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%%%%% Auteur
%%%1
\author{\firstname{R.} \middlename{M.} \lastname{Green}}
\address{Department of Mathematics\\
University of Colorado Boulder, Campus Box 395\\
Boulder, Colorado\\
USA, 80309}
\email{rmg@colorado.edu}
%%%2
\author{\firstname{Tianyuan} \lastname{Xu}}
\address{Department of Mathematics and Statistics\\
Queen's University\\
Kingston, Ontario\\ Canada, K7L 3N6}
\email{tx7@queensu.ca}
%%%%% Sujet
\subjclass{05E10, 20C08}
\keywords{Coxeter groups, Hecke algebras, Lusztig's $\mathbf{a}$-function, fully
commutative elements, heaps, star operations}
%%%%% Gestion
\DOI{10.5802/alco.95}
\datereceived{2018-05-22}
\daterevised{2019-07-10}
\dateaccepted{2019-09-14}
%%%%% Titre et résumé
%\title[Classification of \texorpdfstring{$\mathbf{a}(2)$}{a(2)}-finite Coxeter groups]{Classification of Coxeter groups with finitely many elements of \texorpdfstring{$\mathbf{a}$}{a}-value 2}
\title[Classification of {\bf a}(2)-finite Coxeter groups]{Classification of Coxeter groups with finitely many elements of {\bf a}-value 2}
\begin{abstract}
We consider Lusztig's $\mathbf{a}$-function on Coxeter groups (in the equal parameter
case) and classify all Coxeter groups with finitely many elements of
$\mathbf{a}$-value 2 in terms of Coxeter diagrams.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
\section{Introduction}
This paper concerns Lusztig's $\la$-function on Coxeter groups. The
$\la$-function was first defined for finite Weyl groups via their Hecke
algebras by Lusztig in
\cite{Mu}; subsequently, the definition was extended to affine Weyl groups in~\cite{Lusztig_II} and to arbitrary Coxeter groups in
\cite{LG}. The $\la$-function is intimately related to the study of
Kazhdan--Lusztig cells in Coxeter groups, the construction of Lusztig's asymptotic Hecke algebras, and the representation theory of Hecke
algebras; see, for example,~\cite{Mu}, \cite{Lusztig_II}, \cite{LG}, \cite{Geck_a} and~\cite{Geck_cellular}.
For any Coxeter group $W$ and $w\in W$, $\la(w)$ is a non-negative integer
obtained from the structure constants of the Kazhdan--Lusztig basis of the
Hecke algebra of $W$. While $\la$-values are often difficult to compute
directly, it is known that $\la(w)=0$ if and only if $w$ is the identity
element and that $\la(w)=1$ if and only if $w$ is a non-identity element with a
unique
reduced word (see Proposition~\ref{subregular}). If we define $W$ to be
\emph{$\la(n)$-finite} for $n\in \Z_{\ge 0}$ if $W$ contains
finitely many elements of $\la$-value $n$ and \emph{$\la(n)$-infinite}
otherwise, then it is also known that $W$ is $\la(1)$-finite if and only if each connected component
of the Coxeter diagram of $W$ is a tree
and contains at most one edge of weight higher than 3 (see Proposition~\ref{tree}). The goal of this paper is to obtain a similar classification of
$\la(2)$-finite Coxeter groups in terms of Coxeter diagrams.
Our interest in $\la(2)$-finite Coxeter groups comes from considerations about
the asymptotic Hecke algebra $J$ of $W$. This is an associative algebra
which may be viewed as a ``limit'' of the Hecke algebra of $W$, and each two-sided
Kazhdan--Lusztig cell $E\se W$ gives rise to a subalgebra $J_E$ of $J$ (see
\cite{LG}, Section 18).
%Moreover , $J$ appears as the Grothendieck ring of an interesting multi-tensor
%category $\mathcal{J}$ (in the sense of~\cite{EGNO}), and each $J_E$ appears
%as the Grothendieck ring of a subcategory of $\mathcal{J}$.
