ABSTRACT

Title of Dissertation: A COMBINATORIAL STUDY
OF AFFINE DELIGNE-LUSZTIG VARIETIES

Arghya Sadhukhan
Doctor of Philosophy, 2023

Dissertation Directed by: Professor Jeffrey Adams
Department of Mathematics

We consider affine Deligne-Lusztig varieties Xw(b) and certain unions X(µ, b) in the

affine flag variety of a connected reductive group. They were first introduced by Rapoport

to facilitate the study of mod-p reduction of Shimura varieties and moduli spaces of shtukas.

We improve upon certain existing results in the study of affine Deligne-Lusztig varieties

by weakening the hypothesis to prove them. Such results include a description of generic

Newton points in Iwahori double cosets in the loop group of a split reductive group, covering

relations in the associated Iwahori-Weyl group, and a dimension formula for X(µ, b) in the

case of a quasi-split group. As an application of the work on generic Newton point formula,

we obtain a description of the dimension for X(µ, b) associated with the maximal element b

in its natural range, under a mild hypothesis on µ but no further restrictions on the group.



A COMBINATORIAL STUDY OF
AFFINE DELIGNE-LUSZTIG VARIETIES

by

Arghya Sadhukhan

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2023

Advisory Committee:
Professor Jeffrey Adams, Chair/Advisor
Professor Thomas J. Haines, Co-chair
Professor Xuhua He
Professor Yihang Zhu
Professor Cole Miller, Dean’s representative



© Copyright by
Arghya Sadhukhan

2023



Dedication

To the teachers who inspire, empower, and propel us to new mathematical

horizons.

ii



Acknowledgments

First and foremost, I want to express my heartfelt gratitude to my dissertation advisor

Xuhua He, who despite being on the other side of the world, has made a lasting impact

on my research and provided invaluable guidance throughout this journey. Navigating a

Ph.D. through Zoom meetings presented its challenges, but his dedication, expertise, and

unwavering support have been instrumental in my academic growth.

I am also immensely grateful to my local supervisor Jeffrey Adams, who has been there

for me both mathematically and logistically whenever I needed assistance. His willingness

to offer support, answer my questions, and provide guidance has greatly enriched my grad

school journey. I would like to extend special thanks to Thomas Haines and Michael

Rapoport. Engaging in conversations with them about Shimura varieties and the Langlands

program has broadened my understanding and enhanced the depth of this dissertation.

Additionally, I want to express my appreciation to the professors in the math depart-

ment for offering relevant courses and for their willingness to discuss mathematics outside

of classroom, especially Patrick Brosnan, Harry Tamvakis and Yihang Zhu.

I am grateful to the staff at the Department of Mathematics and the STEM library

at the University of Maryland, especially Haydee, Cristina, Liliana, and Ruth, for their

continued support in navigating various bureaucratic systems seamlessly.

Going back further, I want to acknowledge the teachers and mentors from the days of

my undergraduate and masters studies who have played a pivotal role in shaping my aca-

iii



demic journey, especially Arijit Ganguly, Parthasarathi Mukhopadhyay, and Amritanshu

Prasad. Gratitude is also due to the organizers of the MTTS-2014 summer camp in India,

where I was first exposed to mathematical research, sparking my passion for the subject.

I am deeply thankful for the public education infrastructure in both India and the

United States. Without the opportunities and resources provided by these systems, pur-

suing my academic dreams would not have been possible. Additionally, I want to express

my gratitude for the support received through the Graduate School Summer Research

Fellowship and the Sung Dissertation Fellowship, which have allowed me to focus on my

research.

I am grateful for the friends I have picked up during my Ph.D. journey. To Chengze,

Deric, Dimitri, Gautam, Naren, Prakhar, Rachel, Saurav, Shin, Sohitri, Stavros, Subhayan,

Sze-Hong, Tamoghna and Yiannis, thank you for your warm camaraderie, unwavering

support, and shared experiences. Your friendship has made this journey more meaningful

and enjoyable. I would also like to thank my friends from my undergraduate days, especially

Sovanlal, Srijan and Sudipta for their constant companionship and for being memorable

travel buddies.

During a challenging time post an accident in December 2022, I would like to express

my gratitude to the local bike store, Proteus Bicycles, for their assistance and support. I

would also like to extend my thanks to the American Film Institute theatre in Silver Spring

for providing a much-needed respite and inspiration during the course of this Ph.D. journey.

Additionally, I want to express my appreciation to the authorities in charge of maintain-

ing and preserving the trails, wilderness, and nature near me, especially the Shenandoah

National Park. The serenity and beauty of nature have provided solace and rejuvenation

iv



during the demanding research process. I would also like to express my gratitude to all the

cats I had the privilege to play with, including Jobi-Chan, Django, Elsa, and Cocos.

My deepest appreciation goes to my parents, Amitava and Soma, and my sister,

Somdutta for their mental support and for allowing me to pursue mathematics away from

home for more than a decade. I want to extend a special thank you to my partner, Mita,

for being an anchor throughout the better part of this journey. I look forward to the time

when we finally solve the two-body problem at hand together!

Finally, I would like to express my deepest admiration and gratitude to Robert Lang-

lands for his visionary creation of a grand web of compelling conjectures, and to Robert

Kottwitz for his invaluable contributions in advancing the field through the infusion of

group-theoretic methods into this majestic edifice. To echo the sentiment of Langlands

himself, “certainly the best times were when I was alone with mathematics, free of ambition

and of pretense, and indifferent to the world.”

v



Table of Contents

Preface ii

Foreword ii

Dedication ii

Acknowledgements iii

Table of Contents 1

1 Introduction 2
1.1 Affine Deligne-Lusztig varieties in the affine flag variety . . . . . . . . . . . 3
1.2 Certain union of affine Deligne-Lusztig varieties . . . . . . . . . . . . . . . 6
1.3 Other related work on affine Weyl groups . . . . . . . . . . . . . . . . . . . 11

2 Preliminaries 12
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The σ-conjugacy classes of Ğ . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Affine Deligne-Lusztig varieties . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Demazure product and its variations . . . . . . . . . . . . . . . . . . . . . 20
2.5 Quantum Bruhat graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Generic Newton point and affine Bruhat order 24
3.1 Some combinatorial properties . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Formula for the generic Newton point . . . . . . . . . . . . . . . . . . . . . 27
3.3 Weight of the longest element . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Covering relation in Iwahori-Weyl group . . . . . . . . . . . . . . . . . . . 47

4 A dimension formula for X(µ, b) 64
4.1 Dimension in the quasi-split case . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Expressing bmax via generic Newton point . . . . . . . . . . . . . . . . . . . 70
4.3 Minimal length elements in certain σ0-conjugacy class . . . . . . . . . . . . 79
4.4 Explicit description of the Newton point of bmax . . . . . . . . . . . . . . . 83

5 Some remarks about the weight function in type An 90

Bibliography 99

1



Chapter 1: Introduction

In their seminal paper [DL76], Deligne and Lusztig gave a geometric recipe to con-

struct all the irreducible representations for finite groups of Lie type in terms of l-adic

cohomology of certain algebraic varieties associated with elements of the corresponding

finite Weyl group. On the other hand, the affine Deligne-Lusztig varieties - their coun-

terpart for the affine root system - were first introduced by Rapoport [Rap05] and have

found substantial application in the geometry of Shimura varieties and moduli spaces of

shtukas, and therefore have been a geometric object of recurring interest in the Langlands

program. In [Lan77], Langlands outlines a three-part approach to prove that the Hasse-

Weil ζ-functions of Shimura varieties are related to L-functions of automorphic forms; this

expectation lies at the heart of Langlands program. A key input in this approach is the

description of the geometric points of special fibers of suitable integral models of Shimura

varieties. A conjectural description of such mod-p points was put forth by Langlands and

Rapoport in the cases of good reduction, and subsequently modified by Rapoport and

Kottwitz to include cases of parahoric-level bad reduction, and the geometry of associated

affine Deligne-Lusztig varieties is a key player in proving such result. For instance, infor-

mation about connected components of affine Deligne-Lusztig varieties - a problem that

has generated a lot of attention in the past decade and has only been proved in complete

2



generality in a very recent preprint [GL22] - has played an important role in the version of

the Langlands-Rapoport conjecture proved in [Kis17] and its subsequent strengthening in

[KSZ21].

Due to the combinatorial complexity of the affine Weyl groups, several important

results in the study of affine Deligne-Lusztig varieties (in the affine flag variety) in recent

years have been established only under the so-called superregularity hypothesis. In this

thesis, we weaken the superregularity hypothesis on them and sometimes eliminate it, thus

strengthening these existing results. Additionally, we compute the dimension of a certain

naturally occurring union X(µ, b) of affine Deligne-Lusztig varieties beyond the setting of

quasi-split groups. We now proceed to explain the main results of this thesis in more detail.

1.1 Affine Deligne-Lusztig varieties in the affine flag variety

We refer to Chapter 2 for an explanation of notations and definitions. Let G be a

reductive group over a non-archimedean local field F , with residue field κF and completed

maximal unramified extension F̆ . Denote by σ the Frobenius morphism of F̆ /F and choose

a σ-stable Iwahori subgroup I. Then letting Ğ := G(F̆ ), Ĭ := I(F̆ ), we have two natural

decompositions of the loop group Ğ: namely,

Ğ =
∐

[b]∈B(G)

[b] =
∐
w∈W̃

ĬwĬ.

Here [b] := {g−1bσ(g) : g ∈ Ğ} is the σ-conjugacy class of b, the so-called Kottwitz set B(G)

is the collection of such classes and W̃ is the Iwahori-Weyl group of G. For [b] ∈ B(G)

and w ∈ W̃ , the associated affine Deligne-Lusztig variety Xw(b) is a locally closed, reduced

3



subscheme locally of finite type inside the affine flag variety F lG, with geometric points

given by

Xw(b)(κ̄F ) := {gĬ : g−1bσ(g) ∈ ĬwĬ} ⊂ Ğ/Ĭ.

We list some of the major problems in this field below:

1. For which w, b is Xw(b) nonempty?

2. If non-empty, is Xw(b) equidimensional and can we give a closed formula for its

dimension?

Let us briefly mention why these questions are of interest from the perspective of arithmetic

geometry. Roughly speaking (at least for the Shimura varieties of Hodge type that admits

a moduli interpretation), on the special fiber of a suitable integral model of the Shimura

variety associated to the Shimura datum (G, µ) with Iwahori level structure, there are

two important stratifications: the Newton stratification, coming from grouping together

abelian varieties according to the isogeny class of their p-divisible groups, and the Kottwitz-

Rapoport stratification, induced from the Iwahori orbits on the associated local model; for

an axiomatic approach to these stratifications pertaining to general Shimura varieties, see

[HR17]. The strata are indexed by a certain subset of B(G) in the former case, while in the

latter case the index set is a certain subset of W̃ governed by the Shimura datum. The affine

Deligne-Lusztig varieties capture the delicate interation between these two stratifications;

for instance, Xw(b) ̸= ∅ if and only if the Newton stratum indexed by the element b meets

the Kottwitz-Rapoport stratum indexed by the element w, cf. [Hai05, §12.3]. However, even

this nonemptiness problem has not fully been resolved in complete generality; we mention

4



the work done in [Gör+10], [GHN15], and finally [He21b] for state-of-the-art result, as well

as an interesting conjecture made in [Lim23] in this direction.

An important feature of B(G) is the poset structure on it, defined via closure relation

in Ğ. Recent results of [MV20] highlight certain special elements of W̃ : they show that if

the maximal element [bw] of B(G)w := {[b] ∈ B(G) : Xw(b) ̸= ∅} - which coincides with

the generic σ-conjugacy class in ĬwĬ - satisfies an explicit group-theoretic condition called

cordiality, then the poset B(G)w is saturated; furthermore, all the Newton strata meeting

the fixed Kottwitz-Rapoport stratum indexed by w, as well as the affine Deligne-Lusztig

varieties associated with w exhibits especially well-behaved geometry. In essence, their

theorem gives a condition that can be checked from knowledge of this maximal element bw of

B(G)w, but it provides important information about the shape of the entire poset. In light

of such results, it becomes important to describe bw explicitly in a way that can be used to

check whether w is cordial. By virtue of the concrete parametrization [Kot85] σ-conjugacy

classes via their Kottwitz and Newton points (κ, ν) : B(G) ↪→ π1(G)Γ × (X∗(T )Γ0 ⊗Q)+,

this amounts to asking the following:

Question 1.1.1. Give an explicit closed formula for the generic Newton point ν([bw]).

It turns out that such a description comes from the technical framework of the quan-

tum Bruhat graph: for elements of W̃ that are sufficiently far away from the walls of any

chamber, such a formula was first established in [Mil21]. This combinatorial tool was intro-

duced by Brenti, Fomin and Postnikov in [BFP99] to describe the multiplicative structure

of the quantum cohomology ring of the complex flag manifold. It is obtained by augmenting

the usual Hasse diagram for the Bruhat order on the finite Weyl group W by some quantum

5



edges - certain downward edges that are labeled by the coroot associated to the reflection

used to get from one vertex to the other. With this setup, one can define a certain weight

function wt : W ×W → Z≥0Φ
∨, cf. Section 2.5. Then we have the following result, see

Theorem 3.2.1.

Theorem 1.1.2. Suppose that G is a quasi-simple split group of semi-simple rank n. Let

w = xtλy be an element of W̃ with x, y ∈ W , and depth(λ) > Ξn, where Ξn is certain

explicit linear function of n. Then ν([bw]) = λ− wt(y−1, x).

Here for a dominant cocharacter λ its depth(λ) := min{⟨α, λ⟩ : α simple root} quan-

tifies its distance from the walls of the fundamental Weyl chamber. Hence this result

weakens the hypothesis on the lower bound of the depth from O(n2) in Milićević’s work to

O(n).

The proof of Theorem 1.1.2 crucially employs the Demazure product and its variations

on the finite Weyl group W : this comes from the product in the associated 0-Hecke algebra

and induces a monoid structure on W . Using these tools, we show that the generic Newton

point ν(bw) is given by the Newton point of the maximal translation element below w, and

we identify this element via simple Bruhat order considerations.

1.2 Certain union of affine Deligne-Lusztig varieties

Rapoport in [Rap05] predicted “whereas the individual affine Deligne-Lusztig varieties

are very difficult to understand, the situation seems to change radically when we form a

6



suitable finite union of them.” Here the union refers to

X(µ, b) :=
⋃

w∈Adm(µ)

Xw(b),

where for a dominant cocharacter µ, we define its admissible set as Adm(µ) := {w ∈ W̃ :

w ≤ tx(µ) for some x ∈ W}.

