Master thesis

Simple Lie algebras and singularities

Nicolas Hemelsoet

supervised byProf. 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

graphs from Γ: the resolution graph of the surface C2/Γ, and the McKay graph of the representation

Γ ⊂ SL2(C). The McKay correspondance states that such graphs are the same, more precisely the

resolution graphs of C2/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.

For computing the equations of C2/Γ := Spec(C[x, y]Γ) we need to compute the generators and

relations for the ring C[x, y]Γ. As an example, if Γ = Z/nZ generated by g = diag(ζ, ζ−1) with

ζ a primitive n-th root of unity, the action on C[x, y] is g · x = ζ−1x and g · y = ζy (this is the

contragredient representation). The invariants are u = xn, v = yn and w = xy, the unique relation

1

is uv = wn 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 on C[g] and since G is reductive, by Hilbert’s theorem C[g]G is finitely generated. We get a

morphism of algebraic varieties F : g → g/G (where g/G := Spec(C[g]G) is the categorical quo-

tient) corresponding to the inclusion C[g]G ⊂ C[g]. In the case of g = sln the map is simply

χ : sln → Cn−1,M 7→ (tr(M2), tr(M3), . . . , tr(Mn)). We will see that dim g/G = r where r is the

rank of G. Let x ∈ g be a subregular (meaning that dimZG(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)∩ Sis 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

surface C2/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

space V ∼= R3. In fact, it turns out that this morphism SU2 → GL3(R) has image SO3, because

an element of SU2 preserve norm, and the restriction of the norm of C2 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

Γ ⊂ SU2 by the surjection SU2 → SO3 (topologically this is the covering map S3 → RP 3). Now

admitting the previous facts, to classify finite subgroups of SU2 we can just look at finite subgroups

3

of SO3.

Now let Λ a non-trivial finite subgroup of SO3. Let P be the set of points x ∈ S2 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 .

If N is the number of orbits, then we have N = 1|Λ|∑

g∈Λ |P g| so N = 1|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

each orbit we can write |P | =∑

i |Λ · xi|. Using that |stab(xi)| = |Λ|/|Λ · xi| we have

2

(1− 1

|Λ|

)= N −

∑i

1

|stab(xi)|=∑i

(1− 1

|stab(xi)|

)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 12 ≤ 1− 1

—stab(xi)| ≤ 1, showing that 2 ≤ N ≤ 3.

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.

For each binary tetrahedral group Γ ⊂ SL2(C) we will compute the invariant ring C[x, y]Γ.

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

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.

Returning to binary groups, we have a natural action of Γ on C2, giving the following (contra-

gredient) action on C[x, y] : g · x = (dx − by) and g · y = (ay − cx), where g =(a bc d

). Of course,

computing all the traces would be onerous, so in practice we need another method. We will see

that the ring C[x, y]Γ will always be generated by 3 polynomials, satisfying a unique relation, and

the corresponding surface X ⊂ C3 will have an isolated singularity at the origin.

4

Type An

Here, Γ ∼= Z/nZ and Γ is generated by g =(ζ 00 ζ−1

)where ζ is a primitive n-th root of unity. The

corresponding action on C[x, y] is g ·x = ζ−1x 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.

The invariant ring is therefore C[u, v, w]/(uw−vn). A change of variables u = X+iY, w = X−iYand 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

computing generators of C[x, y]Γ. 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

first idea is that one should look at the action of Γ on P1. We write x for the point [x : 1] ∈ P1,

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,

and then we will be able to compute generators for the invariant ring C[x, y]Γ.

We follow closely the first chapter of [11]. We also recall that there is a 1-1 correspondence

between homogeneous polynomials and effective divisors in P1 (i.e an element in the free abelian

group Z[P1] with all coefficients positive).

