# Representations of Coxeter groups and homology of Coxeter graphs

Hongsheng Hu

###### Abstract

We classify a class of complex representations of an arbitrary Coxeter group
via characters of the integral homology of certain graphs. Such
representations can be viewed as a generalization of the geometric
representation and correspond to the second-highest 2-sided cell in the sense
of Kazhdan-Lusztig. We also give a description of the cell representation
provided by this 2-sided cell, and find out all its simple quotients for
simply laced Coxeter system with no more than one circuit in the Coxeter
graph.

††footnotetext: Keywords: Coxeter groups, IR-representations, homology of
graphs††footnotetext: MSC 2020: 20C15, 20F55

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on dihedral groups and graphs
  3. 3 Backgrounds on Kazhdan-Lusztig cells
  4. 4 Classification of IR-representations
  5. 5 Reducibility and R-representations
  6. 6 Additional properties
  7. 7 General situation: allowing $m_{st}=\infty$
  8. 8 Examples: from $\widetilde{\mathsf{A}}_{n}$ to simply laced type

## §1 Introduction

Let $W=\langle S\mid s^{2}=(st)^{m_{st}}=e,\forall s\neq t\in S\rangle$ be a
Coxeter group of finite rank. Weyl groups and affine Weyl groups are important
examples, playing crucial roles in Lie theory and representation theory. The
representations of (affine) Weyl groups are well understood, while for general
Coxeter groups they are still to be explored.

In the remarkable paper [5], Kazhdan and Lusztig introduced the notion of
2-sided cells for arbitrary Coxeter system $(W,S)$. Each 2-sided cell gives a
representation of $W$, called a cell representation. When Lusztig’s
$\boldsymbol{a}$-function has a uniform bound on $W$, it is well known that
any irreducible representation $V$ corresponds to a unique 2-sided cell
$\mathcal{C}$ in the sense that $V$ is a quotient of the cell representation
provided by $\mathcal{C}$. We recollect these notions briefly in §3.
Incidentally, we obtain a simple proof of the existence of the lowest 2-sided
cell with the boundedness assumption on the function $\boldsymbol{a}$.

On the other hand, elements with $\boldsymbol{a}$-function value $1$ in an
irreducible Coxeter system (i.e. a Coxeter system with connected Coxeter
graph) form a 2-sided cell, denoted by $\mathcal{C}_{1}$. One of the results
in this article is a necessary and sufficient condition for a nontrivial
irreducible representation $V$ to correspond to $\mathcal{C}_{1}$:

$\begin{split}&\text{there is no $v\in V\setminus\\{0\\}$, such that $r\cdot
v=t\cdot v=-v$}\\\ &\text{for some $r\neq t\in S$ with
$m_{rt}<\infty$.}\end{split}$ (A1)

See 3.12 for details. In particular, the geometric representation
$V_{\text{geom}}$ satisfies (A1).

There is an interesting class of representations “between” $V_{\text{geom}}$
and the ones satisfying (A1), namely, the representations $V$ satisfying the
following:

$\begin{split}&\text{$V$ has a basis $\\{\alpha_{s}\mid s\in S\\}$, and for
any $s\in S$ there is a subspace}\\\ &\text{$H_{s}$ of codimension $1$ such
that $s|_{H_{s}}=\operatorname{Id}_{H_{s}}$, and
$s\cdot\alpha_{s}=-\alpha_{s}$.}\end{split}$ (IR)

Let’s refer to such representations as _IR-representations_. The main aim of
§4 is to present a classification of IR-representations assuming
$m_{st}<\infty$ for any pair $s,t\in S$. It turns out that IR-representations
can be classified by some sets of natural numbers and one dimensional
representations of the first homology groups of certain graphs.

In §5, we investigate the reducibility of IR-representations, which is encoded
in a matrix defined by (5.5). Such a criterion is similar to that of the
geometric representation. As an application, we determine all semisimple
R-representations, which are defined to be a slightly larger class than IR-
representations. In §8, we will see that if $(W,S)$ is simply laced (i.e.
$m_{st}=2$ or $3$, $\forall s,t\in S$) and there is no more than one circuit
in its Coxeter graph, all irreducible representations of $W$ corresponding to
$\mathcal{C}_{1}$ are R-representations, thus we have found all irreducible
quotients of the cell representation provided by $\mathcal{C}_{1}$ for such
Coxeter groups.

Further we present some odds and ends in §6. Proposition 6.1 tells us when an
IR-representation has a $W$-invariant bilinear (or sesquilinear) form, and 6.8
discusses the dual of an IR-representation.

The story is similar when we are in the general setting. In §7, we drop the
assumption $m_{st}<\infty$ and present the classification of IR-
representations.

In a sequel paper [3], we use similar methods to construct some infinite
dimensional irreducible representations of a large class of Coxeter groups,
suggesting that the representation theory of general Coxeter groups is quite
different from finite and affine counterparts.

Acknowledgement The author would like to thank professor Nanhua Xi and Tao Gui
for useful discussions. The author is also grateful to professor Si’an Nie for
pointing out an error of 8.3 in a previous draft.

## §2 Preliminaries on dihedral groups and graphs

In the classification of IR-representations, we will investigate the way of
“gluing” representations of dihedral groups, using certain graphs. For
convenience, let’s recall in this section the representations of dihedral
groups and integral homology of graphs. The base field in this article is
assumed to be $\mathbb{C}$ unless otherwise specified. We use $e$ to denote
the unity in a group.

Representations of finite dihedral groups Let $D_{m}:=\langle r,t\mid
r^{2}=t^{2}=(rt)^{m}=e\rangle$ be a finite dihedral group, let
$V_{r,t}:=\mathbb{C}\beta_{r}\oplus\mathbb{C}\beta_{t}$. For natural numbers
$1\leq k\leq m/2$, define the action
$\rho_{k}:D_{m}\to\operatorname{GL}(V_{r,t})$ by

$\displaystyle r\cdot\beta_{r}$ $\displaystyle=-\beta_{r},$ $\displaystyle
r\cdot\beta_{t}$ $\displaystyle=\beta_{t}+2\cos\frac{k\uppi}{m}\beta_{r},$
$\displaystyle t\cdot\beta_{t}$ $\displaystyle=-\beta_{t},$ $\displaystyle
t\cdot\beta_{r}$ $\displaystyle=\beta_{r}+2\cos\frac{k\uppi}{m}\beta_{t}.$

Intuitively, $D_{m}$ acts on the plane via two reflections, the two reflection
axis forming an angle of $k\uppi/m$, see Figure 1.

$\beta_{r}$$\beta_{t}$$r$$t$$\frac{k\uppi}{m}$ Figure 1:
$\rho_{k}:D_{m}\to\operatorname{GL}(V_{r,t})$

If $k<m/2$, $\rho_{k}$ is irreducible. If $m$ is even and $k=m/2$, then
$\rho_{k}=\varepsilon_{r}\oplus\varepsilon_{t}$ splits into a direct sum of
two representations of dimension $1$, where

$\varepsilon_{r}:r\mapsto-1,t\mapsto 1;\quad\quad\varepsilon_{t}:r\mapsto
1,t\mapsto-1.$ (2.1)

Denote by $\mathds{1}$ the trivial representation, by $\varepsilon$ the sign
representation, i.e.

$\varepsilon:r\mapsto-1,t\mapsto-1.$ (2.2)

######  Lemma 2.3.

All irreducible representations of $D_{m}$ are

$\displaystyle\\{\mathds{1},\varepsilon,\rho_{1},\dots,\rho_{\frac{m-1}{2}}\\},\quad\text{if
$m$ is odd};$
$\displaystyle\\{\mathds{1},\varepsilon,\varepsilon_{r},\varepsilon_{t},\rho_{1},\dots,\rho_{\frac{m}{2}-1}\\},\quad\text{if
$m$ is even}.$

######  Remark 2.4.

The $+1$-eigenspaces of $r$ and $t$ in $\rho_{k}$ are both one dimensional.
However, any nonzero vector cannot be fixed by $r$ and $t$ simultaneously.

######  Remark 2.5.

By Schur’s lemma, when $k<m/2$, an endomorphism of $\rho_{k}$ must be a
scalar. This will be frequently used in the following sections.

######  Remark 2.6.

If $d>1$ is a common divisor of $k$ and $m$, then $\rho_{k}$ factors through
$D_{\frac{m}{d}}$:

$\textstyle{D_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{k}}$$\textstyle{\operatorname{GL}(V_{r,t})}$$\textstyle{D_{\frac{m}{d}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\frac{k}{d}}}$

######  Lemma 2.7.

When $1\leq k<m/2$, there exists a bilinear form $\operatorname{B}$ on
$V_{r,t}$, unique up to a $\mathbb{C}$-scalar, invariant under the action of
$D_{m}$ via $\rho_{k}$:

$\operatorname{B}(\beta_{r},\beta_{r})=\operatorname{B}(\beta_{t},\beta_{t})=1,\quad\operatorname{B}(\beta_{r},\beta_{t})=\operatorname{B}(\beta_{t},\beta_{r})=-\cos\frac{k\uppi}{m}.$
(2.8)

In particular, $\operatorname{B}$ is symmetric.

Let $\operatorname{H}:V_{r,t}\times V_{r,t}\to\mathbb{C}$ be the sesquilinear
form on $V_{r,t}$ defined by (2.8), then it is invariant under the action
$\rho_{k}$ of $D_{m}$. If $\operatorname{H}^{\prime}$ is another one invariant
under $\rho_{k}$, then $\operatorname{H}^{\prime}$ is a
$\mathbb{C}^{\times}$-scalar multiple of $\operatorname{H}$. Further,
$\operatorname{H}^{\prime}$ is Hermitian if and only if it is a
$\mathbb{R}^{\times}$-scalar multiple of $\operatorname{H}$.

Representations of the Infinite dihedral group Let $D_{\infty}:=\langle
r,t\mid r^{2}=t^{2}=e\rangle$ be the infinite dihedral group. There are four
representations of dimension 1, namely,
$\mathds{1},\varepsilon,\varepsilon_{r},\varepsilon_{t}$ (defined in the same
way as in (2.1), (2.2)).

Let $\varrho_{x,y}:D_{\infty}\to\operatorname{GL}(V_{r,t})$ be the
representation on $V_{r,t}$ defined by

$\displaystyle r\cdot\beta_{r}$ $\displaystyle=-\beta_{r},$ $\displaystyle
r\cdot\beta_{t}$ $\displaystyle=\beta_{t}+x\beta_{r},$ $\displaystyle
t\cdot\beta_{t}$ $\displaystyle=-\beta_{t},$ $\displaystyle t\cdot\beta_{r}$
$\displaystyle=\beta_{r}+y\beta_{t}.$

where $x,y\in\mathbb{C}$. To distinguish from $\rho_{k}$’s above, we use the
variation symbol $\varrho$.

######  Lemma 2.9.

Suppose $x,y,x^{\prime},y^{\prime}\in\mathbb{C}^{\times}$.

  1. (1)

$\varrho_{x,y}\simeq\varrho_{x^{\prime},y^{\prime}}$ if and only if
$xy=x^{\prime}y^{\prime}$.

  2. (2)

$\varrho_{x,y}$ is irreducible if and only if $xy\neq 4$.

To simplify notation and eliminate ambiguity, for any $z\in\mathbb{C}$, choose
and fix $u=u(z)\in\mathbb{C}$ such that $u^{2}=z$, and if $z\in\mathbb{R}^{+}$
then $u$ is chosen to be positive. Write $\varrho_{z}:=\varrho_{u,u}$. When
$z=4\cos^{2}\frac{k\uppi}{m}$ for some $k,m\in\mathbb{N},1\leq k<\frac{m}{2}$,
$\varrho_{z}$ factors through $D_{m}$. When $z=4$, $\varrho_{4}=\varrho_{2,2}$
is the geometric representation. When $z=0$,
$\varrho_{0}=\varrho_{0,0}\simeq\varepsilon_{r}\oplus\varepsilon_{t}$.

Besides, denote $\varrho_{r}^{t}:=\varrho_{1,0}$ and
$\varrho_{t}^{r}:=\varrho_{0,1}$, then $\varrho_{r}^{t},\varrho_{t}^{r}$ are
both indecomposable. $\varrho_{r}^{t}$ has a sub-representation
$\varepsilon_{r}$ with quotient $\varepsilon_{t}$, while $\varrho_{t}^{r}$ has
a sub-representation $\varepsilon_{t}$ with quotient $\varepsilon_{r}$.

There is an “exotic” representation (which we will not use in this article) of
$D_{\infty}$ defined by
$\varrho_{\varepsilon}^{\mathds{1}}:r\mapsto(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}),t\mapsto(\begin{smallmatrix}-1&1\\\
0&1\end{smallmatrix})$. It has $\varepsilon$ as its sub-representation with
quotient $\mathds{1}$.

