SYMMETRIES OF INTEGRABLE DEFORMATIONS (Combinatorics of Lie Type)
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(2) 12. complex manifold. X. by \mathcal{O}_{X} and \mathcal{M}_{X} respectively. In par‐ \mathcal{O}_{\mathb {P}^{1} and \mathcal{M} \mathcal{M}_{\mathb {P}^{1} for short. Let us denote the ring of convergent (resp. formal) power series of z by \mathbb{C}\{z\} (resp. \mathb {C}[z\mathrm{I} ). Their total quotient fields are written by \mathbb{C}\{\{z\}\} and \mathbb{C}((z)) respectively. on a. ticular when X =\mathbb{P}^{1} ,. write \mathcal{O}. Gauge equivalences. 1.1.. written. are. we. =. =. of differential. equations. We recall gauge ordinary differential equations. transformations of systems of first order linear defined. locally. on. \mathb {P}^{1} and. moreover. recall HukuhararTurrittin‐Levelt normal. forms of local differential equations under formal gauge transformations. Let U be an open subset of \mathb {P}^{1} and z a local coordinate on U. Definition 1.1. with. (gauge transformation).. a. linear differential equation. \displaystyle\frac{d}{dz}\mathrm{Y}=A\mathrm{Y}. A\in M(n, \mathcal{M}(U)). and. \displaystyle\frac{d}{dz}\tilde{\mathrm{Y} =B\ovalbox{\t \smal REJECT} by. equation. For. X\in \mathrm{G}\mathrm{L}(n, \mathcal{M}(U)). ,. define. we. a new. differential. B :=XAX^{-1}+ (\displaystyle \frac{d}{dz}X)X^{-1}.. We call B the. meromorphic gauge transformation of A by X and particular if X\in \mathrm{G}\mathrm{L}(n, \mathcal{O}(U)) we say the holomorhic B=:X[A] transformation. .. Here is. a. we. In. ,. note that if. solution of. Let. us. a. \displaystyle \frac{d}{dz}\mathrm{Y}=A\mathrm{Y} then Ỹ. solution of. B=X[A].. for. take a\in U and choose. the stalks. similarly. vector \mathrm{Y} is. a. \displaystle\frac{d} z Ỹ =B\ovalbox{\t \smal REJECT}. a. local coordinate. z. which is. zero. at. a. write gauge. =. .. xY. Then. \mathcal{O}_{a} and \mathcal{M}_{a} at a can be identified with \mathbb{C}\{z\} and \mathbb{C}\{\{z\}\} We can define holomorphic and meromorphic gauge transformations of a .. local differential. equation. \displaystyle \frac{d}{dz}\mathrm{Y}=A\mathrm{Y} with A\in M(n, \mathcal{M}_{a}). .. define formal gauge. In this case,. we. transformations, namely say X[A] is A X if of formal holomorphic gauge transformation by X\in \mathrm{G}\mathrm{L}(n, \mathbb{C}\ovalbox{\t \smal REJECT} z\mathrm{J}) and formal meromorphic gauge transformation if X\in \mathrm{G}\mathrm{L}(n, \mathbb{C}( z) )\backslash \cdot For a local differential equation \displaystyle \frac{d}{dz}\mathrm{Y}=A\mathrm{Y} with A\in M(n,\mathbb{C}((z))) it is known that there exists a normal form under the formal meromorphic gauge can moreover. we. the. ,. transformations Let \mathcal{P}. as. follows.. :=\displaystyle \bigcup_{s\in \mathb {Z}>0}\mathb {C}( z^{\frac{1}{ $\epsilon$} ). Definition 1.2. ,. the field of Puiseux series.. (HukuhararTu$\iota$\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}‐Levelt. TurTittin‐Levelt normal. form. or. form). By. normal. HTL normal. form. for. short,. Hukuhara‐. we mean an. element of the form. diag. (q_{1}(z^{-\frac{1}{ $\epsilon$}})I_{n1}+R_{1}, \ldots, q_{m}(z^{-\frac{1}{s} )I_{n_{m} +B_{m})z^{-1}. \in M(n, \mathbb{C}((z^{\frac{1}{8} )))\subset M(n, \mathcal{P}). where. q_{i}(t). \in. t\mathbb{C}[t] satisfying. q_{i}. \neq. q_{j} if i. \neq j. ,. and R_{\dot{ $\eta$} \in. M(n_{i}, \mathbb{C}). with. n_{1}+\cdots n_{m}=n. For. the. an. HTL normal form. H\in M(n,\mathcal{P}). irregular part of H Here A\in M(n, \mathcal{P}) by \mathrm{p}\mathrm{r}_{\mathrm{r}\mathrm{e}\mathrm{s} (A) .. .. we. ,. we. call. H_{\mathrm{i}\mathrm{r}\mathrm{r}. :=H-\mathrm{p}\mathrm{r}_{\mathrm{r}\mathrm{e}\mathrm{s} (H)z^{-1}. denote the coefficient matrix of z^{-1} in.
(3) 13. The tial. following. is. a. equations with. fundamental fact of the local formal. theory. of differen‐. irregular singularity.. (Hukuhara‐Turrittin‐Levelt, see [36] for instance). For any M(n, \mathbb{C}((z))) there exist an integer r \in \mathbb{Z}>0 and X \in \mathrm{G}\mathrm{L}(n, \mathbb{C}( z^{\frac{1}{r} ) ) that X[A] is an HTL normal form in M(n, \mathbb{C}( z^{\frac{1}{r} ) ) We call this X[A]. Theorem 1.3. A \in. ,. such. .. the normal form In the above. of A.. theorem,. r=\displaystyle \min. we. ,. assume. the field extension is. minimal, i.e.,. { s| normal form X[A]\in M(n, \mathbb{C}( z^{\frac{1}{ $\epsilon$}}) ) }. H, H'. If two HTL normal forms. A\in M(n, \mathbb{C}((z))). may. \in. then there exists. M(n, \mathbb{C}( z^{\frac{1}{r} ) ). g\in \mathrm{G}\mathrm{L}(n, \mathbb{C}). are. normal forms of. an. such that. g^{-1}H_{\mathrm{i}\mathrm{r}\mathrm{r} g=H_{\mathrm{i}\mathrm{r}\mathrm{r} ', g^{-1}\exp(2 $\pi$\sqrt{-1}k\mathrm{p}\mathrm{r}_{\mathrm{r}\mathrm{e}\mathrm{s} (H) g=\exp(2 $\pi$\sqrt{-1}k\mathrm{p}\mathrm{r}_{\mathrm{r}\mathrm{e}\mathrm{s} (H') for. some. 1.2.. integer k\geq 1. Quiver. ,. [2]. Theorem 6.3 in. see. varieties. In this subsection. for. we. example.. shall introduce. quiver. vari‐. eties.. Representations of quiver and quiver variety. quivers. 1.2.1.. Now let. us. recall repre‐. sentations of. Definition 1.4. sisting in \mathrm{Q}_{0} ,. its. source. For. (quiver).. A quiver. a. s( $\rho$)\in \mathrm{Q}_{0}. and its target. fixed vector. $\alpha$. \in. with the dimension vector. Rep Let. us. \mathrm{Q}=(\mathrm{Q}_{0}, \mathrm{Q}_{1}, s,t). is the. quadruple. con‐. of \mathrm{Q}_{0} , the set of vertices, and \mathrm{Q}_{1} , the set of arrows connecting vertices and two maps s,t:\mathrm{Q}_{1}\rightarrow \mathrm{Q}_{0} , which associate to each arrow $\rho$\in \mathrm{Q}_{1}. t( $\rho$)\in \mathrm{Q}_{0} respectively.. (\mathb {Z}_{\geq 0})^{\mathrm{Q}_{0} $\alpha$. the representation space of the quiver. ,. is. (\displayst le\mathrm{Q}, $\alpha$)=\bigoplus_{$\rho$\in\mathrm{Q}_{1}. recall the double of. a. Homc (\mathb {C}^{$\alpha$_{ $\epsilon$( $\rho$)} , \mathbb{C}^{$\alpha$_{t( $\rho$))} .. quiver Q.. (double quiver). Let \mathrm{Q}=(\mathrm{Q}_{0}, \mathrm{Q}_{1}) be a finite quiver. Then quiver \overline{\mathrm{Q} of \mathrm{Q} is the quiver obtained by adjoining the reverse arrow to each arrow $\rho$:a\rightarrow b Namely \overline{\mathrm{Q} :=(\overline{\mathrm{Q} _{0} :=\mathrm{Q}_{0},\overline{\mathrm{Q} _{1}:=\mathrm{Q}_{1}\cup \mathrm{Q}_{1}^{*}) $\rho$^{*}:b\rightarrow a where \mathrm{Q}_{1}^{*}:=\{$\rho$^{*}:t( $\rho$)\rightarrow s( $\rho$)| $\rho$\in \mathrm{Q}_{1}\}. Definition 1.5. the double. .. Let. us. note that for each. $\rho$\in \mathrm{Q}_{1}. we can. identify. Homc (\mathb {C}^{$\alpha$_{\mathrm{s} (\mathrm{p}), \mathb {C}^{$\alpha$_{i( $\rho$)} )^{*}\cong \mathrm{H}\mathrm{o}\mathrm{r}\mathrm{n}_{\mathrm{C} (\mathb {C}^{$\alpha$_{8($\rho$^{*})} ,\mathb {C}^{$\alpha$_{t($\rho$^{*})} ). by. the trace. pairing. Thus the representation \backslash. space. tified with the cotangent bundle. T^{*}\mathrm{R}\mathrm{e}\mathrm{p}(\mathrm{Q}, $\alpha$)\cong \mathrm{R}\mathrm{e}\mathrm{p}(\overline{\mathrm{Q} , $\alpha$) In this. case. the canonical. symplectic form. is. Rep (\overline{\mathrm{Q} , $\alpha$). .. given by. $\omega$(x,y)=\displaystyle\sum_{$\rho$\in\mathrm{Q}_{1}(\mathrm{t}\mathrm{r}(x_{$\rho$}y_{$\rho$^{*})-\mathrm{t}\mathrm{r}(x_{$\rho$}*y_{$\rho$}). .. can. be iden‐.
