# Grading of affinized Weyl semi-groups of Kac-Moody type

Paul Philippe

► To cite this version:

Paul Philippe. Grading of affinized Weyl semi-groups of Kac-Moody type. 2023. hal-04123289v2

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# Grading of affinized Weyl semi-groups of Kac-Moody type

Paul Philippe

<figure><img src="image_1.png" /></figure>

## To cite this version:

Paul Philippe. Grading of affinized Weyl semi-groups of Kac-Moody type. Pacific Journal of Mathematics, 2024, 332 (2), pp.273-320. 10.2140/pjm.2024.332.273. hal-04123289v2

HAL Id: hal-04123289

https://hal.science/hal-04123289v2

Submitted on 15 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Grading of affinized Weyl semi-groups of Kac-Moody type

Paul PHILIPPE

Université Jean Monnet, Saint-Étienne, France

Institut Camille Jordan, UMR 5208 CNRS

paul.philippe@univ-st-etienne.fr

## Abstract

For any Kac-Moody root datum $\mathcal{D}$, D. Muthiah and D. Orr have defined a partial order on the semidirect product $W_+^a$ of the integral Tits cone with the vectorial Weyl group of $\mathcal{D}$, and a compatible length function. We classify covers for this order and show that this length function defines a $\mathbb{Z}$-grading of $W_+^a$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.

## Contents

<table><tr><td><b>1 Preliminaries</b></td><td><b>7</b></td></tr><tr><td>1.1 Definitions and notation</td><td>7</td></tr><tr><td>1.2 Geometric interpretation</td><td>12</td></tr><tr><td>1.3 Preliminary results</td><td>16</td></tr><tr><td><b>2 Restriction to constant dominance classes</b></td><td><b>20</b></td></tr><tr><td>2.1 Relative length on W</td><td>21</td></tr><tr><td>2.2 Relation with the affinized Bruhat length</td><td>23</td></tr><tr><td>2.3 Vectorial covers with non-constant coweight</td><td>24</td></tr><tr><td><b>3 Properly affine covers</b></td><td><b>28</b></td></tr><tr><td>3.1 A few properties of properly affine covers</td><td>28</td></tr><tr><td>3.2 Another expression for the affinized length difference</td><td>30</td></tr><tr><td>3.3 Almost-dominance in properly affine covers</td><td>34</td></tr><tr><td>3.4 Properly affine covers and quantum roots</td><td>38</td></tr><tr><td>3.5 Conclusion</td><td>44</td></tr></table>
# Grading of affinized Weyl semi-groups of Kac-Moody type

Paul PHILIPPE

Université Jean Monnet, Saint-Étienne, France

Institut Camille Jordan, UMR 5208 CNRS

paul.philippe@univ-st-etienne.fr

## Abstract

For any Kac-Moody root datum $\mathcal{D}$, D. Muthiah and D. Orr have defined a partial order on the semidirect product $W_+^a$ of the integral Tits cone with the vectorial Weyl group of $\mathcal{D}$, and a compatible length function. We classify covers for this order and show that this length function defines a $\mathbb{Z}$-grading of $W_+^a$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.

## Contents

<table><tbody><tr><td><strong>1 Preliminaries</strong></td><td style="text-align: right;"><strong>7</strong></td></tr><tr><td style="padding-left: 2em;">1.1 Definitions and notation</td><td style="text-align: right;">7</td></tr><tr><td style="padding-left: 2em;">1.2 Geometric interpretation</td><td style="text-align: right;">12</td></tr><tr><td style="padding-left: 2em;">1.3 Preliminary results</td><td style="text-align: right;">16</td></tr><tr><td><strong>2 Restriction to constant dominance classes</strong></td><td style="text-align: right;"><strong>20</strong></td></tr><tr><td style="padding-left: 2em;">2.1 Relative length on W</td><td style="text-align: right;">21</td></tr><tr><td style="padding-left: 2em;">2.2 Relation with the affinized Bruhat length</td><td style="text-align: right;">23</td></tr><tr><td style="padding-left: 2em;">2.3 Vectorial covers with non-constant coweight</td><td style="text-align: right;">24</td></tr><tr><td><strong>3 Properly affine covers</strong></td><td style="text-align: right;"><strong>28</strong></td></tr><tr><td style="padding-left: 2em;">3.1 A few properties of properly affine covers</td><td style="text-align: right;">28</td></tr><tr><td style="padding-left: 2em;">3.2 Another expression for the affinized length difference</td><td style="text-align: right;">30</td></tr><tr><td style="padding-left: 2em;">3.3 Almost-dominance in properly affine covers</td><td style="text-align: right;">34</td></tr><tr><td style="padding-left: 2em;">3.4 Properly affine covers and quantum roots</td><td style="text-align: right;">38</td></tr><tr><td style="padding-left: 2em;">3.5 Conclusion</td><td style="text-align: right;">44</td></tr></tbody></table>
# Introduction

## Motivations

### Reductive groups over p-adic fields

Let $\mathbf{G}$ be a split reductive group scheme with the data of a Borel subgroup $\mathbf{B}$ containing a maximal torus $\mathbf{T}$. Let $W = N_{\mathbf{G}}(\mathbf{T})/\mathbf{T}$ be its vectorial Weyl group and $Y$ be its coweight lattice: $Y = \mathrm{Hom}(\mathbb{G}_m, \mathbf{T})$. The action of $W$ on $\mathbf{T}$ induces an action of $W$ on $Y$ and allows to form the semidirect product $W^a = Y \ltimes W$. This group called the extended affinized Weyl group of $\mathbf{G}$, appears naturally in the geometry and the representation theory of $\mathbf{G}$ over discretely valued fields. A foundational work in this regard was done by N. Iwahori and H. Matsumoto in 1965 ([IM65]), when they exhibited a Bruhat decomposition of $\mathbf{G}(\mathbb{Q}_p)$ indexed by $W^a$.

Let $\mathcal{K}$ be a non-archimedean local field with ring of integers $\mathcal{O}_{\mathcal{K}} \subset \mathcal{K}$, uniformizer $\pi \in \mathcal{O}_{\mathcal{K}}$ and residue field $k_{\mathcal{K}} = \mathcal{O}_{\mathcal{K}}/\pi$. Let $G = \mathbf{G}(\mathcal{K})$, let $K = \mathbf{G}(\mathcal{O}_{\mathcal{K}})$ its integral points and let $I$ be its Iwahori subgroup, defined as $I = \{g \in K \mid g \in \mathbf{B}(k_{\mathcal{K}}) \mod \pi\}$. The extended affinized Weyl group can be understood as $N_{\mathbf{G}}(\mathbf{T}(\mathcal{K}))/\mathbf{T}(\mathcal{O}_{\mathcal{K}})$, so admits a lift in $G$. Then, $G$ admits a decomposition in $I$-double cosets indexed by $W^a$, the Iwahori-Matsumoto-Bruhat decomposition:

$$ G = \bigsqcup_{\pi^{\lambda} w \in W^{a}} I \pi^{\lambda} w I. \qquad (0.1) $$

The group $W^a$ is a finite extension of a Coxeter group and thus admits a Bruhat order which arises from the geometry of the homogeneous space $G/I$: for any $\pi^{\lambda}w \in W^a$, $I\pi^{\lambda}wI$ is a subvariety of pure dimension $\ell(\pi^{\lambda}w)$ in $G/I$, and its closure admits a disjoint decomposition in $I$ orbits:

$$ \overline{I\pi^{\lambda}wI} = \bigsqcup_{\pi^{\mu}v \le \pi^{\lambda}w} I\pi^{\mu}vI \qquad (0.2) $$

which extends the Iwahori-Matsumoto decomposition. The connection between the geometry of $G/I$ and the combinatorial structure of $W^a$ is deeper. In particular, R-Kazhdan-Lusztig polynomials introduced by Kazhdan and Lusztig ([KL80]), defined as the number of points of certain intersections in $G/I$, are also given by a recursive formula based on the Bruhat order and the Bruhat length of $W^a$.

These polynomials appear in many topics around reductive groups over local fields, we aim to develop analogous polynomials when $\mathbf{G}$ is replaced by a general Kac-Moody group.

### Extension to Kac-Moody groups

Replace $\mathbf{G}$ by a general split Kac-Moody group. Kac-Moody group functors are entirely defined by the underlying Kac-Moody root datum $\mathcal{D}$, as defined in [Rém02, §2], and reductive groups correspond to root data of finite type. Then the Iwahori-Matsumoto decomposition
# Introduction

## Motivations

### Reductive groups over p-adic fields

Let $\mathbf{G}$ be a split reductive group scheme with the data of a Borel subgroup $\mathbf{B}$ containing a maximal torus $\mathbf{T}$. Let $W = N_{\mathbf{G}}(\mathbf{T})/\mathbf{T}$ be its vectorial Weyl group and $Y$ be its coweight lattice: $Y = \mathrm{Hom}(\mathbb{G}_m, \mathbf{T})$. The action of $W$ on $\mathbf{T}$ induces an action of $W$ on $Y$ and allows to form the semidirect product $W^a = Y \times W$. This group called the extended affinized Weyl group of $\mathbf{G}$, appears naturally in the geometry and the representation theory of $\mathbf{G}$ over discretely valued fields. A foundational work in this regard was done by N. Iwahori and H. Matsumoto in 1965 ([IM65]), when they exhibited a Bruhat decomposition of $\mathbf{G}(\mathbb{Q}_p)$ indexed by $W^a$.

Let $\mathcal{K}$ be a non-archimedean local field with ring of integers $\mathcal{O}_{\mathcal{K}} \subset \mathcal{K}$, uniformizer $\pi \in \mathcal{O}_{\mathcal{K}}$ and residue field $k_{\mathcal{K}} = \mathcal{O}_{\mathcal{K}}/\pi$. Let $G = \mathbf{G}(\mathcal{K})$, let $K = \mathbf{G}(\mathcal{O}_{\mathcal{K}})$ its integral points and let $I$ be its Iwahori subgroup, defined as $I = \{g \in K \mid g \in \mathbf{B}(k_{\mathcal{K}}) \mod \pi\}$. The extended affinized Weyl group can be understood as $N_{\mathbf{G}}(\mathbf{T}(\mathcal{K}))/\mathbf{T}(\mathcal{O}_{\mathcal{K}})$, so admits a lift in $G$. Then, $G$ admits a decomposition in $I$-double cosets indexed by $W^a$, the Iwahori-Matsumoto-Bruhat decomposition:

$$ G = \bigsqcup_{\pi^{\lambda} w \in W^{a}} I \pi^{\lambda} w I. \qquad (0.1) $$

The group $W^a$ is a finite extension of a Coxeter group and thus admits a Bruhat order which arises from the geometry of the homogeneous space $G/I$: for any $\pi^{\lambda}w \in W^a$, $I\pi^{\lambda}wI$ is a subvariety of pure dimension $\ell(\pi^{\lambda}w)$ in $G/I$, and its closure admits a disjoint decomposition in $I$ orbits:

$$ \overline{I\pi^{\lambda}wI} = \bigsqcup_{\pi^{\mu}v \le \pi^{\lambda}w} I\pi^{\mu}vI \qquad (0.2) $$

which extends the Iwahori-Matsumoto decomposition. The connection between the geometry of $G/I$ and the combinatorial structure of $W^a$ is deeper. In particular, $R$-Kazhdan-Lusztig polynomials introduced by Kazhdan and Lusztig ([KL80]), defined as the number of points of certain intersections in $G/I$, are also given by a recursive formula based on the Bruhat order and the Bruhat length of $W^a$.

These polynomials appear in many topics around reductive groups over local fields, we aim to develop analogous polynomials when $\mathbf{G}$ is replaced by a general Kac-Moody group.

### Extension to Kac-Moody groups

Replace $\mathbf{G}$ by a general split Kac-Moody group. Kac-Moody group functors are entirely defined by the underlying Kac-Moody root datum $\mathcal{D}$, as defined in [Rém02, §2], and reductive groups correspond to root data of finite type. Then the Iwahori-Matsumoto decomposition
no longer holds on $G = G(\mathcal{K})$. However there is a partial Iwahori-Matsumoto decomposition: there exists a subsemigroup $G^+$ of $G$ such that:

$$G^+ = \bigsqcup_{\pi^{\lambda} w \in W_+^a} I \pi^{\lambda} w I. \qquad (0.3)$$

The indexing set for this decomposition $W_+^a$ is a subsemigroup of $W^a = Y \times W$, and it appears naturally in other related contexts, for example when trying to construct an Iwahori-Hecke algebra for $G$ [BKP16, BPGR16]. Let us briefly explain how $W_+^a$ is defined:

Let $\Phi$ be the real root system of the root datum $\mathcal{D}$. It is an infinite set (unless $\mathcal{D}$ is reductive) of linear forms on $Y$ coming with a subset of positive roots $\Phi_+ \subset \Phi$ such that $\Phi = \Phi_+ \sqcup -\Phi_+$. Let $Y^{++} = \{\lambda \in Y \mid \forall \alpha \in \Phi_+, \alpha(\lambda) \ge 0\}$ and $Y^+ = W \cdot Y^{++}$. Then $W_+^a$ is defined as $Y^+ \times W$. In the reductive case, $Y^+$ coincides with $Y$ and thus $W_+^a = W^a$. However, $W^a$ can no longer be conceived as a finite extension of a Coxeter system, hence there is a priori no Bruhat order on $W_+^a$, let alone on $W^a$. A well-behaved topology on $G^+/I$ would allow to define an order on $W_+^a$ through the analog of decomposition (0.2), but $G^+/I$ does not seem to have a natural variety, nor even an ind-variety structure.

## An order and two lengths on $W_+^a$

In appendix of their article on the construction of an Iwahori-Hecke algebra for $G$ an affine Kac-Moody group over a p-adic field [BKP16, Appendix B2], A. Braverman, D. Kazhdan and M. Patnaik propose the definition of a preorder on $W_+^a$ which would replace the Bruhat order of $W^a$ and they conjecture that it is a partial order. In [Mut18], D. Muthiah extends the definition of this preorder to any Kac-Moody group $G$, defines a $\mathbb{Z} \oplus \varepsilon\mathbb{Z}$-valued length compatible with this preorder and hence shows that it is an order. In a joint work [MO19], D. Muthiah and D. Orr then show that this length can be evaluated at $\varepsilon = 1$ to obtain a $\mathbb{Z}$-valued length strictly compatible with the order on $W_+^a$.

In order to build a Kazhdan-Lusztig theory of p-adic Kac-Moody groups, we want to understand how close this order is to the Bruhat order of an affine Coxeter group, which properties still hold and which do not. The definition of a $\mathbb{Z}$-length is already a significant step, but many important properties, which are known to hold for Bruhat orders, remain unknown in this context. Several were proved only for Kac-Moody root systems of affine simply laced type using the specific structure of affinized Weyl group of $W$ in this context.

## Choice of vocabulary

The order on $W_+^a$ is often mentioned in the literature as "double affine Bruhat order" and the associated length as "double affine Bruhat length" because it is most studied in the case of $G$ a Kac-Moody group of affine type (in which case $W$ is an affine Weyl group). In this paper we refer to it as "the affinized Bruhat order" and "the affinized Bruhat length", denoted $\ell^a$, because we do not suppose that $W$ is an affine Weyl group. Note that, if $W$ is finite, then the affinized Bruhat length and order on $W_+^a$ are just the ones induced by its Coxeter group structure.
no longer holds on $G = G(\mathcal{K})$. However there is a partial Iwahori-Matsumoto decomposition: there exists a subsemigroup $G^+$ of $G$ such that:

$$G^+ = \bigsqcup_{\pi^{\lambda} w \in W_+^a} I \pi^{\lambda} w I. \qquad (0.3)$$

The indexing set for this decomposition $W_+^a$ is a subsemigroup of $W^a = Y \times W$, and it appears naturally in other related contexts, for example when trying to construct an Iwahori-Hecke algebra for $G$ [BKP16, BPGR16]. Let us briefly explain how $W_+^a$ is defined:

Let $\Phi$ be the real root system of the root datum $\mathcal{D}$. It is an infinite set (unless $\mathcal{D}$ is reductive) of linear forms on $Y$ coming with a subset of positive roots $\Phi_+ \subset \Phi$ such that $\Phi = \Phi_+ \sqcup -\Phi_+$. Let $Y^{++} = \{\lambda \in Y \mid \forall \alpha \in \Phi_+, \alpha(\lambda) \ge 0\}$ and $Y^+ = W \cdot Y^{++}$. Then $W_+^a$ is defined as $Y^+ \times W$. In the reductive case, $Y^+$ coincides with $Y$ and thus $W_+^a = W^a$. However, $W^a$ can no longer be conceived as a finite extension of a Coxeter system, hence there is a priori no Bruhat order on $W_+^a$, let alone on $W^a$. A well-behaved topology on $G^+/I$ would allow to define an order on $W_+^a$ through the analog of decomposition (0.2), but $G^+/I$ does not seem to have a natural variety, nor even an ind-variety structure.

## An order and two lengths on $W_+^a$

In appendix of their article on the construction of an Iwahori-Hecke algebra for $G$ an affine Kac-Moody group over a p-adic field [BKP16, Appendix B2], A. Braverman, D. Kazhdan and M. Patnaik propose the definition of a preorder on $W_+^a$ which would replace the Bruhat order of $W^a$ and they conjecture that it is a partial order. In [Mut18], D. Muthiah extends the definition of this preorder to any Kac-Moody group $G$, defines a $\mathbb{Z} \oplus \varepsilon\mathbb{Z}$-valued length compatible with this preorder and hence shows that it is an order. In a joint work [MO19], D. Muthiah and D. Orr then show that this length can be evaluated at $\varepsilon = 1$ to obtain a $\mathbb{Z}$-valued length strictly compatible with the order on $W_+^a$.

In order to build a Kazhdan-Lusztig theory of p-adic Kac-Moody groups, we want to understand how close this order is to the Bruhat order of an affine Coxeter group, which properties still hold and which do not. The definition of a $\mathbb{Z}$-length is already a significant step, but many important properties, which are known to hold for Bruhat orders, remain unknown in this context. Several were proved only for Kac-Moody root systems of affine simply laced type using the specific structure of affinized Weyl group of $W$ in this context.

## Choice of vocabulary

The order on $W_+^a$ is often mentioned in the literature as "double affine Bruhat order" and the associated length as "double affine Bruhat length" because it is most studied in the case of $G$ a Kac-Moody group of affine type (in which case $W$ is an affine Weyl group). In this paper we refer to it as "the affinized Bruhat order" and "the affinized Bruhat length", denoted $\ell^a$, because we do not suppose that $W$ is an affine Weyl group. Note that, if $W$ is finite, then the affinized Bruhat length and order on $W_+^a$ are just the ones induced by its Coxeter group structure.
## Main result

Our main result is a positive answer to [MO19, Conjecture 1.5] in full generality:

For any partial order $\le$ on a set $X$, we say that $\mathbf{y}$ covers $\mathbf{x}$ if $\mathbf{x} \neq \mathbf{y}$ and $\{\mathbf{z} \in X \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\} = \{\mathbf{x}, \mathbf{y}\}$. A grading of $X$ is a length function $\ell$ on $X$ strictly compatible with $\le$ and such that $\mathbf{y}$ covers $\mathbf{x}$ if and only if $\mathbf{x} \le \mathbf{y}$ and $\ell(\mathbf{y}) - \ell(\mathbf{x}) = 1$. Gradings thus give an easy classification of covers and more generally of saturated chains in $X$. The Bruhat length for a Coxeter group equipped with the Bruhat order is the prototypical example of a grading.

In [MO19], Muthiah and Orr prove that if $\Phi$ is of affine ADE type, the affinized Bruhat length gives a $\mathbb{Z}$-grading of $W_+^a$ for the affinized Bruhat order and conjecture this to be true in general. The main result of this paper is a positive answer to this conjecture:

**Theorem A.** Let $\mathcal{D}$ be any Kac-Moody root datum. Then the affinized length $\ell^a$ on $W_+^a$ defines a $\mathbb{Z}$-grading of $W_+^a$ strictly compatible with the affinized Bruhat order. Otherwise said, let $\mathbf{x}, \mathbf{y} \in W_+^a$ be such that $\mathbf{x} \le \mathbf{y}$. Then

$$ \mathbf{y} \text{ covers } \mathbf{x} \text{ if and only if } \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = 1. \qquad (0.4) $$

Along the way, we obtain several geometric properties of covers for the affinized Bruhat order which we expect to be insightful even if the root datum is reductive (so $W$ is finite and $W_+^a$ is an affine Weyl group) as they only rely on the Coxeter structure of $W$. In particular, we obtain in Proposition 3.20 a classification of covers which generalize results obtained using quantum Bruhat graphs, in the reductive setting by T. Lam and M. Shimozono ([LS10, Proposition 4.4]) and F. Schremmer [Sch23, Proposition 4.5], and in the affine simply laced setting by A. Welch ([Wel22, Theorem 2]).

## Further directions

In an upcoming joint work with A. Hébert, we prove that any element of $W_+^a$ admits a finite number of covers for the affinized Bruhat order. We use this finiteness in the context of masures to define $R$-Kazhdan-Lusztig polynomials, following Muthiah's strategy exposed in [Mut19] and the work on twin masures of N. Bardy-Panse, A. Hébert and G. Rousseau [BPHR22]. Our understanding of covers is useful to compute these $R$-polynomials and we intend to use $R$-polynomials to define $P$-Kazhdan-Lusztig polynomials.

Another interesting (but quite long reach) question is the following: $W_+^a$ appears as the affinization of $W$, which may be taken as an affinized version of a finite Coxeter group. Can we iterate the affinization process, e.g to obtain a valid theory for reductive or Kac-Moody groups on valued fields of higher dimensions?

Finally, little is known on the preorder defined on the whole semidirect product $W^a$, it could be insightful to study it and to connect it to the failure of the full Iwahori-Matsumoto decomposition of $G$.
## Main result

Our main result is a positive answer to [MO19, Conjecture 1.5] in full generality:

For any partial order $\le$ on a set $X$, we say that $\mathbf{y}$ covers $\mathbf{x}$ if $\mathbf{x} \neq \mathbf{y}$ and $\{\mathbf{z} \in X \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\} = \{\mathbf{x}, \mathbf{y}\}$. A grading of $X$ is a length function $\ell$ on $X$ strictly compatible with $\le$ and such that $\mathbf{y}$ covers $\mathbf{x}$ if and only if $\mathbf{x} \le \mathbf{y}$ and $\ell(\mathbf{y}) - \ell(\mathbf{x}) = 1$. Gradings thus give an easy classification of covers and more generally of saturated chains in $X$. The Bruhat length for a Coxeter group equipped with the Bruhat order is the prototypical example of a grading.

In [MO19], Muthiah and Orr prove that if $\Phi$ is of affine ADE type, the affinized Bruhat length gives a $\mathbb{Z}$-grading of $W_+^a$ for the affinized Bruhat order and conjecture this to be true in general. The main result of this paper is a positive answer to this conjecture:

**Theorem A.** Let $\mathcal{D}$ be any Kac-Moody root datum. Then the affinized length $\ell^a$ on $W_+^a$ defines a $\mathbb{Z}$-grading of $W_+^a$ strictly compatible with the affinized Bruhat order. Otherwise said, let $\mathbf{x}, \mathbf{y} \in W_+^a$ be such that $\mathbf{x} \le \mathbf{y}$. Then

$$ \mathbf{y} \text{ covers } \mathbf{x} \text{ if and only if } \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = 1. \qquad (0.4) $$

Along the way, we obtain several geometric properties of covers for the affinized Bruhat order which we expect to be insightful even if the root datum is reductive (so $W$ is finite and $W_+^a$ is an affine Weyl group) as they only rely on the Coxeter structure of $W$. In particular, we obtain in Proposition 3.20 a classification of covers which generalize results obtained using quantum Bruhat graphs, in the reductive setting by T. Lam and M. Shimozono ([LS10, Proposition 4.4]) and F. Schremmer [Sch23, Proposition 4.5], and in the affine simply laced setting by A. Welch ([Wel22, Theorem 2]).

## Further directions

In an upcoming joint work with A. Hébert, we prove that any element of $W_+^a$ admits a finite number of covers for the affinized Bruhat order. We use this finiteness in the context of masures to define $R$-Kazhdan-Lusztig polynomials, following Muthiah's strategy exposed in [Mut19] and the work on twin masures of N. Bardy-Panse, A. Hébert and G. Rousseau [BPHR22]. Our understanding of covers is useful to compute these $R$-polynomials and we intend to use $R$-polynomials to define $P$-Kazhdan-Lusztig polynomials.

Another interesting (but quite long reach) question is the following: $W_+^a$ appears as the affinization of $W$, which may be taken as an affinized version of a finite Coxeter group. Can we iterate the affinization process, e.g to obtain a valid theory for reductive or Kac-Moody groups on valued fields of higher dimensions?

Finally, little is known on the preorder defined on the whole semidirect product $W^a$, it could be insightful to study it and to connect it to the failure of the full Iwahori-Matsumoto decomposition of $G$.
# Organization of the paper

## Proof strategy

The global strategy is to construct a non-trivial chain from $\mathbf{x}$ to $\mathbf{y}$ every time $\mathbf{y} \ge \mathbf{x}$ verifies $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) > 1$. Let $\text{proj}^{Y^+}$ denote the projection $W_+^a = Y^+ \times_W Y^+$. We distinguish two cases which depend on the form of $\mathbf{x}$ and $\mathbf{y}$: The first case is when $\text{proj}^{Y^+}(\mathbf{y})$ lies in the orbit of $\text{proj}^{Y^+}(\mathbf{x})$, we call such covers the vectorial covers. The other case is when $\text{proj}^{Y^+}(\mathbf{y}) \notin W \cdot \text{proj}^{Y^+}(\mathbf{x})$, we call such covers the properly affine covers.

If $\text{proj}^{Y^+}(\mathbf{y}) \in W \cdot \text{proj}^{Y^+}(\mathbf{x})$, we show that the affinized Bruhat order on $\{\mathbf{z} \in W_+^a \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\}$ is, in some sense, a lift of several Bruhat-like orders on $W$. We are then able to construct chains between $\mathbf{x}$ and $\mathbf{y}$ from chains in $W$, we deduce a classification of vectorial covers. The characterization of properly affine covers is, at first glance, more involved. Through a careful study of the relation between the vectorial chamber containing $\text{proj}^{Y^+}(\mathbf{x})$ and the vectorial chamber containing $\text{proj}^{Y^+}(\mathbf{y})$, we show that the length difference $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$ can be rewritten in a more workable form, making clear the conditions for which it is equal to one. Then the difficulty is to build, explicitly, a non-trivial chain every time one of these conditions is not satisfied.

## Organization

Section 1 consists of preliminaries. In Subsection 1.1 we formally define everything we mentioned in this introduction. In particular we give the definition of the affinized Bruhat order and the two affinized Bruhat lengths as they are given in [MO19]. To be more flexible, we chose to define the affinized Bruhat order on the whole affinized Weyl group $W^a = Y^+ \times_W W$, on which it may not be a preorder.

We show, amongst other preliminary results, that we indeed recover the affinized Bruhat order on $W_+^a$ from this preorder in Subsection 1.3.

We also give, in Subsection 1.2, a geometric interpretation of $W_+^a$ and its affinized Bruhat order, which is to be compared with the interpretation of the Bruhat order in the Coxeter complex of a Coxeter group. Even though it is not clearly mentioned in the rest of the paper, this geometric interpretation was very useful to construct chains and understand $W_+^a$.

In Section 2, we prove Theorem A for vectorial covers. We define relative versions of the Bruhat order on $W$ in Subsection 2.1 and we connect theses relative Bruhat orders to the affinized Bruhat length in Subsection 2.2. This is enough to prove Theorem A when $\text{proj}^{Y^+}(\mathbf{y}) = \text{proj}^{Y^+}(\mathbf{x})$ (see Theorem 2.13). Using finer results on parabolic quotients in Subsection 2.3, we extend it to vectorial covers such that $\text{proj}^{Y^+}(\mathbf{y}) \in W \cdot \text{proj}^{Y^+}(\mathbf{x}) \setminus \{\text{proj}^{Y^+}(\mathbf{x})\}$ (see Theorem 2.18).

In Section 3, we deal with properly affine covers. We first show in Subsection 3.1 that these covers are of a very specific form. Namely, if $\mathbf{x} = \pi^{v(\lambda)}w$ with $v, w \in W$ and $\lambda \in Y^{++}$, then $\mathbf{y}$ needs to be of the form $\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w$ or $\pi^{v_{s_\beta}(\lambda+\beta^\vee)}s_{v(\beta)}w$ for some $\beta \in \Phi_+$.
# Organization of the paper

## Proof strategy

The global strategy is to construct a non-trivial chain from $\mathbf{x}$ to $\mathbf{y}$ every time $\mathbf{y} \ge \mathbf{x}$ verifies $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) > 1$. Let $\text{proj}^{Y^+}$ denote the projection $W_+^a = Y^+ \times_W Y^+$. We distinguish two cases which depend on the form of $\mathbf{x}$ and $\mathbf{y}$: The first case is when $\text{proj}^{Y^+}(\mathbf{y})$ lies in the orbit of $\text{proj}^{Y^+}(\mathbf{x})$, we call such covers the vectorial covers. The other case is when $\text{proj}^{Y^+}(\mathbf{y}) \notin W \cdot \text{proj}^{Y^+}(\mathbf{x})$, we call such covers the properly affine covers.

If $\text{proj}^{Y^+}(\mathbf{y}) \in W \cdot \text{proj}^{Y^+}(\mathbf{x})$, we show that the affinized Bruhat order on $\{\mathbf{z} \in W_+^a \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\}$ is, in some sense, a lift of several Bruhat-like orders on $W$. We are then able to construct chains between $\mathbf{x}$ and $\mathbf{y}$ from chains in $W$, we deduce a classification of vectorial covers. The characterization of properly affine covers is, at first glance, more involved. Through a careful study of the relation between the vectorial chamber containing $\text{proj}^{Y^+}(\mathbf{x})$ and the vectorial chamber containing $\text{proj}^{Y^+}(\mathbf{y})$, we show that the length difference $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$ can be rewritten in a more workable form, making clear the conditions for which it is equal to one. Then the difficulty is to build, explicitly, a non-trivial chain every time one of these conditions is not satisfied.