######  Lemma 2.10.

Above are all indecomposable representations of $D_{\infty}$ of dimension $1$
or $2$ (up to isomorphism), listed as

$\\{\mathds{1},\varepsilon,\varepsilon_{r},\varepsilon_{t},\varrho_{r}^{t},\varrho_{t}^{r},\varrho_{\varepsilon}^{\mathds{1}}\\}\cup\\{\varrho_{z}\mid
z\in\mathbb{C}^{\times}\\}.$

All irreducible representations of $D_{m}$ are included, i.e.

$\\{\mathds{1},\varepsilon,\varepsilon_{r},\varepsilon_{t}\\}\cup\\{\varrho_{z}\mid
z\in\mathbb{C}^{\times}\setminus\\{4\\}\\}.$

######  Lemma 2.11.

Let $\varrho$ be an indecomposable representation of $D_{\infty}$ of dimension
2.

  1. (3)

an endomorphism of $\varrho$ must be a scalar;

  2. (4)

there is a $D_{\infty}$-invariant bilinear form $\operatorname{B}$ on
$\varrho$, unique up to a scalar. The representing matrix of
$\operatorname{B}$ (with respect to the basis $(\beta_{r},\beta_{t})$) is

$\begin{pmatrix}1&-\frac{u}{2}\\\ -\frac{u}{2}&1\end{pmatrix},\text{if
}\varrho=\varrho_{z},z\neq 0;\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},\text{if
}\varrho=\varrho_{t}^{r};\begin{pmatrix}0&0\\\ 0&1\end{pmatrix},\text{if
}\varrho=\varrho_{r}^{t}\text{ or }\varrho_{\varepsilon}^{\mathds{1}}.$

Graphs and their homology By definition, a _(undirected) graph_ $G=(S,E)$
consists of a set $S$ of vertices and a set $E$ of edges, elements in $E$
being of the form $\\{s,t\\}$ (unordered), where $s,t\in S$.

For our purpose, we only consider finite graphs without loops and multiple
edges, i.e. there are no edges of the form $\\{s,s\\}$, each pair $\\{s,t\\}$
occurs at most once in $E$, and $S$ is a finite set. Such a graph can be
viewed as a finite simplicial complex in a natural way: vertices as
$0$-simplices, edges as $1$-simplices. A _connected component_ of the graph
$G$ is defined to be that of this simplicial complex.

The chain complex (with coefficient $\mathbb{Z}$) of $G$ looks like:

$0\to\operatorname{C}_{1}(G)\xrightarrow{\partial}\operatorname{C}_{0}(G)\to
0.$

The homology group $\operatorname{H}_{1}(G)=\ker\partial$ is a finitely
generated free abelian group since $\operatorname{C}_{1}(G)$ is such a group.

Call $(s_{1},s_{2},\dots,s_{n})$ a _path_ in $G$ if $\\{s_{i}\\}_{i}\subseteq
S$ and $\\{s_{i},s_{i+1}\\}\in E,\forall i$. If $s_{1}=s_{n}$, say such a path
is a _closed path_. We don’t distinguish the closed paths
$(s_{1},\dots,s_{n-1},s_{1})$ and $(s_{2},\dots,s_{n-1},s_{1},s_{2})$.
Further, if $s_{1},\dots,s_{n-1}$ are distinct in this closed path, call it a
_circuit_.

For a connected graph $G$, a _spanning tree_ is defined to be a subgraph
$T=(S,E_{0})$ with the same vertex set $S$ and an edge set $E_{0}\subseteq E$,
such that $T$ is connected, and there are no circuits in $T$. For general $G$,
choosing a spanning tree for each component of $G$, their union is called a
_spanning forest_ of $G$. Spanning forests always exist, but are not unique in
general.

Fix a spanning forest $G_{0}=(S,E_{0})$ of $G$. For any edge $\mathfrak{e}\in
E\setminus E_{0}$, fixing an orientation of $\mathfrak{e}$, there is a unique
circuit in $(S,E_{0}\cup\\{\mathfrak{e}\\})$ compatible with the orientation
of $\mathfrak{e}$, denoted by $c_{\mathfrak{e}}$, representing an element in
$\operatorname{H}_{1}(G)$.

######  Lemma 2.12.

$\operatorname{H}_{1}(G)=\bigoplus_{\mathfrak{e}\in E\setminus
E_{0}}\mathbb{Z}c_{\mathfrak{e}}$.

In a Coxeter graph $G$, the number $m_{st}$ is regarded as a label on the edge
$\\{s,t\\}$, rather than a multiplicity. Denote again by
$\operatorname{H}_{1}(G)$ the first integral homology of $G$ forgetting labels
on its edges.

## §3 Backgrounds on Kazhdan-Lusztig cells

This section explains partial motivations for considering IR-representations.
Fixing a Coxeter system $(W,S)$ of finite rank, let
$\mathcal{H}:=\mathcal{H}(W,S)$ be a free
$\mathbb{C}[q^{\pm\frac{1}{2}}]$-module with basis
$\\{\widetilde{T}_{w}\\}_{w\in W}$. Define a multiplication on $\mathcal{H}$
by

$\widetilde{T}_{s}\widetilde{T}_{w}:=\begin{cases}\widetilde{T}_{sw},&\mbox{if
}sw>w,\\\
(q^{\frac{1}{2}}-q^{-\frac{1}{2}})\widetilde{T}_{w}+\widetilde{T}_{sw},&\mbox{if
}sw<w,\end{cases}\quad\forall s\in S,w\in W.$

Then $\mathcal{H}$ forms an associative algebra with unity
$\widetilde{T}_{e}$, called the _Hecke algebra_ of $(W,S)$. There is an
involutive $\mathbb{C}$-algebra homomorphism on $\mathcal{H}$ defined by

$\overline{\phantom{a}}:q^{\frac{1}{2}}\mapsto
q^{-\frac{1}{2}},\quad\widetilde{T}_{w}\mapsto\widetilde{T}_{w^{-1}}^{-1}.$

######  Theorem 3.1.

([5, Theorem 1.1]) There is a unique basis $\\{C_{w}\\}_{w\in W}$ of
$\mathcal{H}$,

$C_{w}=\sum_{y\in
W}(-1)^{\ell(w)+\ell(y)}q^{\frac{1}{2}(\ell(w)-\ell(y))}\overline{P_{y,w}}\widetilde{T}_{y},$

such that $\forall y,w\in W$, $\overline{C_{w}}=C_{w}$,
$P_{y,w}\in\mathbb{Z}[q]$, $P_{y,w}=0$ unless $y\leq w$, $P_{w,w}=1$, and if
$y<w$, then $\deg_{q}P_{y,w}\leq\frac{1}{2}(\ell(w)-\ell(y)-1)$.

######  Examples 3.2.

  1. (1)

$C_{e}=\widetilde{T}_{e}$; for $s\in S$,
$C_{s}=\widetilde{T}_{s}-q^{\frac{1}{2}}$.

  2. (2)

If $r\neq t\in S$ with $m_{rt}=m<\infty$, write for simplicity
$r_{k}:=rtr\cdots$ (product of $k$ factors), $t_{k}=trt\cdots$ ($k$ factors),
and $w_{rt}:=r_{m}=t_{m}$. Then

$C_{w_{rt}}=\widetilde{T}_{w_{rt}}-q^{\frac{1}{2}}(\widetilde{T}_{r_{m-1}}+\widetilde{T}_{t_{m-1}})+\dots+(-1)^{i}q^{\frac{i}{2}}(\widetilde{T}_{r_{m-i}}+\widetilde{T}_{t_{m-i}})+\dots+(-1)^{m}q^{\frac{m}{2}}.$
(3.3)

  3. (3)

$P_{y,w}(0)=1$ for all $y\leq w$.

For any $x,y,w\in W$, let $h_{x,y,w}\in{\mathbb{C}[q^{\pm\frac{1}{2}}]}$ such
that $C_{x}C_{y}=\sum_{w\in W}h_{x,y,w}C_{w}$. The following result is useful
but highly nontrivial.

######  Lemma 3.4.

([2, Corollary 1.2]) $\forall x,y,w\in W$, it holds $P_{y,w}\in\mathbb{N}[q]$.
Write $h_{x,y,w}=\sum_{i\in\mathbb{Z}}c_{i}q^{\frac{i}{2}}$, then
$(-1)^{i}c_{i}\in\mathbb{N}$.

For $x,y\in W$, say $y\underset{\mathrm{LR}}{\leqslant}x$ if $\exists
H_{1},H_{2}\in\mathcal{H}$, such that $C_{y}$ has nonzero coefficient in the
expression of $H_{1}C_{x}H_{2}$ with respect to the basis $\\{C_{w}\\}_{w\in
W}$. Say $x\underset{\mathrm{LR}}{\sim}y$ if
$x\underset{\mathrm{LR}}{\leqslant}y$ and
$y\underset{\mathrm{LR}}{\leqslant}x$. It holds that
$\underset{\mathrm{LR}}{\leqslant}$ is a pre-order on $W$, and
$\underset{\mathrm{LR}}{\sim}$ is an equivalence relation on $W$. The
equivalence classes are called _2-sided cells_.

Let $\boldsymbol{a}(w):=\min\\{i\in\mathbb{N}\mid
q^{\frac{i}{2}}h_{x,y,w}\in\mathbb{Z}[q^{\frac{1}{2}}],\forall x,y\in W\\}$,
then $\boldsymbol{a}(w)$ is a well-defined natural number. Say the function
$\boldsymbol{a}$ is _bounded_ if there is $N\in\mathbb{N}$ such that
$\boldsymbol{a}(w)\leq N$ for any $w\in W$.

######  Proposition 3.5.

([7, 8, 9])

  1. (1)

Let $s\in S$, $w\in W$, suppose $sw>w$, then
$C_{s}C_{w}=C_{sw}+\sum\mu_{y,w}C_{y}$ where the sum runs over all $y<w$ such
that $\deg_{q}P_{y,w}=\frac{1}{2}(\ell(w)-\ell(y)-1)$ and $sy<y$, and
$\mu_{y,w}$ is the coefficient of the top-degree term in $P_{y,w}$. The
formula is similar for the case $ws>w$. In particular,
$sw\underset{\mathrm{LR}}{\leqslant}w$ if $sw>w$,
$ws\underset{\mathrm{LR}}{\leqslant}w$ if $ws>w$.

  2. (2)

$\boldsymbol{a}(w)=0$ if and only if $w=e$. For any $s\in S$,
$\boldsymbol{a}(s)=1$.

  3. (3)

If $m_{rt}<\infty$ for some $r\neq t\in S$, let $w=rtr\cdots$ ($m_{rt}$
factors), then $\boldsymbol{a}(w)=m_{rt}$.

  4. (4)

If $y\underset{\mathrm{LR}}{\leqslant}x$, then
$\boldsymbol{a}(y)\geq\boldsymbol{a}(x)$. Thus
$y\underset{\mathrm{LR}}{\sim}x$ yields $\boldsymbol{a}(y)=\boldsymbol{a}(x)$,
and $\boldsymbol{a}(\mathcal{C})$ makes sense for any 2-sided cell
$\mathcal{C}$.

  5. (5)

Suppose $\boldsymbol{a}$ is bounded. If $\boldsymbol{a}(y)=\boldsymbol{a}(x)$
and $y\underset{\mathrm{LR}}{\leqslant}x$, then
$y\underset{\mathrm{LR}}{\sim}x$.

Let $\mathcal{C}$ be a 2-sided cell, define $\mathcal{J}_{\mathcal{C}}$ to be
a free ${\mathbb{C}[q^{\pm\frac{1}{2}}]}$-module with a basis $\\{J_{w}\mid
w\in\mathcal{C}\\}$. The following formula defines an $\mathcal{H}$-module
structure on $\mathcal{J}_{\mathcal{C}}$,

$C_{x}\cdot J_{y}:=\sum_{w\in\mathcal{C}}h_{x,y,w}J_{w},\quad\forall x\in
W,y\in\mathcal{C}.$

By convention, a _representation_ of $\mathcal{H}$ indicates a $\mathbb{C}$
vector space with an $\mathcal{H}$-module structure, where $q^{\frac{1}{2}}$
acts by some $\mathbb{C}^{\times}$-scalar. The following proposition is due to
Lusztig. We present a proof here briefly.

######  Proposition 3.6.

Suppose the function $\boldsymbol{a}$ is bounded, and $V\neq 0$ is an
irreducible representation of $\mathcal{H}$, then there is a unique 2-sided
cell $\mathcal{C}$, such that

  1. (1)

for any $w\in W$, if $C_{w}\cdot V\neq 0$, then it holds
$\mathcal{C}\underset{\mathrm{LR}}{\leqslant}w$;

  2. (2)

there is an element $w\in\mathcal{C}$ such that $C_{w}\cdot V\neq 0$.

In other words, $V$ is a simple quotient of $\mathcal{J}_{\mathcal{C}}$.

