connected sum of so-called “prime” manifolds. In contrast to the simplicity of
the genus expansion in 2d, the prime manifolds are infinite in number, and not
completely and uniquely classified.

Besides the above extensions, there is, of course, always the possibility of
considering models of surfaces with more complicated structures like defects,
foliations, or (as in Balasubramanian:2020jhl ) spin structures.

Finally, the coupling of gravitational regions to non-gravitational TQFT
regions opens other avenues for further study. We find especially interesting
the question of whether a version of the black hole information paradox can be
phrased in this framework, perhaps analogously to the construction in
Almheiri_2020 , or in some different way. Also of interest in this framework
is the question of bulk reconstruction. For example, one could consider TQFT
operators in the bulk gravity region that are appropriately “gravitationally
dressed” so as to be well-defined for spacetimes without fixed topology. (A
simple example of “gravitational dressing” would be to specify a fixed
boundary component that the observable must remain path connected to when we
allow topology to fluctuate.) Then the question arises of whether and how such
bulk operators can be represented once we switch to the dual picture where
gravity is an ensemble of boundary conditions.

## Acknowledgements

We wish to thank Per Kraus and Thomas Dumitrescu for helpful discussions. We
also thank the Mani L. Bhaumik Institute for Theoretical Physics for its
support. SM is grateful to the Alexander S. Onassis Foundation for its
support.

##  Appendix A State sum formulation of Dijkgraaf-Witten theory

Dijkgraaf-Witten theory can be equivalently formulated as a lattice gauge
theory on a triangulated manifold. The outputs of the theory formulated this
way are so-called state-sum expressions for the partition function. In
doughterty20152dimensional , this approach was generalized to DW theory with
defects. Their methods can be straightforwardly generalized to provide an
independent method of obtaining our open/closed DW rules (69)-(73), and at the
same time provide directions for future work.

###  A.1 Review of state sum for DW with defects

Given a surface-curve pair $\Sigma\supset C$, we can define a refine notion of
triangulation as follows.

######  Definition A.1.

A triangulation $T$ of a surface with curve $\Sigma\supset C$ is flag-like if
C is a subcomplex, and if, for every 2-simplex $\sigma\in T$, the intersection
of $\sigma$ and C is either a face (i.e. one vertex or one edge) or is empty.

In other words a flag-like trangulation has to have the curve lie along faces,
and only use triangles like the ones in figure 8.

Figure 8: Basic building blocks of a flag-like traingulation.

In DW theory without defects, the construction proceeds by choosing
appropriate an appropriate group $G$ whose elements will label the edges of
the triangles in the triangulation. In the presence of defects, we need to
pick two finite groups $G$, $H$ and a space $X$ with a right $G$-action and a
left $H$-action. Physically $G$ specifies the degrees of freedom in the bulk,
$H$ the degres of freedom on the defect and $X$ is a way of coupling (or not)
the two together. Then we assign elements of $G$, $H$ and $X$ on edges of the
triangles from figure 8, in a consistent way as shown in fig 9.

Now we can define the partition function of DW theory.

######  Definition A.2.

Let $G$, $H$ be finite groups and $X$ a set equipped with commuting group
action of $G$ on the right and $H$ on the left. Then, for any flag -like
triangulation $T$ of a surface-curve pair $\Sigma\supset C$, the partition
function of untwisted DW theory is defined to be

$\displaystyle Z_{(H,X,G)}(\Sigma\supset
C):=|G|^{-\sigma}|H|^{-c}\kappa_{(H,X,G)}(T)$ (94)

where $\kappa_{(H,X,G)}(T)$ is the number of admissible colorings of the
triangulation, $\sigma$ is the number of bulk vertices, and $c$ is the number
of vertices lying on the defect.

The method of doughterty20152dimensional  for defining the partition function
of 2d DW with 1d defects on a closed manifold $\Sigma$ begins by choosing two
groups $G$, $H$ and a space $X$ on which $G$ has a right action and $H$ has a
left action. The physical interpretation of these choices are as the dofs of
the DW, the dofs on the defects, and a coupling between these dofs,
respectively. Then they introduce a triangulation of $\Sigma$, that is made
out of three basic triangles, see figure 8. A triangulation obtained from
these triangles they call flag-like, and then the partition function of
untwisted DW is essentially the counting of possible group element assignments
on each edge of the triangulation, which is the lattice equivalent of counting
isomorphism classes of bundles,

$\displaystyle Z_{(H,X,G)}(\Sigma\supset
C):=|G|^{-|T^{0}_{0}|}|H|^{-|T^{1}_{0}|}\kappa_{(H,X,G)}(T)$ (95)

where $T^{k}_{n}$ denote the set of n-simplices of $T$ with $k$ vertices on
the curve, and $\kappa_{(H,X,G)}(T)$ is the number of admissible
${(H,X,G)}$-colorings of T. Then they show that this is independent of the
flag-like triangulation chosen, and hence a topological invariant of the
surface-curve pair $\Sigma\supset C$.

Figure 9: Admissible colorings of a flag-like triangulation. Here $f,g,h\in
G$, $x,y\in X$ and $\eta\in H$.

###  A.2 Generalization and importance for our methods

Although, in doughterty20152dimensional  they restrict attention to closed
manifolds $\Sigma$, their methods can be straightforwardly generalized to
manifolds with fixed holonomies and/or parallel transports on their
boundaries. So using (95), we can reproduce all of the DW TQFT rules
(69)-(73). The simplest choice for $X$ is the direct product $X=G\times H$.
With this choice, we label the brane boundaries by parallel transports
$(h,1)\in X$, and since the brane boundaries are dynamical we will sum over
all those $h$. Each non-dynamical, open boundary has to be made up by at least
two different edges, since there is no 2-simplex with two defect vertices and
no defect edge. Let $g\in G$ be the parallel transport specified along a non-
dynamical, open boundary, then we will model this boundary by allowing a non
zero $H$ parallel transport along it, so that the total transport is $(h,g)\in
X$ for some $h\in H$. Other than reproducing (69)-(73), the lattice
perspective helps visualize how a (small) gauge $G$ transformation wouldn’t
act on the boundary, since the brane boundaries are fixed to have parallel
transport $(h,1)$.

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# Some translations in the second lowest two-sided cell
of an affine Weyl group

Yannan Qiu Partially supported by NSFC grant No.11621061.

School of Mathematical Sciences, Zhejiang University

Zhejiang 310058, China

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Beijing 100190, China

<EMAIL_ADDRESS>
Abstract

We are interested in cell properties of translations in affine Weyl groups. We
find out some translations in the second lowest two-sided cell of an affine
Weyl group and use the translations to formulate a refinement of Jianyi Shi’s
conjecture on the number of left cells in the second lowest two-sided cell. We
verify the refinement for an affine Weyl group of type $\tilde{A}_{n-1}$ and
type $\tilde{G}_{2}$.

Keywords: affine Weyl group, extended affine Weyl groups, two-sided cell, left
cell, the second lowest two-sided cell, translations.

##  0 Introduction

Kazhdan-Lusztig cells play an important role in Kazhdan-Lusztig theory. In the
theory an interesting topic is the study of cells in affine Weyl groups. Cell
decomposition in affine Weyl groups has drawn much attention. But the picture
is far more from clear. By now we know the cell decomposition for the
following types : rank two of any type (Lusztig [9]), type $\tilde{A}_{n}$
(Shi [12]), type $\tilde{C}_{3}$ (Bedard [1]), type $\tilde{B}_{3}$ (Du [3]),
type $\tilde{C}_{4},\tilde{D}_{4},\tilde{F}_{4}$ (Shi [16], [17], [18]), type
$\tilde{B}_{4}$ (Zhang [23]), type $\tilde{D}_{5}$ (Chen [2]). For affine Weyl
groups of other types, not much is known about cell decomposition, but the
second highest two-sided cell and the lowest two-sided cell of an affine Weyl
group are well understood (see [8], [13], [15]). Also in [19] Shi studied the
second lowest two-sided cell of an affine Weyl group and gave a conjecture on
the number of left cells in the second lowest two-sided cell.

All translations in an affine Weyl group form a normal subgroup of the affine
Weyl group and the normal subgroup has finite index in the affine Weyl group.
This suggests that cell properties of translations in affine Weyl groups
should be interesting. In [20], Nanhua Xi obtained some cell properties of the
translations and used the properties to prove that each left cell of an affine
Weyl group has only finitely many connected components.

We are interested in cell properties of translations in affine Weyl groups. A
specific question is to find out translations in a given two-sided cell and in
a given left cell of an affine Weyl groups. In this paper we find out some
translations in the second lowest two-sided cell of an affine Weyl group
(Theorems 4.5, 4.8, 4.12, 4.14, 4.17) and use the translations to formulate a
refinement of Jianyi Shi’s conjecture on the number of left cells in the
second lowest two-sided cell (Conjecture 5.2). We verify the refinement for an
affine Weyl group of type $\tilde{A}_{n-1}$ or type $\tilde{G}_{2}$ (Theorems
5.7, 5.6). For type $\tilde{A}_{n-1}$, we can say more. In addition, we
determine all translations in the lowest two-sided cell of an affine Weyl
group (Theorem 3.8).

##  1 Affine Weyl group and extended affine Weyl group

In this section we fix some notations and recall some known facts.

1.1. Let $E$ be an euclidean space of dimension $n$ and $\Phi\subset E$ be an
irreducible root system which spans $E$. Fix a simple root system
$\Delta=\\{\alpha_{1},\ \alpha_{2},\ \cdots,\alpha_{n}\\}$ of $\Phi$. Let
$\Phi^{+}$ and $\Phi^{-}$ be the set of positive roots and the set of negative
roots in $\Phi$ respectively.

For any $\alpha\in\Phi$, the corresponding reflection $s_{\alpha}$ on $E$ is
defined by $s_{\alpha}(x)=x-\langle x,\alpha^{\vee}\rangle\alpha$, where
$\alpha^{\vee}=2\alpha/\langle\alpha,\ \alpha\rangle$, and
$\langle\cdot,\cdot\rangle$ is the inner product on $E$. The reflections
$s_{\alpha}\ (\alpha\in\Phi)$ generate the Weyl group $W_{0}$ of $\Phi$.

Let $Q$ denote the root lattice $\mathbb{Z}\Phi$. Let $P=\\{\lambda\in
E\,|\,\langle\lambda,\alpha^{\vee}\rangle\in\mathbb{Z}\text{\ for all}\
\alpha\in\Phi\\}$ be the weight lattice. Then $Q$ is a subgroup of $P$ with
finite index. The weight lattice $P$ is a free abelian group. Define
$\lambda_{1},\ \lambda_{2},\ ...,\ \lambda_{n}$ by the conditions
$\langle\lambda_{i},{\alpha_{j}}^{\vee}\rangle=\delta_{ij}$ for
$i,j=1,2,...,n$. The weights $\lambda_{1},\ \lambda_{2},\ ...,\ \lambda_{n}$
are called the fundamental dominant weights (with respect to $\Delta$). The
fundamental dominant weights form a basis of $P$.

1.2. For each element $\lambda$ in $P$, define the translation
$t_{\lambda}:E\to E,\ x\mapsto x+\lambda$. Then the translations
$t_{\lambda},\ \lambda\in P$ form a transformation group $X$ of $E$. Let $N$
be the subgroup of $X$ consisting of all $t_{\lambda},\ \lambda\in Q$.
Clearly, the map $\lambda\mapsto t_{\lambda}$ defines a group isomorphism from
$P$ to $X$.

Let $W_{a}$ be the transformation group of $E$ generated by $W_{0}$ and $N$,
and let $W$ be the transformation group of $E$ generated by $W_{0}$ and $X$.
Then $W_{a}$ is an affine Weyl group and $W$ is an extended affine group. We
have $W_{a}=W_{0}\ltimes N$ and $W=W_{0}\ltimes X$. In particular, $W_{a}$ has
finite index in $W$.

Let $-\alpha_{0}$ be the highest short root of $\Phi$. Define
$s_{0}=t_{-\alpha_{0}}s_{\alpha_{0}}$ and $s_{i}=s_{\alpha_{i}},\ 1\leq i\leq
n$. Let $S=\\{s_{0},\ s_{1},\ ...,\ s_{n}\\}$. Then $(W_{a},S)$ and
$(W_{0},S_{0})$ are Coxeter systems, where $S_{0}=\\{s_{1},\ s_{2},\
\cdots,s_{n}\\}$. Let $l:W_{a}\to\mathbb{N}$ be the length function of
$W_{a}$. There is a finite subgroup $\Gamma$ of $W$ such that $W=\Gamma\ltimes
W_{a}$. Following Lusztig we extend the length function to $W$ by setting
$l(\tau w)=l(w)$ for any $\tau\in\Gamma$ and $w\in W_{a}$. Similarly, we
extend the Bruhat order $\leq$ on $W_{a}$ to $W$ by setting $\tau
w\leq\tau^{\prime}u$ if $\tau=\tau^{\prime}$ and $w\leq u$. We denote the
extensions again by $l$ and $\leq$ respectively.

1.3. Iwahori and Matsumoto established a nice formula for the length function
of $W$ (see Proposition 1.23 in [6]):

$l(t_{\lambda}w)=\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
w^{-1}(\alpha)\in{\Phi^{-}}\end{subarray}}{|\langle\lambda,\alpha^{\vee}\rangle-1|}+\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
w^{-1}(\alpha)\in{\Phi^{+}}\end{subarray}}{|\langle\lambda,\alpha^{\vee}\rangle|},\quad\text{for
all }t_{\lambda}\in X,\ \ w\in W_{0}.$ (1)

The above formula is equivalent to the following formula:

$l(wt_{\lambda})=\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
w(\alpha)\in{\Phi^{-}}\end{subarray}}{|\langle\lambda,\alpha^{\vee}\rangle+1|}+\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
w(\alpha)\in{\Phi^{+}}\end{subarray}}{|\langle\lambda,\alpha^{\vee}\rangle|},\qquad\text{for
all }t_{\lambda}\in X,\ \ w\in W_{0}.$ (2)

The subgroup $\Gamma$ actually consists of elements in $W$ with length 0.
Define $x_{i}=t_{\lambda_{i}}$. Then $x_{1},\ ...,\ x_{n}$ form a basis of the
abelian group $X$. By abuse of the term of fundamental dominant weights, we
also call $x_{i}$ a fundamental dominant weight in $X$. Using Iwahori-
Matsumoto’s formula (1) and (2), we see

$\displaystyle l(s_{i}x_{j})=l(x_{j}s_{i})=l(x_{j})+1,\quad if\ 1\leq i\neq
j\leq n,$ (3) $\displaystyle l(s_{i}x_{i})=l(x_{i})+1,\quad
l(x_{i}s_{i})=l(x_{i})-1,\quad if\ 1\leq i\leq n.$ (4)

Let $P^{+}=\\{\lambda\in P\,|\,\langle\lambda,\alpha_{i}^{\vee}\rangle\geq
0\text{\ \ for \ }i=1,2,...,n\\}$ be the set of dominant weights.
Correspondingly set $X^{+}=\\{t_{\lambda}\,|\,\lambda\in P^{+}\\}$. The
elements in $X^{+}$ will also be called dominant weights. Using Iwahori-
Matsumoto’s formula (2) or (1) we see the following three identities.