The interest in studying such a union comes from the fact that (in mixed character-

sitics) they arise as the underlying reduced scheme of a formal moduli space of p-divisible

groups, known as Rapoport-Zink space, cf. [RV14]. Something analogous holds in the

function field case for formal moduli spaces of Shtukas, cf. [Vie18]; in this latter case, µ

can be arbitrary. Rapoport-Zink spaces are local analogs of Shimura varieties, and are

also related to Shimura varieties themselves by the theory of p-adic uniformization. The

problem of describing the l-adic cohomology of Rapoport-Zink spaces may be traced back

to Lubin-Tate theory of formal groups and has had important consequences such as the

proof of the local Langlands conjecture and the local-global compatibility of the Langlands

correspondence in [HT01].

As a first validation of Rapoport’s expectation, let us note that the non-emptiness

pattern of X(µ, b) is completely understood (as opposed to that of Xw(b)): settling the

Kottwitz-Rapoport conjecture in [He16a], He proves that X(µ, b) is non-empty precisely

when [b] lies in the set B(G, µ) of neutrally acceptable σ-conjugacy classes ; this subset

of B(G) is defined by a group theoretic reformulation of Mazur’s inequality between the

Hodge polygon of an F -crystal and the Newton polygon of its underlying F -isocrystal.

As a crucial ingredient toward obtaining a dimension formula for X(µ, b), a useful

7



description of sufficiently large elements in the admissible set Adm(µ) in terms of quantum

Bruhat graph theoretic data is given in [HY21], whenever the depth of µ is O(n2) with

respect to the semisimple rank n. Based on the work on affine Bruhat order (see next

section), we can relax this hypothesis on depth(µ) to O(1); additionally, we exploit the

additivity of the admissible set to upgrade it to a necessary numerical criterion that is

unconditional on µ.

Proposition 1.2.1. Suppose that W is an irreducible Weyl group associated with an affine

Weyl group W̃ . Let µ, λ be dominant cocharacters.

1. Assume that depth(µ) is bigger than a certain constant ΘW that is at most 6 in all

Cartan types, and further that ⟨ρ, µ−λ⟩ < ⌈depth(µ)−ΘW

2
⌉. Then for any two elements

x, y ∈ W , we have xtλy ∈ Adm(µ) if and only if wt(x, y−1) ≤ µ− λ.

2. Assume that µ is only dominant regular, i.e. depth(µ) ≥ 1. Then ⟨ρ,wt(x, y−1)⟩ ≤

⟨ρ, µ− λ⟩ if xtλy ∈ Adm(µ).

This follows from Proposition 3.4.13 and Proposition 4.1.1. Based on Proposition 1.2.1(2),

we show that certain key steps in the proof in loc. sit. of the dimension formula can be

carried out differently to bypass the O(n2) superregularity constraint. Thus we establish

in Section 4.1 the same dimension formula under just a regularity assumption on µ:

Theorem 1.2.2. Suppose that G is a quasi-split group. Let µ be dominant regular and

assume that [b] ∈ B(G, µ). Denote by O the σ-conjugacy class of the longest element w0

in W , and let ℓR(O) be the minimal reflection length of elements in O.

8



1. Let us further require that the Galois average µ⋄ ≥ ν([b]) + 2ρ∨. Then

dimX(µ, b) = ⟨ρ, µ− ν(b)⟩ − 1

2
defG(b) +

1

2
(ℓ(w0)− ℓR(O)).

2. Furthermore, if G is split, we obtain the same result under the hypothesis that µ ≥

ν([b]) + wt(w0, 1) and depth(µ) > 2.

We refer the reader to Section 3.3, Section 2.2.1 and Section 2.3.1 for relevant defi-

nitions of the notions reflection length ℓR, Galois average µ⋄ and defect defG, respectively.

Note that the quasi-split assumption on G is crucial in the above theorem; unlike in the

case of a single affine Deligne-Lusztig variety, here we cannot leverage Proposition 4.2.6

to deduce a dimension formula of X(µ, b) for general G. There is no direct relation be-

tween the admissible set Adm(µ) - and hence X(µ, b) - for an arbitrary reductive group

and its quasi-split inner form. This adds essential difficulties in the study of X(µ, b) for

non-quasi-split groups.

Nevertheless, there has been some partial success in the context of general groups.

Most notably, the investigation into the structure of X(µ, b) for the element b = bmin :=

minB(G, µ) has led to the striking observation that in certain cases, the basic locus admits

a nice description as a union of classical Deligne-Lusztig varieties, the index set and the

closure relations between the strata being encoded in a Bruhat-Tits building attached to

the group theoretic data coming from the Shimura variety; such descriptions have been

used with great success towards applications in the realm of the Langlands program, for

instance in the work toward Zhang’s arithmetic fundamental lemma [RTZ13], as well as the

theory of non-archimedean uniformization of Shimura varieties, first pioneered by Cerednik

9



and Drinfeld, and further developed by Rapoport–Zink [RZ96] and Howard–Pappas [HP17].

However, the dimension problem and more qualitative structural information have remained

elusive for general elements of B(G, µ). Standing at this juncture, it is therefore natural

to look at the other extreme and ask:

Question 1.2.3. What is the dimension of X(µ, b) for b = bmax := maxB(G, µ)?

Note that if G is quasi-split, this maximal element is just the Galois average µ⋄;

hence, the dimension is trivially zero. However, for non-quasi-split groups the description

of this maximal element is more subtle and indirect, see [HN18]. However, this is related

to Question 1.1.1, as bmax = max
x∈W

btxµ . Utilizing a recent result in [He21a] expressing the

generic Newton point in terms of the Demazure power, we can explicitly describe this

maximal element for non-quasi-split groups. Furthermore, combining quantum Bruhat

graph theoretic considerations with explicit computation of certain minimal length elements

in Frobenius-twisted conjugacy classes of W , we prove the following in Section 4.2.5:

Theorem 1.2.4. Assume that the image µad of µ in X∗(Tad)Γ0 has depth at least 2 in every

F̄ -simple factor of Gad. Let b = bmax be the maximal element of B(G, µ). Then

dimX(µ, b) = rkFssG
∗ − rkFssG,

where G∗ is the unique (up to isomorphism) quasi-split inner form of G, and rkFss stands

for semisimple F -rank.

Note that if G ≃ G∗ is quasi-split, this indeed recovers the trivial case mentioned

above, and as such, this theorem also serves as a geometric way to measure how far G is

10



from being quasi-split. A surprising feature of the result is that this dimension does not

depend on µ.

1.3 Other related work on affine Weyl groups

As is already evidenced from our discussion so far, understanding fundamental prop-

erties of the affine Weyl groups is crucial to solving problems related to affine Deligne-

Lusztig varieties. Indeed, the key ingredient in Milićević’s original approach to solving

Question 1.1.1 was to establish that under a lower bound of order O(n2) on the depth,

paths in quantum Bruhat graph encode saturated chains in the Bruhat order on the affine

Weyl group. Even though we can only strengthen her result toward Question 1.1.1 to O(n)

depth hypothesis, we improve this result on the affine Weyl group to one with an O(1)

depth hypothesis. The next result follows from Theorem 3.4.1.

Theorem 1.3.1. Suppose that w = xtλy is an element of W̃ such that depth(λ) is bigger

than a certain constant (at most 6 in all Cartan types). Then we have a precise description

of the cocovers of w, i.e. those elements w′ such that w′ < w and ℓ(w′) = ℓ(w) − 1: such

covering relations in affine Bruhat order are in two-to-one correspondence with edges in the

quantum Bruhat graph:

{cocovers of w = xtλy} ←→ {edges coming into x}
∐
{edges going out of y−1}.

The content appearing in Chapter 3, Section 4.1 and Chapter 5 corresponds to the

paper [Sad23], while the arxiv preprint [Sad22] makes up the rest of Chapter 4.

11



Chapter 2: Preliminaries

2.1 Notations

Let G be a connected reductive group over a non-archimedean local field F . Let

F̆ be the completion of the maximal unramified extension of F and σ be the Frobenius

morphism of F̆ /F . The residue field of F is a finite field Fq and the residue field of F̆ is

the algebraically closed field F̄q. We write Ğ for G(F̆ ). We use the same symbol σ for the

induced Frobenius morphism on Ğ. Let S be a maximal F̆ -split torus of G defined over F ,

which contains a maximal F -split torus. Let A be the apartment of GF̆ corresponding to

SF̆ . We fix a σ-stable alcove a in A, and let Ĭ ⊂ Ğ be the Iwahori subgroup corresponding

to a. Then Ĭ is σ-stable.

Let T be the centralizer of S in G. Then T is a maximal torus. We denote by N the

normalizer of T in G. The Iwahori–Weyl group (associated to S) is defined as

W̃ = N(F̆ )/T (F̆ ) ∩ Ĭ.

For any w ∈ W̃ , we choose a representative ẇ in N(F̆ ); however if there is no possibility

of confusion we will call the lift w too. The action σ on Ğ induces a natural action of σ on

W̃ , which we still denote by σ. We will sometimes identify the element w ∈ W̃ with wa,

12



the (extended) alcove that one obtains as image of the base alcove a under w. Similarly,

we may say w lies in some given chamber to express that wa is an alcove belonging to that

chamber.

We denote by ℓ the length function on W̃ determined by the base alcove a and denote

by S̃ the set of simple reflections in W̃ . Let Wa be the subgroup of W̃ generated by S̃.

Then Waff is an affine Weyl group. Let Ω ⊂ W̃ be the subgroup of length-zero elements

(or equivalently, the stabilizer of a) in W̃ . Then

W̃ = Wa ⋊ Ω.

Since the length function is compatible with the σ-action, the semi-direct product decom-

position W̃ = Wa ⋊ Ω is also stable under the action of σ. Note that Wa is the Coxeter

group associated to S̃, and hence it comes equipped with an associated Bruhat order, which

we denote by ≤. This is extended to W̃ as follows. For two elements w̃1, w̃2 ∈ W̃ , use the

above decomposition to write w̃i = wiζi with wi ∈ Wa, ζi ∈ Ω for i = 1, 2. We then declare

w̃1 ≤ w̃2 if w1 ≤ w2 and ζ1 = ζ2. The following properties of ℓ and ≤ are well-known, e.g.

see [Mil21, Lemma 4.1], [Kna02, exercise 23, Chapter 2] and [BB05, exercise 21, Chapter 2

] respectively.

• Let w = utλv ∈ W̃ , where λ ∈ X∗(T ) is regular dominant and u, v ∈ W . Then

ℓ(w) = ℓ(u) + ℓ(tλ)− ℓ(v) = ℓ(u) + ⟨2ρ, λ⟩ − ℓ(v). (2.1.1)

• For any element x of W , let Inv(x) = {α ∈ Φ+ : xα ∈ −Φ+} be its inversion set,

13



and denote its complement by Inv(x)c, i.e. Inv(x)c = Φ+ \ Inv(x). Then for any two

elements x, y ∈ W , we have

ℓ(xy) = ℓ(x)+ℓ(y)−2|Inv(x)∩Inv(y−1)| = ℓ(x)−ℓ(y)+2|Inv(x)c∩Inv(y−1)|. (2.1.2)

• Let w1, w2, v ∈ W̃ be three elements such that ℓ(wiv) = ℓ(wi) + ℓ(v) for i = 1, 2.

Then

w1 ≥ w2 is equivalent to w1v ≥ w2v. (2.1.3)

Let W = N(F̆ )/T (F̆ ) be the relative Weyl group. We denote by S the subset of S̃

consisting of simple reflections generating W . We let Φ (resp. ∆) denote the set of roots

(resp. simple roots) for W . We write Γ for Gal(F̄ /F ), and write Γ0 for the inertia subgroup

of Γ. Then fixing a special vertex of the base alcove a, we have the splitting

W̃ = X∗(T )Γ0 ⋊W = {tλw;λ ∈ X∗(T )Γ0 , w ∈ W}.

When considering an element λ ∈ X∗(T )Γ0 as an element of W̃ , we write tλ. Note that if

G is not quasi-split over F , then there does not exist a σ-stable special vertex in a and

thus the splitting W̃ = X∗(T )Γ0 ⋊W is not σ-stable.

For an irreducible Weyl group W of rank n, we follow the labeling of roots as in

[Bou02] and we usually write si instead of sαi
, where ∆ = {αi : 1 ≤ i ≤ n}. Let w0

be the longest element in W , and for i ∈ [1, n] we let wi,0 be the longest element of the

parabolic subgroup of W corresponding to ∆ \ {αi}. Let ρ be the dominant weight with

14



⟨α∨, ρ⟩ = 1 for any α ∈ ∆. Let {ϖ∨
i : 1 ≤ i ≤ n} be the set of fundamental coweights.

If ϖ∨
i is minuscule, we denote the image of tϖ

∨
i under the projection W̃ → Ω by τi; then

conjugation by τi is a length preserving automorphism of W̃ , which we denote by Ad(τi).

2.2 The σ-conjugacy classes of Ğ

We say that two elements b, b′ ∈ Ğ are σ-conjugate if there is some g ∈ Ğ such that

b′ = gbσ(g)−1. Let B(G) be the set of σ-conjugacy classes on Ğ. By the work of Kottwitz

in [Kot85] and [Kot97], any σ-conjugacy class [b] is determined by two invariants:

• The Kottwitz point κ([b]) ∈ π1(G)Γ, where π1(G) := X∗(T )/ZΦ∨ is the Borovoi

fundamental group and π1(G)Γ is the set of Γ-coinvariants in π1(G);

• The Newton point ν([b]) ∈ ((X∗(T )Γ0,Q)
+)⟨σ⟩, where X∗(T )Γ0,Q := X∗(T )Γ0 ⊗ Q =

X∗(T )
Γ0 ⊗ Q, and ((X∗(T )Γ0,Q)

+)⟨σ⟩ is defined to be the set of ⟨σ⟩-invariants of the

intersection of X∗(T )Γ0,Q with the set X∗(T )
+
Q of dominant elements in X∗(T )Q.

We denote by ⩽ the dominance order on X∗(T )
+
Q, i.e., for ν, ν ′ ∈ X∗(T )

+
Q, we have

ν ⩽ ν ′ if and only if ν ′−ν is a non-negative (rational) linear combination of positive coroots

over F̆ . The dominance order on X∗(T )
+
Q extends to a partial order on B(G). Namely, for

[b], [b′] ∈ B(G), we say that [b] ⩽ [b′] if and only if κ([b]) = κ([b′]) and ν([b]) ⩽ ν([b′]).

Denote by Jb the σ-centralizer group of b; this is a reductive group over F with

F -rational points given by

Jb(F ) = {g ∈ Ğ | g−1bσ(g) = b}.

15



For any reductive group H over F , we denote by rkssFH the semisimple F -rank of H. The

following result is implicit in [Kot06, §1.9].

Proposition 2.2.1. Let G be quasi-simple over F and assume τ ∈ Ω. Then

rkssF Jτ̇ =| Ad τ ◦ σ orbits on S̃ | −1.

