Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlev\'e Equations

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlev\'e equations is slightly different from (but equivalent to) Okamoto's.


Introduction
First, we briefly explain our main objects in this article. Let • Q be a quiver, i.e., a directed graph, with the set of vertices I (our quivers are always assumed to be finite and have no arrows joining a vertex with itself); • d = (d i ) i∈I ∈ Z I >0 be a collection of positive integers indexed by the vertices.
We think of each number d i as the 'multiplicity' of the vertex i ∈ I, so the pair (Q, d) as a 'quiver with multiplicities'. In this article, we associate to such (Q, d) a holomorphic symplectic manifold N s Q,d (λ, v) having parameters • v = (v i ) i∈I ∈ Z I ≥0 , and call it the quiver variety with multiplicities, because if d i = 1 for all i ∈ I, it then coincides with (the stable locus of) Nakajima's quiver variety M reg ζ (v, w) [21] with w = 0 ∈ Z I ≥0 , ζ = (ζ R , ζ C ) = 0, (λ i,1 ) i∈I ∈ √ −1R I × C I .
As in the case of quiver variety, N s Q,d (λ, v) is defined as a holomorphic symplectic quotient with respect to some algebraic group action (see Section 3). However, the group used here is non-reductive unless d i = 1 or v i = 0 for all i ∈ I. Therefore a number of basic facts in the theory of holomorphic symplectic quotients (e.g. the hyper-Kähler quotient description) cannot be applied to our N s Q,d (λ, v), and for the same reason, they seem to provide new geometric objects relating to quivers.
The definition of N s Q,d (λ, v) is motivated by the theory of Painlevé equations. It is known due to Okamoto's work [23,24,25,26] that all Painlevé equations except the first one have (extended) affine Weyl group symmetries; see the table below, where P J denotes the Painlevé equation of type J (J = II, III, . . . ,VI).

Equations
P VI P V P IV P III P II On the other hand, each of them is known to govern an isomonodromic deformation of rank two meromorphic connections on P 1 [12]; the number of poles and the pole orders of connections remain unchanged during the deformation, and are determined from (if we assume that the connections have only 'unramified' singularities) the type of the Painlevé equation (see e.g. [27]). See the table below, where k 1 + k 2 + · · · + k n means that the connections in the deformation have n poles of order k i , i = 1, 2, . . . , n and no other poles.
In fact, such a relationship can be understood in terms of quiver varieties except in the case of P III . Crawley-Boevey [7] described the moduli spaces of Fuchsian systems (i.e., meromorphic connections on the trivial bundle over P 1 having only simple poles) as quiver varieties associated with 'star-shaped' quivers. In particular, the moduli space of rank two Fuchsian systems having exactly four poles are described as a quiver variety of type D (1) 4 , which is consistent with the above correspondence for P VI . The quiver description in the cases of P II , P IV and P V was obtained by Boalch 1 [4]; more generally, he proved that the moduli spaces of meromorphic connections on the trivial bundle over P 1 having one higher order pole (and possibly simple poles) are quiver varieties.
A remarkable point is that their quiver description provides Weyl group symmetries of the moduli spaces 2 at the same time, because for any quiver, the associated quiver varieties are known to have such symmetry. This is generated by the so-called reflection functors (see Theorem 1.2 below), whose existence was first announced by Nakajima (see [21,Section 9], where he also gave its geometric proof in some important cases), and then shown by several researchers including himself [8,19,22,28].
The purpose of quiver varieties with multiplicities is to generalize their description to the case of P III ; the starting point is the following observation (see Proposition 6.6 for a further generalized, precise statement; see also Remarks 6.4 and 6.5): Here the set of vertices is I = { 0, 1, . . . , n }. Take multiplicities d ∈ Z I >0 with d 0 = 1 and set v ∈ Z I ≥0 by v 0 = 2, v i = 1 (i = 1, . . . , n). Then N s Q,d (λ, v) is isomorphic to the moduli space of stable meromorphic connections on the rank two trivial bundle over P 1 having n poles of order d i , i = 1, . . . , n of prescribed formal type.
On the other hand, to any quiver with multiplicities, we associate a generalized Cartan matrix C as follows: where A is the adjacency matrix of the underlying graph, namely, the matrix whose (i, j) entry is the number of edges joining i and j, and D is the diagonal matrix with entries given by the multiplicities d. It is symmetrizable as DC is symmetric, but may be not symmetric. Now as stated below, our quiver varieties with multiplicities admit reflection functors; this is the main result of this article. Theorem 1.2 (see Section 4). For any quiver with multiplicities (Q, d), there exist linear maps generating actions of the Weyl group of the associated Kac-Moody algebra, such that for any (λ, v) and i ∈ I with λ i,d i = 0, one has a natural symplectomorphism ).
If d i = 1 for all i ∈ I, then the maps F i coincide with the reflection functors.
In the case of star-shaped quivers, the original reflection functor at the central vertex can be interpreted in terms of Katz's middle convolution [14] for Fuchsian systems (see [3, Appendix A]). A similar assertion also holds in the situation of Proposition 1.1; the map F 0 at the central vertex 0 can be interpreted in terms of the 'generalized middle convolution' [1,31] (see Section 6.3).
For instance, consider the star-shaped quivers with multiplicities given below 5 , D 4 , C 2 , A 2 .
Interestingly, this list of Kac-Moody algebras is different from the table given before; however we can clarify the relationship between our description and Boalch's by using a sort of 'shifting trick' established by him (see Section 5.1). This trick, which may be viewed as a geometric phenomenon arising from the 'normalization of the leading coefficient in the principal part of the connection at an irregular singular point', connects two quiver varieties with multiplicities associated to different quivers with multiplicities; more specifically, we prove the following: Section 5). Suppose that a quiver with multiplicities (Q, d) has a pair of vertices (i, j) such that where A = (a ij ) is the adjacency matrix of the underlying graph. Then it determines another quiver with multiplicities (Q,ď) and a map (λ, v) → (λ,v) between parameters such that the following holds: if λ i,d i = 0, then N s Q,d (λ, v) and N sQ ,ď (λ,v) are symplectomorphic to each other.
We call the transformation (Q, d) → (Q,ď), whose precise definition is given in Section 5.2, the normalization. Using this theorem, we can translate the above list of Dynkin diagrams into the original one (see Section 6.4).
There is a close relationship between two Kac-Moody root systems connected via the normalization (see Section 5.3). In particular, we have the following relation between the Weyl groups W ,W associated to (Q, d), (Q,ď): where the semidirect product is taken with respect to some Dynkin automorphism of order 2 (such a Dynkin automorphism canonically exists by the definition of normalization). For instance, in the cases of P V , P IV and P II , we have which mean that our list of Dynkin diagrams for Painlevé equations is a variant of Okamoto's obtained by (partially) extending the Weyl groups.