$\displaystyle X^{+}=\\{x\in X\,|\,l(wx)=l(w)+l(x),\ \forall w\in
W_{0}\\}=\\{x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots
x_{n}^{a_{n}}\,|\,a_{1},a_{2},...,a_{n}\in\mathbb{N}\\}.$ (5) $\displaystyle
l(xy)=l(x)+l(y)\quad\text{for all }\ x,\ y\in X^{+};$ (6) $\displaystyle
l(wx^{k}w^{-1})=kl(x)\quad\text{for all }\ x\in X,\ w\in W_{0},$ (7)

##  2 Kazhdan-Lusztig Cells and Lusztig’s $a$-function

In [7] , Kazhdan and Lusztig introduced the concepts of left, right, two-sided
cell for Coxeter groups. In [9], Lusztig defined the $a$-function
$a:W\rightarrow\mathbb{N}$, which is a useful tool to study cells in Coxeter
groups. The concepts of left, right, two-sided cell for Coxeter groups and the
$a$-function work well for extended affine Weyl groups (see [11] ).

From now on, we keep the notations in section 1. In particular, $W_{0}$,
$W_{a}$, $W$ are the Weyl group, affine Weyl group, the extended affine Weyl
group associated to a root system $\Phi$, respecively. Of course, some
concepts and facts in this section are valid for more general (extended)
Coxeter groups, but those are not used in this paper.

2.1. The preorders $\underset{L}{\leq},\ \underset{R}{\leq},\
\underset{LR}{\leq}$ and the associated equivalence relations
$\underset{L}{\sim},\ \underset{R}{\sim},\ \underset{LR}{\sim}$ on $W_{a}$ are
defined in [7]. The preorders are extended to $W$ by Lusztig in [11] as
follows, for $\tau,\tau^{\prime}\in\Gamma,\ w,u\in W_{a}$, set

$\tau w\underset{L}{\leq}\tau^{\prime}u\Longleftrightarrow\tau=\tau^{\prime},\
w\underset{L}{\leq}u,$ $\tau
w\underset{R}{\leq}\tau^{\prime}u\Longleftrightarrow\tau=\tau^{\prime},\
w\underset{R}{\leq}u,$ $\tau
w\underset{LR}{\leq}\tau^{\prime}u\Longleftrightarrow\tau=\tau^{\prime},\
w\underset{LR}{\leq}u.$

Then the equivalence relations $\underset{L}{\sim},\ \underset{R}{\sim},\
\underset{LR}{\sim}$ on $W_{a}$ are extended to $W$.

Similarly, the $a$-function is extended to $W$ by setting $a(\tau w)=a(w)$ for
any $\tau\in\Gamma,\ w\in W_{a}$. Actually Lusztig defined the preorders and
$a$-function on $W$ using Hecke algebra of $W$.

The equivalence classes with respect to $\underset{L}{\sim}$ ( respectively
$\underset{R}{\sim},\ \underset{R}{\sim}$) are called left cells (
respectively right cells, two-sided cells). And the preorders on $W$ induce
partial orders on the set of left cells (respectively right cells, two-sided
cells) of $W$.

2.2. For any $w\in W$, we associate two subsets of $S$ as follows:

$\mathscr{L}(w)=\\{s\in S\ |\ sw\leq w\\},\qquad\mathscr{R}(w)=\\{s\in S\ |\
ws\leq w\\}.$

According to [7], we have

(a) If $w\underset{L}{\leq}u$ (respectively, $w\underset{R}{\leq}u$), then
$\mathscr{R}(w)\supset\mathscr{R}(u)$ (respectively,
$\mathscr{L}(w)\supset\mathscr{L}(u)$). In particular, if
$w\underset{L}{\sim}u$( respectively, $w\underset{R}{\sim}u$), then
$\mathscr{R}(w)=\mathscr{R}(u)$( respectively,
$\mathscr{L}(w)=\mathscr{L}(u)$).

For convenience, we often write $w\cdot u$ for $wu$ if $l(wu)=l(w)+l(u)$.
Recall that $x_{i}=t_{\lambda_{i}}$.

Lemma 2.3. Let $J\subset\\{1,\ 2,\ \cdots,\ n\\}$ and $x=\prod_{i\in
J}x_{i}^{a_{i}},\ a_{i}\geq 1$, then

(a) $\mathscr{R}(x)=\\{s_{i}\,|\,i\in J\\}$;

(b) $x=(xw_{J})\cdot w_{J}$, i.e. $l(x)=l(xw_{J})+l(w_{J})$, where $w_{J}$ is
the longest element of the parabolic subgroup $W_{J}=\langle s_{i}\,|\,i\in
J\rangle$ of $W$.

###### Proof.

The first result follows from formula (4) and (6), while the second result
follows from the length formula (1) of Iwahori-Matsumoto,

$l(xw_{J})=\sum_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
{w_{J}}^{-1}(\alpha)\in{\Phi}^{-}\end{subarray}}|\langle
x,\alpha^{\vee}\rangle-1|+\sum_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
{w_{J}}^{-1}(\alpha)\in{\Phi}^{+}\end{subarray}}|\langle
x,\alpha^{\vee}\rangle|=l(x)-l(w_{J}).$

∎

2.4. We list some properties of the $a$-function, which were proved in [9].
Let $w,u,v\in W$.

(a) If $w\underset{LR}{\leq}u$, then $a(w)\geq a(u)$. In particular, if
$w\underset{LR}{\sim}u$, then $a(w)=a(u)$. In other words, a-function is
constant on each two-sided cell. Therefore for any integer $i\geq 0$,
$W_{(i)}=\\{w\in W\ |\ a(w)=i\\}$ is a union of some two-sided cells.

(b) If $w\underset{L}{\leq}u$ (respectively, $w\underset{R}{\leq}u$;
$w\underset{LR}{\leq}u$) and $a(w)=a(u)$ , then $w\underset{L}{\sim}u$
(respectively, $w\underset{R}{\sim}u$; $w\underset{LR}{\sim}u$).

(c) For any proper subset $I$ of $S$, if the subgroup $W_{I}$ generated by $I$
is finite, then for the longest element $w_{I}$ of $W_{I}$, we have
$a(w_{I})=l(w_{I})$. If $w=u\cdot w_{I}\cdot v$, then $a(w)\geq a(w_{I})$.

(d) Let ${w_{0}}$ be the longest element of the Weyl group ${W_{0}}$, then for
any $w\in W$, $a(w)\leq l({w_{0}})=a({w_{0}})=:\nu$, where $\nu$ is the number
of positive roots in $\Phi$.

(e) If $w=u\cdot v$, then $w\underset{L}{\leq}v,\ w\underset{R}{\leq}u$, in
particular, $a(w)\geq a(v),\ a(u)$.

For $w,u\in W$ we write $w-u$ if $w\leq u$ and the Kazhdan-Lusztig polynomial
$P_{w,u}$ is of degree $(l(u)-l(w)-1)/2$ or $u\leq w$ and the Kazhdan-Lusztig
polynomial $P_{u,w}$ is of degree $(l(w)-l(u)-1)/2$.

(f) Let $z,z^{\prime}\in W$ be such that $z^{\prime}-z$. If
$\mathscr{R}(z^{\prime})\not\subset\mathscr{R}(z)$ and
$\mathscr{L}(z^{\prime})\not\subset\mathscr{L}(z)$. Assume $a(z)<\infty$, then
$a(z^{\prime})>a(z)$.

Let $\mathcal{D}=\\{w\in{W_{a}}\ |\ 2\deg{P_{e,w}}=l(w)-a(w)\\}$. Then
$\mathcal{D}$ is a finite set and all elements in $\mathcal{D}$ are
involutions. These involutions are called distinguished involutions.

Lusztig proved that $W$ has only finitely many left cells and every left cell
(respectively, right cell) contains a unique distinguished involution (see
[9]).

2.5. According to Lemma 4.2 of [20], for any $x\in X,$ there exists a unique
positive integer $n(x)$ such that

$\displaystyle x^{m}\underset{L}{\sim}x^{n}\quad if\ m,\ n\geq n(x),$ (8)
$\displaystyle x^{n(x)}\underset{L}{\not\sim}x^{n(x)-1}.$ (9)

Since $w\underset{L}{\sim}u$ if and only if $w^{-1}\underset{R}{\sim}u^{-1}$,
the conclusions above are equivalent to for any $x\in X$, there exists a
unique positive integer $n(x)$ such that

$\displaystyle x^{m}\underset{R}{\sim}x^{n}\quad if\ m,\ n\geq n(x),$ (10)
$\displaystyle x^{n(x)}\underset{R}{\not\sim}x^{n(x)-1}.$ (11)

It naturally arises a question : $n(x)=?$. There are some evidences showing
that $n(x)$ is not big. And for an affine Weyl group of type $\tilde{A},\
\tilde{B},\ \tilde{C},\ \tilde{D}$, it is possible that $n(x)$ is no larger
than 3.

In next section we will show that $n(x)=1$ for any translation in an affine
Weyl group of type $\tilde{A}_{n-1}$. For other types, it is difficult to
compute $n(x)$ in general, but for translations in the lowest two-sided cell
and the second lowest two-sided cell we have similar results.

##  3 $n(x)=1$ for type $\tilde{A}_{n-1}$ or the lowest two-sided cell

We keep the notations in section 1. In particular, $W_{a}$ and $W$ stand for
an affine Weyl group and extended affine Weyl group associated to an
irreducible root system $\Phi$ respectively.

3.1. Assume that $W_{a}$ is of type $\tilde{A}_{n-1}$. The Coxeter graph of
$W_{a}$ is as below:

$\tilde{A}_{n-1}:$$0$$1$$2$$n-3$$n-2$$n-1$

Correspondingly, the simple refections in $W_{a}$ are denoted by $s_{1},\
...,s_{n-1},\ s_{0}$. Let $\Pi=\langle\pi\rangle$ be an infinite cyclic group
generated by $\pi$ and let $W=\Pi\ltimes W_{a}$ such that

${s_{1}}\pi=\pi{s_{0}},\ {s_{2}}\pi=\pi{s_{1}},\ \cdots,\
{s_{0}}\pi=\pi{s_{n-1}}.$

Then $W$ is the extended affine Weyl group associated to the general linear
group $GL_{n}(\mathbb{C})$. In $W$, the center $Z$ is generated by $\pi^{n}$.
The quotient group $\tilde{W}:=W/Z$ is just the extended affine Weyl group
associated to a root system of type $\tilde{A}_{n-1}$. There are some
advantages to work with this extended affine Weyl group $W$ since it can be
realized as a permutation group of $\mathbb{Z}$. We explain the realization.
Let

${W_{*}}:=\\{\sigma:\mathbb{Z}\to\mathbb{Z}\,|\,\sigma(i+n)=\sigma(i)+n,\
\sum_{i=1}^{n}(\sigma(i)-i)\equiv 0(\text{mod }n),\ \forall
i\in\mathbb{Z}\\}.$

Then we have

(a) (Lusztig)([22] Lemma 2.1.1) $W$ is isomorphic to $W_{*}$.

3.2. We shall identify $W$ with $W_{*}$. Let $X$ be the subgroup of $W$
consisting of translations, then $X$ is generated by the following
translations $(i=1,\ 2,\ ...,\ n)$:

$\tau_{i}(j)=\begin{cases}j+n,&\quad\text{if }j\equiv i(\text{mod}\ n);\\\
j,&\quad\text{if }j\not\equiv i(\text{mod}\ n).\end{cases}$

The simple reflections $s_{i}$ ($i=0,\ 1,\ 2,\ ...,n-1$) are defined as
follows:

$s_{i}(j)=\begin{cases}j+1,&\quad\text{if }j\equiv i(\text{mod}\ n);\\\
j-1,&\quad\text{if }j\equiv i+1(\text{mod}\ n);\\\ j,&\quad\text{if
}j\not\equiv i,i+1(\text{mod}\ n).\end{cases}$

The element $\pi$ is defined by $\pi(i)=i+1$ for all integer $i$.

The simple reflections $s_{1},\ ...,s_{n-1}$ generate the Weyl group $W_{0}$,
which is isomorphic to the symmetric group of $n$ letters, and the simple
reflections $s_{0},\ s_{1},\ ...,s_{n-1}$ generate the affine Weyl group
$W_{a}$, which is of type $\tilde{A}_{n-1}$.

For $1\leq i\leq n$, set $x_{i}=\tau_{1}\tau_{2}\cdots\tau_{i}$, then $x_{1},\
...,\ x_{n-1}$ are translations corresponding to fundamental dominant weights
$\lambda_{1},\ ...,\ \lambda_{n-1}$ respectively. It is easy to see the
following relations:

$\displaystyle{x_{1}}=\pi{s_{n-1}}{s_{n-2}}\cdots{s_{2}}{s_{1}};$
$\displaystyle{x_{2}}=\pi^{2}({s_{n-2}}{s_{n-3}}\cdots{s_{2}}{s_{1}})({s_{n-1}}{s_{n-2}}\cdots{s_{2}});$
$\displaystyle\cdots$
$\displaystyle{x_{i}}=\pi^{i}({s_{n-i}}{s_{n-i-1}}\cdots{s_{1}})({s_{n-i+1}}{s_{n-i}}\cdots{s_{2}})\cdots({s_{n-1}}{s_{n-2}}\cdots{s_{i}});$
$\displaystyle{x_{n-1}}=\pi^{n-1}{s_{1}}{s_{2}}\cdots{s_{n-2}}{s_{n-1}};$
$\displaystyle{x_{n}}=\pi^{n}.$

One main result of this section is as follows:

Theorem 3.3. Keep the notations above. In particular, $W=W_{*}$ is the
extended affine Weyl group associated to $GL_{n}(\mathbb{C})$. For any $x\in
X$, $w\in W_{0}$, we have

$\displaystyle x\underset{LR}{\sim}wxw^{-1}$ (12) $\displaystyle
x\underset{L}{\sim}x^{m},\quad x\underset{R}{\sim}x^{m},\qquad\text{for any
positive integer m}$ (13)

3.4. To prove the theorem we recall a description of two-sided cells of $W$
proven by Shi and Lusztig independently.

Let $w\in W$, $j_{1},\ j_{2},\ ...,\ j_{k}$ be integers. Call $j_{1},\ j_{2},\
...,\ j_{k}$ a d-antichain of $w$ of length $k$ if the integers satisfy the
following relations:

(1) $j_{k}-n<j_{1}<j_{2}<\cdots<j_{k}$,

(2) $w(j_{k})-n<w(j_{1})<w(j_{2})<\cdots w(j_{k}).$

We call a subset $Y$ of $\mathbb{Z}$ a d-antichain family set of $w$ of index
$q$ if : (1) the elements in $Y$ are noncongruent to each other modulo $n$,
(2) $Y$ is a disjoint union of $q$ d-antichains: $A_{1},\ A_{2},\ ...,\
A_{q}$. We also call $\\{A_{1},\ ...,\ A_{q}\\}$ a d-antichain family of $w$.

Let $d_{i}$ be the maximal one among the cardinalities of all d-antichain
family sets of $w$ of index $i$. Then $d_{1}\leq d_{2}\leq\cdots\leq d_{n}=n$.
According to Theorem 1.5 in [4], we have $d_{1}\geq d_{2}-d_{1}\geq
d_{3}-d_{2}\geq\cdots\geq d_{n}-d_{n-1}$. Then we get a partition of $n$ as

$\mu(w)=(d_{1},\ d_{2}-d_{1},\ \cdots,\ d_{n}-d_{n-1}).$

The following result describes the two-sided cells of $W$ completely ([12],
[10]).

(a) (Shi, Lusztig) For any $w,\ u\in W$, then $w\underset{LR}{\sim}u\
\Longleftrightarrow\mu(w)=\mu(u)$.

