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\newcommand{\lra} {\longrightarrow} %solo per questo papero \newcommand{\Scon} {\text{Sh}(\calS)} \newcommand{\LCF} {{\mathcal LC}_{\mF}} \newcommand{\LCC} {{\mathcal LC}_{\mC}} \newcommand{\duale} {D} \newcommand{\pairing} {< \: , >} \newcommand{\bra} {< \! \!} \newcommand{\ket} {\! \! >} \newcommand{\slice} {\mbox{${\mathcal{S}} _{a,x}$} } \newcommand{\sopraslice} {\mbox{${\widetilde {\mathcal{S}} _{a,x}}$} } \begin{document} \title{A remark on quiver varieties and Weyl groups} \author{Andrea Maffei} \address{Mathematical Science Research Institute, Berkeley} \email{amaffei@mat.uniroma1.it} \thanks{The research was supported by the MSRI, Berkeley} \begin{abstract} In this paper we define an action of the Weyl group on the quiver varieties $M_{m,\grl}(d,v)$ with generic $(m,\grl)$. To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectedness for the smooth quiver variety $M(d,v)$ and normality for $M_0(d,v)$ in the case of a quiver of finite type and $d-v$ a regular weight. \end{abstract} \maketitle In \cite{Na1,Na2} Nakajima defined quiver varieties and show how to use them to give a geometric construction of integrable representation of Kac-Moody algebras. Luckily these varieties can be used also to give a geometric construction of representations of Weyl groups. In \cite{Lu:Q4}, Lusztig constructed a representation of the Weyl group on the homology of quiver varieties. His construction is similar to the construction of Springer representations. In \cite{Na2}, Nakajima gave an construction of isomorphism $\Phi_{\grs,\zeta}(d,v):\goM_{\zeta}(d,v) \lra \goM_{\grs \zeta}(d,\grs(v-d)+d)$ in the case of a quiver of finite type. His construction was analytic and relies on a description of quiver varieties as moduli spaces of instantons on ALE spaces. The main result of this paper is a direct and algebraic construction of these isomorphism which works for a general quiver without simple loops. To do it we also describe a set of generators of the algebra of covariant functions. The paper is organized as follows. In the first section we fix the notation and we give the definition of a quiver variety: $M_{m,\grl}(d,v)$ where $m, \grl$ are two parameter, $d$ is a weight of the algebra associated to the quiver and $v$ an element of the root lattice. We are interested to quiver varieties as algebraic varieties but to explain one of the applications we need to give also the hyperK\"ahler construction of a quiver variety. We use a result of Migliorini \cite{Migliorini} to explain the connection between the two constructions. Algebraic quiver varieties are defined as the $\mathbf{Proj}$ scheme of a ring of covariants. In the second section we describe a set of generators of this ring. In a special case which is not directly to Nakajima's quiver varieties we are also able to give a more precise results and to describe a basis of the vector space of $\chi$-covariants functions. In the third section we use this description to generalize a construction of Lusztig \cite{Lu:Q4}. Namely for any element of the Weyl group we construct an isomorphism $\Phi_{\grs}$ between $M_{m,\grl}(d,v)$ and $M_{\grs m,\grs\grl}(d,\grs(v-d)+d)$ if $m,\grl$ are generic. In the fourth section, following Nakajima \cite{Na1}, we show how to use the action constructed in section 3 (and the connection between the hyperK\"ahler construction and the algebraic construction) to describe an action of the Weyl group on the homology of a class of quiver varieties. This action is different from the one constructed by Lusztig in \cite{Lu:Q4}. In the fifth section we give a result which reduce the study of geometric and algebraic properties of quiver varieties $M_{0,0}(d,v)$ to the case $d-v$ dominant. In the sixth section we prove the normality of the quiver variety $M_0(d,v)$ and the connectedness of $M(d,v)$ in the case of a quiver of finite type and $d-v$ a regular weight. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Notations and definitions} In this section we give the definition of quiver varieties. Except some minor change all definition are due to Nakajima \cite{Na1,Na2}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The graph} Let $(I,H)$ be a finite oriented graph: $I$ is the set of vertices that we suppose of cardinality $n$, $H$ the set of arrows and the orientation is given by the two maps $$ h \longmapsto h_0 \mand h \longmapsto h_1 $$ from $H$ to $I$. We suppose also that: \begin{enumerate} \item $ \forall \, h\in H \quad h_0 \neq h_1$, \item an involution $ h\mapsto \bar h $ of $H$ without fixed points and satisfying ${\bar h}_0 = h_1$ is fixed, \item a map $\gre : H \lra \{-1,1\}$ is given such that $\gre(\bar h) = - \gre(h)$. We define $\grO = \{ h \in H \st \gre(h)= 1\}$ and ${\wbar {\grO}} = \{ h \in H \st \gre(h)= -1\}$. \end{enumerate} Observe that given a symmetric graph without loops is always possible to define $\gre$ and an involution $\bar {\;}$ as above. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{The Cartan matrix and the Weyl group} Let $A$ be the matrix whose entries are the numbers $$a_{ij}= card\{ h \in H \, : \, h_0=i \mand h_1=j \}.$$ We define a generalized symmetric Cartan matrix by $C=2I-A$. Following \cite{Lusztig:QG} an $X,Y$-regular root datum $(I,X,X \cech ,\pairing)$ with Cartan matrix equal to $C$ is defined in the following way: \begin{enumerate} \item $X \cech$ and $X$ are finitely generated free abelian groups, \item $\pairing: X \times X \cech \lra \mZ$ is a perfect bilinear pairing, \item two linearly independent sets $\Pi= \{\gra_i \tc i \in I \} \subset X$ and $\Pi \cech =\{ \gra_i\cech \tc i \in I \} \subset X \cech$ are fixed and we set $Q = \langle \Pi \rangle$ and $Q \cech = \langle \Pi \cech \rangle$, \item $\bra \gra_i \, , \gra_j \cech \ket = c_{ij}$, \item (nonstandard) $\rank X = \rank X \cech = 2n - \rank C$, \item (nonstandard) a linearly independent set $\{ \gro_i \tc i\in I \}$ of $X$ such that $\bra \gro_i , \gra_j \cech \ket = \grd_{ij}$ is fixed. \end{enumerate} Once $C$ is given it is easy to construct a data as above. We call $\goh$ the complexification of $X \cech$ and we observe that through the bilinear pairing $\pairing$ we can identify $\goh^*$ with the complexification of $X$. We observe also that the triple $(\goh,\Pi,\Pi \cech)$ is a realization of the Cartan matrix $C$ (\cite{Kac} pg.1). The Weyl group $W$ attached to $C$ is defined as the subgroup of $Aut(X) \subset GL(\goh^*)$ generated by the reflections \begin{equation}\label{eq:generatoriW} s_i: x \longmapsto x - \! \bra x, \gra_i \cech \ket \gra_i. \end{equation} Observe that the dual action is given by $s_i( y)= y - \! \bra \gra_i, y \ket \gra_i\cech$ and that the lattices $Q$ and $Q\cech$ are stable for these actions. So the annihilator $\overset{\circ}{ Q\cech} =\{x \in X \tc \bra x,y\ket \, = 0 \: \forall y \in Q \cech\}$ is also stable by $W$ and we can consider the action of $W$on the lattice $P=X \, / \, \overset{\circ}{ Q\cech} \isocan \Hom_{\mZ}(Q,\mZ)$ and we call $x \mapsto \wbar x$ the projection from $X$ to $P$. We observe also that this projection is an isomorphism from the lattice $\wt P$, that is not $W$-stable, spanned by $\{ \gro_i \tc i \in I\}$ and $P$. Finally we observe that $$\wbar {\gra}_i = \sum _{j \in I} c_{ij} \wbar {\gro}_j.$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{$d, v$ and the space of all matrices} For the exposition it will be useful to identify the set $I$ with the set of integers $\{ 1, \dots, n \}$. Let $d=(d_1,\dots,d_n)$ and $v=(v_1,\dots,v_n)$ be two $n$-tuples of integers. We also think of $d, v$ as elements of $X$ in the following way: \begin{equation}\label{dvX} d = \sum_{i\in I} d_i {\gro}_i \; \mand \; v= \sum _{i \in I} v_i {\gra}_i\:; \end{equation} and through this identification we define an action of $W$ on $v$. We define also $v\cech = \sum_{i \in I} v_i \gra_i \cech \in Q \cech$. Once $d,v$ are fixed we fix complex vector spaces $D_i$ and $V_i$ of dimensions $d_i$ and $v_i$ and we define the following spaces of maps: \begin{subequations} \label{eq:decompS} \begin{align} S_{\grO}(d,v) &= \bigoplus _{i \in I} \Hom(D_i,V_i) \oplus \bigoplus _{h \in \grO} \Hom(V_{h_0},V_{h_1}),\\ S_{\wbar \grO}(d,v) &=\bigoplus _{i \in I} \Hom(V_i,D_i) \oplus \bigoplus _{h \in \wbar \grO} \Hom(V_{h_0},V_{h_1}), \\ S(d,v)&= S_{\grO}(d,v) \oplus S_{\wbar \grO}(d,v) . \end{align} \end{subequations} More often, when it will not be ambiguous we will write $S_{\grO}, S_{\wbar \grO}$ and $S$ instead of $S_{\grO}(d,v), S_{\wbar \grO}(d,v)$ and $S(d,v)$. For each $h \in H$ (resp. $i\in I$) we define the projection $B_h$ (resp. $\grg_i$ and $\grd_i$) from $S$ to $\Hom(V_{h_0},V_{h_1})$ (resp. $\Hom(D_i,V_i)$ and $\Hom(V_i,D_i)$) with respect to the decomposition described in \eqref{eq:decompS}. When an element $s$ of $S$ is fixed we will often write $B_h$ (resp. $\grg_i$, $\grd_i$) instead of $B_h(s)$ (resp. $\grg_i(s)$ and $\grd_i(s)$). We will also use $\grg$ for $(\grg_1 , \dots , \grg_n)$, $\grd$ for $(\grd_1 , \dots , \grd_n)$ and $B$ for $(B_h)_{h \in H}$ and often we will write an element of $S$ as a triple $(B,\grg,\grd)$. Once $D_i,V_i$ and an element $s$ of $S$ are fixed we define also: \begin{subequations}\label{defTab} \begin{align} T_i &= D_i \oplus \bigoplus _{h \tc h_1 =i} V_{h_0} \, , \label{defT}\\ a_i &= a_i(s) = (\grd_i(s), (B_{\bar h}(s))_{h \tc h_1 =i}): V_i \lra T_i \, ,\label{defa}\\ b_i &= b_i(s) = (\grg_i(s), (\gre(h)B_{ h}(s))_{h \tc h_1 =i}): T_i \lra V_i \, .