Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé 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. Th...

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Дата:	2010
Автор:	Yamakawa, D.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146522
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Цитувати:	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ.

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spelling	irk-123456789-1465222019-02-11T01:24:06Z Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations Yamakawa, D. 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 Painlevé equations is slightly different from (but equivalent to) Okamoto's 2010 Article Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D30; 16G20; 20F55; 34M55 DOI:10.3842/SIGMA.2010.087 http://dspace.nbuv.gov.ua/handle/123456789/146522 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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 Painlevé equations is slightly different from (but equivalent to) Okamoto's
format	Article
author	Yamakawa, D.
spellingShingle	Yamakawa, D. Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Yamakawa, D.
author_sort	Yamakawa, D.
title	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
title_short	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
title_full	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
title_fullStr	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
title_full_unstemmed	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
title_sort	quiver varieties with multiplicities, weyl groups of non-symmetric kac-moody algebras, and painlevé equations
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146522
citation_txt	Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT yamakawad quivervarietieswithmultiplicitiesweylgroupsofnonsymmetrickacmoodyalgebrasandpainleveequations
first_indexed	2025-07-11T00:10:59Z
last_indexed	2025-07-11T00:10:59Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 087, 43 pages Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac–Moody Algebras, and Painlevé Equations Daisuke YAMAKAWA †‡ † Centre de mathématiques Laurent Schwartz, École polytechnique, CNRS UMR 7640, ANR SÉDIGA, 91128 Palaiseau Cedex, France ‡ Department of Mathematics, Graduate School of Science, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: yamakawa@math.kobe-u.ac.jp Received March 19, 2010, in final form October 18, 2010; Published online October 26, 2010 doi:10.3842/SIGMA.2010.087 Abstract. 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 Painlevé equations is slightly different from (but equivalent to) Okamoto’s. Key words: quiver variety; quiver variety with multiplicities; non-symmetric Kac–Moody algebra; Painlevé equation; meromorphic connection; reflection functor; middle convolution 2010 Mathematics Subject Classification: 53D30; 16G20; 20F55; 34M55 1 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 = (di)i∈I ∈ ZI >0 be a collection of positive integers indexed by the vertices. We think of each number di 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 Ns Q,d(λ,v) having parameters • λ = (λi(z))i∈I , where λi(z) = λi,1z −1 + λi,2z −2 + · · ·+ λi,di z−di ∈ z−diC[z]/C[z]; • v = (vi)i∈I ∈ ZI ≥0, and call it the quiver variety with multiplicities, because if di = 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 ∈ ZI ≥0, ζ = (ζR, ζC) = ( 0, (λi,1)i∈I ) ∈ √ −1RI × CI . As in the case of quiver variety, Ns 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 mailto:yamakawa@math.kobe-u.ac.jp http://dx.doi.org/10.3842/SIGMA.2010.087 2 D. Yamakawa non-reductive unless di = 1 or vi = 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 Ns Q,d(λ,v), and for the same reason, they seem to provide new geometric objects relating to quivers. The definition of Ns 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 PJ denotes the Painlevé equation of type J (J = II, III, . . . ,VI). Equations PVI PV PIV PIII PII Symmetries D (1) 4 A (1) 3 A (1) 2 C (1) 2 A (1) 1 On the other hand, each of them is known to govern an isomonodromic deformation of rank two meromorphic connections on P1 [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 k1 + k2 + · · · + kn means that the connections in the deformation have n poles of order ki, i = 1, 2, . . . , n and no other poles. Equations PVI PV PIV PIII PII Connections 1 + 1 + 1 + 1 2 + 1 + 1 3 + 1 2 + 2 4 Roughly speaking, we thus have a non-trivial correspondence between some Dynkin diagrams and rank two meromorphic connections. In fact, such a relationship can be understood in terms of quiver varieties except in the case of PIII. Crawley-Boevey [7] described the moduli spaces of Fuchsian systems (i.e., meromorphic connections on the trivial bundle over P1 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 PVI. The quiver description in the cases of PII, PIV and PV was obtained by Boalch1 [4]; more generally, he proved that the moduli spaces of meromorphic connections on the trivial bundle over P1 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 spaces2 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 Theo- rem 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 PIII; 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): Proposition 1.1. Consider a ‘star-shaped quiver of length one’ · · · 0�� 1 �� 77 oooooooooooo 2 �� GG �� n �� gg OOOOOOOOOOOO 1His description in the case of PV is based on the work of Harnad [10]. 2Actually in each Painlevé case, this action reduces to an action of the corresponding finite Weyl group, which together with ‘Schlesinger transformations’ give the full symmetry; see [5, Section 6]. Quiver Varieties with Multiplicities 3 Here the set of vertices is I = { 0, 1, . . . , n }. Take multiplicities d ∈ ZI >0 with d0 = 1 and set v ∈ ZI ≥0 by v0 = 2, vi = 1 (i = 1, . . . , n). Then Ns Q,d(λ,v) is isomorphic to the moduli space of stable meromorphic connections on the rank two trivial bundle over P1 having n poles of order di, i = 1, . . . , n of prescribed formal type. On the other hand, to any quiver with multiplicities, we associate a generalized Cartan mat- rix C as follows: C = 2Id−AD, 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 si : ZI → ZI , ri : ⊕ i∈I ( z−diC[z]/C[z] ) → ⊕ i∈I ( z−diC[z]/C[z] ) (i ∈ I) generating actions of the Weyl group of the associated Kac–Moody algebra, such that for any (λ,v) and i ∈ I with λi,di 6= 0, one has a natural symplectomorphism Fi : Ns Q,d(λ,v) '−→ Ns Q,d(ri(λ), si(v)). If di = 1 for all i ∈ I, then the maps Fi 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, Ap- pendix A]). A similar assertion also holds in the situation of Proposition 1.1; the map F0 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 1�� 1�� __ ?? ?? ?? 1 �� ?? �� 1 �� ?????? 1�� 2�� 1��// 1�� 1�� XX 22 22 2 3�� 1��// 1��oo 2�� 1��// 2��oo 4�� 1��// Proposition 1.1 says that the associated Ns Q,d(λ,v) with a particular choice of v give the moduli spaces for PVI, PV, PIV, PIII and PII, respectively. On the other hand, the associated Kac–Moody algebras are respectively given by3 D (1) 4 , A (2) 5 , D (3) 4 , C (1) 2 , A (2) 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: 3We follow Kac [13] for the notation of (twisted) affine Lie algebras. 4 D. Yamakawa Theorem 1.3 (see Section 5). Suppose that a quiver with multiplicities (Q,d) has a pair of vertices (i, j) such that di > 1, dj = 1, aik = aki = δjk for any k ∈ I, where A = (aij) is the adjacency matrix of the underlying graph. Then it determines another quiver with multiplicities (Q̌, ď) and a map (λ,v) 7→ (λ̌, v̌) between parameters such that the following holds: if λi,di 6= 0, then Ns Q,d(λ,v) and Ns Q̌,ď (λ̌, v̌) are symplectomorphic to each other. We call the transformation (Q,d) 7→ (Q̌, 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 nor- malization (see Section 5.3). In particular, we have the following relation between the Weyl groups W , W̌ associated to (Q,d), (Q̌, ď): W ' W̌ o Z/2Z, 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 in- stance, in the cases of PV, PIV and PII, we have W ( A (2) 5 ) 'W ( A (1) 3 ) o Z/2Z, W ( D (3) 4 ) 'W ( A (1) 2 ) o Z/2Z, W ( A (2) 2 ) 'W ( A (1) 1 ) o Z/2Z, which mean that our list of Dynkin diagrams for Painlevé equations is a variant of Okamoto’s obtained by (partially) extending the Weyl groups. 2 Preliminaries In this section we briefly recall the definition of Nakajima’s quiver variety [21]. 2.1 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 := ΩtΩ, and extend the map Ω→ Ω, h 7→ 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 = (aij)i,j∈I , called the adjacency matrix, as follows: aij := ]{ edges joining i and j } = ]{h ∈ H \| s(h) = i, t(h) = j }. Let V = ⊕ i∈I Vi be a nonzero finite-dimensional I-graded C-vector space. A representation of Q over V is an element of the vector space RepQ(V) := ⊕ h∈Ω HomC(Vs(h), Vt(h)), Quiver Varieties with Multiplicities 5 and its dimension vector is given by v := dimV ≡ (dim Vi)i∈I . Isomorphism classes of repre- sentations of Q with dimension vector v just correspond to orbits in RepQ(V) with respect to the action of the group GL(V) := ∏ i∈I GLC(Vi) given by g = (gi) : (Bh)h∈Ω 7−→ ( gt(h)Bhg−1 s(h) ) h∈Ω , g ∈ GL(V). We denote the Lie algebra of GL(V) by gl(V); explicitly, gl(V) := ⊕ i∈I glC(Vi). For ζ = (ζi)i∈I ∈ CI , we denote its image under the natural map CI → gl(V) by ζ IdV, and also use the same letter ζ IdV for ζ ∈ C via the diagonal embedding C ↪→ CI . Note that the central subgroup C× ' { ζ IdV \| ζ ∈ C× } ⊂ GL(V) acts trivially on RepQ(V), so we have the induced action of the quotient group GL(V)/C×. Let B = (Bh)h∈Ω ∈ RepQ(V). An I-graded subspace S = ⊕ i∈I Si of V is said to be B- invariant if Bh(Ss(h)) ⊂ St(h) for all h ∈ Ω. If V has no B-invariant subspace except S = 0,V, then B is said to be irreducible. Schur’s lemma4 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 Repirr Q (V) consisting of all irreducible representations over V is proper. 