Simple Lie Algebras and Singularities

Home , Invariant polynomial, McKay graph, Patrick du Val

Master thesis

Simple Lie algebras and singularities

Nicolas Hemelsoet

supervised by Prof. Donna Testerman and Prof. Paul Levy

January 19, 2018 Contents

1 Introduction 1

2 Finite subgroups of SL2(C) and invariant theory 3

3 Du Val singularities 10 3.1 Algebraic surfaces ...... 10 3.2 The resolution graph of simple singularities ...... 16

4 The McKay graphs 21 4.1 Computations of the graphs ...... 21 4.2 A proof of the McKay theorem ...... 26

5 Brieskorn’s theorem 29 5.1 Preliminaries ...... 29 5.2 Quasi-homogenous polynomials ...... 36 5.3 End of the proof of Brieskorn’s theorem ...... 38

6 The Springer resolution 41 6.1 Grothendieck’s theorem ...... 42

6.2 The Springer ﬁber for g of type An ...... 45

7 Conclusion 46

8 Appendix 48

1 Introduction

Let Γ be a binary polyhedral group, that is a ﬁnite subgroup of SL2(C). We can obtain two natural 2 graphs from Γ: the resolution graph of the surface C /Γ, and the McKay graph of the representation Γ ⊂ SL2(C). The McKay correspondance states that such graphs are the same, more precisely the 2 resolution graphs of C /Γ will give all the simply-laced Dynkin diagrams, and the McKay graphs will give all the aﬃne Dynkin diagrams, where the additional vertex is the vertex corresponding to the trivial representation.

2 Γ For computing the equations of C /Γ := Spec(C[x, y] ) we need to compute the generators and Γ −1 relations for the ring C[x, y] . As an example, if Γ = Z/nZ generated by g = diag(ζ, ζ ) with −1 ζ a primitive n-th root of unity, the action on C[x, y] is g · x = ζ x and g · y = ζy (this is the contragredient representation). The invariants are u = xn, v = yn and w = xy, the unique relation

1 n is uv = w which is a singularity of type An−1. Any binary polyhedral group will give similarly a surface with an isolated singularity at the origin, called a Du Val singularity or a simple singularity. The McKay correspondance states that the resolution graph of a Du Val singularity is a Dynkin diagram and that the McKay graph of the corresponding binary group is the corresponding aﬃne Dynkin diagram.

This already gives a link between simple Lie algebras of type ADE and simple singularities, but in fact they arise naturally inside the Lie algebra itself ! More precisely, let g be a complex

Lie algebra of type An,Dn,E6,E7 or E8, and G the connected, simply-connected corresponding complex Lie group. We will look at the adjoint action of G on g. This induces an action of G G on C[g] and since G is reductive, by Hilbert’s theorem C[g] is ﬁnitely generated. We get a G morphism of algebraic varieties F : g → g/G (where g/G := Spec(C[g] ) is the categorical quo- G tient) corresponding to the inclusion C[g] ⊂ C[g]. In the case of g = sln the map is simply n−1 2 3 n χ : sln → C ,M 7→ (tr(M ), tr(M ),..., tr(M )). We will see that dim g/G = r where r is the rank of G. Let x ∈ g be a subregular (meaning that dim ZG(x) = r+2) nilpotent element. Let S be a transverse slice at x, that is a subvariety S of g such that the natural map TxS ⊕ Tx(G · x) → g is ∼ surjective (we identify Txg = g). Then, if N (g) is the nilpotent cone of g, we obtain that N (g) ∩ S is a Du Val singularity of the same type as the Lie algebra ! This gives a connection between simple Lie algebras and simple surface singularities. Moreover, the restriction of F to S is the semiuniversal deformation of the singularity, meaning in some sense that all deformations of this singularity can be obtained by pullback of the map S → g/G. This theorem was conjectured by Grothendieck and showed by Brieskorn for the ADE case. Slodowy extended it to Lie algebras of type BCFG.

∗ The proof of the Brieskorn’s theorem uses an action of C on S and on g/G, with respect to ∗ which the quotient morphism is C -equivariant. This will give us concrete information about the morphism S → g/G by computing the associated weights and we will be able to identify this mor- ∗ phism with the semiuniversal deformation, which also admits an action of C with the same weights.

Next, we will use the Springer resolution to present a nice way of resolving the singular ﬁbers of the map χ : g → g/G. There is a very nice interaction between such resolutions and ﬂag varieties.

Here is how this master project is organized : in section 1 we compute the equation of the 2 surface C /Γ where Γ ⊂ SL2(C) is a ﬁnite group. In section 2 we study the geometry of such surfaces, in particular computing their minimal resolution. In section 3 we compute the McKay graphs of the binary tetrahedral groups. In section 4 we turn to Lie algebras and study the map χ : g → g/G and its restriction to a transverse slice σ : S → g/G, in order to prove Brieskorn’s

2 theorem. In section 5, we present Springer’s resolution of the nilpotent variety, which provides a nice and natural resolution of the Du Val singularities.

2 Finite subgroups of SL2(C) and invariant theory

Let Γ be a ﬁnite subgroup of SL2(C). The list of such groups up to conjugacy is well known, cf [11] , chapter 1 (in fact we follow closely this chapter of the lecture notes by Dolgachev in this subsection, especially when we use relative invariants and Grundformen) or [32] for more details.

By a standard averaging argument, any finite subgroup of SL2(C) is conjugate to a finite subgroup ∼ of SU2. There is a canonical map SU2 → PSU2 = SO3, and in fact any finite group in SU2 is the pullback of a finite subgroup of SO3 (with the exception of cyclic groups of odd order) ! Later we will write explicit generators for each group, so now we just write the list :

• Type An : Γ is cyclic of order n

• Type Dn : Γ has order 4(n − 2), and is called the binary dihedral group of order 4(n − 2).

• Type E6 : Γ has order 24, is called the binary tetrahedral group and is the pullback of

A4 ⊂ SO3.

• Type E7 : Γ has order 48, is called the binary octahedral group and is the pullback of

S4 ⊂ SO3.

• Type E8 : Γ has order 120, is called the binary icosahedral group and is the pullback of

A5 ⊂ SO3.

We will quickly sketch how to ﬁnd this list, since the proof will also give us another useful fact concerning the orbits of the action of Γ on S2. We ﬁrst claim that there is an exact sequence

1 → {±1} → SU2 → SO3 → 1

To show this, a possible way is to use that SU2 is isomorphic to the group of quaternions of norm 1, and then acts by conjugaiton on the ”pure quaternions” ai + bj + ck which form a real vector ∼ 3 space V = R . In fact, it turns out that this morphism SU2 → GL3(R) has image SO3, because 2 an element of SU2 preserve norm, and the restriction of the norm of C on V identify with the usual scalar product. Next, we claim that every finite subgroup in SU2 (except cyclic groups of odd order) is the pullback of a finite group in SO3. This can be proven by showing that finite subgroups

Γ ⊂ SU2 which do not contain −1 are exactly cyclic odd subgroups. We write Γ for the image of 3 3 Γ ⊂ SU2 by the surjection SU2 → SO3 (topologically this is the covering map S → RP ). Now admitting the previous facts, to classify ﬁnite subgroups of SU2 we can just look at ﬁnite subgroups

3 of SO3.

2 Now let Λ a non-trivial ﬁnite subgroup of SO3. Let P be the set of points x ∈ S with stabΛ(x) 6= 1. We will show the following lemma, essential for the classiﬁcation :

Lemma 2.1. Λ acts on P and there are only 2 orbits if Λ is cyclic, and 3 if Λ is not cyclic.

Proof. Let us show that Λ acts on P : if x ∈ P , that is there is h ∈ Λ\{1} with hx = x by definition, then for g ∈ Λ(ghg−1)(gx) = ghx = gx, that is gx ∈ P , so indeed Λ acts on P . 1 P g 1 If N is the number of orbits, then we have N = |Λ| g∈Λ |P | so N = |Λ| (2(|Λ| − 1) + |P |) : indeed, if g = 1 every x ∈ P is fixed, else there are exactly two fixed points. Picking a representative xi in P each orbit we can write |P | = i |Λ · xi|. Using that |stab(xi)| = |Λ|/|Λ · xi| we have

1 X 1 X 1 2 1 − = N − = 1 − |Λ| |stab(x )| |stab(x )| i i i i Now, the left hand side is in [1, 2). On the other hand, the stabilizer has order at least 2 for all orbits, so we have 1 ≤ 1 − 1 ≤ 1, showing that 2 ≤ N ≤ 3. 2 —stab(xi)| If N = 2 then we obtain 2 = |P |, i.e there are two orbits in P and x1, x2 are on the same axis so Λ is cyclic. Conversely, it is clear that if Λ is cylic then there are exactly two orbits.

This gives a good idea of how start the classiﬁcation of ﬁnite subgroup of SO3. Of course a more careful analysis is needed when there are three orbits, see [32] for the complete proof.

Γ For each binary tetrahedral group Γ ⊂ SL2(C) we will compute the invariant ring C[x, y] . It is not obvious but nevertheless true that this ring is finitely generated. In fact, it holds more generally for reductive groups, see 5.1. Of course for finite groups we don’t need the machinery of algebraic groups. A theorem of Noether, in characteristic zero (or if the characteristic of the ground field k does not divide |Γ|), says that the invariant ring k[x]Γ is generated by the traces of P monomials of degree less than |Γ|, where the trace of a monomial x is g∈Γ g · x. In particular k[x]Γ is finitely generated, and Noether’s theorem gives an explicit algorithm to compute generators.

2 Returning to binary groups, we have a natural action of Γ on C , giving the following (contra- a b gredient) action on C[x, y]: g · x = (dx − by) and g · y = (ay − cx), where g = c d . Of course, computing all the traces would be onerous, so in practice we need another method. We will see Γ that the ring C[x, y] will always be generated by 3 polynomials, satisfying a unique relation, and 3 the corresponding surface X ⊂ C will have an isolated singularity at the origin.

4 Type An ∼ ζ 0 Here, Γ = Z/nZ and Γ is generated by g = 0 ζ−1 where ζ is a primitive n-th root of unity. The −1 corresponding action on C[x, y] is g · x = ζ x and g · y = ζy. Since Γ is a subgroup of the diagonal matrices, every monomial is an eigenspace for Γ, so invariant polynomials are sums of invariant monomials. A monomial xkyj is invariant when ζj−k = 1, that is k − j = 0 mod n. Dividing by some power of yn and xn we can assume that 0 ≤ k, j ≤ n. If k 6= 0, n then our monomial is (xy)k. If k = 0 then our monomial is yn and ﬁnally if k = n, j < n our monomial is xn. Finally all the invariant monomials are generated by u = xn, v = xy or w = yn.

n The invariant ring is therefore C[u, v, w]/(uw−v ). A change of variables u = X+iY, w = X−iY and aZ = v where an = −1 gives the relation X2 + Y 2 + Zn = 0 (it seems there is no general agreement for choosing the form of the equation). For a general binary polyhedral group, one can’t expect Γ to be composed of diagonal matrices : we need another method for other binary groups. We will use the relative invariants and the Grundformen, to be deﬁned in the next subsection. Roughly speaking, the Grundformen will play the same role as the monomials in our previous computations : more precisely we will show that any invariant polynomial is a sum of invariant monomials in the Grundformen. After computations of Grundformen we will be able to ﬁnish Γ computing generators of C[x, y] . Surprisingly these relations were already known to Klein !

Disgression on relative invariants and Grundformen

We will present the main tool for computing the invariant ring, namely the Grundformen. The 1 1 ﬁrst idea is that one should look at the action of Γ on P . We write x for the point [x : 1] ∈ P , and [1 : 0] is called ∞. Instead of invariant polynomials, we will look at polynomials with invariant set of zeroes. Such polynomials are exactly homogenous f ∈ C[x, y] with gf = χ(g)f for some ∗ morphism χ :Γ → C and all g ∈ Γ. In this setting, a polynomial f is invariant if and only if χ(g) = 1 for all g ∈ Γ. We will ﬁnd a simple set of generators for the relative invariant polynomials, Γ and then we will be able to compute generators for the invariant ring C[x, y] .

We follow closely the first chapter of [11]. We also recall that there is a 1-1 correspondence 1 between homogeneous polynomials and effective divisors in P (i.e an element in the free abelian 1 group Z[P ] with all coefficients positive).

1 Deﬁnition 2.2. An exceptional orbit is an orbit of Γ (for the induced action on P ) of cardinality smaller than |Γ|.A relative invariant is a homogeneous polynomial f ∈ C[x, y] such that g · f = ∗ χ(g)f for all g ∈ Γ and some morphism χ :Γ → C .A Grundform (plural : Grundformen) is a relative invariant f such that V (f) is an exceptional orbit where V (f) is the set of zeroes of f. 5 The morphism χ is called a character of Γ. For any orbit O, we obtain a natural relative Q invariant fO(x, y) = (t:s)∈O(sx − ty), called the relative invariant associated to O. In fact fO ∈ d1 dn C[x, y] is deﬁned up to a scalar multiple. Any relative invariant can be written f1 . . . fn where n (d1, . . . , dn) ∈ N and f1, . . . , fn are the associated relative invariant of some orbits O1,..., On (again up to a scalar factor).

Now let O be an orbit of cardinality |Γ|, that is a non-exceptional orbit, and fO the associated relative invariant of O. By deﬁnition V (fO) = O and deg(fO) = |Γ| since O is not exceptional. Let f1, f2 be Grundformen, such that the corresponding orbits have cardinalities |Γ|/e1 resp. |Γ|/e2, that is, ei = #(StabΓ(xi)) for some xi ∈ Oi. We write χ1, χ2 for the corresponding characters. We e1 e2 e1 e2 will make the key hypothesis (♣) that χ1 = χ2 . Then, there is a, b ∈ C, such that Φ := af1 +bf2 vanishes on some [t : s] ∈ O. But

e1 e1 e2 e2 e1 e1 e2 Φ(g · (t, s)) = aχ1 f1(s, t) + bχ2 f2(s, t) = χ1 (af1(s, t) + bf2(s, t) ) = 0

So Φ vanishes actually on all O ! In particular, fO is a scalar multiple of Φ since they are both polynomials with same degree and same set of zeroes. We obtained

1 Proposition 2.3. Let Γ be a binary polyhedral group, acting on P in the usual way. Assume that there exists exceptional orbits O1, O2 such that (♣) holds. Then any relative invariant polynomial F is a polynomial in some Grundformen.

Now that we know that under the key hypothesis (♣) the Grundformen generate the ring of relative invariant polynomials, we need to know how to go back to the invariant polynomials. ∗ Fix a character χ :Γ → C and d ∈ N. The relative invariants of degree d with character χ form a vector space Wχ ⊂ C[x, y]d.

We claim that if χ1, . . . , χk are distinct characters then Wχ1 + ··· + Wχk = Wχ1 ⊕ · · · ⊕ Wχk . In- deed, let Φi ∈ Wχi be non-zero elements, and A = c1Φ1 + ··· + ckΦk = 0 for some ci ∈ C. Without loss of generality we may assume that any subset is linearly independent, in particular all the ci are non-zero. If g ∈ Γ is such that χ1(g) 6= χ2(g) we have 0 = gA−χ1(g)A = c2(χ1(g)−χ2(g))Φ2 +··· =

0. The last equality is a linear relation for Φ2,..., Φk contradicting our assumption. We have es- tablished :

Pk Proposition 2.4. Let F = i=1 ciΦi ∈ C[x, y] where Φi ∈ Wχi for all i, and such that χ1, . . . , χk are distinct characters. Then, if F is invariant, each of the Φi are invariant under Γ, that is χi is the identity character. P Proof. Indeed, if g ∈ Γ we have by hypothesis 0 = F − g · F = i ci(1 − χi(g))Φi = 0. Since Φi are linearly independent if follows that 1 = χi(g) for all i and g.