Definition 2.2. An exceptional orbit is an orbit of Γ (for the induced action on P1) 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

invariant fO(x, y) =∏

(t:s)∈O(sx − ty), called the relative invariant associated to O. In fact fO ∈C[x, y] is defined up to a scalar multiple. Any relative invariant can be written fd11 . . . fdnn where

(d1, . . . , dn) ∈ Nn 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

will make the key hypothesis (♣) that χe11 = χe22 . Then, there is a, b ∈ C, such that Φ := afe11 +bfe22

vanishes on some [t : s] ∈ O. But

Φ(g · (t, s)) = aχe11 f1(s, t)e1 + bχe22 f2(s, t)e2 = χe11 (af1(s, t)e1 + bf2(s, t)e2) = 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

Proposition 2.3. Let Γ be a binary polyhedral group, acting on P1 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 :

Proposition 2.4. Let F =∑k

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.

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

χe11 = χe22 (♣)

where ei is the cardinality of the stabilizer of the corresponding orbit.

Type Dn+2

If Γ is the binary dihedral group of order 4n, the generators are g =(ζ 00 ζ−1

)where ζ is a 2n-th

primitive root of unity and h =(

0 ii 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 ex-

ceptional 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

{[λ : 1] ∈ P1 : λn = 1} = O2. Finally, the last exceptional orbit O3 = {[λ : 1] ∈ P1 : λn = −1}.The corresponding Grundformen are Φ1 = xy, Φ2 =

∏λn=1(x − λy) = xn − yn and similarly

Φ3 = xn + yn. The associated characters are χ1(g) = 1, χ1(h) = −1; χ2(g) = −1, χ2(h) = in and

χ3(g) = −1, χ3(h) = −in. Moreover we easily check that e2 = e3 = 4 and χ42 = χ4

3, in fact χ22 = χ2

3

so (♣) is verified.

There is an obvious set of invariant monomials (from the character table) which clearly generate

the Γ-invariant polynomials, given by Φ21,Φ1Φ2

2,Φ1Φ23,Φ

42,Φ

43 and Φ2Φ3 if n is odd. We take F1 =

Φ21 = (xy)2, F2 = Φ2Φ3 = (xn + yn)(xn − yn) = x2n − y2n and F3 = Φ1Φ2

2 = xy(xn − yn)2 and will

show that any other invariant monomials is a polynomial in F1, F2, F3. We have

Φ1Φ23 = xy(xn + yn)2 = xy((xn − yn)2 + 4(xy)n) = F3 + 4F

n+12

1

We just need to show that Φ42 and Φ4

3 are generated by F1, F2 and F3. But

Φ42 = (xn−yn)4 = x4n+y4n−4xnyn(x2n+y2n)+6x2ny2n = (x2n−y2n)2−4xnyn(x2n+y2n−2xnyn) = F 2

2−4Fn−12

1 F3

Finally

Φ43 = (xn + yn)4 = ((xn − yn)2 + 4xnyn)2 = (Φ2

2 + 4Φn3 )2 = F 2

2 + 4F3Fn−12

1 + 16Fn1

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

ators. We observe that

F 23 − F1F

22 = (xy(xn − yn))2 − (xy)2(x2n − y2n)2 = (xy)2((xn − yn)4 − (x2n − y2n)2)

7

= (xy)2(−4x3nyn − 4xny3n + 6x2ny2n + 2x2ny2n) = −4(xy)n+2((xn − yn)2)

but the last term is exactly −4Fn+12

1 F3, so we obtain the relation

F 23 − F1F

22 + 4F3F

n+12

1 = 0

We write this as

(F3 + 2Fn+12

1 )2 − 4Fn+11 − F1F

22 = 0

so replacing F3 by F3 + 2Fn+12

1 and scaling gives

C[x, y]Γ ∼= C[X,Y, Z]/(Z2 +X(Y 2 +Xn))

where X = F1, Y = F2, Z = F3 + 2Fn+12

1 .

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

F1 = Φ21 = (xy)2, F2 = Φ2

2 = (xn − yn)2, F3 = Φ1Φ2Φ3 = xy(xn − yn)(xn + yn)

Fortunately this is easier : indeed Φ1Φ2,Φ1Φ3,Φ2Φ3 are not invariant and Φ23 is invariant, so

we just need to express Φ23 using F1, F2 and F3. But

Φ23 = (xn + yn)2 = (xn − yn)2 − 4xnyn = Φ2

2 − 4Φn1

So we are done and we just need to find the relation that F1, F2 and F3 satisfy. Instead

we slightly change the generator F2 with the transformation F2 → F2 + Fn/21 (n is even). Now