###### Proof:.

By 3.5(5), there exists a 2-sided cell $\mathcal{C}$ satisfying

$\exists w\in\mathcal{C}$, such that $C_{w}\cdot V\neq 0$; $\forall
y\underset{\mathrm{LR}}{\leqslant}\mathcal{C}$ and $y\notin\mathcal{C}$,
$C_{y}\cdot V=0$. (3.7)

Take $v\in V$, such that $C_{w}\cdot v\neq 0$ for some $w\in\mathcal{C}$, then
the map $\varphi:\mathcal{J}_{\mathcal{C}}\to V$, $J_{y}\mapsto C_{y}\cdot v$
is a well-defined surjective homomorphism of $\mathcal{H}$-modules. Thus
$V\simeq\mathcal{J}_{\mathcal{C}}/\ker\varphi$. Suppose $\mathcal{C}^{\prime}$
is another 2-sided cell, and $\exists x\in\mathcal{C}^{\prime}$ such that
$C_{x}\cdot V\neq 0$, then there exists $y\in\mathcal{C}$, such that
$C_{x}\cdot J_{y}\neq 0$, i.e. there is $z\in\mathcal{C}$ such that
$h_{x,y,z}\neq 0$, thus $z\underset{\mathrm{LR}}{\leqslant}x$,
$\mathcal{C}\underset{\mathrm{LR}}{\leqslant}\mathcal{C}^{\prime}$. In
particular, $\mathcal{C}^{\prime}$ doesn’t satisfy (3.7). $\Box$

In the proof, the boundedness of $\boldsymbol{a}$ is only used to find
$\mathcal{C}$ satisfying (3.7). Without this assumption, if such a 2-sided
cell exists, then it is unique.

Immediately we have the following corollary on the lowest 2-sided cell, which
was first studied for affine Weyl groups in [7, 10], and later for general
Coxeter systems in [13] with a slightly stronger assumption.

######  Corollary 3.8.

Suppose $N=\max\\{\boldsymbol{a}(w)\mid w\in W\\}<\infty$, then
$\mathcal{C}_{\text{low}}:=\\{w\in W\mid\boldsymbol{a}(w)=N\\}$ is a 2-sided
cell, and for any $w\in W$ we have
$\mathcal{C}_{\text{low}}\underset{\mathrm{LR}}{\leqslant}w$.

###### Proof:.

Let $\varepsilon:s\mapsto-1,\forall s\in S$ be the sign representation, then
for any $w\in W$, $\varepsilon(C_{w})=(-1)^{\ell(w)}\sum_{y}P_{y,w}(1)$. By
3.2(3) and 3.4, this number is nonzero. Let $\mathcal{C}$ be the 2-sided cell
attached to $\varepsilon$, then 3.6 shows
$\mathcal{C}\underset{\mathrm{LR}}{\leqslant}w$ for any $w$. By 3.5(4), we
have $\mathcal{C}\subseteq\mathcal{C}_{\text{low}}$. The converse inclusion is
deduced from 3.5(5). $\Box$

We shall need the following lemma due to Matsumoto and Tits.

######  Lemma 3.9.

([9, Theorem 1.9]) Let $s_{1}\cdots s_{n}$ and $s_{1}^{\prime}\cdots
s_{n}^{\prime}$ be two reduced expressions of $w$, then we can obtain one from
the other by finite steps of replacement of the form

$\underbrace{rtr\cdots}_{m_{rt}\text{
factors}}=\underbrace{trt\cdots}_{m_{rt}\text{ factors}}\quad(m_{rt}<\infty).$

For irreducible $(W,S)$, there is a 2-sided cell with a simple description,
formulated by Lusztig as following.

######  Lemma 3.10.

  1. (1)

For $w\in W\setminus\\{e\\}$, $\boldsymbol{a}(w)=1$ if and only if $w$ has a
unique reduced expression.

  2. (2)

Let $\mathcal{C}_{1}:=\\{w\in W\mid\boldsymbol{a}(w)=1\\}$. If $(W,S)$ is
irreducible, then $\mathcal{C}_{1}$ is a 2-sided cell. Moreover, for any $x\in
W\setminus\\{e\\}$, we have
$x\underset{\mathrm{LR}}{\leqslant}\mathcal{C}_{1}$.

###### Proof:.

If $w$ has different reduced expressions, we deduce that
$\boldsymbol{a}(w)\geq 2$ from 3.9 and 3.5(1)(3)(4). Conversely, if the
reduced expression of $w$ is unique, suppose $sw<w$, [6, Proposition 3.8]
proved that $w\underset{\mathrm{LR}}{\sim}s$, thus $\boldsymbol{a}(w)=1$ by
3.5(2)(4).

The first part of (2) is proved in [6, Proposition 3.8]. The second part is
deduced from 3.5(1). $\Box$

In general, decompose the Coxeter graph into connected components, say
$S=\sqcup_{i}S_{i}$, then $\mathcal{C}_{1}=\sqcup_{i}\mathcal{C}_{1,i}$ is a
union of 2-sided cells, where $\mathcal{C}_{1,i}=\\{w\in
W\mid\boldsymbol{a}(w)=1,\text{ and $w$ is a product of elements in
$S_{i}$}\\}$.

In the remainder of this section, we specialize $q^{\frac{1}{2}}$ to 1, then
the Hecke algebra $\mathcal{H}$ becomes the group algebra $\mathbb{C}[W]$, and
$\mathcal{J}_{\mathcal{C}}$ becomes a complex representation of $W$. Denote
the image of $C_{w}\in\mathcal{H}$ in $\mathbb{C}[W]$ by $C_{w}$ again.

######  Example 3.11.

Use notations in §2. $D_{m}$ has three 2-sided cells, $\\{e\\}$,
$\\{w_{rt}\\}$ and $\mathcal{C}_{1}$. For $1\leq k<m/2$, $\rho_{k}$ is an
irreducible representation of dimension 2. On this plane, $D_{m}$ consists of
$m$ rotations and $m$ reflections (regard $e$ as the rotation by zero angle).
By (3.3), $\rho_{k}(C_{w_{rt}})$ is the difference of the $m$ rotations and
the $m$ reflections (up to a sign, depends on the parity of $m$). But the sum
of the $m$ rotations is zero, so is the sum of the $m$ reflections, thus
$\rho_{k}(C_{w_{rt}})=0$. One may further check that $C_{w_{rt}}$ acts by zero
on other irreducible representations _except_ $\varepsilon$.

The main result of this section is the following.

######  Theorem 3.12.

Let $V$ be a representation of $W$, then $C_{w}$ acts by zero for all $w\in W$
with $\boldsymbol{a}(w)>1$ if and only if $V$ satisfies the following
condition,

$\begin{split}&\text{there is no $v\in V\setminus\\{0\\}$, such that $r\cdot
v=t\cdot v=-v$}\\\ &\text{for some $r\neq t\in S$ with
$m_{rt}<\infty$.}\end{split}$ (A1)

Call such representations _A1-representations_.

###### Proof:.

Suppose $r\neq t\in S$ with $m_{rt}=m<\infty$, write $w_{rt}=rtr\cdots$ ($m$
factors). Restrict $V$ to the subgroup $D_{m}:=\langle r,t\rangle$, then $V$
(either finite or infinite dimensional) decomposes into irreducible
representations of $D_{m}$. If $v\in V\setminus\\{0\\}$ spans a sign
representation $\varepsilon$, then $C_{w_{rt}}\cdot v\neq 0$ as said in 3.11.
By 3.5(3), $\boldsymbol{a}(w_{rt})=m>1$.

Conversely, suppose $V$ satisfies (A1), then $C_{w_{rt}}\cdot V=0$ for any
pair $r,t$ as above. If $w\in W$, $\boldsymbol{a}(w)>1$, by 3.9 and 3.10,
$w=xw_{rt}y$ for some $r,t$ with $m_{rt}<\infty$, and
$\ell(w)=\ell(x)+m_{rt}+\ell(y)$. Let’s do induction on $\ell(w)$, the
starting point is those elements of the form $w_{rt}$. If $x\neq e$, choose
$s\in S$ such that $sx<x$, then $sw<w$ and $\boldsymbol{a}(sw)\geq 2$. We have
by 3.5(1) $C_{w}=C_{s}C_{sw}-\sum\mu_{z,sw}C_{z}$. In particular, those $z$’s
occurred in the summation satisfy $z\underset{\mathrm{LR}}{\leqslant}sw$, thus
$\boldsymbol{a}(z)\geq\boldsymbol{a}(sw)\geq 2$. Also, note that
$\ell(z)<\ell(w)$, $\ell(sw)<\ell(w)$, by induction hypothesis, $C_{sw}$ and
those $C_{z}$’s act on $V$ by zero, hence so is $C_{w}$. The case $y\neq e$ is
similar. $\Box$

In particular, if $(W,S)$ is irreducible, and $V$ is a nontrivial irreducible
representation of $W$, then it satisfies (A1) if and only if $V$ is a quotient
of $\mathcal{J}_{\mathcal{C}_{1}}$.

## §4 Classification of IR-representations

Let $(W,S)$ be a Coxeter system of finite rank. For simplicity, from here to
§6, we assume:

$m_{rt}<\infty,\quad\forall r,t\in S.$ (4.1)

A representation $V$ of $W$ is said to be an _IR-representation_ if:

$\begin{split}&\text{$V$ has a basis $\\{\alpha_{s}\mid s\in S\\}$, and for
any $s\in S$ there is a subspace}\\\ &\text{$H_{s}$ of codimension $1$ such
that $s|_{H_{s}}=\operatorname{Id}_{H_{s}}$, and
$s\cdot\alpha_{s}=-\alpha_{s}$.}\end{split}$ (IR)

The two letters represent “independent reflections”. Clearly the geometric
representation satisfies (IR), and IR-representations satisfy the condition
(A1).

######  Lemma 4.2.

Let $V$ be an IR-representation. For any pair $r\neq t\in S$,

  1. (1)

$\alpha_{r},\alpha_{t}$ span a sub-representation of the dihedral subgroup
$\langle r,t\rangle$;

  2. (2)

As a representation of the dihedral group $\langle r,t\rangle$, the subspace
$\langle\alpha_{r},\alpha_{t}\rangle$ spanned by $\alpha_{r},\alpha_{t}$ is
isomorphic to $\rho_{k_{rt}}$ for some $1\leq k_{rt}\leq m_{rt}/2$.

###### Proof:.

For each $s\in S$, $V=H_{s}\oplus\mathbb{C}\alpha_{s}$. For any $v\in V$,
write $v=v_{s}+a\alpha_{s}$, where $v_{s}\in H_{s}$, $a\in\mathbb{C}$, then
$s\cdot v=v-2a\alpha_{s}$. (1) is clear. (2) follows from 2.3 and the
assumption (IR). $\Box$

In view of 4.2(2), there exist $a_{r}^{t},a_{t}^{r}\in\mathbb{C}^{\times}$
(but not unique), such that the isomorphism from $\rho_{k_{rt}}$ to
$\langle\alpha_{r},\alpha_{t}\rangle$ sends $\beta_{r}$ to
$a_{r}^{t}\alpha_{r}$ and $\beta_{t}$ to $a_{t}^{r}\alpha_{t}$, i.e.

$r\cdot\alpha_{t}=\alpha_{t}+2\frac{a_{r}^{t}}{a_{t}^{r}}\cos\frac{k_{rt}\uppi}{m_{rt}}\alpha_{r},\quad\quad
t\cdot\alpha_{r}=\alpha_{r}+2\frac{a_{t}^{r}}{a_{r}^{t}}\cos\frac{k_{rt}\uppi}{m_{rt}}\alpha_{t}.$

Thus, the IR-representation is determined by the datum
$(k_{rt},a_{r}^{t})_{r,t\in S,r\neq t}$.

Conversely, given the datum $(k_{rt},a_{r}^{t})_{r,t\in S,r\neq t}$ such that

$k_{rt}\in\mathbb{N},\quad 1\leq k_{rt}=k_{tr}\leq\frac{m_{rt}}{2},\quad
a_{r}^{t}\in\mathbb{C}^{\times},\quad\forall r,t\in S,r\neq t,$

define for each $r\in S$ a linear transformation on the space
$V:=\bigoplus_{s\in S}\mathbb{C}\alpha_{s}$ by:

$r\cdot\alpha_{r}=-\alpha_{r},\quad
r\cdot\alpha_{t}=\alpha_{t}+2\frac{a_{r}^{t}}{a_{t}^{r}}\cos\frac{k_{rt}\uppi}{m_{rt}}\alpha_{r},\quad\forall
t\neq r.$ (4.3)

######  Lemma 4.4.

Formula (4.3) defines an IR-representation structure on $V$.