This proposition is very helpful, because it tells us that to understand which polynomials are invariant, we just need to understand which monomials in the Grundformen are invariant, so in 6 some sense this is close to the ﬁrst method we used for the cyclic group.

Now we will compute the Grundformen for the non-cyclic binary groups. In particular, in order to apply the previous propositions we must check that there exists two diﬀerent exceptional orbits with characters χ1, χ2 such that e1 e2 χ1 = χ2 (♣) where ei is the cardinality of the stabilizer of the corresponding orbit.

Type Dn+2

ζ 0 If Γ is the binary dihedral group of order 4n, the generators are g = 0 ζ−1 where ζ is a 2n-th 0 i primitive root of unity and h = i 0 . To find the exceptional orbits we look at fixed points of g or h. The fixed points of g are 0, ∞; and since h permutes them we obtain a first exceptional orbit O1 = {0, ∞}. The fixed points of h are [1 : 1] and [1 : −1]. For example, we have g · [1 : 1] = [ζ : ζ−1] = [ζ2 : 1] so we see that the orbit of [1 : 1] by G is the set 1 n 1 n {[λ : 1] ∈ P : λ = 1} = O2. Finally, the last exceptional orbit O3 = {[λ : 1] ∈ P : λ = −1}. Q n n The corresponding Grundformen are Φ1 = xy,Φ2 = λn=1(x − λy) = x − y and similarly n n n Φ3 = x + y . The associated characters are χ1(g) = 1, χ1(h) = −1; χ2(g) = −1, χ2(h) = i and n 4 4 2 2 χ3(g) = −1, χ3(h) = −i . Moreover we easily check that e2 = e3 = 4 and χ2 = χ3, in fact χ2 = χ3 so (♣) is verified.

There is an obvious set of invariant monomials (from the character table) which clearly generate 2 2 2 4 4 the Γ-invariant polynomials, given by Φ1, Φ1Φ2, Φ1Φ3, Φ2, Φ3 and Φ2Φ3 if n is odd. We take F1 = 2 2 n n n n 2n 2n 2 n n 2 Φ1 = (xy) ,F2 = Φ2Φ3 = (x + y )(x − y ) = x − y and F3 = Φ1Φ2 = xy(x − y ) and will show that any other invariant monomials is a polynomial in F1,F2,F3. We have

n+1 2 n n 2 n n 2 n 2 Φ1Φ3 = xy(x + y ) = xy((x − y ) + 4(xy) ) = F3 + 4F1

4 4 We just need to show that Φ2 and Φ3 are generated by F1,F2 and F3. But

n−1 4 n n 4 4n 4n n n 2n 2n 2n 2n 2n 2n 2 n n 2n 2n n n 2 2 Φ2 = (x −y ) = x +y −4x y (x +y )+6x y = (x −y ) −4x y (x +y −2x y ) = F2 −4F1 F3

Finally

n−1 4 n n 4 n n 2 n n 2 2 n 2 2 2 n Φ3 = (x + y ) = ((x − y ) + 4x y ) = (Φ2 + 4Φ3 ) = F2 + 4F3F1 + 16F1

Γ This shows that F1,F2,F3 generate C[x, y] . We still need to ﬁnd the relation between the generators. We observe that

2 2 n n 2 2 2n 2n 2 2 n n 4 2n 2n 2 F3 − F1F2 = (xy(x − y )) − (xy) (x − y ) = (xy) ((x − y ) − (x − y ) ) 7 = (xy)2(−4x3nyn − 4xny3n + 6x2ny2n + 2x2ny2n) = −4(xy)n+2((xn − yn)2)

n+1 2 but the last term is exactly −4F1 F3, so we obtain the relation

n+1 2 2 2 F3 − F1F2 + 4F3F1 = 0

We write this as n+1 2 2 n+1 2 (F3 + 2F1 ) − 4F1 − F1F2 = 0 n+1 2 so replacing F3 by F3 + 2F1 and scaling gives

Γ ∼ 2 2 n C[x, y] = C[X,Y,Z]/(Z + X(Y + X ))

n+1 2 where X = F1,Y = F2,Z = F3 + 2F1 .

Now assume that n is even. This time we start with the invariant polynomials

2 2 2 n n 2 n n n n F1 = Φ1 = (xy) ,F2 = Φ2 = (x − y ) ,F3 = Φ1Φ2Φ3 = xy(x − y )(x + y )

2 Fortunately this is easier : indeed Φ1Φ2, Φ1Φ3, Φ2Φ3 are not invariant and Φ3 is invariant, so 2 we just need to express Φ3 using F1,F2 and F3. But

2 n n 2 n n 2 n n 2 n Φ3 = (x + y ) = (x − y ) − 4x y = Φ2 − 4Φ1

So we are done and we just need to ﬁnd the relation that F1,F2 and F3 satisfy. Instead n/2 we slightly change the generator F2 with the transformation F2 → F2 + F1 (n is even). Now 2n 2n F2 = x + y . With these new invariants the relation is

2 n+1 2 F1F2 = 4F1 + F3

A suitable change of variables of the form X = aF1,Y = bF2 and Z = cF3 gives the equation

Z2 = X(Y 2 + Xn)

Type E6

We now consider the binary tetrahedral group BT, generated by 1 g = i 0 , h = 0 i and k = 1 i 0 −i i 0 1 − i 1 −i As usual we will look for the exceptional orbits. We know that 0, ∞ are the fixed points of g. Since g has order 4 we should get 6 points : indeed, we have k · 0 = 1, k · 1 = i and similary k ·∞ = −1, k ·−1 = −i. We obtain the first orbit O1 = {0, 1, −1, i, −i, ∞}. Now let us look at fixed points of k, since ∞ is not fixed we can write p = [x : 1] and we obtain [x : 1] = [x + i : x − i] that is

8 √ 1+i x(x − i) = x + i. This equation has two solutions x1,2 = (1 ± 3). Since k has order 6 we should 2 √ √ 1+i 1−i ﬁnd 4 elements in these orbits, and indeed applying g or h gives O2 = {± (1+ 3), ± (1− 3)} √ √ 2 2 1+i 1−i and O3 = {± (1 − 3), ± (1 + 3)}. Now we can compute the corresponding Grundformen : 2 2 √ √ 4 4 4 2 2 4 4 2 2 4 Φ1 = xy(x − y ), Φ2 = x − 2i 3x y + y and Φ3 = x + 2i 3x y + y . Finally we can compute the characters. For Φ1, we claim that χ1 ≡ 1, that is Φ1 is invariant. It is easy to check for g and h, so we only do it for k. Recall that we use the contragredient action on C[x, y] so i(x + y) x − y kx = − and ky = 1 − i 1 − i Now we compute

i(x + y)) x − y 1 kΦ = − · · ( ) · ((x + y)4 − (x + y)4) 1 1 − i 1 − i (1 − i)4

(x2 − y2) −1 = − · · (8x3y + 8xy3) = (x2 − y2) · (xy(x2 + y2)) = Φ 2 4 1 4 We have gΦl = hΦl = Φl for l = 2, 3. Let us compute χ2(k). Since (1 − i) = −4 we have −1 √ 1 √ kΦ = · ((x + y)4 + (x − y)4 − 2i 3(x + y)2(x − y)2) = (2x2 + 2y2 + 12x2y2 − 2i 3(x2 − y2)2) 2 4 4

√ 1 √ √ 1 √ 1 − i 3 = (1 − i 3)x4 + (3 + i 3)x2y2 + (1 − i 3)y4 = ( )Φ 2 2 2 2 √ 1−i 3 2 2 This shows that χ2(k) = j where j = 2 . In fact, if we modify the sign in front of x y and 2 3 repeat the previous computation we get kΦ3 = j = j . Since j = 1 and |j| = 1. We obtained the following table :

g h k

χ1 1 1 1

χ2 1 1 j 2 χ3 1 1 j

6 6 3 3 From the table we can also see that χ2, χ3 verify (♣), since χ2 = χ3 (in fact χ2 = χ3). We are now ready for computing the invariants : as noticed before, Φ1 is an invariant, as well as Φ2Φ3. 3 3 3 3 Finally, Φ3 and Φ2 are invariant. We will take F1 = Φ1,F2 = Φ2Φ3 and F3 = Φ2 +Φ3 (for symmetry a b c reasons). So if Φ1Φ2Φ3 is invariant, we can assume that a = 0 and e.g that b = 0 by dividing by c Φ1 and Φ2Φ3. On the other hand, Φ3 is invariant only if 3 divides c. So we only need to see that 3 3 Φ2, Φ3 are generated by F1,F2 and F3. But observe the following equality : √ 2 2 4 4 6 6 −1 3 3 F1 = (xy) (x + y ) − 4x y = (12 3i) (Φ2 − Φ3)

9 This gives us that F1,F2,F3 generate the invariant ring, but also the relation : indeed up to 2 3 3 4 3 2 3 3 scaling we can assume that F1 = Φ2 − Φ3 and then F1 = (Φ2 + Φ3) + 4Φ2Φ3 i.e

4 2 3 F1 = F3 + 4F2 and up to scaling once more we get

BT ∼ 2 3 4 C[x, y] = C[X,Y,Z]/(X + Y + Z )

Type E7 and E8

Since the computations for E7 and E8 are diﬃcult, for simplicity we will skip it. The interested reader may consult [11] or [32]. The literature is very rich, especially about the icosahedron since there are relations with the equation of ﬁfth degree. The historical reference is the book by Klein

Conclusion

To summarize, we obtained the following list of algebraic surfaces :

2 2 n+1 • An : x + y + z = 0 (n ≥ 1)

2 2 n−1 • Dn : x + zy + z = 0 (n ≥ 4)

2 3 4 • E6 : x + y + z = 0

2 3 3 • E7 : x + y + yz = 0

2 3 5 • E8 : x + y + z = 0

These surfaces are called Du Val singularities, rational double points, Kleinian singularities, simple singularities ...They have very interesting geometric properties, we will now discuss them.

3 Du Val singularities

3.1 Algebraic surfaces

We now turn to algebraic surfaces. The main result here will be the computation of the resolution graph of all the Du Val singularities. First, we need to deﬁne the blow-up of an algebraic variety and the self-intersection of a curve. The most comprehensive reference about algebraic surfaces I know is [38].

10 Blow-up

Let X be a singular algebraic variety. We want to ﬁnd a good smooth approximation of X.A formal deﬁnition is given by the notion of a resolution of singularities :

Deﬁnition 3.1. Let X,Y be algebraic surfaces over C. A morphism π : Y → X is a resolution of singularities if Y is smooth and f is proper and birational, that is there exist two open set U ⊂ Y,V ⊂ X with V = π(U) such that π : U → V is an isomorphism.

Since we are working over the field of complex numbers we use the topological definition of a proper map, that is for all compact K ⊂ X, its preimage f −1(K) should be compact. There is a more general definition of proper which works for arbitrary field, see [22], chapter II, section 4. Over the complex numbers, the two definitions coincide, but it is non-trivial : the proof is in [21], page 323 in the old edition or 245 in the new, proposition 3.2 in XII.3, ”Comparaison des propriétés de morphismes”. A resolution of singularities always exists for algebraic curves. In fact, over a field of characteristic zero there is always a resolution, by a deep theorem of Hironaka (a proof can be found in [28]). Even if we know the existence, it is not easy in general to compute resolutions. The simplest transformation which could resolve a singularity is a blow-up : later, we will see that the Du Val singularities are exactly the surfaces which can be resolved using only a finite sequence of blow-ups

X˜ → Xn → · · · → X1 → X.

To understand the concept of a blow-up, we start with an informal example, going back to Newton and Puiseux (who gave an algorithm for resolving the singularities of a curve). Consider 2 2 3 2 the cubic curve C : y = x + x in C . The only singular point is (0, 0) and the singularity is locally the union of two smooth ”branches”. So we should try to separate these branches to obtain a picture similar to picture 1 : 2 2 1 To concretely realize such a picture, we could take the following map f : C \{O} → C × P , 2 f(x, y) = ((x, y), [x : y]) where O = (0, 0), thinking of ((x, y), [x : y]) as a point of C along with its direction with respect to the origin. This map is clearly injective. Now if we take a line L(t) = (at, bt), then lim f(L(t)) = (O, [a : b]). This means that if we try to compactify the image, t→0 1 we should add all the (O, [a : b]) for [a : b] ∈ P . This means exactly that under f different tangents √ at O will go to different points. Returning to our example, a Taylor expansion of x2 + x3 around O gives the following two branches y ∼ x + x2/2 + ... and y ∼ −(x + x2/2 + ... ), where ... 2 are higher order terms. So let’s parametrize our first branch B1(t) = (t, t + t /2 + ... ). Clearly, lim B1(t) = (O, [1 : 1]) and similarly lim B2(t) = (O, [1 : −1]). So we could separate tangents t→0 t→0 and the closure of f(C) is a smooth curve. On the other hand, the definition of a resolution of singularity was a map π : Y → X : we need to invert our parametrization. It can be done the 2 1 following way : first define Y to be the closure of f(C\{O}). Now since Y ⊂ C × P we can 11 2 take the first projection π : Y → C . By construction, π : Y → C is a resolution of singularities. 2 Also, the closure Cf2 of f(C \{O} is given by ((x, y), [a : b] | xb = ya). This includes all the points 1 ((x, y), [x : y]) where (x, y) 6= O and the points (0, [s : t]) for any [s : t] ∈ P . They are precisely the points corresponding to ”directions passing through O” and we just replaced the origin by the 2 set of directions passing through it. This gives a map ϕ : Cf2 → C such that ϕ is an isomorphism −1 ∼ 1 2 1 2 outside O and ϕ (O) = P = (O, [s : t]) ⊂ C × P . The map ϕ is called the blow-up of C at O 1 and the exceptional P is called the exceptional divisor of the blow-up, often written E.

3 n Now we will generalize this example to C (it immediately generalizes to C but only need it 3 for n = 3) and give a bit more terminology. The blow-up of C at O is the algebraic variety deﬁned by 2 3 Cf3 = {(`, z) ∈ P × C : z ∈ `}

More precisely, the equations for Cf3 are ! 2 3 u v w {([u : v : w], (x, y, z)) ∈ P × C : rk ≤ 1} x y z

3 This gives 3 quadratic equations : uy = vx, uz = wx and vz = wy. The projection π : Cf3 → C induces an isomorphism 3 ∼ 3 −1 C \O = Cf\π (O)

2 2 where O is the origin. Indeed, if z ∈ C is non-zero there is a unique line ` ∈ P with z ∈ `, namely 12 ` = [z].

−1 ∼ 2 Let E := π (O), then one has E = P , and E is called the exceptional divisor of the blow-up, it represents the set of all directions at O.