2.2 Quiver variety Suppose that a quiver Q and a nonzero finite-dimensional I-graded C-vector space V = ⊕ i∈I Vi are given. We set MQ(V) := RepQ+Q(V) = RepQ(V)⊕ RepQ(V), and regard it as the cotangent bundle of RepQ(V) by using the trace pairing. Introducing the function ε : H → {±1}, ε(h) := { 1 for h ∈ Ω, −1 for h ∈ Ω, we can write the canonical symplectic form on MQ(V) as ω := ∑ h∈Ω tr dBh ∧ dBh = 1 2 ∑ h∈H ε(h) tr dBh ∧ dBh, (Bh)h∈H ∈MQ(V). The natural GL(V)-action on MQ(V) is Hamiltonian with respect to ω with the moment map µ = (µi)i∈I : MQ(V)→ gl(V), µi(B) = ∑ h∈H: t(h)=i ε(h)BhBh (2.1) vanishing at the origin, where we identify gl(V) with its dual using the trace pairing. Definition 2.1. A point B ∈MQ(V) is said to be stable if it is irreducible as a representation of Q + Q. For a GL(V)-invariant Zariski closed subset Z of MQ(V), let Zs 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. 4One can apply Schur’s lemma thanks to the following well-known fact: the category of representations of Q is equivalent to that of an algebra CQ, the so-called path algebra; see e.g. [9]. 6 D. Yamakawa Definition 2.2. For ζ ∈ CI and v ∈ ZI ≥0 \ {0}, taking an I-graded C-vector space V with dimV = v we define Ns Q(ζ,v) := µ−1(−ζ IdV)s/ GL(V), which we call the quiver variety. Remark 2.3. In Nakajima’s notation (see [21] or [22]), Ns Q(ζ,v) is denoted by M reg (0,ζ)(v, 0). 3 Quiver variety with multiplicities 3.1 Definition For a positive integer d, we set Rd := C[[z]]/zdC[[z]], Rd := z−dC[[z]]/C[[z]]. The C-algebra Rd has a typical basis { zd−1, . . . , z, 1 }, with respect to which the multiplication by z in Rd is represented by the nilpotent single Jordan block Jd :=  0 1 0 0 . . . . . . 1 0 0  ∈ End(Cd) = EndC(Rd). The vector space Rd may be identified with the C-dual R∗ d = HomC(Rd, C) of Rd via the pairing Rd ⊗C Rd → C, (f, g) 7→ res z=0 ( f(z)g(z) ) . For a finite-dimensional C-vector space V , we set gd(V ) := gl(V )⊗C Rd = gl(V )[[z]]/zd gl(V )[[z]]. Note that gd(V ) is naturally isomorphic to EndRd (V ⊗C Rd) as an Rd-module; hence it is the Lie algebra of the complex algebraic group Gd(V ) := AutRd (V ⊗C Rd) ' { g(z) = d−1∑ k=0 gkz k ∈ gd(V ) ∣∣∣∣∣ det g0 6= 0 } . The inverse element of g(z) ∈ Gd(V ) is given by taking modulo zd gl(V )[[z]] of the formal inverse g(z)−1 ∈ gl(V )[[z]]. The adjoint action of g(z) is described as (g · ξ)(z) = g(z)ξ(z)g(z)−1 mod zd gl(V )[[z]], ξ(z) ∈ gd(V ). Using the above R∗ d ' Rd and the trace pairing, we always identify the C-dual g∗d(V ) of gd(V ) with gl(V )⊗C Rd = z−d gl(V )[[z]]/ gl(V )[[z]]. Then the coadjoint action of g(z) ∈ Gd(V ) is also described as (g · η)(z) = g(z)η(z)g(z)−1 mod gl(V )[[z]], η(z) = d∑ k=1 ηkz −k ∈ g∗d(V ). The natural inclusion gd(V ) ↪→ EndC(V ⊗C Rd) = EndC(V )⊗C EndC(Rd) is represented by ξ(z) = d−1∑ k=0 ξkz k 7−→ d−1∑ k=0 ξk ⊗ Jk d , Quiver Varieties with Multiplicities 7 whose image is just the centralizer of IdV ⊗ Jd. Accordingly, its transpose can be written as glC(V ⊗C Rd) ' glC(V ⊗C Rd)∗ → g∗d(V ), X 7→ d∑ k=1 trRd [ X ( IdV ⊗ Jk−1 d )] z−k, where trRd : EndC(V ⊗C Rd) = EndC(V ) ⊗C EndC(Rd) → EndC(V ) denotes the trace of the EndC(Rd)-part. Now suppose that a quiver Q and a collection of positive integers d = (di)i∈I ∈ ZI >0 are given. We call the pair (Q,d) as a quiver with multiplicities and di as the multiplicity of the vertex i. Set Rd := ⊕ i∈I Rdi , Rd := ⊕ i∈I Rdi , and for a nonzero finite-dimensional I-graded C-vector space V = ⊕ i∈I Vi, set Vd ≡ V ⊗C Rd := ⊕ i∈I Vi ⊗C Rdi , MQ,d(V) := MQ(Vd) = ⊕ h∈H HomC ( Vs(h) ⊗C Rds(h) , Vt(h) ⊗C Rdt(h) ) , Gd(V) := ∏ i∈I Gdi (Vi), gd(V) := ⊕ i∈I gdi (Vi). The group Gd(V) naturally acts on MQ,d(V) as a subgroup of GL(Vd). Note that the subgroup C× ⊂ GL(Vd) is contained in Gd(V) and acts trivially on MQ,d(V). As in the case of gl(V), for λ = (λi(z))i∈I ∈ Rd we denote its image under the natural map Rd = g∗d(CI) → g∗d(V) by λ IdV. Let ω be the canonical symplectic form on MQ,d(V); ω = 1 2 ∑ h∈H ε(h) tr dBh ∧ dBh, (Bh)h∈H ∈MQ,d(V). Then the Gd(V)-action is Hamiltonian whose moment map µd is given by the composite of the GL(Vd)-moment map µ = (µi) : MQ,d(V) → gl(Vd) (see (2.1) for the definition) and the transpose pr = (pri) of the inclusion gd(V) ↪→ gl(Vd); µd = (µd,i)i∈I : MQ,d(V)→ g∗d(V), µd,i(B) := pri ◦µi(B) = di∑ k=1 ∑ h∈H: t(h)=i ε(h) trRdi [ BhBhNk−1 i ] z−k, where Ni := IdVi ⊗ Jdi . Definition 3.1. A point B ∈ MQ,d(V) is said to be stable if Vd has no nonzero proper B- invariant subspace S = ⊕ i∈I Si such that Si ⊂ Vi ⊗C Rdi is an Rdi -submodule for each i ∈ I. The above stability can be interpreted in terms of the irreducibility of representations of a quiver. Letting Ω̃ := Ω t {`i}i∈I and extending the maps s, t to Ω̃ by s(`i) = t(`i) = i, we obtain a new quiver Q̃ = (I, Ω̃, s, t). Consider the vector space Rep Q̃+Q (Vd) 'MQ,d(V)⊕ gl(Vd) 8 D. Yamakawa associated with the quiver Q̃ + Q = (I, Ω̃ t Ω, s, t). Note that in the above definition, a vector subspace Si ⊂ Vi ⊗ Rdi is an Rdi -submodule if and only if it is invariant under the action of Ni = IdVi ⊗ Jdi , which corresponds to the multiplication by z. Thus letting ι : MQ,d(V) ↪→ Rep Q̃+Q (Vd), B 7→ (B, (Ni)i∈I), (3.1) we see that a point B ∈ MQ,d(V) is stable if and only if its image ι(B) is irreducible as a representation of Q̃ + Q. For a Gd(V)-invariant Zariski closed subset Z of MQ,d(V), let Zs be the subset of all stable points in Z. Proposition 3.2. The group Gd(V)/C× acts freely and properly on Zs. Proof. Note that the closed embedding ι defined in (3.1) is equivariant under the action of Gd(V) ⊂ GL(Vd). Hence the freeness of the Gd(V)/C×-action on Zs follows from that of the GL(Vd)/C×-action on Rep Q̃+Q (Vd)irr and ι(MQ,d(V)s) = ι(MQ,d(V)) ∩ Rep Q̃+Q (Vd)irr, which we have already checked. Furthermore, the above implies that the embedding Zs ↪→ Rep Q̃+Q (Vd)irr induced from ι is closed. Consider the following commutative diagram: Gd(V)/C× × Zs // �� Zs �� GL(Vd)/C× × Rep Q̃+Q (Vd)irr // Rep Q̃+Q (Vd)irr, where the vertical arrows are the maps induced from ι, and the horizonal arrows are the action maps (g, x) 7→ 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]). � Definition 3.3. For λ ∈ Rd and v ∈ ZI ≥0 \ {0}, taking an I-graded C-vector space V with dimV = v we define Ns Q,d(λ,v) := µ−1 d (−λ IdV)s/Gd(V), which we call the quiver variety with multiplicities. We also use the following set-theoretical quotient: Nset Q,d(λ,v) := µ−1 d (−λ IdV)/Gd(V). It is clear from the definition that if (Q,d) is multiplicity-free, i.e., di = 1 for all i ∈ I, then Ns Q,d(λ,v) coincides with the ordinal quiver variety Ns Q(ζ,v) with ζi = res z=0 λi(z). Even when d is non-trivial, for simplicity, we often refer to Ns Q,d(λ,v) just as the ‘quiver variety’. 3.2 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 = (aij)i,j∈I be the adjacency matrix of the underlying graph of Q and set D := (diδij)i,j∈I . Consider the generalized Cartan matrix C = (cij)i,j∈I := 2Id−AD. Quiver Varieties with Multiplicities 9 Note that it is symmetrizable as DC = 2D−DAD is symmetric. Let( g(C), h, {αi}i∈I , {α∨i }i∈I ) be the corresponding Kac–Moody algebra with its Cartan subalgebra, simple roots and simple coroots. As usual we set Q := ∑ i∈I Zαi, Q+ := ∑ i∈I Z≥0αi. The diagonal matrix D induces a non-degenerate invariant symmetric bilinear form ( , ) on h∗ satisfying (αi, αj) = dicij = 2diδij − diaijdj , i, j ∈ I. From now on, we regard a dimension vector v ∈ ZI ≥0 of the quiver variety as an element of Q+ by ZI ≥0 '−→ Q+, v = (vi)i∈I 7→ ∑ i∈I viαi. Let res : Rd → CI be the map defined by res : λ = (λi(z)) 7−→ ( res z=0 λi(z) ) , and for (v, ζ) ∈ Q× CI , let v · ζ := ∑ i∈I viζi be the scalar product. Proposition 3.4. (i) The quiver variety Ns Q,d(λ,v) is a holomorphic symplectic manifold of dimension 2−(v,v) if it is nonempty. (ii) If v · res λ 6= 0, then Nset Q,d(λ,v) = ∅. (iii) If two quivers Q1, Q2 have the same underlying graph, then the associated quiver varieties Ns Q1,d(λ,v), Ns Q2,d(λ,v) are symplectomorphic to each other. Proof. (i) Assume that Ns Q,d(λ,v) is nonempty. Since the action of Gd(V)/C× on the level set µ−1 d (−λ IdV)s is free and proper, the Marsden–Weinstein reduction theorem implies that Ns Q,d(λ,v) is a holomorphic symplectic manifold and dim Ns Q,d(λ,v) = dimMQ,d(V)− 2 dim Gd(V)/C× = ∑ i,j∈I aijdividjvj − 2 ∑ i∈I div 2 i + 2 = tvDADv − 2tvDv + 2 = 2− (v,v). (ii) Assume Nset Q,d(λ,v) 6= ∅ and take a point [B] ∈ Nset Q,d(λ,v). Then we have di∑ k=1 ∑ h∈H: t(h)=i ε(h) trRdi [ BhBhNk−1 i ] z−k = −λi(z) IdV for any i ∈ I. Taking res z=0 ◦ tr of both sides and sum over all i, we obtain∑ h∈H ε(h) tr(BhBh) = − ∑ i∈I vi res z=0 λi(z) = −v · res λ. 10 D. Yamakawa Here the left hand side is zero because∑ h∈H ε(h) tr(BhBh) = ∑ h∈H ε(h) tr(BhBh) = − ∑ h∈H ε(h) tr(BhBh). Hence v · res λ = 0. (iii) By the assumption, we can identify the double quivers Q1 + Q1 and Q2 + Q2. Let H be the set of arrows for them. Then both the sets of arrows Ω1, Ω2 for Q1, Q2 are subsets of H. Now the linear map MQ1,d(V)→MQ2,d(V) = MQ1,d(V) defined by B 7→ B′, B′ h := { Bh if h ∈ Ω1 ∩ Ω2 or h ∈ Ω1 ∩ Ω2, −Bh otherwise, induces a desired symplectomorphism Ns Q1,d(λ,v) '−→ Ns Q2,d(λ,v). � Now fix i ∈ I and set V̂i := ⊕ t(h)=i Vs(h) ⊗C Rds(h) . Then using it we can decompose the vector space MQ,d(V) as MQ,d(V) = Hom(V̂i, Vi ⊗C Rdi )⊕Hom(Vi ⊗C Rdi , V̂i)⊕M(i) Q,d(V), (3.2) where M(i) Q,d(V) := ⊕ t(h),s(h) 6=i Hom(Vs(h) ⊗C Rds(h) , Vt(h) ⊗C Rdt(h) ). According to this decomposition, for a point B ∈MQ,d(V) we put Bi� := ( ε(h)Bh ) t(h)=i ∈ Hom(V̂i, Vi ⊗C Rdi ), B�i := ( Bh ) t(h)=i ∈ Hom(Vi ⊗C Rdi , V̂i), B6=i := ( Bh ) t(h),s(h) 6=i ∈M(i) Q,d(V). We regard these as coordinates for B and write B = (Bi�, B�i, B6=i). Note that the symplectic form can be written as ω = tr dBi� ∧ dB�i + 1 2 ∑ s(h),t(h) 6=i ε(h) tr dBh ∧ dBh, (3.3) and also the i-th component of the moment map can be written as µd,i(B) = pri(Bi�B�i) = di∑ k=1 trRdi [ Bi�B�iN k−1 i ] z−k. Lemma 3.5. Fix i ∈ I and suppose that B satisfies at least one of the following two conditions: (i) B is stable and v 6= αi; (ii) the top coefficient trRdi (Bi�B�iN di−1 i ) of pri(Bi�B�i) is invertible. Then (Bi�, B�i) satisfies Ker B�i ∩Ker Ni = 0, Im Bi� + Im Ni = Vi ⊗C Rdi . (3.4) Quiver Varieties with Multiplicities 11 Proof. First, assume (i) and set S = ⊕ j∈I Sj , Sj := { Ker B�i ∩Ker Ni if j = i, 0 if j 6= i, T = ⊕ j∈I Tj , Tj := { Im Bi� + Im Ni if j = i, Vj ⊗C Rdj if j 6= i. Then both S and T are B-invariant and Nj(Sj) ⊂ Sj , Nj(Tj) ⊂ Tj for all j ∈ I. Since B is stable, we thus have S = 0 or S = Vd, and T = 0 or T = Vd. By the assumption v 6= αi and the definitions of S and T, only the case (S,T) = (0,Vd) occurs. Hence (Bi�, B�i) satisfies (3.4). Next assume (ii). Set A(z) = ∑ Akz −k := pri(Bi�B�i) and Ã :=  0 0 · · · 0 ... ... . . . ... 