mckay correspondence iku nakamura

MCKAY CORRESPONDENCEIKU NAKAMURAAbstract. We discuss the two-dimensional McKay correspondence fromthe view point of Hilbert schemes.0. IntroductionThere is a whole series of apparently unrelated phenomena that are governedby the so-called ADE Dynkin diagram scheme. It is widely believedthat, despite the diverse nature of the objects concerned, there must be somehidden reasons for these coincidences. The ADE Dynkin diagrams provide aclassification of the following types of objects:(1) simple singularities (rational double points) of complex surfaces,(2) finite subgroups of SL(2, C),(3) simple Lie groups and simple Lie algebras,(4) the following three finite simple groups, the derived group F ′ 24 of theFischer F 24 , the Baby monster B and the Monster M are related withE 6 , E 7 and E 8 respectively.Meanwhile there are three outstanding McKay observations. The firstMcKay observation made in November, 1978 was concerned with the so-calledmoonshine, and the second in December, 1978 with the connection betweenthe above items (1) and (2), while the third in February, 1979 with the aboveitem (4). It is the second McKay observation that we discuss in this article,which we refer to as the McKay correspondence. The purpose of this articleis to discuss the McKay correspondence in detail, partially based on [IN99].For a given finite subgroup G of SL(2, C), the McKay correspondence isincorporated into a quiver, called the McKay quiver, in the quotient of thecoinvariant algebra. The McKay quiver gives the Dynkin diagram of theexceptional set of the corresponding simple singularity ADE. Moreover, thenatural strata of the exceptional set are understood via the subquivers of theMcKay quiver, which are easily described by directed Dynkin diagrams (SeeFigures 7-16). The McKay correspondence is also understood as a naturalThe present work is partially supported by the Grant-in-aid (No. 16204001) for ScientificResearch, JSPS.Mathematics subject classification: Primary 14-02, 14B05, 14J17; 17B10, 17B67, 17B68,17C20; Secondary 20C05, 20C15Key words: Simple singularity, ADE, Dynkin diagram, Finite group, Quiver, Hilbertscheme, Quotient singularity, McKay correspondence, Coinvariant algebra1

MCKAY CORRESPONDENCE 31 1 1 1 1A n ✈ ✈ ✈ ... ✈ ✈ (n vertices)✈1 2 2 2 1D n ✈ ✈ ✈ ... ✈ ❅ ❅✈ 1✈2E 6 ✈ ✈ ✈ ✈ ✈1 2 3 2 1✈2E 7 ✈ ✈ ✈ ✈ ✈ ✈2 3 4 3 2 1✈3E 8 ✈ ✈ ✈ ✈ ✈ ✈ ✈2 4 6 5 4 3 2(n vertices)Figure 1. The Dynkin diagrams ADEType G name |G| h (d 1 ,d 2 ,d 3 )A n Z n+1 cyclic n +1 n +1 (2,n+1,n+1)D n D n−2 binary dihedral 4(n − 2) 2n − 2 (4, 2n − 4, 2n − 2)E 6 T binary tetrahedral 24 12 (6, 8, 12)E 7 O binary octahedral 48 18 (8, 12, 18)E 8 I binary icosahedral 120 30 (12, 20, 30)Table 1. Finite subgroups of SL(2, C)( , ) SING can be expressed by a finite graph with simple edges. We rephrasethis as follows: we associate a vertex v(E ′ ) to any irreducible component E ′of E, and join two vertices v(E ′ ) and v(E ′′ ) if and only if (E ′ E ′′ ) SING =1.Thus we have a finite graph with simple edges. We call this graph the dualgraph of E, and denote it by Γ SING (S) or Γ(Irr E).There exists a unique divisor E fund , called the fundamental cycle of X,which is the minimal nonzero effective divisor such that E fund E i ≤ 0 for alli. Let E fund := ∑ ri=1 mSING i E i and E 0 := −E fund . For the simple singularitieswe have E 0 E i = 0 or 1 for any E i ∈ Irr E (except for the case A 1 , whenE 0 E 1 = 2). Thus we can draw a new graph ˜Γ SING by adding the vertex v(E 0 )to Γ SING (S). By abuse of notation we denote Irr E∪{E 0 } by Irr ∗ E. Also for agiven finite subgroup G of SL(2, C), we have a quotient singularity (A 2 /G, 0),which is one of simple singularities so that we have a Dynkin diagram as a dual

MCKAY CORRESPONDENCE 5ρ Tr ρ 1 σ τρ 0 χ 0 1 1 1ρ 1 χ 1 1 1 −1ρ 2 χ 2 2 1 0ρ 3 χ 3 2 −1 0ρ 4 χ 4 1 −1 iρ 5 χ 5 1 −1 −iTable 2. Character table of D 3 (of type D 5 )˜Γ GROUP (G) consisting of the vertices v(ρ) for ρ ∈ Irr G and all the edgesbetween them.For example, let us look at the D 5 case. Let χ j := Tr(ρ j ) be the characterof ρ j . Then from Table 2 we see thatχ 2 (g)χ nat (g) =χ 2 (g)χ 2 (g) =χ 0 (g)+χ 1 (g)+χ 3 (g), for g =1,σ or τ.Hence χ 2 χ nat = χ 0 + χ 1 + χ 3 . General representation theory says that anirreducible representation of G is uniquely determined up to equivalence byits character. Therefore ρ 2 ⊗ρ nat = ρ 0 +ρ 1 +ρ 3 . Hence a ρ2 ,ρ j= 1 for j =0, 1, 3and a ρ2 ,ρ j= 0 for j =2, 4, 5. Similarly, we see thatχ 0 χ nat = χ 2 , χ 1 χ nat = χ 2 ,χ 3 χ nat = χ 2 + χ 4 + χ 5 ,χ 4 χ nat = χ 3 and χ 5 χ nat = χ 3 .In this way we obtain a graph – the extended Dynkin diagram ˜D 5 of Figure 3.Thus we see that there are two completely different ways to obtain the sameextended Dynkin diagram ˜D 5 as ˜Γ SING (A 2 /G, 0) and ˜Γ GROUP (G), while D 5as Γ SING (A 2 /G, 0) and Γ GROUP (G).˜D 5ρ 0 ✈ ρ 2❅ ✈ρ ✈ 3ρ 4✈ ρ1 ✈❅ ❅✈ ρ 5Figure 3. ˜Γ GROUP (D 3 )The same is true in the other cases. Namely the two graphs Γ SING (A 2 /G, 0)and Γ GROUP (G) turn out to be one of the Dynkin diagrams ADE and coincidewith each other, while both ˜Γ SING (A 2 /G, 0) and ˜Γ GROUP (G) are thecorresponding extended Dynkin diagram (See Figure 4). It is also interestingto note that the degrees of the characters deg ρ j = χ j (1) are equal to themultiplicities of the fundamental cycle we computed in section 1.3.

6 IKU NAKAMURAThis is the second observation of McKay that we are going to discuss inthis article.Ã nρ n ρ n−1✈ ✈ ...ρ ✈0ρ 1✈❅ ✈✈✈❅ρ 2 ρ 3✈ ✈ ... ✈ρ ′ 0 ρ ✈❅ 2 ρ 3 ρ n−2ρ˜D✈❅′ n ✈ ✈ ... ✈ n−1ρ′✈ ❅1❅✈ ρ ′ n✈ρ 0✈ρ 2Ẽ 6 ✈ ✈ ✈ ✈ ✈ρ ′ 1 ρ ′ 2 ρ 3 ρ ′′2 ρ ′′1Ẽ 7 ✈ ✈ ✈ ✈ ✈ ✈ ✈ρ ′ 1 ρ ′ 2 ρ ′ 3 ρ 4 ρ 3 ρ 2 ρ 0✈ρ ′′3✈ρ ′′2Ẽ 8 ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ρ ′ 2 ρ ′ 4 ρ 6 ρ 5 ρ 4 ρ 3 ρ 2 ρ 0Figure 4. The extended Dynkin diagrams and representations2. The G-orbit Hilbert schemes2.1. Hilbert schemes. The Hilbert scheme of a given projective (or quasiprojective)scheme X is the scheme parametrizing all the subschemes of X. Moreprecisely, let X be a projective scheme embedded in a projective space P N ,and L the restriction of O È N(1) to X. The Hilbert scheme is the schemerepresenting the functorHilb X : S ↦→{}flat families Y of subschemes of X over S .The Euler–Poincaré characteristic P (m) := ∑ q∈ (−1)q h q (Y s ,L ⊗mY s) (calledthe Hilbert polynomial) of the sheaf L ⊗mY sis constant on each connected componentof S. Therefore the Hilbert scheme decomposes as the disjoint unionof open subsets labelled by Hilbert polynomials.

