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Affine Bruhat order and Demazure products

Published online by Cambridge University Press:  15 April 2024

Felix Schremmer*
Affiliation:
Department of Mathematics and New Cornerstone Science Laboratory, The University of Hong Kong, Pok Fu Lam, Hong Kong, China
*

Abstract

We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points.

Type
Algebra
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

1. Introduction

Let us begin by considering a Coxeter group $(W, S)$ . The Bruhat order on W can be defined by inclusion of reduced words, namely $x_1\leq x_2$ if some reduced word for $x_1$ can be obtained from some fixed reduced word for $x_2$ by deleting any number of letters. This partial order is of central importance for the general theory of Coxeter groups, and it enjoys a number of remarkable properties and applications [Reference Björner and Brenti2, Chapter 2 and beyond]. For example, the Kazhdan–Lusztig polynomials associated with $(W, S)$ satisfy that $P_{u,v}\neq 0$ if and only if $u\leq v$ [Reference Björner and Brenti2, Proposition 5.1.5].

Related to this is the notion of Demazure products. The Demazure product $x_1\ast x_2$ of two elements $x_1, x_2\in W$ is the largest element of the form $x_1' x_2'\in W$ where $x_1'\leq x_1$ and $x_2'\leq x_2$ in the Bruhat order. The Demazure product describes the multiplication in the $0$ -Hecke algebra of $(W, S)$ , cf. [Reference He and Nie12, Section 1.2]. It, too, has a number of remarkable properties and applications.

In this paper, we focus on a specific class of (quasi-)Coxeter groups, namely affine Weyl groups. These groups arise naturally in the context of arithmetic geometry. In a sense, affine Weyl groups are the “simplest” examples of infinite Coxeter groups, so they are also important examples from a pure Coxeter theoretic viewpoint.

If G is a connected reductive group over a non-Archimedian local field F, we get an associated extended affine Weyl group $\widetilde W$ . This group famously occurs as the indexing set of the Iwahori–Bruhat decomposition

$$ \begin{align*} G(\breve F) = \bigsqcup_{x\in \widetilde W} IxI. \end{align*} $$

Here, $\breve F$ is the maximal unramified extension of F, and $I\subseteq G(\breve F)$ is an Iwahori subgroup.

The closure relations of the above decomposition are precisely given by the Bruhat order, that is,

$$ \begin{align*} \overline{IxI} = \bigsqcup_{y\leq x}IyI\subseteq G(\breve F). \end{align*} $$

If $x, y\in \widetilde W$ , the product $IxI\cdot IyI\subseteq G(\breve F)$ will in general not be of the form $IzI$ for any $z\in \widetilde W$ . However, if we pass to closures, we have

$$ \begin{align*} \overline{IxIyI} = \overline{I(x\ast y)I} \end{align*} $$

for the Demazure product.

The Iwahori–Bruhat decomposition has been studied intensively, partly because of its connection to the Bruhat–Tits building [Reference Bruhat and Tits5, Section 4]. Due to this, both the Bruhat order and Demazure products of affine Weyl groups have been used and studied in the past. We mention the definition of admissible sets due to Kottwitz and Rapoport [Reference Kottwitz and Rapoport15, Reference Rapoport, Jacques, Henri, Michael and Marie-France25], the description of generic Newton points in terms of the Bruhat order due to Viehmann [Reference Viehmann31] and the recent works on generic Newton points and Demazure products due to He and Nie [Reference He11, Reference He and Nie12].

The Iwahori Hecke algebra $\mathcal H$ of G, that received tremendous interest starting with the discovery of the Satake isomorphism [Reference Satake27], can be defined as follows: $\mathcal H$ is an algebra over $\mathbb Z[v, v^{-1}]$ , and it is a free $\mathbb Z[v^{\pm 1}]$ module with basis given by $\{T_x\mid x\in \widetilde W\}$ . The multiplication is defined by

$$ \begin{align*} T_x T_y &= T_{xy}\qquad\qquad\qquad\text{ if }\ell(xy) = \ell(x) + \ell(y), \\T_s^2 &=(v -v^{-1})T_s+1~~\text{ if }s\in \widetilde W\text{ is a simple affine reflection}. \end{align*} $$

The multiplication of the Iwahori Hecke algebra is quite complicated and poorly understood. For $x, y\in \widetilde W$ , the product $T_x T_y$ will in general have the form

$$ \begin{align*} T_x T_y = \sum_{z\in \widetilde W}f_{x,y,z}(v-v^{-1})T_z \end{align*} $$

for some polynomials $f_{x,y,z}(X)\in \mathbb Z[X]$ . This product $T_x T_y$ can be seen as a combinatorial model for the multiplication of Iwahori double cosets $IxI\cdot IyI$ in $G(\breve F)$ . Among all $z\in \widetilde W$ such that $f_{x,y,z}\neq 0$ , there is a unique largest one, which is the Demazure product $z = x\ast y$ . We may summarize that understanding Demazure products is a first step towards fully understanding the multiplication in Iwahori Hecke algebras, which is related to important geometric problems. For example, the dimensions of affine Deligne–Lusztig varieties can be expressed in terms of degrees of class polynomials of the Iwahori–Hecke algebra [Reference He10, Theorem 6.1]. In view of this connection, our result on Demazure products is enough to describe generic Newton points associated with the Iwahori–Bruhat decomposition of an algebraic group.

Our main results fully describe the Bruhat order and Demazure products for $\widetilde W$ . We refer to the corresponding sections for the most general statements. To summarize our results roughly, recall that each element $x\in \widetilde W$ can be written as $x = w\varepsilon ^\mu $ , where w is an element of the finite Weyl group W and $\mu $ is an element of an abelian group denoted $X_\ast $ (that can be chosen as the coweight lattice of our root system). By ${\mathrm {wt}} : W\times W\rightarrow X_\ast $ , we denote the weight function of the quantum Bruhat graph, cf. Section 3.

Theorem 1.1. Let $x_1, x_2\in \widetilde W$ , and write them as $x_1 = w_1\varepsilon ^{\mu _1}, x_2 = w_2 \varepsilon ^{\mu _2}$ . Then $x_1\leq x_2$ in the Bruhat order if and only if for each $v_1\in W$ , there exists some $v_2\in W$ satisfying

$$ \begin{align*} v_1^{-1}\mu_1 + {\mathrm{wt}}(v_2\Rightarrow v_1) + {\mathrm{wt}}(w_1 v_1\Rightarrow w_2 v_2)\leq v_2^{-1}\mu_2. \end{align*} $$

For more refined descriptions of the Bruhat order, we refer to Theorems 4.2 and 4.33 as well as Remark 5.23. The order of quantifiers in the above theorem is essential: If one were to instead ask for the analogous condition of the form $\forall v_2 \exists v_1$ , neither implication of Theorem 1.1 would be true. One can easily construct counterexamples by choosing one of the elements $x_1, x_2$ to be $1\in \widetilde W$ and the other one to be of very large length.

The description of Demazure products has the following form:

Theorem 1.2 (Cf. Theorem 5.11).

Let $x_1, x_2\in \widetilde W$ , written as $x_1 = w_1\varepsilon ^{\mu _1}$ and $x_2 = w_2 \varepsilon ^{\mu _2}$ . Then for explicitly described $v_1, v_2\in W$ , we have

$$ \begin{align*} x_1 \ast x_2 = w_1 v_1 v_2^{-1}\varepsilon^{v_2 v_1^{-1}\mu_1 + \mu_2 - v_2{\mathrm{wt}}(v_1\Rightarrow w_2 v_2)}. \end{align*} $$

As an application of our results, we describe the admissible sets as introduced in [Reference Kottwitz and Rapoport15] and [Reference Rapoport, Jacques, Henri, Michael and Marie-France25] as Propositions 4.12 and 4.35. We also get an explicit description of Bruhat covers in $\widetilde W$ (Proposition 4.5) and of the semi-infinite order on $\widetilde W$ (Corollary 4.10). Finally, combining the aforementioned result of Viehmann [Reference Viehmann31] with ideas of He [Reference He11], we present a new description of generic Newton points (Theorem 5.29).

The methods of this paper build upon a previous paper by the same author [Reference Schremmer28]. In particular, the language and results on length functionals as introduced there will be used throughout this paper. To complement the combinatorial prerequisites, this paper introduces and proves a number of new properties of the quantum Bruhat graph in Sections 3 and 5.2. These new results on the quantum Bruhat graph are not only the foundation of our results on the Bruhat order and Demazure products, they also may have potentially further-reaching applications, given the previous usage of the quantum Bruhat graph for quantum cohomology [Reference Postnikov24] or Kirillov–Reshetikhin crystals [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Reference Lenart, Naito, Sagaki, Schilling and Shimozono18]. In addition to the previously studied weight functions of the (parabolic) quantum Bruhat graph, we introduce a new semiaffine weight function.

Note that while both this paper and our previous paper [Reference Schremmer28] provide explicit formulas for generic Newton points, these results are actually complementing rather than overlapping. In terms of logical dependencies, this paper only relies on the discussion of root functionals and length positivity in Section 2.2 of [Reference Schremmer28] and is otherwise independent. Together, both papers cover the contents of the author’s PhD thesis.

2. Affine root system

In this section, we describe the fundamental root-theoretic setup. In the literature, there are several different notions of affine Weyl groups studied in different contexts, so we present a uniform setup that covers all cases. Readers with a combinatorial background are invited to consider any reduced root datum, whereas readers whose background is closer to arithmetic geometry may find more appealing to have the context of an algebraic group, as presented, for example, in [Reference Schremmer28, Section 2.1].

Let $\Phi $ be a reduced crystallographic root system. We choose a basis $\Delta \subseteq \Phi $ and denote the set of positive/negative roots by $\Phi ^{\pm }$ .

Let $X_\ast $ denote an abelian group with a fixed embedding of the coroot lattice $\mathbb Z\Phi ^\vee \subseteq X_\ast $ . The group $X_\ast $ is allowed to have a torsion part. We assume that a bilinear map

$$ \begin{align*} \langle\cdot,\cdot\rangle : X_\ast\otimes \mathbb Z\Phi\rightarrow\mathbb Z \end{align*} $$

has been chosen that extends the natural pairing between $\Phi ^\vee $ and $\Phi $ . For example, both the coroot lattice $X_\ast = \mathbb Z\Phi ^\vee $ and the coweight lattice $X_\ast = {\mathrm {Hom}}_{\mathbb Z}(\mathbb Z\Phi , \mathbb Z)$ are possible choices for $X_\ast $ . We turn $X_\ast $ and $X_\ast \otimes \mathbb Q$ into ordered abelian groups by defining that $\mu _1\leq \mu _2$ if $\mu _2-\mu _1$ is a $\mathbb Z_{\geq 0}$ -linear, resp. $\mathbb Q_{\geq 0}$ -linear, combination of positive coroots. An element $\mu $ in $X_\ast $ or $X_\ast \otimes \mathbb Q$ will be called C-regular for some constant $C>0$ if ${\left \lvert {\langle \mu ,\alpha \rangle }\right \rvert }\geq C$ for all $\alpha \in \Phi $ . Typically, we will not specify the constant and talk of sufficiently regular or superregular elements. An element $\mu $ in $X_\ast $ or $X_\ast \otimes \mathbb Q$ is dominant if $\langle \mu ,\alpha \rangle \geq 0$ for each positive root $\alpha $ .

Denote the Weyl group of $\Phi $ by W and the set of simple reflections by

$$ \begin{align*} S = \{s_\alpha\mid\alpha\in \Delta\}\subseteq W. \end{align*} $$

The Weyl group W acts on $X_\ast $ via the usual convention

$$ \begin{align*} s_\alpha(\mu) = \mu-\langle\mu,\alpha\rangle\alpha^\vee,\qquad \alpha\in \Phi,~\mu\in X_\ast. \end{align*} $$

The semidirect product $\widetilde W:=W\ltimes X_\ast $ is called extended affine Weyl group. Elements in $\widetilde W$ will typically be expressed as $x = w\varepsilon ^\mu \in \widetilde W$ with $w\in W$ and $\mu \in X_\ast $ .

By abuse of notation, we write $\Phi ^+$ for the indicator function of positive roots, that is,

$$ \begin{align*} \Phi^+(\alpha) :=\begin{cases}1,&\alpha\in \Phi^+,\\ 0,&\alpha\in \Phi^-.\end{cases} \end{align*} $$

The following easy facts will be used often, usually without further reference:

Lemma 2.1. Let $\alpha \in \Phi $ .

  1. (a) $\Phi ^+(\alpha ) + \Phi ^+(-\alpha )=1$ .

  2. (b) If $\beta \in \Phi $ and $k,\ell \geq 1$ are such that $k\alpha +\ell \beta \in \Phi $ , we have

    $$ \begin{align*} &0\leq \Phi^+(\alpha)+\Phi^+(\beta)-\Phi^+(k\alpha+\ell\beta)\leq 1. \end{align*} $$

The sets of affine roots, positive affine roots, negative affine roots and simple affine roots are given by

$$ \begin{align*} \Phi_{\mathrm{af}} &:=\Phi\times\mathbb Z, \\\Phi_{\mathrm{af}}^+ &:=(\Phi^+\times \mathbb Z_{\geq 0})\sqcup (\Phi^-\times \mathbb Z_{\geq 1}) = \{(\alpha,k)\in \Phi_{\mathrm{af}}\mid k\geq \Phi^+(-\alpha)\}, \\\Phi_{\mathrm{af}}^- &:=-\Phi_{\mathrm{af}}^+ = \Phi_{\mathrm{af}}\setminus \Phi_{\mathrm{af}}^+= \{(\alpha,k)\in \Phi_{\mathrm{af}}\mid k< \Phi^+(-\alpha)\}, \\\Delta_{\mathrm{af}}&:= \{(\alpha,0)\mid\alpha\in \Delta\}\cup\\&~\{(-\theta,1)\mid\theta\text{ is the longest root of an irreducible component }\Phi'\subseteq \Phi\}\subseteq \Phi_{\mathrm{af}}^+. \end{align*} $$

One checks that the positive affine roots are precisely those affine roots which are a sum of simple affine roots.

The action of $\widetilde W$ on $\Phi _{\mathrm {af}}$ is given by

$$ \begin{align*} (w\varepsilon^\mu)(\alpha,k) := (w\alpha,k-\langle \mu,\alpha\rangle). \end{align*} $$

The length of an element $x=w\varepsilon ^\mu \in \widetilde W$ is defined as

$$ \begin{align*} \ell(x) := \#\{a\in \Phi_{\mathrm{af}}^+\mid xa\in\Phi_{\mathrm{af}}^-\}. \end{align*} $$

Associated to each affine root $a=(\alpha ,k)$ , we have the affine reflection

$$ \begin{align*} r_a = s_\alpha \varepsilon^{k\alpha^\vee}\in \widetilde W. \end{align*} $$

Denote by $W_{\mathrm {af}}\subseteq W$ the subgroup generated by the affine reflections (called affine Weyl group), and write $S_{\mathrm {af}} := \{r_a\mid a\in \Delta _{\mathrm {af}}\}$ (the set of simple affine reflections). It is easy to check that $(W_{\mathrm {af}}, S_{\mathrm {af}})$ is a Coxeter group with length function $\ell $ as defined above, and $W_{\mathrm {af}} = W\ltimes \mathbb Z\Phi ^\vee \subseteq \widetilde W$ .