(4) 14. Thus. we can see. of. Then the. Rep (\overline{\mathrm{Q} , $\alpha$). following. complex symplectic manifold with the. as a. \displaystyle\mathrm{G}:=\prod_{a\in\mathrm{Q}_{0}\mathrm{G}\mathrm{L}($\alpha$_{a},\mathb {C}). map is. a. action. .. moment map;. $\mu$_{$\alpha$}:\displayst le\mathrm{R}\mathrm{e}\mathrm{p}(\overline{\mathrm{Q}, $\alpha$)\rightarow\prod_{a\in\mathrm{Q}_{0}M($\alpha$_{a},\mathb {C}). whose. images ($\mu$_{ $\alpha$}(x)_{a})_{a\in \mathrm{Q}_{0}. are. given by. $\mu$_{$\alpha$}(x)_{a}=\displaystyle\sum_{$\rho$\in\mathrm{Q}_{1} x_{$\rho$}x_{$\rho$^{*-} \sum_{p\in\mathrm{Q}_{1} x_{$\rho$}*x_{$\rho$}. t( $\rho$)=a s( $\rho$)=a. Now. ready. we are. quiver. (quiver variety).. Definition l.ô. $\lambda$=($\lambda$_{a})\in \mathbb{C}^{\mathrm{Q}_{0}. to define. Then. .. varieties.. Let. us. quiver variety. a. \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$) :=$\mu$^{-1}( $\lambda$)//\mathrm{G} Here. \mathbb{C}[$\mu$^{-1}( $\lambda$)]. is the coordinate. take. :=. Specm \mathbb{C}[$\mu$^{-1}( $\lambda$)]^{\mathrm{G} .. ring of. (possibly empty) subspace. $\mu$^{-1}( $\lambda$)^{\mathrm{i}\mathrm{r}\mathrm{r} := { x\in$\mu$^{-1}( $\lambda$)|x. $\mu$^{-1}( $\lambda$). is. Let. .. consider the. us. iiTeducible}.. this space is proper and \mathrm{G}/\mathb {C}^{\times} Thus the symplectic reduction. Then the action of. King [24]).. collection of complex numbers. a. is the affine quotient. on. free. moreover. (see. \mathfrak{M}_{ $\lambda$}^{\mathrm{r}\mathrm{e}\mathrm{g} (\mathrm{Q}, $\alpha$):=$\mu$^{-1}( $\lambda$)^{\mathrm{i}\mathrm{r}\mathrm{r} /\mathrm{G} be. complex manifold with the symplectic structure, i.e., \mathrm{a} complex symplectic manifold. We call this manifold the quiver variety too. can. seen as. \mathrm{R}\mathrm{e}\mathrm{m}\backslash ark. a. 1.7. The above. quiver varieties are special ones of Nakajima quiver enjoy rich geometric properties and applications for repre‐ theory and theoretical physics and so on (see [27] for instance).. varieties which sentation. 1.2.2. Some geometry of quiver symplectic manifold \mathfrak{M}_{ $\lambda$}^{\mathrm{r}\mathrm{e}\mathrm{g} (\mathrm{Q}, $\alpha$). varieties. As is. essary and sufficient condition for by Crawley‐Boevey in [6].. In order to. [19]).. a. Let. explain. \mathrm{Q} be. symmetric. a. we. noted. before,. Thus next. possibly empty. the non‐emptiness. of. the. complex. we see a nec‐. \mathfrak{M}_{$\lambda$}^{\mathrm{r}\mathrm{e}\mathrm{g} (\mathrm{Q}, $\alpha$). obtained. condition, recall the root system of a quiver \mathrm{Q} (cf. quiver. From the Euler form. the. finite. \displaystyle\{$\alpha$, $\beta$\rangle:=\sum_{a\in\mathrm{Q}_{0}$\alpha$_{a}$\beta$_{a}-\sum_{$\rho$\in\mathrm{Q}_{1}$\alpha$_{s($\rho$)}$\beta$_{t($\rho$)},. bilinear form and. quadratic. form. are. defined. by. ( $\alpha$, $\beta$):=\langle $\alpha$, $\beta$\rangle+\{ $\beta$, $\alpha$\rangle,. q( $\alpha$):=\displaystyle \frac{1}{2}( $\alpha$, $\alpha$) and set. p( $\alpha$) :=1-q( $\alpha$). For each vertex. a. ($\epsilon$_{a})_{b}=0, (b\in \mathrm{Q}_{0}\backslash \{a\}) no. edge‐loop, i.e.,. .. Here a,. $\beta$\in \mathb {Z}^{\mathrm{Q}_{0} .. \mathrm{Q}_{0} define. \in. ,. $\epsilon$_{a} \in. \mathb {Z}^{\mathrm{Q}_{0}. (a \in \mathrm{Q}_{0}). so. that. ($\epsilon$_{a})_{a}. =. 1,. We call $\epsilon$_{a} a fundamental root if the vertex a has there is no arrow $\rho$ such that s( $\rho$) =t( $\rho$) =a Denote .. ..
(5) 15. by $\Pi$ the set of fundamental fundamental reflection s_{a} by. roots.. For. s_{a}( $\alpha$) := $\alpha$-( $\alpha,\ \epsilon$_{a})$\epsilon$_{a} W\subset \mathrm{A}\mathrm{u}\mathrm{t}\mathbb{Z}^{\mathrm{Q}_{0} generated by. The group the. group of the. Weyl. Similarly. invariant.. fundamental root $\epsilon$_{a} , define the. a. $\alpha$\in \mathbb{Z}^{\mathrm{Q}_{0} .. for. all fundamental reflections is called. quiver Q. Note that the bilinear form (, ) is W‐ can define the reflection r_{a}:\mathb {C}^{\mathrm{Q}_{0} \rightar ow \mathb {C}^{\mathrm{Q}_{0} by. we. r_{a}( $\lambda$)_{b}:=$\lambda$_{b}-($\epsilon$_{a}, $\epsilon$_{b})$\lambda$_{a} for. $\lambda$\in \mathb {C}^{\mathrm{Q}_{0}. and a,. b\in \mathrm{Q}_{0} Define the .. set of real roots. $\Delta$^{\mathrm{r}\mathrm{e} :=\displaystyle\bigcup_{w\inW}w($\Pi$) Define the F. fundamental. set. .. F\subset \mathbb{Z}^{\mathrm{Q}_{0} by. { $\alpha$\in(\mathbb{Z}_{\geq 0})^{\mathrm{Q}_{0} \backslash \{0\}| ( $\alpha$, $\epsilon$)\leq 0 for all $\epsilon$\in $\Pi$. :=. Then define the set of. by. imaginary. roots. ,. support of. $\alpha$. is. For. a. connected}. by. $\Delta$^{\mathrm{i}\mathrm{m} := \cup w(F\cup-F). .. w\in W. Then the root system is. $\Delta$:=$\Delta$^{\mathrm{r}\mathrm{e} \cup$\Delta$^{\mathrm{i}\mathrm{m} . An element Now. $\Delta$^{+}:= $\alpha$\in $\Delta$\cap(\mathbb{Z}_{\geq 0})^{\mathrm{Q}_{0}. we are. ready. to. is called. ($\lambda$_{a})\in \mathb {C}^{\mathrm{Q}_{0} the set $\Sigma$_{$\lambda$} consists of the (1) $\lambda$\displaystyle \cdot $\alpha$:=\sum_{a\in \mathrm{Q}_{0} $\lambda$_{a}$\alpha$_{a}=0, ,. (2). if there exists. $\lambda$\cdot$\beta$_{i}=0 Theorem 1.8. ,. a. then. positive. a. Crawley‐Boeveys. see. positive. root.. theorem.. roots. decomposition $\alpha$=$\beta$_{1}+$\beta$_{2}+\cdots. ,. with. p( $\alpha$)>p($\beta$_{1})+p($\beta$_{2})+\cdots. (Crawley‐uoevey.. Theorem 1.2 in. quiver and \overline{\mathrm{Q} the double of Q. Let. us. fiư. a. fixed $\lambda$. [6]).. $\beta$_{i}\in$\Delta$^{+}. Let \mathrm{Q} be. dimension vector $\alpha$\in. a. .. .. $\mu$^{-1}( $\lambda$)^{ir $\tau$}. is dense in. Moreover. $\mu$^{-1}( $\lambda$). and. finite. (\mathb {Z}_{\geq 0})^{\mathrm{Q}_{0}. \mathb {C}^{\mathrm{Q}_{0} Then $\mu$^{-1}( $\lambda$)^{irr}\subset Rep (\overline{\mathrm{Q} , $\alpha$) is nonempty if and only if $\Sigma$_{ $\lambda$} Furthermore, in this case $\mu$^{-1}( $\lambda$) is an irreducible algebraic variety and $\lambda$\in. =. satisfying. $\alpha$\in. and. .. Crawley‐Boevey. showed the. following geometric properties. of. quiver varieties. Theorem 1.9. (Crawley‐Boevey Corollary. quiver variety \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$). 2p( $\alpha$) of. is. a. 1.4 in. [6]). If. reduced and irreducible. $\alpha$ \in $\Sigma$_{ $\lambda$} then the variety of dimension. .. Combining these results, we have regular parts of quiver varieties.. the. following non‐emptiness. condition. (Crawley‐uoevey [6]). The quiver variety \mathfrak{M}_{ $\lambda$}^{f}\mathrm{w}(\mathrm{Q}, $\alpha$) is non‐empty if and only if $\alpha$\in$\Sigma$_{ $\lambda$} Furthermore in this case, it is a connected Corollary. 1.10. .. complex symplectic manifold of. dimension. 2p( $\alpha$). ..