3.5. Now we can prove Theorem 3.3.

(1) Firstly, We prove for any $w\in W_{0}$ and $x\in X$,

$wxw^{-1}\underset{LR}{\sim}x.$

Since $W_{0}$ is generated by $s_{1},\ \cdots,\ s_{n-1}$, it is no harm to
assume that $w=s_{i}$ is a simple reflection. By definition (see subsection
3.2), we have

$x(j)=j+\theta_{j}n,\quad\theta_{j}\in\mathbb{Z}.$

Suppose $j_{k}-n<j_{1}<j_{2}<\cdots<j_{k}$ is a d-antichain of $x$, then

$j_{k}+\theta_{j_{k}}n-n<j_{1}+\theta_{j_{1}}n<j_{2}+\theta_{j_{2}}<\cdots<j_{k}+\theta_{j_{k}}n.$

Thus $\theta_{j_{1}}=\theta_{j_{2}}=\cdots=\theta_{j_{k}}$.

To consider the antichains of $x$ and $s_{i}xs_{i}$, we can only consider the
antichains in the set $\\{1,\ 2,\ ...,\ n\\}$. For any $j$ in this set,

$s_{i}xs_{i}(j)=\begin{cases}x(j)=j+\theta_{j}n,\quad&\text{if}\ j\neq i,\
i+1;\\\ i+\theta_{i+1}n,&\text{if}\ j=i;\\\ i+1+\theta_{i}n,&\text{if}\
j=i+1.\end{cases}$

Therefore,

(i) if the d-antichain $j_{1}<j_{2}<\cdots<j_{k}$ of $x$ in $\\{1,\ 2,\
\cdots,\ n\\}$ does not contain $i,\ i+1$, then it is also a d-antichain of
$s_{i}xs_{i}$;

(ii) if the d-antichain $j_{1}<j_{2}<\cdots<j_{k}$ of $x$ in $\\{1,\ 2,\
\cdots,\ n\\}$ contains both $i$ and $i+1$ , then $\theta_{i}=\theta_{i+1}$,
thus it is also a d-antichain of $s_{i}xs_{i}$;

(iii) if the d-antichain $j_{1}<j_{2}<\cdots<j_{k}$ of $x$ in $\\{1,\ 2,\
\cdots,\ n\\}$ contains $i=j_{a}$ without $i+1$, we replace $i$ by $i+1$ to
get a d-antichain of $s_{i}xs_{i}$ whose length is still $k$;

(iv) if the d-antichain $j_{1}<j_{2}<\cdots<j_{k}$ of $x$ in $\\{1,\ 2,\
\cdots,\ n\\}$ contains $i+1=j_{a}$ without $i$, we replace $i+1$ by $i$ to
get a d-antichain of $s_{i}xs_{i}$ whose length is still $k$.

In summary, for a d-antichain family set of $x$ of index $q$, we can obtain a
d-antichain family set of $s_{i}xs_{i}$ of index $q$ correspondingly, and the
two sets are of the same cardinality. As a consequence, we have
$\mu(x)$=$\mu(s_{i}xs_{i})$. According to 3.4 (a), $x$ and $s_{i}xs_{i}$ are
in the same two-sided cell.

(2) Now we prove for any translation $x$ and any positive integer $m$,
$x\underset{L}{\sim}x^{m},\quad x\underset{R}{\sim}x^{m}$. In the case of $x$
being a dominant weight, this result was proved in [22]. Actually, a stronger
result was proved in [22]:

for $J\subset\\{1,2,\cdots,n-1\\},$ set ${x_{J}}:=\prod\limits_{i\in
J}{x_{i}}$.

(v) if every ${a_{i}}\geq 1$, then $\prod\limits_{i\in
J}{x_{i}}^{a_{i}}\underset{R}{\sim}{x_{J}}$;

(vi) if every ${a_{i}}\geq 1$, and $k\in\mathbb{Z}$, then
${x_{n}}^{k}\prod\limits_{i\in J}{x_{i}}^{a_{i}}\underset{R}{\sim}{x_{J}}$;

(vii) ${x_{1}}{x_{2}}\cdots{x_{n-1}}$ lies in the lowest two-sided cell
$W_{(\nu)}$ of $W$.

As is known, there exists $w\in W_{0}$ such that $wxw^{-1}$ is dominant. Then

$wxw^{-1}\underset{LR}{\sim}wx^{m}w^{-1}.$

According to the result given in (1), we have
$x\underset{LR}{\sim}wxw^{-1}\underset{LR}{\sim}wx^{m}w^{-1}\underset{LR}{\sim}x^{m}$.
Theorem 3.3 proves valid.∎

A consequence of Theorem 3.3 is that the $a$-function value of $x$ equals to
the $a$-function value of $x^{m}$ for any translation $x$ in an extended
affine Weyl group of type $\tilde{A}_{n-1}$.

Example 3.6. Let $W_{a}$ be an affine Weyl group of type $\tilde{G}_{2}$
generated by $s_{0},\ s_{1},\ s_{2}$, where $s_{1}$ is the simple reflection
corresponding to the short root. Then $x_{1}=s_{0}s_{1}s_{2}s_{1}s_{2}s_{1}$
and $x_{2}=s_{0}s_{1}s_{2}s_{1}s_{0}s_{2}s_{1}s_{2}s_{1}s_{2}$ and they
generate the normal subgroup $X$ of $W$. It is easy to see

$x_{1}\underset{LR}{\not\sim}{x_{1}}^{2},\quad{x_{1}}^{2}\underset{L}{\sim}{x_{1}}^{m},\quad
x_{2}\underset{L}{\sim}{x_{2}}^{k},\quad
x_{1}x_{2}\underset{L}{\sim}{x_{1}}^{j}{x_{2}}^{k},$

where $j,\ k\geq 1$, $m\geq 2$. So for an affine Weyl group of type
$\tilde{G}_{2}$, there exists $x$ such that $n(x)>1$. Notice that
$s_{1}x_{1}s_{1}\underset{LR}{\not\sim}x_{1}$, thus the assertions of Theorem
3.3 are not true for an affine Weyl group of type $\tilde{G}_{2}$.

3.7. In the rest of this section, $W=W_{0}\ltimes X$ stands for an extended
affine Weyl group associated to an irreducible root system $\Phi$. We prove
that the assertions of Theorem 3.3 remain valid for the lowest two-sided cell
of $W$.

Let $w_{0}$ be the longest element in $W_{0}$. Then $\nu=l(w_{0})=|\Phi^{+}|$
is the upper bound of the $a$-function on $W$ (see [9]).

Recall that $S$ is the set of simple reflections of $W$. For $J\subset S$, the
subgroup $W_{J}$ generated by $J$ is called a parabolic subgroup. If $W_{J}$
is finite, we denote $w_{J}$ the longest element in the subgroup $W_{J}$.

(a) (Shi, [13]) The following set is a two-sided cell of $W$, which is the
lowest one with respect to the partial order $\underset{LR}{\leq}$ on the set
of two-sided cells of $W$:

$\displaystyle W_{(\nu)}=$ $\displaystyle\\{w\in W\ |\ a(w)=\nu\\}$
$\displaystyle=$ $\displaystyle\\{w\cdot w_{J}\cdot u\ |\ w,\ u\in W,\
W_{J}\text{ is\ a\ parabolic\ subgroup\ of\ }W,\ l(w_{J})=l(w_{0})\\}.$

It is natural to call $W_{(\nu)}$ the lowest two-sided cell of $W$. Let
$x_{I}=\prod\limits_{i=1}^{n}x_{i}$ be the product of all fundamental dominant
weights, then $l(x_{I}s_{i})<l(x_{I})$ implies that $x_{I}=w\cdot w_{0}$ by
Lemma 2.3, and we have the following theorem to describe all the translations
in $W_{(\nu)}$.

Theorem 3.8. Let $x\in X\subset W$. Choose $w\in W_{0}$ such that
$wxw^{-1}=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$ is in $X^{+}$. We
have

(a) $x\in W_{(\nu)}$ if and only if every $a_{i}$ is a positive integer;

(b) if $x\in W_{(\nu)}$, then for any $u\in W_{0}$, we have $uxu^{-1}\in
W_{(\nu)}$;

(c) if $x\in W_{(\nu)}$, then for any positive integer $a$, we have $x^{a}\in
W_{(\nu)}$.

3.9. To prove this theorem, we need the coordinate form for $w\in W$ which was
defined by Jianyi Shi in [14]. Be aware that in this article
$s_{0}=t_{-\alpha_{0}}s_{\alpha_{0}}$, in [14]
$s_{0}=s_{\alpha_{0}}t_{-\alpha_{0}}$, where $-\alpha_{0}$ is the highest
short root in $\Phi^{+}$. Since in this article $W$ acts on $E$ on the left
and in [14] $W_{a}$ (or $W$) acts on $E$ on the right, the alcove
corresponding to $w\in W$ in this paper is the same alcove determined by
$w^{-1}$ in [14]. Hence, the coordinate form of $w$ in this article equals to
the coordinate form of $w^{-1}$ in [14]. By definition, the coordinate form of
$w$ is a $\Phi$-tuple ${(k(w,\alpha))}_{\alpha\in\Phi}$ of integers satisfying
the following conditions:

(a) for any $\alpha\in\Phi$, $k(e,\alpha)=0$, where $e$ is the identity
element in $W$;

(b) for any $0\leq i\leq n$,

$k(s_{i},\ \alpha)=\begin{cases}{0},&{\alpha\not=\pm{\alpha_{i}}}\\\ {\mp
1},&{\alpha=\pm{\alpha_{i}}}\end{cases}$

(c) if $w^{\prime}=w\cdot{s_{i}},\ (0\leq i\leq n)$, then
$\forall\alpha\in\Phi$,

$k(w^{\prime},\ \alpha)=k(w,\ \alpha)+k(s_{i},\bar{w}^{-1}(\alpha)),$

where $\bar{w}$ is the image of $w$ under the natural homomorphism $W/X\to
W_{0}$.

For any $\alpha\in\Phi,$ we have $k(w,\ -\alpha)=-k(w,\ \alpha)$, so the
$\Phi$-tuple $(k(w,\ \alpha))_{\alpha\in\Phi}$ is determined by the
$\Phi^{+}$-tuple $(k(w,\ \alpha))_{\alpha\in\Phi^{+}}$ completely. For
convenience we also say that the $\Phi^{+}$-tuple $(k(w,\
\alpha))_{\alpha\in\Phi^{+}}$ is the coordinate form of $w$.

The following result was proved by Jianyi Shi in [14] and [13].

Proposition 3.10. (Shi) Given $w=xu\in W,\ u\in W_{0},\ x\in X$, and the
corresponding coordinate form $(k(w,\ \alpha))_{\alpha\in\Phi^{+}}$, we have:

(a) for any $\alpha\in\Phi^{+}$, $k(w,\ \alpha)=\langle
x,\alpha^{\vee}\rangle+k(u,\ \alpha)$, where

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

(We set $\langle x,\alpha^{\vee}\rangle$ to be
$\langle\lambda,\alpha^{\vee}\rangle$ if $x=t_{\lambda}$.)

(b) $\mathscr{L}(w)=\\{s_{j}\in S\,|\,k(w,\alpha_{j})<0\\}$, and
$\mathscr{R}(w)=\\{s_{j}\in S\,|\,k(w,\ {\bar{w}}(\alpha_{j}))>0\\}$, where
$\bar{w}$ is the image of $w$ under the natural homomorphism $W/X\to W_{0}$.

(c) ${W_{(v)}}:=\\{w\in W\,|\,k(w,\ \alpha)\not=0,\ \forall\alpha\in\Phi\\}$.

3.11. Now we can prove Theorem 3.8. By part (a) and (c) of Proposition 3.10
above, a translation $x\in W$ lies in the lowest two-sided cell ${W_{(v)}}$ if
and only if for any $\alpha\in\Phi$, we have $\langle
x,\alpha^{\vee}\rangle\neq 0$. Then Theorem 3.8 follows.

##  4 Some translations in the second lowest two-sided cell $\Omega_{qr}$

4.1. Keep the notations in section 1. In [19], Jianyi Shi studied the second
lowest two-sided cell of $W_{a}$, which has little difference with the second
lowest two-sided cell of $W$. Following Shi, we denote $\Omega_{qr}$ the
second lowest two-sided cell of $W$. Shi conjectured that $\Omega_{qr}$
contains $|W_{0}|/2$ left cells and proved that the number of left cells in
$\Omega_{qr}$ is at most $|W_{0}|/2$.

In [19], Jianyi Shi gave the table of $a$-function values of $\Omega_{qr}$ as
below:

Table 1

$W_{a}$ | $\tilde{A_{n}}$ | $\tilde{C_{n}}$ | $\tilde{B_{n}}$ | $\tilde{D_{n}}$ | $\tilde{E_{6}}$ | $\tilde{E_{7}}$ | $\tilde{E_{8}}$ | $\tilde{F_{4}}$ | $\tilde{G_{2}}$
---|---|---|---|---|---|---|---|---|---
$a(\Omega_{qr})$ | $\frac{1}{2}(n^{2}-n)$ | $n^{2}-2n+2$ | $n^{2}-n$ | $n^{2}-3n+3$ | 25 | 46 | 91 | 16 | 3

The main aim of this section is to determine some translations in the second
lowest two-sided cell $\Omega_{qr}$. We hope to use the translations to
understand Shi’s conjecture on the number of left cells in $\Omega_{qr}$.

Let $x\in X$ be a translation in $W$. For convenience, we set $\langle
x,\alpha^{\vee}\rangle=\langle\lambda,\alpha^{\vee}\rangle$ if $x=t_{\lambda}$
and $\alpha\in\Phi$. By Proposition 3.10 (c), if $x\in X\cap\Omega_{qr}$, then
there exists some $\alpha\in\Phi$ such that $\langle
x,\alpha^{\vee}\rangle=0$. For $J\subset I=\\{1,2,...,n\\}$, set
$x_{J}=\prod_{i\in J}x_{i}$. Theorem 4.1 in [20] says that there exists some
positive integer $r$ and $1\leq i\leq n$ satisfying

${x_{I-\\{i\\}}}^{r}=\prod_{j\neq i,j\in I}{x_{j}}^{r}\in\Omega_{qr}.$

To go further we firstly establish a technical result.

Lemma 4.2. Assume that the root system $\Phi$ has rank $n$. Let $s_{1},\
\cdots,\ s_{n}$ be the simple reflections in $W_{0}$ and $x_{1},\ \cdots,\
x_{n}$ are the corresponding translations in $W$ (see subsection 1.3). Set
$I_{k}=\\{1,\ \cdots,\ n\\}\backslash\\{k\\}$, $1\leq k\leq n$. Then

$x_{I_{k}}s_{k}=u\cdot w_{0}$

lies in the lowest two-sided cell ${W_{(v)}}$ of $W$, where $\displaystyle
x_{I_{k}}=\prod_{i\in I_{k}}x_{i}$, $w_{0}$ is the longest element in the Weyl
group $W_{0}$.

###### Proof.

By the length formula (1) of Iwahori-Matsumoto,

$l(x_{I_{k}}s_{k})=|\langle
x_{I_{k}},{\alpha_{k}}^{\vee}\rangle-1|+\sum_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
\alpha\not=\alpha_{k}\end{subarray}}|\langle
x_{I_{k}},\alpha^{\vee}\rangle|=l(x_{I_{k}})+1,$
$l(x_{I_{k}}s_{k}w_{0})=\sum_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
\alpha\not=\alpha_{k}\end{subarray}}|\langle
x_{I_{k}},\alpha^{\vee}\rangle-1|+|\langle
x_{I_{k}},{\alpha_{k}}^{\vee}\rangle|=l(x_{I_{k}})+1-l(w_{0}),$

then, $l(x_{I_{k}}s_{k})=l(x_{I_{k}}s_{k}w_{0})+l(w_{0})$. Consequently,
$x_{I_{k}}s_{k}=x_{I_{k}}s_{k}w_{0}\cdot w_{0}$. ∎

Proposition 4.3. Assume that $x\in X\cap\Omega_{qr}$, then for any positive
integer $k$, we have $x^{k}\in\Omega_{qr}$ and $x^{k}\underset{L}{\sim}x,\
x^{k}\underset{R}{\sim}x.$

###### Proof.

According to Proposition 3.10 (c), there exists some $\alpha\in\Phi$ such that
$\langle x,\alpha^{\vee}\rangle=0$. This implies that $\langle
x^{k},\alpha\rangle=k\cdot 0=0$ for any positive integer $k$. Hence
$a(x^{k})\leq a(\Omega_{qr})$.

