\setlength{\parindent}{0pt}

\newcommand{\GU}{\mbox{\rm GU}}
\newcommand{\Sp}{\mbox{\rm Sp}}
\newcommand{\SO}{\mbox{\rm SO}}
\newcommand{\GL}{\mbox{\rm GL}}
\newcommand{\St}{\mbox{\rm St}}
\newcommand{\End}{\mbox{\rm End}}
\newcommand{\Hom}{\mbox{\rm Hom}}
\newcommand{\Infl}{\mbox{\rm Infl}}
\newcommand{\Ind}{\mbox{\rm Ind}}
\newcommand{\Res}{\mbox{\rm Res}}
\newcommand{\Fix}{\mbox{\rm Fix}}
\newcommand{\Defl}{\mbox{\rm Defl}}



%\MATHFRAK
\newcommand{\fA}{\mathfrak{A}}
\newcommand{\fB}{\mathfrak{B}}
\newcommand{\fC}{\mathfrak{C}}
\newcommand{\fD}{\mathfrak{D}}
\newcommand{\fE}{\mathfrak{E}}
\newcommand{\fF}{\mathfrak{F}}
\newcommand{\fG}{\mathfrak{G}}
\newcommand{\fH}{\mathfrak{H}}
\newcommand{\fI}{\mathfrak{I}}
\newcommand{\fJ}{\mathfrak{J}}
\newcommand{\fK}{\mathfrak{K}}
\newcommand{\fL}{\mathfrak{L}}
\newcommand{\fM}{\mathfrak{M}}
\newcommand{\fN}{\mathfrak{N}}
\newcommand{\fO}{\mathfrak{O}}
\newcommand{\fP}{\mathfrak{P}}
\newcommand{\fQ}{\mathfrak{Q}}
\newcommand{\fR}{\mathfrak{R}}
\newcommand{\fS}{\mathfrak{S}}
\newcommand{\fT}{\mathfrak{T}}
\newcommand{\fU}{\mathfrak{U}}
\newcommand{\fV}{\mathfrak{V}}
\newcommand{\fW}{\mathfrak{W}}
\newcommand{\fX}{\mathfrak{X}}
\newcommand{\fY}{\mathfrak{Y}}
\newcommand{\fZ}{\mathfrak{Z}}
\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fc}{\mathfrak{c}}
\newcommand{\fd}{\mathfrak{d}}
\newcommand{\fe}{\mathfrak{e}}
\newcommand{\ff}{\mathfrak{f}}
\newcommand{\ffg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\ffi}{\mathfrak{i}}
\newcommand{\fj}{\mathfrak{j}}
\newcommand{\fk}{\mathfrak{k}}
\newcommand{\fl}{\mathfrak{l}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fn}{\mathfrak{n}}
\newcommand{\fo}{\mathfrak{o}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}}
\newcommand{\fr}{\mathfrak{r}}
\newcommand{\fs}{\mathfrak{s}}
\newcommand{\ft}{\mathfrak{t}}
\newcommand{\fu}{\mathfrak{u}}
\newcommand{\fv}{\mathfrak{v}}
\newcommand{\fw}{\mathfrak{w}}
\newcommand{\fx}{\mathfrak{x}}
\newcommand{\fy}{\mathfrak{y}}
\newcommand{\fz}{\mathfrak{z}}

%\mathscr
\newcommand{\sA}{\mathscr{A}}
\newcommand{\sB}{\mathscr{B}}
\newcommand{\sC}{\mathscr{C}}
\newcommand{\sD}{\mathscr{D}}
\newcommand{\sE}{\mathscr{E}}
\newcommand{\sF}{\mathscr{F}}
\newcommand{\sG}{\mathscr{G}}
\newcommand{\sH}{\mathscr{H}}
\newcommand{\sI}{\mathscr{I}}
\newcommand{\sJ}{\mathscr{J}}
\newcommand{\sK}{\mathscr{K}}
\newcommand{\sL}{\mathscr{L}}
\newcommand{\sM}{\mathscr{M}}
\newcommand{\sN}{\mathscr{N}}
\newcommand{\sO}{\mathscr{O}}
\newcommand{\sP}{\mathscr{P}}
\newcommand{\sQ}{\mathscr{Q}}
\newcommand{\sR}{\mathscr{R}}
\newcommand{\sS}{\mathscr{S}}
\newcommand{\sT}{\mathscr{T}}
\newcommand{\sU}{\mathscr{U}}
\newcommand{\sV}{\mathscr{V}}
\newcommand{\sW}{\mathscr{W}}
\newcommand{\sX}{\mathscr{X}}
\newcommand{\sY}{\mathscr{Y}}
\newcommand{\sZ}{\mathscr{Z}}


