fact, the induced action $\delta$ on $\mathbb{S}$ by $\sigma$ is a fixed
map. It is not varying, unlike $J$ or $w$ in the notation. from [GHN15, 3.3].

######  Definition 2.8 ($(J,w)_{\sigma}$-alcove).

Let $J$ be a $\sigma$-stable subset of $\mathbb{S}$ and $w$ be an element of
$W_{0}$. We say that $x\in\widetilde{W}$ is a $(J,w)_{\sigma}$-alcove if

  * (1)

$w^{-1}x\sigma(w)\in\widetilde{W}_{J}$ and,

  * (2)

for any $a\in w(\Phi^{+}\setminus\Phi_{J}^{+})$, $k(a,x)\geq
k(a,\operatorname{id})$.

Next, the “essential” finite part $\eta_{\sigma}(x)$ of $x$ observed by Reuman
is defined as follows.

######  Definition 2.9 ($\eta_{\sigma}(x)$, [GHN15, 3.6]).

Let $v_{x}\in W_{0}$ be the unique element such that $v_{x}^{-1}x$ is in the
dominant Weyl chamber and $v_{x}^{-1}x=t^{\mu_{x}}w_{x}$ for $\mu_{x}\in
X_{*}(T)_{\Gamma_{0}}^{+}$ and $w_{x}\in W_{0}$. We define

$\eta_{\sigma}(x)=\sigma^{-1}(w_{x})v_{x}.$

Finally, we recall the following from [Bou81, Ch.IV, §1.8. Proposition 7].

######  Definition 2.10 (support and $\sigma$-support).

Given a Coxeter system $(W,S)$ and $w\in W$, the support of $w$ is defined to
be the set of $s\in S$ appearing in some (equivalently each) reduced
expression of $w$ and denoted by $\operatorname{supp}(w)$. When $(W,S)$ is
equipped with an action by $\xi$, the minimal $\xi$-stable set containing
$\operatorname{supp}(w)$ is called the $\xi$-support of $w$ and denoted by
$\operatorname{supp}_{\xi}(w)$.

######  Remark 7.

There are two Coxeter systems $(W_{0},\mathbb{S})$ and
$(W_{a},\widetilde{\mathbb{S}})$ in this paper. We have $\xi=\sigma$ for
$(W_{0},\mathbb{S})$ and $\xi=\omega\sigma$ for
$(W_{a},\widetilde{\mathbb{S}})$ where $\omega$ is an element of $\Omega$
defined in eq. 3. For $x\in W_{a}\rtimes\Omega$ whose projection to $\Omega$
is $\omega_{x}$, we use the notation
$\widetilde{\operatorname{supp}_{\sigma}}(x)$ instead of
$\operatorname{supp}_{\omega_{x}\sigma}(x)$ for simplicity.

###  2.4. On the known result GHN B

Now we recall:

######  Theorem 2.11 (GHN B).

Let $b\in B(G)$ be basic and $x\in\widetilde{W}$ lie in the shrunken Weyl
chambers with $\kappa(b)=\kappa(x)$. Then,

$X_{x}(b)\neq\emptyset$ if and only if the $\sigma$-support of
$\eta_{\sigma}(x)$ is $\mathbb{S}$.

The proof of theorem 2.11 is a combination of the following lemmas.

######  Lemma 2.12 ([GHN15, Proposition 3.6.4 and Proposition 3.6.5]).

Let $b$ be basic. Under the following two assumptions, we have
$X_{x}(b)=\emptyset$.

  1. (1)

$\nu_{\mu_{x}}\neq\nu_{b}$ and

  2. (2)

$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))\neq\mathbb{S}$,

If we assume that $x$ lies in shrunken Weyl chambers, (2) implies (1).

This proves that if $X_{x}(b)\neq\emptyset$ then
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}$ in theorem
2.11.101010However, note that the proof of lemma 2.12 uses [GHN15, Proposition
3.5.1] (“$\sigma$-conjugacy classes never fuse”) which was stated without the
assumption that $b$ is basic but in fact needs that assumption. See [GHN]. For
the reverse direction, we need the following theorem:

######  Proposition 2.13 ([GHN15, Theorem 4.4.7]).

Let $b$ be basic. If $x$ satisfies NLO, then $X_{x}(b)\neq\emptyset$. NLO
means the following: for every pair $(J,w)$ with $\sigma$-stable $J$ and $w\in
W_{0}$ such that $x\in\widetilde{W}$ is a $(J,w)_{\sigma}$-alcove, there
exists $b_{J}\in w\widetilde{W}_{J}\sigma(w)^{-1}$ such that

  1. (1)

$\kappa(b)=\kappa(b_{J})$

  2. (2)

$\nu_{b_{J}}=\nu_{b}$

  3. (3)

$\kappa_{J}(w^{-1}b_{J}\sigma(w))=\kappa_{J}(w^{-1}x\sigma(w))$

If $x$ lies in the shrunken Weyl chamber, one can show that
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}$ and so, by lemma
2.14 below, the only $J$ such that $x$ is a $(J,w)_{\sigma}$-alcove is
$J=\mathbb{S}$. For $J=\mathbb{S}$, we can let $b_{J}$ be any element in the
$\sigma$-straight conjugacy class of $\widetilde{W}$ corresponding to $[b]\in
B(G)$ ([He14, 3.3]) so that $x$ satisfies NLO and $X_{x}(b)\neq\emptyset$.

######  Lemma 2.14 ([GHN15, Proposition 4.1.1]).

Let $x\in\widetilde{W}$ lie in the shrunken Weyl chambers. If $x$ is a
$(J,w)_{\sigma}$-alcove for $\sigma$-stable subset $J$ of $\mathbb{S}$, then
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))\subset J$.

Finally, we have the following lemmas on the properties of dominant
cocharacters and the Weyl group action on roots:

######  Lemma 2.15.

Let $\mu$ be a cocharacter and $w\in W_{0}$. Then, $\mu-w\cdot\mu\in
Q_{\operatorname{supp}(w)}^{\vee}$.

###### Proof.

For any simple reflection $s$, the formula
$s\cdot\mu=\mu-\langle\alpha_{i},\mu\rangle\alpha_{i}^{\vee}$ tells us that
$\mu-s\cdot\mu\in Q_{\\{s\\}}^{\vee}$. Note that $w$ is a product of simple
reflections and so we can think inductively on $\ell(w)$. Considering $\mu-
ws\cdot\mu=\mu-s\cdot\mu+s\cdot\mu-w\cdot(s\cdot\mu)\in
Q_{\\{s\\}}^{\vee}\oplus Q_{\operatorname{supp}(w)}^{\vee}$, it is
straightforward. ∎

The following corollary is stated in [GHN15, 3.5 (1)] without a proof. For
completeness, we give a proof for this.

######  Corollary 2.16.

Let $J$ be a $\sigma$-stable subset of $\mathbb{S}$. If
$x\in\widetilde{W}_{J}$, then $\nu_{v_{x}\cdot\mu_{x}}-\nu_{x}\in
Q_{J,\mathbb{Q}}^{\vee}$.

###### Proof.

The difference is a $\mathbb{Q}$-multiple of
$\sum_{i}(\sigma^{i}(v_{x}\cdot\mu_{x})-(v_{x}w_{x}\sigma)^{i}(v_{x}\cdot\mu_{x}))$.
Noting that $\prod_{0\leq j\leq i-1}\sigma^{j}(v_{x}w_{x})\in W_{J}$, apply
lemma 2.15 to get the conclusion. ∎

Lastly, we prove:

######  Lemma 2.17.

Suppose that the Dynkin diagram of $G$ is $\sigma$-connected. Let $J$ be a
proper $\sigma$-stable subet of $\mathbb{S}$. If $\mu\in
Q^{\vee}_{J,\mathbb{Q}}$ is dominant, then $\mu=0$.

###### Proof.

$Q^{\vee}_{J,\mathbb{Q}}\subset Q^{\vee}_{\mathbb{Q}}=P^{\vee}_{\mathbb{Q}}$,
hence $\mu$ is a $\mathbb{Q}$-linear combination of fundamental coweights. As
it is dominant, all the coefficients are non-negative. If $\mu\neq 0$, then at
least one coefficient is positive so that it is a positive linear combination
of coroots of a connected component by lemma 2.18. Hence, $J$ contains some
connected component completely. As the Dynkin diagram is $\sigma$-connected,
we must have $J=\mathbb{S}$ which is contradiction. ∎

The following Lemma 2.18 has been probably known to experts for a long time.
However, to the best of our knowledge, a reference for this lemma is hard to
locate. So, following Michael Rapoport’s suggestion, we record it here. In
this paper, this is particularly needed to remedy some argument in [GHN15]
using loc.cit. 3.5. (2) which is not correct. See Proposition 4.5.

######  Lemma 2.18.

111111The letter $C$ will be used in this lemma to follow the historical
notations. The letter $C$ appearing in this paper is always referring to the
critical strip, except here. We believe there would be no confusion occurring.