3 0 −1 If X ⊂ C is a surface, let X be the closure of π (X\O) (we take the closure with respect to the euclidean topology). We call X0 the strict transform of X (previously Y in the curve example) and 0 2 E ∩X is called the exceptional curve. Since P can be covered by 3 charts we can cover the blow-up by three charts U1 = Spec(C[x, y/x, z/x]), U2 = Spec(C[x/y, y, z/y]) and U3 = Spec(C[x/z, y/z, z]). 3 For example, C[x, y, z] ⊂ C[x, y/x, z/x] this gives a map U1 → C , (x, y1, z1) 7→ (x, xy1, xz1).

Self-intersection of a curve

We will review some facts from intersection theory on a surface X. A very comprehensive reference is [38], and the classical one is chapter 5 in [22]. We would like to deﬁne an intersection product and if C,D are curves intersecting transversally we would like that C · D := |C ∩ D|. We also want this bilinear extend this to a bilinear product 2 Pic(X) × Pic(X) → Z so in particular we can talk about C for a curve C ⊂ X.

Deﬁnition 3.2. If X is a surface and C,D divisors, we deﬁne C · D := deg(OC (D)). In particular 2 C := deg(OC (D)).

We recall that for a line bundle L, deg(L) := χ(L) − χ(OX ) where χ is the Euler characteristic for sheaves.

For compute this concretely we need the theory of Chern classes, a very comprehensive reference is [?]. We will continue with formal properties before going back to the Du Val singularities.

Definition 3.3. Let L a line bundle on X. Then, L = OX (D) for some divisor D and we define c1(L) = D. This divisor class is called the first Chern class of L.

1 ∗ 2 Remark 3.4. Usually the Chern class is deﬁned as the morphism of groups H (X, OX ) → H (X, C),L =

OX (D) 7→ [D].

In fact define the first Chern class of any vector bundle E, using the associated line bundle Vn det(E) := E (where n = rank(E)). So if E is a vector bundle over X we define c1(E) := c1(det(E)). There are also higher Chern classes but we won’t need it.

Now we list the properties of the ﬁrst Chern class c1 :

13 Proposition 3.5. • Triviality : If E is a trivial vector bundle then c1(E) = 0.

• Tensor product : If L, M are line bundles then c1(L ⊗ M) = c1(L) + c1(M).

• Whitney sum : If 0 → E → F → G → 0 is an exact sequence of vector bundles then

c1(E) + c1(G) = c1(F ).

∗ • Naturality : for any vector bundle E → X and a ﬂat map f : Y → X we have f c1(E) = ∗ c1(f E).

Proof. See [17].

We now look closely at the self-intersection of a closed curve C in a surface X. Intuitively the self-intersection C2 of C should be the number |C ∩C0|, where C0 is a slightly moved 2 2 copy of C. For example, if L ⊂ P is a line, then we should have L = 1 because if we perturb a bit a line we get another line, and two (projective) lines always intersect in one point. In a quadric 1 1 1 0 1 surface Q = P × P , the line L = 0 × P has self-intersection zero as we can take L = ε × P and clearly L ∩ L0 = ∅. On the other hand, sometimes there are ”rigid” curves which can not be moved. The self-intersection can still be computed, and for such curves, it is possible and very frequent that the self-intersection is negative. In fact, continuing our example, it is well known that there are 27 lines on a smooth cubic surface X, and that any such line L satisfies L2 = −1 (in fact, this example is closely related to E6 but this is another story). More generally if L ⊂ X is a line in a 3 2 surface of degree d in P then L = 2 − d. The geometric idea behind the definition of self-intersection is the following : a small deformation of C should correspond to a normal vector field on C, and the zeroes of such a vector field would be exactly the points where C and Cmoved will intersect. A normal vector field is just a section of the line bundle NC/X = OC (C) and we recovered the previous definition of the self-intersection.

We will look at a special resolution of a Du Val singularity, called a minimal resolution. The 2 exceptional curves Ci of this resolution will satisfy Ci = −2 and this property is very important. We give two propositions for motivate the negative self-intersection :

Proposition 3.6 (Castelnovo criterion). Let C a curve on a smooth surface X0. Then, C2 = −1 if and only if C is the exceptional divisor of a blow-up X0 → X where X is a smooth surface.

Proof. This is the theorem 4.51 in [38].

∼ 1 0 2 Proposition 3.7. Let C = P a curve on a smooth surface X . Then, C = −2 if and only if it is the exceptional divisor of a blow-up X0 → X where X is a surface with an isolated quadratic singularity.

14 A singular point on a surface is a quadratic singularity if locally one can write the equation of the singularity f2 + f3 + ··· = 0 where fi are homogenous of degree i and f2 is not identically zero. This will be important later, indeed, if Y → X is a resolution, it is the minimal resolution if and 2 only if Ci = −2 for all exceptional curves Ci. So this is a way of checking if a resolution is minimal. Finally we need two last deﬁnitions before stating the main theorem of this paragraph :

Deﬁnition 3.8. Let Y,X be algebraic surfaces over C and f : Y → X a resolution of singularities. Then, f is minimal if for any other resolution g : Z → X there is a map h : Z → Y with g = h ◦ f.

A well-know criterion ([38]) says that a resolution f : Y → X is minimal if and only if Y has ∼ 1 2 no (−1)-curve, that is curve C = P with C = −1. So it exists whenever arbitrary resolutions exists (and then one can blow-down the other (−1) curve recursively), which is always the case by Hironaka’s theorem. As usual a minimal resolution is unique up to isomorphism.

Deﬁnition 3.9. A resolution f : Y → X of singularities is very good, if the exceptional locus −1 f (0) is a connected union of rational curves Ci, such that Ci ∩ Cj is empty or a point.

Now we state the theorem giving equivalent description of the Du Val singularities :

Theorem 3.10. For an irreducible surface X with a unique isolated point, the following are equivalent :

2 • X is given by C /Γ where Γ is a binary polyhedral group.

• X is a Du Val singularity.

• X can be resolved by a sequence of blow-ups X˜ → Xn → · · · → X, where each step Xk → Xk−1 is the blow-up of a double isolated point.

• The minimal resolution of singularity π : X˜ → X is very good and each exceptional curve Ci ∼ 1 2 satisﬁes Ci = P and Ci = −2.

Proof. Equivalence of the ﬁrst and the second statement was done in the previous section. We will do the implication 2 =⇒ 3) in the next section. For the other implications we refer to [13].

Our next task is to resolve the Du Val singularities by repeated blow-ups. In fact, since the minimal resolution is very good we can even associate a graph to a minimal resolution : the vertex i corresponds to exceptional curve Ci, and there is an edge between i and j if and only if Ci intersects

Cj. Sometimes this is called the dual graph of the resolution.

15 3.2 The resolution graph of simple singularities

Blow-up and the resolution graph of A1

Let us ﬁrst resolve the simplest ADE singularity, namely the quadratic cone given by the equation x2 + y2 + z2 = 0.

The ﬁrst chart of the blow-up is where x 6= 0, we can take as coordinate y1 = y/x, z1 = z/x and 2 2 2 x. We obtain the relations xy1 = y, xz1 = z. In this chart we have the equation x (1+y1 +z1) = 0. Such surface is reducible : x2 = 0 represents the exceptional divisor E. Indeed the projection map is ([u : v : w], x, y, z) 7→ (x, y, z), and in our chart we had the relations y = xv1, z = xw1 (where v1 = v/u, w1 = w/u are aﬃne coordinates) so the projection map is (x, xv1, xw1). This means that the points mapping to zero are exactly the points where x = 0. Notice that the ”natural” equation was x2 = 0 and not x = 0. This is important as it means that C2 = −2, where C is the exceptional curve E ∩ X0 ( see 3.7 ). This said, we will always from now consider the reduced equation, that is 0 2 2 x = 0. The other equation was the smooth surface X : 1+v1 +w1 = 0 which is the strict transform 0 2 2 of X. The exceptional curve E ∩ X is given by 1 + v1 + w1 = 0, x = 0 i.e it is an irreducible conic.

By symmetry, other charts are smooth, so we get indeed that the resolution graph is A1 in this case. Over the real numbers, the resolution is the contraction of a circle lying on a cylinder, which gives a cone.

Resolution graph of An, n ≥ 2

2 2 n+1 Let X be the surface given by the equation x + y + z = 0. Let us take the chart U1 with 2 2 2 n−1 coordinates zx1 = x, zy1 = y and z. The equation for X in U1 is z (x1 + y1 + z ) = 0. This is exactly an An−2 singularity, so we can apply induction hypothesis. For simplicity, we omit the details but here is what is needed to be checked : ﬁrst one should verify that the other charts are smooth, then perform a second blow-up and see how the ﬁrst exceptional curve intersects with the second exceptional curve.

After several blow-ups, we reduce this to the case of A1 already done before or A2, depending on 2 2 3 the parity of n. For A2 the surface equation is x + y + z = 0. A single blow-up is enough for resolve the singularity, and the equation of the strict transform (in the chart with coordinates 2 2 z, x1 = x/z, y1 = y/z) is z + x + y = 0. This means that the exceptional curve is given by z = 0, x2 + y2 = 0, that is we obtain two lines intersecting at one point, and this is indeed the dual graph to A2, as expected. In the appendix we give an alternative proof using toric geometry. We present a picture of the diﬀerent steps for the resolution of an A5 singularity : the red point is the singular locus, and for simplicity we only draw the exceptional curves.

16 C1 C2

Figure 1: Step 1 for the resolution

C3 C4 C1 C2

Figure 2: Step 2 for the resolution

C3 C4 C5 C1 C2

Figure 3: Step 3 for the resolution

Resolution graphs of D4 and D5

Again we will apply an inductive argument, so first we need to solve explicitly the D4 and D5 singularities. 2 3 3 For D4, we can find a change of variables such that the new equation of the surface is x +y +z = 0. ( More precisely, the equation is of the form x2 = −z(y +iz)(y −iz) = abc where a, b, c are three different linear forms in y, z. The new equation is x2 = −y3 − z3 = −(y + z)(y + jz)(y + j2z) = a0b0c0 where j 6= 1, j3 = 1 and a0, b0, c0 are linear forms. So the change of variables is exactly finding 0 M ∈ PGL2(C) with Ma = a , etc. and it is well known that such a transformation exists, by the fundamental theorem of projective geometry).

Let us blow-up this surface with coordinates x, xy1 = y, xz1 = z. We obtain the equation 2 3 3 x (1 + x(y1 + z1)) = 0 and the corresponding surface is smooth, because the exceptional divisor does not intersect the surface in this chart. So we can take coordinates yx2 = x, yz2 = z, y and

17 2 2 3 we obtain the equation y (x2 + y(1 + z2)). The exceptional curve is E : y = x2 = 0 and it intersects 3 three A1 singularities at (0, 0, a) with a = −1. So we can perform a blow-up at each of them and we exactly obtain the graph D4 as the resolution graph.

2 2 4 We now treat the case of D5 : x + zy + z = 0. First, let us look at the chart U1 with 2 2 4 coordinates yx1 = x, yz1 = z, y. The strict transform has equation x1 + yz1 + y z1 = 0, with one singular point at (0, 0, 0). The exceptional curve is C1 : x1 = y = 0. There is another singular point in the chart U2 with coordinates verifying zx3 = x, zy3 = y, z. Returning to U1, we do recognize a 0 4 singularity of type A1. Indeed by a change of variable z1 = z1 + yz1 we obtain the new equation 2 0 0 x1 + y1z1 = 0 which is obviously an A1 singularity. The exceptional curve C is the parabola 2 y2 + x2 = 0, z1 = 0, which intersects the strict transform of C1 : x2 = y2 = 0 at (0, 0, 0). So we just get one more vertex connected to C1 in the resolution graph. The equation of the strict transform 2 2 2 2 y2 2 y4 of X in U2 is x3 + y3z + z = 0. This equation can be written as x3 + (z + 2 ) − 4 = 0 which is easily seen as a A3 singularity, and this concludes the proof.

Resolution graph of Dn

Now we consider the general case x2 + zy2 + zn−1 = 0. The blow-up in the chart with coordinates x, xy1 = y, xz1 = z gives a smooth surface. Now, let’s look at the chart with coordinates y, x2, z2 2 n−3 n−1 verifying yx2 = x, yz2 = z. The strict transform has equation x2 + yz2 + y z2 = 0, which has a singular point at the origin. The exceptional curve C1 is given by y = 0, x2 = 0. We claim that one blow-up is enough for resolving this singularity. For example, in the chart with coordinates 2 2 n−3 2n−5 z2, y3, x3 with z2y3 = y, z2x3 = x2 the strict transform is z2(x3 + y3 + y3 z2 ). The gradient n−4 2n−5 n−3 2n−6 is ∇ = (2x3, 1 + (n − 3)y3 z2 , (2n − 5)y3 z2 ) which is never zero. The exceptional curve 0 2 C1 is given by z2 = 0 = y3 + x3 i.e it is a smooth parabola, and the strict transform of C1 is 0 y3 = 0, x2 = 0 i.e C1 intersects transversally C1. Such points does not belong to the other charts, so the intersection will not change. Now we go back to our ﬁrst equation x2 +zy2 +zn−1 = 0, and look in the last chart with coordinates 2 2 n−3 z, x4, y4 which verify zx4 = x, zy4 = y. The strict transform is x4 + zy4 + z = 0, i.e it is a Dn−1 singularity. The exceptional curve C1 is given in this chart by z = 0, x4 = 0. Now, we need to 0 repeat our previous analysis to the Dn−1 singularity and see how C1 intersects with C2 and C2 in order to apply induction. If we take coordinates y4, x5, z5 with y4x5 = x4, y4z5 = z4 the exceptional 0 0 curve C2 is given by y4 = 0, x5 = 0. It intersects C1 at one point. Also, C2,C2 intersects in one point since the analysis made in the previous paragraph is still valid. Finally, one needs to check that C1 and C2 does not intersect (else we would get a cycle). But C1 does not intersect the chart where C2 appears (the chart with coordinates z, x6, y6 with zx6 = x4, zy6 = y4). So far, we have 0 0 the following scheme for the intersection : C1 intersects C1, which intersect C2, which intersects

18 C2, . . . and we get indeed a linear tree, with two new exceptional curves when passing from Dn to

Dn−1. We can now conclude by induction as we know that the resolution graph for D4 and D5 is

D4 (resp. D5) and that the resolution graph of Dn can be obtained from the resolution graph of

Dn−1 by adding two vertices to the end of the graph.

We know show the real points of the algebraic surface corresponding to D4 and the corresponding surfaces after the two blow-ups.

Resolution graph of E6

Now we consider the equation x2 +y3 +z4 = 0. Two charts for the blow-up are smooth, and the last one is with coordinates z, x1, y1 with relations zx1 = x, zy1 = y and the strict transform is given 2 3 2 by x1 + zy1 + z = 0. We will show that this is an A5 singularity, which reduces to a computation previously done. 2 3 2 2 6 To see this, scaling y1 gives the equation x1 − 2zy1 + z = 0. But this equation is exactly x1 + y1 + 3 2 3 2 2 6 (z − y1) = 0. So a change of variables X = x1,Y = y1,Z = z − y1 gives the equation X + Z + Y which is indeed a singularity of type A5. A little picture of the A5 case shows that indeed we get the E6 conﬁguration.

Resolution graph of E7

The equation is x2 + y3 + yz3 = 0, and again two charts in the blow-up are smooth. The last chart 2 2 3 2 has coordinates z, x1, y1 with zx1 = x, zy1 = y. The strict transform is z (x1 + zy1 + y1z ) = 0.

