# Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

Arghya Sadhukhan  Department of Mathematics, University of Maryland, College
Park, MD 20742<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider affine Deligne-Lusztig varieties $X_{w}(b)$ and
their certain union $X(\mu,b)$ inside the affine flag variety of a reductive
group. Several important results in the study of affine Deligne-Lusztig
varieties have been established under the so-called superregularity
hypothesis. Such results include a description of generic Newton points in
Iwahori double cosets of loop groups, covering relation in associated Iwahori-
Weyl group and dimension formula for $X(\mu,b)$. We show that one can
considerably weaken the superregularity hypothesis and sometimes completely
eliminate it, thus strengthening these existing results.

###### Key words and phrases:

Affine Deligne-Lusztig variety, generic Newton point, affine Weyl group,
dimension formula

###### 2010 Mathematics Subject Classification:

20G25,11G25,20F55

##  1\. Introduction

###  1.1. Motivation

Let $F$ be a nonarchimedean local field and $\breve{F}$ be the completion of
its maximal unramified extension. Let $\sigma$ be the Frobenius automorphism
of $\breve{F}$ over $F$. Let $\mathbf{G}$ be a split connected reductive group
over $F$. Let $\breve{}\mathcal{I}$ be the standard Iwahori subgroup of
$\mathbf{G}(\breve{F})$. Let $w$ be an element in the Iwahori-Weyl group
$\widetilde{W}$ and $[b]$ be a $\sigma$-conjugacy class in $G(\breve{F})$. The
affine Deligne-Lusztig variety associated to this pair $(w,[b])$ is defined to
be a locally closed subscheme of the affine flag variety given by

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

Let $\mu$ be a conjugacy class of coweights of $\breve{G}$ over the algebraic
closure $\bar{F}$ and let $\operatorname{Adm}(\mu)$ be the set of
$\mu$-admissible elements of $\widetilde{W}$; see section 2.4 for more
details. Then we define

$X(\mu,b)=\coprod\limits_{w\in\operatorname{Adm}(\mu)}X_{w}(b).$

This is a closed subscheme of the affine flag variety and it serves as the
group-theoretic model for the Newton stratum corresponding to $[b]$ in the
special fiber of a Shimura variety giving rise to the datum
$(\mathbf{G},\mu)$.

First introduced by Rapoport in [25], the notion of affine Deligne-Lusztig
varieties plays an important role in arithmetic geometry and the Langlands
program. A fundamental problem in the study of $X_{w}(b)$ is the non-emptiness
pattern, i.e. for which choice of $(w,[b])$, $X_{w}(b)$ is nonempty. Note that
$X_{w}(b)\neq\emptyset$ if and only if
$\mathcal{N}_{b,w}:=[b]\cap\breve{}\mathcal{I}w\breve{}\mathcal{I}$ is
nonempty; the latter is the set of geometric points of a locally closed
subscheme of $\breve{}\mathcal{I}w\breve{}\mathcal{I}$; namely, the Newton
stratum associated to $[b]$.

Let $B(G)$ be the set of $\sigma$-conjugacy classes of
$\mathbf{G}(\breve{F})$. The set $B(G)$ is equipped with a natural partial
order defined in terms of the Kottwitz map $\kappa$ and the Newton map $\nu$.
Understanding the non-emptiness pattern of affine Deligne Lusztig varieties
associated to $w\in\widetilde{W}$ naturally lead us to investigate the sub-
poset $B(G)_{w}=\\{[b]:X_{w}(b)\neq\emptyset\\}$. This set contains a unique
maximal element, which we denote by $[b_{w}]$. Recent work by Milićević and
Viehmann in [22] show that whenever the dimension of $X_{w}(b_{w})$ agrees
with its virtual dimension in the sense of [9], the associated Newton strata
inside $\breve{}\mathcal{I}w\breve{}\mathcal{I}$ exhibit well-behaved
geometry. Furthermore, their theorem gives a condition that can be checked
from knowledge of the Newton point $\nu([b_{w}])$ for this maximal element of
the $B(G)_{w}$, but it provides important conclusion about the shape of the
entire poset. In light of these results, it becomes important to understand
the maximal Newton point $\nu([b_{w}])$ for elements $w\in\widetilde{W}$.

An explicit description of $\nu([b_{w}])$ is given by Milićević in [21,
Theorem 3.2] for elements $w\in\widetilde{W}$ that are _suitably far from
walls of any Weyl chamber_. To quantify the later condition, let us define for
any dominant coweight $\lambda$ its depth as

$\text{depth}(\lambda)=\min\\{\langle\alpha,\lambda\rangle:\alpha~{}\text{is a
simple root}\\}.$

This quantity estimates how far $\lambda$ is from the walls of any Weyl
chamber. Then a formula for $\nu([b_{w}])$ obtained in loc. sit. under
‘superregularity’ hypothesis on the dominant translation part $\lambda$ of
$w$, which roughly says that the depth of $\lambda$ is _quadratically large_
with respect to the semisimple rank of $\mathbf{G}$. To establish this result,
Milićević first derives a characterization of the covering relation in
$\widetilde{W}$, again under certain superregularity hypothesis.

This characterization is of independent interest, and more recently it has
been utilized by He and Yu in [12] to deduce a partial description of
admissible sets in $\widetilde{W}$. This latter description is a crucial
ingredient in proving the main result in their paper, which provides a formula
for the dimension of $X(\mu,b)$. However, due to this reliance on the result
about covering relation, such dimension formula is established in loc. sit.
under the hypothesis that $\mu$ is superregular.

###  1.2. Main results

The main purpose of this article is to show that the aforementioned results in
the study of affine Deligne-Lusztig varieties can be improved by weakening the
hypothesis required to prove them.

#### 1.2.1.

An uniform bound is given in [21, Corollary 3.3] in the quasi-simple case,
which says that the description of $\nu([b_{w}])$ provided in the article is
valid whenever the following depth hypothesis is satisfied on $\lambda$:

$depth(\lambda)\geq\begin{cases}8\ell(w_{0}),&\text{if $\mathbf{G}~{}\text{is
of classical type}$;}\\\ 16\ell(w_{0}),&\text{if $\mathbf{G}~{}\text{is of
exceptional type}$.}\\\ \end{cases}$

Here $\ell(w_{0})$ is the length of the longest element $w_{0}$ in the
associated finite Weyl group $W$. However, it is noted in loc. sit. that this
restriction on the lower bound arises merely from the proof method and is
rather superficial, in the sense that depending on the element $w$ one can
often get the same formula for $\nu([b_{w}])$ under a much weaker hypothesis.
In this paper, we show that one can indeed weaken this hypothesis
considerably. For simplicity, we focus on the quasi-simple groups in the
statement of this result.

######  Theorem A (Theorem 4.1).

Suppose that $\mathbf{G}$ is a quasi-simple split group of rank $n$. Let
$w=ut^{\lambda}v$ be an element of its Iwahori-Weyl group $\widetilde{W}$ such
that $\text{depth}(\lambda)>\Xi$, where $\Xi$ is a linear expression of $n$.
Then the maximal Newton point associated to $w$ is given by
$\nu([b]_{w})=\lambda-\text{weight}(v^{-1},u)$.

Here the precise expression of $\Xi$ depends on the group $\mathbf{G}$. We
refer to section 2.7 for the definition of _weight_ and to the statement of
theorem 4.1 and its subsequent remark for the explicit form of $\Xi$ and
relevant discussion on how to handle the case when $\mathbf{G}$ is a split
connected reductive group which is not necessarily quasi-simple.

#### 1.2.2.

Our second main result characterizes the covering relation of Bruhat order for
most of the elements in $\widetilde{W}$.

######  Theorem B (Theorem 6.1).

Assume that $w=ut^{\lambda}v$ be an element of $\widetilde{W}$ such that
$\text{depth}(\lambda)$ is bigger than a certain constant. Let
$r_{\beta}=t^{mu\alpha^{\vee}}s_{u\alpha}$ be an affine reflection for some
positive root $\alpha$ and integer $m$, and let $w^{\prime}=r_{\beta}w$. Then
$w\gtrdot w^{\prime}$ is a covering relation, i.e. $w\geq w^{\prime}$ and
$\ell(w)=\ell(w^{\prime})+1$, if and only if one of the following conditions
holds:

  1. (1)

$m=0$ and $\ell(us_{\alpha})=\ell(u)-1$; in this case,
$w^{\prime}=us_{\alpha}t^{\lambda}v$.

  2. (2)

$m=1$ and $\ell(us_{\alpha})=\ell(u)+\langle 2\rho,\alpha^{\vee}\rangle-1$; in
this case, $w^{\prime}=us_{\alpha}t^{\lambda-\alpha^{\vee}}v$.

  3. (3)

$m=\langle\alpha,\lambda\rangle$ and $\ell(s_{\alpha}v)=\ell(v)+1$; in this
case, $w^{\prime}=ut^{\lambda}s_{\alpha}v$.

  4. (4)

$m=\langle\alpha,\lambda\rangle-1$ and $\ell(s_{\alpha}v)=\ell(v)-\langle
2\rho,\alpha^{\vee}\rangle+1$; in this case,
$w^{\prime}=ut^{\lambda-\alpha^{\vee}}s_{\alpha}v$.

We refer to the statement of Theorem 6.1 and its subsequent remark for the
precise constant lower bound mentioned above. We remark that the same
characterization was first proved in [18, Proposition 4.1] under the
assumption that the depth of the relevant coweight is at least $2|W|+2$. Later
it was improved in [21, Proposition 4.2], where the depth hypothesis is
relaxed to a quadratic lower bound in the quasi-simple case as follows:

$depth(\lambda)\geq\begin{cases}2\ell(w_{0})+2,&\text{if
$\mathbf{G}~{}\text{is not of type }G_{2}$;}\\\ 3\ell(w_{0})+3,&\text{if
$\mathbf{G}~{}\text{is of type }G_{2}$.}\\\ \end{cases}$

#### 1.2.3.

In section 7, we improve upon a result about dimension of $X(\mu,b)$. Based on
recent discovery of the dimension formula for single affine Deligne-Lusztig
variety $X_{w}(b)$ for “sufficiently large” $w$ in [13], He and Yu provide a
formula in [12] for the dimension of $X(\mu,b)$ under the assumption that
$\mu$ is superregular. We show that this formula in fact holds true for all
_dominant regular $\mu$_.

######  Theorem C.

Suppose that G is a split connected reductive group. Let $\mu\in X_{*}(T)$ be
dominant regular. Assume that $[b]\in B(G,\mu)$ with
$\mu\geq\nu([b])+2\rho^{\vee}$. Then

$\dim
X(\mu,b)=\langle\rho,\mu-\nu([b])\rangle-\frac{1}{2}\text{def}_{\textbf{G}}(b)+\frac{1}{2}(\ell(w_{0})-\ell_{R}(w_{0})).$

We refer the reader to section 2.4 and section 2.5 for relevant definitions.

###  1.3. Strategy

We now discuss the outline of proofs of the above results. We refer to section
2 for relevant definitions.

Let us start by summarising our approach to prove theorem A. By employing
certain tools from the theory of Demazure product, we show in section 3.2 that
one can recast the problem of computing the maximal Newton point associated
with an arbitrary element of $\widetilde{W}$ to that of an element lying in
the dominant chamber. Hence, we focus exclusively on such element $w$ . To do
so, we repeatedly apply an observation about Bruhat order on $\widetilde{W}$
from [25] until we reach a dominant translation element below $w$. In order to
carry out this procedure, we impose a certain lower bound on the depth of the
coweight associated to $w$ in section 4.1. We also show that under this depth
hypothesis, the Newton point of $[b_{w}]$ is given by a translation element
below it. We finally confirm that this is indeed the translation element
produced in the process alluded to before. This is achieved by proving a
technical lemma, which employs techniques about downward Demazure product, and
as such brings in another restriction in the depth hypothesis. Taking into
account these two restrictions on the depth, we finally prove the theorem by
appealing to the fact that the result is already known under superregularity
hypothesis in [21].

To prove theorem B, we again (easily) reduce to the case of considering
elements lying in the dominant chamber. Then we estimate a series of relevant
quantities to assert that the coweight associated with the dominant
translation part of the plausible cocover has to remain in the closure of one
of the two specific Weyl chambers in order for covering relation to hold.
Finally, we deal with each of these two cases separately. We note that a
crucial aspect in which our proof differs from that in [21] is that we handle
the cases where this coweight can be possibly singular.

We remark that in the treatment of [21], the covering relation in
$\widetilde{W}$ is in fact the starting point towards obtaining the formula of
maximal Newton point. The strategy in loc. sit. is to apply the
characterization of covering relation on $w$ and its subsequent cocovers
repeatedly, until one reaches the maximal translation element below $w$. One
then carefully keeps track of the cocovers of a given element $w$ to quantify
a depth hypothesis under which this process work. Hence, it is plausible that
one may use an improved version of the covering relation characterization (as
the one established in theorem B) and carry out a similar analysis to obtain
the maximal Newton point formula under a weaker hypothesis on the depth,
_which may even be a constant lower bound hypothesis_. However, we do not
follow this approach here. While we prove theorem A only under a linear bound
hypothesis on depth, our approach exclusively outlines techniques from
Demazure product and _it highlights the fact that the simple observation in
lemma 4.3 lies at the heart of this matter_. We do point out some speculative
assumptions about how one can improve theorem A, see remark 4.9.

Let us now briefly discuss our strategy for proving theorem C. The key
ingredient in eliminating superregularity condition here is the additivity
property of admissible sets established in [11, Theorem 5.1]. This is one of
the crucial results used in proving Kottwitz-Rapoport conjecture on non-
emptiness of $X(\mu,b)$ in loc. sit. We use this additivity to show that
certain steps in the existing proof in [12] can be carried out in a way that
superregularity is no longer required. We point out that the proof philosophy
has a resemblance to that of theorem A, in the sense that we leverage the
results already established in loc. sit. under superregularity hypothesis.

######  Acknowledgement.

I am grateful to my advisor Xuhua He for the suggestion to investigate these
problems as well as numerous helpful discussions over the course of this work.
I would also like to thank Jeffrey Adams and Thomas Haines for many helpful
conversations and valuable feedback. During the preparation of this article, I
was partially supported by a Graduate School Summer Research Fellowship and
NSF grant DMS no. 1801352 through Thomas Haines. Upon completion of writing
this paper, I was informed that Felix Schremmer had independently obtained
similar results. I thank him for his thoughtful comments on a preliminary
draft of the paper, especially for pointing out a minor error in the
manuscript.

##  2\. Preliminary

###  2.1. Notations

Recall that $F$ is a nonarchimedean local field and $\breve{F}$ is the
completion of the maximal unramified extension of $F$. Let $\mathbf{G}$ be a
split connected reductive group over $F$; We write $\breve{G}$ for
$\mathbf{G}(\breve{F})$. Let $\sigma$ be the Frobenius morphism of
$\breve{F}/F$. We use the same symbol $\sigma$ for the induced Frobenius
morphism on $\breve{G}$. Let $B$ be a fixed Borel subgroup with $T$ a split
maximal torus in $B$.

Let $\mathcal{A}$ be the apartment of $\mathbf{G}$ corresponding to $T$. We
fix an alcove $\mathbf{a}$ in $\mathcal{A}$, and let
$\breve{}\mathcal{I}\subset\breve{G}$ be the Iwahori subgroup corresponding to
$\mathbf{a}$. Then $\breve{}\mathcal{I}$ is $\sigma$-stable. We denote by $N$
the normalizer of $T$ in $\mathbf{G}$. The _Iwahori-Weyl group_ (associated to
$T$) is defined as

$\widetilde{W}=N(\breve{F})/T(\breve{F})\cap\breve{}\mathcal{I}.$

Let $W=N(\breve{F})/T(\breve{F})$ be the finite Weyl group. Fixing a special
vertex of the base alcove $\mathbf{a}$, we have the splitting

$\widetilde{W}=X_{*}(T)\rtimes W=\\{t^{\lambda}x:\lambda\in X_{*}(T),x\in
W\\}.$

For any $w\in\widetilde{W}$, we choose a representative in $N(\breve{F})$
which is also denoted by $w$. Note that the action $\sigma$ on $\breve{G}$
induces a natural action of $\sigma$ on $\widetilde{W}$, which is just the
identity automorphism since $\mathbf{G}$ is split. We remark that one can
identify the element $x\in\widetilde{W}$ with $x\mathbf{a}$, the (extended)
alcove that one obtains as image of the base alcove $\mathbf{a}$ under $x$.

Denote by $\tilde{}\mathbb{S}$ the set of simple reflections in
$\widetilde{W}$; these are taken to be reflections along the walls of
$\mathbf{a}$. Let $\mathbb{S}\subset\tilde{}\mathbb{S}$ be the set of simple
reflections in $W$. Let $\Phi$ (resp. $\Phi^{+},\Delta$) be the set of all
roots (resp. positive roots, simple roots). Let $w_{0}$ be the longest element
in $W$. Let $\rho$ be the dominant weight with
$\langle\alpha^{\vee},\rho\rangle=1$ for any $\alpha\in\Delta$ and
$\rho^{\vee}$ be the dominant coweight with
$\langle\rho^{\vee},\alpha\rangle=1$ for any $\alpha\in\Delta$. Let $\theta$
be the highest root. We denote the dominant (rational) Weyl chamber by
$\mathcal{C}^{+}\subset X_{*}(T)_{\mathbb{Q}}$, and the image of this chamber
under some element $x\in W\setminus\\{1\\}$ by $\mathcal{C}_{x}$. For an
irreducible Weyl group $W$, we follow the labeling of roots as in [5] and we
usually write $s_{i}$ instead of $s_{\alpha_{i}}$, where
$\Delta=\\{\alpha_{i}:1\leq i\leq n\\}$. For any $J\subset\tilde{}\mathbb{S}$,
we denote by $W_{J}$ the subgroup of $\widetilde{W}$ generated by to $J$. Let
${}^{J}\widetilde{W}$ (resp. $\widetilde{W}^{J}$) be the set of minimal length
elements in their cosets in $W_{J}\backslash\widetilde{W}$ (resp.
$\widetilde{W}/W_{J}$).