(6) 16. 2. MODULI SPACES. OF STABLE MEROMORPHIC CONNECTIONS ON TRIVIAL BUNDLES. Let. define moduli spaces of meromorphic connections paper [3] (see also [17]).. us. on. trivial bundles. following Boalchs. 2.1. Moduli spaces of. meromorphic. connections. Let. B=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(q_{1}(z^{-1})I_{n_{1}}+R_{1}z^{-1}, \cdots q_{m}(z^{-1})I_{n_{m}}+R_{m}z^{-1})\in \mathrm{G}\mathrm{L}(n, \mathbb{C}((z))) be. an. HTL normal form. The. equivalent class of B under formal holomorphic. gauge transformations is. \mathcal{O}_{B} :=\{X[B]|X\in \mathrm{G}\mathrm{L}(n, \mathbb{C}[z\mathrm{I})\}. Let B. consider another. us. Let. .. us. consider the. equivalent class of projection. B called the truncated orbit of. $\iota$:M(n, \mathbb{C}((z)))\rightarrow M(n, \mathbb{C}((z))/\mathbb{C}[z\mathrm{I}) The map the. $\iota$. Z\in M(n, \mathbb{C}((z))/\mathbb{C}[z\mathrm{I}) of. so. that. \tilde{Z}. Then. \mathrm{G}\mathrm{L}(n, \mathbb{C}[z\mathrm{I}) on M(n, \mathbb{C}((z))/\mathbb{C}[z\mathrm{J} ) from M(n, \mathbb{C}((z))) Namely for g \in \mathrm{G}\mathrm{L}(n, \mathbb{C}[z\mathrm{I}). induces the action of. adjoint action of that. chosen. .. on. define. L(\tilde{Z})=Z. .. We. .. ,. g^{-1}Zg:= $\iota$(g\tilde{Z}g^{-1}) can see. that this is. where. \tilde{Z}\in M(n, \mathbb{C}((z))). is. independent of the choice. regarding B as an element in M(n, \mathbb{C}((z))/\mathbb{C}[z\mathrm{I} ), we define the by the action of \mathrm{G}\mathrm{L}(n, \mathbb{C}[z\mathrm{J}) on M(n, \mathbb{C}((z))/\mathbb{C}[z\mathrm{J} );. trun‐. cated orbit of B. \mathcal{O}_{B}^{\mathrm{t}\mathrm{r}\mathrm{u} :=\{g^{-1}Bg\in M(n, \mathbb{C}( z) /\mathbb{C}[z\mathrm{J})|g\in \mathrm{G}\mathrm{L}(n, \mathbb{C}[z\mathrm{I})\}. Let. consider. us. \mathb {P}^{1}. a. meromorphic. (\mathcal{O}^{n}, \nabla). connection. We write \nabla_{a} \in \mathcal{O}_{B} (resp. \mathcal{O}_{B}^{\mathrm{t}\mathrm{r}\mathrm{u} ) for a \in M(n, \mathbb{C}((z_{a}))) such that \nabla=d-A_{a}dz_{a} near a and. over. \mathcal{O}_{B}^{\mathrm{t}\mathrm{r}\mathrm{u}) Let. .. .. Here z_{a}:=. \left{\begin{ar y}{l z-a\mthrm{i}\ athrm{f}a\in mathb{C}\ mathrm{w}\mathrm{i}\ athrm{}\ athrm{}\mathrm{}\ athrm{}\mathrm{e}\mathrm{s}\ athrm{}\ athrm{a}\mthrm{n}\mathrm{d}\mathrm{a}\mthrm{}\ athrm{d}\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{}\ athrm{d}\mathrm{i}\ athrm{n}\mathrm{a}\mthrm{}\ athrm{e}z\mathrm{o}\mathrm{f}\mathb{C}.\ 1/z\mathrm{i}\ athrm{f}a=\infty \end{ar y}\right.. S=k_{0}a_{0}+\ldots+k_{p}a_{p}. set of. the trivial bundle. on. \mathb {P}^{1} if there exists A_{a} \in A_{a}\in\ovalbox{\t \small REJECT} B (resp. $\iota$(A_{a})\in. meromorphic. be. an. connections. effective divisor. on. on. \mathb {P}^{1}. as. the trivial bundle O^{n}. before. Define. over. a. \mathb {P}^{1}. \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}_{S}^{(n)}:=\{(\mathcal{O}^{n}, \nabla)| \nabla:\mathcal{O}^{n}\rightar ow \mathcal{O}^{n}\otimes$\Omega$_{S} \}. (\mathcal{O}^{n}, \nabla)\in \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}_{S}^{(n)} is stable if there exists nontrivial proper sub‐. We say no space W\subset \mathbb{C}^{n} such that the subbundle \mathcal{W}:=W\otimes \mathcal{O} \subset \mathbb{C}^{n}\otimes \mathcal{O}=\mathcal{O}^{n} is closed under \nabla , i.e.,. \nabla(\mathcal{W})\subset \mathcal{W}\otimes$\Omega$_{S}. Let \mathrm{B}=. forms stable. (B0, . . ., B_{p}) \in M(n, \mathbb{C}((z)))^{p+1}. satisfying \mathrm{o}\mathrm{r}\mathrm{d}(B_{i})=k_{i} meromorphic connections. for all i=0 , on. be. \cdots. ,. a. p. .. collection of HTL normal Then the moduli space of. trivial bundles is. stable,. \mathfrak{M}(\mathrm{B}):=\{(\mathcal{O}^{n},\nabla)\in\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}_{\mathcal{S}^{(n)}\nabla_{a_{l}\in\mathcal{O}_{B_{i}^{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{f}\mathrm{o}\mathrm{r}\^{a}\mathrm{N.}i=\0,.}p/(O\^{mn},a\ntahblar)m: {G}\mathrm{L}(n, \mathb {C}) Here. \mathrm{G}\mathrm{L}(n, \mathbb{C})=\mathrm{G}\mathrm{L}(n, \mathcal{O}(\mathbb{P}^{1}). formations.. acts. on. \mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}_{S}^{(n)}. as. holomorphic. .. gauge trans‐.
(7) 17. identify meromorphic connections on trivial bundles over \mathb {P}^{1} ordinary differential equations on \mathb {P}^{1} Thus we can regard \mathrm{M}(\mathrm{B}) moduli space of meromorphic differential equations on \mathb {P}^{1}, We. can. linear. .. and as a. \mathfrak{M}(\mathrm{B})=. \displayte\{fracd}{z\mathr {Y}=(\sum_{i=1}^p\sum_{$\nu=1}^{k_l}\frac{A_$\nu}^{(i) z-a_{i})^$\nu}-\sum_{2\leq$\nu leqk_{0}A_{$\nu}^{(0)z^{$\nu-2})\mathr {Y}|\sum_{$\nu z^{$\nu}^{k_t}A{R^($\iota$)}=1^{-},\in mathcl{O}_B{i}^\mathr {}\mathr {}\mathr {u}i=0_{;}.,p\mathr {i}\mathr {}\mathr {}\mathr {e}\mathr {d}\mathr {u}\mathr {c}\mathr {i}\mathr {b}1\mathr {e},\ /\mathrm{G}\mathrm{L}(n, \mathb {C}). Here. we. .. set. A_{1}^{(0)}:=-\displaystyle \sum_{i=1}^{p}A_{1}^{(i)}. 2.2. Moduli spaces of connections and realization of the moduli space M(\mathrm{B}) as. a. that. B^{(0)_{1} ,\ldots. ,. B^{(i)}= diag and choose. B^{(p)}. are. written. quiver varieties. We shall give a quiver variety. Let us suppose. by. (q_{1}^{(i)}(z^{-1})I_{n_{1}^{(i)} +R_{1}^{(i)}z^{-1}, \ldots, q_{m^{(i)} ^{(i)}(z^{-1})I_{n_{7n^{(i)} ^{(\mathrm{z})} +R_{m(i)}^{(i)}z^{-1}). complex. numbers. $\xi$_{1}^{[i,j]}. ,. .. .. .. ,. $\xi$_{e_{[$\iota$,j]}^{[i,j]}\cdot\mathrm{s}\mathrm{o} that. e\displaystyle\prod_{k=1}^{[i_{J}] (R_{j}^{(i)}-$\xi$_{k}^{[i,j]})=0 for i=0 ,. .. .. .. ,. p and. j=1. ,. .. ..,. m^{(i)}. .. Set. k_{i} :=-\displaystyle \max_{j=1,\ldots,m^{( $\iota$)} \{\mathrm{o}\mathrm{r}\mathrm{d}(q_{j}^{(i)}(z^{-1}) \} for each i=0 ,. .. ... ,. p. .. Set. I_{\mathrm{i}\mathrm{n}} :=\{i\in\{0, . . .,p\}|m^{(i)}>1\}\cup\{0\} and. I_{\mathrm{r}\mathrm{e}\mathrm{g} :=\{0, . ., p\}\backslash I_{\mathrm{i}\mathrm{r}\mathrm{r} . Here. of. I_{\mathrm{i}\mathrm{r}\mathrm{r} may be. seen as. the set of. regular singular points Then let. us. define. a. irregular singular points and. other than. quiver \mathrm{Q}. as. \infty ,. and. I_{\mathrm{r}\mathrm{e}\mathrm{g}. \infty.. follows. Set. \mathrm{Q}_{0}^{\mathrm{i}$\iota$\mathrm{r}:=\{[i,j] =1i\nI_{\mathrm{i}\mathrm{r}\mathrm{r}.' ,m^{(i)}\, \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g}:=\{[i,j k]|j=1,m_{$\d ag er$i,j]}^{(i)}k=1,e'-1i=0,\cdots\cdot.\cdot.',p \} Then the set of vertices of \mathrm{Q} is the. Qo. disjoint. union. :=\mathrm{Q}_{0}^{\mathrm{i}\mathrm{r}\mathrm{r} \sqcup\mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} ..