The exceptional curve C1 is given by x1 = z = 0, and the only singular point is (0, 0, 0). We 2 3 2 2 claim that x1 + zy1 + y1z is a singularity of type D6. Indeed, we just need to change yz(z + y ) into w(v2 + w2) by a change of variables. We write the last expression as w(v + iw2)(v − iw2) = 2 2 2 2 2 w(w −iw +2iw )(v −iw ). Now, if we let y = 1−i v and z = v −iw the previous equation becomes λy(z − y2)(z) for some nonzero constant λ and we are done.

Resolution graph of E8

Finally let us compute the resolution graph of x2 + y3 + z5 = 0. Again for the ﬁrst blow-up there is only one chart where the strict transform is singular, with coordinates x1, y1, z verifying 2 3 3 zx1 = x, zy1 = y. The strict transform is given by x2 + zy1 + z = 0, the exceptional curve is given by C1 : z = x1 = 0. This is exactly an E7 singularity so we are done.

Conclusion and a little digression

As a conclusion, we give the Dynkin diagram of type An,Dn,E6,E7 and E8.

19 (pictures to include)

We now give a stricking observation, giving a ﬁrst link between surfaces singularities and simple Lie algebras :

Proposition 3.11. Let X be a surface of type ADE and C1,...,Cr the exceptional curves of the minimal resolution. Then, the matrix of the intersection form, that is the matrix M with

Mij = Ci · Cj is the Cartan matrix of the corresponding Dynkin diagram.

Here, Ci · Cj is the intersection product.

As we will see, there is another surprising link between simple Lie algebras and simple singularities. In the next section we will investigate another kind of graph related to G, namely the McKay graph of a binary polyhedral group. Before this, we ﬁnish by a topological digression.

n It is well known that if X ⊂ C is an algebraic variety with a singularity at the origin O, then there is a pair homeomorphism (D,X) =∼ (Cone(S), Cone(X ∩ S)) where D is a small disk around the origin and S = ∂D. In particular, to understand X topologically near O it is enough to understand X ∩ S, called the link of the singularity. This is a manifold if the singularity is isolated. For example, if X is a plane curve, X ∩ S will be a knot. Two famous examples are the cusp y2 = x3 and the union of two lines xy = 0. The link of the cusp is the trefoil knot, and the link of xy = 0 is the Hopf link. If X is an algebraic surface with isolated singularity, X ∩ S will be a manifold of real dimension 3. In his paper [15], Patrick Du Val gave a combinatorial description of the link of the Du Val singularities. The case of E8 is interesting since here X ∩ S is an homology sphere, that is it has the same homology as S3 but it is not diﬀeomorphic to S3. In fact we have :

Proposition 3.12. Let X be a Du Val singularity of type E8. Then, X ∩ S is diﬀeomorphic to the space of all icosahedra inscribed in S2.

2 ∼ Proof. We have C = H, the space of quaternions. So it is enough to understand the quotient H/Γ where Γ is the binary icosahedral group. Clearly, Γ preserves the norm of a vector v ∈ H, so 3 in fact H/Γ is the cone over S/Γ where S = S is the unit sphere in H. On the other hand, we ∼ ∼ have S = SU2 and so S/Γ = SO3/A5 as topological spaces. But SO3/A5 is precisely the space of 2 3 icosahedra inscribed in the unit sphere S ⊂ R .

For more details, [32] is very complete, and the authors of [27] give eight diﬀerent descriptions of the topological space X ∩ S for X a Du Val singularity of type E8. Recently J.Baez also gives a survey of this with special emphasis on the icosahedron, see [4].

20 4 The McKay graphs

4.1 Computations of the graphs

McKay noticed in his paper [33] that the McKay graph of a binary polyhedral group coincides with the affine Dynkin diagram of Lie algebras of type ADE. Such graphs are defined as ”diagrammes de Dynkin complétés”in [6]. We will compute these graphs here. Nice information about this graph can be found in [46], including links with harmonic analysis.

Now we will define the McKay graph of a representation V of a finite group Γ. Let V1,...,Vr be ∼ L ⊕mij the irreducible complex representations of Γ, and mij ∈ N such that V ⊗ Vi = j Vj . We obtain a graph, the set of vertices being the Vi and there are mij edges from Vi to Vj. This graph has various interesting properties, for example it is connected if and only if V is faithful. Now we assume that Γ is a binary polyhedral group. The inclusion Γ ⊂ SL(2) gives a 2-dimensional complex representation V , and we define the McKay graph of Γ to be the McKay graph of V .

For a representation ρ :Γ → GL(V ), let χV be the character of V , i.e χV (g) = tr(ρ(g)). We write χi for χVi . We begin by a remark :

Proposition 4.1. We have mij = mji for all i, j.

Proof. Up to conjugacy we can assume Γ ⊂ SU2. Any element g ∈ SU(2) has real trace, so it follows 1 P 1 P that mij = hχiχV , χji = |Γ| g∈Γ χi(g)χV (g)χj(g) = |Γ| g∈Γ χi(g)χV (g)χj(g) = hχi, χV χji = mji.

McKay graph for An

∗ Let Γ be cyclic of order n, generated by g. Let ζ ∈ C be a primitive n-th root of unity. The i irreducible representations are Vi = C with g ·x = ζ x for i = 0, . . . , n−1 and x ∈ Vi. The inclusion

Γ ⊂ SU(2) splits as V1 ⊕ V−1. So we have Vk ⊗ V = Vk−1 ⊕ Vk+1, and this gives the ﬁrst McKay graph : n vertices forming a cycle, i.e the aﬃne Dynkin diagram of type Afn :

...

1 1 1 1

Figure 4: Type An

We took the following convention : the trivial representation corresponds to the white vertex

21 and the adjacent edges are dotted. In particular, the subgraph constitued of the bold edges and black vertices is the corresponding resolution graph.

McKay graph for type Dn+2

We use the following presentation for Γ :

Γ =∼ ha, x : a2n = 1, x2 = an, xa = a−1xi

It has cardinality 4n, we call it Dn+3 because the corresponding graph will have n + 3 vertices. 2 Note that a generates a normal subgroup isomorphic to Z/2nZ, and that Z(Γ) = {1, x }. Also, any element can be uniquely written as xjak where 0 ≤ j ≤ 1 and 0 ≤ k < 2n. Let us compute the conjugacy classes : as said before we have Z(Γ) = {1, x2}. Since a, x generate Γ and that xakx−1 = a−k so for all 1 ≤ j ≤ n − 1, we obtain a conjugacy class {ak, a−k} for all 1 ≤ k ≤ n − 1. Finally, since (xak)·(alx)·(a−kx−1) = a2k+lx and ak ·alx·a−k = xa−(2k+l) we see that the conjugacy class of alx depends only on the parity of l, so we get two new conjugacy classes {alx : l odd } and {alx : l even }. We obtain 2 + (n − 1) + 2 = n + 3 conjugacy classes, this will be the aﬃne Dynkin diagram of type Dn+2.

Now we will compute AΓ := Γ/[Γ, Γ]. First, [Γ, Γ] ⊂ ha2i, indeed ha2i is normal and Γ/ha2i has order 4, so is abelian. But a2 = [ax, x] so [Γ, Γ] = ha2i. Hence, a presentation of AΓ 2 2 n 4 ∼ is given by ha, x | a = 1, x = a , ax = xai. This is hx | x = 1i = Z/4Z if n is odd and 2 2 ∼ ha, x | ax = xa, a = x = 1i = Z/2Z × Z/2Z if n is even. In both cases we get 4 1-dimensional 2 representations from AΓ and the remaining irreducible representations are Vl, where Vl = C ! ! 0 1 ωl 0 and ρl(x) = , ρl(a) = . We obtain the following character tables, where −1 0 0 ω−l 1 ≤ l, k ≤ n − 1 :

n odd 1 x2 ak ax a2x n even 1 x2 ak ax a2x

χ0 1 1 1 1 1 χ0 1 1 1 1 1

χ−1 1 1 1 −1 −1 χ1,0 1 1 1 −1 −1 k k χi 1 -1 (−1) i −i χ0,1 1 -1 (−1) −1 1 k k χ−i 1 -1 (−1) −i i χ1,1 1 -1 (−1) 1 −1 kl −kl kl −kl χl 2 -2 ω + ω 0 0 χl 2 -2 ω + ω 0 0

Now we can compute the McKay graph, remembering that V1 corresponds to the natural representation Γ ⊂ SU2. For example, if n is odd we get :

• χ1χk = χk−1 + χk+1 for 1 < k < n − 1

• χ1χ1 = χ0 + χ−1 + χ2 22 • χ1χn−1 = χi + χ−i + χn−2

This gives exactly the aﬃne diagram D^n+2, i.e Dn+2 with one new vertex connected only to v2 (with Bourbaki numbering), the supplementary vertex represents the trivial representation.

Remark : since mij = mji we don’t need to compute e.g χ1χ−1 as m1,−1 = 1. Also, the same computations apply with n even, with χ−1 replaced by χ1,0, χi → χ0,1 and χ−i → χ1,1.

1 1

...

1 2 2 2 2 1

Figure 5: Type Dn+2

McKay graph for E6

We will assume the isomorphism between the binary tetrahedral group BT and SL2(F3), see [40], chapter on ﬁnite groups of small order.

To ﬁnd the irreducible representations of BT, we will use the double cover BT → A4.

First let us compute the conjugacy classes, since BT has order 24 we can do it by hand. The center is {±1}. ! ! 1 1 1 − ac a2 If M = , then M G = { | ad−bc = 1}, i.e the conjugacy class of M is given 0 1 −c2 1 + ac ! ! ! 0 1 1 0 −1 1 by {M, , , }. Now we can notice that multiplying by −1, transpose −1 −1 −1 1 −1 0 or both transpose and multiplying by −1 gives 3 other conjugacy classes. The remaining matrices form a unique conjugacy class (they generate a group isomorphic to the quaternions). In the next ∼ table, the column ”image in A4” precise the image of the conjugacy classes of A ∈ SL2(F3) = BT under the surjection BT → A4 :

23 List of elements Order Image in A4 ! 1 0 1 id 0 1 ! −1 0 2 id 0 −1 ! ! ! ! 1 1 0 1 1 0 −1 1 , , , 3 (123) 0 1 −1 −1 −1 1 −1 0 ! ! ! ! −1 −1 0 −1 −1 0 1 −1 , , , 6 (123) 0 −1 1 1 1 −1 1 0 ! ! ! ! 1 0 −1 −1 0 −1 1 −1 , , , 3 (132) 1 1 1 0 1 −1 0 1 ! ! ! ! −1 0 1 1 0 1 −1 1 , , , 6 (132) −1 −1 −1 0 −1 1 0 1 ! ! ! ! ! ! 0 1 0 −1 1 −1 1 1 −1 1 −1 −1 , , , , , 4 (12)(34) −1 0 1 0 −1 −1 1 −1 1 1 −1 1

We recall the character table of A4 (and directly write the corresponding conjugacy classes in BT) :

! ! ! ! ! ! ! 1 0 −1 0 1 1 −1 −1 1 0 −1 0 0 1 0 1 0 −1 0 1 0 −1 1 1 −1 −1 −1 0

χ0 1 1 1 1 1 1 1 2 2 χ1 1 1 ω ω ω ω 1 2 2 χ2 1 1 ω ω ω ω 1

χ6 3 3 0 0 0 0 −1

We have one more character χ3 from the representation G ⊂ SL2(C). Moreover it turns out that χ3χ1 := χ4 and χ3χ2 := χ5 are also irreducible. So the list is complete and we obtain :

24 ! ! ! ! ! ! ! 1 0 −1 0 1 1 −1 −1 1 0 −1 0 0 1 0 1 0 −1 0 1 0 −1 1 1 −1 −1 −1 0

χ0 1 1 1 1 1 1 1 2 2 χ1 1 1 ω ω ω ω 1 2 2 χ2 1 1 ω ω ω ω 1

χ3 2 −2 1 −1 1 −1 0 2 2 χ4 2 −2 ω −ω ω −ω 1 2 2 χ5 2 −2 ω −ω ω −ω 1

χ6 3 3 0 0 0 0 −1

We can now compute the McKay graph : we have

• χ3χ4 = χ1 + χ6

• χ3χ5 = χ2 + χ6

• χ3χ6 = χ3 + χ4 + χ5

So the McKay graph of BT is indeed Ef6 :

1 2 3 2 1

Figure 6: Type E6

Once more, the case E7 and E8 are more complicated so we won’t compute the tables, but we show the corresponding diagrams :

1 2 3 4 3 2 1

Figure 7: Type E7

25 3

2 4 6 5 4 3 2 1

Figure 8: Type E7

4.2 A proof of the McKay theorem

We present now a proof of the McKay theorem without a case-by-case computation. The reference for what follows is [11], chapter about McKay graphs. We will know look at a special class of representations, called admissible representations, introduced by T. Springer.

Deﬁnition 4.2. Let ρ :Γ → GL(V) a representation where Γ is a ﬁnite group. Then, we say that

ρ is admissible if ρ is faithful, its character χV is real and if ρ|Z(Γ) has no trivial summand.

We begin by a well known fact :

∗ Proposition 4.3. et χ : Z → C be a non-trivial morphism of groups, where Z is a ﬁnite group. P Then, z∈Z χ(z) = 0. P P Proof. Indeed, by hypothesis there is w ∈ Z with χ(w) 6= 1. Now, z∈Z χ(z) = z∈z χ(w)χ(z) so P z∈Z χ(z) = 0.

Now we can state and prove the theorem of Springer :

Theorem 4.4. Let (ρ0,V ) be an admissible representation of Γ, with n = dim V , and let χ be its Γ Γ character. Deﬁne F : C × C → C by

(φ, ψ) 7→ hχφ, ψi

1 P where as usual hφ, ψi = |Γ| g∈Γ φ(g)ψ(g). Then, we have the following properties :

Γ 1. The map F is a hermitian form on C .

2. If Vi,Vj are irreducible representations of Γ and χi, χj the corresponding characters then

F (χi, χj) is the multiplicity of Vj in V ⊗ Vi.

3. If χW is the character of an irreducible representation W , then F (χW , χW ) = 0.

Γ 4. For φ ∈ C , we have F (φ, φ) ≤ n(φ, φ) with equality if and only if φ is the character of the regular representation. 26 5. If |Γ| > 2 and χ, χ0 are irreducible characters then 0 ≤ F (χ, χ0) < n

Proof. 1. This is clear because χ is by hypothesis real-valued.

2. This is clear.

3. Now let χW be the character of an irreducible representation W . We want to see that

F (χW , χW ) = 0. We will use that by Schur’s lemma we have |χW (z)| = dim(W ) for any 1 P 2 1 P z ∈ Z(Γ). We have S := F (χW , χW ) = |Γ| g∈Γ |χW (g)| χ(g). We write 1 = |Z(Γ)| z∈Z(Γ) 1 and obtain 1 X X S = ( ((|χ (g)|2) · χ(g))) |Γ| · |Z(Γ)| W z∈Z(Γ) g∈Γ 1 X X = ( ((|χ (zg)|2) · χ(zg))) |Γ| · |Z(Γ)| W z∈Z(Γ) g∈Γ   dim(W ) X X = χ(g)|χ (g)|2 χ(z) |Γ| · |Z(Γ)| W   g∈Γ z∈Z(Γ) (the second line is because for ﬁxed z ∈ Z(Γ), g 7→ zg is a bijection) so it is enough to P check that z∈Z(Γ) χ(z) = 0. It follows from the fact that V|Z(Γ) doesn’t contain any trivial summand combined with the previous proposition.