###  2.2. Length function and Bruhat order on $W$ and $\widetilde{W}$

We denote by $\ell$ the length function on $\widetilde{W}$ determined by the
base alcove $\mathbf{a}$. Let $W_{a}$ be the subgroup of $\widetilde{W}$
generated by $\tilde{}\mathbb{S}$. Then $W_{a}$ is an affine Weyl group. Let
$\Omega\subset\widetilde{W}$ be the subgroup of length-zero elements (or
equivalently, the stabilizer of $\mathbf{a}$) in $\widetilde{W}$. Then

$\widetilde{W}=W_{a}\rtimes\Omega.$

$W_{a}$ is the Coxeter group associated to $\tilde{}\mathbb{S}$, and hence it
comes equipped with an associated Bruhat order, which we denote by $\leq$.
This is extended to $\widetilde{W}$ as follows. For two elements
$\widetilde{w}_{1},\widetilde{w}_{2}\in\widetilde{W}$, use the above
decomposition to write $\widetilde{w}_{i}=w_{i}\zeta_{i}$ with $w_{i}\in
W_{a},\zeta_{i}\in\Omega$ for $i=1,2$. We then declare
$\widetilde{w}_{1}\leq\widetilde{w}_{2}$ if $w_{1}\leq w_{2}$ and
$\zeta_{1}=\zeta_{2}$. The following properties about $\ell$ and $\leq$ are
well-known, e.g. see [21, Lemma 4.1], [14, exercise 23, Chapter 2] and [3,
exercise 21, Chapter 2 ] respectively.

  * •

Let $\lambda\in X_{*}(T)$ be regular dominant, and let
$w=ut^{\lambda}v\in\widetilde{W}$. Then

$\ell(w)=\ell(u)+\ell(t^{\lambda})-\ell(v)=\ell(u)+\langle
2\rho,\lambda\rangle-\ell(v).$ (2.1)

  * •

For any element $x$ of $W$, let
$\text{Inv}(x)=\\{\alpha\in\Phi^{+}:x\alpha\in-\Phi^{+}\\}$. Then for any two
elements $x,y\in W$, we have

$\ell(xy)=\ell(x)+\ell(y)-2|\text{Inv}(x)\cap\text{Inv}(y^{-1})|=\ell(x)-\ell(y)+2|\text{Inv}(x)^{c}\cap\text{Inv}(y^{-1})|.$
(2.2)

  * •

Let $w_{1},w_{2},v\in\widetilde{W}$ be three elements such that
$\ell(w_{i}v)=\ell(w_{i})+\ell(v)$ for $i=1,2$. Then

$w_{1}\geq w_{2}\text{ is equivalent to }w_{1}v\geq w_{2}v.$ (2.3)

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

We define the $\sigma$-conjugation action on $\breve{G}$ by
$g\cdot_{\sigma}g^{\prime}=gg^{\prime}\sigma(g)^{-1}$. Let $B(G)$ be the set
of $\sigma$-conjugacy classes on $\breve{G}$. The classification of the
$\sigma$-conjugacy classes is obtained by Kottwitz in [16] and [17]. Any
$\sigma$-conjugacy class $[b]$ is determined by two invariants:

  * •

Its Kottwitz point $\kappa([b])\in\Omega$, and

  * •

Its Newton point $\nu([b])\in X_{*}(T)_{\mathbb{Q}}$.

We denote by $\leq$ the dominance order on $X_{*}(T)_{\mathbb{Q}}$ i.e., for
$\nu,\nu^{\prime}\in X_{*}(T)_{\mathbb{Q}}$, we say that $\nu\leq\nu^{\prime}$
if $\nu^{\prime}-\nu$ is a non-negative (rational) linear combination of
positive roots. This upgrades to a partial order on the set $B(G)$. We define
$[b_{1}]\preceq[b_{2}]$ if $\kappa([b_{1}])=\kappa([b_{2}])$ and
$\nu([b_{1}])\leq\nu([b_{2}])$. Let
$B(G)_{w}=\\{[b]:[b]\cap\breve{I}w\breve{I}\neq\emptyset\\}$. The subposet
$(B(G)_{w},\preceq)$ has an unique maximal element, which coincides with the
generic $\sigma$-conjugacy class in the double coset $\breve{I}w\breve{I}$. We
denote this maximal element by $[b_{w}]$ and we set $\nu_{w}=\nu([b_{w}])$.

###  2.4. Affine Deligne-Lusztig varieties

Let $\mathcal{F}=\breve{G}/\breve{}\mathcal{I}$ be the _affine flag variety_.
Recall that for any $b\in\breve{G}$ and $w\in\widetilde{W}$, the associated
_affine Deligne-Lusztig variety_ in the affine flag variety is defined as
$X_{w}(b)=\\{g\breve{}\mathcal{I}\in\breve{G}/\breve{}\mathcal{I};g^{-1}b\sigma(g)\in\breve{}\mathcal{I}w\breve{}\mathcal{I}\\}$.
We will also discuss certain finite union of affine Deligne-Lusztig varieties.
Following [25], we define for a dominant $\mu\in X_{*}(T)$ its associated
_admissible set_ as

$\text{Adm}(\mu)=\\{w\in\widetilde{W}:w\leq t^{x(\mu)}\text{ for some }x\in
W\\}.$

We will need the following additivity property of admissible sets.

######  Theorem 2.1.

[11, Theorem 5.1] Let $\mu,\mu^{\prime}\in X_{*}(T)$ be dominant. Then we have

$\text{Adm}(\mu)\cdot\text{Adm}(\mu^{\prime})=\text{Adm}(\mu+\mu^{\prime}).$

For any $b\in\breve{G}$, we set

$X(\mu,b)=\bigcup_{w\in\text{Adm}(\mu)}X_{w}(b).$

The sets $X_{w}(b)$ and $X(\mu,b)$ are subschemes, locally of finite type, of
the affine flag variety (in the usual sense in equal characteristic; in the
sense of Zhu [28] in mixed characteristic). Following [15], we define the set
of _neutrally acceptable elements for $\mu$_ as

$B(G,\mu)=\\{[b]\in B(G):\kappa([b])=\mu,\nu([b])\leq\mu\\}.$

Settling the Kottwitz-Rapoport conjecture made in [15] and [25] about the non-
emptiness pattern for $X(\mu,b)$, He proves the following result in [11].

######  Theorem 2.2.

[11, Theorem A] $X(\mu,b)\neq\emptyset$ if and only if $[b]\in B(G,\mu)$.

###  2.5. Virtual dimension of affine Deligne-Lusztig variety

Note that any element $w\in\widetilde{W}$ can be written in a unique way as
$w=ut^{\lambda}v$ with $\lambda$ dominant, $u,v\in W$ such that
$t^{\lambda}v\in{}^{\mathbb{S}}\widetilde{W}$. In this case, we set
$\eta(w)=vu.$

Let $\mathbf{J}_{b}$ be the reductive group over $F$ with
$\mathbf{J}_{b}(F)=\\{g\in\breve{G};gb\sigma(g)^{-1}=b\\}.$ Then the _defect_
of $b$ is defined by
$\rm{def}_{\mathbf{G}}(b)=\operatorname{rank}_{F}\mathbf{G}-\operatorname{rank}_{F}\mathbf{J}_{b}.$
Here for a reductive group $\mathbf{H}$ defined over $F$,
$\operatorname{rank}_{F}$ is the $F$-rank of the group $\mathbf{H}$.

Following [9, Section 10.2], we define the _virtual dimension_ to be

$d_{w}(b)=\frac{1}{2}\big{(}\ell(w)+\ell(\eta(w))-\rm{def}_{\mathbf{G}}(b)\big{)}-\langle\rho,\nu([b])\rangle.$

The justification of defining such an expression lies in a result proved by He
in [10, Theorem 2.30] that says $\dim X_{w}(b)\leq d_{w}(b)$. Recent work by
Milićević and Viehmann in [22] singles out those $w\in\widetilde{W}$ for which
$\dim X_{w}(b_{w})=d_{w}(b_{w})$ holds; these are called _cordial elements_ ,
and it is shown in loc. sit. that $B(G)_{w}$ exhibits remarkable properties
for such $w$.

###  2.6. Demazure product and its variations

We recall three operations
$*,\rhd,\lhd:\widetilde{W}\times\widetilde{W}\rightarrow\widetilde{W}$. Here
$*$ is the _Demazure product_ and $\lhd,\rhd$ are the left and right _downward
Demazure products_ , respectively. We describe these operations in form of the
following lemma.

######  Lemma 2.3.

[7, Section 2.2] Let $x,y\in\widetilde{W}$.

  1. (1)

The subset $\\{uv:u\leq x,v\leq y\\}$ contains an unique maximal element,
which we denote by $x*y$. Moreover, $x*y=u^{\prime}y=xv^{\prime}$ for some
$u^{\prime}\leq x$ and $v^{\prime}\leq y$ and
$\ell(x*y)=\ell(u^{\prime})+\ell(y)=\ell(x)+\ell(v^{\prime})$.

  2. (2)

The subset $\\{uy:u\leq x\\}$ contains a unique minimal element, which we
denote by $x\rhd y$. Moreover, $x\rhd y=u^{\prime\prime}y$ for some
$u^{\prime\prime}\leq x$ with $\ell(x\rhd y)=\ell(y)-\ell(u^{\prime\prime})$.

  3. (3)

The subset $\\{xv:v\leq y\\}$ contains a unique minimal element, which we
denote by $x\lhd y$. Moreover, $x\lhd y=xv^{\prime\prime}$ for some
$v^{\prime\prime}\leq y$ with $\ell(x\lhd y)=\ell(x)-\ell(v^{\prime\prime})$.

The Demazure product is related to product of closure of two Iwahori double
cosets in $\breve{G}$. More precisely, for any
$w\in\widetilde{W},\overline{\breve{I}w\breve{I}}=\bigcup_{w^{\prime}\leq
w}\breve{I}w^{\prime}\breve{I}$ is a closed admissible subset of $G$ in the
sense of [11]. Then for any $x,y\in\widetilde{W}$, we have

$\overline{\breve{I}x\breve{I}}\cdot\overline{\breve{I}y\breve{I}}=\bigcup_{u\leq
x}\breve{I}x\breve{I}\cdot\bigcup_{v\leq y}\breve{I}y\breve{I}=\bigcup_{w\leq
x*y}\breve{I}w\breve{I}=\overline{\breve{I}(x*y)\breve{I}}.$

The above equation also implies that $*$ is an associative binary operation on
$\widetilde{W}$. We will need the following results about these operations.

######  Lemma 2.4.

[8, Lemma 2 and Lemma 3]

  1. (1)

If $w\geq w^{\prime}$, $v\leq v^{\prime}$ are elements of $\widetilde{W}$,
then $w\rhd v\leq w^{\prime}\rhd v^{\prime}$.

  2. (2)

For any three elements $x,y,z\in\widetilde{W}$, we have $x\rhd(y\rhd
z)=(x*y)\rhd z$. In other words, the action $(\widetilde{W},*)\times
W\rightarrow\widetilde{W},(x,y)\rightarrow x\rhd y$ is a left action of the
monoid $(\widetilde{W},*)$.

###  2.7. Quantum Bruhat graph

We recall the quantum Bruhat graph introduced by Brenti, Fomin and Postnikov
in [6]. By definition, a _quantum Bruhat graph_ $\Gamma_{\Phi}$ is a directed
graph with

  * •

vertices given by the elements of $W$;

  * •

upward edges $w\rightharpoonup ws_{\alpha}$ for some $\alpha\in\Phi^{+}$ with
$\ell(ws_{\alpha})=\ell(w)+1$;

  * •

downward edges $w\rightharpoondown ws_{\alpha}$ for some $\alpha\in\Phi^{+}$
with $\ell(ws_{\alpha})=\ell(w)-\langle 2\rho,\alpha^{\vee}\rangle+1$.

Note that $\ell(s_{\alpha})\leq\langle 2\rho,\alpha^{\vee}\rangle-1$ for any
$\alpha\in\Phi^{+}$, so we have
$\ell(ws_{\alpha})\geq\ell(w)-\ell(s_{\alpha})\geq\ell(w)-\langle
2\rho,\alpha^{\vee}\rangle+1.$ Therefore the condition for downward edges can
be rephrased to saying that

$\ell(ws_{\alpha})=\ell(w)-\ell(s_{\alpha})~{}\text{with}~{}\ell(s_{\alpha})=\langle
2\rho,\alpha^{\vee}\rangle-1.$

Following [19], we call $\alpha\in\Phi^{+}$ to be a _quantum root_ if
$\ell(s_{\alpha})=\langle 2\rho,\alpha^{\vee}\rangle-1$. We have the following
description of quantum roots.

######  Lemma 2.5.

[19, Lemma 4.2] We have that $\alpha\in\Phi^{+}$ is a quantum root if and only
if

  1. (1)

$\alpha$ is a long root, or

  2. (2)

$\alpha$ is a short root, and if
$\alpha=\sum\limits_{\alpha_{i}\in\Delta}c_{i}\alpha_{i}$ then we have
$c_{i}=0$ for any long simple root $\alpha_{i}$.

Here for simply laced root systems we consider all roots to be long. Thus in a
simply laced type, all roots are quantum. Examples of quantum root, in
general, include the simple roots as well as the highest root.

The weight of an upward edge is defined to be $0$ and the weight of a downward
edge $w\rightharpoondown ws_{\alpha}$ is defined to be $\alpha^{\vee}$. The
_weight of a path_ in $\Gamma_{\Phi}$ is defined to be the sum of weights of
the edges in the path. For any $x,y\in W$, we denote by $d_{\Gamma}(x,y)$ the
minimal length among all paths in $\Gamma_{\Phi}$ from $x$ to $y$.

######  Lemma 2.6.

[24, Theorem 2], [6, Lemma 6.7] Let $x,y\in W$. Then

  1. (1)

There exists a directed path in $\Gamma_{\Phi}$ from $x$ to $y$.

  2. (2)

All the shortest paths in $\Gamma_{\Phi}$ from $x$ to $y$ have the same
weight, which we denote by $\text{wt}(x,y)$.

  3. (3)

Any path in $\Gamma_{\Phi}$ from $x$ to $y$ has weight
$\geqslant\text{wt}(x,y)$.

##  3\. Some combinatorial property

###  3.1. Monotonicity of weight function

Let us define the function $\text{wt}:W\rightarrow X_{*}(T)$ by
$\text{wt}(x):=\text{wt}(x,1)$.

######  Lemma 3.1.

Let $x_{1},x_{2}\in W$; if $x_{1}\leq x_{2}$ in Bruhat order, then
$\text{wt}(x_{1})\leq\text{wt}(x_{2})$ in dominance order.

###### Proof.

Assume that $\lambda\in X_{*}(T)^{+}$ is regular. We first note that
$x_{1}\leq x_{2}$ implies $t^{\lambda}x_{1}\geqslant t^{\lambda}x_{2}$. It
suffices to show this when $x_{2}$ is a cover of $x_{1}$, i.e.
$x_{2}=x_{1}s_{\alpha}$ for some $\alpha\in\Phi^{+}$ such that
$\ell(x_{2})=\ell(x_{1})+1$. Then $t^{\lambda}x_{2}=t^{\lambda}x_{1}\cdot
s_{\alpha}$, hence the two elements in question are comparable. Now, eq. 2.1
shows that

$\ell(t^{\lambda}x_{2})=\ell(t^{\lambda})-\ell(x_{2})=\ell(t^{\lambda})-\ell(x_{1})-1=\ell(t^{\lambda}x_{1})-1.$

Hence $t^{\lambda}x_{1}\geqslant t^{\lambda}x_{2}$.

Recall the description of maximal Newton point associated to
$w\in\widetilde{W}$ given in [27].

$\nu_{w}=\max\\{\nu(u):u\leq w,u\in\widetilde{W}\\}.$ (3.1)

By applying this, we get
$\nu_{t^{\lambda}x_{1}}\geqslant\nu_{t^{\lambda}x_{2}}$. Let us now assume
that $\lambda$ is actually superregular; hence the formula for maximal Newton
point in [21, Theorem 3.2] applies in our case and it gives
$\nu_{t^{\lambda}x_{i}}=\lambda-\text{wt}(x_{i}^{-1})=\lambda-\text{wt}(x_{i})$
for $i=1,2$. Thus we get
$\lambda-\text{wt}(x_{1})\geq\lambda-\text{wt}(x_{2})$, whence
$\text{wt}(x_{1})\leqslant\text{wt}(x_{2})$. ∎

###  3.2. Reduction to dominant chamber alcoves

We will now reduce our calculation of maximal Newton point of an arbitrary
element of $\widetilde{W}$ to that of a suitable element lying in the dominant
chamber.

######  Proposition 3.2.

If $\lambda$ is dominant regular, then
$[b_{ut^{\lambda}v}]=[b_{t^{\lambda}(v\lhd u)}]$.

###### Proof.

Note that by eq. 2.1, $\ell(ut^{\lambda}v)=\ell(u)+\ell(t^{\lambda}v)$, hence
$ut^{\lambda}v=u*(t^{\lambda}v)$. Thus
$\overline{\breve{I}ut^{\lambda}v\breve{I}}=\overline{\breve{I}u\breve{I}}\cdot\overline{\breve{I}t^{\lambda}v\breve{I}}$.

Now recall that $\breve{I}$ is $\sigma$-stable, and since $\mathbf{G}$ is
split, $\sigma$ acts trivially on $\widetilde{W}$ too. Hence $\sigma$ fixes
$\overline{\breve{I}w\breve{I}}$ for any $w\in\widetilde{W}$. Therefore, if
$w=w_{1}w_{2}$ is an element with
$w_{1}\in\overline{\breve{I}u\breve{I}},w_{2}\in\overline{\breve{I}t^{\lambda}v\breve{I}}$,
then $w_{1}^{-1}\cdot_{\sigma}w=w_{2}w_{1}$. This shows that
$\breve{G}\cdot_{\sigma}(\overline{\breve{I}u\breve{I}}\cdot\overline{\breve{I}t^{\lambda}v\breve{I}})\subset\breve{G}\cdot_{\sigma}(\overline{\breve{I}t^{\lambda}v\breve{I}}\cdot\overline{\breve{I}u\breve{I}})$.
One obtains the other inclusion in a similar way. This shows that
$\breve{G}\cdot_{\sigma}(\overline{\breve{I}u\breve{I}}\cdot\overline{\breve{I}t^{\lambda}v\breve{I}})=\breve{G}\cdot_{\sigma}(\overline{\breve{I}t^{\lambda}v\breve{I}}\cdot\overline{\breve{I}u\breve{I}}).$

Hence, we conclude that
$\breve{G}\cdot_{\sigma}\overline{\breve{I}ut^{\lambda}v\breve{I}}=\breve{G}\cdot_{\sigma}\overline{\breve{I}(t^{\lambda}v)*u\breve{I}}.$
Since $G\cdot_{\sigma}\overline{\breve{I}w\breve{I}}$ is the set of
$\sigma$-conjugacy classes intersecting $\overline{\breve{I}w\breve{I}}$, this
shows that

$[b_{ut^{\lambda}v}]=[b_{(t^{\lambda}v)*u}].$ (3.2)

Thus it suffices to show that

(a) if $\lambda$ is dominant regular, then $(t^{\lambda}v)*u=t^{\lambda}(v\lhd
u)$.