On the other hand, by 2.4(e) we have $a(x^{k})\geq a(x)=a(\Omega_{qr})$. This
forces that $a(x^{k})=a(x)$. Using 2.4(e) we see that
$x^{a}\underset{L}{\leq}x$ and $x^{k}\underset{R}{\leq}x$. Applying 2.4(b) we
get $x^{k}\underset{L}{\sim}x,\ x^{k}\underset{R}{\sim}x.$ ∎

Then we have $n(x)=1$ for any translation $x\in\Omega_{qr}$.

4.4. Now we are going to give a family of translations in the second lowest
two-sided cell $\Omega_{qr}$ of the extended affine Weyl group $W$ case by
case. Note that the root system $\Phi$ has rank $n$. The simple reflections
$s_{1},\ ...,\ s_{n}$ in $W_{0}$ are numbered as usual. The corresponding
fundamental dominant weights in $X$ are denoted by $x_{1},\ ...,x_{n}$. The
simple reflection in $W\setminus W_{0}$ is denoted by $s_{0}$.

For $1\leq k\leq n$ we set $I_{k}=\\{1,\ 2,\ ...,\ n\\}\setminus\\{k\\}$ and
$x_{I_{k}}=\prod_{i\in I_{k}}x_{i}$.

Theorem 4.5. Assume that $W$ is of type $\tilde{A}_{n}$ or $\tilde{D}_{n}\
(n\geq 4)$ or $\tilde{E}_{n}\ (n=6,7,8)$. Then a translation $x\in X^{+}$ is
in $\Omega_{qr}$ if and only if there exists $1\leq k\leq n$ and positive
integers $a_{i}$ for all $i\in I_{k}$ such that

$x=\prod_{i\in I_{k}}x_{i}^{a_{i}}.$

###### Proof.

Suppose that $x=\prod_{i\in I_{k}}x_{i}^{a_{i}}$, $a_{i}\geq 1$ for all $i\in
I_{k}$. Then $\langle x,{\alpha_{k}}^{\vee}\rangle=0$. By the proof of Theorem
3.8, we know that $x$ is not in the lowest two-sided cell $W_{(\nu)}$ of $W$.
It is sufficient to show $\displaystyle x_{I_{k}}=\prod_{i\in
I_{k}}x_{i}\in\Omega_{qr}$ since $x\underset{LR}{\sim}x_{I_{k}}$.

By Lemma 4.2, $x_{I_{k}}s_{k}=u\cdot w_{0}$, where $w_{0}$ is the longest
element of the Weyl group $W_{0}\subset W$. Thus
$x_{I_{k}}=u\cdot(w_{0}s_{k})$. According to the proof of Proposition 5.7 in
[19] , for an affine Weyl group of type $\tilde{A}_{n},\ \tilde{D}_{n}\ (n\geq
4),\ \tilde{E}_{n}\ (n=6,7,8)$, $w_{0}s_{k}\in\Omega_{qr}$. This forces
$x_{I_{k}}\in\Omega_{qr}$.

Now suppose that $x=\prod_{i=1}^{n}x_{i}^{a_{i}}\in X^{+}$ is in
$\Omega_{qr}$. By Theorem 3.8 (a), there exists some $1\leq k\leq n$ such that
$a_{k}=0$. If in addition, $a_{j}=0$ for some $j\neq k$, then $s_{j}x=xs_{j}$
and this implies that $a(x)<a(s_{j}x)$ by 2.4(f). Since
$k(s_{j}x,\alpha_{k})=0$, by Proposition 3.10, we see $s_{j}x\not\in
W_{(\nu)}$. Thus $a(x)<a(s_{j}x)\leq a(\Omega_{qr})$ and this contradicts
$x\in\Omega_{qr}$. Therefore there is only one $k$ such that $a_{k}=0$ and all
other $a_{i}$ are positive integers. ∎

Corollary 4.6. Assume that $W$ is of type $\tilde{A}_{n}$. Then $x\in X$ is in
$\Omega_{qr}$ if and only if there exists some $w\in W_{0}$ and $1\leq k\leq
n$ such that $wxw^{-1}=\prod_{i\in I_{k}}x_{i}^{a_{i}},\ a_{i}\geq 1$.

###### Proof.

It follows from Theorem 4.5 and Theorem 3.3 directly. ∎

4.7. Now we deal with type $\tilde{B}_{n}$ $(n\geq 3)$.

Let $E$ be the n-dimensional euclidean space with the standard orthogonal
basis $\varepsilon_{1},\ \varepsilon_{2},\ ...,\ \varepsilon_{n}$,
$\Phi=\\{\pm(\varepsilon_{i}\pm\varepsilon_{j}),\ \pm
2\varepsilon_{k}\,|\,1\leq i<j\leq n,\ 1\leq k\leq n\\}.$ The extended affine
Weyl group $W$ associated to $\Phi$ is of type $\tilde{B}_{n}$. The Coxeter
graph of $W$ is as follows ( see [19] ):

$\tilde{B}_{n}:$$2$$0$$1$$3$$n-2$$n-1$$n$

Fix a simple root system as

$\Delta=\\{\varepsilon_{1}-\varepsilon_{2},\ \varepsilon_{2}-\varepsilon_{3},\
...,\ \varepsilon_{n-1}-\varepsilon_{n},\ 2\varepsilon_{n}\\}.$

Then the set of positive roots is

$\Phi^{+}=\\{\varepsilon_{i}\pm\varepsilon_{j},\ 2\varepsilon_{k}\,|\,1\leq
i<j\leq n,\ 1\leq k\leq n\\},$

and the set of negative roots is $\Phi^{-}=-\Phi^{+}.$

Let

$\displaystyle\alpha_{1}$ $\displaystyle=\varepsilon_{1}-\varepsilon_{2},\
\alpha_{2}=\varepsilon_{2}-\varepsilon_{3},\ ...,\
\alpha_{n-1}=\varepsilon_{n-1}-\varepsilon_{n},\ \alpha_{n}=2\varepsilon_{n};$
$\displaystyle\alpha_{ij}$
$\displaystyle=\varepsilon_{i}-\varepsilon_{j},\quad\beta_{ij}=\varepsilon_{i}+\varepsilon_{j},\quad
1\leq i<j\leq n,$ $\displaystyle\gamma_{k}$
$\displaystyle=2\varepsilon_{k},\qquad 1\leq k\leq n.$

Then

$\displaystyle\alpha_{ij}$
$\displaystyle=\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j-1},\quad 1\leq i<j\leq
n;$ $\displaystyle\beta_{ij}$
$\displaystyle=\alpha_{i}+\cdots+\alpha_{j-1}+2\alpha_{j}+\cdots+2\alpha_{n-1}+\alpha_{n},\quad
1\leq i<j\leq n-1;$ $\displaystyle\beta_{in}$
$\displaystyle=\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{n},\quad 1\leq i\leq
n-1;$ $\displaystyle\gamma_{k}$
$\displaystyle=2\alpha_{k}+\alpha_{k+1}+\cdots+2\alpha_{n-1}+\alpha_{n},\quad
1\leq k\leq n-1;$ $\displaystyle\gamma_{n}$ $\displaystyle=\alpha_{n}.$

The Weyl group of the root system $\Phi$ is generated by the simple
reflections $s_{1},\ ...,\ s_{n}$. The corresponding fundamental dominant
weights in $X\subset W$ are denoted by $x_{1},\ ...,\ x_{n}$. As usual,
$s_{0}$ is the simple reflection out of $W_{0}$.

Theorem 4.8. Keep the notations in subsection 4.7, so $W$ is an extended
affine Weyl group of type $\tilde{B}_{n}$. Let

$x=\displaystyle\prod_{i=1}^{n-1}x_{i}^{a_{i}},\qquad a_{1},\ ...,\
a_{n-2}\geq 1,\ a_{n-1}\geq 2.$

Then $x\in\Omega_{qr}$.

4.9. Now we give a proof of Theorem 4.8. By Theorem 3.8 (a), $x$ is not in the
lowest two-sided cell $W_{(\nu)}$ of $W$. Thus $a(x)\leq
n^{2}-n=a(\Omega_{qr})$. Let $J=\\{s_{0},\ s_{1},\ ...,\ s_{n-1}\\}$, then
$W_{J}=\langle s_{i}\,|\,i\in J\rangle$ is a Weyl group of type $D_{n}$ and
its longest element $w_{J}$ is of length $n^{2}-n$. Therefore to prove the
theorem it suffices to show

$x_{1}\cdots x_{n-2}x_{n-1}^{2}=w\cdot w_{J}\cdot u.$

According to Lemma 2.3, $\mathscr{R}(x_{1}\cdots x_{n-2}x_{n-1})=\\{s_{1},\
s_{2},\ ...,\ s_{n-1}\\}$, thus

$x_{1}\cdots x_{n-2}x_{n-1}=v\cdot w_{K},$

where $K=\\{1,\ ...,\ n-1\\}$ and $w_{K}$ is the longest element of
$W_{K}=\langle s_{i}\,|\,i\in K\rangle$

It suffices to prove

$w_{K}x_{n-1}=w_{J}\cdot u.$

We know that $\\{s_{1},\ ...,\ s_{n-1}\\}\subset\mathscr{L}(w_{K}x_{n-1})$. It
will be done if we can show $s_{0}w_{K}x_{n-1}\leq w_{K}x_{n-1}$, or
equivalently, $x_{n-1}^{-1}w_{K}s_{0}\leq x_{n-1}^{-1}w_{K}$.

We need the following two formulas:

(a) $l(x_{n-1}^{-1})=n^{2}-1$,

(b) $l(x^{-1}_{n-1}w_{K})=n^{2}-1+\frac{n(n-1)}{2}$.

We use the length formula (1) of Iwahori-Matsumoto to prove (a) and (b).

Recall the weight lattice of $\Phi$ is $P=\\{\lambda\in E\,|\,\langle\lambda,\
\alpha^{\vee}\rangle\in\mathbb{Z},\text{ for\ any\ }\alpha\in\Phi\\},$ where
$\langle\cdot,\cdot\rangle$ is the inner product in $E$,
$\alpha^{\vee}=2\alpha/\langle\alpha,\alpha).$ The fundamental dominant
weights $\lambda_{1},\ ...,\ \lambda_{n}$ are defined by $\langle\lambda_{i},\
\alpha_{j}^{\vee}\rangle=\delta_{ij},\ 1\leq i,j\leq n.$

By the length formula (1) of Iwahori-Matsumoto, we have

$l(x_{n-1}^{-1})=\sum_{\alpha\in\Phi^{+}}|\langle-\lambda_{n-1},\
\alpha^{\vee}\rangle|=\sum_{\alpha\in\Phi^{+}}|\langle\lambda_{n-1},\
\alpha^{\vee}\rangle|.$

Note that $\alpha_{ij}^{\vee}=\alpha_{ij},\ \ \beta_{ij}^{\vee}=\beta_{ij},\ \
\gamma_{k}^{\vee}=\frac{1}{2}\gamma_{k}$. Hence

$\displaystyle\langle\lambda_{n-1},\ \alpha^{\vee}_{ij}\rangle$
$\displaystyle=\langle\lambda_{n-1},\
\alpha_{ij}\rangle=\begin{cases}{0},\quad&{1\leq i<j\leq n-1}\\\
{1},\quad&{1\leq i<j=n}\end{cases};$ $\displaystyle\langle\lambda_{n-1},\
\beta^{\vee}_{ij}\rangle$ $\displaystyle=\langle\lambda_{n-1},\
\beta_{ij}\rangle=\begin{cases}{2},\quad&{1\leq j\leq n-1}\\\ {1},\quad&{1\leq
i<j=n}\end{cases};$ $\displaystyle\langle\lambda_{n-1},\
\gamma^{\vee}_{k}\rangle$ $\displaystyle=\langle\lambda_{n-1},\
\frac{\gamma_{k}}{2}\rangle=\begin{cases}{1},\quad&{1\leq k\leq n-1}\\\
{0},\quad&{k=n}\end{cases}.$

Thus,

$\displaystyle l(x_{n-1}^{-1})$ $\displaystyle=\sum_{1\leq i<j\leq
n}|\langle\lambda_{n-1},\ \alpha^{\vee}_{ij}\rangle|+\sum_{1\leq i<j\leq
n}|\langle\lambda_{n-1},\ \beta^{\vee}_{ij}\rangle|+\sum_{1\leq k\leq
n}|\langle\lambda_{n-1},\ \gamma^{\vee}_{k}\rangle|$
$\displaystyle=n-1+(2\cdot\frac{(n-1)(n-2)}{2}+n-1)+n-1=n^{2}-1.$

Formula (a) is proved. Formula (b) follows from
$l(x_{n-1}^{-1}w_{K})=l(x_{n-1}^{-1})+l(w_{K})$.

Now we are able to show $x_{n-1}^{-1}w_{I}s_{0}\leq x_{n-1}^{-1}w_{I}$, which
is equivalent to show $l(x_{n-1}^{-1}w_{I}s_{0})=n^{2}-2+\frac{n(n-1)}{2}$.

By definition, $s_{0}=t_{-\alpha_{0}}s_{\alpha_{0}}$, where
$-\alpha_{0}=\varepsilon_{1}+\varepsilon_{2}=\lambda_{2}$ is the highest short
root. Thus

$x_{n-1}^{-1}w_{K}s_{0}=x_{n-1}^{-1}w_{K}t_{\lambda_{2}}s_{\alpha_{0}}=x_{n-1}^{-1}t_{w_{K}(\lambda_{2})}w_{K}s_{\alpha_{0}}.$

We compute $w_{K}(\lambda_{2})$. Notice that $s_{i}(\lambda_{2})=\lambda_{2}$
if $i\neq 2$. Set $L=\\{1,\ 3,\ \cdots,\ n-1\\}$, $w_{L}$ is the longest
element of $W_{L}=\langle s_{k}\,|\,k\in L\rangle$. We have

$w_{K}=s_{n-2}s_{n-3}\cdots s_{2}s_{1}s_{n-1}s_{n-2}\cdots
s_{3}s_{2}w_{L},\quad w_{L}(\lambda_{2})=\lambda_{2}.$

Then

$\displaystyle w_{K}(\lambda_{2})$ $\displaystyle=s_{n-2}s_{n-3}\cdots
s_{2}s_{1}s_{n-1}s_{n-2}\cdots s_{3}s_{2}(\lambda_{2})$
$\displaystyle=s_{n-2}s_{n-3}\cdots
s_{2}s_{1}(\lambda_{2}-\alpha_{2}-\cdots-\alpha_{n-1})$
$\displaystyle=s_{n-2}s_{n-3}\cdots
s_{2}s_{1}(\lambda_{2}-\alpha_{1}-\alpha_{2}-\cdots-\alpha_{n-1})$
$\displaystyle=\lambda_{2}-\alpha_{1}-2\alpha_{2}-\cdots-2\alpha_{n-2}-\alpha_{n-1}$
$\displaystyle=\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-1}+\alpha_{n}-\alpha_{1}-2\alpha_{2}-\cdots-2\alpha_{n-2}-\alpha_{n-1}$
$\displaystyle=\alpha_{n-1}+\alpha_{n}=\varepsilon_{n-1}-\varepsilon_{n}+2\varepsilon_{n}=\varepsilon_{n-1}+\varepsilon_{n}.$

Therefore, by the length formula (1) of Iwahori-Matsumoto, we have

$\displaystyle\ \qquad
l(x_{n-1}^{-1}w_{I}s_{0})=l(x_{n-1}^{-1}t_{\varepsilon_{n-1}+\varepsilon_{n}}w_{I}s_{\alpha_{0}})$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{I}(\alpha)\in{\Phi^{+}}\end{subarray}}|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|+\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{I}(\alpha)\in{\Phi^{-}}\end{subarray}}|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|.$

We need to figure out the positive roots mapped by $s_{\alpha_{0}}w_{I}$ into
$\Phi^{+}$ or $\Phi^{-}$ respectively in detail.