MCKAY CORRESPONDENCE 7The point set Hilb P X classifies the subschemes of X with Hilbert polynomialequal to P . Let U ⊂ X be an open subscheme. Then Hilb U is an opensubscheme of Hilb X consisting of the subschemes of X whose supports arecontained in U. This means that Hilb P U is empty or an open subscheme ofHilb P X for a fixed Hilbert polynomial P .2.2. The Hilbert scheme of n points. In what follows we denote Hilb P Xby Hilb P (X). Write S n (A 2 ) for the nth symmetric product of the affine planeA 2 . This is by definition the quotient of the products of n copies of A 2 by thenatural permutation action of the symmetric group S n on n letters. It is theset of formal sums of n points, in other words, the set of unordered n-tuplesof points. Let P (m) =n for any m ∈ Z, namely let P (m) be a constantpolynomial. We call Hilb n (A 2 ) := Hilb P (A 2 ) the Hilbert scheme of n pointsin A 2 . It is a quasiprojective scheme of dimension 2n. AnyZ ∈ Hilb n (A 2 )isa zero-dimensional subscheme with h 0 (Z, O Z ) = dim(O Z )=n. Suppose thatZ is reduced. Then Z is the union of n distinct points. Since being reducedis an open and generic condition, Hilb n (A 2 ) contains a Zariski open subsetconsisting of formal sums of n distinct points.We have a natural morphism π from Hilb n (A 2 )ontoS n (A 2 ) defined byπ : Z ↦→∑dim(O Z,p )pp∈Supp(Z)which is called the Hilbert-Chow morphism.features of Hilb n (A 2 ) is the following result.One of the most remarkableTheorem 2.3 ([Fogarty68]). Hilb n (A 2 ) is a smooth quasi-projective scheme,and the Hilbert-Chow morphism π : Hilb n (A 2 ) → S n (A 2 ) is a resolution ofsingularities of the symmetric product.We note that smoothness of Hilb n (A 2 ) is peculiar to dim A 2 =2.2.4. The G-orbit Hilbert scheme. For any finite subgroup G of SL(2, C)of order n, we consider the Hilbert-Chow morphism π from Hilb n (A 2 )ontoS n (A 2 ). Since the morphism π : Hilb n (A 2 ) → S n (A 2 )isG-equivariant, wehave a natural morphism between G-fixed point loci. We note that the G-fixed point set of S n (A 2 ) is nothing but A 2 /G because n = |G|. The G-fixedpoint set Hilb n (A 2 ) G in Hilb n (A 2 ) is always nonsingular, but could a priori bedisconnected. There is however a unique irreducible component of Hilb n (A 2 ) Gdominating S n (A 2 ) G , which we denote by Hilb G (A 2 ) and call it the G-orbitHilbert scheme. ItisaG-invariant subscheme of Hilb n (A 2 ) that parametrizesall smoothable G-invariant subschemes of length |G|.Now we recall the following theorem proved in [IN99].Theorem 2.5. Let G ⊂ SL(2, C) be a finite subgroup of order n. Then thereis a unique irreducible component Hilb G (A 2 ) of Hilb n (A 2 ) G dominating A 2 /G,

MCKAY CORRESPONDENCE 9Dynkin diagram emerges naturally (See Figure 6) by extending the McKayquiver in the polynomial ring. We also have a natural irreducible decompositionof a certain universal coherent sheaf on Hilb G (A 2 ). This somewhatsharpens the McKay correspondence (Theorem 3.9).3.2. A stratification of Hilb G (A 2 ) by Irr G. Let G be a finite subgroup ofSL(2, C). In what follows we assume that G is not cyclic because A n -case ismuch easier. As in section 1.4, we write Irr G for the set of all the equivalenceclasses of nontrivial irreducible G-modules, and Irr ∗ G for the union of Irr Gand the trivial one-dimensional G-module.Let E the exceptional set of the minimal resolution, Irr E the set of irreduciblecomponents of E.Any I ∈ X contained in E is a G-invariant ideal of S which contains n byCorollary 2.7. For any ρ, ρ ′ , and ρ ′′ ∈ Irr G, we defineV (I) :=I/(mI + n),E(ρ) := { I ∈ Hilb G (A 2 ); V (I) ⊃ ρ } ,P (ρ, ρ ′ ):= { I ∈ Hilb G (A 2 ); V (I) ⊃ ρ ⊕ ρ ′} ,Q(ρ, ρ ′ ,ρ ′′ ):= { I ∈ Hilb G (A 2 ); V (I) ⊃ ρ ⊕ ρ ′ ⊕ ρ ′′} .Any nonempty stratum will be described in a simple manner by using Dynkindiagrams with directed edges, called McKay subquivers. See subsection 4.8,Figures 7-12.Definition 3.3. Two irreducible G-modules ρ and ρ ′ are said to be adjacentif ρ ⊗ ρ nat ⊃ ρ ′ or vice versa.The Dynkin diagram Γ(Irr G) of Irr G is the graph whose vertices are Irr G,with ρ and ρ ′ joined by a simple edge if and only if ρ and ρ ′ are adjacent.Then the following theorem was proved in [IN99].Theorem 3.4. Let G be a finite subgroup of SL(2, C). Then(1) the map ρ ↦→ E(ρ) is a bijective correspondence between Irr G and Irr E;(2) E(ρ) is a smooth rational curve with E(ρ) 2 = −2 for any ρ ∈ Irr G;(3) P (ρ, ρ ′ ) ≠ ∅ if and only if ρ and ρ ′ are adjacent. In this case P (ρ, ρ ′ ) isa single (reduced) point, at which E(ρ) and E(ρ ′ ) intersect transversally;(4) P (ρ, ρ) =Q(ρ, ρ ′ ,ρ ′′ )=∅ for any ρ, ρ ′ ,ρ ′′ ∈ Irr G.By Theorem 3.4, (3), Γ(Irr G) is the same as the dual graph Γ(Irr E) ofE,in other words, the Dynkin diagram of the singularity (A 2 /G, 0).In what follows we assume that G is not cyclic. We define nonnegativeintegers d(ρ) for any ρ ∈ Irr G as follows. Since G is not cyclic, Γ(Irr G) isstar-shaped with a unique centre. For any ρ ∈ Irr G, we define d(ρ) to be thedistance from the vertex ρ to the centre, where d(ρ) = 0 for the centre ρ. Itis obvious that d(ρ) =d(ρ ′ ) ± 1ifρ and ρ ′ ∈ Irr G are adjacent.

10 IKU NAKAMURALet S = C[x, y]. For any positive integer k let S k be the subspace of homogeneouspolynomials of degree k in S. We say that a G-submodule W ofm/n is homogeneous of degree k if it is generated over C by homogeneouspolynomials of degree k. The G-module m/n splits as a direct sum of irreduciblehomogeneous G-modules. If W is a direct sum of homogeneousG-submodules, then we denote the homogeneous part of W of degree k byS k (W ). For any G-module W in some S k (m/n), we write S j · W for the G-submodule of S k+j (m/n) generated over C by the products of S j (m/n) andW . We denote by W [ρ] the ρ factor of W , that is, the sum of all the copiesof ρ in W ; and similarly, we denote by [W : ρ] the multiplicity of ρ ∈ Irr G ina G-module W .Definition 3.5. The quotient algebra S/n is called the coinvariant algebraof G, denoted by Coinv(G) (or denoted by Coinv(the type of S G ) with thenotation of Table 1). Let h be the Coxeter number of the simple singularityS G . Then we define the very positive part S † := S † (m/n) of S/n to be∑S † :=S k (m/n)[ρ].k> h 2 +d(ρ),ρ∈Irr GWe also define the McKay quiver of G byS McKay (G) = ∑S h2 ±d(ρ)(m/n)[ρ]+S† /S † ,ρ∈Irr Gwhich we also denote by S McKay (the type of S G ). We also define V k (ρ) =S k (m/n)[ρ] and V (ρ) =S McKay (G)[ρ].The following Lemma 3.6 and Lemma 3.7 describe the structure of theMcKay quiver S McKay (G). See [IN99] for the proofs.Lemma 3.6. Let ϕ i be three generators of G-invariants, d i = deg ϕ i and hthe Coxeter number of the simple singularity S G . Then d 1 + d 2 = d 3 +2 andd 3 = h, and moreover S k (m/n) =0for k ≥ d 3 .Lemma 3.7. Assume that G is not cyclic. Let h be the Coxeter number ofthe simple singularity S G . Then as G-modules, we have(1) m/n = ∑ ρ∈Irr G2(deg ρ)ρ;(2) S McKay (G) ≃ ∑ ρ∈Irr G 2ρ;(3) V h−d(ρ)(ρ) ≃ V h +d(ρ)(ρ) =ρ if d(ρ) ≥ 1 and V h (ρ) =ρ ⊕2 if d(ρ) =0;2 2 2(4) S h−k(m/n) ≃ S h +k(m/n) for any k.2 2Using this notation we see by Lemma 3.7, (2) that V (ρ) =V h−d(ρ)(ρ) +2V h+d(ρ)(ρ) ≃ ρ⊕2 and that the subset E(ρ) isP(V (ρ)), the set of all nontrivial2G-submodules of ρ ⊕2 . This is isomorphic to the projective line, or a smoothrational curve by Schur’s lemma. This proves Theorem 3.4, (2).