F2 = x2n + y2n. With these new invariants the relation is

F1F22 = 4Fn+1

1 + F 23

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

g =(i 00 −i

), h =

(0 ii 0

)and k =

1

1− i(

1 i1 −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

x(x− i) = x+ i. This equation has two solutions x1,2 = 1+i2 (1±

√3). Since k has order 6 we should

find 4 elements in these orbits, and indeed applying g or h gives O2 = {±1+i2 (1+

√3),±1−i

2 (1−√

3)}and O3 = {±1+i

2 (1−√

3),±1−i2 (1 +

√3)}. Now we can compute the corresponding Grundformen :

Φ1 = xy(x4− y4), Φ2 = x4− 2i√

3x2y2 + y4 and Φ3 = x4 + 2i√

3x2y2 + y4. 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

kx = − i(x+ y)

1− iand ky =

x− y1− i

Now we compute

kΦ1 = − i(x+ y))

1− i· x− y

1− i·(

(1

(1− i)4) · ((x+ y)4 − (x+ y)4)

)

= −(x2 − y2)

2·(−1

4· (8x3y + 8xy3)

)= (x2 − y2) · (xy(x2 + y2)) = Φ1

We have gΦl = hΦl = Φl for l = 2, 3. Let us compute χ2(k). Since (1− i)4 = −4 we have

kΦ2 =−1

4· ((x+ y)4 + (x− y)4− 2i

√3(x+ y)2(x− y)2) =

1

4(2x2 + 2y2 + 12x2y2− 2i

√3(x2− y2)2)

=1

2(1− i

√3)x4 + (3 + i

√3)x2y2 +

1

2(1− i

√3)y4 = (

1− i√

3

2)Φ2

This shows that χ2(k) = j where j = 1−i√

32 . In fact, if we modify the sign in front of x2y2 and

repeat the previous computation we get kΦ3 = j = j2. Since j3 = 1 and |j| = 1. We obtained the

following table :

g h k

χ1 1 1 1

χ2 1 1 j

χ3 1 1 j2

From the table we can also see that χ2, χ3 verify (♣), since χ62 = χ6

3 (in fact χ32 = χ3

3). We are

now ready for computing the invariants : as noticed before, Φ1 is an invariant, as well as Φ2Φ3.

Finally, Φ33 and Φ3

2 are invariant. We will take F1 = Φ1, F2 = Φ2Φ3 and F3 = Φ32+Φ3

3 (for symmetry

reasons). So if Φa1Φb

2Φc3 is invariant, we can assume that a = 0 and e.g that b = 0 by dividing by

Φ1 and Φ2Φ3. On the other hand, Φc3 is invariant only if 3 divides c. So we only need to see that

Φ32,Φ

33 are generated by F1, F2 and F3. But observe the following equality :

F 21 = (xy)2(x4 + y4)− 4x6y6 = (12

√3i)−1(Φ3

2 − Φ33)

9

This gives us that F1, F2, F3 generate the invariant ring, but also the relation : indeed up to

scaling we can assume that F 21 = Φ3

2 − Φ33 and then F 4

1 = (Φ32 + Φ3)2 + 4Φ3

2Φ33 i.e

F 41 = F 2

3 + 4F 32

and up to scaling once more we get

C[x, y]BT ∼= C[X,Y, Z]/(X2 + Y 3 + Z4)

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 :

• An : x2 + y2 + zn+1 = 0 (n ≥ 1)

• Dn : x2 + zy2 + zn−1 = 0 (n ≥ 4)

• E6 : x2 + y3 + z4 = 0

• E7 : x2 + y3 + yz3 = 0

• E8 : x2 + y3 + z5 = 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 proprietes

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

the cubic curve C : y2 = x2 + x3 in C2. 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 :

To concretely realize such a picture, we could take the following map f : C2\{O} → C2 × P1,

f(x, y) = ((x, y), [x : y]) where O = (0, 0), thinking of ((x, y), [x : y]) as a point of C2 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 limt→0

f(L(t)) = (O, [a : b]). This means that if we try to compactify the image,

we should add all the (O, [a : b]) for [a : b] ∈ P1. 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 . . .

are higher order terms. So let’s parametrize our first branch B1(t) = (t, t + t2/2 + . . . ). Clearly,

limt→0

B1(t) = (O, [1 : 1]) and similarly limt→0

B2(t) = (O, [1 : −1]). So we could separate tangents

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

following way : first define Y to be the closure of f(C\{O}). Now since Y ⊂ C2 × P1 we can

11

take the first projection π : Y → C2. By construction, π : Y → C is a resolution of singularities.