\label{defb} \end{align} \end{subequations} We will identify the dual of space of the $\mC$-linear maps $\Hom (E,F)$ between two finite dimensional vector spaces with $\Hom(F,E)$ through the pairing $\bra \grf , \psi \ket = \Tr ( \grf \comp \psi )$. So we can describe $S$ also as $S_{\grO} \oplus S_{\grO}^* = T^* S_{\grO}$ and we observe that a natural symplectic structure $\gro$ is defined over $S$ by $$ \gro((s_{\grO},s_{\wbar \grO}),(t_{\grO},t_{\wbar \grO})) = \bra s_{\grO},t_{\wbar \grO}\ket -\bra t_{\grO},s_{\wbar \grO}\ket. $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Hermitian structure on $S$} \label{hyperKahlerstructure} We suppose now that the spaces $D_i$, $V_i$ are endowed with hermitian metrics. So we can speak of the adjoint $\grf^*$ of a linear map between these spaces, and we have a positive definite hermitian structure $h$ on $S$ with explicit formula: \begin{align} h( (B,\grg,\grd),(\wt B,\wt \grg,\wt \grd)) &= \sum _{h\in H} \Tr (B_h {\wt B}_{ h}^*) + \sum _{i \in I } \Tr (\grg_i {\wt \grg}_i^* + {\wt \grd}_i^* \grd_i) \label{defh} \\ &= \sum_{i \in I} \Tr(a_i {\wt a}_i^* + {\wt b}_i^* b_i) \notag \end{align} and an associated real and closed symplectic form $\gro_I(s,t)=\Re h (\im s, t) - \Im h(s,t)$. \subsection{Group actions and moment maps}\label{groupactions} We can define an action of the groups $G=GL(V)=\prod GL(V_i)$ and $GL(D)=\prod GL(D_i)$ on the set $S$ in the following way: \begin{align} g (B_h,\grg_i,\grd_i) &= (g_{h_1}B_h g_{h_0}^{-1},g_i \grg _i, \grd_i g_i^{-1}) &&\mfor g=(g_i)\in GL(V), \\ g (B_h,\grg_i,\grd_i) &= (B_h , \grg _i g_i^{-1}, g_i \grd_i ) &&\mfor g=(g_i)\in GL(D). \end{align} Observe that these actions commute and that $\gro$ is $GL(V)$ invariant. Moreover if $U=U(V)=\prod U(V_i)$ is the group of unitary transformations in $GL(V)$ the real symplectic form $\gro_I$ is $U(V)$ invariant. Define $\grm, \grm_I \colon S \lra \gog=\oplus gl(V_i)$ by the following explicit formulas: \begin{align*} \grm_i(B,\grg,\grd)&= \sum_{h \in H \tc h_1=i} \gre(h) B_h B_{\bar h} + \grg_i \grd_i = b_i a_i\, , \\ \grm_{I,i}(B,\grg,\grd)&= \frac{\im}{2} \left(\sum _{h\in H\tc h_1=i} B_h B_h^* - B_{\bar h}^* B_{\bar h} + \grg_i \grg_i^* -\grd_i^* \grd_i \right) =\frac{\im}{2} ( b_i b_i^* - a_i^* a_i), \end{align*} If we identify $\gog^*=\Hom_{\mC}(\gog,\mC)$ (resp. $\gou^* = \Hom_{\mR} (\gou , \mR)$) with $\gog=\oplus gl(V_i)$ (resp. $\gou$) through the pairing $\bra (x_i)\, , (y_i) \ket = \sum _i \Tr (x_i y_i)$, we can observe that $\grm$ is a moment map for the action of $G$ on the symplectic manifold $(S, \gro)$ and that $\grm_I$ is a moment map for the action of $U$ on the symplectic manifold $(S, \gro_I)$. It is common to group all these moment maps together and to define an hyperK\"ahler moment map $$ \wt \grm = (\grm_I, \grm): S \lra \gou \oplus \gog = (\mR \oplus \mC) \otimes_{\mR} \gou. $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Quiver varieties as hyperK\"ahler quotients} \label{definitionquiver} Let $\grz_i = (\xi_i,\grl_i) \in \mR \oplus \mC$ and $\grz = (\grz_1,\dots,\grz_N)$. We define: $$ \goL_{\grz}(d,v) = \{s \in S \tc \grm (s) - \grl_i \Id_{V_i} = 0 \,\mand\, \grm_{I,i} (s) - \im \xi_i \Id_{V_i} = 0 \}. $$ We observe that $\goL_{\grz}(d,v)$ is stable for the action of $U(V)$, so, at least as a topological Hausdorff space we can define the \emph{quiver variety of type} $\grz$ as $$ \goM_{\grz} (d,v) = \goL_{\grz}(d,v) / U(V). $$ It will be convenient to define also $\goM_{\grz} (d,v)=\vuoto$ if $d,v \in \mZ^n $ and there exists $i$ such that $v_i <0$ or $d_i <0$ for some $i$. We call $\goZ = \mR^n \oplus \mC^n$ and we observe that we can identify it to $(\mR \oplus \mC) \otimes _{\mZ}P$ through: \begin{equation}\label{zetaRCP} (\xi_1,\dots,\xi_n,\grl_1,\dots,\grl_n) \longleftrightarrow \sum _{i \in I} (\xi_i , \grl_i) \wbar{\gro}_i. \end{equation} In particular we consider an action of the Weyl group $W$ on $\goZ$ through this identification. \begin{oss}\label{oss:centrogoZ} There is a surjective map from $\goZ$ to $Z_U(\gou) \oplus Z_G(\gog)$: $$ (\xi_1,\dots,\xi_n,\grl_1,\dots,\grl_n) \lra \sum _{i \in I} ( \im \xi_i,\grl_i) \Id_{V_i} $$ Observe that $\goL_{\zeta}$ is the fiber of $\wt \mu$ over the image of $\zeta$ in $Z_U(\gou) \oplus Z_G(\gog)$. \end{oss} \begin{oss}\label{oss:riduzioneanonzero} If $v , d \geq 0$ define: $I^* = \{ i \in I \st v_i \neq 0\}$, $H^* = \{ h \in H \st h_0,h_1 \in I^*\}$, $\gre^* = \gre\bigr|_{H^*}$, $v^* = (v_i)_{i \in I^*}$, $d^* =(d_i )_{i \in I^*}$, and $\zeta^*= (\grz_i)_{i \in I^*}$ then it is clear that $$ \goL_ {\grz^*}(d^*,v^*) \isocan \goL_{\grz}(d,v) \;\mand\; \goM_ {\grz^*}(d^*,v^*) \isocan \goM_{\grz}(d,v). $$ \end{oss} Except for the last equivalence which is trivial in our case the following is a general well known fact (\cite{GIT} ch. 8). \begin{lem} \label{regolaritamu} Let $s \in \goL_{\zeta}$ then \begin{gather*} d{\wt {\grm}}_s \text{ is surjective} \iff d\grm_s \text{ is surjective} \iff d\grm_I \text{ is surjective } \iff \\ \iff \dim \mathrm{Stab}_G\{s\}= 0 \iff \mathrm{Stab}_G\{s\}=\{1_G\} \end{gather*} \end{lem} \begin{dfn} If $u \in \mZ^n = Q\cech$ $A\subset Q\cech$ we define $$ \calH_u=\{\grz =(\xi,\grl)\in \goZ \tc \bra \xi \, , u \cech \ket= \bra \grl \, , u \cech \ket=0 \} \; \mand \; \calH_A= \bigcup_{u\in A} H_u. $$ Let now $U_v = \{ u \in \mN^n-\{0\} \msuchthat$ $0 \leq u_i \leq v_i \}$ and $\calH = \calH_{U_v}$. $\calH$ is a union of a finite number of real subspace of $\goZ$ of codimension $3$. \end{dfn} \begin{lem}[Nakajima, \cite{Na1}] \label{lem:regmu} If $\grz \in \goZ - \calH $ and $\wt {\grm}(s) = \grz$ then $\mathrm{Stab}_G\{s\}={1_G}$. \end{lem} As a consequence of the above lemma and general results on on hyperK\"ahler manifolds (for example \cite{Hit} or \cite{HKLR}) we obtain the following corollary. \begin{cor}\label{dimensioniquiver} If $\grz \in \goz - \calH$ then if it is not empty $\goM_{\grz} (d,v)$ is a smooth hyperK\"ahler manifold of real dimension $2 \bra 2d-v , v \cech \ket$. \end{cor} %If there exists $s \in S$ such that %$d\wt{\grm}_s $ is surjective then %for all $\grz \in \goz - \calH$, $\goM_{\grz} (d,v)$ is a smooth %hyperK\"ahler manifold of real dimension %$2 \bra 2d-v , v \cech \ket$. %\begin{proof} %By the above lemmas and general results on hyperK\"ahler manifolds %(for example \cite{Hit} or \cite{HKLR}) it's enough to prove that %$\wt {\grm}$ is surjective. %It is convenient to consider $\wt \grm = (\grm_I , \grm_J , \grm_K)\colon %S \lra \mR^3 \otimes \gou$. It is clear that we have an action of $SO(3,\mR)$ %on the three complex structures and that for any triple of such structures we have %\begin{alignat*}{3} %\grm_I(Ip)&=\grm_I(p)&\quad \grm_I(Jp)&=-\grm_I(p) % &\quad \grm_I(Kp)&=-\grm_I(p)\\ %\grm_J(Ip)&=-\grm_J(p)&\quad \grm_J(Jp)&=\grm_J(p) % &\quad \grm_J(Kp)&=-\grm_J(p)\\ %\grm_K(Ip)&=-\grm_K(p)&\quad \grm_K(Jp)&=-\grm_K(p) % &\quad \grm_K(Kp)&=\grm_K(p) %\end{alignat*} %%%%%So $\wt{\grm} (s_0)+\wt{\grm} (s_1)+\wt{\grm} (s_2)+\wt{\grm} (s_3)=0$ and that %%%%%%$\Im f$ is a cone. %Hence we can assume that $\wt \grm(s) = (\xi_I,0,0)\in \mR^3 \otimes \gou$. %So $\wt \grm(Js)= -\wt \grm(s)$ and moreover $d\wt \grm_{Js}$ is also surjective. %Therefore $\Im f$ contains an open cone $C$ and its opposite %$-C$ %Since $\wt \grm $ is an homogeneous polynomial %of degree two it follows that it is surjective. %\end{proof} %\begin{oss} %This proposition is not prooved in \cite{Na1} but this proof seems to %me necessary. We observe also that %$\bra 2d-v,v \cech \ket \geq 0$ it is not enough to garantee that %$\goM_{\grz}\neq \vuoto$. Indeed if $\grz=(\xi_i,\grl_i) \in \goz - H$ and %$\grl_i \neq 0$ it follows from $b_ia_i=\grl_i \Id$ that $b_i$ is surjective and %that $d_i + \sum _j a_{ij}v_j \geq v_i$. %\end{oss} %Since $ \goz - \calH$ is connected we have also: %\begin{cor} %If $\grz,\grz^{\prime} \in \goz - \calH$ then $\goM_{\grz} (d,v)$ %and $\goM_{\grz^{\prime}} (d,v)$ are isomorphic as $C^{\infinito}$-manifold. %\end{cor} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Geometric invariant theory and moment map} \label{ssec:GIT} In this section we explain the relation between the moment map and the GIT quotient proved by Kempf, Ness \cite{KeNe}, Kirwan \cite{GIT} and others. To be more precise we need a generalization of their results in the case of an action on an affine variety proved by Migliorini \cite{Migliorini}. Let $X$ be an affine variety over $\mC$ and $G$ a reductive group acting on $X$. We can assume that $X$ is a closed sub-variety of a vector space $V$ where $G$ acts linearly. Let $h$ be an hermitian form on $V$ invariant by the action of a maximal compact group $U$ of $G$ and define a real $U$-invariant symplectic form on $V$ by $$\eta (x,y) = \Re h(\im x,y).$$ Then we can define a moment map $\nu : V \lra \gou^*=\Hom_{\mR}(\gou,\mR) $: $$\bra \nu(x), u \ket = \tfrac{1}{2} \eta ( u \cdot x , x).$$ We observe that the real symplectic form $\eta$ restricted to a complex sub-manifold is always non degenerate and that $\mu$ restricted to the non singular locus of $X$ is a moment map for the action of $U$ on $X$. Now let $\chi$ be a multiplicative character of $G$. We observe that for all $g \in U$ we have $|\chi(g)|=1$ so $\im \,d\chi : u \lra \mR$. In particular we can think to $\im\,d\chi$ as an element of $\gou ^*$. Moreover we observe that it is invariant by the dual adjoint action, hence it makes sense to consider the quotient: $$\goM = \nu^{-1}(i\,d\chi) / U.$$ As we saw our variety are a particular case of this construction. \medskip On the other side we can consider the GIT quotient. Let us remind the definition. If $\grf$ is a character of $G$ we consider the line bundle $L_{\grf} = V \times \mC$ on $V$ with the following $G$-linearization: $$ g(x,z)= (g\cdot x,\grf(g)z). $$ An invariant section of $L_{\grf}$ is determined by an algebraic function $f:V\lra \mC$ such that $f(gx)=\grf(g)f(x)$ for all $g \in G$ and $x \in V$. We use the same symbol $L_{\grf}$ also for the restriction of $L_{\grf}$ to $X$. Given a rational action of $G$ on $\mC$-vector space $A$ we define \begin{gather*} A_{\grf,n} = \{ a \in A \tc g\cdot a = \grf^{-n}(g) a \text{ for all } g \in G\},\\ A_{\grf} = \bigoplus _{n=0}^{\infty} A_{\grf,n} \quad \text{ as a graded vector space.} \end{gather*} Hence we have that $H^0(X,L_{\grf})^G = \mC[X]_{\grf,1}$. We observe that if $I$ is the ideal of algebraic function on $V$ vanishing on $X$ then $$H^0(X,L_{\grf})^G = \frac{H^0(V,L_{\grf})^G}{I_{\grf,1}}.$$ This last fact can be proved easily for example averaging a $\grf$ equivariant function $f$ on $X$ in the following way: $$\tilde f (v) = \int_U \grf^{-1}(u) f(u\cdot v) \, du .$$ \begin{dfn} A point $x$ of $X$ is said to be $\chi$-\emph{semi-stable} if there exist $n>0$ and $f \in H^0(X,L_{\chi}^{\otimes n})^G$ such that $f(x) \neq 0$. We observe that by the remark above a point of $X$ is $\chi$-semi-stable if and only if is $\chi$-semi-stable as a point of $V$. We call $X^{ss}_{\chi}$ (resp. $V^{ss}_{\chi}$) the open subset of $\chi$-semi-stable points of $X$ (resp. $V$). \end{dfn} \begin{prp}[ \cite{GIT,Newstead}] There exists a good quotient of $X^{ss}_{\chi}$ by the action of $G$ and we have that $$ X^{ss}_{\chi} //G = \mathrm{Proj}\, \mC[X]_{\chi}.