MCKAY CORRESPONDENCE 113.8. The ideals n X and π ∗ m. Since (A 2 /G) × ( 2 /G) X ≃ X, X is a closedsubscheme of (A 2 /G) × X, which is defined by an ideal I X of O 2 /G ⊗ O X .Let n X be the ideal of O 2 ⊗ O X generated by I X . Let Z univ be the universalsubscheme of A 2 × X, and I univ the ideal sheaf of O 2 ×X defining Z univ . Sincewe have a commutative diagrampr 2Z univ −−−→ X⏐ pr 1⏐↓⏐π↓A 2φ−−−→ A 2 /G,the morphism φ × id X : A 2 × X → A 2 /G × X sends Z univ into (A 2 /G) × ( 2 /G)X ≃ X. This implies that n X = I X O 2 ⊗ O X ⊂ I univ . Now we defineV := V (I univ )=I univ /mI univ + n X .Let m be the maximal ideal of O 2 /G of the unique singular point. Wenote n =Γ(A 2 ,φ ∗ m). We also note π ∗ (m) ∩ I univ = {0}. In fact, supposeπ ∗ (F ) ∈ I univ . Then π ∗ (F )=0onZ univ . Since Z univ is surjective over A 2 /G,F = 0. It follows π ∗ (m) ∩ I univ = {0}.Since π ∗ m := mO X is the defining ideal of E fund by [Artin66], we see thatV is a finite O Efund -module. See section 6 for the proof. However we prove alittle strongerTheorem 3.9. The O 2 × X-module V is a finite O E -module; as an O 2 × X-module with G-action, we have an isomorphismV≃ ⊕ρ ⊗ O E(ρ) (−1).ρ∈Irr GIn section 6 we will prove Theorem 3.9. In order to prepare for the proofof Theorem 3.9, we recall in section 4 the proof of Theorem 3.4 and relatedconstructions in the case of D 5 in full detail. In the cases D n (n ≠ 5) and E 6we give only a sketch of proofs of Theorem 3.9. Though we give almost noproof for E 7 because we need more notation, we remark that we can prove itin the same manner. It remains to check E 8 .4. The Simple Singularity D 5In this section we explain the case of D 5 in full detail.4.1. The binary dihedral group D 3 . The simple singularity D 5 is thequotient singularity of A 2 by the binary dihedral group G := D 3 of order 12,which is generated by σ and τ:σ =( )ɛ 00 ɛ −1 , τ =( )0 1,−1 0where ɛ := e 2πi/6 . We have σ 6 = τ 4 =1,σ 3 = τ 2 and τστ −1 = σ −1 . Thering of G-invariants in C[x, y] is generated by three elements A 6 := x 6 + y 6 ,

12 IKU NAKAMURAA 8 := xy(x 6 − y 6 ) and A 4 := x 2 y 2 . The quotient A 2 /G is isomorphic to thehypersurface 4A 4 4 + A2 8 − A 4A 2 6 =0.k S k S k (S/n)0 ρ 0 ρ 01 ρ 2 ρ 22 ρ 1 + ρ 3 (ρ 1 )+ρ 33 ρ 2 + ρ 4 + ρ 5 (ρ 2 + ρ 4 + ρ 5 )4 [ρ 0 ]+2ρ 3 (2ρ 3 )5 2ρ 2 + ρ 4 + ρ 5 (ρ 2 + ρ 4 + ρ 5 )6 [ρ 0 ]+2ρ 1 +2ρ 3 (ρ 1 )+ρ 37 3ρ 2 + ρ 4 + ρ 5 ρ 2Table 3. Irreducible decompositions of S and Coinv(D 5 )k V k (ρ) ⊂ S k (S/n) equiv.class0 {1} ρ0 ρ 01 {x, y} ρ2 ρ 22 {xy} ρ1 ⊕{x 2 ,y 2 } ρ3 (ρ 1 )+ρ 33 {x 2 y, −xy 2 } ρ2 (ρ 2 + ρ 4 + ρ 5 )⊕{x 3 + iy 3 } ρ4 ⊕{x 3 − iy 3 } ρ54 {y 4 ,x 4 } ρ3 ⊕{x 3 y, −xy 3 } ρ3 (2ρ 3 )5 {y 5 , −x 5 } ρ2 (ρ 2 + ρ 4 + ρ 5 )⊕{xy(x 3 − iy 3 )} ρ4 ⊕{xy(x 3 + iy 3 )} ρ56 {x 6 − y 6 } ρ1 ⊕{xy 5 , −x 5 y} ρ3 (ρ 1 )+ρ 37 {xy 6 ,x 6 y} ρ2 ρ 2Table 4. Coinv(D 5 )degree 1 2 3 4equiv.class ρ 2 (ρ 1 )+ρ 3 (ρ 2 + ρ 4 + ρ 5 )degree 7 6 5 (2ρ 3 )equiv.class ρ 2 (ρ 1 )+ρ 3 (ρ 2 + ρ 4 + ρ 5 )Table 5. Dual pairs of Coinv(D 5 )

MCKAY CORRESPONDENCE 134.2. The McKay quiver, tables and figures. We consider the case of D 5 .By an elementary computation we have the irreducible decomposition of thecoinvariant algebra Coinv(D 5 ) in Table 3. As Tables 3-5 indicate, Coinv(D 5 )consists of dual pairs. The McKay quiver S McKay (D 5 ) of Coinv(D 5 ), thatis, the irreducible factors in the parentheses in Tables 3 - 5 consist of dualpairs, exactly one pair for each equivalence class ρ ∈ Irr G except ρ 3 , whileV 4 (ρ 3 )=2ρ 3 is self-dual.V 2 (ρ 1 )⊕V 6 (ρ 1 )✛✲V 3 (ρ 2 )⊕V 5 (ρ 2 )✛✲V 4 (ρ 3 )✁✁✁✕✁✁✁✁☛ ✁✁❆❑❆❆❆❆❆ ❆❆❯V 3 (ρ 4 )⊕V 5 (ρ 4 )V 3 (ρ 5 )⊕V 5 (ρ 5 )Figure 5. The McKay quiver of D 5V 4 (ρ 0 )⊕V 6 (ρ 0 )V 2 (ρ 1 )⊕V 6 (ρ 1 )❆❑❆❆❑❆❆❆❆❆❆✁✕✁✁ ✁✁✁ ✁✁✁☛V 3 (ρ 2 )⊕V 5 (ρ 2 )✛✲V 4 (ρ 3 )✁✁✁✕✁✁✁✁☛ ✁✁❆❑❆❆❆❆❆ ❆❆❯V 3 (ρ 4 )⊕V 5 (ρ 4 )V 3 (ρ 5 )⊕V 5 (ρ 5 )Figure 6. The extended McKay quiver of D 5The coinvariant algebra Coinv(D 5 ) admits a quiver structure induced frommultiplication of the symmetric algebra. This induces a quiver structure onS McKay (D 5 ), which we call the McKay quiver of D 5 . It yields naturally acorresponding Dynkin diagram of the minimal resolution X := Hilb G (A 2 ),as we see later. In other words, the McKay quiver of D 5 gives the Dynkindiagram D 5 of the exceptional set E of X by replacing each pair of arrows

14 IKU NAKAMURAby an edge. To describe the strata of E precisely we also introduce thesubquivers Quiv(ρ) and Quiv(ρ, ρ ′ ), which are Dynkin diagrams with certaindirected edges. See Figures 7-16. These enable us to specify which of theG-invariants A 4 , A 6 and A 8 generates the ideal π ∗ (n) along each E(ρ).Moreover in order to obtain the extended Dynkin diagram ˜D 5 it seemsnatural to add to Figure 5 the G-submodules V 4 (ρ 0 ):=S 4 [ρ 0 ] and V 6 (ρ 0 ):=S 6 [ρ 0 ] as in Figure 6, where V 4 (ρ 0 ) and V 6 (ρ 0 ), the parts of Table 3 in thebracket, are generated by the generators A 4 = x 2 y 2 and A 6 = x 6 + y 6 ofG-invariants respectively.We note that the arrows between ρ 2 to ρ 0 are exceptional in the sense thatboth the arrows between them are directed from ρ 2 to ρ 0 , while in any othercases, say for ρ 1 and ρ 2 , both directions are taken by arrows between them.As its consequence deg V 4 (ρ 0 ) + deg V 6 (ρ 0 )=10> 8=2+6=3+5=4+4,the sum of degrees for the other pairs or the Coxeter number of D 5 .4.3. The subset E(ρ 1 ).4.3.1. We first classify I ∈ E(ρ 1 ). Now we recallV 2 (ρ 1 )={xy}, V 6 (ρ 1 )={x 6 − y 6 },V 3 (ρ 2 )={x 2 y, xy 2 }, V 5 (ρ 2 )={x 5 ,y 5 }.For any nonzero G-submodule W of V 2 (ρ 1 ) ⊕ V 6 (ρ 1 ) with W ≠ V 6 (ρ ′ 1 ), letI(W ):=SW + n where S = C[x, y]. The G-module W is generated byxy + t(x 6 − y 6 ) for some t ∈ C. Then since S 8 ⊂ n by Lemma 3.6, we seeS 2 W + n = S 2 · (xy + t(x 6 − y 6 )) + n = S 2 · xy + n,S k W + n = S k V 2 (ρ 1 )+n for any k ≥ 2By Table 4, we see in S/nS 2 V 2 (ρ 1 )={x 3 y, x 2 y 2 ,xy 3 } = {x 3 y, −xy 3 } = V 4 (ρ 3 ),S 3 V 2 (ρ 1 )={x 4 y, xy 4 } = V 5 (ρ 4 ) ⊕ V 5 (ρ 5 ),S 4 V 2 (ρ 1 )=V 6 (ρ 3 ),S 5 V 2 (ρ 1 )=V 7 (ρ 2 ).It follows thatS 1 W + S 5 W + n = S 1 V 2 (ρ 1 )+S 5 V 2 (ρ 1 )+n.Hence we see5∑5∑I(W )/n = W + S k W + n/n = W + S k V 2 (ρ 1 )+n/nk=1k=1≃ W + ρ 2 + ρ 3 +(ρ 4 + ρ 5 )+ρ 3 + ρ 2 ≃ ∑deg(ρ)ρ.ρ∈Irr GThus we have S/I(W ) ≃ S/n ⊖ I(W )/n ≃ ∑ ρ∈Irr ∗ Gdeg(ρ)ρ. Hence I(W )belongs to Hilb |G| (A 2 ) G−inv .