2.2.1 The straight σ-conjugacy classes

Note that the action of σ on Ğ gives rise to an action on W̃ , still denoted by σ. The

set of σ-conjugacy classes in W̃ is denoted by B(W̃ , σ).

Let w ∈ W̃ , n ∈ N. The n-th σ-twisted power of w is defined by

wσ,n = wσ(w)σ2(w) · · ·σn−1(w).

Note that this is nothing but the image of the n-th power of wσ ∈ W̃ ⋊ ⟨σ⟩ under the

quotient map W̃ ⋊ ⟨σ⟩ → W̃ , since (wσ)n = wσ,nσn. Then by definition, w is called a

σ-straight element if ℓ(wσ,n) = nℓ(w), for all n ∈ N. A σ-conjugacy class of W̃ is called

straight if it contains a σ-straight element, and we denote the collection of such straight

σ-conjugacy classes by B(W̃ , σ)str. We have a map Ψ : B(W̃ , σ) → B(G), coming from

the assignment w → ẇ. We can also define Kottwitz and Newton maps, denoted by the

same symbols κ, ν resp. from B(W̃ , σ) with the same targets as before, cf. [He14, §1.7],

[Gör+10, §7.2]. The importance of the straight σ-conjugacy classes is illustrated in the

following result.

16



Theorem 2.2.2. [He14, Theorem 3.7] The restriction of Ψ induces a bijective map Ψ :

B(W̃ )str → B(G). Moreover, We have the following commutative diagram

B(W̃ )str
Ψ //

(κ,ν) ((

B(G)

(κ,ν)uu

π1(G)Γ × ((X∗(T )Γ0,Q)
+)⟨σ⟩

Given w ∈ W̃ , define

B(G)w := {[b] ∈ B(G) : [b] ∩ ĬwĬ ̸= ∅}.

It is easy to see that B(G)w has an unique maximal element [bw], which coincides with the

generic σ-conjugacy class in ĬwĬ. We will write νw to denote ν([bw]).

Let {µ} be a conjugacy class of cocharacters over F̄ . Choose µ be a dominant repre-

sentative of {µ} and denote by µ its image in X∗(T )Γ0 .

We also have the set of neutrally acceptable elements, cf. [KR03]

B(G, µ) = {[b] ∈ B(G) | κ([b]) = µ♮, ν([b]) ≤ µ⋄}.

Here µ♮ denotes the common image of µ ∈ {µ} in π1(G)Γ, and µ⋄ denotes the average of

the σ0-orbit of µ. Here σ0 denotes the L-action of σ, see [GHN20, definition 2.1] for details.

The set B(G, µ) inherits a partial order from B(G). Since the Kottwitz map κ is constant

on B(G, µ), we may view it as a subset of X∗(T )Γ0,Q via the Newton map. The following

description of B(G, µ) is obtained in [HN18, Theorem 1.1 & Lemma 2.5]. To state the

17



result, for each σ0 orbit O on S, we set ϖO =
∑
i∈O

ϖi.

Theorem 2.2.3. [HN18] (1) Let v ∈ X∗(T )Γ0,Q. Then v ∈ B(G, µ) if and only if σ0(v) = v

is dominant, and for any σ0-orbit O on S with ⟨v, αi⟩ ≠ 0 for each (or equivalently, some)

i ∈ O, we have that ⟨µ+ σ(0)− v,ϖO⟩ ∈ Z and ⟨µ− v,ϖO⟩ ≥ 0.

(2) The set B(G, µ) contains a unique maximal element.

2.3 Affine Deligne-Lusztig varieties

For b ∈ Ğ and w ∈ W̃ , the associated affine Deligne-Lusztig variety Xw(b) is a locally

closed, reduced subscheme locally of finite type inside the affine flag variety F lG, with

geometric points given by

Xw(b)(κ̄F ) := {gĬ : g−1bσ(g) ∈ ĬwĬ} ⊂ F lG(κ̄F ) = Ğ/Ĭ.

We will also discuss certain finite unions of affine Deligne-Lusztig varieties. As before, we

denote by µ the image of a dominant cocharacter µ in X∗(T )Γ0 Following [Rap05], we then

define the associated admissible set as

Adm(µ) = {w ∈ W̃ : w ≤ tx(µ) for some x ∈ W}.

We will need the following additivity property of admissible sets.

Theorem 2.3.1. [He16a, Theorem 5.1][HH17, Theorem 1.4] Let µ, µ′ ∈ X∗(T ) be domi-

nant. Then we have

Adm(µ) · Adm(µ′) = Adm(µ+ µ′).

18



For any b ∈ Ğ, we set

X(µ, b) =
⋃

w∈Adm(µ)

Xw(b).

We remark that Xw(b) and X(µ, b) are subschemes of the affine flag variety in the

usual sense in equal characteristic, and in the sense of Zhu [Zhu17], Bhatt and Scholze

[BS17] in mixed characteristic. Settling the Kottwitz-Rapoport conjecture made in [KR03]

and [Rap05] about the non-emptiness pattern for X(µ, b), He proves the following result in

[He16a].

Theorem 2.3.2. [He16a, Theorem A] X(µ, b) ̸= ∅ if and only if [b] ∈ B(G, µ).

2.3.1 Virtual dimension of affine Deligne-Lusztig variety

Suppose that G is quasi-split; then σ preserves the set of finite simple roots and hence

acts on W . Note that w ∈ W̃ can be written in a unique way as w = utλv with λ dominant,

u, v ∈ W such that tλv is a minimum length element in the right W -coset in W̃ determined

by w. In this case, we set ησ(w) = σ−1(v)u.

Let b ∈ Ğ. The defect of b is defined by defG(b) = rankF G− rankF Jb.

Following [He14, §10.1], we then define the virtual dimension for Xw(b) to be

dw(b) =
1

2

(
ℓ(w) + ℓ(η(w))− defG(b)

)
− ⟨ρ, ν([b])⟩.

The justification of defining such an expression lies in a result proved by He in [He14,

Corollary 10.4] that says dimXw(b) ≤ dw(b) whenever κ(w) = κ([b]). Recent work by

Milićević and Viehmann in [MV20] singles out those w ∈ W̃ for which dimXw(bw) = dw(bw)

19



holds; these are called cordial elements. It is shown in loc. sit. that B(G)w exhibits

remarkable properties for such w, see [MV20, Theorem 1.1] for a discussion of this.

2.4 Demazure product and its variations

We now discuss three operations ∗,�,� : W̃ × W̃ → W̃ . Here ∗ is the Demazure

product and �,� are the left and right downward Demazure products, respectively. We

describe these operations in form of the following lemma.

Lemma 2.4.1. [HL15, Section 2.1] Let x, y ∈ W̃ .

1. The subset {uv : u ≤ x, v ≤ y} contains a unique maximal element, which we denote

by x ∗ y. Moreover, x ∗ y = u′y = xv′ for some u′ ≤ x and v′ ≤ y and ℓ(x ∗ y) =

ℓ(u′) + ℓ(y) = ℓ(x) + ℓ(v′).

2. The subset {uy : u ≤ x} contains a unique minimal element, which we denote by

x� y. Moreover, x� y = u′′y for some u′′ ≤ x with ℓ(x� y) = ℓ(y)− ℓ(u′′).

3. The subset {xv : v ≤ y} contains a unique minimal element, which we denote by

x� y. Moreover, x� y = xv′′ for some v′′ ≤ y with ℓ(x� y) = ℓ(x)− ℓ(v′′).

The Demazure product is related to product of closure of two Iwahori double cosets

in Ğ. More precisely, for any w ∈ W̃ , ĬwĬ =
⋃

w′≤w

Ĭw′Ĭ is a closed admissible subset of G

in the sense of [He16a]. Then for any x, y ∈ W̃ , we have

ĬxĬ · ĬyĬ =
⋃
u≤x

ĬuĬ ·
⋃
v≤y

ĬvĬ =
⋃

w≤x∗y

ĬwĬ = Ĭ(x ∗ y)Ĭ .

20



The above equation also implies that ∗ is an associative binary operation on W̃ . We

will need the following results about these operations.

Lemma 2.4.2. [He09, Lemma 2 and Lemma 3]

1. If w ≥ w′, v ≤ v′ are elements of W̃ , then w � v ≤ w′ � v′.

2. For any three elements x, y, z ∈ W̃ , we have x� (y� z) = (x∗y)� z. In other words,

the action (W̃ , ∗)× W̃ → W̃ , (x, y)→ x� y is a left action of the monoid (W̃ , ∗).

2.5 Quantum Bruhat graph

We recall the quantum Bruhat graph introduced by Brenti, Fomin and Postnikov in

[BFP99]. By definition, a quantum Bruhat graph ΓΦ is a directed graph with

• vertices given by the elements of W ;

• upward edges w ⇀ wsα for some α ∈ Φ+ with ℓ(wsα) = ℓ(w) + 1;

• downward edges w ⇁ wsα for some α ∈ Φ+ with ℓ(wsα) = ℓ(w)− ⟨2ρ, α∨⟩+ 1.

Note that by [BFP99, Lemma 4.3], ℓ(sα) ≤ ⟨2ρ, α∨⟩ − 1 for any α ∈ Φ+, so we have

ℓ(wsα) ≥ ℓ(w)− ℓ(sα) ≥ ℓ(w)− ⟨2ρ, α∨⟩+ 1. Therefore the condition for downward edges

can be rephrased to saying that

ℓ(wsα) = ℓ(w)− ℓ(sα) with ℓ(sα) = ⟨2ρ, α∨⟩ − 1.

Following [Len+15], we call α ∈ Φ+ to be a quantum root if ℓ(sα) = ⟨2ρ, α∨⟩− 1. We

have the following description of quantum roots.

21



Lemma 2.5.1. [Len+15, Lemma 4.2] We have that α ∈ Φ+ is a quantum root if and only

if

1. α is a long root, or

2. α is a short root, and if α =
∑

αi∈∆
ciαi then we have ci = 0 for any long simple root

αi.

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.

We now recall some graph theoretic notions related to ΓΦ; we refer to [Mil21, section

3.1] for more details. The weight of an upward edge is defined to be 0 and the weight

of a downward edge w ⇁ wsα is defined to be α∨. For two elements w,w′, a directed

path between them is defined to be concatenation of directed edges joining a sequence of

vertices in ΓΦ. The weight of a path in ΓΦ is defined to be the sum of weights of the edges

in the path. The length of path is defined to be the number of edges present in it. For any

x, y ∈ W , we denote by dΓ(x, y) the minimal length among all paths in ΓΦ from x to y.

Any path between x and y affording dΓ(x, y) as its length is called a shortest path between

them.

Lemma 2.5.2. [Pos05, Lemma 1], [BFP99, Lemma 6.7] Let x, y ∈ W . Then

1. There exists a directed path (consisting of possibly both upward and downward edges)

in ΓΦ from x to y.

22



2. Any two shortest paths in ΓΦ from x to y have the same weight, which we denote by

wt(x, y).

3. Any path in ΓΦ from x to y has weight ⩾ wt(x, y).

23



Chapter 3: Generic Newton point and affine Bruhat order

For a split connected reductive group, an explicit description of νw was first derived

by Milićević in [Mil21, Theorem 3.2] for elements w ∈ W̃ with superregular dominant

translation part. More precisely, a uniform bound is given in [Mil21, Corollary 3.3] in the

quasi-simple case, which says that the description of νw in loc. sit. is valid whenever the

following depth hypothesis is satisfied by λ:

depth(λ) ≥


8ℓ(w0), if G is of classical type;

16ℓ(w0), if G is of exceptional type.

To establish this result, Milićević first derives in [Mil21, Proposition 4.2] a characteriza-

tion of the covering relations in W̃ , again under certain superregularity hypothesis of the

following nature:

depth(λ) ≥


2ℓ(w0) + 2, if G is not of type G2;

3ℓ(w0) + 3, if G is of type G2.

This characterization is of independent interest, and more recently it has been utilized by

He and Yu in [HY21] to deduce a partial description of the admissible sets in W̃ . In this

24



chapter, we improve upon these results by weakening the superregularity hypothesis to

prove them.

3.1 Some combinatorial properties

We start by proving a monotonicity property for the weight function. Let us define

the function wt : W → X∗(T ) by wt(x) := wt(x, 1).

Lemma 3.1.1. Let x1, x2 ∈ W ; if x1 ≤ x2 in Bruhat order, then wt(x1) ≤ wt(x2) in

dominance order.

Proof. It suffices to show this when x2 is a cover of x1, i.e. x2 = x1sα for some α ∈ Φ+

such that ℓ(x2) = ℓ(x1)+1. We construct a path from x1 to 1 by concatenating the upward

edge x1 ⇀ x2 with a path of shortest length from x2 to 1. Since the first upward edge has

weight 0, the weight of this particular path is wt(x2). Appealing to Lemma 2.5.2(3), we

then get wt(x2) ≥ wt(x1).

We will now reduce our calculation of the maximal Newton point of an arbitrary

element of W̃ to that of a suitable element lying in the dominant chamber.

Proposition 3.1.2. If λ is dominant regular, then [butλv] = [btλ(v�σ(u)].

Proof. Note that by Equation (2.1.1), ℓ(utλv) = ℓ(u)+ ℓ(tλv), hence utλv = u∗ (tλv). Thus

ĬutλvĬ = ĬuĬ · ĬtλvĬ.

Now, recall that Ĭ is σ-stable. Therefore, if w = w1w2 is an element with w1 ∈

ĬuĬ, w2 ∈ ĬtλvĬ, then w−1
1 ·σw = w2σ(w1). This shows that Ğ ·σ (ĬuĬ · ĬtλvĬ) ⊂ Ğ ·σ (ĬtλvĬ ·

25



Ĭσ(u)Ĭ). One obtains the other inclusion similarly. This shows that Ğ ·σ (ĬuĬ · ĬtλvĬ) =

Ğ ·σ (ĬtλvĬ · ĬuĬ).

Hence, we conclude that Ğ ·σ ĬutλvĬ = Ğ ·σ Ĭ(tλv) ∗ σ(u)Ĭ .

Since G ·σ ĬwĬ is the set of σ-conjugacy classes intersecting ĬwĬ, by appealing to

[Vie14, Corollary 5.6] we get that

[butλv] = [b(tλv)∗σ(u)]. (3.1.1)

Thus it suffices to show that

(a) if λ is dominant regular, then (tλv) ∗ u′ = tλ(v � u′) for elements u′, v ∈ W .

By Lemma 2.4.2(2), we only need to argue the case when u′ is a simple reflection

s ∈ S. Then this statement is equivalent to the assertion that ℓ(vs) = ℓ(v)− 1 if and only

if ℓ(tλvs) = ℓ(tλv) + 1. This follows directly from a computation using Equation (2.1.1).