$$ Moreover $\mathrm{Proj} \,\mC[X]_{\chi}$ is a finitely generated $\mC$-algebra and a natural projective map $$\pi : X^{ss}_{\chi} //G \lra X//G = \mathrm{Spec} \, \mC[X]^G$$ is defined. \end{prp} In the case of $\chi \coinc 1$ the following fact is well known: $$\mathrm{Proj}\, \mC[X]_{\chi}= \mathrm{Spec}\, \mC[X]^G = \nu^{-1}(0)/U.$$ The following result is less well known, and its proof requires some adjustment of the classical proof for the case $\chi \coinc 1$ (see for example an appendix of \cite{Migliorini} or \cite{TatoPhD} ). \begin{prp}[Migliorini, \cite{Migliorini}]\label{prp:KNM1} Let $x \in X$ then $$ \exists g \in G \st \nu(gx) = \im d\chi \iff Gx \text{ is a closed orbit in } X^{ss}_{\chi}. $$ \end{prp} \begin{prp}[Migliorini, \cite{Migliorini}]\label{prp:KNM2} The inclusion $\nu^{-1}(\im d\chi) \subset X^{ss}_{\chi}$ induces an homeomorphism $$ \nu^{-1}(\im d\chi) / U \isocan X^{ss}_{\chi}//G . $$ \end{prp} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Quiver varieties as algebraic varieties} If $m= (m _1,\dots, m _n) \in \mZ^N$ we define a character $\chi_m$ of $G_v$ by $\chi_m = \prod_{i \in I} \det^{m_i}_{GL(V_I)}$. If $\grl = (\grl_1,\dots,\grl_n ) \in \mC^n$ and $m=(m_1,\dots , m_n) \in \mZ^n$ then we define the varieties: \begin{align*} \grL_{\grl}(d,v) &= \{ s \in S \st \grm_i(s) - \grl_i \Id_{V_i} =0 \text{ for all }i\}, \\ \grL_{m,\grl}(d,v) &= \{ s \in \grL_{\grl}(d,v) \tc s \text{ is } \chi_m-\text{semi-stable}\}. \end{align*} and the associated \emph{quiver varieties} \begin{align*} M_{m,\grl}(d,v) &= \grL_{m,\grl}(d,v) /\!/ G_v \, \mand \, \\ M^0_{\grl}(d,v) &= M_{0,\grl}(d,v) = \grL_{\grl}(d,v) /\!/G_v \end{align*} We call $p_{m, \grl}^{d,v} \colon \grL_{m,\grl}(d,v) \lra M_{m,\grl}(d,v)$ the quotient map. Observe that the inclusion $\grL_{m,\grl}(d,v) \subset \grL_{\grl}(d,v)$ induces a projective morphism $$ \pi_{m,\grl}^{d,v} : M_{m,\grl}(d,v) \lra M^0_{\grl}(d,v). $$ Finally it will be convenient to define $M_{m,\grl}(d,v)=\vuoto$ if $d,v \in \mZ^n$ and $v_i <0$ or $d_i<0$ for some $i$. We identify $\mZ^n$ with $P$ and $Z=\mC^n$ with $\mC \otimes_{\mZ} P$ through $$ (m_1,\dots,m_n) \longrightarrow \sum m_i \wbar \omega_i \;\mand\; (\grl_1,\dots,\grl_n) \longrightarrow \sum \grl_i \wbar \omega_i. $$ \begin{oss} As in \ref{oss:centrogoZ} we have a surjective map from $Z$ to $Z(\gog)$ and $\grL_{\grl}(d,v)$ is the fiber of $\mu$ over the image of $\grl$ in $Z(\gog)$. \end{oss} \begin{oss} Remark \ref{oss:riduzioneanonzero} holds without changes also in this case. \end{oss} \begin{oss}Observe that $P \oplus Z \subset \goZ$. Observe also that the map $m \lra \chi_m$ define a surjective morphism from $P$ to $\Hom(G_v,\mC^*)$ and that the following diagram commute: $$ \xymatrix{ P \ar@{^{(}->}[d] \ar[rr]& & \Hom(G,\mC^*) \ar@{}[r]|>>>>>{\ni} \ar[d]& \chi \ar[d]\\ \mR^n \ar[rr] && {Z(\gou) \isocan (\gou^*)^U} \ar@{}[r]|>>>>{\ni} & {\im d\chi} } $$ In particular we can apply \ref{prp:KNM2} to the action of $G_v$ on $\grL_{\grl}(d,v)$ and we obtain: $$ \goM_{(m,\grl)}(d,v) \isocan M_{m,\grl}(d,v). $$ \end{oss} \begin{prp}\label{prp:1.14} Let $(m,\grl) \notin \calH$ and $s \in \grL_{m,\grl}(d,v)$ then $\mathrm{Stab}_{G_v}(s) = \{1\}$. \end{prp} \begin{proof} As we have already claimed it is enough to prove $\dim \mathrm{Stab}_G(s)=\{1\}$. We know that there is a good quotient of $\grL_{m,\grl}(d,v)$ so it is enough to prove that any closed orbit has maximal dimension. By Proposition \ref{prp:KNM1} if $G_vs$ is closed in $\grL_{m,\grl}(d,v)$ then there exists $g \in G_v$ such that $\mu_I(gs) = \im d \chi$. The thesis follows now form $(m,\grl) \notin \calH$ and Lemma \ref{lem:regmu}. \end{proof} \begin{cor} If $(m,\grl) \notin \calH$ and $M_{m,\grl}(d,v) \neq \vuoto$ then it is a smooth algebraic variety of dimension $\bra v\cech , 2d-v \ket$. \end{cor} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Path algebra and $b$-path algebra} To describe functions on quiver varieties we need some notation about the path algebra. \begin{dfn} \label{defpath} A \emph{path} $\gra$ in our graph is a sequence $h^{(m)}\dots h^{(1)}$ such that $h^{(i)} \in H$ and $h^{(i)}_1 = h^{(i+1)}_0$ for $i =1,\dots,m-1$. We define also $\gra_0=h^{(1)}_0$, $\gra_1= h^{(m)}_1$ and we say that the length of $\gra$ is $m$. If $\gra_0=\gra_1$ we say that $\gra$ is a closed path. We consider also the empty paths $\vuoto_i$ for $i\in I$ and we define $(\vuoto_i)_0=(\vuoto_i)_1=i$. The product of path is defined in the obvious way. A $b$-\emph{path} $[\grb]$ in our graph is a sequence $[ i_{m+1}^{r_{m+1}} \gra^{(m)}i_{m}^{r_m}\dots\gra^{(1)}i_1^{r_1} ]$, that we write between square brackets such that $i_j \in I$, $\gra^{(j)}$ are $B$-path, $r_j \in \mN$ and $\gra^{(j)}_0 = i_j $ and $\gra^{(j)}_1= i_{j+1} $ for $j=1,\dots,m$. We consider also the ``empty'' $b$-paths indexed by elements of $I$: $[\vuoto_i]$. We define $[\grb]_0 = i_1$, $[\grb]_1 = i_{m+1}$ and $[\vuoto_i]_0=[\vuoto_i]_1=i$. The length of $[\grb]$ is $\sum_{j=1}^{m+1} r_j +\sum_{j=1}^{m} length(\gra^{j})$ and the product of $b$-paths is defined in the obvious way: $$ [\grb]\cdot[\grb^{\prime}]= \begin{cases} 0 &\text{if } [\grb^{\prime}]_1 \neq [\grb]_0 \\ {[\grb \grb^{\prime}]} &\text{if } [\grb^{\prime}]_1 = [\grb]_0 =i \end{cases} $$ Given a path $\gra= h^{(m)}\dots h^{(1)}$ and a $b$-path $\grb =[ i_{m+1}^{r_{m+1}} \gra^{(m)}\dots$ $\dots i_1^{r_1} ]$ we define an evaluation of $\gra$ and $\grb$ on $S$ in the following way: if $s=(B,\grg,\grd) \in S $ then \begin{align*} \vuoto_i(s)& = 0 \in \Hom(V_i,V_i)\;\mand\; [\vuoto_i](s) = 0 \in \Hom(V_i,V_i),\\ \gra(s) &= B_{h^{(m)}} \comp \dots \comp B_{h^{(1)}} \in \Hom (V_{\gra_0},V_{\gra_1}),\\ \grb(s) &= (\grg_{i_{m+1}}\comp\grd_{i_{m+1}})^{r_{m+1}} \comp \gra^{(m)}(s) \comp (\grg_{i_{m}}\comp \grd_{i_{m}})^{r_{m}} \comp \dots \comp \\ &\quad \circ \dots \circ \gra^{(1)}(s) \comp (\grg_{i_{1}}\comp \grd_{i_{1}})^{r_{1}} \in \Hom (D_{\grb_0},D_{\grb_1}). \end{align*} The path algebra $\calR$ is the vector space spanned by paths with the product induced by the product of paths. If $i,j \in I$ we say that an element in $\calR$ is of type $(i,j)$ if it is in the linear span of the paths with source in $i$ and target in $j$. The $b$-path algebra $\calQ$ is the vector space spanned by $b$-paths with the product induced by the product of $b$-paths described above. If $i,j \in I$ we say that an element in $\calR$ is of type $(i,j)$ if it is in the linear span of the $b$-paths with source in $i$ and target in $j$. \end{dfn} \begin{oss} We observe that the evaluation on $S$ is a morphism of algebra from $\calR$ to the algebra defined by the morphisms of the category of vector spaces. Moreover if $f$ is of type $(i,j)$ we observe that $f(s) \in \Hom (V_i,V_j)$. \end{oss} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Generators of the projective ring of a quiver variety} In this section we want to describe a set of generators of the graded ring $\mC[S] _{\chi}$ and by consequence of the projective ring of a quiver variety $\mC[\grL_{\grl}]_{\chi}$. More precisely we will give a set of generators as $\mC[S]^G$-module of its $l$-homogeneous part: $\mC[S] _{\chi,l}$. This result is a generalization of the one obtained by Lusztig in the case of invariants: $\chi \coinc 1$. First of all recall his result. \begin{teo}[Lusztig, \cite{Lu:Q3} theorem 1.3]\label{generatoriinvarianti} The ring $\mC[S]^G$ is generated by the polynomials: $$ s \longmapsto \Tr\left( \gra(s) \right) \;\mand\; s \longmapsto \grf\left(\grd_{\grb_1}(s)\grb(s)\grg_{\grb_0}(s)\right) $$ for $\gra$ a closed path, $\grb$ a path and $\grf \in \left(\Hom(D_{\grb_0},D_{\grb_1}) \right)^*$. \end{teo} \subsubsection{Determinants} \label{sssec:det} To describe our result we do first some general remark. Forget for a moment our quiver, and suppose to have a finite set of finite dimensional vector spaces $X_1,\dots,X_k$ of dimensions $u_1,\dots,u_k$ and a pair of nonnegative integers $(m_i^+, m_i^-)$ for each of them. Finally let $m^+, m^-$ two nonnegative integers such that $$ N=\sum_{i=1}^k m_i^+ u_i + m^+ = \sum_{i=1}^k m_i^- u_i + m^-,$$ and two vector spaces $M^+$ and $M^-$ of dimension $m^+,m^-$. Construct the vector spaces: $$ Y = \bigoplus _{i=1}^k \mC^{m_i^-} \otimes X_i \oplus M^-, \qquad Z = \bigoplus _{i=1}^k \mC^{m_i^+} \otimes X_i \oplus M^+ $$ and observe that $\dim Y = \dim Z = N$. Define an action of the general linear group $GL(X_i)$ of $X_i$ on $Y$ by $$ g_i\cdot(\sum_{j=1}^k v_j \otimes x_j + m) = \sum_{j\neq i} v_j \otimes x_j + m+ v_i \otimes g_i x_i, $$ and also a similar action on $Z$. Hence the vector space $\Hom(Y,Z)$ acquires a natural structure of $G_X = \prod_{i=1}^k GL(X_i)$ module. If we choose an isomorphism $\grs$ between $\Hom(\bigwedge^NY,\bigwedge^NZ)$ and $\mC$ we can define a function $det$ on $\Hom(Y,Z)$ by $$ det ( A) = \grs \left( \bigwedge^nA \right). $$ For simplicity we do not emphasize the role of $\grs$ on this definition, so strictly speaking, $\det$ is a function defined only up to a nontrivial constant factor. We observe also that $ \bigwedge^nY \isocan \left({\bigwedge^{u_i}X_1}\right)^{\otimes m_1^-} \otimes \dots \otimes \left({\bigwedge^{u_k}X_k}\right)^{\otimes m_k^-} \otimes \bigwedge^{m^-}M^- $ \noindent (and similarly for $Z$) so an isomorphism $\grs$ is determined if we choose orientations, or basis, of $X_j, M^+,M^-$. Finally observe that for any $g = (g_j) \in G_X$ we have $$\det( g \cdot A) = \prod_{i=1}^k (det_{GL(X_i)}(g_i))^{m_i^+ - m_i^-} \; \det(A).