While $J$ has been interpreted geometrically for Weyl and affine Weyl
groups by Bezrukavnikov et al. in
\cite{Bez1}, \cite{Bez3} and~\cite{Bez2}, it is not well understood for
other Coxeter groups, and one approach to understand $J$ in these cases is to
start with the subalgebras $J_E$ in the case $E$ is finite, whence $J_E$ is a
\emph{multi-fusion ring} in the sense of~\cite{EGNO}. As the $\la$-function is known to be
constant on each cell, the presence of $\la(2)$-finite groups in our
classification that are not Weyl groups or affine Weyl groups potentially offers
interesting examples of multi-fusion rings of the form $J_E$ where $E$ is a
cell of $\la$-value 2. (For a study of algebras of the form $J_E$ where $E$ is a cell of
$\la$-value 1, see~\cite{Xu}.)
We now state our main results. For any (undirected) graph $G$, we define a \emph{cycle} in $G$ to be a sequence
$C=(v_1,v_2,\ldots, v_n,v_1)$ involving $n$ distinct vertices such that $n\ge 3$ and $\{v_1,v_2\},
\ldots$, $\{v_{n-1},v_n\}$ and $\{v_n,v_1\}$ are all edges in $G$, and we
say $G$ is \emph{acyclic} if it contains no
cycle. Our first main theorem is the following.
\begin{theo}
\label{first theorem}
Let $W$ be an irreducible Coxeter group with Coxeter diagram $G$.
\begin{enumerate}
\item\label{theo1.1_1} If $G$ contains a cycle, then $W$ is $\la(2)$-finite if and only if
$G$ is a complete graph.
\item\label{theo1.1_2} If $G$ is acyclic, then
$W$ is $\la(2)$-finite if and only if $G$ is one of the graphs in Figure
~\ref{fig:finite E}, where $n$ denotes the number of vertices in a graph whenever it appears as
a subscript in the label of the graph and there are $q$ and $r$ vertices
strictly to the left and the right of the trivalent vertex in $E_{q,r}$.
\begin{figure}[htb]\centering
\begin{tikzpicture}
\node(1) {$A_n$};
\node[main node] (11) [right=0.5cm of 1] {};
\node[main node] (12) [right=1cm of 11] {};
\node[main node] (13) [right=1cm of 12] {};
%\node (14) [right=0.5cm of 13] {$\cdots$};
\node[main node] (15) [right=1.5cm of 13] {};
\node[main node] (16) [right=1cm of 15] {};
\node (17) [right=0.5cm of 16] {$(n\ge 1)$};
\path[draw]
(11)--(12)--(13)
(15)--(16);
\path[draw,dashed]
(13)--(15);
\node(2) [below=0.5cm of 1] {$B_n$};
\node[main node] (21) [right=0.5cm of 2] {};
\node[main node] (22) [right=1cm of 21] {};
\node[main node] (23) [right=1cm of 22] {};
%\node (24) [right=0.5cm of 23] {$\cdots$};
\node[main node] (25) [right=1.5cm of 23] {};
\node[main node] (26) [right=1cm of 25] {};
\node (27) [right=0.5cm of 26] {$(n\ge 2)$};
\path[draw]
(21) edge node [above] {$4$} (22)
(22)--(23)
(25)--(26);
\path[draw,dashed]
(23)--(25);
\node (c) [below=1cm of 2] {$\tilde C_n$};
\node[main node] (c1) [right=0.5cm of c] {};
\node[main node] (c2) [right=1cm of c1] {};
\node[main node] (c3) [right=1cm of c2] {};
%\node (c4) [right=0.5cm of c3] {$\cdots$};
\node[main node] (c5) [right=1.5cm of c3] {};
\node[main node] (c6) [right=1cm of c5] {};
\node (c7) [right=0.5cm of c6] {$(n\ge 5)$};
\path[draw]
(c1) edge node [above] {$4$} (c2)
(c2)--(c3)
(c5) edge node [above] {$4$} (c6);
\path[draw,dashed]
(c3)--(c5);
\node (3) [below=1cm of c] {$E_{q,r}$};
\node[main node] (31) [right=0.4cm of 3] {};
\node[main node] (32) [right=1cm of 31] {};
%\node (33) [right=0.5cm of 32] {$\cdots$};