## Organization

Section 1 consists of preliminaries. In Subsection 1.1 we formally define everything we mentioned in this introduction. In particular we give the definition of the affinized Bruhat order and the two affinized Bruhat lengths as they are given in [MO19]. To be more flexible, we chose to define the affinized Bruhat order on the whole affinized Weyl group $W^a = Y^+ \times_W Y^+$, on which it may not be a preorder.

We show, amongst other preliminary results, that we indeed recover the affinized Bruhat order on $W_+^a$ from this preorder in Subsection 1.3.

We also give, in Subsection 1.2, a geometric interpretation of $W_+^a$ and its affinized Bruhat order, which is to be compared with the interpretation of the Bruhat order in the Coxeter complex of a Coxeter group. Even though it is not clearly mentioned in the rest of the paper, this geometric interpretation was very useful to construct chains and understand $W_+^a$.

In Section 2, we prove Theorem A for vectorial covers. We define relative versions of the Bruhat order on $W$ in Subsection 2.1 and we connect theses relative Bruhat orders to the affinized Bruhat length in Subsection 2.2. This is enough to prove Theorem A when $\text{proj}^{Y^+}(\mathbf{y}) = \text{proj}^{Y^+}(\mathbf{x})$ (see Theorem 2.13). Using finer results on parabolic quotients in Subsection 2.3, we extend it to vectorial covers such that $\text{proj}^{Y^+}(\mathbf{y}) \in W \cdot \text{proj}^{Y^+}(\mathbf{x}) \setminus \{\text{proj}^{Y^+}(\mathbf{x})\}$ (see Theorem 2.18).

In Section 3, we deal with properly affine covers. We first show in Subsection 3.1 that these covers are of a very specific form. Namely, if $\mathbf{x} = \pi^{v(\lambda)}w$ with $v, w \in W$ and $\lambda \in Y^{++}$, then $\mathbf{y}$ needs to be of the form $\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w$ or $\pi^{v_{s_\beta}(\lambda+\beta^\vee)}s_{v(\beta)}w$ for some $\beta \in \Phi_+$.
The strategy is then to get enough necessary conditions on $v, w, \lambda, \beta$ for $\mathbf{y}$ to cover $\mathbf{x}$, in order to obtain a simplified expression for $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$. Proposition 3.3 gives a first result in this direction. In Subsection 3.2 we fully exploit this strategy to obtain Expression (3.15) for the length difference.

Finally, in Subsections 3.3 and 3.4, we construct various chains from $\mathbf{x}$ to $\mathbf{y}$ to prove that the quantities appearing in Expression (3.15) need to be minimal when $\mathbf{y}$ covers $\mathbf{x}$, which allows us to conclude the argument in Subsection 3.5.

## Acknowledgements

I am especially grateful to Auguste Hébert for his help in understanding the affinized Bruhat order and his feedback throughout the conception of this paper, as well as to Stéphane Gaussent for his supervision and his careful rereading. I would also like to thank the anonymous referee for his help and corrections.
The strategy is then to get enough necessary conditions on $v, w, \lambda, \beta$ for $\mathbf{y}$ to cover $\mathbf{x}$, in order to obtain a simplified expression for $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$. Proposition 3.3 gives a first result in this direction. In Subsection 3.2 we fully exploit this strategy to obtain Expression (3.15) for the length difference.

Finally, in Subsections 3.3 and 3.4, we construct various chains from $\mathbf{x}$ to $\mathbf{y}$ to prove that the quantities appearing in Expression (3.15) need to be minimal when $\mathbf{y}$ covers $\mathbf{x}$, which allows us to conclude the argument in Subsection 3.5.

## Acknowledgements

I am especially grateful to Auguste Hébert for his help in understanding the affinized Bruhat order and his feedback throughout the conception of this paper, as well as to Stéphane Gaussent for his supervision and his careful rereading. I would also like to thank the anonymous referee for his help and corrections.
# 1 Preliminaries

## 1.1 Definitions and notation

Let $\mathcal{D} = (A, X, Y, (\alpha_i)_{i \in I}, (\alpha_i^{\vee})_{i \in I})$ be a Kac-Moody root datum as defined in [Rém02, §8].
It is a quintuplet such that:

1. $I$ is a finite indexing set and $A = (a_{ij})_{(i,j) \in I \times I}$ is a generalized Cartan matrix.

2. $X$ and $Y$ are two dual free $\mathbb{Z}$-modules of finite rank, we write $\langle \cdot, \cdot \rangle$ the duality bracket.

3. $(\alpha_i)_{i \in I}$ (resp. $(\alpha_i^{\vee})_{i \in I}$) is a family of linearly independent elements of $X$ (resp. $Y$):
the simple roots (resp. simple coroots).

4. For all $(i, j) \in I^2$ we have $\langle \alpha_i^{\vee}, \alpha_j \rangle = a_{ij}$.

### 1.1.1 Vectorial Weyl group

For every $i \in I$ set $s_i \in \mathrm{Aut}_{\mathbb{Z}}(X) : x \mapsto x - \langle \alpha_i^{\vee}, x \rangle \alpha_i$. The generated group $W = \langle s_i | i \in I \rangle$ is the **vectorial Weyl group** of the Kac-Moody root datum.

The duality bracket $\langle \cdot, \cdot \rangle$ induces a contragredient action of $W$ on $Y$, explicitly $s_i(y) = y - \langle y, \alpha_i \rangle \alpha_i^{\vee}$. The bracket is then $W$-invariant.

The vectorial Weyl group $W$ is a Coxeter group with set of simple reflections $S = \{s_i | i \in I\}$, in particular it has a Bruhat order $<$ and a length function $\ell$ compatible with the Bruhat order. We refer to [BB05] for general definitions and properties of Coxeter groups. A **reflection** in a Coxeter group is any element conjugated to a simple reflection.

### 1.1.2 Real roots

Let $\Phi = W \cdot \{\alpha_i | i \in I\}$ be the set of real roots of $\mathcal{D}$, it is a root system in the classical sense, but possibly infinite. In particular let $\Phi_+ = \bigoplus_{i \in I} \mathbb{N}\alpha_i \cap \Phi$ be the set of positive real roots, then $\Phi = \Phi_+ \sqcup -\Phi_+$, we write $\Phi_- = -\Phi_+$ the set of negative roots.

The set $\Phi^{\vee} = W \cdot \{\alpha_i^{\vee} | i \in I\}$ is the set of **coroots**, and its subset $\Phi_+^{\vee} = \bigoplus_{i \in I} \mathbb{N}\alpha_i^{\vee} \cap \Phi^{\vee}$ is the set of **positive coroots**.

To each root $\beta$ corresponds a unique coroot $\beta^{\vee}$: if $\beta = w(\alpha_i)$ then $\beta^{\vee} = w(\alpha_i^{\vee})$. This map $\beta \mapsto \beta^{\vee}$ is well defined, bijective between $\Phi$ and $\Phi^{\vee}$ and sends positive roots to positive coroots. Note that $\langle \beta^{\vee}, \beta \rangle = 2$ for all $\beta \in \Phi$.

Moreover to each root $\beta$ we associate a reflection $s_{\beta} \in W$: If $\beta = w(\pm \alpha_i)$ then $s_{\beta} := ws_i w^{-1}$. Explicitly it is the map $Y \to Y$ defined by $s_{\beta}(x) = x - \langle \beta^{\vee}, x \rangle \beta$. For any $\beta \in \Phi$ we have $s_{\beta} = s_{-\beta}$ and the map $\beta \mapsto s_{\beta}$ forms a bijection between the set of positive roots and the set $\{ws_i w^{-1} | (w, i) \in W \times I\}$ of reflections of $W$.
# 1 Preliminaries

## 1.1 Definitions and notation

Let $\mathcal{D} = (A, X, Y, (\alpha_i)_{i \in I}, (\alpha_i^{\vee})_{i \in I})$ be a Kac-Moody root datum as defined in [Rém02, §8].
It is a quintuplet such that:

1. $I$ is a finite indexing set and $A = (a_{ij})_{(i,j) \in I \times I}$ is a generalized Cartan matrix.

2. $X$ and $Y$ are two dual free $\mathbb{Z}$-modules of finite rank, we write $\langle \cdot, \cdot \rangle$ the duality bracket.

3. $(\alpha_i)_{i \in I}$ (resp. $(\alpha_i^{\vee})_{i \in I}$) is a family of linearly independent elements of $X$ (resp. $Y$):
the simple roots (resp. simple coroots).

4. For all $(i, j) \in I^2$ we have $\langle \alpha_i^{\vee}, \alpha_j \rangle = a_{ij}$.

### 1.1.1 Vectorial Weyl group

For every $i \in I$ set $s_i \in \mathrm{Aut}_{\mathbb{Z}}(X) : x \mapsto x - \langle \alpha_i^{\vee}, x \rangle \alpha_i$. The generated group $W = \langle s_i | i \in I \rangle$ is the **vectorial Weyl group** of the Kac-Moody root datum.

The duality bracket $\langle \cdot, \cdot \rangle$ induces a contragredient action of $W$ on $Y$, explicitly $s_i(y) = y - \langle y, \alpha_i \rangle \alpha_i^{\vee}$. The bracket is then $W$-invariant.

The vectorial Weyl group $W$ is a Coxeter group with set of simple reflections $S = \{s_i | i \in I\}$, in particular it has a Bruhat order $<$ and a length function $\ell$ compatible with the Bruhat order. We refer to [BB05] for general definitions and properties of Coxeter groups. A **reflection** in a Coxeter group is any element conjugated to a simple reflection.

### 1.1.2 Real roots

Let $\Phi = W \cdot \{\alpha_i | i \in I\}$ be the set of real roots of $\mathcal{D}$, it is a root system in the classical sense, but possibly infinite. In particular let $\Phi_+ = \bigoplus_{i \in I} \mathbb{N}\alpha_i \cap \Phi$ be the set of positive real roots, then $\Phi = \Phi_+ \sqcup -\Phi_+$, we write $\Phi_- = -\Phi_+$ the set of negative roots.

The set $\Phi^{\vee} = W \cdot \{\alpha_i^{\vee} | i \in I\}$ is the set of **coroots**, and its subset $\Phi_+^{\vee} = \bigoplus_{i \in I} \mathbb{N}\alpha_i^{\vee} \cap \Phi^{\vee}$ is the set of **positive coroots**.

To each root $\beta$ corresponds a unique coroot $\beta^{\vee}$: if $\beta = w(\alpha_i)$ then $\beta^{\vee} = w(\alpha_i^{\vee})$. This map $\beta \mapsto \beta^{\vee}$ is well defined, bijective between $\Phi$ and $\Phi^{\vee}$ and sends positive roots to positive coroots. Note that $\langle \beta^{\vee}, \beta \rangle = 2$ for all $\beta \in \Phi$.

Moreover to each root $\beta$ we associate a reflection $s_{\beta} \in W$: If $\beta = w(\pm \alpha_i)$ then $s_{\beta} := ws_i w^{-1}$. Explicitly it is the map $Y \to Y$ defined by $s_{\beta}(x) = x - \langle \beta^{\vee}, x \rangle \beta$. For any $\beta \in \Phi$ we have $s_{\beta} = s_{-\beta}$ and the map $\beta \mapsto s_{\beta}$ forms a bijection between the set of positive roots and the set $\{ws_i w^{-1} | (w, i) \in W \times I\}$ of reflections of $W$.
### 1.1.3 Inversion sets

For any $w \in W$, let $\mathrm{Inv}(w) = \Phi_+ \cap w^{-1}(\Phi_-) = \{\alpha \in \Phi_+ | w(\alpha) \in \Phi_-\}$. These sets are strongly connected to the Bruhat order, as by [Kum02, 1.3.13], for all $\alpha \in \Phi_+$.

$$ \alpha \in \mathrm{Inv}(w) \iff ws_{\alpha} < w \iff s_{\alpha}w^{-1} < w^{-1}. \qquad (1.1) $$

Moreover, they are related to the Bruhat length: $\ell(w) = |\mathrm{Inv}(w)|$ ([Kum02, 1.3.14]).

### 1.1.4 Fundamental chamber and Tit's cone

We define the (closed) integral fundamental chamber by $Y^{++} = \{\lambda \in Y | \langle \lambda, \alpha_i \rangle \ge 0 \forall i \in I\}$. If $\lambda \in Y^{++}$, we say that it is a **dominant coweight**. Then, the integral $\mathrm{Tit}_i$ cone is $Y^+ := \bigcup_{w \in W} w(Y^{++})$. It is a convex cone of $Y$, in particular it is a semigroup for the group operation of $Y$, and it is equal to $Y$ if and only if $W$ is finite, if and only if $\Phi$ is finite, if and only if $A$ is of finite type (see [Kum02, 1.4.2]).

The integral fundamental chamber $Y^{++}$ is a fundamental domain for the action of $W$ on $Y^+$, and for any $\lambda \in Y^+$ we define $\lambda^{++}$ to be the unique element of $Y^{++}$ in its $W$-orbit.

There is a height function on $Y^+$, defined as follows:

**Definition 1.1.** Let $(\Lambda_i)_{i \in I}$ be a set of fundamental weights, that is to say $\langle \alpha_i^\vee, \Lambda_i \rangle = \delta_{ij}$ for any $i, j \in I$. We fix it once and for all. Let $\rho = \sum_{i \in I} \Lambda_i$. Then for any $\lambda \in Y$ define the **height** of $\lambda$ as:

$$ \mathrm{ht}(\lambda) = \langle \lambda, \rho \rangle. \qquad (1.2) $$

The height depends on the choice of fundamental weights, but its restriction to $Q^\vee = \bigoplus_{i \in I} \mathbb{Z}\alpha_i^\vee$ does not: $\mathrm{ht}(\sum_{i \in I} n_i \alpha_i^\vee) = \sum_{i \in I} n_i$.

### 1.1.5 Parabolic subgroups, minimal coset representatives

For $\lambda \in Y^+$, let $\Phi_\lambda$ denote the set $\{\alpha \in \Phi | \langle \lambda, \alpha \rangle = 0\}$ and $W_\lambda = \mathrm{Stab}_W(\lambda)$. We say that $\lambda$ is **regular** if $\Phi_\lambda = 0$, or equivalently if $W_\lambda = 1_W$. More generally we say that $\lambda$ is **spherical** if $W_\lambda$ is finite.

Let $v \in W$ be such that $\lambda = v\lambda^{++}$. Then $W_\lambda v = vW_{\lambda^{++}}$ and, since $\lambda^{++}$ is dominant, $W_{\lambda^{++}}$ is a standard parabolic subgroup, that is a group of the form $W_J = \langle s | s \in J \rangle$ where $J \subset S$ is a set of simple reflections. More precisely, $J = \{s \in S | s(\lambda^{++}) = \lambda^{++}\}$.

By standard Coxeter group theory (see for instance [BB05, Section 2.2]), for any $u \in W$, the left coset $uW_{\lambda^{++}} = uW_J$ has a unique representative of minimal length which we denote $u^J$, and one has a decomposition $u = u^J u_J$ with $u_J \in W_J$ such that:

$$ \ell(u) = \ell(u^J) + \ell(u_J). \qquad (1.3) $$

**Notation 1.2.** 1. For any $J \subset S$, we denote by $W^J$ the set of minimal length representatives for $W_J$-cosets in $W$:

$$ w \in W^J \iff \forall \tilde{w} \in W_J, \ell(w\tilde{w}) > \ell(w) \iff \forall s \in J, \ellлик(s) > \ell(w). \qquad (1.4) $$
### 1.1.3 Inversion sets

For any $w \in W$, let $\mathrm{Inv}(w) = \Phi_+ \cap w^{-1}(\Phi_-) = \{\alpha \in \Phi_+ | w(\alpha) \in \Phi_-\}$. These sets are strongly connected to the Bruhat order, as by [Kum02, 1.3.13], for all $\alpha \in \Phi_+$.

$$ \alpha \in \mathrm{Inv}(w) \iff ws_{\alpha} < w \iff s_{\alpha}w^{-1} < w^{-1}. \qquad (1.1) $$

Moreover, they are related to the Bruhat length: $\ell(w) = |\mathrm{Inv}(w)|$ ([Kum02, 1.3.14]).

### 1.1.4 Fundamental chamber and Tit's cone

We define the (closed) integral fundamental chamber by $Y^{++} = \{\lambda \in Y | \langle \lambda, \alpha_i \rangle \ge 0 \forall i \in I\}$. If $\lambda \in Y^{++}$, we say that it is a **dominant coweight**. Then, the integral $\mathrm{Tit}_i$ cone is $Y^+ := \bigcup_{w \in W} w(Y^{++})$. It is a convex cone of $Y$, in particular it is a semigroup for the group operation of $Y$, and it is equal to $Y$ if and only if $W$ is finite, if and only if $\Phi$ is finite, if and only if $A$ is of finite type (see [Kum02, 1.4.2]).

The integral fundamental chamber $Y^{++}$ is a fundamental domain for the action of $W$ on $Y^+$, and for any $\lambda \in Y^+$ we define $\lambda^{++}$ to be the unique element of $Y^{++}$ in its $W$-orbit. There is a height function on $Y^+$, defined as follows:

**Definition 1.1.** Let $(\Lambda_i)_{i \in I}$ be a set of fundamental weights, that is to say $\langle \alpha_i^\vee, \Lambda_i \rangle = \delta_{ij}$ for any $i, j \in I$. We fix it once and for all. Let $\rho = \sum_{i \in I} \Lambda_i$. Then for any $\lambda \in Y$ define the **height** of $\lambda$ as:

$$ \mathrm{ht}(\lambda) = \langle \lambda, \rho \rangle. \qquad (1.2) $$

The height depends on the choice of fundamental weights, but its restriction to $Q^\vee = \bigoplus_{i \in I} \mathbb{Z}\alpha_i^\vee$ does not: $\mathrm{ht}(\sum_{i \in I} n_i \alpha_i^\vee) = \sum_{i \in I} n_i$.

### 1.1.5 Parabolic subgroups, minimal coset representatives

For $\lambda \in Y^+$, let $\Phi_\lambda$ denote the set $\{\alpha \in \Phi | \langle \lambda, \alpha \rangle = 0\}$ and $W_\lambda = \mathrm{Stab}_W(\lambda)$. We say that $\lambda$ is **regular** if $\Phi_\lambda = 0$, or equivalently if $W_\lambda = 1_W$. More generally we say that $\lambda$ is **spherical** if $W_\lambda$ is finite.

Let $v \in W$ be such that $\lambda = v\lambda^{++}$. Then $W_\lambda v = vW_{\lambda^{++}}$ and, since $\lambda^{++}$ is dominant, $W_{\lambda^{++}}$ is a standard parabolic subgroup, that is a group of the form $W_J = \langle s | s \in J \rangle$ where $J \subset S$ is a set of simple reflections. More precisely, $J = \{s \in S | s(\lambda^{++}) = \lambda^{++}\}$.

By standard Coxeter group theory (see for instance [BB05, Section 2.2]), for any $u \in W$, the left coset $uW_{\lambda^{++}} = uW_J$ has a unique representative of minimal length which we denote $u^J$, and one has a decomposition $u = u^J u_J$ with $u_J \in W_J$ such that:

$$ \ell(u) = \ell(u^J) + \ell(u_J). \qquad (1.3) $$

**Notation 1.2.** 1. For any $J \subset S$, we denote by $W^J$ the set of minimal length representatives for $W_J$-cosets in $W$:

$$ w \in W^J \iff \forall \tilde{w} \in W_J, \ell(w\tilde{w}) > \ell(w) \iff \forall s \in J, \ellлик(s) > \ell(w). \qquad (1.4) $$
If $\lambda \in Y^{++}$ is such that $W_\lambda = W_J$, then we may use $W^\lambda$ as an alternative notation for $W^J$.

2. For any $\lambda \in Y^+$ (not necessarily dominant), we denote by $v^\lambda$ the minimal length element in $W$ which verifies $\lambda = v^\lambda \lambda^{++}$:

$$v^\lambda = \min\{v \in W \mid \lambda = v\lambda^{++}\}. \qquad (1.5)$$

In other words, for any $u \in W$ such that $\lambda = u\lambda^{++}$, we have $v^\lambda = u^J$, where $J$ is the set of simple reflections such that $W_J = W_{\lambda^{++}}$.

### 1.1.6 Affinized Weyl semigroup

The action of $W$ on $Y$ allows to form the semidirect product $Y \times W$, which we denote $W^a$. We denote its elements by $\pi^\lambda w$ with $\lambda \in Y, w \in W$.

By definition, $Y^+ \subset Y$ is stable by the action of $W$ on $Y$, therefore we can form $W_+^a = Y^+ \times W$ which is a subsemigroup of $W^a$. This semigroup is called the affinized Weyl semigroup. In [MO19] Muthiah and Orr define a Bruhat order and an associated length function on $W_+^a$ which we aim to study in this article.

Denote by $\text{proj}^{Y^+}: W_+^a \to Y^+$ the canonical projection, which sends $\pi^\lambda w$ onto $\lambda$. Moreover denote by $\text{proj}^{Y^{++}}: W_+^a \to Y^{++}$ the projection to $Y^{++}$: $\text{proj}^{Y^{++}}(\mathbf{x}) = (\text{proj}^{Y^+}(\mathbf{x}))^{++}$. Let us call $\text{proj}^{Y^+}(\mathbf{x})$ the **coweight** of $\mathbf{x}$, and $\text{proj}^{Y^{++}}(\mathbf{x})$ its **dominance class**.

### 1.1.7 Affinized roots

Let $\Phi^a = \Phi \times \mathbb{Z}$ be the **set of affinized roots** and denote by $\beta + n\pi$ the affinized root $(\beta, n)$. The affinized root $\beta + n\pi$ is said to be positive if $n > 0$ or ($n = 0$ and $\beta \in \Phi_+$) and we write $\Phi_+^a$ for the set of positive affinized roots. We have $\Phi^a = \Phi_+^a \sqcup -\Phi_+^a$.

The semidirect product $W^a$ acts on $\Phi^a$ by:

$$\pi^{\lambda}w(\beta + n\pi) = w(\beta) + (n + \langle \lambda, w(\beta) \rangle)\pi. \qquad (1.6)$$

For any $n \in \mathbb{Z}$, its sign is denoted sgn($n$) $\in \{-1, +1\}$, with the convention that sgn($0$) = +1. Note that $|n| = \text{sgn}(n)n$. We also define the sign of an affinized root: sgn($\beta + n\pi$) $\in \{-1, +1\}$ and sgn($\beta + n\pi$) = +1 if and only if $\beta + n\pi \in \Phi_+^a$.

For $n \in \mathbb{Z}$ and $\beta \in \Phi_+$, set:

$$\beta[n] = \text{sgn}(n)\beta + |n|\pi \in \Phi_+^a \qquad (1.7)$$

$$s_{\beta[n]} = \pi^{n\beta^{\vee}} s_{\beta}. \qquad (1.8)$$

We also define $\beta[n] \in \Phi_+^a$ for $\beta \in \Phi_-$, by $\beta[n] = (-\beta)[-n]$, and $s_{\beta[n]} = s_{-\beta[-n]} = \pi^{n\beta^{\vee}} s_{\beta}$. The affinized root $\beta[n]$ is therefore the positive affinized root within the pair $\{\beta+n\pi, -(\beta+n\pi)\}$. Note that $s_{\beta[0]}$ is the vectorial reflection $s_\beta$.
If $\lambda \in Y^{++}$ is such that $W_{\lambda} = W_{J}$, then we may use $W^{\lambda}$ as an alternative notation for $W^{J}$.

2. For any $\lambda \in Y^{+}$ (not necessarily dominant), we denote by $v^{\lambda}$ the minimal length element in $W$ which verifies $\lambda = v^{\lambda}\lambda^{++}$:

$$v^{\lambda} = \min\{v \in W \mid \lambda = v\lambda^{++}\}. \qquad (1.5)$$

In other words, for any $u \in W$ such that $\lambda = u\lambda^{++}$, we have $v^{\lambda} = u^{J}$, where $J$ is the set of simple reflections such that $W_{J} = W_{\lambda^{++}}$.

### 1.1.6 Affinized Weyl semigroup

The action of $W$ on $Y$ allows to form the semidirect product $Y \times W$, which we denote $W^a$. We denote its elements by $\pi^{\lambda}w$ with $\lambda \in Y, w \in W$.

By definition, $Y^{+} \subset Y$ is stable by the action of $W$ on $Y$, therefore we can form $W_{+}^{a} = Y^{+} \times W$ which is a subsemigroup of $W^{a}$. This semigroup is called the affinized Weyl semigroup. In [MO19] Muthiah and Orr define a Bruhat order and an associated length function on $W_{+}^{a}$ which we aim to study in this article.

Denote by $\text{proj}^{Y^{+}}: W_{+}^{a} \to Y^{+}$ the canonical projection, which sends $\pi^{\lambda}w$ onto $\lambda$. Moreover denote by $\text{proj}^{Y^{++}}: W_{+}^{a} \to Y^{++}$ the projection to $Y^{++}$: $\text{proj}^{Y^{++}}(\mathbf{x}) = (\text{proj}^{Y^{+}}(\mathbf{x}))^{++}$. Let us call $\text{proj}^{Y^{+}}(\mathbf{x})$ the **coweight** of $\mathbf{x}$, and $\text{proj}^{Y^{++}}(\mathbf{x})$ its **dominance class**.

### 1.1.7 Affinized roots

Let $\Phi^{a} = \Phi \times \mathbb{Z}$ be the **set of affinized roots** and denote by $\beta + n\pi$ the affinized root $(\beta, n)$. The affinized root $\beta + n\pi$ is said to be positive if $n > 0$ or ($n = 0$ and $\beta \in \Phi_{+}$) and we write $\Phi_{+}^{a}$ for the set of positive affinized roots. We have $\Phi^{a} = \Phi_{+}^{a} \sqcup -\Phi_{+}^{a}$.

The semidirect product $W^{a}$ acts on $\Phi^{a}$ by:

$$\pi^{\lambda}w(\beta + n\pi) = w(\beta) + (n + \langle \lambda, w(\beta) \rangle)\pi. \qquad (1.6)$$

For any $n \in \mathbb{Z}$, its sign is denoted sgn($n$) $\in \{-1, +1\}$, with the convention that sgn(0) = +1. Note that $|n| = \text{sgn}(n)n$. We also define the sign of an affinized root: sgn($\beta + n\pi$) $\in \{-1, +1\}$ and sgn($\beta + n\pi$) = +1 if and only if $\beta + n\pi \in \Phi_{+}^{a}$.

For $n \in \mathbb{Z}$ and $\beta \in \Phi_{+}$, set:

$$\beta[n] = \text{sgn}(n)\beta + |n|\pi \in \Phi_{+}^{a} \qquad (1.7)$$

$$s_{\beta[n]} = \pi^{n\beta^{\vee}} s_{\beta}. \qquad (1.8)$$

We also define $\beta[n] \in \Phi_{+}^{a}$ for $\beta \in \Phi_{-}$, by $\beta[n] = (-\beta)[-n]$, and $s_{\beta[n]} = s_{-\beta[-n]} = \pi^{n\beta^{\vee}} s_{\beta}$. The affinized root $\beta[n]$ is therefore the positive affinized root within the pair $\{\beta+n\pi, -(\beta+n\pi)\}$. Note that $s_{\beta[0]}$ is the vectorial reflection $s_{\beta}$.
### 1.1.8 Bruhat order on $W_+$

Recall Braverman, Kazhdan and Patnaik's definition of the Bruhat order < introduced in [BKP16, Section B. 2]: Let $\mathbf{x} \in W_+$ and let $\beta[n] \in \Phi_+$ be such that $\mathbf{x}s_{\beta[n]} \in W_+^a$. Then,

$$ \mathbf{x} < \mathbf{x}s_{\beta[n]} \iff \operatorname{sgn}(\beta + n\pi) = \operatorname{sgn}(\mathbf{x}(\beta + n\pi)) \iff \mathbf{x}(\beta[n]) \in \Phi_+^a. \quad (1.9) $$

Explicitly, if $\mathbf{x} = \pi^{\lambda}w \in W_+^a$, the right-hand side condition writes out:

$$ \operatorname{sgn}(n)(n + \langle \lambda, w(\beta) \rangle) > 0 \\ \text{or } n = -\langle \lambda, w(\beta) \rangle \text{ and } \operatorname{sgn}(n)w(\beta) > 0. $$

Then we extend this relation by transitivity, which makes it a preorder. Originally, Braverman, Kazhdan and Patnaik defined it only for affine vectorial Weyl groups, but the definition extends to any vectorial Weyl group and Muthiah showed in [Mut18] that it is an order on $W_+$ in general.

### 1.1.9 Extension to $W^a$

As the whole semidirect product $W^a$ acts on $\Phi_+$, Formula (1.9) makes sense for any $\mathbf{x} \in W^a$, and in this paper we define < on $W^a$, as the closure by transitivity of the relation defined through (1.9) for $\mathbf{x} \in W^a$. We show in the next section that if $\mathbf{x} < \mathbf{y}$ and $\mathbf{y} \in W_+$, then $\mathbf{x} \in W_+^a$. This ensures that the restriction of the $W^a$-preorder to $W_+^a$ coincides with Braverman, Kazhdan, Patnaik's order on $W_+^a$. However < may not be an order on $W^a$.

### 1.1.10 Bruhat order through a right action

In this paper, we consider multiplication by reflections on the left. To switch between the right and left actions note that:

$$ s_{\beta[n]}\pi^{\lambda}w = \pi^{s_{\beta}\lambda+n\beta^{\vee}}s_{\beta}w = \pi^{\lambda+(n-\langle\lambda,\beta\rangle)\beta^{\vee}}ws_{w^{-1}(\beta)} = \pi^{\lambda}ws_{w^{-1}(\beta)[n-\langle\lambda,\beta\rangle]}. \quad (1.10) $$

In particular:

$$ s_{\beta[0]}\pi^{\lambda}w = s_{\beta}\pi^{\lambda}w = \pi^{s_{\beta}\lambda}s_{\beta}w \text{ and } s_{\beta[\langle\lambda,\beta\rangle]}\pi^{\lambda}w = \pi^{\lambda}s_{\beta}w = \pi^{\lambda}ws_{w^{-1}(\beta)}. \quad (1.11) $$

Using Formula (1.10), the affinized Bruhat order can be recovered using a right action of $W^a$ on $\Phi_+$.