###### Proof:.

First of all, $r^{2}$ acts by identity. Now consider the action of
$(rt)^{m_{rt}}$. On $\langle\alpha_{r},\alpha_{t}\rangle$, $(rt)^{m_{rt}}$
acts by identity since the two dimensional subspace forms the representation
$\rho_{k_{rt}}$. For $s\neq r,t$ (if exists), write
$U:=\langle\alpha_{s},\alpha_{r},\alpha_{t}\rangle$ be the $3$ dimensional
subspace. By construction, there are vectors $v_{1}$ in
$\langle\alpha_{r},\alpha_{s}\rangle$ with nonzero $\alpha_{s}$-coefficient,
and $v_{2}$ in $\langle\alpha_{r},\alpha_{t}\rangle$ with nonzero
$\alpha_{t}$-coefficient, both staying invariant under $r$. Thus $v_{1}$ and
$v_{2}$ span a $2$ dimensional subspace of $U$. Similarly, there are $v_{3}$
and $v_{4}$ spanning a $2$ dimensional subspace of $U$ staying invariant under
$t$. By dimension reason, there is a nonzero $v\in U$ fixed by $r$ and $t$
simultaneously. By 2.4,
$v\notin\mathbb{C}\alpha_{r}\oplus\mathbb{C}\alpha_{t}$. Thus,
$\alpha_{s}=x_{v}v+x_{r}\alpha_{r}+x_{t}\alpha_{t}$ for some
$x_{v},x_{r},x_{t}\in\mathbb{C}$. It follows
$(rt)^{m_{rt}}\cdot\alpha_{s}=\alpha_{s}$. Now we see that $V$ is a
representation of $W$ via the actions defined above.

From the argument, we also see that for each $r\in S$ there are $|S|-1$
linearly independent vectors stay invariant under $r$, thus this
representation satisfies (IR). $\Box$

We need to determine when two different data $(k_{rt},a_{r}^{t})_{r,t}$ and
$(l_{rt},b_{r}^{t})_{r,t}$ define isomorphic representations. Denote by
$V_{1}=\bigoplus_{s\in S}\mathbb{C}\alpha_{s}$ and $V_{2}=\bigoplus_{s\in
S}\mathbb{C}\alpha_{s}^{\prime}$ the two IR-representations defined by the two
data respectively. By 4.2, it obviously holds:

######  Lemma 4.5.

$V_{1}\simeq V_{2}$ only if $k_{rt}=l_{rt}$, $\forall r,t\in S$.

Suppose $V_{1},V_{2}$ are isomorphic via
$\varphi:V_{1}\xrightarrow{\sim}V_{2}$. Define a new Coxeter system
$(\widetilde{W},S)$ with the same generator set $S$, and the order of $rt$
being $\widetilde{m}_{rt}:=m_{rt}/d_{rt}$, where
$d_{rt}:=\gcd(m_{rt},k_{rt})$. Denote by $\widetilde{G}=(S,E)$ the Coxeter
graph of $(\widetilde{W},S)$. We have a canonical homomorphism
$W\twoheadrightarrow\widetilde{W}$. By 2.6, the representations $V_{1}$ and
$V_{2}$ of $W$ factor through $\widetilde{W}$. As an IR-representation of
$\widetilde{W}$, $V_{1}$ is defined by the datum
$(\widetilde{k}_{rt},a_{r}^{t})_{r,t}$, where
$\widetilde{k}_{rt}:=k_{rt}/d_{rt}$. Similar for $V_{2}$.

The chain group $\operatorname{C}_{1}(\widetilde{G})$ is a free abelian group
generated by oriented edges $(r,t)$. The datum $(k_{rt},a_{r}^{t})_{r,t}$
gives a one dimensional representation of
$\operatorname{C}_{1}(\widetilde{G})$ defined by

$(r,t)\mapsto\frac{a_{t}^{r}}{a_{r}^{t}}.$ (4.6)

This restricts to the subgroup $\operatorname{H}_{1}(\widetilde{G})$, denoted
by $\chi_{1}$. Similarly, $(r,t)\mapsto b_{t}^{r}/b_{r}^{t}$ defines another
character of $\operatorname{H}_{1}(\widetilde{G})$, denoted by $\chi_{2}$.

######  Lemma 4.7.

$\chi_{1}=\chi_{2}$.

###### Proof:.

Recall that $\varphi:V_{1}\to V_{2}$ is an isomorphism. Let
$(s_{1},s_{2},\dots,s_{n},s_{1})$ be a circuit in $\widetilde{G}$, write
$\alpha_{i}:=\alpha_{s_{i}}$ and $a_{i}^{j}:=a_{s_{i}}^{s_{j}}$ for
convenience, similar for $m_{ij},\widetilde{m}_{ij},k_{ij},$
$\alpha_{i}^{\prime},b_{i}^{j}$, etc. By assumption,
$\varphi(\alpha_{1})=x\alpha_{1}^{\prime}$ for some $x\in\mathbb{C}^{\times}$.
Since ${s_{1},s_{2}}$ is an edge of $\widetilde{G}$, we have $k_{12}\neq
m_{12}/2$, thus $\langle\alpha_{1},\alpha_{2}\rangle$ and
$\langle\alpha_{1}^{\prime},\alpha_{2}^{\prime}\rangle$ form isomorphic
irreducible representations $\rho_{k_{12}}$ of $\langle s_{1},s_{2}\rangle$.
By Schur’s lemma, we have
$\varphi(\alpha_{2})=x(a_{1}^{2}b_{2}^{1}/a_{2}^{1}b_{1}^{2})\alpha_{2}^{\prime}$.
Do this recursively along the circuit, finally we get

$\varphi(\alpha_{1})=x\frac{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}a_{n}^{1}b_{2}^{1}b_{3}^{2}\cdots
b_{n}^{n-1}b_{1}^{n}}{a_{2}^{1}a_{3}^{2}\cdots
a_{n}^{n-1}a_{1}^{n}b_{1}^{2}b_{2}^{3}\cdots
b_{n-1}^{n}b_{n}^{1}}\alpha_{1}^{\prime}.$

Hence, compared with $\varphi(\alpha_{1})=x\alpha_{1}^{\prime}$,

$\frac{a_{2}^{1}a_{3}^{2}\cdots a_{n}^{n-1}a_{1}^{n}}{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}a_{n}^{1}}=\frac{b_{2}^{1}b_{3}^{2}\cdots
b_{n}^{n-1}b_{1}^{n}}{b_{1}^{2}b_{2}^{3}\cdots b_{n-1}^{n}b_{n}^{1}}.$

The two sides are the images of the circuit under $\chi_{1}$ and $\chi_{2}$
respectively. $\Box$

Fixing those numbers $(k_{rt})_{r,t}$, we have a well defined map
$\Theta=\Theta((k_{rt})_{r,t})$ from the set of isomorphic classes

$\\{\text{IR-representations defined by }(k_{rt},a_{r}^{t})_{r,t}\mid
a_{r}^{t}\in\mathbb{C}^{\times},\forall r,t\in S\\}/\simeq$

to the set $\\{$characters of $\operatorname{H}_{1}(\widetilde{G})\\}$ by 4.7.

######  Lemma 4.8.

$(k_{rt})_{r,t}$ fixed as above, $\Theta$ is surjective.

###### Proof:.

Fix a spanning forest $\widetilde{G}_{0}=(S,E_{0})$ of $\widetilde{G}=(S,E)$.
For any $\mathfrak{e}\in E\setminus E_{0}$, choose an orientation of
$\mathfrak{e}$, let $c_{\mathfrak{e}}$ be defined as in §2, then by 2.12,

$\operatorname{H}_{1}(\widetilde{G})=\bigoplus_{\mathfrak{e}\in E\setminus
E_{0}}\mathbb{Z}c_{\mathfrak{e}}.$

Giving a character $\chi$ of $\operatorname{H}_{1}(\widetilde{G})$ is
equivalent to giving numbers
$\\{x_{\mathfrak{e}}\in\mathbb{C}^{\times}\mid\mathfrak{e}\in E\setminus
E_{0}\\}$, and assign $\chi(c_{\mathfrak{e}})=x_{\mathfrak{e}}$.

For any distinct $r,t\in S$, let

$a_{r}^{t}=\begin{cases}x_{(t,r)},&\mbox{if }\\{t,r\\}\in E\setminus
E_{0},\mbox{ and is oriented as }(t,r),\\\ 1,&\mbox{otherwise.}\\\
\end{cases}$

Then, under $\Theta$, the datum $(k_{rt},a_{r}^{t})_{r,t}$ gives the character
$\chi$. $\Box$

######  Lemma 4.9.

$(k_{rt})_{r,t}$ fixed as above, $\Theta$ is injective.

###### Proof:.

Suppose $\Theta(V_{1})=\Theta(V_{2})$, where $V_{1}=\bigoplus_{s\in
S}\mathbb{C}\alpha_{s}$, $V_{2}=\bigoplus_{s\in
S}\mathbb{C}\alpha_{s}^{\prime}$ are IR-representations defined by two data
$(k_{rt},a_{r}^{t})_{r,t}$ and $(k_{rt},b_{r}^{t})_{r,t}$ respectively. In
arbitrary component of $\widetilde{G}$, choose a vertex $s_{1}$. For any other
vertex $s$ in the same component, choose a path $(s_{1},s_{2},\dots,s_{n}=s)$
connecting $s_{1}$ and $s$. Use notations $\alpha_{i},a_{i}^{j}$, etc as in
the proof of 4.7. Define a linear map $\varphi:V_{1}\to V_{2}$ by

$\displaystyle\alpha_{1}$ $\displaystyle\mapsto\alpha_{1}^{\prime},$
$\displaystyle\alpha_{n}$ $\displaystyle\mapsto\frac{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}b_{2}^{1}b_{3}^{2}\cdots b_{n}^{n-1}}{a_{2}^{1}a_{3}^{2}\cdots
a_{n}^{n-1}b_{1}^{2}b_{2}^{3}\cdots b_{n-1}^{n}}\alpha_{n}^{\prime}.$

We need to show $\varphi$ is well-defined, i.e. $\varphi(\alpha_{n})$ is
independent of the choice of the path (but $\varphi$ does depend on the choice
of $s_{1}$ in each component). For another path connecting $s_{1}$ and
$s_{n}=s$, say $(s_{1},s_{p},s_{p-1},\dots,s_{n+1},s_{n})$, $p\geq n$, the two
paths form a closed path, i.e. $(s_{1},\dots
s_{n},s_{n+1},\dots,s_{p},s_{1})$. Since $\Theta(V_{1})=\Theta(V_{2})$, we
have

$\frac{a_{2}^{1}\cdots a_{n}^{n-1}a_{n+1}^{n}\cdots
a_{p}^{p-1}a_{1}^{p}}{a_{1}^{2}\cdots a_{n-1}^{n}a_{n}^{n+1}\cdots
a_{p-1}^{p}a_{p}^{1}}=\frac{b_{2}^{1}\cdots b_{n}^{n-1}b_{n+1}^{n}\cdots
b_{p}^{p-1}b_{1}^{p}}{b_{1}^{2}\cdots b_{n-1}^{n}b_{n}^{n+1}\cdots
b_{p-1}^{p}b_{p}^{1}}.$

Rewrite the equation,

$\frac{a_{1}^{2}a_{2}^{3}\cdots a_{n-1}^{n}b_{2}^{1}b_{3}^{2}\cdots
b_{n}^{n-1}}{a_{2}^{1}a_{3}^{2}\cdots a_{n}^{n-1}b_{1}^{2}b_{2}^{3}\cdots
b_{n-1}^{n}}=\frac{a_{1}^{p}a_{p}^{p-1}\cdots
a_{n+1}^{n}b_{p}^{1}b_{p-1}^{p}\cdots b_{n}^{n+1}}{a_{p}^{1}a_{p-1}^{p}\cdots
a_{n}^{n+1}b_{1}^{p}b_{p}^{p-1}\cdots b_{n+1}^{n}}.$

This indicates that $\varphi(\alpha_{n})$ is independent of the choice of the
path.

Clearly, $\varphi$ is a linear isomorphism. It remains to verify that
$\varphi$ is a homomorphism of representations. It suffices to check
$\varphi(s\cdot\alpha_{t})=s\cdot\varphi(\alpha_{t}),\forall s,t\in S$. If
$s=t$, this is obvious by definition. If $s$ and $t$ are distinct and not
adjacent in $\widetilde{G}$, i.e. $\widetilde{m}_{st}=2$, then
$s\cdot\alpha_{t}=\alpha_{t}$ and
$s\cdot\alpha_{t}^{\prime}=\alpha_{t}^{\prime}$, the equation also holds. Now
assume $\widetilde{m}_{st}\geq 3$, then $s$ and $t$ are in the same connected
component of $\widetilde{G}$. Suppose
$\varphi(\alpha_{s})=x\alpha_{s}^{\prime}$, $x\in\mathbb{C}^{\times}$, then

$\varphi(\alpha_{t})=x\frac{a_{s}^{t}b_{t}^{s}}{a_{t}^{s}b_{s}^{t}}\alpha_{t}^{\prime}.$

It is then straightforward to verify
$\varphi(s\cdot\alpha_{t})=s\cdot\varphi(\alpha_{t})$ by definition. $\Box$

Combine the discussions above, we have proved:

######  Theorem 4.10.

Isomorphic classes of IR-representations of $(W,S)$ one-to-one correspond to
the set of data (where $\widetilde{G}$ is determined by $(k_{rt})_{r,t}$)

$\Bigl{\\{}\bigl{(}(k_{rt})_{r\neq t\in S},\chi\bigr{)}\Bigm{|}1\leq
k_{rt}=k_{tr}\leq\frac{m_{rt}}{2},\forall r,t;\phantom{a}\chi\text{ is a
character of }\operatorname{H}_{1}(\widetilde{G})\Bigr{\\}}.$

######  Remark 4.11.

  1. (1)

Take all $k_{rt}$ to be $1$, and $\chi$ to be the trivial character, then the
IR-representation obtained is just the geometric representation
$V_{\text{geom}}$.