(8) 18. Also set. \mathrm{Q}_{1}^{0\rightarowI_{\mathrm{i}\mathrm{r}\mathrm{r} :=\{$\rho$_{[i,j]}^{[0,j]}:[0,j]\rightarow[i,j']| =1,m^{(i)}j=1,.\cdot.\cdot.',m^{(0)}i\nI_{i\mathrm{r}\mathrm{r}\backsla h\{0\},'\}. 1\leq j<j'\leq m^{(i)},. :=\left\{\begin{ar ay}{l} [k]\ $\rho$_{[i,j][i,j']}: [i,j]\rightar ow[i,j'] \end{ar ay}\right. \mathrm{Q}_{1}^{1\mathrm{e}\mathrm{g}^{(i)} :=\{$\rho$_{[i,j,k]} [i,j, k]\rightarrow[i,j, k-1] \mathrm{Q}_{1}^{B}. (i). 1\leq k\leq d_{\dot{ $\eta$}}(j,j') j=1. :. ,. .. k=2 ,. .. .. .. .. m^{(i)},. ,. .. . ,. e[i,j]-1. . \mathrm{Q}_{1}^{1\mathrm{e}\mathrm{g}^{(i)}\rightar owB^{(i)}. :=\{$\rho$_{[i,j,1]}: [i,j, 1]\rightarrow[i,j]|j=1, . . . , m^{(i)}\}, \mathrm{Q}_{1}^{1\mathrm{e}\mathrm{g}^{(i)}\rightar ow 0}:=\{$\rho$_{[0_{)}j]}^{[i,1,1]}: [i, 1, 1]\rightar ow[0,j]|i\in I_{\mathrm{r}\mathrm{e}\mathrm{g} , j=1, . . , m^{(0)}\}.. Here. d_{i}(j,j'). :=\deg_{\mathbb{C}[z]}(q_{j}^{(i)}(z)-q_{j}^{(i)}(z))-2. of is the union. Then the set of. arrows. \mathrm{Q}. disjoint. \mathrm{Q}_{1:=\mathrm{Q}_{1^0\rightarowI_{\mathrm{i}\mathrm{}\mathrm{} \sqcup_{i\n} sqcup_{I \mathrm{i}\mathrm{}\mathrm{} (\mathrm{Q}_{1^B}(i)\mathrm{u}\mathrm{Q}_{1^ \mathrm{e}\mathrm{g}^(i)}\rightarowB^{($\iota$)}\sqcup\mathrm{Q}_{1^ \mathrm{e}\mathrm{g}^($\iota$)} \mathrm{u}\sqcup(\mathrm{Q}_{1^ \mathrm{e}\mathrm{g}^($\iota$)}\rightarow0}\sqcup\mathrm{Q}_{1^ \mathrm{e}\mathrm{g}^($\iota$)} iEI_{r\mathrm{e}\mathrm{g} Let. $\alpha$=($\alpha$_{a})_{a\in \mathrm{Q}_{0} \in \mathbb{Z}^{\mathrm{Q}_{0} $\alpha$_{[i,j]}. :=n_{j}^{(i)}. and. $\alpha$_{[i,j,k]}. $\lambda$=($\lambda$_{a})_{a\in \mathrm{Q}_{0} \in \mathbb{C}^{\mathrm{Q}_{0}. Also define. :=. \displaystyle\prod_{l=1}^{k}(R_{j}^{(i)}-$\xi$_{l}^{[i,j]}). rank. :=-$\xi$_{1}^{[i,j]}. for. i\in I_{\mathrm{i}\mathrm{r}\mathrm{r} \backslash \{0\}, j=1. $\lambda$_{[0,j]}. :=-$\xi$^{[0,j]}-\displaystyle\sum_{i\nI_{\mathrm{r}\mathrm{e}\mathrm{g} $\xi$_{1}^{[i,1]}. for. j=1. :=$\xi$_{k}^{[i,j]}-$\xi$_{k+1}^{[i,j]}. for. Also define. a. \mathcal{L}=. sublattice of. ,. .. i=0 ,. .. .. .,. ... ... ,. .,. m^{(i)},. m^{(0)}, ,. p,. j=1. ,. ... .. ,. m^{(i)},. k=1, \cdots , e_{[i,j]}-1.. \mathb {Z}^{\mathrm{Q}_{0} ,. { $\beta$\displayst le\in\mathb {Z}^{0 }|\sum_{j=1}^{m^{(0)}$\beta$_{[0,j]}=\sum_{j=1}^{m^{(i)}$\beta$_{[i,j]}. for all. i\in I_{\mathrm{i}\mathrm{r}\mathrm{r} \backslash \{0\}. \mathcal{L}^{+}=\mathcal{L}\cap(\mathbb{Z}_{\geq 0})^{\mathrm{Q}_{0} . \mathfrak{M}(\mathrm{B}). .. by. $\lambda$_{[i,j]}. $\lambda$_{[i,j,k]}. Set. be the vector,. }.. a quiver variety. Now we shall give an identification of of the quiver variety \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$) Before seeing this, subspace \mathrm{M}(\mathrm{B}) we introduce \mathcal{L} ‐irreducible representations in $\mu$^{-1}( $\lambda$) which are defined by a weaker condition than the irreducibility.. 2.2.1.. with. and. a. Definition 2.1. .. ( \mathcal{L}‐irreducible).. subrepresentation be \mathcal{L} ‐irreducible. Then. we. have the. If. x. \in. $\mu$^{-1}( $\lambda$). has. no. $\mu$^{-1}( $\lambda$). with. \dim y\in \mathcal{L}. following bijection. from. SPt(B). \{0\}\neq y_{\neq}^{\subset}x. quiver variety \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$). .. in. nontrivial proper then x is said to. ,. onto. a. subset of the.
(9) 19. Theorem 2.2. (Theorem. [14]).. 5.14 in. There exists. a. bijection. $\Phi$_{\mathrm{B} :\mathfrak{M}(\mathrm{B})\rightar ow \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{dif} where. \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{dif}:=. As. an. \{x in$\mu$^{-1}($\lambda$)|\det(X_{$\rho$_{[,g]}^{1_l}^{0_J\mathrm{J})_{1\leqj\leqm^{(0)}\neq0,i'\nI_{ir\mathrm{r}\backslah\{0 } ,xis\mathcal{L}-i_{W}educible1\leqj'\leqm^{(n)}\/ mathrm{G}.. analogy. of. Corollary. by Crawley‐Boevey,. 1.10. \mathfrak{M}(\mathrm{B})\neq\emptyset. necessary and sufficient condition for. of. $\beta$\in \mathcal{L}^{+} satisfying (1) $\beta$ is a positive root of \mathrm{Q} and $\beta$\cdot $\lambda$=0, (2) for any decomposition $\beta$=$\beta$_{1}+\cdots+$\beta$_{r} roots of. \mathrm{Q} satisfying $\beta$_{i}\cdot $\lambda$=0. ,. where. (Non‐emptiness of moduli spaces. \mathfrak{M}(\mathrm{B})\neq\emptyset if and only if $\alpha$\in$\Sigma$_{ $\lambda$}^{dif}.. Theorem 2.3. Let. [23].. us. recall the. spectral type which. is. set. a. a. $\Sigma$_{ \lambda$}^{\mathrm{d}\mathrm{i}\mathrm{f}. consists. are. positive. $\beta$_{i}\in \mathcal{L}^{+}. have. we. p( $\beta$)>p($\beta$_{1})+\cdots+p(\sqrt{}r) moduli space. determin. we can. Define. .. .. Theorem 0.9 in. [14]).. The. in Section 1.2 in. already appeared. Consider the inductive limit. \displaystyle \mathb {Z}^{\infty}:=\lim_{\rightar ow}\mathb {Z}^{n} defined i=1 ,. 2,. by inclusions .. Defimition 2.4 \mathrm{B} is the. \ni. (al,. \cdots. ,. a_{i} ) \mapsto. (al,. \cdots. ,. a_{i}, 0 ) \in. \mathbb{Z}^{i+1} for. (spectral type. and index of. rigidity).. The. spectral type. of. pair. where \mathrm{m}_{ $\alpha$}=. isfies. $\phi$_{i,i+1}:\mathbb{Z}^{i}. \cdots. (i=0,\ldots,p. ((m_{[i,j,1]}, \ldots , m1^{i,j,e_{[i,g]}]})) 1<\lrcorner\leq m^{(i)} \in\oplus_{i=0}^{\mathrm{p} \oplus_{j=1}^{m^{(i)} \mathb {Z}^{\infty}. which sat‐. 0\leq i\leq p. \displaystyle \sum_{j=1}^{m^{(0)} \sum_{k=1}^{e_{[0,j]} m_{[0,j k]}=\cdots=\sum_{j=1}^{m^{(\mathrm{p})} \sum_{k=1^{\mathrm{I} }^{e_{\mathrm{l}p} Jm_{[p,j k]} is defined by m[i,j,k] := $\alpha$- $\alpha$. where. and. $\alpha$[i,j,e_{[J]}\dot{\mathrm{f} ,]=0. The index. $\alph$_{[i,j0]}=\left{\begin{ar y}{l $\alph$_{[i,j]}&\mathrm{i}\mathrm{f}i\nI_{\mathrm{i}\mathrm{}\mathrm{},\ sum_{k=1[0,k]}^{m (0)_{$\alph$} &\mathrm{i}\mathrm{f}i\nI_{\mathrm{}\mathrm{e}\mathrm{g} \end{ar y}\right.. .. Sometimes. of rigidity. we. write. idxm For convenience. number. d_{i}(j,j')+1. we. is. \mathrm{m}_{ $\alpha$}=(\mathrm{m}_{ $\alpha$}, d_{i}(j,j'). of \mathrm{m}_{ $\alpha$} is defined. the sequences m[i,j,1], m[i,j,2] ,. ... .. and. by. :=2q( $\alpha$). introduce the. expressed by. for short.. .. following. notation for. the number of. \mathrm{m}. The each. .. parentheses (). m_{[i,j',1]}, m_{[i,j',2]}. ,. .. \cdots. For. between. instance, if. \mathrm{m}_{ $\beta$}=\cdots m_{[i,j,1]}m_{[i,j,2]}\ldots m_{[i,j,l_{J}\mathfrak{g},]}))((m_{[i,j',1]}m_{[i,j',2]}\cdots. ,.