**Proposition 1.3.** Let $\pi^{\lambda}w \in W^a$ and $(\beta, n) \in \Phi \times \mathbb{Z} \setminus (\Phi_- \times \{0\})$. Then

$$ s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w \iff \operatorname{sgn}(n)w^{-1}(\beta) + (|n| - \operatorname{sgn}(n)\langle\lambda, \beta\rangle)\pi \in \Phi_+^a. \quad (1.12) $$

**Remark 1.4.** The root appearing in the right-hand side of Equivalence (1.12) is the affinized root $(\pi^{\lambda}w)^{-1}(\beta[n])$.
### 1.1.8 Bruhat order on $W_+$

Recall Braverman, Kazhdan and Patnaik's definition of the Bruhat order < introduced in [BKP16, Section B. 2]: Let $x \in W_+$ and let $\beta[n] \in \Phi_+$ be such that $xs_{\beta[n]} \in W_+^a$. Then,

$$x < xs_{\beta[n]} \iff \operatorname{sgn}(\beta + n\pi) = \operatorname{sgn}(x(\beta + n\pi)) \iff x(\beta[n]) \in \Phi_+^a. \quad (1.9)$$

Explicitly, if $x = \pi^\lambda w \in W_+$, the right-hand side condition writes out:

$$\operatorname{sgn}(n)(n + \langle \lambda, w(\beta) \rangle) > 0$$

$$\text{or } n = -\langle \lambda, w(\beta) \rangle \text{ and } \operatorname{sgn}(n)w(\beta) > 0.$$

Then we extend this relation by transitivity, which makes it a preorder. Originally, Braverman, Kazhdan and Patnaik defined it only for affine vectorial Weyl groups, but the definition extends to any vectorial Weyl group and Muthiah showed in [Mut18] that it is an order on $W_+$ in general.

### 1.1.9 Extension to $W^a$

As the whole semidirect product $W^a$ acts on $\Phi_+$, Formula (1.9) makes sense for any $x \in W^a$, and in this paper we define < on $W^a$, as the closure by transitivity of the relation defined through (1.9) for $x \in W^a$. We show in the next section that if $x < y$ and $y \in W_+$, then $x \in W_+^a$. This ensures that the restriction of the $W^a$-preorder to $W_+^a$ coincides with Braverman, Kazhdan, Patnaik's order on $W_+^a$. However < may not be an order on $W^a$.

### 1.1.10 Bruhat order through a right action

In this paper, we consider multiplication by reflections on the left. To switch between the right and left actions note that:

$$s_{\beta[n]}\pi^{\lambda}w = \pi^{s_{\beta}\lambda+n\beta^{\vee}}s_{\beta}w = \pi^{\lambda+(n-\langle\lambda,\beta\rangle)\beta^{\vee}}ws_{w^{-1}(\beta)} = \pi^{\lambda}ws_{w^{-1}(\beta)[n-\langle\lambda,\beta\rangle]}. \quad (1.10)$$

In particular:

$$s_{\beta[0]}\pi^{\lambda}w = s_{\beta}\pi^{\lambda}w = \pi^{s_{\beta}\lambda}s_{\beta}w \text{ and } s_{\beta[\langle\lambda,\beta\rangle]}\pi^{\lambda}w = \pi^{\lambda}s_{\beta}w = \pi^{\lambda}ws_{w^{-1}(\beta)}. \quad (1.11)$$

Using Formula (1.10), the affinized Bruhat order can be recovered using a right action of $W^a$ on $\Phi_+$.

**Proposition 1.3.** Let $\pi^{\lambda}w \in W^a$ and $(\beta, n) \in \Phi \times \mathbb{Z} \setminus (\Phi_- \times \{0\})$. Then

$$s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w \iff \operatorname{sgn}(n)w^{-1}(\beta) + (|n| - \operatorname{sgn}(n)\langle\lambda, \beta\rangle)\pi \in \Phi_+^a. \quad (1.12)$$

**Remark 1.4.** The root appearing in the right-hand side of Equivalence (1.12) is the affinized root $(\pi^{\lambda}w)^{-1}(\beta[n])$.
Proof. Let $\pi^{\lambda}w \in W^a$ and $\beta + n\pi \in \Phi^a$. Then by Formulas (1.9) and (1.10),

$$s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w \iff \operatorname{sgn}(\beta + n\pi) = \operatorname{sgn}(w^{-1}(\beta) + (n - \langle\lambda, \beta\rangle)\pi).$$

If $(\beta, n) \notin \Phi_- \times \{0\}$, then $\beta[n] = \operatorname{sgn}(n)(\beta + n\pi)$ so this is equivalent to $\operatorname{sgn}(n)(w^{-1}(\beta) + (n - \langle\lambda, \beta\rangle)\pi)$ being a positive affinized root, which is Equivalence (1.12). $\square$

Note that Formula (1.12) is no longer correct if $\beta \in \Phi_-$ and $n = 0$, in which case it needs to be applied to $(-\beta)[0]$. Applying reflections on the left is better suited for the geometric interpretation we will give in Subsection 1.2.

### 1.1.11 Terminology on partially ordered sets

For $p \le q \in \mathbb{Z}$, we denote by $[[p, q]]$ the set $\{r \in \mathbb{Z} \mid p \le r \le q\}$. If $p > q$, then $[[p, q]]$ is another notation for $[[q, p]]$. We also write $[[p, q]]$ for $[[p, q] \setminus \{p, q\}$.

Let $(\mathcal{P}, \le)$ be a partially ordered set. For $\mathbf{x}, \mathbf{y} \in \mathcal{P}$, we say that $\mathbf{x}$ and $\mathbf{y}$ are comparable if $\mathbf{x} \le \mathbf{y}$ or $\mathbf{y} \le \mathbf{x}$. We say that $\mathbf{y}$ covers $\mathbf{x}$ if $\mathbf{x} \ne \mathbf{y}$ and $\{\mathbf{z} \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\} = \{\mathbf{x}, \mathbf{y}\}$, we then write $\mathbf{x} \triangleleft \mathbf{y}$. If $\mathcal{P} = W_a^+$, covers $\mathbf{x} \triangleleft \mathbf{y}$ such that $\operatorname{proj}^{Y++}(\mathbf{y}) = \operatorname{proj}^{Y++}(\mathbf{x})$ are called **vectorial covers** and covers which are not vectorial covers are called **properly affine covers**.

A **chain** from $\mathbf{x}$ to $\mathbf{y}$ is a finite sequence $(\mathbf{x}_0, \dots, \mathbf{x}_n)$ such that $\mathbf{x}_0 = \mathbf{x}, \mathbf{x}_n = \mathbf{y}$ and $\mathbf{x}_k \le \mathbf{x}_{k+1}$ for all $k \in [0, n-1]$. If $\mathcal{P} = W_a^+$ (resp. if $\mathcal{P}$ is a Coxeter group), we add the condition that $\mathbf{x}_{k+1}\mathbf{x}_k^{-1}$ is an affinized reflection (resp. a reflection). A chain is **saturated** if $\mathbf{x}_k \triangleleft \mathbf{x}_{k+1}$ for all $k \in [0, n-1]$. We say that a subset $\mathcal{C}$ of $\mathcal{P}$ is **convex** if, for all $\mathbf{x}, \mathbf{y} \in \mathcal{C}$ and $\mathbf{z} \in \mathcal{P}$,

$$\mathbf{x} \le \mathbf{z} \le \mathbf{y} \implies \mathbf{z} \in \mathcal{C}. \qquad (1.13)$$

Equivalently, $\mathcal{C}$ is convex if and only if any chain from one element of $\mathcal{C}$ to another is contained in $\mathcal{C}$.

Let $\ell: \mathcal{P} \to A$ be a function with values in a totally ordered set $A$, we say that it is **order-preserving** if, for all $\mathbf{x}, \mathbf{y} \in \mathcal{P}$,

$$\mathbf{x} \le \mathbf{y} \implies \ell(\mathbf{x}) \le \ell(\mathbf{y}). \qquad (1.14)$$

We say that $\ell$ is a **strictly compatible** ($A$-valued) length function if:

$$\mathbf{x} < \mathbf{y} \iff \mathbf{x}, \mathbf{y} \text{ are comparable and } \ell(\mathbf{x}) < \ell(\mathbf{y}). \qquad (1.15)$$

We say that a strictly compatible length function $\ell$ defines a **grading** of $\mathcal{P}$ if:

$$\mathbf{x} \triangleleft \mathbf{y} \iff \mathbf{x} \le \mathbf{y} \text{ and } \ell(\mathbf{x}) \triangleleft \ell(\mathbf{y}). \qquad (1.16)$$

In particular if $A = \mathbb{Z}$ or $A = \mathbb{N}$, this rewrites

$$\mathbf{x} \triangleleft \mathbf{y} \iff \mathbf{x} \le \mathbf{y} \text{ and } \ell(\mathbf{y}) = \ell(\mathbf{x}) + 1. \qquad (1.17)$$

For instance, the Bruhat length on a Coxeter group $W$ is strictly compatible with the Bruhat order, and defines a $\mathbb{N}$-valued grading of $W$. Muthiah and Orr associated length functions strictly compatible with the Bruhat order on $W_+^a$, generalizing the classical Bruhat length on Coxeter groups. We now formally introduce these lengths.
Proof. Let $\pi^{\lambda}w \in W^a$ and $\beta + n\pi \in \Phi^a$. Then by Formulas (1.9) and (1.10),

$$s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w \iff \operatorname{sgn}(\beta + n\pi) = \operatorname{sgn}(w^{-1}(\beta) + (n - \langle\lambda, \beta\rangle)\pi).$$

If $(\beta, n) \notin \Phi_- \times \{0\}$, then $\beta[n] = \operatorname{sgn}(n)(\beta + n\pi)$ so this is equivalent to $\operatorname{sgn}(n)(w^{-1}(\beta) + (n - \langle\lambda, \beta\rangle)\pi)$ being a positive affinized root, which is Equivalence (1.12). $\square$

Note that Formula (1.12) is no longer correct if $\beta \in \Phi_-$ and $n = 0$, in which case it needs to be applied to $(-\beta)[0]$. Applying reflections on the left is better suited for the geometric interpretation we will give in Subsection 1.2.

### 1.1.11 Terminology on partially ordered sets

For $p \le q \in \mathbb{Z}$, we denote by $[[p, q]]$ the set $\{r \in \mathbb{Z} \mid p \le r \le q\}$. If $p > q$, then $[[p, q]]$ is another notation for $[[q, p]]$. We also write $[[p, q]]$ for $[[p, q] \setminus \{p, q\}$.

Let $(\mathcal{P}, \le)$ be a partially ordered set. For $\mathbf{x}, \mathbf{y} \in \mathcal{P}$, we say that $\mathbf{x}$ and $\mathbf{y}$ are comparable if $\mathbf{x} \le \mathbf{y}$ or $\mathbf{y} \le \mathbf{x}$. We say that $\mathbf{y}$ covers $\mathbf{x}$ if $\mathbf{x} \ne \mathbf{y}$ and $\{\mathbf{z} \mid \mathbf{x} \le \mathbf{z} \le \mathbf{y}\} = \{\mathbf{x}, \mathbf{y}\}$, we then write $\mathbf{x} \triangleleft \mathbf{y}$. If $\mathcal{P} = W_a^+$, covers $\mathbf{x} \triangleleft \mathbf{y}$ such that $\operatorname{proj}^{Y^{++}}(\mathbf{y}) = \operatorname{proj}^{Y^{++}}(\mathbf{x})$ are called **vectorial covers** and covers which are not vectorial covers are called **properly affine covers**.

A **chain** from $\mathbf{x}$ to $\mathbf{y}$ is a finite sequence $(\mathbf{x}_0, \dots, \mathbf{x}_n)$ such that $\mathbf{x}_0 = \mathbf{x}, \mathbf{x}_n = \mathbf{y}$ and $\mathbf{x}_k \le \mathbf{x}_{k+1}$ for all $k \in [0, n-1]$. If $\mathcal{P} = W_a^+$ (resp. if $\mathcal{P}$ is a Coxeter group), we add the condition that $\mathbf{x}_{k+1}\mathbf{x}_k^{-1}$ is an affinized reflection (resp. a reflection). A chain is **saturated** if $\mathbf{x}_k \triangleleft \mathbf{x}_{k+1}$ for all $k \in [0, n-1]$. We say that a subset $\mathcal{C}$ of $\mathcal{P}$ is **convex** if, for all $\mathbf{x}, \mathbf{y} \in \mathcal{C}$ and $\mathbf{z} \in \mathcal{P}$,

$$\mathbf{x} \le \mathbf{z} \le \mathbf{y} \implies \mathbf{z} \in \mathcal{C}. \qquad (1.13)$$

Equivalently, $\mathcal{C}$ is convex if and only if any chain from one element of $\mathcal{C}$ to another is contained in $\mathcal{C}$.

Let $\ell: \mathcal{P} \to A$ be a function with values in a totally ordered set $A$, we say that it is **order-preserving** if, for all $\mathbf{x}, \mathbf{y} \in \mathcal{P}$,

$$\mathbf{x} \le \mathbf{y} \implies \ell(\mathbf{x}) \le \ell(\mathbf{y}). \qquad (1.14)$$

We say that $\ell$ is a **strictly compatible** ($A$-valued) length function if:

$$\mathbf{x} < \mathbf{y} \iff \mathbf{x}, \mathbf{y} \text{ are comparable and } \ell(\mathbf{x}) < \ell(\mathbf{y}). \qquad (1.15)$$

We say that a strictly compatible length function $\ell$ defines a **grading** of $\mathcal{P}$ if:

$$\mathbf{x} \triangleleft \mathbf{y} \iff \mathbf{x} \le \mathbf{y} \text{ and } \ell(\mathbf{x}) \triangleleft \ell(\mathbf{y}). \qquad (1.16)$$

In particular if $A = \mathbb{Z}$ or $A = \mathbb{N}$, this rewrites

$$\mathbf{x} \triangleleft \mathbf{y} \iff \mathbf{x} \le \mathbf{y} \text{ and } \ell(\mathbf{y}) = \ell(\mathbf{x}) + 1. \qquad (1.17)$$

For instance, the Bruhat length on a Coxeter group $W$ is strictly compatible with the Bruhat order, and defines a $\mathbb{N}$-valued grading of $W$. Muthiah and Orr associated length functions strictly compatible with the Bruhat order on $W_+^a$, generalizing the classical Bruhat length on Coxeter groups. We now formally introduce these lengths.
### 1.1.12 Length functions on $W_+^a$

**Definition 1.5.** The affinized length function is the map $W_+^a \to \mathbb{Z} \oplus \varepsilon\mathbb{Z}$ defined by:

$$ \ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \varepsilon(|\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle < 0\}|). \quad (1.18) $$

The affinized length with integral values is the affinized length function on which we set $\varepsilon = 1$:

$$ \ell^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + (|\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle < 0\}|). \quad (1.19) $$

**Theorem 1.6.** [Mut18, Theorem 4.24] [MO19, Theorem 3.6]

The affinized length function and the affinized length function with integral values are strictly compatible with the affinized Bruhat order on $W_+^a$. In other words, for any $\mathbf{x} \in W_+^a$ and $\beta[n] \in \Phi_+^a$,

$$ \mathbf{x}s_{\beta[n]} > \mathbf{x} \iff \ell_{\varepsilon}^{a}(\mathbf{x}s_{\beta[n]}) > \ell_{\varepsilon}^{a}(\mathbf{x}) \iff \ell^{a}(\mathbf{x}s_{\beta[n]}) > \ell^{a}(\mathbf{x}). \quad (1.20) $$

In particular the affinized Bruhat order is a partial order.

In what follows we will mostly use $\ell^a$ and rarely mention $\ell_{\varepsilon}^a$, we now refer to $\ell^a$ as the affinized Bruhat length.

## 1.2 Geometric interpretation

We introduced everything in a very algebraic way, but there is a strong geometric intuition behind root systems, vectorial Weyl groups and the vectorial Bruhat order, developed for instance in the context of buildings in [Ron89]. There is also a geometrical interpretation of the Bruhat order on $W_+^a$ which we develop in this paragraph, it takes place in the standard apartment of the measure associated to a Kac-Moody group with underlying Kac-Moody datum $\mathcal{D}$.

Let $V = Y \otimes_{\mathbb{Z}} \mathbb{R}$, the lattice $X$ embeds in its dual $V^\vee$ and the vectorial Weyl group $W$ acts naturally on it. Inside $V$ we have the (closed) fundamental chamber $C_f^v = \{v \in V | \langle v, \alpha_i \rangle \ge 0\}$ and the Tits cone $\mathcal{T} = W \cdot C_f^v$. A vectorial chamber is a set of the form $w \cdot C_f^v$ for $w \in W$. Since the interior of $C_f^v$ has trivial stabiliser in $W$, the set of chambers is in natural bijection with $W$ by $w \mapsto C_w^v := w \cdot C_f^v$.

To each root $\beta \in \Phi_+$ let $M_\beta = \{x \in V | \langle x, \beta \rangle = 0\}$, it is an hyperplane of $V$ and, if $\beta = w(\alpha_i)$ with $\alpha_i$ a simple root, then $C_w^v \cap C_{w\beta_i}^v \subset M_\beta \cap \mathcal{T}$. The intersection $C_w^v \cap C_{w\beta_i}^v$ is called the panel of type $s_i$ of $w$.

We can put a structure of simplicial complex on $\mathcal{T}$, for which chambers are the cells of maximal rank and panels are the cells of maximal rank within non-chambers. This simplicial complex is a realization of the Coxeter complex of $(W, S)$. Each wall splits
### 1.1.12 Length functions on $W_+^a$

**Definition 1.5.** The affinized length function is the map $W_+^a \to \mathbb{Z} \oplus \varepsilon\mathbb{Z}$ defined by:

$$ \ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \varepsilon(|\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle < 0\}|). \quad (1.18) $$

The affinized length with integral values is the affinized length function on which we set $\varepsilon = 1$:

$$ \ell^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + (|\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) | \langle\lambda, \alpha\rangle < 0\}|). \quad (1.19) $$

**Theorem 1.6.** [Mut18, Theorem 4.24] [MO19, Theorem 3.6]

The affinized length function and the affinized length function with integral values are strictly compatible with the affinized Bruhat order on $W_+^a$. In other words, for any $\mathbf{x} \in W_+^a$ and $\beta[n] \in \Phi_+^a$,

$$ \mathbf{x}s_{\beta[n]} > \mathbf{x} \iff \ell_{\varepsilon}^{a}(\mathbf{x}s_{\beta[n]}) > \ell_{\varepsilon}^{a}(\mathbf{x}) \iff \ell^{a}(\mathbf{x}s_{\beta[n]}) > \ell^{a}(\mathbf{x}). \quad (1.20) $$

In particular the affinized Bruhat order is a partial order.

In what follows we will mostly use $\ell^a$ and rarely mention $\ell_{\varepsilon}^a$, we now refer to $\ell^a$ as the affinized Bruhat length.

## 1.2 Geometric interpretation

We introduced everything in a very algebraic way, but there is a strong geometric intuition behind root systems, vectorial Weyl groups and the vectorial Bruhat order, developed for instance in the context of buildings in [Ron89]. There is also a geometrical interpretation of the Bruhat order on $W_+^a$ which we develop in this paragraph, it takes place in the standard apartment of the measure associated to a Kac-Moody group with underlying Kac-Moody datum $\mathcal{D}$.

Let $V = Y \otimes_{\mathbb{Z}} \mathbb{R}$, the lattice $X$ embeds in its dual $V^\vee$ and the vectorial Weyl group $W$ acts naturally on it. Inside $V$ we have the (closed) fundamental chamber $C_f^v = \{v \in V | \langle v, \alpha_i \rangle \ge 0\}$ and the Tits cone $\mathcal{T} = W \cdot C_f^v$. A vectorial chamber is a set of the form $w \cdot C_f^v$ for $w \in W$. Since the interior of $C_f^v$ has trivial stabiliser in $W$, the set of chambers is in natural bijection with $W$ by $w \mapsto C_w^v := w \cdot C_f^v$.

To each root $\beta \in \Phi_+$ let $M_\beta = \{x \in V | \langle x, \beta \rangle = 0\}$, it is an hyperplane of $V$ and, if $\beta = w(\alpha_i)$ with $\alpha_i$ a simple root, then $C_w^v \cap C_{w\beta_i}^v \subset M_\beta \cap \mathcal{T}$. The intersection $C_w^v \cap C_{w\beta_i}^v$ is called the panel of type $s_i$ of $w$.

We can put a structure of simplicial complex on $\mathcal{T}$, for which chambers are the cells of maximal rank and panels are the cells of maximal rank within non-chambers. This simplicial complex is a realization of the Coxeter complex of $(W, S)$. Each wall splits
the Tits cone in two parts, and separate the set of vectorial chambers in two: say that $C_w^v$ is on the positive side of $M_\beta$ if $w^{-1}(\beta) > 0$. In particular since $\beta$ is a positive root, the positive side is always the one which contain the dominant chamber.

Then the vectorial Bruhat order can be interpreted by: $s_\beta w > w$ if and only if, when we split $\mathcal{T}$ along $M_\beta$ the chambers $C_w^v$ and $C_f^v$ are in the same connected component of $\mathcal{T}$, that is to say $C_w^v$ is on the positive side of $M_\beta$.

Moreover $\mathrm{Inv}(w^{-1})$ the inversion set of $w^{-1}$ can be interpreted as the set of walls separating the chamber $C_w^v = w \cdot C_f^v$ from the fundamental chamber $C_f^v$.

In Figure 1.1 we represent the Tits cone and its structure for a root system of rank 2 with Cartan matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$, which is of indefinite type. The Tits cone is colored in blue, and the vectorial chamber $C_w^v$ is labelled by $w$. It is an approximation since $W$ is infinite.

<figure><img src="image_2.png" /><figcaption>Figure 1.1: The Tits cone for a root system of Cartan Matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$.</figcaption></figure>
the Tits cone in two parts, and separate the set of vectorial chambers in two: say that $C_w^v$ is on the positive side of $M_β$ if $w^{-1}(β) > 0$. In particular since $β$ is a positive root, the positive side is always the one which contain the dominant chamber.

Then the vectorial Bruhat order can be interpreted by: $s_βw > w$ if and only if, when we split $\mathcal{T}$ along $M_β$ the chambers $C_w^v$ and $C_f^v$ are in the same connected component of $\mathcal{T}$, that is to say $C_w^v$ is on the positive side of $M_β$.

Moreover $\mathrm{Inv}(w^{-1})$ the inversion set of $w^{-1}$ can be interpreted as the set of walls separating the chamber $C_w^v = w \cdot C_f^v$ from the fundamental chamber $C_f^v$.

In Figure 1.1 we represent the Tits cone and its structure for a root system of rank 2 with Cartan matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$, which is of indefinite type. The Tits cone is colored in blue, and the vectorial chamber $C_w^v$ is labelled by $w$. It is an approximation since $W$ is infinite.

<figure><img src="image_3.png" /><figcaption>Figure 1.1: The Tits cone for a root system of Cartan Matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$.</figcaption></figure>
Let us now turn to the interpretation of the $W_+^a$-Bruhat order. Let $\mathbb{A}$ be a real affine space with direction $V$, we call $\mathbb{A}$ the (standard) affine apartment associated to $\mathcal{D}$. The tangent space of $\mathbb{A}$ is canonically isomorphic to $T\mathbb{A} = \mathbb{A} \times V$, with, for any $x \in \mathbb{A}$, $T_x\mathbb{A} = \{x\} \times V$.

The semigroup $W_+^a$ has an affine action on $\mathbb{A}$, given by $\pi^\lambda w(x) = -\lambda + w(x)$, it induces an action on the tangent space $T\mathbb{A}$ given by $\pi^\lambda w((x, v)) = (-\lambda + w(x), w(v))$. To any positive affinized root $\beta[n] \in \Phi_+^a$ corresponds an affine hyperplane:

$$M_{\beta[n]} = \{x \in \mathbb{A} \mid \langle x, \beta \rangle + n = 0\}, \qquad (1.21)$$

the **affine wall** associated to the affinized root $\beta[n]$. For any $x \in M_{\beta[n]}$ we have $T_x M_{\beta[n]} = \{x\} \times M_\beta \subset T_x \mathbb{A}$.

For any $\pi^\lambda w \in W_+^a$ let

$$C_{\pi^\lambda w} = \{-\lambda\} \times C_w^v \subset T_{-\lambda} \mathbb{A} \subset T\mathbb{A}, \qquad (1.22)$$

we call it the **alcove of type $\pi^\lambda w$**. Mirroring the classical situation, $C_0 = \{0\} \times C_f^v$ is a fundamental domain for the action of $W_+^a$ on $Y^+ \times \mathcal{T} \subset T\mathbb{A}$ and $W_+^a$ acts on $\{C_x \mid x \in W_+^a\}$ simply transitively. Affine walls separate naturally the set of alcoves in two and we call the side containing $C_0$ the positive side.

Then the $W_+^a$-Bruhat order can be interpreted geometrically:

$$s_{\beta[n]} \pi^{\lambda} w > \pi^{\lambda} w \iff C_{\pi^{\lambda} w} \text{ is on the positive side of } M_{\beta[n]}. \qquad (1.23)$$

We give an illustration of the affine apartment in Figure 1.2 below.
Let us now turn to the interpretation of the $W_+^a$-Bruhat order. Let $\mathbb{A}$ be a real affine space with direction $V$, we call $\mathbb{A}$ the (standard) affine apartment associated to $\mathcal{D}$. The tangent space of $\mathbb{A}$ is canonically isomorphic to $T\mathbb{A} = \mathbb{A} \times V$, with, for any $x \in \mathbb{A}$, $T_x\mathbb{A} = \{x\} \times V$.

The semigroup $W_+^a$ has an affine action on $\mathbb{A}$, given by $\pi^\lambda w(x) = -\lambda + w(x)$, it induces an action on the tangent space $T\mathbb{A}$ given by $\pi^\lambda w((x, v)) = (-\lambda + w(x), w(v))$. To any positive affinized root $\beta[n] \in \Phi_+^a$ corresponds an affine hyperplane:

$$M_{\beta[n]} = \{x \in \mathbb{A} \mid \langle x, \beta \rangle + n = 0\}, \qquad (1.21)$$

the **affine wall** associated to the affinized root $\beta[n]$. For any $x \in M_{\beta[n]}$ we have $T_x M_{\beta[n]} = \{x\} \times M_\beta \subset T_x \mathbb{A}$.

For any $\pi^\lambda w \in W_+^a$ let

$$C_{\pi^\lambda w} = \{-\lambda\} \times C_w^v \subset T_{-\lambda} \mathbb{A} \subset T\mathbb{A}, \qquad (1.22)$$

we call it the **alcove of type $\pi^\lambda w$**. Mirroring the classical situation, $C_0 = \{0\} \times C_f^v$ is a fundamental domain for the action of $W_+^a$ on $Y^+ \times \mathcal{T} \subset T\mathbb{A}$ and $W_+^a$ acts on $\{C_x \mid x \in W_+^a\}$ simply transitively. Affine walls separate naturally the set of alcoves in two and we call the side containing $C_0$ the positive side.

Then the $W_+^a$-Bruhat order can be interpreted geometrically:

$$s_{\beta[n]} \pi^\lambda w > \pi^\lambda w \iff C_{\pi^\lambda w} \text{ is on the positive side of } M_{\beta[n]}. \qquad (1.23)$$

We give an illustration of the affine apartment in Figure 1.2 below.
<figure><img src="image_3.png" /><figcaption>Figure 1.2: The affine apartment for a root system of Cartan Matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$.</figcaption></figure>

In Figure 1.2 we represent the affine apartment for the same root datum as in Figure 1.1. The blue polygons represent the local Tits cones at three different points: the origin, $-\lambda \in -Y^+$ and $-\mu$, which is the image of $-\lambda$ by the reflection along the wall $M_{s_1(\alpha_2)[2]}$ (represented in green).

We have highlighted three alcoves: In green the alcove $C_0$; in red the alcove $C = C_{\pi^{\lambda}s_1s_2}$ and in yellow $D = C_{\pi^{\mu}s_1}$ which is the image of $C$ by $s_{s_1(\alpha_2)[2]}$. We see that $D$ is on the same side of $M_{s_1(\alpha_2)[2]}$ as the fundamental alcove $C_0$, thus $\pi^{\lambda}s_1s_2 = s_{s_1(\alpha_2)[2]}(\pi^{\mu}s_1) > \pi^{\mu}s_1$.

Note that $-\lambda$ lies in the negative vectorial chamber $-s_2C_f^v$, that is to say that $s_2\lambda$ is dominant. Therefore $\pi^{\lambda}s_2$ is the minimal length element of $\pi^{\lambda}W$. We will make this more explicit in Section 2.2.

### 1.2.1 Notation for segments

For any two elements $x, y \in V = Y \otimes_{\mathbb{Z}} \mathbb{R}$, we introduce the following notation:

$$[x, y] = \{tx + (1-t)y \mid 0 \le t \le 1\}$$

$$[x, y] = \{tx + (1-t)y \mid 0 < t < 1\}.$$

Note in particular that, if $x \in Y$ and $y = x + n\beta^{\vee}$ for $n \in \mathbb{Z}$ and $\beta \in \Phi$, then for any $m \in [0, n]$ we have $x + m\beta^{\vee} \in [x, y] \cap Y$.
<figure><img src="image_4.png" /><figcaption>Figure 1.2: The affine apartment for a root system of Cartan Matrix $\begin{pmatrix} 2 & -3 \\ -2 & 2 \end{pmatrix}$.</figcaption></figure>

In Figure 1.2 we represent the affine apartment for the same root datum as in Figure 1.1. The blue polygons represent the local Tits cones at three different points: the origin, $-\lambda \in -Y^+$ and $-\mu$, which is the image of $-\lambda$ by the reflection along the wall $M_{s_1(\alpha_2)[2]}$ (represented in green).