$$ \subsubsection{Description of generators}\label{sssec:generators} We go back now to our quiver and we describe a set of covariant polynomials on $S$. Any character $\chi$ of the group $G_v=GL(V)$ is of the form $\chi=\chi_m = \prod_{i \in I} det^{m_i}_{GL(V_i)}$. We fix such a character and we define \begin{align*} I^+ &= \{ i \in I \tc m _i >0 \} \mand \wt m_i = m _i \mif i\in I^+,\\ I^0 &= \{ i \in I \tc m _i =0 \} \mand \wt m_i = 0 \mif i\in I^0,\\ I^- &= \{ i \in I \tc m _i <0 \} \mand \wt m_i = - m _i \mif i\in I^-. \end{align*} We use now the construction explained in \ref{sssec:det} in the case $X_i= V_i$ and $m_i^+ - m_i^- = m_i$. We choose ordered sets $A=(a_1, \dots,a_{m^-}) \subset (\bigcup D_i)^{m^-}$ and $B=(b_1, \dots, b_{m^+}) \subset (\bigcup D_i^*)^{m^+}$ and we define a function $I:A,B \lra I$ by $a \in D_{I(a)}$, $b \in D^*_{I(b)}$. In the framework described above it is then possible to set $M^- = \bigoplus_{i=1}^{m^-} \mC_{a_i}$ and $M^+ = \bigoplus_{i=1}^{m^+} \mC_{b_i}$. In particular we have $$ Y = \bigoplus_{i\in I} \bigoplus_{h=1}^{m_i^-} V_i^{(h)} \oplus \bigoplus_{i=1}^{m^-} \mC_{a_i}, \qquad Z = \bigoplus_{i\in I} \bigoplus_{k=1}^{m_i^+} V_i^{[k]} \oplus \bigoplus_{i=1}^{m^+} \mC_{b_i} $$ where $V_i^{(l)}, V_i^{[l]}$ are isomorphic copies of $V_i$. We choose now elements of the $b$-path algebra as follows: \begin{enumerate} \item for any $i,j \in I$ and for any $1 \leq h \leq m^-_i$, $1 \leq k \leq m^+_j$ we choose an element $\gra^{i,h}_{j,k}$ of the $b$-path algebra of type $(i,j)$, \item for any $i \in I$, $1 \leq h \leq m^-_i$ and for any $1\leq l \leq m^+$ we choose an element $\gra^{i,h}_l$ of the $b$-path algebra of type $(i,I(b_l))$, \item for any $1 \leq l \leq m^-$ and for any $j \in I$, $1 \leq k \leq m^+_j$ we choose an element $\gra^l_{j,k}$ of the $b$-path algebra of type $(I(a_l),j)$, \item for any $1 \leq l \leq m^-$ and for any $1 \leq l'\leq m^-$ we choose an element $\gra^l_{l'}$ of the $b$-path algebra of type $(I(a_l),I(b_{l'}))$. \end{enumerate} We call such a data $\Delta = (\{ (m_i^+,m_i^-) \}_{i\in I} ,(m^+,m^-),A,B, \gra^{i,h}_{j,k},\gra^{i,h}_l,\gra^l_{j,k},\gra^l_{l'})$ a $\chi$-data and we attach to it a $\chi$-covariant function on $S$: $$ f_{\Delta} (s) = \det \left( \Psi_{\Delta} (s)\right) $$ where $\Psi_{\Delta} $ is a linear map from $Y$ to $Z$ defined by \begin{align*} [\Psi_{\Delta}] ^{V_i^{(h)}}_{V_j^{[k]}} (s) &= \gra^{i,h}_{j,k}(s), \\ [\Psi_{\Delta}] ^{V_i^{(h)}}_{\mC_{b_l}} (s) &= b_l \comp \grd_{I(b_l)}\comp \gra^{i,h}_l (s) , \\ [\Psi_{\Delta}] ^{\mC_{a_l}}_{V_j^{[k]}} (s) &= \gra^l_{j,k} (s) \comp \grg_{I(a_l)} \bigr|_{\mC a_l}, \\ [\Psi_{\Delta}] ^{\mC_{a_l}}_{\mC_{b_{l'}}}(s) &= b_{l'} \comp \grd_{I(b_{l'})} \comp \gra^l_{l'}(s)\comp\grg_{I(a_l)}\bigr|_{\mC a_l}. \end{align*} The function $f_{\Delta}$ are a set of generators as $\mC[S]^G$-module of $\mC[S]_{\chi,1}$, but we will need to define a smaller set of generators. To define this set we give a notion of good $\Delta$. \begin{dfn} A data $\Delta$ as above is said to be $\chi$-\emph{good} if it satisfies the following conditions: \begin{enumerate} \item $m_i^+ + m_i^- = \wt m_i$ for all $i \in I$, \item $\gra^{l}_{l'} = 0$ for all $l,l'$, \item $\gra^*_*$ is an element of the path algebra (and not just an element of the $b$-path algebra which is obviously bigger), \item $card\{(j,k) \tc \gra^{i,h}_{j,k} \neq 0 \} + card\{l\tc \gra^{i,h}_l \neq 0\} \leq v_i$ for all $i,h$, \item $card\{(i,h) \tc \gra^{i,h}_{j,k} \neq 0 \} + card\{l\tc \gra^l_{j,k} \neq 0\} \leq v_j$ for all $j,k$, \item for all $l$ there exists at most one pair $(i,h)$ such that $\gra^{i,h}_l \neq 0$, \item for all $l$ there exists at most one pair $(j,k)$ such that $\gra^l_{j,k} \neq 0$. \end{enumerate} \end{dfn} For the applications the only important point will be the first one. \begin{prp} \label{prp:covarianti} The set of polynomials $f_{\Delta}$ with $\Delta$ $\chi$-good generates $\mC[S]_{\chi,1}$ as a $\mC[S]^{G_v}$-module. \end{prp} \begin{oss} Prof. Weyman said me that in the case $D=0$ a similar proposition has been proved by him and Derksen for arbitrary characteristic in the case of a quiver without loops: \cite{DeWe}. \end{oss} \subsection{Some remark on the invariant theory of $GL(n)$} \label{ssec:invGL} If $V$ is a finite dimensional representation of a linearly reductive Lie group $G$ and $S$ is a simple representation of $S$ we write $V[S]$ for the $S$-isotipic component of type $S$ of $V$. We now fix $n$ and we make some remark on the representations of $GL(n)$. To any partition of height less or equal to $n$ we associate an irreducible representation of $GL(n)$ in the usual way. If we multiply these representations by a power of the inverse of determinant representation we obtain a complete list of irreducible representations of $GL(n)$. If $\grl$ is a partition we call $\grl$ the transpose partition as usual and we define $\grl^{op}=( \grl_1 - \grl_n , \grl_1 - \grl_{n-1}, \dots , \grl_1 - \grl_1)$. We call $\grd$ the determinant representation of $GL(n)$ and $\gre= 1^n$ the associated partition. Finally we call $V$ the natural representation. \begin{lem}\label{lem:invGL1} \begin{enumerate} \item ${L_{\grl}}^* = \grd^{-\grl_1} \otimes L_{\grl^{op}}$, \item $\Hom_{GL(n)}(\grd^m, L_{\grl} \otimes L_{\mu}) = \begin{cases} \mC &\mif \grl = \mu^{op} +(m-\mu_1)\gre, \\ 0 &\text{ otherwise,}\\ \end{cases} $ \item $\Hom_{GL(n)}(\grd^m, L_{\grl} \otimes L^*_{\mu}) = \begin{cases} \mC &\mif \grl = \mu +m\gre, \\ 0 &\text{ otherwise.} \end{cases}$ \end{enumerate} \end{lem} \begin{proof} We prove only 2). \begin{align*} Hom_{GL(n)}(\grd^m, L_{\grl} \otimes L_{\mu}) &= Hom_{GL(n)}(\grd^m\otimes {L_{\mu}}^*, L_{\grl} ) \\ &= Hom_{GL(n)}(\grd^{m-\mu_1} \otimes {L_{\mu^{op}}}, L_{\grl} ) \end{align*} If $m \geq \mu_1$ the last group is isomorphic to $Hom_{GL(n)}( L_{\mu^{op}+(m-\mu_1) \gre}, L_{\grl} )$ and if $m< \mu_1$ is isomorphic to $Hom_{GL(n)}( {L_{\mu^{op}}}; L_{\grl+ (\mu_1 -m)\gre})$. In any case the thesis follows. \end{proof} We want now to describe $\Hom_{GL(n)} \left(\grd^m,V^{\otimes i} \otimes \left( V^* \right)^{\otimes j} \right)$. To do it we will use Schur-duality. Remind that the irreducible representations of the groups $S_m$ are parametrized by partitions of $m$ and call $S_{\grl}$ the irreducible representation associated with $\grl$. Consider now the action of $S_m$ on $V^{\otimes m}$ given by permuting the factors. This action commute with the $GL(n)$ action. Schur duality asserts that the action of the group $S_m \times GL(n)$ on $V^{\otimes m}$ decomposes in the following way: $$V^{\otimes m} = \bigoplus_{\substack{\grl \vdash m \\ ht(\grl)\leq n}} S_{\grl} \otimes L_{\grl}. $$ We describe a set of elements of $\Hom_{GL(n)} \left(\grd^m,V^{\otimes i} \otimes \left( V^* \right)^{\otimes j} \right)$. Let $m$ be a nonnegative integers a choose a permutation $\grs$ of $\{1,\dots,i+mn\}$. To $\grs$ we associate maps: \begin{align*} \Phi_{\grs} &: \left(V^{\otimes i} \otimes (V^*)^{\otimes i}\right)^{GL(n)} \lra V^{\otimes i+mn} \otimes (V^*)^{\otimes i} [\grd^m] \\ \Psi_{\grs} &: \left(V^{\otimes i} \otimes (V^*)^{\otimes i}\right)^{GL(n)} \lra V^{\otimes i} \otimes (V^*)^{\otimes i+mn} [\grd^{-m}] \end{align*} by \begin{align*} \Phi_{\grs} (t \otimes s) &= \grs(o\otimes \dots \otimes o\otimes t) \otimes s \\ \Psi_{\grs} (t \otimes s) &= t \otimes \grs(o^* \otimes \dots \otimes o^*\otimes s) \end{align*} where $o $ is a nonzero vector in $\bigwedge^n V$ and $o^*$ is a non zero vector in $\bigwedge^n V^*$ and $t \in V^{\otimes i}$, $s \in (V^*)^{\otimes i} $. \begin{lem}\label{lem:invGL2} 1) If $i \neq j+mn$ then $$\Hom_{GL(N)} \left(\grd^m,V^{\otimes i} \otimes \left( V^* \right)^{\otimes j} \right) = 0.$$ 2) If $m>0$ then $$ V^{\otimes i+mn} \otimes ( V^* )^{\otimes i} [\grd^m] = \sum _{\grs} \Im \Phi_{\grs}.$$ 3) If $m>0$ then $$ V^{\otimes i} \otimes ( V^* )^{\otimes i+mn} [\grd^{-m}] = \sum _{\grs} \Im \Psi_{\grs}.$$ \end{lem} \begin{proof} 1) follows directly from lemma \ref{lem:invGL1}. 2) Let $M=V^{\otimes i+mn} \otimes ( V^* )^{\otimes i}[\grd^m]$ and $N = (V^{\otimes i} \otimes ( V^* )^{\otimes i})^G$. $N$ is a $S_i \times S_i$ module, $M$ is a $S_{i+mn} \times S_i$- module and the maps $\Phi_{\calI,\grs}$ are equivariant with respect the $S_i$ action on $(V^* )^{\otimes i}$. In particular it is enough to prove that if $\grl $ is a partition of $i$, $M_{\grl}$ is the $S_{\grl}$-isotipic component of $M$ w.r.t. the $S_i$ action and $N_{\grl}$ the $S_{\grl}$-isotipic component of $N$ w.r.t. the $S_i$ action on $(V^* )^{\otimes i}$ then $$ M_{\grl} = \sum _{\grs} \Phi_{\grs}(N_{\grl}). $$ By point 3 of lemma\ref{lem:invGL1} we have that \begin{align*} M &= \bigoplus_{\substack{\grl \vdash i \\ ht(\grl)\leq n} } \left( S_{\grl+m\gre} \otimes L_{\grl+m\gre} \right) \otimes \left( S_{\grl} \otimes L_{\grl} \right)^* [\grd^m] \\ &= \bigoplus_{\substack{\grl \vdash i \\ ht(\grl)\leq n} } S_{\grl+m\gre} \otimes S_{\grl} \otimes \left( \grd^m \otimes \left( L_{\grl} \otimes {L_{\grl}}^*\right)^G\right) \end{align*} In particular $M_{\grl} = S_{\grl+m\gre} \otimes S_{\grl} $ is an irreducible representation of $S_{i+mn}\times S_i$. Observe $\sum _{\grs} \Phi(N_{\grl})$ is a $S_{i+mn}\times S_i$-submodule of $M_{\grl}$ and that it is clearly nonzero. So $M_{\grl}=\sum _{\grs} \Phi_{\grs}(N_{\grl})$ as claimed. The proof of 3) is equal to the previous one. \end{proof} We want now to give a slightly different formulation of the lemma above. Let $M = V^{\otimes i} \otimes (V^*)^{\otimes j}$ we want to describe $ M^*_{\grd^m} = \{ \grf \in M^* \st g \cdot \grf = \grd^{-m}(g) \grf \}$. Of course this problem is completely equivalent to the previous one. What we want to do is to reformulate in a more convenient way for our purposes the description of a set of generators of $M^*_{\grd^m}$. Let $m \geq 0$ and choose $\calI = \{ I_1,\dots,I_m \} $ a collection of $m$ disjoint subsets of $\{1, \dots ,i+mn\}$ of cardinality $n$. Let $I_j = \{ i_{j1}< \dots < i_{jn} \}$ and $\{1, \dots ,i+mn\} - \bigcup \calI = \{ j_1 <\dots >> V_i' @>{a_i'}>> T_i @>{b_i}>> V_i @>>> 0, \end{CD} \end{equation} \item $a_i'(s')b_i'(s') = a_i(s)b_i(s) - \grl_i \Id_{T_i}$, \item $s \in \grL_{\grl}(d,v)$ and $s' \in \grL_{\grl'}(d,v')$. \end{enumerate} \end{dfn} \begin{lem} Let $(s,s') \in S_i(d,v) \times S_i(d,v')$ and suppose that it satisfies conditions 1), 2), 3), 4) , 5) above then: \begin{enumerate} \item $s \in \grL_{\grl}(d,v) \iff s' \in \grL_{\grl'}(d,v')$, \item if $\mu_j(s) - \grl_j \Id_{V_j}=0$ for all $j\neq i$ then $s \in \grL_{\grl}(d,v)$ , \item if $\mu_j(s') = \grl_j \Id_{V_j'}$ for all $j\neq i$ then $s' \in \grL_{\grl'}(d,v')$. \end{enumerate} \end{lem} \begin{proof} 2) We have to prove $b_ia_i - \grl_i\Id _{V_i}=0$ and by condition 4) it is enough to prove $b_ia_ib_i = \grl_i b_i$. So $b_ia_ib_i = b_i(a_i'b_i'-\grl_i) = \grl_i b_i$ by conditions 4) and 5). The proof of 3) is equal to the proof of 2). We prove the implication $\then$ in 1). By 2) and 3) it is enough to prove that $b_j'a_j' = \grl_j'$ for $j\neq i$. {\allowdisplaybreaks \begin{align*} b_j'a_j' &= \sum_{h_1=j} \gre(h) B_h' B'_{\bar h} + \grg_j' \grd_j' = \\ & = \sum_{h_1=j, h_0 \neq i} \gre(h) B_h B_{\bar h} + \grg_j \grd_j + \sum_{h_1=j, h_0 = i} \gre(h) B_h' B_{\bar h}' = \\ & = b_j a_j + \sum_{h_1=j, h_0 = i} \gre(h) \left( B_h' B_{\bar h}' - B_h B_{\bar h} \right) \\ & = b_j