MCKAY CORRESPONDENCE 154.3.2. Next we prove that I(V 2 (ρ 1 )) belongs to Hilb G (A 2 ). For this purposewe consider local deformations of I(V 2 (ρ 1 )) in Hilb |G| (A 2 ) G−inv . LetI = I(V 2 (ρ 1 )). Then I = xyS + A 6 S because n is generated by A 4 = x 2 y 2 ,A 6 = x 6 + y 6 and A 8 = xy(x 6 − y 6 ). General deformation theory saysthat G-equivariant local deformations of I are captured by the ρ 0 -part ofHom S (I/I 2 , S/I). We see(S/I)[ρ 1 ]={xy, x 6 − y 6 } = V 2 (ρ 1 )+V 6 (ρ 1 ), (S/I)[ρ 0 ]={1} = C.Since I/I 2 is generated by the elements xy and A 6 = x 6 +y 6 , the G-moduleHom S (I/I 2 , S/I)[ρ 0 ] is spanned by the elements ψ 1 and ψ 2 :ψ 1 (xy) =x 6 − y 6 ,ψ 1 (A 6 )=0,ψ 2 (xy) =0,ψ 2 (A 6 )=1.In fact, let φ ∈ Hom S (I/I 2 , S/I)[ρ 0 ]. Let a = xy be a generator of V 2 (ρ 1 ).Then we may assume φ(a) =s(x 6 − y 6 ). Then we have (x 6 − y 6 )φ(a) =φ(A 8 )+s 0 φ(A 2 6 − 4A3 4 )=φ(A 8) because A j A k ∈ I 2 , while (x 6 − y 6 )φ(a) =s(A 2 6 − 4A3 4 )=0inS/I because A k ∈ I. Hence φ(A 8 ) = 0. Moreover,φ(A 4 )=φ(xya) =sA 8 = 0. Since Hom S (I/I 2 , S/I)[ρ 0 ] is two-dimensionaland I is generated by a and A 6 , it follows that letting φ(A 6 ) = t, thenφ = sψ 1 + tψ 2 , whence ψ 1 and ψ 2 span Hom S (I/I 2 , S/I)[ρ 0 ].Thus the tangent space T [I] (Hilb |G| (A 2 )) G−inv at the point [I] is spanned by∂∂s and ∂ ∂t . In other words, s and t are local (regular) parameters of HilbG (A 2 )at I. This implies the following. Let C[[s, t]] be the formal power series ringof two variables s and t, and R = C[[s, t]][x, y]. Now we define I to be theideal of R generated by the elementsxy + t(x 6 − y 6 ),A 6 + s, A 4 + tA 8 ,A 8 + s 2 t +4t 4 A 3 8 .Let φ 1 (s, t) be a power series with initial term −s 2 t satisfying the equationφ 1 + s 2 t +4t 4 φ 3 1 =0,which comes from A 2 8 = A 4A 2 6 − 4A4 4 . The power series φ 1 is uniquely determinedby this property. Then I is also generated by the elementsWe note thatxy + t(x 6 − y 6 ),A 6 + s, A 4 + tφ 1 (s, t),A 8 − φ 1 (s, t).S/I =5⊕{x k ,y k }⊕{x 6 − y 6 }.k=1By the upper semi-continuity R/I is generated over C[[s, t]] by S/I (regardedas a G-submodule of S). Then it is almost clear that R/I is a freeC[[s, t]]-module with the same basis S/I because of the forms of the fourgenerators of I. Hence R/I gives a C[[s, t]]-flat family of zero-dimensionalsubschemes deforming Z := Spec(S/I). It is obvious that the support of thesubscheme over C((s, t)) is away from the origin (if necessary by restricting

16 IKU NAKAMURAthe deformation to a small disc of the (s, t) space near the origin in the complextopology). In other words, Z is deformed into a G-invariant reducedsubscheme of A 2 . Therefore Z, hence I, belongs to Hilb G (A 2 ).4.3.3. Since I belongs to Hilb G (A 2 ), any deformation I ′ of I over a connectedbase belongs to Hilb G (A 2 ) because Hilb G (A 2 ) is connected. In particular,I(W ) defined above and all I ′ we are going to construct in this (sub)sectionbelong to Hilb G (A 2 ).4.3.4. Next we consider the case where W = V 6 (ρ 1 ). Let W ∞ = V 6 (ρ 1 ) andW t =(xy + t(x 6 − y 6 ))S + n. Then lim t→∞ W t = W ∞ in P 1 = P(V (ρ 1 )) =P(V 2 (ρ 1 ) ⊕ V 6 (ρ 1 )). Now we compute lim t→∞ I(W t ) in Hilb G (A 2 ). For any t,hence for t = ∞ we have5∑5∑I(W t )/n = W t + S k W t + n/n = W t + S k V 2 (ρ 1 )+n/n,k=1I(W ∞ )/n = V 6 (ρ 1 )+k=15∑S k V 2 (ρ 1 )+n/n ≃ ∑k=1ρ∈Irr Gdeg(ρ)ρ.Since I(W ∞ ) is a limit of I(W t ), to be more precise, we have a flat familyover P 1 deforming I(W t )intoI(W ∞ ), the limit I(W ∞ ) belongs to Hilb G (A 2 ).We note thatV (I(W ∞ )) = V 6 (ρ 1 )+S 1 V 2 (ρ 1 )=V 6 (ρ 1 )+S 3 (ρ 2 ).Hence I(W ∞ ) belongs to the subset P (ρ 1 ,ρ 2 ) of Hilb G (A 2 ).4.3.5. Now we shall prove the converse. Let I ∈ Hilb |G| (A 2 ) G−inv . SupposeS/I ≃ C[G] and ρ 1 ⊂ V (I). Then n ⊂ I by (the same reason as) Corollary 2.7.This implies that a nonzero G-submodule W of V 2 (ρ 1 ) ⊕ V 6 (ρ 1 ) is containedin V (I), and in I as part of generators of I. IfW ≠ V 6 (ρ 1 ), then I containsI(W ) defined above. Since I(W )/n ≃ I/n, we have I = I(W ). SupposeW = W ∞ = V 6 (ρ 1 ), or equivalently V 6 (ρ 1 ) ⊂ V (I). Since W is part ofgenerators of I, V 5 (ρ 2 ) is not contained in I. There are only V 3 (ρ 2 ) andV 5 (ρ 2 )inS(m/n)[ρ 2 ]. Hence V (I) ⊃{x 2 y +ty 5 ,xy 2 −tx 5 } for some t, whencet{x 6 , −y 6 } + n ⊂ S 1 V (I) +n ⊂ I because x 2 y 2 ∈ n. If t ≠ 0, this impliesthat {x 6 , −y 6 } = V 6 (ρ 1 ) is not part of generators of I, which contradictsthe assumption W = V 6 (ρ 1 ). It follows that t = 0, and V 3 (ρ 2 ) is containedin I. Let I(W ) = I(W ∞ )=I(V 6 (ρ 1 )) := V 6 (ρ 1 )S + V 3 (ρ 2 )S + n. ThenI(W ∞ ) ⊂ I. As we saw above, I(W ∞ ) ∈ Hilb G (A 2 ) and I(W ∞ )/n ≃ C[G].Hence I(W ∞ )/n ≃ I/n, whence I = I(W ∞ ).Thus we have proved that if S/I ≃ C[G] and V (I) ⊃ ρ 1 , then I = I(W )for some nonzero G-submodule W ⊂ V 2 (ρ 1 ) ⊕ V 6 (ρ 1 ).