We have highlighted three alcoves: In green the alcove $C_0$; in red the alcove $C = C_{\pi^{\lambda}s_1s_2}$ and in yellow $D = C_{\pi^{\mu}s_1}$ which is the image of $C$ by $s_{s_1(\alpha_2)[2]}$. We see that $D$ is on the same side of $M_{s_1(\alpha_2)[2]}$ as the fundamental alcove $C_0$, thus $\pi^{\lambda}s_1s_2 = s_{s_1(\alpha_2)[2]}(\pi^{\mu}s_1) > \pi^{\mu}s_1$.

Note that $-\lambda$ lies in the negative vectorial chamber $-s_2C_f^v$, that is to say that $s_2\lambda$ is dominant. Therefore $\pi^{\lambda}s_2$ is the minimal length element of $\pi^{\lambda}W$. We will make this more explicit in Section 2.2.

### 1.2.1 Notation for segments

For any two elements $x, y \in V = Y \otimes_{\mathbb{Z}} \mathbb{R}$, we introduce the following notation:

$$[x, y] = \{tx + (1-t)y \mid 0 \le t \le 1\}$$

$$[x, y] = \{tx + (1-t)y \mid 0 < t < 1\}.$$

Note in particular that, if $x \in Y$ and $y = x + n\beta^{\vee}$ for $n \in \mathbb{Z}$ and $\beta \in \Phi$, then for any $m \in [0, n]$ we have $x + m\beta^{\vee} \in [x, y] \cap Y$.
## 1.3 Preliminary results

Since the affinized Bruhat order is generated on $W_+$ by the relations $s_{\beta[n]}\mathbf{x} > \mathbf{x} \iff \ell^a(s_{\beta[n]}\mathbf{x}) > \ell^a(\mathbf{x})$ for affinized roots $\beta[n] \in \Phi_+$, covers are always of this form. In the rest of the paper, we always apply affinized reflections on the left.

**Lemma 1.7.** Let $\pi^\lambda w \in W^a$ and $\beta[n] \in \Phi_+^a$. Write $\pi^\mu w'$ for $s_{\beta[n]}\pi^\lambda w$ and suppose that $(\pi^\lambda w)^{-1}(\beta[n]) \in \Phi_+^a$. Then $\lambda \in [\mu, s_\beta\mu]$. In particular

$$ \mu \in Y^+ \implies \lambda \in Y^+. \qquad (1.24) $$

*Proof.* Explicitly:

$$ \pi^\mu w' = \pi^{n\beta^\vee} s_\beta \cdot \pi^\lambda w = \pi^{s_\beta\lambda + n\beta^\vee} s_\beta w. $$

Thus $\mu = s_\beta\lambda + n\beta^\vee = \lambda + (n - \langle\lambda, \beta\rangle)\beta^\vee$ and $s_\beta\mu = \lambda - n\beta^\vee$.

Moreover, since $(\pi^\lambda w)^{-1}(\beta[n]) \in \Phi_+^a$, by Formula (1.12):

$$ |n| - \operatorname{sgn}(n)(\lambda, \beta) = \operatorname{sgn}(n)(n - (\lambda, \beta)) \ge 0. $$

Therefore, unless $n - (\lambda, \beta) = 0$, $n$ and $n - (\lambda, \beta)$ have same sign and thus, $\lambda = s_\beta\mu + n\beta^\vee = \mu - (n - (\lambda, \beta))\beta^\vee$ lies in $[s_\beta\mu, \mu]$. If $n - (\lambda, \beta) = 0$ then $\mu = \lambda$ and the result remains true.

The Tits cone $\mathcal{T}$ is convex ([Kum02, Proposition 1.4.2c]) and $W$-stable, so if $\mu \in \mathcal{T}$, then $[\mu, s_\beta\mu]$ is contained in $\mathcal{T}$ for any $\beta \in \Phi$. Therefore in the situation above, if $\mu \in Y^+ = \mathcal{T} \cap Y$, then $\lambda \in [\mu, s_\beta\mu] \cap Y \subset \mathcal{T} \cap Y = Y^+$, and thus $\mu \in Y^+ \implies \lambda \in Y^+$. $\square$

We directly obtain from Lemma 1.7 the following result.

**Proposition 1.8.** The affinized Bruhat order defined on $W_+$ coincides with the restriction of the preorder defined through (1.9) on the whole semidirect product $W^a$.

### 1.3.1 Properties of the height function

We give here a few elementary results on the height function, which will be useful in our study of the affinized Bruhat length. They are also used in [MO19, Section 3].

**Proposition 1.9.** For any $w \in W$,

$$ \rho - w^{-1}(\rho) = \sum_{\gamma \in \operatorname{Inv}(w)} \gamma. \qquad (1.25) $$

*Proof.* This is [Kum02, 1.3.22 Corollary 3], we prove it by induction on the length of $w$.

1. If $w$ is a simple reflection $s_\alpha$ then $\operatorname{Inv}(s_\alpha) = \{\alpha\}$ and $\rho - s_\alpha(\rho) = \langle\alpha^\vee, \rho\rangle\alpha = \alpha$ since $\langle\alpha^\vee, \rho\rangle = 1$ by definition of $\rho$.
## 1.3 Preliminary results

Since the affinized Bruhat order is generated on $W_+$ by the relations $s_{\beta[n]}\mathbf{x} > \mathbf{x} \iff \ell^a(s_{\beta[n]}\mathbf{x}) > \ell^a(\mathbf{x})$ for affinized roots $\beta[n] \in \Phi_+$, covers are always of this form. In the rest of the paper, we always apply affinized reflections on the left.

**Lemma 1.7.** Let $\pi^\lambda w \in W^a$ and $\beta[n] \in \Phi_+^a$. Write $\pi^\mu w'$ for $s_{\beta[n]}\pi^\lambda w$ and suppose that $(\pi^\lambda w)^{-1}(\beta[n]) \in \Phi_+^a$. Then $\lambda \in [\mu, s_\beta\mu]$. In particular

$$ \mu \in Y^+ \implies \lambda \in Y^+. \qquad (1.24) $$

*Proof.* Explicitly:

$$ \pi^\mu w' = \pi^{n\beta^\vee} s_\beta \cdot \pi^\lambda w = \pi^{s_\beta\lambda + n\beta^\vee} s_\beta w. $$

Thus $\mu = s_\beta\lambda + n\beta^\vee = \lambda + (n - \langle\lambda, \beta\rangle)\beta^\vee$ and $s_\beta\mu = \lambda - n\beta^\vee$.

Moreover, since $(\pi^\lambda w)^{-1}(\beta[n]) \in \Phi_+^a$, by Formula (1.12):

$$ |n| - \operatorname{sgn}(n)\langle\lambda, \beta\rangle = \operatorname{sgn}(n)(n - \langle\lambda, \beta\rangle) \ge 0. $$

Therefore, unless $n - \langle\lambda, \beta\rangle = 0$, $n$ and $n - \langle\lambda, \beta\rangle$ have same sign and thus, $\lambda = s_\beta\mu + n\beta^\vee = \mu - (n - \langle\lambda, \beta\rangle)\beta^\vee$ lies in $[s_\beta\mu, \mu]$. If $n - \langle\lambda, \beta\rangle = 0$ then $\mu = \lambda$ and the result remains true.

The Tits cone $\mathcal{T}$ is convex ([Kum02, Proposition 1.4.2c]) and $W$-stable, so if $\mu \in \mathcal{T}$, then $[\mu, s_\beta\mu]$ is contained in $\mathcal{T}$ for any $\beta \in \Phi$. Therefore in the situation above, if $\mu \in Y^+ = \mathcal{T} \cap Y$, then $\lambda \in [\mu, s_\beta\mu] \cap Y \subset \mathcal{T} \cap Y = Y^+$, and thus $\mu \in Y^+ \implies \lambda \in Y^+$. $\square$

We directly obtain from Lemma 1.7 the following result.

**Proposition 1.8.** The affinized Bruhat order defined on $W_+$ coincides with the restriction of the preorder defined through (1.9) on the whole semidirect product $W^a$.

### 1.3.1 Properties of the height function

We give here a few elementary results on the height function, which will be useful in our study of the affinized Bruhat length. They are also used in [MO19, Section 3].

**Proposition 1.9.** For any $w \in W$,

$$ \rho - w^{-1}(\rho) = \sum_{\gamma \in \mathrm{Inv}(w)} \gamma. \qquad (1.25) $$

*Proof.* This is [Kum02, 1.3.22 Corollary 3], we prove it by induction on the length of $w$.

1. If $w$ is a simple reflection $s_\alpha$ then $\mathrm{Inv}(s_\alpha) = \{\alpha\}$ and $\rho - s_\alpha(\rho) = \langle\alpha^\vee, \rho\rangle\alpha = \alpha$ since $\langle\alpha^\vee, \rho\rangle = 1$ by definition of $\rho$.
2. Suppose the result true for elements of length $n$, and suppose that $\ell(w) = n+1$ then write $w = w_1 s_\alpha$ for $\alpha$ a simple root and $w_1$ an element of length $n$. Then

$$ \rho - w(\rho) = \rho - w_1(\rho) + w_1(\rho - s_\alpha(\rho)) = \sum_{\gamma \in \mathrm{Inv}(w_1^{-1})} \gamma + w_1(\alpha) $$

and since $\mathrm{Inv}(w^{-1}) = \mathrm{Inv}(w_1^{-1}) \sqcup \{w_1(\alpha)\}$ we get the result for $w$.

□

**Corollary 1.10.** For any positive root $\beta \in \Phi_+$ we have

$$ 2\mathrm{ht}(\beta^{\vee}) = \sum_{\gamma \in \mathrm{Inv}(s_{\beta})} \langle \beta^{\vee}, \gamma \rangle. \qquad (1.26) $$

Moreover all the terms in the sum are positive.

*Proof.* Let $\beta \in \Phi_+$ be a positive root. Note that $-s_\beta(\beta^\vee) = \beta^\vee$ and thus $\langle \beta^\vee, \rho \rangle = \langle -s_\beta(\beta^\vee), \rho \rangle = \langle \beta^\vee, -s_\beta(\rho) \rangle$. Therefore by Proposition 1.9,

$$ 2\mathrm{ht}(\beta^{\vee}) = 2\langle \beta^{\vee}, \rho \rangle = \langle \beta^{\vee}, \rho - s_{\beta}(\rho) \rangle = \sum_{\gamma \in \mathrm{Inv}(s_{\beta})} \langle \beta^{\vee}, \gamma \rangle. $$

Moreover for any $\gamma \in \mathrm{Inv}(s_\beta)$, by definition $\gamma \in \Phi_+$ and $s_\beta(\gamma) = \gamma - \langle \beta^\vee, \gamma \rangle \beta^\vee \in \Phi_-$ so, since $\beta$ is a positive root, the coefficient $\langle \beta^\vee, \gamma \rangle$ is necessarily positive. □

**Corollary 1.11.** Let $\mu \in Y^+$ and $u \in W$ be such that $\mu = u(\mu^{++})$. Then

$$ \mathrm{ht}(\mu^{++}) = \mathrm{ht}(\mu) - \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \mu, \tau \rangle. \qquad (1.27) $$

Moreover the terms in this sum are non-positive and

$$ \mathrm{ht}(\mu) \le \mathrm{ht}(\mu^{++}). \qquad (1.28) $$

The inequality is strict unless $\mu$ is dominant.

*Proof.* By definition $\mathrm{ht}(\mu^{++}) = \langle u^{-1}(\mu), \rho \rangle = \langle \mu, u(\rho) \rangle$, and by Proposition 1.9:

$$ \mathrm{ht}(\mu^{++}) = \langle \mu, u(\rho) \rangle = \langle \mu, \rho - \sum_{\tau \in \mathrm{Inv}(u^{-1})} \tau \rangle = \mathrm{ht}(\mu) - \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \mu, \tau \rangle. $$

Moreover, for any $\tau \in \Phi$, we have $\langle \mu, \tau \rangle = \langle \mu^{++}, u^{-1}(\tau) \rangle$, so $\tau \in \mathrm{Inv}(u^{-1}) \implies \langle \mu, \tau \rangle \le 0$ and the terms of the above sum are all non-positive, we deduce Formula (1.28). If $\mu$ is not dominant, then there exists $\tau \in \Phi_+$ such that $\langle \mu, \tau \rangle < 0$, and thus

$$ \mathrm{ht}(\mu) < \mathrm{ht}(s_{\tau}\mu) \le \mathrm{ht}(\mu^{++}). $$

□
2. Suppose the result true for elements of length $n$, and suppose that $\ell(w) = n+1$ then write $w = w_1 s_\alpha$ for $\alpha$ a simple root and $w_1$ an element of length $n$. Then

$$ \rho - w(\rho) = \rho - w_1(\rho) + w_1(\rho - s_\alpha(\rho)) = \sum_{\gamma \in \text{Inv}(w_1^{-1})} \gamma + w_1(\alpha) $$

and since $\text{Inv}(w^{-1}) = \text{Inv}(w_1^{-1}) \sqcup \{w_1(\alpha)\}$ we get the result for $w$.

□

**Corollary 1.10.** For any positive root $\beta \in \Phi_+$ we have

$$ 2\text{ht}(\beta^{\vee}) = \sum_{\gamma \in \text{Inv}(s_{\beta})} \langle \beta^{\vee}, \gamma \rangle. \qquad (1.26) $$

*Moreover all the terms in the sum are positive.*

*Proof.* Let $\beta \in \Phi_+$ be a positive root. Note that $-s_\beta(\beta^\vee) = \beta^\vee$ and thus $\langle \beta^\vee, \rho \rangle = \langle -s_\beta(\beta^\vee), \rho \rangle = \langle \beta^\vee, -s_\beta(\rho) \rangle$. Therefore by Proposition 1.9,

$$ 2\text{ht}(\beta^{\vee}) = 2\langle \beta^{\vee}, \rho \rangle = \langle \beta^{\vee}, \rho - s_{\beta}(\rho) \rangle = \sum_{\gamma \in \text{Inv}(s_{\beta})} \langle \beta^{\vee}, \gamma \rangle. $$

Moreover for any $\gamma \in \text{Inv}(s_\beta)$, by definition $\gamma \in \Phi_+$ and $s_\beta(\gamma) = \gamma - \langle \beta^\vee, \gamma \rangle \beta^\vee \in \Phi_-$ so, since $\beta$ is a positive root, the coefficient $\langle \beta^\vee, \gamma \rangle$ is necessarily positive. □

**Corollary 1.11.** Let $\mu \in Y^+$ and $u \in W$ be such that $\mu = u(\mu^{++})$. Then

$$ \text{ht}(\mu^{++}) = \text{ht}(\mu) - \sum_{\tau \in \text{Inv}(u^{-1})} \langle \mu, \tau \rangle. \qquad (1.27) $$

*Moreover the terms in this sum are non-positive and*

$$ \text{ht}(\mu) \le \text{ht}(\mu^{++}). \qquad (1.28) $$

*The inequality is strict unless $\mu$ is dominant.*

*Proof.* By definition $\text{ht}(\mu^{++}) = \langle u^{-1}(\mu), \rho \rangle = \langle \mu, u(\rho) \rangle$, and by Proposition 1.9:

$$ \text{ht}(\mu^{++}) = \langle \mu, u(\rho) \rangle = \langle \mu, \rho - \sum_{\tau \in \text{Inv}(u^{-1})} \tau \rangle = \text{ht}(\mu) - \sum_{\tau \in \text{Inv}(u^{-1})} \langle \mu, \tau \rangle. $$

Moreover, for any $\tau \in \Phi$, we have $\langle \mu, \tau \rangle = \langle \mu^{++}, u^{-1}(\tau) \rangle$, so $\tau \in \text{Inv}(u^{-1}) \implies \langle \mu, \tau \rangle \le 0$ and the terms of the above sum are all non-positive, we deduce Formula (1.28). If $\mu$ is not dominant, then there exists $\tau \in \Phi_+$ such that $\langle \mu, \tau \rangle < 0$, and thus

$$ \text{ht}(\mu) < \text{ht}(s_{\tau}\mu) \le \text{ht}(\mu^{++}). $$

□
Amongst other things, Corollary 1.11 directly implies the following result, which was first indicated to the author by Hébert and Muthiah.

**Lemma 1.12.** Let $\lambda \in Y^+$ and $\beta \in \Phi_+$ such that $s_\beta\lambda \neq \lambda$. Suppose that $\mu \in ]\lambda, s_\beta\lambda[$. Then $\mathrm{ht}(\mu^{++}) < \mathrm{ht}(\lambda^{++})$.

*Proof.* Note that we do not suppose $\mu \in Y$, the height function is extended to $V = Y \otimes_Z \mathbb{R}$ linearly. Let $t \in ]0, 1[$ be such that $\mu = t\lambda + (1-t)s_\beta\lambda$, let $v \in W$ be such that $\mu^{++} = v\mu$. Then $\mathrm{ht}(\mu^{++}) = \mathrm{ht}(v\mu) = \mathrm{ht}(v\lambda) + (1-t)\mathrm{ht}(v\beta\lambda)$. By Corollary 1.11, $\mathrm{ht}(v\lambda) \leq \mathrm{ht}(\lambda^{++})$ and $\mathrm{ht}(v\beta\lambda) \leq \mathrm{ht}(\lambda^{++})$ and, since $s_\beta\lambda \neq \lambda$, at least one of the two inequality is strict. We deduce $\mathrm{ht}(\mu^{++}) < \mathrm{ht}(\lambda^{++})$. $\square$

**Proposition 1.13.** Let $\mathbf{x} \in W_+^a$ and $\beta[n] \in \Phi_+^a$ such that $s_{\beta[n]}\mathbf{x} \in W_+^a$. Then

$$ \mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{x}) \iff n \in \{0, \langle \mathrm{proj}^{Y^{+}}(\mathbf{x}), \beta \rangle\}. \quad (1.29) $$

*Proof.* To simplify notation, let $\lambda \in Y^+$ denote $\mathrm{proj}^{Y^+}(\mathbf{x})$. If $n \in \{0, \langle \lambda, \beta \rangle\}$ then by Formula (1.11), $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}) \in \{s_\beta(\lambda), \lambda\}$ and therefore it has same dominance class.

Conversely, if $n \in ]0, \langle \lambda, \beta \rangle[$ then $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}) = s_\beta(\lambda) + n\beta^\vee \in ]\lambda, s_\beta(\lambda)[$, and if $n \notin ]0, \langle \lambda, \beta \rangle[$ then $\lambda \in ]s_\beta(\lambda) + n\beta^\vee, \lambda - n\beta^\vee[=$] $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x})$, $s_\beta(\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}))[$.

Either way by Lemma 1.12, $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x})) \neq \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x}))$ and in particular $\mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x}) \neq \mathrm{proj}^{Y^{++}}(\mathbf{x})$. $\square$

**Remark 1.14.** Note that if $n = \langle \lambda, \beta \rangle$, then by Formula (1.11), $s_{\beta[n]}\pi^\lambda w = \pi^\lambda w_{w^{-1}(\beta)}$. Therefore Proposition 1.13 indicates that, if $\mathbf{y} = s_{\beta[n]}\mathbf{x}$, then $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$ if and only if $\mathbf{y}$ is obtained from $\mathbf{x}$ by applying a vectorial reflection either on the left hand side or on the right hand side. This justifies the terminology for vectorial covers and properly affine covers.

**Proposition 1.15.** Let $\mathbf{x}, \mathbf{y} \in W_+^a$ and suppose that $\mathbf{x} \leq \mathbf{y}$. Then

$$ \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) \leq \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y})), \quad (1.30) $$

with equality if and only if $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$.

In particular the function $\mathrm{ht} \circ \mathrm{proj}^{Y^{++}}: W_+^a \to \mathbb{Z}$ is order-preserving.

*Proof.* It is enough to prove it for cover relations, if $\mathbf{y} = s_{\beta[n]}\mathbf{x}$ for some $\beta[n] \in \Phi_+^a$. In that case, by Lemma 1.7 we have $\mathrm{proj}^{Y^+}(\mathbf{x}) \in [\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y}))]$. If $\mathrm{proj}^{Y^+}(\mathbf{x}) \in \{\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta\mathrm{proj}^{Y^+}(\mathbf{y})\}$ then they have the same dominance class: $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$ and we obtain the equality case.

Otherwise, $\mathrm{proj}^{Y^+}(\mathbf{x}) \in ]\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y}))[$, necessarily $s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y})) \neq \mathrm{proj}^{Y^+}(\mathbf{y})$ and by Lemma 1.12 we deduce $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) < \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y}))$. $\square$

**Corollary 1.16.** For any $\lambda^{++} \in Y^{++}$, the set $\{\mathbf{x} \in W_+^a | \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\}$ is convex for the affinized Bruhat order.
Amongst other things, Corollary 1.11 directly implies the following result, which was first indicated to the author by Hébert and Muthiah.

**Lemma 1.12.** Let $\lambda \in Y^+$ and $\beta \in \Phi_+$ such that $s_\beta\lambda \neq \lambda$. Suppose that $\mu \in ]\lambda, s_\beta\lambda[$. Then $\mathrm{ht}(\mu^{++}) < \mathrm{ht}(\lambda^{++})$.

*Proof.* Note that we do not suppose $\mu \in Y$, the height function is extended to $V = Y \otimes_Z \mathbb{R}$ linearly. Let $t \in ]0, 1[$ be such that $\mu = t\lambda + (1-t)s_\beta\lambda$, let $v \in W$ be such that $\mu^{++} = v\mu$. Then $\mathrm{ht}(\mu^{++}) = \mathrm{ht}(v\mu) = \mathrm{ht}(v\lambda) + (1-t)\mathrm{ht}(v\beta\lambda)$. By Corollary 1.11, $\mathrm{ht}(v\lambda) \leq \mathrm{ht}(\lambda^{++})$ and $\mathrm{ht}(v\beta\lambda) \leq \mathrm{ht}(\lambda^{++})$ and, since $s_\beta\lambda \neq \lambda$, at least one of the two inequality is strict. We deduce $\mathrm{ht}(\mu^{++}) < \mathrm{ht}(\lambda^{++})$. $\square$

**Proposition 1.13.** Let $\mathbf{x} \in W_+^a$ and $\beta[n] \in \Phi_+^a$ such that $s_{\beta[n]}\mathbf{x} \in W_+^a$. Then

$$ \mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{x}) \iff n \in \{0, \langle \mathrm{proj}^{Y^{+}}(\mathbf{x}), \beta \rangle\}. \quad (1.29) $$

*Proof.* To simplify notation, let $\lambda \in Y^+$ denote $\mathrm{proj}^{Y^+}(\mathbf{x})$. If $n \in \{0, \langle \lambda, \beta \rangle\}$ then by Formula (1.11), $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}) \in \{s_\beta(\lambda), \lambda\}$ and therefore it has same dominance class.

Conversely, if $n \in ]0, \langle \lambda, \beta \rangle[$ then $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}) = s_\beta(\lambda) + n\beta^\vee \in ]\lambda, s_\beta(\lambda)[$, and if $n \notin ]0, \langle \lambda, \beta \rangle[$ then $\lambda \in ]s_\beta(\lambda) + n\beta^\vee, \lambda - n\beta^\vee[=$] $\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x})$, $s_\beta(\mathrm{proj}^{Y^+}(s_{\beta[n]}\mathbf{x}))[$.

Either way by Lemma 1.12, $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x})) \neq \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x}))$ and in particular $\mathrm{proj}^{Y^{++}}(s_{\beta[n]}\mathbf{x}) \neq \mathrm{proj}^{Y^{++}}(\mathbf{x})$. $\square$

**Remark 1.14.** Note that if $n = \langle \lambda, \beta \rangle$, then by Formula (1.11), $s_{\beta[n]}\pi^\lambda w = \pi^\lambda w_{w^{-1}(\beta)}$. Therefore Proposition 1.13 indicates that, if $\mathbf{y} = s_{\beta[n]}\mathbf{x}$, then $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$ if and only if $\mathbf{y}$ is obtained from $\mathbf{x}$ by applying a vectorial reflection either on the left hand side or on the right hand side. This justifies the terminology for vectorial covers and properly affine covers.

**Proposition 1.15.** Let $\mathbf{x}, \mathbf{y} \in W_+^a$ and suppose that $\mathbf{x} \leq \mathbf{y}$. Then

$$ \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) \leq \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y})), \quad (1.30) $$

with equality if and only if $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$.

In particular the function $\mathrm{ht} \circ \mathrm{proj}^{Y^{++}}: W_+^a \to \mathbb{Z}$ is order-preserving.

*Proof.* It is enough to prove it for cover relations, if $\mathbf{y} = s_{\beta[n]}\mathbf{x}$ for some $\beta[n] \in \Phi_+^a$. In that case, by Lemma 1.7 we have $\mathrm{proj}^{Y^+}(\mathbf{x}) \in [\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y}))]$. If $\mathrm{proj}^{Y^+}(\mathbf{x}) \in \{\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta\mathrm{proj}^{Y^+}(\mathbf{y})\}$ then they have the same dominance class: $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$ and we obtain the equality case.

Otherwise, $\mathrm{proj}^{Y^+}(\mathbf{x}) \in ]\mathrm{proj}^{Y^+}(\mathbf{y}), s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y}))[$, necessarily $s_\beta(\mathrm{proj}^{Y^+}(\mathbf{y})) \neq \mathrm{proj}^{Y^+}(\mathbf{y})$ and by Lemma 1.12 we deduce $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) < \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y}))$. $\square$

**Corollary 1.16.** For any $\lambda^{++} \in Y^{++}$, the set $\{\mathbf{x} \in W_+^a \mid \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\}$ is convex for the affinized Bruhat order.
**Proof.** By Proposition 1.15 the function $W_+^a \to \mathbb{Z}: \mathbf{x} \mapsto \mathrm{ht} \circ \mathrm{proj}^{Y^{++}}$ is compatible with the affinized Bruhat order. Suppose that $\mathbf{x}, \mathbf{y} \in W_+^a$ satisfy $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$ and $\mathbf{x} \le \mathbf{y}$. Let $\mathbf{z} \in W_+^a$ be such that $\mathbf{x} \le \mathbf{z} \le \mathbf{y}$. Then by Proposition 1.15, $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) \le \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{z})) \le \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y})) = \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x}))$. By the equality case in Proposition 1.15, we deduce $\mathrm{proj}^{Y^{++}}(\mathbf{z}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$. $\square$

**Remark 1.17.** Note that, for $\lambda \in Y^+$, the set $\{\mathbf{x} \in W_+^a | \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\}$ is the double W-orbit of $\pi^\lambda$:

$$ \{\mathbf{x} \in W_+^a | \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\} = W\pi^\lambda W. \qquad (1.31) $$

We show in Section 2 that the right W-orbits $\pi^\lambda W$ are also convex for the affinized Bruhat order.

We end this section with several metric properties of Coxeter groups, the results stated are proved in the context of Coxeter complexes and buildings in [Ron89].

### 1.3.2 Metric properties of Coxeter groups

On any Coxeter group $(W_0, S_0)$ we define a map $d: W_0 \times W_0 \to W_0$ by $d(v, w) = v^{-1}w$, called the **vectorial distance** of $W_0$, it is $W_0$-invariant: $d(uv, uw) = d(v, w)$ for any $u, v, w \in W_0$. We also define $d^N = \ell \circ d$ where $\ell$ is the Bruhat length on $(W_0, S_0)$ (note that $\ell$ and $d^N$ depend on the set of simple reflections $S_0$, but the vectorial distance does not). These maps have properties analogous to the standard distance axioms, which justify the name (see [Ron89, Chapter 3 §1]).

An unfolded gallery (resp. a gallery) in $W_0$ from $w$ to $v$ is a sequence $w = w_0, \dots, w_n = v$ such that $d^N(w_i, w_{i+1}) = 1$ (resp. $d^N(w_i, w_{i+1}) \in \{0, 1\}$) for all $i \in [0, n-1]$. A gallery is said to be minimal if its length $n$ is equal to $d^N(w_1, w_n)$, a minimal gallery is necessarily unfolded. We refer to [Ron89, Chapter 2] for properties of minimal galleries, but note that if $(w_0, \dots, w_n)$ is a minimal gallery then $d^N(w_0, w_i) = i$ and thus $(w_0, \dots, w_i)$ is a minimal gallery from $w_0$ to $w_i$. Moreover since the distance is $W_0$ invariant, $(vw_0, \dots, vw_n)$ is also a minimal gallery for any $v \in W_0$. The next Lemma is a reformulation of [Ron89, Proposition 2.8].

**Lemma 1.18.** Let $(W_0, S_0)$ be a Coxeter system and let $v_1, v_2, w \in W_0$ be such that $v_2$ is not on a minimal gallery from $v_1$ to $w$. Then there is a reflection $r \in W_0$ such that $d(v_1, rw) > d(v_1, w)$ and $d(v_2, rw) < d(v_2, w)$.

**Proof.** If $v_2$ is not on a minimal gallery from $v_1$ to $w$, by [Ron89, Proposition 2.8] there exists a root $\alpha$ (seen as a half-apartment: $\alpha = \{u \in W_0 | \ell(u) < \ell(s_\alpha u)\}$) such that $v_1, w \in \alpha$ and $v_2 \notin \alpha$. Then consider the folding along $\alpha$, defined by:

$$ \forall u \in W_0, \rho_\alpha(u) = \begin{cases} s_\alpha u & \text{if } u \notin \alpha \\ u & \text{otherwise.} \end{cases} $$
*Proof.* By Proposition 1.15 the function $W_+^a \to \mathbb{Z}: \mathbf{x} \mapsto \mathrm{ht} \circ \mathrm{proj}^{Y^{++}}$ is compatible with the affinized Bruhat order. Suppose that $\mathbf{x}, \mathbf{y} \in W_+^a$ satisfy $\mathrm{proj}^{Y^{++}}(\mathbf{x}) = \mathrm{proj}^{Y^{++}}(\mathbf{y})$ and $\mathbf{x} \le \mathbf{y}$. Let $\mathbf{z} \in W_+^a$ be such that $\mathbf{x} \le \mathbf{z} \le \mathbf{y}$. Then by Proposition 1.15, $\mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x})) \le \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{z})) \le \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{y})) = \mathrm{ht}(\mathrm{proj}^{Y^{++}}(\mathbf{x}))$. By the equality case in Proposition 1.15, we deduce $\mathrm{proj}^{Y^{++}}(\mathbf{z}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$. $\square$

**Remark 1.17.** Note that, for $\lambda \in Y^+$, the set $\{\mathbf{x} \in W_+^a | \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\}$ is the double W-orbit of $\pi^\lambda$:

$$ \{\mathbf{x} \in W_+^a | \mathrm{proj}^{Y^{++}}(\mathbf{x}) = \lambda^{++}\} = W\pi^\lambda W. \qquad (1.31) $$

We show in Section 2 that the right W-orbits $\pi^\lambda W$ are also convex for the affinized Bruhat order.