(10) 20. then the double. d_{\dot{ $\eta$}}(j,j'\rangle=1. For. parenthesis )) ( ( between m_{[i,j,1]}\ldots. ,. and. means. m[i,j',1]\cdots. example, put p=1,. (m^{(0)}, m^{(1)})=(2,3). (eeeee)=(1,2,1,1,2) (d_{0}(1,2), d_{1}(1,2), d_{1}(2,3), d_{1}(1,3))=(0,0,1,1) ,. ,. .. Then. \mathrm{m}=( m_{[i,j,1]}, \ldots, m_{[i,j,l_{i_{J} ,]}) _{0\leq i\leq p ,1\leq j\leq k_{i}. is written. by. (m_{[0,1,1]})(m_{[0,2,1]}m_{[0,2,2]}) , ((m_{[1,1,1]})(m_{[1,2,1]}))((m_{[1,3,1]}m_{[1,3,2]})). .. Integrable deformation. Let us introduce integrable admissible fam‐ following Boalch [3] and Yamakawa [41]. Let $\Gamma$ be a contractible complex manifold and a_{i}: $\Gamma$\rightar ow \mathbb{P}^{1}\times $\Gamma$, i=0 p, holomorphic sections of the fiber bundle $\pi$:\mathbb{P}^{1}\times $\Gamma$\rightar ow $\Gamma$ Moreover assume 2.3.. ilies of connections. ,. .. .. .,. .. that. a_{i}(t)\neq a_{j}(t). \mathb {P}_{t}^{1} :=\mathbb{P}^{1}\times\{t\}. in each fiber. \mathbb{C}\cup\{\infty\} Let. that. so. Moreover. .. a_{0}(s)=\infty. if. we. and d_{ $\Gamma$}z=0. i\neq j. fix. standard coordinate. a. the trivial bundle. on. z:\mathbb{P}_{t}^{1}\cong. \mathb {P}^{1} \times $\Gamma$\rightarrow $\Gamma$.. set. us. (z,t)\mapsto\left\{ begin{ar ay}{l} 1/z&(i=0)\ z-a_{i}(t)&(i\neq0) \end{ar ay}\right.. z_{i}:\mathbb{P}^{1}\times $\Gamma$\rightar ow $\Gamma$ ; for i=0 ,. \cdots. ,. p. .. Let. consider. us. a. family \mathrm{B}(t)=(B^{(i)}(t))_{i=0,\ldots,p} of collections. of HTL normal forms of the forms. B^{(i)}(t)=. (q_{1}^{(i)}(t, z_{i}^{-1})I_{n_{1}^{(i)} +R_{1}^{(i)}(t)z_{i}^{-1},.\cdots, q_{m^{( $\iota$)} ^{(i)}(t, z_{i}^{-1})I_{n^{(i)} m^{(i)} +R_{m^{(i)} ^{(i)}(t)z_{i}^{-1}). diag. Here all. $\Gamma$ \ni. mappings. t. \mapsto. q_{j}^{(i)}(t, z). M(n_{j}^{(i)}, \mathbb{C}) depend smoothly t\in $\Gamma$ \overline{q}_{j}^{(i)}(t, z^{-1}))-2 We say that \mathrm{B}(t) is on. .. of HTL normal forms if i=0 , Let Then. .. .. .. ,. p and. (\mathrm{B}(t) _{t\in $\Gamma$} as we saw. j,j'=1 be. ,. d_{i}(t;j,j') .. ..,. .. an. and. m^{(i)}. family. in Remark. $\lambda$\in \mathb {C}^{\mathrm{Q}_{0} independently. R_{j}^{(i)}(t) d_{i}(t;j,j') :=\deg_{\mathbb{C}[z]}(q_{j}^{(i)}(t, z)-. \mathbb{C}[z]. Define. admissible. an. \in. .. and $\Gamma$ \ni. familyl. admissible. R_{j}^{(i)}(t). are. t. \in. \mapsto. of the collections. independent of. t for all. of collections of HTL normal forms.. refinvariance,. we can. of t\in $\Gamma$ such that. we. \mathrm{Q}, $\alpha$\in \mathbb{Z}^{\mathrm{Q}_{0} isomorphisms. find quiver. have. and. $\Phi$_{\mathrm{B}(t)}:\mathfrak{M}(\mathrm{B}(t) \rightar ow\sim \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} for all t\in $\Gamma$ resonant if. 0,. .. ... ,. .. We further say that the admissible. eigenvalues. p and. j=1. ,. .. .. .. ,. of. R_{j}^{(i)}(t). m^{(i)}. ,. which is. $\lambda$_{[i,j,k]} \not\in \mathbb{Z}\backslash \{0\} lThis. is. a. never. differ. by. equivalent. for all. family (\mathrm{B}(t) _{t\in $\Gamma$} any. to the. integer. condition,. [i,j k]\in \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} .. little stronger condition than that in. [41].. is. non‐. for each i. =.
(11) 21. Definition 2.5. (admissible family).. meromorphic connections is called the followings are satisfied:. (1) (2) (3). the admissible. We have. family (\mathrm{B}(t) _{t\in $\Gamma$} 0,. =. .. ... A_{i}(t, z_{i})dz_{i}, A_{i}(t, z_{i}) ists. a. an. (\mathcal{O}_{\mathb {P}_{t}^{1} ^{n}, \nabla_{t})\in \mathfrak{M}(\mathrm{B}(t). For each i. Then the. holomorphic. ,. admissible. family. with. (\mathrm{B}(t) _{t\in $\Gamma$}. of if. is non‐resonant.. M ( n, \mathb {C} ((zi))). $\Gamma$ , let. near. z_{i}. write \nabla_{t} dThen there ex‐. us. =. 0. =. \hat{g_{i} : $\Gamma$\rightar ow \mathrm{G}\mathrm{L}(n, \mathbb{C}[z_{i}\mathrm{I}). map. ( \mathcal{O}_{\mathb {P}_{t}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$}. for all t\in $\Gamma$.. p and fixed t \in \in. family. .. such that. A_{i}(t, z_{\hat{i}})=\hat{g}_{i}(t)[B(i)(t)]. As. we see. this. triple. (\mathrm{B}(t) _{t\in $\Gamma$}. .. above, the. we can. spectral. define the. We call the number. the dimension of the admissible \cdot. Definition 2.6. .. We call. ( \mathcal{O}_{\mathb {P}_{t}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$}. family. 2p( $\alpha$) =\dim(\mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)). with. \dim(\mathfrak{M}(\mathrm{B}(t))). =. ,. family.. (integrable family).. Let. ( \mathcal{O}_{\mathb {P}_{t}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$}. be. an. admissible. (\mathrm{B}(t) _{t\in $\Gamma$} If there exists a flat meromorphic connection \hat{\nabla} on poles on ưi=0^{a_{i}( $\Gamma$)} such that \hat{\nabla}|_{\mathb {P}_{t}^{1} =\nabla_{t} then we say that the. family. with. \mathcal{O}_{\mathb {P}^{1}\times$\Gam a$}^{n}. with. family. ( \mathcal{O}_{\mathb {P}_{t}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$} integrable. ( \mathcal{O}_{\mathb {P}_{t}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$}.. flat. triple (\mathrm{Q}, $\lambda$, $\alpha$) from (\mathrm{B}(t) _{t\in $\Gamma$}. data of the admissible. .. ,. is. \cdot. In this. case. such. (\mathcal{O}_{\mathb {P}^{1}\times $\Gam a$}^{n},\hat{\nabla}). is called. a. extension of. 3. MIDDLE CONVOLUTIONS, WEYL GROUPS. AND INTEGRABLE. DEFORMATIONS. In this. section, we see the relationship between middle convolutions and of quivers and give a classification of their symmetries in certain lower dimensional cases. And we see the symmetries of integrable families as an application.. Weyl groups. 3.1. Middle convolution and reflection functor. Let. M(\mathrm{B}). Set. where. us. take. and write. (\mathcal{O}^{n}, \nabla)\in. \displaystyle\nabla=d-(\sum_{i=1}^{p}\sum_{$\nu$=1}^{k_{i}\frac{A_{$\nu$}^{(i)}{(z-a_{i})^{$\nu$}-\sum_{2\leq$\nu$\leqk_{0}A_{$\nu$}^{(0)}z^{$\nu$-2})dz. \displaystyle\mathrm{A}=(\sum_{j=1}^{k_{i}A_{j}^{(i)}z^{-j})_{0\leqi\leqp}\in\prod_{i=0}^{p}\mathcal{O}_{B(\dot{\mathrm{t}). A_{1}^{(0)}:=-\displaystyle \sum_{i=1}^{p}A_{1}^{(i)} J_{i}. .. Set. :=\{[i,j]|j=1, . .. , m^{(i)}\}. for i=0 ,. .. ... ,. p. and. Then. \displaystyle\mathcal{J}:=\prod_{i=0}^{p}J_{i}.. define an operation called middle convolution for meromorphic connections on trivial bundles over \mathb {P}^{1} , see [10], [9],[37], [1],[22],[35],and [40]. we can.