a_j + \sum_{h_0=j, h_1 = i} \left( B_{\bar h} \gre(h) B_{h}- B_{\bar h}' \gre(h) B'_h \right) \\ & = b_j a_j + \sum_{h_0=j, h_1 = i} \left( [a_ib_i]^{V_{h_0}}_{V_{h_0}} - [a'_ib'_i]^{V_{h_0}}_{V_{h_0}} \right) \\ & = \grl_j + \sum_{h_0=j, h_1 = i} \grl_i = \grl_j' \end{align*} } The proof of the converse is completely analogous. \end{proof} \begin{lem} Let $\grl \in Z_G$, $(s,s') \in Z^{\grl}_i(d,v)$ and $\gra $ be an element of the path algebra algebra of type $(\gra_0,\gra_1)$ then \begin{enumerate} \item if $\gra_0,\gra_1\neq i$ there exists an element $\gra'$ of the $b$-path algebra of type $(\gra_0,\gra_1)$ such that $\gra'(s')=\gra(s)$, \item if $\gra_1\neq i$ there exists an element $\gra'$ of the $b$-path algebra of type $(\gra_0,\gra_1)$ such that $\gra'(s') \grg '_{\gra_0} =\gra(s)\grg_{\gra_0}$, \item if $\gra_0\neq i$ there exists an element $\gra'$ of the $b$-path algebra of type $(\gra_0,\gra_1)$ such that $\grd '_{\gra_1}\gra'(s') =\grd _{\gra_1}\gra(s)$, \item there exists an element $\gra'$ of the $b$-path algebra of type $(\gra_0,\gra_1)$ such that $\grd '_{\gra_1}\gra'(s') \grg '_{\gra_0} = \grd _{\gra_1}\gra(s)\grg_{\gra_0}$, \end{enumerate} \end{lem} \begin{proof} By induction on the length of $\gra$ we can reduce the proof of this lemma to the following identities that are a consequence of condition 5) in definition \ref{dfn:generatorsweyl}: \begin{align*} B'_hB'_k &= \begin{cases} B_hB_k &\mif h \neq \bar k \\ B_hB_k - \grl_i &\mif k = \bar h \end{cases} \\ \grd '_iB'_k &= \grd _iB_k \\ B'_h \grg '_i &= B_h \grg_i \\ \grd '_i \grg _i ' &= \grd _i \grg_i -\grl_i \end{align*} for $h,k$ such that $h_0 = i = k_1$. \end{proof} \begin{lem} Let $(s,s') \in Z^{\grl}_i(d,v)$ and suppose $m_i \geq 0$ or $\grl_i\neq 0$ then $$ s \text{ is } \chi_m \text{ semistable } \iff s' \text{ is } \chi_{m'} \text{ semistable } $$ \end{lem} \begin{proof}We prove only $\then$. Let's do first the case $m_i \geq 0$. If $s$ is $\chi_m$ semi-stable, then there exists $\grD= \{ A,B,\gra^{*}_{*} \}$ $m$-good such that $f_{\grD}(s) \neq 0$. Using the notation in \ref{sssec:generators} we have $\grf_{\grD} = \det \Psi_{\grD}$ where $ \Psi_{\grD} : Y \lra Z$ is a linear map. In our case we can write $Z$ as $\mC^{m_i} \otimes V_i \oplus \wt Z $ and we observe that no $V_i$ summands appear in $Y$ or $\wt Z$. Now we construct a new data $\grD'= \{ A',B',{\gra'}^*_* \}$ such that $f_{\grD'}(s') \neq 0 $ and $f_{\grD'}$ a $\chi'$-covariant polynomial. Our strategy will be the following: we substitute each $V_i$ with the space $T_i$ in the space $Z$ and we add $m_i$ copies of $V_i'$ to $Y$. Let's do it more precise: first of all the new data will not be $m'$ good so we have to define ${m'}_j^+ $ and ${m'}_j^-$: \begin{enumerate} \item ${m'}_i^+ = 0$ and ${m'}_i^- = m_i = m_i^+$, \item ${m'_j}^- = m_j^-$ and ${m'_j}^+ = m_j^+ + a_{ij} m_i^+$ for all $j\neq i$, \item ${m'}^- = m^- $ and ${m'}^+ = m^+ + d_i m_i^+$. \end{enumerate} Observe that ${m'_j}^+ - {m'_j}^- = m _j'$ for all $j$ so our data will furnish a $\chi'$ equivariant function. Moreover if we define $$ Z' = \mC^{m_i} \otimes V_i \oplus \wt Z \;\mand\; Y' = \mC^{m_i} \otimes T_i \oplus Y $$ we observe that they have the numbers of $V'_j$, $\mC_a$, $\mC_b$ factors specified by ${m'}$. Now we construct the new data $\grD'$ in such a way that with respect to the decompositions above we have: \begin{align*} [\Psi_{\grD}(s)]_{\mC^{m_i}\otimes V_i \oplus \wt Z}^{Y} & = \begin{pmatrix} ( \Id\otimes b_i ) \comp \pi\\ \Phi \end{pmatrix},\\ [\Psi_{\grD'}(s')]_{\mC^{m_i}\otimes T_i \oplus \wt Z}^{\mC^{m_i}\otimes V_i' \oplus Y} & = \begin{pmatrix} \Id \otimes a'_i & \pi \\ 0 & \Phi \end{pmatrix}. \end{align*} If we construct a data with this property we observe that $\Psi_{\grD}(s)$ is an isomorphism if and only if $\Psi_{\grD'}(s')$ is an isomorphism. Hence $f_{\grD}(s) \neq 0$ implies $f_{\grD'}(s') \neq 0$ and the lemma is proved. To construct the new data we choose a basis $e_1,\dots,e_{d_i}$ of $D_i$ and we define the other elements of the data according to the following rules \begin{enumerate} \item $A' =A$, \item if $B=(b_1,\dots,b_{m^+})$ we set $B'=(b_1,\dots,b_{m^+}, \underbrace{e_1,\dots,e_1}_{m_1\;times},\dots , \underbrace{e_{d_i}, \dots e_{d_i}} _{m_1\;times})$, \item ${\gra'}^{j_1,h_1}_{j_2,h_2}$ for $j_1 \neq i$ and $h_2 \leq m^+_{j_2}$ is an element constructed according to case 1) in the previous lemma, \item ${\gra'}^{a}_{j_2,h_2}$ for $h_2 \leq m^+_{j_2}$ is an element constructed according to case 2) in the previous lemma, \item ${\gra'}^{j_1,h_1}_{b_l}$ for $j_1 \neq i$ and $l \leq m^+$ is an element constructed according to case 3) in the previous lemma. \item ${\gra'}^{i,h}_{b_l} = {\gra'}^{i,h}_{j,k}=0$ if $l \leq m^+$ and $k \leq m_j^+$. \end{enumerate} In this way we guarantee that the projection of $\Psi_{\grD'} (s') $ onto $\wt Z$ is equal to $\bigl( 0\; \Phi \bigr)$. To define the remaining part of the new data we do not give details on the indexes, but we explain how to construct it. It is clear that we can choose ${\gra'}^{i,h}_*$ for the remaining indexes * in such a way that the projection of $\Psi_{\grD'} (s')\bigr|_{ \mC^{m_i}\otimes V_i'}$ on $\mC^{m_i} \otimes T_i$ is equal to $\Id \otimes a'_i$. Finally we observe that a path $\grb$ from $V_j$ to $V_i$ with $j\neq i$ has to go through a summand of $T_i$ so there exists a path $\gra$ such that $\grb(s)= b_i \comp \gra(s)$. Now we use the previous lemma to change $\gra$ with a $\gra'$ such that $\grb(s) = b_i \comp \gra'(s')$. More generally if $\grb$ is an element of the path algebra of type $(j,i)$ with $j\neq i$ then there exists an element of the $b$-path algebra $\gra'$ such that $\grb(s) = b_i\comp \gra'(s)$. In this way we define the elements of the $b$-path algebra connecting summands of $Y$ and summands of $\mC^{m_i}\otimes T_i$. In the case $m_i<0$ we proceed in a similar way: we choose $\grD$ $m$-good and we have $$ Y = \mC^{-m_i} \otimes V_i \oplus \wt Y, \quad Y'= \mC^{-m_i} \otimes T_i \oplus \wt Y, \quad Z'= \mC^{-m_i} \otimes V'_i \oplus Z. $$ As in the previous case we can find a new data $\grD'$ such that: \begin{align*} [\Psi_{\grD}(s)] _{ Z} ^{\mC^{-m_i}\otimes V_i \oplus \wt Y} &= \begin{pmatrix} \pi \comp ( \Id \otimes a_i ) & \Phi \end{pmatrix},\\ [\Psi_{\grD'}(s')] _{\mC^{-m_i}\otimes V_i' \oplus Z} ^{\mC^{-m_i}\otimes T_i \oplus \wt Y} & = \begin{pmatrix} \Id \otimes b'_i & 0 \\ \pi & \Phi \end{pmatrix}. \end{align*} Now to conclude that $\Psi_{\grD'}(s')$ is an isomorphism if $\Psi_{\grD}(s)$ is we need to know that $b'_i$ is an epimorphism and this is not guarantee by $(s,s') \in Z^{\grl}_i(d,v)$. But if $\grl_i \neq 0$ then, since $b_i'a_i'= -\grl_i$, we have that $b_i'$ is surjective. \end{proof} \begin{dfn} Let $p$ (resp. $p'$) be the projections of $Z_i^{\grl}(d,v)$ on $\grL_{\grl}(d,v) \subset S(d,v)$ (resp. $\grL_{\grl'}(d,v') \subset S(d,v')$). Suppose that $m_i > 0$ or $\grl_i \neq 0$ then we define $$ Z_i^{m,\grl}= p^{-1}\bigl( \grL_{m,\grl}(d,v) \bigr) = {p'}^{-1}\bigl( \grL_{m',\grl'}(d,v') \bigr). $$ We define also $$ G_{i,v} = \prod _{j \neq i} GL(V_j) \times GL(V_i) \times GL(V_i'). $$ Observe that there are natural projections from $G_{i,v}$ to $G_v$ and $G_{v'}$, therefore there are natural actions of $G_{i,v}$ on $S_i(d,v)$, $S_i(d,v')$. Observe that here is a natural action of $G_{i,v}$ on $Z_i^{\grl}$ and $Z_i^{m,\grl}$ such that the projections $p$, $p'$ are equivariant. \end{dfn} \begin{lem} \label{lem:abmonoepi} Let $s \in \grL_{\grl,m}(d,v)$ then \begin{enumerate} \item if $\grl_i \neq 0$ then $b_i$ is epic and $ a_i$ is mono, \item if $m_i>0$ then $b_i$ is epic, \item if $m_i<0$ then $a_i$ is mono. \end{enumerate} \end{lem} \begin{proof} If $\grl_i \neq 0$ then the result is clear by $b_ia_i = \grl_i$. Suppose now that $\grl_i = 0 $ and $m_i >0$. Let $U_i = \Im b_i$ and let $V_i = U_i \oplus W_i$. Define now a one parameter subgroup $g(t)$ of $G_V$ in the following way: $$ [g_i(t)]^{U_i \oplus W_i}_{U_i\oplus W_i} = \begin{pmatrix} 1 & 0 \\ 0 & t^{-1} \end{pmatrix} \mand g_j \coinc 1 \mfor j \neq i $$ Since $\Im b_i \subset U_i$ we have that there exists the limit $\lim_{t\to 0 } g(t) \cdot s = s_{0}$. Let now $n>0$ and $f$ a $\chi^n$-covariant function on $S$ such that $f(s)\neq 0$. Then $$ f(s_0) = \lim_{t \to 0} f(g(t) \cdot s) = \lim_{t \to 0} {\det}_{GL(V_i)}^{nm_i} f(s) = \lim_{t \to 0} t^{-nm_i\dim W_i} f(s) $$ So we must have $\dim W_i =0$. The proof of the third case is completely similar to this one. \end{proof} \begin{lem}[see also Lusztig \cite{Lu:Q4}] %1) If $\grl_i \neq 0 $ then %\begin{enumerate} % \item $p: Z_i^{\grl}(d,v) \lra \grL_{\grl}(d,v)$ is a principal % $Gl(V_i')$ bundle, % \item $p': Z_i^{\grl}(d,v) \lra \grL_{\grl'}(d,v')$ is a principal % $Gl(V_i)$ bundle. %\end{enumerate} If $m_i>0$ or $\grl_i \neq 0 $ then \begin{enumerate} \item $p: Z_i^{m,\grl}(d,v) \lra \grL_{m,\grl}(d,v)$ is a principal $GL(V_i')$ bundle, \item $p': Z_i^{m,\grl}(d,v) \lra \grL_{m',\grl'}(d,v')$ is a principal $GL(V_i)$ bundle. \end{enumerate} \end{lem} \begin{proof} Lusztig's proof extend to this case without changes. Let's prove for example 1. We have to prove: $i.$ that the action on the fiber is free, $ii.$ that it is transitive. First of all we observe that by the previous lemma if $s \in \grL_{m,\grl}$ then $b_i(s)$ is epic. In particular there exists $a_i' : V_i' \lra T_i$ such that sequence \eqref{eq:seqes:defsi} is exact, and clearly $a_i'$ is univoquely determined up to the action of $GL(V_i')$, moreover this action is free. So $i.$ and $ii.$ reduce to the following fact: if $s \in \grL_{m,\grl}$ and $a_i'$ is such that sequence \eqref{eq:seqes:defsi} is exact, then there exists a unique, $b_i'$ such that $a_i'b_i'=a_ib_i -\grl_i$. Since $a_i'$ is mono the unicity is clear. To prove the existence we observe that it is equivalent to $\Im a_i' \supset \Im (a_ib_i - \grl_i)$. But the last statement is clear since we have: $\Im a_i' = \ker b_i$ and $ b_i(a_ib_i - \grl_i) = 0$. \end{proof} \begin{prp} \label{prp:isoMsi} If $m_i>0$ or $\grl_i \neq 0$ then the projections $p$, $p'$ induces algebraic isomorphisms $\bar p$ , $\bar p'$: $$ \begin{CD} \grL_{m,\grl}(d,v) /\!/ G_v @<{\circa}<{\bar p}< Z_i^{m,\grl}(d,v) /\!/ G_{i,v} @>{\circa}>{\bar p '}> \grL_{m',\grl'}(d,v') /\!