We end this section with several metric properties of Coxeter groups, the results stated are proved in the context of Coxeter complexes and buildings in [Ron89].

### 1.3.2 Metric properties of Coxeter groups

On any Coxeter group $(W_0, S_0)$ we define a map $d: W_0 \times W_0 \to W_0$ by $d(v, w) = v^{-1}w$, called the **vectorial distance** of $W_0$, it is $W_0$-invariant: $d(uv, uw) = d(v, w)$ for any $u, v, w \in W_0$. We also define $d^N = \ell \circ d$ where $\ell$ is the Bruhat length on $(W_0, S_0)$ (note that $\ell$ and $d^N$ depend on the set of simple reflections $S_0$, but the vectorial distance does not). These maps have properties analogous to the standard distance axioms, which justify the name (see [Ron89, Chapter 3 §1]).

An unfolded gallery (resp. a gallery) in $W_0$ from $w$ to $v$ is a sequence $w = w_0, \dots, w_n = v$ such that $d^N(w_i, w_{i+1}) = 1$ (resp. $d^N(w_i, w_{i+1}) \in \{0, 1\}$) for all $i \in [0, n-1]$. A gallery is said to be minimal if its length $n$ is equal to $d^N(w_1, w_n)$, a minimal gallery is necessarily unfolded. We refer to [Ron89, Chapter 2] for properties of minimal galleries, but note that if $(w_0, \dots, w_n)$ is a minimal gallery then $d^N(w_0, w_i) = i$ and thus $(w_0, \dots, w_i)$ is a minimal gallery from $w_0$ to $w_i$. Moreover since the distance is $W_0$ invariant, $(vw_0, \dots, vw_n)$ is also a minimal gallery for any $v \in W_0$. The next Lemma is a reformulation of [Ron89, Proposition 2.8].

**Lemma 1.18.** Let $(W_0, S_0)$ be a Coxeter system and let $v_1, v_2, w \in W_0$ be such that $v_2$ is not on a minimal gallery from $v_1$ to $w$. Then there is a reflection $r \in W_0$ such that $d(v_1, rw) > d(v_1, w)$ and $d(v_2, rw) < d(v_2, w)$.

*Proof.* If $v_2$ is not on a minimal gallery from $v_1$ to $w$, by [Ron89, Proposition 2.8] there exists a root $\alpha$ (seen as a half-apartment: $\alpha = \{u \in W_0 | \ell(u) < \ell(s_\alpha u)\}$) such that $v_1, w \in \alpha$ and $v_2 \notin \alpha$. Then consider the folding along $\alpha$, defined by:

$$ \forall u \in W_0, \rho_\alpha(u) = \begin{cases} s_\alpha u & \text{if } u \notin \alpha \\ u & \text{otherwise.} \end{cases} $$
It reduces the vectorial distance (see [Ron89, §2]) hence

$$d(v_1, w) = d(\rho_\alpha(v_1), \rho_\alpha(s_\alpha w)) < d(v_1, s_\alpha w), \\ d(v_2, s_\alpha w) = d(s_\alpha v_2, w) = d(\rho_\alpha(v_2), \rho_\alpha(w)) < d(v_2, w).$$

□

Recall that for $J \subset S$, $W_J$ is the subgroup generated by the set of simple reflections $J$. The Coxeter system $(W_J, J)$ is an example of Coxeter system for which we will use Lemma 1.18. For any $w \in W$, the coset $wW_J$ is convex, in the sense that, if $w_1, w_2 \in wW_J$ then any minimal gallery from $w_1$ to $w_2$ lies in $wW_J$ (see [Ron89, Lemma 2.10]).

**Definition 1.19.** For any $J \subset S$ and $v, w \in W$, the projection of $w$ on $wW_J$ is the unique element of $wW_J$ which reaches $\min_{\tilde{v} \in vW_J} d^N(w, \tilde{v})$. It is denoted $\text{proj}_{vW_J}(w)$. Moreover any minimal gallery from $v$ to an element of $wW_J$ goes through $\text{proj}_{vW_J}(w)$ (see [Ron89, Theorem 2.10]).

## 2 Restriction to constant dominance classes

In this section, we study the affinized Bruhat order restricted to a dominance class, that is to say, for a given $\lambda^{++} \in Y^{++}$, we study the restriction of the affinized Bruhat order to the subset $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++}) = W\pi^{\lambda^{++}}W$. By Corollary 1.16 these are convex subsets for the affinized Bruhat order. We start by showing that, for any $\lambda \in Y^+$, the subset $\pi^\lambda W = (\text{proj}^{Y^+})^{-1}(\lambda)$ of $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++})$ is also convex for the affinized Bruhat order.

**Lemma 2.1.** Let $\lambda \in Y^+$, recall Notation 1.2 for $v^\lambda$. Then $\text{Inv}((v^\lambda)^{-1}) \cap \Phi_\lambda = \emptyset$. In particular for any $\beta \in \Phi_+$,

$$\text{ht}(\lambda) < \text{ht}(s_\beta \lambda) \iff \langle \lambda, \beta \rangle < 0 \iff s_\beta v^\lambda < v^\lambda. \qquad (2.1)$$

*Proof.* Let $\lambda \in Y^+$ and $\alpha \in \text{Inv}((v^\lambda)^{-1}) \cap \Phi_\lambda$, then since $\alpha \in \Phi_\lambda$, $s_\alpha$ fixes $\lambda$, that is $s_\alpha \in W_\lambda$. Moreover $(v^\lambda)^{-1}(\alpha) < 0$ so $s_\alpha v^\lambda < v^\lambda$, this contradicts the minimality of $v^\lambda$ (note that, as $W_\lambda v^\lambda = v^\lambda W_{\lambda^{++}}, v^\lambda$ is also the minimal representative for the right coset $W_\lambda v^\lambda$). Hence $\text{Inv}((v^\lambda)^{-1}) \cap \Phi_\lambda = \emptyset$, therefore any $\beta \in \text{Inv}((v^\lambda)^{-1})$ verifies $\langle \lambda, \beta \rangle \neq 0$.

For $\beta \in \Phi_+$, $\langle \lambda, \beta \rangle = \langle \lambda^{++}, (v^\lambda)^{-1}(\beta) \rangle$. Since $\lambda^{++}$ is dominant, if this is negative then $\beta \in \text{Inv}((v^\lambda)^{-1})$, and since $\text{Inv}((v^\lambda)^{-1})$ and $\Phi_\lambda$ are disjoint, the converse is also true. Since $\beta \in \text{Inv}((v^\lambda)^{-1}) \iff s_\beta v^\lambda < v^\lambda$ we deduce the second equivalence in Formula (2.1). Moreover $\text{ht}(s_\beta \lambda) = \text{ht}(\lambda) - \langle \lambda, \beta \rangle \text{ht}(\beta^\vee)$ by linearity of the height function, and since $\text{ht}(\beta^\vee) > 0$, the first equivalence in Formula (2.1) is clear. □

**Remark 2.2.** The fact that $\text{Inv}((v^\lambda)^{-1}) \cap \Phi_\lambda = \emptyset$ is visible geometrically in the Coxeter complex of $W$, in which $\Phi_\lambda$ is the set of walls containing $\lambda$, $\text{Inv}(v^{-1})$ is the set of walls separating $C_f^v$ and $C_v^v$. The chamber $C_{v^\lambda}^v$ is the closest chamber from the fundamental chamber amongst the chambers containing $\lambda$ in their closure, in other words $v^\lambda = \text{proj}_{W_\lambda}(1_W)$.
It reduces the vectorial distance (see [Ron89, §2]) hence

$$d(v_1, w) = d(\rho_\alpha(v_1), \rho_\alpha(s_\alpha w)) < d(v_1, s_\alpha w), \\ d(v_2, s_\alpha w) = d(s_\alpha v_2, w) = d(\rho_\alpha(v_2), \rho_\alpha(w)) < d(v_2, w).$$

□

Recall that for $J \subset S$, $W_J$ is the subgroup generated by the set of simple reflections $J$. The Coxeter system $(W_J, J)$ is an example of Coxeter system for which we will use Lemma 1.18. For any $w \in W$, the coset $wW_J$ is convex, in the sense that, if $w_1, w_2 \in wW_J$ then any minimal gallery from $w_1$ to $w_2$ lies in $wW_J$ (see [Ron89, Lemma 2.10]).

**Definition 1.19.** For any $J \subset S$ and $v, w \in W$, the projection of $w$ on $wW_J$ is the unique element of $wW_J$ which reaches $\min_{\tilde{v} \in vW_J} d^N(w, \tilde{v})$. It is denoted $\text{proj}_{vW_J}(w)$. Moreover any minimal gallery from $v$ to an element of $wW_J$ goes through $\text{proj}_{vW_J}(w)$ (see [Ron89, Theorem 2.10]).

## 2 Restriction to constant dominance classes

In this section, we study the affinized Bruhat order restricted to a dominance class, that is to say, for a given $\lambda^{++} \in Y^{++}$, we study the restriction of the affinized Bruhat order to the subset $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++}) = W\pi^{\lambda^{++}}W$. By Corollary 1.16 these are convex subsets for the affinized Bruhat order. We start by showing that, for any $\lambda \in Y^{+}$, the subset $\pi^{\lambda}W = (\text{proj}^{Y^{+}})^{-1}(\lambda)$ of $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++})$ is also convex for the affinized Bruhat order.

**Lemma 2.1.** Let $\lambda \in Y^{+}$, recall Notation 1.2 for $v^{\lambda}$. Then $\text{Inv}((v^{\lambda})^{-1}) \cap \Phi_{\lambda} = \emptyset$. In particular for any $\beta \in \Phi_{+}$,

$$\text{ht}(\lambda) < \text{ht}(s_{\beta}\lambda) \iff \langle\lambda, \beta\rangle < 0 \iff s_{\beta}v^{\lambda} < v^{\lambda}. \qquad (2.1)$$

*Proof.* Let $\lambda \in Y^{+}$ and $\alpha \in \text{Inv}((v^{\lambda})^{-1}) \cap \Phi_{\lambda}$, then since $\alpha \in \Phi_{\lambda}$, $s_{\alpha}$ fixes $\lambda$, that is $s_{\alpha} \in W_{\lambda}$. Moreover $(v^{\lambda})^{-1}(\alpha) < 0$ so $s_{\alpha}v^{\lambda} < v^{\lambda}$, this contradicts the minimality of $v^{\lambda}$ (note that, as $W_{\lambda}v^{\lambda} = v^{\lambda}W_{\lambda^{++}}$, $v^{\lambda}$ is also the minimal representative for the right coset $W_{\lambda}v^{\lambda}$). Hence $\text{Inv}((v^{\lambda})^{-1}) \cap \Phi_{\lambda} = \emptyset$, therefore any $\beta \in \text{Inv}((v^{\lambda})^{-1})$ verifies $\langle\lambda, \beta\rangle \neq 0$.

For $\beta \in \Phi_{+}$, $\langle\lambda, \beta\rangle = \langle\lambda^{++}, (v^{\lambda})^{-1}(\beta)\rangle$. Since $\lambda^{++}$ is dominant, if this is negative then $\beta \in \text{Inv}((v^{\lambda})^{-1})$, and since $\text{Inv}((v^{\lambda})^{-1})$ and $\Phi_{\lambda}$ are disjoint, the converse is also true. Since $\beta \in \text{Inv}((v^{\lambda})^{-1}) \iff s_{\beta}v^{\lambda} < v^{\lambda}$ we deduce the second equivalence in Formula (2.1). Moreover $\text{ht}(s_{\beta}\lambda) = \text{ht}(\lambda) - \langle\lambda, \beta\rangle\text{ht}(\beta^{\vee})$ by linearity of the height function, and since $\text{ht}(\beta^{\vee}) > 0$, the first equivalence in Formula (2.1) is clear. □

**Remark 2.2.** The fact that $\text{Inv}((v^{\lambda})^{-1}) \cap \Phi_{\lambda} = \emptyset$ is visible geometrically in the Coxeter complex of $W$, in which $\Phi_{\lambda}$ is the set of walls containing $\lambda$, $\text{Inv}(v^{-1})$ is the set of walls separating $C_{f}^{v}$ and $C_{v}^{v}$. The chamber $C_{v^{\lambda}}^{v}$ is the closest chamber from the fundamental chamber amongst the chambers containing $\lambda$ in their closure, in other words $v^{\lambda} = \text{proj}_{W_{\lambda}}(1_{W})$.
**Proposition 2.3.** Suppose that $\pi^{\lambda}w \in W_{+}^{a}$ and $r \in W$ is a reflection such that $r\lambda \neq \lambda$. Then

$$ \pi^{r\lambda}rw > \pi^{\lambda}w \iff rv^{\lambda} < v^{\lambda}. \qquad (2.2) $$

Moreover for any $\lambda^{++} \in Y^{++}$, the restriction of the function $\mathrm{ht} \circ \mathrm{proj}^{Y^{+}}$ to $(\mathrm{proj}^{Y^{++}})^{-1}(\lambda^{++})$ is order-preserving.

**Proof.** Suppose that $r \in W$ is a reflection which does not fix $\lambda$. By definition there exists a positive root $\beta \in \Phi_{+}$ such that $r = s_{\beta}$ and, since $r$ does not fix $\lambda$, $\langle \lambda, \beta \rangle \neq 0$. Note that $\pi^{r\lambda}rw = s_{\beta[0]}\pi^{\lambda}w$ so, using Formula (1.12), we have

$$ \pi^{r\lambda}rw > \pi^{\lambda}w \iff -\langle \lambda, \beta \rangle > 0 \iff \langle \lambda, \beta \rangle < 0. $$

By Lemma 2.1 this is equivalent to $rv^{\lambda} < v^{\lambda}$, and to $\mathrm{ht}(\lambda) < \mathrm{ht}(r\lambda)$. This is enough to obtain Equivalence (2.2). Moreover by convexity of $(\mathrm{proj}^{Y^{++}})^{-1}(\lambda^{++})$ (see Corollary 1.16) and by Proposition 1.13 it also implies that $\mathrm{ht} \circ \mathrm{proj}^{Y^{+}} : (\mathrm{proj}^{Y^{++}})^{-1}(\lambda^{++}) \to \mathbb{Z}$ is order-preserving. $\square$

Note that the function $\mathrm{ht} \circ \mathrm{proj}^{Y^{+}}$ is not order-preserving on the whole semi-group $W_{+}^{a}$. For example suppose that $\lambda \in Y^{++}$ and $\beta \in \Phi_{+}$ are such that $\lambda + \beta^{\vee}$ is also dominant. Then we can check that $\pi^{s_{\beta}(\lambda)} < \pi^{s_{\beta}(\lambda)-\beta^{\vee}}s_{\beta}$ whereas $\mathrm{ht}(s_{\beta}(\lambda) - \beta^{\vee}) < \mathrm{ht}(s_{\beta}(\lambda))$. Proposition 2.3 implies convexity of left W-cosets:

**Corollary 2.4.** Let $\lambda \in Y^{+}$, then the set $\pi^{\lambda}W = \{\mathbf{x} \in W_{+}^{a} | \mathrm{proj}^{Y^{+}}(\mathbf{x}) = \lambda\}$ is convex for the affinized Bruhat order.

**Proof.** Let $\mathbf{x}, \mathbf{y} \in \pi^{\lambda}W$ such that $\mathbf{x} < \mathbf{y}$ and let $\mathbf{x} = \mathbf{x}_{0} < \mathbf{x}_{1} < \dots \mathbf{x}_{n} = \mathbf{y}$ be a chain from $\mathbf{x}$ to $\mathbf{y}$, in particular for all $k \in [0, n-1]$, let $\beta_{k}[n_{k}] \in \Phi_{+}^{a}$ be such that $\mathbf{x}_{k+1} = s_{\beta_{k}[n_{k}]\mathbf{x}}$. For $k \in [0, n]$, write $\mathbf{x}_{k} = \pi^{\lambda_{k}}w_{k}$ with $\lambda_{k} \in Y^{+}$, $w_{k} \in W$. By convexity of $(\mathrm{proj}^{Y^{++}})^{-1}(\lambda^{++})$, $\mathrm{proj}^{Y^{++}}$ is constant along the chain, therefore by Proposition 1.13, for all $k \in [0, n-1]$ we have $\lambda_{k+1} \in \{\lambda_{k}, s_{\beta_{k}}(\lambda_{k})\}$. From Proposition 2.3 we deduce that $v^{\lambda_{k+1}} \leq v^{\lambda_{k}}$. Since $\lambda_{0} = \lambda_{n} = \lambda$, $v^{\lambda_{0}} = v^{\lambda_{n}} = v^{\lambda}$ and thus $v^{\lambda_{k}} = v^{\lambda}$, so $\lambda_{k} = \lambda$ for all $k \in [0, n]$. Hence $\pi^{\lambda}W$ is convex. $\square$

## 2.1 Relative length on W

We define a relative length and a relative Bruhat order on W, which naturally arises in the study of the affinized length $\ell^{a}$ on $W_{+}^{a}$. This connection was already observed by Muthiah and Orr in [MO18].

**Definition 2.5.** For any $v, w \in W$ let:

$$ \ell_v(w) = |\mathrm{Inv}(w^{-1}) \setminus \mathrm{Inv}(v^{-1})| - |\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1})|. \qquad (2.3) $$

This is a signed version of the Bruhat length, in particular $\ell_{1} = \ell$.
**Proposition 2.3.** Suppose that $\pi^{\lambda}w \in W_{+}^{a}$ and $r \in W$ is a reflection such that $r\lambda \neq \lambda$. Then

$$ \pi^{r\lambda}rw > \pi^{\lambda}w \iff rv^{\lambda} < v^{\lambda}. \qquad (2.2) $$

Moreover for any $\lambda^{++} \in Y^{++}$, the restriction of the function $ht \circ \text{proj}^{Y^{+}}$ to $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++})$ is order-preserving.

**Proof.** Suppose that $r \in W$ is a reflection which does not fix $\lambda$. By definition there exists a positive root $\beta \in \Phi_{+}$ such that $r = s_{\beta}$ and, since $r$ does not fix $\lambda$, $\langle \lambda, \beta \rangle \neq 0$. Note that $\pi^{r\lambda}rw = s_{\beta[0]}\pi^{\lambda}w$ so, using Formula (1.12), we have

$$ \pi^{r\lambda}rw > \pi^{\lambda}w \iff -\langle \lambda, \beta \rangle > 0 \iff \langle \lambda, \beta \rangle < 0. $$

By Lemma 2.1 this is equivalent to $rv^{\lambda} < v^{\lambda}$, and to $ht(\lambda) < ht(r\lambda)$. This is enough to obtain Equivalence (2.2). Moreover by convexity of $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++})$ (see Corollary 1.16) and by Proposition 1.13 it also implies that $ht \circ \text{proj}^{Y^{+}} : (\text{proj}^{Y^{++}})^{-1}(\lambda^{++}) \to \mathbb{Z}$ is order-preserving. $\square$

Note that the function $ht \circ \text{proj}^{Y^{+}}$ is not order-preserving on the whole semi-group $W_{+}^{a}$. For example suppose that $\lambda \in Y^{++}$ and $\beta \in \Phi_{+}$ are such that $\lambda + \beta^{\vee}$ is also dominant. Then we can check that $\pi^{s_{\beta}(\lambda)} < \pi^{s_{\beta}(\lambda)-\beta^{\vee}}s_{\beta}$ whereas $ht(s_{\beta}(\lambda) - \beta^{\vee}) < ht(s_{\beta}(\lambda))$. Proposition 2.3 implies convexity of left W-cosets:

**Corollary 2.4.** Let $\lambda \in Y^{+}$, then the set $\pi^{\lambda}W = \{\mathbf{x} \in W_{+}^{a} | \text{proj}^{Y^{+}}(\mathbf{x}) = \lambda\}$ is convex for the affinized Bruhat order.

**Proof.** Let $\mathbf{x}, \mathbf{y} \in \pi^{\lambda}W$ such that $\mathbf{x} < \mathbf{y}$ and let $\mathbf{x} = \mathbf{x}_{0} < \mathbf{x}_{1} < \dots \mathbf{x}_{n} = \mathbf{y}$ be a chain from $\mathbf{x}$ to $\mathbf{y}$, in particular for all $k \in [0, n-1]$, let $\beta_{k}[n_{k}] \in \Phi_{+}^{a}$ be such that $\mathbf{x}_{k+1} = s_{\beta_{k}[n_{k}]\mathbf{x}}$. For $k \in [0, n]$, write $\mathbf{x}_{k} = \pi^{\lambda_{k}}w_{k}$ with $\lambda_{k} \in Y^{+}$, $w_{k} \in W$. By convexity of $(\text{proj}^{Y^{++}})^{-1}(\lambda^{++})$, $\text{proj}^{Y^{++}}$ is constant along the chain, therefore by Proposition 1.13, for all $k \in [0, n-1]$ we have $\lambda_{k+1} \in \{\lambda_{k}, s_{\beta_{k}}(\lambda_{k})\}$. From Proposition 2.3 we deduce that $v^{\lambda_{k+1}} \leq v^{\lambda_{k}}$. Since $\lambda_{0} = \lambda_{n} = \lambda$, $v^{\lambda_{0}} = v^{\lambda_{n}} = v^{\lambda}$ and thus $v^{\lambda_{k}} = v^{\lambda}$, so $\lambda_{k} = \lambda$ for all $k \in [0, n]$. Hence $\pi^{\lambda}W$ is convex. $\square$

## 2.1 Relative length on W

We define a relative length and a relative Bruhat order on W, which naturally arises in the study of the affinized length $\ell^{a}$ on $W_{+}^{a}$. This connection was already observed by Muthiah and Orr in [MO18].

**Definition 2.5.** For any $v, w \in W$ let:

$$ \ell_v(w) = |\text{Inv}(w^{-1}) \setminus \text{Inv}(v^{-1})| - |\text{Inv}(w^{-1}) \cap \text{Inv}(v^{-1})|. \qquad (2.3) $$

This is a signed version of the Bruhat length, in particular $\ell_{1} = \ell$.
We associate an order to $\ell_v$ by setting, for any element $w \in W$ and any reflection $r \in W$: $w <_v wr$ if and only if $\ell_v(w) < \ell_v(wr)$ and then let $<_v$ be the order generated by these relations, it is strictly compatible with $\ell_v$. In particular $<_1$ is the classical Bruhat order.

As does the Bruhat length, the lengths $\ell_v$ have a geometric interpretation in the Coxeter complex associated to $(W, S)$. For $M$ a wall of the Coxeter complex and $w \in W$, let $\varepsilon_w(M) = -1$ if $M$ separates $C_f^v$ and $C_w^v$, and $\varepsilon_w(M) = +1$ otherwise. Then

$$ \ell_v(w) = \sum_{M \in \varepsilon_w^{-1}(-1)} \varepsilon_{\ell_v(M)}. \qquad (2.4) $$

We will use this relative length to give an alternative definition of the affinized length. Let us first give an explicit formula for $\ell_v$ depending only on the classical length $\ell = \ell_1$.

**Lemma 2.6.** If $sv > v$ with $v \in W$ and $s$ a simple reflection then for any $w \in W$, $\ell_{sv}(w) = \ell_v(sw) - 1$.

*Proof.* For any $w \in W$, the map $\gamma \mapsto s\gamma$ defines a bijection:

$$ \mathrm{Inv}(w^{-1}) \setminus \{\alpha_s\} \cong \mathrm{Inv}(w^{-1}s) \setminus \{\alpha_s\}. $$

Moreover because $sv > v$, $\alpha_s \in \mathrm{Inv}(v^{-1}s)$ and $\alpha_s \notin \mathrm{Inv}(v^{-1})$.

Therefore

$$ |\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}s) \setminus \{\alpha_s\}| = |\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| $$

and

$$ |\mathrm{Inv}(w^{-1}) \setminus \mathrm{Inv}(v^{-1}s)| = |\mathrm{Inv}(w^{-1}s) \setminus (\mathrm{Inv}(v^{-1}) \cup \{\alpha_s\})|. $$

1. If $\alpha_s \in \mathrm{Inv}(w^{-1})$ then $\alpha_s \notin \mathrm{Inv}(w^{-1}s)$ and $\ell_{sv}(w) = |\mathrm{Inv}(w^{-1}s) \setminus \mathrm{Inv}(v^{-1})| - (|\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| + 1) = \ell_v(sw) - 1$.

2. If $\alpha_s \notin \mathrm{Inv}(w^{-1})$ then $\alpha_s \in \mathrm{Inv}(w^{-1}s)$ and $\ell_{sv}(w) = (|\mathrm{Inv}(w^{-1}s) \setminus \mathrm{Inv}(v^{-1})| - 1) - |\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| = \ell_v(sw) - 1$.

□

**Proposition 2.7.** For all $v, w \in W$ the relative length $\ell_v(w)$ is given by

$$ \ell_v(w) = \ell(v^{-1}w) - \ell(v). \qquad (2.5) $$

*Proof.* Since $\ell = \ell_1$, we take a reduced expression for $v$ and apply Lemma 2.6 recursively to get the result. □

**Corollary 2.8.** For any $v \in W$, the relative length $\ell_v$ is a grading of $(W, <_v)$.
We associate an order to $\ell_v$ by setting, for any element $w \in W$ and any reflection $r \in W$: $w <_v wr$ if and only if $\ell_v(w) < \ell_v(wr)$ and then let $<_v$ be the order generated by these relations, it is strictly compatible with $\ell_v$. In particular $<_1$ is the classical Bruhat order.

As does the Bruhat length, the lengths $\ell_v$ have a geometric interpretation in the Coxeter complex associated to $(W, S)$. For $M$ a wall of the Coxeter complex and $w \in W$, let $\varepsilon_w(M) = -1$ if $M$ separates $C_f^v$ and $C_w^v$, and $\varepsilon_w(M) = +1$ otherwise. Then

$$ \ell_v(w) = \sum_{M \in \varepsilon_w^{-1}(-1)} \varepsilon_{\ell_v(M)}. \qquad (2.4) $$

We will use this relative length to give an alternative definition of the affinized length. Let us first give an explicit formula for $\ell_v$ depending only on the classical length $\ell = \ell_1$.

**Lemma 2.6.** If $sv > v$ with $v \in W$ and $s$ a simple reflection then for any $w \in W$, $\ell_{sv}(w) = \ell_v(sw) - 1$.

*Proof.* For any $w \in W$, the map $\gamma \mapsto s\gamma$ defines a bijection:

$$ \mathrm{Inv}(w^{-1}) \setminus \{\alpha_s\} \cong \mathrm{Inv}(w^{-1}s) \setminus \{\alpha_s\}. $$

Moreover because $sv > v$, $\alpha_s \in \mathrm{Inv}(v^{-1}s)$ and $\alpha_s \notin \mathrm{Inv}(v^{-1})$.

Therefore

$$ |\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}s) \setminus \{\alpha_s\}| = |\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| $$

and

$$ |\mathrm{Inv}(w^{-1}) \setminus \mathrm{Inv}(v^{-1}s)| = |\mathrm{Inv}(w^{-1}s) \setminus (\mathrm{Inv}(v^{-1}) \cup \{\alpha_s\})|. $$

1. If $\alpha_s \in \mathrm{Inv}(w^{-1})$ then $\alpha_s \notin \mathrm{Inv}(w^{-1}s)$ and $\ell_{sv}(w) = |\mathrm{Inv}(w^{-1}s) \setminus \mathrm{Inv}(v^{-1})| - (|\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| + 1) = \ell_v(sw) - 1$.

2. If $\alpha_s \notin \mathrm{Inv}(w^{-1})$ then $\alpha_s \in \mathrm{Inv}(w^{-1}s)$ and $\ell_{sv}(w) = (|\mathrm{Inv}(w^{-1}s) \setminus \mathrm{Inv}(v^{-1})| - 1) - |\mathrm{Inv}(w^{-1}s) \cap \mathrm{Inv}(v^{-1})| = \ell_v(sw) - 1$.

□

**Proposition 2.7.** For all $v, w \in W$ the relative length $\ell_v(w)$ is given by

$$ \ell_v(w) = \ell(v^{-1}w) - \ell(v). \qquad (2.5) $$

*Proof.* Since $\ell = \ell_1$, we take a reduced expression for $v$ and apply Lemma 2.6 recursively to get the result. □

**Corollary 2.8.** For any $v \in W$, the relative length $\ell_v$ is a grading of $(W, <_v)$.
*Proof.* Let $v, w, w' \in W$. By proposition 2.7, $\ell_v(w') - \ell_v(w) = \ell(v^{-1}w') - \ell(v^{-1}w)$ and $w'$ covers $w$ for $<_v$ if and only if $v^{-1}w'$ covers $v^{-1}w$ for the (standard) Bruhat order. Since the Bruhat length is a grading of $(W, <_1)$ (see [BB05, Theorem 2.2.6]), $v^{-1}w'$ covers $v^{-1}w$ if and only if $\ell(v^{-1}w') - \ell(v^{-1}w) = 1$ and $v^{-1}w' = v^{-1}wr$ for some reflection $r \in W$. Hence $w'$ covers $w$ for $<_v$ if and only if $\ell_v(w') - \ell_v(w) = 1$ and $w' = wr$ for some reflection $r \in W$: $\ell_v$ is a grading of $(W, <_v)$. $\square$

The order $<_v$ also has a geometric interpretation which will be important later on, it is given by the following corollary.

**Corollary 2.9.** For any root $\alpha \in \Phi$ and elements $w, v \in W$, we have that $w <_v s_\alpha w$ if and only if, in the Coxeter complex of $W$, $C_w^v$ and $C_v^v$ are on the same side of the wall $M_\alpha$.