By lemma 2.4, we just need to argue the case when $u$ is a simple reflection
$s\in\mathbb{S}$. Then this statement is equivalent to the assertion that
$\ell(vs)=\ell(v)-1$ if and only if
$\ell(t^{\lambda}vs)=\ell(t^{\lambda}v)+1$. This follows directly from a
computation using eq. 2.1. Hence we are done. ∎

######  Corollary 3.3.

Let $x,y\in W$. Then $\text{wt}(x,y)=\text{wt}(x^{-1}\lhd y)$.

###### Proof.

Suppose that $\lambda\in X_{*}(T)^{+}$ is superregular. Applying the formula
established in [21, Theorem 3.2] to $w:=xt^{\lambda}y$ and
$w^{\prime}:=t^{\lambda}(y\lhd x)$, we get that
$\nu_{w}=\lambda-\text{wt}(y^{-1},x)$ and
$\nu_{w^{\prime}}=\lambda-\text{wt}((y\lhd x)^{-1})=\lambda-\text{wt}(y\lhd
x)$. Since $\nu_{w}=\nu_{w^{\prime}}$ by proposition 3.2, we get
$\text{wt}(y^{-1},x)=\text{wt}(y\lhd x)$. Now replacing the pair $(x,y)$ by
$(y,x^{-1})$, we get the conclusion. ∎

###  3.3. Reformulation of weight in quantum Bruhat graph

We now give another interpretation of weight of a shortest path in the quantum
Bruhat graph.

######  Definition 3.4.

For an element $x\in W$, we say that $x=s_{\beta_{1}}\cdots s_{\beta_{k}}$ is
a reduced quantum reflection decomposition if

  1. (1)

$\\{\beta_{i}:1\leq i\leq k\\}$ is a collection of (not necessarily simple)
quantum roots, i.e. $l(s_{\beta_{i}})=\langle
2\rho,\beta_{i}^{\vee}\rangle-1.$

  2. (2)

$\ell(x)=\sum_{i=1}^{k}\ell(s_{\beta_{i}})$.

  3. (3)

Subject to the first two conditions, $k$ is minimal.

In this case, we say that $k$ is the minimal reduced length of $x$ in terms of
reflections associated to quantum roots, and write $\ell_{\downarrow}(x)=k$.
Note that $d_{\Gamma}(x,1)=\ell_{\downarrow}(x)$.

The choice of notation is suggested from the fact that

{Reduced quantum reflection decompositions for $x$}

↕

{Shortest paths in $\Gamma_{\Phi}$ from $x$ to $1$ that uses only downwards
edges}

By [22, Proposition 4.11], the weight of any path in the later set is equal to
$\text{wt}(x)$. Therefore, the problem of computing $\text{wt}(x)$ reduces to
one of finding a suitable decomposition for $x$: if $x=s_{\beta_{1}}\cdots
s_{\beta_{k}}$ is such a decomposition then
$\text{wt}(x)=\sum_{i=1}^{k}\beta_{i}^{\vee}$. In view of corollary 3.3, we
can thus reformulate $\text{wt}(x,y)$ in terms of a decomposition of
$x^{-1}\lhd y$.

##  4\. Formula for the maximal Newton point

The goal of this section is to prove the following result.

######  Theorem 4.1.

Assume that $\mathbf{G}$ is a quasi-simple split group of rank $n$. Let
$w=t^{\lambda}x$ be an element of its Iwahori-Weyl group such that $\lambda$
is dominant and $\text{depth}(\lambda)>\Xi$, where $\Xi$ is an integer
depending on $n$. Then the maximal Newton point associated to $w$ is given by
$\nu_{w}=\lambda-\text{wt}(x)$.

For each Cartan type, the lower bound $\Xi$ for the depth mentioned above is
given in the following table.

Type | $A_{n}$ | $B_{n}/C_{n}$ | $D_{n}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---
$\Xi$ | $3n+1$ | $6n-2$ | $6n-6$ | $23$ | $33$ | $57$ | $23$ | $9$

######  Remark 4.2.

We note that one can handle the case of a split connected reductive group
$\mathbf{G}$ via a routine reduction procedure, which we discuss now. In that
case, we have

$W=W_{1}\times\cdots\times W_{l},$

where $W_{i}$ are irreducible Weyl groups. This gives rise to a partition of
$m$, where $m$ is the semisimple rank of $\mathbf{G}$. Any element $x$ of $W$
is of the form $(x_{1},\cdots,x_{l})$. Similarly, we write $\lambda$ as
$(\lambda_{1},\cdots,\lambda_{l})$. We have $w_{0}=(w_{0,1},\cdots,w_{0,l})$,
where $w_{0,i}$ is the longest element of $W_{i}$. In the same way, we write
the sum of positive roots as $(2\rho_{1}^{\vee},\cdots,2\rho_{l}^{\vee})$ and
the highest root as $(\theta_{1},\cdots,\theta_{l})$.

We observe that while dealing with the quasi-simple case, we introduce a
restriction on depth in two stages - as found in section 4.1 and section 4.4.
By the last paragraph we therefore need to require for each $i$

$\text{depth}(\lambda_{i})\geq\Xi_{i},$

where $\Xi_{i}$ is the lower bound given for the Weyl group $W_{i},$ to be
read off from the table above. Note that $\Xi_{i}$ is then some linear
expression of $m$, depending on the type of $i$-th factor and the partition of
$m$. We remark that under such hypothesis, theorem 4.1 applies to each quasi
simple factor of $\mathbf{G}$ and thus we need to justify the formula for such
groups.

By proposition 3.2, we know that $\nu_{ut^{\lambda}v}=\nu_{t^{\lambda}(v\lhd
u)}$. Then by theorem 4.1 it follows that $\nu_{t^{\lambda}(v\lhd
u)}=\lambda-\text{wt}(v\lhd u)$. By corollary 3.3 we then conclude that
$\nu_{w}=\lambda-\text{wt}(v^{-1},u)$. This concludes our discussion.

###  4.1. Proposed linear bound on depth

For the rest of section 4, we assume that $W$ is an irreducible Weyl group for
the root system of Cartan type $X_{n}$. We first establish an upper bound for

$\mathcal{M}=\mathcal{M}_{X_{n}}:=\max\\{\langle\alpha_{i},\text{wt}(x)\rangle:\alpha_{i}\in\Delta,x\in
W\\}$

This is based on the explicit computation of $\text{wt}(w_{0})$ for the
longest element $w_{0}$ in each type, which we defer to the next section for
the sake of continuity.

By lemma 3.1, we have $\\{\text{wt}(x):x\in W\\}\subset\\{\nu\in
X_{*}(T):\nu\leq\text{wt}(w_{0})\\}$. Therefore,

$\mathcal{M}\leq\max\\{\langle\alpha_{i},\nu\rangle:\nu\in
X_{*}(T),\nu\leq\text{wt}(w_{0}),\alpha_{i}\in\Delta\\}.$

It is easy to see that maximum of the latter set is realized at
$\nu=\kappa_{l}\alpha_{l}^{\vee}$, where $\kappa_{l}\alpha_{l}^{\vee}$ is a
summand of $\text{wt}(w_{0})$ such that
$\kappa_{l}=\max\\{\kappa_{j}:\kappa_{j}\alpha_{j}^{\vee}~{}\text{is a summand
of $\text{wt}(w_{0})$}\\}$. Since the maximum value of
$\langle\alpha_{i},\alpha_{l}^{\vee}\rangle$ is $2$ in every Cartan type, this
shows that $\mathcal{M}_{X_{n}}$ is bounded above by twice the largest
coefficient in the expression of $\text{wt}(w_{0})$ in each type $X_{n}$. This
is tabulated in the list below, where we denote by
$\widetilde{}\mathcal{M}=\widetilde{}\mathcal{M}_{X_{n}}$ the upper bound
obtained in this manner.

Type | $A_{n}$ | $B_{n}/C_{n}$ | $D_{n}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---
$\widetilde{}\mathcal{M}$ | $n+1$ | $2n$ | $2n$ | $12$ | $16$ | $28$ | $12$ | $4$

###  4.2. Proof of the easier inequality

We need the following result about Bruhat order on $\widetilde{W}$.

######  Lemma 4.3.

[25, Remark 3.9] Let $\lambda\in X_{*}(T)$ be dominant. Let $\beta\in\Phi^{+}$
such that $\lambda-\beta^{\vee}$ is dominant. Then
$t^{\lambda-\beta^{\vee}}\leq t^{\lambda}s_{\beta}\leq t^{\lambda}$ in
$\widetilde{W}$.

We start by establishing the easier inequality.

######  Proposition 4.4.

Let $w=t^{\lambda}x\in\widetilde{W}$ such that
$\text{depth}(\lambda)\geq\widetilde{}\mathcal{M}$. Then $w\geq
t^{\lambda-\text{wt}(x)}$ and therefore $\nu_{w}\geq\lambda-wt(x)$.

###### Proof.

Suppose that $x=s_{\beta_{1}}\cdots s_{\beta_{l}}$ is a reduced quantum
reflection decomposition with $\ell_{\downarrow}(x)=l>1$, and therefore
$\text{wt}(x)=\sum\limits_{i=1}^{l}\beta_{i}^{\vee}$. Let us first note that:

(a) $s_{\beta_{1}}\cdots s_{\beta_{k}}$ is a reduced quantum reflection
decomposition for $x_{k}:=xs_{\beta_{l}}\cdots s_{\beta_{k+1}}$ for all $k\leq
l$.

Suppose otherwise; then we can find a reduced quantum reflection decomposition
of the form $x_{k}=s_{\gamma_{1}}\cdots s_{\gamma_{j}}$ with $j<k$. But then
$x=x_{k}s_{\beta_{k+1}}\cdots s_{\beta_{l}}=s_{\gamma_{1}}\cdots
s_{\gamma_{j}}s_{\beta_{k+1}}\cdots s_{\beta_{l}}$ satisfies length
additivity:

$\ell(x)=\ell(x_{k})+\sum\limits_{i=k+1}^{l}\beta_{i}^{\vee}=\sum\limits_{i=1}^{j}\gamma_{i}^{\vee}+\sum\limits_{i=k+1}^{l}\beta_{i}^{\vee}.$

Note that all associated roots in this new decomposition of $x$ are quantum,
and it has $j+l-k<l$ factors, hence it contradicts the fact that
$\ell_{\downarrow}(x)=l$. So statement (a) is proved.

To prove the proposition, we induct on $\ell_{\downarrow}(x)$. We note our
depth hypothesis ensures that $\lambda-\beta^{\vee}$ is dominant for all
$\beta\in\Phi^{+}$: for $\alpha_{i}\in\Delta$, we have
$\langle\alpha_{i},\lambda-\beta^{\vee}\rangle=\langle\alpha_{i},\lambda\rangle-\langle\alpha_{i},\beta^{\vee}\rangle>0$,
since maximum value of $\langle\alpha^{\vee},\alpha_{i}\rangle$ equals $2$ in
every Cartan type, except for $G_{2}$ \- in which case it equals $3$, cf. [5,
Chapter 6, Section 1, no. 3]. Therefore lemma 4.3 covers the base case, i.e.
when $x=s_{\beta}$ for some $\beta\in\Phi^{+}$.

Let us now apply induction hypothesis to $xs_{\beta_{l}}=s_{\beta_{1}}\cdots
s_{\beta_{l-1}}$. Since
$\text{wt}(xs_{\beta_{l}})=\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}$ by virtue
of statement (a), this gives

$t^{\lambda}s_{\beta_{1}}\cdots s_{\beta_{l-1}}\geq
t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}}.$ (4.1)

To perform the induction step, we apply a property of Bruhat order as
described in eq. 2.3. We set

$w_{1}=t^{\lambda}x,w_{2}=t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}}s_{\beta_{l}},v=s_{\beta_{l}}$

and check the required length additivity. Note that
$\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}$ is dominant under our depth
hypothesis: for $\alpha_{i}\in\Delta$,

$\langle\alpha_{i},\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}\rangle=\langle\alpha_{i},\lambda\rangle-\langle\alpha_{i},\text{wt}(xs_{\beta_{l}})\rangle>\langle\alpha_{i},\lambda\rangle-\mathcal{M}\geq
0,$

therefore the length formula in eq. 2.1 applies. We see that

  1. (1)

$\ell(w_{1}v)=\ell(t^{\lambda}s_{\beta_{1}}\cdots
s_{\beta_{l-1}})=\ell(t^{\lambda})-\sum\limits_{i=1}^{l-1}\ell(s_{\beta_{i}})=\ell(t^{\lambda})-\sum\limits_{i=1}^{l}\ell(s_{\beta_{i}})+\ell(s_{\beta_{l}})=\ell(w_{1})+\ell(v).$

  2. (2)

$\ell(w_{2}v)=\ell(t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}})=\ell(t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}})-\ell(s_{\beta_{l}})+\ell(s_{\beta_{l}})=\ell(w_{2})+\ell(v).$

In presence of eq. 4.1, we therefore get $w_{1}\geq w_{2}$, i.e.
$t^{\lambda}x\geq
t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}}s_{\beta_{l}}$. Finally, we
note that $\lambda-\text{wt}(x)=\lambda-\sum\limits_{i=1}^{l}\beta_{i}^{\vee}$
is dominant (by similar argument as before) and thus lemma 4.3 applies to give
$t^{\lambda-\sum\limits_{i=1}^{l-1}\beta_{i}^{\vee}}s_{\beta_{l}}\geq
t^{\lambda-\sum\limits_{i=1}^{l}\beta_{i}^{\vee}}$. Combining these
inequalities, we get $t^{\lambda}x\geq t^{\lambda-\text{wt}(x)}$. Appealing to
eq. 3.1, we thus get $\nu_{w}\geq\lambda-\text{wt}(x)$. This finishes the
proof. ∎

######  Lemma 4.5.

Let $w=t^{\lambda}x\in\widetilde{W}$ such that
$\text{depth}(\lambda)>\widetilde{}\mathcal{M}$. Then
$\nu_{w}=\max\\{\gamma^{+}:t^{\gamma}\leq w\\}$.

###### Proof.

By eq. 3.1, we can assume that $\nu_{w}=\nu(t^{y\mu}z)$ for some
$t^{y\mu}z\in\widetilde{W},w\geq t^{y\mu}z$. Suppose that $z\neq 1$, therefore
the order of $z$ equals $m\geq 2$. Then
$\nu(t^{y\mu}z)=(\frac{1}{m}\sum\limits_{i=1}^{m}z^{i}(y\mu))^{+}$. Note that
$\zeta\cdot\sum\limits_{i=1}^{m}z^{i}(y\mu)=\sum\limits_{i=1}^{m}z^{i}(y\mu)$
for all $\zeta\in W$, so the element $\sum\limits_{i=1}^{m}z^{i}(y\mu)$ lies
on the wall of some Weyl chamber, cf. [5, Chapter 5, Section 3.3, Remark 3].
Therefore $(\frac{1}{m}\sum\limits_{i=1}^{m}z^{i}(y\mu))^{+}$ lies on the wall
of $\mathcal{C}^{+}$, and hence $\nu_{w}$ is singular.

By proposition 4.4, we can assume that $\nu_{w}=\lambda-\epsilon$, with
$\epsilon\leq\text{wt}(x)\leq\text{wt}(w_{0})$. Hence, there is some
$\alpha_{i}\in\Delta$ such that $\langle\alpha_{i},\lambda-\epsilon\rangle=0$,
and thus $\langle
a_{i},\lambda\rangle=\langle\alpha_{i},\epsilon\rangle\leq\widetilde{M}$,
which is a contradiction to hypothesis on $\text{depth}(\lambda)$. Therefore,
$\nu_{w}$ cannot be singular and hence $z=1$. ∎

###  4.3. Toward computing a downward Demazure product

We need to understand the largest translation dominated by $w$ to determine
its associated generic Newton point $\nu_{w}$. In view of proposition 4.4, let
us denote this translation element by $t^{y\gamma}$ for some $y\in W$ and
$\gamma\geq\lambda-\text{wt}(x)$. Of course, this would be equality if
$\lambda$ is superregular. We want to leverage this result to establish such
equality with some linear bound on depth here. The following lemma lay the
path towards proving that.

######  Lemma 4.6.

If $\lambda-2\rho^{\vee}$ is dominant, we have $t^{\lambda}x\geq t^{y\gamma}$
if and only if $t^{\lambda-2\rho^{\vee}}\geq t^{-2\rho^{\vee}}\rhd
t^{y\gamma}$.

###### Proof.

Note that if $\lambda-2\rho^{\vee}$ is dominant,
$\ell(t^{\lambda-2\rho^{\vee}})=\langle
2\rho,\lambda-2\rho^{\vee}\rangle=\ell(t^{\lambda})-\ell(t^{-2\rho^{\vee}})$;
also, since $\lambda\geq\lambda-2\rho^{\vee}$ are dominant coweights,
$t^{\lambda}\geq t^{\lambda-2\rho^{\vee}}$, cf. [25, proof of Proposition
3.5]. Hence, $t^{-2\rho^{\vee}}\rhd t^{\lambda}=t^{\lambda-2\rho^{\vee}}$.

Now, it suffices to show the following:

(a) If $a,b$ are two elements in a Coxeter group and $s$ is a simple
reflection, then $a\geq b$ if and only if $s\rhd a\geq s\rhd b$.

This follows from the well known lifting property of Bruhat order. Note that
we only need to consider the case when $s\rhd a=sa$, i.e. $a\geq sa$. Then
there are two further cases.

  1. (1)

$s\rhd b=b$: in this case, $sb\geq b$. Hence $a\geq b$ is equivalent to
$sa\geq b$, and hence $s\rhd a\geq s\rhd b$.

  2. (2)

$s\rhd b=sb$: in this case, $b\geq sb$. Hence $a\geq b$ is equivalent to
$sa\geq sb$, and hence $s\rhd a\geq s\rhd b$.