Hence we are done.

Corollary 3.1.3. Let x, y ∈ W . Then wt(x, y) = wt(x−1 � y).

Proof. Suppose that λ ∈ X∗(T )
+ is superregular. Applying the formula for maximal New-

ton point established in [Mil21, Theorem 3.2] to w := xtλy and w′ := tλ(y � x), we get

that νw = λ− wt(y−1, x) and νw′ = λ− wt((y � x)−1) = λ− wt(y � x). Since νw = νw′ by

Proposition 3.1.2, we get wt(y−1, x) = wt(y�x). Now replacing the pair (x, y) by (y, x−1),

we get the conclusion.

We now close this section by giving another interpretation of the weight of a shortest

path in the quantum Bruhat graph.

26



Definition 3.1.4. For an element x ∈ W , we say that x = sβ1 · · · sβk
is a reduced quantum

reflection decomposition if

1. {βi : 1 ≤ i ≤ k} is a collection of (not necessarily simple) quantum roots, i.e.

l(sβi
) = ⟨2ρ, β∨

i ⟩ − 1.

2. ℓ(x) =
∑k

i=1 ℓ(sβi
).

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 ℓ↓(x) = k.

The choice of notation is suggested from the fact that

{Reduced quantum reflection decompositions for x}xy
{Shortest paths in ΓΦ from x to 1 that uses only downward edges}

By [MV20, Proposition 4.11], the weight of any path in the latter set is equal to wt(x) and

dΓ(x, 1) = ℓ↓(x). Therefore, the problem of computing wt(x) reduces to one of finding a suit-

able decomposition for x: if x = sβ1 · · · sβk
is such a decomposition then wt(x) =

∑k
i=1 β

∨
i .

In view of Corollary 3.1.3, we can thus reformulate wt(x, y) in terms of a decomposition of

x−1 � y.

3.2 Formula for the generic Newton point

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

27



Theorem 3.2.1. Assume that G is a quasi-simple split group of rank n. Let w = tλx be

an element of its Iwahori-Weyl group such that λ is dominant and depth(λ) > Ξ, where Ξ

is an integer depending on n. Then the maximal Newton point associated to w is given by

νw = λ− wt(x).

For each Cartan type, the lower bound Ξ for the depth mentioned above is given in

the following table.

Type An Bn/Cn Dn E6 E7 E8 F4 G2

Ξ 3n+ 1 6n− 2 6n− 6 23 33 57 23 9

Remark 3.2.2. The statement of Theorem 3.2.1 seems weaker than the statement of

Theorem 1.1.2 in the sense that the former gives a formula only for certain elements of

W̃ (associated to a quasi-simple split group G) in the dominant chamber for whereas the

latter promises to handle elements in arbitrary chambers (for arbitrary split groups). Hence

let us first explain how Theorem 3.2.1 implies Theorem 1.1.2 - for quasi-simple groups; in

the next remark, we further remove this condition on the group. By Proposition 3.1.2, we

know that

νutλv = νtλ(v�σ(u).

Under the hypothesis on λ, this element tλ(v � σ(u)) lies in the dominant chamber. Then

by Theorem 3.2.1 it follows that

νtλ(v�σ(u) = λ− wt(v � σ(u)).

By Corollary 3.1.3 we finally deduce that νutλv = λ − wt(v−1, σ(u)). This concludes our

28



discussion.

Remark 3.2.3. One can handle the case of a connected reductive split group G via a

routine reduction procedure, which we discuss now. In that case, we have

W = W1 × · · · ×Wl,

where Wi are irreducible Weyl groups associated to F -simple factors of Gad. This gives

rise to a partition of m, where m is the semisimple rank of G. Any element x of W is of

the form (x1, · · · , xl). Similarly, we write λ as (λ1, · · · , λl). We have w0 = (w0,1, · · · , w0,l),

where w0,i is the longest element of Wi. In the same way, we write the sum of positive roots

as (2ρ∨1 , · · · , 2ρ∨l ) and the highest root as (θ1, · · · , θl).

We observe that while dealing with the quasi-simple case, we introduce a restriction

on depth in two stages - as found in Section 3.2.1 and Section 3.2.4. By the last paragraph

we therefore need to require for each i

depth(λi) ≥ Ξi,

where Ξi is the lower bound given for the Weyl group Wi, to be read off from the table

above. Note that Ξi 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 hypotheses, Theorem 3.2.1

applies to each quasi-simple factor of Gad and thus we need to justify the formula for such

groups.

Remark 3.2.4. Let us note that Theorem 3.2.1 has since been strengthened due to the

29



work of multiple authors, by weakening the bound on depth of the relevant coweight. In

[HN21], He and Nie prove the formula for generic Newton point under the assumption that

the relevant cocharacter is of depth at least 2. Finally, Schremmer completely solves the

problem for arbitrary cocharacter in [Sch22b] by removing the hypothesis on depth, albeit

the description gets more complicated in this situation.

We now give a technical layout of the proof. As mentioned before, under the regularity

assumption on its associated dominant coweight as in Proposition 3.1.2, we only need to

handle elements w ∈ W̃ that lie in the dominant chamber. In Proposition 3.2.6, we show

that w is bigger than a certain translation element, thus giving us a lower bound for νw.

This holds under a certain depth hypothesis proposed in Section 3.2.1, which relies on

explicit calculation in Section 3.3 of wt(w0) for the longest element w0 in each Cartan type.

Using this lower bound, we also show in Lemma 3.2.7 that νw is given by Newton point of

some translation tγ smaller than w. Note that if this translation can be shown to be lying

in the dominant chamber (i.e. γ is dominant), we are practically done - as one can then

multiply the inequality tλy ≥ tγ by a large enough dominant translation element, thereby

making the translation parts in the resulting elements superregular and hence suitable for

applying the existing formula for νw in [Mil21, Theorem 3.2]. However, we cannot show

this and rather bypass this problem by converting the Bruhat inequality tλy ≥ tγ to one

involving elements whose translation parts are dominant. This is done in Lemma 3.2.9 by

applying techniques from the theory of the Demazure product on W̃ . Then we can carry

out the aforementioned trick of artificially adding a large translation part, applying [Mil21,

Theorem 3.2] and then finally subtracting it - as shown in Section 4.2.5.

30



3.2.1 Proposed linear bound on depth

For the rest of Section 3.2, we assume that W is an irreducible Weyl group for the

root system of Cartan type Xn. We first establish an upper bound for

M =MXn := max{⟨αi,wt(x)⟩ : αi ∈ ∆, x ∈ W}

This is based on an explicit computation of wt(w0) for the longest element w0 in each type,

which we defer to the next section for the sake of continuity.

By Lemma 3.1.1, we have {wt(x) : x ∈ W} ⊂ {ν ∈ ZΦ∨ : ν ≤ wt(w0)}. Therefore,

M≤ max{⟨αi, ν⟩ : ν ∈ ZΦ∨, ν ≤ wt(w0), αi ∈ ∆}.

It is easy to see that maximum of the latter set is realized at ν = κlα
∨
l , where κlα

∨
l is a

summand of wt(w0) such that κl = max{κj : κjα
∨
j is a summand of wt(w0)}. Since the

maximum value of ⟨αi, α
∨
l ⟩ is 2 in every Cartan type, this shows that MXn is bounded

above by twice the largest coefficient in the expression of wt(w0) in each type Xn. This is

tabulated in the list below, where we denote by M̃ = M̃Xn the upper bound obtained in

this manner.

Type An Bn/Cn Dn E6 E7 E8 F4 G2

M̃ n+ 1 2n 2n 12 16 28 12 4

31



3.2.2 Proof of the easier inequality

We need the following result about Bruhat order on W̃ .

Lemma 3.2.5. [Rap05, Remark 3.9] Let λ ∈ X∗(T ) be dominant. Let β ∈ Φ+ such that

λ− β∨ is dominant. Then tλ−β∨ ≤ tλsβ ≤ tλ in W̃ .

We start by establishing the easier inequality.

Proposition 3.2.6. Let w = tλx ∈ W̃ such that depth(λ) ≥ M̃. Then w ≥ tλ−wt(x) and

therefore νw ≥ λ− wt(x).

Proof. Suppose that x = sβ1 · · · sβl
is a reduced quantum reflection decomposition with

ℓ↓(x) = l > 1, and therefore wt(x) =
l∑

i=1

β∨
i . Let us first note that:

(a) sβ1 · · · sβk
is a reduced quantum reflection decomposition for xk := xsβl

· · · sβk+1
for all k ≤ l.

Suppose otherwise; then we can find a reduced quantum reflection decomposition of the

form xk = sγ1 · · · sγj with j < k. But then x = xksβk+1
· · · sβl

= sγ1 · · · sγjsβk+1
· · · sβl

satisfies length additivity:

ℓ(x) = ℓ(xk) +
l∑

i=k+1

β∨
i =

j∑
i=1

γ∨
i +

l∑
i=k+1

β∨
i .

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 ℓ↓(x) = l. So statement (a) is

proved.

32



To prove the proposition, we argue by induction on ℓ↓(x). We note our depth hy-

pothesis ensures that λ−β∨ is dominant for all β ∈ Φ+: for αi ∈ ∆, we have ⟨αi, λ−β∨⟩ =

⟨αi, λ⟩−⟨αi, β
∨⟩ > 0, since maximum value of ⟨αi, β

∨⟩ equals 2 in every Cartan type, except

for G2 - in which case it equals 3, cf. [Bou02, Chapter VI, Section 1, no. 3]. Therefore

Lemma 3.2.5 covers the base case, i.e. when x = sβ for some β ∈ Φ+.

Let us now apply the induction hypothesis to xsβl
= sβ1 · · · sβl−1

. Since wt(xsβl
) =

l−1∑
i=1

β∨
i by virtue of statement (a), this gives

tλsβ1 · · · sβl−1
≥ t

λ−
l−1∑
i=1

β∨
i
. (3.2.1)

To perform the induction step, we apply a property of Bruhat order as described in Equa-

tion (2.1.3). We set

w1 = tλx,w2 = t
λ−

l−1∑
i=1

β∨
i
sβl

, v = sβl

and check the required length additivity. Note that λ−
l−1∑
i=1

β∨
i is dominant under our depth

hypothesis: for αi ∈ ∆,

⟨αi, λ−
l−1∑
i=1

β∨
i ⟩ = ⟨αi, λ⟩ − ⟨αi,wt(xsβl

)⟩ > ⟨αi, λ⟩ − M̃ ≥ 0,

therefore the length formula in Equation (2.1.1) applies. We see that

1. ℓ(w1v) = ℓ(tλsβ1 · · · sβl−1
) = ℓ(tλ)−

l−1∑
i=1

ℓ(sβi
) = ℓ(tλ)−

l∑
i=1

ℓ(sβi
)+ℓ(sβl

) = ℓ(w1)+ℓ(v).

2. ℓ(w2v) = ℓ(t
λ−

l−1∑
i=1

β∨
i
) = ℓ(t

λ−
l−1∑
i=1

β∨
i
)− ℓ(sβl

) + ℓ(sβl
) = ℓ(w2) + ℓ(v).

In presence of Equation (3.2.1), we therefore get w1 ≥ w2, i.e. tλx ≥ t
λ−

l−1∑
i=1

β∨
i
sβl

. Finally,

33



we note that λ−wt(x) = λ−
l∑

i=1

β∨
i is dominant (by similar argument as before) and thus

Lemma 3.2.5 applies to give t
λ−

l−1∑
i=1

β∨
i
sβl
≥ t

λ−
l∑

i=1
β∨
i
. Combining these inequalities, we get

tλx ≥ tλ−wt(x). By [Vie14, Corollary 5.6], we have

νw = max{ν(u) : u ≤ w, u ∈ W̃}. (3.2.2)

Appealing to Equation (3.2.2), we thus get νw ≥ λ− wt(x). This finishes the proof.

Lemma 3.2.7. Let w = tλx ∈ W̃ such that depth(λ) > M̃. Then νw = max{γ+ : tγ ≤ w}.

Proof. By Equation (3.2.2), we can assume that νw = ν(tµz) for some tµz ∈ W̃ = X∗(T )⋊

W , such that w ≥ tµz. Suppose that z ̸= 1, therefore the order of z equals m ≥ 2. Then

ν(tµz) = ( 1
m

m∑
i=1

zi(µ))+. Note that z ·
m∑
i=1

zi(µ) =
m∑
i=1

zi(µ), so the element
m∑
i=1

zi(µ) lies on

the wall of some Weyl chamber, cf. [Bou02, Chapter V, Section 3.3, Remark 3]. Therefore

( 1
m

m∑
i=1

zi(µ))+ lies on the wall of C+, and hence νw is singular.

By Proposition 3.2.6, we can assume that νw = λ − ϵ, with ϵ ≤ wt(x) ≤ wt(w0).

Hence, there is some αi ∈ ∆ such that ⟨αi, λ − ϵ⟩ = 0, and thus ⟨ai, λ⟩ = ⟨αi, ϵ⟩ ≤ M̃ ,

which is a contradiction to hypothesis on depth(λ). Therefore, νw cannot be singular and

hence z = 1.

3.2.3 Toward computing a downward Demazure product

We need to understand the largest translation dominated by w to determine its asso-

ciated generic Newton point νw. In view of Proposition 3.2.6, let us denote this translation

element by tyγ for some y ∈ W and γ ≥ λ− wt(x). Of course, this would be equality if λ

34



is superregular. We want to leverage this result to establish such equality with some linear

bound on depth here. The following lemma lays the path towards proving that.

Lemma 3.2.8. If λ−2ρ∨ is dominant, we have tλx ≥ tyγ if and only if tλ−2ρ∨ ≥ t−2ρ∨�tyγ.

Proof. Note that if λ− 2ρ∨ is dominant, ℓ(tλ−2ρ∨) = ⟨2ρ, λ− 2ρ∨⟩ = ℓ(tλ)− ℓ(t−2ρ∨); also,

since λ ≥ λ − 2ρ∨ are dominant coweights, tλ ≥ tλ−2ρ∨ , cf. [Rap05, proof of Proposition

3.5]. Hence, t−2ρ∨ � tλ = tλ−2ρ∨ .

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 ≥ b

if and only if s� a ≥ s� b.

This follows from the well known lifting property of Bruhat order. Note that we only

need to consider the case when s� a = sa, i.e. a ≥ sa. Then there are two further cases.

1. s � b = b: in this case, sb ≥ b. Hence a ≥ b is equivalent to sa ≥ b, and hence

s� a ≥ s� b.

2. s � b = sb: in this case, b ≥ sb. Hence a ≥ b is equivalent to sa ≥ sb, and hence

s� a ≥ s� b.