1 P 2 1 P 2 1 P 2 4. We have F (φ, φ) = |Γ| g∈Γ χ(g)|φ(g)| . Now |F (φ, φ)| = |Γ| | g∈Γ χ(g)|φ(g)| ≤ |Γ| g∈Γ |χ(g)||φ(g) | ≤ P 2 n g∈Γ |φ(g)| . If there is equality, this implies that |χ(g)| = n if φ(g) 6= 0. But the ﬁrst inequality implies that all the non-zero χ(g) should have the same sign (or be zero), but since χ(1) = n and that χ is real, this means that any other g with |χ(g)| = n should have

ρV (g) = idV . Since V was faithful this is impossible except if g = 1 so indeed we obtain that if there is equality then χ is the character of the regular representation.

5. Let χi, χj be distinct irreducible characters. We already proved that F (χi, χj) ≥ 0. Let Γ Γ G : C × C → C be the bilinear form deﬁned by (φ, ψ) 7→ nhφ, ψ) − F (φ, ψ). Then, by the previous point G is positive semi-deﬁnite. Applying Cauchy-Schwartz to G gives

F (χi, χj) ≤ n, with equality iﬀ there is a linear dependant combinaison of χreg, χi and χj. Since |Γ| > 2 this is impossible unless i = j.

Deﬁnition 4.5. Let (V, ρ) be an admissible representation of a ﬁnite group Γ and F be as in

4.4. Let (χ1, . . . , χm) be the irreducible character of Γ. Then, we deﬁne a m × m matrix Aij :=

F (χi, χj) − nδij. The matrix is called the Springer-McKay matrix of (Γ, ρ).

27 The following corollary is immediate :

Corollary 4.6. The Springer-McKay matrix enjoys the following properties :

1. A is a symmetric integral matrix.

2. The diagonal elements are equal to −n, and else verify 0 ≤ aij < n.

3. A is a negative semi-deﬁnite matrix, and the kernel is generated by (dim(χ1),..., dim(χn))

As a second corollary we deduce that the Springer-McKay matrix of a binary polyhedral group is exactly the affine Cartan matrix of the corresponding Lie algebra. Indeed, in the chapter 4 of [26], Kac classifies matrices which are semi-negative definite, integral, have diagonal entries −2 and has kernel of dimension 1. Finally a weaker condition than symmetry is required, namely that aij = 0 =⇒ aji = 0. Such matrices corresponds to affine untwisted Dynkin diagram. If as in our case A is symmetric we obtain only untwisted affine Dynkin diagrams, with the exception of the twisted A2 system but it doesn’t appears as the McKay graph of a binary polyhedral group. Now we just say a few words about root system for Kac-Moody algebras. In the root system for the finite-dimensional semisimple case, any root is W -conjugate to a simple root. This is not true in the infinite dimensional case, roots which are W -conjugate to a simple root are called real roots, and other roots are called singular or imaginary roots. The space of imaginary root can have arbitrary dimension, however for affine root system, there is an imaginary root δ such that all imaginary roots are multiple of δ. In this case we can write δ = α0 +α ˆ whereα ˆ is the heighest root.

Conclusion

The McKay correspondance shows an unexpected connexion between simple Lie algebra and binary polyhedral groups. We conclude by few remarks from the paper [45]. Let X be a Du Val singularity 2 and Γ be the corresponding binary group. Let Z be the scheme theoretic-ﬁber of 0 ∈ C /Γ under ˜ 2 P the map X → C /Γ. Then, Z is supported on the exceptional locus so Z = diCi for some di ∈ Z.

Proposition 4.7. If we ﬁx a natural bijection (i.e coming from a isomorphism of graphs) between the irreducible representations of Γ and the exceptional curves Ci, then di = dim ρi.

In fact, a little modification of this construction gives a geometric interpretation for the affine 3 node in the McKay graph. Let H a generic linear hyperplane in C and H := H ∩ X where X is a Du Val singularity. We have π∗(H) = T + Z where T is the strict transform of H. Then, one can verify that the T ·Ci iff there is an edge between vi and v0 in the McKay graph. So we can naturally extend the resolution graph with a new vertex corresponding to T , and this new resolution graph is exactly the McKay graph of Γ.

28 To verify the previous proposition, we give the deﬁnition of a other cycle called the numerical cycle, which is easiest to manipulate. For rational singularities both cycles are actually equal, see the chapter about surface singularities in [38].

Proposition 4.8. Let X be a singular surface with an isolated singularity and f : Y → X a resolution. There exists a minimal eﬀective divisor Zf such that for all exceptional curves Ci,

Zf Ci ≤ 0, we call Zf the numerical cycle of f.

We also observe that the longest root has the same coeﬃcients as the fundamental cycle. We obtain the following table :

Simple Lie algebras Binary groups Simple surfaces singularities Root system Irreducible representations Exceptional curves

(αi,αj ) 2 2δij − F (χi, χj) −Ci · Cj (αi,αi) Longest root Regular representation Fundamental cycle

5 Brieskorn’s theorem

In this section we follow the paper of Slodowy [41], the paper being itself a condensed version of his book about simple groups and singularities. The main idea is the following : let G be a semisimple Lie group and g its Lie algebra. Let S be a transverse slice at x ∈ g to the adjoint action of G (see 5.15 for the definition of a transverse slice) where x is nilpotent and subregular. Then S ∩ N (g) will be shown to be a Du Val singularity of the same type as g (where N (g) is the nilpotent cone) ! The adjoint quotient χ : S → h/W will be the semiuniversal deformation of N (g) ∩ S (to be defined later). First let us define the main heroes of the section :

Deﬁnition 5.1. Let g a semisimple Lie algebra, a regular element x ∈ g is an element with dim ZG(x) = r, where r = dim h where h is a Cartan subalgebra. An element is subregular if dim ZG(x) = r + 2.

Regular elements are dense in g, and subregular elements are important as they will be naturally equipped with a Du Val singularity as we will see later.

5.1 Preliminaries

Invariant theory

For this part, the reference is Mukai’s introductory book on geometric invariant theory (GIT for short) and moduli [35]. Let R = C[x1, . . . , xn] and G a reductive algebraic group acting on R. We

29 n n write X = C = Spec(R). Usually, G acts on V = C and we take the corresponding action on Sym(V ∗) = R. By reductive we mean that the unipotent radical of G is trivial. Semisimple groups are reductive. We will use the following property :

Theorem 5.2. Let G a reductive group over a ﬁeld of characteristic zero. Then, any surjective morphism of G-modules V → W induces a surjection V G → W G.

This theorem is a corollary of the following theorem :

Theorem 5.3. Let G a reductive algebraic group over a ﬁeld of characteristic zero. Then, any ﬁnite dimensional representation of G is completely reducible.

Proof. This is theorem 22.138 in [34].

Indeed, if the previous theorem holds, then a surjection V → W induces a surjection of all isotypic components of V onto the corresponding component of W , in particular there is indeed a surjection V G → W G.

By Hilbert’s theorem ([35], Theorem 4.51), the ring of invariant functions RG is ﬁnitely gener- n G ated and we deﬁne C /G := Spec(R ), called the categorical quotient. Once and for all, we choose G n k a minimal set of generators f1, . . . , fk of R giving a map χ : C → C , x 7→ (f1(x), . . . , fk(x)). If x, y are in the same orbit then χ(x) = χ(y). The converse is not true, but we will see that X/G parametrizes closed orbits in X.

Theorem 5.4. Let χ : X → X/G the map corresponding to the inclusion RG ⊂ R. Then, χ(x) = χ(y) if and only if G · x ∩ G · y is non-empty.

Proof. Let O, O0 two diﬀerent orbits of the G-action on X. Let I,I0 be the corresponding ideals. Assume that O ∩ O0 = ∅. By Hilbert’s Nullstellensatz I + I0 = R. Now, I and I0 are subrepresen- tations of G since O and O0 are G-invariant. The map I ⊕ I0 → R is a map of G-modules and G is reductive so I ∩ RG ⊕ I0 ∩ RG → RG is surjective by the previous theorem. Then there exists G 0 0 G 0 f ∈ I ∩ R , f ∈ I ∩ R with f + f = 1. In particular f|O = 0 and f|O0 = 1 so χ separates O and O0. Conversely, if two orbits intersect, then any G-invariant function is constant on both orbits, in particular χ(x) = χ(y).

This also shows that there is a unique closed G-orbit in each ﬁber of the map X → X/G. Uniqueness is clear by the previous theorem, an orbit of minimal dimension is necessarily closed. For more informations about this, a good reference is the book by Mukai [35].

30 Corollary 5.5. Let G a ﬁnite group acting on a variety X. Then, there is a natural bijection G between the closed points of Spec(C[X] ) and the orbits of the G-action on X.

Now consider G a semisimple, simply-connected algebraic group over C and g its Lie algebra. We will look at the adjoint action : first we will look at an arbitrary simple connected, simply- connected group G, and in the next paragraph specialize to SLn where arguments are easier. If h G W is a Cartan subalgebra of g, then restriction induces a map C[g] → C[h] . Chevalley’s restriction theorem shows that such map is an isomorphism. Also, the Chevalley-Shephard-Todd theorem G ([8]) states that if G is a finite reflection group acting on a vector space V then Spec(C[V ] ) is an affine space of the same dimension as V . In type An this is just the usual fundamental theorem of symmetric polynomials.

We want to understand geometric properties of the map χ : g → g/G. We will ﬁrst show the Chevalley restriction isomorphism which give an isomorphism g/G =∼ h/W . First, since W =∼

NG(T )/T where T = exp(h), a G-invariant polynomial is W -invariant when restricted to h so we G W obtain a map res : C[g] → C[h] .

G W Theorem 5.6. The map res : C[g] → C[h] is an isomorphism.

Remark 5.7. Since any regular semi-simple element is conjugated to an element of h and regular elements are dense in g, we conclude immediately that res is injective. So the diﬃcult part is the surjectivity.

Proof of the Chevalley restriction theorem. As said before we only need to prove surjectivity. Let + + λ ∈ P be a dominant weight and let σλ : g → gl(L (λ)) be the corresponding irreducible m G representation. If fλ,m(x) = tr(σλ(x) ), then clearly fλ,m ∈ C[g] so it is enough to show that W + W W these functions span C[h] . For this, notice that ch(L (λ)) ∈ (ZP ) ⊂ (CP ) where CP is the group algebra of P and ZP the integral group ring. Indeed, we need to show that there is an action of W on the weight spaces. If φ : g → Aut(V ) is a g module, then we can take a Chevalley ∼ basis ea, fa. Since they are nilpotent, exp ad(ea) ∈ Int(g) = G. So G also acts by exp φ(ea) and exp φ(eb). Now we define νa = exp(φ(ea)) exp(−φ(fa)) exp(φ(ea)). We claim that the νa generate the Weyl group W and that we have νaVµ = Vνa(µ). From this, it is clear that the Weyl group just permutes the weight spaces, and in particular the character is invariant. + P µ P m If ch(L (λ)) = aµ,λe then we have res(fλ,m)(h) = aµ,λµ(h) , so to know the restriction of the fλ,µ to h we only need to know the character of σλ. The exponential is not polynomial but we µ P k still have an embedding C[P ] → C[[h]], e → k µ /k!. + W + So this is enough to show that the ch(L (λ)) form a basis of (CP ) , where λ ∈ P , and that W W C[P ] = C[h] (we will show it by looking at each homogeneous components). 31 + W W To show that ch(L (λ)) generates (CP ) , we first notice that a basis of (CP ) is given by the P wµ Sµ := w∈W e for µ ∈ P . On the other hand for any weight there is a unique dominant weight W in the W -orbit of µ : in particular the Sλ generates C[P ] for λ dominant. Now we can write + P ch(L (λ)) = Sλ + µ<λ aµ,λSµ by W -invariance and by induction this shows that the characters W W of irreducible representations spans C[P ] . Now if f ∈ (C[[h]] )m, it is easy to find f ∈ C[P ] with pm(f) = f because if we also truncate the image of the exponential, the image contains all the monomials µm by looking at eaµ when a is an integer.

Proposition 5.8. The map χ : g → h/W is given by x 7→ (G · xs) ∩ h, where x = xs + xn is the

Jordan-Chevalley decomposition of x, xs is semi-simple, xn nilpotent and [xs, xn] = 0.

We conclude by giving more information about the geometry of g/G, including two theorems by Kostant (cf [29]) :

Theorem 5.9 (Kostant). The quotient map χ is a flat map (see 5.11 for a definition of flat morphism) and all the fibers are normal.

Theorem 5.10 (Kostant’s criterion). An element x ∈ g is regular if and only if dxχ : g → h/W has maximal rank.

The ﬁrst theorem is extremely useful : in particular it implies that all the ﬁbers have the same dimension (see [22] chapter III, section 9).

Example : Ar

We now look at g = slr+1, the Lie algebra of complex matrices with trace zero and will to verify some of the properties we stated for a general semisimple Lie algebra g. Our Cartan subalgebra h will be the diagonal matrices. Here W is the symmetric group, permuting the r ﬁrst entries of ∼ G ∼ W h ∈ h. We will verify the isomorphism g/G = h/W , i.e the isomorphism C[g] = C[h] . We follow the strategy of [12] (see the section 1.4 ”Conjugacy classes of matrices”).

Let s1, . . . , sr, sr+1 be the symmetric elementary polynomials in the eigenvalues, i.e X sl(M) = λi1 (M) . . . λil (M) 1≤i1<···

32 r+1 r−1 use the companion matrix of t + x1t + ··· + xr, given by   0 ...... 0 −xr    1 0 ...... 0 −xr−1      0 1 0 ...... 0 −xr−2 Ax =     ......    ··· ... 0 −x   1  · ... 0 1 0

r i where x = (x1, . . . , xr) ∈ C . Notice that si(Ax) = (−1) xi−1(Ax) for i = 2, . . . , r + 1. We now SL show that the si generate the invariant ring : let p ∈ C[slr+1] r+1 . To p we can associate a polynomial in r variables by f(x1, . . . , xr) = p(Ax), where Ax is the companion matrix of the polynomial r+1 r−1 r t + x1t + ··· + xr. Let P ∈ C[slr+1] defined by P (A) = f(−s2(A),..., (−1) sr+1(A)). By definition, P (Ax) = p(Ax) for any companion matrix Ax and P is a polynomial in s2, . . . , sr+1. So for show that the si generate the invariant ring it would be enough to check that P = p on all −1 C[slr+1]. But by hypothesis p is SLr+1-invariant, so in fact p and P agree on U = {gAxg | x ∈ r C , g ∈ SLr+1}. To conclude the proof we should show that U is dense in slr+1, this is very classic but let us recall the argument : first, the set of matrices with distinct eigenvalues in dense in slr+1, so it’s enough to show any matrix with distinct eigenvalues is conjugated to the companion matrix of its characteristic polynomial.

Indeed, let A a matrix with r distinct eigenvalues. Let v1, . . . , vr be the associated eigenvectors. r Then, w1 = v1 + ··· + vr, wk := Awk−1 form a basis of C . Indeed, the determinant of the matrix B with Bvi = wi is exactly the Vandermonde matrix of (λ1, . . . , λr), in particular non-zero. r−1 Now, the matrix of A in the basis (w1, Aw1,...,A w1) is a companion matrix so the claim follows.