Preliminaries
In this section we briefly recall the definition of Nakajima's quiver variety [21].

Quiver
Recall that a (finite) quiver is a quadruple Q = (I, Ω, s, t) consisting of two finite sets I, Ω (the set of vertices, resp. arrows) and two maps s, t : Ω → I (assigning to each arrow its source, resp. target). Throughout this article, for simplicity, we assume that our quivers Q have no arrow h ∈ Ω with s(h) = t(h).
For given Q, we denote by Q = (I, Ω, s, t) the quiver obtained from Q by reversing the orientation of each arrow; the set Ω = { h | h ∈ Ω } is just a copy of Ω, and s(h) := t(h), t(h) := s(h) for h ∈ Ω. We set H := Ω ⊔ Ω, and extend the map Ω → Ω, h → h to an involution of H in the obvious way. The resulting quiver Q + Q = (I, H, s, t) is called the double of Q.
The underlying graph of Q, which is obtained by forgetting the orientation of each arrow, determines a symmetric matrix A = (a ij ) i,j∈I , called the adjacency matrix, as follows: Let V = i∈I V i be a nonzero finite-dimensional I-graded C-vector space. A representation of Q over V is an element of the vector space and its dimension vector is given by v := dim V ≡ (dim V i ) i∈I . Isomorphism classes of representations of Q with dimension vector v just correspond to orbits in Rep Q (V) with respect to the action of the group GL(V) := i∈I GL C (V i ) given by We denote the Lie algebra of GL(V) by gl(V); explicitly, gl(V) := i∈I gl C (V i ). For ζ = (ζ i ) i∈I ∈ C I , we denote its image under the natural map C I → gl(V) by ζ Id V , and also use the same letter ζ Id V for ζ ∈ C via the diagonal embedding C ֒→ C I . Note that the central then B is said to be irreducible. Schur's lemma 4 implies that the stabilizer of each irreducible B is just the central subgroup C × ⊂ GL(V), and a standard fact in Mumford's geometric invariant theory [20,Corollary 2.5] (see also [16]) implies that the action of GL(V)/C × on the subset Rep irr Q (V) consisting of all irreducible representations over V is proper.

Quiver variety
Suppose that a quiver Q and a nonzero finite-dimensional I-graded C-vector space V = i∈I V i are given. We set and regard it as the cotangent bundle of Rep Q (V) by using the trace pairing. Introducing the function we can write the canonical symplectic form on M Q (V) as The natural GL(V)-action on M Q (V) is Hamiltonian with respect to ω with the moment map vanishing at the origin, where we identify gl(V) with its dual using the trace pairing.
For a GL(V)-invariant Zariski closed subset Z of M Q (V), let Z s be the subset of all stable points in Z. It is a GL(V)-invariant Zariski open subset of Z, on which the group GL(V)/C × acts freely and properly.
which we call the quiver variety.

Definition
For a positive integer d, we set The C-algebra R d has a typical basis { z d−1 , . . . , z, 1 }, with respect to which the multiplication by z in R d is represented by the nilpotent single Jordan block .
For a finite-dimensional C-vector space V , we set Note that g d (V ) is naturally isomorphic to End R d (V ⊗ C R d ) as an R d -module; hence it is the Lie algebra of the complex algebraic group ]. The adjoint action of g(z) is described as Using the above R * d ≃ R d and the trace pairing, we always identify the C-dual g ]. Then the coadjoint action of g(z) ∈ G d (V ) is also described as whose image is just the centralizer of Id V ⊗ J d . Accordingly, its transpose can be written as Now suppose that a quiver Q and a collection of positive integers d = (d i ) i∈I ∈ Z I >0 are given. We call the pair (Q, d) as a quiver with multiplicities and d i as the multiplicity of the vertex i. Set and for a nonzero finite-dimensional The group G d (V) naturally acts on M Q,d (V) as a subgroup of GL(V d ). Note that the subgroup and acts trivially on M Q,d (V). As in the case of gl(V), for λ = (λ i (z)) i∈I ∈ R d we denote its image under the natural map Let ω be the canonical symplectic form on M Q,d (V); Then the G d (V)-action is Hamiltonian whose moment map µ d is given by the composite of the GL(V d )-moment map µ = (µ i ) : M Q,d (V) → gl(V d ) (see (2.1) for the definition) and the transpose pr = (pr i ) of the inclusion g d (V) ֒→ gl(V d ); The above stability can be interpreted in terms of the irreducibility of representations of a quiver. Letting Ω := Ω ⊔ {ℓ i } i∈I and extending the maps s, t to Ω by s(ℓ i ) = t(ℓ i ) = i, we obtain a new quiver Q = (I, Ω, s, t). Consider the vector space associated with the quiver Q + Q = (I, Ω ⊔ Ω, s, t). Note that in the above definition, a vector subspace S i ⊂ V i ⊗ R d i is an R d i -submodule if and only if it is invariant under the action of N i = Id V i ⊗ J d i , which corresponds to the multiplication by z. Thus letting which we have already checked. Furthermore, the above implies that the embedding Z s ֒→ Rep Q+Q (V d ) irr induced from ι is closed. Consider the following commutative diagram: where the vertical arrows are the maps induced from ι, and the horizonal arrows are the action maps (g, x) → g · x. Since the bottle horizontal arrow is proper and both vertical arrows are closed, the properness of the top horizontal arrow follows from well-known basic properties of proper maps (see e.g. [11,Corollary 4.8]).
which we call the quiver variety with multiplicities.
We also use the following set-theoretical quotient:

Properties
Here we introduce some basic properties of quiver varieties with multiplicities. First, we associate a symmetrizable Kac-Moody algebra to a quiver with multiplicities (Q, d). Let A = (a ij ) i,j∈I be the adjacency matrix of the underlying graph of Q and set D := (d i δ ij ) i,j∈I . Consider the generalized Cartan matrix Note that it is symmetrizable as DC = 2D − DAD is symmetric. Let be the corresponding Kac-Moody algebra with its Cartan subalgebra, simple roots and simple coroots. As usual we set The diagonal matrix D induces a non-degenerate invariant symmetric bilinear form ( , ) on h * satisfying From now on, we regard a dimension vector v ∈ Z I ≥0 of the quiver variety as an element of Q + by Let res : R d → C I be the map defined by and for (v, ζ) ∈ Q × C I , let v · ζ := i∈I v i ζ i be the scalar product.  (ii) If v · res λ = 0, then N set Q,d (λ, v) = ∅. (iii) If two quivers Q 1 , Q 2 have the same underlying graph, then the associated quiver varieties N s Proof . (i) Assume that N s Q,d (λ, v) is nonempty. Since the action of G d (V)/C × on the level set µ −1 d (−λ Id V ) s is free and proper, the Marsden-Weinstein reduction theorem implies that N s Q,d (λ, v) is a holomorphic symplectic manifold and for any i ∈ I. Taking res z=0 • tr of both sides and sum over all i, we obtain Here the left hand side is zero because Hence v · res λ = 0.
(iii) By the assumption, we can identify the double quivers Q 1 + Q 1 and Q 2 + Q 2 . Let H be the set of arrows for them. Then both the sets of arrows Ω 1 ,

Now fix i ∈ I and set
Then using it we can decompose the vector space According to this decomposition, for a point B ∈ M Q,d (V) we put We regard these as coordinates for B and write B = (B i , B i , B =i ). Note that the symplectic form can be written as and also the i-th component of the moment map can be written as Lemma 3.5. Fix i ∈ I and suppose that B satisfies at least one of the following two conditions: Proof . First, assume (i) and set Then both S and T are B-invariant and N j (S j ) ⊂ S j , N j (T j ) ⊂ T j for all j ∈ I. Since B is stable, we thus have S = 0 or S = V d , and T = 0 or T = V d . By the assumption v = α i and the definitions of S and T, only the case (S, Then we have Now note that both Ker N i and Coker N i are naturally isomorphic to V i , and the natural injection Thus we have The following lemma is a consequence of results obtained in [31]: Lemma 3.6. Suppose that the set is nonempty. Then the quotient of it modulo the action of G d i (V i ) is a smooth complex manifold having a symplectic structure induced from tr dB i ∧dB i , and is symplectomorphic to a G d i ( V i )coadjoint orbit via the map given by Proof . Take any point (B i , B i ) in the above set and let O be the . By Proposition 4, (a), Theorem 6 and Lemma 3 in [31], there exist such that the quotient modulo the natural G N -action of the set where pr N is the transpose of the inclusion Lie G N ֒→ gl(W ), is a smooth manifold having a symplectic structure induced from tr dX ∧ dY , and is symplectomorphic to Indeed, this is the case thanks to Proposition 4, (c) and Theorem 6 (the uniqueness assertion) in [31].
Note that in the above lemma, the assumption The following lemma tells us that if the top coefficient of λ i (z) is nonzero, then the converse is true and the corresponding coadjoint orbit can be explicitly described: Then the set Z i in Lemma 3.6 is nonempty and the coadjoint orbit contains an element of the form

Ref lection functor
In this section we construct reflection functors for quiver varieties with multiplicities.

Main theorem
Recall that the Weyl group W (C) of the Kac-Moody algebra g(C) is the subgroup of GL(h * ) generated by the simple reflections The fundamental relations for the generators s i , i ∈ I are where the numbers m ij are determined from c ij c ji as the table below (we use the convention r ∞ = Id for any r) We will define a W (C)-action on the parameter space R d × Q for the quiver variety. The action on the second component Q is given by just the restriction of the standard action on h * , namely, The action on the first component R d is unusual. We define r i ∈ GL(R d ) by Proof . The relations r 2 i = Id, i ∈ I are obvious. To check the relation (r i r j ) m ij = Id for i = j, first note that the transpose of s i : Q → Q relative to the scalar product is given by Then we easily see that and hence that Therefore we may assume that res λ = 0. Set λ ′ ≡ (λ ′ k (z)) := (r i r j ) m ij (λ). Then we have If m ij is odd, by the definition we have c ij c ji = 1. In particular, i = j and This implies d i = d j = 1 and hence that λ i (z) = λ j (z) = 0.
The main result of this section is as follows: Then there exists a bijection such that F 2 i = Id and the restriction gives a symplectomorphism We call the above map F i the i-th reflection functor.