∎

In section 4.4 we will focus on computing the downward Demazure product that
appears in lemma 4.6. Before proceeding any further to do that, let us first
sketch what the answer should look like. We first apply lemma 2.4 with
$w=t^{-2\rho^{\vee}},w^{\prime}=w_{0}$ and $v=v^{\prime}=t^{y\mu}$; if we
require that $\lambda$ is of large enough depth (in a sense made precise
below) so that $\gamma$ is dominant regular, we get $w_{0}\rhd
yt^{\gamma}y^{-1}=t^{\gamma}y^{-1}\geq t^{-2\rho^{\vee}}\rhd t^{y\gamma}$ . We
next apply lemma 2.4 with $v^{\prime}=t^{y\gamma},v=t^{\gamma}y^{-1}$ and
$w=w^{\prime}=t^{-2\rho^{\vee}}$. If we now require $\lambda$ to be of
sufficiently large depth so that $\gamma-2\rho^{\vee}$ is also dominant, we
get $t^{-2\rho^{\vee}}\rhd t^{y\gamma}\geq t^{-2\rho^{\vee}}\rhd
t^{\gamma}y^{-1}=t^{\gamma-2\rho^{\vee}}y^{-1}$. Combining these, we see that
whenever $\lambda$ has “large” depth (to be specified later) we have

$t^{\gamma}y^{-1}\geq t^{-2\rho^{\vee}}\rhd t^{y\gamma}\geq
t^{\gamma-2\rho^{\vee}}y^{-1}.$

Therefore, it is natural to expect that this downward Demazure product depends
only on $y$ whenever $\lambda$ has a “large” depth. In the next section, we
quantify this depth condition and show that we indeed get the desired
conclusion.

###  4.4. A technical lemma

For each irreducible Cartan type $X_{n}$, we let
$\mathcal{S}=\mathcal{S}_{X_{n}}$ be twice the sum of all coefficients
appearing in the expression of the highest root in terms of simple roots; in
other words, $\mathcal{S}=\langle\theta,2\rho^{\vee}\rangle$, where $\theta$
is the highest root. This quantity will become relevant for our next lemma. We
record this integer for each type in the table below.

Type | $A_{n}$ | $B_{n}/C_{n}$ | $D_{n}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---
$\mathcal{S}$ | $2n$ | $4n-2$ | $4n-6$ | $11$ | $17$ | $29$ | $11$ | $5$

Note that $\Xi=\widetilde{}\mathcal{M}+\mathcal{S}$ is the final lower bound
that appears in theorem 4.1. We now prove a key lemma that would help us
achieve the harder inequality.

######  Lemma 4.7.

Let $\text{depth}(\lambda)>\Xi$. We continue to assume that
$w=t^{\lambda}x\geq t^{y\gamma}$ such that
$\gamma=\nu_{w}\geq\lambda-\text{wt}(x)$. Then there exists a coweight
$\mu_{y}$ depending only on $y$ such that $t^{-2\rho^{\vee}}\rhd t^{\gamma
y}=t^{\gamma-\mu_{y}}y^{-1}$.

###### Proof.

We begin by noting that we can compute this downward Demazure product from a
specific length additive decomposition of $t^{y\gamma}$. More precisely, we
have the following desiderata

  * •

(D1) a decomposition $t^{y\gamma}=a_{1}t^{\mu_{1}}b_{1}\cdot
a_{2}t^{\mu_{2}}b_{2}$, where $\mu_{i}$ is dominant and
$t^{\mu_{i}}b_{i}\in~{}^{\mathbb{S}}\widetilde{W}$ for $i=1,2$, and

  * •

(D2)
$\ell(t^{y\gamma})=\ell(a_{1}t^{\mu_{1}}b_{1})+\ell(a_{2}t^{\mu_{2}}b_{2})$,
and

  * •

(D3) $a_{1}t^{\mu_{1}}b_{1}$ is the largest element dominated by
$t^{2\rho^{\vee}}$, subject to the first two conditions.

Lemma 2.3 asserts the existence of such decomposition, and then we have
$t^{-2\rho^{\vee}}\rhd t^{\gamma y}=a_{2}t^{\mu_{2}}b_{2}$. We now proceed to
identify these elements $a_{i}t^{\mu_{i}}b_{i}$ in the following three steps.

####  4.4.1. Relating relevant coweights

We first determine how the coweights associated to translation part of these
elements are related. By (D1),
$t^{y\gamma}=a_{1}b_{1}a_{2}t^{a_{2}^{-1}b_{1}^{-1}\mu_{1}+\mu_{2}}b_{2}$, so
we have $\gamma=(a_{2}^{-1}b_{1}^{-1}\mu_{1}+\mu_{2})^{+}$, i.e. there exists
some $v\in W$ such that $\gamma=v(a_{2}^{-1}b_{1}^{-1}\mu_{1}+\mu_{2})$. We
will now show that $v=1$.

Assume the contrary and let $\beta\in\text{Inv}(v)$. Now,
$\mu_{2}=v^{-1}\gamma-a_{2}^{-1}b_{1}^{-1}\mu_{1}$ is dominant, therefore

$\langle\beta,v^{-1}\gamma-a_{2}^{-1}b_{1}^{-1}\mu_{1}\rangle\geq 0.$

By (D3), we have $t^{2\rho^{\vee}}\geq a_{1}t^{\mu_{1}}b_{1}$, thus
$2\rho^{\vee}\geq\mu_{1}$; by a geometric characterization of dominance order,
e.g. see [1, Lemma 12.14], this gives

$\mu_{1}=\sum\limits_{\zeta\in W}a_{\zeta}\zeta\cdot 2\rho^{\vee}~{}\text{for
some}~{}a_{\zeta}\in\mathbb{R}_{\geq 0}~{}\text{such
that}~{}\sum\limits_{\zeta\in W}a_{\zeta}=1.$

Substituting this expression of $\mu_{1}$ in the pairing above and using its
$W$-invariance, we get

$\langle v\beta,\gamma\rangle-\sum_{\zeta\in
W}a_{\zeta}\langle\zeta^{-1}b_{1}a_{2}\beta,2\rho^{\vee}\rangle\geq 0.$

Let $\beta^{\prime}=-v\beta\in\Phi^{+}$. Note that
$\zeta^{-1}b_{1}a_{2}\beta\geq-\theta$ for any element $\zeta\in W$, and
therefore
$\langle\zeta^{-1}b_{1}a_{2}\beta,2\rho^{\vee}\rangle\geq-\langle\theta,2\rho^{\vee}\rangle$.
Putting all these together, we obtain

$\langle\beta^{\prime},\gamma\rangle=-\sum_{\zeta\in
W}a_{\zeta}\langle\zeta^{-1}b_{1}a_{2}\beta,2\rho^{\vee}\rangle\leq\sum\limits_{\zeta\in
W}a_{\zeta}\langle\theta,2\rho^{\vee}\rangle=\mathcal{S}.$

Now recall that $\gamma=\nu_{w}=\lambda-\epsilon$ is dominant with
$\epsilon\leq\text{wt}(x)\leq\text{wt}(w_{0})$. Choose a simple root
$\alpha\leq\beta^{\prime}$. Then we get
$\langle\alpha,\gamma\rangle\leq\langle\beta^{\prime},\gamma\rangle\leq\mathcal{S}$,
and therefore
$\langle\alpha,\lambda\rangle\leq\mathcal{S}+\langle\alpha,\epsilon\rangle\leq\mathcal{S}+\widetilde{}\mathcal{M}$.
Since this yields a contradiction to the depth hypothesis of $\lambda$, we are
done. Therefore $v=1$ and $\gamma=a_{2}^{-1}b_{1}^{-1}\mu_{1}+\mu_{2}$.

####  4.4.2. Showing that certain coweights are regular

Note that $\gamma$ is regular, since for any simple root $\alpha_{i}$ we have
$\langle\alpha_{i},\gamma\rangle=0\implies\langle\alpha_{i},\lambda\rangle=\langle\alpha_{i},\epsilon\rangle\leq\widetilde{}\mathcal{M}$,
thereby contradicting the depth hypothesis imposed on $\lambda$. Hence,
knowing that the translation parts in
$t^{y\gamma}=a_{1}b_{1}a_{2}t^{a_{2}^{-1}b_{1}^{-1}\mu_{1}+\mu_{2}}b_{2}$ are
equal allows us to relate the finite Weyl group components from both sides.
Namely, we have $y=a_{1}b_{1}a_{2}=b_{2}^{-1}$. We can simplify these
relations further in the following way. We observe that $\mu_{2}$ is regular,
since otherwise we can find a simple root $\alpha$ such that
$\langle\alpha,\gamma\rangle=\sum_{\zeta\in
W}a_{\zeta}\langle\alpha,\zeta\cdot
a_{2}^{-1}b_{1}^{-1}2\rho^{\vee}\rangle=\sum_{\zeta\in W}\langle
b_{1}a_{2}\zeta\alpha,2\rho^{\vee}\rangle\leq\sum_{\zeta\in
W}a_{\zeta}\langle\theta,2\rho^{\vee}\rangle=\mathcal{S}$; but this would
yield contradiction to the imposed depth condition as before.

####  4.4.3. Showing that the product describes a dominant chamber alcove

We can now show that $a_{2}=1$. Recall that $t^{-2\rho^{\vee}}\rhd t^{\gamma
y}=a_{2}t^{\mu_{2}}b_{2}$. Let us rewrite
$t^{-2\rho^{\vee}}=w_{0}t^{2\rho^{\vee}}w_{0}=w_{0}t^{2\rho^{\vee}}w_{0}$ and
apply $w_{0}\rhd$ to this equation. This gives

$w_{0}\rhd\\{(w_{0}t^{2\rho^{\vee}}w_{0})\rhd
t^{y\gamma}\\}=w_{0}\rhd(a_{2}t^{\mu_{2}}b_{2}).$

We now apply lemma 2.4 on the left hand side, and use the fact $\mu_{2}$ is
regular in the right hand side. We get

$\\{w_{0}*(w_{0}t^{2\rho^{\vee}}w_{0})\\}\rhd t^{y\gamma}=(w_{0}\rhd
a_{2})t^{\mu_{2}}b_{2}.$

Note that $w_{0}t^{2\rho^{\vee}}w_{0}=w_{0}*t^{2\rho^{\vee}}w_{0}$, and hence
we can use associativity of operation $*$ to rewrite the element in the
parenthesis on the left hand side as
$w_{0}*(w_{0}*t^{2\rho^{\vee}}w_{0})=(w_{0}*w_{0})*t^{2\rho^{\vee}}w_{0}=w_{0}*t^{2\rho^{\vee}}w_{0}=w_{0}t^{2\rho^{\vee}}w_{0}=t^{-2\rho^{\vee}}$.
Therefore, we finally deduce

$t^{-2\rho^{\vee}}\rhd t^{y\gamma}=t^{\mu_{2}}b_{2}.$

This proves our claim.

Therefore, the relations arising from (D1) are

$\gamma=b_{1}^{-1}\mu_{1}+\mu_{2},y=a_{1}b_{1}=b_{2}^{-1}.$ (4.2)

Let us now utilize (D2). It says that we have length additivity in the
decomposition $t^{y\gamma}=a_{1}t^{\mu_{1}}b_{1}\cdot t^{\mu_{2}}b_{2}$, i.e.

$\langle\gamma,2\rho\rangle=\ell(a_{1})+\langle\mu_{1},2\rho\rangle-\ell(b_{1})+\langle\mu_{2},2\rho\rangle-\ell(b_{2}).$

By use of eq. 4.2, the last equation reduces to

$\ell(a_{1}b_{1})=\ell(a_{1})-\ell(b_{1})+\langle\mu_{1}-b_{1}^{-1}\mu_{1},2\rho\rangle.$
(4.3)

Therefore we are led to consider the following subset of $\widetilde{W}$

$\Omega_{y}=\\{at^{\mu}b\in\widetilde{W}:y=ab,at^{\mu}b\leq
t^{2\rho^{\vee}},\ell(ab)=\ell(a)-\ell(b)+\langle\mu-b^{-1}\mu,2\rho\rangle\\}.$

Note that the definition of $\Omega_{y}$ encapsulates all information amongst
$a_{1},b_{1},\mu_{1}$ coming out of the first two items in our desiderata. By
(D3), the element $a_{1}t^{\mu_{1}}b_{1}$ is therefore uniquely specified as
the maximal element of $\Omega_{y}$. Hence, the element $b_{1}^{-1}\mu_{1}$
depends only on $y$. Denoting it by $\mu_{y}$, we see that eq. 4.2 implies
$\mu_{2}=\gamma-\mu_{y}$. This finishes the proof. ∎

###  4.5. Finishing the proof

###### Proof of Theorem 4.1.

By combining lemma 4.6 and lemma 4.7, we observe that our standing assumption
$t^{\lambda}x\geq t^{y\gamma}$ implies

$t^{\lambda-2\rho^{\vee}}x\geq t^{\gamma-\mu_{y}}y^{-1}.$

Let us now choose $\lambda^{\prime}$ to be large enough so that
$\lambda+\lambda^{\prime}$ is superregular, e.g. take
$\lambda^{\prime}=n\rho^{\vee}$ for large $n$. We note that

  1. (1)

$\ell(t^{\lambda^{\prime}}\cdot
t^{\lambda-2\rho^{\vee}}x)=\ell(t^{\lambda+\lambda^{\prime}-2\rho^{\vee}}x)=\langle
2\rho,\lambda+\lambda^{\prime}-2\rho^{\vee}\rangle-\ell(x)=\langle
2\rho,\lambda^{\prime}\rangle+\langle
2\rho,\lambda-2\rho^{\vee}\rangle-\ell(x)=\ell(t^{\lambda^{\prime}})+\ell(t^{\lambda-2\rho^{\vee}}x).$

  2. (2)

$\ell(t^{\lambda^{\prime}}\cdot
t^{\gamma-\mu_{y}}y^{-1})=\ell(t^{\lambda^{\prime}+\gamma-\mu_{y}}y^{-1})=\langle
2\rho,\lambda^{\prime}+\gamma-\mu_{y}\rangle-\ell(y^{-1})=\langle
2\rho,\lambda^{\prime}\rangle+\langle
2\rho,\gamma-\mu_{y}\rangle-\ell(y^{-1})=\ell(t^{\lambda^{\prime}})+\ell(t^{\gamma-\mu_{y}}y^{-1}).$

Hence, we can apply eq. 2.3 in this situation, and get

$t^{\lambda+\lambda^{\prime}-2\rho^{\vee}}x\geq
t^{\gamma+\lambda^{\prime}-\mu_{y}}y^{-1}.$

Now, we use the ‘only if’ part of lemma 4.6 to deduce from the previous
equation

$t^{\lambda+\lambda^{\prime}}x\geq t^{y(\gamma+\lambda^{\prime})}.$

Therefore, by eq. 3.1 we have
$\nu_{t^{\lambda+\lambda^{\prime}}x}\geq\nu_{t^{y(\gamma+\lambda^{\prime})}}$.
Since the formula for the maximal Newton point in [21, Theorem 3.2] applies to
elements of $\widetilde{W}$ having $\lambda+\lambda^{\prime}$ as its dominant
translation part, we get that
$\lambda+\lambda^{\prime}-\text{wt}(x)\geq\gamma+\lambda^{\prime}$, whence
$\lambda-\text{wt}(x)\geq\gamma=\nu_{w}$. Combining this with proposition 4.4,
we get $\nu_{w}=\lambda-\text{wt}(x)$. This finishes the proof. ∎

###  4.6. A remark about Bruhat order on $\widetilde{W}$

######  Proposition 4.8.

Suppose that $\mathbf{G}$ is a quasi-simple split group. Assume
$\text{depth}(\lambda)>\Xi$. Then $ut^{\lambda}v\geq t^{y\gamma}$ for some
dominant $\gamma$ implies that $t^{\lambda}(v\lhd u)\geq t^{\gamma}$.

###### Proof.

Note that $ut^{\lambda}v\geq t^{y\gamma}$ implies
$\nu_{ut^{\lambda}v}\geq\nu_{t^{y\gamma}}$ by eq. 3.1. By theorem 4.1 and
corollary 3.3, we then have $\lambda-\text{wt}(v\lhd u)\geq\gamma$. By
proposition 4.4, we have that $t^{\lambda}(v\lhd u)\geq
t^{\lambda-\text{wt}(v\lhd u)}$. Since $\lambda-\text{wt}(v\lhd u),\gamma$ are
both dominant, we also have that $t^{\lambda-\text{wt}(v\lhd u)}\geq
t^{\gamma}$, cf. [25, proof of Proposition 3.5]. Putting together the last two
inequalities, we finally get $t^{\lambda}(v\lhd u)\geq t^{\gamma}$. ∎

######  Remark 4.9.

Note that if $u=1$, this says that $t^{\lambda}v\geq t^{y\gamma}$ implies
$t^{\lambda}v\geq t^{\gamma}$. In other words, if some $W$-conjugate of a
dominant translation element lies below an element in the dominant chamber, so
does the dominant translation element itself. We point out that in the
presence of proposition 4.4 and the fact that maximal Newton point formula is
available for elements with superregular translation part, this is equivalent
to theorem 4.1. Hence, an independent proof of proposition 4.8 would help us
get rid of the bound imposed in section 4.4; similarly, an improved estimate
for $\mathcal{M}$ would weaken the final depth hypothesis required in theorem
4.1.

##  5\. Weight of the longest element

In this section, we list out reduced quantum reflection decomposition for the
longest element $w_{0}$ in Weyl group $W$ of each irreducible Cartan type. For
an element $x\in W$, the reflection length of $x$ is the smallest number $l$
such that $x$ can be written as a product of $l$ reflections in $W$. We denote
by $\ell_{R}(x)$ the reflection length of $x$. Note that in general
$\ell_{R}(x)\leq\ell_{\downarrow}(x)$, and strict inequality can occur.

We enumerate reflection length for $w_{0}$ for all the simple groups below.

Type | $A_{n}$ | $B_{n}/C_{n}$ | $D_{n}$ | $E_{6}$ | $E_{7}$ | $E_{8}$ | $F_{4}$ | $G_{2}$
---|---|---|---|---|---|---|---|---
$\ell_{R}(w_{0})$ | $\lceil\frac{n}{2}\rceil$ | $n$ | $2\lfloor\frac{n}{2}\rfloor$ | $4$ | $7$ | $8$ | $4$ | $2$

We compute $\text{wt}(w_{0})$ by exhibiting suitable decomposition of $w_{0}$
in each type, and on the way we see that
$\ell_{R}(w_{0})=\ell_{\downarrow}(w_{0})$. We first write down expression of
the longest element affording its reflection length; for the classical types,
we extract this from [4], and for the exceptional ones we can check this
directly by hand or using a computer algebra system such as [26]. Once we find
these decompositions, we see that they only involve reflections corresponding
to quantum roots; furthermore, these expressions do satisfy the length
additivity condition as well. Taken together, these observations establish
that we have found reduced quantum reflection decomposition for $w_{0}$ in
these types. Finally, we add up the coroots associated to the reflections
appearing in each such decompositions and deduce $\text{wt}(w_{0})$ from that.
See remark 5.1 for an alternative recipe for computing $\text{wt}(w_{0})$.