G ∼ W This shows that for g = slr+1 we have C[g] = C[h] . In fact a bit more is true : the poly- ∼ r nomials s2, . . . , sr+2 are algebraically independant, i.e h/W = C . Indeed, any polynomial is the characteristic polynomial of its companion matrix, this shows that the set {(s2(A), . . . , sr+2(A)) : r A ∈ sln} = C . In particular, indeed any relation f(s2, . . . , sr+2) = 0 would implies that f is zero r on C i.e zero itself, so indeed the si are algebraically independant. Here it is easy to see that X/G is very far from parametrizing orbits : any nilpotent element x will verify χ(x) = 0 but x is conjugate to y if and only if they have the same normal Jordan form. Since two matrices over C are conjugate if and only if they have the same Jordan normal forms, the orbits are parametrized by uples of complex numbers λ1, . . . , λj (1 ≤ j ≤ r + 1) and the set of partitions m1 = σ1,1 + ··· + k1σ1,k1 , . . . , mj = σj,1 + ··· + kjσj,kj where mi is the multiplicity of the λi, and σi,l is the number of Jordan blocks of size l with eigenvalue λi. If x is nilpotent we have j = 1 so nilpotent orbits for slr+1 are parametrized by partitions of r + 1. The closed orbit corresponds to 0, and the open dense orbit corresponds to the partition with σ1,r+1 = 1, i.e there is

33 a single Jordan block. Such an element is called a regular element. The ”next” elements are called subregular elements, there are two Jordan blocks, of sizes 1 and r.

Now, in order to state the theorem by Brieskorn and Slodowy we need the notion of deformations from singularity theory.

Deformations and semiuniversal deformations

n Let X ⊂ C be an algebraic variety with 0 ∈ X. We would like to look at a ”smaller and smaller” n neighborhood of X around 0. Formally, we look at all the couples (V,Y ) where V ⊂ C is open, Y is a closed algebraic subvariety of V and 0 ∈ Y . We say that (V,Y ) and (V 0,Y 0) are equivalent if Y ∩ W = Y 0 ∩ W for some open set W ⊂ V ∩ V with 0 ∈ W . An equivalence class is called a germ of a variety. The deﬁnition easily generalizes to any point x ∈ X, in this case we write (X, x) for the corresponding germ of the variety.

Definition 5.11. A ring morphism φ : R → S is flat if S is flat as an R-module (the structure being induced by φ). A morphism f : X → Y of algebraic varieties is flat if for all x ∈ X, the ∗ induced morphism of local rings fx : OY,f(x) → OX,x is a flat morphism. We say that f is faithfully flat if f is surjective and flat.

We now come to the main deﬁnition :

Definition 5.12. Let X0 an algebraic variety and x0 ∈ X0. A deformation of (X0, x0) is a flat −1 ∼ morphism χ :(X, x) → (U, u) between germs, with (χ (u), x) = (X0, x0). U is called the base of the deformation. An isomorphism of deformations is an isomorphism X =∼ X0, where X,X0 are deformations over the same base, and the isomorphism commutes with the projection, and is the identity when restricted to X0. If φ : T → U is a finite morphism, the induced deformation is the pullback of X by φ.

Remark 5.13 (More about flatness). The intuition for a flat morphism f : X → U is that the fibers do not vary ”too much”. For example they should have the same dimension (see [22] for a proof and more properties of flat morphisms). For example, the blow-up π : X˜ → X is not a flat morphism. Here is an example of a deformation : let X be defined by {(x, y, t): y2 = x(x − 1)(x − t)}. The 2 2 morphism f : X → C, (x, y, t) 7→ t is a deformation of the nodal curve y = x (x − 1). Indeed, 2 the corresponding map is C[t] ⊂ C[x, y, t]/(y − x(x − 1)(x − t)) = M and M is a flat C[t]-module since it is a free module, a basis being xiyj for j ∈ {0, 1} and i ≥ 0. This implies that f is flat (for details see the chapter about localization in [3]). The fibers are affine elliptic curves, with only two singulars fibers over 0 and 1. More generally, a well-known criterion says that if X,Y are smooth, 34 then a morphism f : X → Y is flat if and only if the fibers of f have the same dimension. Another good case to keep in mind is if the morphism f : Y → X is projective (that is, there is m m an injection g : Y → P × X such that f = p2 ◦ g where p2 : P × X → X is the projection. In particular the fibers are now naturally embedded in the projective space). Such a morphism is flat if and only if the fibers have a constant Hilbert polynomial, in particular the same dimension.

Deﬁnition 5.14. Let (X0, x0) be a germ of variety. A deformation χ : X → U of (X0, x0) 0 0 is semiuniversal if any other deformation χ : X → T of (X0, x0) is induced from a morphism f :(T, t) → (U, u). Moreover, if f1, f2 are two such morphisms, then dtf1 = dtf2.

In our particular case, i.e when X is the zero set of a convergent (in the usual sense) series f ∈ C{x, y, z} with an isolated singularity, the semiuniversal deformation exists and can be described 1 as follows : let T = C{x, y, z}/(f, ∂xf, ∂yf, ∂zf), where C{x, y, z} is the ring of convergent power series. Since f has an isolated singularity, T 1 is a ﬁnite-dimensional vector space (we will give a 1 proof of this fact later, see 5.21). Let b1, . . . , br ∈ C{x, y, z} be the lifts of a basis of T with br = 1. Then, the semiuniversal deformation is the morphism of germs r−1 X (x, y, z, u1, . . . , ur−1) 7→ (f(x, y, z) + uibi), u1, . . . , ur) i=1 The proof of this fact is non-trivial, see [2], chapter 1, section 8. 3 2 For example, for g = sl2 the semiuniversal deformation is χ : C → C, (x, y, z) 7→ x + yz. More generally, let us describe concretely the semiuniversal deformation of the singularity x2 + y2 + n+1 1 ∼ n ∼ n 1 z = 0. We have T = C{x, y, z}/(f, 2x, 2y, (n + 1)z ) = C[z]/(z ). A basis of T is given by 1, z, z2, . . . , zn−1. The semiuniversal deformation is

2 2 n+1 n−1 (x, y, z, u1, . . . , un−1) 7→ (x + y + z + u1z + ··· + un−1z , u1, . . . , un)

Statement of the theorem

Before stating Brieskorn’s theorem (1970), conjectured by Grothendieck, we need a last deﬁnition.

Deﬁnition 5.15. We say that S is a transversal slice at x ∈ g if S is a locally closed subvariety of g verifying dim S = codim(G · x), such that the map G × S → g, (g, s) 7→ Ad(g)s is a submersion, i.e its diﬀerential at x is surjective.

Transverse slices are in some sense ”orthogonal” to the action of the group : for example if G is a Lie group acting freely on a manifold X, then the space of orbits X/G is smooth, and a local parametrization of X/G is given by a transverse slice. This is the famous ”slice theorem” : there is also a version of this theorem in algebraic geometry, called the Luna slice theorem (which we will not require).

We can now state the celebrated Brieskorn theorem (see the ﬁrst chapter in [41]) : 35 Theorem 5.16 (Brieskorn). Let g be a semisimple Lie algebra over C of type Ar,Dr or Er, and x a subregular nilpotent element of g. Then, if S is a transversal slice at x, the restriction of χ : g → h/W to S is the semiuniversal deformation of the corresponding simple singularity.

Notice that as a special case of this theorem, we get an isomorphism of germs N (g) ∩ S =∼ X, where X is the corresponding Du Val singularity. This was conjectured by Grothendieck, and proved by Brieskorn in the case of ADE type Lie algebras, Slodowy generalized this to type BCFG. The ﬁrst method envisaged was to use the Springer resolution but it was not known how to compute the self-intersection of the curves, until the thesis of H. Esnault [23].

5.2 Quasi-homogenous polynomials

∗ In this subsection we will give a proof of Brieskorn’s theorem, for this we introduce actions of C on S ∩ N (g) and h and show that χ : g → h/W is equivariant with respect to this action : weights of this action will be fully determined by g and will determine the singularity and the deformation.

The transversal slice

We will explicitly describe the transversal slice S, using the following lemma :

Lemma 5.17 (Jacobson-Morozov). Let x be a nilpotent element in a complex reductive Lie algebra 0 0 g. Then, there is h, y ∈ g such that (h, x, y) is an sl2-triple . (In fact, any other triple (h , x, y ) is conjugated by an element of ZG(x)).

Proof. [1], p.195.

In particular, if x is a nilpotent element and (x, y, h) a corresponding sl2-triple we get a struc- ∼ ture of sl2-module on g and g = ⊕iVni where each Vni is a irreducible representation of highest weight ni − 1. The tangent space of G · x at x is given by [g, x]. This a linear complement of the ∼ L span of the lowest weight spaces. Indeed, for simplicty assume that g = V (n) = −n−1≤i≤n−1 Wi is an irreducible sl2 representation of dimension n (we keep the convention from [41] to write V (n)), where Wk is the space of weight k. Then, [x, Wi] = Wi+2, so [x, g] = W−(n−3) ⊕ · · · ⊕ Wn−1. This shows that a linear complement to Tx(G·x) is given by W−n−1, i.e by {z ∈ g :[y, z] = 0}. The same ∼ Lr0 argument applies in general, where g = i=1 V (ni) and we get that S := x + zg(y) is a transversal slice to G · x at x. It is certainly locally closed since it is an aﬃne subspace, and by construction the natural map S ⊕ [g, x] → g is surjective.

∗ The usual scalar multiplication C × g → g does not preserve S since it is an aﬃne subspace.

On the other hand, we can again use the sl2-module structure of g : the action of h ⊂ sl2 ⊂ g gives 0 ∗ ∗ P 2 Pr −ni+1 an action of the torus C ρ : C × S → S given by ρ(t)(x + i ciei) = t x + i=1 t ciei, where

36 2 −1 the ei are the lowest weight vectors (so ni is the weight). We get an action j(t) = t ρ(t ) which

P P ni+1 stabilize S, indeed j(t) · (x + i ciei) = x + i t ciei. According to the Chevalley restriction theorem and the Chevalley-Shephard-Todd theorem, the r adjoint quotient χ : g → h/W can be realized as a morphism g → C , (χ1, . . . , χr) where χi are G homogeneous generators of degree dj of C[g] . 2 2 Since each χi is G-invariant, it is H = exp(h) invariant so χi(t · s) = χi(t ρ(t)s) = χi(t s) =

2di ∗ ∼ r ∗ t χi(s). So if C acts with weight 2di on h/W = C , the map σ : S → h/W becomes C - equivariant, where σ = χ|S.

∼ m ∼ n ∗ Deﬁnition 5.18. Let V = C ,W = C be vector spaces with action of C with weights (a1, . . . , am) ∗ and (b1, . . . , bn). If χ : V → W is a C -equivariant morphism, we say that χ is of type (a1, . . . , am; b1, . . . , bn).

2 For example, if we take V = W = C with weight (1, 2) for V and (2, 3) for W , then (u, v) 7→ (u2 + v, uv) is of type (1, 2; 2, 3). In particular, the previous discussion shows that χ is of type (n1 + 1, . . . , nr0 + 1; 2d1,..., 2dr)(ni are the weights of g seen as a sl2-module). Now we 0 look at the special case when x is a subregular nilpotent element, that is dim ZG(x) = r = r + 2.

In this case, it is possible to compute the numbers di and ni introduced before. The di are called the exponent of the Lie algebra (for compute di one could take di = ni but with x regular, see [44]

ni+1 for a proof of this fact). For simplicity, we compute instead wi := 2 . In fact (again see [44]) we have wi = di for i = 1, . . . , r − 1 so we only need to compute all the di and wr, wr+1 and wr+2 :

Type d1 d2 d3 d4 d5 d6 ... dr wr wr+1 wr+2 r+1 r+1 Ar 2 3 ...... r + 1 1 2 2 Br 2 4 ...... 2r 1 r r

Cr 2 4 ...... 2r 2 r − 1 r

Dr 2 4 ...... 2r − 4 r 2r − 2 2 r − 2 r − 1

E6 2 5 6 8 9 12 3 4 6

E7 2 6 8 10 12 18 4 6 9

E8 2 8 12 14 18 20 24 30 6 10 15

F4 2 6 8 12 3 4 6

G2 2 6 2 2 3

Example 5.19. For g = sln+1, we have αi = ei − ei+1, 1 ≤ i ≤ n. Any positive root is on the form

αi + ··· + αj. Thus we have d1 = 2, d2 = 3, . . . , dr = r + 1. For wi we take ht(α2) = 0. We get that d1 = w1, . . . , dr−1 = wr−1.

Example 5.20. Let g be of type B6, namely so7. With its Dynkin diagram, we can directly compute the set of positive roots and obtain the following table : 37 Height Positive roots Number of positive roots

1 α1, α2, α3, α4, α5, α6 6

2 α1 + α2, . . . , α5 + α6 5

3 α1 + α2 + α3, . . . , α4 + α5 + α6, α5 + 2α6 5

4 α1 + ··· + α4, . . . , α3 + ··· + α6, α4 + α5 + 2α6 4

5 α1 + ··· + α5, α2 + ··· + α6, α3 + α4 + α5 + 2α6, α4 + 2α5 + 2α6 4

6 α1 + ··· + α6, α2 + ··· + α5 + 2α6, α3 + α4 + 2α5 + 2α6 3

7 α1 + ··· + α5 + 2α6, α2 + α3 + α4 + 2α5 + 2α6, α3 + 2α4 + 2α5 + 2α6 3

8 α1 + ··· + α4 + 2α5 + 2α6, α2 + α3 + 2(α4 + α5 + α6) 2

9 α1 + α2 + α3 + 2(α4 + α5 + α6), α2 + 2(α3 + ··· + α6) 2

10 α1 + α2 + 2(α3 + α4 + α5 + α6) 1

11 α1 + 2(α2 + α3 + α4 + α5 + α6) 1 12 ∅ 0

G ∼ W This give us generators for C[g] = C[h] in degree 2, 4, 6, 8, 10 and 12 as expected from the previous table. For wr, wr+1 and wr+2, we just need to re-organize the roots, keeping in mind that now height(α6) = 0.

5.3 End of the proof of Brieskorn’s theorem

Identiﬁcation of the singularities

We now turn to singularity theory, in order to indentify the Du Val singularity. We begin by two lemmas :

3 Lemma 5.21. Let (X, 0) a germ of a surface in C , given by the equation f = 0. If X has an isolated singularity then

dimC C{x, y, z}/(f, ∂xf, ∂yf, ∂zf) < ∞

Proof. We can consider C{x, y, z}/J as the global section of the quotient sheaf C{x, y, z}/J, which has only support at the origin. By the ﬁniteness theorem of Serre-Cartan (cf [7]), the cohomology spaces Hi(X,F ) are ﬁnite dimensional for any coherent sheaf F and compact analytic variety X. Since a point is compact we get the desired result by taking i = 0.

Remark 5.22. In fact this is an equivalence : if X does not have an isolated singularity, then X should be singular along a curve C which is locally the intersection of two surfaces given by g1 = 0 and g2 = 0. If J is the Jacobian ideal J = hf, fx, fy, fzi, our hypothesis says that C ⊂ Z(J) i.e

J ⊂ hg1, g2i, so dimC C{x, y, z}/J = ∞.

38 Here is the second lemma, for the proof we refer to the book by Slodowy, Lemma 1, in section 8.3 (page 125).

Lemma 5.23. Let χ : g → h/W be the quotient map, and σ : S → h/W the restriction of F to a transversal slice at x, where as usual x is a subregular nilpotent element. Then, the diﬀerential dxσ : S → h/W has rank r − 1.