MCKAY CORRESPONDENCE 19The subset E(ρ 2 ) is proved as before to be a smooth rational curve withself-intersection −2. We also see thatI 2 (V 3 (ρ 2 )) =I 2 (V 5 (ρ 2 )) =lim I 2(W )=V 3 (ρ 2 )S + V 6 (ρ 1 )S + n,W →V 3 (ρ 2 )lim I 2 (W )=V 5 (ρ 2 )S + S 1 V 3 (ρ 2 )S + n.W →V 5 (ρ 2 )4.4.4. Now we focus on the intersection point P (ρ 1 ,ρ 2 ) of two rational curvesE(ρ 1 ) and E(ρ 2 ). The computation of limits shows thatI 1 (V 6 (ρ 1 )) =I 2 (V 3 (ρ 2 )) =lim I 1 (W )={V 6 (ρ 1 )+S 1 V 2 (ρ 1 )}S + nW →V 6 (ρ 1 )= {V 6 (ρ 1 )+V 3 (ρ 2 )}S + n,lim I 2 (W )={V 3 (ρ 2 )S + S 1 V 5 (ρ 2 )}S + nW →V 3 (ρ 2 )= {V 3 (ρ 2 )+V 6 (ρ 1 )}S + n.The reason why I 1 (V 6 (ρ 1 )) = I 2 (V 3 (ρ 2 )) holds true is just the factS 1 V 2 (ρ 1 )=V 3 (ρ 2 ),S 1 V 5 (ρ 2 )=V 6 (ρ 1 ) mod n,where V 2 (ρ 1 ) is the dual partner of V 6 (ρ 1 ), while V 3 (ρ 2 ) is the dual partner ofV 5 (ρ 2 ). Since S 1 ≃ ρ nat , the natural representation of G, this is part of theMcKay rule of irreducible decompositions by tensoring with ρ nat , relevant toρ 1 and ρ 2 : ρ 1 ρ nat = ρ 2 , ρ 2 ρ nat = ρ 1 + ···.4.5. The subset E(ρ 3 ). The subset E(ρ 3 ) is computed in the same manneras before. Let W be a nonzero irreducible G-submodule of V 4 (ρ 3 ). Then wedefine in the same manner as before⎧(S 1 V 3 (ρ 2 )[ρ 3 ]+V 5 (ρ 2 ))S + n if W = S 1 V 3 (ρ 2 )[ρ 3 ],⎪⎨(S 1 V 3 (ρ 4 )+V 5 (ρ 4 ))S + n if W = S 1 V 3 (ρ 4 ),I 3 (W )=(S 1 V 3 (ρ 5 )+V 5 (ρ 5 ))S + n if W = S 1 V 3 (ρ 5 ),⎪⎩SW + notherwise,7∑= W + S k (m/n)+n for any W.k=5Then one can prove in the same manner as beforeE(ρ 3 )={I 3 (W ); ρ 3 ≃ W ⊂ V 4 (ρ 3 )},P (ρ 2 ,ρ 3 )={I 3 (S 1 V 3 (ρ 2 ))} = {S 1 V 3 (ρ 2 )S + V 5 (ρ 2 )S + n},P (ρ 3 ,ρ 4 )={I 3 (S 1 V 3 (ρ 4 ))} = {S 1 V 3 (ρ 4 )S + V 5 (ρ 4 )S + n},P (ρ 3 ,ρ 5 )={I 3 (S 1 V 3 (ρ 5 ))} = {S 1 V 3 (ρ 5 )S + V 5 (ρ 5 )S + n}.

24 IKU NAKAMURA{0}⊕V 6 (ρ 1 )✲✛V 3 (ρ 2 )⊕{0}✲✛0S 1 V 3 (ρ 2 )❣❣ 0 ❣ ✲ ❣ ✒❅ ❅❘ ❣✁ ✁✁✕❆❆❆❯V 5 (ρ 4 )V 5 (ρ 5 )Figure 11. Quiv(ρ 1 ,ρ 2 )❣❣❣ 0 ❣ ✲ ❣ ✒ ❣ ✛ ❣ 0 ❣ ✒❅ ❅ ❅❘ ❣❅❘ ❣❣❣0❣ ✛ ❣ ✛ ❣ ❣ ✛ ❣ ✛ ❣ ✒❅ ❅❘ ❣❅ 0❣Figure 12. Quiv(ρ, ρ ′ ) for D 5V 4 (ρ 0 )⊕{0}V 2 (ρ 1 )⊕V 6 (ρ 1 )❆❑❆❆❑❆❆❆✁ ✁✕ ✁✁✁✁☛V 3 (ρ 2 )⊕{0}✲✛0S 1 V 3 (ρ 2 )❣❣❅■❅ ❣ ✲ ❣ ✒❣❝ ✒ ❅ ❅❘ ❣✁ ✁✕✁V 5 (ρ 4 )❆❆❆❯V 5 (ρ 5 )Figure 13. The extended Quiv(ρ 1 )4.9. The extended McKay quiver revisited. The subquiver Quiv(ρ 1 )extends itself in the extended McKay quiver by adding an arrow from V 3 (ρ 2 )to V 4 (ρ 0 ) as Figure 13 indicates.This implies the following. By Table 4, V 3 (ρ 2 )=S 1 V 2 (ρ 1 ) and V 4 (ρ 0 ) ⊂S 1 V 3 (ρ 2 ). The subquiver Quiv(ρ 1 ) does not generate V 6 (ρ 0 )inI 1 (W ), which

26 IKU NAKAMURAk S k V k0 ρ 0 01 ρ 2 ρ 22 ρ 3 ρ 33 ρ ′ 2 + ρ ′′2 ρ ′ 2 + ρ ′′24 ρ ′ 1 + ρ′′ 1 + ρ 3 (ρ ′ 1 + ρ′′ 1 )+ρ 35 ρ 2 + ρ ′ 2 + ρ′′ 2 (ρ 2 + ρ ′ 2 + ρ′′ 2 )6 [ρ 0 ]+2ρ 3 (2ρ 3 )7 2ρ 2 + ρ ′ 2 + ρ′′ 2 (ρ 2 + ρ ′ 2 + ρ′′ 2 )8 [ρ 0 ]+ρ ′ 1 + ρ′′ 1 +2ρ 3 (ρ ′ 1 + ρ′′ 1 )+ρ 39 ρ 2 +2ρ ′ 2 +2ρ ′′2 ρ ′ 2 + ρ ′′210 ρ ′ 1 + ρ′′ 1 +3ρ 3 ρ 311 2ρ 2 +2ρ ′ 2 +2ρ′′ 2 ρ 212 2ρ 0 + ρ ′ 1 + ρ′′ 1 +3ρ 3 0Table 7. Irreducible decompositions of S and Coinv(E 6 )V 6 (ρ 0 ) ⊕ V 8 (ρ 0 )✻✻V 5 (ρ 2 ) ⊕ V 7 (ρ 2 )V 4 (ρ ′ 1 )⊕V 8 (ρ ′ 1)✲✛V 5 (ρ ′ 2 )⊕V 7 (ρ ′ 2)✲✛✻❄V 6 (ρ 3 )✛✲V 5 (ρ ′′2 )⊕V 7 (ρ ′′2)✛✲V 4 (ρ ′′1 )⊕V 8 (ρ ′′1)Figure 14. The extended McKay quiver of E 65.2. Symmetric tensors modulo n. Let S m be the space of homogeneouspolynomials in x and y of degree m. The G-modules S m and S m (S/n) viaρ nat decompose into irreducible G-submodules. We define a G-submodule ofm/n by V i (ρ j ):=S i (m/n)[ρ j ] the sum of all copies of ρ in S i (m/n), which wealways choose as a homogeneous G-submodule of S i . For a G-module W wedefine W [ρ] to be the sum of all the copies of ρ in W . It is known by [Klein],p. 51 that there are G-invariant polynomials A 6 , A 8 , A 2 6 and A 12 respectivelyof degrees 6, 8, 12 and 12. We may assume that A 6 = T , A 8 = W andA 12 = U := ϕ 3 +ψ 3 . Let V 6 (ρ 0 )=S 6 [ρ 0 ]={A 6 } and V 8 (ρ 0 )=S 8 [ρ 0 ]={A 8 },

MCKAY CORRESPONDENCE 27❣✻❝❣ ✲ ❣ ✲ ❣ ✲ ❣ ✲ ❣❣✻❣ ✛ ❝ ❣ ✲ ❣ ✲ ❣ ✲ ❣❣✻❣ ✛ ❣ ✛ ❝ ❣ ✲ ❣ ✲ ❣❣✻❣ ✛ ❣ ✛ ❣ ✛ ❝ ❣ ✲ ❣❣✻❣ ✛ ❣ ✛ ❣ ✛ ❣ ✛ ❝ ❣❝ ❣❣ ✛ ❣ ✛❄❣ ✲ ❣ ✲ ❣Figure 15. Quiv(ρ) for E 6❣✻❣ 0 ❣ ✲ ❣ ✲ ❣ ✲ ❣❣✻❣ ✛ ❣ 0 ❣ ✲ ❣ ✲ ❣❣0❣ ✛ ❣ ✛ ❣ ✲ ❣ ✲ ❣❣✻❣ ✛ ❣ ✛ ❣ 0 ❣ ✲ ❣❣✻❣ ✛ ❣ ✛ ❣ ✛ ❣ 0 ❣Figure 16. Quiv(ρ, ρ ′ ) for E 6where U 2 =4W 3 − 27T 4 . See [IN99], subsection 14.3 for the notation. Wecaution that A 12 in [IN99] is different from the present one.The decomposition of S k and S k (m/n) for small values of k are given inTable 7. The factors of S k (m/n) in the parentheses are those in S McKay (G).We see by Table 7 that V 6±d(ρ) (ρ) ≃ ρ ⊕2 if d(ρ) = 0, or ρ if d(ρ) ≥ 1. We alsosee that S 6−k (m/n) ≃ S 6+k (m/n) for any k. Thus Lemma 3.7 for E 6 followsfrom Table 7 immediately. The (extended) McKay quiver and subquivers ofE 6 are given in Figures 14-16. See also Figure 4.6. Proof of Theorem 3.9Now we prove Theorem 3.9 mainly for D 5 .6.1. The sheaf V. Let Z univ be the universal subscheme of A 2 of G-orbitsparameterized by Hilb G (A 2 ), and X = Hilb G (A 2 ). The natural morphism