  1. (a)

Type $A_{n}$: here

$w_{0}=\begin{cases}s_{\alpha_{1}+\cdots+\alpha_{2k}}s_{\alpha_{2}+\cdots+\alpha_{2k-1}}\cdots
s_{\alpha_{k}+\alpha_{k+1}},&\text{if $n=2k$;}\\\
s_{\alpha_{1}+\cdots+\alpha_{2k+1}}s_{\alpha_{2}+\cdots+{\alpha_{2k}}}\cdots
s_{\alpha_{k+1}},&\text{if $n=2k+1$.}\end{cases}$

Hence,

$\text{wt}(w_{0})=\begin{cases}\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+\cdots+(k-1)\alpha_{k-1}^{\vee}+k\alpha_{k}^{\vee}+k\alpha_{k+1}^{\vee}+(k-1)\alpha_{k+2}^{\vee}+\cdots+\alpha_{2k}^{\vee},&\text{if
$n=2k$;}\\\
\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+\cdots+k\alpha_{k}^{\vee}+(k+1)\alpha_{k+1}^{\vee}+k\alpha_{k+2}^{\vee}+\cdots+\alpha_{2k+1}^{\vee},&\text{if
$n=2k+1$.}\end{cases}$

  2. (b)

Type $B_{n}$: here

$w_{0}=\begin{cases}s_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{2k}}s_{\alpha_{1}}s_{\alpha_{3}+2\alpha_{4}+\cdots+2\alpha_{2k}}s_{\alpha_{3}}\cdots
s_{\alpha_{2k-1}+2\alpha_{2k}}s_{\alpha_{2k-1}},&\text{if $n=2k$;}\\\
s_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{2k+1}}s_{\alpha_{1}}s_{\alpha_{3}+2\alpha_{4}+\cdots+2\alpha_{2k+1}}s_{\alpha_{3}}\cdots
s_{\alpha_{2k-1}+2\alpha_{2k}+2\alpha_{2k+1}}s_{\alpha_{2k+1}},&\text{if
$n=2k+1$.}\end{cases}$

Hence,

$\text{wt}(w_{0})=\begin{cases}2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+4\alpha_{4}^{\vee}+\cdots+2(k-1)\alpha_{2k-3}^{\vee}+2(k-1)\alpha_{2k-2}^{\vee}\\\
+2k\alpha_{2k-1}^{\vee}+k\alpha_{2k}^{\vee},&\text{if $n=2k$;}\\\
2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+4\alpha_{4}^{\vee}+\cdots+2k\alpha_{2k-1}^{\vee}+2k\alpha_{2k}^{\vee}+(k+1)\alpha_{2k+1}^{\vee},&\text{if
$n=2k+1$.}\end{cases}$

  3. (c)

Type $C_{n}$: here
$w_{0}=s_{2\alpha_{1}+\cdots+2\alpha_{n-1}+\alpha_{n}}s_{2\alpha_{2}+\cdots+2\alpha_{n-1}+\alpha_{n}}\cdots
s_{2\alpha_{n-1}+\alpha_{n}}s_{\alpha_{n}}$, and thus

$\text{wt}(w_{0})=\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+\cdots+n\alpha_{n}^{\vee}.$

  4. (d)

Type $D_{n}$: here

$w_{0}=\begin{cases}s_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{2k-2}+\alpha_{2k-1}+\alpha_{2k}}s_{\alpha_{1}}s_{\alpha_{3}+2\alpha_{4}+\cdots+2\alpha_{2k-2}+\alpha_{2k-1}+\alpha_{2k}}\\\
s_{\alpha_{3}}\cdots
s_{\alpha_{2k-3}+2\alpha_{2k-2}+\alpha_{2k-1}+\alpha_{2k}}s_{\alpha_{2k-3}}s_{\alpha_{2k}},&\text{if
$n=2k$;}\\\
s_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{2k-2}+\alpha_{2k-1}+\alpha_{2k}}s_{\alpha_{1}}s_{\alpha_{3}+2\alpha_{4}+\cdots+2\alpha_{2k-2}+\alpha_{2k-1}+\alpha_{2k}}\\\
s_{\alpha_{3}}\cdots
s_{\alpha_{2k-3}+2\alpha_{2k-2}+2\alpha_{2k-1}+\alpha_{2k}+\alpha_{2k+1}}s_{\alpha_{2k-1}+\alpha_{2k}+\alpha_{2k+1}}s_{\alpha_{2k-1}},&\text{if
$n=2k+1$.}\end{cases}$

Hence,

$\text{wt}(w_{0})=\begin{cases}2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+4\alpha_{4}^{\vee}+\cdots+(2k-2)\alpha_{2k-3}^{\vee}+(2k-2)\alpha_{2k-2}^{\vee}+k\alpha_{2k-1}^{\vee}\\\
+k\alpha_{2k}^{\vee},&\text{if $n=2k$;}\\\
2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+4\alpha_{4}^{\vee}+\cdots+(2k-2)\alpha_{2k-3}^{\vee}+(2k-2)\alpha_{2k-2}^{\vee}+2k\alpha_{2k-1}^{\vee}\\\
+k\alpha_{2k}^{\vee}+k\alpha_{2k+1}^{\vee},&\text{if $n=2k+1$.}\end{cases}$

  5. (e)

Type $E_{6}$: here
$w_{0}=s_{\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}}s_{\alpha_{1}+\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6}}s_{\alpha_{3}+\alpha_{4}+\alpha_{5}}s_{\alpha_{4}}$,
and thus

$\text{wt}(w_{0})=2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+6\alpha_{4}^{\vee}+4\alpha_{5}^{\vee}+2\alpha_{6}^{\vee}.$

  6. (f)

Type $E_{7}$: here
$w_{0}=s_{2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}}s_{\alpha_{2}+\alpha_{3}+2\alpha_{4}+2\alpha_{5}+2\alpha_{6}+\alpha_{7}}s_{\alpha_{2}+\alpha_{3}+2\alpha_{4}+\alpha_{5}}s_{\alpha_{2}}$
$s_{\alpha_{3}}s_{\alpha_{5}}s_{\alpha_{7}}$, and thus

$\text{wt}(w_{0})=2\alpha_{1}^{\vee}+5\alpha_{2}^{\vee}+6\alpha_{3}^{\vee}+8\alpha_{4}^{\vee}+7\alpha_{5}^{\vee}+4\alpha_{6}^{\vee}+3\alpha_{7}^{\vee}.$

  7. (g)

Type $E_{8}$: here
$w_{0}=s_{2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}}s_{2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}}$
$s_{\alpha_{2}+\alpha_{3}+2\alpha_{4}+2\alpha_{5}+2\alpha_{6}+\alpha_{7}}s_{\alpha_{2}+\alpha_{3}+2\alpha_{4}+\alpha_{5}}s_{\alpha_{2}}s_{\alpha_{3}}s_{\alpha_{5}}s_{\alpha_{7}}$,
and thus

$\text{wt}(w_{0})=4\alpha_{1}^{\vee}+8\alpha_{2}^{\vee}+10\alpha_{3}^{\vee}+14\alpha_{4}^{\vee}+12\alpha_{5}^{\vee}+8\alpha_{6}^{\vee}+6\alpha_{7}^{\vee}+2\alpha_{8}^{\vee}.$

  8. (h)

Type $F_{4}$: here
$w_{0}=s_{2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}}s_{\alpha_{2}+2\alpha_{3}+2\alpha_{4}}s_{\alpha_{2}+2\alpha_{3}}s_{\alpha_{2}}$,
and thus

$\text{wt}(w_{0})=2\alpha_{1}^{\vee}+6\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+2\alpha_{4}^{\vee}.$

  9. (i)

Type $G_{2}$: here $w_{0}=s_{3\alpha_{1}+2\alpha_{2}}s_{\alpha_{1}}$, and thus

$\text{wt}(w_{0})=2\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}.$

######  Remark 5.1.

We note that the weights of longest element found above appear in a rather
different context, cf. [20]. In that paper, Lusztig points out that the
coroots appearing as a summand of $\text{wt}(w_{0})$ as above form a so-called
_cascade_ (terminology due to Kostant). He lists out these coroots and their
sums in loc. sit. section 1.2 and section 1.8. In fact, a map $xr_{x}$ is
defined from the set of involutions $\mathcal{I}_{W}$ in an irreducible Weyl
group $W$ in loc. sit. via the notion of cascade, and this is crucially used
in constructing certain lifts of involutions in associated reductive groups.
We compare these two maps defined on the set of involutions in section 8.

##  6\. Covering relation in affine Weyl group

The goal of this section is to prove the following result.

######  Theorem 6.1.

Suppose that $W$ is an irreducible finite Weyl group. Assume that

$depth(\lambda)\geq\begin{cases}3,&\text{if $W\text{is of simply laced
type}$;}\\\ 4,&\text{if $W\text{is of non-simply laced type but not of
type}~{}G_{2}$;}\\\ 6,&\text{if $W\text{is of type}~{}G_{2}$.}\end{cases}$
(6.1)

Then $w:=t^{\lambda}y\gtrdot w^{\prime}:=t^{m\alpha^{\vee}}s_{\alpha}w$ if and
only if one of the following holds.

  1. (1)

$m=1$ and $\ell(s_{\alpha})=\langle 2\rho,\alpha^{\vee}\rangle-1$; in this
case, $w^{\prime}=s_{\alpha}t^{\lambda-\alpha^{\vee}}y$.

  2. (2)

$m=\langle\alpha,\lambda\rangle$ and $\ell(s_{\alpha}y)=\ell(y)+1$; in this
case, $w^{\prime}=t^{\lambda}s_{\alpha}y$.

  3. (3)

$m=\langle\alpha,\lambda\rangle-1$, $\ell(s_{\alpha}y)=\ell(y)-\langle
2\rho,\alpha^{\vee}\rangle+1$; in this case,
$w^{\prime}=t^{\lambda-\alpha^{\vee}}s_{\alpha}y$.

######  Remark 6.2.

We explain how theorem 6.1 implies theorem B. Let us first note that we can
again reduce to the case of irreducible factors as in remark 4.2. Notice that
if $\lambda$ is regular, $\ell(ut^{\lambda}v)=\ell(u)+\ell(t^{\lambda}v)$ by
eq. 2.1. Choose reduced expressions $u=s_{i_{1}}\cdots s_{i_{l}}$ and
$t^{\lambda}v=s_{j_{1}}\cdots s_{j_{k}}\zeta$, where
$s_{i_{p}},s_{j_{q}}\in\mathbb{S}$ for $1\leq p\leq l,1\leq q\leq k$ and
$\zeta\in\Omega$. Then $ut^{\lambda}v=s_{i_{1}}\cdots s_{i_{l}}s_{j_{1}}\cdots
s_{j_{k}}\zeta$ is a reduced expression, and a cocover of $ut^{\lambda}v$ must
be of the form of either $s_{i_{1}}\cdots\widehat{s_{i_{m}}}\cdots
s_{i_{l}}s_{j_{1}}\cdots s_{j_{k}}\zeta$ for some $p\in[1,l]$ or
$s_{i_{1}}\cdots s_{i_{l}}s_{j_{1}}\cdots\widehat{s_{j_{n}}}\cdots
s_{j_{k}}\zeta$ for some $q\in[1,k]$. In other words, any cocover of
$ut^{\lambda}v$ is obtained by

  * •

(C1) either multiplying a cocover of $u$ with $t^{\lambda}v$, or

  * •

(C2) multiplying $u$ with a cocover of $t^{\lambda}v$.

The procedure listed in (C1) is easy to describe. A cocover of $u$ is of the
form $us_{\alpha}$ for some $\alpha\in\Phi^{+}$ such that
$\ell(us_{\alpha})=\ell(u)-1$. This corresponds to case (1) in theorem B. We
claim that the three other cases there, i.e. (2)-(4), come from procedure
described in (C2), and hence they correspond to those listed in theorem 6.1.
Clearly, the third and fourth case in theorem B corresponds respectively to
the second and third case in theorem 6.1. Finally, suppose that
$w^{\prime}:=s_{a}t^{\lambda-\alpha^{\vee}}v$ is a cocover of
$w:=t^{\lambda}v$ such that $uw^{\prime}$ is a cocover of $uw$. By eq. 2.1,

$\ell(uw^{\prime})=\ell(us_{a})+\ell(t^{\lambda-\alpha^{\vee}})-\ell(v)=\ell(us_{a})+\langle
2\rho,\lambda-\alpha^{\vee}\rangle-\ell(v).$

We use dominance of $\lambda-\alpha^{\vee}$ in the last equality; this is true
as
$\langle\alpha_{i},\lambda-a^{\vee}\rangle=\langle\alpha_{i},\lambda\rangle-\langle\alpha_{i},\alpha^{\vee}\rangle\geq
0$ for any simple root $a_{i}$ \- since the maximum value of
$\langle\alpha_{i},\alpha^{\vee}\rangle$ equals $2$ in every Cartan type,
except for $G_{2}$ \- in which case it equals $3$, cf. [5, Chapter VI, Section
3, no. 1]. Note that

$\ell(us_{\alpha})\leq\ell(u)+\ell(s_{a})\leq\ell(u)+\langle
2\rho,\alpha^{\vee}\rangle-1.$ (6.2)

Since $\ell(ut^{\lambda}v)=\ell(u)+\langle 2\rho,\lambda\rangle-\ell(v)$, the
cocover condition $uw\gtrdot uw^{\prime}$ gives

$\ell(u)+\langle 2\rho,\lambda\rangle-\ell(v)-1=\ell(us_{a})+\langle
2\rho,\lambda-\alpha^{\vee}\rangle-\ell(v)\leq\ell(u)+\langle
2\rho,\alpha^{\vee}\rangle-1+\langle
2\rho,\lambda-\alpha^{\vee}\rangle-\ell(v).$

Hence we deduce that all of the inequalities in eq. 6.2 must be equality, and
thus we get $\ell(us_{\alpha})=\ell(u)+\langle 2\rho,\alpha^{\vee}\rangle-1$.
This is exactly the situation listed in the second item in theorem B.

We will only focus on proving the necessity of the above conditions. For the
sufficiency part, the argument in [21, Section 4] toward the end of
proposition $4.2$ can be applied.

We divide our discussion of the proof in the following subsections.

###  6.1. Some useful inequalities

In this subsection, we lay out an estimate of some relevant quantities that we
shall use in the next subsection. We shall resume the assumption on the depth
of $\lambda$ stated in theorem 6.1 throughout our discussion after the first
lemma below.

######  Lemma 6.3.

Suppose $w\gtrdot w^{\prime}$. Then we must have
$w^{\prime}=t^{m\alpha^{\vee}}s_{\alpha}w$ for some affine root $\alpha+m$
with $1\leq m\leq\langle\alpha,\lambda\rangle$.

###### Proof.

Let $w=s_{i_{1}}\cdots s_{i_{l}}$ be a reduced expression, where
$s_{i_{j}}\in\tilde{}\mathbb{S}$. This describes a reduced gallery from
$\mathbf{a}$ to $w\mathbf{a}$ via the chain of adjacent alcoves
$\mathbf{a}\rightarrow s_{i_{1}}\mathbf{a}\rightarrow\cdots\rightarrow
s_{i_{1}}\cdots s_{i_{l}}\mathbf{a}$. Since the gallery is reduced, it cannot
cross any hyperplane twice - hence all the alcoves lie in the dominant
chamber. Let $r_{j}$ be the affine reflection with respect to the common wall
between $\mathbf{a}_{j-1}:=s_{i_{1}}\cdots s_{i_{j-1}}\mathbf{a}$ and
$\mathbf{a}_{j}:=s_{i_{1}}\cdots s_{i_{j}}\mathbf{a}$, that is
$r_{j}=s_{i_{1}}\cdots s_{i_{j-1}}s_{i_{j}}s_{i_{j-1}}\cdots s_{i_{1}}$.

Now suppose that $w^{\prime}=s_{i_{1}}\cdots\widehat{s_{i_{k}}}\cdots
s_{i_{l}}$ is a cocover; as before, let $\\{\mathbf{a}_{j}^{\prime}:1\leq
j\leq l-1\\}$ be the collection of alcoves describing the reduced gallery
corresponding to this expression of $w^{\prime}$ and let $r_{j}^{\prime}$ be
the associated affine reflections. Then we see that $r_{j}=r_{j}^{\prime}$ and
$\mathbf{a}_{j}=\mathbf{a}_{j}^{\prime}$ for $1\leq j\leq k-1$, and the rest
of the gallery for $w^{\prime}$ is the reflection of portion of the gallery
for $w$ from $\mathbf{a}_{k+1}$ onward with respect to the hyperplane
corresponding to $r_{k}$. This is because for $j\geq k$ we have

$\mathbf{a}_{j-1}^{\prime}=s_{i_{1}}\cdots s_{i_{k-1}}s_{i_{k+1}}\cdots
s_{i_{j}}\mathbf{a}=s_{i_{1}}\cdots s_{i_{k-1}}s_{i_{k}}s_{i_{k-1}}\cdots
s_{i_{1}}(s_{i_{1}}\cdots s_{i_{j}}\mathbf{a})=r_{k}(\mathbf{a}_{j}).$

In particular, $w^{\prime}=r_{k}w$.

Therefore, to create a cocover of $w$, we must reflect $w\mathbf{a}$ with
respect to some hyperplane $H_{\alpha-m}$ lying between $\mathbf{a}$ and
itself. Since the number of hyperplanes between the alcove $w\mathbf{a}$ and
the wall $H_{\alpha}$ is $\langle\alpha,\lambda\rangle$, this restricts the
possibility of $m$ to asserted values above. ∎

Now,
$w^{\prime}=t^{m\alpha^{\vee}}s_{\alpha}w=s_{\alpha}t^{\lambda-m\alpha^{\vee}}y$.
By lemma 6.3, the coweight associated to the translation part of $w^{\prime}$
belongs to following list of coweights:

$\lambda-\alpha^{\vee},\lambda-2\alpha^{\vee},\cdots,\lambda-(\langle\alpha,\lambda\rangle-1)\alpha^{\vee}=s_{\alpha}(\lambda-\alpha^{\vee}),\lambda-\langle\alpha,\lambda\rangle\alpha^{\vee}=s_{\alpha}(\lambda).$
(6.3)

Let us now define

$k=k_{\lambda,\alpha}:=\text{max}\\{m:\lambda-m\alpha^{\vee}\in\overline{\mathcal{C}^{+}},1\leq
m\leq\langle\alpha,\lambda\rangle,m\in\mathbb{Z}\\}.$

We now give a lower bound for $k$.