Denoting the subgroup of length zero elements of $\widetilde W$ by $\Omega \leq \widetilde W$ , we get a semidirect product decomposition $\widetilde W = \Omega \ltimes W_{\mathrm {af}}$ .

The Bruhat order on $W_{\mathrm {af}}$ is the usual Coxeter-theoretic notion. We define the Bruhat order on $\widetilde W$ by declaring that

$$ \begin{align*} \omega_1 x_1\leq \omega_2 x_2\iff \left(\omega_1=\omega_2\text{ and }x_1\leq x_2\in W_{\mathrm{af}}\right), \end{align*} $$

where $\omega _1, \omega _2\in \Omega $ and $x_1, x_2\in W_{\mathrm {af}}$ . Equivalently, this is the partial order on $\widetilde W$ generated by the relations $x< xr_a$ for $x\in \widetilde W$ and $a\in \Phi _{\mathrm {af}}$ such that $\ell (x)<\ell (xr_a)$ .

We will occasionally denote the classical part of an affine root $a = (\alpha ,k)$ or an element $x = w\varepsilon ^\mu \in \widetilde W$ by

$$ \begin{align*} {\mathrm{cl}}(a) = \alpha\in \Phi,\qquad {\mathrm{cl}}(x) = w\in W. \end{align*} $$

We need the language of length functionals from [Reference Schremmer28, Section 2.2]. We recall the basic definitions here and refer to the cited paper for some geometric intuition and fundamental properties.

Definition 2.2. Let $x = w\varepsilon ^\mu \in \widetilde W$ .

  1. (a) For $\alpha \in \Phi $ , we define the length functional of x by

    $$ \begin{align*} \ell(x,\alpha) := \langle\mu,\alpha\rangle + \Phi^+(\alpha) - \Phi^+(w\alpha). \end{align*} $$
  2. (b) An element $v\in W$ is called length positive for x, written as $v\in {\mathrm {LP}}(x)$ , if every positive root $\alpha \in \Phi ^+$ satisfies $\ell (x,v\alpha )\geq 0$ .

  3. (c) If $v\in W$ is not length positive for x and $\alpha \in \Phi ^+$ satisfies $\ell (x,v\alpha )<0$ , we call $vs_\alpha \in W$ an adjustment of v for $\ell (x,\cdot )$ .

The name “length functional” comes from the fact that the length of x can be expressed as the sum of all positive values $\ell (x,\alpha )$ for $\alpha \in \Phi $ .

We prove in [Reference Schremmer28, Lemma 2.3] that iteratively adjusting any given $v\in W$ yields a length positive element for x. The following characterization of length positive elements will frequently come in handy:

Lemma 2.3 [Reference Schremmer28, Corollary 2.11].

Let $x = w\varepsilon ^\mu \in \widetilde W$ and $v\in W$ . Then

$$ \begin{align*} \ell(x)\geq \langle v^{-1}\mu,2\rho\rangle - \ell(v) + \ell(wv). \end{align*} $$

Equality holds if and only if v is length positive for x.

The length functional can be used to characterize the shrunken Weyl chambers [Reference Schremmer28, Proposition 2.15]: The element $x \in \widetilde W$ lies in a shrunken Weyl chamber if and only if $\ell (x,\alpha )\neq 0$ for all $\alpha \in \Phi $ , which is equivalent to saying that ${\mathrm {LP}}(x)$ contains only one element.

3. Quantum Bruhat graph

In this section, we recall the definition of quantum Bruhat graphs and study its weight functions. Before turning to the abstract theory of these graphs, we will discuss the situation of root systems of type $A_n$ as a motivational example.

For each simple affine root $a = (\alpha ,k)\in \Delta _{\mathrm {af}}$ , we define a coweight $\omega _a \in \mathbb Q \Phi ^\vee $ as follows: For $\beta \in \Delta $ , we define

$$ \begin{align*} \langle \omega_a,\beta\rangle = \begin{cases}1,&\alpha=\beta,\\ 0,&\alpha\neq\beta.\end{cases} \end{align*} $$

In particular, $\omega _a = 0$ if $\alpha \notin \Delta $ .

Let now $x_1 = w_1\varepsilon ^{\mu _1}, x_2 = w_2 \varepsilon ^{\mu _2}\in \widetilde W$ . By [Reference Björner and Brenti2, Theorem 8.3.7], we have

$$ \begin{align*} x_1\leq x_2\iff \forall a,a'\in \Delta_{{\mathrm{af}}}:~(\mu_1 + \omega_a - w_1^{-1}\omega_{a'})^{{\mathrm{dom}}} \leq (\mu_2 + \omega_a -w_2^{-1}\omega_{a'})^{{\mathrm{dom}}}. \end{align*} $$

Here, we write $\nu ^{\mathrm {dom}}\in X_\ast $ for the unique dominant element in the W-orbit of $\nu \in X_\ast $ .

Suppose that $\mu _1$ and $\mu _2$ are sufficiently regular such that we find $v_1, v_2\in W$ with

$$ \begin{align*} \forall a, a'\in \Delta_{{\mathrm{af}}}:~(\mu_i + \omega_a - w_i^{-1}\omega_{a'})^{{\mathrm{dom}}} = v_i^{-1}(\mu_i + \omega_a - w_i^{-1}\omega_{a'}). \end{align*} $$

Then we conclude

$$ \begin{align*} x_1\leq x_2\iff& \forall a, a':~v_1^{-1}(\mu_1 + \omega_a - w_1^{-1}\omega_{a'})\leq v_2^{-1}(\mu_2 + \omega_a - w_2^{-1}\omega_{a'})\\ {\iff}&{v_1^{-1}}\mu_1 + \sup_{a\in \Delta_{{\mathrm{af}}}} (v_1^{-1}\omega_a - v_2^{-1}\omega_a) + \sup_{a'\in \Delta_{{\mathrm{af}}}} (w_2 v_2)^{-1} \omega_{a'} - (w_1 v_1)^{-1}\omega_{a'}\leq v_2^{-1}\mu_2. \end{align*} $$

So if we define

(3.1) $$ \begin{align} {\mathrm{wt}}(v_1\Rightarrow v_2) := \sup_{a\in \Delta_{{\mathrm{af}}}} (v_2^{-1}\omega_a - v_1^{-1}\omega_a), \end{align} $$

we can conclude a version of our result on the Bruhat order (Theorem 1.1).

Indeed, formula (3.1) holds true for root systems of type $A_n$ , but not for any other root system. Many properties of the weight function are easier to prove for type $A_n$ , where an explicit formula exists, so it is helpful to keep this example in mind.

We refer to a paper of Ishii [Reference Ishii14] for explicit formulas for the weight functions of all classical root systems (while he discusses explicit criteria for the semi-infinite order, these can be translated to explicit formulas for the weight function as outlined above in the $A_n$ case).

3.1. (Parabolic) quantum Bruhat graph

We start with a discussion of the quantum roots in $\Phi ^+$ .

Lemma 3.1. Let $\alpha \in \Phi ^+$ . Then

$$ \begin{align*} \ell(s_\alpha) \leq \langle \alpha^\vee,2\rho\rangle-1. \end{align*} $$

Equality holds if and only if for all $\alpha \neq \beta \in \Phi ^+$ with $s_\alpha (\beta )\in \Phi ^-$ , we have $\langle \alpha ^\vee ,\beta \rangle = 1$ .

Roots satisfying the equivalent properties of Lemma 3.1 are called quantum roots. We see that all long roots are quantum (so in a simply laced root system, all roots are quantum). Moreover, all simple roots are quantum.

The first inequality of Lemma 3.1 is due to [Reference Brenti, Fomin and Postnikov4, Lemma 4.3]. By [Reference Braverman, Maulik and Okounkov3, Lemma 7.2], we have the following more explicit (but somehow less useful for us) result: A short root $\alpha $ is quantum if and only if $\alpha $ is a sum of short simple roots.

Proof of Lemma 3.1.

We calculate

$$ \begin{align*} \langle \alpha^\vee, 2\rho\rangle = \frac 12\left(\langle \alpha^\vee, 2\rho\rangle + \langle s_\alpha(\alpha^\vee), s_\alpha(2\rho)\rangle\right) = \frac 12\langle \alpha^\vee, 2\rho - s_\alpha(2\rho)\rangle. \end{align*} $$

Let

$$ \begin{align*} I := \{\beta\in \Phi^+\mid s_\alpha(\beta)\in \Phi^-\}. \end{align*} $$

Then $s_\alpha (I) = -I$ and $s_\alpha (\Phi ^+\setminus I) = \Phi ^+\setminus I$ . It follows that

$$ \begin{align*} 2\rho - s_\alpha(2\rho) &= \sum_{\beta \in I} \left(\beta - s_\alpha(\beta)\right) + \sum_{\beta\in \Phi^+\setminus I} \left(\beta-s_\alpha(\beta)\right) \\ &=2\sum_{\beta\in I} \beta. \end{align*} $$

Therefore, we obtain

$$ \begin{align*} \langle \alpha^\vee, 2\rho\rangle = \sum_{\beta\in I}\langle\alpha^\vee, \beta\rangle. \end{align*} $$

Certainly, $\alpha \in I$ . Hence,

$$ \begin{align*} \langle \alpha^\vee, 2\rho\rangle = 2+\sum_{\substack{\alpha \neq \beta\in \Phi^+\\s_\alpha(\beta)\in \Phi^-}}\langle\alpha^\vee, \beta\rangle. \end{align*} $$

Now, if $\alpha ,\beta \in \Phi ^+$ and $s_\alpha (\beta ) = \beta -\langle \alpha ^\vee , \beta \rangle \alpha \in \Phi ^-$ , we get $\langle \alpha ^\vee , \beta \rangle \geq 1$ . We conclude

$$ \begin{align*} \langle \alpha^\vee, 2\rho\rangle = 2+\sum_{\substack{\alpha \neq \beta\in \Phi^+\\s_\alpha(\beta)\in \Phi^-}}\langle\alpha^\vee, \beta\rangle\geq 2 + \#\{\beta\in \Phi^+\setminus\{\alpha\}\mid s_\alpha(\beta)\in \Phi^-\} = 1+\ell(s_\alpha). \end{align*} $$

All claims of the lemma follow immediately from this.

The parabolic quantum Bruhat graph as introduced by Lenart–Naito–Sagaki–Schilling–Schimozono [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17] is a generalization of the classical construction of the quantum Bruhat graph by Brenti–Fomin–Postnikov [Reference Brenti, Fomin and Postnikov4]. To avoid redundancy, we directly state the definition of the parabolic quantum Bruhat graph, even though we will be mostly concerned with the (ordinary) quantum Bruhat graph.

Fix a subset $J\subseteq \Delta $ . We denote by $W_J$ the Coxeter subgroup of W generated by the reflections $s_\alpha $ for $\alpha \in J$ . We let

$$ \begin{align*} W^J = \{w\in W\mid w(J)\subseteq \Phi^+\}. \end{align*} $$

For each $w\in W$ , let $w^J\in W^J$ and $w_J\in W_J$ be the uniquely determined elements with $w = w^J \cdot w_J$ [Reference Björner and Brenti2, Proposition 2.4.4].

We write $\Phi _J = W_J(J)$ for the root system generated by J. The sum of positive roots in $\Phi _J$ is denoted $2\rho _J$ . The quotient lattice $\mathbb Z\Phi ^\vee / \mathbb Z \Phi _J^\vee $ is ordered by declaring $\mu _1+\Phi _J^\vee \leq \mu _2 + \Phi _J^\vee $ if the difference $\mu _2 - \mu _1 + \Phi _J^\vee $ is equal to a sum of positive coroots modulo $\Phi _J^\vee $ .

Definition 3.2.

  1. (a) The parabolic quantum Bruhat graph associated with $W^J$ is a directed and $(\mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee )$ -weighted graph, denoted ${\mathrm {QB}}(W^J)$ . The set of vertices is given by $W^J$ . For $w_1, w_2\in W^J$ , we have an edge $w_1\rightarrow w_2$ if there is a root $\alpha \in \Phi ^+\setminus \Phi _J$ such that $w_2 = (w_1s_\alpha )^J$ and one of the following conditions is satisfied:

    1. (B) $\ell (w_2) = \ell (w_1)+1$ or

    2. (Q) $\ell (w_2) = \ell (w_1) + 1 - \langle \alpha ^\vee ,2\rho - 2\rho _J\rangle $ .

    Edges of type (B) are Bruhat edges and have weight $0\in \mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee $ . Edges of type (Q) are quantum edges and have weight $\alpha ^\vee \in \mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee $ .

  2. (b) A path in ${\mathrm {QB}}(W^J)$ is a sequence of adjacent edges

    $$ \begin{align*} p : w = w_1\rightarrow w_2\rightarrow\cdots \rightarrow w_{\ell+1} = w'. \end{align*} $$
    The length of p is the number of edges, denoted $\ell (p)\in \mathbb Z_{\geq 0}$ . The weight of p is the sum of its edges’ weights, denoted ${\mathrm {wt}}(p)\in \mathbb Z\Phi ^\vee /\mathbb Z\Phi ^\vee _J$ . We say that p is a path from w to $w'$ .
  3. (c) If $w, w'\in W^J$ , we define the distance function by

    $$ \begin{align*} d_{{\mathrm{QB}}(W^J)}(w\Rightarrow w') = \inf\{\ell(p)\mid p\text{ is a path in }{\mathrm{QB}}(W^J)\text{ from }w\text{ to }w'\}\in \mathbb Z_{\geq 0}\cup\{\infty\}. \end{align*} $$
    A path p from w to $w'$ of length $d_{{\mathrm {QB}}(W^J)}(w\Rightarrow w')$ is called shortest.
  4. (d) The quantum Bruhat graph of W is the parabolic quantum Bruhat graph associated with $J=\emptyset $ , denoted ${\mathrm {QB}}(W) := {\mathrm {QB}}(W^\emptyset )$ . We also shorten our notation to

    $$ \begin{align*} d(w\Rightarrow w') := d_{{\mathrm{QB}}(W)}(w\Rightarrow w'). \end{align*} $$

Remark 3.3. Let us consider the case $J=\emptyset $ , that is, the quantum Bruhat graph. If $w \in W$ and $\alpha \in \Delta $ , then $w\rightarrow ws_\alpha $ is always an edge of weight $\alpha ^\vee \Phi ^+(-w\alpha )$ .

The quantum edges are precisely the edges of the form $w\rightarrow ws_\alpha $ , where $\alpha $ is a quantum root and $\ell (ws_\alpha ) = \ell (w) - \ell (s_\alpha )$ .