Proof of the main theorem
Fix i ∈ I and suppose that the top coefficient and the set Z i given in Lemma 3.6. Lemma 3.5 and the assumption λ i,d i = 0 imply that any

By Lemma 3.7, it is nonempty if and only if
, Lemmas 3.6 and 3.7 imply that and consider the associated symplectic vector space M Q,d (V ′ ). Note that V ′ i = V i . Thus by interchanging the roles of V and V ′ , λ i and −λ i in Lemmas 3.6 and 3.7, we obtain an isomorphism which is characterized as follows: if by Lemma 3.5.
Taking the residue of both sides of (4.3), we have On the other hand, (4.2) means that Note that Therefore the image under pr j of both sides of (4.5) gives The result follows.
Hence S is B-invariant. Clearly N j (S j ) ⊂ S j for all j ∈ I. Therefore the stability condition for B implies that S = 0 or S = V d . First, assume S = 0. Then S ′ j = S j = 0 for j = i, and hence has a nonzero cokernel because it is nilpotent. However it implies Im Proof of Theorem 4.2. As the map F i is clearly which preserves the stability by Lemma 4.4. We easily obtain the relation F 2 i = Id by noting that F i is induced from the scalar shift By Lemma 3.6 and (4.3), we have  Remark 4.6. It is known (see e.g. [19]) that if d i = 1 for all i ∈ I, then the reflection functors F i satisfy relations (4.1). We expect that this fact is true for any (Q, d).

Application
In this subsection we introduce a basic application of reflection functors.
be the projection. Then Lemma 3.5 together with the assumption v = α i implies that B i ι is injective and πB i is surjective. On the other hand, (3.5) and the assumption for λ i (z) imply that which is equivalent to (v, α i ) ≤ 0. Now applying Crawley-Boevey's argument in [6, Lemma 7.3] to our quiver varieties with multiplicities, we obtain the following: then v is a positive root of g(C).
Proof . Assume N s Q,d (λ, v) = ∅ and that v is not a real root. We show that v is an imaginary root using [13,Theorem 5.4]; namely, show that there exists w ∈ W (C) such that w(v) has a connected support and (w(v), α i ) ≤ 0 for any i ∈ I.
Assume that there is i ∈ I such that (v, α i ) > 0. The above lemma implies that the top coefficient of λ i (z) is nonzero, which together with Theorem 4.2 implies that N s Q,d (r i (λ), s i (v)) = ∅. In particular we have s i (v) ∈ Q + , and further v − s i (v) ∈ Z >0 α i by the assumption (v, α i ) > 0. We then replace (λ, v) with (r i (λ), s i (v)), and repeat this argument. As the components of v decrease, it eventually stops after finite number of steps, and we finally obtain a pair (λ, v) ∈ R d × Q + such that (v, α i ) ≤ 0 for all i ∈ I. Additionally, the property N s Q,d (λ, v) = ∅ clearly implies that the support of v is connected. The result follows.

Normalization
In this section we give an application of Boalch's 'shifting trick' to quiver varieties with multiplicities.

Shifting trick
Definition 5.1. Let (Q, d) be a quiver with multiplicities. A vertex i ∈ I is called a pole vertex if there exists a unique vertex j ∈ I such that d j = 1, a ik = a ki = δ jk for any k ∈ I.
The vertex j is called the base vertex for the pole i. If furthermore d i > 1, the pole i ∈ I is said to be irregular.
Let i ∈ I be a pole vertex with the base j ∈ I.
In what follows we assume that the top coefficient of λ i (z) is nonzero. As the set N set Recall the isomorphism given in the previous section: where O is the G d i (V j )-coadjoint orbit through the element of the form Let us decompose Λ(z) as according to the decomposition The above is naturally dual to the Lie algebra b d i (V j ) of the unipotent subgroup The coadjoint action of g(z) ∈ B d i (V j ) is given by be the Levi subgroup associated to the decomposition V j = V i ⊕ V j /V i . The results in this section is based on the following two facts: Lemma 5.2. The orbitǑ is invariant under the conjugation action by K, and there exists a K-equivariant algebraic symplectomorphism sending Λ 0 (z) ∈Ǒ to the origin.
induces a bijection between (i) the (set-theoretical) symplectic quotient ofǑ × M by the diagonal K-action at the level − res Furthermore, under this bijection a point in the space (i) represents a free K-orbit if and only if the corresponding point in the space (ii) represents a free GL(V j )-orbit, at which the two symplectic forms are intertwined.
Lemma 5.3 is what we call 'Boalch's shifting trick'. We directly check the above two facts in Appendix A.
Remark 5.4. Let Λ 1 , Λ 2 , . . . , Λ k ∈ End(V ) be mutually commuting endomorphisms of a Cvector space V , and suppose that Λ 2 , . . . , Λ k are semisimple. To such endomorphisms we associate which is called a normal form. Let Σ ⊂ g * k (C) be the subset consisting of all residue-free elements λ(z) = k j=2 λ j z −j with (λ 2 , . . . , λ k ) being a simultaneous eigenvalue of (Λ 2 , . . . , Λ k ), and let V = λ∈Σ V λ be the eigenspace decomposition. Then we can express Λ(z) as It is known that any A(z) ∈ g * k (V ) whose leading term is regular semisimple is equivalent to some normal form under the coadjoint action.
Note that Λ(z) treated in Lemmas 5.2 and 5.3 is a normal form. A generalization of Lemma 5.2 for an arbitrary normal form has been announced in [4, Appendix C]. Lemma 5.3 is known in the case where Λ(z) is a normal form whose leading term is regular semisimple [2]; however, as mentioned in [4], the arguments in [2, Section 2] needed to prove this fact can be generalized to the case where Λ(z) is an arbitrary normal form.
On the other hand, by Lemma 5.2, the symplectic quotient of the space (i) by the action of k =i,j G d k (V k ) coincides with the symplectic quotient of by the action of