Let $\bar{C}$ be the closed Weyl chamber defined as $\\{\mu\in
V:\langle\mu,\alpha\rangle\geq 0\text{ for all }\alpha\in\Delta\\}$ and
$C^{\vee}$ be the obtuse Weyl chamber defined as $\\{\mu\in
V:\mu=\sum_{i\in\mathbb{S}}c_{i}\alpha_{i}^{\vee},\leavevmode\nobreak\
c_{i}\geq 0\\}$. Then $\bar{C}\setminus\\!\\{\vec{0}\\}$ is contained in the
interior of $C^{\vee}$. In other words, a fundamental coweight is a positive
linear combination of simple coroots.

###### Proof.

The fundamental coweights $\varpi_{i}^{\vee}$ form a basis of $V$ and they
satisfy $\langle\varpi_{i}^{\vee},\alpha_{j}\rangle=\delta_{ij}$. Hence,
$\bar{C}\setminus\\{\vec{0}\\}=\\{\sum_{i\in\mathbb{S}}d_{i}\varpi_{i}^{\vee}:\text{
not all }d_{i}\text{'s are }0\\}$. Hence, it is enough to show that
$\langle\varpi_{i}^{\vee},\varpi_{j}\rangle\equalscolon\pi_{ij}$, the
coefficient of $\alpha_{j}^{\vee}$ in the expression of $\varpi_{i}^{\vee}$,
is positive.

First of all, the coefficients from [Bou81, Ch.VI, §4.5-§4.13, (VI)] prove the
dual version. For our version, recall the Cartan matrix $A$ whose entries are
$A_{ij}=\langle\alpha_{i}^{\vee},\alpha_{j}\rangle$. Then,
$\delta_{ik}=\langle\varpi_{i}^{\vee},\alpha_{k}\rangle=\langle\sum_{j}\pi_{ij}\alpha_{j}^{\vee},\alpha_{k}\rangle=\sum_{j}\pi_{ij}A_{jk}$.
Therefore, $\pi_{ij}$’s are the entries of the inverse of $A$, which we know
to be positive by [OV90, Reference Chapter. §2. Table 2] (note that one should
take their transpose). For a more general situation, see [LT92, 5.]. ∎

######  Remark 8.

$b$ being basic seems to be related to the following fact: Assume that the
Dynkin diagram of $G$ is $\delta$-connected. Let $J$ be a proper
$\delta$-stable subset of $\mathbb{S}$. If $\lambda$, $\lambda^{\prime}\in V$
such that $\langle\lambda,\alpha\rangle\geq 0$ for all $a\in J$,
$\lambda^{\prime}$ is central and
$\lambda-\lambda^{\prime}\in{Q_{J}^{\vee}}\otimes\mathbb{Q}$, then
$\lambda=\lambda^{\prime}$.

##  3\. Proof of main theorems

###  3.1. Closed subsets of a root system

[closedset] Lemma 3.1 [Bou81] Chapitre 6, Proposition 19 tells us that the
positive-like set is equivalent to the radical closed subset. [ĐCH94, Theorem
4]: In the introduction, they prove that if $P\cup(-P)=\Phi$ (called
parabolic) and $P$ is closed, then it is actually invertible. However, in our
situation, only $\Psi^{c}\cap\Phi^{+}$ is closed, so we cannot apply the
theorem.

######  Definition 3.1.

A subset $\Psi$ of the set of roots $\Phi$ is called121212The terms closed and
parabolic are from [Bou81, Ch.VI, §1.7. Définition 4.] and the term radical is
from [ĐCH94, 1.].

  1. (1)

closed if $\alpha,\beta\in\Psi$ implies
$\\{\alpha+\beta\\}\cap\Phi\subset\Psi$,

  2. (2)

radical if $\Psi\cap-\Psi=\emptyset$, and

  3. (3)

parabolic if $\Psi\cup-\Psi=\Phi$.

Being closed or radical are invariant under any $W_{0}$-action. An example of
a radical closed subset is any subset of all positive roots or
$W_{0}$-conjugate of that. In fact, this is essentially the only example.

######  Lemma 3.2.

For a radical closed subset $\Psi$, there exists $w\in W_{0}$ such that
$\Psi\subset w\Phi^{+}$.

###### Proof.

[Sop02, Proposition 3] (cf. [Bou81, Ch.VI, §1.7. Proposition 22]) ∎

The following proposition is stronger than lemma 3.2 and crucial to generalize
our work to multiple critical strips case. The proof relies on the
classification of parabolic subsets.

######  Proposition 3.3.

Let $\Psi_{r}\subset\Psi_{p}$ be two closed subsets of $\Phi$ such that
$\Psi_{r}$ is radical and $\Psi_{p}$ is parabolic. Then, there exists $w\in
W_{0}$ such that

$w\Psi_{r}\subset\Phi^{+}\subset w\Psi_{p}.$

###### Proof.

We call a closed subset to be invertible if the complement (in the full
$\Phi$) is also closed. A parabolic subset is an invertible subset and all
invertible subsets are $W_{0}$-conjugate to
$\Phi_{J}\cup(\Phi^{+}\setminus\Phi_{J^{\prime}}^{+})$ where $J\subset
J^{\prime}\subset\mathbb{S}$ and $J\bot(J^{\prime}\setminus J)$ by [ĐCH94,
Lemma 1 and Theorem 4]. Only when $J=J^{\prime}$, it is parabolic.

Hence, $\Psi_{p}=w_{0}(\Phi^{+}\cup\Phi_{J}^{-}$) for some $w_{0}\in W_{0}$
and $J\subset\mathbb{S}$. Now, consider $w_{0}^{-1}\Psi_{r}\cap\Phi_{J}$ which
is radical because it belongs to a radcial subset $w_{0}^{-1}\Psi_{r}$ and
closed because it is the intersection of two closed subsets. Hence, there
exists $w_{1}\in W_{J}$ such that $w_{0}^{-1}\Psi_{r}\cap\Phi_{J}\subset
w_{1}\Phi_{J}^{+}$ by lemma 3.2.

Note that $\Psi_{r}\subset\Psi_{p}=w_{0}(\Phi^{+}\cup\Phi_{J}^{-})$ so that

$w_{1}^{-1}w_{0}^{-1}\Psi_{r}=w_{1}^{-1}(w_{0}^{-1}\Psi_{r}\cap\Phi_{J})\cup
w_{1}^{-1}(w_{0}^{-1}\Psi_{r}\cap(\Phi^{+}\setminus\Phi_{J}^{+}))$

But from above the first part belongs to $W_{J}^{+}$. The second set belongs
to $w_{1}^{-1}(\Phi^{+}\setminus\Phi_{J}^{+})$ which belongs to $\Phi^{+}$ as
$w_{1}\in W_{J}$. Obviously,
$w_{1}^{-1}w_{0}^{-1}\Psi_{p}=w_{1}^{-1}(\Phi^{+}\cup\Phi_{J}^{-})\supset\Phi^{+}$.
∎

Now, let $\Phi_{x}$ be the set of positive roots $\alpha$ such that $x\in
C_{v_{x}\alpha}$. The following is the most important observation towards our
main theorem.

######  Proposition 3.4.

$\Phi^{+}\setminus\Phi_{x}$ is a radical closed subset.

###### Proof.

It is enough to prove that $\Psi_{x}$ satisfies the following:

$\text{for }\alpha,\beta\in\Phi^{+}\text{, if }\alpha+\beta\in\Psi_{x}\text{
then }\alpha\in\Psi_{x}\text{ or }\beta\in\Psi_{x}.$

However, for a positive root $\gamma$, $\gamma\in\Psi_{x}$ is equivalent to
$k(\gamma,t^{\mu_{x}}w_{x})=k(v_{x}\gamma,\operatorname{id})=0$. Applied to
$\gamma=\alpha+\beta$, we get
$k(\alpha+\beta,t^{\mu_{x}}w_{x})=k(v_{x}\alpha+v_{x}\beta,\operatorname{id})=0$.
Using corollary 2.6 (2) and that $t^{\mu_{x}}w_{x}$ lies in the dominant Weyl
chamber, we get $k(\alpha,t^{\mu_{x}}w_{x})=k(\beta,t^{\mu_{x}}w_{x})=0$.
Moreover, by corollary 2.6 (2) again,
$0=\delta_{v_{x}\alpha}+\delta_{v_{x}\beta}+1$ or
$0=\delta_{v_{x}\alpha}+\delta_{v_{x}\beta}$. Hence, $\delta_{v_{x}\alpha}=0$
or $\delta_{v_{x}\beta}=0$, that is, $\alpha\in\Psi_{x}$ or
$\beta\in\Psi_{x}$. ∎

###  3.2. The set $W_{x}$

We define $W_{x}$ mentioned in Conjecture 1.