\node[main node] (34) [right=1.5cm of 32] {};
%\node (37) [right=0.5cm of 34] {$\cdots$};
\node[main node] (38) [right=1.5cm of 34] {};
\node[main node] (39) [right=1cm of 38] {};
\node (40) [right=0.5cm of 39] {$(q,r\ge 1)$};
\node[main node] (a) [above=0.8cm of 34] {};
\path[draw]
(34)--(a)
(31)--(32)
(38)--(39);
\path[draw,dashed]
(32)--(34)--(38);
\node(4) [below=0.5cm of 3] {${F}_n$};
\node[main node] (41) [right=0.5cm of 4] {};
\node[main node] (42) [right=1cm of 41] {};
\node[main node] (43) [right=1cm of 42] {};
%\node (44) [right=0.5cm of 43] {$\cdots$};
\node[main node] (45) [right=1.5cm of 43] {};
\node[main node] (46) [right=1cm of 45] {};
\node (47) [right=0.5cm of 46] {$(n\ge 4)$};
\path[draw]
(41)--(42)
(42) edge node [above] {$4$} (43)
(45)--(46);
\path[draw,dashed]
(43)--(45);
\node(h) [below=0.5cm of 4] {$H_n$};
\node[main node] (h1) [right=0.45cm of h] {};
\node[main node] (h2) [right=1cm of h1] {};
\node[main node] (h3) [right=1cm of h2] {};
%\node (h4) [right=0.5cm of h3] {$\cdots$};
\node[main node] (h5) [right=1.5cm of h3] {};
\node[main node] (h6) [right=1cm of h5] {};
\node (h7) [right=0.5cm of h6] {$(n\ge 3)$};
\path[draw]
(h1) edge node [above] {$5$} (h2)
(h2)--(h3)
(h5)--(h6);
\path[draw,dashed]
(h3)--(h5);
\node(i) [below=0.5cm of h] {$I_2(m)$};
\node[main node] (i1) [right=0.25cm of i] {};
\node[main node] (i2) [right=1cm of i1] {};
\node (i3) [right=0.5cm of i2] {$(5\le m\le \infty)$};
\path[draw]
(i1) edge node [above] {$m$} (i2);
\end{tikzpicture}
\caption{Irreducible $\la(2)$-finite Coxeter groups with acyclic diagrams.}
%\caption{}
\label{fig:finite E}
\end{figure}
\end{enumerate}
\end{theo}
\begin{rema}
\label{DE}
When $q=1$, $E_{q,r}$ coincides with the Coxeter diagram for the
Weyl group $D_{r+3}$. When $q=2$ and $r=2,3,4$, $E_{q,r}$ coincides with
the Coxeter diagram of the Weyl group $E_6, E_7$ and $E_8$, respectively. More generally,
for any larger value of $r$, $E_{2,r}$ coincides with $E_{r+4}$ in the
notation of~\cite{FC}, which is considered an extension of the type
$E$ Coxeter diagrams. In Section~\ref{case 2}, we will recall a result from
~\cite{FC} which uses the notations $D_n$ and $E_n$.
\end{rema}
Our second main theorem reduces the classification of reducible
$\la(2)$-finite Coxeter groups to that of irreducible Coxeter groups in the
following sense:
\begin{theo}
\label{second theorem}
Let $W$ be a reducible Coxeter group with Coxeter diagram $G$. Let $G_1,G_2,\ldots, G_n$ be the
connected components of $G$, and let $W_1,W_2,\ldots, W_n$ be their
corresponding Coxeter groups, respectively. Then the following are equivalent.
\begin{enumerate}
\item\label{theo1.3_1} $W$ is $\la(2)$-finite.
\item\label{theo1.3_2} The number $n$ is finite, \ie $G$ has finitely many connected components, and
$W_i$ is both $\la(1)$-finite and
$\la(2)$-finite for each $1\le i\le n$.
\item\label{theo1.3_3} The number $n$ is finite, and for each $1\le i\le n$, $G_i$ is a graph of the form $A_n (n\ge 1), B_n
(n\ge 2),
E_{q,r} (q,r\ge 1), F_n (n\ge 4), H_n(n\ge 3)$ or $I_2(m) (5\le m\le
\infty)$, \ie
$G_i$ is a graph from Figure~\ref{fig:finite E} other than $\tilde{C}_n
(n\ge 5)$.
%\begin{itemize}
%\item $A_n (n\ge 1)$;
%\item $B_n (n\ge 2)$;
%\item $E_{q,r} (q,r\ge 1)$;
%\item $F_n (n\ge 4)$;
%\item $H_n (n\ge 3)$;
%\item $I_2(m) (5\le m\le \infty)$.