Proposition 3.4 [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Proposition 8.1] and [Reference Lenart, Naito, Sagaki, Schilling and Shimozono18, Lemma 7.2].

Consider $w, w'\in W^J$ .

  1. (a) The graph ${\mathrm {QB}}(W^J)$ is strongly connected, that is, there exists a path from w to $w'$ in ${\mathrm {QB}}(W^J)$ .

  2. (b) Any two shortest paths from w to $w'$ have the same weight, denoted

    $$ \begin{align*} {\mathrm{wt}}_{{\mathrm{QB}}(W^J)}(w\Rightarrow w')\in \mathbb Z\Phi^\vee/\mathbb Z\Phi_J^\vee. \end{align*} $$
  3. (c) Any path p from w to $w'$ has weight ${\mathrm {wt}}(p)\geq {\mathrm {wt}}_{{\mathrm {QB}}(W^J)}(w\Rightarrow w')\in \mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee $ .

  4. (d) The image of

    $$ \begin{align*} {\mathrm{wt}}(w\Rightarrow w') := {\mathrm{wt}}_{{\mathrm{QB}}(W)}(w\Rightarrow w')\in \mathbb Z\Phi^\vee \end{align*} $$
    under the canonical projection $\mathbb Z\Phi ^\vee \rightarrow \mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee $ is given by ${\mathrm {wt}}_{{\mathrm {QB}}(W^J)}(w\Rightarrow w')$ .

One interpretation of the weight function is that it measures the failure of the inequality $w_1 W_J\leq w_2 W_J$ in the Bruhat order on $W/W_J$ (cf. [Reference Björner and Brenti2, Section 2.5]): Indeed, $w_1W_J\leq w_2 W_J$ if and only if ${\mathrm {wt}}_{{\mathrm {QB}}(W^J)}(w_1\Rightarrow w_2)=0$ .

We have the following converse to part (c) of Proposition 3.4:

Lemma 3.5 (Cf. [Reference Milićević and Viehmann21, Formula 4.3]).

Let $w_1, w_2\in W^J$ . For any path p from $w_1$ to $w_2$ in ${\mathrm {QB}}(W^J)$ , we have

$$ \begin{align*} \langle {\mathrm{wt}}(p),2\rho-2\rho_J\rangle = \ell(p) + \ell(w_1) - \ell(w_2). \end{align*} $$

In particular,

$$ \begin{align*} \langle {\mathrm{wt}}_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow w_2),2\rho-2\rho_J\rangle = d_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow w_2) + \ell(w_1) - \ell(w_2), \end{align*} $$

and p is shortest if and only if ${\mathrm {wt}}(p) = {\mathrm {wt}}_{{\mathrm {QB}}(W^J)}(w_1\Rightarrow w_2)$ .

Proof. Note that if $p: w_1\rightarrow w_2 = (w_1s_\alpha )^J$ is an edge in ${\mathrm {QB}}(W^J)$ , then by definition,

$$ \begin{align*} \ell(w_2) = \ell(w_1) + 1 - \langle {\mathrm{wt}}(p),2\rho-2\rho_J\rangle. \end{align*} $$

In general, iterate this observation for all edges of p.

The weights of nonshortest paths do not add more information:

Lemma 3.6. Let $\mu \in \mathbb Z\Phi ^\vee /\mathbb Z\Phi _J^\vee $ and $w_1, w_2\in W$ . Then $\mu \geq {\mathrm {wt}}_{{\mathrm {QB}}(W^J)}(w_1\Rightarrow w_2)$ if and only if there is a path p from $w_1$ to $w_2$ in ${\mathrm {QB}}(W^J)$ of weight $\mu $ .

Proof. By part (d) of Proposition 3.4, it suffices to consider the case $J=\emptyset $ , that is, the quantum Bruhat graph.

The if condition is part (c) of Proposition 3.4. It remains to show the only if condition. Note that for each $w\in W$ and $\alpha \in \Delta $ , we get a “silly path” of the form

$$ \begin{align*} w\rightarrow ws_\alpha\rightarrow w \end{align*} $$

in ${\mathrm {QB}}(W)$ . Precisely one of the edges is quantum with weight $\alpha ^\vee $ , and the other one is Bruhat with weight $0$ .

If $\mu \geq {\mathrm {wt}}(w_1\Rightarrow w_2)$ , we may compose a shortest path from $w_1$ to $w_2$ with suitably chosen silly paths as above to obtain a path from $w_1$ to $w_2$ of weight $\mu $ .

Lemma 3.7 [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Lemma 7.7].

Let $J\subseteq \Delta $ , $w_1, w_2\in W^J$ and $a = (\alpha ,k)\in \Delta _{\mathrm {af}}$ such that $w_2^{-1}\alpha \in \Phi ^-$ .

  1. (a) We have an edge $(s_\alpha w_2)^J\rightarrow w_2$ in ${\mathrm {QB}}(W^J)$ of weight $-kw_2^{-1}\alpha ^\vee \in \mathbb Z\Phi ^\vee /\mathbb Z\Phi ^\vee _J$ .

  2. (b) If $w_1^{-1}\alpha \in \Phi ^+$ , then the above edge is part of a shortest path from $w_1$ to $w_2$ , that is,

    $$ \begin{align*} d_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow w_2) = d_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow (s_\alpha w_2)^J) + 1. \end{align*} $$
  3. (c) If $w_1^{-1}\alpha \in \Phi ^-$ , we have

    $$ \begin{align*} d_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow w_2) &= d_{{\mathrm{QB}}(W^J)}((s_\alpha w_1)^J\Rightarrow (s_\alpha w_2)^J), \\ {\mathrm{wt}}_{{\mathrm{QB}}(W^J)}(w_1\Rightarrow w_2) &= {\mathrm{wt}}_{{\mathrm{QB}}(W^J)}((s_\alpha w_1)^J\Rightarrow (s_\alpha w_2)^J) + k(w_1^{-1}\alpha^\vee - w_2^{-1}\alpha^\vee). \end{align*} $$

We can use this lemma to reduce the calculation of weights ${\mathrm {wt}}(w_1\Rightarrow w_2)$ to weights of the form ${\mathrm {wt}}(w\Rightarrow 1)$ : If $w_2\neq 1$ , we find a simple root $\alpha \in \Delta $ with $w_2^{-1}\alpha \in \Phi ^-$ . Then

$$ \begin{align*} {\mathrm{wt}}(w_1\Rightarrow w_2) &= \begin{cases}{\mathrm{wt}}(w_1\Rightarrow s_\alpha w_2),&w_1^{-1}\alpha\in \Phi^+,\\ {\mathrm{wt}}(s_\alpha w_1\Rightarrow s_\alpha w_2),&w_1^{-1}\alpha \in \Phi^-,\end{cases} \\ &={\mathrm{wt}}(\min(w_1, s_\alpha w_1), s_\alpha w_2). \end{align*} $$

For an alternative proof of this reduction, cf. [Reference Sadhukhan26, Corollary 3.3].

The quantum Bruhat graph has a number of useful automorphisms.

Lemma 3.8. Let $w_1, w_2\in W$ , and let $w_0\in W$ be the longest element.

  1. (a) ${\mathrm {wt}}(w_0 w_1\Rightarrow w_0 w_2) = {\mathrm {wt}}(w_2 \Rightarrow w_1)$ .

  2. (b) ${\mathrm {wt}}(w_0 w_1 w_0\Rightarrow w_0 w_2 w_0) = -w_0{\mathrm {wt}}(w_1 \Rightarrow w_2)$ .

  3. (c) ${\mathrm {wt}}(w_1\Rightarrow 1) = {\mathrm {wt}}(w_1^{-1}\Rightarrow 1)$ .

Proof. Part (a) follows from [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Proposition 4.3].

For part (b), observe that we have an automorphism of $\Phi $ given by $\alpha \mapsto -w_0\alpha $ . The induced automorphism of W is given by $w\mapsto w_0 w w_0$ . Since the function ${\mathrm {wt}}(\cdot \Rightarrow \cdot )$ is compatible with automorphisms of $\Phi $ , we get the claim.

Now, for (c), consider a reduced expression

$$ \begin{align*} w_0 w_1 = s_1\cdots s_q. \end{align*} $$

Then, iterating Lemma 3.7, we get

$$ \begin{align*} {\mathrm{wt}}(w_1\Rightarrow 1) \underset{\text{(a)}}=&\ {\mathrm{wt}}(w_0\Rightarrow w_0 w_1) = {\mathrm{wt}}(w_0\Rightarrow s_1\cdots s_q) \\=&\ {\mathrm{wt}}(s_1w_0\Rightarrow s_2\cdots s_q)=\cdots = {\mathrm{wt}}(s_q\cdots s_1 w_0\Rightarrow 1) \\=&\ {\mathrm{wt}}((w_0 w_1)^{-1}w_0\Rightarrow 1) = {\mathrm{wt}}(w_1^{-1}\Rightarrow 1).\\[-36pt] \end{align*} $$

3.2. Lifting the parabolic quantum Bruhat graph

For sufficiently regular elements of the extended affine Weyl group, the Bruhat covers in $\widetilde W$ are in a one-to-one correspondence with edges in the quantum Bruhat graph [Reference Lam and Shimozono16, Proposition 4.4]. This result is very useful for deriving properties about the quantum Bruhat graph. Moreover, our strategy to prove our results on the Bruhat order will be to reduce to this superregular case.

The result of Lam and Shimozono has been generalized by Lenart et al. [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Theorem 5.2], and the extra generality of the latter result will be useful for us. Throughout this section, let $J\subseteq \Delta $ be any subset.

Definition 3.9 [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17].

  1. (a) Define

    $$ \begin{align*} (W^J)_{\mathrm{af}} :=& \{x\in W_{\mathrm{af}}\mid \forall \alpha\in \Phi_J:~\ell(x,\alpha)=0\}, \\ \widetilde{(W^J)} :=& \{x\in \widetilde W\mid \forall \alpha\in \Phi_J:~\ell(x,\alpha)=0\}. \end{align*} $$
  2. (b) Let $C>0$ be any real number. We define $\Omega _J^{-C}$ to be the set of all elements $x = w\varepsilon ^\mu \in \widetilde {(W^J)}$ such that

    $$ \begin{align*} \forall \alpha\in \Phi^+\setminus \Phi_J:~\langle\mu,\alpha\rangle\leq -C. \end{align*} $$
    Similarly, we say $x\in \Omega _J^C$ if
    $$ \begin{align*} \forall \alpha\in \Phi^+\setminus \Phi_J:~\langle\mu,\alpha\rangle\geq C. \end{align*} $$
  3. (c) For elements $x, x'\in \widetilde W$ , we write $x\lessdot x'$ and call $x'$ a Bruhat cover of x if $\ell (x') = \ell (x)+1$ and $x^{-1} x'$ is an affine reflection in $\widetilde W$ .

  4. (d) For $\mu ,\mu '\in X_\ast $ , we write $\mu '\leq \mu\ \pmod {\Phi _J^\vee }$ if the difference $\mu -\mu '+\mathbb Z\Phi _J^\vee $ is a sum of positive coroots in the quotient group $X_\ast / \mathbb Z\Phi _J^\vee $ . This is, according to our convention, equivalent to $\mu -\mu '+\mathbb Z\Phi _J^\vee \geq 0+\mathbb Z\Phi _J^\vee $ in $\mathbb Z\Phi / \mathbb Z\Phi _J^\vee $ .

Theorem 3.10 [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Theorem 5.2].

There is a constant $C>0$ depending only on $\Phi $ such that the following holds:

  1. (a) If $x = w\varepsilon ^\mu \lessdot x' = w'\varepsilon ^{\mu '}$ is a Bruhat cover with $x\in \Omega ^{-C}_J$ and $x' \in \widetilde {(W^J)}$ , there exists an edge $(w')^J\rightarrow w^J$ in ${\mathrm {QB}}(W^J)$ of weight $\mu - \mu '+\mathbb Z \Phi _J^\vee $ .

  2. (b) If $x = w\varepsilon ^\mu \in \Omega _J^{-C}$ and $\tilde w'\rightarrow w^J$ is an edge in ${\mathrm {QB}}(W^J)$ of weight $\omega $ , then there exists a unique element $x\lessdot x' = w'\varepsilon ^{\mu '}\in \widetilde {(W^J)}$ with $\tilde w' = (w')^J$ and $\mu \equiv \mu ' + \omega\ \pmod {\mathbb Z \Phi _J^\vee }$ .

This theorem‘lifts’ ${\mathrm {QB}}(W^J)$ into the Bruhat covers of $\Omega _J^{-C}$ for sufficiently large C.

The theorem is originally formulated only for $(W^J)_{\mathrm {af}}$ , but the generalization to $\widetilde {(W^J)}$ is straightforward.

With a bit of bookkeeping, we can compare paths in ${\mathrm {QB}}(W^J)$ (i.e., sequences of edges) with the Bruhat order on $\Omega _J^{-C}$ (i.e., sequences of Bruhat covers).

Lemma 3.11. Let $C_1>0$ be any real number. Then there exists some $C_2>0$ such that for all $x = w\varepsilon ^{\mu }\in \Omega _J^{C_2}$ and $x' = w'\varepsilon ^{\mu '}\in \widetilde {(W^J)}$ with $\ell (x^{-1}x')\leq C_1$ , we have

$$ \begin{align*} x\leq x'\iff \mu - {\mathrm{wt}}(w'\Rightarrow w)\leq \mu'\quad \pmod{\Phi_J^\vee}. \end{align*} $$

Proof. Let $C>0$ be a constant sufficiently large for the conclusion of Theorem 3.10 to hold. We see that if $x_1\lessdot x_2$ is any cover in $\Omega _J^{-C}$ , then there are only finitely many possibilities for $x_1^{-1} x_2$ , so the length $\ell (x_1^{-1} x_2)$ is bounded. We fix a bound $C'>0$ for this length.

We can pick $C_2>0$ such that for all $x_1 = w\varepsilon ^\mu \in \Omega _J^{-C_2}$ and $x_2\in \widetilde W^J$ with $\ell (x_1^{-1}x_2)\leq C_1 C'$ , we must at least have $x_2 \in \Omega _J^{-C}$ .

We now consider elements $x = w\varepsilon ^\mu \in \Omega _J^{-C_2}$ and $x' = w'\varepsilon ^{\mu '}\in \widetilde W^J$ with $\ell (x^{-1} x')\leq C_1$ .

First, suppose that $x\leq x'$ . We find elements $x= x_1\lessdot x_2\lessdot \cdots \lessdot x_k = x'$ . Note that $k = \ell (x') - \ell (x) \leq \ell (x^{-1} x')\leq C_1$ . By choice of $C'$ , we conclude that $\ell (x^{-1} x_i)\leq C' i\leq C'C_1$ for $i=1,\dotsc ,k$ . Thus, $x_i \in \Omega _J^{-C}$ .