######  Definition 3.5.

Given $x\in\widetilde{W}$, the subset $W_{x}\subset W_{0}$ is the set of $r$’s
such that

$r(\Phi^{+}\setminus\Phi_{x})\subset\Phi^{+},$

or equivalently, by taking the negation and the complement,
$r^{-1}\Phi^{+}\subset\Phi^{+}\cup-\Phi_{x}$.

######  Remark 9.

If $x$ lies in a shrunken Weyl chamber then $\Phi_{x}=\emptyset$, so
$W_{x}=\\{\operatorname{id}_{W_{0}}\\}$. For $x$ lying in exactly one critical
strip, we have $\Phi_{x}=\\{\alpha_{x}\\}$ for some unique
$\alpha_{x}\in\Phi^{+}$. In fact, it should be a simple root by Theorem 4.1.
Hence,

$W_{x}=\\{r\in
W_{0}:r^{-1}\Phi^{+}\subset\Phi^{+}\cup\\{-\alpha_{x}\\}\\}=\\{id,s_{x}\\}$

where $s_{x}$ is the simple reflection corresponding to $\alpha_{x}$.

We note that $W_{x}$ is not necessarily a subgroup, but the following lemma
suggests some structure on $W_{x}$.

######  Lemma 3.6.

The set $W_{x}$ is left-closed in the sense that if $w\in W_{x}$ and
$\ell(sw)<\ell(w)$ for a simple reflection $s$, then $sw\in W_{x}$.

###### Proof.

We know that $w$ can be written as $sw^{\prime}$ where
$\ell(sw^{\prime})=\ell(w^{\prime})+1$. Then
$w^{\prime-1}\Phi^{+}=w^{-1}s(\Phi^{+})=w^{-1}(\Phi^{+}\cup\\{-\alpha_{s}\\}\setminus\\{\alpha_{s}\\})=w^{-1}\Phi^{+}\cup\\{-w\alpha_{s}\\}\setminus\\{w\alpha_{s}\\}$.
However, as $w^{-1}s<w^{-1}$, $-w^{-1}\alpha_{s}\in\Phi^{+}$ so that
$w^{-1}\Phi^{+}\cup\\{-w\alpha_{s}\\}\setminus\\{w\alpha_{s}\\}\subset
w^{-1}\Phi^{+}\cup\Phi^{+}\subset\Phi^{+}\cup-\Phi_{x}$ as $w\in W_{x}$. ∎

###  3.3. Lemma 2.14 in general

As before, $b$ is basic. We will express $x$ as $v_{x}t^{\mu_{x}}w_{x}$ where
$v_{x}\in W_{0}$ is the unique element such that $v_{x}^{-1}x$ is in the
dominant Weyl chamber.

######  Lemma 3.7.

If $x$ is a $(J,w)_{\sigma}$-alcove for a $\sigma$-stable subset $J$ of
$\mathbb{S}$, then $\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1}\in W_{J}$ for some
$r\in W_{x}$.

Similarly as before, we get the following corollary:

######  Corollary 3.8.

Let $x$ be an arbitrary element in $\widetilde{W}$ and $b\in\breve{G}$ such
that $\kappa(x)=\kappa(b)$. If
$\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})=\mathbb{S}$
for all $r\in W_{x}$, then $X_{x}(b)\neq\emptyset$.

###### Proof of lemma 3.7.

Suppose that there exists $a\in w(\Phi^{+}\setminus\Phi_{J}^{+})$ such that
$v_{x}^{-1}a\in-(\Phi^{+}\setminus\Phi_{x})$. Then, by definition of
$\Phi_{x}$, $-a$ satisfies $k(-a,x)\neq k(-a,\operatorname{id})$, that is,
$k(a,x)\neq k(a,\operatorname{id})$. Hence,
$k(v_{x}^{-1}a,t^{\mu_{x}}w_{x})\gneq k(a,\operatorname{id})\geq-1$ and so
$v_{x}^{-1}a$ should be a positive root as $t^{\mu_{x}}w_{x}$ is in the
dominant Weyl chamber. This contradicts to the assumption. Hence,
$v_{x}^{-1}a\in\Phi^{+}\cup-\Phi_{x}$.

This implies that
$v_{x}^{-1}w(\Phi^{+}\setminus\Phi_{J}^{+})\subset\Phi^{+}\cup-\Phi_{x}$, or
equivalently, $\Phi^{+}\setminus\Phi_{x}\subset
v_{x}^{-1}w(\Phi^{+}\cup\Phi_{J}^{-})$. By proposition 3.4,
$\Psi_{r}\colonequals\Phi^{+}\setminus\Phi_{x}$ and $\Psi_{p}\colonequals
v_{x}^{-1}w(\Phi^{+}\cup\Phi_{J}^{-})$ fit into the assumptions in proposition
3.3 so that there exists $r\in W_{0}$ such that

$r(\Phi^{+}\setminus\Phi_{x})\subset\Phi^{+}\subset
rv_{x}^{-1}w(\Phi^{+}\cup\Phi_{J}^{-}).$

Taking the complement and the negation, we get
$rv_{x}^{-1}w(\Phi^{+}\setminus\Phi_{J}^{+})\subset\Phi^{+}$ for some $r\in
W_{x}$ by definition 3.5. Therefore, $rv_{x}^{-1}w\in W_{J}$ for some $r\in
W_{x}$ and consequently $\sigma(w^{-1})\sigma(v_{x})\sigma(r^{-1})\in W_{J}$
as $J$ is $\sigma$-stable. Moreover, $w^{-1}v_{x}w_{x}\sigma(w)\in W_{J}$
since $x$ is a $(J,w)_{\sigma}$-alcove. Multiplying them together, we get
$rw_{x}\sigma(v_{x})\sigma(r^{-1})\in W_{J}$ so that
$\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1}\in W_{J}$. ∎

###  3.4. Lemma 2.12 under some restriction

For $r\in W_{x}$, let us denote by $J_{r,x}$ the $\sigma$-stable subset
$\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})$.

######  Lemma 3.9.

$x$ is a $(J_{r,x},v_{x}r^{-1})_{\sigma}$-alcove.

###### Proof.

The first condition of definition 2.8 can be checked easily because the finite
part is $rv_{x}^{-1}v_{x}w_{x}\sigma(v_{x}r^{-1})\in W_{J_{x,r}}$. For
definition 2.8 (2), we need to compare
$k(v_{x}r^{-1}\alpha,v_{x}t^{\mu_{x}}w_{x})$ and
$k(v_{x}r^{-1}\alpha,\operatorname{id})$ for all
$\alpha\in\Phi^{+}\setminus\Phi_{J_{x,r}}^{+}$. The first one is
$k(r^{-1}\alpha,t^{\mu_{x}}w_{x})$ which is $\geq 0$ if
$r^{-1}\alpha\in\Phi^{+}$ and $0\geq k(v_{x}r^{-1}\alpha,\operatorname{id})$
always. Hence, we only need to consider the case $r^{-1}\alpha\in\Phi^{-}$.
However, $r(\Phi^{+}\setminus\Phi_{x})\subset\Phi^{+}$ implies (again by the
negation complement) $r^{-1}\Phi^{+}\subset\Phi^{+}\cup-\Phi_{x}$. Therefore,
we have $r^{-1}\alpha\in-\Phi_{x}$. By the definition of $\Phi_{x}$, we have
the same $k$-values. ∎

######  Proposition 3.10.

Let $x\in\widetilde{W}$ be an element with $\ell(x)\gg 0$. If
$J_{r,x}\neq\mathbb{S}$ for some $r\in W_{x}$, then $X_{x}(b)=\emptyset$.

###### Proof.

By the previous lemma, $x$ is a $(J_{r,x},vr^{-1})_{\sigma}$-alcove so [GHN15,
Proposition 3.6.4] tells us that

$\nu_{r\cdot\mu_{x}}\in Q_{J_{r,x},\mathbb{Q}}^{\vee}.$

Note that, inductively computing, when $r=s_{i_{m}}\cdots s_{i_{1}}$,

$r\cdot\mu_{x}=\mu_{x}-\sum_{j=1}^{k}\langle s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}},\mu_{x}\rangle\alpha_{i_{j}}^{\vee}$ (4)

so that

$\nu_{r\cdot\mu_{x}}=\nu_{\mu_{x}}-\nu_{\langle\alpha_{1},\mu_{x}\rangle\alpha_{1}^{\vee}+\cdots+\langle
s_{1}\cdots s_{k-1}\alpha_{k},\mu_{x}\rangle\alpha_{k}^{\vee}}.$

Assume that the expression of $r$ is reduced, then $s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}}\in\Phi^{+}$. Moreover, $rs_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}}\in\Phi^{-}$. Hence, $s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}}\in\Phi^{+}\cap r^{-1}\Phi^{-}\subset\Phi_{x}$
implying that $\langle s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}},\mu_{x}\rangle=0$ or $1$. Therefore, the
$\sigma$-average of the sum of coroots on the right hand side is bounded.
However, for $x$ such that $\ell(x)\gg 0$, if
$\mu_{x}=\sum_{i}c_{i}\varpi_{i}^{\vee}$, then $\sum_{i}c_{i}\gg 0$ and, by
lemma 2.18, the coefficients in the linear combination of coroots are large
enough so that $\nu_{\mu_{x}}-\nu_{\sum_{j=1}^{k}\langle s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}},\mu_{x}\rangle\alpha_{i_{j}}^{\vee}}$ has all
positive coefficients. This is a contradiction. ∎

Combining proposition 3.10 and corollary 3.8, we get

######  Theorem 3.11.

Let $b\in B(G)$ be basic and suppose that $x\in\widetilde{W}$ satisfies
$\kappa(b)=\kappa(x)$ and $\ell(x)\gg 0$. Then

$X_{x}(b)\neq\emptyset$ if and only if
$\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})=\mathbb{S}$
for all $r\in W_{x}$.