%\end{itemize}
\end{enumerate}
\end{theo}
Of the claims in the theorems, Part~\eqref{theo1.1_2} of Theorem~\ref{first theorem}, \ie
the classification of $\la(2)$-finite Coxeter groups with acyclic Coxeter diagrams, turns
out to require the most amount of work. We describe our strategy for its
proof below.
%We need to show that for $W$ to be $\la(2)$-finite, it is sufficient and
%necessary for $G$ to be a graph from Figure~\ref{fig:finite E}.
A key fact we shall use is that each element of
$\la$-value 2 in a Coxeter group must be \emph{fully commutative} in the sense of Stembridge
(see Section~\ref{fc}). This implies, in particular, that we may
associate to any element $w$ with $\la(w)=2$ a poset called its \emph{heap}, a notion
well-defined for any fully commutative element. Heaps of fully commutative
elements will be a fundamental tool for this paper.
In showing that
$W$ is $\la(2)$-finite if $G$ is a graph in Figure~\ref{fig:finite E}, the
full commutativity of elements of $\la$-value 2 will reduce our work to the
cases $G=I_2(\infty), G=\tilde C_n$ or
$G=E_{q,r}$ where $\min(q,r)\ge 3$. Indeed, thanks to a result of Stembridge's in
\cite{FC}, $W$ contains finitely many fully
commutative elements if $G$ is any other graph from Figure~\ref{fig:finite E},
so $W$ must be $\la(2)$-finite in these cases. It will be easy to show that $W$ is
$\la(2)$-finite when $G=I_2(\infty)$, and the case $G=\tilde C_n$ will also be
easy thanks to a result of Ernst from~\cite{Ernst} on the
generalized Temperley--Lieb algebra of type $\tilde C_n$, therefore the only
case requiring more work is $G=E_{q,r}$ where $\min(q,r)\ge 3$. We will prove $W$ is
$\la(2)$-finite in this case via a series of lemmas in Section~\ref{case 2},
using arguments that involve heaps.
To show that $G$ must be a graph in Figure~\ref{fig:finite E} if $W$ is
$\la(2)$-finite, we first prove that $W$ would be $\la(2)$-infinite whenever $G$
contains certain subgraphs, then show that to avoid these subgraphs $G$ has to
be in Figure~\ref{fig:finite E}. For each of these subgraphs, we will construct
an infinite family of fully commutative elements that we call ``witnesses'' and verify that they have
$\la$-value 2. We will use three methods for these verifications:
%\vspace{-1em}
\begin{enumerate}%[leftmargin=2em]
\item\label{sect1_1} First, we
recall a powerful result of Shi from~\cite{Shi} that says each fully
commutative element $w$ in a Weyl or affine Weyl group satisfies
$\la(w)=n(w)$, where $n$ is a statistic defined using heaps. We prove
the same result for \emph{star reducible} groups (in the sense of
\cite{star_reducible}, see Proposition~\ref{star reducible a}), and use these results to show our witnesses have
$\la$-value 2 by showing they have $n$-value 2.
\item\label{sect1_2} In our second method, we recall
that the $\la$-function is constant on each two-sided Kazhdan--Lusztig cells of
$W$ and that each cell is closed under the so-called \emph{generalized star
operations} (see Section~\ref{star}),
then show our witnesses have $\la$-value 2 by relating them to elements of
$\la$-value 2 by these operations.
\item\label{sect1_3}
In our third and most technical method, we
again show our witnesses have $\la$-value~2 by showing they are in the same
cell as some other element of $\la$-value~2, but the proof will require more
careful arguments involving certain leading coefficients, or
``$\mu$-coefficients'', from Kazhdan--Lusztig polynomials.
\end{enumerate}
\looseness-1
The rest of the paper is organized as follows. In Section~\ref{Sect2}, we briefly recall
the background on
Coxeter groups and Hecke algebras leading to the definition of the $\la$-function, as well as the
definition and some properties of Kazhdan--Lusztig cells. In Section~\ref{sec:tools}, we introduce our main
technical tools for computing and verifying $\la$-values, namely, generalized
star operations and heaps of fully commutative elements. Sections~\ref{sufficient} and~\ref{sec:necessity} prove
the sufficiency and necessity of the diagram criterion of Theorem~\ref{first theorem}, respectively. The first three subsections of Section~\ref{sec:necessity}
contain a list of lemmas on the subgraphs that $G$ must avoid for $W$ to be
$\la(2)$-finite. The lemmas are grouped according to the method of verifying
the witnesses' $\la$-values, with~\ref{mu arguments} being the most technical
part of the paper. We prove Theorem~\ref{second theorem} in Section~\ref{reducible}.