28 IKU NAKAMURAπ : X → A 2 /G is known by Theorem 2.5 to be the minimal resolution. Wedefine I univ to be the ideal sheaf of O 2 ×X defining Z univ in A 2 × X. Then wehave an exact sequence 0 → I univ → O 2 ×X → O Zuniv → 0.We define I X to be the defining ideal of X as a subscheme of X ≃(A 2 /G) × ( 2 /G) X of (A 2 /G) × X, n X := I X O 2 ×X andV (I univ ):=I univ /mI univ + n X ,which we denote by V. The sheaf V is a finite O 2 ⊗O X -module supported by{0}×E because mO 2 = O 2 outside the origin and {0}×X∩Supp(Z univ ) red ={0}×E. It is clear that mV = n X V = 0 from the definition of V.Let m be the maximal ideal of the unique singular point of A 2 /G. By thedefinition of n X , φ ∗ F = π ∗ F mod n X for any F ∈ m.We prove next π ∗ (m)V = 0. Let H ∈ I univ and a ∈ π ∗ m. Then there aresome F k ∈ O X and A k ∈ m such that a = ∑ k F kπ ∗ (A k ). Since n X ⊂ I univ(See subsection 3.8), we have aH ≡ ∑ ∑k F kφ ∗ (A k )H in I univ /n X . Howeverk F kφ ∗ (A k )H ∈ mI univ , which proves aH =0inV. It follows π ∗ (m)V =0.Since π ∗ m is the ideal of O X defining the fundamental cycle E fund of E, thisimplies that V is a finite O Efund -module.Since E(ρ) is a subscheme of E fund , V⊗O E(ρ) is a finite O E(ρ) -module andwe have a natural homomorphism(2)V→ ∑V⊗O E(ρ) .ρ∈Irr GWe prove this is an isomorphism in subsections 6.2 and 6.3.6.2. Freeness outside Sing(E). First we prove that (2) is an isomorphismat a nonsingular point of E. Let S = C[x, y].Let I ∈ E(ρ) \ Sing(E). Then the ideal I is generated by a nonzero irreducibleG-submodule W of V (ρ) =V h−d(ρ) (ρ)+V h+d(ρ) (ρ) and n by section 4or by [IN99], Theorem 10.7.First we consider the case D 5 .6.2.1. Let ρ = ρ 1 . Then W ≠ V 6 (ρ 1 ). As Hilb G (A 2 ) is nonsingular and twodimensional,the tangent space T [I] (Hilb G (A 2 )) is exactly two-dimensional.By subsection 4.3Hom S (I/I 2 , S/I)[ρ 0 ] = Hom (W, V 6 (ρ 1 )) ⊕ Hom (V 6 (ρ 0 ),V 0 (ρ 0 )).We note that I is generated by W and A 6 by subsection 4.9. SinceT [I] (E(ρ 1 )) = Hom (W, V 6 (ρ 1 )), the parameter t of Hom (V 6 (ρ 0 ),V 0 (ρ 0 ))gives a defining equation of E(ρ 1 ).By subsection 4.3 the ideal I univ of Z univ is over E(ρ 1 ) \ Sing(E) generatedby xy+s(x 6 −y 6 ) and A 6 +t. Since A 6 +t ∈ n X , the quotient V is S⊗C[s, t]/tfreeof rank one, hence O E -free of rank one.

30 IKU NAKAMURASince T [I] (E(ρ 2 )) = Hom (W, V 5 (ρ 2 )), the parameter t of Hom (W, V 1 (ρ 2 ))gives the equation of E(ρ 2 ) locally along E(ρ 2 ) \ Sing(E). Now we prove thatthe ideal I univ is generated by the elementsB 1 := x 2 y + sy 5 − tx, B 2 := −xy 2 − sx 5 − ty, A := A 4 − η,where η := π ∗ A 4 is a power series of t with initial term t 2 satisfying s 2 η 2 −η +t 2 = 0. In fact, Noetherian property shows that I univ is generated by B 1 , B 2and some elements with initial terms being A k . We see 2A 4 +sA 6 = yB 1 −xB 2 ,sA 8 +2tA 4 = −xy(yB 1 + xB 2 ) and A 8 − tA 6 = x 5 B 1 + y 5 B 2 . Hence all ofthese belong to I univ . Since π ∗ (m) ∩ I univ = {0}, we have π ∗ (2A 4 + sA 6 )=π ∗ (A 8 − tA 6 ) = 0, and therefore the ideal π ∗ m =(π ∗ A 4 ,π ∗ A 6 ,π ∗ A 8 )ofO Xis generated by π ∗ A 4 because s is invertible over E(ρ 2 ) \ Sing(E). We noteE(ρ 2 ) \ Sing(E) ≃ Spec C[s, s −1 ]. It is easy to infer from A 2 8 = A 4A 2 6 − 4A4 4that η satisfies s 2 η 2 − η + t 2 =0.Since −t 2 = s 2 η 2 − η and A 4 − η ∈ n X , we have in VtB 1 = tx 2 y + sty 5 +(s 2 A 2 4 − A 4)x= −xyB 1 − sy 4 B 2 =0,tB 2 = xyB 2 + sx 4 B 1 =0.Hence V is an O E -module with B 1 and B 2 generators. By section 4.4 thatV (I) is of rank two for any I ∈ E(ρ 2 ) \ Sing(E), hence for each s ∈ C, s ≠0.This implies that V⊗S[s, s −1 ,t]/(t) isS[s, s −1 ]-free of rank two.6.2.5. As the final case of D 5 , we consider ρ 3 . Let I = I 3 (W ), W ≠ S 1 V 3 (ρ k )(k =2, 4, 5). Then we seeHom S (I/I 2 , S/I)[ρ 0 ] = Hom (W, V 4 (ρ 3 )/W ) ⊕ Hom (W, V 2 (ρ 3 )).The proof in this case is however rather tricky. Let b 1 = y 4 + is 0 x 3 y andb 2 = x 4 − is 0 xy 3 . By the condition W ≠ S 1 V 3 (ρ k ), we have s 0 ≠ ±1, ∞. Letφ ∈ Hom S (I/I 2 , S/I)[ρ 0 ]. Since (S/I)[ρ 3 ]={x 3 y, −xy 3 }⊕V 2 (ρ 3 ), we mayassume φ(b 1 )=isx 3 y + tx 2 and φ(b 2 )=−isxy 3 + ty 2 . First we note thatφ(b i ) ∈ (S 4 + S 2 )(m/I), whence φ(S 2 b i ) ∈ (S 6 + S 4 )(m/I) =S 4 (m/I). Hencewe see φ(y 2 b 1 )=isx 3 y 3 +tA 4 = tA 4 =0,φ(x 2 b 2 )=−isx 3 y 3 +tA 4 = tA 4 =0.Similarly φ(x 2 b 1 )=isx 5 y + tx 4 = tx 4 , φ(xyb 1 )=tx 3 y, φ(xyb 2 )=txy 3 andφ(y 2 b 2 )=ty 4 .Then φ(xy 3 b 1 − x 3 yb 2 )=−φ(A 8 )+2is 0 φ(A 2 4 )=−φ(A 8), while φ(xy 3 b 1 −x 3 yb 2 )=xyφ(y 2 b 1 ) − xyφ(x 2 b 2 ) = 0. Hence φ(A 8 ) = 0. Similarly φ(A 6 )=φ(y 2 b 1 + x 2 b 2 )=2tA 4 = 0. We also see that (1 − s 2 0 )φ(x2 y 4 )=φ(x 2 b 1 −is 0 xyb 2 )=tx 4 − is 0 txy 3 = tb 2 ∈ W ⊂ I. Hence (1 − s 2 0 )φ(x2 y 4 )=0inS/I,whence φ(x 2 y 4 ) = 0 because 1 − s 2 0 ≠ 0. Similarly we see φ(x4 y 2 ) = 0. Itfollows that {x 2 ,y 2 }φ(A 4 ) = 0. Hence φ(A 4 ) = 0 because 0 ≠ {x 2 ,y 2 }⊂(S/I)[ρ 2 ]. This proves φ(A k ) = 0 for any A k . Hence Hom S (I/I 2 , S/I) =Hom (W, V 4 (ρ 3 )/W ) ⊕ Hom (W, V 2 (ρ 3 )).