###### Proof of Theorem 1.2.

Note that $\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$
condition in theorem 1.2 (1) is satisfied for $x$ such that $\ell(x)\gg 0$. As
there are only finitely many $x$ with $\ell(x)$ bounded by some number,
conjecture 1 holds.∎

###### Proof of Theorem 1.2 (2).

If $x=v_{x}t^{\mu_{x}}w_{x}$ is a translation element, we have
$w_{x}=v_{x}^{-1}$. In this case, $\Phi_{x}$ consists of $\alpha\in\Phi^{+}$
such that
$0\leq\langle\alpha,\mu_{x}\rangle+\delta_{v_{x}\alpha}=\delta_{v_{x}\alpha}$
so that $\langle\alpha,\mu_{x}\rangle=0$ and $v_{x}\alpha\in\Phi^{+}$ for all
$\alpha\in\Phi_{x}$. For $x=v_{x}t^{\mu_{x}}$, we have
$w_{x}=\operatorname{id}$ so that $\alpha\in\Phi_{x}$ satisfies
$\langle\alpha,\mu_{x}\rangle+\delta_{\alpha}=\delta_{v_{x}\alpha}$ so that
$\langle\alpha,\mu_{x}\rangle=0$ and $v_{x}\alpha\in\Phi^{+}$ as well.

Hence, in both cases, we do not have $\alpha\in\Phi_{x}$ such that
$\langle\alpha,\mu_{x}\rangle=1$. Therefore, in the proof of proposition 3.10,
all the terms $\langle s_{i_{1}}\cdots
s_{i_{j-1}}\alpha_{i_{j}},\mu_{x}\rangle\alpha_{i_{j}}^{\vee}$ are zeros.
Therefore, $\nu_{r\cdot\mu_{x}}$ is dominant still so that
$J_{r,x}\neq\mathbb{S}$ is a contradiction assuming $X_{x}(b)\neq\emptyset$. ∎

###### Proof of Theorem 1.3.

By lemma 3.7 and lemma 3.9, $x$ is a $(J,w)_{\sigma}$-alcove if and only if
$J=J_{r,x}$. Theorem 3.11 implies that, for $\ell(x)\gg 0$,
$X_{x}(b)\neq\emptyset$ if and only if $J_{r,x}=\mathbb{S}$ for all $r\in
W_{x}$.∎

##  4\. One critical strip

###  4.1. Finding a simple root

We will prove the following:

######  Theorem 4.1.

Let $b\in B(G)$ be basic and $x\in\widetilde{W}$ lie in exactly one critical
strip $C_{a}$ for some $a\in\Phi$. Then $v_{x}^{-1}a$ is a simple root and
$X_{x}(b)\neq\emptyset$ if and only if $\kappa_{G}(b)=\kappa_{G}(x)$ and

both $\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))$ and
$\operatorname{supp}_{\sigma}(\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x})$ are
$\mathbb{S}$,

where $s_{x}$ is the simple reflection corresponding to $v_{x}^{-1}a$.

This is slightly stronger than Theorem 1.2 (1). More precisely, compared to
Theorem 1.2 (1), Theorem 4.1 has additional claims that $v_{x}^{-1}a$ is a
simple root and has no restriction
‘$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$’. In
fact, one can easily show that $v_{x}^{-1}a$ must be a simple root (lemma 4.2)
from proposition 3.4. The absence of the restriction
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$ is the
main reason why Theorem 4.1 is stronger. It is explained in lemma 4.6.

######  Lemma 4.2.

Suppose that $x$ belongs to exactly one critical strip associated to $a$.
Then, $v_{x}^{-1}a$ is a simple root.

###### Proof.

By proposition 3.4, $\Phi_{x}$ is a singleton such that
$\Phi^{+}\setminus\Phi_{x}$ is closed. As simple roots generate $\Phi^{+}$, we
know $\Phi_{x}$ should contain a simple root.∎

We will denote by $\alpha_{x}$ the unique positive simple root such that $x\in
C_{v_{x}\alpha_{x}}$.

###  4.2. Lemma 2.12 for one strip

Here, we prove that if
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))\neq\mathbb{S}$ or
$\operatorname{supp}_{\sigma}(s_{x}\eta_{\sigma}(x)\sigma(s_{x}))\neq\mathbb{S}$,
then $X_{x}(b)=\emptyset$.

######  Proposition 4.3.

Under the assumption of theorem 4.1, if
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))\neq\mathbb{S}$ then
$X_{x}(b)=\emptyset$.

###### Proof.

By Lemma 2.12, it is enough to show that $\nu_{\mu_{x}}\neq\bar{\nu}_{b}$. We
follow the strategy of [GHN15, Proposition 3.6.5] here.

Let $n$ be the order of $W_{0}\rtimes\langle\sigma\rangle$ and suppose that
$\nu_{\mu_{x}}$ is central. Then
$0=\langle\nu_{\mu_{x}},\beta\rangle=\frac{1}{n}\langle\mu_{x}+\sigma(\mu_{x})+\cdots+\sigma^{n-1}(\mu_{x}),\beta\rangle$,
but $\sigma^{i}(\mu_{x})$ are all dominant. Hence, if $\beta$ is positive then
$\langle\mu_{x},\beta\rangle=0$. Consider the case $\beta$ being the maximal
root, we get $\mu_{x}$ central. This implies that $w_{x}=\operatorname{id}$
and $x=v_{x}t^{\mu_{x}}$. Note that $\eta_{\sigma}(x)=v_{x}$.

As we assume that there is only one critical strip containing $x$, it means
that the number of $a\in\Phi^{+}$ such that
$k(a,v_{x}t^{\mu_{x}})=k(a,\operatorname{id})=0$ is $1$. But,
$k(a,v_{x}t^{\mu_{x}})=\delta_{v_{x}^{-1}a}$. It means that $|\Phi^{+}\cap
v_{x}\Phi^{+}|=|\Phi^{+}|-\ell(v_{x})=1$ so that $\ell(v_{x})=|\Phi^{+}|-1$.
However, if
$\ell(v_{x})\leq|\Phi_{\operatorname{supp}_{\sigma}(v_{x})}^{+}|<|\Phi^{+}|-1$.
∎

Now we consider the case
$\operatorname{supp}_{\sigma}(\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x})\neq\mathbb{S}$.

######  Lemma 4.4.

Under the assumption of theorem 4.1, denote by $J_{x}$ the set
$\operatorname{supp}_{\sigma}(\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x})$. Then
$x$ is a $(J_{x},v_{x}s_{x})_{\sigma}$-alcove.

###### Proof.

We have $\Phi_{x}=\\{\alpha_{x}\\}$, so $s_{x}\in W_{x}$ because
$s_{x}\Phi^{+}=\Phi^{+}\cup\\{-\alpha_{x}\\}\setminus\\{\alpha_{x}\\}$. Now,
the conclusion follows from lemma 3.9. ∎

######  Proposition 4.5.

If $J_{x}\neq\mathbb{S}$ in lemma 4.4, then $X_{x}(b)=\emptyset$.

###### Proof.

$x$ is a $(J_{x},v_{x}s_{x})_{\sigma}$-alcove. Hence, it is enough to prove
that

$\nu_{\lambda^{\prime}}-\overline{\nu}_{b}\in
Q_{J_{x}}^{\vee}\otimes\mathbb{Q}$

leads to a contradiction where $\lambda^{\prime}\colonequals s_{x}\mu_{x}$
following the proof of [GHN15, Proposition 3.6.4]. However,
$\lambda^{\prime}=s_{x}\mu_{x}$ is either $\mu_{x}-\alpha_{x}^{\vee}$ or
$\mu_{x}$. The latter case is already proved131313Note that, as mentioned
before, the proof in loc.cit. is incorrect when referring to ‘Section 3.5 (2)’
and you need to use lemma 2.17 instead. in loc.cit..