Normalization
The observation in the previous subsection leads us to define the following: Definition 5.5. Let i ∈ I be an irregular pole vertex of a quiver with multiplicities (Q, d) and j ∈ I be the base vertex for i. Then defineď = (ď k ) ∈ Z I >0 by d i := 1,ď k := d k for k = i, and letQ = (I,Ω, s, t) be the quiver obtained from (Q, d) as the following: (i) first, delete a unique arrow joining i and j; then (ii) for each arrow h with t(h) = j, draw an arrow from s(h) to i; (iii) for each arrow h with s(h) = j, draw an arrow from i to t(h); (iv) finally, draw d i − 2 arrows from j to i.
The transformation (Q, d) → (Q,ď) is called the normalization at i.
The adjacency matrixǍ = (ǎ kl ) of the underlying graph ofQ satisfieš The number of edges joining the two vertices are d − 2. If d = 3, the Kac-Moody algebra associated to (Q, d) is of type G 2 , while the one associated to (Q,ď) is of type A 2 . If d = 4, the Kac-Moody algebra associated to (Q, d) is of type A 2 , while the one associated to (Q,ď) is of type A Here we assume d > 1 and the number of vertices is n ≥ 3. The vertex on the far left is an irregular pole, at which we can perform the normalization and the resulting (Q,ď) has the underlying graph with multiplicities drawn below If d = 2, then the Kac-Moody algebra associated to (Q, d) is of type C n , while the one associated to (Q,ď) is of type A 3 if n = 3 and of type D n if n > 3. If (d, n) = (3, 3), the Kac-Moody algebra associated to (Q, d) is of type D The associated Kac-Moody algebra is of type D 3 . If n ≥ 4, the resulting (Q,ď) has the underlying graph with multiplicities drawn below The associated Kac-Moody algebra is of type A The associated Kac-Moody algebra is of type A 3 . If n > 4, the resulting (Q,ď) has the underlying graph with multiplicities drawn below The associated Kac-Moody algebra is of type D In the situation discussed in the previous subsection, letV = kV k be the I-graded vector space defined by Then we see that the group in (5.2) coincides with Gď(V). Furthermore, the following holds: Taking the cotangent bundle, we thus see that MQ ,ď (V) coincides with (5.1).

Setv := dimV anď
Then the value given in Now we state the main result of this section.
Theorem 5.8. Let i ∈ I be an irregular pole vertex of a quiver with multiplicities (Q, d) and j ∈ I be the base vertex for i. Let (Q,ď) be the quiver with multiplicities obtained by the normalization of (Q, d) at i.
is nonzero. Then the quiver varieties N s Q,d (λ, v) and N sQ ,ď (λ,v) are symplectomorphic to each other.
Proof . We have already constructed a bijection between N set Q,d (λ, v) and N seť Q,ď (λ,v). Thanks to Lemma 5.3, in order to prove the assertion it is sufficient to check that the bijection maps . It immediately follows from the three lemmas below.
is stable if and only if the corresponding (A(z), Q,d (V) satisfies the following condition: if a collection of subspaces Proof . This is similar to Lemma 4.4. First, assume that B is stable and that a collection of subspaces S k ⊂ V k ⊗ C R d k , k = i satisfies (5.5). We define and set S : Next assume that the pair (A(z), B =i ) satisfies the condition in the statement. Let S = k S k be a B-invariant subspace of V d satisfying N k (S k ) ⊂ S k for all k ∈ I. Then clearly the collection S k , k = i satisfies (5.5), and hence S k = 0 (k = i) or S k = V k ⊗ C R d k (k = i). If S k = 0, k = i, we have B i (S i ) = 0, which implies S i = 0 since Ker B i ∩ Ker N i = 0 by Lemma 3.5 and N i | S i is nilpotent. Dualizing the argument, we easily see that Proof . In Appendix A, we show that all the block components of A 0 l relative to the decomposition V j =V i ⊕V j are described as a (non-commutative) polynomial in B ′ h over h ∈ H with (t(h), s(h)) = (i, j) or (j, i), and vice versa (see Remark A.3, where A 0 is denoted by B and B ′ h for such h are denoted by a ′ k , b ′ k ). Hence an I-graded subspace S ofVď satisfies (5.6) if and only if it is B ′ -invariant and N k (S k ) ⊂ S k for k = i, j. Proof . By definition we have so the 'if' part is clear. To prove the 'only if' part, note that if a collection of subspaces S k ⊂ V k ⊗ C R d k , k = i satisfies (5.5), then in particular S j is preserved by the action of Now the result immediately follows.