By Theorem 3.10, we get a path from $(w')^J$ to $w^J$ of weight $\mu -\mu '+\mathbb Z\Phi _J^\vee $ . Thus,

$$ \begin{align*}{\mathrm{wt}}(w_1\Rightarrow w_2)\leq \mu-\mu'\quad \pmod{\Phi_J^\vee},\end{align*} $$

which is the estimate we wanted to prove.

Now, suppose conversely that we are given $\mu - {\mathrm {wt}}(w'\Rightarrow w)\geq \mu '\ \pmod {\Phi _J^\vee }$ . By Lemma 3.6, we find a path $(w')^J = w_1\rightarrow w_2\rightarrow \cdots \rightarrow w_k = w^J$ in ${\mathrm {QB}}(W^J)$ of weight $\mu -\mu '+\mathbb Z\Phi _J^\vee $ . Since $\mu -\mu '$ is bounded in terms of $C_1$ , the length k of this path is bounded in terms of $C_1$ as well. By adding another lower bound for $C_2$ , we can guarantee that each such path $w_1\rightarrow \cdots \rightarrow w_k$ can indeed be lifted to $\Omega _J^{-C}$ , proving that $x\leq x'$ .

We find working with superdominant instead superantidominant coweights a bit easier, so let us restate the lemma for $\Omega _J^C$ instead of $\Omega _J^{-C}$ .

Corollary 3.12. Let $C_1>0$ be any real number. Then there exists some $C_2>0$ such that for all $x = w\varepsilon ^\mu \in \Omega _J^{C_2}$ and $x' = w'\varepsilon ^{\mu '}\in \widetilde {(W^J)}$ with $\ell (x^{-1} x')\leq C_1$ , we have

$$ \begin{align*} x\leq x'\iff\mu + {\mathrm{wt}}(w\Rightarrow w')\leq \mu'\quad \pmod{\Phi_J^\vee}. \end{align*} $$

Proof. Let $w_0(J)\in W_J$ be the longest element. Let $C_2>0$ such that the conclusion of the previous Lemma is satisfied.

If $x\in \Omega _J^{C_2}$ , then $x w_0(J) w_0 \in \Omega _{-w_0(J)}^{-C_2}$ . Moreover, $w_0(J)w_0$ is a length positive element for x, so $\ell (x w_0(J) w_0) = \ell (x) + \ell (w_0(J) w_0)$ . Choosing $C_2$ appropriately, we similarly may assume $x' \in \Omega ^{C}_{J}$ for some $C>0$ and obtain $\ell (x'w_0(J)w_0) = \ell (x') + \ell (w_0(J)w_0)$ . Then, with the right choice of constants and using the automorphism $\alpha \mapsto -w_0\alpha $ of $\Phi $ , we get

$$ \begin{align*}x\leq x'\iff&xw_0(J)w_0\leq x'w_0(J)w_0\\{\iff}&w_0 w_0(J)\mu - \mathrm{wt}(w' w_0(J) w_0\Rightarrow ww_0(J) w_0)\geq w_0 w_0(J)\mu'\quad \pmod{\Phi_{-w_0(J)}^\vee}\\{\iff}&w_0(J)\mu + \mathrm{wt}(w_0 w'w_0(J)\Rightarrow w_0 ww_0(J))\leq w_0(J)\mu'\quad \pmod{\Phi_J^\vee}\\\underset{[17,\ \text{Proof 4.3}]}{\iff}&w_0(J)\mu + \mathrm{wt}(w\Rightarrow w')\leq w_0(J)\mu'\quad \pmod{\Phi_J^\vee}. \end{align*} $$

Since $w_0(J)\mu \equiv \mu\ \pmod {\Phi _J^\vee }$ , we get the desired conclusion.

3.3. Computing the weight function

We already saw in Lemma 3.7 how to find for all $w_1, w_2\in W$ an element $w \in W$ such that ${\mathrm {wt}}(w_1\Rightarrow w_2) = {\mathrm {wt}}(w\Rightarrow 1)$ . It remains to find a method to compute these weights. First, we note that we only need to consider quantum edges for this task.

Lemma 3.13 [Reference Milićević and Viehmann21, Proposition 4.11].

For each $w\in W$ , there is a shortest path from w to $1$ in ${\mathrm {QB}}(W)$ consisting only of quantum edges.

So we only need to find for each $w\in W\setminus \{1\}$ a quantum edge $w\rightarrow w'$ in ${\mathrm {QB}}(W)$ with $d(w'\Rightarrow 1) = d(w\Rightarrow 1)-1$ . In this section, we present a new method to obtain such edges.

Definition 3.14. Let $w\in W$ .

  1. (a) The set of inversions of w is

    $$ \begin{align*} {\mathrm{inv}}(w) := \{\alpha\in \Phi^+\mid w^{-1}\alpha\in \Phi^-\}. \end{align*} $$
  2. (b) An inversion $\gamma \in {\mathrm {inv}}(w)$ is a maximal inversion if there is no $\alpha \in {\mathrm {inv}}(w)$ with $\alpha \neq \gamma \leq \alpha $ . Here, $\gamma \leq \alpha $ means that $\alpha -\gamma $ is a sum of positive roots.

    We write $\mathrm{max}\ \mathrm{inv}(w)$ for the set of maximal inversions of w.

Remark 3.15. If $\theta \in {\mathrm {inv}}(w)$ is the longest root of an irreducible component of $\Phi $ , then certainly $\theta \in \mathrm{max}\ \mathrm{inv}(w)$ . In this case, everything we want to prove is already shown in [Reference Lenart, Naito, Sagaki, Schilling and Shimozono17, Section 5.5]. Our strategy is to follow their arguments as closely as possible while keeping the generality of maximal inversions.

Lemma 3.16. Let $w\in W$ and $\gamma \in \mathrm{max}\ \mathrm{inv}(w)$ . Then $w\rightarrow s_\gamma w$ is a quantum edge.

Proof. Note that $s_\gamma w = ws_{-w^{-1}\gamma }$ . We have to show that $-w^{-1}\gamma $ is a quantum root and that

$$ \begin{align*} \ell(ws_{-w^{-1}\gamma}) = \ell(w) - \ell(s_{-w^{-1}\gamma}). \end{align*} $$

Step 1. We show that $-w^{-1}\gamma $ is a quantum root using Lemma 3.1. So pick an element $-w^{-1}\gamma \neq \beta \in \Phi ^+$ with $s_{-w^{-1}\gamma }(\beta )\in \Phi ^-$ . We want to show that $\langle -w^{-1}\gamma ^\vee ,\beta \rangle =1$ .

Note that

$$ \begin{align*} s_{-w^{-1}\gamma}(\beta) = \beta+\langle -w^{-1}\gamma^\vee,\beta\rangle w^{-1}\gamma. \end{align*} $$

In particular, $k := \langle -w^{-1}\gamma ^\vee ,\beta \rangle>0$ . It follows from the theory of root systems that

$$ \begin{align*} \beta_i := \beta +i w^{-1}\gamma\in \Phi,\qquad i=0,\dotsc,k. \end{align*} $$

Since $\beta _0 = \beta \in \Phi ^+$ and $\beta _{k} = s_{-w^{-1}\gamma }(\beta )\in \Phi ^-$ , we find some $i\in \{0,\dotsc ,k-1\}$ with $\beta _i\in \Phi ^+$ and $\beta _{i+1}\in \Phi ^-$ . We show that $k\leq 1$ as follows:

  • Suppose $w\beta _i \in \Phi ^+$ . Then $w\beta _{i+1} = w\beta _i+\gamma> \gamma $ . In particular, $w\beta _{i+1}\in \Phi ^+$ . We see that $w\beta _{i+1}\in {\mathrm {inv}}(w)$ , contradicting maximality of $\gamma $ .

  • Suppose $w\beta _{i+1}\in \Phi ^-$ . Then $-w\beta _i = -w\beta _{i+1}+\gamma> \gamma $ . In particular, $-w\beta _{i}\in \Phi ^+$ . We see that $-w\beta _{i}\in {\mathrm {inv}}(w)$ , contradicting maximality of $\gamma $ .

  • Suppose $i\geq 1$ . Then $\gamma - w\beta _i = -w\beta _{i-1}\in \Phi $ . We already proved $w\beta _i\in \Phi ^-$ , so $-w\beta _i\in {\mathrm {inv}}(w)$ . Since also $\gamma \in {\mathrm {inv}}(w)$ , we conclude $\gamma <-w\beta _{i-1}\in {\mathrm {inv}}(w)$ , contradicting the maximality of $\gamma $ .

  • Suppose $i\leq k-2$ . Then $w\beta _{i+2} = w\beta _{i+1} + \gamma \in \Phi $ . Since both $\gamma $ and $w\beta _{i+1}$ are in ${\mathrm {inv}}(w)$ , we conclude that $\gamma < w\beta _{i+2}\in {\mathrm {inv}}(w)$ , which is a contradiction to the maximality of $\gamma $ .

In summary, we conclude $0=i\geq k-1$ , thus $k\leq 1$ . This shows $\langle -w^{-1}\gamma ^\vee ,\beta \rangle =1$ .

Step 2. We show that

$$ \begin{align*} \ell(ws_{-w^{-1}\gamma}) = \ell(w) - \ell(s_{-w^{-1}\gamma}). \end{align*} $$

Suppose this is not the case. Then we find some $\alpha \in \Phi ^+$ such that $w\alpha \in \Phi ^+$ and $s_{-w^{-1}\gamma }(\alpha ) \in \Phi ^-$ . As we saw before, $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle =1$ , so $s_{-w^{-1}\gamma }(\alpha ) = \alpha + w^{-1}\gamma \in \Phi ^-$ . Now, consider the element $ws_{-w^{-1}\gamma }(\alpha ) = w\alpha + \gamma \in \Phi $ . Since $w\alpha \in \Phi ^+$ by assumption, we have $ws_{-w^{-1}\gamma }(\alpha )>\gamma $ , in particular $ws_{-w^{-1}\gamma }(\alpha )\in \Phi ^+$ . We conclude $ws_{-w^{-1}\gamma }(\alpha )\in {\mathrm {inv}}(w)$ , yielding a final contradiction to the maximality of $\gamma $ .

Lemma 3.17. Let $w\in W$ and $\alpha \in \Phi ^+$ such that $w\rightarrow ws_\alpha $ is a quantum edge. Let moreover $-w\alpha \neq \gamma \in \mathrm{max}\ \mathrm{inv}(w)$ . Then $\gamma \in \mathrm{max}\ \mathrm{inv}(ws_\alpha )$ and $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle \geq 0$ .

Proof. We first show $\gamma \in {\mathrm {inv}}(ws_\alpha )$ , that is, $s_\alpha w^{-1}\gamma \in \Phi ^-$ .

Aiming for a contradiction, we thus suppose that

$$ \begin{align*} s_\alpha (-w^{-1}\gamma) = \langle \alpha^\vee,w^{-1}\gamma\rangle \alpha - w^{-1}\gamma \in \Phi^-. \end{align*} $$

Then $-w^{-1}\gamma $ is a positive root whose image under $s_\alpha $ is negative. Since $\alpha $ is quantum, we conclude $\langle \alpha ^\vee ,-w^{-1}\gamma \rangle =1$ . Thus, $-\alpha - w^{-1}\gamma \in \Phi ^-$ . Consider the element

$$ \begin{align*}w(\alpha+w^{-1}\gamma) = \gamma+w\alpha \in \Phi.\end{align*} $$

We distinguish the following cases:

  • If $\gamma +w\alpha \in \Phi ^-$ , we get $\gamma <-w\alpha \in {\mathrm {inv}}(w)$ , contradicting maximality of $\gamma $ .

  • If $\gamma +w\alpha \in \Phi ^+$ , we compute

    $$ \begin{align*} ws_\alpha(-w^{-1}\gamma) = -(ws_\alpha w^{-1})\gamma = -s_{w\alpha}(\gamma) = -(\gamma+w\alpha)\in \Phi^-. \end{align*} $$
    In other words, the positive root $-w^{-1}\gamma \in \Phi ^+$ gets mapped to negative roots both by $s_\alpha $ and by $ws_\alpha \in W$ . This is a contradiction to $\ell (w) = \ell (ws_\alpha ) + \ell (s_\alpha )$ (since $w\rightarrow ws_\alpha $ was supposed to be a quantum edge).

In any case, we get a contradiction. Thus, $\gamma \in {\mathrm {inv}}(ws_\alpha )$ .

The quantum edge condition $w\rightarrow ws_\alpha $ implies $\ell (w) = \ell (ws_\alpha ) + \ell (s_\alpha )$ , so ${\mathrm {inv}}(ws_\alpha )\subset {\mathrm {inv}}(w)$ . Because $\gamma $ is maximal in ${\mathrm {inv}}(w)$ and $\gamma \in {\mathrm {inv}}(ws_\alpha )\subseteq {\mathrm {inv}}(w)$ , it follows that $\gamma $ must be maximal in ${\mathrm {inv}}(ws_\alpha )$ as well.

Finally, we have to show $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle \geq 0$ . If this was not the case, then we would get

$$ \begin{align*} \gamma <s_\gamma(-w\alpha) = -w\alpha + \langle w^{-1}\gamma^\vee,\alpha\rangle \gamma\in {\mathrm{inv}}(w), \end{align*} $$

again contradicting maximality of $\gamma $ .

Proposition 3.18. Let $w\in W$ and $\gamma \in \mathrm{max}\ \mathrm{inv}(w)$ . Then

$$ \begin{align*} {\mathrm{wt}}(w\Rightarrow 1) = {\mathrm{wt}}(s_\gamma w\Rightarrow 1) - w^{-1}\gamma^\vee. \end{align*} $$

Proof. Since the estimate

$$ \begin{align*} {\mathrm{wt}}(w\Rightarrow 1)\leq&{\mathrm{wt}}(w\Rightarrow s_\gamma w) + {\mathrm{wt}}(s_\gamma w\Rightarrow 1) \\\leq&-w^{-1}\gamma^\vee + {\mathrm{wt}}(s_\gamma w\Rightarrow 1) \end{align*} $$

follows from [Reference Schremmer28, Lemma 4.3], all we have to show is the inequality “ $\geq $ ”.

For this, we use induction on $\ell (w)$ . If $1\neq w\in W$ , we find by Lemma 3.13 some quantum edge $w\rightarrow ws_\alpha $ with ${\mathrm {wt}}(w\Rightarrow 1) = {\mathrm {wt}}(ws_\alpha \Rightarrow 1)+\alpha ^\vee $ . If $\alpha =-w^{-1}\gamma $ , we are done.