  2. (2)

This classification of IR-representations also works over the base field
$\mathbb{R}$.

  3. (3)

Ignoring the labels on edges, the graph $\widetilde{G}$ is a subgraph of $G$
(the Coxeter graph of $(W,S)$) with the same vertex set. An edge $\\{r,t\\}$
in $G$ disappears in $\widetilde{G}$ if and only if $m_{rt}$ is even, and
$k_{rt}$ is chosen to be $m_{rt}/2$. It may happen that $G$ is connected but
$\widetilde{G}$ has several components.

From the proof of 4.9, we have the following corollary.

######  Corollary 4.12.

Let $V$ be an IR-representation, $\widetilde{G}$ be the corresponding graph.
Let $g$ be the number of connected components of $\widetilde{G}$, then
$\operatorname{End}_{W}(V)\simeq\mathbb{C}^{\oplus g}$. In particular, if
$g=1$, any endomorphism of $V$ is a scalar multiplication.

## §5 Reducibility and R-representations

Recall that if $(W,S)$ is reducible, the geometric representation
$V_{\text{geom}}$ decomposes canonically into a direct sum of pieces
corresponding to components of the Coxeter graph $G$. When $(W,S)$ is
irreducible, the reducibility and structure of $V_{\text{geom}}$ can be
described by the following proposition.

######  Proposition 5.1.

([1, Ch.V §4 no.7]) Assume $(W,S)$ is irreducible.

  1. (1)

There is a unique up to scalar $W$-invariant bilinear form $\operatorname{B}$
on $V_{\text{geom}}$.

  2. (2)

$V_{\text{geom}}$ is irreducible if and only if the bilinear form
$\operatorname{B}$ is non-degenerate.

  3. (3)

When $V_{\text{geom}}$ is reducible, it has a maximal sub-representation
$V_{0}$ with trivial $W$-action, and the quotient $V_{\text{geom}}/V_{0}$ is
irreducible.

In this section we will see that IR-representations admit a similar criterion.
As an application, a larger class of representations are classified.

Reducibility of IR-representations Let $V$ be an IR-representation of $(W,S)$
defined by datum $(k_{rt},a_{r}^{t})_{r,t}$, and $\widetilde{G}$ be the
Coxeter graph attached to the numbers $(k_{rt})_{r,t}$, with corresponding
Coxeter group $\widetilde{W}$, see §4. If $S=\sqcup_{i}S_{i}$ is the
decomposition of $\widetilde{G}$ into connected components, let $W_{i}$ and
$\widetilde{W}_{i}$ denote the subgroup of $W$ and $\widetilde{W}$ generated
by $S_{i}$ respectively. The following are obvious.

######  Lemma 5.2.

  1. (1)

$\widetilde{W}=\prod_{i}\widetilde{W}_{i}$.

  2. (2)

$V=\bigoplus_{i}V_{i}$ where $V_{i}=\bigoplus_{s\in
S_{i}}\mathbb{C}\alpha_{s}$ is a sub-representation of $W$.

  3. (3)

Each $V_{i}$ is an IR-representation of the Coxeter system $(W_{i},S_{i})$,
defined by the datum $(k_{rt},a_{r}^{t})_{r,t\in S_{i},r\neq t}$, and factors
through $\widetilde{W}_{i}$.

  4. (4)

For $j\neq i$, $W_{j}$ acts trivially on $V_{i}$.

Thus, we can focus our study on each direct summand.

######  Proposition 5.3.

Suppose $\widetilde{G}$ is connected, and $U\subsetneq V$ is a sub-
representation.

  1. (1)

If $U\neq 0$, the action of $W$ on $U$ is trivial.

  2. (2)

If $V$ is reducible, then $V$ has a maximal sub-representation $V_{0}$ with
trivial $W$-action, and the quotient $V/V_{0}$ is irreducible. $V$ is
indecomposable.

###### Proof:.

Take $v\in U$, suppose $s\cdot v\neq v$ for some $s\in S$, then $v-s\cdot
v\in\mathbb{C}^{\times}\alpha_{s}$ by the proof of 4.2, thus $\alpha_{s}\in
U$. If $r\in S$ is adjacent to $s$ in $\widetilde{G}$, then $\alpha_{r}\in U$
since $\langle\alpha_{s},\alpha_{r}\rangle$ forms an irreducible
representation of $\langle s,r\rangle$. Inductively, we obtain $U=V$ which is
a contradiction. (2) follows from (1). $\Box$

Label elements in $S$ to be $S=\\{s_{1},s_{2},\dots,s_{n}\\}$, and use
notations $\alpha_{i},k_{ij},a_{i}^{j},$ etc as in the proof of 4.7. Write
$v=\sum_{j}x_{j}\alpha_{j}$, $x_{j}\in\mathbb{C}$, then

$\displaystyle s_{i}\cdot v$ $\displaystyle=-x_{i}\alpha_{i}+\sum_{j\neq
i}x_{j}\Bigl{(}\alpha_{j}+2\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}\alpha_{i}\Bigr{)}$
$\displaystyle=\Bigl{(}-x_{i}+\sum_{j\neq
i}2x_{j}\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}\Bigr{)}\alpha_{i}+\sum_{j\neq
i}x_{j}\alpha_{j}.$

If $v$ is fixed by all $s_{i}$, then

$x_{i}-\sum_{j\neq
i}x_{j}\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}=0,\quad\forall
i=1,\dots,n.$ (5.4)

$V$ is reducible if and only if there exists nonzero vector $v\in V$ fixed by
each $s_{i}$, if and only if the set of equations (5.4) has a nonzero solution
for variables $\\{x_{i}\\}_{i}$, if and only if the following $n\times n$
matrix is singular:

$A:=\begin{pmatrix}1&-\frac{a_{1}^{2}}{a_{2}^{1}}\cos\frac{k_{12}\uppi}{m_{12}}&\dots\\\
-\frac{a_{2}^{1}}{a_{1}^{2}}\cos\frac{k_{12}\uppi}{m_{12}}&1&\\\
\vdots&&\ddots\end{pmatrix}$ (5.5)

(All diagonal elements are $1$, and for any $i\neq j$ the element at
$(i,j)$-position is
$-\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}$. Note that $A$ is
not symmetric in general.) The co-rank of $A$ equals to the dimension of the
solution space, as well as the dimension of the maximal sub-representation of
$V$. To conclude:

######  Proposition 5.6.

Suppose $\widetilde{G}$ is connected.

  1. (1)

$V$ is indecomposable. $V$ is irreducible if and only if the matrix $A$
defined in (5.5) is invertible.

  2. (2)

If $V$ is reducible, and $V_{0}$ is the maximal sub-representation, then the
quotient $V/V_{0}$ is irreducible of dimension $\operatorname{rank}(A)$.

In general, $V$ is semisimple if and only if $A$ is invertible.

######  Remark 5.7.

Fixing $(k_{rt})_{r,t}$ such that $\widetilde{G}$ is connected and adapt the
choice in 4.8, the corresponding IR-representations are parameterized by a
torus $(\mathbb{C}^{\times})^{N}$, where $N$ is the rank of
$\operatorname{H}_{1}(\widetilde{G})$. Those reducible representations form a
subvariety of $(\mathbb{C}^{\times})^{N}$.

R-representations In the condition (IR), we require that those “reflection
vectors” $\\{\alpha_{s}\\}_{s}$ are linearly independent. Now let’s remove
this restriction, but we still want to stay in the (A1) assumption, so we
define _R-representations_ to be representations $V$ of $W$ satisfying the
following condition:

$\begin{split}&\text{$V$ is spanned by nonzero vectors $\\{\alpha_{s}\mid s\in
S\\}$, and $\forall s\in S$, $\exists$ a subspace}\\\ &\text{$H_{s}$ of
codimension $1$ such that $s|_{H_{s}}=\operatorname{Id}_{H_{s}}$, and
$s\cdot\alpha_{s}=-\alpha_{s}$.}\\\ &\text{Moreover, $\forall r\neq t\in S$
with $m_{rt}<\infty$, $\alpha_{r}$ and $\alpha_{t}$ are linearly
independent.}\end{split}$ (R)

Note that at present we require $m_{rt}<\infty$ for any $r,t$ (see the
beginning of §4), so each pair $\alpha_{r},\alpha_{t}$ are not proportional.

By definition, R-representations form a larger class than IR-representations.
But in fact they don’t produce too much more.

######  Theorem 5.8.

Let $V$ be an R-representation.

  1. (1)

There is a unique (up to isomorphism) IR-representation $V^{\prime}$ such that
$V$ is a quotient of $V^{\prime}$, say, $\pi:V^{\prime}\twoheadrightarrow V$,
and $W$ acts on $\ker\pi$ trivially.

  2. (2)

Isomorphism classes of semisimple R-representations one-to-one correspond to
isomorphism classes of IR-representations, thus to the set of data
$\\{((k_{rt})_{r,t},\chi)\\}$ in 4.10. In particular, simple R-representations
correspond to those data such that $\widetilde{G}$ is connected.

###### Proof:.

For such $V$, $V=\sum_{s\in S}\mathbb{C}\alpha_{s}$, and $\forall s\in S$,
$V=H_{s}\oplus\mathbb{C}\alpha_{s}$. By assumption (4.1), for any $r\neq t\in
S$, $\langle\alpha_{r},\alpha_{t}\rangle$ is two dimensional, and the same
argument in 4.2 still works. Thus, we can extract the datum
$(k_{rt},a_{r}^{t})_{r,t}$ from $V$ as in §4. Let $V^{\prime}=\bigoplus_{s\in
S}\mathbb{C}\alpha_{s}^{\prime}$ be the IR-representation defined by this
datum, and $\pi:V^{\prime}\to V$ be the surjective linear map defined by
$\alpha_{s}^{\prime}\mapsto\alpha_{s}$. One may easily verify that $\pi$ is a
homomorphism of representations, thus $V$ is a quotient of $V^{\prime}$.

Suppose $V^{\prime\prime}=\bigoplus_{s\in
S}\mathbb{C}\alpha_{s}^{\prime\prime}$ is another IR-representation defined by
$(l_{rt},b_{r}^{t})_{r,t}$, and
$\pi^{\prime\prime}:V^{\prime\prime}\twoheadrightarrow V$ is a surjective
homomorphism. After rescaling, we may assume
$\pi^{\prime\prime}(\alpha_{s}^{\prime\prime})=\alpha_{s}$, $\forall s$.
Restrict to the dihedral subgroup $\langle r,t\rangle$, it must hold
$k_{rt}=l_{rt}$, since
$\langle\alpha_{r}^{\prime\prime},\alpha_{t}^{\prime\prime}\rangle$ and
$\langle\alpha_{r},\alpha_{t}\rangle$ should be isomorphic as representations
of $\langle r,t\rangle$. Next, consider
$\pi^{\prime\prime}(r\cdot\alpha_{t}^{\prime\prime})=r\cdot\alpha_{t}$, if
$k_{rt}\neq m_{rt}/2$, then we obtain
$a_{r}^{t}/a_{t}^{r}=b_{r}^{t}/b_{t}^{r}$ by (4.3). Thus the two data define
the same character of $\operatorname{H}_{1}(\widetilde{G})$, and
$V^{\prime}\simeq V^{\prime\prime}$ by 4.10, where $\widetilde{G}$ is the
Coxeter graph attached to $(k_{rt})_{r,t}$. This proves the uniqueness part in
(1).

Let $S=\sqcup_{i}S_{i}$ be the decomposition of $\widetilde{G}$ into
components, then $V^{\prime}=\bigoplus_{i}V_{i}^{\prime}$ where
$V_{i}^{\prime}=\bigoplus_{s\in S_{i}}\mathbb{C}\alpha_{s}^{\prime}$ is a sub-
representation in $V^{\prime}$, as in 5.2. Write $V_{i,0}^{\prime}$ to be the
maximal sub-representation of $V_{i}^{\prime}$ (may be zero). Note that
$\ker\pi$ is a sub-representation of $V^{\prime}$ but $\ker\pi$ can not
contain any $V_{i}^{\prime}$. Suppose $0\neq v\in\ker\pi$, write
$v=\oplus_{i}v_{i}$ where each $v_{i}\in V_{i}^{\prime}$. Take $s\in S_{j}$,
for any $i\neq j$, we have $s\cdot v_{i}=v_{i}$. If $s\cdot v_{j}\neq v_{j}$,
the argument in 5.3 shows $V_{j}^{\prime}\subseteq\ker\pi$, which is
impossible. Thus for each $i$, $v_{i}\in V_{i,0}^{\prime}$, $W$ acts on
$\ker\pi$ trivially. (1) is proved.