0 0 · · · 0 Adi Adi−1 · · · A1  ∈ EndC(Vi ⊗C Cdi) = EndC(Vi ⊗C Rdi ). Then we have trRdi (ÃNk−1) = Ak = trRdi (Bi�B�iN k−1), k = 1, 2, . . . , di, i.e., Ã−Bi�B�i ∈ Ker pri. Here, since the group Gdi (Vi) coincides with the centralizer of Ni in GLC(Vi ⊗C Rdi ), we have Ker pri = Im adNi . Hence there is C ∈ EndC(Vi ⊗C Rdi ) such that Bi�B�i = Ã + [Ni, C]. Now note that both KerNi and CokerNi are naturally isomorphic to Vi, and the natural injection ι : KerNi → Vi ⊗C Rdi and projection π : Vi ⊗C Rdi → CokerNi can be respectively identified with the following block matrices: IdV 0 ... 0  : Vi → Vi ⊗C Rdi , ( 0 0 · · · 0 IdV ) : Vi ⊗C Rdi → Vi. Thus we have πBi�B�iι = π(Ã + [Ni, C])ι = πÃι = Adi . (3.5) By the assumption Adi is invertible. Hence πBi� is surjective and B�iι is injective. � The following lemma is a consequence of results obtained in [31]: Lemma 3.6. Suppose that the set Zi := { (Bi�, B�i) ∈ Hom(V̂i, Vi ⊗C Rdi )⊕Hom(Vi ⊗C Rdi , V̂i) \| pri(Bi�B�i) = −λi(z) IdVi , (Bi�, B�i) satisfies (3.4) } is nonempty. Then the quotient of it modulo the action of Gdi (Vi) is a smooth complex manifold having a symplectic structure induced from tr dBi�∧dB�i, and is symplectomorphic to a Gdi (V̂i)- coadjoint orbit via the map given by Φi : (Bi�, B�i) 7−→ −B�i(z −Ni)−1Bi� ∈ g∗di (V̂i). 12 D. Yamakawa Proof. Take any point (Bi�, B�i) in the above set and let O be the Gdi (V̂i)-coadjoint orbit through Φi(Bi�, B�i). By Proposition 4, (a), Theorem 6 and Lemma 3 in [31], there exist • a finite-dimensional C-vector space W ; • a nilpotent endomorphism N ∈ End(W ) with Ndi = 0; • a coadjoint orbit ON ⊂ (Lie GN )∗ of the centralizer GN ⊂ GL(W ) of N , such that the quotient modulo the natural GN -action of the set{ (Y, X) ∈ Hom(V̂i,W )⊕Hom(W, V̂i) ∣∣∣∣∣ prN (Y X) ∈ ON , Ker X ∩Ker N = 0, Im Y + Im N = V̂i } , where prN is the transpose of the inclusion Lie GN ↪→ gl(W ), is a smooth manifold having a symplectic structure induced from tr dX ∧ dY , and is symplectomorphic to O via the map (Y, X) 7→ X(z IdW −N)−1Y . Note that if W = Vi ⊗C Rdi , N = Ni and ON is a single element λi(z) IdVi , then we obtain the result by the coordinate change (Y, X) = (Bi�,−B�i). 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 Zi 6= ∅ implies dim Vi ≤ dim V̂i; indeed, if (Bi�, B�i) ∈ Zi, then B�i\|Ker Ni is injective by condition (3.4), and hence dim Vi = dim Ker Ni = rank (B�i\|Ker Ni) ≤ dim V̂i. The following lemma tells us that if the top coefficient of λi(z) is nonzero, then the converse is true and the corresponding coadjoint orbit can be explicitly described: Lemma 3.7. Suppose dim Vi ≤ dim V̂i and that the top coefficient λi,di of λi(z) is nonzero. Then the set Zi in Lemma 3.6 is nonempty and the coadjoint orbit contains an element of the form Λi(z) = ( λi(z) IdVi 0 0 0 IdV ′ i ) , where Vi is regarded as a subspace of V̂i and V ′ i ⊂ V̂i is a complement of it. Proof. Suppose dim Vi ≤ dim V̂i and that the top coefficient of λi(z) is nonzero. We set Bi� :=  0 0 ... ... 0 0 IdVi 0  : V̂i = Vi ⊕ V ′ i → Vi ⊗C Rdi , B�i := − ( λi,di IdVi · · · λi,1 IdVi 0 · · · 0 ) : Vi ⊗C Rdi → V̂i, where λi,k denotes the coefficient in λi(z) of z−k. Then we have trRdi (Bi�B�iN k−1 i ) = −λi,k IdVi , k = 1, 2, . . . , di, i.e., pri(Bi�B�i) = −λi(z) IdVi . The assumption λi,di 6= 0 and Lemma 3.5 imply that (Bi�, B�i) satisfies (3.4). Hence (Bi�, B�i) ∈ Zi. Moreover we have Φi(Bi�, B�i) = −B�i(z −Ni)−1Bi� = − ∑ k=1 B�iN k−1 i Bi� z−k = ∑ k=1 ( λi,k IdVi 0 0 0 IdV ′ i ) z−k = Λi(z). � Quiver Varieties with Multiplicities 13 4 Reflection functor In this section we construct reflection functors for quiver varieties with multiplicities. 4.1 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 si(β) := β − 〈β, α∨i 〉αi = β − 2(β, αi) (αi, αi) αi, i ∈ I, β ∈ h∗. The fundamental relations for the generators si, i ∈ I are s2 i = Id, (sisj)mij = Id, i, j ∈ I, i 6= j, (4.1) where the numbers mij are determined from cijcji as the table below (we use the convention r∞ = Id for any r) cijcji 0 1 2 3 ≥ 4 mij 2 3 4 6 ∞ We will define a W (C)-action on the parameter space Rd × 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, si : v = ∑ i∈I viαi 7−→ v − 〈v, α∨i 〉αi = v − ∑ j∈I cijvjαi. The action on the first component Rd is unusual. We define ri ∈ GL(Rd) by ri(λ) = λ′ ≡ (λ′j(z)), λ′j(z) := { −λi(z) if j = i, λj(z)− z−1cij res z=0 λi(z) if j 6= i. Lemma 4.1. The above ri, i ∈ I satisfy relations (4.1). Proof. The relations r2 i = Id, i ∈ I are obvious. To check the relation (rirj)mij = Id for i 6= j, first note that the transpose of si : Q→ Q relative to the scalar product is given by tsi : CI → CI , tsi(ζ) = ζ − ζi ∑ j∈I cijαj . Now let λ ∈ Rd. We decompose it as λ = λ0 + res(λ) z−1, res λ0 = 0. Then we easily see that ri(res(λ) z−1) = tsi(res(λ)) z−1, and hence that (rirj)mij (λ) = (rirj)mij (λ0) + res(λ) z−1. 14 D. Yamakawa Therefore we may assume that res λ = 0. Set λ′ ≡ (λ′k(z)) := (rirj)mij (λ). Then we have λ′k(z) = { (−1)mijλk(z) if k = i, j, λk(z) if k 6= i, j. If mij is odd, by the definition we have cijcji = 1. In particular, i 6= j and aijdjajidi = cijcji = 1. This implies di = dj = 1 and hence that λi(z) = λj(z) = 0. � The main result of this section is as follows: Theorem 4.2. Let λ = (λi(z)) ∈ Rd and suppose that the top coefficient λi,di of λi(z) for fixed i ∈ I is nonzero. Then there exists a bijection Fi : Nset Q,d(λ,v) '−→ Nset Q,d(ri(λ), si(v)) such that F2 i = Id and the restriction gives a symplectomorphism Fi : Ns Q,d(λ,v) '−→ Ns Q,d(ri(λ), si(v)). We call the above map Fi the i-th reflection functor. 4.2 Proof of the main theorem Fix i ∈ I and suppose that the top coefficient λi,di of λi(z) is nonzero. Recall the decomposi- tion (3.2) of MQ,d(V): MQ,d(V) = Hom(V̂i, Vi ⊗C Rdi )⊕Hom(Vi ⊗C Rdi , V̂i)⊕M(i) Q,d(V), and the set Zi given in Lemma 3.6. Lemma 3.5 and the assumption λi,di 6= 0 imply that any B = (Bi�, B�i, B6=i) ∈ µ−1 d,i(−λi(z) IdVi) satisfies condition (3.4). Thus we have µ−1 d,i(−λi(z) IdVi) = Zi ×M(i) Q,d(V). By Lemma 3.7, it is nonempty if and only if vi ≤ dim V̂i = ∑ j aijdjvj = 2vi − ∑ j cijvj , i.e., the i-th component of si(v) is non-negative. We assume this condition, because otherwise both Nset Q,d(λ,v) and Nset Q,d(ri(λ), si(v)) are empty (since si(v) /∈ ZI ≥0). Fix a C-vector space V ′ i of dimension dim V̂i − dim Vi and an identification V̂i = Vi ⊕ V ′ i . As the group Gdi (Vi) acts trivially on M(i) Q,d(V), Lemmas 3.6 and 3.7 imply that µ−1 d,i(−λi(z) IdVi)/Gdi (Vi) = Zi/Gdi (Vi)×M(i) Q,d(V) ' O ×M(i) Q,d(V), where O is the Gdi (V̂i)-coadjoint orbit through Λ(z) = ( λi(z) IdVi 0 0 0 IdV ′ i ) . Quiver Varieties with Multiplicities 15 Now let us define an I-graded C-vector space V′ with dimV′ = si(v) by V′ = ⊕ j∈I V ′ j , V ′ j := { V ′ i if j = i, Vj if j 6= i, and consider the associated symplectic vector space MQ,d(V′). Note that V̂ ′ i = V̂i. Thus by interchanging the roles of V and V′, λi and −λi in Lemmas 3.6 and 3.7, we obtain an isomorphism µ−1 d,i(λi(z) IdV ′ i )/Gdi (V ′ i ) ' O′ ×M(i) Q,d(V′) = O′ ×M(i) Q,d(V), where O′ is the Gdi (V̂i)-coadjoint orbit through( 0 IdVi 0 0 −λi(z) IdV ′ i ) = Λ(z)− λi(z) Id V̂i , i.e., O′ = O − λi(z) Id V̂i . Hence the scalar shift O '−→ O − λi(z) Id V̂i induces an isomorphism F̃i : µ−1 d,i(−λi(z) IdVi)/Gdi (Vi) '−→ µ−1 d,i(λi(z) IdV ′ i )/Gdi (V ′ i ), which is characterized as follows: if F̃i[B] = [B′], B = (Bi�, B�i, B6=i), B′ = (B′ i�, B′ �i, B ′ 6=i), one has B6=i = B′ 6=i, (4.2) −B′ �i(z −N ′ i) −1B′ i� = −B�i(z −Ni)−1Bi� − λi(z) Id V̂i , (4.3) where N ′ i := IdV ′ i ⊗C Jdi ∈ EndC(V ′ i ⊗C Rdi ). Note that Ker B′ �i ∩Ker N ′ i = 0, Im B′ i� + Im N ′ i = V ′ i ⊗C Rdi (4.4) by Lemma 3.5. Lemma 4.3. If µd(B) = −λ IdV, then µd(B′) = −ri(λ) IdV′. Proof. Let λ′ = (λ′j(z)) := ri(λ). The identity µd,i(B′) = λi(z) IdV ′ i is clear from the construc- tion. We check µd,j(B′) = −λ′j(z) IdV ′ j for j 6= i. Taking the residue of both sides of (4.3), we have B′ �iB ′ i� = B�iBi� + λi,1 Id V̂i , which implies that ε(h)B′ h B′ h = ε(h)BhBh + λi,1 IdVs(h) if t(h) = i. On the other hand, (4.2) means that B′ h = Bh whenever t(h), s(h) 6= i. Thus for j 6= i, we obtain∑ t(h)=j ε(h)B′ hB′ h = ∑ h : i→j ε(h)B′ hB′ h + ∑ t(h)=j,s(h) 6=i ε(h)B′ hB′ h = ∑ h : i→j ( ε(h)BhBh − λi,1 IdVj ) + ∑ t(h)=j,s(h) 6=i ε(h)BhBh 16 D. Yamakawa = ∑ t(h)=j ε(h)BhBh − aijλi,1 IdVj . (4.5) Note that prj(IdVj ) = dj∑ k=1 trRdj (Nk−1 j )z−k = djIdVjz −1. Therefore the image under prj of both sides of (4.5) gives µd,j(B′) = µd,j(B)− aijλi,1 pri(IdVj ) = µd,j(B) + cijλi,1 IdVjz −1. The result follows. � Lemma 4.4. If B is stable, then so is B′. Proof. Suppose that there exists a B′-invariant subspace S′ = ⊕ j S′j⊂V′ d such that N ′ j(S ′ j)⊂S′j . We define an I-graded subspace S = ⊕ j Sj of Vd by Sj :=  di∑ k=1 Nk−1 i Bi� ( Ŝ′i ) if j = i, S′j if j 6= i, where Ŝ′i := ⊕ t(h)=i S ′ s(h) = Ŝi. Then Bi� ( Ŝi ) ⊂ Si and B�i(Si) = di∑ k=1 B�iN k−1 i Bi� ( Ŝ′i ) = di∑ k=1 ( B′ �i(N ′ i) k−1B′ i� − λi,k )( Ŝ′i ) ⊂ Ŝ′i = Ŝi. Hence S is B-invariant. Clearly Nj(Sj) ⊂ Sj for all j ∈ I. Therefore the stability condition for B implies that S = 0 or S = Vd. First, assume S = 0. Then S′j = Sj = 0 for j 6= i, and hence B′ �i(S ′ i) ⊂ Ŝ′i = 0, i.e., S′i ⊂ Ker B′ �i. If S′i is nonzero, then the kernel of the restriction N ′ i \|S′i is nonzero because it is nilpotent. However it implies KerB′ �i ∩ Ker N ′ i 6= 0, which contradicts to (4.4). Hence S′i = 0. Next assume S = Vd. Then S′j = Sj = Vj ⊗C Rdj for j 6= i, and hence S′i ⊃ B′ i� ( Ŝ′i ) = Im B′ i�. If V ′ i /S′i is nonzero, then the endomorphism of V ′ i /S′i induced from N ′ i has a nonzero cokernel because it is nilpotent. However it implies Im B′ i� + Im N ′ i 6= V ′ i ⊗C Rdi , which contradicts to (4.4). Hence S′i = V ′ i ⊗C Rdi . � Proof of Theorem 4.2. As the map F̃i is clearly ∏ j 6=i Gdj (Vj)-equivariant, Lemma 4.3 implies that it induces a bijection Fi : Nset Q,d(λ,v)→ Nset Q,d(ri(λ), si(v)), [B] 7→ [B′], which preserves the stability by Lemma 4.4. We easily obtain the relation F2 i = Id by noting that Fi is induced from the scalar shift O → O′ = O−λi(z) Id V̂i and the i-th component of ri(λ) is −λi. Consider the restriction Fi : Ns Q,d(λ,v)→ Ns Q,d(ri(λ), si(v)), [B] 7→ [B′]. By Lemma 3.6 and (4.3), we have tr dBi� ∧ dB�i = tr dB′ i� ∧ dB′ �i, because the scalar shift O → O′ is a symplectomorphism. Substituting it and (4.2) into (3.3), we see that the above map Fi is a symplectomorphism. � Quiver Varieties with Multiplicities 17 Remark 4.5. It is clear from (4.2), (4.3) and (4.4) that if dj = 1 for all j ∈ I, then Fi coincides with the original i-th reflection functor for quiver varieties (see conditions (a), (b1) and (c) in [18, Section 3]). Remark 4.6. It is known (see e.g. [19]) that if di = 1 for all i ∈ I, then the reflection functors Fi satisfy relations (4.1). We expect that this fact is true for any (Q,d). 4.3 Application In this subsection we introduce a basic application of reflection functors. Lemma 4.7. Let (λ,v) ∈ Rd ×Q+, i ∈ I. Suppose that the top coefficient of λi(z) is zero and v 6= αi. Then Ns Q,d(λ,v) 6= ∅ implies (v, αi) ≤ 0. Proof. Take any point [B] ∈ Ns Q,d(λ,v). Let ι : KerNi → Vi ⊗C Rdi be the inclusion and π : Vi ⊗C Rdi → CokerNi be the projection. Then Lemma 3.5 together with the assumption v 6= αi implies that B�iι is injective and πBi� is surjective. On the other hand, (3.5) and the assumption for λi(z) imply that Vi ' Ker Ni B�iι // V̂i πBi� // CokerNi ' Vi is a complex. Thus we have 0 ≤ dim V̂i − 2 dim Vi = ∑ j aijdjvj − 2vi, which is equivalent to (v, αi) ≤ 0. � Now applying Crawley-Boevey’s argument in [6, Lemma 7.3] to our quiver varieties with multiplicities, we obtain the following: Proposition 4.8. If Ns Q,d(λ,v) 6= ∅, then v is a positive root of g(C). Proof. Assume Ns Q,d(λ,v) 6= ∅ 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), αi) ≤ 0 for any i ∈ I. Assume that there is i ∈ I such that (v, αi) > 0. The above lemma implies that the top coeffi- cient of λi(z) is nonzero, which together with Theorem 4.2 implies that Ns Q,d(ri(λ), si(v)) 6= ∅. In particular we have si(v) ∈ Q+, and further v− si(v) ∈ Z>0αi by the assumption (v, αi) > 0. We then replace (λ,v) with (ri(λ), si(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) ∈ Rd ×Q+ such that (v, αi) ≤ 0 for all i ∈ I. Additionally, the property Ns Q,d(λ,v) 6= ∅ clearly implies that the support of v is connected. The result follows. � 5 Normalization In this section we give an application of Boalch’s ‘shifting trick’ to quiver varieties with multi- plicities. 18 D. Yamakawa 5.1 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 dj = 1, aik = aki = δjk for any k ∈ I. The vertex j is called the base vertex for the pole i. If furthermore di > 1, the pole i ∈ I is said to be irregular. Let i ∈ I be a pole vertex with the base j ∈ I. Then V̂i = Vj ⊗C Rdj = Vj . In what follows we assume that the top coefficient of λi(z) is nonzero. As the set Nset Q,d(λ,v) is empty unless dim Vi ≤ dim V̂i = dim Vj , we also assume Vi ⊂ Vj and fix an identification Vj ' Vi ⊕ Vj/Vi. Recall the isomorphism given in the previous section: µ−1 d,i(−λi(z) IdVi)/Gdi (Vi) ' O ×M(i) Q,d(V), where O is the Gdi (Vj)-coadjoint orbit through the element of the form Λ(z) = ( λi(z) IdVi 0 0 0 IdVj/Vi ) . Let us decompose Λ(z) as Λ(z) = Λ0(z) + z−1 res z=0 Λ(z) according to the decomposition g∗di (Vj) = b∗di (Vj)⊕ z−1 gl(Vj), where b∗di (Vj) := Ker [ res z=0 : g∗di (Vj)→ gl(Vj) ] ' z−di gl(Vj)[[z]]/z−1 gl(Vj)[[z]]. The above is naturally dual to the Lie algebra bdi (Vj) of the unipotent subgroup Bdi (Vj) := { g(z) ∈ Gdi (Vj) \| g(0) = IdVj } . The coadjoint action of g(z) ∈ Bdi (Vj) is given by (g · η)(z) = g(z)η(z)g(z)−1 mod z−1 gl(Vj)[[z]], η(z) = dj∑ k=2 ηkz −k ∈ b∗di (Vj). Now consider the Bdi (Vj)-coadjoint orbit Ǒ through Λ0(z). Let K := GL(Vi)×GL(Vj/Vi) ⊂ GL(Vj) be the Levi subgroup associated to the decomposition Vj = Vi ⊕ Vj/Vi. The results in this section is based on the following two facts: Lemma 5.2. The orbit Ǒ is invariant under the conjugation action by K, and there exists a K-equivariant algebraic symplectomorphism Ǒ ' Hom(Vj/Vi, Vi)⊕(di−2) ⊕Hom(Vi, Vj/Vi)⊕(di−2) sending Λ0(z) ∈ Ǒ to the origin. Quiver Varieties with Multiplicities 19 Lemma 5.3. Let M be a holomorphic symplectic manifold with a Hamiltonian action of GL(Vj) and a moment map µM : M → gl(Vj). Then for any ζ ∈ C, the map Ǒ ×M → g∗di (Vj)×M, (B(z), x) 7→ ( B(z)− z−1µM (x)− z−1ζ IdVj , x ) induces a bijection between (i) the (set-theoretical) symplectic quotient of Ǒ ×M by the diagonal K-action at the level − res z=0 Λ(z)− ζ IdVj ; and (ii) that of O ×M by the diagonal GL(Vj)-action at the level −ζ IdVj . 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(Vj)-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 Λ1,Λ2, . . . ,Λk ∈ End(V ) be mutually commuting endomorphisms of a C- vector space V , and suppose that Λ2, . . . ,Λk are semisimple. To such endomorphisms we asso- ciate Λ(z) := k∑ j=1 Λjz−j ∈ g∗k(V ), 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 λjz−j with (λ2, . . . , λk) being a simultaneous eigenvalue of (Λ2, . . . ,Λk), and let V = ⊕ λ∈Σ Vλ be the eigenspace decomposition. Then we can express Λ(z) as Λ(z) = ⊕ λ∈Σ ( λ(z) IdVλ + Γλ z ) , Γλ = Λ1\|Vλ ∈ End(Vλ). 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. We apply Lemma 5.3 to the case where M = M(i) Q,d(V), ζ = res z=0 λj(z). In this case, the symp- lectic quotient of the space (ii) by the action of ∏ k 6=i,j Gdk (Vk) turns out to be µ−1 d (−λ IdV)/Gd(V) = Nset Q,d(λ,v). On the other hand, by Lemma 5.2, the symplectic quotient of the space (i) by the action of ∏ k 6=i,j Gdk (Vk) coincides with the symplectic quotient of Hom(Vj/Vi, Vi)⊕(di−2) ⊕Hom(Vi, Vj/Vi)⊕(di−2) ⊕M(i) Q,d(V) (5.1) by the action of GL(Vi)×GL(Vj/Vi)× ∏ k 6=i,j Gdk (Vk), (5.2) at the level given by − ( res z=0 ( λi(z) + λj(z) ) , res z=0 λj(z), (λk(z))k 6=i,j ) . (5.3) 20 D. Yamakawa 5.2 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 define ď = (ďk) ∈ ZI >0 by ďi := 1, ďk := dk for k 6= i, and let Q̌ = (I, Ω̌, 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 di − 2 arrows from j to i. The transformation (Q,d) 7→ (Q̌, ď) is called the normalization at i. The adjacency matrix Ǎ = (ǎkl) of the underlying graph of Q̌ satisfies ǎkl = ǎlk =  di − 2 if (k, l) = (i, j), ajl if k = i, l 6= j, akl if k, l 6= i. Example 5.6. (i) Suppose that (Q,d) has the graph with multiplicities given below d�� 1�� Here we assume d > 1. The left vertex is an irregular pole, at which we can perform the normalization and the resulting (Q̌, ď) has the underlying graph with multiplicities drawn below 1�� 1�� d−2 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 G2, while the one associated to (Q̌, ď) is of type A2. If d = 4, the Kac–Moody algebra associated to (Q,d) is of type A (2) 2 , while the one associated to (Q̌, ď) is of type A (1) 1 . (ii) Suppose that (Q,d) has the graph with multiplicities given below d�� 1�� 1�� · · · 1�� 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 (Q̌, ď) has the underlying graph with multiplicities drawn below 1�� 1 ��MMMMMMMMM 1 ��qqqqqqqqqd−2 · · · 1�� Quiver Varieties with Multiplicities 21 If d = 2, then the Kac–Moody algebra associated to (Q,d) is of type Cn, while the one associated to (Q̌, ď) is of type A3 if n = 3 and of type Dn if n > 3. If (d, n) = (3, 3), the Kac–Moody algebra associated to (Q,d) is of type D (3) 4 , while the one associated to (Q̌, ď) is of type A (1) 2 . (iii) Suppose that (Q,d) has the graph with multiplicities given below 2�� 1�� 1�� · · · 1�� 2�� Here the number of vertices is n ≥ 3. The associated Kac–Moody algebra is of type C (1) n−1. It has two irregular poles. Let us perform the normalization at the vertex on the far right. If n = 3, the resulting (Q̌, ď) has the underlying graph with multiplicities drawn below 2 �� 1��qqqqqqqqq 1��MMMMMMMMM = 1�� 2�� 1�� The associated Kac–Moody algebra is of type D (2) 3 . If n ≥ 4, the resulting (Q̌, ď) has the underlying graph with multiplicities drawn below 2�� 1�� · · · 1�� 1�� 1��22 22 2 The associated Kac–Moody algebra is of type A (2) 2n−3. The vertex on the far left is still an irregular pole, at which we can perform the normalization again. If n = 4, the resulting (Q̌, ď) has the underlying graph with multiplicities drawn below 1��1 �� 1��1 ��????????????? �� = 1��1 �� 1��1 �� The associated Kac–Moody algebra is of type A (1) 3 . If n > 4, the resulting (Q̌, ď) has the underlying graph with multiplicities drawn below 1 �� 1��22 22 2 1 �� · · · 1�� 1�� 1��22 22 2 The associated Kac–Moody algebra is of type D (1) n−1. In the situation discussed in the previous subsection, let V̌ = ⊕ k V̌k be the I-graded vector space defined by V̌j := Vj/Vi, V̌k := Vk for k 6= j. Then we see that the group in (5.2) coincides with Gď(V̌). Furthermore, the following holds: 22 D. Yamakawa Lemma 5.7. The symplectic vector space in (5.1) coincides with MQ̌,ď(V̌). Proof. The definitions of Q̌, ď, V̌ imply RepQ̌(V̌ď) = ⊕ h∈Ω̌ h : j→i Hom(Vj/Vi, Vi) ⊕ ⊕ k 6=i,j  ⊕ h∈Ω̌ h : k→i Hom(Vk ⊗C Rdk , Vi)⊕ ⊕ h∈Ω̌ h : i→k Hom(Vi, Vk ⊗C Rdk )  ⊕ ⊕ k 6=i,j  ⊕ h∈Ω̌ h : k→j Hom(Vk ⊗C Rdk , Vj/Vi)⊕ ⊕ h∈Ω̌ h : j→k Hom(Vj/Vi, Vk ⊗C Rdk )  ⊕ ⊕ k,l 6=i,j ⊕ h∈Ω̌ h : k→l Hom(Vk ⊗C Rdk , Vl ⊗C Rdl ) = Hom(Vj/Vi, Vi)⊕(di−2) ⊕ ⊕ k 6=i,j  ⊕ h∈Ω h : k→j Hom(Vk ⊗C Rdk , Vj)⊕ ⊕ h∈Ω h : j→k Hom(Vj , Vk ⊗C Rdk )  ⊕ ⊕ k,l 6=i,j ⊕ h∈Ω h : k→l Hom(Vk ⊗C Rdk , Vl ⊗C Rdl ) = Hom(Vj/Vi, Vi)⊕(di−2) ⊕ ⊕ h∈Ω s(h),t(h) 6=i Hom(Vs(h) ⊗C Rds(h) , Vt(h) ⊗C Rdt(h) ). Taking the cotangent bundle, we thus see that MQ̌,ď(V̌) coincides with (5.1). � Set v̌ := dim V̌ and λ̌ = (λ̌k(z)) ∈ Rď, λ̌k(z) :=  z−1 res z=0 ( λi(z) + λj(z) ) if k = i, z−1 res z=0 λj(z) if k = j, λk(z) if k 6= i, j. Then the value given in (5.3) coincides with −λ̌. Note that v̌ · res λ̌ = vi res z=0 ( λi(z) + λj(z) ) + (vj − vi) res z=0 λj(z) + ∑ k 6=i,j vk res z=0 λk(z) = v · res λ. (5.4) 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 (Q̌, ď) be the quiver with multiplicities obtained by the normalization of (Q,d) at i. Take (λ,v) ∈ Rd × Q+ such that the top coefficient λi,di of λi(z) is nonzero. Then the quiver varieties Ns Q,d(λ,v) and Ns Q̌,ď (λ̌, v̌) are symplectomorphic to each other. Proof. We have already constructed a bijection between Nset Q,d(λ,v) and Nset Q̌,ď (λ̌, v̌). Thanks to Lemma 5.3, in order to prove the assertion it is sufficient to check that the bijection maps Ns Q,d(λ,v) onto Ns Q̌,ď (λ̌, v̌). It immediately follows from the three lemmas below. � Quiver Varieties with Multiplicities 23 Lemma 5.9. A point B ∈ µ−1 d (−λ Idv) is stable if and only if the corresponding (A(z), B6=i) =( di∑ l=1 Alz −l, B6=i ) ∈ O ×M(i) Q,d(V) satisfies the following condition: if a collection of subspaces Sk ⊂ Vk ⊗C Rdk , k 6= i satisfies Nk(Sk) ⊂ Sk for k 6= i, j; Bh(Ss(h)) ⊂ St(h) for h ∈ H with t(h), s(h) 6= i; (5.5) Al(Sj) ⊂ Sj for l = 1, . . . , di, then Sk = 0 (k 6= i) or Sk = Vk ⊗C Rdk (k 6= i). Proof. This is similar to Lemma 4.4. First, assume that B is stable and that a collection of subspaces Sk ⊂ Vk ⊗C Rdk , k 6= i satisfies (5.5). We define Si := di∑ l=1 N l−1 i Bi�(Sj), and set S := ⊕ k∈I Sk ⊂ Vd. Then Ni(Si) ⊂ Si, Bi�(Sj) ⊂ Si and B�i(Si) = ∑ l B�iN l−1 i Bi�(Sj) = ∑ l Al(Sj) ⊂ Sj imply that S is B-invariant. Since B is stable, we thus have S = 0 or S = Vd. Next assume that the pair (A(z), B6=i) satisfies the condition in the statement. Let S = ⊕ k Sk be a B-invariant subspace of Vd satisfying Nk(Sk) ⊂ Sk for all k ∈ I. Then clearly the collection Sk, k 6= i satisfies (5.5), and hence Sk = 0 (k 6= i) or Sk = Vk ⊗C Rdk (k 6= i). If Sk = 0, k 6= i, we have B�i(Si) = 0, which implies Si = 0 since KerB�i ∩ Ker Ni = 0 by Lemma 3.5 and Ni\|Si is nilpotent. Dualizing the argument, we easily see that Si = Vi ⊗C Rdi if Sk = Vk ⊗C Rdk , k 6= i. � Lemma 5.10. A point B′ ∈ µ−1 ď (−λ̌ Idv̌) is stable if and only if the corresponding (A0(z), B6=i) =( di∑ l=2 A0 l z −l, B6=i ) ∈ Ǒ ×M(i) Q,d(V) satisfies the following condition: if an I-graded subspace S = ⊕ k Sk of V̌ď = V̌ ⊗C Rď satisfies Nk(Sk) ⊂ Sk for k 6= i, j; Bh(Ss(h)) ⊂ St(h) for h ∈ H with (t(h), s(h)) 6= (i, j), (j, i); (5.6) A0 l (Si ⊕ Sj) ⊂ Si ⊕ Sj for l = 2, . . . , di, then S = 0 or S = V̌ď. Proof. In Appendix A, we show that all the block components of A0 l relative to the decompo- sition Vj = V̌i ⊕ V̌j 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 A0 is denoted by B and B′ h for such h are denoted by a′k, b′k). Hence an I-graded subspace S of V̌ď satisfies (5.6) if and only if it is B′-invariant and Nk(Sk) ⊂ Sk for k 6= i, j. � Lemma 5.11. Let (A0(z), B6=i) ∈ Ǒ ×M(i) Q,d(V) and let (A(z), B6=i) ∈ O ×M(i) Q,d(V) be the corresponding pair under the map given in Lemma 5.3. Then (A0(z), B6=i) satisfies the condition in Lemma 5.10 if and only if (A(z), B6=i) satisfies the one in Lemma 5.9. 24 D. Yamakawa Proof. By definition we have A(z) = A0(z)− z−1 ∑ t(h)=j, s(h) 6=i ε(h)BhBh − λj(z) IdVj , so the ‘if’ part is clear. To prove the ‘only if’ part, note that if a collection of subspaces Sk ⊂ Vk ⊗C Rdk , k 6= i satisfies (5.5), then in particular Sj is preserved by the action of Adi = A0 di = λi,di IdV̌i ⊕ 0 IdV̌j , and hence is homogeneous relative to the decomposition Vj = V̌i ⊕ V̌j ; Sj = (Sj ∩ V̌i)⊕ (Sj ∩ V̌j). Now the result immediately follows. � 5.3 Weyl groups Let (Q,d) be a quiver with multiplicities having an irregular pole vertex i ∈ I with base j ∈ I, and let (Q̌, 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 (Q̌, 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, αk, sk, 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 by Č, Ď, ȟ, Q̌, α̌k, šk, the similar objects associated to (Q̌, ď). Let ϕ : Q→ Q̌ be the linear map defined by v 7→ v̌ = v− viα̌j . The same letter is also used on the matrix representing ϕ with respect to the simple roots. Lemma 5.12. The identity tϕĎČϕ = DC holds. Proof. To prove it, we express the matrices in block form with respect to the decomposition of the index set I = {i} t {j} t (I \ {i, j}). First, ϕ is expressed as ϕ =  1 0 0 −1 1 0 0 0 Id  . By the properties of i and j, the matrices D and A are respectively expressed as D = di 0 0 0 1 0 0 0 D′  , A = 0 1 0 1 0 ta 0 a A′  , where D′ (resp. A′) is the sub-matrix of D (resp. A) obtained by restricting the index set to I \ {i, j}, and a = (akj)k 6=i,j . By the definition of the normalization, the matrices Ď and Ǎ are then respectively expressed as Ď = 1 0 0 0 1 0 0 0 D′  , Ǎ =  0 di − 2 ta di − 2 0 ta a a A′  . Now we check the identity. We have DC = 2D−DAD = 2di 0 0 0 2 0 0 0 2D′ −  0 di 0 di 0 taD′ 0 D′a D′A′D′  Quiver Varieties with Multiplicities 25 = 2di −di 0 −di 2 −taD′ 0 −D′a 2Id−D′A′D′  . On the other hand, ĎČ = 2Ď− ĎǍĎ = 2 0 0 0 2 0 0 0 2D′ −  0 di − 2 taD′ di − 2 0 taD′ D′a D′a D′A′D′  =  2 2− di −taD′ 2− di 2 −taD′ −D′a −D′a 2Id−D′A′D′  . Hence tϕĎČϕ = 1 −1 0 0 1 0 0 0 Id  2 2− di −taD′ 2− di 2 −taD′ −D′a −D′a 2Id−D′A′D′  1 0 0 −1 1 0 0 0 Id  =  di −di 0 2− di 2 −taD′ −D′a −D′a 2Id−D′A′D′  1 0 0 −1 1 0 0 0 Id  = 2di −di 0 −di 2 −taD′ 0 −D′a 2Id−D′A′D′  = DC. � The above lemma implies that the map ϕ preserves the inner product. Furthermore it also implies rank Č = rankC, which means dim h = dim ȟ. Thus we can extend ϕ to an isomorphism ϕ̃ : 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 matrix Č. Hence it defines an involution of W (Č), or equivalently, a homomorphism Z/2Z→ Aut(W (Č)). Proposition 5.13. Under the isomorphism ϕ̃, the Weyl group W (C) associated to C is iso- morphic to the semidirect product W (Č) o Z/2Z of the one associated to Č and Z/2Z by the permutation σ. Proof. By the construction of ϕ̃ we have ϕ̃(αk) = { α̌i − α̌j if k = i, α̌k if k 6= i. As ϕ̃ preserves the inner product, the above implies that for k 6= i, the map ϕ̃skϕ̃ −1 coincides with the reflection šk relative to ϕ̃(αk) = α̌k, and the map ϕ̃siϕ̃ −1 coincides with the reflection relative to ϕ̃(αi) = α̌i−α̌j . Note that since the matrix ĎČ is invariant under the permutation σ, we have (α̌i + α̌j , α̌i − α̌j) = 0, (α̌k, α̌i − α̌j) = 0, k 6= i, j, which imply that ϕ̃siϕ̃ −1(α̌k) = α̌σ(k) for any k ∈ I. Hence the map (ϕ̃siϕ̃ −1)šk(ϕ̃siϕ̃ −1)−1, which is the reflection relative to ϕ̃siϕ̃ −1(α̌k), coincides with šσ(k) for each k. Now the result immediately follows. � 26 D. Yamakawa We can easily check that res(λ̌) = tϕ−1 ( res(λ) ) for λ ∈ Rd. Note that the action of W (Č) on Rď naturally extends to an action of W (Č) o Z/2Z. We see from the above relation that the map Rd → Rď, λ 7→ λ̌ is equivariant, and hence so is the map Rd × Q → Rď × Q̌, (λ,v) 7→ (λ̌, v̌), with respect to the isomorphism W (C) ' W (Č) o Z/2Z given in Proposition 5.13. 6 Naive moduli of meromorphic connections on P1 This final section is devoted to study moduli spaces of meromorphic connections on the trivial bundle over P1 with some particular type of singularities. 6.1 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 cor- responding ‘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 k1, k2, . . . , kn; • mutually distinct points t1, t2, . . . , tn in C. Then consider a system du dz = A(z) u(z), A(z) = n∑ i=1 ki∑ j=1 Ai,j (z − ti)j , Ai,j ∈ End(V ) of linear ordinary differential equations with rational coefficients. It has a pole at ti of order at most ki for each i, and (possibly) a simple pole at ∞ with residue − ∑ i Ai,1. We identify such a system with its coefficient matrix A(z), which may be regarded as an element of ⊕ i g ∗ ki (V ) via A(z) 7→ (Ai), Ai(z) := ∑ k Ai,jz −j . After Boalch [2], we introduce the following (the terminologies used here are different from his): Definition 6.1. For a system A(z) = (Ai) ∈ ⊕ i g ∗ ki (V ) and each i = 1, . . . , n, the Gki (V )- coadjoint orbit through Ai is called the truncated formal type of A(z) at ti. For given coadjoint orbits Oi ⊂ g∗ki (V ), i = 1, . . . , n, the set Mset(O1, . . . ,On) := { A(z) ∈ n∏ i=1 Oi ∣∣∣∣∣ n∑ i=1 res z=ti A(z) = 0 } / GL(V ) is called the naive moduli space of systems having a pole of truncated formal type Oi at each ti, i = 1, . . . , n. Note that ∏ iOi is a holomorphic symplectic manifold, and the map n∏ i=1 Oi → n⊕ i=1 g∗ki (V ), A(z) 7→ n∑ i=1 res z=ti A(z) Quiver Varieties with Multiplicities 27 is a moment map with respect to the simultaneous GL(V )-conjugation action. Hence the set Mset(O1, . . . ,On) is a set-theoretical symplectic quotient. It is also useful to introduce the following ζ-twisted naive moduli space: Mset ζ (O1, . . . ,On) := { A(z) ∈ n∏ i=1 Oi ∣∣∣∣∣ n∑ i=1 res z=ti A(z) = −ζ IdV } / GL(V ) (ζ ∈ C). Definition 6.2. A system A(z) ∈ ⊕ i g ∗ ki (V ) is said to be irreducible if there is no nonzero proper subspace S ⊂ V preserved by all the coefficient matrices Ai,j . If A(z) ∈ ⊕ i g ∗ ki (V ) 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 ∏ iOi is proper. Definition 6.3. For ζ ∈ C, the holomorphic symplectic manifold Mirr ζ (O1, . . . ,On) := A(z) ∈ n∏ i=1 Oi ∣∣∣∣∣∣ A(z) is irreducible,∑ i res z=ti A(z) = −ζ IdV  / GL(V ) is called the ζ-twisted naive moduli space of irreducible systems having a pole of truncated formal type Oi at each ti, i = 1, . . . , n. In the 0-twisted (untwisted) case, we simply write Mirr 0 (O1, . . . ,On) ≡Mirr(O1, . . . ,On). If we have a specific element Λi(z) ∈ Oi for each i, the following notation is also useful: Mset ζ (Λ1, . . . ,Λn) ≡Mset ζ (O1, . . . ,On), Mirr ζ (Λ1, . . . ,Λn) ≡Mirr ζ (O1, . . . ,On). 