In Section 3.2.4 we will focus on computing the downward Demazure product that

appears in Lemma 3.2.8. Before proceeding any further to do that, let us first sketch

what the answer should look like. We first apply Lemma 2.4.2 with w = t−2ρ∨ , w′ = w0

and v = v′ = tyµ; if we require that λ is of large enough depth (in a sense made precise

below) so that γ is dominant regular, we get w0 � ytγy−1 = tγy−1 ≥ t−2ρ∨ � tyγ . We

35



next apply Lemma 2.4.2 with v′ = tyγ, v = tγy−1 and w = w′ = t−2ρ∨ . If we now require

λ to be of sufficiently large depth so that γ − 2ρ∨ is also dominant, we get t−2ρ∨ � tyγ ≥

t−2ρ∨ � tγy−1 = tγ−2ρ∨y−1. Combining these, we see that whenever λ has “large” depth (to

be specified later) we have

tγy−1 ≥ t−2ρ∨ � tyγ ≥ tγ−2ρ∨y−1.

Therefore, it is natural to expect that this downward Demazure product depends only

on y whenever λ has a “large” depth. In the next section, we quantify this depth condition

and show that we indeed get the desired conclusion.

3.2.4 A technical lemma

For each irreducible Cartan type Xn, we let S = SXn be twice the sum of all coef-

ficients appearing in the expression of the highest root in terms of simple roots; in other

words, S = ⟨θ, 2ρ∨⟩, where θ 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 An Bn/Cn Dn E6 E7 E8 F4 G2

S 2n 4n− 2 4n− 6 11 17 29 11 5

Note that Ξ = M̃ + S is the final lower bound that appears in Theorem 3.2.1. We

now prove a key lemma that would help us achieve the harder inequality.

Lemma 3.2.9. Let depth(λ) > Ξ. We continue to assume that w = tλx ≥ tyγ such that

36



γ = νw ≥ λ − wt(x). Then there exists a coweight µy depending only on y such that

t−2ρ∨ � tyγ = tγ−µyy−1.

Proof. We begin by noting that we can compute this downward Demazure product from

a specific length additive decomposition of tyγ. More precisely, we have the following

desiderata

• (D1) a decomposition tyγ = a1t
µ1b1 · a2tµ2b2, where µi is dominant and tµibi ∈ SW̃

for i = 1, 2;

• (D2) ℓ(tyγ) = ℓ(a1t
µ1b1) + ℓ(a2t

µ2b2);

• (D3) a1t
µ1b1 is the largest element dominated by t2ρ

∨
, subject to the first two condi-

tions.

Lemma 2.4.1 asserts the existence of such decomposition, and then we have t−2ρ∨ � tγy =

a2t
µ2b2. We now proceed to identify these elements ait

µibi in the following three steps.

3.2.4.1 Relating relevant coweights

We first determine how the coweights associated to translation part of these elements

are related. By (D1), tyγ = a1b1a2t
a−1
2 b−1

1 µ1+µ2b2, so we have γ = (a−1
2 b−1

1 µ1 + µ2)
+, i.e.

there exists some v ∈ W such that γ = v(a−1
2 b−1

1 µ1 + µ2). We will now show that v = 1.

Assume the contrary and let β ∈ Inv(v). Now, µ2 = v−1γ − a−1
2 b−1

1 µ1 is dominant,

therefore

⟨β, v−1γ − a−1
2 b−1

1 µ1⟩ ≥ 0.

37



By (D3), we have t2ρ
∨ ≥ a1t

µ1b1, thus 2ρ∨ ≥ µ1; by a geometric characterization of

dominance order, e.g. see [AB83, Lemma 12.14], this gives

µ1 =
∑
ζ∈W

aζζ · 2ρ∨ for some aζ ∈ R≥0 such that
∑
ζ∈W

aζ = 1.

Substituting this expression of µ1 in the pairing above and using its W -invariance,

we get

⟨vβ, γ⟩ −
∑
ζ∈W

aζ⟨ζ−1b1a2β, 2ρ
∨⟩ ≥ 0.

Let β′ = −vβ ∈ Φ+. Note that ζ−1b1a2β ≥ −θ for any element ζ ∈ W , and therefore

⟨ζ−1b1a2β, 2ρ
∨⟩ ≥ −⟨θ, 2ρ∨⟩. Putting all these together, we obtain

⟨β′, γ⟩ = −
∑
ζ∈W

aζ⟨ζ−1b1a2β, 2ρ
∨⟩ ≤

∑
ζ∈W

aζ⟨θ, 2ρ∨⟩ = S.

Now recall that γ = νw = λ−ϵ is dominant with ϵ ≤ wt(x) ≤ wt(w0). Choose a simple

root α ≤ β′. Then we get ⟨α, γ⟩ ≤ ⟨β′, γ⟩ ≤ S, and therefore ⟨α, λ⟩ ≤ S + ⟨α, ϵ⟩ ≤ S + M̃.

Since this yields a contradiction to the depth hypothesis of λ, we are done. Therefore v = 1

and γ = a−1
2 b−1

1 µ1 + µ2.

3.2.4.2 Showing that certain coweights are regular

Note that γ is regular, since for any simple root αi we have ⟨αi, γ⟩ = 0 =⇒ ⟨αi, λ⟩ =

⟨αi, ϵ⟩ ≤ M̃, thereby contradicting the depth hypothesis imposed on λ. Hence, knowing

that the translation parts in tyγ = a1b1a2t
a−1
2 b−1

1 µ1+µ2b2 are equal allows us to relate the finite

Weyl group components from both sides. Namely, we have y = a1b1a2 = b−1
2 . We next

38



simplify these relations further in the following way. Let us first observe that µ2 is regular,

since otherwise we can find a simple root α such that ⟨α, γ⟩ =
∑

ζ∈W aζ⟨α, ζ ·a−1
2 b−1

1 2ρ∨⟩ =∑
ζ∈W ⟨b1a2ζα, 2ρ∨⟩ ≤

∑
ζ∈W aζ⟨θ, 2ρ∨⟩ = S; but this would yield contradiction to the

imposed depth condition as before.

3.2.4.3 Showing that the product describes a dominant chamber alcove

We can now show that a2 = 1. Recall that t−2ρ∨ � tyγ = a2t
µ2b2. Let us rewrite

t−2ρ∨ = w0t
2ρ∨w0 and apply w0� to the former equation. This gives

w0 � {(w0t
2ρ∨w0)� tyγ} = w0 � (a2t

µ2b2).

We now apply Lemma 2.4.2(2) on the left hand side, and use the fact that µ2 is

regular in the right hand side. We get

{w0 ∗ (w0t
2ρ∨w0)}� tyγ = (w0 � a2)t

µ2b2.

Note that w0t
2ρ∨w0 = w0 ∗ t2ρ

∨
w0, and hence we can use associativity of operation

∗ to rewrite the element in the parenthesis on the left hand side as w0 ∗ (w0 ∗ t2ρ
∨
w0) =

(w0 ∗ w0) ∗ t2ρ
∨
w0 = w0 ∗ t2ρ

∨
w0 = w0t

2ρ∨w0 = t−2ρ∨ . Therefore, we finally deduce

t−2ρ∨ � tyγ = tµ2b2.

This proves our claim.

39



Therefore, the relations arising from (D1) are

γ = b−1
1 µ1 + µ2, y = a1b1 = b−1

2 . (3.2.3)

Let us now utilize (D2). It says that we have length additivity in the decomposition

tyγ = a1t
µ1b1 · tµ2b2, i.e.

⟨γ, 2ρ⟩ = ℓ(a1) + ⟨µ1, 2ρ⟩ − ℓ(b1) + ⟨µ2, 2ρ⟩ − ℓ(b2).

By use of Equation (3.2.3), the last equation reduces to

ℓ(a1b1) = ℓ(a1)− ℓ(b1) + ⟨µ1 − b−1
1 µ1, 2ρ⟩. (3.2.4)

Therefore we are led to consider the following subset of W̃ :

Ωy := {atµb ∈ W̃ : y = ab, atµb ≤ t2ρ
∨
, ℓ(ab) = ℓ(a)− ℓ(b) + ⟨µ− b−1µ, 2ρ⟩}.

Note that the definition of Ωy encapsulates all information amongst a1, b1, µ1 coming

out of the first two items in our desiderata. By (D3), the element a1t
µ1b1 is therefore

uniquely specified as the maximal element of Ωy. Hence, the element b−1
1 µ1 depends only

on y. Denoting it by µy, we see that Equation (3.2.3) implies µ2 = γ − µy. This finishes

the proof.

40



3.2.5 Finishing the proof

Proof of Theorem 3.2.1. By combining Lemma 3.2.8 and Lemma 3.2.9, we observe that

our standing assumption tλx ≥ tyγ implies

tλ−2ρ∨x ≥ tγ−µyy−1.

Let us now choose λ′ to be large enough so that λ + λ′ is superregular, e.g. take

λ′ = nρ∨ for large n. We note that

1. ℓ(tλ
′ · tλ−2ρ∨x) = ℓ(tλ+λ′−2ρ∨x) = ⟨2ρ, λ+ λ′− 2ρ∨⟩ − ℓ(x) = ⟨2ρ, λ′⟩+ ⟨2ρ, λ− 2ρ∨⟩ −

ℓ(x) = ℓ(tλ
′
) + ℓ(tλ−2ρ∨x).

2. ℓ(tλ
′ · tγ−µyy−1) = ℓ(tλ

′+γ−µyy−1) = ⟨2ρ, λ′ + γ − µy⟩ − ℓ(y−1) = ⟨2ρ, λ′⟩ + ⟨2ρ, γ −

µy⟩ − ℓ(y−1) = ℓ(tλ
′
) + ℓ(tγ−µyy−1).

Hence, we can apply Equation (2.1.3) in this situation, and get

tλ+λ′−2ρ∨x ≥ tγ+λ′−µyy−1.

Now, we apply Lemma 3.2.8 again to deduce from the previous equation

tλ+λ′
x ≥ ty(γ+λ′).

Therefore, by Equation (3.2.2) we have νtλ+λ′x ≥ νty(γ+λ′) . Since the formula for the

maximal Newton point in [Mil21, Theorem 3.2] applies to elements of W̃ having λ+λ′ as its

41



dominant translation part, we get that λ+λ′−wt(x) ≥ γ+λ′, whence λ−wt(x) ≥ γ = νw.

Combining this with Proposition 3.2.6, we get νw = λ−wt(x). This finishes the proof.

3.2.6 A remark about Bruhat order on W̃

Proposition 3.2.10. Suppose that G is a quasi-simple split group. Assume depth(λ) > Ξ.

Then utλv ≥ tyγ for some dominant γ implies that tλ(v � u) ≥ tγ.

Proof. Note that utλv ≥ tyγ implies νutλv ≥ νtyγ by Equation (3.2.2). By Theorem 3.2.1

and Corollary 3.1.3, we then have λ − wt(v � u) ≥ γ. By Proposition 3.2.6, we have

that tλ(v � u) ≥ tλ−wt(v�u). Since λ − wt(v � u), γ are both dominant, we also have

that tλ−wt(v�u) ≥ tγ, cf. [Rap05, proof of Proposition 3.5]. Putting together the last two

inequalities, we finally get tλ(v � u) ≥ tγ.

Remark 3.2.11. Note that if u = 1, this says that tλv ≥ tyγ implies tλv ≥ tγ. 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 3.2.6 and the fact that maximal Newton point formula is avail-

able for elements with superregular translation part, this is equivalent to Theorem 3.2.1.

Hence, an independent proof of Proposition 3.2.10 would help us get rid of the bound im-

posed in Section 3.2.4; similarly, an improved estimate forM would weaken the final depth

hypothesis required in Theorem 3.2.1.

42



3.3 Weight of the longest element

In this section, we give a reduced quantum reflection decomposition for the longest

element w0 in Weyl group W of each irreducible Cartan type. For an element x ∈ 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 ℓR(x) the reflection length of x. Note that in general

ℓR(x) ≤ ℓ↓(x), and strict inequality can occur. We enumerate reflection length for w0 for

all the Cartan types below.

Type An Bn/Cn Dn E6 E7 E8 F4 G2

ℓR(w0) ⌈n
2
⌉ n 2⌊n

2
⌋ 4 7 8 4 2

We compute wt(w0) by exhibiting suitable decomposition of w0 in each type, and

on the way we see that ℓR(w0) = ℓ↓(w0). We first write down expression of the longest

element affording its reflection length; for the classical types, we extract this from [Bla09],

and for the exceptional ones we can check this directly by hand or using a computer algebra

system such as [TheYY]. 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 w0 in these types. Finally, we

add up the coroots associated to the reflections appearing in each such decompositions and

deduce wt(w0) from that. See Remark 3.3.1 for an alternative recipe for computing wt(w0).

43



(a) Type An: here

w0 =


sα1+···+α2k

sα2+···+α2k−1
· · · sαk+αk+1

, if n = 2k;

sα1+···+α2k+1
sα2+···+α2k

· · · sαk+1
, if n = 2k + 1.

Hence,

wt(w0) =



α∨
1 + 2α∨

2 + · · ·+ (k − 1)α∨
k−1 + kα∨

k + kα∨
k+1 + (k − 1)α∨

k+2

+ · · ·+ α∨
2k, if n = 2k;

α∨
1 + 2α∨

2 + · · ·+ kα∨
k + (k + 1)α∨

k+1 + kα∨
k+2

+ · · ·+ α∨
2k+1, if n = 2k + 1.

(b) Type Bn: here

w0 =


sα1+2α2+···+2α2k

sα1sα3+2α4+···+2α2k
sα3 · · · sα2k−1+2α2k

sα2k−1
, if n = 2k;

sα1+2α2+···+2α2k+1
sα1sα3+2α4+···+2α2k+1

sα3 · · · sα2k−1+2α2k+2α2k+1
sα2k+1

, if n = 2k + 1.

Hence,

wt(w0) =



2α∨
1 + 2α∨

2 + 4α∨
3 + 4α∨

4 + · · ·+ 2(k − 1)α∨
2k−3 + 2(k − 1)α∨

2k−2

+2kα∨
2k−1 + kα∨

2k, if n = 2k;

2α∨
1 + 2α∨

2 + 4α∨
3 + 4α∨

4 + · · ·+ 2kα∨
2k−1 + 2kα∨

2k + (k + 1)α∨
2k+1, if n = 2k + 1.

44



(c) Type Cn: here w0 = s2α1+···+2αn−1+αns2α2+···+2αn−1+αn · · · s2αn−1+αnsαn , and thus

wt(w0) = α∨
1 + 2α∨

2 + · · ·+ nα∨
n .