/ G_{v'} \end{CD} $$ \end{prp} \begin{proof} This proposition is a straightforward consequence of the previous lemma and the following general fact (see for example \cite{GIT} Proposition 0.2): let $G$ be an algebraic groups over $\mC$ and $X$, $Y$ two irreducible algebraic variety over $\mC$; if $G$ acts on $X$ and $\grf:X \lra Y$ is such that for all $y \in Y$ the fiber $X_y$ contains exactly one $G$-orbit then $\grf $ is a categorical quotient. If we apply this lemma to the projection $p$, (resp. $p'$) and to the group $GL(V_i')$ (resp. $GL(V_i)$) we obtain the required result. \end{proof} We can use this proposition to define the action of the generators of the Weyl group. \begin{dfn} Let $i,\grl,m,d,v,\grl',m',v'$ be as above, and suppose $d_j \geq 0$, $ v_j,v_j' \geq 0 $ for all $j$ then we define an isomorphism of algebraic variety $$ \Phi^{d,v}_{s_i,\grl,m}\colon M_{m,\grl}(d,v) \lra M_{m',\grl'}(d,v') $$ in the following way: \begin{enumerate} \item if $m_i>0$ or $\grl_i \neq 0$ then we set $\Phi^{d,v}_{s_i,\grl,m} = \bar p' {\bar p}^{-1}$, \item if $m_i<0$ then we exchange the role of $v,v'$ in the previous construction: more precisely we observe that $m_i' >0$ so we can define $\Phi^{d,v'}_{s_i,\grl',m'}\colon M_{m',\grl'}(d,v') \lra M_{m,\grl}(d,v)$ and we define $\Phi^{d,v}_{s_i,\grl,m} = \bigl(\Phi^{d,v'}_{s_i,\grl',m'}\bigr)^{-1}$. \end{enumerate} \end{dfn} \begin{oss}\label{oss:defisosi} To see that $\Phi^{d,v}_{s_i,\grl,m}$ is univoquely defined we have to verify that if $\grl_i \neq 0$ and $m_i<0$ the two definitions above coincide. This fact reduces easily to the following remark: if $\grl_i \neq 0$ then $$ (s,s') \in Z_i^{\grl}(d,v) \iff (s',s) \in Z_i^{\grl'}(d,v'). $$ Let us prove, for example, the $\then$ part. Since $a_i b_i = a_i'b_i' + \grl_i = a_i'b_i' - \grl_i ' $ the only thing we have to verify is that the sequence $$ \begin{CD} 0 @>>> V_i @>{a_i}>> T_i @>{b_i'}>> V_i' @>>> 0 \end{CD} $$ is exact. The surjectivity of $b_i'$ and the injectivity of $a_i$ are a consequence of $\grl_i \neq 0$. Since $\dim T_i = \dim V_i + \dim V_i'$ we need only to prove that $b_i'a_i =0$. Observe that $b_i'a_i =0$ if and only if $ a_i'b_i'a_i =0$ since also $a_i'$ is injective. Finally $ a_i'b_i'a_i = (a_ib_i-\grl_i) a_i = 0$. \end{oss} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Preliminaries} We saw how to define $$ \Phi^{d,v}_{s_i,m,\grl}\colon M_{m,\grl} \bigl(d,v\bigr)\lra M_{s_i(m),s_i(\grl)}\bigl( s_i(d,v) \bigr) $$ in the case that $(\grl_i,m_i ) \neq 0$ and $d,v,s_iv \geq 0$. To define an action of the Weyl group we have now to guarantee that coxeter relations hold. We will prove these relations in the next paragraph. Before doing it we observe that we have to guarantee some conditions on $m,\grl$ such that we will be able to define $\Phi_{s_i,\grs m,\grs\grl}^{\grs(d,v)}$ for any element $\grs \in W$: this condition will be $(m,\grl) \in \calG_v$ (\ref{dfn:22}). We have also to say something about the case $d_i <0$ or $v_i<0$ for some $i \in I$. In the case that $d_i <0$ for some $i$ then $M_{m,\grl}\bigl( \grs(d,v) \bigr)= \vuoto $ for all $\grs,m,\grl$ by the very definition, so there is nothing to define. The second trivial case is $d=v=0$. Indeed in this case we have $M_{\grs m,\grs \grl}\bigl(\grs(d,v)\bigr) = \{ 0 \}$ so the definition is trivial. The other two cases are treated in the two lemmas below. In the following we fix $d$ such that $d_i \geq 0 $ for all $i$. It will be convenient to define an affine action of $W$ on $Q$ by $\grs \cdot v = \grs(v-d) +d$. \begin{lem} Let $d\geq 0 $ and $(m,\grl) \in \calG_v$ if there exists $\grs$ such that $\grs \cdot v \not \geq 0$ then $M_{m,\grl}(d,v) = \vuoto$. \end{lem} \begin{proof} Suppose that $\grs$ is an element of minimal length such that $\grs \cdot v \not \geq 0$ and let $l = \ell(\grs)$. We prove the lemma by induction on $l$. The case $l=0$ is trivial. \noindent \emph{Initial step: $l=1$.} If $s_i \cdot v \not \geq 0$ then we have $0 \leq d_i + \sum a_{ij} v_j < v_i$. Hence $\dim T_i < \dim V_i $, $u = ( 0,\dots,\overset{i}{1},0,\dots) \in \tilde U_v$ and $(\grl_i,m_i) \neq 0$. So $M_{m,\grl}(d,v) = \vuoto$ by lemma \ref{lem:abmonoepi}. \noindent \emph{Inductive step: if $l\geq2$ then $l-1 \then l$.} Let $\grs = \tau s_i$ with $\ell(\tau) = l-1$ and $v' = s_i \cdot v$, $\grl'= s_i \grl$, $m ' = s_i m$. By induction $M_{m',\grl'}(d,v') = \vuoto$ and, since $l \geq 2 $, $v' \geq 0$. If $(m_i,\grl_i) \neq 0$ then we can apply Proposition \ref{prp:isoMsi} and we obtain $M_{m,\grl}(d,v)\isocan M_{m',\grl'}(d,v') = \vuoto$. If $(m_i,\grl_i)=0$ then $u = ( 0,\dots,\overset{i}{1},0,\dots) \notin \tilde U_v$, hence $v_i =0$. Moreover $\grl ' = \grl$ and $m' = m$ so $(m_i',\grl_i')=0$ and $u = ( 0,\dots,\overset{i}{1},0,\dots) \notin U_{v'}$. Hence $v_i'=0$ so $v'=v$ and $\tau v \not \geq 0$ against the minimality of $\grs$. \end{proof} \begin{lem}\label{lem:sisipuodefinire} Let $(I,H)$ be connected, $(m,\grl ) \in \calG_v$ and suppose $d\geq 0$ and $\grs \cdot v \geq 0$ for all $\grs \in W$. If there exists $i \in I, \grs \in W$ such that $\grs(m,\grl) = ( m',\grl')$ and $(m_i',\grl_i')=0$ then $d=v=0$. \end{lem} \begin{proof} Without loss of generality we can assume $\grs =1$. \noindent \emph{First step}: $v_i =0$. This is clear since otherwise $u = ( 0,\dots,\overset{i}{1},0,\dots) \in U_v$. \noindent \emph{Second step}: $d_i =0$ and $v_j=0$ for all $j$ such that $a_{ij}$. Let $v'= s_i\cdot v$ and observe that $s_i\grl =\grl$ and $s_im=m$. Then as in first step we have $0 =v_i' = d_i + \sum_{j}a_{ij}v_j$ from which the claim follows. Let now $ W' = \bra \{s_j \st a_{ij} \neq 0 \mand j\neq i\} \ket $. If $(d,v) \neq 0$ then there exists $j \in I$ and $\grs \in W'$ such that $a_{ij} \neq 0$ and $$ n = d_j + \sum_{h\in I} a_{jh}\Tilde v _h >0. $$ where $\Tilde v=\grs\cdot v$. Since $(\grs \grl)_i = \grl_i = 0 = m_i = (\grs m)_i$ we can assume $\grs = 1$. Let now $v' = s_is_j \cdot v$, $ \grl' =s_is_j \grl$ and $m ' = s_is_j m$, we have: \begin{align*} v'_i &= a_{ij}n & \grl'_i&= -a_{ij}\grl_j & m_i&= -a_{ij}m_j \\ v'_j &= n & \grl'_j&= (a_{ij}^2-1) \grl_j & m_j&= (a_{ij}^2-1)m_j . \end{align*} Hence $u= (0,\dots,\overset{j}{a_{ij}},0,\dots, \overset{i}{a_{ij}^2-1},0,\dots)\in U_{v'}$ and $\bra u \cech ,\grl' \ket = \bra u\cech, m' \ket =0$ against $(m,\grl) \in \calG_v$. \end{proof} \begin{oss} the analogous lemma in the case of $\calG= \{(m,\grl,v) \st (m,grl) \notin H^{\infty} \}$ are more simple. \end{oss} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Relations} In this section we define an isomorphism of algebraic variety $$ \Phi_{\grs,m,\grl}^{d,v} \colon M_{m,\grl}\bigl(d,v\bigr) \lra M_{\grs m,\grs\grl}\bigl(\grs(d,v)\bigr). $$ in the case $(m,\grl) \in \calG _v$ or $(m,\grl) \notin H^{\infty}$. In the case $d \not \geq 0$ or in the case in which there exists $\grs \in W$ such that $\grs v \not \geq 0$ or in the case $d=v=0$ we have seen in the previous section that there is nothing to define or that the definition is trivial. In the remaining cases we observe that for all $\tau, i$ we have $(\tau(m)_i , \tau(\grl)_i) \neq 0$ by lemma \ref{lem:sisipuodefinire}. Hence we can define $\Phi_{\grs,m,\grl}^{d,v}$ by induction on $\ell(\grs)$ by the formula \begin{equation} \label{eq:defphisigma} \Phi_{\grs,m,\grl}^{d,v} = \Phi_{s_i,\tau m,\tau \grl}^{\tau(d,v)} \comp \Phi_{\tau , m,\grl}^{d,v}. \end{equation} Of course we have to prove that this definition is well given by checking Coxeter relations: $$ s_i^2 = \Id, \quad s_is_j=s_js_i \;\mif a_{ij}=0 \; \mand \; s_is_js_i = s_js_is_j\;\mif a_{ij}=1 $$ which in our situation take the following form: \begin{subequations} \begin{gather} \Phi_{s_i,s_i\grl,s_im}^{s_i(d,v)} \comp \Phi_{s_i,\grl,m}^{d,v} = \Id \label{eq:cox1}\\ \Phi_{s_i,s_j\grl,s_jm}^{s_j(d,v)} \comp \Phi_{s_j,\grl,m}^{d,v} = \Phi_{s_j,s_i\grl,s_im}^{s_i(d,v)} \comp \Phi_{s_i,\grl,m}^{d,v} \label{eq:cox2}\\ \Phi_{s_i,s_js_i\grl,s_js_im}^{s_js_i(d,v)} \comp \Phi_{s_j,s_i\grl,s_im}^{s_i(d,v)} \comp \Phi_{s_i,\grl,m}^{d,v} = \Phi_{s_j,s_is_j\grl,s_is_jm}^{s_is_j(d,v)} \comp \Phi_{s_i,s_j\grl,s_jm}^{s_j(d,v)} \comp \Phi_{s_j,\grl,m}^{d,v}. \label{eq:cox3} \end{gather} \end{subequations} The first of the two equations is clear by the very definition and remark \ref{oss:defisosi}. The second equation is trivial. We need to prove the third equation. We will need the following two simple lemmas of linear algebra which proofs are trivial. \begin{lem}\label{lem:25} Let $V,W,X,Y,Z$ be finite dimensional vector spaces and $\gra,\grb,\grg,\grd,\gre,\grf$ linear maps between them as in the diagrams below. The diagram $$ \begin{CD} 0 @>>> V @>{\begin{pmatrix} \gra \\ \grb \\ \grg \end{pmatrix}}>> W \oplus X \oplus Y @>{\begin{pmatrix} \grd & 0 & -1 \\ 0 & \gre & \grf \end{pmatrix}}>> Y \oplus Z @>>> 0 \end{CD} $$ is exact if and only if the diagram $$ \begin{CD} 0 @>>> V @>{\begin{pmatrix} \gra \\ \grb \end{pmatrix}}>> W \oplus X @>{\begin{pmatrix} \grf \grd & \gre \end{pmatrix}}>> Z @>>> 0 \end{CD} $$ is exact and $\grg = \grd\gra$. \end{lem} \begin{lem}\label{lem:26} Let $U,V,W,X,Y,Z$ be finite dimensional vector spaces and $\gra,\grb,\grg,\grd,\gre,\grf,\psi,\rho,\grs$ linear maps between them as in the diagrams below such that $\psi \oplus \rho : W \oplus X \lra Z$ is an epimorphism. Then the diagram $$ \begin{CD} 0 \lra U @>{\begin{pmatrix} \gra \\ \grb \\ \grg \end{pmatrix}}>> V \oplus W \oplus X @>{\begin{pmatrix} \grd & 0 & 1 \\ \gre & \grf & 0 \\ 0 & \psi & \rho \end{pmatrix}}>> X \oplus Y \oplus Z @>{\begin{pmatrix}\rho & \grs & -1 \end{pmatrix}}>> Z \lra 0 \end{CD} $$ is exact if and only if $\grg = - \grd \gra$ , $\psi=\grs\phi$ , $\rho\grd + \grs\gre =0$ and the diagram $$ \begin{CD} 0 @>>> U @>{\begin{pmatrix} \gra \\ \grb \end{pmatrix}}>> V \oplus W @>{\begin{pmatrix} \gre & \grf \end{pmatrix}}>> Y @>>> 0 \end{CD} $$ is exact. \end{lem} We fix now and $i,j$ such that $a_{ij}=1$ and we verifies \eqref{eq:cox3}. Let \begin{align*} \grl ' & = s_i \grl & m' &=s_i m & v' & =s_i v \\ \grlsec & = s_j \grl ' & \msec &=s_j m' & \vsec & =s_j v' \\ \grlter & = s_i \grlsec & \mter &=s_i \msec & \vter &=s_i \vsec \\ \Tilde {\grl} & = s_j \grl & \Tilde m &=s_j m & \Tilde v & = s_j v \\ {\Tilde {\Tilde {\grl}}} & = s_i {\Tilde {\grl}} & {\Tilde {\Tilde m}} &=s_i {\Tilde m} & {\Tilde {\Tilde v}} & = s_i {\Tilde v } \end{align*} First of all we observe that since relation \eqref{eq:cox1} holds we can assume that: \begin{enumerate} \item $\grl_i \neq 0$ or $m_i > 0$ and $\grl_j \neq 0 $ or $m_j >0$, \item $\grl'_j \neq 0$ or $m'_j > 0$ and $\Tilde{\grl} _i \neq 0 $ or $\Tilde m _i >0$, \item $\grlsec_i \neq 0$ or $\msec_i > 0$ and $\Tilde{ \Tilde{\grl}}_j \neq 0 $ or $\Tilde{\Tilde m}_j >0$. \end{enumerate} Define \begin{align*} Z_{iji} & = \{ (\ster,s) \in \grL_{\mter,\grlter} (d,\vter) \times \grL_{m,\grl}(d,v) \st \exists \ssec \in S(d,\vsec), \\ & \;\qquad \mand s' \in S(d,v') \msuchthat (\ster,\ssec) \in Z_i^{\msec,\grlsec}(d,\vsec), \\ & \;\qquad (\ssec,s')\in Z_j^{m',\grl'}(d,v') \mand (s',s) \in Z_i^{m,\grl}(d,v) \} \\ Z_{jij} & = \{ (\ster,s) \in \grL_{\mter,\grlter} (d,\vter) \times \grL_{m,\grl}(d,v) \st \exists \Tilde{\Tilde{s}} \in S(d,\Tilde{\Tilde{v}}), \\ & \;\qquad \mand \Tilde s \in S(d,\Tilde v ) \msuchthat (\ster,\Tilde{\Tilde s}) \in Z_j^{\Tilde{\Tilde m},\Tilde{\Tilde{\grl} } }(d,\Tilde{\Tilde v}), \\ & \;\qquad (\Tilde{\Tilde s},\Tilde s )\in Z_i^{\Tilde m,\Tilde{\grl}}(d,\Tilde v ) \mand (\Tilde s,s) \in Z_j^{m,\grl}(d,v) \} \end{align*} Observe that $(\ster,s) \in Z_{iji} \iff p^{d,\vter}_{\mter.