Also, the closure C2 of f(C2\{O} is given by ((x, y), [a : b] | xb = ya). This includes all the points

((x, y), [x : y]) where (x, y) 6= O and the points (0, [s : t]) for any [s : t] ∈ P1. They are precisely

the points corresponding to ”directions passing through O” and we just replaced the origin by the

set of directions passing through it. This gives a map ϕ : C2 → C2 such that ϕ is an isomorphism

outside O and ϕ−1(O) ∼= P1 = (O, [s : t]) ⊂ C2 × P1. The map ϕ is called the blow-up of C2 at O

and the exceptional P1 is called the exceptional divisor of the blow-up, often written E.

Now we will generalize this example to C3 (it immediately generalizes to Cn but only need it

for n = 3) and give a bit more terminology. The blow-up of C3 at O is the algebraic variety defined

by

C3 = {(`, z) ∈ P2 × C3 : z ∈ `}

More precisely, the equations for C3 are

{([u : v : w], (x, y, z)) ∈ P2 × C3 : rk

(u v w

x y z

)≤ 1}

This gives 3 quadratic equations : uy = vx, uz = wx and vz = wy. The projection π : C3 → C3

induces an isomorphism

C3\O ∼= C3\π−1(O)

where O is the origin. Indeed, if z ∈ C2 is non-zero there is a unique line ` ∈ P2 with z ∈ `, namely

12

` = [z].

Let E := π−1(O), then one has E ∼= P2, and E is called the exceptional divisor of the blow-up,

it represents the set of all directions at O.

If X ⊂ C3 is a surface, let X ′ be the closure of π−1(X\O) (we take the closure with respect to the

euclidean topology). We call X ′ the strict transform of X (previously Y in the curve example) and

E∩X ′ is called the exceptional curve. Since P2 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]).

For example, C[x, y, z] ⊂ C[x, y/x, z/x] this gives a map U1 → C3, (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

Pic(X)× Pic(X)→ Z so in particular we can talk about C2 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

C2 := 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.

Remark 3.4. Usually the Chern class is defined as the morphism of groups H1(X,O∗X)→ H2(X,C), L =

OX(D) 7→ [D].

In fact define the first Chern class of any vector bundle E, using the associated line bundle

det(E) :=∧nE (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∩C ′|, where C ′ is a slightly moved

copy of C. For example, if L ⊂ P2 is a line, then we should have L2 = 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

surface Q = P1 × P1, the line L = 0× P1 has self-intersection zero as we can take L′ = ε× P1 and

clearly L∩L′ = ∅. 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

surface of degree d in P3 then L2 = 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

exceptional curves Ci of this resolution will satisfy C2i = −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 X ′. Then, C2 = −1

if and only if C is the exceptional divisor of a blow-up X ′ → X where X is a smooth surface.

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

Proposition 3.7. Let C ∼= P1 a curve on a smooth surface X ′. Then, C2 = −2 if and only if

it is the exceptional divisor of a blow-up X ′ → 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

only if C2i = −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

no (−1)-curve, that is curve C ∼= P1 with C2 = −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

f−1(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 equiv-

alent :

• X is given by C2/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

satisfies Ci ∼= P1 and C2i = −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

x. We obtain the relations xy1 = y, xz1 = z. In this chart we have the equation x2(1+y21 +z2

1) = 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 ∩X ′ ( see 3.7 ). This said, we will always from now consider the reduced equation, that is

x = 0. The other equation was the smooth surface X ′ : 1+v21 +w2

1 = 0 which is the strict transform

of X. The exceptional curve E ∩X ′ is given by 1 + v21 +w2

1 = 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

Let X be the surface given by the equation x2 + y2 + zn+1 = 0. Let us take the chart U1 with

coordinates zx1 = x, zy1 = y and z. The equation for X in U1 is z2(x21 + y2

1 + zn−1) = 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

the parity of n. For A2 the surface equation is x2 + y2 + z3 = 0. A single blow-up is enough

for resolve the singularity, and the equation of the strict transform (in the chart with coordinates

z, x1 = x/z, y1 = y/z) is z + x2 + y2 = 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

C2C1

Figure 1: Step 1 for the resolution

C4C3C2C1

Figure 2: Step 2 for the resolution

C4C3C2C1

C5

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.