%\mathcal
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cLR}{\mathcal{LR}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}


%field
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}

%\mathbf
\newcommand{\ba}{\mathbf{a}}
\newcommand{\bb}{\mathbf{b}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bd}{\mathbf{d}}
%\newcommand{\be}{\mathbf{e}}
\newcommand{\bff}{\mathbf{f}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\bi}{\mathbf{i}}
\newcommand{\bj}{\mathbf{j}}
\newcommand{\bk}{\mathbf{k}}
\newcommand{\bl}{\mathbf{l}}
\newcommand{\bm}{\mathbf{m}}
\newcommand{\bn}{\mathbf{n}}
\newcommand{\bo}{\mathbf{o}}
\newcommand{\bp}{\mathbf{p}}
\newcommand{\bq}{\mathbf{q}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\bt}{\mathbf{t}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}


%Greek
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\si}{\sigma}
\newcommand{\la}{\lambda}
\newcommand{\ga}{\gamma}
\newcommand{\eps}{\varepsilon}
%\newcommand{\th}{\theta}
\newcommand{\vphi}{\varphi}
\newcommand{\ka}{\kappa}
\newcommand{\ze}{\zeta}
\newcommand{\om}{\omega}
\newcommand{\de}{\delta}


%GREEK
\newcommand{\Al}{\Alpha}
\newcommand{\Be}{\Beta}
\newcommand{\Si}{\Sigma}
\newcommand{\La}{\Lambda}
\newcommand{\Ga}{\Gamma}
\newcommand{\Eps}{\Varepsilon}
%\newcommand{\th}{\theta}
\newcommand{\Vphi}{\Varphi}
\newcommand{\Ka}{\Kappa}
\newcommand{\Ze}{\Zeta}
\newcommand{\Om}{\Omega}
\newcommand{\De}{\Delta}

\newcommand{\bla}{\boldsymbol{\la}}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bpi}{\boldsymbol{\pi}}
\newcommand{\bsi}{\boldsymbol{\si}}


%% tilde
\newcommand{\ta}{\tilde{a}}
\newcommand{\tb}{\tilde{b}}
\newcommand{\tc}{\tilde{c}}
\newcommand{\td}{\tilde{d}}
\newcommand{\te}{\tilde{e}}
\newcommand{\tf}{\tilde{f}}
\newcommand{\tg}{\tilde{g}}
%\newcommand{\th}{\tilde{h}}
\newcommand{\ti}{\tilde{i}}
\newcommand{\tj}{\tilde{j}}
\newcommand{\tk}{\tilde{k}}
\newcommand{\tl}{\tilde{l}}
\newcommand{\tm}{\tilde{m}}
\newcommand{\tn}{\tilde{n}}
%\newcommand{\to}{\tilde{o}}
\newcommand{\tp}{\tilde{p}}
\newcommand{\tq}{\tilde{q}}
\newcommand{\tr}{\tilde{r}}
\newcommand{\ts}{\tilde{s}}
\newcommand{\tv}{\tilde{v}}

\newcommand{\tE}{\tilde{E}}
\newcommand{\tF}{\tilde{F}}

%%dot
\newcommand{\dw}{\dot{w}}

%% dot+greek
\newcommand{\dtau}{\dot{\tau}}


%% dot+bold
\newcommand{\dbs}{\dot{\bs}}
\newcommand{\dbla}{\dot{\bla}}



\newcommand{\dsB}{\dot{\sB}}


%% special : tilde+dot
\newcommand{\tde}{\tilde{\dot{e}}}
\newcommand{\tdf}{\tilde{\dot{f}}}
\newcommand{\tdE}{\tilde{\dot{E}}}
\newcommand{\tdF}{\tilde{\dot{F}}}

%% special : tilde+bold
\newcommand{\tbla}{\tilde{\bla}}
\newcommand{\tbmu}{\tilde{\bmu}}
\newcommand{\tbs}{\tilde{\bs}}
\newcommand{\tbr}{\tilde{\br}}

%% special : bold+greek
\newcommand{\bbe}{{\boldsymbol{\be}}}
\newcommand{\bal}{{\boldsymbol{\al}}}