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 P1 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 ∗ ki (V ) is irreducible if and only if the associated vector bundle with meromorphic connection (P1 × 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 Γλ is non-resonant, i.e., no two distinct eigenvalues of Γλ 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 g(z) ∈ AutC[[z]](C[[z]] ⊗ V ) which makes d − A(z) dz into d−Λ(z) dz (see [31, Remark 18]). In this sense the truncated formal type of Λ(z) actually prescribe a formal type. Hence, if each Oi ⊂ g∗ki (V ) contains some normal form with non- resonant residue parts, then the naive moduli spaceMset(O1, . . . ,On) gives the moduli space of meromorphic connections on the trivial bundle P1 × V having a pole of prescribed formal type at each ti. 6.2 Star-shaped quivers of length one In some special case, the naive moduli space Mirr(O1, . . . ,On) can be described as a quiver variety. Suppose that for each i = 1, . . . , n, the coadjoint orbit Oi contains an element of the form Ξi(z) = ( ξi(z) IdVi 0 0 ηi(z) IdV ′ i ) 28 D. Yamakawa for some vector space decomposition V = Vi ⊕ V ′ i and distinct ξi, ηi ∈ g∗ki (C). Let di be the pole order of λi := ξi − ηi. Note that Ξi is a particular example of normal forms introduced in Remark 5.4, and it has non-resonant residue parts (see Remark 6.5) if and only if di > 1 or res z=0 (ξi − ηi) /∈ Z. Also, note that n∑ i=1 tr res z=0 Ξi(z) = 0 is a necessary condition for the non- emptiness of Mset(Ξ1, . . . ,Ξn) = Mset(O1, . . . ,On). Indeed, if some A(z) gives a point in Mset(Ξ1, . . . ,Ξn), then 0 = n∑ i=1 tr res z=ti A(z) = n∑ i=1 tr res z=0 Ξi(z), since the function tr ◦ resz=0 : g∗ki (V )→ C is invariant under the coadjoint action for each i. Set I := { 0, 1, . . . , n } and let Q = (I, Ω, s, t) be the ‘star-shaped quiver with n legs of length one’ as drawn below · · · 0�� 1 �� 77 oooooooooooo 2 �� GG �� n �� gg OOOOOOOOOOOO We set V0 := V , d0 := 1, which together with the above Vi, di give an I-graded C-vector space V = ⊕ i Vi and multiplicities d = (di) ∈ ZI >0. Also we set λ0(z) := z−1 n∑ i=1 res z=0 ηi(z) ∈ R1, which together with the above λi gives an element λ = (λi) ∈ Rd. Note that n∑ i=1 tr res z=0 Ξi(z) = n∑ i=1 [ (dim Vi) res z=0 λi(z) + (dim V ) res z=0 ηi(z) ] = n∑ i=1 (dim Vi) res z=0 λi(z) + (dim V ) res z=0 λ0(z) = v · res λ, (6.1) where v := dimV. Hence n∑ i=1 tr res z=0 Ξi(z) = 0 if and only if v · res λ = 0, which is a necessary condition for the non-emptiness of Nset Q,d(λ,v) (Proposition 3.4). Proposition 6.6. There exists a bijection from Nset Q,d(λ,v) to Mset(Ξ1, . . . ,Ξn), which maps Ns Q,d(λ,v) symplectomorphically onto Mirr(Ξ1, . . . ,Ξn). Proof. Set ζ := res z=0 λ0(z) = n∑ i=1 res z=0 ηi(z). Then the scalar shift with ηi induces a bijection Mset(Ξ1, . . . ,Ξn)→Mset ζ (Λ1, . . . ,Λn), where Λi(z) := Ξi(z)− ηi(z) IdV = ( λi(z) IdVi 0 0 0 IdV ′ i ) ∈ g∗di (V ) ⊂ g∗ki (V ), (6.2) and it preserves the irreducibility. As Λi has the pole order di, the Gki (V )-action on Λi reduces to the Gdi (V )-action via the natural projection Gki (V )→ Gdi (V ), so that the orbit Gki (V )·Λi = Gdi (V ) · Λi is a Gdi (V )-coadjoint orbit. This replacement of order has no effect on the naive moduli space. Quiver Varieties with Multiplicities 29 By the definition of Q, we have V̂i = V0 ⊗C R1 = V for each i > 0 and MQ,d(V) = n⊕ i=1 Mi, Mi := Hom(V, Vi ⊗C Rdi )⊕Hom(Vi ⊗C Rdi , V ). Now consider the sets Zi ⊂ Mi, i > 0 given in Lemma 3.6. Since the top coefficients of λi ∈ g∗di (C), i > 0 are nonzero, Lemma 3.5 implies that for each i > 0, any point in µ−1 d,i(λi(z) IdVi) satisfies condition (3.4). Hence n⋂ i=1 µ−1 d,i(λi(z) IdVi) = n∏ i=1 Zi. Since dim Vi ≤ dim V for all i > 0, Lemmas 3.6 and 3.7 imply that the map Φ = (Φi) : MQ,d(V)→ n⊕ i=1 g∗di (V ), B 7→ ( −B�i(z IdVi⊗Rdi −Ni)−1Bi� ) (6.3) induces a symplectomorphism n⋂ i=1 µ−1 d,i(λi(z) IdVi)/ n∏ i=1 Gdi (Vi) = n∏ i=1 (Zi/Gdi (Vi)) '−→ n∏ i=1 Gdi (V ) · Λi, which is clearly GL(V )-equivariant. Note that n∑ i=1 res z=0 ( −B�i(z IdVi⊗Rdi −Ni)−1Bi� ) = − n∑ i=1 B�iBi� = res z=0 µd,0(B). Taking the (set-theoretical) symplectic quotient by the GL(V )-action at −ζ IdV , we thus obtain a bijection from Nset Q,d(λ,v) to Mset ζ (Λ1, . . . ,Λn). The proof of what it maps Ns Q,d(λ,v) onto Mirr ζ (Λ1, . . . ,Λn) is quite similar to Lemma 5.9. First, assume that a point B ∈ µ−1 d (−λ IdV) is stable. Let Φ(B) = ( di∑ l=1 Ai,lz −l ) , and assume further that a subspace S0 ⊂ V is invariant under all Ai,l. We define Si := di∑ l=1 N l−1 i Bi�(S0), i > 0, and set S := ⊕ i∈I Si ⊂ Vd. Then Ni(Si) ⊂ Si, Bi�(S0) ⊂ Si and B�i(Si) = ∑ l B�iN l−1 i Bi�(S0) = ∑ l Ai,l(S0) ⊂ S0 imply that S is B-invariant. Since B is stable, we thus have S = 0 or S = Vd, and in particular, S0 = 0 or S0 = V , which shows that the system Φ(B) is irreducible. Conversely, assume that the system Φ(B) = ( ∑ l Ai,lz −l) is irreducible. Let S = ⊕ i Si be a B-invariant subspace of Vd satisfying Ni(Si) ⊂ Si for all i ∈ I. Then S0 is invariant under all Ai,l, and hence S0 = 0 or S0 = V . If S0 = 0, then for each i > 0, we have B�i(Si) = 0, which implies Si = 0 since KerB�i ∩ Ker Ni = 0 by Lemma 3.5 and Ni\|Si is nilpotent. Dualizing the argument, we easily see that Si = Vi ⊗C Rdi , i > 0 if S0 = V . Hence S = 0 or S = Vd, which shows that B is stable. � 30 D. Yamakawa Conversely, let Q = (I, Ω, s, t) be as above and suppose that an I-graded C-vector space V =⊕ i Vi and multiplicities d = (di) are given. Suppose further that they satisfy dim Vi ≤ dim V0 and d0 = 1. Set V := V0, and fix a C-vector space V ′ i of dimension dim V − dim Vi together with an identification V ' Vi ⊕ V ′ i for each i > 0. Also, for each λ ∈ Rd, set ζ := res z=0 λ0 and let Λi be as in (6.2). Then the above proof also shows that the map Φ given in (6.3) induces a bijection Nset Q,d(λ,v) → Mset ζ (Λ1, . . . ,Λn) mapping Ns Q,d(λ,v) symplectomorphically ontoMirr ζ (Λ1, . . . ,Λn). 6.3 Middle convolution Recall the map given in (6.3); Φ = (Φi) : MQ,d(V)→ n⊕ i=1 g∗di (V ), B 7→ ( −B�i(z IdVi⊗Rdi −Ni)−1Bi� ) . Noting V̂0 = ⊕n i=1 Vi ⊗Rdi , we set T := n⊕ i=1 (ti IdVi⊗Rdi + Ni) ∈ End(V̂0). Using the natural inclusion ιi : Vi ⊗Rdi → V̂0 and projection πi : V̂0 → Vi ⊗Rdi , we then have (z Id V̂0 − T )−1 = n∑ i=1 ιi(z − ti −Ni)−1πi = n∑ i=1 ki∑ j=1 (z − ti)−jιiN j−1 i πi. Thus we can write the systems Φ(B) as Φ(B) = − n∑ i=1 B�i ( (z − ti) IdVi⊗Rdi −Ni )−1 Bi� = B0�(z Id V̂0 − T )−1B�0. (6.4) 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 P1 and generalized by Arinkin [1] for irregular D- modules. 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]: Proposition 6.7 ([31, Propositions 1 and 2]). Under the assumption V 6= 0, for any sys- tem A(z) with poles at ti, i = 1, 2, . . . , n and possibly a simple pole at∞, there exists a quadruple (W,T,X, Y ) consisting of • a finite-dimensional C-vector space W ; • an endomorphism T ∈ End(W ) with eigenvalues ti, i = 1, 2, . . . , n; • a pair of homomorphisms (X, Y ) ∈ Hom(W,V )⊕Hom(V,W ), such that X(z IdW − T )−1Y = A(z), (6.5) Ker Xi ∩Ker Ni = 0, Im Yi + Im Ni = V, (6.6) Quiver Varieties with Multiplicities 31 where Ni is the nilpotent part of T restricted on its generalized ti-eigenspace Wi := Ker(T − ti IdW )dim W , and (Xi, Yi) ∈ Hom(Wi, V ) ⊕ Hom(V,Wi) is the block component of (X, Y ) with respect to the decomposition W = ⊕ i Wi. 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 fTf−1 = T ′, X = X ′f, fY = Y ′. The above enables us to define the middle convolution. For a system A(z) = (Ai) ∈⊕n i=1 g∗ki (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 IdW ) and let • Xζ : W → V ζ be the projection; • Y ζ : V ζ →W be the injection induced from Y X + ζ IdW . Now we define mcζ(A) := Xζ(z IdW − T )−1Y ζ ∈ n⊕ i=1 g∗ki (V ζ). By virtue of Proposition 6.7, the equivalence class of mcζ(A) under constant gauge transforma- tions 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 (V̂0, T, B0�, B�0) satisfies (6.5) and (6.6) for A(z) = Φ(B). Now assume λ0(z) 6= 0 and consider the middle convolution mcζ(A) with ζ := res z=0 λ0. By the definition, the triple (V ζ , Bζ 0�, Bζ �0) satisfies Bζ �0B ζ 0� = B�0B0� + ζ Id V̂0 , (6.7) Ker Bζ �0 = 0, Im Bζ 0� = V ζ , (6.8) i.e., it provides a full-rank decomposition of the matrix B�0B0�+ζ Id V̂0 . Recall that such a triple already appeared in Section 4; conditions (4.3) and (4.4) for the 0-th reflection functor F0 imply that if we take an I-graded C-vector space V′ = ⊕ i V ′ i with dimV′ = s0(v) as in Section 4.2 and a representative B′ ∈ MQ,d(V′) of F0[B] ∈ Ns Q,d(r0(λ), s0(v)), then the triple (V ′ 0 , B ′ 0�, B′ �0) also satisfies (6.7) and (6.8) (note that d0 = 1 and N0 = 0). By the uniqueness of the full-rank decomposition, we then see that there exists an isomorphism f : V ζ → V ′ 0 such that B′ 0� = fBζ 0�, B′ �0 = Bζ �0f −1, and hence Φ(B′) = B′ 0�(z Id V̂0 − T )−1B′ �0 = fBζ 0�(z Id V̂0 − T )−1Bζ �0f −1 = f mcζ(A)f−1. The arguments in the previous subsection for V′, λ′ := r0(λ) show that Φ: MQ,d(V′) →⊕n i=1 g∗di (V ′ 0) induces a bijection between Nset Q,d(r0(λ), s0(v)) andMset −ζ(Λ ′ 1, . . . ,Λ ′ n), where Λ′i(z) = ( λ′i(z) IdVi 0 0 0 IdV ′′ i ) ∈ g∗di (V ′ 0), V ′ 0 ' Vi ⊕ V ′′ i . We have now proved the following: 5In [31], an explicit construction of the quadruple (W, T, X, Y ) is given so that the middle convolution mcζ(A) is well-defined as a system, not as a gauge equivalence class. 