Otherwise, $\gamma \in \mathrm{max}\ \mathrm{inv}(ws_\alpha )$ and $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle \geq 0$ by the previous lemma. By induction, we have

(3.2) $$ \begin{align} {\mathrm{wt}}(w\Rightarrow 1)&={\mathrm{wt}}(ws_\alpha \Rightarrow 1)+\alpha^\vee \notag\\&= {\mathrm{wt}}(s_\gamma ws_\alpha\Rightarrow 1) + \alpha^\vee - (ws_\alpha)^{-1}\gamma^\vee. \end{align} $$

By Lemma 3.16, we get the following three quantum edges:

This allows for the following computation:

(3.3) $$ \begin{align} \ell(s_\gamma ws_\alpha) =&\,\ell(ws_\alpha) + 1-\langle -(ws_\alpha)^{-1}\gamma^\vee,2\rho\rangle \notag\\=&\,\ell(w) + 2-\langle \alpha^\vee,2\rho\rangle - \langle -w^{-1}\gamma^\vee - \langle -w^{-1}\gamma^\vee,\alpha\rangle \alpha^\vee,2\rho\rangle \notag\\=&\,\ell(s_\gamma w) + 1 +(\langle -w^{-1}\gamma^\vee,\alpha\rangle-1)\langle \alpha^\vee,2\rho\rangle. \end{align} $$

We now distinguish several cases depending on the value of $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle \in \mathbb Z_{\geq 0}$ .

  • Case $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle =0$ . In this case, we get a quantum edge $s_\gamma w\rightarrow s_\gamma w s_\alpha $ by Equation (3.3). Evaluating this in Equation (3.2), we get

    $$ \begin{align*} {\mathrm{wt}}(w\Rightarrow 1)&={\mathrm{wt}}(s_\gamma w s_\alpha\Rightarrow 1)+\alpha^\vee - (ws_\alpha)^{-1}\gamma^\vee \\ &\geq{\mathrm{wt}}(s_\gamma w\Rightarrow 1) - s_\alpha w^{-1}\gamma^\vee \\ &={\mathrm{wt}}(s_\gamma w\Rightarrow 1) - w^{-1}\gamma^\vee. \end{align*} $$
  • Case $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle =1$ . In this case, we get a Bruhat edge $s_\gamma w\rightarrow s_\gamma w s_\alpha $ by Equation (3.3). Evaluating this in Equation (3.2), we get

    $$ \begin{align*} {\mathrm{wt}}(w\Rightarrow 1)& ={\mathrm{wt}}(s_\gamma w s_\alpha\Rightarrow 1)+\alpha^\vee - (ws_\alpha)^{-1}\gamma^\vee \\ &\geq{\mathrm{wt}}(s_\gamma w\Rightarrow 1) + \alpha^\vee - s_\alpha w^{-1}\gamma^\vee \\ &={\mathrm{wt}}(s_\gamma w\Rightarrow 1) - w^{-1}\gamma^\vee. \end{align*} $$
  • Case $\langle -w^{-1}\gamma ^\vee ,\alpha \rangle \geq 2$ . We get

    $$ \begin{align*} \ell(s_\gamma ws_\alpha) &\leq \ell(s_\gamma w) + \ell(s_\alpha) \underset{\text{L}3.1}\leq \ell(s_\gamma w) + \langle \alpha^\vee,2\rho\rangle -1 \\ &<\ell(s_\gamma w) + \ell(s_\alpha) \leq \ell(s_\gamma w) +1+\left(\langle -w^{-1}\gamma^\vee,\alpha\rangle-1\right) \langle \alpha^\vee,2\rho\rangle. \end{align*} $$
    This is a contradiction to Equation (3.3).

In any case, we get a contradiction or the required conclusion, finishing the proof.

Remark 3.19.

  1. (a) By Lemma 3.5, it follows that concatenating the quantum edge $w\rightarrow s_\gamma w$ with a shortest path $s_\gamma w\Rightarrow 1$ yields indeed a shortest path from w to $1$ . Thus, iterating Proposition 3.18, we get a shortest path from w to $1$ .

  2. (b) If $w\in W^J$ and $\gamma \in \mathrm{max}\ \mathrm{inv}(w)$ , we do not in general have a quantum edge $w\rightarrow (s_\gamma w)^J$ in ${\mathrm {QB}}(W^J)$ . However, we can concatenate a shortest path from w to $(s_\gamma w)^J$ (which will have weight $-w^{-1}\gamma ^\vee + \mathbb Z \Phi _J^\vee $ ) with a shortest path from $(s_\gamma w)^J$ to $1$ in ${\mathrm {QB}}(W^J)$ to get a shortest path from w to $1$ .

3.4. Semiaffine quotients

We saw that for $w_1, w_2\in W$ and $J\subseteq \Delta $ , we can assign a weight to the cosets $w_1 W_J$ and $w_2 W_J$ in $\mathbb Z\Phi ^\vee / \mathbb Z\Phi _J^\vee $ . In this section, we consider left cosets $W_J w$ instead. This is pretty straightforward if $J\subseteq \Delta $ ; however, it is more interesting if J is instead allowed to be a subset of $\Delta _{{\mathrm {af}}}$ . The quotient of the finite Weyl group by a set of simple affine roots will be called semiaffine quotient.

In this section, we introduce the resulting semiaffine weight function. This new function generalizes properties of the ordinary weight function. We have the following two motivations to study it:

  • For root systems of type $A_n$ , we can explicitly express the weight function using formula (3.1):

    $$ \begin{align*} {\mathrm{wt}}(v_2\Rightarrow v_1) = \sup_{a\in \Delta_{{\mathrm{af}}}} (v_2^{-1}\omega_a - v_1^{-1}\omega_a). \end{align*} $$
    Using the semiaffine weight function, we can prove a generalization of this formula, expressing the weight ${\mathrm {wt}}(v_2\Rightarrow v_1)$ as a supremum of semiaffine weights (Lemmas 3.29 and 4.34)
  • There is a close relationship between the quantum Bruhat graph and the Bruhat order of the extended affine Weyl group $\widetilde W$ . Now, Deodhar’s lemma [Reference Deodhar7] is an important result on the Bruhat order of general Coxeter groups. Translating the statement of Deodhar’s lemma to the quantum Bruhat graph yields exactly the semiaffine weight function.

    Conversely, using the semiaffine weight function and Deodhar’s lemma, we can generalize our result on the Bruhat order in Section 4.3.

In this article, the results of this section are only used in Section 4.3, whose results are not used later. A reader who is not interested in the aforementioned applications is thus invited to skip these two sections.

Definition 3.20. Let $J\subseteq \Delta _{{\mathrm {af}}}$ be any subset.

  1. (a) We denote by $\Phi _J$ the root system generated by the roots

    $$ \begin{align*} {\mathrm{cl}}(J) := \{{\mathrm{cl}}( a)\mid a\in J\} = \{\alpha\mid (\alpha,k)\in J\}. \end{align*} $$
  2. (b) We denote by $W_J$ the Weyl group of the root system $\Phi _J$ , that is, the subgroup of W generated by $\{s_\alpha \mid \alpha \in {\mathrm {cl}}(J)\}$ .

  3. (c) Similarly, we denote by $(\Phi _{\mathrm {af}})_J\subseteq \Phi _J$ the (affine) root system generated by J, and by $\widetilde W_J$ the Coxeter subgroup of $W_{\mathrm {af}}$ generated by the reflections $r_{ a}$ with $ a\in J$ .

  4. (d) We say that J is a spherical subset of $\Delta _{{\mathrm {af}}}$ if no connected component of the affine Dynkin diagram of $\Phi _{{\mathrm {af}}}$ is contained in J, that is, if $\widetilde W_J$ is finite.

Lemma 3.21. Let $J\subseteq \Delta _{{\mathrm {af}}}$ be a spherical subset.

  1. (a) ${\mathrm {cl}}(J)$ is a basis of $\Phi _J$ . The map $(\Phi _{\mathrm {af}})_J\rightarrow \Phi _J, (\alpha ,k)\mapsto \alpha $ is bijective.

  2. (b) Writing $\Phi _J^+$ for the positive roots of $\Phi _J$ with respect to the basis ${\mathrm {cl}}(J)$ , we get a bijection

    $$ \begin{align*} \Phi_J^+\rightarrow (\Phi_{\mathrm{af}})_J^+,\quad \alpha\mapsto (\alpha,\Phi^+(-\alpha)). \end{align*} $$

Proof.

  1. (a) Consider the Cartan matrix

    $$ \begin{align*} C_{\alpha,\beta} := \langle \alpha^\vee,\beta\rangle,\qquad \alpha,\beta\in {\mathrm{cl}}(J). \end{align*} $$
    This must be the Cartan matrix associated to a certain Dynkin diagram, namely the subdiagram of the affine Dynkin diagram of $\Phi _{{\mathrm {af}}}$ with set of nodes given by J. We know that this must coincide with the Dynkin diagram of a finite root system by the fact that J is spherical. Hence, $C_{\bullet ,\bullet }$ is the Cartan matrix of a finite root system. Both claims follow immediately from this observation.
  2. (b) Let $\varphi $ denote the map

    $$ \begin{align*} \varphi : \Phi_J^+\rightarrow \Phi_{\mathrm{af}}^+,\qquad \alpha\mapsto (\alpha,\Phi^+(-\alpha)). \end{align*} $$
    By (a), the map is injective. For each root $\alpha \in {\mathrm {cl}}(J)$ , we certainly have $\varphi (\alpha )\in \Phi _J^+$ .

    Now, for an inductive argument, suppose that $\alpha \in \Phi _J^+, \beta \in {\mathrm {cl}}(J)$ and $\alpha +\beta \in \Phi $ satisfy $\varphi (\alpha )\in \Phi _J^+$ . We want to show that $\varphi (\alpha +\beta )\in \Phi _J^+$ .

    We have $(\alpha ,\Phi ^+(-\alpha )),(\beta ,\Phi ^+(-\beta ))\in \Phi _J^+$ , hence

    $$ \begin{align*} (\alpha+\beta,\Phi^+(-\alpha) + \Phi^+(-\beta))\in \Phi_J^+. \end{align*} $$
    Hence, it suffices to show that $\Phi ^+(-\alpha ) + \Phi ^+(-\beta ) = \Phi ^+(-\alpha -\beta )$ .

    If $\beta \in \Delta $ , this is clear. Hence, we may assume that $\beta =-\theta $ , where $\theta $ is the longest root of the irreducible component of $\Phi $ containing $\alpha ,\beta $ . Then $\alpha -\theta \in \Phi $ implies $\alpha \in \Phi ^+$ and $\alpha -\theta \in \Phi ^-$ . We see that $\Phi ^+(-\alpha ) + \Phi ^+(\theta ) = \Phi ^+(-\alpha +\theta )$ holds true.

The parabolic subgroup $\widetilde W_J\subseteq W_{\mathrm {af}}$ allows the convenient decomposition of $W_{\mathrm {af}}$ as $W_{\mathrm {af}} = \widetilde W_J \cdot {}^{J}{}{W_{\mathrm {af}}}$ [Reference Björner and Brenti2, Proposition 2.4.4]. We get something similar for $W_J\subseteq W$ .

Definition 3.22. Let $J\subseteq \Delta _{{\mathrm {af}}}$ .

  1. (a) By $\Phi _J^+$ , we denote the set of positive roots in $\Phi _J$ with respect to the basis ${\mathrm {cl}}(J)$ . By abuse of notation, we also use $\Phi ^+_J$ as the symbol for the indicator function of $\Phi ^+_J$ , that is,

    $$ \begin{align*} \Phi^+_J(\alpha) :=\begin{cases}1,&\alpha\in \Phi^+_J,\\ 0,&\alpha\in \Phi\setminus \Phi^+_J.\end{cases}. \end{align*} $$
  2. (b) We define

    $$ \begin{align*} {}^J{}W :=& \{w\in W\mid \forall b\in J: w^{-1}{\mathrm{cl}}(b)\in \Phi^+\} \\=& \{w\in W\mid \forall \beta\in \Phi_J^+: w^{-1}\beta\in \Phi^+\}. \end{align*} $$
  3. (c) For $w\in W$ , we put

    $$ \begin{align*} {}^J{}\ell(w) :=\#\{\beta\in \Phi^+_J\mid w^{-1}\beta \in \Phi^-\}. \end{align*} $$

Lemma 3.23. If $w\in W$ and $\beta \in \Phi ^+_J$ satisfy $w^{-1}\beta \in \Phi ^-$ , then

$$ \begin{align*} {}^J{}\ell(s_\beta w)<{}^J{}\ell(w). \end{align*} $$

Proof. Write

$$ \begin{align*} I := \{\beta\neq \gamma\in \Phi^+_J\mid s_\beta(\gamma)\notin \Phi^+_J\}. \end{align*} $$

Then

$$ \begin{align*} {}^J{}\ell(s_\beta w) =&\, \#\{\gamma\in \Phi^+_J\mid w^{-1}s_\beta(\gamma)\in \Phi^-\} \\=&\,\#\{\gamma\in \Phi^+_J\setminus (I\cup\{\beta\})\mid w^{-1}s_\beta(\gamma)\in \Phi^-\} + \#\{\gamma\in I\mid w^{-1}s_\beta(\gamma)\in \Phi^-\}. \end{align*} $$

Since $s_\beta $ permutes the set $\Phi ^+_J\setminus (I\cup \{\beta \})$ , we get

$$ \begin{align*} \ldots =& \,\#\{\gamma\in \Phi^+_J\setminus (I\cup\{\beta\})\mid w^{-1}\gamma\in \Phi^-\} + \#\{\gamma\in I\mid w^{-1}s_\beta(\gamma)\in \Phi^-\}. \end{align*} $$

Note that if $\gamma \in I$ , then $\langle \beta ^\vee ,\gamma \rangle>0$ and thus

$$ \begin{align*} w^{-1}s_\beta(\gamma) = w^{-1}\gamma - \langle \beta^\vee,\gamma\rangle w^{-1}\beta> w^{-1}\gamma. \end{align*} $$

We obtain

$$ \begin{align*} &\#\{\gamma\in \Phi^+_J\setminus (I\cup\{\beta\})\mid w^{-1}\gamma\in \Phi^-\} + \#\{\gamma\in I\mid w^{-1}s_\beta(\gamma)\in \Phi^-\} \\\leq&\,\#\{\gamma\in \Phi^+_J\setminus (I\cup\{\beta\})\mid w^{-1}\gamma\in \Phi^-\} + \#\{\gamma\in I\mid w^{-1}\gamma\in \Phi^-\} \\=&\,{}^J{}\ell(w)-1.\\[-38pt] \end{align*} $$

Lemma 3.24. Let $J\subseteq \Delta _{{\mathrm {af}}}$ be a spherical subset. Then there exists a uniquely determined map ${}^J{}\pi : W\rightarrow {}^J{}W\times \mathbb Z\Phi ^\vee $ with the following two properties:

  1. (1) For all $w\in {}^J{}W$ , we have ${}^J{}\pi (w) = (w,0)$ .