Now, let us consider the case $s_{x}\mu_{x}=\mu_{x}-\alpha_{x}^{\vee}$. We
have

$\nu_{s_{x}\mu_{x}}-\bar{\nu}_{b}\in Q^{\vee}_{J_{x},\mathbb{Q}}.$

Let $\varpi_{i}^{\vee}$ be the fundamental coweights and $\varpi_{i}$ be the
fundamental weights. For simplicity, denote by $\mathcal{O}$ the
$\sigma$-orbit of $s_{x}$ and by $\alpha_{\mathcal{O}}$ (resp.
$\varpi_{\mathcal{O}}$ or $\varpi_{\mathcal{O}}^{\vee}$) the sum of the
elements in the $\sigma$-orbit of $\alpha_{x}$ (resp. $\varpi_{\mathcal{O}}$
or $\varpi_{\mathcal{O}}^{\vee}$).

Taking the inner product with $\varpi_{\mathcal{O}}$, we get $0$ on the right
hand side. On the left hand side, we have
$\langle\varpi_{\mathcal{O}},s_{x}\mu_{x}\rangle$. It is $\langle
s_{x}\varpi_{\mathcal{O}},\mu_{x}\rangle=\langle\varpi_{\mathcal{O}}-\alpha_{x},\mu_{x}\rangle=\langle\varpi_{\mathcal{O}},\mu_{x}\rangle-1$.
However, if $\mu_{x}=\sum c_{i}\varpi_{i}^{\vee}$ where
$c_{i}\in\mathbb{Z}_{\geq 0}$.

Let $D$ be the connected component containing $\alpha_{x}$. If we restrict to
$D$ of the value $\langle\varpi_{\mathcal{O}},\mu_{x}\rangle$, we have
$\langle\varpi_{x},\varpi_{x}^{\vee}+\sum_{i\neq
x}c_{i}\varpi_{i}^{\vee}\rangle$. Note that
$\langle\varpi_{i},\varpi_{j}^{\vee}\rangle$ is the $(i,j)$-entry of the
inverse of the Cartan matrix of $D$ which are all positive. In most cases,
this already exceeds $1$ by [OV90, Reference Chapter. §2. Table 2]. Note that
$\langle\varpi_{x},\varpi_{x}^{\vee}\rangle$ is the corresponding entry on the
diagonal of the inverse matrix.

There are two cases left in large:
$\langle\varpi_{x},\varpi_{x}^{\vee}\rangle=1$ or
$\langle\varpi_{x},\varpi_{x}^{\vee}\rangle<1$. The first case is when $x$ is
the unique vertex of degree=1 (with a single edge) in $B_{\geq}2$,
$C_{\geq}2$, $D_{\geq}4$ or the middle vertex of $A_{3}=D_{3}$. The second
case is when $x$ is one of two degree=1 vertices in $A_{n}$. As they contain a
little tedious computation, we put it off to section 5.1. ∎

Combining proposition 4.3 and proposition 4.5, we get

$X_{x}(b)\neq\emptyset\text{ only if
}\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\operatorname{supp}_{\sigma}(\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x})=\mathbb{S}.$

###### Proof of theorem 4.1.

Note again that $\Phi_{x}=\\{\alpha_{x}\\}$ so that
$W_{x}=\\{\operatorname{id}_{W_{0}},s_{x}\\}$. Now, combine corollary 3.8 and
section 4.2. ∎

###  4.3. On the condition
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$

Finally, the following lemma explains why the assumption
$\widetilde{\operatorname{supp}_{\sigma}}(x)=\widetilde{\mathbb{S}}$ is not
necessary in Theorem 1.2 (1). Note that
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\widetilde{\mathbb{S}}$
implies that $\nu_{x}$ is central.

######  Lemma 4.6.

Let $x$ be in exactly one critical strip and suppose $\nu_{x}$ is central.
Then both $\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))$ and
$\operatorname{supp}_{\sigma}(\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x})$ are
$\mathbb{S}$.

###### Proof.

Note that $\nu_{x}=v_{x}\cdot\nu_{v_{x}^{-1}x\sigma(v_{x})}$ and
$v_{x}^{-1}x\sigma(v_{x})=t^{\mu_{x}}w_{x}\sigma(v_{x})=t^{\mu_{x}}\sigma(\eta_{\sigma}(x))$141414Caution.
This is not necessarily in the dominant Weyl chamber.. Hence,
$\nu_{t^{\mu_{x}}\sigma(\eta_{\sigma}(x))}$ is also central. On the other
hand, similarly to corollary 2.16, we have
$\nu_{\mu_{x}}-\nu_{t^{\mu_{x}}\sigma(\eta_{\sigma}(x))}\in
Q_{\operatorname{supp}_{\sigma}(\sigma(\eta_{\sigma}(x))),\mathbb{Q}}^{\vee}$
by applying lemma 2.15. As $\mu_{x}$ is dominant, $\nu_{\mu_{x}}$ is dominant.
If $\mu_{x}$ is non-central, then $\nu_{\mu_{x}}$ is also non-central and so
lemma 2.17 enforces
$\operatorname{supp}_{\sigma}(\sigma(\eta_{\sigma}(x)))=\mathbb{S}$. Similarly
as above, we have

$\nu_{s_{x}\cdot\mu_{x}}-\nu_{t^{s_{x}\cdot\mu_{x}}s_{x}w_{x}\sigma(v_{x})\sigma(s_{x})}\in
Q_{\operatorname{supp}_{\sigma}(s_{x}\sigma(\eta_{\sigma}(x))\sigma(s_{x})),\mathbb{Q}}^{\vee}.$

However,
$t^{s_{x}\cdot\mu_{x}}s_{x}w_{x}\sigma(v_{x})\sigma(s_{x})=(v_{x}s_{x})^{-1}x\sigma(v_{x}s_{x})$,
but $\nu_{x}$ is central so the second term is central. We can now repeat the
proof of proposition 4.5.

When $\mu_{x}$ is central, we have $x=v_{x}t^{\mu_{x}}$. We can repeat
proposition 4.3 to show that
$\operatorname{supp}_{\sigma}(\eta_{\sigma}(x))=\mathbb{S}$. Moreover, the one
critical strip assumption tells us that $w_{0}v_{x}^{-1}$ is a simple
reflection $s$ (corresponding to $\alpha\in\Phi^{+}$). Then,
$\alpha_{x}=v_{x}^{-1}\alpha$ and so $v_{x}\alpha_{x}\in\Phi^{+}$, that is,
$\ell(v_{x}s_{x})=\ell(v_{x})+1=\ell(w_{0})$. Hence, $v_{x}=w_{0}s_{x}$ and
$\sigma^{-1}(s_{x})\eta_{\sigma}(x)s_{x}=\sigma^{-1}(s_{x})w_{0}$ whose
support is $\mathbb{S}$. ∎

##  5\. Some computations and applications

In this section, we show computations for possibly exceptional cases
(mentioned in proposition 4.5) when $x$ belongs to exactly one critical strip.
Moreover, we have some remarks on type $A_{n}$ case and we prove Theorem 1.4.

###  5.1. Completion of the proof for the one critical strip case

###### Proof of proposition 4.5 (cont’d).

We separate into two cases. (Note that we use [OV90, Reference Chapter. §2.
Table 2] here.)

Case 1-1. $x$ is the vertex of degree $1$ (with a single edge) in $B_{\neq
2},C_{\neq 2},D_{\geq 4}$.

$\langle\varpi_{x},\varpi_{x}^{\vee}\rangle$ is already $1$. So, $c_{i}$’s are
all zero, that is, $\mu_{x}=\varpi_{x}^{\vee}$. For simplicity, we use the
notation $*_{1}$ instead of $*_{x}$ where $*=s,\alpha,\varpi^{\vee}$ in this
paragraph. For any positive root $\alpha$ with no support at $\alpha_{1}$, we
have $x\not\in C_{v_{x}\alpha}$ and
$\langle\alpha,\varpi_{1}^{\vee}\rangle=0$. So we have
$v_{x}\alpha\in\Phi^{-}$ and $w_{x}^{-1}\alpha\in\Phi^{+}$ by lemma 2.5. As
$x\in C_{v_{x}\alpha_{1}}$ and $\langle\alpha_{1},\varpi_{1}^{\vee}\rangle=1$,
we have $v_{x}\alpha_{1}\in\Phi^{+}$ and $w_{x}^{-1}\alpha_{1}\in\Phi^{-}$.
Note that $s_{2}\alpha_{1}=\alpha_{1}+\alpha_{2}$ is different from
$\alpha_{1}$ and in $\Phi^{+}$. Moreover, $\langle
s_{2}\alpha_{1},\varpi_{1}^{\vee}\rangle=1$. Due to the uniqueness assumption
of critical strips, either $v_{x}(s_{2}\alpha_{1})\in\Phi^{-}$ or
$w_{x}^{-1}(s_{2}\alpha_{1})\in\Phi^{+}$ by lemma 2.5.