For D4, we can find a change of variables such that the new equation of the surface is x2+y3+z3 = 0.

( More precisely, the equation is of the form x2 = −z(y+iz)(y−iz) = abc where a, b, c are three dif-

ferent linear forms in y, z. The new equation is x2 = −y3 − z3 = −(y+ z)(y+ jz)(y+ j2z) = a′b′c′

where j 6= 1, j3 = 1 and a′, b′, c′ are linear forms. So the change of variables is exactly finding

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

x2(1 + x(y31 + z3

1)) = 0 and the corresponding surface is smooth, because the exceptional divi-

sor does not intersect the surface in this chart. So we can take coordinates yx2 = x, yz2 = z, y and

17

we obtain the equation y2(x22 +y(1+z3

2)). The exceptional curve is E : y = x2 = 0 and it intersects

three A1 singularities at (0, 0, a) with a3 = −1. So we can perform a blow-up at each of them and

we exactly obtain the graph D4 as the resolution graph.

We now treat the case of D5 : x2 + zy2 + z4 = 0. First, let us look at the chart U1 with

coordinates yx1 = x, yz1 = z, y. The strict transform has equation x21 + yz1 + y2z4

1 = 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

singularity of type A1. Indeed by a change of variable z′1 = z1 + yz41 we obtain the new equation

x21 + y1z

′1 = 0 which is obviously an A1 singularity. The exceptional curve C ′ is the parabola

y2 + x22 = 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

of X in U2 is x23 + y2

3z + z2 = 0. This equation can be written as x23 + (z + y2

2 )2 − y4

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

verifying yx2 = x, yz2 = z. The strict transform has equation x22 + yz2 + yn−3zn−1

2 = 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

z2, y3, x3 with z2y3 = y, z2x3 = x2 the strict transform is z22(x2

3 + y3 + yn−33 z2n−5

2 ). The gradient

is ∇ = (2x3, 1 + (n− 3)yn−43 z2n−5

2 , (2n− 5)yn−33 z2n−6

2 ) which is never zero. The exceptional curve

C ′1 is given by z2 = 0 = y3 + x23 i.e it is a smooth parabola, and the strict transform of C1 is

y3 = 0, x2 = 0 i.e C ′1 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

z, x4, y4 which verify zx4 = x, zy4 = y. The strict transform is x24 + zy2

4 + zn−3 = 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

repeat our previous analysis to the Dn−1 singularity and see how C1 intersects with C2 and C ′2 in

order to apply induction. If we take coordinates y4, x5, z5 with y4x5 = x4, y4z5 = z4 the exceptional

curve C ′2 is given by y4 = 0, x5 = 0. It intersects C1 at one point. Also, C2, C′2 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

the following scheme for the intersection : C ′1 intersects C1, which intersect C ′2, 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 toD4 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

by x21 + zy3

1 + z2 = 0. We will show that this is an A5 singularity, which reduces to a computation

previously done.

To see this, scaling y1 gives the equation x21− 2zy3

1 + z2 = 0. But this equation is exactly x21 + y6

1 +

(z−y31)2 = 0. So a change of variables X = x1, Y = y1, Z = z−y3

1 gives the equation X2 +Z2 +Y 6

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

has coordinates z, x1, y1 with zx1 = x, zy1 = y. The strict transform is z2(x21 + zy3

1 + y1z2) = 0.

The exceptional curve C1 is given by x1 = z = 0, and the only singular point is (0, 0, 0). We

claim that x21 + zy3

1 + y1z2 is a singularity of type D6. Indeed, we just need to change yz(z + y2)

into w(v2 + w2) by a change of variables. We write the last expression as w(v + iw2)(v − iw2) =

w(w− iw2 +2iw2)(v− iw2). Now, if we let y = 21−iv and z = v− iw2 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

zx1 = x, zy1 = y. The strict transform is given by x22 + zy3

1 + z3 = 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 singular-

ities. 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.