%% hat
\newcommand{\hw}{\widehat{w}}
\newcommand{\hv}{\widehat{v}}
\newcommand{\hP}{\widehat{P}}
\newcommand{\hPprim}{\widehat{P'}}

%arrows
\newcommand{\ra}{\rightarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\lar}{\leftarrow}
\newcommand{\llar}{\longleftarrow}
\newcommand{\Lar}{\Leftarrow}
\newcommand{\eq}{\Leftrightarrow}
\newcommand{\eqdef}{\xLeftrightarrow{\text{def}}}
\newcommand{\da}{\downarrow}
\newcommand{\llra}{\longleftrightarrow}



%Special command
\newcommand{\Ue}{\mathcal{U}'_q (\widehat{\mathfrak{sl}_e})}
\newcommand{\Ul}{\mathcal{U}'_p (\widehat{\mathfrak{sl}_\ell})}
\newcommand{\Uinf}{\mathcal{U}_q ({\mathfrak{sl}_\infty})}
\newcommand{\Ueprime}{\mathcal{U}_q' (\widehat{\mathfrak{sl}_e})}
\newcommand{\sle}{\widehat{\mathfrak{sl}_e}}
\newcommand{\sll}{\widehat{\mathfrak{sl}_l}}
\newcommand{\bemptyset}{\boldsymbol{\emptyset}}
\newcommand{\charac}{\text{\rm char}}
\newcommand{\ds}{\displaystyle}
\newcommand{\mand}{\quad\text{and}\quad}
\newcommand{\Irr}{\text{\rm Irr}}
\newcommand{\wt}{\text{\rm wt}}
\newcommand{\id}{\text{\rm Id}}
\newcommand{\hla}{\widehat{\la}}
\newcommand{\w}{\wedge}
\newcommand{\chl}{\mathrm{ch}_{L}}
\newcommand{\chll}{\mathrm{ch}_{LL}}


\newcommand{\res}{\text{\rm res}}
\newcommand{\cont}{\text{\rm cont}}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
% \renewcommand{\theenumi}{\arabic{enumi}}

\newcommand{\moinsun}{$-$1}
\newcommand{\moinsdeux}{$-$2}
\newcommand{\moinstrois}{$-$3}
\newcommand{\moinsquatre}{$-$4}
\newcommand{\moinscinq}{$-$5}
\newcommand{\moinssix}{$-$6}
\newcommand{\dix}{10}
\newcommand{\onze}{11}
\newcommand{\douze}{12}
\newcommand{\treize}{13}
\newcommand{\quatorze}{14}
\newcommand{\quinze}{15}
\newcommand{\seize}{16}
\newcommand{\dixsept}{17}
\newcommand{\dixhuit}{18}
\newcommand{\dixneuf}{19}
\newcommand{\vingt}{20}
\newcommand{\vingtun}{21}
\newcommand{\vingtdeux}{22}
\newcommand{\vingttrois}{23}
\newcommand{\vingtquatre}{24}
\newcommand{\vingtcinq}{25}


\newcommand{\m}{\text{-}}

\def\labelitemi{--}


\newenvironment{prf}{{\bf Proof.}}{\hfill $\Box$ \\[-1.0ex]}

\newtheorem{num}{Notation}[section]

\newtheorem{defi}[num]{Definition}
\newtheorem*{defi*}{Definition}
\newtheorem{defis}[num]{Definitions}
\newtheorem{numb}[num]{}
\newtheorem{thm}[num]{Theorem}
\newtheorem*{thm*}{Theorem}
\newtheorem{lem}[num]{Lemma}
\newtheorem*{lem*}{Lemma}
\newtheorem{prop}[num]{Proposition}
\newtheorem*{prop*}{Proposition}
\newtheorem{prope}[num]{Property}
\newtheorem*{prope*}{Property}
\newtheorem{cor}[num]{Corollary}
\newtheorem*{cor*}{Corollary}
\newtheorem*{ques*}{Question}
\newtheorem{conj}[num]{Conjecture}
\newtheorem*{conj*}{Conjecture}
% \newtheorem{xmpl}[num]{Example}
% \newtheorem*{xmpl*}{Example}
% \newtheorem{rem}[num]{Remark}
\newtheorem*{rem*}{Remark}
\newtheorem{speech}[num]{}
\pagenumbering{arabic}


\theoremstyle{remark}
\newtheorem{exa}[num]{Example}
\newtheorem{rem}[num]{Remark}

\raggedbottom