  1. (1)

The latter case: $w_{x}^{-1}(\alpha_{1}+\alpha_{2})\in\Phi^{+}$ so that
$w_{x}^{-1}s_{1}\alpha\in\Phi^{+}$ for all positive simple roots $\alpha$.
Hence, $w_{x}=s_{1}$ so that $\sigma^{-1}(s_{1})\eta_{\sigma}(x)s_{1}=vs_{1}$.
As $J_{x}=\mathbb{S}\setminus{\mathcal{O}}$, $v_{x}\in
W_{\mathbb{S}\setminus{\mathcal{O}}}s_{1}$. However, $\Phi^{+}\ni
v_{x}\alpha_{1}\in
W_{\mathbb{S}\setminus{\mathcal{O}}}(-\alpha_{1})\subset\Phi^{-}$ as
$1\in{\mathcal{O}}$. Contradiction.

  2. (2)

The first case: Apply the same argument to $w_{0}v_{x}$ instead of
$w_{x}^{-1}$ and get $v_{x}=w_{0}s_{1}$. Then
$\sigma^{-1}(s_{1})\eta_{\sigma}(x)s_{1}=\sigma^{-1}(s_{1}w_{x})w_{0}$.
Consider $\sigma(w_{0})w_{x}^{-1}s_{1}\alpha_{1}$ which is in $\Phi^{-}$ by
above, but $\sigma(w_{0})w_{x}^{-1}s_{1}\in
W_{\mathbb{S}\setminus{\mathcal{O}}}$ so it should be a positive root which is
a contradiction.

Case 1-2. $x$ is the middle point in $A_{3}$.

$\mu_{x}=\varpi_{2}^{\vee}$. Similarly as before, $v_{x}\alpha_{2}\in\Phi^{+}$
and $w_{x}^{-1}\alpha_{2}\in\Phi^{-}$. It means that $w_{x}\in
s_{2}W_{\\{1,3\\}}$ or $w_{x}=s_{2}s_{1}s_{3}s_{2}$. Hence, $s_{2}w_{x}\in
W_{\\{1,3\\}}$ or $s_{2}w_{x}=s_{1}s_{3}s_{2}$ and
$\sigma^{-1}(s_{2})\eta_{\sigma}(x)s_{2}=\sigma^{-1}(s_{2}w_{x})v_{x}s_{2}$.
As it belongs to $W_{\mathbb{S}\setminus{\mathcal{O}}}$, we have $v_{x}\in
W_{\mathbb{S}\setminus{\mathcal{O}}}s_{2}$ so that
$v_{x}\alpha_{2}\in\Phi^{-}$ which is a contradiction or $v_{x}\in
s_{2}W_{\mathbb{S}\setminus{\mathcal{O}}}s_{2}$. In the latter case, note that
$v_{x}$ is supported at $s_{1}$ because $v_{x}\alpha_{1}\in\Phi^{-}$ and so
$v_{x}\alpha_{2}\in\Phi^{-}$ from $v_{x}\in
s_{2}W_{\mathbb{S}\setminus{\mathcal{O}}}s_{2}$.

Case 1-3. $x$ is the vertex with an outward arrow in $B_{2}=C_{2}$.

May assume $B_{2}$ and $\mu_{x}=\varpi_{1}^{\vee}$. This is same as Case 1-1
as $s_{2}\alpha_{1}=\alpha_{1}+\alpha_{2}$.

Case 2. $x$ is a vertex of degree $1$ in $A_{n}$.

As the sum is $1$, $D\cap{\mathcal{O}}=\\{x\\}$ necessarily and
$\mu_{x}=\varpi_{x}^{\vee}+\varpi_{y}^{\vee}$ where $y$ is a vertex (possibly
in another connected component) of degree $1$ not in ${\mathcal{O}}$.

For simplicity, we will call $D^{\prime}$ the connected component containing
$\varpi_{y}^{\vee}$ and use $\varpi_{1,D}^{\vee}$ instead of
$\varpi_{x}^{\vee}$ and $\varpi_{n,D^{\prime}}^{\vee}$ instead of
$\varpi_{y}^{\vee}$. Moreover, $\sigma$ does not shuffle $1$ and $n$ in
(distinct) connected components.

  1. (1)

The case $D\neq D^{\prime}$: The proof is exactly the same as Case 1-1.

  2. (2)

The case $D=D^{\prime}$: $\mu_{x}=\varpi_{1}^{\vee}+\varpi_{n}^{\vee}$ in the
same connected component. The proof is similar. In a similar way, we have
$v_{x}\alpha\in\Phi^{-}$ and $w_{x}^{-1}\alpha\in\Phi^{+}$ for $\alpha$ not
supported at both $\alpha_{1}$ and $\alpha_{n}$. For $\alpha=\alpha_{1}$, we
have the opposite situation. For the remaining case, it should not be the
opposite situation.

The method is essentially the same. Note that the goal in the above was to
show that $w_{x}=s_{1}$ or $v_{x}=w_{0}s_{1}$. Here, the goal is to show that
$w_{x}=s_{1}\cdots s_{m}$ for some $m\leq n$ or $v_{x}=w_{0}s_{m}\cdots s_{1}$
for some $m\leq n$. Using that the roots are of the form
$\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j}$, one can prove that. We skip the
proof.

∎

###  5.2. Sharper bounds for Theorem 1.2 (3)

######  Lemma 5.1.

Let $w$ be an element in $W_{0}(A_{n})$ with $\operatorname{supp}(w)$
connected. Assume that there exists a unique simple refleciton $s_{m}$ such
that $\ell(ws_{m})<\ell(w)$. Then,
$w\alpha_{m}=-\sum_{k\in\operatorname{supp}(w)}\alpha_{k}$, the highest root
supported at $\operatorname{supp}(w)$.

###### Proof.

We will prove $\alpha_{\operatorname{supp}(w)}\not\in w\Phi^{+}\cup w\Phi^{-}$
which is a contradiction.

For $\alpha\in\Phi^{+}$ supported at $m$, $w\alpha\geq
w\alpha_{m}\gneq-\alpha_{\operatorname{supp}(w)}$. For $\alpha\in\Phi^{+}$ not
supported at $m$, $w\alpha$ is a positive root. Hence,
$w\Phi^{+}\not\ni-\alpha_{\operatorname{supp}(w)}$, or equivalently,
$\alpha_{\operatorname{supp}(w)}\not\in w\Phi^{-}$.

Let $J_{l}$ and $J_{r}$ to be two connected components of
$\operatorname{supp}(w)\setminus\\{m\\}$ (one of them could be empty). Then,
there are only three possibilities for $\alpha$ such that
$w\alpha=\alpha_{\operatorname{supp}(w)}$: $\alpha_{J_{l}}$ or
$\alpha_{J_{r}}$ or $\alpha_{\operatorname{supp}(w)}$. This is because
$w\alpha_{i}$’s are all positive except for $m$ so that adding $w\alpha_{i}$
for $i\neq m$ gives a higher root.

In the case of $A_{n}$, the Dynkin diagram is symmetric, so we may assume that
$w^{-1}\alpha_{\operatorname{supp}(w)}=\alpha_{J_{l}}$ or
$\alpha_{\operatorname{supp}(w)}$. Now consider the vertex $j_{l}$ adjacent to
$J_{l}$ which is not $m$. Then, $w\alpha_{j_{l}}$ is a positive root
containing $\alpha_{j_{l}}$ which becomes a root once added to
$w\alpha_{J_{l}}$ or $w\alpha_{\operatorname{supp}(w)}$ which is
$\alpha_{\operatorname{supp}(w)}$. Noting that $j_{l}$ is the only vertex not
equal to $m$ which is adjacent to $J_{l}$, we can conclude that
$w\alpha_{j_{l}}=\alpha_{j_{l}}$. Hence, $w$ fixes all the $\alpha_{i}$’s for
$i\leq j_{l}$ implying that (by the fact below) $\operatorname{supp}(w)$ does
not contain $j_{l}+1\in J_{l}$ which is a contradiction.

The fact aforementioned is:

> If $w\in W_{0}(A_{n})$ satisfies $w\alpha_{i}=\alpha_{i}$ for all $1\leq
> i\leq m$ for $m\geq 1$. Then $i\not\in\operatorname{supp}(w)$ for all $1\leq
> i\leq m+1$.

∎

Denote by $Q^{\vee}_{>0}$ the set of positive $\mathbb{Q}$-linear sums of all
simple coroots. Note that $\nu_{\mu_{x}}\in Q^{\vee}_{>0}$ unless $\mu_{x}$ is
central.

Suppose that $r\in W_{x}$ and $s$ is a simple reflection such that $sr\in
W_{x}$ and $sr>r$. Considering eq. 4, we know that
$sr\cdot\mu=r\cdot\mu-\langle r^{-1}\alpha,\mu\rangle\alpha^{\vee}$ where
$\alpha$ is the simple positive root corresponding to $s$. Moreover, as $sr\in
W_{x}$, $\langle r^{-1}\alpha,\mu\rangle=0$ or $1$. Hence, as long as $sr\in
W_{x}$ (assuming $sr>r$), we know that $sr\cdot\mu$ is the same as $r\cdot\mu$
or $r\cdot\mu-\alpha^{\vee}$.