######  Lemma 6.4.

We have that

$k\geq\begin{cases}1,&\text{if $W\text{is of simply laced type}$;}\\\
2,&\text{if $W\text{is of non-simply laced type}$.}\end{cases}$

###### Proof.

By the definition of $k$, there is a real number $\widetilde{m}$ with
$k\leq\widetilde{m}<k+1$ such that $\lambda-\widetilde{m}\alpha^{\vee}$ lies
on at least one of the walls of $\mathcal{C}^{+}$. It follows that for some
simple root $\alpha_{i}$, we have
$\langle\alpha_{i},\lambda-\widetilde{m}\alpha^{\vee}\rangle=0$. Therefore,
$\widetilde{m}=\frac{\langle\alpha_{i},\lambda\rangle}{\langle\alpha_{i},\alpha^{\vee}\rangle}$
\- hence giving
$k=\lfloor\frac{\langle\alpha_{i},\lambda\rangle}{\langle\alpha_{i},\alpha^{\vee}\rangle}\rfloor$.
Recalling the estimate about maximum value of
$\langle\alpha^{\vee},\alpha_{i}\rangle$ in each Cartan type, we get the
desired bound on $k$ from the depth condition on $\lambda$. ∎

Our next lemma estimates the length of the translation elements corresponding
to the coweights in the list (6.1). This proof closely follows a part of the
proof of [21, Proposition 4.2]. We include it here for completeness’ sake.

######  Lemma 6.5.

Suppose that for some integer $m\in[1,\langle\alpha,\lambda\rangle]$,
$\lambda-m\alpha^{\vee}$ lies in the closure of a chamber different from
$\mathcal{C}^{+}$ or $\mathcal{C}_{s_{\alpha}}$. Then
$\ell(t^{\lambda-m\alpha^{\vee}})\leq\ell(t^{\lambda-k\alpha^{\vee}})$.

###### Proof.

Following [18], define the function $f:\mathbb{R}\rightarrow\mathbb{R}_{\geq
0}$ by linearly extending the function
$\widetilde{f}:\mathbb{Z}\rightarrow\mathbb{Z}_{\geq 0}$ defined by

$\widetilde{f}(m)=\ell(t^{\lambda-m\alpha^{\vee}}).$

More precisely, $f$ is the function associated to the graph obtained by
joining $\widetilde{f}(m)$ and $\widetilde{f}(m+1)$ by the line passing
through them for every $m\in\mathbb{Z}$. It is easy to see that $f$ is a
convex function, cf. [18], proof of proposition 4; in fact, it is a piece-wise
linear function, and is given by a single expression linear in $m$ as long as
$\lambda-m\alpha^{\vee}$ is in the same chamber. Since
$\lambda,\lambda-\alpha^{\vee}$ are both dominant due to the imposed depth
hypothesis, we see that $f(1)=\langle
2\rho,\lambda-\alpha^{\vee}\rangle,f(\langle\alpha,\lambda\rangle)=\langle
2\rho,\lambda\rangle$.

We now show that $f$ is decreasing around $1$ and increasing around
$\langle\alpha,\lambda\rangle$. For example, if $m\in(0,\frac{3}{2})$ and
$\alpha_{i}\in\Delta$ then by our earlier discussion about maximum value of
$\langle\alpha_{i},\alpha^{\vee}\rangle$ we have

$\langle\alpha_{i},\lambda-m\alpha^{\vee}\rangle=\langle\alpha_{i},\lambda\rangle-m\langle\alpha_{i},\alpha^{\vee}\rangle\geq\begin{cases}\text{depth}(\lambda)-3,&\text{if
$W\text{is not of type}~{}G_{2}$;}\\\
\text{depth}(\lambda)-\frac{9}{2},&\text{if $W\text{is of
type}~{}G_{2}$.}\end{cases}$

Hence, our depth hypothesis ensures that
$\langle\alpha_{i},\lambda-m\alpha^{\vee}\rangle\geq 0$. Therefore,
$\lambda-m\alpha^{\vee}$ is in $\overline{\mathcal{C}^{+}}$ whenever
$m\in(0,\frac{3}{2})$, and thus

$f(m)=\langle 2\rho,\lambda-m\alpha^{\vee}\rangle=\langle
2\rho,\lambda\rangle-m\langle 2\rho,\alpha^{\vee}\rangle$ (6.4)

is clearly decreasing in this neighbourhood. Similarly, we note that if
$m\in(\langle\alpha,\lambda\rangle-\frac{1}{2},\langle\alpha,\lambda\rangle+\frac{1}{2})$
and $\alpha_{i}\in\Delta$ we have

$\displaystyle\langle\alpha_{i},s_{\alpha}(\lambda-m\alpha^{\vee})\rangle=\langle\alpha_{i},\lambda\rangle-(\langle\alpha,\lambda\rangle-m)\langle\alpha_{i},\alpha^{\vee}\rangle\geq\begin{cases}\text{depth}(\lambda)-1,&\text{if
$W\text{is not of type}~{}G_{2}$;}\\\
\text{depth}(\lambda)-\frac{3}{2},&\text{if $W\text{is of
type}~{}G_{2}$.}\end{cases}$

Again, our depth hypothesis ensures that
$\langle\alpha_{i},s_{\alpha}(\lambda-m\alpha^{\vee})\rangle>0$. Hence
$\lambda-m\alpha^{\vee}\in\mathcal{C}_{s_{\alpha}}$ whenever
$m\in(\langle\alpha,\lambda\rangle-\frac{1}{2},\langle\alpha,\lambda\rangle+\frac{1}{2})$,
and thus

$f(m)=\langle 2\rho,s_{a}(\lambda-m\alpha^{\vee})\rangle=\langle
2\rho,\lambda\rangle-(\langle\alpha,\lambda\rangle-m)\langle
2\rho,\alpha^{\vee}\rangle=\langle 2\rho,s_{a}(\lambda)\rangle+m\langle
2\rho,\alpha^{\vee}\rangle$ (6.5)

is clearly increasing in the neighbourhood.

Since the intervals $(0,\frac{3}{2})$ and
$(\langle\alpha,\lambda\rangle-\frac{1}{2},\langle\alpha,\lambda\rangle+\frac{1}{2})$
are disjoint, by convexity of $f$ we infer the global shape of the graph of
$f$. Namely, $f(m)$ steadily decreases and is defined by eq. 6.4 as long as
$\lambda-m\alpha^{\vee}\in\overline{\mathcal{C}^{+}}$; as
$\lambda-m\alpha^{\vee}$ traverses through other chambers with value of $m$
increasing, it (weakly) decreases further until it reaches local minimum, then
it starts increasing; and finally, once $\lambda-m\alpha^{\vee}$ enters
$\overline{\mathcal{C}_{s_{\alpha}}}$, $f(m)$ is defined by eq. 6.5 and
steadily increases. Therefore, if we define

$k^{\prime}=k^{\prime}_{\lambda,\alpha}:=\text{min}\\{m:\lambda-m\alpha^{\vee}\in\overline{\mathcal{C}_{s_{\alpha}}},1\leq
m\leq\langle\alpha,\lambda\rangle,m\in\mathbb{Z}\\},$

we must have $f(m)\leq\text{max}\\{(f(k),f(k^{\prime})\\}$ for all
$m\in[k,k^{\prime}]\cap\mathbb{Z}$. Since
$\lambda-k\alpha^{\vee}=s_{\alpha}(\lambda-k^{\prime}\alpha^{\vee})$ by
symmetry, we have $f(k)=f(k^{\prime})=\ell(t^{\lambda-k\alpha^{\vee}})$. Hence
we are done. ∎

Recall from lemma 6.3 that $m\leq\langle\alpha,\lambda\rangle$. We now define
$z\in W$ to be such that
$\lambda-m\alpha^{\vee}\in\overline{\mathcal{C}_{z}}$. Note that if
$\lambda-m\alpha^{\vee}$ is singular, $z$ is not uniquely specified by this
condition. However, the precise choice of $z$ would be immaterial in what
follows, and henceforth pick one such $z$ satisfying the above condition. Let
us denote the left and right weak Bruhat order by $\prec_{\text{left}}$ and
$\prec_{\text{right}}$ respectively.

######  Lemma 6.6.

We have $z\prec_{\text{right}}s_{\alpha}$. As a consequence,
$\ell(s_{\alpha}z)+\ell(z)=\ell(s_{\alpha})$.

###### Proof.

It suffices to show that $\text{Inv}(z^{-1})\subset\text{Inv}(s_{\alpha})$,
because this is equivalent to $z^{-1}\prec_{\text{left}}s_{\alpha}$, which in
turn is equivalent to the claim above.

We start by noting that for $\beta\in\text{Inv}(z^{-1})$, we have
$\langle\beta,\lambda-m\alpha^{\vee}\rangle\leq 0$. This follows from
rewriting $\langle\beta,\lambda-m\alpha^{\vee}\rangle=\langle
z^{-1}\beta,z^{-1}(\lambda-m\alpha^{\vee})\rangle$ and noting that
$z^{-1}(\lambda-m\alpha^{\vee})$ is a dominant coweight and
$z^{-1}\beta\in-\Phi^{+}$. Therefore we get $\langle\beta,\lambda\rangle\leq
m\langle\beta,\alpha^{\vee}\rangle$. Since $\lambda$ dominant regular,
$\langle\beta,\lambda\rangle>0$ and hence
$\langle\beta,\alpha^{\vee}\rangle>0$ as well. Dividing out both side of the
inequality by $\langle\beta,\alpha^{\vee}\rangle$, we get
$m\geq\frac{\langle\beta,\lambda\rangle}{\langle\beta,\alpha^{\vee}\rangle}$.
Combining this with lemma 6.3, we obtain

$\langle\alpha,\lambda\rangle\geq\frac{\langle\beta,\lambda\rangle}{\langle\beta,\alpha^{\vee}\rangle}.$

But then
$\langle\beta,\alpha^{\vee}\rangle\langle\alpha,\lambda\rangle\geq\langle\beta,\lambda\rangle$,
whence $\langle
s_{\alpha}(\beta),\lambda\rangle=\langle\beta-\langle\beta,\alpha^{\vee}\rangle\alpha,\lambda\rangle\leq
0$. Since $\lambda$ is dominant regular, this gives
$s_{\alpha}(\beta)\in-\Phi^{+}$, hence $\beta\in\text{Inv}(s_{\alpha})$. Thus
we have shown $\text{Inv}(z^{-1})\subset\text{Inv}(s_{\alpha})$.

Finally, let us note that

$\ell(s_{\alpha}z)+\ell(z)=\ell(s_{\alpha})-\ell(z)+2|\text{Inv}(s_{\alpha})^{c}\cap\text{Inv}(z^{-1})|+\ell(z)=\ell(s_{\alpha})+2|\text{Inv}(s_{\alpha})^{c}\cap\text{Inv}(z^{-1})|=\ell(s_{\alpha}).$

This finishes the proof. ∎

###  6.2. Two distinct possibilities

In this subsection we further pin down the possible values of $z$. Recall that
$z^{-1}(\lambda-m\alpha^{\vee})$ is dominant; let us temporarily denote this
coweight by $\mu$. Let $J\subset\mathbb{S}$ be such that
$W_{J}=\text{Stab}(\mu)$. Abbreviate $\zeta=z^{-1}y$. By standard fact about
Coxeter groups, $\zeta$ has an unique factorization as
$\zeta=\zeta_{J}\zeta^{J}$ with $\zeta_{J}\in W_{J}$ and
$\zeta^{J}\in~{}^{J}W$. Note that

$w^{\prime}=s_{\alpha}t^{\lambda-m\alpha^{\vee}}y=s_{\alpha}zt^{\mu}z^{-1}y=s_{\alpha}zt^{\mu}\zeta_{J}\zeta^{J}=s_{\alpha}z\zeta_{J}t^{\mu}\zeta^{J}.$
(6.6)

######  Lemma 6.7.

The alcove $t^{\mu}\zeta^{J}\mathbf{a}$ is in the dominant chamber.

###### Proof.

Define
$\bar{\omega}:=\frac{1}{n}\sum_{i=1}^{n}\frac{\omega_{i}^{\vee}}{n_{i}}$,
where $n_{i}$ is the coefficient of $\alpha_{i}$ in the highest root $\theta$.
Since $\mu+\zeta^{J}\bar{\omega}$ is centroid of the alcove
$t^{\mu}\zeta^{J}\mathbf{a}$, we can make the following observation:
$t^{\mu}\zeta^{J}\mathbf{a}$ is in the dominant chamber if and only if
$\mu+\zeta^{J}\bar{\omega}$ is dominant.

We now show that the latter statement holds true. Note that if $\alpha_{i}\in
J$, then
$\langle\alpha_{i},\mu+y^{J}\bar{\omega}\rangle=\langle(y^{J})^{-1}\alpha_{i},\bar{\omega}\rangle>0$,
since by definition
$s_{i}y^{J}>y^{J}\implies(y^{J})^{-1}s_{i}>(y^{J})^{-1}\implies(y^{J})^{-1}\alpha_{i}\in\Phi^{+}$.
Now let $\alpha_{i}\in\Delta\setminus J$, then
$\langle\alpha_{i},\mu\rangle\geq 1$; since
$(y^{J})^{-1}\alpha_{i}\geq-\theta$, we have
$\langle\alpha_{i},\mu+y^{J}\bar{\omega}\rangle\geq
1+\langle-\theta,\bar{\omega}\rangle\geq 1-1=0$. This proves the claim. ∎

Applying the previous lemma, we get from eq. 6.6

$\ell(w^{\prime})=\ell(s_{\alpha}z\zeta_{J})+\langle
2\rho,\mu\rangle-\ell(\zeta^{J}).$

Therefore the cocover condition gives

$\displaystyle\langle
2\rho,\lambda\rangle-\ell(y)-1=\ell(s_{\alpha}z\zeta_{J})+\langle
2\rho,\mu\rangle-\ell(\zeta^{J}).$

In other words, we get

$\ell(t^{\lambda})-\ell(t^{\mu})=1+\ell(s_{\alpha}z\zeta_{J})+\ell(y)-\ell(\zeta^{J}).$
(6.7)

Write $y=z\cdot z^{-1}y=z\zeta_{J}\zeta^{J}$. Note that eq. 2.2 then gives

$\ell(y)=\ell(z\cdot\zeta_{J}\zeta^{J})=\ell(z)+\ell(\zeta_{J})+\ell(\zeta^{J})-2|\text{Inv}(z)\cap\text{Inv}((\zeta^{J})^{-1}\zeta_{J}^{-1})|.$

Similarly,

$\ell(s_{a}z\zeta_{J})=\ell(s_{\alpha}z)-\ell(\zeta_{J})+2|\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(\zeta_{J}^{-1})|$

Hence we can rewrite the right hand side of eq. 6.7 as

$\displaystyle
1+\ell(s_{\alpha}z)+\ell(z)+2\\{|\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(\zeta_{J}^{-1})|-|\text{Inv}(z)\cap\text{Inv}((\zeta^{J})^{-1}\zeta_{J}^{-1})|\\}.$

Since $\zeta_{J}^{-1}\prec_{\text{left}}(\zeta^{J})^{-1}\zeta_{J}^{-1}$, we
have that
$\text{Inv}(\zeta_{J}^{-1})\subset\text{Inv}((\zeta^{J})^{-1}\zeta^{-1})$.
Combining this with the fact that $|A|-|B|\leq|A\setminus B|$ for two sets
$A,B$, we therefore conclude that

$\ell(t^{\lambda})-\ell(t^{\mu})\leq
1+\ell(s_{\alpha}z)+\ell(z)+2|\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(z)^{c}\cap\text{Inv}(\zeta_{J}^{-1})|.$
(6.8)

######  Lemma 6.8.

We have
$\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(z)^{c}\cap\text{Inv}(\zeta_{J}^{-1})=\emptyset$.

###### Proof.

The argument here is similar to the proof of lemma 6.6. Suppose
$\beta\in\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(z)^{c}\cap\text{Inv}(\zeta_{J}^{-1})\subset\text{Inv}(s_{\alpha}z)^{c}\cap\text{Inv}(z)^{c}\cap\Phi_{J}^{+}$.
Thus $\langle\beta,z^{-1}(\lambda-m\alpha^{\vee})\rangle=0$, hence $\langle
z\beta,\lambda\rangle=m\langle z\beta,\alpha^{\vee}\rangle$; since $z\beta$ is
a positive root, both sides are positive and

$m=\frac{\langle z\beta,\lambda\rangle}{\langle
z\beta,\alpha^{\vee}\rangle}\leq\langle\alpha,\lambda\rangle.$

Therefore, $\langle z\beta,\lambda\rangle-\langle
z\beta,\alpha^{\vee}\rangle\langle\alpha,\lambda\rangle=\langle
z\beta,s_{\alpha}(\lambda)\rangle\leq 0$. Since
$\beta\in\text{Inv}(s_{\alpha}z)^{c}$, we have
$s_{\alpha}(z\beta)\in\Phi^{+}$. Hence we have $\langle
z\beta,s_{\alpha}(\lambda)\rangle=\langle
s_{\alpha}(z\beta),\lambda\rangle\leq 0$, but that is a contradiction since
$\lambda$ is dominant regular. This shows that the purported set must be empty
and we are done. ∎

Therefore, combining eq. 6.8 with lemma 6.6 and lemma 6.8 gives

$\ell(t^{\lambda})-\ell(t^{\mu})\leq 1+\ell(s_{\alpha})$ (6.9)

Now, suppose that $z\neq 1,s_{\alpha}$. By lemma 6.5 this gives

$\ell(t^{\lambda})-\ell(t^{\mu})\geq\ell(t^{\lambda})-\ell(t^{\lambda-k\alpha^{\vee}})=k\langle
2\rho,\alpha^{\vee}\rangle.$

Combining this with eq. 6.9, we get

$k\langle 2\rho,\alpha^{\vee}\rangle\leq 1+\ell(s_{\alpha})\leq\langle
2\rho,\alpha^{\vee}\rangle$

This gives $k=1$. For the non-simply laced types, this is a contradiction with
previously established lower bound in lemma 6.4. Therefore in such cases, we
get that $z\in\\{1,s_{\alpha}\\}$.