With this lemma we can identify the singularity of the ﬁber σ−1(0) by direct computation.

Let W1,...,Wr+2 be coordinates on S with weight w1, . . . , wr+2. We write (σ1, . . . , σr) for the components of σ. We know by comparing degrees in the previous table computing the di and the wi that χr does not have the same degree as any of the Wi, in particular χr does not contains a linear term in the W , so ∂χr = 0 for all i. It follows that for all j < r we should have at least one i ∂Wi ∂χj ij with 6= 0 : comparing degree, since wi = di for i < r and that the di are strictly increasing ∂Wij we should have χi = Wi + Ri where Ri has no linear terms. This shows that up to local analytic equivalence the map is of the form χi = Wi for i < r + 1. Therefore, the singularity is locally isomorphic to C[Wr,Wr+1,Wr+2]/(χr) where by χr we mean χr(0, 0,..., 0,Wr,Wr+1,Wr+2).

From now on, for simplicity we name the variables x = Wr, y = Wr+1 and z = Wr+2. To verify that N (g) ∩ S = χ−1(0) is the expected Du Val singularity we just need to do a case-by-case computation, as an example we will do it for the exceptional types.

For the E6 case, we have generators x, y and z of degree 6, 4 and 3. Since χr is of degree 12 we 2 3 2 2 have χr = ax + by + cz + dx z. ∼ 2 We claim that c 6= 0. Indeed, if c = 0 a little computation give C{x, y, z}/J = C{x, z}/(x , xz) which has inﬁnite dimension as a vector space as it contains the monomials zk for k ≥ 0. Therefore d2 4 3 2 d 2 by 5.21, c 6= 0 and a small computation gives χr = (a − 4c )z + by + c(x + 2c z) . A change of variable (and relabelling constants) gives az4 + by3 + cx2 = 0. Using again that the singularity is isolated, abc 6= 0 so we can assume a = b = c = 1 which gives indeed the E6 singularity.

Suprisingly, E7 and E8 are the easiest cases. Indeed, there are only three monomials which can 3 3 2 2 3 5 appear : z y, y , x for E7 and x , y and z for E8. Since the singularity is an isolated singularity any of these monomials should appears, and we get exactly a singularity of the corresponding type.

Identiﬁcation of the semiuniversal deformation

Again we need two lemmas taken from [42], chapter 8, before being able to identify the map σ : S → h/W with the semiuniversal deformation constructed in 5.14. We begin by

∗ Lemma 5.24. Let V,U be complex vector spaces and φ : V → U a C -equivariant morphism of type

(a1, . . . , an, b1, . . . , bn). Assume that d0φ is of rank n. Then, up to reordering, a1 = b1, . . . , an = bn, and moreover φ is an isomorphism with a polynomial inverse.

39 Proof. We know that φ is a polynomial in v1, . . . , vn with appropriate weights. Since d0φ has full rank, then this means that d0φ is linear and equivariant, i.e that up to reordering the weights are equal and have same multiplicities. ∼ Lm ∗ Now, let V = j=1 Vj be the decomposition of V into C eigenspaces of weight λj with λ1 < λ2 < ∼ Lm ... , with dim Vj = #{1 ≤ i ≤ n : ai = λj}, and similarly for U = j=1 Uj.

First, we look at `1 := φ|V1 : V1 → U1 : this is an invertible linear map. We can write φ|V1⊕V2 :

V1 ⊕ V2 → U1 ⊕ U2 as (`1(v1), `2(v2) + p2(v1)) where p2 is a polynomial in v1 ∈ V1 and `2 is linear −1 in v2. But v1 = l1 (u1) and `2 is invertible by hypothesis that d0φ has maximal rank, so again we can ﬁnd a polynomial inverse to φ|V1⊕V2 . By induction we can ﬁnd an inverse.

Now we can conclude :

∗ Lemma 5.25. Let φ : V → U a morphism of type (d1, . . . , dn; d1, . . . , dn) of two C -vector spaces, whose ﬁber φ−1(0) is zero-dimensional. If the weights are strictly positive then φ is an isomorphism.

Proof. If φ is not an isomorphism, d0φ has rank less than n by the ﬁrst lemma. By [42] lemma 2 p.122, we can assume that it has rank 0. Now, in particular, if a is the smallest weight of V , then the corresponding component of φ should be identically zero : it follows that the ﬁber has positive dimension.

After these lemmas we can show that χ : S → h/W is the semiuniversal deformation. Indeed, by construction the semiuniversal deformation X → U of an ADE singularity is equivariant of type r+2 r (w1, . . . , wr+2; d1, . . . , dr). So if π : C = X → U = C is the semiuniversal deformation, we ∗ obtain a C -equivariant deformation φ : h/W → U such that the following diagram commutes :

Φ S / X

χ π φ h/W / U

where Φ is induced by the universal property of the semiuniversal deformation. We need a last technical lemma, namely that there is a unique fiber of the map σ : S → h/W isomorphic to X0, −1 namely σ (0). More precisely, any fiber Xt verifies the following property : its resolution graph can be obtained from the resolution graph of X0 by deleting some vertices and edges, see the appendix. This shows that φ−1(0) is 0 and by 5.25 φ is an isomorphism. Since Φ was the pullback map it follows as well that the deformation χ : S → h/W is the semiuniversal deformation, concluding the proof.

2 3 4 Example 5.26 (Type E6). The equation of type E6 is given by x + y + z = 0. We need to ﬁnd a basis of C{x, y, z}/J where J is the Jacobian ideal, this is given by u0 = 1, u1 = z, u2 = y, u3 =

40 2 2 z , u4 = yz, u5 = yz (we order by weights). The weights on (x, y, z) are (6, 4, 3), and the total degree is 12. The morphism is

2 3 4 2 2 χ :(x, y, z, u1, u2, u3, u4, u5) 7→ (x + y + z + u1z + u2y + u3z + u4yz + u5yz , u1, u2, u3, u4, u5)

The ﬁrst component χ1 is of degree 12 : this means that u1z should be of degree 12, that is u1 is of degree 9. Similarly, we ﬁnd that the weights of (u2, u3, u4, u5) are (8, 6, 5, 2). So we obtain a morphism of type (6, 4, 3, 9, 8, 6, 5, 2; 12, 9, 8, 6, 5, 2) which coincides with the weights previously computed.

6 The Springer resolution

We follow again [41], section ”Group-theoretic resolution”. The previous section focused on the semiuniversal deformation of a Du Val singularity, now we will resolve all the Du Val singularities at one time.

Definition 6.1. If χ : X → U is a flat morphism, a strong simultaneous resolution of X is a commutative diagram ψ Y / X χ θ U such that ψ is proper and surjective, and induces for all u ∈ U a resolution of singularities −1 −1 ψu : θ (u) → χ (u). We also require that θ is smooth, that is that the differential dyθ is surjective for all y ∈ Y .

Such a morphism is hard to obtain, so we will use a simpler version :

Definition 6.2. Let X → U be a flat morphism. A simultaneous resolution for X is a strong simultaneous resolution for X ×U V → V where V → U is a surjective finite map.

Strong simultaneous resolution seems more natural but there are topological obstructions to the existence of such a map, in fact this is the content of the last section of [41]. Such a resolution interacts nicely with the transverse slices S constructed before, and in fact will give the minimal resolution of the associated Du Val singularity computed before. For this, it will be crucial to be able to compute the self-intersection of the exceptional curves. It was ﬁrst done by H. Esnault in her master thesis, but we will reproduce a simpliﬁed argument given by Slodowy in [41], pages 40-41.

41 6.1 Grothendieck’s theorem

We will ﬁrst present Springer’s theorem resolving the nilpotent cone N (g) of a semisimple Lie algebra g. We begin by a lemma :

Lemma 6.3. Let x ∈ N (g), where g is semisimple. Then the following are equivalent :

• x is regular, that is dim ZG(x) = r where r is the rank of g.

• x is contained in exactly one Borel subalgebra.

• If {α1, . . . , αm} are a set positive roots and παi : g → gαi the corresponding projection, then

παi (x) 6= 0 for all i = 1, . . . , m.

Proof. (to complete)

We are now in good position to state and prove Springer’s theorem. We just need a few more notations : ﬁx a maximal torus and a Borel subgroup T ⊂ B ⊂ G. Write b = Lie(B) = h ⊕ n where h = Lie(T ) and n is the nilradical of b.

B Theorem 6.4. Let Ne = G × n. Then, the map ψ0 : Ne → N , (g, n) 7→ Ad(g)n is a resolution of singularities.

Proof. First, G×B n is a vector bundle over the ﬂag variety G/B =: B, with ﬁber n : in particular Ne is smooth. We need to check that ψ0 is proper and birational. For this, we will use the isomorphism Ne =∼ {(A, x) ∈ B × N (g): x ∈ A}. This is an application of the lemma 8.5 in the appendix, with

E = n,F = N (g),G = G and H = B. Now it is easy to see that under this identiﬁcation ψ0 is just the projection onto the second component. This shows that ψ0 is proper since B is a projective variety (this is a non-trivial result, see [25]). Moreover, if x is a regular element, then by the previous lemma the preimage of x under ψ0 is only constitued of a point. Since regular elements are dense in N (g) this shows that indeed ψ0 is birational (here we use that a bijective morphism f : X → Y is an isomorphism if Y is normal, this is a corollary of Zariski main’s theorem, see [?]).

Now we turn to Grothendieck’s theorem, which gives us a simultaneous resolution for the quotient map χ : g → h/W . Let ˜b = G ×B b. Consider the following diagram :

ψ b˜ / g

θ χ φ h / h/W where ψ(g, b) = Ad(g)b and θ((g, b)) = h where we wrote (g, b) = (e, h + n).

42 Theorem 6.5 (Grothendieck). The diagram above is a simultaneous resolution of singularities for χ : g → h/W .

Proof. First let us show that θ is well defined. We have b · (g, h + n) = (gb−1, Ad(b)(h + n)) = (gb−1, h + Ad(b)n + n0) since Ad(B)h = h + n0 for all h ∈ h and some nilpotent element n0, so θ is well defined. The fact that the diagram is commutative is clear since the semisimple part of ad(g)(h + n) is conjugated to h. In general, if h ∈ h and n is nilpotent, the semisimple part of h + n is not h, but here by construction [h, n] = 0, so the semi-simple part of Ad(g)(h + n) is conjugated to h. The morphism b˜ → h decomposes as ˜b → G ×B h −→B∼ × h → h and each of these maps are smooth, so θ is also smooth. The same argument used before applies for the properness of ψ. So we essentially −1 −1 need to show that ψh : θ (h) → χ (h) is a resolution, where h is a lift of h ∈ h/W . For this, we will use associated bundle for reduce this case to the Springer resolution. In the appendix, (see [?]) we proved the isomorphism χ−1(h) =∼ G ×Z(h) N (z(h)). In particular we obtain a G-equivariant map θ−1(h) → G/Z(h), and by 8.6 we have an G-isomorphism θ−1(h) =∼ G ×Z(h) F where F is the fiber over eZ(h) of the map θ−1(h) 7→ G/Z(h). Concretely this is the set of (g, h + n) ∈ G ×B (h + n) such that the semisimple part of Ad(g)(h + n) is h. Up to conjugacy by an element b ∈ B we can already assume that ad(b)(h + n) has semisimple part h, i.e that gb−1 ∈ Z(h). This means that the fiber is exactly the bundle Z(h) ×B (h)n(h) where B(h) is a Borel subgroup of Z(h) and n(h) the nilradical of b(h) := Lie(B(h)). Now we are done. Indeed, consider the Springer resolution for G0 = Z(h):

B(h) ψ0(h): Z(h) × n(h) → N (z(h))

Then, transform this map into a map of associated B-bundles :

B B B(h) B G × ψ0(h): G × (Z(h) × n(h)) → G × N (z(h))

−1 −1 Since the previous map is nothing but ψh : θ (h) → χ (h) this concludes the proof.

In fact, with this proof one could also resolve the semiuniversal deformation but this is more complicated so we refer once more to [42].

Now let x subregular and S a transverse slice at x. We want to show that the resolution −1 Se ∩ θ (0) → S0 = N (g) ∩ S coincides with the minimal resolution constructed in the section 3. The fact that the exceptional divisor is the expected one was proved by Steinberg and Tits (see

[44]). In the next section we will compute the exceptional divisor for type An . According to 3.10 we just need to compute the self-intersection of the exceptional curves Ci. This is the object of the next theorem, but first we need some preliminary results from differential geometry about normal bundles. 43 Definition 6.6. Let Y ⊂ X two smooth algebraic varieties. The normal bundle NY/X a vector bundle on Y defined by NY/X = (TX|Y )/T Y (where TX,TY are the tangent bundles of X,Y ).

We can extend this deﬁnition if Y is singular.

Let us assume that we have a map f : X → Y and Z ⊂ Y is transverse to f(X). In this case, we have the following diagram : f −1(Z) / X

f i Z / Y and it is natural to ask how the normal bundle of f −1(Z) in X is related to the normal bundle of Z in Y . This is given by the following theorem :

Theorem 6.7. In the setting as above we have an isomorphism

∗ ∼ f NZ/Y = Nf −1(Z)/X

Remark 6.8. We will use this theorem in two special cases : ﬁrst, when f is a smooth map between two algebraic varieties, typically the projection of an associated bundle π : G ×H F → G/H. The other interesting case is when f : X → Y is the inclusion of a subvariety transverse to Z.

We continue with two other propositions :

Proposition 6.9. Let X ⊂ Y ⊂ Z manifolds, then there is an exact sequences of vector bundles over Z :

0 → NX/Y → NX/Z → (NY/X )|Z → 0

Proposition 6.10. Let X,Y be algebraic varieties and f : X → Y a smooth map. Moreover, let −1 dim(Y ) m D ⊂ F = f (y) be a subvariety in a ﬁber, then we have ND/F ⊕ C = ND/X , where C is the trivial bundle of rank dim(Y ). In particular, c1(ND/F )) = c1(ND/X ).

Finally, we can compute the self-intersection of an exceptional curve.

Theorem 6.11. Let C an irreducible component of the exceptional divisor, then C2 = −2 (we take the self-intersection in X˜ := S˜ ∩ θ−1(0)).

Proof. We need to compute deg(c1(NC/X˜ )). If Y ⊂ X we write c1(Y,X) for c1(NY/X ). The idea ∗ 1 of the proof is to reduce this computation to the computation of c1(E,T P ) where E is the zero section.

First since C ⊂ X˜ ⊂ G ×B n we can apply 6.9 and obtain the exact sequence

0 → NC/X˜ → NC/G×B n → NX/G˜ ×B n → 0 44 B B We will show that c1(X,G˜ × n) = 0 so c1(C, X˜) = c1(C,G × n). For this we use the square

X˜ / Ne

f i X / N (g) and the exact sequence

0 → NX/N (g) → NX/g → NN (g)/g → 0

So it is enough to see that c1(NX/n(g)) = 0. By the short exact sequence we can just verify that

NX/g and NN (g)/g are zero. This is because they are both aﬃne complete intersections so the B normal bundle is trivial, and ﬁnally c1(C, X˜) = c1(C,G × n).