  2. (2) For all $w\in W$ and $\beta \in \Phi _J^+$ where we write ${}^J{}\pi (w) = (w', \mu )$ , we have

    $$ \begin{align*} {}^J{}\pi(s_\beta w) = (w', \mu + \Phi^+(-\beta)w^{-1}\beta^\vee) \end{align*} $$
    and $w\mu \in \mathbb Z {\mathrm {cl}}(J)$ .

Proof. We fix an element $\lambda \in \mathbb Z\Phi ^\vee $ that is dominant and sufficiently regular (the required regularity constant follows from the remaining proof).

For $w\in W$ , we consider the element $w\varepsilon ^\lambda \in \widetilde W$ . Then there exist uniquely determined elements $w'\varepsilon ^{\lambda '}\in {}^J{}{W_{\mathrm {af}}}$ and $y \in \widetilde W_J$ such that

$$ \begin{align*} w\varepsilon^\lambda = y\cdot w' \varepsilon^{\lambda'}. \end{align*} $$

We define ${}^J{}\pi (w) := (w', \lambda -\lambda ')$ and check that it has the required properties.

  1. (0) $w'\in {}^J{}W$ : Since $\widetilde W_J$ is a finite group, we may assume that $\lambda '$ is superregular and dominant as well. For $(\alpha ,k)\in J$ , we have

    $$ \begin{align*} (w'\varepsilon^{\lambda'})^{-1}(\alpha,k) = ((w')^{-1}\alpha, k+\langle \lambda', (w')^{-1} \alpha\rangle)\in \Phi^+_{{\mathrm{af}}} \end{align*} $$
    because $w'\varepsilon ^{\lambda '}\in {}^J{}{W_{\mathrm {af}}}$ . Since $\lambda '$ is superregular and dominant, we have
    $$ \begin{align*} ((w')^{-1}\alpha, k+\langle \lambda', (w')^{-1} \alpha\rangle)\in \Phi^+_{{\mathrm{af}}}\iff (w')^{-1}\alpha\in \Phi^+. \end{align*} $$
    This proves $w'\in {}^J{}W$ .
  2. (1) If $w\in {}^J{}W$ , then ${}^J{}\pi (w) = (w,0)$ : The proof of (0) shows that $w\varepsilon ^\lambda \in {}^J{}{W_{\mathrm {af}}}$ so that $w\varepsilon ^\lambda = w'\varepsilon ^{\lambda '}$ .

  3. (2) Let $w\in W$ and $\beta \in \Phi _J^+$ . We have to show

    $$ \begin{align*} {}^J{}\pi(s_\beta w) = (w', \lambda - \lambda' + \Phi^+(-\beta)w^{-1}\beta^\vee). \end{align*} $$
    Put
    $$ \begin{align*} b := (\beta, \Phi^+(-\beta))\in \Phi_{{\mathrm{af}}}^+. \end{align*} $$
    By Lemma 3.21, we have $ b \in (\Phi _{\mathrm {af}})_J^+$ . The projection of
    $$ \begin{align*} r_{ b}w\varepsilon^\lambda = s_\beta w \varepsilon^{\lambda + \Phi^+(-\beta)w^{-1}\beta^\vee} \in \widetilde W_J\cdot w\varepsilon^\lambda \end{align*} $$
    onto ${}^J{}{W_{\mathrm {af}}}$ must again be $w'\varepsilon ^{\lambda '}$ . We obtain
    $$ \begin{align*} {}^J{}\pi(s_\beta w) = (w', \lambda + \Phi^+(-\beta)w^{-1}\beta^\vee-\lambda') \end{align*} $$
    as desired.

    For the second claim, it suffices to observe that

    $$ \begin{align*} \varepsilon^{w(\lambda-\lambda')} = w\varepsilon^\lambda \varepsilon^{-\lambda'} w^{-1} = y w' \varepsilon^{\lambda'} \varepsilon^{-\lambda'} w^{-1} = y \underbrace{w' w^{-1}}_{\in W_J}\in \widetilde W_J. \end{align*} $$

The fact that ${}^J{}\pi $ is uniquely determined (in particular, independent of the choice of $\lambda $ ) can be seen as follows: If $w\in {}^J{}W$ , then ${}^J{}\pi (w)$ is determined by (1). Otherwise, we find $\beta \in \Phi _J^+$ with $w^{-1}\beta \in \Phi ^-$ . We multiply w on the left with $s_\beta $ , and iterate this process, until we obtain an element in ${}^J{}W$ . This process will terminate after at most ${}^J{}\ell (w)$ steps with an element in ${}^J{}W$ . Now, for each of these steps, we can use property (2) to determine the value of ${}^J{}\pi (w)$ .

We call the set ${}^J{}W$ a semiaffine quotient of W, as it is a quotient of a finite Weyl group by a set of affine roots. The map ${}^J{}\pi $ is the semiaffine projection. We now introduce the semiaffine weight function.

Lemma 3.25. Let $w_1, w_2\in W$ and $J\subseteq \Delta $ be a spherical subset. Write

$$ \begin{align*} {}^J{}\pi(w_1) = (w_1', \mu_1),\qquad {}^J{}\pi(w_2) = (w_2', \mu_2). \end{align*} $$

Then

$$ \begin{align*} {\mathrm{wt}}(w_1'\Rightarrow w_2') - \mu_1+\mu_2 = {\mathrm{wt}}(w_1'\Rightarrow w_2) -\mu_1 \leq {\mathrm{wt}}(w_1\Rightarrow w_2). \end{align*} $$

Proof. We first show the equation

$$ \begin{align*} {\mathrm{wt}}(w_1'\Rightarrow w_2')+\mu_2 = {\mathrm{wt}}(w_1'\Rightarrow w_2). \end{align*} $$

Induction by ${}^J{}\ell (w_2)$ . The statement is trivial if $w_2\in {}^J{}W$ . Otherwise, we find some $\alpha \in {\mathrm {cl}}(J)$ with $w_2^{-1}\alpha \in \Phi ^-$ . Because $(w_1')^{-1}\alpha \in \Phi ^+$ , we obtain from Lemma 3.7 that

$$ \begin{align*} {\mathrm{wt}}(w_1'\Rightarrow w_2) =& {\mathrm{wt}}(w_1'\Rightarrow s_\alpha w_2) - \Phi^+(-\alpha)w_2^{-1}\alpha^\vee. \end{align*} $$

By Lemma 3.24, we have

$$ \begin{align*} {}^J{}\pi(s_\alpha w_2) = (w^{\prime}_2, \mu_2 + \Phi^+(-\alpha)w_2^{-1}\alpha^\vee). \end{align*} $$

Using the inductive hypothesis, we get

$$ \begin{align*} {\mathrm{wt}}(w_1'\Rightarrow w_2) &= {\mathrm{wt}}(w_1'\Rightarrow s_\alpha w_2) - \Phi^+(-\alpha)w_2^{-1}\alpha^\vee \\ &={\mathrm{wt}}(w_1'\Rightarrow w_2') + \mu_2 + \Phi^+(-\alpha)w_2^{-1}\alpha^\vee - \Phi^+(-\alpha)w_2^{-1}\alpha^\vee \\ &={\mathrm{wt}}(w_1'\Rightarrow w_2') + \mu_2. \end{align*} $$

This finishes the induction.

It remains to prove the inequality

$$ \begin{align*} {\mathrm{wt}}(w_1'\Rightarrow w_2)-\mu_1\leq {\mathrm{wt}}(w_1\Rightarrow w_2). \end{align*} $$

The argument is entirely analogous, using [Reference Schremmer28, Lemma 4.3] in place of Lemma 3.7.

Definition 3.26. Let $w_1, w_2\in W$ and $J\subseteq \Delta _{\mathrm {af}}$ be a spherical subset. We write

$$ \begin{align*} {}^J{}\pi(w_1) = (w_1', \mu_1),\qquad {}^J{}\pi(w_2) = (w_2', \mu_2). \end{align*} $$
  1. (a) We define the semiaffine weight function by

    $$ \begin{align*} {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) := {\mathrm{wt}}(w_1'\Rightarrow w_2') - \mu_1+\mu_2 = {\mathrm{wt}}(w_1'\Rightarrow w_2) -\mu_1\in \mathbb Z\Phi^\vee. \end{align*} $$
  2. (b) If $\beta \in \Phi _J$ and $(\beta ,k)\in (\Phi _{\mathrm {af}})_J$ is the image of $\beta $ under the bijection of Lemma 3.21, we define $\chi _J(\beta ) := -k$ .

    If $\beta \in \Phi \setminus \Phi _J$ , we define $\chi _J(\beta ) := \Phi ^+(\beta )$ .

    In other words, for $\beta \in \Phi $ , we have

    $$ \begin{align*} \chi_J(\beta) = \Phi^+(\beta) - \Phi^+_J(\beta). \end{align*} $$

Example 3.27. Suppose that $\Phi $ is irreducible of type $A_2$ with basis $\alpha _1, \alpha _2$ . Let $J = \{(-\theta ,1)\} = \{(-\alpha _1-\alpha _2,1)\}$ such that $\Phi _J^+ = \{-\theta \} = \{-\alpha _1-\alpha _2\}$ . We want to compute ${}^J{} {\mathrm {wt}}(1\Rightarrow s_1 s_2)$ (writing $s_i := s_{\alpha _i}$ ).

Observe that ${}^J{}\pi (1) = (s_\theta ,\theta ^\vee )$ . Hence,

$$ \begin{align*} {}^J{}{\mathrm{wt}}(1\Rightarrow s_1) &= {\mathrm{wt}}(s_\theta\Rightarrow s_1 s_2) - \theta^\vee \\ &={\mathrm{wt}}(s_1 s_2 s_1\Rightarrow s_1 s_2) - \alpha_1^\vee-\alpha_2^\vee = -\alpha_2^\vee. \end{align*} $$

Unlike the usual weight function, the value ${}^J{} {\mathrm {wt}}(w_1\Rightarrow w_2)$ no longer needs to be a sum of positive coroots. In general for root systems of type $A_n$ , we have

$$ \begin{align*} {}^J{}{}{\mathrm{wt}}(w_1\Rightarrow w_2) = \sup_{\alpha\in \Delta_{\mathrm{af}}\setminus J} (w_1^{-1}\omega_a - w_2^{-1}\omega_a). \end{align*} $$

Lemma 3.28. Let $w_1, w_2, w_3\in W$ , and let $J\subseteq \Delta $ be a spherical subset.

  1. (a) The semiaffine weight function satisfies the triangle inequality,

    $$ \begin{align*} {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_3)\leq {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) + {}^J{}{\mathrm{wt}}(w_2\Rightarrow w_3). \end{align*} $$
  2. (b) If $\alpha \in \Phi _J$ , we have

    $$ \begin{align*} {}^J{}{\mathrm{wt}}(s_\alpha w_1\Rightarrow w_2) &= {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) + \chi_J(\alpha)w_1^{-1}\alpha^\vee,\\ {}^J{}{\mathrm{wt}}(w_1\Rightarrow s_\alpha w_2) &= {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) - \chi_J(\alpha)w_2^{-1}\alpha^\vee. \end{align*} $$
  3. (c) If $\beta \in \Phi ^+$ , we have

    $$ \begin{align*} {}^J{}{\mathrm{wt}}(w_1s_\beta\Rightarrow w_2)\leq& {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) + \chi_J(w_1\beta)\beta^\vee,\\ {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2s_\beta)\leq& {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) + \chi_J(-w_2\beta)\beta^\vee. \end{align*} $$

Proof. Part (a) follows readily from the definition. Let us prove part (b). We focus on the first identity, as the proof of the second identity is analogous.

Up to replacing $\alpha $ by $-\alpha $ , which does not change the reflection $s_\alpha $ nor the value of

$$ \begin{align*} \chi_J(\alpha)w_1^{-1}\alpha^\vee, \end{align*} $$

we may assume $\alpha \in \Phi _J^+$ . Now, write

$$ \begin{align*} {}^J{}\pi(w_1) = (w_1', \mu_1),\qquad {}^J{}\pi(w_2) = (w_2', \mu_2). \end{align*} $$

Then ${}^J{}\pi (s_\alpha w_1) = (w_1', \mu _1 + \Phi ^+(-\alpha )w_1^{-1}\alpha ^\vee )$ . Thus,

$$ \begin{align*} {}^J{}{\mathrm{wt}}(s_\alpha w_1\Rightarrow w_2) &= {\mathrm{wt}}(w_1'\Rightarrow w_2') - \mu_1 - \Phi^+(-\alpha)w_1^{-1}\alpha^\vee + \mu_2 \\ &={}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) - \Phi^+(-\alpha)w_1^{-1}\alpha^\vee \\ &= {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) +\chi_J(\alpha)w_1^{-1}\alpha^\vee \end{align*} $$

as $\alpha \in \Phi _J^+$ .

Now, we prove part (c). Again, we only show the first inequality. If $w_1\beta \in \Phi _J$ , the inequality follows from part (b). Otherwise, we use (a) and [Reference Schremmer28, Lemma 4.3] to compute

$$ \begin{align*} {}^J{}{\mathrm{wt}}(w_1s_\beta\Rightarrow w_2)\leq~\,& {}^J{}{\mathrm{wt}}(w_1s_\alpha\Rightarrow w_1) + {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) \\\underset{\text{L}3.25}\leq&{\mathrm{wt}}(w_1s_\alpha\Rightarrow w_1) + {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) \\\leq~\,&\Phi^+(w\alpha)\alpha^\vee + {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2) \\=~\,&\chi_J(w\alpha)\alpha^\vee + {}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2). \end{align*} $$

This finishes the proof.

Lemma 3.29. Let $w_1, w_2\in W$ and $J\subseteq \Delta _{{\mathrm {af}}}$ be spherical. Suppose that, for all $\alpha \in \Phi ^+_J$ , at least one of the following conditions is satisfied:

$$ \begin{align*}w_1^{-1}\alpha\in \Phi^+\text{ or }w_2^{-1}\alpha\in \Phi^-.\end{align*} $$

Then ${}^J{} {\mathrm {wt}}(w_1\Rightarrow w_2) = {\mathrm {wt}}(w_1\Rightarrow w_2)$ .

Proof. We show the claim via induction on ${}^J{}\ell (w_1)$ . If $w_1\in {}^J{}W$ , then the claim follows from Lemma 3.25.

Otherwise, we find some $\alpha \in {\mathrm {cl}}(J)$ with $w_1^{-1}\alpha \in \Phi ^-$ . By assumption, also $w_2^{-1}\alpha \in \Phi ^-$ . Using Lemma 3.7, we get

$$ \begin{align*} {\mathrm{wt}}(w_1\Rightarrow w_2) &={\mathrm{wt}}(s_\alpha w_1\Rightarrow s_\alpha w_2) + \chi_J(\alpha)w_1^{-1}\alpha^\vee - \chi_J(\alpha)w_2^{-1}\alpha^\vee. \end{align*} $$

Since ${}^J{}\ell (s_\alpha w_1)<{}^J{}\ell (w_1)$ by Lemma 3.23, we want to show that $(s_\alpha w_1,s_\alpha w_2)$ also satisfy the condition stated in the lemma.