\grlter}(\ster) = \Phi_{s_i}\Phi_{s_j}\Phi_{s_i}(p^{d,v}_{m,\grl}(s))$ and that $(\ster,s) \in Z_{jij} \iff p^{d,\vter}_{\mter.\grlter}(\ster) = \Phi_{s_j}\Phi_{s_i}\Phi_{s_j}(p^{d,v}_{m,\grl}(s))$. So relation \eqref{eq:cox3} is equivalent to $Z_{iji}=Z_{jij}$. Let now $R_i = D_i \oplus \bigoplus_{h\st h_1=i, h_0 \neq j} V_{h_0} $, $R_j = D_i \oplus \bigoplus_{h\st h_1=i, h_0 \neq j} V_{h_0} $ and observe that $T_i = R_i \oplus V_j$ and $T_j = R_j \oplus V_i$. Let $k$ be the only element of $H$ such that $k_0=j$ and $k_1 = i$. Let $\gre = \gre(k)$. Define also $A=A(s)= B_k(s)$, $B=B(s)=B_{\bar k}(s)$ and for $l =i ,j $ and $\{l',l\}=\{i,j\}$ set $c_l=c_l(s)= \pi_{R_l}^{R_l\oplus V_{l'}} a_l(s)$ and $d_l=d_l(s)= b_l(s)\bigr|_{R_l} $. Let now $(s,s\ter) \in \grL_{\grl}(d,v) \times \grL_{\grl\ter}(d,v\ter)$ and set $A^* =A(s^*)$, $ B^* =B(s^*)$, $ c_l^* = c_l(s^*)$ and $ d_l^* = d_l(s^*)$ for $l \in \{i,j\}$ and $* \in \{\;, \ter \}$. If we apply lemmas \ref{lem:25} and \ref{lem:26} to our situation we obtain the following result: $(s,\ster) \in Z_{iji} $ of and only there exists vector spaces $V_i',V_j',V_i\sec ,V_j\sec $ and linear maps $A',B',c_i',d_i',c_j',d_j', A\sec , B\sec , c_i\sec , d_i\sec , c_j\sec , d _j\sec$ such that: \begin{enumerate} \item $\dim V_l^* =v_l^*$ for $l \in \{i,j\} $ and $* \in \{ ',\sec\}$, \item for each $*\in \{\, ' , \,\sec\}$ and $l \in \{ i, j\}$ $A^* \in \Hom (V^*_i,V^*_j)$, $B^* \in \Hom(V^*_j , V^* _i)$, $c_l \in \Hom (V^*_l,R^*_l)$ and $d_l \in \Hom (R^*_l,V^*_l)$, \item $V_j \ter = V_j \sec$, $c_j \ter = c_j \sec$, $d_j \ter = d_j \sec$ and \begin{align*} c_i\ter d_i \ter &= c_i d_i -\grl_i -\grl_j & c_i\ter B\ter& =c_i'B\sec \\ A\ter d_i\ter &=A\sec d_i'& \gre A\ter B\ter&= \gre A\sec B\sec - \grl_j \end{align*} \item $V_i \sec = V_i '$, $c_i \sec = c_i '$, $d_i \sec = d_i'$ and \begin{align*} c_j\sec d_j \sec &= c_j d_j -\grl_i -\grl_j & c_j\sec A\sec & =c_j A' \\ B\sec d_j\sec &= B' d_j & \gre A \sec B \sec&= \gre A' B' + \grl_i + \grl_j \end{align*} \item $V_j' = V_j $, $c_j' = c_j$, $d_j ' = d_j$ and \begin{align*} c_i' d_i ' &= c_i d_i -\grl_i & c_i' B'& =c_i B \\ A' d_i' &=A d_i & \gre A' B'&= \gre A B - \grl_i \end{align*} \item $ \gre c_i' B\sec A\ter + c_i' d_i' c_i \ter =0$ and $\gre A' B\sec = d_ja_j\sec$, \item the following diagrams are exact $$ \begin{CD} 0@>>> V_i\ter @>{\begin{pmatrix} c_i \ter \\ c_j\sec A\ter \end{pmatrix}}>> R_i \oplus R_j @>{\begin{pmatrix} Ad_i& d_j \end{pmatrix}}>> V_j @>>> 0 \\ 0@>>> V_j\sec @>{\begin{pmatrix} c_j \sec \\ c_i'B\sec \end{pmatrix}}>> R_j \oplus R_i @>{\begin{pmatrix} Bd_j& d_i \end{pmatrix}}>> V_i @>>> 0 \\ 0@>>> V_i' @>{\begin{pmatrix} c_i' \\ A' \end{pmatrix}}>> R_i \oplus V_j @>{\begin{pmatrix} d_i & \gre B \end{pmatrix}}>> V_i @>>> 0 \end{CD} $$ \end{enumerate} \begin{oss} The first condition in point 6) is equivalent to $\gre B\sec A\ter = d_i' c_i \ter$. Indeed this condition is certainly sufficient. To prove the necessity observe that by the injectivity of $a_i' = (c_i '\;A')^t$ it is enough to prove $ \gre c_i' B\sec A\ter + c_i' d_i' c_i \ter =0$ and $ \gre A' B\sec A\ter + A' d_i' c_i \ter =0$. The first equation is the first condition in point 6) and the second one is a consequence of $ \gre A'B\sec =d_j c_i \ter $, $A'd_i'=Ad_i$ and the exactness of the first sequence. \end{oss} \begin{oss} The condition $(s,s\ter) \in Z_{jij}$ can be expressed in a similar way. In the previous conditions we have only to change $i$ with $j$ and $\gre$ with $-\gre$. \end{oss} We will prove now $Z_{iji} \subset Z_{jij}$. To do it we suppose that $A', \dots, d_j \sec$ are given as above and we construct $\Tilde A ,\Tilde B, \Tilde c_i , \Tilde d_i ,\Tilde c_j, \Tilde d_j, \Tilde{\Tilde {A}} , \Tilde{\Tilde { B}} ,\Tilde{\Tilde { c}}_i ,\Tilde{\Tilde { d}}_i , \Tilde{\Tilde { c}}_j ,\Tilde{\Tilde { d}} _j$ such that they satisfy the conditions. for $(s,s\ter) \in Z_{jij}$. \noindent {First step: construction of $\Tilde A, \Tilde B, \Tilde c_i, \Tilde c_j , \Tilde d_i, \Tilde d_j$}. Choose $\Tilde s$ such that $(\Tilde s, s ) \in Z_j^{\chi,\grl}$ and define $\Tilde A = A(\Tilde s)$, $\Tilde B = B( \Tilde s)$, $\Tilde c_l = c_l(\Tilde s)$ and $\Tilde d_l = d_l(\Tilde s)$ for $l \in \{i,j\}$ . Now I claim that there exists unique $\Tilde {\Tilde A} : V_i\ter \lra \Tilde V_j$ and $\Tilde {\Tilde B} : \Tilde V_j \lra V_i\ter$ such that: $$ \begin{cases} \Tilde c_j \Tilde{\Tilde A} = c_j\sec A\ter \\ \Tilde B \Tilde{\Tilde A} = - \gre d_i c_i \ter \end{cases} \;\mand\; \begin{cases} \Tilde{\Tilde A} \Tilde{\Tilde B} = \Tilde A \Tilde B - \gre \grl_i - \gre \grl_j \\ c_i \ter \Tilde{\Tilde B} = c_i \Tilde B \end{cases} $$ \noindent \emph{Unicity of $\Tilde{\Tilde A}$}: since the map $\Tilde a_j = (\Tilde c_j \;\; - \gre \Tilde B )^t$ is injective the unicity is clear. \noindent \emph{Existence of $\Tilde{\Tilde A}$}: to prove the existence of $\Tilde{\Tilde A}$ is enough to prove: $$ \Im \begin{pmatrix} c_j \sec A \ter \\ - \gre d_i c_i \ter \end{pmatrix} \subset \Im \begin{pmatrix} \Tilde c_j \\ \Tilde B \end{pmatrix} = \ker \begin{pmatrix} d_j & -\gre A \end{pmatrix}. $$ So the thesis follows from $d_jc_j\sec A\ter + A d_i c_i \ter =0$. Let now $\Tilde {\Tilde a} _i = (c_i \ter \;\; \Tilde {\Tilde A })^t$. I claim that $\Tilde {\Tilde a} _i $ is injective and that $\Im \Tilde {\Tilde a} _i = \ker ( d_i \;\; \gre \Tilde B)= \ker \Tilde b _i$. First of all observe that since $\Tilde m_i >0$ or $ \grl_i \neq 0 $, $\Tilde b_i$ is surjective. Observe also that $$ \begin{pmatrix} \Tilde c_j & 0 \\ 0 & \Id_{V_i\ter} \end{pmatrix} \comp \begin{pmatrix} \Tilde {\Tilde A} \\ c_i \ter \end{pmatrix} = \begin{pmatrix} c_j \sec A\ter \\ c_i \ter \end{pmatrix}. $$ So $\Tilde {\Tilde a}_i$ is injective as claimed. Now since $\dim R_i + \dim \Tilde V_j = \dim V_i \ter + \dim V_i$ to prove the last part of the claim it is enough to check that $\Tilde b_i \Tilde{\Tilde a}_i=0$. Indeed $$ \Tilde b_i \Tilde{\Tilde a}_i = d_i c_i \ter +\gre \Tilde B \Tilde{\Tilde A} =0. $$ \noindent \emph{Unicity of $\Tilde{\Tilde B}$}: this is a consequence of $\Tilde {\Tilde a}_i$ injective. \noindent \emph{Existence of $\Tilde{\Tilde B}$}: As for the existence of $\Tilde{\Tilde A}$ this is equivalent to $$ \Im \begin{pmatrix} c_i \Tilde B \\ \Tilde A \Tilde B -\gre \grl_i - \gre \grl_j \end{pmatrix} \subset \Im \begin{pmatrix} c_i \ter \\ \Tilde{\Tilde A} \end{pmatrix} = \ker \begin{pmatrix} d_i & \gre \Tilde B \end{pmatrix}. $$ So the thesis follows from $ \gre \Tilde B \Tilde A \Tilde B - \grl_i \Tilde B - \grl_j \Tilde B + d_i c_i \Tilde B =0$. Finally we set \begin{align*} \Tilde{\Tilde V}_i & = V_i \ter & \Tilde{\Tilde c}_i &=c_i \ter & \Tilde{\Tilde d}_i&=d_i\ter \\ \Tilde{\Tilde V}_i & =\Tilde V_j & \Tilde{\Tilde c}_j &=\Tilde c_j & \Tilde{\Tilde d}_j&=\Tilde d_j. \end{align*} The verification of all the conditions is now straightforward. The inclusion $Z_{jij}\subset Z_{iji}$ can be proved similarly and equation \eqref{eq:azioneweyl} is clear by definition. So Proposition \ref{prp:azioneweyl} is proved. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{A representation of the Weyl group} In this section, following Nakajima \cite{Na1}, we show how to use the above action to construct an action of the Weyl group on the homology of quiver varieties. Maybe this action is related with the one constructed by Slodowy in the case of flag varieties (\cite{Slodowy}, ch.4). First we recall some general about the action of the Weyl group. Let $Z\cech = Q\cech \otimes _{\mZ} \mC$ and $Z= Q \otimes _{\mZ}P$. On $Z$, $Z\cech$ there is a natural action of $W$. \begin{lem} For all $ u\in Z\cech$ the set $Wu$ is discrete. \end{lem} \begin{lem} Consider the action of $W$ on $\mP(Z\cech)$. If $p \in \mP(Z\cech)$ then $$ \overline{Wp} \text{ is countable}. $$ \end{lem} If $p \in \mP(Z\cech)$ we define $H_p= \{x \in P\otimes _{\mZ}\mC \st \bra x , p \ket =0\}$ \begin{lem} If $p \in \mP(Z\cech)$ then $$ \overline{W H_p} = \bigcup_{q \in \overline{Wp}} H_q. $$ \end{lem} We define $\calH = \overline{W\calH_v \cup \calH_U}$ and $\calR = \goz - \calH$. By the previous lemmas $\calH$ is the union of a countable number of real codimension $3$ subspaces in $\goZ$ and in particular $\calR$ is simply connected. We need also the following definition \begin{align*} K &= \{ u \in \mZ^I \st -\sum{u_i\gra_i} \text{ is dominant and supp}\, u \; \text{ is connected } \} \\ P_0 &= \{ p \in P \st p \text{ is dominant and } \bra u \cech , p \ket \geq 2 \text{ for all } u \in K \}. \end{align*} Now we choose $d,v$ such that $\bar d =\bar v$. \begin{lem}\label{lem:lemma38} If $\bar d \in P_0$ then $\tilde {\grm}$ is surjective and is a locally trivial bundle over $\calR$. \end{lem} \begin{proof} By Proposition 10.5 and Corollary 10.6 in \cite{Na2} there exists a closed orbit $Gs$ in $\grL_0(d,v)$ with trivial stabilizer. Then by Proposition \ref{prp:KNM1} there exists $t \in Gs$ such that $\tilde {\grm}(t) = 0$ and by lemma \ref{lem:regmu} and \ref{regolaritamu} $d\tilde {\mu}_t$ is surjective. Now the surjectivity follows by homogeneity. The local triviality over $\calR$ follows also from lemma \ref{lem:regmu} and \ref{regolaritamu}. \end{proof} Now consider \begin{align*} R & = \{\grl \in Z \st (0,\grl) \notin \calR \}, \\ \grL(d,v) & = \{ (\grl,s) \in Z \times S \st s \in \grL_{\grl} \} , \\ M(d,v) &= \grL(d,v) /\!