MCKAY CORRESPONDENCE 31We know by the subquiver Quiv(ρ 3 ) that I is generated by W and A 4 .Wehave B 1 , B 2 and A as generators of I univ as follows :B 1 = y 4 + isx 3 y + tx 2 ,B 2 = x 4 − isxy 3 + ty 2 ,A= A 4 − π ∗ (A 4 ),where π ∗ (A 4 )=t2. We note A1−s 2 6 +2tA 4 and A 8 − 2isA 2 4 ∈ I univ . Since t isthe parameter of Hom (W, V 2 (ρ 3 )), E(ρ 3 ) is defined by t =0.Now we prove in S[s, t]tB 1 = y 2 B 2 + isxyB 1 − (1 − s 2 )x 2 A,tB 2 = x 2 B 1 − isxyB 2 − (1 − s 2 )y 2 A.1Hence tB 1 = tB 2 = 0. This shows that V is C[s, ]-free of rank two,1−s 21where E(ρ 3 ) \ Sing(E) = Spec C[s, ].1−s 2This completes the proof of freeness of V over E \ Sing(E) in the D 5 -case.6.2.6. It is clear that one can generalize the above arguments to D n for theother n. To settle the E 6 and E 7 -cases, we need to discuss three-dimensionalor four-dimensional representations ρ. Since the discussion below on thethree-dimensional representation ρ 3 of E 6 shows the general features of thearguments for the proof sufficiently, we take up only ρ 3 of E 6 in order to avoidthe overwhelming notation for E 7 .6.2.7. We consider the E 6 -case. Let G be the binary tetrahedral group T,and ρ 3 the unique three-dimensional representation of G. Let A 6 , A 8 and A 12be the homogeneous generators of the ring of G-invariants :A 6 = T = p 1 p 2 p 3 ,A 8 = W = ϕψ, A 12 = U = ϕ 3 + ψ 3 ,where U 2 =4W 3 − 27T 4 . See [IN99], subsection 14.3 for the notation. Wenote ϕ 3 − ψ 3 = 3(2ω +1)T 2 and that both ϕ 3 and ψ 3 are G-invariants.Then any point I = I 3 (W ) ∈ E(ρ 3 ) \ Sing(E) is given by an irreducibleG-submodule of V 6 (ρ 3 ) with W ≠ S 1 V 5 (ρ 2 ),S 1 V 5 (ρ ′ 2 ),S 1V 5 (ρ ′′2 ) under the notationof section 5. Then we seeHom S (I/I 2 , S/I) = Hom (W, V 6 (ρ 3 )/W ) ⊕ Hom (W, V 4 (ρ 3 )).Moreover a versal deformation I univ of I is generated by six elementsB 1 ,B 2 ,B 3 ,A 6 − π ∗ (A 6 ),A 8 − π ∗ (A 8 ),A 12 − π ∗ (A 12 ),where with the notation of [IN99], subsection 14.3,B 1 = p 1 (ϕ + ωψ)+sp 1 ϕ + tp 2 p 3 + up 1 ,B 2 = ωp 2 (ϕ + ω 2 ψ)+sωp 2 ϕ − tp 3 p 1 + up 2 ,B 3 = ω 2 p 3 (ϕ + ψ)+sω 2 p 3 ϕ + tp 1 p 2 + up 3 ,where s, t and u are parameters. We note thatV 6 (ρ 3 )={p 1 ϕ, ωp 2 ϕ, ω 2 p 3 ϕ}⊕{p 1 ψ, ω 2 p 2 ψ, ωp 3 ψ},V 4 (ρ 3 )={p 2 p 3 , −p 3 p 1 ,p 1 p 2 },V 2 (ρ 3 )={p 1 ,p 2 ,p 3 }.

32 IKU NAKAMURAThen we see p 1 B 1 − ω 2 p 2 B 2 + ωp 3 B 3 = ω(1 − ω)(ψ 2 − ωuϕ). Hence ψ 2 −ωuϕ ∈ I univ , whence ψ 3 −ωuW ∈ I univ . Similarly we see p 1 B 1 −p 2 B 2 +p 3 B 3 =(ω − 1)sW − 3tT, which belongs to I univ . We also have (1 − ω 2 ){(1 + s)ϕ 2 −ωuψ} = p 1 B 1 − ωp 2 B 2 + ω 2 p 3 B 3 ∈ I univ . Hence (1 + s)ϕ 3 − ωuW ∈ I univ .From π ∗ m ∩ I univ = {0} it follows thatIt follows thatπ ∗ ψ 3 = ωuπ ∗ W, (1 + s)π ∗ ϕ 3 = ωuπ ∗ W,(ω − 1)sπ ∗ W =3tπ ∗ T,π ∗ ϕ 3 − π ∗ ψ 3 = 3(2ω +1)π ∗ T 2 .π ∗ W = ω2 u 21+s ,π∗ ϕ 3 =u 3(1 + s) 2 ,π∗ ψ 3 = u31+s ,3π ∗ T =(1− ω 2 u 2 s)(1 + s)t , (1 − ω)su = t2 .Though the relation (1 − ω)su = t 2 looks singular at P (ρ 2 ,ρ 3 ), the points = t = 0 of Hilb G (A 2 ), it is not singular at all because s and t/s are regularparameters at P (ρ 2 ,ρ 3 ).The condition W ≠ S 1 V 5 (ρ 2 ),S 1 V 5 (ρ ′ 2),S 1 V 5 (ρ ′′2) implies s(1 + s) ≠0,s ≠∞. The parameters s, t and u are related by (1 − ω)su = t 2 . Hence π ∗ m =(π ∗ T )=(u 2 /t) =(t 3 ) along E(ρ 3 ) \ Sing(E), whence E fund =3E(ρ 3 ) there.We see mod n XtB 1 = tp 1 (ϕ + ωψ)+stp 1 ϕ + t 2 p 2 p 3 + tup 1= tp 1 (ϕ + ωψ)+stp 1 ϕ + t 2 p 2 p 3 − 3ω(1 + s)p 1 T= −ω 2 (s − ω)p 3 B 2 +(s − ω 2 )p 2 B 3 ,tB 2 = ω(s − ω)p 1 B 3 − (s − ω 2 )p 3 B 1 ,tB 3 = −ω(s − ω)p 2 B 1 +(s − ω 2 )p 1 B 2 .1This proves tB i =0inV. Hence V is S[s, ]-free of rank three. Thiss(1+s)completes the proof of freeness of V over E(ρ 3 ) \ Sing(E) for E 6 .6.2.8. We explain very briefly the most complicated case of E 7 , that is, theρ 4 -case. The finite group G involved is the binary octahedral group O, andthe invariant ring of G is generated by homogeneous polynomials of degree 8,12, and 18, where we note that 18 is also the Coxeter number of E 7 .Any point I = I 4 (W ) ∈ E(ρ 4 ) \ Sing(E) is given by an irreducible G-submodule of V 9 (ρ 4 ) with W ≠ S 1 V 8 (ρ ′′2 ),S 1V 8 (ρ 3 ),S 1 V 8 (ρ ′ 3 ) under the notationof section 5. Then we seeHom S (I/I 2 , S/I) = Hom (W, V 9 (ρ 4 )/W ) ⊕ Hom (W, V 7 (ρ 4 )).The versal deformation I univ is generated over E(ρ 4 ) \ Sing(E) by five elementsB 1 ,B 2 ,B 3 ,B 4 ,A := A 8 − π ∗ (A 8 ), where A 8 = W is the same as W of

MCKAY CORRESPONDENCE 33E 6 and the elements B i are of the formB i = B i1 + sB i2 + tB i3 + uB i4 + vB i5such thatV 9 (ρ 4 )={B i1 ,B i2 ; i =1, 2, 3, 4},V 13−2k (ρ 4 )={B ik ; i =1, 2, 3, 4} (k =3, 4, 5).See [IN99], Table 13 for ρ 4 -factors of S McKay (G). Moreover over E(ρ 4 ) \Sing(E), u (resp. v) is a unit multiple of t 2 (or resp. t 3 ), while π ∗ (A 8 )isaunit multiple of t 4 , and π ∗ m =(A 8 )=(t 4 ). Then we can prove tB i =0inVin the same manner as before. In fact, the proof goes roughly as follows. Theterm tB 1 is the sum of B ik , whose last term tvB i5 is a multiple of t 4 . HencetvB i5 can be replaced mod n X by a unit multiple of A 8 B i5 . Then we see thattB 1 − tvB i5 + (the unit multiple of A 8 B i5 ) is a sum of B j over m. In otherwords, tB 1 is a sum of B j and A over m. Since ρ 4 is irreducible, this impliesthat tB i is also a sum of B j and A over m for any i. Thus we can prove thatV is O E(ρ4 )-free over E(ρ 4 ) \ Sing(E).We can write down precisely the versal deformations of I ∈ E(ρ) \ Sing(E)similarly for any ρ of E 7 . By this, we can prove freeness of V along E\Sing(E).We omit the details of E 7 -case because we need more notation.6.3. Isomorphism at I(ρ, ρ ′ ). In this subsection we prove that (2) is anisomorphism at any singular point I := I(ρ, ρ ′ )ofE.6.3.1. First we consider the pair ρ = ρ 1 and ρ ′ = ρ 2 in the D 5 -case. ThenI univ is given in (1). It is clear that V is generated by those elements whosespecializations at s = t = 0 are just the generators of I belonging to V 6 (ρ 1 )+V 3 (ρ 2 ). Let B = x 6 −y 6 +sxy, C 1 = x 2 y +ty 5 − st x and C 2 = −xy 2 −tx 5 − st y. 2The ideal π ∗ (m) is generated by φ 2 , hence by s 2 t. We first seetB = −(yC 1 + xC 2 )=0 inV.Next we prove sC 1 =0inV. We compute mod mI univ + n X :sC 1 = sx 2 y + sty 5 − s2 t2 x= xB − x(x 6 − y 6 )+sty 5 − s2 t2 x= −xA 6 +2xy 6 + sty 5 − s2 t2 x (∵ xB ∈ mI univ)= −x(A 6 + s2 t2 + t3 2 A2 6 )+2xy6 + sty 5 + t3 2 xA2 6=2xy 6 + sty 5 +2txA 2 4 = −2y4 C 2 =0,and similarly sC 2 = 0. This provesV = O E(ρ1 )B + O E(ρ2 )C 1 + O E(ρ2 )C 2 .