(12) 22. Thus form the connection \nabla. \mathcal{O}^{n'}. trivial bundle. Suppose. (1) (2). over. choose \mathrm{i}\in \mathcal{J}. we can. If \nabla is If \nabla is. stable, stable,. define. we can. \mathrm{m}\mathrm{c}\mathrm{i}(\nabla). connection. a new. \mathb {P}^{1} satisfying the following properties.. then. so. \mathrm{m}\mathrm{c}_{\mathrm{i} (\nabla). that. on a. $\xi$_{\mathrm{i} \neq 0.. is stable.. \mathrm{m}\mathrm{c}_{\mathrm{i} \circ \mathrm{m}\mathrm{c}_{\mathrm{i} (\mathrm{A})\sim \mathrm{A}, i.e., there exists g\in \mathrm{G}\mathrm{L}(n, \mathbb{C}) such that. \mathrm{m}\mathrm{c}_{\mathrm{i} \circ \mathrm{m}\mathrm{c}_{\mathrm{i} (\mathrm{A})=g\mathrm{A}g^{-1} :=(gA_{i}(z^{-1})g^{-1})_{0\leq i\leq p}. 3.2. Middle convolutions. on representations of a quiver. We shall analogy of the reflection functors for the subspace \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \subset \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$) by using middle convolutions. For \mathrm{i}= ([i,j_{i}])_{0\leq i\leq p} \in \mathcal{J} let us define $\epsilon$ \mathrm{i}\in \mathb {Z}^{\mathrm{Q}_{0} by. define. an. ,. ($\epsilon$_{\mathrm{i})_a:=\left{\begin{ar y}{l 1\mathrm{i}\ athrm{f}a=[i,j_{}]i\nI_{\mathrm{i}\ athrm{}\ athrm{},\ 0\mathrm{o}\mathrm{}\ athrm{}\mathrm{e}\mathrm{}\ athrm{w}\mathrm{i}\ athrm{s}\mathrm{e}. \end{ar y}\right. We note that $\epsilon$_{\mathrm{i} for \mathrm{i}\in J. are. positive real. roots of. Q. Let. us. define. s_{\mathrm{i} ( $\beta$):= $\beta$-( $\beta,\ \epsilon$_{\mathrm{i} )$\epsilon$_{\mathrm{i}. $\beta$\in \mathb {Z}^{\mathrm{Q}_{0}. for \mathrm{i}\in J and. .. Also define. r_{\mathrm{i} ( $\mu$). for. $\mu$\in \mathb {C}^{\mathrm{Q}_{0} by. r_{\mathrm{i}($\mu$)_{[i,j]}:=\left\{ begin{ar y}{l $\mu$[i,j]&\mathrm{i}\mathrm{f}[i,j]\neq[0,j_{0}],\ $\mu$_{[0,j_{0}]-2$\mu$_{\mathrm{i} &\mathrm{i}\mathrm{f}[i,j]=[0,j_{0}], \end{ar y}\right. r_{\mathrm{i}($\mu$)_{[i,jk]}:=\left\{ begin{ar y}{l $\mu$_{[i,jk]}&\mathrm{i}\mathrm{f}[i,j k]\neq[i,j_{i},1]\ $\mu$_{[i,j_{i},1]+$\mu$_{\mathrm{i} &\mathrm{i}\mathrm{f}[i,j k]=[i,j_{i},1]. \end{ar y}\right. Then ations. we can see. on. quiver. that the middle convolutions induce the. Theorem 3.1. Let. \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{dif}. variety can. take. exists. a. following. oper‐. varieties. us. \mathfrak{M}(\mathrm{B}) \neq \emptyset. consider. \mathrm{i}=([i,j_{i}])\in \mathcal{J}. so. bijection. and the. corresponding quiver bijection Suppose that we that $\lambda$_{\mathrm{i} :=\displaystyle \sum_{i\in I_{l\mapsto} $\lambda$_{[i,j_{i}]} =-$\xi$_{\mathrm{i}}\neq 0 Then there. under the. in Theorem 2.2.. .. s_{\mathrm{i} :\mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{dif}\rightar ow \mathfrak{M}_{r_{\mathrm{i} ($\lambda$')}(\mathrm{Q}, s_{\mathrm{i} ( $\alpha$) ^{dif} 3.3. The lattice \mathcal{L}. 2.3,. if. as a. Kac‐Moody. \mathfrak{M}(\mathrm{B}) \cong 9\mathfrak{N}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \neq\emptyset. set of roots in. \mathb {Z}^{\mathrm{Q}_{0}. .. then. ,. This inclines. us. root lattice. As a. to. can. be. \langle s_{a} |. a\in J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \rangle. seen as \mathrm{a}. .. acts. on. \mathcal{L}. .. \mathcal{L}\cap $\Delta$. see. of roots of the lattice \mathcal{L} which may not be It can be checked that L is generated by W^{\mathrm{m}\mathrm{c}}=. we saw. in Theorem. must. be in \mathcal{L}\cap $\Delta$ where $\Delta$ is the. a. true. analogy of Kac‐Moody root as an. \{$\epsilon$_{a} | a \in J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \}. This may lead. root lattice with the set of. simple. us. roots. over. the set. lattice. \mathb {Z} and. to believe that \mathcal{L}. \{$\epsilon$_{a}|a\in J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \}. Weyl group W^{\mathrm{m}\mathrm{c} However elements in \{$\epsilon$_{a}|a\in J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \} are not independent over \mathb {Z} in general. Thus we shall introduce a new lattice \hat{\mathcal{L} of and the. ..
(13) 23. which \mathcal{L}. can. be. quotient. Let. seen as a. us. note that. ($\epsilon$_{\mathrm{i},$\epsilon$_{\mathrm{i}')=2-\displayst le\sum_{0\leqi\leqp,j_{i}\neqj_{i}'(d_{i}(j_{i},j_{i}')+2). (1). ,. ($\epsilon$_{\mathrm{i},$\epsilon$_{[i,jk]})=\left{\begin{ar y}{l -1&\mathrm{i}\mathrm{f}j=_{i}\mathrm{a}\mathrm{n}\mathrm{d}k=1,\ 0&\mathrm{o}\mathrm{t}\mathrm{}\mathrm{e}\mathrm{}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}, \end{ar y}\right.. (2). (3). ( $\epsilon \epsilon$)=. \left{begin{ary}l 2&\mathr{i}\mathr{f}[i,jk]=',jk\ -1&\mathr{i}\mathr{f}(i,j)=' \mathr{}\mathr{n}\mathr{d}|k-'=1,\ 0&\mathr{o}\mathr{}\mathr{}\mathr{e}\mathr{}\mathr{w}\mathr{i}\mathr{s}\mathr{e} \nd{ary}\ight.. for \mathrm{i}, \mathrm{i}' \in \mathcal{J} and [i,j, k], [i',j', k'] \in \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} generated by the set of indeterminate. .. Thus. we. consider. a new. lattice. \hat{\mathcal{L}. C=\{c_{a}|a\in J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \}, symmetric bilinear form (, ) on \hat{\mathcal{L} in accordance with equations (3). Then \hat{\mathcal{L} becomes a symmetric Kac‐Moody root lattice and projection. and define. (1), (2) we. a. and. have. a. \hat{L}\rightar ow \mathcal{L} where for. $\gamma$=\displaystyle \sum_{c\in C}$\gamma$_{c}c\in\hat{\mathcal{L}. ,. the image. - -( $\gamma$)=($\beta$_{a})_{a\in \mathrm{Q}_{0}. is. given by. $\beta$_{[i,j]}=\displaystyle\sum_{\ mathrm{i}=([i,j_{l}])\inJ|j_{i}=j\}$\gam a$_{\mathrm{c}_{\mathrm{i} ,. $\beta$_{[i,j,k]}=$\gamma$_{c_{[i,g,k]} . Then Theorem 3.6 in. [13]. shows that : maps the. Weyl. group of. \hat{\mathcal{L} to. W^{\mathrm{m}\mathrm{c} .. Namely say that \hat{L} is a lift of \mathcal{L} to a Kac‐Moody root lattice with the Weyl group W^{\mathrm{m}\mathrm{c} . The kernel of : is a big space in general. Thus if we consider the inverse we can. image of an element $\beta$\in \mathcal{L} as follows. Fix $\beta$\in \mathcal{L} and and. ,. it is convenient to restrict: to. (\mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g})_{$\beta$} :=\mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} \cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\beta$). some. J_{ $\beta$}:= { ([i,j_{i}])\in \mathcal{J}|$\beta$_{[i,j_{i}]}\neq 0. set .. smaller space. for all. i\in I_{\mathrm{i}\mathrm{r}\mathrm{r} }. Then define. (\mathcal{J}\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} )_{ $\beta$}:=J_{ $\beta$}\cup(\mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} )_{ $\beta$} and. a. sublattice and. subgroup. \displaystle\hat{\mathcal{L}_{$\beta$}:=\sum_{\a in(J\cup\mathrm{Q}_{0^1\mathrm{e}\mathrm{g})_{$\beta$}\ mathb {Z}c_{a},. W_{ $\beta$}^{\mathrm{m}\mathrm{c} :=\{s_{a}|a\in(\mathcal{J}\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} )_{ $\beta$}\rangle. Denote the set of all. of :. on. \hat{\mathcal{L}_{$\beta$}. by - - $\beta$.. positive elements. in. \hat{\mathcal{L}_{$\beta$} by \hat{\mathcal{L}_{$\beta$}^{+}. .. We write the restriction.