Weyl groups
Let (Q, d) be a quiver with multiplicities having an irregular pole vertex i ∈ I with base j ∈ I, and let (Q,ď) be the one obtained by the normalization of (Q, d) at i. In this subsection we discuss on the relation between the two Weyl groups associated to (Q, d) and (Q,ď).
Recall our notation for objects relating to the Kac-Moody algebra; C = 2Id − AD is the generalized Cartan matrix associated to (Q, d), and h, Q, α k , s k , the Cartan subalgebra, the root lattice, the simple roots, and the simple reflections, of the corresponding Kac-Moody algebra g(C). In what follows we denote byČ,Ď,ȟ,Q,α k ,š k , the similar objects associated to (Q,ď).
Let ϕ : Q →Q be the linear map defined by v →v = v − v iαj . The same letter is also used on the matrix representing ϕ with respect to the simple roots. By the properties of i and j, the matrices D and A are respectively expressed as where D ′ (resp. A ′ ) is the sub-matrix of D (resp. A) obtained by restricting the index set to I \ {i, j}, and a = (a kj ) k =i,j . By the definition of the normalization, the matricesĎ andǍ are then respectively expressed aš Now we check the identity. We have On the other hand, The above lemma implies that the map ϕ preserves the inner product. Furthermore it also implies rankČ = rank C, which means dim h = dimȟ. Thus we can extend ϕ to an isomorphism ϕ : h * →ȟ * preserving the inner product.
Note that by the definition of normalization, the permutation of the indices i and j, which we denote by σ, has no effect on the matrixČ. Hence it defines an involution of W (Č), or equivalently, a homomorphism Z/2Z → Aut(W (Č)).
Proposition 5.13. Under the isomorphismφ, the Weyl group W (C) associated to C is isomorphic to the semidirect product W (Č) ⋊ Z/2Z of the one associated toČ and Z/2Z by the permutation σ.
We can easily check that Note that the action of W (Č) on Rď naturally extends to an action of W (Č) ⋊ Z/2Z. We see from the above relation that the map R d → Rď, λ →λ is equivariant, and hence so is the map given in Proposition 5.13.
6 Naive moduli of meromorphic connections on P 1 This final section is devoted to study moduli spaces of meromorphic connections on the trivial bundle over P 1 with some particular type of singularities.

Naive moduli
When constructing the moduli spaces of meromorphic connections, one usually fix the 'formal type' of singularities. However, we fix here the 'truncated formal type', and consider the corresponding 'naive' moduli space. Actually in generic case, such a naive moduli space gives the moduli space in the usual sense, which will be explained in Remark 6.5. Fix n ∈ Z >0 and • a nonzero finite-dimensional C-vector space V ; • positive integers k 1 , k 2 , . . . , k n ; • mutually distinct points t 1 , t 2 , . . . , t n in C.
Then consider a system of linear ordinary differential equations with rational coefficients. It has a pole at t i of order at most k i for each i, and (possibly) a simple pole at ∞ with residue − i A i,1 . We identify such a system with its coefficient matrix A(z), which may be regarded as an element of i g * After Boalch [2], we introduce the following (the terminologies used here are different from his): For given coadjoint orbits O i ⊂ g * k i (V ), i = 1, . . . , n, the set is called the naive moduli space of systems having a pole of truncated formal type O i at each t i , i = 1, . . . , n.
Note that i O i is a holomorphic symplectic manifold, and the map is a moment map with respect to the simultaneous GL(V )-conjugation action. Hence the set M set (O 1 , . . . , O n ) is a set-theoretical symplectic quotient. It is also useful to introduce the following ζ-twisted naive moduli space: is irreducible, Schur's lemma shows that the stabilizer of A(z) with respect to the GL(V )-action is equal to C × , and furthermore one can show that the action on the set of irreducible systems in i O i is proper. Definition 6.3. For ζ ∈ C, the holomorphic symplectic manifold is called the ζ-twisted naive moduli space of irreducible systems having a pole of truncated formal type O i at each t i , i = 1, . . . , n. In the 0-twisted (untwisted) case, we simply write If we have a specific element Λ i (z) ∈ O i for each i, the following notation is also useful: Remark 6.4. Recall that a holomorphic vector bundle with meromorphic connection (E, ∇) over a compact Riemann surface is stable if for any nonzero proper subbundle F ⊂ E preserved by ∇, the inequality deg F/ rank F < deg E/ rank E holds. It is easy to see that if the base space is P 1 and E is trivial, then (E, ∇) is stable if and only if it has no nonzero proper trivial subbundle F ⊂ E preserved by ∇. This implies that a system A(z) ∈ i g * k i (V ) is irreducible if and only if the associated vector bundle with meromorphic connection (P 1 × V, d − A(z) dz) is stable. Remark 6.5. Let us recall a normal form Λ(z) introduced in Remark 5.4. Assume that each Γ λ is non-resonant, i.e., no two distinct eigenvalues of Γ λ differ by an integer. Then one can show that an element A(z) ∈ g * k (V ) is equivalent to Λ(z) under the coadjoint action if and only if there is a formal gauge transformation [31,Remark 18]). In this sense the truncated formal type of Λ(z) actually prescribe a formal type. Hence, if each O i ⊂ g * k i (V ) contains some normal form with nonresonant residue parts, then the naive moduli space M set (O 1 , . . . , O n ) gives the moduli space of meromorphic connections on the trivial bundle P 1 × V having a pole of prescribed formal type at each t i .

