\section{ Preliminaries for root system and coroot system }\label{sec:2}

Let \( A=(a_{i,j})_{i,j \in I } \) be a
 (not necessarily symmetrizable) Cartan matrix 
 of a Kac-Moody Lie algebra \cite{Kac}\cite{MP}.
%We do not assume the symmetrizability of \( A \).
We denote the set of real numbers by \( \RR \).
%\denote{\( \RR \)}{the field of real numbers}
Let \( \h \) be an \( \RR \) -vector space % such that \( \dim_{ \RR } \h \geq \# I \) ,
 and \( \h^{*} \) the dual space of \( \h \)
 %endowed with
 and \( \langle , \rangle : \h^{*} \times \h \rightarrow \RR \) the cannonical bilinear form.
 %which is nondegenerate.
%Furthurmore,
We suppose
 the existence of linearly independent subsets
 \( \Pi := \set{ \alpha_i }{ i \in I } \subset \h^{*} \)
 and \( \Pi^\vee := \set{ \alpha_i^{\vee} }{ i \in I } \subset \h \)
 such that \( \langle \alpha_j , \alpha_i^\vee \rangle = a_{i,j} \).
An element \( \lambda \in \h^{*} \) is said to be an \emph{integral weight} if
\begin{equation*}
  \langle \lambda ,\, \alpha_i^{\vee} \rangle \in \ZZ , \quad i \in I
  .
\end{equation*}
The set of integral weights is denoted by \( P \).
For each \( i \in I \), we define the \emph{simple reflection} \( s_i \in GL( \h^{*} ) \) by:
\begin{equation*}
  s_i : \lambda \mapsto \lambda - \langle \lambda ,\, \alpha_i^{\vee} \rangle \alpha_i
  , \quad \lambda \in \h^{*}
  .
\end{equation*}
The group \( W \) generated by \( \set{ s_i }{ i \in I } \) is called the \emph{Weyl group},
 which acts on \( \h \) by:
% satisfying the relations \( s_i^2 = ( s_i s_j )^{m_{i,j}} = 1 \) ,
% where ,for \( i \neq j \) , \( m_{i,j}=2,3,4,6, \aut \infty \) ,
% according to whether \( a_{i,j} a_{j,i} =0,1,2,3, \aut \geq 4 \) .
\begin{equation*}
  \langle w( \lambda ) , w( h ) \rangle = \langle \lambda , h \rangle,
  \quad w \in W , \lambda \in \h^{*} , h \in \h
  .
\end{equation*}
We define the \emph{real root system} (resp. \emph{real coroot system})
 by \( \rroot := W \Pi \) (resp. \( \rroot^\vee := W \Pi^\vee \)).
The \emph{dual} \( \beta^\vee \in \rroot^\vee \) of a root \( \beta \in \rroot \) is defined by
\begin{equation*}
  \beta^\vee = w (\alpha_i^\vee) , \qquad (\beta = w(\alpha_i), w \in W, \alpha_i \in \Pi).
\end{equation*}
This is independent from choice of \( w \in W \) and \( \alpha_i \in \Pi \).
In general, we have
\begin{equation*}
  (w(\beta))^\vee = w(\beta^\vee), \qquad (\beta \in \rroot, w \in W).
\end{equation*}
The map \( \rroot \ni \beta \mapsto \beta^\vee \in \rroot^\vee \) is a bijection.
For \( R \subseteq \rroot \), we put \( R^\vee := \set{ \beta^\vee }{ \beta \in R } \subseteq \rroot^\vee \).
For each \( \beta \in \rroot \), we define \( s_{\beta} \in W \) by:
\begin{equation*}
  s_{\beta}( \lambda ) = \lambda - \langle \lambda ,\, \beta^\vee \rangle \beta, \quad \lambda \in \h^{*} ,
  \quad \text{ or, equivalently, by } \quad
\end{equation*}
 or, equivalently, by
%\begin{equation*}
%  s_{\beta}( h ) = h - \langle \beta ,\, h \rangle \beta^\vee, \quad h \in \h .
%\end{equation*}
We denote:
\begin{equation*}
  Q^\vee := \bigoplus_{i \in I} \ZZ \alpha_i^\vee ,
  \qquad
  \et
  \qquad
  Q_+^\vee := \bigoplus_{i \in I} \NN \alpha_i^\vee ,
\end{equation*}
where \( \NN \) is the set of non-negative integers.
We have \( \rroot^\vee \subseteq Q^\vee \).
Put \( \rroot_+^\vee := \rroot^\vee \cap Q_+^\vee \), and \( \rroot_-^\vee := \rroot^\vee \cap (-Q_+^\vee) \).
We have \( \rroot^\vee = \rroot_+^\vee \amalg \rroot_-^\vee \).
For \( \beta^\vee,\gamma^\vee \in \rroot^\vee \),
we denote \( \beta^\vee \leq \gamma^\vee \) if
\begin{equation*}
  \gamma^\vee - \beta^\vee \in Q_+^\vee .
\end{equation*}
This order is called the \emph{ordinary order} of coroots.