It is well known that if X ⊂ Cn 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.

Proof. We have C2 ∼= 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

in fact H/Γ is the cone over S/Γ where S = S3 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

icosahedra inscribed in the unit sphere S2 ⊂ R3.

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 completes” 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

the irreducible complex representations of Γ, and mij ∈ N such that V ⊗ Vi ∼=⊕

j V⊕mij

j . 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

that mij = 〈χiχV , χj〉 = 1|Γ|∑

g∈Γ χi(g)χV (g)χj(g) = 1|Γ|∑

g∈Γ χi(g)χV (g)χj(g) = 〈χi, χV χj〉 =

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

irreducible representations are Vi = C with g ·x = ζix 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 An :

1 11 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 ∼= 〈a, x : a2n = 1, x2 = an, xa = a−1x〉

It has cardinality 4n, we call it Dn+3 because the corresponding graph will have n + 3 vertices.

Note that a generates a normal subgroup isomorphic to Z/2nZ, and that Z(Γ) = {1, x2}. 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] ⊂ 〈a2〉, indeed 〈a2〉 is normal and Γ/〈a2〉has order 4, so is abelian. But a2 = [ax, x] so [Γ,Γ] = 〈a2〉. Hence, a presentation of AΓ

is given by 〈a, x | a2 = 1, x2 = an, ax = xa〉. This is 〈x | x4 = 1〉 ∼= Z/4Z if n is odd and

〈a, x | ax = xa, a2 = x2 = 1〉 ∼= Z/2Z × Z/2Z if n is even. In both cases we get 4 1-dimensional

representations from AΓ and the remaining irreducible representations are Vl, where Vl = C2

and ρl(x) =

(0 1

−1 0

), ρl(a) =

(ωl 0

0 ω−l

). We obtain the following character tables, where

1 ≤ l, k ≤ n− 1 :

n odd 1 x2 ak ax a2x

χ0 1 1 1 1 1

χ−1 1 1 1 −1 −1

χi 1 -1 (−1)k i −iχ−i 1 -1 (−1)k −i i

χl 2 -2 ωkl + ω−kl 0 0

n even 1 x2 ak ax a2x

χ0 1 1 1 1 1

χ1,0 1 1 1 −1 −1

χ0,1 1 -1 (−1)k −1 1

χ1,1 1 -1 (−1)k 1 −1

χl 2 -2 ωkl + ω−kl 0 0

Now we can compute the McKay graph, remembering that V1 corresponds to the natural rep-

resentation 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 Dn+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.

2 22 2

. . .

1

1

1

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}.

If M =

(1 1

0 1

), then MG = {

(1− ac a2

−c2 1 + ac

)| ad−bc = 1}, i.e the conjugacy class of M is given

by {M,

(0 1

−1 −1

),

(1 0

−1 1

),

(−1 1

−1 0

)}. Now we can notice that multiplying by −1, transpose

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

0 1

)1 id(

−1 0

0 −1

)2 id(

1 1

0 1

),

(0 1

−1 −1

),

(1 0

−1 1

),

(−1 1

−1 0

)3 (123)(

−1 −1

0 −1

),

(0 −1

1 1

),

(−1 0

1 −1

),

(1 −1

1 0

)6 (123)(

1 0

1 1

),

(−1 −1

1 0

),

(0 −1

1 −1

),

(1 −1

0 1

)3 (132)(

−1 0

−1 −1

),

(1 1

−1 0

),

(0 1

−1 1

),

(−1 1

0 1

)6 (132)(

0 1

−1 0

),

(0 −1

1 0

),

(1 −1

−1 −1

),

(1 1

1 −1

),

(−1 1

1 1

),

(−1 −1

−1 1

)4 (12)(34)

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

BT) :