*Proof.* We have $\ell_v(s_\alpha w) - \ell_v(w) = \ell(v^{-1}s_\alpha w) - \ell(v^{-1}w) = d^N(v, s_\alpha w) - d^N(v, w)$. By the definition of the Coxeter complex this is positive if and only if $C_v^v$ and $C_w^v$ are on the same side of the wall $M_\alpha$. $\square$

Therefore $<_v$ can be interpreted as a shift of the classical Bruhat order, corresponding geometrically to taking $C_v^v$ as fundamental chamber in the Coxeter complex.

## 2.2 Relation with the affinized Bruhat length

We relate the affinized Bruhat order and the relative order defined in subsection 2.1, we start by an alternative expression for the affinized Bruhat length.

**Proposition 2.10.** For any coweight $\lambda = v\lambda^{++} \in Y^+$, for any $w \in W$,

$$|\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\}| = \ell_{v^\lambda}(w). \quad (2.6)$$

Therefore

$$\ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \varepsilon\ell_{v^{\lambda}}(w), \quad (2.7)$$

$$\ell^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \ell_{v^{\lambda}}(w). \quad (2.8)$$

*Proof.* For $\lambda \in Y^+$ and $v \in W$ such that $\lambda = v\lambda^{++}$, then, $\alpha \in \Phi_+$ verifies $\langle \lambda, \alpha \rangle \ge 0$ if and only if $\alpha \in \Phi_\lambda \cup (\Phi_+ \setminus \mathrm{Inv}(v^{-1}))$, so:

$$\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\} = (\mathrm{Inv}(w^{-1}) \setminus \mathrm{Inv}(v^{-1})) \sqcup (\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}) \\ \mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) = \{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\} \sqcup (\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}).$$

Therefore

$$|\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\}| = \ell_v(w) + 2|\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}|$$

By Lemma 2.1 we deduce Formula (2.6). $\square$

<figure><img src="image_4.png" /></figure>
*Proof.* Let $v, w, w' \in W$. By proposition 2.7, $\ell_v(w') - \ell_v(w) = \ell(v^{-1}w') - \ell(v^{-1}w)$ and $w'$ covers $w$ for $<_v$ if and only if $v^{-1}w'$ covers $v^{-1}w$ for the (standard) Bruhat order. Since the Bruhat length is a grading of $(W, <_1)$ (see [BB05, Theorem 2.2.6]), $v^{-1}w'$ covers $v^{-1}w$ if and only if $\ell(v^{-1}w') - \ell(v^{-1}w) = 1$ and $v^{-1}w' = v^{-1}wr$ for some reflection $r \in W$. Hence $w'$ covers $w$ for $<_v$ if and only if $\ell_v(w') - \ell_v(w) = 1$ and $w' = wr$ for some reflection $r \in W$: $\ell_v$ is a grading of $(W, <_v)$. $\square$

The order $<_v$ also has a geometric interpretation which will be important later on, it is given by the following corollary.

**Corollary 2.9.** For any root $\alpha \in \Phi$ and elements $w, v \in W$, we have that $w <_v s_\alpha w$ if and only if, in the Coxeter complex of $W$, $C_w^v$ and $C_v^v$ are on the same side of the wall $M_\alpha$.

*Proof.* We have $\ell_v(s_\alpha w) - \ell_v(w) = \ell(v^{-1}s_\alpha w) - \ell(v^{-1}w) = d^N(v, s_\alpha w) - d^N(v, w)$. By the definition of the Coxeter complex this is positive if and only if $C_v^v$ and $C_w^v$ are on the same side of the wall $M_\alpha$. $\square$

Therefore $<_v$ can be interpreted as a shift of the classical Bruhat order, corresponding geometrically to taking $C_v^v$ as fundamental chamber in the Coxeter complex.

## 2.2 Relation with the affinized Bruhat length

We relate the affinized Bruhat order and the relative order defined in subsection 2.1, we start by an alternative expression for the affinized Bruhat length.

**Proposition 2.10.** For any coweight $\lambda = v\lambda^{++} \in Y^+$, for any $w \in W$,

$$|\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\}| = \ell_{v^\lambda}(w). \quad (2.6)$$

Therefore

$$\ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \varepsilon\ell_{v^{\lambda}}(w), \quad (2.7)$$

$$\ell^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\lambda^{++}) + \ell_{v^{\lambda}}(w). \quad (2.8)$$

*Proof.* For $\lambda \in Y^+$ and $v \in W$ such that $\lambda = v\lambda^{++}$, then, $\alpha \in \Phi_+$ verifies $\langle \lambda, \alpha \rangle \ge 0$ if and only if $\alpha \in \Phi_\lambda \cup (\Phi_+ \setminus \mathrm{Inv}(v^{-1}))$, so:

$$\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\} = (\mathrm{Inv}(w^{-1}) \setminus \mathrm{Inv}(v^{-1})) \sqcup (\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}) \\ \mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) = \{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\} \sqcup (\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}).$$

Therefore

$$|\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle \ge 0\}| - |\{\alpha \in \mathrm{Inv}(w^{-1}) \mid \langle \lambda, \alpha \rangle < 0\}| = \ell_v(w) + 2|\mathrm{Inv}(w^{-1}) \cap \mathrm{Inv}(v^{-1}) \cap \Phi_{\lambda}|$$

By Lemma 2.1 we deduce Formula (2.6). $\square$

<figure><img src="image_5.png" /></figure>
**Remark 2.11.** By combining Lemma 1.11 with Proposition 2.10, we obtain the formulas already given by Muthiah and Orr in [MO19, Proposition 3.10].

**Corollary 2.12.** Let $\lambda \in Y^+$ and $w \in W$. Suppose that $\pi^\mu w' = s_{\beta[n]}\pi^\lambda w$ for some affinized root $\beta[n] \in \Phi^a$ such that $\mu^{++} = \lambda^{++}$. Then

$$ \pi^{\mu} w' > \pi^{\lambda} w \iff \ell_{v^{\mu}}(w') > \ell_{v^{\lambda}}(w). \qquad (2.9) $$

Moreover for any $\lambda \in Y^+$ and $w, w' \in W$,

$$ \pi^{\lambda} w < \pi^{\lambda} w' \iff w <_{v^{\lambda}} w'. \qquad (2.10) $$

In particular, $\pi^\lambda v^\lambda$ is the minimal element of $\pi^\lambda W$.

**Proof.** Equivalence (2.9) is a direct consequence of Formula (2.8) and strict compatibility of the affinized Bruhat length and the affinized Bruhat order. It implies by iteration that a chain for the relative order $<_{v^\lambda}$ from $w$ to $w'$ lifts to a chain for the affinized Bruhat order from $\pi^\lambda w$ to $\pi^\lambda w'$. Conversely, by Corollary 2.4 $\text{proj}^{Y^+}$ is constant along any chain from $\pi^\lambda w$ to $\pi^\lambda w'$, and therefore the projection on $W$ of a chain from $\pi^\lambda w$ to $\pi^\lambda w'$ induces a chain for the relative Bruhat order $<_v$ from $w$ to $w'$. $\square$

We deduce a partial version of Theorem A, for vectorial covers with constant coweight.

**Theorem 2.13.** Let $x, y \in W_+^a$ be such that $\text{proj}^{Y^+}(x) = \text{proj}^{Y^+}(y)$ and $x \le y$. Then

$$ x \triangleleft y \iff \ell^a(y) = \ell^a(x) + 1. \qquad (2.11) $$

More precisely, if $x = \pi^\lambda w$ then $y = \pi^\lambda rw$ for some reflection $r \in W$ such that $rw$ covers $w$ for the relative Bruhat order $<_{v^\lambda}$.

**Proof.** By Equivalence (2.10), $\pi^\lambda w'$ covers $\pi^\lambda w$ if and only if $w'$ covers $w$ for the relative Bruhat order $<_{v^\lambda}$. By Corollary 2.8, this is equivalent to $\ell_{v^\lambda}(w') = \ell_{v^\lambda}(w) + 1$, Therefore by Formula (2.8) we deduce that $x \triangleleft y \implies \ell^a(y) = \ell^a(x) + 1$. The converse is immediate by strict compatibility of the affinized Bruhat length (Theorem 1.6). $\square$

## 2.3 Vectorial covers with non-constant coweight

The aim of this section is to prove Theorem A for vectorial covers with non-constant coweight.

Beforehand, we need a few results on parabolic decomposition. The first lemma is an adaptation of a standard result on minimal coset representatives (see [BB05, Theorem 2.5.5]), and the second is proved by P-E. Chaput, L. Fresse and T. Gobet in [CFG21].

**Lemma 2.14.** Let $J$ be a subset of $S$, recall Notation 1.2 for $W^J$. Let $v$ be an element of $W^J$ and $u$ be any element of $W$ such that $u < v$. Then, there is a saturated chain for the Bruhat order:

$$ u = u_0 \triangleleft u_1 \triangleleft \cdots \triangleleft u_N = v \qquad (2.12) $$

such that, for any $i \in [1, N]$, $u_{i-1}^{-1}u_i$ does not belong to $W_J$.
**Remark 2.11.** By combining Lemma 1.11 with Proposition 2.10, we obtain the formulas already given by Muthiah and Orr in [MO19, Proposition 3.10].

**Corollary 2.12.** Let $\lambda \in Y^+$ and $w \in W$. Suppose that $\pi^\mu w' = s_{\beta[n]}\pi^\lambda w$ for some affinized root $\beta[n] \in \Phi^a$ such that $\mu^{++} = \lambda^{++}$. Then

$$ \pi^{\mu} w' > \pi^{\lambda} w \iff \ell_{v^{\mu}}(w') > \ell_{v^{\lambda}}(w). \qquad (2.9) $$

Moreover for any $\lambda \in Y^+$ and $w, w' \in W$,

$$ \pi^{\lambda} w < \pi^{\lambda} w' \iff w <_{v^{\lambda}} w'. \qquad (2.10) $$

In particular, $\pi^\lambda v^\lambda$ is the minimal element of $\pi^\lambda W$.

**Proof.** Equivalence (2.9) is a direct consequence of Formula (2.8) and strict compatibility of the affinized Bruhat length and the affinized Bruhat order. It implies by iteration that a chain for the relative order $<_{v^\lambda}$ from $w$ to $w'$ lifts to a chain for the affinized Bruhat order from $\pi^\lambda w$ to $\pi^\lambda w'$. Conversely, by Corollary 2.4 $\text{proj}^{Y^+}$ is constant along any chain from $\pi^\lambda w$ to $\pi^\lambda w'$, and therefore the projection on $W$ of a chain from $\pi^\lambda w$ to $\pi^\lambda w'$ induces a chain for the relative Bruhat order $<_v$ from $w$ to $w'$. $\square$

We deduce a partial version of Theorem A, for vectorial covers with constant coweight.

**Theorem 2.13.** Let $x, y \in W_+^a$ be such that $\text{proj}^{Y^+}(x) = \text{proj}^{Y^+}(y)$ and $x \le y$. Then

$$ x \triangleleft y \iff \ell^a(y) = \ell^a(x) + 1. \qquad (2.11) $$

More precisely, if $x = \pi^\lambda w$ then $y = \pi^\lambda rw$ for some reflection $r \in W$ such that $rw$ covers $w$ for the relative Bruhat order $<_{v^\lambda}$.

**Proof.** By Equivalence (2.10), $\pi^\lambda w'$ covers $\pi^\lambda w$ if and only if $w'$ covers $w$ for the relative Bruhat order $<_{v^\lambda}$. By Corollary 2.8, this is equivalent to $\ell_{v^\lambda}(w') = \ell_{v^\lambda}(w) + 1$. Therefore by Formula (2.8) we deduce that $x \triangleleft y \implies \ell^a(y) = \ell^a(x) + 1$. The converse is immediate by strict compatibility of the affinized Bruhat length (Theorem 1.6). $\square$

## 2.3 Vectorial covers with non-constant coweight

The aim of this section is to prove Theorem A for vectorial covers with non-constant coweight.

Beforehand, we need a few results on parabolic decomposition. The first lemma is an adaptation of a standard result on minimal coset representatives (see [BB05, Theorem 2.5.5]), and the second is proved by P-E. Chaput, L. Fresse and T. Gobet in [CFG21].

**Lemma 2.14.** Let $J$ be a subset of $S$, recall Notation 1.2 for $W^J$. Let $v$ be an element of $W^J$ and $u$ be any element of $W$ such that $u < v$. Then, there is a saturated chain for the Bruhat order:

$$ u = u_0 \triangleleft u_1 \triangleleft \cdots \triangleleft u_N = v \qquad (2.12) $$

such that, for any $i \in [1, N]$, $u_{i-1}^{-1}u_i$ does not belong to $W_J$.
*Proof.* If $v$ covers $u$, it is clear since $u < v$ is a saturated chain, and as $v$ is a minimal coset representative, $u^{-1}v \notin W_J$. By induction it thus suffices, for a general pair $(u, v)$, to construct $u_1 \in W$ such that $u_1$ covers $u$, $u^{-1}u_1 \notin W_J$ and $u_1 < v$, the rest of the chain is obtained by induction. Let $s_1 \dots s_n$ be a reduced expression of $v$. Since $u < v$, there exists a reduced expression of $u$ obtained from $s_1 \dots s_n$ by deleting letters $s_{i_1}, \dots, s_{i_N}$. Choose one such that $i_N$ is minimal. Then let $t \in W$ be the reflection defined by $t = s_n \dots s_{i_N+1} s_{i_N} s_{i_N+1} \dots s_n$. We show that $u_1 = ut$ satisfies the desired properties.

1. By construction, an expression of $ut$ is obtained from $s_1 \dots s_n$ by deleting the $N-1$ letters $s_{i_1}, \dots, s_{i_{N-1}}$. Therefore $ut < v$.

2. Since an expression of $vt$ is obtained from $s_1 \dots s_n$ by deleting $s_{i_N}$, $vt < v$, and since $v$ is the minimal coset representative of $vW_J$, $t$ does not belong to $W_J$.

3. It remains to show that $ut$ covers $u$. By the first point, we have that $\ell(ut) \le \ell(u)+1$, so it suffices to show that $ut \not\le u$. Suppose by contradiction that $ut < u$. Then, by the strong exchange property, an expression of $ut$ is obtained from $u$ by deleting one letter $s_p$ of the reduced expression $s_1 \dots \check{s}_{i_1} \dots \check{s}_{i_N} \dots s_n$ (where $\check{s}_i$ denotes a letter $s_i$ taken away from the expression $s_1 \dots s_n$).

(a) Suppose that $p > i_N$. Then $t$ can also be written $s_n \dots s_{p+1}s_p s_{p+1} \dots s_n$, and $v = (vt)t = s_1 \dots s_{i_N} \dots \check{s}_p \dots s_n$, which contradicts the hypothesis that $s_1 \dots s_n$ is reduced.

(b) Suppose now that there is $d \le N-1$ such that $i_d < p < i_{d+1}$ (with the convention that $i_0 = -1$). Then $t = s_n \dots \check{s}_{i_N} \dots \check{s}_{i_{d+1}} \dots s_p \dots \check{s}_{i_{d+1}} \dots \check{s}_{i_N} \dots s_n$, and $u = (ut)t$ can be written from $s_1 \dots s_n$ by deleting the terms of indices $i_1, \dots, i_{N-1}$ and $p < i_N$, but not $i_N$. This contradicts the minimality of $i_N$.

□

**Definition 2.15.** For $v, w \in W$, we write $v \le_R w$ for:

$$v \le_R w \iff \ell(w) = \ell(v) + \ell(wv^{-1}). \qquad (2.13)$$

**Remark 2.16.** The relation $\le_R$ is called the weak Bruhat order and it is related to minimal galleries: $v \le_R w$ if and only if there is a minimal gallery from 1 to $w^{-1}$ going through $v^{-1}$.

Recall that for $J \subset S$ and $x \in W$, $(x^J, x_J)$ denotes the unique pair of $W^J \times W_J$ such that $x = x^J \cdot x_J$.

**Lemma 2.17.** [CFG21, Lemma 8.11] Let $J \subset S$ be a subset of simple reflections. Let $u$ be an element of $W$ and $t$ be a reflection of $W \setminus W_J$ such that $ut$ covers $u$. Then $(ut)_J \le_R u_J$. In other words $((ut)_J)^{-1}$ lies on a minimal gallery from 1 to $(u_J)^{-1}$.
*Proof.* If $v$ covers $u$, it is clear since $u < v$ is a saturated chain, and as $v$ is a minimal coset representative, $u^{-1}v \notin W_J$. By induction it thus suffices, for a general pair $(u, v)$, to construct $u_1 \in W$ such that $u_1$ covers $u$, $u^{-1}u_1 \notin W_J$ and $u_1 < v$, the rest of the chain is obtained by induction. Let $s_1 \dots s_n$ be a reduced expression of $v$. Since $u < v$, there exists a reduced expression of $u$ obtained from $s_1 \dots s_n$ by deleting letters $s_{i_1}, \dots, s_{i_N}$. Choose one such that $i_N$ is minimal. Then let $t \in W$ be the reflection defined by $t = s_n \dots s_{i_N+1}s_{i_N}s_{i_N+1} \dots s_n$. We show that $u_1 = ut$ satisfies the desired properties.

1. By construction, an expression of $ut$ is obtained from $s_1 \dots s_n$ by deleting the $N-1$ letters $s_{i_1}, \dots, s_{i_{N-1}}$. Therefore $ut < v$.

2. Since an expression of $vt$ is obtained from $s_1 \dots s_n$ by deleting $s_{i_N}$, $vt < v$, and since $v$ is the minimal coset representative of $vW_J$, $t$ does not belong to $W_J$.

3. It remains to show that $ut$ covers $u$. By the first point, we have that $\ell(ut) \le \ell(u)+1$, so it suffices to show that $ut \not\le u$. Suppose by contradiction that $ut < u$. Then, by the strong exchange property, an expression of $ut$ is obtained from $u$ by deleting one letter $s_p$ of the reduced expression $s_1 \dots \check{s}_{i_1} \dots \check{s}_{i_N} \dots s_n$ (where $\check{s}_i$ denotes a letter $s_i$ taken away from the expression $s_1 \dots s_n$).

(a) Suppose that $p > i_N$. Then $t$ can also be written $s_n \dots s_{p+1}s_p s_{p+1} \dots s_n$, and $v = (vt)t = s_1 \dots s_{i_N} \dots \check{s}_p \dots s_n$, which contradicts the hypothesis that $s_1 \dots s_n$ is reduced.

(b) Suppose now that there is $d \le N-1$ such that $i_d < p < i_{d+1}$ (with the convention that $i_0 = -1$). Then $t = s_n \dots \check{s}_{i_N} \dots \check{s}_{i_{d+1}} \dots s_p \dots \check{s}_{i_{d+1}} \dots \check{s}_{i_N} \dots s_n$, and $u = (ut)t$ can be written from $s_1 \dots s_n$ by deleting the terms of indices $i_1, \dots, i_{N-1}$ and $p < i_N$, but not $i_N$. This contradicts the minimality of $i_N$.

□

**Definition 2.15.** For $v, w \in W$, we write $v \le_R w$ for:

$$v \le_R w \iff \ell(w) = \ell(v) + \ell(wv^{-1}). \qquad (2.13)$$

**Remark 2.16.** The relation $\le_R$ is called the weak Bruhat order and it is related to minimal galleries: $v \le_R w$ if and only if there is a minimal gallery from $1$ to $w^{-1}$ going through $v^{-1}$.

Recall that for $J \subset S$ and $x \in W$, $(x^J, x_J)$ denotes the unique pair of $W^J \times W_J$ such that $x = x^J \cdot x_J$.

**Lemma 2.17.** [CFG21, Lemma 8.11] Let $J \subset S$ be a subset of simple reflections. Let $u$ be an element of $W$ and $t$ be a reflection of $W \setminus W_J$ such that $ut$ covers $u$. Then $(ut)_J \le_R u_J$. In other words $((ut)_J)^{-1}$ lies on a minimal gallery from $1$ to $(u_J)^{-1}$.
**Theorem 2.18.** Let $\mathbf{x}, \mathbf{y} \in W_+^a$ be such that $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$ and $\mathbf{x} \le \mathbf{y}$. Then

$$ \mathbf{x} \triangleleft \mathbf{y} \iff \ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1. \qquad (2.14) $$

More precisely, write $\mathbf{x} = \pi^{\lambda}w$. Let $J$ be the set of simple reflections stabilizing $\lambda^{++}$ and let $v \in W^J$ be such that $\lambda = v\lambda^{++}$ (so $v = v^{\lambda}$ with Notation 1.2). Then, if $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \mathrm{proj}^{Y^+}(\mathbf{x})$, there exists a unique reflection $r \in W$ such that:

1. The reflection $r$ does not stabilize $\lambda$ and $\mathbf{y} = \pi^{r\lambda}rw$.

2. For the Bruhat order on $W$, $v$ covers $rv$.

3. Set $u = rv$, so $u_J \in W_J$ denotes the element $(rv)_J$ and $rvu_J^{-1} \in W^J$. Then $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$ in $W$.

*Proof.* If $\mathrm{proj}^{Y^+}(\mathbf{y}) = \mathrm{proj}^{Y^+}(\mathbf{x})$ then Equivalence 2.14 is given by Theorem 2.13. Moreover if $\ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1$ and $\mathbf{y} \ge \mathbf{x}$ then, by strict compatibility of $\ell^a$, $\mathbf{y}$ covers $\mathbf{x}$. We are thus reduced to prove that, if $\mathbf{y}$ covers $\mathbf{x}$ with $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \mathrm{proj}^{Y^+}(\mathbf{x})$ and $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$, then $\ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1$.

Write $\mathbf{x} = \pi^{\lambda}w$ and $v = v^{\lambda}$. By definition of the affinized Bruhat order, if $\mathbf{x} \triangleleft \mathbf{y}$ then $\mathbf{y}$ is of the form $s_{\beta[n]}\mathbf{x}$ for some $\beta[n] \in \Phi_+^a$.

Let $\mathbf{y} = s_{\beta[n]}\mathbf{x} \in W_+^a$ with $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \lambda^{++}$ and $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \lambda$, in particular $n \neq \langle\lambda, \beta\rangle$. By Proposition 1.13, $n = 0$ so $\mathbf{y} = \pi^{r\lambda}rw$ for the reflection $r = s_{\beta}$ which does not stabilize $\lambda$. Let us write $u = rv$, note that $u^J = v^{r\lambda}$. By Formula (2.8),

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \ell_{uJ}(rw) - \ell_v(w). \qquad (2.15) $$

By definition, $rv = u^J u_J$ with, by Equality (1.3), $\ell(rv) = \ell(u_J) + \ell(u^J)$. We compute

$$ \begin{aligned} \ell_{uJ}(rw) - \ell_v(w) &= \ell((rv(u_J)^{-1})^{-1}rw) - \ell(v^{-1}w) + \ell(v) - \ell(u^J) \\ &= \ell(u_J v^{-1}w) - \ell(v^{-1}w) + \ell(v) + \ell(u_J) - \ell(u) \\ &= (\ell(v) - \ell(u)) + (\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))). \end{aligned} \qquad (2.16) $$

From Equations (2.15) and (2.16), we deduce

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = (\ell(v) - \ell(u)) + (\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))). \qquad (2.17) $$

In Expression (2.17), by the triangle inequality and since $d^{\mathbb{N}}(v, vu_J^{-1}) = \ell(u_J)$, the second term $\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))$ is non-negative, and it is equal to 0 if and only if $d^{\mathbb{N}}(v, w) = d^{\mathbb{N}}(v, vu_J^{-1}) + d^{\mathbb{N}}(vu_J^{-1}, w)$, so if and only if $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$.

Recall Definition 1.19 of $\mathrm{proj}_{vW_J}(w)$. Since $vu_J^{-1}$ lies in $vW_J$, a minimal gallery from $vu_J^{-1}$ to $w$ goes through $\mathrm{proj}_{vW_J}(w)$. Thus $\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))$ is equal to zero if and only if $u_J^{-1}$ is on a minimal gallery from 1 to $v^{-1}\mathrm{proj}_{vW_J}(w)$ in $W_J$.

Let us first suppose that this is not the case, we want to deduce that $\mathbf{y}$ does not cover $\mathbf{x}$. We thus want to produce a non trivial chain from $\pi^{\lambda}w$ to $\pi^{r\lambda}rw$. By Lemma 1.18,
**Theorem 2.18.** Let $\mathbf{x}, \mathbf{y} \in W_+^a$ be such that $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$ and $\mathbf{x} \le \mathbf{y}$. Then

$$ \mathbf{x} \triangleleft \mathbf{y} \iff \ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1. \qquad (2.14) $$

More precisely, write $\mathbf{x} = \pi^{\lambda}w$. Let $J$ be the set of simple reflections stabilizing $\lambda^{++}$ and let $v \in W^J$ be such that $\lambda = v\lambda^{++}$ (so $v = v^{\lambda}$ with Notation 1.2). Then, if $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \mathrm{proj}^{Y^+}(\mathbf{x})$, there exists a unique reflection $r \in W$ such that:

1. The reflection $r$ does not stabilize $\lambda$ and $\mathbf{y} = \pi^{r\lambda}rw$.

2. For the Bruhat order on $W$, $v$ covers $rv$.

3. Set $u = rv$, so $u_J \in W_J$ denotes the element $(rv)_J$ and $rvu_J^{-1} \in W^J$. Then $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$ in $W$.

*Proof.* If $\mathrm{proj}^{Y^+}(\mathbf{y}) = \mathrm{proj}^{Y^+}(\mathbf{x})$ then Equivalence 2.14 is given by Theorem 2.13. Moreover if $\ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1$ and $\mathbf{y} \ge \mathbf{x}$ then, by strict compatibility of $\ell^a$, $\mathbf{y}$ covers $\mathbf{x}$. We are thus reduced to prove that, if $\mathbf{y}$ covers $\mathbf{x}$ with $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \mathrm{proj}^{Y^+}(\mathbf{x})$ and $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \mathrm{proj}^{Y^{++}}(\mathbf{x})$, then $\ell^a(\mathbf{y}) = \ell^a(\mathbf{x}) + 1$.

Write $\mathbf{x} = \pi^{\lambda}w$ and $v = v^{\lambda}$. By definition of the affinized Bruhat order, if $\mathbf{x} \triangleleft \mathbf{y}$ then $\mathbf{y}$ is of the form $s_{\beta[n]}\mathbf{x}$ for some $\beta[n] \in \Phi_+^a$.

Let $\mathbf{y} = s_{\beta[n]}\mathbf{x} \in W_+^a$ with $\mathrm{proj}^{Y^{++}}(\mathbf{y}) = \lambda^{++}$ and $\mathrm{proj}^{Y^+}(\mathbf{y}) \neq \lambda$, in particular $n \neq \langle\lambda, \beta\rangle$. By Proposition 1.13, $n = 0$ so $\mathbf{y} = \pi^{r\lambda}rw$ for the reflection $r = s_{\beta}$ which does not stabilize $\lambda$. Let us write $u = rv$, note that $u^J = v^{r\lambda}$. By Formula (2.8),

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \ell_{uJ}(rw) - \ell_v(w). \qquad (2.15) $$

By definition, $rv = u^J u_J$ with, by Equality (1.3), $\ell(rv) = \ell(u_J) + \ell(u^J)$. We compute

$$ \begin{aligned} \ell_{u^J}(rw) - \ell_v(w) &= \ell((rv(u_J)^{-1})^{-1}rw) - \ell(v^{-1}w) + \ell(v) - \ell(u^J) \\ &= \ell(u_J v^{-1}w) - \ell(v^{-1}w) + \ell(v) + \ell(u_J) - \ell(u) \\ &= (\ell(v) - \ell(u)) + (\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))). \end{aligned} \qquad (2.16) $$

From Equations (2.15) and (2.16), we deduce

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = (\ell(v) - \ell(u)) + (\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))). \qquad (2.17) $$

In Expression (2.17), by the triangle inequality and since $d^{\mathbb{N}}(v, vu_J^{-1}) = \ell(u_J)$, the second term $\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))$ is non-negative, and it is equal to 0 if and only if $d^{\mathbb{N}}(v, w) = d^{\mathbb{N}}(v, vu_J^{-1}) + d^{\mathbb{N}}(vu_J^{-1}, w)$, so if and only if $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$.

Recall Definition 1.19 of $\mathrm{proj}_{vW_J}(w)$. Since $vu_J^{-1}$ lies in $vW_J$, a minimal gallery from $vu_J^{-1}$ to $w$ goes through $\mathrm{proj}_{vW_J}(w)$. Thus $\ell(u_J) - (d^{\mathbb{N}}(v, w) - d^{\mathbb{N}}(vu_J^{-1}, w))$ is equal to zero if and only if $u_J^{-1}$ is on a minimal gallery from 1 to $v^{-1}\mathrm{proj}_{vW_J}(w)$ in $W_J$.

Let us first suppose that this is not the case, we want to deduce that $\mathbf{y}$ does not cover $\mathbf{x}$. We thus want to produce a non trivial chain from $\pi^{\lambda}w$ to $\pi^{r\lambda}rw$. By Lemma 1.18,
there is a reflection $t \in W_J$ such that $d^{W_J}(1, tv^{-1}\mathrm{proj}_{vW_J}(w)) > d^{W_J}(1, v^{-1}\mathrm{proj}_{vW_J}(w))$ and $d^{W_J}(u_J^{-1}, tv^{-1}\mathrm{proj}_{vW_J}(w)) < d^{W_J}(u_J^{-1}, v^{-1}\mathrm{proj}_{vW_J}(w))$. In W, this implies $d(vt, w) > d(v, w)$ and $d(vtu_J^{-1}, w) < d(vu_J^{-1}, w)$.