(d) Type Dn: here

w0 =



sα1+2α2+···+2α2k−2+α2k−1+α2k
sα1sα3+2α4+···+2α2k−2+α2k−1+α2k

sα3 · · · sα2k−3+2α2k−2+α2k−1+α2k
sα2k−3

sα2k
, if n = 2k;

sα1+2α2+···+2α2k−2+α2k−1+α2k
sα1sα3+2α4+···+2α2k−2+α2k−1+α2k

sα3 · · · sα2k−3+2α2k−2+2α2k−1+α2k+α2k+1
sα2k−1+α2k+α2k+1

sα2k−1
, if n = 2k + 1.

Hence,

wt(w0) =



2α∨
1 + 2α∨

2 + 4α∨
3 + 4α∨

4 + · · ·+ (2k − 2)α∨
2k−3

+(2k − 2)α∨
2k−2 + kα∨

2k−1 + kα∨
2k, if n = 2k;

2α∨
1 + 2α∨

2 + 4α∨
3 + 4α∨

4 + · · ·+ (2k − 2)α∨
2k−3 + (2k − 2)α∨

2k−2

+2kα∨
2k−1 + kα∨

2k + kα∨
2k+1, if n = 2k + 1.

(e) Type E6: here w0 = sα1+2α2+2α3+3α4+2α5+α6sα1+α3+α4+α5+α6sα3+α4+α5sα4 , and thus

wt(w0) = 2α∨
1 + 2α∨

2 + 4α∨
3 + 6α∨

4 + 4α∨
5 + 2α∨

6 .

(f) Type E7: here w0 = s2α1+2α2+3α3+4α4+3α5+2α6+α7sα2+α3+2α4+2α5+2α6+α7sα2+α3+2α4+α5sα2

45



sα3sα5sα7 , and thus

wt(w0) = 2α∨
1 + 5α∨

2 + 6α∨
3 + 8α∨

4 + 7α∨
5 + 4α∨

6 + 3α∨
7 .

(g) Type E8: here w0 = s2α1+3α2+4α3+6α4+5α5+4α6+3α7+2α8s2α1+2α2+3α3+4α4+3α5+2α6+α7

sα2+α3+2α4+2α5+2α6+α7sα2+α3+2α4+α5sα2sα3sα5sα7 , and thus

wt(w0) = 4α∨
1 + 8α∨

2 + 10α∨
3 + 14α∨

4 + 12α∨
5 + 8α∨

6 + 6α∨
7 + 2α∨

8 .

(h) Type F4: here w0 = s2α1+3α2+4α3+2α4sα2+2α3+2α4sα2+2α3sα2 , and thus

wt(w0) = 2α∨
1 + 6α∨

2 + 4α∨
3 + 2α∨

4 .

(i) Type G2: here w0 = s3α1+2α2sα1 , and thus

wt(w0) = 2α∨
1 + 2α∨

2 .

Remark 3.3.1. We note that the weight of longest element found above appears in a

rather different context, cf. [Lus18]. In that paper, Lusztig points out that the coroots

appearing as a summand of wt(w0) 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 x 7→ rx is defined from the set of involutions IW in an irreducible Weyl

group W in loc. sit. and this is crucially used in constructing certain lifts of involutions in

46



associated reductive groups. We compare this map defined on the set of involutions with

our weight function in Chapter 5.

3.4 Covering relation in Iwahori-Weyl group

The main result of this section characterizes the covering relation of Bruhat order for

most of the elements in W̃ .

Theorem 3.4.1. Suppose that G is a split reductive group and w = utλv be an element

of W̃ such that depth(λ) is bigger than a certain constant. Let rβ = tmuα∨
suα be an affine

reflection for some positive root α and integer m, and let w′ = rβw. Then w ⋗ w′ is a

covering relation, i.e. w ≥ w′ and ℓ(w) = ℓ(w′) + 1, if and only if one of the following

conditions holds:

1. m = 0 and ℓ(usα) = ℓ(u)− 1; in this case, w′ = usαt
λv.

2. m = 1 and ℓ(usα) = ℓ(u) + ⟨2ρ, α∨⟩ − 1; in this case, w′ = usαt
λ−α∨

v.

3. m = ⟨α, λ⟩ and ℓ(sαv) = ℓ(v) + 1; in this case, w′ = utλsαv.

4. m = ⟨α, λ⟩ − 1 and ℓ(sαv) = ℓ(v)− ⟨2ρ, α∨⟩+ 1; in this case, w′ = utλ−α∨
sαv.

In fact, it suffices to show the following result instead; see the remarks following the

theorem below.

Theorem 3.4.2. Suppose that W̃ is an Iwahowi Weyl group of a quasi-simple split reductive

group. Let tmα∨
sα be an affine reflection, with (α,m) ∈ Φ+ × Z and assume w = tλy is an

47



element of W̃ such that

depth(λ) ≥



3, if W is of simply laced type;

4, if W is of non-simply laced type but not of type G2;

6, if W is of type G2.

(3.4.1)

Then w′ := tmα∨
sαw is a cocover of w1 if and only if one of the following holds.

1. m = 1 and ℓ(sα) = ⟨2ρ, α∨⟩ − 1; in this case, w′ = sαt
λ−α∨

y.

2. m = ⟨α, λ⟩ and ℓ(sαy) = ℓ(y) + 1; in this case, w′ = tλsαy.

3. m = ⟨α, λ⟩ − 1 and ℓ(sαy) = ℓ(y)− ⟨2ρ, α∨⟩+ 1; in this case, w′ = tλ−α∨
sαy.

Remark 3.4.3. We explain how Theorem 3.4.2 implies Theorem 3.4.1. Let us first note

that we can again reduce to the case of irreducible factors as in Remark 3.2.3. Note that

if λ is regular, ℓ(utλv) = ℓ(u) + ℓ(tλv) by Equation (2.1.1). Choose reduced expressions

u = si1 · · · sil and tλv = sj1 · · · sjkζ, where sip , sjq ∈ S for 1 ≤ p ≤ l, 1 ≤ q ≤ k and ζ ∈ Ω.

Then utλv = si1 · · · silsj1 · · · sjkζ is a reduced expression, and a cocover of utλv must be of

the form of either si1 · · · ŝim · · · silsj1 · · · sjkζ for some p ∈ [1, l] or si1 · · · silsj1 · · · ŝjn · · · sjkζ

for some q ∈ [1, k]. In other words, any cocover of utλv is obtained by

• (C1) either multiplying a cocover of u with tλv, or

• (C2) multiplying u with a cocover of tλv.

1for two elements w,w′ of W̃ , we say that w′ is a cocover of w if w is a cover of w′, i.e. w > w′ and
ℓ(w′) = ℓ(w)− 1. We also shorthand these last two conditions by writitng w ⋗ w′.

48



The procedure listed in (C1) is easy to describe. A cocover of u is of the form usα for some

α ∈ Φ+ such that ℓ(usα) = ℓ(u)− 1. This corresponds to case (1) in theorem B. We claim

that the three other cases there, i.e. (2)-(4), come from the procedure described in (C2),

and hence they correspond to those listed in Theorem 3.4.2. Clearly, the third and fourth

case in theorem B corresponds respectively to the second and third case in Theorem 3.4.2.

Finally, suppose that w′ := sat
λ−α∨

v is a cocover of w := tλv such that uw′ is a cocover of

uw. By Equation (2.1.1),

ℓ(uw′) = ℓ(usa) + ℓ(tλ−α∨
)− ℓ(v) = ℓ(usa) + ⟨2ρ, λ− α∨⟩ − ℓ(v).

We use dominance of λ − α∨ in the last equality; this is true as ⟨αi, λ − a∨⟩ = ⟨αi, λ⟩ −

⟨αi, α
∨⟩ ≥ 0 by our earlier discussion about the maximum value of ⟨αi, α

∨⟩. Note that

ℓ(usα) ≤ ℓ(u) + ℓ(sa) ≤ ℓ(u) + ⟨2ρ, α∨⟩ − 1. (3.4.2)

Since ℓ(utλv) = ℓ(u) + ⟨2ρ, λ⟩ − ℓ(v), the cocover condition uw ⋗ uw′ gives

ℓ(u) + ⟨2ρ, λ⟩ − ℓ(v)− 1 = ℓ(usa) + ⟨2ρ, λ− α∨⟩ − ℓ(v)

≤ ℓ(u) + ⟨2ρ, α∨⟩ − 1 + ⟨2ρ, λ− α∨⟩ − ℓ(v).

Note that

ℓ(u)+⟨2ρ, λ⟩−ℓ(v)−1 = ℓ(usa)+⟨2ρ, λ−α∨⟩−ℓ(v) ≤ ℓ(u)+⟨2ρ, α∨⟩−1+⟨2ρ, λ−α∨⟩−ℓ(v).

49



Hence we deduce that all of the inequalities in Equation (3.4.2) must be equality, and thus

we get ℓ(usα) = ℓ(u) + ⟨2ρ, α∨⟩ − 1. This is exactly the situation listed in the second item

in theorem B.

Remark 3.4.4. We note that in a recent preprint [Sch22a][proposition 4.5], Schremmer

gives several equivalent criteria that describes the covering relation, without any assumption

on the depth of the relevant coweight.

We will only focus on proving the necessity of the above conditions. For the sufficiency

part, the argument in [Mil21, Section 4.1] toward the end of proposition 4.2 can be applied.

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

3.4.1 Some useful inequalities

In this subsection, we lay out estimate of some relevant quantities that we shall

use in the next subsection. We shall resume the assumption on the depth of λ stated in

Theorem 3.4.2 throughout our discussion after the first lemma below. Note that we can

assume throughout that w ∈ Wa, i.e. the length zero component of w is 1 ∈ Ω.

Lemma 3.4.5. Let w = tλy be an element of W̃ with λ dominant and w ∈ SW̃ . Suppose

w⋗w′. Then we must have w′ = tmα∨
sαw for some affine reflection tmα∨

sα with 1 ≤ m ≤

⟨α, λ⟩.

Proof. Let w = si1 · · · sil be a reduced expression, where sij ∈ S̃. This describes a reduced

gallery from a to wa via the chain of adjacent alcoves a→ si1a→ · · · → si1 · · · sila. Since

the gallery is reduced, it cannot cross any hyperplane twice - hence all the alcoves lie in

50



the dominant chamber. Let rj be the affine reflection with respect to the common wall

between aj−1 := si1 · · · sij−1
a and aj := si1 · · · sija, that is rj = si1 · · · sij−1

sijsij−1
· · · si1 .

Now suppose that w′ = si1 · · · ŝik · · · sil is a cocover; as before, let {a′
j : 1 ≤ j ≤ l− 1}

be the collection of alcoves describing the reduced gallery corresponding to this expression

of w′ and let r′j be the associated affine reflections. Then we see that rj = r′j and aj = a′
j

for 1 ≤ j ≤ k − 1, and the rest of the gallery for w′ is the reflection of portion of the

gallery for w from ak+1 onward with respect to the hyperplane corresponding to rk. This

is because for j ≥ k + 1 we have

a′
j−1 = si1 · · · sik−1

sik+1
· · · sija = si1 · · · sik−1

siksik−1
· · · si1(si1 · · · sija) = rk(aj).

In particular, w′ = rkw.

Therefore, to create a cocover of w, we must reflect wa with respect to some hy-

perplane Hα−m lying between a and itself. Since the number of hyperplanes between the

alcove wa and the wall Hα is ⟨α, λ⟩, this restricts the possibility of m to asserted values

above.

Now, w′ = tmα∨
sαw = sαt

λ−mα∨
y. By Lemma 3.4.5, the coweight associated to the

translation part of w′ belongs to following list of coweights:

λ− α∨, λ− 2α∨, · · · , λ− (⟨α, λ⟩ − 1)α∨ = sα(λ− α∨), λ− ⟨α, λ⟩α∨ = sα(λ). (3.4.3)

51



Let us now define

k = kλ,α := max{m : λ−mα∨ ∈ C+, 1 ≤ m ≤ ⟨α, λ⟩,m ∈ Z}.

For a chamber Cx, we denote its closure by Cx. We now give a lower bound for k.

Lemma 3.4.6. We have that

k ≥


1, if W is of simply laced type;

2, if W is of non-simply laced type.

Proof. By the definition of k, there is a real number m̃ with k ≤ m̃ < k + 1 such that

λ− m̃α∨ lies on at least one of the walls of C+. It follows that for some simple root αi, we

have ⟨αi, λ− m̃α∨⟩ = 0. Therefore, m̃ = ⟨αi,λ⟩
⟨αi,α∨⟩ - hence giving k = ⌊ ⟨αi,λ⟩

⟨αi,α∨⟩⌋. Recalling the

estimate about maximum value of ⟨αi, α
∨⟩ in each Cartan type, we get the desired bound

on k from the depth condition on λ.

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 [Mil21,

Proposition 4.2]. We include it here for completeness’ sake.

Lemma 3.4.7. Suppose that for some integer m ∈ [1, ⟨α, λ⟩], λ−mα∨ lies in the closure

of a chamber different from C+ or Csα. Then ℓ(tλ−mα∨
) ≤ ℓ(tλ−kα∨

).

Proof. Following [LS10], define the function f : R→ R≥0 by linearly extending the function

f̃ : Z→ Z≥0 defined by

f̃(m) = ℓ(tλ−mα∨
).

52



More precisely, f is the function associated to the graph obtained by joining f̃(m) and

f̃(m + 1) by the line passing through them for every m ∈ Z. It is easy to see that f

is a convex function, cf. [LS10], 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 λ−mα∨ is in the same

chamber. Since λ, λ− α∨ are both dominant due to the imposed depth hypothesis, we see

that f(1) = ⟨2ρ, λ− α∨⟩, f(⟨α, λ⟩) = ⟨2ρ, λ⟩.

We now show that f is decreasing around 1 and increasing around ⟨α, λ⟩. For example,

if m ∈ (0, 3
2
) and αi ∈ ∆ then by our earlier discussion about maximum value of ⟨αi, α

∨⟩

we have

⟨αi, λ−mα∨⟩ = ⟨αi, λ⟩ −m⟨αi, α
∨⟩ ≥


depth(λ)− 3, if W is not of type G2;

depth(λ)− 9
2
, if W is of type G2.

Hence, our depth hypothesis ensures that ⟨αi, λ−mα∨⟩ ≥ 0. Therefore, λ−mα∨ is in C+

whenever m ∈ (0, 3
2
), and thus

f(m) = ⟨2ρ, λ−mα∨⟩ = ⟨2ρ, λ⟩ −m⟨2ρ, α∨⟩ (3.4.4)

is clearly decreasing in this neighbourhood. Similarly, we note that if m ∈ (⟨α, λ⟩ −

1
2
, ⟨α, λ⟩+ 1

2
) and αi ∈ ∆ we have

⟨αi, sα(λ−mα∨)⟩ = ⟨αi, λ⟩ − (⟨α, λ⟩ −m)⟨αi, α
∨⟩ ≥


depth(λ)− 1, if W is not of type G2;

depth(λ)− 3
2
, if W is of type G2.