Star-shaped quivers of length one
In some special case, the naive moduli space M irr (O 1 , . . . , O n ) can be described as a quiver variety. Suppose that for each i = 1, . . . , n, the coadjoint orbit O i contains an element of the form . Let d i be the pole order of λ i := ξ i − η i . Note that Ξ i is a particular example of normal forms introduced in Remark 5.4, and it has non-resonant residue parts (see Remark 6.5) which together with the above λ i gives an element λ = (λ i ) ∈ R d . Note that where v := dim V. Hence Proposition 6.6. There exists a bijection from N set Proof . Set ζ := res z=0 λ 0 (z) = n i=1 res z=0 η i (z). Then the scalar shift with η i induces a bijection and it preserves the irreducibility. As Λ i has the pole order d i , the G k i (V )-action on Λ i reduces to the G d i (V )-action via the natural projection This replacement of order has no effect on the naive moduli space.
By the definition of Q, we have V i = V 0 ⊗ C R 1 = V for each i > 0 and Now consider the sets Z i ⊂ M i , i > 0 given in Lemma 3.6. Since the top coefficients of λ i ∈ g * d i (C), i > 0 are nonzero, Lemma 3.5 implies that for each i > 0, any point in Since dim V i ≤ dim V for all i > 0, Lemmas 3.6 and 3.7 imply that the map Taking the (set-theoretical) symplectic quotient by the GL(V )-action at −ζ Id V , we thus obtain a bijection from N set Q,d (λ, v) to M set ζ (Λ 1 , . . . , Λ n ). The proof of what it maps N s Q,d (λ, v) onto M irr ζ (Λ 1 , . . . , Λ n ) is quite similar to Lemma 5.9.
First, assume that a point A i,l z −l , and assume further that a subspace S 0 ⊂ V is invariant under all A i,l . We define Since B is stable, we thus have S = 0 or S = V d , and in particular, S 0 = 0 or S 0 = V , which shows that the system Φ(B) is irreducible. Conversely, assume that the system Φ(B) = ( l A i,l z −l ) is irreducible. Let S = i S i be a B-invariant subspace of V d satisfying N i (S i ) ⊂ S i for all i ∈ I. Then S 0 is invariant under all A i,l , and hence S 0 = 0 or S 0 = V . If S 0 = 0, then for each i > 0, we have B i (S i ) = 0, which implies S i = 0 since Ker B i ∩ Ker N i = 0 by Lemma 3.5 and N i | S i is nilpotent. Dualizing the argument, we easily see that Conversely, let Q = (I, Ω, s, t) be as above and suppose that an I-graded C-vector space V = i V i and multiplicities d = (d i ) are given. Suppose further that they satisfy dim V i ≤ dim V 0 and d 0 = 1. Set V := V 0 , and fix a C-vector space i for each i > 0. Also, for each λ ∈ R d , set ζ := res z=0 λ 0 and let Λ i be as in (6.2). Then the above proof also shows that the map Φ given in (6.3) induces a bijection N set

Middle convolution
Recall the map given in (6.3); Using the natural inclusion ι i : Thus we can write the systems Φ(B) as Such an expression of systems has been familiar since Harnad's work [10], and is in fact quite useful to formulate the so-called middle convolution [31], which was originally introduced by Katz [14] for local systems on a punctured P 1 and generalized by Arinkin [1] for irregular Dmodules. Let us define the generalized middle convolution according to [31]. First, we introduce the following fact, which is a refinement of Woodhouse and Kawakami's observation [30,15]: • an endomorphism T ∈ End(W ) with eigenvalues t i , i = 1, 2, . . . , n; • a pair of homomorphisms (X, Y ) ∈ Hom(W, V ) ⊕ Hom(V, W ), such that 6) where N i is the nilpotent part of T restricted on its generalized t i -eigenspace is the block component of (X, Y ) with respect to the decomposition W = i W i . Moreover the choice of (W, T, X, Y ) is unique in the following sense: if two quadruples (W, T, X, Y ) and (W ′ , T ′ , X ′ , Y ′ ) satisfy (6.5) and (6.6), then there exists an isomorphism f : W → W ′ such that The above enables us to define the middle convolution. For a system A(z) = (A i ) ∈ n i=1 g * k i (V ), take a quadruple (W, T, X, Y ) satisfying (6.5) and (6.6). Then for given ζ ∈ C, set V ζ := W/ Ker(Y X + ζ Id W ) and let • X ζ : W → V ζ be the projection; By virtue of Proposition 6.7, the equivalence class of mc ζ (A) under constant gauge transformations depends only on that of A(z). We call it the middle convolution of A(z) with ζ. 5 Let us come back to our situation. The expression (6.4) and Lemma 3.5 (which we apply for all i > 0) imply that the quadruple ( V 0 , T, B 0 , B 0 ) satisfies (6.5) and (6.6) for A(z) = Φ(B). Now assume λ 0 (z) = 0 and consider the middle convolution mc ζ (A) with ζ := res z=0 λ 0 . By the definition, the triple (V ζ , B ζ 0 , B ζ 0 ) satisfies Ker i.e., it provides a full-rank decomposition of the matrix B 0 B 0 +ζ Id V 0 . Recall that such a triple already appeared in Section 4; conditions (4.3) and (4.4) for the 0-th reflection functor F 0 imply that if we take an I-graded C-vector space V ′ = i V ′ i with dim V ′ = s 0 (v) as in Section 4.2 and a representative B ′ ∈ M Q,d (V ′ ) of F 0 [B] ∈ N s Q,d (r 0 (λ), s 0 (v)), then the triple (V ′ 0 , B ′ 0 , B ′ 0 ) also satisfies (6.7) and (6.8) (note that d 0 = 1 and N 0 = 0). By the uniqueness of the full-rank decomposition, we then see that there exists an isomorphism f : The arguments in the previous subsection for V ′ , λ ′ := r 0 (λ) show that Φ : We have now proved the following: Proposition 6.8. Let (Q, d), λ, v be as in Proposition 6.6, and assume ζ := res z=0 λ 0 is nonzero.
Under the above notation, one then has the following commutative diagram: Next, consider the reflection functors . Then condition (4.2) implies which together with (4.3) shows that the two systems Φ(B) and Φ ′ (B) are related via Proposition 6.9. Let (Q, d), λ, v be as in Proposition 6.6, and set ζ := res z=0 λ 0 . For i = 1, 2, . . . , n, one then has the following commutative diagram: where the bottom horizontal arrow is given by the shift Remark 6.10. In [10], Harnad considered two meromorphic connections having the following symmetric description: where V , W are finite-dimensional C-vector spaces, S, T are regular semisimple endomorphisms of V , W respectively, and (X, Y ) ∈ Hom(W, V ) ⊕ Hom(V, W ) such that both (W, T, X, Y ) and (V, S, Y, X) satisfy (6.6). These have an order 2 pole at z = ∞ and simple poles at the eigenvalues of T , S respectively. He then proved that the isomonodromic deformations of the two systems are equivalent. After his work, such a duality, called the Harnad duality, was established in more general cases by Woodhouse [30]. Note that if S = 0, we have ∇ ′ = d + z −1 P Q dz. Hence on the 'dual side', the operation mc ζ corresponds to just the scalar shift by z −1 ζ dz. This interpretation enables us to generalize the middle convolution further; see [31].