We write $\alpha>0$ if $\alpha$ is a positive root and write $\alpha<0$ if
$\alpha$ is a negative root. Assume $1\leq i<j\leq n,\ 1\leq k\leq n$, it is
easy to get

$\displaystyle s_{\alpha_{0}}w_{I}(\alpha_{ij})$
$\displaystyle=s_{\alpha_{0}}w_{I}(\varepsilon_{i}-\varepsilon_{j})=s_{\alpha_{0}}(\varepsilon_{j}-\varepsilon_{i})$
$\displaystyle=\begin{cases}{\varepsilon_{j}-\varepsilon_{i}<0},\quad&{i,\
j\not\in\\{1,\ 2\\}}\\\ {\varepsilon_{2}-\varepsilon_{1}<0},\quad&{(i,\
j)=(1,\ 2)}\\\ {\varepsilon_{2}+\varepsilon_{j}>0},\quad&{i=1,\ j>2}\\\
{\varepsilon_{1}+\varepsilon_{j}>0},\quad&{i=2,\ j>2}.\end{cases}$

$\displaystyle s_{\alpha_{0}}w_{I}(\beta_{ij})$
$\displaystyle=s_{\alpha_{0}}w_{I}(\varepsilon_{i}+\varepsilon_{j})=s_{\alpha_{0}}(\varepsilon_{n-i+1}+\varepsilon_{n-j+1})$
$\displaystyle=\begin{cases}{\varepsilon_{n-i+1}+\varepsilon_{n-j+1}>0},\quad&{i,\
j\not\in\\{n-1,\ n\\}}\\\ {-\varepsilon_{1}-\varepsilon_{2}<0},\quad&{(i,\
j)=(n-1,\ n)}\\\ {-\varepsilon_{2}+\varepsilon_{n-i+1}<0},\quad&{1\leq i\leq
n-2,\ j=n}\\\ {-\varepsilon_{1}+\varepsilon_{n-i+1}<0},\quad&{1\leq i\leq
n-2,\ j=n-1}.\end{cases}$ $\displaystyle s_{\alpha_{0}}w_{I}(\gamma_{k})$
$\displaystyle=s_{\alpha_{0}}w_{I}(2\varepsilon_{k})=s_{\alpha_{0}}(2\varepsilon_{n-k+1})$
$\displaystyle=\begin{cases}{2\varepsilon_{n-k+1}>0},\quad&{1\leq k\leq
n-2}\\\ {-2\varepsilon_{2}<0},\quad&{k=n}\\\
{-2\varepsilon_{1}<0},\quad&{k=n-1}.\end{cases}$

Now we compute $|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|$ and
$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|$ in accordance with $s_{\alpha_{0}}w_{I}(\alpha)>0$ or
$<0$.

Case 1: $\alpha\in\Phi^{+},\ s_{\alpha_{0}}w_{I}(\alpha)>0,.$

(1) When $\alpha=\alpha_{1j}=\varepsilon_{1}-\varepsilon_{j},\ \ j\geq 3$, we
have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|=\begin{cases}{0},\quad&{1\leq j\leq n-2}\\\
{1},\quad&{j=n-1}\\\ {2},\quad&{j=n}.\end{cases}$

Thus, these positive roots contribute 3 to the sum in the length formula.

(2) When $\alpha=\alpha_{2j}=\varepsilon_{2}-\varepsilon_{j},\ \ j\geq 3$, we
have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|=\begin{cases}{0},\quad&{1\leq j\leq n-2}\\\
{1},\quad&{j=n-1}\\\ {2},\quad&{j=n}.\end{cases}$

Thus, these positive roots contribute 3 to the sum in the length formula.

(3) When $\alpha=\beta_{ij}=\varepsilon_{i}+\varepsilon_{j},\ \
i,j\not\in\\{n-1,\ n\\}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|=2.$

Thus, these positive roots contribute $2\cdot\frac{(n-2)(n-3)}{2}=(n-2)(n-3)$
to the sum in the length formula.

(4) When $\alpha=\gamma_{k}=2\varepsilon_{k},\ \ k\neq n-1,\ n$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle|=1.$

Thus, these positive roots contribute $n-2$ to the sum in the length formula.

Case 2: $\alpha\in\Phi^{+},\ s_{\alpha_{0}}w_{I}(\alpha)<0$,

(1) When $\alpha=\alpha_{ij}=\varepsilon_{1}-\varepsilon_{j},\ \ i,\
j\not\in\\{1,2\\}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=\begin{cases}{1},\quad&{3\leq i<j\leq n-2}\\\
{3},\quad&{3\leq i\leq n-2,\ j=n}\\\ {2},\quad&{i=n-1,\ j=n}\\\
{2},\quad&{3\leq i\leq n-2,\ j=n-1}.\end{cases}$

Thus, these positive roots contribute
$\frac{(n-4)(n-5)}{2}+3(n-4)+2+2(n-4)=5n-18+\frac{(n-4)(n-5)}{2}$ to the sum
in the length formula.

(2) When $\alpha=\varepsilon_{1}-\varepsilon_{2}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=1.$

Thus, this positive root contributes 1 to the sum in the length formula.

(3) When $\alpha=\varepsilon_{n-1}+\varepsilon_{n}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=0.$

Thus, this positive root contributes 0 to the sum in the length formula.

(4) When $\alpha=\beta_{in}=\varepsilon_{i}+\varepsilon_{n},\ \ 1\leq i\leq
n-2$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=1.$

Thus, these positive roots contribute $n-2$ to the sum in the length formula.

(5) When $\alpha=\beta_{i,n-1}=\varepsilon_{i}+\varepsilon_{n-1},\ \ 1\leq
i\leq n-2$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=2.$

Thus, these positive roots contribute $2(n-2)$ to the sum in the length
formula.

(6) When $\alpha=\gamma_{n}=2\varepsilon_{n}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=0.$

Thus, this positive root contributes 0 to the sum in the length formula.

(7) When $\alpha=\gamma_{n-1}=2\varepsilon_{n-1}$, we have

$|\langle-\lambda_{n-1}+\varepsilon_{n-1}+\varepsilon_{n},\
\alpha^{\vee}\rangle-1|=1.$

Thus, this positive root contributes 1 to the sum in the length formula.

Add up all the numbers in Case 1 and Case 2, finally we get:

$\displaystyle l(x_{n-1}^{-1}w_{I}s_{0})=$ $\displaystyle
3+3+(n-2)(n-3)+(n-2)$ $\displaystyle\
+5n-18+\frac{(n-4)(n-5)}{2}+1+0+(n-2)+2(n-2)+0+1$ $\displaystyle=$
$\displaystyle n^{2}-2+\frac{n(n-1)}{2}=l(x_{n-1}^{-1}w_{I})-1.$

which is exactly what we need, and the theorem follows.∎

######  Remark.

It is difficult to prove the theorem by using the reduced expressions of the
fundamental dominant weights directly. Actually Jinhao Guo discussed the
reduced expressions in section 3.2.2 in his PhD thesis[5], the expressions
seem complicated for application except for a few cases. The reduced
expressions obtained by Guo are deduced from the formulas in [8]:

$\displaystyle x_{1}=\tau s_{1}s_{2}s_{3}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{3}s_{2}s_{1},$ $\displaystyle x_{2}=s_{1}x_{1}s_{1}x_{1},$ $\displaystyle
x_{3}=s_{2}s_{1}x_{1}s_{1}s_{2}x_{2},$ $\displaystyle.........,$
$\displaystyle x_{n-1}=s_{n-2}\cdots s_{2}s_{1}x_{1}s_{1}s_{2}\cdots
s_{n-2}x_{n-2},$ $\displaystyle x_{n}=s_{n-1}s_{n-2}\cdots
s_{2}s_{1}x_{1}s_{1}s_{2}\cdots s_{n-2}s_{n-1}x_{n-1},$

where $\tau\in\Omega$ satisfies $\tau s_{0}=s_{1}\tau,\ \tau s_{i}=s_{i}\tau$,
for $i=2,\ 3,\ \cdots,\ n$.

4.10. Now we deal with type $\tilde{C}_{n}$ $(n\geq 2)$.

Let $E$ be the n-dimensional euclidean space with the standard orthogonal
basis $\varepsilon_{1},\ \varepsilon_{2},\ ...,\ \varepsilon_{n}$,
$\Phi=\\{\pm(\varepsilon_{i}\pm\varepsilon_{j}),\ \varepsilon_{k}\,|\,1\leq
i<j\leq n,\ 1\leq k\leq n\\}.$ The extended affine Weyl group associated to
$\Phi$ is of type $\tilde{C}_{n}\ (n\geq 2)$. The Coxeter graph of $W$ is as
follows ( see [19]):

$\tilde{C_{n}}:$$0$$1$$2$$n-2$$n-1$$n$

Fix a simple root system as

$\Delta=\\{\varepsilon_{1}-\varepsilon_{2},\ \varepsilon_{2}-\varepsilon_{3},\
...,\ \varepsilon_{n-1}-\varepsilon_{n},\ \varepsilon_{n}\\}.$

Then the set of positive roots is

$\Phi^{+}=\\{\varepsilon_{i}\pm\varepsilon_{j},\ \varepsilon_{k}\,|\,1\leq
i<j\leq n,\ 1\leq k\leq n\\},$

and the set of negative roots is $\Phi^{-}=-\Phi^{+}.$ Let

$\displaystyle\alpha_{1}$ $\displaystyle=\varepsilon_{1}-\varepsilon_{2},\
\alpha_{2}=\varepsilon_{2}-\varepsilon_{3},\ ...,\
\alpha_{n-1}=\varepsilon_{n-1}-\varepsilon_{n},\ \alpha_{n}=\varepsilon_{n};$
$\displaystyle\alpha_{ij}$
$\displaystyle=\varepsilon_{i}-\varepsilon_{j},\quad\beta_{ij}=\varepsilon_{i}+\varepsilon_{j},\quad
1\leq i<j\leq n,$ $\displaystyle\gamma_{k}$
$\displaystyle=\varepsilon_{k},\qquad 1\leq k\leq n.$

Then

$\displaystyle\alpha_{ij}$
$\displaystyle=\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j-1},\quad 1\leq i<j\leq
n;$ $\displaystyle\beta_{ij}$
$\displaystyle=\alpha_{i}+\cdots+\alpha_{j-1}+2\alpha_{j}+\cdots+2\alpha_{n-1}+2\alpha_{n},\quad
1\leq i<j\leq n;$ $\displaystyle\gamma_{k}$
$\displaystyle=\alpha_{k}+\alpha_{k+1}+\cdots+\alpha_{n-1}+\alpha_{n},\quad
1\leq k\leq n.$

The fundamental dominant weights $\lambda_{1},\ \cdots,\ \lambda_{n}$ in the
weight lattice $P$ of $\Phi$ are well known:

$\displaystyle\lambda_{i}$
$\displaystyle=\alpha_{1}+2\alpha_{2}+\cdots+(i-1)\alpha_{i-1}+i(\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{n}),\quad
1\leq i\leq n-1,$ $\displaystyle\lambda_{n}$
$\displaystyle=\frac{1}{2}(\alpha_{1}+2\alpha_{2}+\cdots+n\alpha_{n}).$

In terms of the standard orthogonal basis we have

$\displaystyle\lambda_{i}$
$\displaystyle=\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{i},\quad
1\leq i\leq n-1,$ $\displaystyle\lambda_{n}$
$\displaystyle=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}+\cdots+\varepsilon_{n}).$

Note that $t_{\lambda_{i}}=x_{i}$. Using the length formula (1) of Iwahori-
Matsumoto, we get

(a) $l(x_{i})=i(2n-i+1),\quad 1\leq i\leq n-1$;

(b) $l(x_{n})=\frac{n(n+1)}{2}$.

The following identities can be found in [8]:

$\displaystyle x_{1}=s_{0}s_{1}s_{2}s_{3}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{3}s_{2}s_{1}$ $\displaystyle x_{2}=s_{1}x_{1}s_{1}x_{1}$ $\displaystyle
x_{3}=s_{2}s_{1}x_{1}s_{1}s_{2}x_{2}$ $\displaystyle.........$ $\displaystyle
x_{n-1}=s_{n-2}\cdots s_{2}s_{1}x_{1}s_{1}s_{2}\cdots s_{n-2}x_{n-2}$
$\displaystyle x_{n}=\tau(s_{n}s_{n-1}\cdots s_{2}s_{1})(s_{n}s_{n-1}\cdots
s_{2})\cdots(s_{n}s_{n-1})(s_{n}),$

where $\tau\in\Omega$ satisfies $\tau s_{0}=s_{n}\tau,\ \tau
s_{i}=s_{n-i}\tau$, for $i=2,3,...,n-1$.

We can determine reduced expressions of the fundamental dominant weights by
these formulas. Jinhao Guo has obtained a reduced expression for each $x_{i}$
(see Lemma 3.3.2 in [5]). But those reduced expressions are not easy to use
for the discussion here. Hence, we display a different reduced expression for
these $x_{i}$.

Proposition 4.11. Keep the notations in subsection 4.10. Then $x_{1},\ ...,\
x_{n}$ are fundamental dominant weights in the extended affine Weyl group $W$
of type $\tilde{C}_{n}$. We claim that the following expressions are reduced:

$\displaystyle x_{1}=s_{0}s_{1}s_{2}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{2}s_{1};$ $\displaystyle x_{2}=s_{0}s_{1}s_{2}\cdots
s_{n-1}s_{n}s_{n-1}\cdots s_{2}s_{0}s_{1}s_{2}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{2}=(s_{0}s_{1}s_{2}\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{2})^{2};$
$\displaystyle............$ $\displaystyle x_{i}=(s_{0}s_{1}s_{2}\cdots
s_{n-1}s_{n}s_{n-1}\cdots s_{i})^{i},\quad 1\leq i\leq n-1;$ $\displaystyle
x_{n}=\tau(s_{n}s_{n-1}\cdots s_{2}s_{1})(s_{n}s_{n-1}\cdots
s_{2})\cdot\cdots\cdot(s_{n}s_{n-1})s_{n}.$

where $\tau\in\Omega$ satisfies $\tau s_{0}=s_{n}\tau,\ \tau
s_{i}=s_{n-i}\tau$, for $i=2,3,...,n-1$.

###### Proof.

The above expressions for $x_{1}$ and $x_{n}$ are given in [8]. By the length
formulas 4.10 (a) and 4.10 (b) we know that above expressions for $x_{1}$ and
$x_{n}$ are reduced.

Now we show that the expression is reduced for each $x_{i},\ 1\leq i\leq n-1$.
We use induction on $i$. It is known that for $i=1$, the expression is
reduced. Suppose the assertion is true for $i-1$.