53



Again, our depth hypothesis ensures that ⟨αi, sα(λ − mα∨)⟩ > 0. Hence λ − mα∨ ∈ Csα

whenever m ∈ (⟨α, λ⟩ − 1
2
, ⟨α, λ⟩+ 1

2
), and thus

f(m) = ⟨2ρ, sa(λ−mα∨)⟩ = ⟨2ρ, λ⟩−(⟨α, λ⟩−m)⟨2ρ, α∨⟩ = ⟨2ρ, sa(λ)⟩+m⟨2ρ, α∨⟩ (3.4.5)

is clearly increasing in the neighbourhood.

Since the intervals (0, 3
2
) and (⟨α, λ⟩ − 1

2
, ⟨α, λ⟩ + 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 Equation (3.4.4) as long as λ−mα∨ ∈ C+; as λ−mα∨ 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 λ −mα∨ enters Csα , f(m) is defined

by Equation (3.4.5) and steadily increases. Therefore, if we define

k′ = k′
λ,α := min{m : λ−mα∨ ∈ Csα , 1 ≤ m ≤ ⟨α, λ⟩,m ∈ Z},

we must have f(m) ≤ max{(f(k), f(k′)} for allm ∈ [k, k′]∩Z. Since λ−kα∨ = sα(λ−k′α∨)

by symmetry, we have f(k) = f(k′) = ℓ(tλ−kα∨
). Hence we are done.

Recall from Lemma 3.4.5 that m ≤ ⟨α, λ⟩. We now define z ∈ W to be such that

λ−mα∨ ∈ Cz. Note that if λ−mα∨ is singular, z is not uniquely specified by this condition.

However, the precise choice of z would be immaterial in what follows, and henceforth fix

one such z satisfying the above condition. Let us denote the left and right weak Bruhat

order by ≺left and ≺right respectively.

Lemma 3.4.8. We have z ≺right sα. As a consequence, ℓ(sαz) + ℓ(z) = ℓ(sα).

54



Proof. It suffices to show that Inv(z−1) ⊂ Inv(sα), because this is equivalent to z−1 ≺left sα,

which in turn is equivalent to the claim above.

We start by noting that for β ∈ Inv(z−1), we have ⟨β, λ −mα∨⟩ ≤ 0. This follows

from rewriting ⟨β, λ − mα∨⟩ = ⟨z−1β, z−1(λ − mα∨)⟩ and noting that z−1(λ − mα∨) is

a dominant coweight and z−1β ∈ −Φ+. Therefore we get ⟨β, λ⟩ ≤ m⟨β, α∨⟩. Since λ

dominant regular, ⟨β, λ⟩ > 0 and hence ⟨β, α∨⟩ > 0 as well. Dividing out both side of the

inequality by ⟨β, α∨⟩, we get m ≥ ⟨β,λ⟩
⟨β,α∨⟩ . Combining this with Lemma 3.4.5, we obtain

⟨α, λ⟩ ≥ ⟨β, λ⟩
⟨β, α∨⟩

.

But then ⟨β, α∨⟩⟨α, λ⟩ ≥ ⟨β, λ⟩, whence ⟨sα(β), λ⟩ = ⟨β − ⟨β, α∨⟩α, λ⟩ ≤ 0. Since

λ is dominant regular, this gives sα(β) ∈ −Φ+, hence β ∈ Inv(sα). Thus we have shown

Inv(z−1) ⊂ Inv(sα).

Finally, let us note that

ℓ(sαz)+ℓ(z) = ℓ(sα)−ℓ(z)+2|Inv(sα)c∩Inv(z−1)|+ℓ(z) = ℓ(sα)+2|Inv(sα)c∩Inv(z−1)| = ℓ(sα).

This finishes the proof.

3.4.2 Two distinct possibilities

In this subsection we further pin down the possible values of z. Recall that z−1(λ−

mα∨) is dominant; let us temporarily denote this coweight by µ. Let J ⊂ ∆ be such

that Stab(µ) = WJ , the associated standard parabolic subgroup. Abbreviate ζ = z−1y.

55



By standard fact about Coxeter groups, ζ has an unique factorization as ζ = ζJζ
J with

ζJ ∈ WJ and ζJ ∈ JW . Note that

w′ = sαt
λ−mα∨

y = sαzt
µz−1y = sαzt

µζJζ
J = sαzζJt

µζJ . (3.4.6)

Lemma 3.4.9. The alcove tµζJa is in the dominant chamber.

Proof. Define ω̄ := 1
n

∑n
i=1

ω∨
i

ni
, where ni is the coefficient of αi in the highest root θ. Since

µ+ ζJ ω̄ is centroid of the alcove tµζJa, we can make the following observation: tµζJa is in

the dominant chamber if and only if µ+ ζJ ω̄ is dominant.

We now show that the latter statement holds true. Note that if αi ∈ J , then ⟨αi, µ+

ζJ ω̄⟩ = ⟨(ζJ)−1αi, ω̄⟩ > 0, since by definition siζ
J > ζJ , or equivalently (ζJ)−1si > (ζJ)−1,

thereby giving (ζJ)−1αi ∈ Φ+. Now let αi ∈ ∆ \ J , then ⟨αi, µ⟩ ≥ 1; since (ζJ)−1αi ≥ −θ,

we have ⟨αi, µ+ ζJ ω̄⟩ ≥ 1 + ⟨−θ, ω̄⟩ = 1− 1 = 0. This proves the claim.

Applying the previous lemma, we get from Equation (3.4.6)

ℓ(w′) = ℓ(sαzζJ) + ⟨2ρ, µ⟩ − ℓ(ζJ).

Therefore the cocover condition gives

⟨2ρ, λ⟩ − ℓ(y)− 1 = ℓ(sαzζJ) + ⟨2ρ, µ⟩ − ℓ(ζJ).

56



In other words, we get

ℓ(tλ)− ℓ(tµ) = 1 + ℓ(sαzζJ) + ℓ(y)− ℓ(ζJ). (3.4.7)

Write y = z · z−1y = zζJζ
J . Note that Equation (2.1.2) then gives

ℓ(y) = ℓ(z · ζJζJ) = ℓ(z) + ℓ(ζJ) + ℓ(ζJ)− 2|Inv(z) ∩ Inv((ζJ)−1ζ−1
J )|.

Similarly,

ℓ(sazζJ) = ℓ(sαz)− ℓ(ζJ) + 2|Inv(sαz)c ∩ Inv(ζ−1
J )|

Hence we can rewrite the right hand side of Equation (3.4.7) as

1 + ℓ(sαz) + ℓ(z) + 2{|Inv(sαz)c ∩ Inv(ζ−1
J )| − |Inv(z) ∩ Inv((ζJ)−1ζ−1

J )|}.

Since ζ−1
J ≺left (ζ

J)−1ζ−1
J , we have that Inv(ζ−1

J ) ⊂ Inv((ζJ)−1ζ−1). Combining this

with the fact that |A| − |B| ≤ |A \B| for two sets A,B, we therefore conclude that

ℓ(tλ)− ℓ(tµ) ≤ 1 + ℓ(sαz) + ℓ(z) + 2|Inv(sαz)c ∩ Inv(z)c ∩ Inv(ζ−1
J )|. (3.4.8)

Lemma 3.4.10. We have Inv(sαz)
c ∩ Inv(z)c ∩ Inv(ζ−1

J ) = ∅.

Proof. The argument here is similar to the proof of Lemma 3.4.8. Suppose β ∈ Inv(sαz)
c∩

Inv(z)c∩ Inv(ζ−1
J ) ⊂ Inv(sαz)

c∩ Inv(z)c∩Φ+
J . Thus ⟨β, z−1(λ−mα∨)⟩ = 0, hence ⟨zβ, λ⟩ =

57



m⟨zβ, α∨⟩. Since zβ is a positive root, both sides are positive and

m =
⟨zβ, λ⟩
⟨zβ, α∨⟩

≤ ⟨α, λ⟩.

Therefore, ⟨zβ, λ⟩ − ⟨zβ, α∨⟩⟨α, λ⟩ = ⟨zβ, sα(λ)⟩ ≤ 0. Since β ∈ Inv(sαz)
c, we have

sα(zβ) ∈ Φ+. Hence we have ⟨zβ, sα(λ)⟩ = ⟨sα(zβ), λ⟩ ≤ 0, but that is a contradiction

since λ is dominant regular. This shows that the purported set must be empty and we are

done.

Therefore, combining Equation (3.4.8) with Lemma 3.4.8 and Lemma 3.4.10 gives

ℓ(tλ)− ℓ(tµ) ≤ 1 + ℓ(sα). (3.4.9)

Now, suppose that z ̸= 1, sα. By Lemma 3.4.7 this gives

ℓ(tλ)− ℓ(tµ) ≥ ℓ(tλ)− ℓ(tλ−kα∨
) = k⟨2ρ, α∨⟩.

Combining this with Equation (3.4.9), we get

k⟨2ρ, α∨⟩ ≤ 1 + ℓ(sα) ≤ ⟨2ρ, α∨⟩

This gives k = 1. For the non-simply laced types, this is a contradiction with pre-

viously established lower bound in Lemma 3.4.6. Therefore in such cases, we get that

z ∈ {1, sα}.

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

58



system:

if ⟨β, α∨⟩ = 2 for a fixed coroot α, then β = α.

This can be checked directly in type An and Dn, where the positive roots are given by

{ei − ej : 1 ≤ i < j ≤ n} and {ei ± ej : 1 ≤ i < j ≤ n} ∪ {ei + en : 1 ≤ i < n} respectively.

Since root systems of type E6, E7 arise as subsystem of type E8, we just give an argument

for root system of type E8 to conclude. Note that for root system of type E8, the positive

roots are of two kinds - given by {±ei + ej : 1 ≤ i < j ≤ 8}, and {1
2
(e8 +

∑7
i=1(−1)ν(i)ei):∑7

i=1 ν(i) ∈ 2Z}. We can easily see that pairing between a root γ from the first set with

a coroot α∨ corresponding to elements of either sets cannot be 2 unless γ = α. The

remaining possibility is ⟨1
2
(e8 +

∑7
i=1(−1)ν1(i)ei),

1
2
(e8 +

∑7
i=1(−1)ν2(i)ei)⟩ = 2, but that

gives ⟨e8 +
∑7

i=1(−1)ν1(i)ei, e8 +
∑7

i=1(−1)ν2(i)ei⟩ = 8 - therefore forcing ν1 = ν2.

We now resume the discussion about estimating k and restrict ourselves to the sim-

ply laced types. Recall that k = 1. This means that λ − α∨ ∈ C+, but λ − 2α∨ /∈ C+;

hence there exists β ∈ Φ+ such that ⟨β, λ− 2α∨⟩ ≤ 0. By the depth condition, this yields

3 ≤ ⟨β, λ⟩ ≤ 2⟨β, α∨⟩. Therefore ⟨β, α∨⟩ = 2, and ⟨β, λ⟩ is equal to either 3 or 4. But

this gives β = α, and hence ⟨α, λ⟩ is either 3 or 4. If ⟨α, λ⟩ = 3, the relevant coweights

are λ − α∨, λ − 2α∨ = sα(λ − α∨), λ − 3α∨ = sα(λ); the first one is in C+ by the depth

condition, and the last two are in Csα . If ⟨α, λ⟩ = 4, then the relevant coweights are

λ−α∨, λ− 2α∨, λ− 3α∨ = sα(λ−α∨), λ− 4α∨ = sα(λ); the first one is in C+, the last two

are in Csα and λ− 2α∨ lies on the wall Hα, so it lies in both C+ and Csα .

We summarize the content of this subsection as follows.

59



Proposition 3.4.11. If sαt
λ−mα∨

y is a cocover of tλy, then λ − mα∨ is either in C+ or

Csα.

3.4.3 Finishing the proof

Proof of Theorem 6.1. We deal with the two cases found above.

1. When λ−mα∨ is in C+: in this case, Equation (3.4.9) gives us

⟨2ρ, λ⟩ − ⟨2ρ, λ−mα∨⟩ ≤ 1 + ℓ(sα) ≤ ⟨2ρ, α∨⟩.

Therefore we get m ≤ 1; hence m = 1 and 1 + ℓ(sα) = ⟨2ρ, α∨⟩. This corresponds to

the first case in Theorem 3.4.2.

2. When λ−mα∨ is in Csα : in this case, Equation (3.4.9) gives us

⟨2ρ, λ⟩ − ⟨2ρ, sα(λ−mα∨)⟩ ≤ 1 + ℓ(sα). (3.4.10)

Since ℓ(sa) ≤ ⟨2ρ, α∨⟩ − 1, we thus get

(⟨α, λ⟩ −m)⟨2ρ, α∨⟩ ≤ ⟨2ρ, α∨⟩.

Therefore we get ⟨α, λ⟩ − m ≤ 1. Since m ⩽ ⟨α, λ⟩, this means that either m =

⟨α, λ⟩ − 1 or m = ⟨α, λ⟩.

Suppose first that it is the former case. Then substituting this value of m in Equa-

tion (3.4.10) yields ℓ(sα) = ⟨2ρ, α∨⟩ − 1. Note that λ−mα∨ = sα(λ− α∨) is regular

60



(since λ−α∨ is regular dominant due to the depth hypothesis) and thus J = ∅; substi-

tuting µ = sα(λ−α∨), z = sα, ζJ = 1, ζJ = sαy in Equation (3.4.7) therefore produces

⟨2ρ, α∨⟩ = 1+ℓ(y)−ℓ(sαy), whence we get ℓ(sαy) = ⟨2ρ, α∨⟩−1−ℓ(y) = ℓ(sα)−ℓ(y).

This corresponds to the third case in Theorem 3.4.2.

Now, if it is the latter case we can carry out a similar substitution in Equation (3.4.7)

to get 0 = 1+ ℓ(y)− ℓ(sαy); Equation (3.4.10) does not yield any information in this

case. This gives ℓ(say) = ℓ(y) + 1 and hence it corresponds to the second case in

Theorem 3.4.2. This completes the proof.

Remark 3.4.12. We know a posteriori that only the first and the last two choices from

the string of coweights in Equation (3.4.3) are viable - so we must necessarily put a depth

condition on λ to ensure that λ − α∨ is dominant; in that case, the first coweight is in

the dominant chamber and the last two are in Csα . In other words, the least stringent

hypothesis on λ under which one can expect to prove a result like this would be that λ−α∨

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 λ− α∨ 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.

61



3.4.4 Admissible subsets of W̃

A useful description of the admissible set is given in [HY21] in terms of weight of

minimal length paths in the quantum Bruhat graph. This relies on the characterization

of covering relation in W̃ as established in [Mil21, Proposition 4.2] and hence as such

it necessarily brings into picture the superregularity condition, cf. [HY21, 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 3.4.13. Suppose that W is an irreducible Weyl group. Assume that

depth(µ) ≥



3, if W is of simply laced type;

4, if W is of non-simply laced type but not of type G2.