Before going forward, we make the following observation about a simply laced
root system:

if $\langle\beta,\alpha^{\vee}\rangle=2$ for a fixed coroot $\alpha$, then
$\beta=\alpha$.

This can be checked directly in type $A_{n}$ and $D_{n}$, where the positive
roots are given by $\\{e_{i}-e_{j}:1\leq i<j\leq n\\}$ and $\\{e_{i}\pm
e_{j}:1\leq i<j\leq n\\}\cup\\{e_{i}+e_{n}:1\leq i<n\\}$ respectively. Since
root systems of type $E_{6},E_{7}$ arise as subsystem of type $E_{8}$, we just
give an argument for root system of type $E_{8}$ to conclude. Note that for
root system of type $E_{8}$, the positive roots are of two kinds - given by
$\\{\pm e_{i}+e_{j}:1\leq i<j\leq 8\\}$, and
$\\{\frac{1}{2}(e_{8}+\sum_{i=1}^{7}(-1)^{\nu(i)}e_{i})$:
$\sum_{i=1}^{7}\nu(i)\in 2\mathbb{Z}\\}$. We can easily see that pairing
between a root $\gamma$ from the first set with a coroot $\alpha^{\vee}$
corresponding to elements of either sets cannot be $2$ unless $\gamma=\alpha$.
The remaining possibility is
$\langle\frac{1}{2}(e_{8}+\sum_{i=1}^{7}(-1)^{\nu_{1}(i)}e_{i}),\frac{1}{2}(e_{8}+\sum_{i=1}^{7}(-1)^{\nu_{2}(i)}e_{i})\rangle=2$,
but that gives $\langle
e_{8}+\sum_{i=1}^{7}(-1)^{\nu_{1}(i)}e_{i},e_{8}+\sum_{i=1}^{7}(-1)^{\nu_{2}(i)}e_{i}\rangle=8$
\- therefore forcing $\nu_{1}=\nu_{2}$.

We now resume the discussion about estimating $k$ and restrict ourselves to
the simply laced types. Recall that $k=1$. This means that
$\lambda-\alpha^{\vee}\in\overline{\mathcal{C}^{+}}$, but
$\lambda-2\alpha^{\vee}\notin\overline{\mathcal{C}^{+}}$; hence there is a
$\beta\in\Phi^{+}$ such that $\langle\beta,\lambda-2\alpha^{\vee}\rangle\leq
0$. By the depth condition, this yields $3\leq\langle\beta,\lambda\rangle\leq
2\langle\beta,\alpha^{\vee}\rangle$. Therefore
$\langle\beta,\alpha^{\vee}\rangle=2$, and $\langle\beta,\lambda\rangle$ is
equal to either $3$ or $4$. But this gives $\beta=\alpha$, and hence
$\langle\alpha,\lambda\rangle$ is either $3$ or $4$. If
$\langle\alpha,\lambda\rangle=3$, the relevant coweights are
$\lambda-\alpha^{\vee},\lambda-2\alpha^{\vee}=s_{\alpha}(\lambda-\alpha^{\vee}),\lambda-3\alpha^{\vee}=s_{\alpha}(\lambda)$;
the first one is in $\mathcal{C}^{+}$ by the depth condition, and the last two
are in $\mathcal{C}_{s_{\alpha}}$. If $\langle\alpha,\lambda\rangle=4$, then
the relevant coweights are
$\lambda-\alpha^{\vee},\lambda-2\alpha^{\vee},\lambda-3\alpha^{\vee}=s_{\alpha}(\lambda-\alpha^{\vee}),\lambda-4\alpha^{\vee}=s_{\alpha}(\lambda)$;
the first one is in $\mathcal{C}^{+}$, the last two are in
$\mathcal{C}_{s_{\alpha}}$ and $\lambda-2\alpha^{\vee}$ lies on the wall
$H_{\alpha}$, so it lies in both $\overline{\mathcal{C}^{+}}$ and
$\overline{\mathcal{C}_{s_{\alpha}}}$.

We summarize the content of this subsection as follows.

######  Proposition 6.9.

If $s_{\alpha}t^{\lambda-m\alpha^{\vee}}y$ is a cocover of $t^{\lambda}y$,
then $\lambda-m\alpha^{\vee}$ is either in $\overline{\mathcal{C}^{+}}$ or
$\overline{\mathcal{C}_{s_{\alpha}}}$.

###  6.3. Finishing the proof

###### Proof of Theorem 6.1.

We deal with the two cases found above.

  1. (1)

When $\lambda-m\alpha^{\vee}$ is in $\overline{\mathcal{C}^{+}}$: in this
case, eq. 6.9 gives us

$\displaystyle\langle 2\rho,\lambda\rangle-\langle
2\rho,\lambda-m\alpha^{\vee}\rangle\leq 1+\ell(s_{\alpha})\leq\langle
2\rho,\alpha^{\vee}\rangle.$

Therefore we get $m\leq 1$; hence $m=1$ and $1+\ell(s_{\alpha})=\langle
2\rho,\alpha^{\vee}\rangle$. This corresponds to the first case in theorem
6.1.

  2. (2)

When $\lambda-m\alpha^{\vee}$ is in $\overline{\mathcal{C}_{s_{\alpha}}}$: in
this case, eq. 6.9 gives us

$\langle 2\rho,\lambda\rangle-\langle
2\rho,s_{\alpha}(\lambda-m\alpha^{\vee})\rangle\leq 1+\ell(s_{\alpha}).$
(6.10)

Since $\ell(s_{a})\leq\langle 2\rho,\alpha^{\vee}\rangle-1$, we thus get

$(\langle\alpha,\lambda\rangle-m)\langle 2\rho,\alpha^{\vee}\rangle\leq\langle
2\rho,\alpha^{\vee}\rangle.$

Therefore we get $\langle\alpha,\lambda\rangle-m\leq 1$. Since
$m\leqslant\langle\alpha,\lambda\rangle$, this means that either
$m=\langle\alpha,\lambda\rangle-1$ or $m=\langle\alpha,\lambda\rangle$.

Suppose first that it is the former case. Then substituting this value of $m$
in eq. 6.10 yields $\ell(s_{\alpha})=\langle 2\rho,\alpha^{\vee}\rangle-1$.
Note that $\lambda-m\alpha^{\vee}=s_{\alpha}(\lambda-\alpha^{\vee})$ is
regular (since $\lambda-\alpha^{\vee}$ is regular dominant due to the depth
hypothesis) and thus $J=\emptyset$; substituting
$\mu=s_{\alpha}(\lambda-\alpha^{\vee}),z=s_{\alpha},\zeta_{J}=1,\zeta^{J}=s_{\alpha}y$
in eq. 6.7 therefore produces $\langle
2\rho,\alpha^{\vee}\rangle=1+\ell(y)-\ell(s_{\alpha}y)$, whence we get
$\ell(s_{\alpha}y)=\langle
2\rho,\alpha^{\vee}\rangle-1-\ell(y)=\ell(s_{\alpha})-\ell(y)$. This
corresponds to the third case in theorem 6.1.

Now, if it is the latter case we can carry out a similar substitution in eq.
6.7 to get $0=1+\ell(y)-\ell(s_{\alpha}y)$; eq. 6.10 does not yield any
information in this case. This gives $\ell(s_{a}y)=\ell(y)+1$ and hence it
corresponds to the second case in theorem 6.1. This completes the proof.

∎

######  Remark 6.10.

We know a posteriori that only the first and the last two choices from the
string of coweights in eq. 6.3 are viable - so we must necessarily put a depth
condition on $\lambda$ to ensure that $\lambda-\alpha^{\vee}$ is dominant; in
that case, the first coweight is in the dominant chamber and the last two are
in $\mathcal{C}_{s_{\alpha}}$. In other words, the least stringent hypothesis
on $\lambda$ under which one can expect to prove a result like this would be
that $\lambda-\alpha^{\vee}$ is dominant. The depth condition that we impose
above for the simply laced types is almost as weak an assumption as this, in
the sense that we require $\lambda-\alpha^{\vee}$ to be _dominant regular_.
Calculations in small rank seem to provide evidence for our this speculation.
In other words, we suspect that the depth hypothesis can be lowered to $2$
uniformly in all the cases.

###  6.4. Admissible subsets of $\widetilde{W}$

A useful description of the admissible set is given in [12] in terms of weight
of minimal length paths in the quantum Bruhat graph. This relies on the
characterization of covering relation in $\widetilde{W}$ as established in
[21, Proposition 4.2] and hence as such it necessarily brings into picture the
superregularity condition, cf. [12, Proposition 3.3]. Since we can strengthen
the result about covering relation, we automatically get the following
improvement in the description of the admissible set.

######  Proposition 6.11.

Suppose that $W$ is an irreducible Weyl group. Assume that

$depth(\mu)\geq\begin{cases}3,&\text{if $W\text{is of simply laced type}$;}\\\
4,&\text{if $W\text{is of non-simply laced type but not of type}~{}G_{2}$.}\\\
6,&\text{if $W\text{is of type}~{}G_{2}$.}\end{cases}$

Let $\lambda$ be dominant and assume that

$\langle\rho,\mu-\lambda\rangle<\begin{cases}\lceil\frac{\text{depth}(\mu)-3}{2}\rceil,&\text{if
$W$ is of simply laced type;}\\\
\lceil\frac{\text{depth}(\mu)-4}{2}\rceil,&\text{if $W$ is of non-simply laced
type but not of type $G_{2}$;}\\\
\lceil\frac{\text{depth}(\mu)-6}{2}\rceil,&\text{if $W$ is of type
$G_{2}$.}\end{cases}$

Then $xt^{\lambda}y\in\operatorname{Adm}(\mu)$ if and only if
$\text{weight}(x,y^{-1})\leq\mu-\lambda$.

The proof is of this proposition is identical to the argument made in [12],
cf. section 3.3 and proposition 3.3 in loc. sit. and hence we do not repeat it
here. The additional ingredient is theorem 6.1, and the conditions on $\mu$
and $\lambda$ are a direct reflection of that.

##  7\. Dimension formula for $X(\mu,b)$

In this section, we show that the dimension formula provided in [12, Theorem
6.1] holds in the absence of the superregularity condition imposed there. Our
argument closely follows the proof scheme established in [12]. Therefore, we
replace most proofs with references to the corresponding arguments made in
loc. sit. and only point out how to bypass certain steps that will enable us
to drop the superregularity condition.

######  Proposition 7.1.

Suppose that $\mu$ is regular. Then

$d_{\text{Adm}(\mu)}(b)=\langle\rho,\mu-\nu([b])\rangle-\frac{1}{2}\mathrm{def}_{\mathbf{G}}(b)+\frac{1}{2}\ell(w_{0})-\frac{1}{2}\min\\{d_{\Gamma}(x,xw_{0}):x\in
W\\}.$ (7.1)

###### Proof.

We first show that

(i) if $xt^{\lambda}y\in\text{Adm}(\mu)$, then
$xt^{\lambda+\mu^{\prime}}y\in\text{Adm}(\mu+\mu^{\prime})$ for any dominant
$\mu^{\prime}$.

We write $xt^{\lambda+\mu^{\prime}}y=xt^{\lambda}yy^{-1}t^{\mu^{\prime}}y$,
and note that $y^{-1}t^{\mu^{\prime}}y\in\text{Adm}(\mu^{\prime})$ by
definition. Hence the claim follows from an application of theorem 2.1.

Now we argue that statement (a) in the course of the proof in loc. sit. holds
true in the absence of the superregularity condition. In other words, we need
to show that

(ii) if $xt^{\lambda}y\in\text{Adm}(\mu)$ where $\mu$ is regular, then
$\langle\rho,\text{weight}(x,y^{-1})\rangle\leq\langle\rho,\mu-\lambda\rangle$.

Let $xt^{\lambda}y\in\text{Adm}(\mu)$. Choose $\mu^{\prime}$ to be
superregular, e.g. $\mu^{\prime}=n\rho^{\vee}$ for large enough $n$. Apply the
statement in (i) to obtain
$xt^{\lambda+\mu^{\prime}}\in\text{Adm}(\mu+\mu^{\prime})$. Now,
$\mu+\mu^{\prime}$ is superregular and thus statement (a) in loc. sit. applies
to give us

$\langle\rho,\text{weight}(x,y^{-1})\rangle\leq\langle\rho,(\mu+\mu^{\prime})-(\lambda+\mu^{\prime})\rangle=\langle\rho,\mu-\lambda\rangle.$

This finishes the proof of (ii). Therefore, one can use this enhanced version
of statement (a) to get the established upper bound (i.e. the expression in
the right hand side of eq. 7.1) for $d_{\text{Adm}}(\mu)$ by resorting to the
same proof method in loc. sit.

In the remaining part of the proof, the authors construct an explicit
$w\in\text{Adm}(\mu)\cap\mathcal{C}_{x}$ for some $x\in W$ and argue that

$d_{w}(b)=\langle\rho,\mu-\nu([b])\rangle-\frac{1}{2}\text{def}_{\textbf{G}}(b)+\frac{1}{2}\ell(w_{0})-\frac{1}{2}d_{\Gamma}(x,xw_{0}).$
(7.2)

To prove this, the authors employ the description of $\text{Adm}(\mu)$
established in proposition 3.3 in loc. sit. and hence it relies on the
superregularity hypothesis. However, one can bypass this simply by choosing a
different candidate. Namely, let $w=xt^{\mu}(xw_{0})^{-1}$ for some $x\in W$
such that $xw_{0}\geq x$. Note that section 5.6 of loc. sit. provides an
explicit construction of such an element $x\in W$. Then we have $w\leq
xw_{0}t^{\mu}(xw_{0})^{-1}$ by a combination of eq. 2.1 and eq. 2.3. This is
where we use that $\mu$ is regular. Hence, $w\in\text{Adm}(\mu)$. As has been
shown in loc. sit., it now follows from definition that eq. 7.2 holds true for
this element. This completes the proof that eq. 7.1 holds for dominant regular
$\mu$. ∎

###### Proof of theorem C.

We note that once the formula for $d_{\text{Adm}(\mu)}(b)$ in eq. 7.1 is
established, the remaining part of [12] focuses on showing

$\min\\{d_{\Gamma}(x,xw_{0}):x\in W\\}=\ell_{R}(w_{0}).$

However, this is a calculation done purely on the finite Weyl group $W$, and
the superregularity condition does not appear in establishing this.

Finally, the characterization of admissible sets is used once more in the
proof of the theorem in section 6 in loc. sit. to show that
$xt^{\mu}w_{0}x^{-1}\in\text{Adm}(\mu)$ for certain element $x\in W$
satisfying $xw_{0}\geq x$. However, this follows directly as we have shown
above. Hence, the superregularity hypothesis on $\mu$ can be avoided
completely. This finishes the proof of the theorem. ∎

######  Remark 7.2.

Note that since we work with only split groups in this article, we assume
$\sigma=\text{id}$ in quoting statements from loc. sit. However, the
conclusion of theorem C holds for quasi-split groups as well. The additional
ingredient in handling this general case is the existence of an element $x\in
W$ such that $\sigma(x)w_{0}\geq x$. This can be obtained using [12, Theorem
5.1] in the following way. Let $\mathcal{O}$ be the $\sigma$-conjugacy class
of $w_{0}$ and define
$\ell(\mathcal{O})=\min\\{\ell_{R}(w):w\in\mathcal{O}\\}$. Then the
aforementioned theorem asserts that for any finite Coxeter group $W$ and a
length preserving graph automorphism on $W$, we have that

$\ell(w_{0})-\ell_{R}(\mathcal{O})=2\max\\{\ell(x):x\leq\sigma(x)w_{0}\\}.$

Therefore, the existence of our required element is equivalent to the
assertion that $\ell(w_{0})>\ell_{R}(\mathcal{O})$. The latter statement is
true in all types arising from quasi-split groups, cf. section 5.1 in loc.
sit.

######  Remark 7.3.

We remark that [12, Remark 6.2] gives an example where $\mu$ is singular and
$W$ is of type $C_{4}$, and they point out that the dimension formula in
theorem C fails in this case. Therefore, the hypothesis in theorem C is
optimal. However, we do not know if the formula for $d_{\text{Adm}}(\mu)$
given in proposition 7.1 is valid in the absence of regularity condition on
$\mu$.

##  8\. A remark about the weight function

Finally, we would like to explore any possible connection between the function
wt and the notion of cascades. The goal of this section is to prove the
following result.

######  Proposition 8.1.

Let $W$ be an irreducible Weyl group. Then $\text{wt}(x)=r_{x}$ for any
involution $x$ in $W$ if and only if $W$ is of type $A_{n}$.

Here $r_{x}\in\mathbb{Z}\Phi^{\vee}$ is associated to $x$ as per the
construction in [20, Section 1.8], which we recall in the next subsection.

###  8.1. Definition of $r_{x}$

Let us briefly summarize the setup from [20], for more details we refer to
section 1.1 and 1.8 in loc. sit. Let $W$ be an irreducible Weyl group $W$ and
we denote the set of involutions in it by $\mathcal{I}_{W}$. For an element
$x\in\mathcal{I}_{W}$, set $Y_{x}=\\{\lambda\in
X_{*}(T)_{\mathbb{R}}:x(\lambda)=-\lambda\\}$ and define
$\Phi_{x}^{\vee}=\Phi^{\vee}\cap Y_{x}$. One can similarly define
$(X_{x},\Phi_{x})$ using the action of $W$ on $X^{*}(T)_{\mathbb{R}}$. It is
then proved in loc. sit. that $(X_{x},Y_{x},\Phi_{x},\Phi_{x}^{\vee})$ forms a
root system. Furthermore, $\Phi_{x}^{\vee+}=\Phi_{x}^{\vee}\cap\Phi^{\vee+}$
is a set of positive coroots for this system.

Then one inductively defines the following subsets of $X_{x}$. Namely, let
$\mathcal{E}_{x,1}$ be the set of maximal elements in $\Phi_{x}^{+}$ with
respect to the usual dominance order. Then for $i\geq 2$, let
$\mathcal{E}_{x,i}$ to be the maximal elements of

$\\{\alpha\in\Phi_{x}^{+}:\langle\alpha,\beta^{\vee}\rangle=0\text{ for all
}\beta\in\mathcal{E}_{x,1}\cup\cdots\cup\mathcal{E}_{x,i-1}\\}.$

Finally, one defines $\mathcal{E}^{\vee}_{x}=\bigcup_{i\geq
1}\\{\alpha^{\vee}:\alpha\in\mathcal{E}_{x,i}\\}$. Using this set, the
following element of $\mathbb{Z}\Phi^{\vee}$ is defined in loc. sit.