Notice for $1\leq k\neq i\leq n$, we have $s_{k}x_{i}=x_{i}s_{k}$. Therefore,

$\displaystyle x_{i}$ $\displaystyle=s_{i-1}\cdots
s_{2}s_{1}x_{1}s_{1}s_{2}\cdots s_{i-1}x_{i-1}$ $\displaystyle=s_{i-1}\cdots
s_{2}s_{1}s_{0}s_{1}s_{2}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{i+1}s_{i}(s_{0}s_{1}\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{i}s_{i-1})^{i-1}$
$\displaystyle=s_{1}s_{2}\cdots s_{i-1}x_{i}s_{i-1}\cdots s_{2}s_{1}$
$\displaystyle=s_{0}s_{1}s_{2}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{i+1}s_{i}(s_{0}s_{1}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{i}s_{i-1})^{i-1}s_{i-1}\cdots s_{2}s_{1}.$

By commutation relation $s_{k}s_{k+1}s_{k}=s_{k+1}s_{k}s_{k+1},\ 1\leq k\leq
n-2$ and the induction hypothesis, it is easy to see for $1\leq k\leq i-1$,

$s_{0}s_{1}\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{i}s_{i-1}(s_{0}s_{1}\cdots
s_{n-1}s_{n}s_{n-1}\cdots s_{i})^{i-1-k}s_{k}=(s_{0}s_{1}\cdots
s_{n-1}s_{n}s_{n-1}\cdots s_{i})^{i-k}.$

Thus

$x_{i}=s_{i-1}\cdots s_{2}s_{1}x_{1}s_{1}s_{2}\cdots
s_{i-1}x_{i-1}=(s_{0}s_{1}\cdots s_{n-1}s_{n}s_{n-1}\cdots s_{i})^{i}.$

According to 4.10 (a), the number of simple reflections in the expression
above is equal to $l(x_{i})$, this means that the expression is reduced. The
proposition follows. ∎

Theorem 4.12. Keep the notations in 4.10. Then $W$ is an extended affine Weyl
group of type $\tilde{C}_{n}\ (n\geq 2)$ and $x_{1},\ ...,\ x_{n}$ are
fundamental dominant weights in $W$. Let $J=\\{2,\ 3,\ \cdots,\ n\\}$,
$x_{J}=\displaystyle\prod_{i\in J}x_{i}$. Then $x_{I}\in\Omega_{qr}$. In
particular, $x=\prod_{i\in J}x_{i}^{a_{i}}\in\Omega_{qr},$ if all
$a_{i}^{\prime}s$ are positive integers.

###### Proof.

By Theorem 3.8 (a), $x_{J}$ is not in the lowest two-sided cell $W_{(\nu)}$ of
$W$. Therefore $a(x)\leq(n-1)^{2}+1=a(\Omega_{qr})$.

Now we show $x_{J}=w\cdot w_{K}\cdot u$, where $K=J\cup\\{0\\}$ and $w_{K}$ is
the longest element in $W_{K}=\\{s_{i}\,|\,i\in K\\}$. Let
$y=x_{n}x_{n-1}\cdots x_{3},\ z:=s_{0}s_{1}\cdots s_{n-1}s_{n}s_{n-1}\cdots
s_{3}s_{2}$. Using Proposition 4.11 we get $x_{J}=y{z}^{2}$. According to
Lemma 2.3, $\mathscr{R}(y)=\\{s_{3},\ s_{4},\ ...,\ s_{n}\\}$. Note that for
any $3\leq i\leq n$ we have $s_{i}z=zs_{i}$. So
$\mathscr{R}(yz)\supset\\{s_{i}\,|\,i\in J\\}$. In addition, $s_{0}z<z$. Thus
$x_{J}=y{z}^{2}=w\cdot w_{K}\cdot u$. Since
$a(w_{k})=l(w_{K})=a(\Omega_{qr})$, we see $w_{K}\in\Omega_{qr}$. The theorem
then follows from $w_{K}\in\Omega_{qr}$ and $x_{J}\not\in W_{(\nu)}$. ∎

4.13 Now we deal with type $\tilde{F}_{4}$.

Let $E$ a the 4-dimensional euclidean space with the standard orthogonal basis
$\varepsilon_{1},\ \varepsilon_{2},\ \varepsilon_{3},\ \varepsilon_{4}$,
$\Phi=\\{\pm(\varepsilon_{i}\pm\varepsilon_{j}),\ \pm\varepsilon_{k},\
\pm\frac{1}{2}(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})\,|\,1\leq
i<j\leq 4,\ 1\leq k\leq 4\\}.$ As before, $W$ is the extended affine Weyl
group associated to $\Phi$, which equals $W_{a}$ in this case, $W_{0}$ is the
corresponding Weyl group, $X$ is the subgroup consisting of all translations
in $W$. The simple refections $s_{0},s_{1},\cdots,s_{4}$ in $W$ are numbered
as usual.

The Coxeter graph of $W$ is as follows ( see [19] ):

$\tilde{F}_{4}:$$0$$1$$2$$3$$4$

Fix a simple root system as

$\Delta=\\{\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}),\
\varepsilon_{4},\ \varepsilon_{3}-\varepsilon_{4},\
\varepsilon_{2}-\varepsilon_{3}\\}.$

Then the set of positive roots is

$\Phi^{+}=\\{\varepsilon_{i}\pm\varepsilon_{j},\ \varepsilon_{k},\
\frac{1}{2}(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})\,|\,1\leq
i<j\leq 4,\ 1\leq k\leq 4\\},$

and the set of negative roots is $\Phi^{-}=-\Phi^{+}.$ Let

$\alpha_{1}=\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}),\
\alpha_{2}=\varepsilon_{4},\ \alpha_{3}=\varepsilon_{3}-\varepsilon_{4},\
\alpha_{4}=\varepsilon_{2}-\varepsilon_{3}.$

Notice that for any short root $\alpha$, $\alpha^{\vee}=2\alpha$, and for any
long root $\beta$, $\beta^{\vee}=\beta$.

The fundamental dominant weights in the weight lattice $P$ are:

$\displaystyle\lambda_{1}$
$\displaystyle=2\alpha_{1}+3\alpha_{2}+2\alpha_{3}+\alpha_{4}=\varepsilon_{1},$
$\displaystyle\lambda_{2}$
$\displaystyle=3\alpha_{1}+6\alpha_{2}+4\alpha_{3}+2\alpha_{4}=2\varepsilon_{1}-\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}),$
$\displaystyle\lambda_{3}$
$\displaystyle=4\alpha_{1}+8\alpha_{2}+6\alpha_{3}+3\alpha_{4}=2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3},$
$\displaystyle\lambda_{4}$
$\displaystyle=2\alpha_{1}+4\alpha_{2}+3\alpha_{3}+2\alpha_{4}=\varepsilon_{1}+\varepsilon_{2}.$

Recall that $x_{i}=t_{\lambda_{i}}$.

Theorem 4.14. Keep the notations in subsection 4.13. Then $W$ is an affine
Weyl group of type $\tilde{F}_{4}$ and $x_{1},\ ...,\ x_{4}$ are the
fundamental dominant weights in $W$. Set
$x=x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}},\ a_{1},a_{2}\geq 1,\ a_{3}\geq 2$,
then $x\in\Omega_{qr}$.

4.15. Proof of Theorem 4.14. By Theorem 3.8 (a), $x$ is not in the lowest two-
sided cell $W_{(\nu)}$ of $W$. Thus $a(x)\leq 16=a(\Omega_{qr})$. Let
$J=\\{0,\ 1,\ 2,\ 3\\}$, then $W_{J}=<s_{i}\,|\,i\in J>$ is a Weyl group of
type $B_{4}$, whose longest element $w_{J}$ is of length
$16=4^{2}=a(\Omega_{qr})$. It suffices to show

$x_{1}x_{2}x_{3}^{2}=w\cdot w_{J}\cdot u.$

By Lemma 2.3 (1), $\mathscr{R}(x_{1}x_{2}x_{3})=\\{s_{1},\ s_{2},\ s_{3}\\}$,

$x_{1}x_{2}x_{3}=v\cdot w_{K},$

where $w_{K}$ is the longest element of the group $W_{K}=\langle
s_{1},s_{2},s_{3}\rangle$. We only need to prove

$w_{K}x_{3}=w_{J}\cdot u.$

Since $\\{s_{1},\ s_{2},\ s_{3}\\}\subset\mathscr{L}(w_{I}x_{3})$, it suffices
to show $s_{0}w_{K}x_{3}\leq w_{I}x_{3}$, or equivalently,
$x_{3}^{-1}w_{K}s_{0}\leq x_{3}^{-1}w_{K}$.

We have

(a) $l(x_{3}^{-1})=42$,

(b) $l(x^{-1}_{3}w_{K})=51$.

In fact, using the length formula (1) of Iwahori-Matsumoto, we get

$l(x_{3}^{-1})=\sum_{\alpha\in\Phi^{+}}|\langle-\lambda_{3},\
\alpha^{\vee}\rangle|=\sum_{\alpha\in\Phi^{+}}|\langle\lambda_{3},\
\alpha^{\vee}\rangle|.$

By direct computations, we see

$l(x_{3}^{-1})=l(x_{3})=42,\quad l(x_{3}^{-1}w_{K})=42+9=51.$

Now we show $x_{3}^{-1}w_{I}s_{0}\leq x_{3}^{-1}w_{K}$. We need to prove
$l(x_{3}^{-1}w_{K}s_{0})=50$.

By definition, $s_{0}=t_{-\alpha_{0}}s_{\alpha_{0}}$, where
$-\alpha_{0}=\varepsilon_{1}=\lambda_{1}$ is the highest short root. Then

$x_{3}^{-1}w_{K}s_{0}=x_{3}^{-1}w_{K}t_{\varepsilon_{1}}s_{\alpha_{0}}=x_{3}^{-1}t_{w_{K}(\varepsilon_{1})}w_{K}s_{\alpha_{0}}.$

We compute $w_{K}(\varepsilon_{1})$. Since
$s_{i}(\varepsilon_{1})=\varepsilon_{1}$ if $i=2,3,4$ and
$w_{K}=s_{1}s_{2}s_{3}s_{2}s_{1}s_{2}s_{3}s_{2}s_{3}$, we have

$\displaystyle w_{K}(\varepsilon_{1})$
$\displaystyle=s_{1}s_{2}s_{3}s_{2}s_{1}(\varepsilon_{1})=s_{1}s_{2}s_{3}s_{2}(\varepsilon_{1}-\alpha_{1})$
$\displaystyle=s_{1}s_{2}s_{3}(\varepsilon_{1}-\alpha_{1}-\alpha_{2})=s_{1}s_{2}(\varepsilon_{1}-\alpha_{1}-\alpha_{2}-\alpha_{3})$
$\displaystyle=s_{1}(\varepsilon_{1}-\alpha_{1}-2\alpha_{2}-\alpha_{3})$
$\displaystyle=\varepsilon_{1}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$
$\displaystyle=\varepsilon_{1}-(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4})-2\varepsilon_{4}-(\varepsilon_{3}-\varepsilon_{4})$
$\displaystyle=\varepsilon_{2}.$

By the length formula (1) of Iwahori-Matsumoto, we get

$\displaystyle\ \qquad
l(x_{3}^{-1}w_{K}s_{0})=l(x_{3}^{-1}t_{\varepsilon_{2}}w_{K}s_{\alpha_{0}})$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{K}(\alpha)\in{\Phi^{+}}\end{subarray}}|\langle-\lambda_{3}+\varepsilon_{2},\
\alpha^{\vee}\rangle|+\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{K}(\alpha)\in{\Phi^{-}}\end{subarray}}|\langle-\lambda_{3}+\varepsilon_{2},\
\alpha^{\vee}\rangle-1|$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{K}(\alpha)\in{\Phi^{+}}\end{subarray}}|\langle-2\varepsilon_{1}-\varepsilon_{3},\
\alpha^{\vee}\rangle|+\sum_{\begin{subarray}{c}\alpha\in{\Phi^{+}}\\\
s_{\alpha_{0}}w_{K}(\alpha)\in{\Phi^{-}}\end{subarray}}|\langle-2\varepsilon_{1}-\varepsilon_{3},\
\alpha^{\vee}\rangle-1|.$

We need to figure out the positive roots mapped by $s_{\alpha_{0}}w_{K}$ into
$\Phi^{+}$ or $\Phi^{-}$ respectively in detail.

We write $\alpha>0$ if $\alpha$ is positive root, and write $\alpha<0$ when
$\alpha$ is negative root. Since $w_{K}(\alpha_{i})=-\alpha_{i}$ if $i=2,\ 3,\
4$, we see

$s_{\alpha_{0}}(w_{K}):\ \varepsilon_{1}\mapsto\varepsilon_{2},\
\varepsilon_{2}\mapsto-\varepsilon_{1},\
\varepsilon_{3}\mapsto-\varepsilon_{3},\
\varepsilon_{4}\mapsto-\varepsilon_{4}.$

It is easy to get

$\displaystyle
s_{\alpha_{0}}w_{K}(\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4}))=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})>0,$
$\displaystyle s_{\alpha_{0}}w_{K}(\varepsilon_{1})=\varepsilon_{2}>0,$
$\displaystyle s_{\alpha_{0}}w_{K}(\varepsilon_{k})=-\varepsilon_{k}>0,\ k=3,\
4,$ $\displaystyle
s_{\alpha_{0}}w_{K}(\varepsilon_{1}\pm\varepsilon_{j})=\varepsilon_{2}\mp\varepsilon_{j}>0,\
j=3,\ 4,$ $\displaystyle
s_{\alpha_{0}}w_{K}(\varepsilon_{1}-\varepsilon_{2})=\varepsilon_{1}+\varepsilon_{2}>0,\
$ $\displaystyle
s_{\alpha_{0}}w_{K}(\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4}))=\frac{1}{2}(-\varepsilon_{1}+\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})<0,$
$\displaystyle s_{\alpha_{0}}w_{K}(\varepsilon_{2})=-\varepsilon_{1}<0,$
$\displaystyle
s_{\alpha_{0}}w_{K}(\varepsilon_{2}\pm\varepsilon_{j})=-\varepsilon_{1}\mp\varepsilon_{j}<0,\
\ j=3,\ 4,$ $\displaystyle
s_{\alpha_{0}}w_{K}(\varepsilon_{1}-\varepsilon_{2})=\varepsilon_{1}+\varepsilon_{2}>0,\
$ $\displaystyle
s_{\alpha_{0}}w_{K}(\varepsilon_{3}\pm\varepsilon_{4})=-\varepsilon_{3}\mp\varepsilon_{4}<0.$

Now we compute $|\langle-2\varepsilon_{1}-\varepsilon_{3},\
\alpha^{\vee}\rangle|$and $|\langle-2\varepsilon_{1}-\varepsilon_{3},\
\alpha^{\vee}\rangle-1|$ in accordance with $s_{\alpha_{0}}w_{K}(\alpha)>0$ or
$<0$ respectively.

Case 1: $\alpha\in\Phi^{+},\ s_{\alpha_{0}}w_{K}(\alpha)>0$.

(1) When
$\alpha=\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}+\varepsilon_{3}\pm\varepsilon_{4})$,
we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=3$

Thus, the two positive roots contribute 6 to the sum in the length formula.

(2) When
$\alpha=\frac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}\pm\varepsilon_{4})$,
we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=1$

Thus, the two positive roots contribute 2 to the sum in the length formula.

(3) When $\alpha=\varepsilon_{1}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=4.$

Thus, this positive root contributes 4 to the sum in the length formula.