/ G_v \;\mand\; p : M(d,v) \lra Z \text{ the projection} \\ \goL(d,v) & = \{ (\zeta,s) \in \goZ \times S \st s \in \goL_{\zeta} \} , \\ \goM(d,v) &= \goL(d,v) / U(V) \;\mand\; \tilde p : \goM(d,v) \lra \goZ \text{ the projection} \end{align*} We have the following commutative diagram $$ \xymatrix{ (\grl,s) \ar@{}[r]|{\in} \ar[d] & M(d,v) \ar[r]^p \ar[d]^{\mi_M} & Z \ar[d] & \grl \ar@{}[l]|{\ni} \ar[d]\\ (0,\grl,s) \ar@{}[r]|{\in} & \goM(d,v) \ar[r]^{\tilde p} & \goZ & (0,\grl) \ar@{}[l]|{\ni} } $$ By Proposition \ref{prp:KNM2} the diagram is a pull back and by lemma \ref{lem:lemma38} $p$ and $\tilde p$ are locally trivial over $R$, $\calR$. We call $M_R = p^{-1}(R)$ and $\goM_{\calR}= \tilde p ^{-1} (\calR)$. Now consider the complex $\calF = R p_* (\mZ_{M_{R}}) = \mi_M^{-1} R\tilde p _* (\mZ_{\goM_{\calR}}) $ which is cohomologically a locally constant complex. We observe now that $\Pi_1(\calR)$ is trivial so $ R\tilde p _* (\mZ_{\goM_{\calR}})$ is isomorphic to cohomologically constant complex on $\calR$ so it is $\calF$ on $R$. In particular for any $x,y\in R$ we have a canonically isomorphism $$ \psi^i_{x,y} : H^i(\calF_x) \lra H^i(\calF_y). $$ Now observe that by Proposition \ref{prp:azioneweyl} there is an action on $W$ on $R, M_R$ and that $p$ is equivariant with respect to this action. So we can define a $W$ action on $H^i(M_{0,\grl}(d,v),\mZ)$ by $$ \grs (c) = \psi^i_{\grs\grl,\grl}\comp H^i(\Phi^{d,v}_{\grs,0,\grl}) (c) $$ for any $\grs \in W$. To verify that this is an action we have only to verify that $$ \psi^i_{\grs\grl^2,\grl^1}\comp H^i(\Phi^{d,v}_{\grs,0,\grl^2}) \psi^i_{\grl^1,\grl^2}(c) =\psi^i_{\grs\grl^1,\grl^1}\comp H^i(\Phi^{d,v}_{\grs,0,\grl^1}) (c). $$ Since $R$ is connected and $H^i(M,\mZ)$ is discrete this is clear. S o we have proved the following corollary. \begin{cor} If $d=v$ and $(0,\grl) \in \calR$ then there is an action of $W$ on $H^i(M_{0,\grl}(d,v),\mZ)$. \end{cor} \begin{oss} If $m_+=(1,\dots,1)$ and $\grl =0$ it is easy to see that $d\grm_s$ is surjetive for all $s \in \grL_{m_+,0}(d,v)$. Then by lemma \ref{lem:lemma38} there is a canonical isomorphism $H_*(M_{m_+,0}(d,v)) \isocan H_*(M_{0,\grl}(d,v))$ if $(0,\grl) \in \calR$. So by Nakajima's Theorem (Theorem 10.2 \cite{Na2}) it is natural to make the following conjecture: \end{oss} \begin{con} Let $top = \frac{1}{2} \dim H_*(M_{0,\grl}(d,v))$ then $$ H^{top}\bigl(M_{(0,\grl)}(d,v), \mC \bigr) \isocan \bigl(L_d \bigr)_0 $$ where $\bigl(L_d \bigr)_0$ is the $0$-weight space of the Kac-Moody algebra associated to the quiver of highest weight $\sum_i d_i \bar {\omega}_i$. \end{con} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Reduction to the dominant case} \label{riduzioneM0} As a consequence of Proposition \ref{prp:azioneweyl} we see that if $(m,\grl) \in \calG_v$ then there exists $\grs \in W$ and $v' = \grs \cdot v$ such that $d- v'$ is dominant and $M_{\grs m , \grs \grl}(d,v') \isocan M_{m,\grl}(d,v)$. We generalize now this result to arbitrary $\grl$. On $Q$ we consider the following order: $v' \leq v$ if and only if $v_i' \leq v_i$. We consider now the following construction: let $v' \leq v$ and fix an embedding $V'_i \incluso V_i$ and a complement $U_i$ of $V'$ in $V_i$, then we can define a map $\wt{\mj} \colon S(d,v') \lra S(d,v)$ through: \begin{equation}\label{defembeddingj} \wt{\mj}(B^{\prime},\grg^{\prime},\grd^{\prime})= \left( \begin{pmatrix} B^{\prime} & 0 \\ 0 & 0 \end{pmatrix} , \begin{pmatrix} \grg^{\prime} \\ 0 \end{pmatrix} , \begin{pmatrix} \grd^{\prime} & 0 \end{pmatrix}\right) \end{equation} where the matrices of the new triple represents the maps described through the decomposition $V_i=V'_i \oplus U_i$. Suppose now that $(m_i,\grl_i)=0$ for all $i$ such that $v_i'\neq v_i$. Then it is easy to see that this map restrict to a map $\mj_r : \grL_{m,\grl}(d,v') \lra \grL_{m,\grl}(d,v)$ and so induces a map $\mj^{v'}_v=\mj \colon M_{0,\grl}(d,v') \lra M_{0,\grl}(d,v)$. \begin{lem} \label{lemmaj} $\mj$ is a closed immersion \end{lem} \begin{proof} We prove that the map $\mj^{\sharp} : \mC[\grL_{\grl}(d,v)]^{G(v)} \lra \mC[\grL_{\grl}(d,v')]^{G(v' )}$ is surjective. By proposition \ref{generatoriinvarianti} this follows by the following two identities: $$ \Tr\left(\gra\left(\mj (s) \right) \right) = \Tr\left(\gra (s) \right) \; \mand \; \grb\left(\mj (s) \right) = \grb (s) $$ for each $B$-path $\gra$ and for each admissible path $\grb$. \end{proof} \begin{lem} \label{lemmariduzioneM0} If $2 v_i > d_i + \sum_{j \in I} a_{ij}v_j$ and $v' = v -\gra_i$ then $\mj$ is an isomorphism of algebraic varieties \end{lem} \begin{proof} It's enough to prove that $\mj$ is surjective. Let $s=(B,\grg,\grd) \in \grL_0(d,v)$ and consider the sequence (see \eqref{defTab} for the notation) : $$ \begin{CD} T_i @>{b_i}>> V_i @>{a_i}>> T_i. \end{CD} $$ Since $b_ia_i =0 $ and $2 \dim V_i > \dim T_i$ we have that $b_i$ is not surjective or that $a_i$ is not injective. Suppose that $b_i$ is not surjective, then up to the action of $G_v$ we can assume that $\Im b_i \subset v'_i$. Then, for $t \in \mC^*$ consider $g_t=(g_{j,t}) \in G_v$ with $$ g_i = \begin{pmatrix} \Id_{v'_i} & 0 \\ 0 & t^{-1} \end{pmatrix} \; \mand \; g_j = \Id_{V_j} \mfor j \neq i. $$ Then \begin{enumerate} \item $g_{i,t} B_h = B_h $ if $h_1 =i$ and $g_i \grg_i =\grg_i$, since $\Im B_h , \Im \grg_i \subset \Im b_i \subset v'_i$, \item $\exists\, \lim_{t \to 0} B_h g_{i,t}^{-1} = B_h $ if $h_0 =i$ and $ \grd_i g_i^{-1} =\grd_i$ \end{enumerate} So $\exists\, \lim_{t \to 0} g_t s = s^{\prime}$ and it is clear that $s^{\prime} \in \wt{\mj}(\grL_0(d,v'))$ and that $p_0(s)= p_0(s^{\prime})\in \Im \mj$. If $b_j$ is surjective and $a_i$ is not injective the argument is similar. \end{proof} \begin{prp}\label{prp:riduzioneM0}For all $\grl$ and for all $d\geq 0, v\geq 0$ there exists $v'$ and $\grs \in W$ such that $\bar d - \bar v ' $ is dominant and $$ M_{0 ,\grs \grl}(d,v') \isocan M_{0,\grl}(d,v). $$ \end{prp} \begin{proof} We prove this proposition by induction on the order $\leq$ on $Q$. \noindent{First step: $v=0$}. If $v=0$ we can take $v'=v$ and $\grs =1$. \noindent{Inductive step.} If $d-v$ is not dominant then there exists $i$ such that $2v_i > d_i + \sum a_{ij}v_j$. If $ \grl_i \neq 0 $ we observe that $s_i v= v' < v$ (that is $v' \leq v$ and $v'\neq v$) and that $M_{s_i m,s_i \grl}(d,v')\isocan M_{m,\grl}(d,v)$ and so we can apply the inductive hypothesis. If $\grl_i = 0$ we apply the previous lemma and the inductive hypothesis. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{On normality and connectedness of quiver variety in the finite type case} In this section we restrict our attention to the case of quiver varieties of finite type and to the case $m=(1,\dots,1)$ and $\grl =0$ and we fix $d,v$. By remark \ref{oss:riduzioneanonzero} we can assume without loss of generality that $v_i >0$ for all $i$. We would like to prove the following conjecture: \begin{con} $M_{m,0}(d,v)$ is connected and $M_{0,0}(d,v)$ is normal. \end{con} \begin{oss} If $\tilde m = (m_1,\dots,m_n) \in\mN_+^I$ it is easy to see that $\grL_{\tilde m,0}= \grL_{m,0}$ ao in particular $M_{m,0}(d,v)$ is smooth. Instead $M_{0,0}$ is a cone so it is clearly connected. \end{oss} By proposition \ref{prp:riduzioneM0}it is enough to prove the theorem in the case $d-v$ dominant. Unfortunately I'm not able to prove the conjecture only in the case $d-v$ regular: $\bra d-v,\gra_i\ket >0$ for all $i$. To prove the conjecture in this case we will use the following stratification introduced by Lusztig in \cite{Lu:Q4}. \begin{dfn} For any $s\in S$ and $i\in I$ let $$ V_i^+ = V_i^+(s) = \sum_{\gra \text{ a }B-\text{path}\st \gra_1=i} \Im (\gra(s)\grg_{\gra_0}) $$ If $v'=(v_1',\dots,v_n') \in \mN^n$ we define $$ \grL^{v'} = \{ s\in \grL_0(d,v) \st \dim V_i^+ (s) = v_i'\}. $$ \end{dfn} Observe that $\grL^v = \grL_{m_+,0}(d,v)$. To prove our result we will use the following lemma of Lusztig. \begin{lem}[Lusztig: \cite{Lu:Q4} Proposition 4.5 and Proposition 5.3] \label{lem:Lusztigdim}If $0\leq v'_i \leq v_i$ for each $i$ then $$ \dim \grL^{v'}(d,v) = \dim S - \sum_{i\in I} \dim gl(V_i) - \bra (v-v')\cech , d-v \ket - \tfrac{1}{2} \bra (v-v')\cech , v-v' \ket $$ \end{lem} Our result follows trivially from the following lemma. \begin{lem} 1) If $d-v$ is dominant then $\grL_{0,0}(d,v)$ is a complete intersection. \noindent 2) If $d-v$ is regular then $\grL_{0,0}$ is normal and irreducible and $\grL_{m_+,0}(d,v)$ is connected. \end{lem} \begin{proof} Observe that $\grL_{0,0}(d,v) =\mu ^{-1}(0)$ so each irreducible component of $\grL_{0,0}(d,v)$ must have dimension at least $\dim S - \sum_i \dim gl(V_i) = \delta_V$. Suppose now that $d-v$ is dominant. By Nakajima's theorem (\cite{Na2} Theorem 10.2) $M_{m_+,0}$ is not empty. Observe also that by Proposition \ref{prp:1.14} $\grL_{m_+,0}(d,v)$ is a smooth subset of $\grL_{0,0}(d,v)$ of dimension $\delta_V$. It is well known that $\grL_{m_+,0} (d,v) = \grL^v$. Hence $$ \grL_{0,0}(d,v) - \grL_{m_+,0} (d,v) = \bigcup_{v'\leq v \mand v' \neq v} \grL^{v'}. $$ By the lemma above we have that if $v' \leq v$ and $v' \neq v$ $\dim \grL^{v'} < \delta_V$. So $\grL_{m_+,0}(d,v)$ must be dense in $\grL_{0,0}(d,v)$ and $\grL_{0,0}(d,v)$ is a complete intersection. Moreover if $d-v$ is regular we have that $\dim \grL^{v'} < \delta_V - 1$ so the singular locus has codimension at least two and normality and irreducibility follows. Finally by our discussion it is clear that if $\grL_{m_+,0}(d,v)$ is disconnected then $\grL_{0,0}(d,v)$ is not irreducible. \end{proof} \begin{oss} In the lemma we can substitute $\grL_{m_+,0}(d,v)$ with any other subset $Reg$ of regular points in $\grL(d,v)$. In this way is indeed possible to improve a little bit the theorem but Crawley-Boevey explained me that this strategy cannot work in general because there are cases where $d-v$ is dominant and $\grL_{0,0}(d,v)$ is not normal. It should be also pointed out that Crawley-Boevey proved the connectedness in complete generality (\cite{CrBo}). He said me that is also able to prove normality for a much bigger class of quiver varieties. \end{oss} \begin{thebibliography}{10} \bibitem{CrBo} W.~Crawley-Boevey, \emph{Geometry of the moment map for representations of quivers}, preprint available at {\tt{http://www.amsta.leeds.ac.uk/\~{ }pmtwc}}. \bibitem{DeWe} H. ~Derksen and J. ~Weyman, \emph{Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients}, to appear in J. Amer. Math. 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