(14) 24. 3.3.1. A. classification of spectral types.. mental set of the root lattice. Let. us. define. an. \mathb {Z}^{\mathrm{Q}_{0} ,. analogue of funda‐. \displaystyle\tilde{F}:=\{$\beta$\in\mathcal{L}^{+}\backslash\{0\}($\beta$,)\leq0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}a\in\mathcal{J}\cup\mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g}\sup^{$\epsilon$_{a} \} port. called \mathcal{L} ‐fundamental set. Then. o. we can see. \mathrm{f} $\beta$ \mathrm{i}\mathrm{s}\mathrm{c}onnected. \tilde{F}. that. be. can. seen as a. fundar. mental domain under the action of the group W^{\mathrm{m}\mathrm{c} Namely, we can show that quiver varieties with imaginary roots as dimension vectors can be re‐ .. duced to Thus. \mathfrak{M}_{$\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f}. we. introduce the. us. of. shape. the action of W^{\mathrm{m}\mathrm{c} .. classification of elements in the set. a. \tilde{F}. .. First let. $\beta$\in \mathcal{L}.. (shape).. Definition 3.2. $\alpha$\in\tilde{F} by. with. shall consider. a Kac‐Moody root lattice L=\oplus_{i\in I}\mathbb{Z}\mathrm{a}_{i} and Dynkin diagram of the support of a we attach each coefficient m_{i} of a to the vertex corresponding to $\alpha$_{i} then we obtain the diagram with the coefficients, which we call the shape of $\alpha$.. $\alpha$=\displaystyle \sum_{i\in I}m_{i}$\alpha$_{i}\in L. Fix. For the. .. ,. ,. For. if. example,. support. ,. By using. this. the. define. we. Deflnition 3.3. For in. \mathb {Z}_{\geq0}^{\mathrm{Q}_{0}. Let. .. us. shapes. of elements in \mathcal{L}. ,. We say that $\beta$ \in such that. i\in\{1, \cdots, p\}. diagram with coefficients. $\beta$\in \mathcal{L} the shape of $\beta$. - $\beta$\subset\hat{\mathcal{L} _{ $\beta$}.. $\beta$_{[i,j]} \neq 0\}. with the. $\alpha$=m_{1}$\alpha$_{i_{1}}+m_{2}$\alpha$_{i_{2}}+m_{3}$\alpha$_{i_{3}}\in L. is reduced if it. as. follows.. \#\{j |$\beta$_{[i,j]} \neq 0\}=1. shapes of. e_{[i,j_{i}]}. =1 where. Ht. { (B_{i})\in\oplus M(n, \mathbb{C}[z^{-1}])\infty. :=\displaystle\bigcup_{n=1}^{\infty}. Ht (n). \mathrm{m}. is. effective. \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \neq basic if. Also. \mathrm{m}. is. .. are. HTL normal. .. forms}. .. \emptyset and. $\alpha$\in\tilde{F}. all B_{i}. (fundamental spectral type).. Definition 3.4 We say that. j_{i}\in\{j | \mathfrak{M}(\mathrm{B}). consider the set of all nonempty moduli spaces. Set. \mathrm{H}\mathrm{t}^{(n)}:=. elements. that there exists. happens. and. of the. is. is the set of. never. diagram. \mathrm{m}. we. =. Let. \mathrm{m}. be. a. spectral type.. if there exists \mathrm{B} \in Ht such that. \mathrm{m}_{ $\alpha$}. A. .. spectral type. \mathrm{m}. =. \mathfrak{M}(\mathrm{B}). \cong. \mathrm{m}_{ $\alpha$} is said to be. say that \mathrm{m} is reduced if $\alpha$ is reduced. We say that is effective, basic and reduced. By the shape of \mathrm{m},. fundamental if \mathrm{m} the shape of $\alpha$.. we mean. Then. we can. Theorem 3.5 there exist. show the. (Theorem. only finite. following 8 in. number. finiteness of basic. [16]).. Let. spectral types.. fix an integer q\in 2\mathbb{Z}_{\leq 0} Then of fundamental spectral types \mathrm{m} satisfying us. .. idxm =q. Let. us see. the. cases. q=0 and. -2 for. example.. The first. case. is. q=0..
(15) 25. Theorem 3.6. satisfying. We. (Theorem. idxm =0. 9 in. are one. [16]). Shapes of fundamental spectral types the. of. \mathrm{m}. following.. simply write sets \{x_{a}|a\in \mathbb{Z}\} and \{x\} by x_{a}(a\in \mathbb{Z}) and x respectively. first 4 star shaped graphs, corresponding spectral types are given in ,. For the. Remark 3. 7 below.. If. \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \neq\emptyset. shapes of any. $\alpha$ , we can. w\in W_{ $\alpha$}^{\mathrm{m}\mathrm{c}. .. and. $\alpha$\in\tilde{F}. check that. $\alpha$. with. q( $\alpha$). =0 , then. is invariant under. Then. by the above list of W_{ $\alpha$}^{\mathrm{m}\mathrm{c} i.e., w( $\alpha$)= $\alpha$ for ,. s_{a}:\mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \rightar ow \mathfrak{M}_{r_{a}( $\lambda$)}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} for each. a\in(\mathcal{J}\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} )_{ $\alpha$} r_{a}. :. defines. a. W_{ $\alpha$}^{\mathrm{m}\mathrm{c} ‐action. \displaystyle \sum_{a\in(J\cup \mathrm{Q}_{0}^{1\mathrm{e}\mathrm{g} )_{$\alpha$} \mathb {C}c_{a}. \mapsto^{\rightarow}. on. the parameter space. \displaystle\sum_{a\in(mathcl{J}\cup\mathrm{Q}_0^{1\mathrm{e}\mathrm{g})_{$\alph$}\mathb{C}c_{a}r ($\lambda$). see also Proposition 3.7 in [13]. Here if $\lambda$_{a}=0 i.e., s_{a} on \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} is not well‐defined, we formally set s_{a} =\mathrm{i}\mathrm{d} and r_{a}=\mathrm{i}\mathrm{d} By the above theorem, W_{ $\alpha$}^{\mathrm{m}\mathrm{c} is isomorphic to one of the Weyl groups of the following types, ,. .. E_{8}^{(1)}, E_{7}^{(1)}, E_{6}^{(1)}, D_{4}^{(1)}, A_{3}^{(1)}, A_{2}^{(1)}, A_{1}^{(1)}, A_{1}^{(1)}\times A_{1}^{(1)}. Remark 3.7. In the above list. star‐shaped diagrams.. lows. Consider. a. shape. m_{(i,j+1)}:=n_{i,j}-n_{i,j+1},. For these. ofshapes, we omit the spectral types for cases spectral types are obtained as fol‐. and put m_{(i,1)} :=n_{0}-n_{i,1},. m_{(i,0)}:=\displaystyle \sum_{0\leq k\leq p}n_{k,1}-n_{0}. and. m_{(0)}. :=\displaystyle \sum_{i=0}^{p}n_{i,1}-.
(16) 26. n_{0}. .. Then the. to the. shape corresponds. m_{(0,1)}m_{(0,2)}\ldots, m_{(1,1)}m_{(1,2)}\ldots. ,. ... .. following ,. 5 types.. m(p,1)^{m}(p,2). \cdots. (m_{(0,2)}m_{(0,3)}\ldots)\ldots(m_{(p,2)}m_{(p,3)} m_{(i,0)}m_{(i,1)}\ldots, (m_{(0,2)}m_{(0,3)}\ldots)\ldots(m_{(i-1,2)}\ldots)(m_{(i+1,2)}. . .) ((m_{(i,1)^{m}(i,2)}.. ((mm\ldots)\ldots(m_{(i-2,2)}\ldots)(m_{(i+1,2)}.. .) ((n0)) ((m_{(0,2)}m_{(0,3)}.. .) (mm_{(,3)} m_{(0)}n_{0},. .. .. .. .. Next let. us see. Theorem 3.8 \mathrm{m}. satisfying. the. case. (Theorem. idx \mathrm{m}=-2. .. .. .. .. .,. .. .. .. q=-2. 10 in. [16]). Shapes of fundamental spectral types. are one. of. the. following..
(17) 27. simply denote the sets \{x_{a} | a \in \mathbb{Z}\} and \{y\} by x_{a} (a \in \mathbb{Z}) and respectively. For the spectral types of the star shaped graphsf see Remark. Here y,. we.