Therefore, if $\nu_{r^{\prime}\cdot\mu_{x}}\in Q^{\vee}_{J,\mathbb{Q}}$ for
some $J\subsetneq\mathbb{S}$ and $r^{\prime}\in W_{x}$, we can find a minimal
$r_{0}\in W_{0}$ such that
$\ell(r^{\prime}r_{0}^{-1})+\ell(r_{0})=\ell(r^{\prime})$ (so that $r_{0}\in
W_{x}$ by lemma 3.6) and $r_{0}\cdot\mu\not\in Q^{\vee}_{>0}$ in the sense
that $sr_{0}\cdot\mu\in Q^{\vee}_{>0}$ for any simple reflection $s$ such that
$sr_{0}<r_{0}$. Note that it is not necessarily unique and we make any choice.

The minimality assumption tells us that any reduced expression
$s_{i_{k}}\cdots s_{i_{1}}$ of $r_{0}$ has the same $s_{i_{k}}$ (the first
simple reflection should be constant). Moreover, $\langle s_{i_{1}}\cdots
s_{i_{k-1}}\alpha_{i_{k}},\mu_{x}\rangle=1$ considering eq. 4. Finally,
$\operatorname{supp}(r_{0})$ should be connected. Now, the precise statement
of Theorem 1.2 (3) is the following. By lemma 5.1, this implies that
$r_{0}^{-1}\alpha_{i_{k}}=-\alpha_{\operatorname{supp}(r_{0})}$ and so
$\langle\alpha_{\operatorname{supp}(r_{0})},\mu_{x}\rangle=1$.

######  Proposition 5.2.

In type $A_{n}$, suppose that $\mu_{x}\in Q^{\vee}$ and assume
$\langle\varpi_{1},\mu_{x}\rangle>1$, $\langle\varpi_{n},\mu_{x}\rangle>1$.
Then Conjecture 1 holds.

###### Proof.

Let $r_{0}$ be a minimal element chosen before and $m$ be the unique number
such that $r_{0}\alpha_{m}<0$. Similar to proposition 3.10, it is enough to
show that $r_{0}\cdot\mu_{x}\in Q^{\vee}_{\neq\mathbb{S},\mathbb{Q}}$ is a
contradiction. Note that $\mu_{x}\in Q^{\vee}$, so we can assume
$r_{0}\cdot\mu_{x}\in Q_{\neq\mathbb{S}}^{\vee}$. Due to the minimality of
$r_{0}$, we have that
$\mathbb{S}\setminus\operatorname{supp}(r_{0}\cdot\mu_{x})=\\{m\\}$. Moreover,
for each $i\in\operatorname{supp}(r_{0}\cdot\mu_{x})$,
$\alpha_{i}^{\vee}$-coefficient of $r_{0}\cdot\mu_{x}$ is a positive integer.
Therefore, if $m\neq 1$ or $n$, we have
$\langle\alpha_{m},r_{0}\cdot\mu_{x}\rangle\leq-2$. However,
$r_{0}^{-1}\alpha_{m}\in-\Phi_{x}$ so that $\langle
r_{0}^{-1}\alpha_{m},\mu_{x}\rangle=-1$ which is a contradiction.

When $m=1$ (resp. $n$), note that $r_{0}\cdot\mu_{x}\in Q_{J}^{\vee}$ (where
$J=\mathbb{S}\setminus\\{m\\}$) is weakly dominant in the sense of [Nie18,
Proposition 3.1]. Therefore, $r_{0}\cdot\mu$ and $\mu$ fits into the setting
in [Nie18, Lemma 5.9] so that $\mu-r_{0}\cdot\mu$ is the sum of some positive
roots orthogonal to each other. However, no two positive roots containing
$\alpha_{1}$ (resp. $\alpha_{n}$) can be orthogonal. This implies that the
$\alpha_{1}^{\vee}$(resp. $\alpha_{n}^{\vee}$)-coefficient of $\mu$ is either
that of $r_{0}\cdot\mu$ (which is $0$) or one larger than that. So it must be
$1$. This contradicts to the assumption. ∎

We remark that the above proof works for $D_{n}$ and $E_{n}$ types but need
one more assumption that $\langle\varpi_{n-1},\mu_{x}\rangle>1$ where $n-1$ is
the vertex of degree $1$ other than the vertices $1$ and $n$.

###  5.3. Application: Cordial elements and generic $\sigma$-conjugacy class

We will summarize some related facts first. For more details, we refer to
[MV20].

Given $x\in\widetilde{W}$, let $B(G)_{x}$ be the set of $[b]\in B(G)$ such
that $I\dot{x}I\cap[b]\neq\emptyset$. Then, $B(G)_{x}$ contains a unique
maximal element called generic $\sigma$-conjugacy class and denoted by
$[b_{x}]$.

######  Definition 5.3 (Cordial element).

$x\in\widetilde{W}$ is called cordial if

$\ell(x)-\ell(\eta_{\sigma}(x))=\langle
2\rho,\nu_{x}\rangle-\operatorname{def}(b_{x}),$

where $2\rho$ is the sum of all positive coroots and
$\operatorname{def}(b_{x})$ is the difference between the rank of $G$ and
$J_{b_{x}}$, the $\sigma$-centralizer of $b_{x}$ in $\breve{G}$.

As an example, $x$ in the antidominant Weyl chamber is cordial by loc.cit.
Theorem 1.2.

######  Lemma 5.4.

Let $x$ be a cordial element. Then, $B(G)_{x}$ is saturated in the following
sense:

> Suppose that $[b_{1}]$, $[b_{2}]\in B(G)_{x}$ satisfy $[b_{1}]\leq[b_{2}]$.
> Then, for any $[b]\in B(G)$ satisfying $[b_{1}]\leq[b]\leq[b_{2}]$, we have
> $[b]\in B(G)_{x}$.

Here, $\leq$ is the partial ordering defined in $B(G)$.

Hence, if the minimal and maximal elements are known, we have the full
description of $B(G)_{x}$ which implies the complete classification of the
nonemptiness of $X_{x}(b)$ for a fixed $x$.

###### Proof of Theorem 1.4.

By [He21, Proposition 4.2], $x=vt^{\mu}$ for a dominant $\mu$ is cordial and
the generic $\sigma$-conjugacy class is $[t^{\mu}]$. By lemma 5.4, we only
need to describe the minimal element in $B(G)_{x}$.

For a central $\mu$, we note that $[t^{\mu}]$ is the minimal among the
elements whose image under the Kottwitz map is $\kappa_{G}(t^{\mu})$. As it is
minimal and maximal at the same time, $B(G)_{x}=\\{[t^{\mu}]\\}=B(G,\mu)$.

Let $\mu$ be non-central. We proved
$\Phi_{x}=\\{\alpha\in\Phi^{+}:\langle\alpha,\mu\rangle=0\text{ and
}v\alpha\in\Phi^{+}\\}$ in the last part of section 3.4. Hence, $\mu$ is fixed
by the reflection with respect to $H_{\alpha}$ for $\alpha\in\Phi_{x}$. Now,
we get

$\displaystyle W_{x}$ $\displaystyle=\\{r=s_{m}\cdots s_{1}\in
W_{0}:\alpha_{1},s_{1}\alpha_{2},\cdots,s_{1}\cdots
s_{m-1}\alpha_{m}\in\Phi_{x}\\}$ $\displaystyle=\\{r\in
W_{0}:\operatorname{supp}(r)\subset W_{0}(\mu)\text{ and
}v\alpha_{1},\cdots,vs_{1}\cdots s_{m-1}\alpha_{m}\in\Phi^{+}\\}$
$\displaystyle=\\{r\in W_{0}(\mu):\ell(vr^{-1})=\ell(v)+\ell(r)\\}$

Now, by Theorem 1.2 (2), $X_{x}(b_{\text{b}})\neq\emptyset$ for the basic
element $b_{\text{b}}$ satisfying $\kappa_{G}(x)=\kappa_{G}(b_{\text{b}})$ if
$\operatorname{supp}_{\sigma}(\sigma^{-1}(r)\eta_{\sigma}(x)r^{-1})=\mathbb{S}$
for all $r\in W_{x}$. In such a case, by lemma 5.4, $B(G)_{x}=\\{[b]\in
B(G):[b_{\text{b}}]\leq[b]\leq[b_{x}]=[t^{\mu}]\\}=\\{[b]\in
B(G):[b]\leq[t^{\mu}]\\}=B(G,\mu)$. ∎

As $\langle\alpha_{\operatorname{supp}(r_{0})},\mu_{x}\rangle=1$,
$\operatorname{supp}(r_{0})\cap\\{1\leq i\leq
n:\langle\alpha_{i},\mu_{x}\rangle\geq 1\\}$ is a singleton, say $\\{m\\}$. By
the assumption that

$i_{k}\in\operatorname{supp}(r_{0})\subset\\{i_{k-1}+1,\cdots,i_{k+1}-1\\}$.
By assumption, we have $r_{0}\cdot\mu_{x}$ has non-positive coefficient for
$\alpha_{k}$.