34 IKU NAKAMURASince V is O E(ρ) -free of rank deg ρ over E(ρ)\Sing(E), the upper-semicontinuityshows that O E(ρ1 )B is O E(ρ1 )-free of rank deg ρ 1 (= 1) at s = t = 0, whileO E(ρ2 )C 1 + O E(ρ2 )C 2 is O E(ρ2 )-free of rank deg ρ 2 (= 2) at s = t = 0. Thisproves that (2) is an isomorphism at I(ρ 1 ,ρ 2 ):s = t =0.6.3.2. If ρ = ρ 2 and ρ ′ = ρ 3 in the D 5 -case, then I univ is generated by thoseelements whose specializations at s = t = 0 belong to V 5 (ρ 1 )+S 1 V 3 (ρ 2 ).Let R = C[[s, t]][x, y]. By subsection 4.7, the versal deformation I univ of Iis given by I 3 , which is, as the ideal of R, generated by the elementsB 1 := y 5 + sx 2 y + λx, B 2 := −x 5 − sxy 2 + λy,C 1 := x 3 y + ty 4 + stx 2 ,C 2 := −xy 3 + tx 4 + sty 2 ,A 6 +2sA 4 ,A 8 − 2λA 4 ,A 4 − tλ,where λ =s2 t. Let A := A1+t 2 4 − tλ. We will check sC 1 = sC 2 = 0 andtB 1 = tB 2 =0inV. In fact, in S[[s, t]] we haveIt follows thattB 1 = yC 1 − xA, tB 2 = −xC 2 − yA,sC 1 =(1+t 2 )xB 1 − txyC 1 + y 2 C 2 ,sC 2 =(1+t 2 )yB 2 − txyC 2 + x 2 C 1 .V = O E(ρ1 )B + O E(ρ2 )C 1 + O E(ρ2 )C 2 .Since V is O E(ρ) -free of rank deg ρ over E(ρ)\Sing(E), the upper-semicontinuityshows that O E(ρ2 )B 1 + O E(ρ2 )B 2 is O E(ρ2 )-free of rank deg ρ 2 at s = t =0,while O E(ρ3 )C 1 + O E(ρ3 )C 2 is O E(ρ3 )-free of rank deg ρ 3 at s = t = 0. Thisproves that (2) is an isomorphism at I(ρ 2 ,ρ 3 ):s = t =0.6.3.3. In this subsubsection we consider one of the most complicated caseof E 6 where ρ = ρ 2 and ρ ′ = ρ 3 with deg(ρ 2 ) = 2 and deg(ρ 3 ) = 3. Forthe notation see Figure 4. Let I := P (ρ 2 ,ρ 3 ) and W =(S 1 V 5 (ρ 2 ))[ρ 3 ]. ThenI := I(ρ 2 ,ρ 3 )=W + ∑ 11k=7 S k + n and V (I) =W ⊕ V 7 (ρ 2 ). MoreoverHom S (I/I 2 , S/I) = Hom (W, V 6 (ρ 3 /W )) ⊕ Hom (V 7 (ρ 2 ),V 5 (ρ 2 )).The versal deformation I univ of I(ρ 2 ,ρ 3 ) is generated byB 1 ,B 2 ,B 3 ,C 1 ,C 2 ,A,where B i (i =1, 2, 3) is the same as in subsubsection 6.2.8, andC 1 := −s 2 ϕ + vγ 1 + wx, C 2 := s 1 ϕ + vγ 2 + wy,A := T − π ∗ (T )=T − 12ω +1 sw = T − 11+s s2 v 3 .

MCKAY CORRESPONDENCE 35The parameters s and v are regular parameters of Hilb G (A 2 )atI. Theother parameters are related as follows :t = −(1 − ω)sv, u =(1− ω)sv 2 2ω +1,w = −1+s sv3 ,whence u = −tv, uv = −ω 2 (1 + s)w and (2ω +1)π ∗ (T )=sw.Then we see in S[s, v],vB 1 = −ω 2 (1 + s)(xC 1 − yC 2 )+ 11 − ω (ωp 2B 3 − p 3 B 2 ),vB 2 = −ω 2 (1 + s)(xC 1 + yC 2 ) − 11 − ω (ωp 3B 1 − p 1 B 3 ),vB 3 = −ω 2 (1 + s)(yC 1 + xC 2 )+ 11 − ω (ωp 1B 2 − p 2 B 1 ),sC 1 = 11 − ω (yB 1 + yB 2 − xB 3 ) − (2ω +1)xA,sC 2 = 11 − ω (xB 1 − xB 2 + yB 3 ) − (2ω +1)yA.This proves V = V⊗O E(ρ2 ) + V⊗O E(ρ3 ).6.3.4. In general, let I ∈ P (ρ, ρ ′ ) be a singular point of E. Then as we sawabove, we have an isomorphismV [I] = V [I] ⊗ O E(ρ) ⊕V [I] ⊗ O E(ρ ′ )locally at I. This proves the global isomorphism over EV≃⊕V⊗O E(ρ) .ρ∈Irr(G)6.4. The sheaf V⊗O E(ρ) . It remains to prove V⊗O E(ρ) ≃ ρ ⊗ O E(ρ) (−1).6.4.1. Let V be a two-dimensional C-vector space, and P(V ) the projectiveline of one-dimensional subspaces of V . There is a universal family of onedimensionalsubspaces of V parametrized by P(V ), which we denote W univ .This is a line bundle (an invertible sheaf) on P(V ). There is an exact sequenceof O È(V ) -modules:0 → W univ → V ⊗ O È(V ) → V ⊗ O È(V ) /W univ → 0.This implies that W univ ≃ O È(V ) (−1) because the bundle is twisted linearly.Let V (ρ) := S McKay (G)[ρ] = V h−d(ρ) + V h+d(ρ) ≃ ρ ⊕2 , and P(V (ρ)) theprojective line of nonzero irreducible G-submodules of V (ρ). Then V (ρ) ≃ρ ⊗ V and P(V (ρ)) ≃ P(V ). It is obvious that ρ ⊗ W univ yields a universalfamily W univ (ρ) of nonzero G-submodules of V (ρ) parametrized by P(V (ρ)) (≃P(V )). We see W univ (ρ) ≃ ρ ⊗ O È(V (ρ)) (−1).

36 IKU NAKAMURA6.4.2. Let I ∈ E(ρ). Then by identifying E(ρ) with P(V ), on any Zariskiopen subset U := Spec A of E(ρ) we have∑I univ ⊗ A = W univ (ρ)+V ⊗ A + S † ⊗ A,V ⊂Quiv(ρ),v≄ρand by subsection 6.3V⊗A = W univ (ρ) ⊗ A = W univ (ρ).In other words, V⊗O E(ρ) ≃ W univ (ρ) ≃ ρ ⊗ O E(ρ) (−1).This completes the proof of Theorem 3.9.References[Artin66] M. Artin, On isolated rational singularities of surfaces. Amer. J. Math. 88(1966) 129–136.[Bourbaki] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4,5 et 6 Masson, Paris–New York,1981.[Fogarty68] J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90(1968) 511–521.[Fujiki83] A. Fujiki, On primitively symplectic compact Kähler V -manifolds, Progressin Mathematics, Birkhäuser 39 (1983) 71–250.[GSV83] G .Gonzalez-Sprinberg, J. Verdier, Construction géométrique de la correspondencede McKay, Ann. scient. Éc. Norm. Sup. 16 (1983) 409–449.[IN99] Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends inalgebraic geometry (Proc. of the July 1996 Warwick European Alg. Geom.Conf.), Cambridge University Press (1999) 151-233.[Klein] F. Klein, Vorlesungen über das Ikosaeder, Birkhäuser B.G. Teubner, 1993.[McKay80] J. McKay, Graphs, singularities, and finite group, Proc. Symp. Pure Math.,AMS 37, 1980, pp. 183–186.[Nakamura01] I. Nakamura, Hilbert schemes of abelian group orbits, Jour. Alg. Geom. 10(2001), 757-779.[Schur07] I. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durchgebrochene lineare Substitutionen, J. Reine Angew. Math. 132 (1907) 85–137.[Slodowy80] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Note inMath. 815, Springer-Verlag, 1980.