We have shown that $\ker\pi\subseteq
V_{0}^{\prime}:=\bigoplus_{i}V_{i,0}^{\prime}$. On one hand, by 5.6,
$V^{\prime}/V_{0}^{\prime}$ is a semisimple R-representation for any IR-
representation $V^{\prime}$. On the other hand, if $\ker\pi\subsetneq
V_{0}^{\prime}$, suppose $V_{j,0}^{\prime}\nsubseteq\ker\pi$, then
$\pi(V_{j}^{\prime})$ is an indecomposable but reducible sub-representation of
$V$. Thus $V$ is not semisimple. (2) follows. $\Box$

######  Remark 5.9.

  1. (1)

Let $v=\oplus_{i}v_{i}\in\ker\pi$ as in the proof, it may happen that
$v_{i}\notin\ker\pi$. In fact, take any subspace $K$ of
$\bigoplus_{i}V_{i,0}^{\prime}$, the quotient $V^{\prime}/K$ is an
R-representation of $W$.

  2. (2)

Let $\pi_{1}:V^{\prime}\twoheadrightarrow V_{1}$,
$\pi_{2}:V^{\prime}\twoheadrightarrow V_{2}$, where $V^{\prime}$ is an IR-
representation and $V_{1},V_{2}$ are R-representations. Suppose
$\widetilde{G}$ is connected, the same argument of 4.12 shows
$\operatorname{End}_{W}(V_{1})=\mathbb{C}$ and
$\operatorname{Hom}_{W}(V^{\prime},V_{1})\simeq\mathbb{C}$. It yields that
$V_{1}\simeq V_{2}$ if and only if $\ker\pi_{1}=\ker\pi_{2}$. Thus for such
$V^{\prime}$, the isomorphism classes of quotients of $V^{\prime}$ is
parameterized by subspaces of the maximal sub-representation of $V^{\prime}$
(a union of Grassmannians). See 8.4 for an example for the case
$\widetilde{G}$ is disconnected.

## §6 Additional properties

$W$-invariant bilinear form First we determine when an IR-representation
admits a $W$-invariant bilinear form. By 5.2, we may focus on a direct summand
of $V$.

######  Proposition 6.1.

Let $V$ be an IR-representation defined by datum $(k_{rt},a_{r}^{t})_{r,t}$,
and $\chi$ be the corresponding character of
$\operatorname{H}_{1}(\widetilde{G})$. Assume $\widetilde{G}$ is connected.

  1. (1)

$V$ has a nonzero $W$-invariant bilinear form if and only if
$\chi(\operatorname{H}_{1}(\widetilde{G}))\subseteq\\{\pm 1\\}$.

  2. (2)

$V$ has a nonzero $W$-invariant sesquilinear form if and only if
$\chi(\operatorname{H}_{1}(\widetilde{G}))\subseteq S^{1}$.

###### Proof:.

Suppose there is a nonzero $W$-invariant bilinear form $\operatorname{B}$ on
$V$. For any path $(s_{1},s_{2},\dots,s_{n})$ in $\widetilde{G}$, suppose
$\operatorname{B}(\alpha_{1},\alpha_{1})=x$ (here $\alpha_{1}=\alpha_{s_{1}}$,
similarly for other notations), then by 2.7 and the construction of $V$, it
holds $\operatorname{B}(\alpha_{2},\alpha_{2})=x(a_{1}^{2}/a_{2}^{1})^{2}$. Do
this recursively along this circuit, finally we get

$\operatorname{B}(\alpha_{n},\alpha_{n})=x\left(\frac{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}}{a_{2}^{1}a_{3}^{2}\cdots a_{n}^{n-1}}\right)^{2}.$ (6.2)

If $x=0$, then $\operatorname{B}(\alpha_{s},\alpha_{s})=0$ for any $s\in S$
since $\widetilde{G}$ is connected. As a result, if $s,t\in S$ such that
$\widetilde{m}_{st}\geq 3$, then
$\operatorname{B}(\alpha_{s},\alpha_{t})=\operatorname{B}(\alpha_{t},\alpha_{s})=0$
by 2.7. If $\widetilde{m}_{st}=2$, then
$\operatorname{B}(\alpha_{s},\alpha_{t})=\operatorname{B}(s\cdot\alpha_{s},s\cdot\alpha_{t})=-\operatorname{B}(\alpha_{s},\alpha_{t})$
yields $\operatorname{B}(\alpha_{s},\alpha_{t})=0$. Therefore
$\operatorname{B}=0$ which is a contradiction. Thus $x\neq 0$. Suppose now
$s_{1}=s_{n}$, then $\operatorname{B}(\alpha_{n},\alpha_{n})=x$, (6.2) implies
the image of the closed path under $\chi$ lies in $\\{\pm 1\\}$.

Conversely, suppose $\chi(\operatorname{H}_{1}(\widetilde{G}))\subseteq\\{\pm
1\\}$. Choose and fix a vertex $s_{1}$ in $\widetilde{G}$, define
$\operatorname{B}(\alpha_{1},\alpha_{1}):=1$. For another vertex $s$, choose a
path $(s_{1},s_{2},\dots,s_{n}=s)$, in view of 2.7, define

$\operatorname{B}(\alpha_{s},\alpha_{s}):=\left(\frac{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}}{a_{2}^{1}a_{3}^{2}\cdots a_{n}^{n-1}}\right)^{2}.$

For $t\neq s$, define

$\displaystyle\operatorname{B}(\alpha_{t},\alpha_{s})=\operatorname{B}(\alpha_{s},\alpha_{t}):=-\frac{a_{s}^{t}}{a_{t}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\operatorname{B}(\alpha_{s},\alpha_{s})$
$\displaystyle\bigl{(}\text{also equals to
}-\frac{a_{t}^{s}}{a_{s}^{t}}\cos\frac{k_{st}\uppi}{m_{st}}\operatorname{B}(\alpha_{t},\alpha_{t})\bigr{)}.$

$\operatorname{B}$ is well-defined, i.e.
$\operatorname{B}(\alpha_{s},\alpha_{s})$ is independent of the choice of the
path. This is because, choosing another path
$(s_{1},s_{p},s_{p-1},\dots,s_{n+1},s_{n}=s)$, $p\geq n$, the two paths form a
circuit $(s_{1},\dots,s_{n},s_{n+1},\dots,s_{p},s_{1})$, and our assumption
says

$\pm 1=\frac{a_{1}^{2}\cdots a_{n-1}^{n}a_{n}^{n+1}\cdots
a_{p-1}^{p}a_{p}^{1}}{a_{2}^{1}\cdots a_{n}^{n-1}a_{n+1}^{n}\cdots
a_{p}^{p-1}a_{1}^{p}},\text{ thus }\left(\frac{a_{1}^{2}a_{2}^{3}\cdots
a_{n-1}^{n}}{a_{2}^{1}a_{3}^{2}\cdots
a_{n}^{n-1}}\right)^{2}=\left(\frac{a_{1}^{p}a_{p}^{p-1}\cdots
a_{n+1}^{n}}{a_{p}^{1}a_{p-1}^{p}\cdots a_{n}^{n+1}}\right)^{2}.$

Now we need to check $\operatorname{B}$ is $W$-invariant. It suffices to check
$\operatorname{B}(s\cdot\alpha_{r},s\cdot\alpha_{t})=\operatorname{B}(\alpha_{r},\alpha_{t})$
for $s,r,t\in S$. If $s=r=t$, this is obvious. If $s=r\neq t$, or $s=t\neq r$,
or $s\neq r=t$, everything happens in a dihedral world, the equation holds by
2.7. If $s,r,t$ are distinct,

$\displaystyle\operatorname{B}(s\cdot\alpha_{r},s\cdot\alpha_{t})$
$\displaystyle=\operatorname{B}\Bigl{(}\alpha_{r}+2\frac{a_{s}^{r}}{a_{r}^{s}}\cos\frac{k_{sr}\uppi}{m_{sr}}\alpha_{s},\alpha_{t}+2\frac{a_{s}^{t}}{a_{t}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\alpha_{s}\Bigr{)}$
$\displaystyle=\operatorname{B}(\alpha_{r},\alpha_{t})+2\frac{a_{s}^{t}}{a_{t}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\operatorname{B}(\alpha_{r},\alpha_{s})$
$\displaystyle\mathrel{\phantom{=}}+2\frac{a_{s}^{r}}{a_{r}^{s}}\cos\frac{k_{sr}\uppi}{m_{sr}}\operatorname{B}(\alpha_{s},\alpha_{t})+4\frac{a_{s}^{t}a_{s}^{r}}{a_{t}^{s}a_{r}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\cos\frac{k_{sr}\uppi}{m_{sr}}\operatorname{B}(\alpha_{s},\alpha_{s})$
$\displaystyle=\operatorname{B}(\alpha_{r},\alpha_{t}).$

The proof of (2) is similar to the argument above. $\Box$

######  Remark 6.3.

  1. (1)

From the proof we see that the $W$-invariant bilinear form $\operatorname{B}$
(if exists) is symmetric and is unique up to a $\mathbb{C}^{\times}$-scalar.

  2. (2)

If we don’t assume $\widetilde{G}$ is connected, write $V=\bigoplus_{i}V_{i}$
as in 5.2, then 6.1 is applied on each $V_{i}$.

  3. (3)

By 6.1, there are only finitely many IR-representations of $W$ admitting an
invariant bilinear form. The geometric representation is one of them.

Let $V$ be an IR-representation with connected $\widetilde{G}$. Suppose $V$ is
not irreducible, let $V_{0}$ be the maximal sub-representation. Assume further
$V$ admits a nonzero $W$-invariant bilinear form $\operatorname{B}$. Let
$v=\sum_{s}x_{s}\alpha_{s}\in V_{0}$, $x_{s}\in\mathbb{C}$, then

$\operatorname{B}(v,\alpha_{r})=x_{r}\operatorname{B}(\alpha_{r},\alpha_{r})-\sum_{s\neq
r}x_{s}\frac{a_{r}^{s}}{a_{s}^{r}}\cos\frac{k_{sr}\uppi}{m_{sr}}\operatorname{B}(\alpha_{r},\alpha_{r})=0,\quad\forall
r\in S.$ (6.4)

The second equality is due to (5.4). Let $V_{1}$ be an R-representation which
is a quotient of $V$, $\pi:V\twoheadrightarrow V_{1}$ be the projection, then
$\ker\pi\subseteq V_{0}$. Formula (6.4) tells us the bilinear form
$\operatorname{B}$ descents to $V_{1}$.

Conversely, let $V_{1}$ be an R-representation with connected $\widetilde{G}$.
Suppose $V_{1}$ has a nonzero $W$-invariant bilinear form $\operatorname{B}$,
by the same argument of 6.1, we shall see that
$\chi(\operatorname{H}_{1}(\widetilde{G}))\subseteq\\{\pm 1\\}$, and such
bilinear form is unique up to a scalar. Also, $\operatorname{B}$ can be lifted
to $V$, the corresponding IR-representation.

Dual of an IR-representation Let $V=\bigoplus_{s\in S}\mathbb{C}\alpha_{s}$ be
an IR-representation defined by datum $(k_{rt},a_{r}^{t})_{r,t}$, and
$\\{\alpha_{s}^{*}\\}_{s\in S}$ be the dual basis of $\\{\alpha_{s}\\}_{s}$ in
$V^{*}:=\operatorname{Hom}_{\mathbb{C}}(V,\mathbb{C})$. $V^{*}$ forms a
representation of $W$ via $(s\cdot\alpha_{t}^{*})(-)=\alpha_{t}^{*}(s\cdot-)$,
$\forall s\in S$. A direct computation shows

$\displaystyle s\cdot\alpha_{s}^{*}$
$\displaystyle=-\alpha_{s}^{*}+\sum_{t\neq
s}2\frac{a_{s}^{t}}{a_{t}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\alpha_{t}^{*},$
(6.5) $\displaystyle s\cdot\alpha_{t}^{*}$
$\displaystyle=\alpha_{t}^{*},\quad\forall t\neq s.$ (6.6)

From this we can see that the unique (up to a scalar) $-1$-eigenvector of $s$
is

$\gamma_{s}:=\alpha_{s}^{*}-\sum_{t\neq
s}\frac{a_{s}^{t}}{a_{t}^{s}}\cos\frac{k_{st}\uppi}{m_{st}}\alpha_{t}^{*}.$
(6.7)

######  Proposition 6.8.

  1. (1)

$\\{\gamma_{s}\\}_{s\in S}$ is linearly independent if and only if the matrix
$A$ defined in (5.5) is invertible, if and only if $V$ is semisimple.

  2. (2)

When $V$ is semisimple, $V^{*}=\bigoplus_{s}\mathbb{C}\gamma_{s}$ is an IR-
representation of $W$. Suppose $((k_{rt})_{r,t},\chi)$ is the datum
corresponding to $V$, then $V^{*}$ corresponds to the datum
$((k_{rt})_{r,t},\chi^{*})$, here $\chi^{*}$ is the dual character,
$\chi^{*}(-)=\chi(-)^{-1}$.