(4) When $\alpha=\varepsilon_{1}+\varepsilon_{3}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=3.$

Thus, this positive root contributes 3 to the sum in the length formula.

(5) When $\alpha=\varepsilon_{1}+\varepsilon_{4},\
\varepsilon_{1}-\varepsilon_{2},\ \varepsilon_{1}-\varepsilon_{4}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=2.$

Thus, the three positive roots contribute 6 to the sum in the length formula.

(6) When $\alpha=\varepsilon_{1}-\varepsilon_{3}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle|=1.$

Thus, this positive root contributes 1 to the sum in the length formula.

Case 2: $\alpha\in\Phi^{+},\ s_{\alpha_{0}}w_{K}(\alpha)<0$.

(1) When
$\alpha=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\pm\varepsilon_{4})$,
we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=4.$

Thus, the two positive roots contribute 8 to the sum in the length formula.

(2) When
$\alpha=\frac{1}{2}(\varepsilon_{1}+\varepsilon_{2}-\varepsilon_{3}\pm\varepsilon_{4})$,
we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=2.$

Thus, the two positive roots contribute 4 to the sum in the length formula.

(3) When $\alpha=\varepsilon_{2},\ \varepsilon_{4}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=1.$

Thus, the two positive roots contribute 2 to the sum in the length formula.

(4) When $\alpha=\varepsilon_{3}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=3.$

Thus, this positive root contributes 3 to the sum in the length formula.

(5) When $\alpha=\varepsilon_{1}+\varepsilon_{2}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=3.$

Thus, this positive root contributes 3 to the sum in the length formula.

(6) When $\alpha=\varepsilon_{2}+\varepsilon_{3},\
\varepsilon_{3}\pm\varepsilon_{4}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=2.$

Thus, the three positive roots contribute 6 to the sum in the length formula.

(7) When $\alpha=\varepsilon_{2}\pm\varepsilon_{4}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=1.$

Thus, the two positive roots contribute 2 to the sum in the length formula.

(8) When $\alpha=\varepsilon_{2}-\varepsilon_{3}$, we have

$|\langle-2\varepsilon_{1}-\varepsilon_{3},\ \alpha^{\vee}\rangle-1|=0.$

Thus, this positive root contributes 0 to the sum in the length formula.

Adding up all the numbers in Case 1 and Case 2 above, finally we get:

$\displaystyle l(x_{3}^{-1}w_{K}s_{0})=$ $\displaystyle 6+2+4+3+6+1$
$\displaystyle+8+4+3+2+3+6+2+0$ $\displaystyle=$ $\displaystyle
50=l(x_{3}^{-1}w_{K})-1,$

which is exactly what we need. Then Theorem 4.14 is proved.∎

4.16. Now we consider type $\tilde{G}_{2}$.

Let $E$ be the 3-dimensional euclidean space with the standard orthogonal
basis $\varepsilon_{1},\ \varepsilon_{2},\ \varepsilon_{3}$,
$\Phi=\pm\\{(\varepsilon_{1}\pm\varepsilon_{2}),\
(\varepsilon_{1}\pm\varepsilon_{3}),\ (\varepsilon_{2}\pm\varepsilon_{3}),\
2\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3},\
2\varepsilon_{2}-\varepsilon_{1}-\varepsilon_{3},2\varepsilon_{3}-\varepsilon_{1}-\varepsilon_{2}\\}.$
Choose $\Delta=\\{\varepsilon_{1}-\varepsilon_{2},\
-2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\\}$ to be the set of simple
roots. In this case, the weight lattice and the root lattice are equal, so the
extended affine Weyl group $W$ equals the affine Weyl group $W_{a}$. We number
the simple reflections $s_{1},\ s_{2},\ s_{0}$ as usual. Let $x_{1},\ x_{2}$
be the corresponding fundamental dominant weights in $W$.

The Coxeter graph of $W$ is as follows([19]):

$\widetilde{G}_{2}:$$0$$1$$2$

Theorem 4.17. Keep the notations in 4.16. So $W$ is an affine Weyl group of
type $\tilde{G}_{2}$ and $x_{1},\ x_{2}$ are fundamental dominant weights in
$W$. Then a translation $x\in X^{+}$ is in $\Omega_{qr}$ if and only if
$x={x_{1}}^{m}$ with any interger $m\geq 2$.

###### Proof.

By Theorem 3.8 (a), $x\in\Omega_{qr}$ if and only if either $x={x_{1}}^{m}$ or
$x={x_{2}}^{m}$ with some positive integer $m$. In Example 3.6,
${x_{2}}^{m}\underset{L}{\sim}x_{2}$ for any positive integer $m$, and
$x_{2}=s_{0}s_{1}s_{2}s_{1}s_{0}s_{2}s_{1}s_{2}s_{1}s_{2}\underset{L}{\sim}s_{0}s_{1}s_{2}s_{1}s_{0}s_{2}\underset{R}{\sim}s_{0}s_{2}$
by star actions, thus
$a({x_{2}}^{m})=2<a(\Omega_{qr}),{x_{2}}^{m}\not\in\Omega_{qr}$. In addition,
$x_{1}\not\underset{L}{\sim}{x_{1}}^{m}$ while
${x_{1}}^{m}\underset{L}{\sim}{x_{1}}^{2}$ with $m\geq 2$ . Thus, it suffices
to show ${x_{1}}^{2}$ is in $\Omega_{qr}$ in the following.

Note that $s_{1}s_{0}s_{1}\in\Omega_{qr},\ a(\Omega_{qr})=3$. Since
$x_{1}=s_{0}s_{1}s_{2}s_{1}s_{2}s_{1}$, we see $x_{1}^{2}\in\Omega_{qr}$.
Therefore, if $m\geq 2$, then $x_{1}^{m}\in\Omega_{qr}$. ∎

##  5 A conjecture and some related results

5.1. In [19], Shi proposed a conjecture saying that the number $n_{qr}$ of
left cells in the second lowest two-sided cell $\Omega_{qr}$ is half of the
cardinality of the Weyl group $W_{0}$ and he proved $n_{qr}\leq|W_{0}|/2$. He
also verified his conjecture for the following cases(loc.cit.): type
$\tilde{A}_{n}$ and affine Weyl groups of rank $\leq 4$. Jinhao Guo verified
the conjecture for type $\tilde{B}_{5}$(see [5]).

In this section we state a conjecture and provide some evidences. The
conjecture could be regarded as a refinement of Shi’s conjecture on the number
of left cells in the second lowest two-sided cell $\Omega_{qr}$.

Conjecture 5.2. Keep the notations in section 1. Then $W$ is an extended
affine Weyl group associated to irreducible root system $\Phi$, $W_{0}$ is the
corresponding Weyl group, $X^{+}=\\{t_{\lambda}\,|\,\lambda\in P^{+}\\}\subset
W$. $x\in X^{+}$. If $x$ is in the second lowest two-sided cell $\Omega_{qr}$
of $W$, then

(a) for any $w\in W_{0}$, we have $wxw^{-1}\in\Omega_{qr}$;

(b) if $w,\ u\in W_{0}$, and $wxw^{-1}\neq uxu^{-1}$, then
$wxw^{-1}\underset{R}{\not\sim}uxu^{-1}$;

(c) for any right cell $\Gamma$ in $\Omega_{qr}$ , there must exist some $w\in
W_{0}$ such that $wxw^{-1}\in\Gamma$. In particular, $\Omega_{qr}$ contains
$\frac{|W_{0}|}{2}$ right cells.

If the conjecture is true, using the results in section 4 we then obtain some
representatives for each left cell in $\Omega_{qr}$. We will show that a weak
version of Conjecture 5.2 (a) is true and Conjecture 5.2 is true for type
$\tilde{A}_{n}$ and type $\tilde{G}_{2}$.

Lemma 5.3. Keep the notations in section 1. Then $W$ is the extended affine
Weyl group of the irreducible root system $\Phi$ and $X^{+}$ is the set of
dominant weights in $W$. Assume that $x\in X^{+}$ is in the the second lowest
two-sided cell $\Omega_{qr}$. Then for any $w\in W_{0}$ such that
$\mathscr{R}(wx)=\mathscr{R}(x)$, we have $wx\in\Omega_{qr}$.

###### Proof.

By Proposition 3.10 (c), there exists $\alpha_{i}\in{\Phi}^{+}$ such that
$k(x,\alpha_{i})=\langle x,\ {\alpha_{i}}^{\vee}\rangle=0$. By Proposition
3.10 (a), $k(x^{-1}w^{-1},\alpha_{i})=\langle
x^{-1},{\alpha_{i}}^{\vee}\rangle+k(w^{-1},\alpha_{i})=-\langle
x,{\alpha_{i}}^{\vee}\rangle+k(w^{-1},\alpha_{i})=k(w^{-1},\alpha_{i})$. By
the length formula (1) of Iwahori-Matsumoto, we get

$\displaystyle l(xs_{i})$ $\displaystyle=|\langle x,\
{\alpha_{i}}^{\vee}\rangle-1|+\sum\limits_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
\alpha\not=\alpha_{i}\end{subarray}}|\langle x,\ {\alpha}^{\vee}\rangle|$
$\displaystyle=1+\sum\limits_{\begin{subarray}{c}\alpha\in{\Phi}^{+}\\\
\alpha\not=\alpha_{i}\end{subarray}}|\langle x,\ {\alpha}^{\vee}\rangle|$
$\displaystyle=1+l(x).$

This implies $xs_{i}=s_{i}x$. If $k(w^{-1},\alpha_{i})\neq 0$, by Proposition
3.10 (a), $w(\alpha_{i})\in{\Phi}^{-}$, i.e. $ws_{i}<w$. Then
$s_{i}\in\mathscr{R}(wx)$, which contradicts $\mathscr{R}(wx)=\mathscr{R}(x)$.
So we must have $w(\alpha_{i})\in\Phi^{+}$. By Proposition 3.10 (a) and (c),
we get $k(x^{-1}w^{-1},\alpha_{i})=0$. This implies that $x^{-1}w^{-1}$ is not
in $W_{(\nu)}$, which is equivalent to $wx\not\in W_{(\nu)}$.

Since $wx=w\cdot x,\ x\in\Omega_{qr}$, so $a(wx)\geq a(x)$. Consequently,
$wx\in\Omega_{qr},\ a(wx)=a(x)$. The lemma follows. ∎

The following result shows that a weak version of Conjecture 5.2 (a) is true.

Theorem 5.4. Let $W$ be the extended affine Weyl group of an irreducible root
system $\Phi$ of rank $n$, $W_{0}\subset W$ be the Weyl group of $\Phi$, $X$
be the set of translations (isomorphic to weight lattice), $X^{+}$ be the set
of dominant weights, and $x_{1},\ x_{2},\ \cdots,\ x_{n}$ be the fundamental
dominant weights. Set $I_{k}=\\{1,\ 2,\ \cdots,\ n\\}\setminus\\{k\\}$. Assume
that $x=\prod_{i\in I_{k}}{x_{i}}^{a_{i}}\in X^{+}\cap\Omega_{qr}$. Then for
any $w\in W_{0}$, we have

$wxx_{I_{k}}w^{-1}\in\Omega_{qr},$

where $x_{I_{k}}=\prod_{i\in{I_{k}}}x_{i}$.

###### Proof.

By Theorem 3.8 (a), $wxx_{I_{k}}w^{-1}\not\in W_{(\nu)}$. Note that for
$s=s_{k}$ we have $sxs=x$ and $sx_{I_{k}}s=x_{I_{k}}$. So it is no harm to
assume $ws\geq w$. By the length formula (1) of Iwahori-Matsumoto,

$x_{I_{K}}=(x_{I_{K}}w^{-1})\cdot w.$

Thus $wxx_{I_{K}}w^{-1}=(wx)\cdot(x_{I_{K}}w^{-1})\underset{R}{\leq}wx$.
According to Lemma 5.3, $a(wx{x_{I_{K}}}w^{-1})\geq a(wx)=a(x)$. This forces
that $a(wx{x_{I_{K}}}w^{-1})=a(wx)$ and $wxx_{I_{K}}w^{-1}\in\Omega_{qr}$. ∎

In the following, we will prove Conjecture 5.2 is true for an (extended)
affine Weyl group of $\tilde{G}_{2}$ or $\tilde{A}_{n}$.

Proposition 5.5. ([21], 11.2) Let $W_{a}$ be an affine Weyl group of type
$\tilde{G}_{2}$ generated by $s_{0},s_{1},s_{2}$, where
$s_{0}s_{2}=s_{2}s_{0}$. Then the second lowest two-sided cell of $W$ is

$\Omega_{qr}=\\{u(i,\ j,\ k)|1\leq i,j\leq 6,k\in\mathbb{N}\\},$

where $u(i,\ j,\ k)=u_{i}s_{0}s_{1}s_{0}(s_{2}s_{1}s_{0})^{k}{u_{j}}^{-1}$,
$u_{1}=e,\ u_{2}=s_{2},\ u_{3}=s_{1}s_{2},\ u_{4}=s_{2}s_{1}s_{2},\
u_{5}=s_{1}s_{2}s_{1}s_{2},\ u_{6}=s_{0}s_{1}s_{2}s_{1}s_{2}$. Moreover,

(a) $\mathcal{D}\cap\Omega_{qr}=\\{u(i,\ i,\ 0)|1\leq i\leq 6\\}$ is the set
of distinguished involutions in $\Omega_{qr}$,

(b) $u(i,\ j,\ k)\underset{L}{\sim}u(m,\ n,\ k^{\prime})$ if and only if
$j=n,$

$u(i,\ j,\ k)\underset{R}{\sim}u(m,\ n,\ k^{\prime})$ if and only if $i=m$.

Theorem 5.6. Conjecture 5.2 is true for an affine Weyl group of
$\tilde{G}_{2}$.

###### Proof.

Keep the notations in Proposition 5.5. For simplicity, in the proof we shall
write $i_{1}i_{2}\cdots i_{m}$ for reduced expression
$s_{i_{1}}s_{i_{2}}\cdots s_{i_{m}}$.

Let $x$ be a dominant weight in the second lowest two-sided cell
$\Omega_{qr}$, then by Theorem 4.17, $x={x_{1}}^{a}=(012121)^{a}$ for some
integer $a\geq 2$. Since $s_{2}x=xs_{2}$, we may assume $ws_{2}>w,\ w\in
W_{0}$ when considering $wxw^{-1}$, i.e. $w\in\\{e,1,21,121,2121,12121\\}$,
and we denote these six elements as $w_{i},\ 1\leq i\leq 6$ correspondingly.
We have

(1)

$\displaystyle w_{1}x{w_{1}}^{-1}$ $\displaystyle=x$
$\displaystyle=012121012121(012121)^{a-2}$
$\displaystyle=(01212)010(2121)(012121)^{a-2}$
$\displaystyle=u_{6}(010)(210)^{2(a-2)}{u_{5}}^{-1}=u(6,\ 5,\ 2(a-2)).$

(2)

$\displaystyle w_{2}x{w_{2}}^{-1}$
$\displaystyle=1012121(012121)^{a-2}(01212)$
$\displaystyle=(010)(210)^{2(a-1)}212$
$\displaystyle=u_{1}(010)(210)^{2(a-1)}{u_{4}}^{-1}=u(1,\ 4,\ 2(a-1))$

(3)

$\displaystyle w_{3}x{w_{3}}^{-1}$
$\displaystyle=21012121(012121)^{a-2}(0121)$
$\displaystyle=2(010)(210)^{2(a-1)}21$
$\displaystyle=u_{2}(010)(210)^{2(a-1)}{u_{3}}^{-1}=u(2,\ 3,\ 2(a-1))$

(4)

$\displaystyle w_{4}x{w_{4}}^{-1}$
$\displaystyle=121012121(012121)^{a-2}(012)$
$\displaystyle=12(010)(210)^{2(a-1)}2$
$\displaystyle=u_{3}(010)(210)^{2(a-1)}{u_{2}}^{-1}=u(3,\ 2,\ 2(a-1))$

(5)

$\displaystyle w_{5}x{w_{5}}^{-1}$
$\displaystyle=2121012121(012121)^{a-2}(01)$
$\displaystyle=212(010)(210)^{2(a-1)}e$
$\displaystyle=u_{4}(010)(210)^{2(a-1)}{u_{1}}^{-1}=u(4,\ 1,\ 2(a-1))$

(6)

$\displaystyle w_{6}x{w_{6}}^{-1}$
$\displaystyle=12121012121(012121)^{a-2}(0)$
$\displaystyle=1212(010)(210)^{2(a-2)}21210$
$\displaystyle=u_{5}(010)(210)^{2(a-2)}{u_{6}}^{-1}=u(5,\ 6,\ 2(a-2))$

By Proposition 5.5, one can get for any $w\in W_{0}$,
$wxw^{-1}\in\Omega_{qr}$, and when $ws_{2}>w$ and $us_{2}>u$, $w\not=u$, we
have $wxw^{-1}\underset{R}{\not\sim}uxu^{-1}$. Part (a) and (b) of Conjecture
5.2 are proved. Part (c) of Conjecture 5.2 then follows from Proposition 5.5.
∎

Theorem 5.7. Conjecture 5.2 is true for an extended affine Weyl group of type
$\tilde{A}_{n}$.