Now we will use a reduction trick : we can assume that the image of the map C ⊂ G ×B n → G is P/B where P = Pα is the minimal parabolic subgroup corresponding to a simple root α : this is done in [44] for the general case : later we will see this explicitely for the type An. Now we apply 6.10 to the map G ×B n → G/P , with D = C, which by hypothesis is contained in the ﬁber over eP . This gives B B c1(C,G × n) = c1(C,P × n)

The map C → G ×B n → P/B is an isomorphism, in particular we can consider C as a section of B B B the bundle P × n, so c1(P × n) = c1(C,P × n)

Now we use the exact sequence 0 → nP → n → n/nP → 0 which gives an exact sequence of bundles (over P/B) B B B 0 → P × nP → P × n → P × n/nP → 0

The first bundle is trivial by 8.4 since the action of B is induced by an action of P . So finally B B we just need to compute c1(P × n/nP ). This reduces immediately to c1(SL2(C) × C) where the action is with weight 2 on C, that is we obtained O(−2) which finishes the proof.

6.2 The Springer ﬁber for g of type An

∗ For this subsection we explicitely compute the Springer ﬁber of the map T B → N (g) for type An.

Let g = sln, a subregular element x ∈ N (g) has two Jordan blocks, one of size n − 1 and one of

45 n size 1. We will take a basis v1, . . . , vn of C such that   0 1 0 ...... 0    0 0 1 0 ...... 0       0 0 0 1 0 ...... 0      x = ......    ......       0 ...... 0 1 0    0 ...... 0 0 0 The Springer fiber is the set of Borel subalgebras b such that x ∈ b, which can be interpreted as the set of complete flags F1 ⊂ F2 ⊂ ...Fn−1 such that xFi ⊂ Fi (in fact, since x the only eigenvalue is 0 this implies that xFi ⊂ Fi−1). Let us write Fi = hw1, . . . , wii. Of course, if ui ∈ Fi, 0 P 0 then wi = wi + j≤i−1 λjuj determines the same flag, so we will not distinguish wk, wk if they are n equal in C /Fk−1.

First, if y is a regular element, i.e of the form y(v1) = 0 and y(vk) = vk−1 for k ≥ 2 the fiber should be a point as the Springer resolution is an isomorphism outside the singular locus. This is indeed what happens : any flag stabilized by y should have F1 = hv1i, which forces V2 = hv1, v2i since we should have y(w2) ∈ F1 and by induction we get wk = vk. So finally there is only one flag stabilized by y namely the canonical flag. The difference now with our subregular element x is that vn is another element in the kernel of x, so we can ”modify” the flag at the stage k : instead of adding vk, we can add vk + avn. After such modification the flag is rigid, i.e uniquely determined. 1 This gives a copy of P for every such modification and the dual graph is as expected An. We omit the formal proof and give directly the Springer fiber :

Proposition 6.12. The Springer ﬁber of type An is the union of the curves C1,...,Cn, where each

Ck is as rational curve, an explicit isomorphism being

n [a : b] 7→ hv1i ⊂ hv1, v2i ⊂ · · · ⊂ hv1, v2, . . . avk + bvni ⊂ hv1, . . . , avk + bvn, vki ⊂ · · · ⊂ C

7 Conclusion

As a general conclusion, we will give an example and sketch how to generalize the theorem of Brieskorn to other simple Lie algebras. After, we will give references for more advanced related topics about McKay correspondance and Brieskorn’s theorem .

Brieskorn’s theorem extended

We will now look at symmetries of Dynkin diagram : the idea is that if ∆0 is a non-simply laced Dynkin diagram, then we can associate to ∆0 a simply laced Dynkin diagram ∆ and a group 46 G ⊂ Aut(∆) such that ∆0 = ∆/G. For do this, we need a key condition : two connected vertices can’t be in the same G-orbit. In an equivalent way, two roots in the same orbit should be orthogonal. This is a nice idea but it is not immediately clear how to apply this construction to the corresponding root system, and details are in [?] : the construction is called folding a Dynkin diagram.

For example, folding D4 under the S3 symmetry gives G2 diagram. Now, what should be a

G2-singularity ? We would like to have a D4 singularity, and a Z/3Z symmetry subgroup. The equation x2 + y3 + z3 = 0 indeed admits such a action, for example (y, z) 7→ (jy, z) where j3 = 1. This action also permutes the exceptional components of the resolution, and possible (equivariant) semiuniversal should leave an invariant Dynkin diagram ∆ˆ ⊂ ∆. There are two such diagrams :

3A1 and A1. We write down the explicit semiuniversal deformation (we take the same convention as Slodowy and take the equation z2 = x2 − 3xy2 ):

4 2 2 2 2 2 2 χ : C → C , (x, y, z, t) 7→ (z − x + 3xy + t(x + y ), t)

For more details about this (in particular concerning the Lie algebras) the best reference is again the article by Slodowy [41].

Further directions

We now state several directions one could take from this project.

Starting with a ﬁnite group Γ ⊂ SL2(C), we saw a connection between the representation theory 2 of Γ and the geometry of the space C /Γ. For further developments of the McKay correspondence an excellent survey is [39]. We already quoted [45] which use K-theory. For relation with quivers, symplectic geometry and Hilbert schemes see [11], especially the last chapters. Finally, the McKay correspondence was interpreted as an equivalence of derived categories in [?].

A natural thing to do is to take a transverse slice to an arbitrary nilpotent element x. If x is not regular or subregular then S ∩ N (g) is not an isolated singularity, but we can replace N (g) by an arbitrary nilpotent orbit closure and we will still get a symplectic singularity (this notion was introduced by Beauville in the paper [5]). A particularly interesting special case is when O is a nilpotent orbit containing x in its closure, and which is adjacent to G·x in the partial order. Then S ∩O is an isolated singularity. Its dimension is dim O − dimG · x, which is even and often equals 2. The pair (Gx,˙ O) is called a minimal degeneration. In fact the codimension 2 case and for classical Lie algebras, it turns out that the singularity S ∩ O is either a du Val singularity (of type Am or Dm) or a union of two du Val singularities of type A2k−1, meeting transversally at the common singular point.

47 The recent paper [19] determines all of the singularities S ∩ O associated to minimal degener- ations in exceptional type Lie algebras. They are mostly du Val singularities or minimal nilpotent orbit closures, but other interesting special cases appear.

Finally there is more to say about deformations of singularities : there has been a lot of study of so-called ”noncommutative deformations” of du Val singularities. We deﬁne a noncommutative Γ deformation of C[V ] to be a Z-ﬁltered associative algebra A whose corresponding graded algebra Γ is isomorphic to C[V ] .

There are natural noncommutative deformations of arbitrary Slodowy slices, called ﬁnite W - algebras and introduced 15 years ago by Premet in [36]. These noncommutative deformations induce deformations of all of the (singular) intersections S ∩ O. In particular when S is transverse to a subregular nilpotent, we get noncommutative deformations of S ∩ N (g). There is a conjecture by Premet in 2002, saying that the ﬁnite W -algebras are isomorphic to certain algebras constructed in mathematical physics.

Finally in the important paper [10] various noncommutative deformations of du Val singularities are constructed. The idea is to form the smash product C < x, y > #CΓ and quotient by the ideal hxy − yx − λ∠ where λ ∈ Z[CΓ]. Call this algebra Hλ. Then eHλe is a noncommutative defor- Γ 1 P mation of C[x, y] where e = |Γ| g∈Γ g. The Crawley-Boevey-Holland algebras were a forerunner to symplectic reﬂection algebras, which have been another major area of study in representation theory in the last 15 years.

Conclusion

Simple surface singularities, Lie algebras and ﬁnite groups of SL2(C) are fascinating yet very different subject. The fact that the Dynkin diagram all connect them together is very beautiful, and we will ﬁnish with a citation borrowed from [18] :

”If we needed to make contact with an alien civilization and show them how sophisticated our civilization is, perhaps showing them Dynkin diagrams would be the best choice!”

8 Appendix

We ﬁrst present an alternative computation for the resolution graph using toric geometry. Then, we use associated bundles for show some isomorphisms of varieties we used in the section about

48 Springer’s resolution.

Toric geometry

In this section, we will use toric geometry for compute the resolution graph of An and also for 1 show that the total space of the resolution of the quadratic cone is the cotangent bundle of P ∗ 1 where the resolution : f : T P → X is just the contraction of the zero section. It implies 2 ∗ 1 that E = −2 where E is the zero section of T P , and since X is the nilpotent cone of sl2 this fact will be essential for the Springer resolution. An excellent reference for toric geometry is [9], in particular the chapter about toric surfaces contains some part about the McKay correspondance.

As usual let’s ﬁrst see the case of A1 : indeed we can take the cone σ generated by v = ∨ 2 2 ∨ −e1 + e2, w = e1 + e2. We have σ = σ (identitying R and (R ) with scalar product) and the 2 ∨ generator of the dual lattice Z ∩ σ are v, w and u = e2, with relations v + w = 2e2. This gives 2 the algebra C[x, y, z]/(xz = y ), so this is indeed the quadratic cone X. For resolving the singularity of Uσ = X we will subdivide σ by adding the ray R+e2. Let us call 2 Y the corresponding toric variety : Y is obtained by gluing two copies of C with coordinates a = x−1, b = xy and c = x, d = x−1y. The gluing map is (a, b) 7→ (a−1, a2b) which is exactly ∗ 1 ∼ the gluing map for T P = O(−2). Moreover, from this description we can also see that the ﬁber coordinates are a, d so b = 0 = c is the zero section of O(−2). From the toric description of the blow-up, we obtain that the zero section is the exceptional divisor.

2 Now let us do the general case, and let σ ⊂ R a strongly convex rational polyhedral cone, with basis v1, v2. By an SL2(Z) change of variable we can assume that v2 = e2. Substracting a suitable multiple of v2 we can assume that v1 = me1 − ke2 with 0 ≤ k < m. 2 ∼ Proposition 8.1. The toric variety Xσ is the quotient of C by the following action of Z/mZ = k µm : ζ · (x, y) = (ζx, ζ y).

In particular, taking k = m − 1 gives the usual action of Z/mZ, and the invariant polynomials m m m are a = x , b = y , c = xy with the relation ab = c i.e this is an Am singularity. A change of basis f1 = e2, f2 = e1 − e2 gives new generators u1 = f1 + (k + 1)f2, u2 = f2.

Proposition 8.2. A resolution of singularity of a toric surface is given by a fan Σ obtained by a sequence of subdivisions the cone σ, such for all cones σ0 ∈ Σ are smooth (i.e the two primitives 0 2 vectors of σ generates Z ). Every ray with primitive vector v correspond to an exceptional irreducible rational curve Ev, and two such curves intersects if and only if the rays share a commun cone. Moreover, if a ray with primitive vector v has neighborhood w1, w2 there is a unique k ∈ Z 2 with w1 + w2 = kv, and we have Ev = −k. 49 In particular, adding a ray Ei with primitive vector passing by (1, i) for 1 ≤ i ≤ k + 1 gives a resolution : the explicit description of the intersection also gives that this resolution has dual graph to An.

Principal G-bundles and the ﬁber of the adjoint quotient

In this section we use the concept of G-bundle for identify different varieties together. This will give crucial identifications for the Springer resolution and will give a nice description of the fiber of the morphism χ : g → h/W . In fact, it is the main ingredient in the proof of the following theorem :

Theorem 8.3. Let g a Lie algebra with Dynkin diagram ∆, then σ−1(0) is the only ﬁber of type ∆.

This theorem is extremely important for identify the semiuniversal deformation. Now we state and proof everything we need about principal G-bundles. The following is nothing but the pages 19-20 in [41] with more details added. Since this part is a bit abstract, for keeping the intuition a good case to keep in mind is when G a simple algebraic group, H a Borel subgroup, F = N (g) the nilpotent cone of g and E ⊂ N (g) the nilradical n of b.

Let G be an algebraic group. A triple (X, B, π) where X,B are algebraic varieties and π : X → B is a morphism is called a G-principal bundle if there is an action of G on X such that the action −1 preserves the ﬁber, and for all b ∈ B there is a G-equivariant isomorphism φb : G → π (b). −1 Such isomorphism is by deﬁnition exactly a choice of an element φb(eG) ∈ π (b). If such a choice comes from a map σ : B → X (called a section) then we obtain an isomorphism B × G =∼ X, (b, g) 7→ (gσ(b)). So we should think to a G-principal bundle as a generalization of a product, and at least locally it is isomorphic to a product.

If F is a variety on which H acts, we can form the bundle G ×H F = (G × F )/H where h(g, f) = (gh−1, hf). An element of G ×H F should be written [(g, f)] but we will omit the bracket if there is no source of confusion. There is a projection G ×H F → G/H and the ﬁber over gH is isomorphic to F (this works for any such bundle X ×H Y as long as H acts freely on X).

Lemma 8.4. H-action on F was the restriction a G-action we obtain an isomorphism G ×H F =∼ G/H × F , by the map (g, f) 7→ (gH, gf). Moreover, if E ⊂ F is H-stable we get an embedding G ×H E ⊂ G/H × F .

Proof. Consider the map G × F → G/H × F (g, f) 7→ (gH, gf). The image of h · (g, f) is (gh−1H, gh−1 · hf) = (gH, f) so we obtain a map GH × F → G/H × F . Injectivity always 50 H works : if (g1H, g1f1) = (g2H, g2f2) for some (gi, fi) ∈ G × F , then there is some h ∈ H with −1 −1 g1h = g2. Now (g1, f1) = (g1h, h f1) = (g2, g2 g1f1) = (g2, f2) since g2f2 = g1f1 by hypothesis. For surjectivity we use that (g, g−1f) has image (gH, f) (here we used that the action of H is the restriction of a G-action.

Lemma 8.5. Let G, H, F, E as before and assume that H is the stabilizer of E, that is the set of g ∈ G with gE = E. Write E = G/H, it is exactly the set of conjugate of E by G. Then, there is an embedding G ×H E → E × F , and the image is described by all the couples (E0, f) with f ∈ E0.

Proof. The embedding G ×H E → E × F is simply the map (g, e) 7→ (gE, ge). It is clear that the image of this map is exactly the set of couples (E0, f) where E0 is conjugate to E and f ∈ E0.

This setting appears in the resolution of Springer and give an embedding G ×B n ,→ B × N (g) where B is the ﬂag variety.

Now we turn to the next lemma, necessary for the Grothendieck’s theorem and for the description of the ﬁbers of the map σ : S → h/W .

Lemma 8.6. Let X be a G-variety, and π : X → G/H a G-equivariant morphism. Then, we have an isomorphism of bundles X =∼ G ×H F where F = π−1(eH).

Proof. We have an embedding G ×H F → G/H × X, (g, f) 7→ (gH, fg) by the previous lemma. In fact, its image is exactly the graph of X → G/H.

Again, let χ : g → h/W be the quotient map. We already saw in 5.4 that there is a unique closed orbit in every ﬁber, constitued of the semi-simple elements. Now, if h ∈ h/W , and h ∈ h a −1 lift of h, we can identify the closed orbit of χ (h) with G/ZG(h). The map x → xs restricts to a map σ : χ−1(h) 7→ G/Z(h). By the previous lemma, we obtain an isomorphism χ−1(h) =∼ G×Z(h) F where F = σ−1(Z(h)). Hence, F = {h + n | n ∈ N (g), [h, n] = 0} = h + N (z(h)). Conclusion :

Proposition 8.7. We have an isomorphism χ−1(h) =∼ G ×ZG(h) N (z(h))

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