(1 0

0 1

) (−1 0

0 −1

) (1 1

0 1

) (−1 −1

0 −1

) (1 0

1 1

) (−1 0

−1 −1

) (0 1

−1 0

)χ0 1 1 1 1 1 1 1

χ1 1 1 ω ω ω2 ω2 1

χ2 1 1 ω2 ω2 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

0 1

) (−1 0

0 −1

) (1 1

0 1

) (−1 −1

0 −1

) (1 0

1 1

) (−1 0

−1 −1

) (0 1

−1 0

)χ0 1 1 1 1 1 1 1

χ1 1 1 ω ω ω2 ω2 1

χ2 1 1 ω2 ω2 ω ω 1

χ3 2 −2 1 −1 1 −1 0

χ4 2 −2 ω −ω ω2 −ω2 1

χ5 2 −2 ω2 −ω2 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 E6 :

2 231 1

1

2

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 :

2 431 3 2 1

2

Figure 7: Type E7

25

4 562 4 3 2 1

3

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.

Then,∑

z∈Z χ(z) = 0.

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∑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 = dimV , and let χ be its

character. Define F : CΓ × CΓ → C by

(φ, ψ) 7→ 〈χφ, ψ〉

where as usual 〈φ, ψ〉 = 1|Γ|∑

g∈Γ φ(g)ψ(g).

Then, we have the following properties :

1. The map F is a hermitian form on CΓ.

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.

4. For φ ∈ CΓ, 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, χ′ are irreducible characters then 0 ≤ F (χ, χ′) < 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

z ∈ Z(Γ). We have S := F (χW , χW ) = 1|Γ|∑

g∈Γ |χW (g)|2χ(g). We write 1 = 1|Z(Γ)|

∑z∈Z(Γ) 1

and obtain

S =1

|Γ| · |Z(Γ)|(∑

z∈Z(Γ)

∑g∈Γ

((|χW (g)|2) · χ(g)))

=1

|Γ| · |Z(Γ)|(∑

z∈Z(Γ)

∑g∈Γ

((|χW (zg)|2) · χ(zg)))

=dim(W )

|Γ| · |Z(Γ)|∑g∈Γ

χ(g)|χW (g)|2 ∑z∈Z(Γ)

χ(z)

(the second line is because for fixed z ∈ Z(Γ), g 7→ zg is a bijection) so it is enough to

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.

4. We have F (φ, φ) = 1|Γ|∑

g∈Γ χ(g)|φ(g)|2. Now |F (φ, φ)| = 1|Γ| |∑

g∈Γ χ(g)|φ(g)|2 ≤ 1|Γ|∑

g∈Γ |χ(g)||φ(g)2| ≤n∑

g∈Γ |φ(g)|2. 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

G : CΓ × CΓ → C be the bilinear form defined by (φ, ψ) 7→ n〈φ, 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 where α 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

and Γ be the corresponding binary group. Let Z be the scheme theoretic-fiber of 0 ∈ C2/Γ under

the map X → C2/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

node in the McKay graph. Let H a generic linear hyperplane in C3 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,

ZfCi ≤ 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

2(αi,αj)(αi,αi)

2δij − F (χi, χj) −Ci · CjLongest 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

dimZG(x) = r, where r = dim h where h is a Cartan subalgebra. An element is subregular if

dimZG(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

write X = Cn = Spec(R). Usually, G acts on V = Cn 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 →WG.

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 →WG.

By Hilbert’s theorem ([35], Theorem 4.51), the ring of invariant functions RG is finitely gener-

ated and we define Cn/G := Spec(RG), called the categorical quotient. Once and for all, we choose

a minimal set of generators f1, . . . , fk of RG giving a map χ : Cn → Ck, 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,O′ two different orbits of the G-action on X. Let I, I ′ be the corresponding ideals.

Assume that O ∩O′ = ∅. By Hilbert’s Nullstellensatz I + I ′ = R. Now, I and I ′ are subrepresen-

tations of G since O and O′ are G-invariant. The map I ⊕ I ′ → R is a map of G-modules and G

is reductive so I ∩ RG ⊕ I ′ ∩ RG → RG is surjective by the previous theorem. Then there exists

f ∈ I ∩RG, f ′ ∈ I ′ ∩RG with f + f ′ = 1. In particular f|O = 0 and f|O′ = 1 so χ separates O and

O′.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

between the closed points of Spec(C[X]G) 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

is a Cartan subalgebra of g, then restriction induces a map C[g]G → C[h]W . Chevalley’s restriction

theorem shows that such map is an isomorphism. Also, the Chevalley-Shephard-Todd theorem

([8]) states that if G is a finite reflection group acting on a vector space V then Spec(C[V ]G) 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

obtain a map res : C[g]G → C[h]W .