In order to verify Conjecture 5.2 (b) is true for an extended affine Weyl
group of type $\tilde{A}_{n}$, we need recall some definitions and results due
to Shi[12].

Definition 5.8. ([12], Definition 7.2.1) Let $(W_{a},S)$ be an affine Weyl
group of type $\tilde{A}_{n}$, we say a set $M\subset W_{a}$ is left connected
if, for any $x,\ y\in M$, there exists a sequence $y_{0}=x,\ y_{1},\ \cdots,\
y_{t}=y$ in $M$ such that for every $j$ satisfying $1\leq j\leq t$,
$y_{j}=s_{i_{j}}y_{j-1}$ for some $s_{i_{j}}\in S$. We say $M$ is right
connected if $M^{-1}=\\{w^{-1}\,|\,w\in M\\}$ is left connected.

Theorem 5.9.  ([12], Theorem 18.2.1) Let $(W_{a},S)$ be an affine Weyl group
of type $\tilde{A}_{n}$, then

(a) any left cell in $W_{a}$ is left connected and is also a maximal left
connected component in the two-sided cell containing it;

(b) any right cell in $W_{a}$ is right connected and is also a maximal right
connected component in the two-sided cell containing it;

Lemma 5.10. ([12], Lemma 18.2.5) Let $(W,S)$ be an extended affine Weyl group
of type $\tilde{A}_{n}$, $\Theta$ be a two-sided cell in $W$. Then

(a) any left connected set of $\Theta$ is contained in some left cell of $W$,

(b) any right connected set of $\Theta$ is contained in some right cell of
$W$.

Lemma 5.11. Let $(W,S)$ be an extended affine Weyl group of type
$\tilde{A}_{n}$, $W_{0}\subset W$ be the Weyl group,
$x=\prod_{i\in{I_{k}}}{x_{i}}^{a_{i}}\in X^{+}\cap\Omega_{qr}$ with $a_{i}\geq
1,\ I_{k}=\\{1,\ 2,\ \cdots,\ n\\}\setminus\\{k\\}$. Then for any $w\in W_{0}$
with $ws_{k}>w$, we have $wx\in\Omega_{qr}$.

###### Proof.

For any $w\in W_{0}$ satisfying $ws_{k}>w$ in $W_{0}$, we have
$\mathscr{R}(wx)=\mathscr{R}(x)$. Then by Lemma 5.3, $wx\in\Omega_{qr}$. ∎

Corollary 5.12. Let $W,\ S,\ W_{0},\ x,\ w$ be as in the Lemma 5.11. Let
$w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{j}}$ be a reduced expression,
$w_{l}:=s_{i_{l}}s_{i_{l+1}}\cdots s_{i_{j}},\ 1\leq l\leq j$, then
$wx{w_{l}}^{-1}\in\Omega_{qr}$. In particular,
$wx\underset{R}{\sim}wx{w_{l}}^{-1}$.

###### Proof.

By Theorem 3.3, $wxw^{-1}\underset{LR}{\sim}x$. Thus $wxw^{-1}\in\Omega_{qr}$.
Since
$wx=wx{w_{l}}^{-1}\cdot{w_{l}}=wxw^{-1}\cdot(w{w_{l}}^{-1})\cdot{w_{l}}$,
according to 2.4(e),
$wx\underset{R}{\leq}wx{w_{l}}^{-1}\underset{R}{\leq}wxw^{-1}$, and $a(wx)\geq
a(wx{w_{l}}^{-1})\geq a(wx{w_{l-1}}^{-1})\geq\cdots\geq a(wxw^{-1})$. Since
$wx,wxw^{-1}\in\Omega_{qr}$, for any $1\leq l\leq j$, we have
$a(wx{w_{l}}^{-1})=a(wx),\ wx{w_{l}}^{-1}\in\Omega_{qr}$. Hence
$wx\underset{R}{\sim}wx{w_{l}}^{-1}$. ∎

5.13. Proof of Theorem 5.7: Let $(W,S)$ be an extended affine Weyl group of
type $\tilde{A}_{n}$, $x$ be a dominant weight in $\Omega_{qr}$. Part (a) of
Conjecture 5.2 for type $\tilde{A}_{n}$ is a special case of Theorem 3.3.

Now we prove part (b) of Conjecture 5.2. According to Theorem 4.5, there
exists $1\leq k\leq n$ such that $x=\prod_{i\in I_{k}}x_{i}^{a_{i}},\ \forall
a_{i}\geq 1$, where $I_{k}=\\{1,\ 2,\ ...,\ n\\}\setminus\\{k\\}$. Assume that
$w,u\in W_{0}$ such that $wxw^{-1}\neq uxu^{-1}$. Since $s_{k}xs_{k}=x$, it is
no harm to assume that $ws_{k}>w$ and $us_{k}>u$. By Corollary 5.12,
$wx\underset{R}{\sim}wxw^{-1}$ and $\ ux\underset{R}{\sim}uxu^{-1}$. If
$wxw^{-1}\underset{R}{\sim}uxu^{-1}$, then $wx\underset{R}{\sim}ux$. According
to Theorem 5.9, $wx$ and $ux$ are in the same right connected set. As a
result, there exists $v\in W$ such that $wx=uxv$. Then $u^{-1}wxv^{-1}=x$.
Recall that $X$ is a normal subgroup of $W=W_{0}\ltimes X$, so $u^{-1}w=v$ and
it commutes with $x$. This forces $v=e$ or $s_{k}$. If $v=s_{k}$, then
$w=us_{k}$, which contradicts the assumption of $ws_{k}>w,\ us_{k}>u$. Thus
$v$ has to be equal to $e$ and $w=u$. This contradicts the assumption
$wxw^{-1}\neq uxu^{-1}$, so we must have
$wxw^{-1}\not\underset{R}{\sim}uxu^{-1}$ if $wxw^{-1}\neq uxu^{-1}$.
Conjecture 5.2 (b) is proved for type $\tilde{A}_{n}$.

By Theorem 14.4.5 in [12] , $\Omega_{qr}$ has $|W_{0}|/2$ right cells. There
are exactly $|W_{0}|/2$ elements in the set $\\{w\in W_{0}|ws_{k}>w\\}$. By
the argument for part (b) of Conjecture 5.2, we see that if $w,u\in W_{0}$,
$ws_{k}>w,\ us_{k}>u$, and $w\neq u$, then
$wxw^{-1}\not\underset{R}{\sim}uxu^{-1}$. Hence any right cell in
$\Omega_{qr}$ contains some $wxw^{-1}$. This proves part (c) of Conjecture 5.2
for type $\tilde{A}_{n}$.

Theorem 5.7 is proved.∎

Now we show that the Conjecture 5.2 is true for the lowest two-sided cell of
an extended affine Weyl group.

Definition 5.14. ([15], section 2) A $\Phi$-tuple
$X=({X_{\alpha}})_{\alpha\in\Phi}$ over the set $\\{+,-,\bigcirc\\}$ is called
a sign type of type $\Phi$ if the set $\\{{X_{\alpha}},{X_{-\alpha}}\\}$ is
either $\\{\bigcirc,\bigcirc\\}$ or $\\{+,-\\}$ for any $\alpha\in\Phi$. We
see that a sign type $({X_{\alpha}})_{\alpha\in\Phi}$ is entirely determined
by the $\Phi^{+}$ -tuple. We shall identify
$({X_{\alpha}})_{\alpha\in\Phi^{+}}$ with $({X_{\alpha}})_{\alpha\in\Phi}$ and
call $({X_{\alpha}})_{\alpha\in\Phi^{+}}$ a sign type. Let
$\bar{\mathscr{S}}=\bar{\mathscr{S}}(\Phi)$ be the set of all sign types of
type $\Phi$. Let

$\mathcal{G}_{1}=\left\\{\begin{matrix}+&+&+&\bigcirc&-&+&+&\bigcirc&\bigcirc&-\\\
+\;+,&+\;\bigcirc,&+\;-,&+\;-,&+\;-,&\bigcirc\;+,&\bigcirc\;\bigcirc,&\bigcirc\;\bigcirc,&\bigcirc\;-,&\bigcirc\;-,\\\
\end{matrix}\right.\\\ \left.\begin{matrix}+&\bigcirc&-&\bigcirc&-&-\\\
-\;+,&-\;+,&-\;+,&-\;\bigcirc,&-\;\bigcirc,&-\;-\\\ \end{matrix}\right\\}.$
(14)

$\mathcal{G}_{2}=\left\\{\begin{matrix}\bigcirc&\bigcirc&\bigcirc&-&-&-&+&\bigcirc&\bigcirc&\bigcirc\\\
\bigcirc\;\bigcirc&+\;\bigcirc&+\;\bigcirc&\bigcirc\;\bigcirc&-\;\bigcirc&-\;\bigcirc&+\;\bigcirc&\bigcirc\;-&\bigcirc\;-&+\;-\\\
\bigcirc,&\bigcirc,&+,&\bigcirc,&\bigcirc,&-,&+,&\bigcirc,&-,&\bigcirc,\\\
\end{matrix}\right.\\\
\qquad\qquad\qquad\qquad\qquad\left.\begin{matrix}\bigcirc&+&+&-&+&+&+&-&-\\\
-\;-&\bigcirc\;-&+\;-&-\;-&-\;-&+\;-&+\;-&\bigcirc\;+&\bigcirc\;+\\\
-,&-,&\bigcirc,&-,&-,&-,&+,&\bigcirc,&+,\\\ \end{matrix}\right.\\\
\left.\begin{matrix}-&\bigcirc&-&-&-&+\\\ -\;+&+\;+&+\;+&-\;+&-\;+&+\;+\\\
\bigcirc,&+,&+,&+,&-,&+\\\ \end{matrix}\right\\}.$ (15)

For any subsystem $\Phi^{\prime}$ of $\Phi$,
$\Phi^{\prime+}=\Phi^{+}\cap\Phi^{\prime}$ is a positive subsystem of $\Phi$.

Given an indecomposable positive subsystem $\Phi^{\prime+}$ of $\Phi$ of rank
2, we say that a sign type $({X_{\alpha}})_{\alpha\in\Phi^{\prime+}}$ is
admissible if we have one of the following cases.

(1) $\Phi^{\prime+}$ has type $A_{2}$, say
$\Phi^{\prime+}=\\{\alpha,\beta,\alpha+\beta\\}$. Then

$X_{\alpha+\beta}$

$X_{\alpha}\qquad X_{\beta}$

belongs to $\mathcal{G}_{1}$.

(2) $\Phi^{\prime+}$ has type $B_{2}$, say
$\Phi^{\prime+}=\\{\alpha,\beta,\alpha+\beta,2\alpha+\beta\\}$. Then

$X_{\alpha+\beta}$

$X_{\alpha}\qquad\qquad X_{\beta}$

$X_{2\alpha+\beta}$

belongs to $\mathcal{G}_{2}$.

We say that a sign type $({X_{\alpha}})_{\alpha\in\Phi}$ is admisssible if for
any indecomposable positive subsystem $\Phi^{{}^{\prime}+}$ of $\Phi$ of rank
2, the sign type $({X_{\alpha}})_{\alpha\in\Phi^{{}^{\prime}+}}$ is
admissible. Let $\mathscr{S}=\mathscr{S}(\Phi)$ be the set of all admissible
sign types of $\bar{\mathscr{S}}$.

Define a map $\zeta:{W_{a}}\longrightarrow\mathscr{S}$ by sending
$w=(k(w,\alpha))_{\alpha\in\Phi}$ to ${X_{w}}=(X_{\alpha})_{\alpha\in\Phi}$
such that for any $\alpha\in\Phi$,

$\displaystyle k(w,\alpha)>0\Leftrightarrow X_{\alpha}=+,$ (16) $\displaystyle
k(w,\alpha)=0\Leftrightarrow X_{\alpha}=\bigcirc,$ (17) $\displaystyle
k(w,\alpha)<0\Leftrightarrow X_{\alpha}=-.$ (18)

Theorem 5.15. ([15], Theorem 2.1) $\zeta({W_{a}})=\mathscr{S}(\Phi)$ for any
indecomposable root system $\Phi$.

Fact: The elements in $W_{a}$ having the same sigh type form an equivalence
class of $W$, we call it an ST-class of $W_{a}$.

$\bf{Theorem5.16.}$ Let $x\in W_{(\nu)}$ be a translation. Then

(a) for any $w\in W_{0}$, $wxw^{-1}\in W_{(\nu)}$;

(b) if $w,u\in W_{0}$, and $w\neq u$,
$wxw^{-1}\not\underset{R}{\sim}uxu^{-1}$;

(c) for any right cell $\Gamma$ in $W_{(\nu)}$, there must exist some $w\in
W_{0}$ such that $wxw^{-1}\in\Gamma$.

###### Proof.

(a) follows from Theorem 3.8.

In [15] Corollary 1.2, Shi proved each left cell of $W_{(\nu)}$ is an ST-class
of $W$. For different elements $w$ and $u$ in $W_{0}$, there must exist some
$\alpha\in\Phi$, such that $w^{-1}(\alpha)\in\Phi^{+}$, and
$u^{-1}(\alpha)\in\Phi^{-}$. We may assume that $x$ is dominant. Then
$k(wx^{-1}w^{-1},\alpha)=\langle x^{-1},(w^{-1}(\alpha))^{\vee}\rangle<0$, and
$k(ux^{-1}u^{-1},\alpha)=\langle x^{-1},(u^{-1}(\alpha))^{\vee}\rangle>0$.
Hence $wxw^{-1}\not\underset{L}{\sim}ux^{-1}u^{-1}$, equivalently,
$wxw^{-1}\not\underset{R}{\sim}uxu^{-1}$. (b) is proved.

In [15], Shi proved that $W_{(\nu)}$ contains exactly $|W_{0}|$ left(
respectively, right) cells. And (b) shows that for a fixed translation $x\in
W_{(\nu)}$, the $|W_{0}|$ elements $wxw^{-1}$ with $w\in W_{0}$ are in
different right cells, therefore, (c) is true. ∎

## Acknowledgement

I would like to thank Professor Nanhua Xi for his supervision on this paper.

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