For each \( w \in W \), we define a set \( \inv{w} \, ( \subseteq \rroot_+ ) \) by:
\begin{equation*}
  \inv{w} := \set{ \gamma \in \rroot_+ }{ w^{-1}(\gamma)<0 }
  .
\end{equation*}
%For \( w, w' \in W \), we have:
%\begin{equation*}
%  \inv{w} = \inv{w'}
%  \Leftrightarrow w = w'
%  ,
%\end{equation*}

%For \( \beta , \gamma \in \rroot \), we have:
%\begin{equation*}
%  \langle \beta ,\, \gamma^\vee \rangle = 0
%  \Leftrightarrow \langle \gamma ,\, \beta^\vee \rangle = 0 ,
%\end{equation*}
% and
%\begin{equation*}
%  \langle \beta ,\, \gamma^\vee \rangle > 0
%  \Leftrightarrow \langle \gamma ,\, \beta^\vee \rangle > 0 .
%\end{equation*}















\begin{fullversion}

\begin{prop}\label{prop:comparison}
  Let \( \beta^\vee ,\, \gamma^\vee \in \rroot^\vee \).
  Suppose \( \langle \gamma ,\, \beta^\vee \rangle \geq 1 \).
  Then:
  \begin{enumerate}
    \item
    \( \gamma^\vee - \beta^\vee \) is a (possibly imaginary) coroot.
    \item
    we have either
    \( \beta^\vee < \gamma^\vee \), \( \beta^\vee = \gamma^\vee \), or \( \beta^\vee > \gamma^\vee \).
  \end{enumerate}
  In part (1), we note that we regard \( 0 \in Q^\vee \) as an imaginary coroot (see \cite{MP}).
\end{prop}
\begin{proof}
  We note that \( \beta^\vee \) and \( \gamma^\vee \) are real coroots. 
  We have a \( \gamma^\vee \)-chain of \( \beta^\vee \):
  \begin{equation*}
    s_\gamma(\beta^\vee) \!=\! \beta^\vee \!-\! \langle \gamma ,\, \beta^\vee \rangle \gamma^\vee \,,\,
    \beta^\vee \!-\! ( \langle \gamma ,\, \beta^\vee \rangle \!-\! 1 ) \gamma^\vee \,,\,
    \cdots \,,\,
    \beta^\vee \!-\! \gamma^\vee \,,\,
    \beta^\vee \,.
  \end{equation*}
  This is a sequence of (possibly imaginay) coroots (see \cite{MP}).
  Hence, we have either
  \( \beta^\vee - \gamma^\vee >0 \), \( \beta^\vee - \gamma^\vee =0 \), or \( \beta^\vee - \gamma^\vee <0 \).
  This proves part (1) and (2).
\end{proof}
\begin{Rem}
  Proposition \ref{prop:comparison} is a generalization of \cite[Proposition 5.3]{CHFstandard}.
\end{Rem}
\end{fullversion}