32 D. Yamakawa Proposition 6.8. Let (Q,d), λ, v be as in Proposition 6.6, and assume ζ := res z=0 λ0 is nonzero. Under the above notation, one then has the following commutative diagram: Nset Q,d(λ,v) F0 // Φ �� Nset Q,d(r0(λ), s0(v)) Φ �� Mset ζ (Λ1, . . . ,Λn) mcζ //Mset −ζ(Λ ′ 1, . . . ,Λ ′ n). Next, consider the reflection functors Fi for i > 0. Let [B′] = Fi[B]. Then condition (4.2) implies B′ �j(z IdVj⊗Rdj −Nj)−1B′ j� = B�j(z IdVj⊗Rdj −Nj)−1Bj�, j 6= 0, i, which together with (4.3) shows that the two systems Φ(B) and Φ′(B) are related via Φ(B′) = Φ(B)− λi(z − ti) IdV . Proposition 6.9. Let (Q,d), λ,v be as in Proposition 6.6, and set ζ := res z=0 λ0. For i = 1, 2, . . . , n, one then has the following commutative diagram: Nset Q,d(λ,v) Fi // Φ �� Nset Q,d(ri(λ), si(v)) Φ �� Mset ζ (Λ1, . . . ,Λn) −λi(z−ti) IdV //Mset ζ+res z=0 λi (Λ1, . . . ,Λi − λi(z) IdV , . . . ,Λn), where the bottom horizontal arrow is given by the shift A(z) 7→ A(z)− λi(z − ti) IdV . Remark 6.10. In [10], Harnad considered two meromorphic connections having the following symmetric description: ∇ = d− ( S + X(z IdW − T )−1Y ) dz, ∇′ = d + ( T + Y (z IdV − S)−1X ) dz, 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−1PQ dz. Hence on the ‘dual side’, the operation mcζ corresponds to just the scalar shift by z−1ζ dz. This interpretation enables us to generalize the middle convolution further; see [31]. 6.4 Examples: rank two cases The case dim V = 2 is most important because in this case a generic element in g∗ki (V ) can be transformed into an element of the form Ξi(z) = ξi(z) ⊕ ηi(z) for some distinct ξi, ηi ∈ g∗ki (C). The dimension of Mirr(Ξ1, . . . ,Ξn) can be computed as dimMirr(Ξ1, . . . ,Ξn) = dim Ns Q,d(λ,v) = 2− (v,v) = 2 n∑ i=1 di − 6, if it is nonempty. Quiver Varieties with Multiplicities 33 First, consider the case dimMirr(Ξ1, . . . ,Ξn) = 0. The above formula implies that the tuple (d1, . . . , dn) must be one of the following (up to permutation on indices): (1, 1, 1), (2, 1), (3). The corresponding (Q,d) have the underlying graphs with multiplicities given by the picture below 1�� 1�� 1�� 1��22 22 2 2�� 1�� 1�� 3�� 1�� The associated Kac–Moody algebras are respectively given by D4, C3, G2. From Example 5.6, we see that the effect of normalization on these quivers with multiplicities is given as follows: D4 ← C4, C3 → A3, G2 → A2, where the arrows represent the process of normalization. Next consider the case dimMirr(Ξ1, . . . ,Ξn) = 2. Then the tuple (d1, . . . , dn) 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 1�� 1��?? ?? ?? 1 �� 1 ��?????? 1�� 2�� 1�� 1�� 1��22 22 2 3�� 1�� 1�� 2�� 1�� 2�� 4�� 1�� The associated Kac–Moody algebras are respectively given by D (1) 4 , A (2) 5 , D (3) 4 , C (1) 2 , A (2) 2 . (6.10) From Example 5.6, we see that the effect of normalization on these quivers with multiplicities is given as follows: D (1) 4 ← A (2) 7 ← C (1) 4 , A (1) 3 ← A (2) 5 ← C (1) 3 , D (3) 4 → A (1) 2 , C (1) 2 → D (2) 3 , A (2) 2 → A (1) 1 , 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: D (1) 4 , A (1) 3 , A (1) 2 , C (1) 2 , A (1) 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. 34 D. Yamakawa Remark 6.11. In all the cases appearing in (6.9), we can check that Mirr(Ξ1, . . . ,Ξn) is nonempty if and only if n∑ i=1 tr res z=0 Ξi = 0 (recall that the ‘only if’ part is always true). We sketch the proof of the ‘if’ part below. If (d1, . . . , dn) 6= (2, 2), the naive moduli space Mirr(Ξ1, . . . ,Ξn) is isomorphic to an ordinal quiver variety Ns Q(ζ,v) for some extended Dynkin quiver Q and ζ, v as discussed above. The formulas (5.4) and (6.1) imply ζ ·v = n∑ i=1 tr res z=0 Ξi. Furthermore, since the expected dimension of Ns Q(ζ,v) is two, we have (v,v) = 0, which implies that v is a (positive) imaginary root (see [13, Proposition 5.10]). In fact, v is the minimal positive imaginary root δ because at least one of its components is equal to one. It is known [17] that if ζ · δ = 0, then Ns Q(ζ, δ) is a deformation of a Kleinian singularity, which is indeed nonempty6. Now assume (d1, . . . , dn) = (2, 2) and 2∑ i=1 tr res z=0 Ξi = 0. Let λi(z) = λi,2z −2 + λi,1z −1, Λi(z), ζ be as in the proof of Proposition 6.6, and for instance, set A1(z) := ( 2λ1,2 −2λ1,2 λ1,2 −λ1,2 ) z−2 + ( λ1,1 + ζ −λ1,1 − ζ ζ −ζ ) z−1 = ( 2 1 1 1 ){ Λ1(z) + ( 0 0 ζ − λ1,1 0 ) z−1 }( 2 1 1 1 )−1 , A2(z) := Λ2(z) + ( 0 λ1,1 + ζ −ζ 0 ) z−1. For each i, using the assumption λi,2 6= 0 and the formula( 1 az bz 1 ) · Λi = Λi(z) + ( 0 bλi,2 −aλi,2 0 ) z−1, a, b ∈ C, we easily see that Ai(z) is contained in the G2(C2)-coadjoint orbit through Λi(z). Furthermore, the assumption ∑ i tr res z=0 Ξi = 0 implies λ1,1 + λ2,1 = −2ζ, and hence res z=0 A1(z) + res z=0 A2(z) = ( λ1,1 + λ2,1 + ζ 0 0 −ζ ) = −ζ IdC2 . The assumption λi,2 6= 0 also implies that the top coefficients of A1(z), A2(z) have no common eigenvector, which shows that the system (A1, A2) is irreducible. Therefore the system (A1 + η1(z) IdC2 , A2 + η2(z) IdC2) gives a point in Mirr(Ξ1,Ξ2). Remark 6.12. Our list (6.10) of Dynkin diagrams is obtained from Sasano’s on [29, p. 352] by taking the transpose of the generalized Cartan matrices. It is an interesting problem to ask the relation between our symmetries and Sasano’s. A Appendix on normalization In this appendix, we prove Lemmas 5.2 and 5.3. Recall the situation discussed in Section 5.1; i ∈ I is a fixed pole vertex with base j, and O is the Gdi (Vj)-coadjoint orbit through Λ(z) = ( λi(z) IdVi 0 0 0 IdVj/Vi ) , Vj ' Vi ⊕ Vj/Vi, where the top coefficient λi,di of λi(z) is assumed to be nonzero. Its ‘normalized orbit’ Ǒ is the Bdi (Vj)-coadjoint orbit through the residue-free part Λ0 of Λ. 6As a more direct proof, one can check that if ζ · δ = 0, then (ζ, δ) satisfies the necessary and sufficient condition for the non-emptiness of Ns Q(ζ,v) given in [6, Theorem 1.2]. Quiver Varieties with Multiplicities 35 A.1 Proof of Lemma 5.2 We check that the Bdi (Vj)-coadjoint orbit Ǒ is invariant under the conjugation action by K, and is K-equivariantly symplectomorphic to the symplectic vector space Hom(Vj/Vi, Vi)⊕(di−2) ⊕Hom(Vi, Vj/Vi)⊕(di−2). Note that all the coefficients of Λ0 are fixed by K, and that the subset Bdi (Vj) ⊂ Gdi (Vj) is invariant under the conjugation by constant matrices. Hence for any k ∈ K and g(z) ∈ Bdi (Vj), k ( g · Λ0 ) k−1 = ( kgk−1 ) · ( kΛ0k−1 ) = ( kgk−1 ) · Λ0 ∈ Ǒ, i.e., Ǒ is invariant under the conjugation by K. Let us calculate the stabilizer of Λ0(z) with respect to the coadjoint Bdi (Vj)-action. Suppose that g(z) ∈ Bdi (Vj) stabilizes Λ0(z). By the definition, we then have g(z)Λ0(z) = Λ0(z)g(z) mod z−1 gl(Vj)[[z]]. (A.1) Write g(z) = ( G11(z) G12(z) G21(z) G22(z) ) according to the decomposition Vj = Vi ⊕ Vj/Vi, and let λ0 i (z) be the residue-free part of λi(z). Then [ Λ0(z), g(z) ] = [( λ0 i IdVi 0 0 0 IdVj/Vi ) , ( G11 G12 G21 G22 )] = ( 0 λ0 i G12 −λ0 i G21 0 ) . Therefore (A.1) is equivalent to λ0 i (z)f(z) ∈ z−1C[[z]] for all the matrix entries f(z) = di−1∑ k=1 fkz k of G12(z) and G21(z). We can write the above condition as λi,di λi,di−1 · · · λi,2 0 λi,di · · · λi,3 ... . . . . . . ... 0 · · · 0 λi,di   fdi−1 fdi−2 ... f1  ∈ C  1 0 ... 0  . Since λi,di 6= 0, this means fk = 0 for all k = 1, 2, . . . , di − 2. Hence the stabilizer is given by{ g(z) = IdVj + di−1∑ k=1 gkz k ∣∣∣∣∣ gk ∈ Lie K, k = 1, . . . , di − 2, gdi−1 ∈ gl(Vj) } . The above implies that the orbit Ǒ is naturally isomorphic to( gl(Vj)/ Lie K )⊕(di−2) ' Hom(Vj/Vi, Vi)⊕(di−2) ⊕Hom(Vi, Vj/Vi)⊕(di−2). Let us denote an element of the vector space on the right hand side by (a1, . . . , adi−2, b1, . . . , bdi−2), ak ∈ Hom(Vj/Vi, Vi), bk ∈ Hom(Vi, Vj/Vi), 36 D. Yamakawa and set a(z) := ∑ akz k, b(z) := ∑ k bkz k. Then the isomorphism is explicitly given by (ak, bk) di−2 k=1 7−→ g · Λ0 ∈ Ǒ, g(z) := ( IdVi a(z) b(z) IdVj/Vi ) ∈ Bdi (Vj). (A.2) It is clearly K-equivariant. Let us calculate the Kirillov–Kostant–Souriau symplectic form ωǑ on Ǒ in terms of the coordinates (a, b). Let (δla, δlb), l = 1, 2 be two tangent vectors at (a, b). Then the corresponding tangent vectors at g · Λ0 ∈ Ǒ are given by vl = [ δlg · g−1, gΛ0g−1 ] mod z−1 gl(Vj)[[z]] ∈ b∗di (Vj), where δlg := ( 0 δla(z) δlb(z) 0 ) ∈ bdi (Vj), l = 1, 2. By the definition, we have ωǑ(v1, v2) = tr res z=0 ( gΛ0g−1[δ1g · g−1, δ2g · g−1] ) = tr res z=0 ( Λ0[g−1δ1g, g−1δ2g] ) = tr res z=0 ( [Λ0, g−1δ1g]g−1δ2g ) . (A.3) Using the obvious formula g(z)−1 = ( IdVi a(z) b(z) IdVj/Vi )−1 = ( (IdVi − ab)−1 −a(IdVj/Vi − ba)−1 −b(IdVi − ab)−1 (IdVj/Vi − ba)−1 ) , (A.4) we have g−1δ1g = ( (IdVi − ab)−1 −a(IdVj/Vi − ba)−1 −b(IdVi − ab)−1 (IdVj/Vi − ba)−1 )( 0 δ1a(z) δ1b(z) 0 ) = ( (IdVi − ab)−1δ1b (IdVi − ab)−1δ1a (IdVj/Vi − ba)−1δ1b −b(IdVi − ab)−1δ1a ) , and hence [Λ0, g−1δ1g] = ( 0 λ0 i (IdVi − ab)−1δ1a −λ0 i (IdVj/Vi − ba)−1δ1b 0 ) . Substituting it into (A.3), we obtain ωǑ(v1, v2) = tr res z=0 [ λ0 i (IdVi − ab)−1δ1a (IdVj/Vi − ba)−1δ2b ] − tr res z=0 [ λ0 i (IdVj/Vi − ba)−1δ1b (IdVi − ab)−1δ2a ] , i.e., ωǑ = tr res z=0 [ λ0 i (IdVi − ab)−1 da ∧ (IdVj/Vi − ba)−1 db ] . (A.5) Now we set a′k := res z=0 [ zkλ0 i (IdVi − ab)−1a ] , b′k := bk, k = 1, . . . , di − 2. (A.6) Quiver Varieties with Multiplicities 37 Using (IdVi − ab)−1 = ∑ l≥0 (ab)l, we see that a′k is the sum of matrices λi,m(ap1bq1)(ap2bq2) · · · (apl bql )ar over all l ≥ 0 and m, p1, . . . , pl, q1, . . . , ql, r with m = k + ∑ pj + ∑ qj + r + 1. Note that the indices for a, b satisfy r ≤ m− k − 1 ≤ di − k − 1, pj , qj ≤ m− k − r − 1 < di − k − 1, and r = di − k − 1 only when m = di and l = 0. Thus we can write a′k = λi,di adi−k−1 + fk(a1, . . . , adi−k−2, b1, . . . , bdi−k−2) for some non-commutative polynomial fk. Since λi,di 6= 0, the above implies that one can uniquely determine (ak, bk) di−2 k=1 from (a′k, b ′ k) di−2 k=1 in an algebraic way. Hence (ak, bk) di−2 k=1 7→ (a′k, b ′ k) di−2 k=1 is a biregular map. By the definition, it is clearly K-equivariant. Let us calculate the 1-form di−2∑ k=1 tr da′k ∧ db′k. First, we have d [ (IdVi − ab)−1a ] = d(IdVi − ab)−1 · a + (IdVi − ab)−1 da = (IdVi − ab)−1 d(ab)(IdVi − ab)−1a + (IdVi − ab)−1 da = (IdVi − ab)−1 da [ b(IdVi − ab)−1a + IdVj/Vi ] + (IdVi − ab)−1adb(IdVi − ab)−1a. Note that the obvious equality b(IdVi − ab) = (IdVj/Vi − ba)b implies b(IdVi − ab)−1 = (IdVj/Vi − ba)−1b. Thus we have d [ (IdVi − ab)−1a ] = (IdVi − ab)−1 da [ (IdVj/Vi − ba)−1ba + IdVj/Vi ] + (IdVi − ab)−1adb (IdVi − ab)−1a = (IdVi − ab)−1 da (IdVj/Vi − ba)−1 + (IdVi − ab)−1adb (IdVi − ab)−1a, and hence tr ( λ0 i d [ (IdVi − ab)−1a ] ∧ db ) = tr [ λ0 i (IdVi − ab)−1 da ∧ (IdVj/Vi − ba)−1 db ] + tr [ λ0 i (IdVi − ab)−1adb ∧ (IdVi − ab)−1adb ] = tr [ λ0 i (IdVi − ab)−1 da ∧ (IdVj/Vi − ba)−1 db ] . The above and (A.5) imply that the 1-form di−2∑ k=1 tr da′k ∧ db′k coincides with ωǑ; indeed, di−2∑ k=1 tr da′k ∧ db′k = di−2∑ k=1 tr res z=0 ( zkλ0 i d [ (IdVi − ab)−1a ] ∧ dbk ) = res z=0 tr ( λ0 i d [ (IdVi − ab)−1a ] ∧ db ) = res z=0 tr [ λ0 i (IdVi − ab)−1 da ∧ (IdVj/Vi − ba)−1 db ] = ωǑ. Hence the map (a′k, b ′ k) di−2 k=1 7→ g · Λ0 is a K-equivariant symplectomorphism Hom(Vj/Vi, Vi)⊕(di−2) ⊕Hom(Vi, Vj/Vi)⊕(di−2) ' Ǒ. Since this sends the origin to Λ0, Lemma 5.2 follows. 