Finally, in Section~\ref{conclusion} we briefly discuss several open problems naturally arising from
this paper, as well as their connections to other works in the literature. The
problems include generalizing the aforementioned result of Shi to arbitrary
Coxeter groups, enumerating
elements of $\la$-value 2 in $\la(2)$-finite Coxeter groups, and the
classification of $\la(3)$-finite Coxeter groups.
\section{Preliminaries}\label{Sect2}
The $\la$-function arises from the Kazhdan--Lusztig theory of Coxeter groups. We
review the relevant basic notions and facts in this section. Besides defining $\la$, we will recall the definitions of Kazhdan--Lusztig cells and the
``$\mu$-coefficients'' of Kazhdan--Lusztig polynomials. Both these notions will be key
to the proofs of our main theorems.
\subsection{Coxeter groups}
\label{Coxeter def}
Throughout the article, $W$ shall denote a Coxeter
group with a finite generating set $S$ and Coxeter matrix $M=[m(s,t)]_{s,t\in S}$. Thus,
$m(s,s)=1$ for all $s\in S$, $m(s,t)=m(t,s)\in \Z_{\ge 2}\cup \{\infty\}$ for all
distinct $s,t\in S$, and $W$ is generated by $S$ subject to the relations
$(st)^{m(s,t)}=1$ for all $s,t$ for which $m(s,t)$ is finite.
%We do not presume that the generating set $S$ is finite before we characterize
%when $W$ is $\la$(2)-finite. Instead, it will be evident after we establish Lemma
%\ref{infinite path} and Theorem~\ref{second theorem} that for $W$ to be
%$\la$(2)-finite, the set $S$ must be finite.
The defining data of each Coxeter group can be encoded via its \emph{Coxeter diagram}.
This is the weighted, undirected graph with vertex set $S$ and edge set
$\{\{s,t\}:s,t\in S,m(s,t)\ge 3\}$ such that each edge $\{s,t\}$ has weight
$m(s,t)$. Each edge is labelled by its weight except when the weight is $3$.
A Coxeter group is called \emph{irreducible} if its Coxeter diagram is
connected; otherwise the group is \emph{reducible}. Note that any reducible
Coxeter group $W$ with Coxeter diagram $G$ is isomorphic to the direct product
of the Coxeter groups encoded by the connected components of $G$.
Let $S^*$ be the free monoid generated by $S$. For any $w\in W$, we define the
\emph{length} of $w$, written $l(w)$, to be the minimum length of all words in
$S^*$ that express $w$. We call any such minimum-length word a \emph{reduced word} of
$w$. For any distinct $s,t\in S$, we call the relation
\[
sts\cdots=tst\cdots
\]
where both sides have $m(s,t)$ factors a \emph{braid relation}. Since
$s^2=(ss)^{m(s,s)}=1$ for all $s\in S$, the
braid relation is equivalent to
the relation $(st)^{m(s,t)}=1$ from the definition of $W$. When $m(s,t)=2$, we call the relation $st=ts$ a \emph{commutation
relation}, for $s$ and $t$ commute.
We can now recall the useful Matsumoto--Tits Theorem.
\begin{prop}[\cite{Tits};~{\cite[Theorem 1.9]{LG}}]
\label{Matsumoto-Tits}
Let $w\in W$. Then any pair of reduced words of $w$ can be obtained from each
other by a finite sequence of braid relations.
\end{prop}
For more basic notions and facts about Coxeter groups such as
the Bruhat order and its subword property, see~\cite{BB}.
\subsection{The \texorpdfstring{$\mathbf{a}$}{a}-function}
Let $W$ be an arbitrary Coxeter group. We recall Lusztig's definition of the function
$\la: W\ra \Z_{\ge 0}$ below.
Let $\cala=\Z[v,v\inverse]$. Following~\cite{LG}, we define the \emph{Hecke
algebra} of $W$ to be the unital $\cala$-algebra $H$ generated by the set
$\{T_s:s\in S\}$ subject to the relations
\begin{equation}
(T_s-v)(T_s+v\inverse)=0
\label{eq:our normalization}
\end{equation}
for all $s\in S$ and
\[
T_sT_tT_s\cdots = T_tT_sT_t\cdots
\]
for all $s,t\in S$, where both sides have $m(s,t)$ factors.