(18) 28. for fundamental spectral types. 3.7. Here. \mathrm{m}=\mathrm{m}_{ $\alpha$}. W^{inv}:=\langle s_{a}|s_{a}( $\alpha$)= $\alpha$, a\in(\mathcal{J}\cup \mathrm{Q}_{0}^{leg})_{ $\beta$}\rangle\subset W_{ $\alpha$}^{\mathrm{m}\mathrm{c} . Plain circles in the. s_{a}( $\beta$)= $\beta$ As well. $\alpha$\in\tilde{F}. Dynkin diagrams correspond to simple correspond to s_{a}( $\beta$)\neq $\beta$.. the. as. such that. q=0. case. q( $\alpha$)=-2. in the. ,. has. a. Integrable deformations. below connects the W^{\mathrm{m}\mathrm{c} ‐action The. following. for. [5]. [11]. and Boalch. collections. e. ,. integrable deformations. by Haraoka‐Filipuk in [11] for Fuchsain. (\mathcal{O}_{\mathb {P}_{\mathrm{t}^{1}^{n}\nabla_{t})_{t\in$\Gam a$}. (\mathrm{B}(t) _{t\in $\Gamma$}. vists. an. and the. admissible. spectral. Definition 3.10. We say that. (\mathrm{Q}, $\lambda$, $\alpha$). is. fundamental. be. an. (\mathrm{Q}, $\lambda$, $\alpha$). data. integrable deformation. spectral data (\mathrm{Q}, r_{\mathrm{i} ( $\lambda$), s_{\mathrm{i} ( $\alpha$)) such that. with. on. Theorem 3.11. Let. admissible. where. an. for. each \mathrm{i} \in. (\mathcal{O}_{\mathrm{F}_{t}^{1}^{n}\nabla_{t}^{\mathrm{i})_{t\in$\Gam a$}. (\mathrm{B}(t) _{t\in $\Gamma$}. and. [15]).. q( $\alpha$). \leq 0. Then. .. be. as. with. Suppose. ( O_{\mathb {P}_{t}^{1} , \nabla_{t}) _{t\in $\Gamma$}. condition,. see. admissible. integral deformation by. a. following.. can. finite. ((O^{n}, \nabla_{t}))_{t\in $\Gamma$}. be. spectral data (\mathrm{Q}, $\lambda$, $\alpha$) generic (for the precise. and the. that $\lambda$ is. ( \mathcal{O}_{\mathb {P}_{i}^{1} ^{n},\nabla_{t}) _{t\in$\Gam a$}. is reduced.. a. in Theorem 3.9. Let. (\mathrm{B}(t) _{t\in $\Gamma$}. with the. all t\in $\Gamma$.. integral family. $\alpha$\in\tilde{F}\cap$\Sigma$_{$\lambda$}^{\mathrm{d}\mathrm{i}\mathrm{f} and. 0. integrable. Then. .. \nabla_{t}^{\mathrm{i} \cong \mathrm{m}\mathrm{c}_{\mathrm{i} (\nabla_{t}) for. admissible. when. integrable family. $\Sigma$_{$\lambda$}^{dif}. \in. $\alpha$. [41]. admissible. Then Theorem refreduction and Theorem 3.9 show the. an. in. (B^{(0)}(t))_{irr}\equiv. of HTL normal forms which satisfies that that. with. J there. and. (Yamakawa. Corollary 3.17 in [41]. cf. Haraoka‐Filipuk [5]). Let (\mathrm{B}(t) _{t\in $\Gamma$} be an non‐resonant admissible family of. and \mathrm{p}\mathrm{r}_{res}(B^{(0)}(t) \dot{u} invertible. Let. family. \mathfrak{M}(\mathrm{B}). simply‐laced \mathrm{Q} with I_{\mathrm{i}\mathrm{r}\mathrm{r} =\{0\} and Yamakawa. for. general Q.. Theorem 3.9. \displaystle\sum_{a\in(J\cup\mathrm{Q}_0^{1\mathrm{e}\mathrm{g})_{$\alpha$}\mathb {C}c_{a}r_{a}($\lambda$). \mapsto^{\rightarow}. and middle convolutions. The theorem on. theorem is obtained. cases, Boalch in. of q= -2, \mathfrak{M}_{ $\lambda$}(\mathrm{Q}, $\alpha$)^{\mathrm{d}\mathrm{i}\mathrm{f} \neq\emptyset with on the parameter space. case. W^{\mathrm{i}\mathrm{n}\mathrm{v} ‐action. \displaystyle \sum_{a\in(J\cup \mathrm{Q}^{1\mathrm{e}\mathrm{g} )_{ $\alpha$} \mathb {C}c_{a}. :. r_{a}. 3.4.. roots c_{a} such that. and dotted circles. be reduced to. iteration. of. a. fundamental. middle convolutions. and additions. For. an. (\mathrm{Q}, $\lambda$, \mathrm{a}). ,. admissible we. integrable family. call \mathrm{m}_{ $\alpha$} the. ( O_{\mathb {P}_{t}^{1} ^{n}, \nabla_{t}) _{t\in $\Gamma$}. with. a. spectral. data. spectral type and also $\lambda$ the spectral parameter.. Theorem 3.12. Let tral types. Proof.. us fix an integer d\in 2\mathbb{Z}>0 There exists only finite spec‐ offundamental admissible integrable deformations of dimension d.. This. directly. \square. follows from Theorems 3.5 and 3.9.. We have the classification of. spectral types. of admissible deformations of. dimension d=2 and 4. Theorem 3.13.. Spectral types of fundamental admissible integrable defor‐. dimension d. mations. of. 3.8) for. d= 2. W^{\mathrm{m}\mathrm{c} ‐actions. =. 2, 4. are. listed in Theorem 3.6. (resp. (resp. W_{inv} ‐action) for d=4 ).. Moreover. (resp.. Theorem. generic spectral parameters. d=2. (resp.. d=4).. have.
(19) 29. In. [23], Kawakami,. Nakamura and Sakai considered isomonodromic de‐. formations of linear differential equations which obtained by the confluent process from Fuchsian differential equations with 4 accessory parameters. classified. Oshima in. by. [31].. And. they. gave. explicit Hamiltonian equations. of the isomonodromic deformations after Sakais computation in the Fuch‐ sian cases (see [33]). Then under the above identification of spectral types, Theorem 3.8 shows that the list of. [23]. is the. spectral types appeared in their paper spectral types of dimension 4.. list of fundamental. complete. Theorem 3.14. Under the above identification of spectral types, if. we. ex‐. clude the. spectral types corresponding to differential equations which have only 3 regular singular points and no other singularities, then the list of spectral types appeared in Section 1.3 of [23] is the complete list of spectral types of fundamental integrable deformations of dimension 4. Moreover Theorem 3.13 in. [23]. have. assures. that. W^{\mathrm{i}\mathrm{n}\mathrm{v}_{-} symmetries listed. integrable deformations considered. in Theorem 3.8.. REFERENCES. [1]. Arinkin, Rigid irregular. D.. connections. on. \mathb {P}^{1} Compos. Math., 146, ,. no.. 5. (2010),. 1323‐1338.. [2]. D. Babbitt and V.. [4] [5]. P.. [6]. W.. Varadarajan, Formal reduction theory of meromorphic differential equations: a group theoretic Wiew, Pacific J. Math. 109 (1983), no. 1, 1‐80. [3] P. Boalch, Symplectic manifolds and isomonodromic deformations, Adv. Math. 163. (2) (2001),. P.. 137‐205.. Boalch, Irregular connections and Kac‐Moody root systems, 200S, axXiv:0806.1060. Boalch, Simply‐laced isomonodromy systems, Publ. Math. IHES, 116, no. 1 (2012),. 1‐68.. Crawley‐Boevey, Geometry of the moment map for representations of quivers, Compos. Math. 126, no.3 (2001), 257‐293. [7] W. Crawley‐uoevey, On matrices in prescribed conjugacy dasses with no common invariant subspace and sum zero, Duke Math. J. 118, no. 2 (2003), 33k352. [8] W. Crawley‐uoevey, M. P. Holland, Noncommutative deformations of Kleinian sigu‐. larities, Duke Math. J. 92 (1998), no.3, 605‐635. Dettweiler, S. Reiter, An algorithm of Katz and its application to the inverse Galou problem, J. Symbolic Comput., 30, no. 6 (2000), 761‐798. [10] M. Dettweiler, S. Reiter, Middle convolution of FhLchsian systems and the construction of rigid differential systems, J. AlgeUra, 318, no. 1 (2007), 1‐24.. [9]. [11]. M.. Y. Haraoka and G.. Filipuk, Middle. J. Lond. Math. Soc. [12]. J.. Harnad,. (2). 76. Dual isomonodromic. Comm. Math. Phys. 166. [13] [14] [15] [16] [17] [18]. K.. no.. defotmations. (1994), no.2,. deformation for Ruchsian systems,. 2, 438‐450. and moment maps to. loop algebras,. 337‐365.. Hiroe, Lenear differential equations. (2013), K.. convolution and. (2007),. on. \mathb {P}^{1} and. root. systems, J. Algebra, 382,. 1‐38.. Hiroe, Linear differential equations on the Riemann sphere and representations of quivers, to appear in Duke Math. J., (2016). K. Hiroe, Moduli spaces of meromorphic connections, quiver varieties, and integrable deformations, in 4‐dimensional Painlev6‐type equaitons, to appear in MSJ Memoirs. K. Hiroe, T. Oshima, A classification of roots of symmetric Kac‐Moody root systems and its application, in Symmetries, integrable systems and representations, Springer‐ Verlag, 2013, 195‐241. K. Hiroe, D. Yamakawa, Moduli spaces of meromorphic connection and quiver vari‐ eties, Adv. Math., 266 (2014), 120‐151. M. Inaba, M.‐H. Saito. Moduli of unramified irregular singular parabolic connections on a smooth projective curve, Kyoto J. Math. 53 (2013), no.2, 433‐482..
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