38 D. Yamakawa A.2 Proof of Lemma 5.3 First, we show the following lemma: Lemma A.1. Let Ǒ → Lie K, B(z) 7→ −ΓB ∈ Lie K be the K-moment map sending Λ0 to zero. Then for any B(z) ∈ Ǒ, there exists g(z) ∈ Bdi (Vj) such that g(z)B(z)g(z)−1 = Λ0(z) + z−1ΓB mod gl(Vj)[[z]]. Proof. Let B(z) = di∑ k=2 Bkz −k ∈ Ǒ, and let a(z), b(z), g(z) be as in (A.2) such that B = g · Λ0. By the definition of the Bdi (Vj)-action, we then have g(z)−1B(z)g(z) = Λ0(z) + z−1Γ mod gl(Vj)[[z]] (A.7) for some Γ ∈ gl(Vj). According to the decomposition Vj = Vi ⊕ Vj/Vi, we write it as Γ = ( Γ11 Γ12 Γ21 Γ22 ) , and set ΓB := ( Γ11 0 0 Γ22 ) , U := ( 0 λ−1 i,di Γ12 −λ−1 i,di Γ21 0 ) , u(z) := IdVj + Uzdi−1. Note that ΓB ∈ Lie K. Let Λdi be the top coefficient of Λ0(z). Then U satisfies [Λdi , U ] = ( 0 λi,di · λ−1 i,di Γ12 (−λi,di ) · −λ−1 i,di Γ21 0 ) = Γ− ΓB, and hence u(z)g(z)−1B(z)g(z)u(z) = u(z)(Λ0(z) + z−1Γ)u(z)−1 mod gl(Vj)[[z]] = Λ0(z) + z−1Γ + z−1[U,Λdi ] mod gl(Vj)[[z]] = Λ0(z) + z−1ΓB mod gl(Vj)[[z]]. Now we explicitly describe ΓB in terms of the coordinates (a′k, b ′ k) di−2 k=1 , which shows that B 7→ −ΓB is a K-moment map. Note that the constant term of g(z) is the identity, and hence it acts trivially on z−1 gl(Vj)[[z]]/ gl(Vj)[[z]] by conjugation. Therefore (A.7) implies B(z) = g(z)(Λ0(z) + z−1Γ)g(z)−1 mod gl(Vj)[[z]] = g(z)Λ0(z)g(z)−1 + z−1Γ mod gl(Vj)[[z]]. Substituting (A.4) into the above equality, we have B(z) = ( λ0 i (IdVi − ab)−1 −λ0 i a(IdVj/Vi − ba)−1 bλ0 i (IdVi − ab)−1 −bλ0 i a(IdVj/Vi − ba)−1 ) + Γ z mod gl(Vj)[[z]]. (A.8) Quiver Varieties with Multiplicities 39 Note that B(z) and λ0 i (z) have no residue parts. Looking at the block diagonal part of the above and taking the residue, we thus obtain Γ11 = − res z=0 [ λ0 i (IdVi − ab)−1 ] = − ∞∑ l=0 res z=0 [ λ0 i (ab)l ] = − ∞∑ l=1 res z=0 [ λ0 i (ab)l ] = − res z=0 [ λ0 i (IdVi − ab)−1ab ] = − ∑ k res z=0 [ zk λ0 i (IdVi − ab)−1a ] bk = − ∑ k a′kb ′ k, and similarly, Γ22 = res z=0 [ bλ0 i a(IdVj/Vi − ba)−1 ] = ∑ k b′ka ′ k. Hence ΓB = − ∑ k ( a′kb ′ k 0 0 −b′ka ′ k ) , 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)−1B(z)g(z), so that it depends algebraically on ak, bk. Hence u(z)g(z)−1 also depends algebraically on ak, bk. This means that one can choose g(z) in the assertion of Lemma A.1 so that it depends algebraically on B ∈ Ǒ. Remark A.3. In the above proof, let us write B(z) = ( B11(z) B12(z) B21(z) B22(z) ) . Then (A.8) implies B11(z) = λ0 i (IdVi − ab)−1 mod z−1 gl(Vi)[[z]], B12(z) = −λ0 i a(IdVj/Vi − ba)−1 mod Hom(Vj/Vi, Vi)⊗ z−1C[[z]], B21(z) = bλ0 i (IdVi − ab)−1 mod Hom(Vi, Vj/Vi)⊗ z−1C[[z]], B22(z) = −bλ0 i a(IdVj/Vi − ba)−1 mod z−1 gl(Vj/Vi)[[z]]. Note that λ0 i (IdVi − ab)−1a has pole order di − 1 and λ0 i (IdVi − ab)−1a = di−2∑ k=1 a′kz −k−1 mod Hom(Vj/Vi, Vi)⊗ z−1C[[z]]. Set a′(z) := di−2∑ k=1 a′kz −k−1. Using the obvious formulas a(IdVj/Vi − ba)−1 = (IdVi − ab)−1a and (IdVi − ab)−1 = IdVi + (IdVi − ab)−1ab, we can then rewrite the above four equalities as B11(z) = λ0 i IdVi + a′b′ mod z−1 gl(Vi)[[z]], (A.9) B12(z) = −a′, (A.10) B21(z) = λ0 i b ′ + b′a′b′ mod Hom(Vi, Vj/Vi)⊗ z−1C[[z]], (A.11) B22(z) = −b′a′ mod z−1 gl(Vj/Vi)[[z]], 40 D. Yamakawa 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 B21(z) = b′(z)B11(z) mod Hom(Vi, Vj/Vi)⊗ z−1C[[z]]. Writing Bij = ∑ k Bij,kz −k, we then have ( B21,di−1 · · · B21,2 ) = ( b′1 · · · b′di−2 )  B11,di B11,di−1 · · · B11,3 0 B11,di · · · B11,4 ... . . . . . . ... 0 · · · 0 B11,di  . Note that (A.9) also shows B11,di = λi,di IdVi . Hence the block matrix on the far right is invertible, and therefore we can express b′k as b′k = di−1∑ l=2 B21,lFlk(B11,3, . . . , B11,di−1) with some non-commutative polynomial Flk. Proof of Lemma 5.3. We give a proof of Lemma 5.3. Let ϕ : Ǒ ×M → g∗di (Vj) ×M be the map defined in its statement; ϕ(B(z), x) = (A(z), x), A(z) := B(z)− µM (x) + ζ IdVj z , which is clearly equivariant under the conjugation by K. Now suppose that (B(z), x) ∈ Ǒ ×M satisfies the moment map condition µ̌(B, x) := −ΓB + µM (x) = − res z=0 Λ(z)− ζ IdVj . By Lemma A.1, there exists g(z) ∈ Bdi (Vj) such that B(z) = g(z) ( Λ0(z) + z−1ΓB ) g(z)−1 mod gl(Vj)[[z]]. (A.12) Noting that the constant term g(0) of g(z) is the identity, we obtain A(z) = g(z) ( Λ0(z) + ΓB z ) g(z)−1 − µM (x) + ζ IdVj z mod gl(Vj)[[z]] = g(z)Λ0(z)g(z)−1 + ΓB − µM (x)− ζ IdVj z mod gl(Vj)[[z]] = g(z)Λ0(z)g(z)−1 + res z=0 Λ z mod gl(Vj)[[z]] = g(z) ( Λ0(z) + res z=0 Λ z ) g(z)−1 mod gl(Vj)[[z]] = g(z)Λ(z)g(z)−1 mod gl(Vj)[[z]], which implies A(z) ∈ O. Since B(z) has no residue, we have res z=0 A(z) = −µM (x) − ζ IdVj , in other words, the value of the GL(Vj)-moment map µ : O ×M → gl(Vj), (A, x) 7→ res z=0 A(z) + µM (x) Quiver Varieties with Multiplicities 41 at ϕ(B, x) is −ζ IdVj . Hence ϕ induces a map between the symplectic quotients ϕ : µ̌−1 ( − res z=0 Λ− ζ IdVj ) /K −→ µ−1(−ζ IdVj )/ GL(Vj). We show that the above map is bijective. Suppose that (B, x), (B′, x′) ∈ µ̌−1(− res z=0 Λ−ζ IdVj ) and g ∈ GL(Vj) satisfy g · ϕ(B, x) = ϕ(B′, x′). Then g · x = x′ and g ( B(z)− µM (x) + ζ IdVj z ) g−1 = B′(z)− µM (x′) + ζ IdVj z = B′(z)− gµM (x)g−1+ ζ IdVj z = B′(z)− g µM (x) + ζ IdVj z g−1. Hence gB(z)g−1 = B′(z). Since B,B′ ∈ Ǒ, their top coefficients are Λdi = λi,di IdVi ⊕ 0 IdVj/Vi , whose centralizer is GL(Vi) × GL(Vj/Vi) = K. By comparing the top coefficients of gB(z)g−1, B′(z), we thus obtain g ∈ K, and hence (B, x) and (B′, x′) lie in the same K-orbit. To prove the surjectivity, suppose that (A, x) ∈ µ−1(−ζ IdVj ) is given. By using the GL(Vj)- action if necessary, we may assume that A = g · Λ for some g(z) ∈ Bdi (Vj) (if A = g · Λ for g(z) ∈ Gdi (Vj), we replace (A, x) with g(0)−1 ·(A, x)). Let B(z) ∈ b∗di (Vj) be the residue-free part of A(z). Taking modulo z−1 gl(Vj)[[z]] of A = g ·Λ, we then have B = g ·Λ0 ∈ Ǒ. Furthermore, the moment map condition for (A, x) implies A(z) = B(z) + res z=0 A z = B(z)− µM (x) + ζ IdVj z . Hence (B, x) = ϕ(A, x). This shows that ϕ is surjective. We have proved that ϕ is bijective. Furthermore, by letting (B, x) = (B′, x′) in the proof of the injectivity, we see that the stabilizer of ϕ(B, x) with respect to the GL(Vj)-action is contained in that of (B, x) with respect to the K-action. The converse is clear from the K-equivariance of ϕ, and hence the two stabilizers coincide. In particular, free K-orbits correspond to free GL(Vj)-orbits via ϕ, which is the second assertion of Lemma 5.3. Finally, we show that ϕ preserves the symplectic structure at points representing free orbits. Let (B, x) be a point in the level set µ̌−1(− res z=0 Λ− ζ IdVj ) whose stabilizer is trivial (so the level set is smooth at (B, x)), and let (A, x) = ϕ(B, x). We take g(z) ∈ Bdi (Vj) satisfying (A.12) so that it depends smoothly on B, which is possible as mentioned in Remark A.2. Then the argument just after (A.12) shows A = g · Λ, and furthermore, the smoothness of g implies that for any tangent vector (δB, v) at (B, x), there exists δg ∈ bdi (Vj) such that δB = [δg · g−1, B] mod z−1 gl(Vj)[[z]], δA = [δg · g−1, A] mod gl(Vj)[[z]], where (δA, v) = ϕ∗(δB, v) is the corresponding tangent vector at (A, x). Now let (δiB, vi), i = 1, 2 be two tangent vectors at (B, x) and δiA, δig as above. Let ωO (resp. ωM ) be the symplectic form on O (resp. M). By the definition, we have ωO(δ1A, δ2A) = tr res z=0 ( A[δ1g · g−1, δ2g · g−1] ) . Since δig has no constant term, we have [δ1g · g−1, δ2g · g−1] ∈ z2 gl(Vj)[[z]], which implies tr res z=0 ( A[δ1g · g−1, δ2g · g−1] ) = tr res z=0 ( B[δ1g · g−1, δ2g · g−1] ) = ωǑ(δ1B, δ2B), and hence ωO(δ1A, δ2A) + ωM (v1, v2) = ωǑ(δ1B, δ2B) + ωM (v1, v2). This shows the assertion. � 42 D. Yamakawa Acknowledgements I am grateful to Philip Boalch for stimulating conversations, and to Professor Hiraku Nakajima for valuable comments. 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Ann., to appear, arXiv:0911.3863. http://arxiv.org/abs/0902.1702 http://dx.doi.org/10.1619/fesi.51.351 http://dx.doi.org/10.1619/fesi.51.351 http://arxiv.org/abs/0704.2393 http://dx.doi.org/10.1016/j.geomphys.2006.09.009 http://arxiv.org/abs/nlin.SI/0601003 http://dx.doi.org/10.1007/s00208-010-0517-3 http://arxiv.org/abs/0911.3863 1 Introduction 2 Preliminaries 2.1 Quiver 2.2 Quiver variety 3 Quiver variety with multiplicities 3.1 Definition 3.2 Properties 4 Reflection functor 4.1 Main theorem 4.2 Proof of the main theorem 4.3 Application 5 Normalization 5.1 Shifting trick 5.2 Normalization 5.3 Weyl groups 6 Naive moduli of meromorphic connections on P1 6.1 Naive moduli 6.2 Star-shaped quivers of length one 6.3 Middle convolution 6.4 Examples: rank two cases A Appendix on normalization A.1 Proof of Lemma 5.2 A.2 Proof of Lemma 5.3 References

Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations

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