Examples: rank two cases
The case dim V = 2 is most important because in this case a generic element in g * k i (V ) can be transformed into an element of the form Ξ i (z) = ξ i (z) ⊕ η i (z) for some distinct ξ i , η i ∈ g * k i (C). The dimension of M irr (Ξ 1 , . . . , Ξ n ) can be computed as if it is nonempty.
The corresponding (Q, d) have the underlying graphs with multiplicities given by the picture below The associated Kac-Moody algebras are respectively given by From Example 5.6, we see that the effect of normalization on these quivers with multiplicities is given as follows: where the arrows represent the process of normalization.
Next consider the case dim M irr (Ξ 1 , . . . , Ξ n ) = 2. Then the tuple (d 1 , . . . , d n ) must be one of the following (up to permutation on indices): (1, 1, 1, 1), (2, 1, 1), (3, 1), (2, 2), (4). (6.9) The corresponding (Q, d) have the underlying graphs with multiplicities given by the picture below The associated Kac-Moody algebras are respectively given by 2 . (6.10) From Example 5.6, we see that the effect of normalization on these quivers with multiplicities is given as follows: 3 , A where the arrows represent the process of normalization. Hence by performing the normalization if necessary, we obtain the following list of (untwisted) affine Lie algebras: 1 , which is well-known as the list of Okamoto's affine Weyl symmetry groups of the Painlevé equations of type VI, V, . . . , II, as mentioned in Introduction.
A.1 Proof of Lemma 5.2 We check that the B d i (V j )-coadjoint orbitǑ is invariant under the conjugation action by K, and is K-equivariantly symplectomorphic to the symplectic vector space Note that all the coefficients of Λ 0 are fixed by K, and that the subset is invariant under the conjugation by constant matrices. Hence for any k ∈ K and g(z) ∈ B d i (V j ), k g · Λ 0 k −1 = kgk −1 · kΛ 0 k −1 = kgk −1 · Λ 0 ∈Ǒ, i.e.,Ǒ is invariant under the conjugation by K. Let us calculate the stabilizer of Λ 0 (z) with respect to the coadjoint B d i (V j )-action. Suppose that g(z) ∈ B d i (V j ) stabilizes Λ 0 (z). By the definition, we then have according to the decomposition V j = V i ⊕ V j /V i , and let λ 0 i (z) be the residue-free part of λ i (z). Then Therefore (A.1) is equivalent to for all the matrix entries f (z) = d i −1 k=1 f k z k of G 12 (z) and G 21 (z). We can write the above condition as  Since λ i,d i = 0, this means f k = 0 for all k = 1, 2, . . . , d i − 2. Hence the stabilizer is given by g k z k g k ∈ Lie K, k = 1, . . . , d i − 2, g d i −1 ∈ gl(V j ) .
The above implies that the orbitǑ is naturally isomorphic to Let us denote an element of the vector space on the right hand side by for some Γ ∈ gl(V j ). According to the decomposition V j = V i ⊕ V j /V i , we write it as Γ = Γ 11 Γ 12 Γ 21 Γ 22 , and set Note that Γ B ∈ Lie K. Let Λ d i be the top coefficient of Λ 0 (z). Then U satisfies and hence u(z)g(z) −1 B(z)g(z)u(z) = u(z)(Λ 0 (z) + z −1 Γ)u(z) −1 mod gl(V j )[[z]] Now we explicitly describe Γ B in terms of the coordinates (a ′ k , b ′ k ) d i −2 k=1 , which shows that B → −Γ B is a K-moment map. Note that the constant term of g(z) is the identity, and hence it acts trivially on z Substituting (A.4) into the above equality, we have Note that B(z) and λ 0 i (z) have no residue parts. Looking at the block diagonal part of the above and taking the residue, we thus obtain and similarly, which gives the minus of the K-moment map vanishing at a ′ k , b ′ k = 0.
Remark A.2. The matrix Γ in (A.7) is characterized by Γ = res z=0 g(z) −1 B(z)g(z), so that it depends algebraically on a k , b k . Hence u(z)g(z) −1 also depends algebraically on a k , b k . This means that one can choose g(z) in the assertion of Lemma A.1 so that it depends algebraically on B ∈Ǒ. Then (A.8) implies Note that λ 0 i (Id V i − ab) −1 a has pole order d i − 1 and Set a ′ (z) := a ′ k z −k−1 . Using the obvious formulas a(Id V j /V i − ba) −1 = (Id V i − ab) −1 a and (Id V i − ab) −1 = Id V i + (Id V i − ab) −1 ab, we can then rewrite the above four equalities as which give the explicit description of B in terms of the coordinates (a ′ k , b ′ k ). Conversely, we can describe (a ′ , b ′ ) in terms of B using the above. Indeed, (A.10) determines a ′ , and (A.9) and (A.11) imply Writing B ij = k B ij,k z −k , we then have with some non-commutative polynomial F lk .