It is well-known
that $H$ has a \emph{standard basis} $\{T_w:w\in W\}$ where
$T_w=T_{s_1}\cdots T_{s_q}$ for any reduced word $s_1\cdots s_q$ ($s_1,\ldots,
s_q\in S$) of $w$, as well as a \emph{Kazhdan--Lusztig basis} $\{C_w:w\in W\}$
with remarkable properties (see~\cite{EW}). Now let $h_{x,y,z} (x,y,z\in W)$ be the elements of $\cala$
such that
\[
C_xC_y=\sum_{z\in W} h_{x,y,z}C_z
\]
for all $x,y$. By Lemma 13.5 of~\cite{LG}, for each $z\in W$, there exists a
unique integer $\la(z)\ge 0$ that satisfies the conditions
\begin{enumerate}
\item $h_{x,y,z}\in v^{\la(z)}\Z[v\inverse]$ for all $x,y\in W$,
\item $h_{x,y,z}\not\in v^{\la(z)-1}\Z[v\inverse]$ for some $x,y\in W$.
\end{enumerate}
This defines the function $\la: W\ra\Z_{\ge 0}$.
Elements of $\la$-value 0 or 1 are well understood in the following sense.
\begin{prop}[{\cite[Proposition~13.7]{LG}}, {\cite[Corollary~4.10]{Xu}}]
\label{subregular}
Let $W$ be an arbitrary Coxeter group, and let $1_W$ be the identity of
$W$. For all $w\in W$, we have
\begin{enumerate}
\item $\la(w)=0$ if and only if
$w=1_W$.
\item $\la(w)=1$ if and only if $w\neq 1_W$ and $w$ has a
unique reduced word.
\end{enumerate}
\end{prop}
\noindent Here, the set of non-identity elements with a unique reduced word is
known to be a \emph{two-sided Kazhdan--Lusztig
cell} (which we will define in the next subsection), and is sometimes called the \emph{subregular cell} (see~\cite{subregular}
and~\cite{Xu}).
%It would be interesting to know if a similar description in terms
%of reduced words exists for the set of elements of $\la$-value 2 in an
%arbitrary Coxeter group.
The following result classifies $\la(1)$-finite Coxeter groups, \ie Coxeter
groups with finite subregular cells, in
terms of Coxeter diagrams. We will use it in the proof of Theorem
\ref{second theorem} in Section~\ref{reducible}.
\begin{prop}[{\cite[Proposition~3.8]{subregular}}]
\label{tree}
Let $W$ be an irreducible Coxeter group with Coxeter diagram $G$. Then $W$ is
$\la(1)$-finite if and
only if $G$ is a tree and there is at most one edge of weight higher than 3
in $G$.
\end{prop}
\noindent
Besides the identity element and the elements in the subregular cell, it is
also easy to compute the $\la$-values
of products of commuting generators in a
Coxeter group, thanks to the following two results of Lusztig.
\begin{prop}[{\cite[Section 14]{LG}}]
\label{a local}
Let $W$ be a Coxeter group with generating set $S$. Let $I\se S$ and let $W_I$ be the subgroup of $W$ generated by
$I$. If $w\in W_I$, then $\la(w)$ computed in terms of $W_I$ is equal to
$\la(w)$ computed in terms of $W$.
\end{prop}
\begin{rema}
The above statement appears as part of Conjecture 14.2 in~\cite{LG}. However, it is
known to hold in the setting of this paper, which is called the
\emph{equal parameter} or the \emph{split} case (see~\cite{LG}, Section 15).
The same remark applies to Proposition~\ref{constant}.
\end{rema}
\begin{prop}[{\cite[Proposition~13.8]{LG}}]
\label{a for longest}
Let $W$ be a finite Coxeter group, and let
$w_0$ be the longest element of $W$. Then $\la(w_0)=l(w_0)$.
\end{prop}
\begin{coro}
\label{product a}
Let $W$ be a Coxeter group with generating set $S$. Let $I=\{s_1,s_2,\ldots,
s_k\}$
be a subset of $S$ such that $m(s_i,s_j)=2$ for all $1\le iy,
\end{cases}\\
C_y C_s&=
\begin{cases}
(v+v\inverse) C_y&\quad \text{if}\quad ysy.