$r_{x}=\sum\limits_{\beta\in\mathcal{E}^{\vee}_{x}}\beta.$

The following result gives a characterization of $r_{x}$ constructed in this
way.

######  Theorem 8.2.

[20, Theorem 0.2] There is a unique map
$\mathcal{I}_{W}\rightarrow\mathbb{Z}\Phi$, $xr_{x}$ such that (i)-(iii) below
hold.

(i) $r_{1}=0$, $r_{s_{\alpha}}=\alpha^{\vee}$ for any $\alpha\in\Delta$;

(ii) for any $x\in\mathcal{I}_{W}$ and $\alpha\in\Delta$ such that
$s_{\alpha}x\neq xs_{\alpha}$, we have
$s_{\alpha}(r_{x})=r_{s_{\alpha}xs_{\alpha}}$;

(iii) for any $x\in\mathcal{I}_{W}$ and $\alpha\in\Delta$ such that
$s_{a}x=xs_{a}$, we have $r_{s_{\alpha}x}=r_{x}+\mathcal{N}\alpha^{\vee}$
where $\mathcal{N}\in\\{-1,0,1\\}$.

If in addition $G$ is simply laced we have $\mathcal{N}\in\\{-1,1\\}$.

###  8.2. The notion of $W$-depth

We now recall some feature about a statistic defined on Coxeter groups that
will be relevant for us in the next subsection. For a Coxeter system
$(W,\mathbb{S})$ and a set of positive roots $\Phi^{+}$ in it, [3, Section
4.6] defines _$W$ -depth_111In fact, this is called _depth_ in [3], [2] and
[23]; we alter the terminology here to avoid any confusion with the notion of
depth of a coweight defined earlier. of an element $\beta\in\Phi^{+}$ by
$\text{dp}(\beta):=\min\\{k:x\cdot\beta\in-\Phi^{+}\text{ for some }x\in
W\text{ with }\ell(x)=k\\}.$

It is a classically known fact that one can use this to give a partial order
on the set of roots, cf. [3]. In [23], Petersen and Tenner extend this concept
to define the following function on $W$ (still denoted by dp)

$\text{dp}(x):=\min\\{\sum\limits_{i}\text{dp}(\beta_{i}):x=s_{\beta_{1}}\cdots
s_{\beta_{k}},\beta_{i}\in\Phi^{+}\\}.$

It is easy to see that
$\text{dp}(s_{\beta})=\text{dp}(\beta)=\frac{1}{2}(\ell(s_{\beta})+1)$ for any
positive root $\beta$. Hence we have that
$\text{dp}(s_{\beta})\leq\langle\rho,\beta^{\vee}\rangle$, and equality occurs
if and only if $\beta$ is a quantum root. In particular, if $x$ is an element
of a Weyl group of simply laced type, we have

$\text{dp}(x)=\min\\{\sum\limits_{i}\langle\rho,\beta_{i}^{\vee}\rangle:x=s_{\beta_{1}}\cdots
s_{\beta_{k}},\beta_{i}\in\Phi^{+}\\}.$ (8.1)

Following [2], we say that _the $W$-depth of $x$ is realized by a reduced
factorization of $x$_ if there exists an expression $x=s_{\beta_{1}}\cdots
s_{\beta_{k}}$ with $\beta_{i}\in\Phi^{+}$, such that
$\ell(x)=\sum\limits_{i=1}^{k}\ell(s_{\beta_{i}})$ and
$\text{dp}(x)=\sum\limits_{i=1}^{k}\text{dp}(s_{\beta_{i}})$.

We also recall the notion of _reduced reflection length_ 222In the same vein,
we alter the terminology $\ell_{R}$ used in loc. sit. for the definition of
this length function and call it $\ell_{\text{red}}$. $\ell_{\text{red}}$
introduced in loc. sit. Essentially, its definition is parallel to definition
3.4, without the quantum root proviso - i.e. it is defined using a
decomposition that satisfies only the last two conditions in that definition.

It is proved in [23] that the $W$-depth of every element in a classical finite
Coxeter group is realized by reduced factorization. This observation plays a
central role in the following result.

######  Proposition 8.3.

[2, Proposition 6.6] If the $W$-depth of $x$ is realized by a reduced
factorization then $\text{dp}(x)=\frac{1}{2}(\ell(x)+\ell_{\text{red}}(x))$;
in particular, this equality holds whenever $W$ is a classical finite Coxeter
group.

###  8.3. Relation between wt and dp

######  Lemma 8.4.

Let $W$ be an irreducible Weyl group of type $A_{n}$ or $D_{n}$. Then we have
$\langle\rho,\text{wt}(x)\rangle=\text{dp}(x)$ for any element $x\in W$.

###### Proof.

Note that for any irreducible Weyl group $W$ and for any two elements $x,y\in
W$, the following holds true by [12, Section 4.2, Statement (a)]:

$\ell(y)=\ell(x)-\langle 2\rho,\text{wt}(x,y)\rangle+d_{\Gamma}(x,y).$

Letting $y=1$, we get

$\langle\rho,\text{wt}(x)\rangle=\frac{1}{2}(\ell(x)+d_{\Gamma}(x,1))=\frac{1}{2}(\ell(x)+\ell_{\downarrow}(x))$
(8.2)

Since all roots are quantum in a group of simply laced type, we have
$\ell_{\downarrow}(x)=\ell_{\text{red}}(x)$. Now, we get the desired
conclusion appealing to proposition 8.3. ∎

######  Remark 8.5.

Note that in general we have $\langle\rho,\text{wt}(x)\rangle\geq\text{dp}(x)$
just by appealing to the definition. If $W$ is an irreducible Weyl group of
non-simply laced type, the existence of an element $x\in W$ with
$\ell_{\downarrow}(x)>\ell_{\text{red}}$ ensures that strict inequality do
occur. We refer to remark 8.9 for explicit example of such elements. As noted
in [2, Question 6.5], it is hard to verify by a computer whether the $W$-depth
of all elements are realized by reduced factorization, even for a group of
type $E_{7}$. Hence it is not clear if lemma 8.4 holds true for the groups of
type $E_{6},E_{7},E_{8}$ as well.

###  8.4. Weight of an involution

It is classically known that if $x$ is an involution in an irreducible Weyl
group of rank $n$, then it can be written as a product of at most
$\ell_{R}(x)$ commuting reflections, where $\ell_{R}(x)\leq n$. This fact is
needed in the statement of proposition 8.7 below. We also need the following
lemma.

######  Lemma 8.6.

[25, Lemma 3.3] Let $P$ be a $W$-stable convex polygon in
$X_{*}(T)_{\mathbb{R}}$. Let $v\in\mathbf{a}$ and assume that $w_{1},w_{2}$
are two elements of $\widetilde{W}$ such that $w_{1}\leq w_{2}$. Then we have
$w_{1}v-v\in P$, if $w_{2}v-v\in P$.

In our case, we are going to let $P$ to be $P_{\zeta}:=\text{Conv}(W\zeta)$
for certain dominant $\zeta$; this is the convex hull of points in its
$W$-orbit. Recall that if $\theta=\sum\limits_{i=1}^{n}m_{i}\alpha_{i}$, then
the vertices of $\mathbf{a}$ are given by
$\\{\frac{\varpi^{\vee}}{m_{i}}:1\leq i\leq n\\}$, where
$\\{\varpi_{i}^{\vee}:1\leq i\leq n\\}$ is the set of fundamental coweights.

######  Proposition 8.7.

Suppose that $W$ is a Weyl group of type $A_{n}$. Let $x\in W$ be an
involution. Suppose that $x=s_{\beta_{1}}\cdots s_{\beta_{k}}$, where $k\leq
n$ and $\\{\beta_{i}:1\leq i\leq k\\}$ is a set of orthogonal positive roots.
Then

$\text{wt}(x)=\sum\limits_{i=1}^{k}\beta_{i}^{\vee}.$

###### Proof.

Let us first note that it suffices to show
$\text{wt}(x)\geq\sum\limits_{i=1}^{k}\beta_{i}^{\vee}$. Indeed, we then have

$\langle\rho,\text{wt}(x)\rangle\geq\langle\rho,\sum\limits_{i=1}^{k}\beta_{i}^{\vee}\rangle.$

By lemma 8.4 and eq. 8.1, the above must be an equality. Therefore, writing
$\text{wt}(x)-\sum\limits_{i=1}^{k}\beta_{i}^{\vee}=\sum\limits_{\alpha\in\Delta}n_{\alpha}\alpha^{\vee}$
with $n_{\alpha}\in\mathbb{Z}_{\geq 0}$, we get
$\sum\limits_{\alpha\in\Delta}n_{\alpha}=0$, thereby showing that
$n_{\alpha}=0$ for each $\alpha\in\Delta$. Hence we have the desired equality.

Let us choose $\lambda\in X_{*}(T)$ such that $\text{depth}(\lambda)\geq 2n$.
By proposition 4.4, we then have $t^{\lambda}x\geq t^{\lambda-\text{wt}(x)}$.
An easy computation shows that

$t^{\lambda}x\varpi_{j}^{\vee}-\varpi_{j}^{\vee}=\lambda-\sum\limits_{i=1}^{k}\langle\beta_{i},\varpi_{j}^{\vee}\rangle\beta_{i}^{\vee}=\lambda-\sum\limits_{\beta_{i}\geq\alpha_{j}}\beta_{i}^{\vee}.$

We set
$\zeta_{j}:=\lambda-\sum\limits_{\beta_{i}\geq\alpha_{j}}\beta_{i}^{\vee}$.
Note that at most $n$ coroots are subtracted in the expression of $\zeta_{j}$,
and since the maximum value of $\langle\alpha,\beta_{i}^{\vee}\rangle$ is $2$
for any simple root $\alpha$, the depth hypothesis on $\lambda$ ensures that
$\zeta_{j}$ is dominant. Applying lemma 8.6, we get that
$\lambda-\text{wt}(x)\in P_{\zeta_{j}}$. Since $\lambda-\text{wt}(x)$ is
dominant, we conclude that $\lambda-\text{wt}(x)\leq\zeta_{j}$, i.e.

$\text{wt}(x)\geq\sum\limits_{\beta_{i}\geq\alpha_{j}}\beta_{i}^{\vee}.$ (8.3)

Since eq. 8.3 is valid for any $j=1,\cdots,n$, we get that

$\text{wt}(x)\geq\bigvee\limits_{j=1}^{n}(\sum\limits_{\beta_{i}\geq\alpha_{j}}\beta_{i}^{\vee}),$

where $\bigvee$ stands for the join operation. It is easy to see that that
this join is equal to $\sum\limits_{i=1}^{k}\beta_{i}^{\vee}$. Hence we are
done. ∎

######  Remark 8.8.

We note that [2, Example 2.12] gives an example of such an element $x$ where
the conclusions of proposition 8.7 and proposition 8.1 fail. Namely, let $W$
be a Weyl group of type $D_{4}$. Then

$x=s_{4}s_{2}s_{3}s_{1}s_{2}s_{4}s_{2}$

is an involution such that $\ell_{R}(x)=3$. The only way to write $x$ as a
product of $3$ commuting reflections is
$x=s_{\alpha_{1}+\alpha_{2}+\alpha_{4}}s_{\alpha_{2}+\alpha_{3}+\alpha_{4}}s_{2}$,
and thus by [20, Section 1.8, statement (a)] we conclude that $r_{x}$ is the
sum of coroots corresponding to these roots. Hence if $\text{wt}(x)=r_{x}$, we
would get $\text{dp}(x)=7$ by lemma 8.4. However, [2, Theorem 2.9] shows that
$\text{dp}(x)=6$.

Since the root systems of $E_{n}$ for $n=6,7,8$ all contain a subsystem
isomorphic to that of $D_{4}$, the conclusions of proposition 8.7 and
proposition 8.1 fail in these cases as well. By our discussion about the
computation of $r_{x}$, we also see that $\text{wt}(x)\neq r_{x}$.

######  Remark 8.9.

We now give some explicit examples of elements in irreducible Weyl groups of
non-simply laced type, for which the conclusions of proposition 8.1 and lemma
8.4 fail.

(i) Let $W$ be a Weyl group of type $C_{3}$ and let $x$ be the reflection
element in $W$ corresponding to a non-quantum root
$\beta=\alpha_{1}+2\alpha_{2}+\alpha_{3}$, i.e.
$x=s_{2}s_{3}s_{1}s_{2}s_{3}s_{1}s_{2}$. We claim that

(a) $x=s_{2\alpha_{2}+\alpha_{3}}s_{1}s_{2\alpha_{2}+\alpha_{3}}$ is a reduced
quantum reflection decomposition of $x$.

Indeed, this is a length additive decomposition using only quantum roots. We
note that $\ell_{\downarrow}(x)\neq 2$ since the right hand side of eq. 8.2
must be an integer; since $\beta$ is a non-quantum root, we cannot have
$\ell_{\downarrow}(x)=1$. Hence,
$\ell_{\downarrow}(x)=3>1=\ell_{\text{red}}(x)$. However, note that in this
case
$\text{wt}(x)=(\alpha_{2}^{\vee}+\alpha_{3}^{\vee})+\alpha_{1}^{\vee}+(\alpha_{2}^{\vee}+\alpha_{3}^{\vee})$,
which does match with
$r_{x}=\beta^{\vee}=\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+2\alpha_{3}^{\vee}$.

Now, considering $W$ to be the Weyl group associated to a root system of type
$B_{3}$, let $x^{\prime}$ be the reflection element corresponding to a non-
quantum root $\alpha_{1}+\alpha_{2}+\alpha_{3}$. Direct computation shows that
there is only one way to write $x^{\prime}$ as a product of $3$ reflections in
a length additive manner: $x^{\prime}=s_{1}s_{\alpha_{2}+\alpha_{3}}s_{1}$,
but this uses a non-quantum root $\alpha_{2}+\alpha_{3}$. Hence, its reduced
quantum reflection decomposition is $x=s_{1}s_{2}s_{3}s_{2}s_{1}$, thereby
giving $\ell_{\downarrow}(x)=5>1=\ell_{\text{red}}(x)$

(ii) Now, let $W$ be a Weyl group of type $B_{4}$ and pick the element
$y=s_{4}s_{3}s_{4}s_{2}s_{3}s_{4}s_{1}s_{2}s_{3}s_{4}s_{2}$. We claim that

(b)
$y=s_{\alpha_{3}+2\alpha_{4}}s_{2}s_{3}s_{1}s_{\alpha_{2}+\alpha_{3}+2\alpha_{4}}$
is a reduced quantum reflection decomposition of $y$.

Indeed, this is a length additive decomposition using only quantum roots.
Direct computation shows that there are only two ways to express $y$ as a
product of $3$ reflections; namely, we have
$y=s_{\alpha_{1}+\alpha_{2}+\alpha_{3}+2\alpha_{4}}s_{\alpha_{2}+2\alpha_{3}+2\alpha_{4}}s_{2}$
and
$y=s_{\alpha_{1}+\alpha_{2}+\alpha_{3}+2\alpha_{4}}s_{\alpha_{2}+\alpha_{3}+\alpha_{4}}s_{\alpha_{3}+\alpha_{4}}$,
but none of these are length additive decomposition. This rules out the
possibility $\ell_{\downarrow}(y)=3$. Since $y$ is not a reflection, we deduce
that the claim is true. In this case,
$\text{wt}(y)=\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}+3\alpha_{3}^{\vee}+2\alpha_{4}^{\vee}$.
An easy computation about the -$1$-eigenspace of $y$ shows that
$r_{x}=\alpha_{1}^{\vee}+3\alpha_{2}^{\vee}+3\alpha_{3}^{\vee}+2\alpha_{4}^{\vee}$.

Since the root system of type $B_{4}$ occurs as a subsystem of type $F_{4}$,
this example works in that case as well.

(iii) Finally, for a Weyl group of type $G_{2}$ we let $z=s_{2}s_{1}s_{2}$.
This is the reflection element of $W$ corresponding to a non-quantum root
$\gamma=\alpha_{1}+\alpha_{2}$. We have
$\ell_{\downarrow}(z)=3>1=\ell_{\text{red}}(z)$. Also, here
$\text{wt}(z)=\alpha_{1}^{\vee}+2\alpha_{2}^{\vee}$ and
$r_{z}=\gamma^{\vee}=\alpha_{1}^{\vee}+3\alpha_{2}^{\vee}$.

In view of the above remarks, we only need to justify that
$\text{wt}(x)=r_{x}$ for any involution $x$ in a Weyl group of type $A_{n}$.

###### Proof of Proposition 9.1.

We need to show that the properties (i)-(iii) in theorem 8.2 are satisfied by
wt as well. Note that (i) is trivially true. To check (ii), let us choose a
decomposition $x=s_{\beta_{1}}\cdots s_{\beta_{k}}$ as described in
proposition 8.7. Then $s_{\alpha}xs_{\alpha}=s_{s_{\alpha}(\beta_{1})}\cdots
s_{s_{\alpha}(\beta_{k})}$ is a decomposition of similar kind for
$s_{\alpha}xs_{\alpha}$. By proposition 8.7, we have
$r_{s_{\alpha}xs_{\alpha}}=\sum\limits_{i=1}^{k}s_{\alpha}(\beta_{i}^{\vee})=s_{\alpha}(r_{x})$.

Finally, we argue that property (iii) is satisfied by wt for any $x\in W$,
where $W$ is any irreducible Weyl group. Suppose first that $s_{\alpha}x>x$.
Then we can form a path from $s_{\alpha}x$ to $1$ by concatenating the edge
$s_{a}x\rightharpoondown x$ with a path from $x$ to $1$ of shortest length
that only uses downwards edges, whence
$\alpha^{\vee}+\text{wt}(x)\geq\text{wt}(s_{\alpha}x)$. By lemma 3.1, we also
have $\text{wt}(s_{\alpha}x)\geq\text{wt}(x)$. Combining these two
inequalities, we get that $\text{wt}(s_{\alpha}x)$ is either $\text{wt}(x)$ or
$\text{wt}(x)+\alpha^{\vee}$. Similarly, if $s_{\alpha}x<x$, we can show that
$\text{wt}(s_{\alpha}x)$ is either $\text{wt}(x)$ or
$\text{wt}(x)-\alpha^{\vee}$.

This completes the proof. ∎

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