  3. (3)

In general case, $\\{\gamma_{s}\\}_{s\in S}$ spans a sub-representation
$V_{1}:=\sum_{s\in S}\mathbb{C}\gamma_{s}$ of $V^{*}$. Moreover, $V_{1}$ is an
R-representation isomorphic to the semisimple quotient of the IR-
representation corresponding to the datum $((k_{rt})_{r,t},\chi^{*})$. The
action of $W$ on the quotient $V^{*}/V_{1}$ is trivial.

###### Proof:.

By (6.7), the transition matrix from $\\{\alpha_{s}^{*}\\}_{s}$ to
$\\{\gamma_{s}\\}_{s}$ is $A^{\mathrm{T}}$, the transpose of $A$, thus
$\\{\gamma_{s}\\}_{s}$ is linearly independent if and only if $A$ is
invertible. The second part of (1) is due to 5.6.

When $\\{\gamma_{s}\\}_{s}$ is linearly independent, clearly
$V^{*}=\bigoplus_{s}\mathbb{C}\gamma_{s}$, and $V^{*}$ is an IR-
representation. A direct computation shows

$s\cdot\gamma_{t}=\gamma_{t}+2\frac{a_{t}^{s}}{a_{s}^{t}}\cos\frac{k_{st}\uppi}{m_{st}}\gamma_{s},$
(6.9)

thus the datum $(k_{rt},b_{r}^{t})_{r,t}$ defines $V^{*}$, where
$b_{r}^{t}=a_{t}^{r}$. Therefore $V^{*}$ corresponds to the character
$\chi^{*}$ of $\operatorname{H}_{1}(\widetilde{G})$. (2) is proved.

In general, let $V^{\prime}$ be the IR-representation defined by
$(k_{rt},b_{r}^{t})_{r,t}$, then $A^{\mathrm{T}}$ is the matrix of
$V^{\prime}$ defined as in (5.5). Notice that $V_{1}$ is an R-representation,
and $\dim V_{1}$ equals to the rank of $A^{\mathrm{T}}$, thus $V_{1}$ is the
semisimple quotient of $V^{\prime}$. By (6.5) and (6.7), we have
$s\cdot\alpha_{s}^{*}=\alpha_{s}^{*}-2\gamma_{s}$. Combined with (6.6), we see
that $W$ acts on $V^{*}/V_{1}$ trivially. (3) is proved. $\Box$

## §7 General situation: allowing $m_{st}=\infty$

From now on, we consider arbitrary Coxeter system $(W,S)$ of finite rank with
Coxeter graph $G$, dropping the assumption that $m_{rt}<\infty$, $\forall
r,t\in S$. For any pair $r\neq t\in S$, define a parameter set

$P_{rt}:=P_{m_{rt}}=\begin{cases}\\{\rho_{k}\mid k\in\mathbb{N},1\leq
k\leq\frac{m_{rt}}{2}\\},&\mbox{if }m_{rt}<\infty;\\\
\\{\varrho_{r}^{t},\varrho_{t}^{r}\\}\cup\\{\varrho_{z}\mid
z\in\mathbb{C}\\},&\mbox{if }m_{rt}=\infty.\end{cases}$

(The notations $\varrho$’s are defined in §2.) Note that $P_{\infty}$ consists
of all IR-representations of $D_{\infty}$. Suppose we are given
$\delta_{rt}\in P_{rt}$ for any pair $r\neq t\in S$, such that
$\delta_{rt}=\delta_{tr}$ representing the same representation of $\langle
r,t\rangle$, define a Coxeter system $(\widetilde{W},S)$ with Coxeter graph
$\widetilde{G}$ as follows.

  1. (1)

If $m_{rt}<\infty$ and $\delta_{rt}=\rho_{k}$, let
$\widetilde{m}_{rt}=m_{rt}/d$ where $d=\operatorname{g.c.d.}(m_{rt},k)$;

  2. (2)

If $m_{rt}=\infty$, $\delta_{rt}=\varrho_{z}$ and
$z=4\cos^{2}\frac{k\uppi}{m}$ for some co-prime $k,m\in\mathbb{N}$, $1\leq
k<m/2$, let $\widetilde{m}_{rt}=m$;

  3. (3)

If $m_{rt}=\infty$, $\delta_{rt}=\varrho_{0}$, let $\widetilde{m}_{rt}=2$;

  4. (4)

Otherwise, let $\widetilde{m}_{rt}=\infty$.

######  Theorem 7.1.

Isomorphism classes of IR-representations of $(W,S)$ one-to-one correspond to
the set of data

$\Bigl{\\{}\bigl{(}(\delta_{rt})_{r\neq t\in
S},\chi\bigr{)}\Bigm{|}\delta_{rt}=\delta_{tr}\in P_{rt},\forall
r,t;\phantom{a}\chi\text{ is a character of
}\operatorname{H}_{1}(\widetilde{G})\Bigr{\\}}.$

###### Proof:.

Let $V=\bigoplus_{s}\mathbb{C}\alpha_{s}$ be an IR-representation of $W$,
4.2(1) still holds. For any pair $r\neq t\in S$, there is an element
$\delta_{rt}\in P_{rt}$, such that $\langle\alpha_{r},\alpha_{t}\rangle$ forms
a representation of $\langle r,t\rangle$ isomorphic to $\delta_{rt}$. There
exist $a_{r}^{t},a_{t}^{r}\in\mathbb{C}^{\times}$ such that the isomorphism
sends $a_{r}^{t}\alpha_{r}$ to $\beta_{r}$, $a_{t}^{r}\alpha_{t}$ to
$\beta_{t}$. The structure of $V$ is determined by the datum
$(\delta_{rt},a_{r}^{t})_{r,t}$, since the action $r\cdot\alpha_{t}$ has been
clear, for any $r,t$.

Conversely, given datum $(\delta_{rt},a_{r}^{t})_{r,t}$ in which
$\delta_{rt}=\delta_{tr}\in P_{rt}$, $a_{r}^{t}\in\mathbb{C}^{\times}$,
identify $\langle\alpha_{r},\alpha_{t}\rangle$ and $\delta_{rt}$ via
$a_{r}^{t}\alpha_{r}\mapsto\beta_{r}$, $a_{t}^{r}\alpha_{t}\mapsto\beta_{t}$,
then $\langle r,t\rangle$ acts on $\langle\alpha_{r},\alpha_{t}\rangle$. The
same argument in 4.4 shows that this defines a $W$-action on
$V:=\bigoplus_{s}\mathbb{C}\alpha_{s}$.

For a datum $(\delta_{rt},a_{r}^{t})_{r,t}$ and the corresponding
$\widetilde{G}$, define $\chi$ as in §4. The arguments in 4.7, 4.8, 4.9 still
work. The theorem follows. $\Box$

######  Theorem 7.2.

Let $V=\sum_{s}\mathbb{C}\alpha_{s}$ be an R-representation, then $V$ is a
quotient of a unique IR-representation
$V^{\prime}=\bigoplus_{s}\mathbb{C}\alpha_{s}^{\prime}$, say,
$\pi:V^{\prime}\twoheadrightarrow V$. Moreover, $W$ acts on $\ker\pi$
trivially.

###### Proof:.

Note that we are allowing $m_{rt}$ to be $\infty$, so $\alpha_{r}$ and
$\alpha_{t}$ may be proportional. If this happens, choose
$\delta_{rt}=\varrho_{4}$ (i.e. the geometric representation), and choose
$a_{r}^{t},a_{t}^{r}\in\mathbb{C}^{\times}$ such that
$a_{r}^{t}\alpha_{r}+a_{t}^{r}\alpha_{t}=0$. Except this case, the proof is
nearly the same as that of 5.8(1). $\Box$

The matrix A Let $V$ be an IR-representation. In view of 7.2, we wish to find
sub-representations in $V$ with trivial $W$-action. Write
$S=\\{s_{1},\dots,s_{n}\\}$, and use notations
$P_{ij},\delta_{ij},\varrho_{i}^{j}$, etc in the same way as before. Let
$v=\sum_{j}x_{j}\alpha_{j}\in V$, then

$\displaystyle s_{i}\cdot v$ $\displaystyle=-x_{i}\alpha_{i}+\sum_{s_{j}\in
S_{1}}x_{j}\Bigl{(}\alpha_{j}+2\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}\alpha_{i}\Bigr{)}+\sum_{s_{j}\in
S_{2}}x_{j}\alpha_{j}$ $\displaystyle\mathrel{\phantom{=}}+\sum_{s_{j}\in
S_{3}}x_{j}\Bigl{(}\alpha_{j}+\frac{a_{i}^{j}}{a_{j}^{i}}\alpha_{i}\Bigr{)}+\sum_{s_{j}\in
S_{4}}x_{j}\Bigl{(}\alpha_{j}+u_{ij}\frac{a_{i}^{j}}{a_{j}^{i}}\alpha_{i}\Bigr{)}$
(7.3) $\displaystyle=\Big{(}-x_{i}+\sum_{s_{j}\in
S_{1}}2x_{j}\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}}+\sum_{s_{j}\in
S_{3}}x_{j}\frac{a_{i}^{j}}{a_{j}^{i}}+\sum_{s_{j}\in
S_{4}}x_{j}u_{ij}\frac{a_{i}^{j}}{a_{j}^{i}}\Big{)}\alpha_{i}+\sum_{j\neq
i}x_{j}\alpha_{j},$

where

$\displaystyle S_{1}$ $\displaystyle:=\\{s_{j}\in S\mid m_{ij}<\infty\\},\quad
1\leq k_{ij}\leq\frac{m_{ij}}{2}\text{ such that }\delta_{ij}=\rho_{k_{ij}},$
$\displaystyle S_{2}$ $\displaystyle:=\\{s_{j}\in S\mid
m_{ij}=\infty,\delta_{ij}=\varrho_{j}^{i}\\},$ $\displaystyle S_{3}$
$\displaystyle:=\\{s_{j}\in S\mid
m_{ij}=\infty,\delta_{ij}=\varrho_{i}^{j}\\},$ $\displaystyle S_{4}$
$\displaystyle:=\\{s_{j}\in S\mid
m_{ij}=\infty,\delta_{ij}=\varrho_{z_{ij}},z_{ij}\in\mathbb{C}\\},$
$\displaystyle\quad u_{ij}$ $\displaystyle:=u(z_{ij})\text{ chosen in
\S\ref{Dm-and-graph}}.$

We have $S=S_{1}\sqcup S_{2}\sqcup S_{3}\sqcup S_{4}\sqcup\\{s_{i}\\}$.
Similar to (5.5), let $A=(A_{ij})\in\mathrm{M}_{n}(\mathbb{C})$ be a square
matrix with element $A_{ij}$ at $(i,j)$-position defined by (here
$S_{1}=S_{1}(i),\dots$ depend on $i$)

$A_{ij}=\begin{cases}1,&\mbox{if }i=j,\\\
-\frac{a_{i}^{j}}{a_{j}^{i}}\cos\frac{k_{ij}\uppi}{m_{ij}},&\mbox{if }s_{j}\in
S_{1},\\\ 0,&\mbox{if }s_{j}\in S_{2},\\\
-\frac{a_{i}^{j}}{2a_{j}^{i}},&\mbox{if }s_{j}\in S_{3},\\\
-\frac{u_{ij}a_{i}^{j}}{2a_{j}^{i}},&\mbox{if }s_{j}\in S_{4},.\end{cases}$

From (7), the null space of $A$ is the subspace of $V$ with trivial
$W$-action, as we saw in §5.

Other reducibility Compared with 5.3 and 5.6, there are some different
phenomena when considering reducibility of IR-representations in this general
setting. For example, $\varrho_{r}^{t}$ of $D_{\infty}$ have sub-
representations with nontrivial group action.

Suppose $V$ is an IR-representation. For convenience, let’s append more
information to the graph $\widetilde{G}$. For any pair $r\neq t\in S$, if
$\delta_{rt}=\varrho_{r}^{t}$, assign a direction $t\to r$ to the edge
$\\{r,t\\}$. Symmetrically, $r\to t$ if $\delta_{rt}=\varrho_{t}^{r}$. For
other edges $\\{r,t\\}$ in $\widetilde{G}$
($3\leq\widetilde{m}_{rt}\leq\infty$), both directions are assigned, i.e.
$r\to t$ and $t\to r$. These directions have the following meaning: if $r\to
t$ then $\alpha_{t}$ belongs to the sub-representation generated by
$\alpha_{r}$.

The directions generate a pre-order on $S$: say
$r\underset{\mathrm{d}}{\leqslant}t$ if there is a sequence
$r=s_{1},s_{2},\dots,s_{n}=t$ in $S$ such that $s_{i+1}\to s_{i}$ for all $i$.
The corresponding equivalence relation is denoted by
$\underset{\mathrm{d}}{\sim}$: say $r\underset{\mathrm{d}}{\sim}t$ if
$r\underset{\mathrm{d}}{\leqslant}t$ and $t\underset{\mathrm{d}}{\leqslant}r$.
Immediately we have the following fact.

######  Corollary 7.4.

Given an IR-representation $V$.