Let $\tilde{w} = vtv^{-1}w$. We compute:

$$ \ell_v(\tilde{w}) - \ell_v(w) = d^{\mathbb{N}}(vt, w) - d^{\mathbb{N}}(v, w) > 0 $$

$$ \ell_{uJ}(rw) - \ell_{uJ}(r\tilde{w}) = d^{\mathbb{N}}(vu_J^{-1}, w) - d^{\mathbb{N}}(vtu_J^{-1}, w) > 0 $$

$$ \ell_{uJ}(r\tilde{w}) - \ell_v(\tilde{w}) = \ell(v) - \ell(u) + \ell(u_J) - (d^{\mathbb{N}}(v, \tilde{w}) - d^{\mathbb{N}}(vu, \tilde{w})) > 0. $$

Hence by Lemma 2.3 and Corollary 2.12,

$$ \pi^{\lambda}w < \pi^{\lambda}\tilde{w} < \pi^{r\lambda}r\tilde{w} < \pi^{r\lambda}rw. \qquad (2.18) $$

Suppose now that $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$, then by Formula (2.17), $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \ell(v) - \ell(rv)$. If this is equals to 1 then $\mathbf{y}$ covers $\mathbf{x}$; it remains to show the converse. Suppose that $\ell(v) - \ell(rv) = N > 1$. Let

$$ rv = u_0 \triangleleft u_1 \triangleleft \cdots \triangleleft u_N = v \qquad (2.19) $$

be a saturated chain in W from rv to v given by Lemma 2.14 and, for $i \in [1, N]$ let $\beta_i \in \Phi_+$ be such that $u_i = s_{\beta_i}u_{i-1}$, so $u_i = s_{\beta_i} \cdots s_{\beta_1}u \ge rv$. Note in particular that

$$ \ell(u_i) = \ell(rv) + i = \ell(v) - N + i. \qquad (2.20) $$

Let us show that it induces a chain for the affinized Bruhat order

$$ \pi^{\lambda}w = s_{\beta_{N}[0]} \cdots s_{\beta_{1}[0]}\pi^{r\lambda}rw < s_{\beta_{N-1}[0]} \cdots s_{\beta_{1}[0]}\pi^{r\lambda}rw < \cdots < \pi^{r\lambda}rw. \qquad (2.21) $$

Since $s_{\beta_i[0]} \cdots s_{\beta_1[0]}\pi^{r\lambda}rw = \pi^{u_i\lambda^{++}}s_{\beta_i} \cdots s_{\beta_1}rw$, by Formula (2.8) it is enough to verify

$$ \forall i \in [0, n], \quad \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) = \ell_v(w) + N - i. \qquad (2.22) $$

We compute

$$ \begin{aligned} \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) &= \ell((u_i(u_i)^{-1})^{-1}s_{\beta_i} \cdots s_{\beta_1}rw) - \ell(u_i^J) \\ &= \ell((u_i)_J v^{-1}w) - \ell(u_i^J). \end{aligned} \qquad (2.23) $$

Since the saturated chain $u_0 < u_1 < \cdots < u_N$ is obtained from Lemma 2.14, $u_i$ covers $u_{i-1}$ such that the reflection $u_{i-1}^{-1}u_i$ does not belong to $W_J$, so by Lemma 2.17, $(u_i)_J \le R (u_{i-1})_J$ and by iteration we have $(u_i)_J \le R (u_0)_J = u_J$. Otherwise said, $(u_i)_J^{-1}$ is on a minimal gallery from 1 to $u_J^{-1}$, therefore $v(u_i)_J^{-1}$ is on a minimal gallery from $v$ to $vu_J^{-1}$, hence on a minimal gallery from $v$ to $w$. We deduce

$$ \ell((u_i)_J v^{-1}w) = \ell(v^{-1}w) - \ell((u_i)_J). \qquad (2.24) $$

Combining Formulas (2.23) and (2.24) we obtain

$$ \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) = \ell(v^{-1}w) - \ell((u_i)_J) - \ell(u_i^J). \qquad (2.25) $$

Moreover,

$$ \ell((u_i)_J) + \ell(u_i^J) = \ell(u_i) = \ell(v) - (N - i) \qquad (2.26) $$

by Formulas (1.3) and (2.20). Combining Formulas (2.25) and (2.26), we deduce Formula (2.22). This concludes the proof. $\square$
there is a reflection $t \in W_J$ such that $d^{W_J}(1, tv^{-1}\mathrm{proj}_{vW_J}(w)) > d^{W_J}(1, v^{-1}\mathrm{proj}_{vW_J}(w))$ and $d^{W_J}(u_J^{-1}, tv^{-1}\mathrm{proj}_{vW_J}(w)) < d^{W_J}(u_J^{-1}, v^{-1}\mathrm{proj}_{vW_J}(w))$. In W, this implies $d(vt, w) > d(v, w)$ and $d(vtu_J^{-1}, w) < d(vu_J^{-1}, w)$.

Let $\tilde{w} = vtv^{-1}w$. We compute:

$$ \ell_v(\tilde{w}) - \ell_v(w) = d^{\mathbb{N}}(vt, w) - d^{\mathbb{N}}(v, w) > 0 $$

$$ \ell_{uJ}(rw) - \ell_{uJ}(r\tilde{w}) = d^{\mathbb{N}}(vu_J^{-1}, w) - d^{\mathbb{N}}(vtu_J^{-1}, w) > 0 $$

$$ \ell_{uJ}(r\tilde{w}) - \ell_v(\tilde{w}) = \ell(v) - \ell(u) + \ell(u_J) - (d^{\mathbb{N}}(v, \tilde{w}) - d^{\mathbb{N}}(vu, \tilde{w})) > 0. $$

Hence by Lemma 2.3 and Corollary 2.12,

$$ \pi^{\lambda}w < \pi^{\lambda}\tilde{w} < \pi^{r\lambda}r\tilde{w} < \pi^{r\lambda}rw. \qquad (2.18) $$

Suppose now that $vu_J^{-1}$ is on a minimal gallery from $v$ to $w$, then by Formula (2.17), $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \ell(v) - \ell(rv)$. If this is equals to 1 then $\mathbf{y}$ covers $\mathbf{x}$; it remains to show the converse. Suppose that $\ell(v) - \ell(rv) = N > 1$. Let

$$ rv = u_0 \triangleleft u_1 \triangleleft \cdots \triangleleft u_N = v \qquad (2.19) $$

be a saturated chain in W from rv to v given by Lemma 2.14 and, for $i \in [1, N]$ let $\beta_i \in \Phi_+$ be such that $u_i = s_{\beta_i}u_{i-1}$, so $u_i = s_{\beta_i} \cdots s_{\beta_1}u \ge rv$. Note in particular that

$$ \ell(u_i) = \ell(rv) + i = \ell(v) - N + i. \qquad (2.20) $$

Let us show that it induces a chain for the affinized Bruhat order

$$ \pi^{\lambda}w = s_{\beta_{N}[0]} \cdots s_{\beta_{1}[0]}\pi^{r\lambda}rw < s_{\beta_{N-1}[0]} \cdots s_{\beta_{1}[0]}\pi^{r\lambda}rw < \cdots < \pi^{r\lambda}rw. \qquad (2.21) $$

Since $s_{\beta_i[0]} \cdots s_{\beta_1[0]}\pi^{r\lambda}rw = \pi^{u_i\lambda^{++}}s_{\beta_i} \cdots s_{\beta_1}rw$, by Formula (2.8) it is enough to verify

$$ \forall i \in [0, n], \quad \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) = \ell_v(w) + N - i. \qquad (2.22) $$

We compute

$$ \begin{aligned} \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) &= \ell((u_i(u_i)^{-1})^{-1}s_{\beta_i} \cdots s_{\beta_1}rw) - \ell(u_i^J) \\ &= \ell((u_i)_J v^{-1}w) - \ell(u_i^J). \end{aligned} \qquad (2.23) $$

Since the saturated chain $u_0 < u_1 < \cdots < u_N$ is obtained from Lemma 2.14, $u_i$ covers $u_{i-1}$ such that the reflection $u_{i-1}^{-1}u_i$ does not belong to $W_J$, so by Lemma 2.17, $(u_i)_J \le R (u_{i-1})_J$ and by iteration we have $(u_i)_J \le R (u_0)_J = u_J$. Otherwise said, $(u_i)_J^{-1}$ is on a minimal gallery from 1 to $u_J^{-1}$, therefore $v(u_i)_J^{-1}$ is on a minimal gallery from $v$ to $vu_J^{-1}$, hence on a minimal gallery from $v$ to $w$. We deduce

$$ \ell((u_i)_J v^{-1}w) = \ell(v^{-1}w) - \ell((u_i)_J). \qquad (2.24) $$

Combining Formulas (2.23) and (2.24) we obtain

$$ \ell_{u_i^J}(s_{\beta_i} \cdots s_{\beta_1}rw) = \ell(v^{-1}w) - \ell((u_i)_J) - \ell(u_i^J). \qquad (2.25) $$

Moreover,

$$ \ell((u_i)_J) + \ell(u_i^J) = \ell(u_i) = \ell(v) - (N - i) \qquad (2.26) $$

by Formulas (1.3) and (2.20). Combining Formulas (2.25) and (2.26), we deduce Formula (2.22). This concludes the proof. $\square$
# 3 Properly affine covers

## 3.1 A few properties of properly affine covers

We now turn to the case of covers $\pi^{\lambda}w < \pi^{\mu}w'$ in $W_{+}^{a}$ with $\mu^{++} \neq \lambda^{++}$. Such covers are called properly affine covers.

By Formula (1.12), if $\pi^{\mu}s_{\beta}w = s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w$ with $\beta[n] \in \Phi_{+}^{a}$, then $n \in \mathbb{Z} \setminus \mathbb{Z}[0, \langle \lambda, \beta \rangle]$ then $s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w$ however, $s_{\beta[n]}\pi^{\lambda}w$ may not be in $W_{+}^{a}$ as $\lambda + \mathbb{Z}\beta^{\vee} \not\subset Y^{+}$. The limit cases $n \in \{0, \langle \lambda, \beta \rangle\}$ correspond to $\lambda^{++} = \mu^{++}$ dealt with in the previous section.

We first show that properly affine covers occur only for minimal $n$, in the following sense.

**Proposition 3.1.** Let $\lambda \in Y^{+}$ and $w \in W$, let $\beta \in \Phi$ and $n \in \mathbb{Z}$. Let us denote $\sigma = \text{sgn}(\langle \lambda, \beta \rangle) \in \{1, -1\}$. If $\pi^{\mu}w' = s_{\beta[n]}\pi^{\lambda}w \triangleright \pi^{\lambda}w$ is a cover with $\lambda^{++} \neq \mu^{++}$, then $n \in \{-\sigma, \langle \lambda, \beta \rangle + \sigma\}$.

*Proof.* For any $\nu \in Y^{+}$ if we identify the Coxeter complex of $W$ with the positive Tits cone $\mathcal{T} \subset \mathbb{A}$, $C_{\nu}^{\nu}$ is the closest vectorial chamber, from the fundamental chamber, containing $\nu$ in its closure. All the elements of $\lambda + \sigma\mathbb{Z}_{>0}\beta^{\vee}$ are on the same side of $M_{\beta}$, hence by Corollary 2.9, for any two such $\nu, \nu' \in \lambda + \sigma\mathbb{Z}_{>0}\beta^{\vee}$ and any $w \in W$,

$$w <_{\nu} s_{\beta}w \iff w <_{\nu'} s_{\beta}w. \qquad (3.1)$$

Suppose first that $n \in \langle \lambda, \beta \rangle + \sigma\mathbb{Z}_{>1}$ and let $\mu = \lambda + (n - \langle \lambda, \beta \rangle)\beta^{\vee}$, Then

1. If $w <_{\nu^{\mu}} s_{\beta}w$, we have the chain

$$\pi^{\lambda}w < s_{\beta[\langle \lambda, \beta \rangle + \sigma]}\pi^{\lambda}w = \pi^{\lambda + \sigma\beta^{\vee}}s_{\beta}w < \pi^{\mu}w < \pi^{\mu}s_{\beta}w. \qquad (3.2)$$

The second inequality comes from $\pi^{\mu}w = s_{\beta[n+\sigma]}\pi^{\lambda+\sigma\beta^{\vee}}s_{\beta}w$ and Formula (1.12), the third comes from Equivalence (2.10).

2. Else $s_{\beta}w <_{\nu^{\mu}} w$, so by Equivalence (3.1), $s_{\beta}w <_{\nu^{\lambda+\sigma\beta^{\vee}}} w$ and we have the chain

$$\pi^{\lambda}w < s_{\beta[\langle \lambda, \beta \rangle + \sigma]}\pi^{\lambda}w = \pi^{\lambda + \sigma\beta^{\vee}}s_{\beta}w < \pi^{\lambda + \sigma\beta^{\vee}}w < \pi^{\mu}s_{\beta}w. \qquad (3.3)$$

Here the second inequality comes from Equivalence (2.10). The third comes from $\pi^{\mu}s_{\beta}w = s_{\beta[n+\sigma]}\pi^{\lambda+\sigma\beta^{\vee}}w$ and Formula (1.12).

Either way, for $n \in \langle \lambda, \beta \rangle + \sigma\mathbb{Z}_{>1}$, $s_{\beta[n]}\pi^{\lambda}w$ does not cover $\pi^{\lambda}w$. For $n \in -\sigma\mathbb{Z}_{>1}$ the argument is similar, because all the elements of $s_{\beta}\lambda - \sigma\mathbb{Z}_{>0}\beta^{\vee}$ are on the same side of $M_{\beta}$, in particular on the side of $\mu = s_{\beta}\lambda + n\beta^{\vee}$.

1. If $w <_{\nu^{\mu}} s_{\beta}w$, we have a chain

$$\pi^{\lambda}w < s_{\beta[-\sigma]}\pi^{\lambda}w = \pi^{s_{\beta}(\lambda + \sigma\beta^{\vee})}s_{\beta}w < \pi^{\mu}w < \pi^{\mu}s_{\beta}w. \qquad (3.4)$$
# 3 Properly affine covers

## 3.1 A few properties of properly affine covers

We now turn to the case of covers $\pi^{\lambda}w < \pi^{\mu}w'$ in $W_{+}^{a}$ with $\mu^{++} \neq \lambda^{++}$. Such covers are called properly affine covers.

By Formula (1.12), if $\pi^{\mu}s_{\beta}w = s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w$ with $\beta[n] \in \Phi_{+}^{a}$, then $n \in \mathbb{Z} \setminus \lfloor 0, \langle \lambda, \beta \rangle \rfloor$. Conversely, if $n \in \mathbb{Z} \setminus \lfloor 0, \langle \lambda, \beta \rangle \rfloor$ then $s_{\beta[n]}\pi^{\lambda}w > \pi^{\lambda}w$ however, $s_{\beta[n]}\pi^{\lambda}w$ may not be in $W_{+}^{a}$ as $\lambda + \mathbb{Z}\beta^{\vee} \not\subset Y^{+}$. The limit cases $n \in \{0, \langle \lambda, \beta \rangle\}$ correspond to $\lambda^{++} = \mu^{++}$ dealt with in the previous section.

We first show that properly affine covers occur only for minimal $n$, in the following sense.

**Proposition 3.1.** Let $\lambda \in Y^{+}$ and $w \in W$, let $\beta \in \Phi$ and $n \in \mathbb{Z}$. Let us denote $\sigma = \text{sgn}(\langle \lambda, \beta \rangle) \in \{1, -1\}$. If $\pi^{\mu}w' = s_{\beta[n]}\pi^{\lambda}w \triangleright \pi^{\lambda}w$ is a cover with $\lambda^{++} \neq \mu^{++}$, then $n \in \{-\sigma, \langle \lambda, \beta \rangle + \sigma\}$.

*Proof.* For any $\nu \in Y^{+}$ if we identify the Coxeter complex of $W$ with the positive Tits cone $\mathcal{T} \subset \mathbb{A}$, $C_{\nu}^{\nu}$ is the closest vectorial chamber, from the fundamental chamber, containing $\nu$ in its closure. All the elements of $\lambda + \sigma\mathbb{Z}_{>0}\beta^{\vee}$ are on the same side of $M_{\beta}$, hence by Corollary 2.9, for any two such $\nu, \nu' \in \lambda + \sigma\mathbb{Z}_{>0}\beta^{\vee}$ and any $w \in W$,

$$w <_{\nu} s_{\beta}w \iff w <_{\nu'} s_{\beta}w. \qquad (3.1)$$

Suppose first that $n \in \langle \lambda, \beta \rangle + \sigma\mathbb{Z}_{>1}$ and let $\mu = \lambda + (n - \langle \lambda, \beta \rangle)\beta^{\vee}$, Then

1. If $w <_{\nu^{\mu}} s_{\beta}w$, we have the chain

$$\pi^{\lambda}w < s_{\beta[\langle \lambda, \beta \rangle + \sigma]}\pi^{\lambda}w = \pi^{\lambda + \sigma\beta^{\vee}}s_{\beta}w < \pi^{\mu}w < \pi^{\mu}s_{\beta}w. \qquad (3.2)$$

The second inequality comes from $\pi^{\mu}w = s_{\beta[n+\sigma]}\pi^{\lambda+\sigma\beta^{\vee}}s_{\beta}w$ and Formula (1.12), the third comes from Equivalence (2.10).

2. Else $s_{\beta}w <_{\nu^{\mu}} w$, so by Equivalence (3.1), $s_{\beta}w <_{\nu^{\lambda+\sigma\beta^{\vee}}} w$ and we have the chain

$$\pi^{\lambda}w < s_{\beta[\langle \lambda, \beta \rangle + \sigma]}\pi^{\lambda}w = \pi^{\lambda + \sigma\beta^{\vee}}s_{\beta}w < \pi^{\lambda + \sigma\beta^{\vee}}w < \pi^{\mu}s_{\beta}w. \qquad (3.3)$$

Here the second inequality comes from Equivalence (2.10). The third comes from $\pi^{\mu}s_{\beta}w = s_{\beta[n+\sigma]}\pi^{\lambda+\sigma\beta^{\vee}}w$ and Formula (1.12).

Either way, for $n \in \langle \lambda, \beta \rangle + \sigma\mathbb{Z}_{>1}$, $s_{\beta[n]}\pi^{\lambda}w$ does not cover $\pi^{\lambda}w$. For $n \in -\sigma\mathbb{Z}_{>1}$ the argument is similar, because all the elements of $s_{\beta}\lambda - \sigma\mathbb{Z}_{>0}\beta^{\vee}$ are on the same side of $M_{\beta}$, in particular on the side of $\mu = s_{\beta}\lambda + n\beta^{\vee}$.

1. If $w <_{\nu^{\mu}} s_{\beta}w$, we have a chain

$$\pi^{\lambda}w < s_{\beta[-\sigma]}\pi^{\lambda}w = \pi^{s_{\beta}(\lambda+\sigma\beta^{\vee})}s_{\beta}w < \pi^{\mu}w < \pi^{\mu}s_{\beta}w. \qquad (3.4)$$
2. Else $s_\beta w <_{v^\mu} w$ so $s_\beta w <_{v^{s_\beta(\lambda+\sigma\beta^\vee)}} w$ and we have a chain

$$ \pi^{\lambda} w < s_{\beta[-\sigma]} \pi^{\lambda} w = \pi^{s_{\beta}(\lambda+\sigma\beta^{\vee})} s_{\beta} w < \pi^{s_{\beta}(\lambda+\sigma\beta^{\vee})} w < \pi^{\mu} s_{\beta} w. \quad (3.5) $$

Hence the only possible covers (with varying coweights) are for $n \in \{-\sigma, \langle\lambda, \beta\rangle + \sigma\}$. $\square$

**Remark 3.2.** To follow up on Remark 1.14, note that by Equation (1.10), we have $s_{\beta[\sigma+\langle\lambda, \beta\rangle]}\pi^\lambda w = \pi^\lambda w s_{w^{-1}(\beta)[\sigma]}$ (where $\sigma = \operatorname{sgn}(\langle\lambda, \beta\rangle)$). Therefore Proposition 3.1 tells us that, if $\mathbf{y}$ covers $\mathbf{x}$ in $W_+^a$, then $\mathbf{y}$ is obtained from $\mathbf{x}$ applying an affinized reflection $s_{\tilde{\beta}[n]}$ either on the left or on the right, with $n \in \{-1, 0, 1\}$.

This is still far from a sufficient condition and many cases of potential covers can still be eliminated. We give another necessary condition for $\pi^\mu s_\beta w = s_{\beta[n]}\pi^\lambda w > \pi^\lambda w$ to be a cover, this is a generalisation of the chains produced in the proof of Theorem 2.18:

**Proposition 3.3.** Let $\pi^\mu s_\beta w = s_{\beta[n]}\pi^\lambda w > \pi^\lambda w$ with $\mu^{++} \neq \lambda^{++}$. Suppose that $s_\beta v^\mu$ is not on a minimal gallery from $w$ to $v^\lambda$. Then $\pi^\mu s_\beta w > \pi^\lambda w$ is not a cover.

*Proof.* We express the difference of $\varepsilon$-length using Formula (2.7):

$$ \ell_{\varepsilon}^{a}(\pi^{\mu}s_{\beta}w) - \ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\mathrm{ht}(\mu^{++} - \lambda^{++}) + \varepsilon(\ell_{v^{\mu}}(s_{\beta}w) - \ell_{v^{\lambda}}(w)). \quad (3.6) $$

If there exists a reflection $r \in W$ such that $\ell_{v^\lambda}(rw) > \ell_{v^\lambda}(w)$ and $\ell_{v^\mu}(s_\beta rw) < \ell_{v^\mu}(s_\beta w)$ then using Formula (2.7) to compute the length $\ell_\varepsilon^a$, we have a chain

$$ \pi^{\lambda} w < \pi^{\lambda} rw < \pi^{\mu} s_{\beta} rw < \pi^{\mu} s_{\beta} w. \quad (3.7) $$

Since $\ell_v(rw) - \ell_v(w) = \ell(v^{-1}rw) - \ell(v^{-1}w)$ for $v, r, w \in W$, Lemma 1.18 guarantees the existence of $r$, which proves the proposition. $\square$

In Figure 3.1 below, we give an example of a chain constructed this way in the $A_1$-affine case (with Cartan matrix $\begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$).

In this example, $\alpha$ and $\beta$ are the simple roots of a $A_1$-affinized root system, and we have chosen $\lambda, w$ and $\beta[n]$ such that $v^\lambda = s_\alpha$, $v^\mu = s_\alpha s_\beta$ and $w = s_\beta$. $\pi^\lambda w$ corresponds to the alcove $C_1$ in light blue, and its image $\pi^\mu s_\beta w$ by $s_{\beta[6]}$ corresponds to $C_4$. Since $r = s_\beta s_\alpha s_\beta$ verifies $d(v^\lambda, rw) = s_\alpha s_\beta s_\alpha > d(v^\lambda, w) = s_\alpha s_\beta$, and $d(s_\beta v^\mu, rw) = s_\beta < s_\beta s_\alpha = d(s_\beta v^\mu, w)$, there is a chain $\pi^\lambda w < \pi^\lambda rw < \pi^\mu s_\beta rw < \pi^\mu s_\beta w$ which corresponds to the sequence of alcoves $(C_1, C_2, C_3, C_4)$ on the figure.

**Remark 3.4.** Let $v_0, w_0 \in W$, $\mu_0 \in Y$ and $\alpha_0 \in \Phi$. To produce chains, note that Formula (1.12) applied with the affinized reflection $s_{v_0(\alpha_0)[m+\langle\mu_0, \alpha_0\rangle]}$ to $\pi^{v_0(\mu_0)}w_0$ gives

$$ \forall m \in \mathbb{Z} \setminus [-\langle \mu_0, \alpha_0 \rangle, 0], \quad \pi^{v_0(\mu_0+m\alpha_0^\vee)} s_{v_0(\alpha_0)} w_0 > \pi^{v_0(\mu_0)} w_0. \quad (3.8) $$

Moreover, applying the affinized reflection $s_{v_0(\alpha_0)[-m]}$ to $\pi^{v_0(\mu_0)}w_0$ instead we obtain

$$ \forall m \in \mathbb{Z} \setminus [-\langle \mu_0, \alpha_0 \rangle, 0], \quad \pi^{v_0 s_{\alpha_0}(\mu_0+m\alpha_0^\vee)} s_{v_0(\alpha_0)} w_0 > \pi^{v_0(\mu_0)} w_0. \quad (3.9) $$
2. Else $s_\beta w <_{v^\mu} w$ so $s_\beta w <_{v^{s_\beta(\lambda+\sigma\beta^\vee)}} w$ and we have a chain

$$ \pi^{\lambda} w < s_{\beta[-\sigma]} \pi^{\lambda} w = \pi^{s_{\beta}(\lambda+\sigma\beta^{\vee})} s_{\beta} w < \pi^{s_{\beta}(\lambda+\sigma\beta^{\vee})} w < \pi^{\mu} s_{\beta} w. \quad (3.5) $$

Hence the only possible covers (with varying coweights) are for $n \in \{-\sigma, \langle\lambda, \beta\rangle + \sigma\}$. $\square$

**Remark 3.2.** To follow up on Remark 1.14, note that by Equation (1.10), we have $s_{\beta[\sigma+\langle\lambda, \beta\rangle]}\pi^\lambda w = \pi^\lambda w s_{w^{-1}(\beta)[\sigma]}$ (where $\sigma = \operatorname{sgn}(\langle\lambda, \beta\rangle)$). Therefore Proposition 3.1 tells us that, if $\mathbf{y}$ covers $\mathbf{x}$ in $W_+^a$, then $\mathbf{y}$ is obtained from $\mathbf{x}$ applying an affinized reflection $s_{\tilde{\beta}[n]}$ either on the left or on the right, with $n \in \{-1, 0, 1\}$.

This is still far from a sufficient condition and many cases of potential covers can still be eliminated. We give another necessary condition for $\pi^\mu s_\beta w = s_{\beta[n]}\pi^\lambda w > \pi^\lambda w$ to be a cover, this is a generalisation of the chains produced in the proof of Theorem 2.18:

**Proposition 3.3.** Let $\pi^\mu s_\beta w = s_{\beta[n]}\pi^\lambda w > \pi^\lambda w$ with $\mu^{++} \neq \lambda^{++}$. Suppose that $s_\beta v^\mu$ is not on a minimal gallery from $w$ to $v^\lambda$. Then $\pi^\mu s_\beta w > \pi^\lambda w$ is not a cover.

*Proof.* We express the difference of $\varepsilon$-length using Formula (2.7):

$$ \ell_{\varepsilon}^{a}(\pi^{\mu}s_{\beta}w) - \ell_{\varepsilon}^{a}(\pi^{\lambda}w) = 2\operatorname{ht}(\mu^{++} - \lambda^{++}) + \varepsilon(\ell_{v^{\mu}}(s_{\beta}w) - \ell_{v^{\lambda}}(w)). \quad (3.6) $$

If there exists a reflection $r \in W$ such that $\ell_{v^\lambda}(rw) > \ell_{v^\lambda}(w)$ and $\ell_{v^\mu}(s_\beta rw) < \ell_{v^\mu}(s_\beta w)$ then using Formula (2.7) to compute the length $\ell_\varepsilon^a$, we have a chain

$$ \pi^{\lambda}w < \pi^{\lambda}rw < \pi^{\mu}s_{\beta}rw < \pi^{\mu}s_{\beta}w. \quad (3.7) $$

Since $\ell_v(rw) - \ell_v(w) = \ell(v^{-1}rw) - \ell(v^{-1}w)$ for $v, r, w \in W$, Lemma 1.18 guarantees the existence of $r$, which proves the proposition. $\square$

In Figure 3.1 below, we give an example of a chain constructed this way in the $A_1$-affine case (with Cartan matrix $\begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}$).

In this example, $\alpha$ and $\beta$ are the simple roots of a $A_1$-affinized root system, and we have chosen $\lambda, w$ and $\beta[n]$ such that $v^\lambda = s_\alpha$, $v^\mu = s_\alpha s_\beta$ and $w = s_\beta$. $\pi^\lambda w$ corresponds to the alcove $C_1$ in light blue, and its image $\pi^\mu s_\beta w$ by $s_{\beta[6]}$ corresponds to $C_4$. Since $r = s_\beta s_\alpha s_\beta$ verifies $d(v^\lambda, rw) = s_\alpha s_\beta s_\alpha > d(v^\lambda, w) = s_\alpha s_\beta$, and $d(s_\beta v^\mu, rw) = s_\beta < s_\beta s_\alpha = d(s_\beta v^\mu, w)$, there is a chain $\pi^\lambda w < \pi^\lambda rw < \pi^\mu s_\beta rw < \pi^\mu s_\beta w$ which corresponds to the sequence of alcoves $(C_1, C_2, C_3, C_4)$ on the figure.

**Remark 3.4.** Let $v_0, w_0 \in W$, $\mu_0 \in Y$ and $\alpha_0 \in \Phi$. To produce chains, note that Formula (1.12) applied with the affinized reflection $s_{v_0(\alpha_0)[m+\langle\mu_0, \alpha_0\rangle]}$ to $\pi^{v_0(\mu_0)}w_0$ gives

$$ \forall m \in \mathbb{Z} \setminus [-\langle \mu_0, \alpha_0 \rangle, 0], \quad \pi^{v_0(\mu_0+m\alpha_0^\vee)} s_{v_0(\alpha_0)} w_0 > \pi^{v_0(\mu_0)} w_0. \quad (3.8) $$

Moreover, applying the affinized reflection $s_{v_0(\alpha_0)[-m]}$ to $\pi^{v_0(\mu_0)}w_0$ instead we obtain

$$ \forall m \in \mathbb{Z} \setminus [-\langle \mu_0, \alpha_0 \rangle, 0], \quad \pi^{v_0 s_{\alpha_0}(\mu_0+m\alpha_0^\vee)} s_{v_0(\alpha_0)} w_0 > \pi^{v_0(\mu_0)} w_0. \quad (3.9) $$
<figure><img src="image_5.png" /><figcaption>Figure 3.1: Example of a chain constructed as in Proposition 3.3.</figcaption></figure>

For $m \in \mathbb{Z} - \langle \mu_0, \alpha_0 \rangle, 0 \in \mathbb{Z}$ the inequalities are reversed. The limit cases $m \in \{-\langle \mu_0, \alpha_0 \rangle, 0\}$ need to be treated more carefully, they depend on the sign of the root $v_0(\alpha_0)$ (because Formula (1.12) holds for the affinized reflection $s_{v_0(\alpha_0)[0]}$ only if $v_0(\alpha_0) \in \Phi_+$), on the sign of $\langle \mu_0, \alpha_0 \rangle$ and on the vectorial element $w_0$.