\end{cases}
\end{align*}
\end{prop}
\begin{rema}
\label{symmetry}
It is known that $\mu_{x,y}=\mu_{x\inverse,y\inverse}$ for any $x,y\in W$
(see Section 5.6 and Corollary 6.5 of~\cite{LG}),
therefore the last formula in the proposition also holds with $\mu_{x,y}$ in
place of $\mu_{x\inverse,y\inverse}$.
\end{rema}
\begin{rema}
\label{conversion}
The paper~\cite{KL} uses a normalization of the Hecke algebra that is
different from ours, namely, it uses the relation $(T_s+1)(T_s-q)=0$ in place
of our Equation~\eqref{eq:our normalization}. Consequently, the
Kazhdan--Lusztig polynomials $P_{x,y}$ obtained in~\cite{KL}, which are
polynomials in $q$, do not exactly
agree with our Kazhdan--Lusztig polynomial $p_{x,y}$. However, it is
straightforward to
check that we may convert $P_{x,y}$ to $p_{x,y}$ by first substituting
$q$ by $v^2$ in $P_{x,y}$ and then multiplying the result by $v^{l(x)-l(y)}$.
In particular, our definition of the numbers $\mu_{x,y}$ agrees with that in
~\cite{KL}.
\end{rema}
\noindent The $\mu$-coefficients are often called the ``leading coefficients
of Kazhdan--Lusztig polynomials'' in the literature. Note that the elements $C_s (s\in S)$ generate $H$ by Proposition~\ref{KL basis mult}, so in a sense the $\mu$-coefficients control the
multiplication of the Kazhdan--Lusztig basis elements in the Hecke algebra. As
such, they also lead to an alternative characterization of the relations
$\prec_L$ and $\prec_R$: for each $y\in W$, define the \emph{left descent set} and
\emph{right descent set} of $y$ to be the sets
\[
\mathcal{L}(y)=\{s\in S:syy$, and $ys\le_R y$ if $ys>y$;
\item\label{coro2.13_2} If there exist elements $u,v\in W$ such that $y=uxv$ and
$l(y)=l(u)+l(x)+l(v)$ where $l$ is the length function on
$W$ \textup{(}see Section~\ref{Coxeter def} for the definition of
$l$\textup{)}, then
we have $y\le_{LR}x$ and $\la(y)\ge \la(x)$.
\end{enumerate}
\end{coro}
\begin{proof}
This is a simple corollary of Propositions~\ref{KL basis
mult} and~\ref{constant}. Note that~\eqref{coro2.13_2} follows from repeated application of~\eqref{coro2.13_1} and
Proposition~\ref{constant}, hence it suffices to prove~\eqref{coro2.13_1}. Suppose
$sy>y$. Then $D_{sy}(C_sC_y)=1$ by Proposition~\ref{KL basis mult},
therefore $sy\prec y$ and $sy\le_L y$ by definition. Similarly,
we have $ys\le_R y$ if $ys>y$. \end{proof}
\begin{prop}[{\cite[Proposition~5.4]{LG}}]
\label{degree bound}
Let $x,y\in W$. If $x\le y$, then $p_{x,y}=v^{-l(y)+l(x)}\mod
v^{-l(y)+l(x)+1}\Z[v]$.
\end{prop}
\begin{coro}
\label{difference 1}
Let $x,y\in W$. If $x\le y$ and $l(x)=l(y)-1$, then $p_{x,y}=v\inverse$
and hence $\mu_{x,y}=1$.
\end{coro}
\begin{proof}
This is immediate from the
well-known fact that $p_{x,y}\in \Z[v\inverse]$ (see \cite[Section~5.3]{LG}) and Proposition~\ref{degree bound}.
\end{proof}
%\begin{prop}[\cite{KL}, Statement (2.3.e)]
%\label{neighbor prec} Let $x,y\in W$ and $s\in S$.
%\begin{enumerate}
%\item
%Suppose $xx$. Then
%$x\prec_L y$ if and only if $y=sx$. Moreover, this implies
%$\mu(x,y)=1$.
%\item
%Suppose $xx$. Then
%$x\prec_R y$ if and only if $y=xs$. Moreover, this implies
%$\mu(x,y)=1$.
%\end{enumerate}
%\end{prop}
\begin{prop}[{\cite[Fact 5]{Warrington}}]
\label{extremal}
Let $x,y\in W$ be such that $l(x)