  1. (1)

If all elements in $S$ are equivalent with respect to
$\underset{\mathrm{d}}{\sim}$, then the action of $W$ on any sub-
representation of $V$ is trivial.

  2. (2)

Let $I\subsetneq S$ be an equivalent class with respect to
$\underset{\mathrm{d}}{\sim}$, then
$\bigoplus_{s\underset{\mathrm{d}}{\leqslant}I}\mathbb{C}\alpha_{s}$ is a sub-
representation of $V$.

Bilinear forms Different from 6.1, it may happen that there is no nonzero
$W$-invariant bilinear form on an IR-representation $V$ even when
$\chi(\operatorname{H}_{1}(\widetilde{G}))\subseteq\\{\pm 1\\}$. For example,
$S=\\{s,t,r\\}$, $m_{st}=\infty$, $m_{rt}=m_{sr}=3$, let
$\delta_{st}=\varrho_{s}^{t}$. Suppose $\operatorname{B}$ is a $W$-invariant
bilinear form on $V$. From 2.11(4), it holds
$\operatorname{B}(\alpha_{s},\alpha_{s})=0$. By 2.7, it yields
$\operatorname{B}(\alpha_{r},\alpha_{r})=0$ and then
$\operatorname{B}(\alpha_{t},\alpha_{t})=0$. $\operatorname{B}$ has to be
zero.

If for any pair $r\neq t\in S$ it holds $\delta_{rt}\neq\varrho_{r}^{t}$ and
$\varrho_{t}^{r}$, then 6.1 still works.

## §8 Examples: from $\widetilde{\mathsf{A}}_{n}$ to simply laced type

######  Example 8.1.

The Coxeter system $(W,S)$ of type $\widetilde{\mathsf{A}}_{n}$ is defined by
$\langle s_{0},s_{1},\dots,s_{n}\mid s_{i}^{2}=(s_{i}s_{i+1})^{3}=1,\forall
i\rangle$ (regard $n+1$ as $0$). Its Coxeter graph $G$ is

$s_{1}$$s_{2}$$s_{n-1}$$s_{n}$$s_{0}$$\dots$

In the data defining IR-representations, for any distinct pair $i,j$, the
condition $1\leq k_{ij}\leq m_{ij}/2$ forces $k_{ij}=1$, thus the graph
$\widetilde{G}=G$. The homology group $\operatorname{H}_{1}(G)$ is isomorphic
to $\mathbb{Z}$ generated by the circuit $c=(s_{0},s_{1},\dots,s_{n},s_{0})$.
Giving a character $\chi$ of $\operatorname{H}_{1}(G)$ is equivalent to giving
a number $x\in\mathbb{C}^{\times}$ and assigning $\chi(c)=x$. Thus, all IR-
representations of $W$ is parameterized by $\mathbb{C}^{\times}$.

Let $a_{0}^{n}=x$, and $a_{n}^{0}=a_{0}^{1}=a_{1}^{0}\dots=1$, the square
matrix $A$ (of size $n+1$, defined in (5.5)) is

$A=\begin{pmatrix}1&-\frac{1}{2}&0&\dots&0&-\frac{x}{2}\\\
-\frac{1}{2}&1&-\frac{1}{2}&&&0\\\ 0&-\frac{1}{2}&\ddots&\ddots&&\vdots\\\
\vdots&&\ddots&\ddots&-\frac{1}{2}&0\\\ 0&&&-\frac{1}{2}&1&-\frac{1}{2}\\\
-\frac{1}{2x}&0&\dots&0&-\frac{1}{2}&1\end{pmatrix}$

One may use induction on $n$ to compute the determinant of $A$,

$\det A=\frac{2-x-x^{-1}}{2^{n+1}}.$

Denote by $V_{x}$ the IR-representation given by $x$. Then, by 5.6, $V_{x}$ is
irreducible if and only if $x\neq 1$. When $x=1$, $V_{x}\simeq
V_{\text{geom}}$.

######  Proposition 8.2.

Let $(W,S)$ be of type $\widetilde{\mathsf{A}}_{n}$. Suppose $V$ is a
nontrivial irreducible A1-representation of $W$, then either $V$ is the simple
quotient of $V_{\text{geom}}$, or $V=V_{x}$ for some
$x\in\mathbb{C}\setminus\\{0,1\\}$. (See also [11, Ch.12] for the version of
affine Hecke algebras.)

###### Proof:.

First of all, $V$ is finite dimensional (see, for example, [4] or [12]).
Restricted to the dihedral group $\langle s_{0},s_{1}\rangle\simeq D_{3}$, $V$
decomposes into a direct sum of copies of $\mathds{1},\varepsilon$ and
$\rho_{1}$. Let $V_{-}^{i}$ denote the subspace of $-1$-eigenvectors of
$s_{i}$. The condition (A1) forces that $\varepsilon$ doesn’t occur in $V$,
thus $V_{-}^{0}\cap V_{-}^{1}=0$ and $\dim V_{-}^{0}=\dim V_{-}^{1}$. Similar
result holds for any pair $i,i+1$.

If $V$ is nontrivial, then $V_{-}^{0}\neq 0$. Take a basis of $V_{-}^{0}$, say
$\\{\alpha_{0,j}\\}_{j=1,\dots,r}$. Let

$\alpha_{1,j}:=s_{1}\cdot\alpha_{0,j}-\alpha_{0,j},\quad j=1,\dots,r,$

then $\\{\alpha_{1,j}\\}_{j}$ is a basis of $V_{-}^{1}$, and for any $j$,
$\\{\alpha_{0,j},\alpha_{1,j}\\}$ spans a representation of $\langle
s_{0},s_{1}\rangle$ isomorphic to $\rho_{1}$. Apply similar considerations
inductively on $\langle s_{1},s_{2}\rangle,\dots,\langle
s_{n-1},s_{n}\rangle$, we obtain a basis $\\{\alpha_{i,j}\\}_{j}$ of
$V_{-}^{i}$ ($1\leq i\leq n$), such that $\forall 1\leq i\leq n,1\leq j\leq
r$, $\\{\alpha_{i-1,j},\alpha_{i,j}\\}$ spans a $\rho_{1}$ of $\langle
s_{i-1},s_{i}\rangle$.

Now consider $\langle s_{n},s_{0}\rangle$. Write

$\alpha_{0,j}^{\prime}:=s_{0}\cdot\alpha_{n,j}-\alpha_{n,j},\quad
j=1,\dots,r,$

then $\\{\alpha_{n,j},\alpha_{0,j}^{\prime}\\}$ spans a $\rho_{1}$ of $\langle
s_{n},s_{0}\rangle$ for any $j$, and $\\{\alpha_{0,j}^{\prime}\\}_{j}$ is a
basis of $V_{-}^{0}$ as well. Thus there is a matrix
$X\in\operatorname{GL}_{r}(\mathbb{C})$ such that

$(\alpha_{0,1}^{\prime},\dots,\alpha_{0,r}^{\prime})=(\alpha_{0,1},\dots,\alpha_{0,r})\cdot
X.$

Let $v=(x_{1},\dots,x_{r})^{\mathrm{T}}\in\mathbb{C}^{r}$ be an eigenvector of
$X$ of with eigenvalue $x\in\mathbb{C}^{\times}$. Write

$\alpha_{i}:=\sum_{1\leq j\leq r}x_{j}\alpha_{i,j},\quad 0\leq i\leq
n,\quad\text{and }\alpha_{0}^{\prime}:=\sum_{1\leq j\leq
r}x_{j}\alpha_{i,j}^{\prime},$

then for $1\leq i\leq n$, $\\{\alpha_{i-1},\alpha_{i}\\}$ span a $\rho_{1}$
for $\langle s_{i-1},s_{i}\rangle$. At last,
$\\{\alpha_{n},\alpha_{0}^{\prime}\\}$ span a $\rho_{1}$ for $\langle
s_{n},s_{0}\rangle$. We have

$\alpha_{0}^{\prime}=(\alpha_{0,1}^{\prime},\dots,\alpha_{0,r}^{\prime})\cdot
v=(\alpha_{0,1},\dots,\alpha_{0,r})\cdot X\cdot
v=x(\alpha_{0,1},\dots,\alpha_{0,r})\cdot v=x\alpha_{0}.$

Consider the subspace $V^{\prime}$ spanned by
$\\{\alpha_{0},\alpha_{1},\dots,\alpha_{n}\\}$, we claim that $V^{\prime}$
forms a sub-representation satisfying (R). For any pair $i,i^{\prime}$ such
that $s_{i},s_{i^{\prime}}$ are not adjacent in $G$, we have
$s_{i}s_{i^{\prime}}=s_{i^{\prime}}s_{i}$, and hence
$s_{i}\cdot\alpha_{i^{\prime}}\in V_{-}^{i^{\prime}}$. Then
$s_{i}\cdot\alpha_{i^{\prime}}-\alpha_{i^{\prime}}\in V_{-}^{i}\cap
V_{-}^{i^{\prime}}$, thus $s_{i}\cdot\alpha_{i^{\prime}}=\alpha_{i^{\prime}}$.
Now we can see that for any $i$, $s_{i}$ leaves $V^{\prime}$ stable, and has
only one $-1$-eigenvector in $V^{\prime}$ (up to scalars). Thus, $V^{\prime}$
is an R-representation. By the construction of $V^{\prime}$ and the
classification of R-representations, $V^{\prime}$ is isomorphic to $V_{x}$ if
$x\neq 1$, to $V_{\text{geom}}$ or its simple quotient if $x=1$. The
proposition is proved. $\Box$

This proposition says, for type $\widetilde{\mathsf{A}}_{n}$ any nontrivial
irreducible A1-representation is an R-representations. The proof can be
generalized a little bit, as follows.

######  Theorem 8.3.

Let $(W,S)$ be an irreducible Coxeter system with simply laced Coxeter graph
$G$. Suppose the rank of $\operatorname{H}_{1}(G)$ is $0$ or $1$, then any
nontrivial irreducible A1-representation is an R-representation. In
particular, they are finite dimensional. If $\operatorname{H}_{1}(\Gamma)$ has
rank $0$, such representation is unique.

###### Proof:.

Let $V$ be an irreducible A1-representation of $W$, denote by $V_{-}^{s}$ the
subspace of $-1$-eigenvectors of $s$. If
$\operatorname{H}_{1}(G)\simeq\mathbb{Z}$, let $c$ be a circuit generating
$\operatorname{H}_{1}(G)$, and $s$ be a vertex in $c$. Similar to the proof of
8.2, the circuit $c$ gives a linear automorphism $X$ of $V_{-}^{s}$. Now
$V_{-}^{s}$ becomes a module of the Laurent polynomial ring $\mathbb{C}[X^{\pm
1}]$. By Hilbert’s Nullstellensatz, any simple module of $\mathbb{C}[X^{\pm
1}]$ is one dimensional. Therefore, if $\dim V_{-}^{s}>1$, then there is a
proper submodule $V_{0}\subsetneq V_{-}^{s}$. This subspace $V_{0}$ generates
a proper sub-representation of $W$ in $V$, contradicting the irreducibility of
$V$. Hence $\dim V_{-}^{s}=1$, and $V_{-}^{s}$ generates an R-representation.

If $G$ is a tree, fixing $s\in S$, any nonzero vector in $V_{-}^{s}$ generates
a sub-representation, which is an R-representation. Such a representation is
unique according to 5.8. $\Box$

Motivated by this proof, for a Coxeter graph $G$ with at least two circuits,
we may use an infinite dimensional irreducible representation of the
fundamental group of $G$ (which is a non-abelian free group) to construct an
infinite dimensional irreducible representation of the corresponding Coxeter
group. We will discuss this in a subsequent paper, see [3].

######  Example 8.4.

This example is a supplement to 5.9(2). When $\widetilde{G}$ is not connected,
the quotients of an IR-representation by two different subspace may be
isomorphic. Suppose $\widetilde{G}$ is of type
$\widetilde{\mathsf{A}}_{2}\times\widetilde{\mathsf{A}}_{2}$, i.e.

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$$s_{6}$

and let $V$ be the corresponding geometric representation. Then $V$ is
isomorphic to the direct sum of two copies of the geometric representations of
$\widetilde{\mathsf{A}}_{2}$. Let $v_{1}=\alpha_{1}+\alpha_{2}+\alpha_{3}$,
$v_{2}=\alpha_{4}+\alpha_{5}+\alpha_{6}$, then $V_{0}:=\langle
v_{1},v_{2}\rangle$ is the maximal sub-representation in $V$ with trivial
group action. The two quotient $V/\langle v_{1}\rangle$ and $V/\langle
v_{2}\rangle$ are clearly non-isomorphic. However, let $V_{1}=V/\langle
v_{1}+v_{2}\rangle$, $V_{2}=V/\langle v_{1}+2v_{2}\rangle$, then
$\alpha_{i}\mapsto\alpha_{i},i=1,2,3$, $\alpha_{j}\mapsto 2\alpha_{j},j=4,5,6$
defines an isomorphism from $V_{1}$ to $V_{2}$.

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Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
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