For this, let $\beta \in \Phi _J^+$ . If $\beta = \alpha $ , then $(s_\alpha w_1)^{-1}\alpha = -w_1^{-1}\alpha \in \Phi ^+$ by choice of $\alpha $ . Now, assume that $\beta \neq \alpha $ so that $s_\alpha \beta \in \Phi ^+_J$ . By the assumption on $w_1$ and $w_2$ , we must have $w_1^{-1}s_\alpha (\beta )\in \Phi ^+$ or $w_2^{-1} s_\alpha (\beta )\in \Phi ^-$ . In other words, we have

$$ \begin{align*} (s_\alpha w_1)^{-1}\beta\in \Phi^+\text{ or }(s_\alpha w_2)^{-1}\beta \in \Phi^-. \end{align*} $$

This shows that $(s_\alpha w_1, s_\alpha w_2)$ satisfy the desired properties.

By the inductive hypothesis and Lemma 3.28, we get

$$ \begin{align*} &{\mathrm{wt}}(s_\alpha w_1\Rightarrow s_\alpha w_2) + \chi_J(\alpha) w_1^{-1}\alpha^\vee - \chi_J(\alpha)w_2^{-1}\alpha^\vee\\ &= {}^J{}{\mathrm{wt}}(s_\alpha w_1\Rightarrow s_\alpha w_2) + \chi_J(\alpha) w_1^{-1}\alpha^\vee - \chi_J(\alpha)w_2^{-1}\alpha^\vee \\ &={}^J{}{\mathrm{wt}}(w_1\Rightarrow w_2). \end{align*} $$

This completes the induction and the proof.

4. Bruhat order

The Bruhat order on $\widetilde W$ is a fundamental Coxeter-theoretic notion that has been studied with great interest, for example, [Reference Björner and Brenti1, Reference Kottwitz and Rapoport15, Reference Rapoport, Jacques, Henri, Michael and Marie-France25, Reference Lenart, Naito, Sagaki, Schilling and Shimozono17]. In this section, we present new characterizations of the Bruhat order on $\widetilde W$ .

The structure of this section is as follows: In Section 4.1, we state our main criterion for the Bruhat order as Theorem 4.2 and discuss some of its applications. We then prove this criterion in Section 4.2. Finally, Section 4.3 will cover some consequences of Deodhar’s lemma (cf. [Reference Deodhar7]) and feature an even more general criterion.

4.1. A criterion

Definition 4.1. Let $x = w\varepsilon ^\mu \in \widetilde W$ . A Bruhat-deciding datum for x is a tuple $(v, J_1,\dotsc ,J_m)$ , where $v\in W$ and $J_\bullet $ is a finite collection of arbitrary subsets $J_1,\dotsc ,J_m\subseteq \Delta $ with $m\geq 1$ , satisfying the following two properties:

  1. (1) The element v is length positive for x, that is, $\ell (x,v\alpha )\geq 0$ for all $\alpha \in \Phi ^+$ .

  2. (2) Writing $J := J_1\cap \cdots \cap J_m$ , we have $\ell (x,v\alpha )=0$ for all $\alpha \in \Phi _J$ .

The name Bruhat-deciding is justified by the following result.

Theorem 4.2. Let $x = w\varepsilon ^\mu , x' = w'\varepsilon ^{\mu '}\in \widetilde W$ . Fix a Bruhat-deciding datum $(v, J_1,\dotsc ,J_m)$ for x. Then the following are equivalent:

  1. (1) $x\leq x'$ .

  2. (2) For all $i=1,\dotsc ,m$ , there exists an element $v^{\prime }_i\in W$ such that

    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(v^{\prime}_i\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow w'v^{\prime}_i)\leq (v^{\prime}_i)^{-1}\mu'\quad \pmod{\Phi_{J_i}^\vee}. \end{align*} $$

We again use the shorthand notation $\mu _1\leq \mu _2\ \pmod {\Phi ^\vee _J}$ for $\mu _1-\mu _2+\mathbb Z\Phi ^\vee _J\leq 0+\mathbb Z\Phi ^\vee _J$ in $\mathbb Z\Phi ^\vee /\mathbb Z\Phi ^\vee _J$ .

This theorem is the main result of this section. We give a proof in Section 4.2.

First, let us remark that the construction of a Bruhat-deciding datum is easy. It suffices to choose any length positive element v for x, and then $(v,\emptyset )$ is Bruhat-deciding.

The inequality of Theorem 4.2 is only interesting for $v\in {\mathrm {LP}}(x)$ and $v_i'\in {\mathrm {LP}}(x')$ , as explained by the following lemma in conjunction with [Reference Schremmer28, Lemma 2.3].

Lemma 4.3. Let $x = w\varepsilon ^\mu , x' = w'\varepsilon ^{\mu '}\in \widetilde W$ . Suppose we are given elements $v, v'\in W$ , a subset $J\subseteq \Delta $ and a positive root $\alpha \in \Phi ^+$ .

  1. (a) Assume $\ell (x,v\alpha )<0$ . Then the inequality

    $$ \begin{align*} (vs_\alpha)^{-1}\mu + {\mathrm{wt}}(v'\Rightarrow vs_\alpha) + {\mathrm{wt}}(wvs_\alpha\Rightarrow w'v')\leq (v')^{-1}\mu'\quad \pmod{\Phi_J^\vee} \end{align*} $$
    implies
    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(v'\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow w'v')\leq (v')^{-1}\mu'\quad \pmod{\Phi_J^\vee}. \end{align*} $$
  2. (b) Assume $\ell (x',v\alpha )<0$ . Then the inequality

    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(v'\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow w'v')\leq (v')^{-1}\mu'\quad \pmod{\Phi_J^\vee} \end{align*} $$
    implies
    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(v's_\alpha\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow w'v's_\alpha)\leq (v's_\alpha)^{-1}\mu'\quad \pmod{\Phi_J^\vee}. \end{align*} $$

Proof.

  1. (a) We have

    $$ \begin{align*}(v')^{-1}\mu'&\geq(vs_\alpha)^{-1}\mu + \mathrm{wt}(v'\Rightarrow vs_\alpha) + \mathrm{wt}(wvs_\alpha\Rightarrow w'v')\\ &\geq v^{-1}\mu - \langle v^{-1}\mu,\alpha\rangle \alpha^\vee + \mathrm{wt}(v'\Rightarrow v) - \mathrm{wt}(vs_\alpha\Rightarrow v) \\&\quad + \mathrm{wt}(wv\Rightarrow w'v') - \mathrm{wt}(wv\Rightarrow wvs_\alpha)\\ &\underset{(\ast)}\geq v^{-1}\mu - \langle v^{-1}\mu,\alpha\rangle \alpha^\vee + \mathrm{wt}(v'\Rightarrow v) - \Phi^+(v\alpha)\alpha^\vee\\&\quad + \mathrm{wt}(wv\Rightarrow w'v') - \Phi^+(-wv\alpha)\alpha^\vee\\ &=v^{-1}\mu + \mathrm{wt}(v'\Rightarrow v) + \mathrm{wt}(wv\Rightarrow w'v') - (\ell(x,v\alpha)+1)\alpha^\vee\\ &\geq v^{-1}\mu + \mathrm{wt}(v'\Rightarrow v) + \mathrm{wt}(wv\Rightarrow w'v')\quad \pmod{\Phi_J^\vee}. \end{align*} $$
    The inequality $(\ast )$ is [Reference Schremmer28, Lemma 4.3].
  2. (b) The calculation is completely analogous.

Proof of Theorem 1.1 using Theorem 4.2.

We use the notation of Theorem 1.1. In view of Lemma 4.3 and [Reference Schremmer28, Lemma 2.3], the condition

(*) $$ \begin{align} \exists v_2\in W:~ v_1^{-1}\mu_1 + {\mathrm{wt}}(v_2\Rightarrow v_1) + {\mathrm{wt}}(w_1 v_1\Rightarrow w_2 v_2)\leq v_2^{-1}\mu_2 \end{align} $$

is true for all $v_1\in {\mathrm {LP}}(x)$ iff it is true for all $v_1\in W$ . We see that asking condition $(\ast )$ for all $v_1\in W$ is equivalent to asking condition (2) of Theorem 4.2 for each Bruhat-deciding datum of the form $(v_1,\emptyset )$ with $v_1\in {\mathrm {LP}}(x_1)$ . In this sense, Theorem 4.2 implies Theorem 1.1.

If $x'$ is in a shrunken Weyl chamber, there is a canonical choice for $v'$ .

Corollary 4.4. Let $x = w\varepsilon ^\mu $ and $x' = w'\varepsilon ^{\mu '}$ . Assume that $x'$ is in a shrunken Weyl chamber and that $v'$ is the length positive element for $x'$ . Pick any length positive element v for x. Then $x\leq x'$ if and only if

$$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(v'\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow w'v')\leq (v')^{-1}\mu'. \end{align*} $$

Proof. $(v,\emptyset )$ is a Bruhat-deciding datum for x. By Lemma 4.3 and [Reference Schremmer28, Corollary 2.4], the inequality in Theorem 4.2 (2) is satisfied by some $v'\in W$ iff it is satisfied by the unique length positive element $v'$ for $x'$ .

We now show how Theorem 4.2 can be used to describe Bruhat covers in $\widetilde W$ . The following proposition generalizes the previous results of Lam–Shimozono [Reference Lam and Shimozono16, Proposition 4.1] and Milićević [Reference Milićević20, Proposition 4.2].

Proposition 4.5. Let $x = w\varepsilon ^\mu , x' = w'\varepsilon ^{\mu '}\in \widetilde W$ and $v\in {\mathrm {LP}}(x)$ . Then the following are equivalent:

  1. (a) $x\lessdot x'$ , that is, $x < x'$ and $\ell (x) = \ell (x')-1$ .

  2. (b) There exists some $v'\in {\mathrm {LP}}(x')$ such that

    1. (b.1) $v^{-1}\mu + {\mathrm {wt}}(v'\Rightarrow v) + {\mathrm {wt}}(wv\Rightarrow w'v') = (v')^{-1}\mu '$ and

    2. (b.2) $d(v'\Rightarrow v) + d(wv\Rightarrow w'v')=1$ .

  3. (c) There is a root $\alpha \in \Phi ^+$ satisfying at least one of the following conditions:

    1. (c.1) There exists a Bruhat edge $v' := s_\alpha v\rightarrow v$ in ${\mathrm {QB}}(W)$ with $x' = xs_\alpha $ and $v'\in {\mathrm {LP}}(x')$ .

    2. (c.2) There exists a quantum edge $v' := s_\alpha v\rightarrow v$ in ${\mathrm {QB}}(W)$ with $v^{-1}\alpha \in \Phi ^+, x' = xr_{(-\alpha ,1)}$ and $v'\in {\mathrm {LP}}(x')$ .

    3. (c.3) There exists a Bruhat edge $wv\rightarrow s_\alpha wv$ in ${\mathrm {QB}}(W)$ such that $x' = s_\alpha x$ and $v\in {\mathrm {LP}}(x')$ .

    4. (c.4) There exists a quantum edge $wv\rightarrow s_\alpha wv$ in ${\mathrm {QB}}(W)$ with $(wv)^{-1}\alpha \in \Phi ^-$ , $x' = r_{(-\alpha ,1)} x$ and $v\in {\mathrm {LP}}(x')$ .

  4. (d) There exists a root $\alpha \in \Phi ^+$ satisfying at least one of the following conditions:

    1. (d.1) We have $w' = ws_\alpha , \mu ' = s_\alpha (\mu ), \ell (s_\alpha v) = \ell (v)-1$ and for all $\beta \in \Phi ^+$ :

      $$ \begin{align*} \ell(x,v\beta) + \Phi^+(s_\alpha v\beta) - \Phi^+(v\beta)\geq 0. \end{align*} $$
    2. (d.2) We have $w' = ws_\alpha , \mu ' = s_\alpha (\mu )-\alpha ^\vee , \ell (s_\alpha v) = \ell (v)-1+\langle v^{-1}\alpha ^\vee ,2\rho \rangle $ and for all $\beta \in \Phi ^+$ :

      $$ \begin{align*} \ell(x,v\beta) + \langle \alpha^\vee,v\beta\rangle + \Phi^+(s_\alpha v\beta) - \Phi^+(v\beta)\geq 0. \end{align*} $$
    3. (d.3) We have $w' = s_\alpha w, \mu ' = \mu , \ell (s_\alpha wv) = \ell (wv)+1$ and for all $\beta \in \Phi ^+$ :

      $$ \begin{align*} \ell(x,v\beta) + \Phi^+(wv\beta) - \Phi^+(s_\alpha wv\beta)\geq 0. \end{align*} $$
    4. (d.4) We have $w' = s_\alpha w, \mu ' = \mu -w^{-1}\alpha ^\vee , \ell (s_\alpha wv) = \ell (wv)+1+\langle (wv)^{-1}\alpha ^\vee ,2\rho \rangle $ and for all $\beta \in \Phi ^+$ :

      $$ \begin{align*} \ell(x,v\beta) + \langle \alpha^\vee,wv\beta\rangle + \Phi^+(wv\beta) - \Phi^+(s_\alpha wv\beta)\geq 0. \end{align*} $$

Proof. (a) $\iff $ (b): We start with a key calculation for $v'\in {\mathrm {LP}}(x')$ :

$$ \begin{align*} &\langle (v')^{-1}\mu' - {\mathrm{wt}}(v'\Rightarrow v) - {\mathrm{wt}}(wv\Rightarrow w'v') - v^{-1}\mu,2\rho\rangle \\ \underset{\text{L}3.5} =&\ \langle (v')^{-1}\mu,2\rho\rangle - d(v'\Rightarrow v) -\ell(v') + \ell(v) \\ &\qquad-d(wv\Rightarrow w'v') - \ell(wv) + \ell(w'v') - \langle v^{-1}\mu,2\rho\rangle \\ &\underset{\text{L}2.3} = \ell(x') - \ell(x) - d(v'\Rightarrow v) - d(wv\Rightarrow w'v'). \end{align*} $$

First, assume that (a) holds, that is, $x\lessdot x'$ . By Theorem 4.2 and Lemma 4.3, we find $v'\in {\mathrm {LP}}(x')$ such that

$$ \begin{align*} (v')^{-1}\mu' - {\mathrm{wt}}(v'\Rightarrow v) - {\mathrm{wt}}(wv\Rightarrow w'v') - v^{-1}\mu\geq 0. \end{align*} $$

By the above key calculation, we see that

$$ \begin{align*} \ell(x') \geq \ell(x) + d(v'\Rightarrow v) + d(wv\Rightarrow w'v'), \end{align*} $$

where equality holds if and only if (b.1) is satisfied. Note that $x\lessdot x'$ implies that $x^{-1}x'$ must be an affine reflection, thus $w\neq w'$ . We see that $v\neq v'$ or $wv\neq w'v'$ , thus in particular

$$ \begin{align*} \ell(x)+1= \ell(x') \geq \ell(x) + d(v'\Rightarrow v) + d(wv\Rightarrow w'v')\geq \ell(x)+1. \end{align*} $$

Since equality must hold, we get (b.1) and (b.2).