In the following, we let

$\mu_{x}=\sum_{j=1}^{n}c_{j}\varpi_{j}^{\vee}.$

######  Proposition 5.5.

If

$\max\\{q-p:1\leq p<q\leq n\text{ s.t. }\begin{array}[]{cc}c_{j}=0&\text{ for
}p<j<q\\\ c_{j}\neq 0&\text{ for }j=p,q\end{array}\\}$

is less than $\langle\varpi_{1},\mu_{x}\rangle$ and
$\langle\varpi_{n},\mu_{x}\rangle$ then Conjecture 1 holds.

###### Proof.

$\langle\varpi_{m},\mu_{x}\rangle>\langle\varpi_{m},\mu_{x}-r_{0}\cdot\mu_{x}\rangle$.

$\sum_{j=1}^{k-1}c_{i_{j}}i_{j}(n+1-m)+\sum_{j=k+1}c_{i_{j}}m(n+1-i_{j})=(\sum_{j=1}^{k-1}c_{i_{j}}i_{j}+\min\\{m,i_{k}\\}+\sum_{j=k+1}^{N-1}c_{i_{j}}m)(n+1)-m\sum_{j=1}^{N}c_{i_{j}}i_{j}$.

$\langle\varpi_{m}-r_{0}^{-1}\varpi_{m},\varpi_{a}^{\vee}\rangle\leq\min$.

Note that
$\langle\varpi_{1},\mu_{x}\rangle=\sum_{j=1}^{N}c_{i_{j}}-\frac{\sum_{j=1}^{N}c_{i_{j}}i_{j}}{n+1}$
and
$\langle\varpi_{n},\mu_{x}\rangle=\frac{\sum_{j=1}^{N}c_{i_{j}}i_{j}}{n+1}$ ∎

$\mu_{x}=\sum c_{i}\varpi_{i}^{\vee}$ then $\operatorname{supp}(r_{0})$
contains exactly one $i$ with $c_{i}=1$ and all other $i$’s in
$\operatorname{supp}(r_{0})$ satisfy $c_{i}=0$. Ignoring the indices with
$c_{i}=0$, we may re-write

One can modify these arguments to give some bound for other type of root
systems. However, this method will not take care of those finite many cases
left because we have the following examples:

In $A_{8}$, consider $\varpi_{3}^{\vee}$.

Note that, for $\mu_{x}\in Q^{\vee}$, if
$\widetilde{\operatorname{supp}_{\sigma}}(x)\neq\mathbb{S}_{0}$ then either
$\mu_{x}$ is zero (central) and $\langle\varpi_{1},\mu_{x}\rangle=1$ and
$\langle\varpi_{n},\mu_{x}\rangle=1$.

###### Proof.

Let $r_{\text{min}}$ be the minimal length element of $W_{x}$ such that
$\nu_{r_{\text{min}}\mu_{x}}\in Q_{J,\mathbb{Q}}^{\vee}$ for some
$J\subsetneq\mathbb{S}$ (it is automatically $\sigma$-stable). If
$r_{\text{min}}=s_{k}\cdots s_{1}$ in its reduced expression,

$s_{k}\cdots s_{1}\mu_{x})=s_{k-1}\cdots
s_{1}\mu_{x}-\langle\alpha_{k},s_{k-1}\cdots s_{1}\mu_{x}\rangle=s_{k-1}\cdots
s_{1}\mu_{x}-\langle s_{1}\cdots
s_{k-1}\alpha_{k},\mu_{x}\rangle\alpha_{k}^{\vee}$. By the reducedness of the
expression, $s_{1}\cdots s_{k-1}\alpha_{k}\in\Phi^{+}$ and, on the other hand,
it is $s_{1}\cdots s_{k}(-\alpha_{k})\in r_{\text{min}}^{-1}\Phi^{-}$. As
$r_{\text{min}}\in W_{x}$, we get $s_{1}\cdots s_{k-1}\alpha_{k}\in\Phi_{x}$.
Inductively thinking, we get $r_{\text{min}}$ must start with $s_{k}$ and

Think: We do the $A_{n}$ with coroots $\mu_{x}$ case with $\sigma$ trivial.

Hence,
$\langle\varpi_{\mathcal{O}},r_{\text{min}}\mu_{x}\rangle=\langle\varpi_{\mathcal{O}},\mu_{x}\rangle-\sum\langle\varpi_{\mathcal{O}},\alpha_{i}^{\vee}\rangle\langle
s_{1}\cdots s_{i-1}\alpha_{i},\mu_{x}\rangle$.

How about $\alpha_{\mathcal{O}}$?

$\langle\alpha_{\mathcal{O}},r\mu_{x}\rangle=\langle\alpha_{\mathcal{O}},\mu_{x}\rangle-\sum\langle\alpha_{\mathcal{O}},\alpha_{i}^{\vee}\rangle\langle
s_{1}\cdots s_{i-1}\alpha_{i},\mu_{x}\rangle$.

When $\mu_{x}$ is a coroot, it is very simple because we can consider all
${\mathcal{O}}$ and then LHS$>0$ and so $\langle
2\rho,\mu_{x}\rangle>1+1+\cdots+1$.

Maybe, better to use $\varpi_{1}$.
$\langle\varpi_{1},\varpi_{k}^{\vee}+\text{sth}\rangle>1$

Or, maybe, we can consider when one of the coefficients go $\leq 0$.

The sum part is an integer. Hence, $\mu_{x}=\varpi_{m}^{\vee}+\sum_{i\neq
m}c_{i}\varpi_{i}^{\vee}$ where $c_{i}\in\mathbb{Z}$ implies that
$\langle\varpi_{m},\mu_{x}\rangle=\sum_{i=m}\langle s_{1}\cdots
s_{i-1}\alpha_{i},\mu_{x}\rangle$.

Think: The point is that $\langle\varpi_{\mathcal{O}},\mu_{x}$ has to be an
integer. However, if there are too many $i$’s such that $c_{i}\geq 1$, there
is some bound on $\ell(r)$ or to be precise on the number of
$\alpha\in\Phi_{x}$ such that $\langle\alpha,\mu_{x}\rangle=1$. Hence, the
right hand side cannot be too big so to say.

Think: Or maybe, you can use the fact that if $\eta_{\sigma}(x)t^{\mu_{x}}$
gives the empty set, then $x$ also should be. He Theorem 10.3 maybe? You only
need to compare the $\Phi_{x}$ set. It looks not easy to relate the $W_{x}$
set, but worth trying. The worst case scenario, you can prove this thing for
the easiest case $vt^{\mu}$ form or $vt^{\mu}s$ form.

$r=s_{2}s_{3}s_{4}s_{5}\cdots s_{n}s_{1}s_{2}s_{3}\cdots s_{n-1}$.

$r^{-1}=s_{n-1}\cdots s_{1}s_{n}\cdots s_{2}$.

$s_{321432}=324132=3$

$R_{w^{-1}}=\\{\alpha\in\Phi^{+}:w^{-1}\alpha<0\\}=\Phi^{+}\cap w\Phi^{-}$.

$R_{w}=\Phi^{+}\cap w^{-1}\Phi^{-}=-wR_{w^{-1}}$.

Think: For the roots corresponding to $r$, see [Papi]. $R_{r}\subset\Phi_{x}$
is our condition. Moreover, $R_{r}$ and $\Phi^{+}\setminus R_{r}$ are both
closed. Also, I think we can reduce to the problem where $k$ is the only
vertex with $\varpi_{k}^{\vee}$ alive and bounded by other
$\varpi_{j}^{\vee}$’s so that the case reduces to $A_{n}$ in most cases.
∎

Think: Several conjectures: (1) Let $r\in S_{n}$ be a connected element. Then,
$R_{r^{-1}}$ is contains the highest root supported at the support of $r$. (2)
Assume that $\mu_{x}\in Q^{\vee}$. (3)

We can allow ‘more than one’ critical strips case as well:

##  6\. Suggestions and future works

###  6.1. Applications toward the dimension formula

We would like to mention that in $2$-dimensional root system cases, the above
conjecture can be proven case by case by combining the inductive dimension
formula and our main result as there are only finitely many alcoves that
belong to (more than) two critical strips and we already have the behavior for
them from the figures of [GHKR]’s.

###  6.2. Nonbasic $b$

We would like to point out that the following theorem can be generalized to
the cases where $w$ is not necessarily lying in a shrunken Weyl chamber.

######  Theorem 6.1.

Suppose $w\in\widetilde{W}$ lies in a shrunken Weyl chamber. If
$\kappa(w)=\kappa(b)$, $\lambda_{w}-\nu_{b}$ is a positive combination of
simple coroots, and $\lambda_{w}^{\flat\flat}\geq\nu_{b}$, then we have the
nonemptiness pattern and the dimension formula.

Note that one only needs to give a lower bound for the dimension because $\dim
X_{w}(b)\leq d_{w}(b)$ is known in full generality.

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