## 3.2 Another expression for the affinized length difference

Outside of the case of vectorial covers dealt with in Theorems 2.13 and 2.18, if we write $\mathbf{x} = \pi^{v\lambda}w$ with $\lambda \in Y^{++}$, $v, w \in W$ with $v$ of minimal length in $vW_\lambda$, by Proposition 3.1 the only covers are of the form $\mathbf{y} \in \{\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w, \pi^{v\beta(\lambda+\beta^\vee)}s_{v(\beta)}w\}$ for some $\beta \in \Phi_+$, so the rest of this paper is dedicated to covers of this sort.

**Notation 3.5.** From now on, unless stated otherwise, we use the following notation:

1. $\lambda \in Y^{++}$ is a dominant coweight

2. $\beta \in \Phi_+$ is a positive root

3. $v \in W^\lambda$ is the minimal representative of a $W_\lambda$-coset.

4. $w \in W$ is any element of $W$.

5. $\mathbf{x} = \pi^{v(\lambda)}w$ and $\mathbf{y} \in \{\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w, \pi^{v\beta(\lambda+\beta^\vee)}s_{v(\beta)}w\}$ are elements of $W^+$.
<figure><img src="image_6.png" /><figcaption>Figure 3.1: Example of a chain constructed as in Proposition 3.3.</figcaption></figure>

For $m \in \mathbb{Z} - \langle \mu_0, \alpha_0 \rangle, 0 \in \mathbb{Z}$ the inequalities are reversed. The limit cases $m \in \{-\langle \mu_0, \alpha_0 \rangle, 0\}$ need to be treated more carefully, they depend on the sign of the root $v_0(\alpha_0)$ (because Formula (1.12) holds for the affinized reflection $s_{v_0(\alpha_0)[0]}$ only if $v_0(\alpha_0) \in \Phi_+$), on the sign of $\langle \mu_0, \alpha_0 \rangle$ and on the vectorial element $w_0$.

## 3.2 Another expression for the affinized length difference

Outside of the case of vectorial covers dealt with in Theorems 2.13 and 2.18, if we write $\mathbf{x} = \pi^{v\lambda}w$ with $\lambda \in Y^{++}$, $v, w \in W$ with $v$ of minimal length in $vW_\lambda$, by Proposition 3.1 the only covers are of the form $\mathbf{y} \in \{\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w, \pi^{v\beta(\lambda+\beta^\vee)}s_{v(\beta)}w\}$ for some $\beta \in \Phi_+$, so the rest of this paper is dedicated to covers of this sort.

**Notation 3.5.** From now on, unless stated otherwise, we use the following notation:

1. $\lambda \in Y^{++}$ is a dominant coweight

2. $\beta \in \Phi_+$ is a positive root

3. $v \in W^\lambda$ is the minimal representative of a $W_\lambda$-coset.

4. $w \in W$ is any element of $W$.

5. $\mathbf{x} = \pi^{v(\lambda)}w$ and $\mathbf{y} \in \{\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w, \pi^{v\beta(\lambda+\beta^\vee)}s_{v(\beta)}w\}$ are elements of $W^+$.
In particular, $\lambda$ is a dominant coweight. This choice is made in order to avoid the heavier notation $\lambda^{++}$. Recall from Notation 1.2 that $W^{\lambda}$ is the set of minimal coset representatives of $W/W_{\lambda}$, where $W_{\lambda}$ is the standard parabolic subgroup $\{w \in W | w(\lambda) = \lambda\}$.

In this subsection, we give another expression for the length difference $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$.

The two following lemmas give information on the vectorial chamber of $v(\lambda + \beta^{\vee})$.

**Lemma 3.6.** Let $\lambda \in Y^{++}$ be a dominant coweight and let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^{\vee} \in Y^+$. Let $u \in W$ be such that $\lambda + \beta^{\vee}$ belongs to the vectorial chamber $C_u^v$, that is to say $u^{-1}(\lambda + \beta^{\vee}) \in Y^{++}$. Then

$$ \ell(s_{\beta}u) = \ell(s_{\beta}) + \ell(u). \qquad (3.10) $$

*Proof.* Let $s_{\tau_1} \dots s_{\tau_n}$ be a reduced expression of $u$, so that $\ell(u) = n$ and

$$ \mathrm{Inv}(u^{-1}) = \{\tau_1, s_{\tau_1}(\tau_2), \dots, s_{\tau_1} \dots s_{\tau_{n-1}}(\tau_n)\}. $$

We show that $s_{\tau_{k+1}} \dots s_{\tau_1} s_{\beta} > s_{\tau_k} \dots s_{\tau_1} s_{\beta}$ for all $k \in [0, n-1]$.

For any $\alpha \in \mathrm{Inv}(u^{-1})$ we have $\langle \lambda + \beta^{\vee}, \alpha \rangle \le 0$ (because $\lambda + \beta^{\vee} \in C_u^v$). Since $\lambda$ is dominant this implies $\langle \beta^{\vee}, \alpha \rangle \le 0$.

Let $k \in [0, n-1]$. Then $\langle \beta^{\vee}, s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) \rangle \le 0$, so: $s_{\beta}(s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1})) = s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) - \langle \beta^{\vee}, s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) \rangle \beta$ is a positive root as a sum of positive roots. Thus $s_{\tau_{k+1}} \dots s_{\tau_1} s_{\beta} > s_{\tau_k} \dots s_{\tau_1} s_{\beta}$ for any $k \in [0, n-1]$ and therefore $\ell(s_{\beta}u) = \ell(u^{-1}s_{\beta}) = n + \ell(s_{\beta}) = \ell(s_{\beta}) + \ell(u)$. $\square$

**Lemma 3.7.** Let $\lambda \in Y^{++}$ be a dominant coweight and let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^{\vee} \in Y^+$. Let $v \in W^{\lambda}$, $w \in W$ and let $u$ denote the element $v^{\lambda+\beta^{\vee}}$.

Then, if $\pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w$ (resp. $\pi^{v s_{\beta}(\lambda+\beta^{\vee})} s_{v(\beta)} w$) covers $\mathbf{x} = \pi^{v(\lambda)} w$,

$$ \ell(vu) = \ell(v) + \ell(u) \quad (\text{resp. } \ell vs_{\beta}u = \ell(v) + \ell(s_{\beta}u) \text{ and } \ell vs_{\beta} = \ell(v) + \ell(s_{\beta})). \quad (3.11) $$

*Proof.* To simplify the notation, write $W_J$ for $W_{(\lambda+\beta^{\vee})}$. Note that, with the notation of Definition 1.19, $vu = \mathrm{proj}_{vuW_J}(v)$ since $u$ is the element of minimal length in $uW_J$.

Suppose by contradiction that $\pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w$ covers $\mathbf{x}$ with $\ell(vu) < \ell(v) + \ell(u)$. Then $d^{\mathbb{N}}(1, vu) = \ell(vu) < d^{\mathbb{N}}(1, v) + d^{\mathbb{N}}(v, vu) = \ell(v) + \ell(u)$, so $v$ is not on a minimal gallery from 1 to $vu$. Therefore by Lemma 1.18, there is a reflection $r \in W$ such that $d(1, rvu) > d(1, vu)$ and $d(1, rv) < d(1, v)$, that is to say $rv < v$ and $rvu > vu$.

By minimality of $u$, $r$ is not in $vuW_J(vu)^{-1}$: otherwise $rvu \in vuW_J$ verifies $d(v, rvu) = d rv, vu) < d(v, vu)$, because foldings reduce the vectorial distance and $v, vu$ are on different sides of the wall $M_r$ associated to $r$.

Since $vu$ is the projection of $v$ on $vuW_J$ which is convex (see [Ron89, Lemma 2.10]), and since the wall $M_r$ separates $v$ and $vu$, any element of $vuW_J$ is on the same side of the wall $M_r$ as $vu$, so $rvu\tilde{u} > vu\tilde{u}$ for any $\tilde{u} \in W_J$. In particular, let $\tilde{u} \in W_J$ be such that $rvu\tilde{u}$ is the minimal coset representative of $rvuW_J$. Then by Proposition 2.3, since $rvu\tilde{u} > vu\tilde{u}$, we have

$$ \pi^{rv(\lambda+\beta^{\vee})} rs_{v(\beta)} w = \pi^{rvu\tilde{u}((\lambda+\beta^{\vee})^{++})} rs_{v(\beta)} w < \pi^{vu\tilde{u}((\lambda+\beta^{\vee})^{++})} s_{v(\beta)} w = \pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w. \quad (3.12) $$
In particular, $\lambda$ is a dominant coweight. This choice is made in order to avoid the heavier notation $\lambda^{++}$. Recall from Notation 1.2 that $W^{\lambda}$ is the set of minimal coset representatives of $W/W_{\lambda}$, where $W_{\lambda}$ is the standard parabolic subgroup $\{w \in W | w(\lambda) = \lambda\}$.

In this subsection, we give another expression for the length difference $\ell^a(\mathbf{y}) - \ell^a(\mathbf{x})$.

The two following lemmas give information on the vectorial chamber of $v(\lambda + \beta^{\vee})$.

**Lemma 3.6.** Let $\lambda \in Y^{++}$ be a dominant coweight and let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^{\vee} \in Y^+$. Let $u \in W$ be such that $\lambda + \beta^{\vee}$ belongs to the vectorial chamber $C_u^v$, that is to say $u^{-1}(\lambda + \beta^{\vee}) \in Y^{++}$. Then

$$ \ell(s_{\beta}u) = \ell(s_{\beta}) + \ell(u). \qquad (3.10) $$

*Proof.* Let $s_{\tau_1} \dots s_{\tau_n}$ be a reduced expression of $u$, so that $\ell(u) = n$ and

$$ \mathrm{Inv}(u^{-1}) = \{\tau_1, s_{\tau_1}(\tau_2), \dots, s_{\tau_1} \dots s_{\tau_{n-1}}(\tau_n)\}. $$

We show that $s_{\tau_{k+1}} \dots s_{\tau_1} s_{\beta} > s_{\tau_k} \dots s_{\tau_1} s_{\beta}$ for all $k \in [0, n-1]$.

For any $\alpha \in \mathrm{Inv}(u^{-1})$ we have $\langle \lambda + \beta^{\vee}, \alpha \rangle \le 0$ (because $\lambda + \beta^{\vee} \in C_u^v$). Since $\lambda$ is dominant this implies $\langle \beta^{\vee}, \alpha \rangle \le 0$.

Let $k \in [0, n-1]$. Then $\langle \beta^{\vee}, s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) \rangle \le 0$, so: $s_{\beta}(s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1})) = s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) - \langle \beta^{\vee}, s_{\tau_1} \dots s_{\tau_k}(\tau_{k+1}) \rangle \beta$ is a positive root as a sum of positive roots. Thus $s_{\tau_{k+1}} \dots s_{\tau_1} s_{\beta} > s_{\tau_k} \dots s_{\tau_1} s_{\beta}$ for any $k \in [0, n-1]$ and therefore $\ell(s_{\beta}u) = \ell(u^{-1}s_{\beta}) = n + \ell(s_{\beta}) = \ell(s_{\beta}) + \ell(u)$. $\square$

**Lemma 3.7.** Let $\lambda \in Y^{++}$ be a dominant coweight and let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^{\vee} \in Y^+$. Let $v \in W^{\lambda}$, $w \in W$ and let $u$ denote the element $v^{\lambda+\beta^{\vee}}$.

Then, if $\pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w$ (resp. $\pi^{v s_{\beta}(\lambda+\beta^{\vee})} s_{v(\beta)} w$) covers $\mathbf{x} = \pi^{v(\lambda)} w$,

$$ \ell(vu) = \ell(v) + \ell(u) \quad (\text{resp. } \ell vs_{\beta}u = \ell(v) + \ell(s_{\beta}u) \text{ and } \ell vs_{\beta} = \ell(v) + \ell(s_{\beta})). \qquad (3.11) $$

*Proof.* To simplify the notation, write $W_J$ for $W_{(\lambda+\beta^{\vee})^{++}}$. Note that, with the notation of Definition 1.19, $vu = \mathrm{proj}_{vuW_J}(v)$ since $u$ is the element of minimal length in $uW_J$.

Suppose by contradiction that $\pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w$ covers $\mathbf{x}$ with $\ell(vu) < \ell(v) + \ell(u)$. Then $d^{\mathbb{N}}(1, vu) = \ell(vu) < d^{\mathbb{N}}(1, v) + d^{\mathbb{N}}(v, vu) = \ell(v) + \ell(u)$, so $v$ is not on a minimal gallery from 1 to $vu$. Therefore by Lemma 1.18, there is a reflection $r \in W$ such that $d(1, rvu) > d(1, vu)$ and $d(1, rv) < d(1, v)$, that is to say $rv < v$ and $rvu > vu$.

By minimality of $u$, $r$ is not in $vuW_J(vu)^{-1}$: otherwise $rvu \in vuW_J$ verifies $d(v, rvu) = d rv, vu) < d(v, vu)$, because foldings reduce the vectorial distance and $v, vu$ are on different sides of the wall $M_r$ associated to $r$.

Since $vu$ is the projection of $v$ on $vuW_J$ which is convex (see [Ron89, Lemma 2.10]), and since the wall $M_r$ separates $v$ and $vu$, any element of $vuW_J$ is on the same side of the wall $M_r$ as $vu$, so $rvu\tilde{u} > vu\tilde{u}$ for any $\tilde{u} \in W_J$. In particular, let $\tilde{u} \in W_J$ be such that $rvu\tilde{u}$ is the minimal coset representative of $rvuW_J$. Then by Proposition 2.3, since $rvu\tilde{u} > vu\tilde{u}$, we have

$$ \pi^{rv(\lambda+\beta^{\vee})} rs_{v(\beta)} w = \pi^{rvu\tilde{u}((\lambda+\beta^{\vee})^{++})} rs_{v(\beta)} w < \pi^{vu\tilde{u}((\lambda+\beta^{\vee})^{++})} s_{v(\beta)} w = \pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)} w. \qquad (3.12) $$
Therefore by Proposition 2.3 for the left and right hand side inequalities and Formula (3.8) applied with $(\mu_0, \alpha_0, v_0, w_0, m) = (\lambda, \beta, rv, rw, 1)$ for the middle one, we have a chain

$$ \pi^{v(\lambda)}w < \pi^{rv(\lambda)}rw < \pi^{rv(\lambda+\beta^\vee)}s_{rv(\beta)}rw = \pi^{rv(\lambda+\beta^\vee)}rs_{v(\beta)}w < \pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w. \quad (3.13) $$

Therefore if $\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w$ covers $\mathbf{x}$ then $\ell(vu) = \ell(v) + \ell(u)$.

Now assume by contradiction that $\pi^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w$ covers $\pi^\lambda w$ with $\ell(\text{vs}_\beta u) < \ell(v) + \ell(s_\beta u)$. Then, similarly there is a reflection $r \in W$ such that $rv < v$ and $rvs_\beta u\tilde{u} > \text{vs}_\beta u\tilde{u}$. By Proposition 2.3 for the left and right hand side inequalities and Formula (3.9) applied with $(\mu_0, \alpha_0, v_0, w_0, m) = (\lambda, \beta, rv, rw, 1)$ for the middle one, we have a chain

$$ \pi^{v(\lambda)}w < \pi^{rv(\lambda)}rw < \pi^{rvs_\beta(\lambda+\beta^\vee)}s_{rv(\beta)}rw = \pi^{rvs_\beta(\lambda+\beta^\vee)}rs_{v(\beta)}w < \pi^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w. \quad (3.14) $$

We deduce that, if $\pi^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w$ covers $\mathbf{x}$ then $\ell(\text{vs}_\beta u) = \ell(v) + \ell(s_\beta u)$. By Lemma 3.6 this is $\ell(v) + \ell(s_\beta) + \ell(u)$, by the triangle inequality we deduce that $\ell(v) + \ell(s_\beta) \ge \ell(\text{vs}_\beta) \ge \ell(\text{vs}_\beta u) - \ell(u) = \ell(v) + \ell(s_\beta)$ and we obtain the second equality in this case. $\square$

**Proposition 3.8.** Let $\lambda \in Y^{++}$, $v \in W^\lambda$, $w \in W$. Let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^\vee \in Y^+$ and let $u$ denote $v^{\lambda+\beta^\vee} \in W^{(\lambda+\beta^\vee)++}$.

Suppose that $\mathbf{y} \in \{\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w, \pi^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w\}$ covers $\mathbf{x} = \pi^{v(\lambda)}w$. Then

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \left( 2\mathrm{ht}(\beta^\vee) - \ell(s_\beta) \right) - 2 \left( \ell(u) + \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^\vee, \tau \rangle \right). \quad (3.15) $$

*Proof.* Let $W_J$ denote the standard parabolic subgroup $W_{(\lambda+\beta^\vee)++}$. Recall that $u = v^{\lambda+\beta^\vee}$ is the minimal element of $W$ such that $u((\lambda + \beta^\vee)^{++}) = \lambda + \beta^\vee$, so it is the minimal representative of the coset $uW_J$. By Proposition 2.10 we have

$$ \ell^a(\pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w) - \ell^a(\pi^{v(\lambda)}w) = 2\mathrm{ht}((\lambda + \beta^\vee)^{++}) - 2\mathrm{ht}(\lambda) + \ell_{v^{v(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w) \quad (3.16) $$

$$ \ell^a(\pi^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w) - \ell^a(\pi^{v(\lambda)}w) = 2\mathrm{ht}((\lambda + \beta^\vee)^{++}) - 2\mathrm{ht}(\lambda) + \ell_{v^{vs_\beta(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w). \quad (3.17) $$

We unwrap these formulas with the help of previous results.

1. In the case $\mathbf{y} = \pi^{v(\lambda+\beta^\vee)}s_{v(\beta)}w$, let $\tilde{u} \in W_J$ be such that $vu\tilde{u} = (vu)^J = v^{v(\lambda+\beta^\vee)}$. The term $\ell_{v^{v(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w)$ rewrites as $\ell((u\tilde{u})^{-1}s_\beta v^{-1}w) - \ell(vu\tilde{u}) - \ell(v^{-1}w) + \ell(v)$. Since $\mathbf{y} > \mathbf{x}$ is a covering, by Proposition 3.3, $\text{vs}_\beta u\tilde{u} = s_{v(\beta)}(vu)^J$ is on a minimal gallery from $v$ to $w$, so $\ell(v^{-1}w) = \ell((\text{vs}_\beta u\tilde{u})^{-1}w) + \ell(s_\beta u\tilde{u})$. Moreover by Lemma 3.6, $\ell(s_\beta u\tilde{u}) = \ell(s_\beta) + \ell(u\tilde{u})$ and, by Lemma 3.7, $\ell(vu) = \ell(v) + \ell(u)$. Finally, by Equation (1.3), since $u = u^J = v^{\lambda+\beta^\vee}$ and $vu\tilde{u} = (vu)^J = v^{v(\lambda+\beta^\vee)}$, we have $\ell(u\tilde{u}) = \ell(u) + \ell(\tilde{u})$ and $\ell(vu) = \ell(vu\tilde{u}) + \ell(\tilde{u})$. Thus

$$ \begin{aligned} \ell_{v^{v(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w) &= \ell((u\tilde{u})^{-1}s_\beta v^{-1}w) - \ell(v^{-1}w) - \ell(vu\tilde{u}) + \ell(v) \\ &= -\ell(s_\beta u\tilde{u}) - \ell(vu) + \ell(\tilde{u}) + \ell(v) \\ &= -\ell(s_\beta) - \ell(u\tilde{u}) - \ell(u) + \ell(\tilde{u}) \\ &= -\ell(s_\beta) - 2\ell(u). \end{aligned} \quad (3.18) $$
Therefore by Proposition 2.3 for the left and right hand side inequalities and Formula (3.8) applied with $(\mu_0, \alpha_0, v_0, w_0, m) = (\lambda, \beta, rv, rw, 1)$ for the middle one, we have a chain

$$ \pi^{v(\lambda)}w < \pi^{rv(\lambda)}rw < \pi^{rv(\lambda+\beta^{\vee})}s_{rv(\beta)}rw = \pi^{rv(\lambda+\beta^{\vee})}rs_{v(\beta)}w < \pi^{v(\lambda+\beta^{\vee})}s_{v(\beta)}w. \quad (3.13) $$

Therefore if $\pi^{v(\lambda+\beta^{\vee})}s_{v(\beta)}w$ covers $\mathbf{x}$ then $\ell(vu) = \ell(v) + \ell(u)$.

Now assume by contradiction that $\pi^{vs_{\beta}(\lambda+\beta^{\vee})}s_{v(\beta)}w$ covers $\pi^{\lambda}w$ with $\ell(\text{vs}_{\beta}u) < \ell(v) + \ell(s_{\beta}u)$. Then, similarly there is a reflection $r \in W$ such that $rv < v$ and $rvs_{\beta}u\tilde{u} > vs_{\beta}u\tilde{u}$. By Proposition 2.3 for the left and right hand side inequalities and Formula (3.9) applied with $(\mu_0, \alpha_0, v_0, w_0, m) = (\lambda, \beta, rv, rw, 1)$ for the middle one, we have a chain

$$ \pi^{v(\lambda)}w < \pi^{rv(\lambda)}rw < \pi^{rvs_{\beta}(\lambda+\beta^{\vee})}s_{rv(\beta)}rw = \pi^{rvs_{\beta}(\lambda+\beta^{\vee})}rs_{v(\beta)}w < \pi^{vs_{\beta}(\lambda+\beta^{\vee})}s_{v(\beta)}w. \quad (3.14) $$

We deduce that, if $\pi^{vs_{\beta}(\lambda+\beta^{\vee})}s_{v(\beta)}w$ covers $\mathbf{x}$ then $\ell(\text{vs}_{\beta}u) = \ell(v) + \ell(s_{\beta}u)$. By Lemma 3.6 this is $\ell(v) + \ell(s_{\beta}) + \ell(u)$, by the triangle inequality we deduce that $\ell(v) + \ell(s_{\beta}) \ge \ell(\text{vs}_{\beta}) \ge \ell(\text{vs}_{\beta}u) - \ell(u) = \ell(v) + \ell(s_{\beta})$ and we obtain the second equality in this case. $\square$

**Proposition 3.8.** Let $\lambda \in Y^{++}$, $v \in W^{\lambda}$, $w \in W$. Let $\beta \in \Phi_+$ be a positive root such that $\lambda + \beta^{\vee} \in Y^+$ and let $u$ denote $v^{\lambda+\beta^{\vee}} \in W^{(\lambda+\beta^{\vee})++}$.

Suppose that $\mathbf{y} \in \{\pi^{v(\lambda+\beta^{\vee})}s_{v(\beta)}w, \pi^{vs_{\beta}(\lambda+\beta^{\vee})}s_{v(\beta)}w\}$ covers $\mathbf{x} = \pi^{v(\lambda)}w$. Then

$$ \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) = \left( 2\mathrm{ht}(\beta^{\vee}) - \ell(s_{\beta}) \right) - 2 \left( \ell(u) + \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^{\vee}, \tau \rangle \right). \quad (3.15) $$

*Proof.* Let $W_J$ denote the standard parabolic subgroup $W_{(\lambda+\beta^{\vee})++}$. Recall that $u = v^{\lambda+\beta^{\vee}}$ is the minimal element of $W$ such that $u((\lambda + \beta^{\vee})^{++}) = \lambda + \beta^{\vee}$, so it is the minimal representative of the coset $uW_J$. By Proposition 2.10 we have

$$ \ell^a(\pi^{v(\lambda+\beta^{\vee})} s_{v(\beta)}w) - \ell^a(\pi^{v(\lambda)}w) = 2\mathrm{ht}((\lambda + \beta^{\vee})^{++}) - 2\mathrm{ht}(\lambda) + \ell_{v^{v(\lambda+\beta^{\vee})}}(s_{v(\beta)}w) - \ell_v(w) \quad (3.16) $$

$$ \ell^a(\pi^{vs_{\beta}(\lambda+\beta^{\vee})} s_{v(\beta)}w) - \ell^a(\pi^{v(\lambda)}w) = 2\mathrm{ht}((\lambda + \beta^{\vee})^{++}) - 2\mathrm{ht}(\lambda) + \ell_{v^{vs_{\beta}(\lambda+\beta^{\vee})}}(s_{v(\beta)}w) - \ell_v(w). \quad (3.17) $$

We unwrap these formulas with the help of previous results.

1. In the case $\mathbf{y} = \pi^{v(\lambda+\beta^{\vee})}s_{v(\beta)}w$, let $\tilde{u} \in W_J$ be such that $vu\tilde{u} = (vu)^J = v^{v(\lambda+\beta^{\vee})}$. The term $\ell_{v^{v(\lambda+\beta^{\vee})}}(s_{v(\beta)}w) - \ell_v(w)$ rewrites as $\ell((u\tilde{u})^{-1}s_{\beta}v^{-1}w) - \ell(vu\tilde{u}) - \ell(v^{-1}w) + \ell(v)$. Since $\mathbf{y} > \mathbf{x}$ is a covering, by Proposition 3.3, $\text{vs}_{\beta}u\tilde{u} = s_{v(\beta)}(vu)^J$ is on a minimal gallery from $v$ to $w$, so $\ell(v^{-1}w) = \ell((\text{vs}_{\beta}u\tilde{u})^{-1}w) + \ell(s_{\beta}u\tilde{u})$. Moreover by Lemma 3.6, $\ell(s_{\beta}u\tilde{u}) = \ell(s_{\beta}) + \ell(u\tilde{u})$ and, by Lemma 3.7, $\ell(vu) = \ell(v) + \ell(u)$. Finally, by Equation (1.3), since $u = u^J = v^{\lambda+\beta^{\vee}}$ and $vu\tilde{u} = (vu)^J = v^{v(\lambda+\beta^{\vee})}$, we have $\ell(u\tilde{u}) = \ell(u) + \ell(\tilde{u})$ and $\ell(vu) = \ell(vu\tilde{u}) + \ell(\tilde{u})$. Thus

$$ \begin{aligned} \ell_{v^{v(\lambda+\beta^{\vee})}}(s_{v(\beta)}w) - \ell_v(w) &= \ell((u\tilde{u})^{-1}s_{\beta}v^{-1}w) - \ell(v^{-1}w) - \ell(vu\tilde{u}) + \ell(v) \\ &= -\ell(s_{\beta}u\tilde{u}) - \ell(vu) + \ell(\tilde{u}) + \ell(v) \\ &= -\ell(s_{\beta}) - \ell(u\tilde{u}) - \ell(u) + \ell(\tilde{u}) \\ &= -\ell(s_{\beta}) - 2\ell(u). \end{aligned} \quad (3.18) $$
2. In the second case, let $\tilde{u} \in W_J$ be such that $vs_\beta u\tilde{u} = (vs_\beta u)^J = v^{vs_\beta(\lambda+\beta^\vee)}$, then $\ell_{v^{vs_\beta(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w)$ rewrites as $\ell((u\tilde{u})^{-1}v^{-1}w) - \ell(vs_\beta u\tilde{u}) - \ell(v^{-1}w) + \ell(v)$. By Proposition 3.3, $\ell((u\tilde{u})^{-1}v^{-1}w) = \ell(v^{-1}w) - \ell(u\tilde{u})$. By Equation (1.3), $\ell(u\tilde{u}) = \ell(u) + \ell(\tilde{u})$ and $\ell(vs_\beta u\tilde{u}) = \ell(vs_\beta u) - \ell(\tilde{u})$. By Lemma 3.7 and 3.6, $\ell(vs_\beta u) = \ell(v) + \ell(s_\beta u) = \ell(v) + \ell(s_\beta) + \ell(u)$.

Thus in this case:

$$ \begin{aligned} \ell_{v^{vs_\beta(\lambda+\beta^\vee)}}(s_{v(\beta)}w) - \ell_v(w) &= \ell((u\tilde{u})^{-1}v^{-1}w) - \ell(vs_\beta u\tilde{u}) - \ell(v^{-1}w) + \ell(v) \\ &= \ell(v^{-1}w) - \ell(u\tilde{u}) - (\ell(vs_\beta u) - \ell(\tilde{u})) - \ell(v^{-1}w) + \ell(v) \\ &= -\ell(s_\beta) - 2\ell(u). \end{aligned} \quad (3.19) $$

3. By Lemma (1.11) we have

$$ \begin{aligned} 2\mathrm{ht}((\lambda + \beta^{\vee})^{++}) &= 2(\mathrm{ht}(\lambda + \beta^{\vee}) - \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^{\vee}, \tau \rangle) \\ &= 2(\mathrm{ht}(\lambda) + \mathrm{ht}(\beta^{\vee}) - \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^{\vee}, \tau \rangle). \end{aligned} \quad (3.20) $$

By reinjecting Expressions (3.18), (3.20) in Formula (3.16), and Expressions (3.19), (3.20) in Formula (3.17) we obtain either way:

$$ \begin{aligned} \ell^a(\mathbf{y}) - \ell^a(\mathbf{x}) &= 2\mathrm{ht}(\lambda) + 2\mathrm{ht}(\beta^\vee) - 2 \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^\vee, \tau \rangle - 2\mathrm{ht}(\lambda) - \ell(s_\beta) - 2\ell(u) \\ &= \left( 2\mathrm{ht}(\beta^\vee) - \ell(s_\beta) \right) - 2 \left( \ell(u) + \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^\vee, \tau \rangle \right). \end{aligned} $$

□

Using Lemma 1.10, it is easy to see that $2\mathrm{ht}(\beta^\vee) - \ell(s_\beta)$ is always positive and that, on the contrary, $\ell(u) + \sum_{\tau \in \mathrm{Inv}(u^{-1})} \langle \lambda + \beta^\vee, \tau \rangle$ is always non-positive. Therefore, the length difference is equal to 1 if and only if in the right-hand side of Equation (3.15), the first term is equal to 1 and the second term cancels out. This motivates the following definitions.

**Definition 3.9.** A coweight $\mu \in Y^+$ is **almost dominant** if and only if

$$ \forall \tau \in \Phi_+, \langle \mu, \tau \rangle \geq -1. \quad (3.21) $$

A root $\beta \in \Phi_+$ is **a quantum root** if and only if

$$ \ell(s_\beta) = 2\mathrm{ht}(\beta^\vee) - 1. \quad (3.22) $$