Now, assume conversely that (b) holds. By (b.1) and Theorem 4.2, we see that $x<x'$ . Now, using the key calculation and (b.2), we get $\ell (x') =\ell (x)+1$ .

(b) $\iff $ (c): The condition (b.2) means that either $v=v'$ and $wv\rightarrow w'v'$ is an edge in ${\mathrm {QB}}(W)$ , or $wv = w'v'$ and $v'\rightarrow v$ is an edge. If we now distinguish between Bruhat and quantum edges, we get the explicit conditions of (c) (or (d)).

Let us first assume that (b) holds. We distinguish the following cases:

  1. (1) $wv = w'v'$ and $v'\rightarrow v$ is a Bruhat edge: Then we can write $v' = s_\alpha v$ for some $\alpha \in \Phi ^+$ with $v^{-1}\alpha \in \Phi ^-$ . Now, the condition $wv = w'v'$ implies $w' = ws_\alpha $ . Condition (b.1) implies $v^{-1}\mu = (v')^{-1}\mu '$ , so $\mu ' = s_\alpha (\mu )$ . We get (c.1).

  2. (2) $wv = w'v'$ and $v'\rightarrow v$ is a quantum edge: Then we can write $v' = s_\alpha v$ for some $\alpha \in \Phi ^+$ with $v^{-1}\alpha \in \Phi ^+$ . Now, the condition $wv = w'v'$ implies $w' = ws_\alpha $ . Condition (b.1) implies $v^{-1}\mu +v^{-1}\alpha ^\vee = (v')^{-1}\mu '$ , so $\mu ' = s_\alpha (\mu )-\alpha ^\vee $ . We get (c.2).

  3. (3) $v = v'$ and $wv\rightarrow w'v'$ is a Bruhat edge: Then we can write $w'v' = s_\alpha wv$ for some $\alpha \in \Phi ^+$ with $(wv)^{-1}\alpha \in \Phi ^-$ . Now, the condition $v=v'$ implies $w' = s_\alpha w$ . Condition (b.1) implies $v^{-1}\mu = (v')^{-1}\mu $ , so $\mu ' = \mu $ . We get (c.3).

  4. (4) $v=v'$ and $wv\rightarrow w'v'$ is a quantum edge: Then we can write $w'v' = s_\alpha wv$ for some $\alpha \in \Phi ^+$ with $(wv)^{-1}\alpha \in \Phi ^-$ . Now, the condition $v=v'$ implies $w' = s_\alpha w$ . Condition (b.1) implies $v^{-1}\mu -(wv)^{-1}\alpha ^\vee = (v')^{-1}\mu $ , so $\mu ' = \mu -w^{-1}\alpha ^\vee $ . We get (c.4).

Reversing the calculations above shows that (c) $\implies $ (b).

For (c) $\iff $ (d), we just explicitly rewrite the conditions for length positivity of $v'$ , and the definition of edges in the quantum Bruhat graph.

Remark 4.6. If the translation part $\mu $ of $x=w\varepsilon ^\mu $ is sufficiently regular, the estimates for the length function of x in part (d) of Proposition 4.5 are trivially satisfied. Writing ${\mathrm {LP}}(x) = \{v\}$ , we get a one-to-one correspondence

$$ \begin{align*} \{\text{Bruhat covers of }x\}\leftrightarrow \{\text{edges }?\rightarrow v\}\sqcup\{\text{edges }wv\rightarrow ?\}. \end{align*} $$

We obtain the following useful technical observation from Proposition 4.5:

Corollary 4.7. Let $x \in \widetilde W$ , $v\in {\mathrm {LP}}(x)$ and $(\alpha ,k)\in \Delta _{\mathrm {af}}$ with $\ell (x,\alpha )=0$ . If $v^{-1}\alpha \in \Phi ^+$ , then $s_\alpha v\in {\mathrm {LP}}(x)$ .

Proof. Since $x(\alpha ,k)\in \Phi ^+$ by [Reference Schremmer28, Lemma 2.9], we have $x<xr_a$ . Since a is a simple affine root, we must have $x\lessdot xr_a$ . So one of the four possibilities (c.1)–(c.4) of Proposition 4.5 must be satisfied.

If (c.3) or (c.4) are satisfied, we get $v\in {\mathrm {LP}}(x')$ . Since $x' = x r_a$ is a length additive product, [Reference Schremmer28, Lemma 2.13] shows $s_\alpha v\in {\mathrm {LP}}(x)$ , finishing the proof.

Now, assume that (c.1) is satisfied. Then $x' = xs_\beta $ for some $\beta \in \Phi ^+$ means $k=0$ and $\alpha =\beta $ . Now, $v^{-1}\alpha \in \Phi ^+$ means that $\ell (s_\alpha v)> \ell (v)$ , so $s_\alpha v\rightarrow v$ cannot be a Bruhat edge.

Finally, assume that (c.2) is satisfied. Then $x' = xr_{(-\beta ,1)}$ for some $\beta \in \Phi ^+$ means that $k=1$ and $\alpha =-\beta \in \Phi ^-$ . Then $s_\alpha v\rightarrow v$ cannot be a quantum edge, as $\ell (s_\alpha v)<\ell (v)$ .

We get the desired claim or a contradiction, finishing the proof.

As a second application, we discuss the semi-infinite order on $\widetilde W$ as introduced by Lusztig [Reference Lusztig19]. It plays a role for certain constructions related to the affine Hecke algebra, cf. [Reference Lusztig19, Reference Naito and Watanabe22].

Definition 4.8. Let $x = w\varepsilon ^\mu \in \widetilde W$ .

  1. (a) We define the semi-infinite length of x as

    $$ \begin{align*} \ell^{\frac\infty 2}(x) := \ell(w) +\langle \mu,2\rho\rangle. \end{align*} $$
  2. (b) We define the semi-infinite order on $\widetilde W$ to be the order $<^{\frac \infty 2}$ generated by the relations

    $$ \begin{align*} \forall x\in \widetilde W, a\in \Phi_{\mathrm{af}}:~x<^{\frac \infty 2}xr_a\text{ if }\ell^{\frac\infty 2}(x)\leq \ell^{\frac\infty 2}(xr_a). \end{align*} $$

We have the following link between the semi-infinite order and the Bruhat order:

Proposition 4.9 [Reference Naito and Watanabe22, Proposition 2.2.2].

Let $x_1, x_2\in \widetilde W$ . There exists a number $C>0$ such that for all $\lambda \in \mathbb Z\Phi ^\vee $ satisfying the regularity condition $\langle \lambda ,\alpha \rangle> C$ for every positive root $\alpha $ , we have

$$ \begin{align*}x_1\leq^{\frac\infty 2}x_2{\iff} x_1\varepsilon^\lambda \leq x_2\varepsilon^\lambda. \end{align*} $$

Corollary 4.10. Let $x_1 = w_1\varepsilon ^{\mu _1}, x_2 = w_2\varepsilon ^{\mu _2}\in \widetilde W$ . Then $x_1\leq ^{\frac \infty 2}x_2$ if and only if

$$ \begin{align*} \mu_1 + {\mathrm{wt}}(w_1\Rightarrow w_2)\leq \mu_2. \end{align*} $$

Proof. Let $\lambda $ be as in Proposition 4.9. Choosing $\lambda $ sufficiently large, we may assume that $x_1\varepsilon ^{\lambda }$ and $x_2\varepsilon ^{\lambda }$ are superregular with ${\mathrm {LP}}(x_1\varepsilon ^\lambda ) = {\mathrm {LP}}(x_2\varepsilon ^\lambda )=\{1\}$ . Now, $x_1\varepsilon ^\lambda \leq x_2\varepsilon ^\lambda $ if and only if

$$ \begin{align*} \mu_1 + {\mathrm{wt}}(w_1\Rightarrow w_2)\leq \mu_2, \end{align*} $$

by Corollary 4.4.

We finish this section with another application of our Theorem 4.2, namely a discussion of admissible and permissible sets in $\widetilde W$ , as introduced by Kottwitz and Rapoport [Reference Kottwitz and Rapoport15].

Definition 4.11. Let $x = w\varepsilon ^\mu \in \widetilde W$ and $\lambda \in X_\ast $ a dominant coweight.

  1. (a) We say that x lies in the admissible set defined by $\lambda $ , denoted $x\in {\mathrm {Adm}}(\lambda )$ , if there exists $u\in W$ such that $x\leq \varepsilon ^{u\lambda }$ with respect to the Bruhat order on $\widetilde W$ .

  2. (b) The fundamental coweight associated with $a =(\alpha ,k)\in \Delta _{\mathrm {af}}$ is the uniquely determined element $\omega _a\in \mathbb Q\Phi ^\vee $ such that for each $\beta \in \Delta $ ,

    $$ \begin{align*} \langle \omega_a,\beta\rangle = \begin{cases}1,&a = (\beta,0),\\ 0,&a\neq (\beta,0).\end{cases} \end{align*} $$

    In particular, $\omega _a=0$ iff $k\neq 0$ .

  3. (c) Let $a = (\alpha ,k)\in \Delta _{\mathrm {af}}$ , and denote by $\theta \in \Phi ^+$ the longest root of the irreducible component of $\Phi $ containing $\alpha $ . The normalized coweight associated with a is

    $$ \begin{align*} \widetilde \omega_a = \begin{cases}0,&k\neq 0,\\ \frac{1}{\langle \omega_a,\theta\rangle}\omega_a,&k=0.\end{cases} \end{align*} $$
  4. (d) We say that x lies in the permissible set defined by $\lambda $ , denoted $x\in {\mathrm {Perm}}(\lambda )$ , if $\mu \equiv \lambda\ \pmod {\Phi ^\vee }$ and for every simple affine root $a\in \Delta _{\mathrm {af}}$ , we have

    $$ \begin{align*} (\mu + \widetilde\omega_a - w^{-1}\widetilde\omega_a)^{{\mathrm{dom}}}\leq \lambda\text{ in }X_\ast\otimes\mathbb Q. \end{align*} $$

It is shown in [Reference Kottwitz and Rapoport15] that the admissible set is always contained in the permissible set and that equality holds for the groups ${\mathrm {GL}}_n$ and ${\mathrm {GSp}}_{2n}$ if $\lambda $ is minuscule (i.e., a fundamental coweight of some special node). It is a result of Haines and Ngô [Reference Haines and Ngô8] that ${\mathrm {Adm}}(\lambda )\neq {\mathrm {Perm}}(\lambda )$ in general. We show how the latter result can be recovered using our methods.

Proposition 4.12 (Cf. [Reference He and Yu13, Prop. 3.3]).

Let $x = w\varepsilon ^\mu \in \widetilde W$ and $\lambda \in X_\ast $ a dominant coweight. Then the following are equivalent:

  1. (1) $x\in {\mathrm {Adm}}(\lambda )$ .

  2. (2) For all $v\in W$ , we have

    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(wv\Rightarrow v)\leq \lambda. \end{align*} $$
  3. (3) For some $v\in {\mathrm {LP}}(x)$ , we have

    $$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(wv\Rightarrow v)\leq \lambda. \end{align*} $$

Proof. (1) $\implies $ (2): Suppose that $x\in {\mathrm {Adm}}(\lambda )$ , so $x\leq \varepsilon ^{u\lambda }$ for some $u\in W$ . Let also $v\in W$ . By Lemma 4.15, we find $\tilde u\in W$ such that

$$ \begin{align*} v^{-1}\mu + {\mathrm{wt}}(\tilde u\Rightarrow v) + {\mathrm{wt}}(wv\Rightarrow \tilde u)\leq \tilde u^{-1}u\lambda. \end{align*} $$

Thus,

$$ \begin{align*}v^{-1}\mu+\mathrm{wt}(wv\Rightarrow v)\leq&v^{-1}\mu + \mathrm{wt}(\tilde u\Rightarrow v) + \mathrm{wt}(wv\Rightarrow \tilde u)\\\leq &\tilde u^{-1}u\lambda\\\leq&(\tilde u^{-1}u\lambda)^{\mathrm{dom}} = \lambda. \end{align*} $$

Since (2) $\implies $ (3) is trivial, it remains to show (3) $\implies $ (1). So let $v\in {\mathrm {LP}}(x)$ satisfy $v^{-1}\mu + {\mathrm {wt}}(wv\Rightarrow v)\leq \lambda $ . By Theorem 4.2, we immediately get $x\leq \varepsilon ^{v\lambda }$ , showing (1).

Lemma 4.13. Let $x = w\varepsilon ^\mu \in \widetilde W$ and $\lambda \in X_\ast $ a dominant coweight. Then the following are equivalent:

  1. (1) $x\in {\mathrm {Perm}}(\lambda )$ .

  2. (2) For all $v\in W$ , we have

    $$ \begin{align*} v^{-1}\mu + \sup_{a\in \Delta_{\mathrm{af}}}\left( v^{-1}\widetilde\omega_a - (wv)^{-1}\widetilde\omega_a\right)\leq \lambda. \end{align*} $$

If moreover x lies in a shrunken Weyl chamber, the conditions are equivalent to

  1. (3) For the uniquely determined $v\in {\mathrm {LP}}(x)$ , we have

    $$ \begin{align*} v^{-1}\mu + \sup_{a\in \Delta_{\mathrm{af}}} \left(v^{-1}\widetilde\omega_a - (wv)^{-1}\widetilde\omega_a\right)\leq \lambda. \end{align*} $$

Proof. We have

$$ \begin{align*} \text{(1)}\iff&\forall a\in \Delta_{\mathrm{af}}:~\left(\mu +\widetilde\omega_a - w^{-1}\widetilde\omega_a\right)^{{\mathrm{dom}}}\leq\lambda \\\iff&\forall a\in \Delta_{\mathrm{af}}, v\in W:~v^{-1}\left(\mu + \widetilde\omega_a - w^{-1}\widetilde\omega_a\right)\leq\lambda \\\iff&\forall v\in W:\sup_{a\in \Delta_{\mathrm{af}}}v^{-1}\left(\mu + \widetilde\omega_a- w^{-1}\widetilde\omega_a\right)\leq\lambda \\\iff&\text{(2)}. \end{align*} $$

Now, assume that x is in a shrunken Weyl chamber, ${\mathrm {LP}}(x) = \{v\}$ and $a\in \Delta _{\mathrm {af}}$ . We claim that

$$ \begin{align*} \left(\mu + \widetilde\omega_a - w^{-1}\widetilde\omega_a\right)^{{\mathrm{dom}}} = v^{-1}\left(\mu + \widetilde\omega_a - w^{-1}\widetilde\omega_a\right). \end{align*} $$

Once this claim is proved, the equivalence (1) $\iff $ (3) follows.

It remains to show that