6, if W is of type G2.

Let λ be dominant and assume that

⟨ρ, µ− λ⟩ <



⌈depth(µ)−3
2

⌉, if W is of simply laced type;

⌈depth(µ)−4
2

⌉, if W is of non-simply laced type but not of type G2;

⌈depth(µ)−6
2

⌉, if W is of type G2.

Then xtλy ∈ Adm(µ) if and only if wt(x, y−1) ≤ µ− λ.

The proof of this proposition is identical to the argument made in [HY21], cf. section

3.3 and proof of proposition 3.3 in there and hence we do not repeat it here. The additional

62



ingredient is Theorem 3.4.2, and the conditions on µ and λ are a direct reflection of that.

63



Chapter 4: A dimension formula for X(µ, b)

The primary goal of this chapter is to investigate the dimension of the union X(µ, b)

of affine Deligne-Lusztig varieties. Such a geometric object is the group-theoretic model

for the Newton stratum associated with the element [b] lying B(G, µ). We remark that in

the situations where the He-Rapoport axioms in [HR17] hold, this sought-after dimension

is equal to the dimension of the associated Newton stratum minus the dimension of the

corresponding central leaf, see [He16a, §2.12]. The said union of affine Deligne-Lusztig

varieties can be defined purely in terms of group-theoretic data without any reference to

Shimura varieties, and as before we resort to this latter setup. In Section 4.1, we obtain a

dimension formula in the case of a quasi-split group, improving an earlier work of [HY21].

Let us now place ourselves in the context of a general group G, and let bmin be the

unique minimal (equivalently, basic) element of B(G, µ). Then the work of Görtz, He

and Rapoport in [GHR22] gives characterizations of when X(µ, bmin) can be of minimal

dimension zero, or of maximal dimension ⟨2ρ, µ⟩; the model cases of such phenomena are

the Lubin-Tate case and the Drinfeld case, respectively. Let us also remark that in [GHN20],

Görtz, He and Nie provide a sharp lower bound for the dimension of X(µ, bmin), and they

are also able to classify completely when this becomes an equality. Besides these results,

there is no dimension formula available for X(µ, b), nor is there any conjectural description.

64



Standing at this juncture, it is therefore natural to investigate the dimension problem for

the other extremal element of B(G, µ), namely bmax. In the rest of this chapter, we compute

this dimension with a mild depth hypothesis on µ.

4.1 Dimension in the quasi-split case

In this section, we show that the dimension formula provided in [HY21, Theorem 6.1]

holds in the absence of the superregularity condition imposed there. Our argument closely

follows the proof scheme established in [HY21]. 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. We first prove

an enhanced version of [HY21, Proposition 4.4]. First, we set dAdm(µ)(b) = max
w∈Adm(µ)

dw(b).

Proposition 4.1.1. Assume that G is quasi-split. Let µ be regular. Then

dAdm(µ)(b) = ⟨ρ, µ− ν([b])⟩ − 1

2
defG(b) +

1

2
ℓ(w0)−

1

2
min{dΓ(x, σ(x)w0) : x ∈ W}. (4.1.1)

Proof. We first show that

(i) if xtλy ∈ Adm(µ), then xtλ+µ′
y ∈ Adm(µ+ µ′) for any dominant µ′.

We write xtλ+µ′
y = xtλyy−1tµ

′
y, and note that y−1tµ

′
y ∈ Adm(µ′) by definition.

Hence the claim follows from an application of Theorem 2.3.1.

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 xtλy ∈ Adm(µ) where µ is regular, then ⟨ρ,wt(x, y−1)⟩ ≤ ⟨ρ, µ− λ⟩.

65



Let xtλy ∈ Adm(µ). Choose µ′ to be superregular, e.g. µ′ = nρ∨ for large enough n.

Apply the statement in (i) to obtain xtλ+µ′
y ∈ Adm(µ + µ′). Now, µ + µ′ is superregular

and thus statement (a) in loc. sit. applies to give us

⟨ρ,wt(x, y−1)⟩ ≤ ⟨ρ, (µ+ µ′)− (λ+ µ′)⟩ = ⟨ρ, µ− λ⟩.

This finishes the proof of (ii). Therefore, one can use this enhanced version of state-

ment (a) to get the established upper bound (i.e. the expression in the right hand side of

Equation (4.1.1)) for dAdm(µ) by resorting to the same proof method in loc. sit.

In the remaining part of the proof, the authors construct an explicit w ∈ Adm(µ)∩Cx

for some x ∈ W and argue that

dw(b) = ⟨ρ, µ− ν([b])⟩ − 1

2
defG(b) +

1

2
ℓ(w0)−

1

2
dΓ(x, σ(x)w0). (4.1.2)

To prove this, they employ the description of Adm(µ) 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 = xtµ(σ(x)w0)
−1 for some x ∈ W

such that σ(x)w0 ≥ x. Let us first explain how this can be obtained using [HY21, Theorem

5.1].

Let O be the σ-conjugacy class of w0 and define ℓR(O) = min{ℓR(w) : w ∈ O}.

Then the aforementioned theorem asserts that for any finite Coxeter group W and a length

66



preserving graph automorphism on W , we have that

ℓ(w0)− ℓR(O) = 2max{ℓ(x) : x ≤ σ(x)w0}.

Therefore, the existence of our required element is equivalent to the assertion that ℓ(w0) >

ℓR(O). The latter statement is true in all types arising from quasi-split groups, cf. section

5.1 in loc. sit.

Then we have w ≤ σ(x)w0t
µ(σ(x)w0)

−1 by a combination of Equation (2.1.1) and

Equation (2.1.3). This is where we use that µ is regular. Hence, w ∈ Adm(µ). As has been

shown in loc. sit., it now follows from definition that Equation (4.1.2) holds true for this

element. This completes the proof that Equation (4.1.1) holds for dominant regular µ.

Let us record an immediate consequence of the above result.

Corollary 4.1.2. Suppose that µ is regular. Then

dimX(µ, b) ≤ ⟨ρ, µ− ν([b])⟩ − 1

2
defG(b) +

1

2
ℓ(w0)−

1

2
min{dΓ(x, σ(x)w0) : x ∈ W}.

Proof of Theorem 1.2.2[(1). ] We note that once the formula for dAdm(µ)(b) in Equation (4.1.1)

is established, the remaining part of [HY21] focuses on showing

min{dΓ(x, σ(x)w0) : x ∈ W} = ℓR(O).

However, this is a calculation done purely on the finite Weyl group W , and the superregu-

larity condition does not appear in establishing this.

67



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 xtµw0σ(x)
−1 ∈ Adm(µ) for certain element

x ∈ W satisfying σ(x)w0 ≥ x. However, this follows directly as we have shown above.

Hence, the superregularity hypothesis on µ can be avoided completely. This finishes the

proof.

Remark 4.1.3. We remark that [HY21, Remark 6.2] gives an example where µ is singular

and W is of type C4, and they point out that the dimension formula fails in this case.

Therefore, the depth hypothesis on µ is optimal, subject to the other condition. However,

we do not know if the formula for dAdm(µ) given in Proposition 4.1.1 is valid in the absence

of regularity condition on µ.

When G is split, we can prove the following statement.

Theorem 4.1.4. The same assertion as above holds under the hypothesis that µ has depth

at least 3 and µ ≥ ν([b]) + wt(w0, 1).

Note that wt(w0, 1) is substantially smaller than 2ρ∨, cf. Section 3.3; for instance,

in type A2n, we have wt(w0, 1) = ϖ∨
n + ϖ∨

n+1. We remark that unlike our above proof

of the dimension formula in the quasi-split case, the proof of Theorem 4.1.4 is completely

independent of the arguments in [HY21].

Basically, the restriction µ⋄ ≥ ν(b) + 2ρ∨ enters in the proof of [HY21][Theorem 6.1]

because the dimension for the affine Deligne-Lusztig variety associated to w := xtµw0σ(x)
−1

- the element chosen in the proof of Theorem 6.1. in loc. sit. - can be asserted to be equal

to dw(b) by [He21a] only under that stated restriction. Note that it is then shown for this

element, we have dw(b) = dAdm(µ)(b) - by which one concludes that dimX(µ, b) = dAdm(µ)(b).

68



In other words, for this particular choice of w, the associated affine Deligne-Lusztig variety

has a known dimension by design, and also makes it to the top dimensional component of

X(µ, b).

Proof. Instead of working with the element w as above, let us consider the element w′ :=

w0t
µ−wt(w0,1). We claim that

(a) w′ ∈ Adm(µ).

Indeed, proceeding as in the proof of Proposition 3.2.6 and utilizing the decomposition

of w0 in Section 3.3, we have

tµ−wt(w0,1) ≤ tµw0, and hence w′ ≤ w0t
µw0.

This last inequality needs depth at least 3 to ensure that µ − wt(w0, 1) is dominant

regular, as well the coweights appearing in the intermediate steps (arising from application

of the proof technique in Proposition 3.2.6) are dominant. Hence, w′ ∈ Adm(µ).

Now, we claim that

(b) dw′(b) = dAdm(µ)(b).

To that end, let us compute

dw′(b) =
1

2
{ℓ(w′) + ℓ(w0)− def(b)− ⟨2ρ, ν([b])⟩}

=
1

2
{ℓ(w0) + ⟨2ρ, µ− wt(w0, 1)⟩+ ℓ(w0)− def(b)− ⟨2ρ, ν([b])⟩}

=
1

2
{⟨2ρ, µ− ν(b)⟩ − def(b)}+ {ℓ(w0)− ⟨ρ,wt(w0, 1)⟩}.

The claim then would follow if we can show that the part in the second curly braces

69



is equal to ℓ(w0) − ℓR(w0). Note that Equation (5.0.2) gives ⟨2ρ,wt(w0, 1)⟩ = ℓ(w0) +

d(w0, 1), but the explicit decomposition of w0 in Section 3.3 shows that d(w0, 1) = ℓR(w0).

Combining, we have ℓ(w0)−⟨ρ,wt(w0, 1)⟩ = ℓ(w0)− 1
2
(ℓ(w0)+ℓR(w0)) =

1
2
{ℓ(w0)−ℓR(w0)}.

Finally, we have that

(c) dimXw′(b) = dw′(b), whenever Xw′(b) ̸= ∅.

By [MV20, Theorem 1.2] every element in the antidominant chamber is cordial.

Therefore w′ is cordial, and then above claim follows from [MV20, Corollary 3.17].

Finally, note further that the basic element in B(G) satisfying κ(b) = κ(w′) is indeed

the minimal element ofB(G)w′ by [GHN16][Theorem B]. Hence, B(G)w′ = {[b] : [b] ≤ [bw′ ]}.

By [He21b, Theorem 4.2], we have ν([bw′ ]) = µ − wt(w0, 1). By Equation (3.2.2), this in

turn enforces the condition µ ≥ ν([b]) + wt(w0, 1) in order to ensure that Xw′(b) ̸= ∅. We

are done.

4.2 Expressing bmax via generic Newton point

In the rest of this chapter, we focus on the dimension problem for X(µ, b) associated

to the maximal element b = bmax of B(G, µ). We first identify bmax as the maximum

of generic σ-conjugacy classes associated with the maximal translation elements in the

µ-admissible set. Then we proceed to express such generic σ-conjugacy classes in two

distinct ways, first in terms of σ-twisted Demazure power and then via a reduction to

the quasi-split case. Such considerations allow us to express the dimension of individual

affine Deligne-Lusztig varieties associated to such generic σ-conjugacy classes in terms of

certain statistics on the quantum Bruhat graph. Further analysis in Section 4.2.5 shows

70



that dimX(µ, bmax) lies between the length and the reflection length of certain minimal

length elements of some Frobenius-twisted conjugacy classes in the finite Weyl group. We

then show that such minimal length elements are, in fact, partial Coxeter elements via

a case-by-case calculation in Section 4.3, and hence their length (which is equal to their

reflection length) matches the asserted dimension in Theorem 1.2.4, thereby finishing the

proof. Throughout the rest of this chapter, we make use of the following strengthened

formula for the generic Newton point (for quasi-split groups) as established in [HN21].

Theorem 4.2.1. [HN21, Proposition 3.1] Suppose that G is quasi-split and adjoint over

F . Let x, y ∈ W and µ ∈ X∗(T )
+
Γ0

be such that depth(µ) ≥ 2. Then νxtµy is the average of

the σ-orbit of µ− wt(y−1, σ(x)).

4.2.1 Dimension of a generic affine Deligne-Lusztig variety

Following [He21a], we define the n-th σ-twisted Demazure power of w by setting

w∗σ,n = w ∗ σ(w) ∗ σ2(w) ∗ · · · ∗ σn−1(w).

We need the following result, which on the one hand describes the dimension of a single

affine Deligne-Lusztig variety associated with generic σ-conjugacy class, and on the other

hand quantifies such generic σ-conjugacy class in terms of σ-twisted Demazure power. For

the first equality below, see [He16a, Theorem 2.23] and [MV20, Lemma 3.2], whereas for

the second equality, see [He21a, Theorem 0.1].

Theorem 4.2.2. Let w ∈ W̃ . Then dimXw(bw) = ℓ(w)− ⟨2ρ, νw⟩ = ℓ(w)− lim
n→∞

ℓ(w∗σ,n)
n

.

71



Note that there can be elements w ∈ W̃ that are not pure translations but still

satisfy bw = bmax; this can be seen e.g. using the Deligne-Lusztig reduction method, see

[He14, Proposition 4.2]. However, if w contributes to top dimensional components - i.e.,

dimX(µ, bmax) = dimXw(bmax) - then w is necessarily a translation element of the form

txµ for some x ∈ W , by the first equality in Theorem 4.2.2.

Now, again by an application of the first equality in Theorem 4.2.2 we see that

dimXtxµ(btxµ) is a decreasing function of btxµ ; therefore, finding top dimensional Xw(bmax)

inside X(µ, bmax) boils down to understanding elements x ∈ W such that dimXtxµ(btxµ) is

minimized, and this minimum value of the dimension is indeed the dimension of X(µ, bmax).

We approach the problem of computing dimXtxµ(btxµ) below in two different ways. In the

rest of this chapter, we omit the underline for simplicity and write µ instead of µ throughout.

4.2.2 A standard reduction

Let G be a connected reductive group over F , and let Gad be its adjoint group. Let

Tad be the image of T in Gad, and denote by µad the image of µ in X∗(Tad)Γ0 . Similarly for

any b ∈ Ğ, we denote by bad its image in Ğad. By [Kot97, Proposition 4.10], we then have

an isomorphism of posets

B(G, µ)→ B(Gad, µad), via [b] 7→ [bad].

Next, we have a decomposi