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

representation. If fλ,m(x) = tr(σλ(x)m), then clearly fλ,m ∈ C[g]G so it is enough to show that

these functions span C[h]W . For this, notice that ch(L+(λ)) ∈ (ZP )W ⊂ (CP )W 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 expφ(ea) and

expφ(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.

If ch(L+(λ)) =∑aµ,λe

µ then we have res(fλ,m)(h) =∑aµ,λµ(h)m, 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

still have an embedding C[P ]→ C[[h]], eµ →∑

k µk/k!.

So this is enough to show that the ch(L+(λ)) form a basis of (CP )W , where λ ∈ P+, and that

C[P ]W = C[h]W (we will show it by looking at each homogeneous components).

31

To show that ch(L+(λ)) generates (CP )W , we first notice that a basis of (CP )W is given by the

Sµ :=∑

w∈W ewµ for µ ∈ P . On the other hand for any weight there is a unique dominant weight

in the W -orbit of µ : in particular the Sλ generates C[P ]W for λ dominant. Now we can write

ch(L+(λ)) = Sλ +∑

µ<λ aµ,λSµ by W -invariance and by induction this shows that the characters

of irreducible representations spans C[P ]W . Now if f ∈ (C[[h]]W )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

h ∈ h. We will verify the isomorphism g/G ∼= h/W , i.e the isomorphism C[g]G ∼= C[h]W . 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

sl(M) =∑

1≤i1<···<il≤r+1

λi1(M) . . . λil(M)

where λ1(M), . . . , λr+1(M) are the eigenvalues of M . For example, we have sr+1(M) = det(M)

and s1(M) = tr(M) ≡ 0 since M ∈ slr+1.

Now, we claim that the invariant functions in C[slr+1]SLr+1 are generated by s2, . . . , sr+2. We will

32

use the companion matrix of tr+1 + x1tr−1 + · · ·+ xr, given by

Ax =

0 . . . . . . 0 −xr1 0 . . . . . . 0 −xr−1

0 1 0 . . . . . . 0 −xr−2

. . . . . .

· · · . . . 0 −x1

· . . . 0 1 0

where x = (x1, . . . , xr) ∈ Cr. Notice that si(Ax) = (−1)ixi−1(Ax) for i = 2, . . . , r + 1. We now

show that the si generate the invariant ring : let p ∈ C[slr+1]SLr+1 . To p we can associate a polyno-

mial in r variables by f(x1, . . . , xr) = p(Ax), where Ax is the companion matrix of the polynomial

tr+1 + x1tr−1 + · · · + xr. Let P ∈ C[slr+1] defined by P (A) = f(−s2(A), . . . , (−1)rsr+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

C[slr+1]. But by hypothesis p is SLr+1-invariant, so in fact p and P agree on U = {gAxg−1 | x ∈Cr, 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.

Then, w1 = v1 + · · · + vr, wk := Awk−1 form a basis of Cr. Indeed, the determinant of the ma-

trix B with Bvi = wi is exactly the Vandermonde matrix of (λ1, . . . , λr), in particular non-zero.

Now, the matrix of A in the basis (w1, Aw1, . . . , Ar−1w1) is a companion matrix so the claim follows.

This shows that for g = slr+1 we have C[g]G ∼= C[h]W . In fact a bit more is true : the poly-

nomials s2, . . . , sr+2 are algebraically independant, i.e h/W ∼= Cr. Indeed, any polynomial is the

characteristic polynomial of its companion matrix, this shows that the set {(s2(A), . . . , sr+2(A)) :

A ∈ sln} = Cr. In particular, indeed any relation f(s2, . . . , sr+2) = 0 would implies that f is zero

on Cr 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

Let X ⊂ Cn be an algebraic variety with 0 ∈ X. We would like to look at a ”smaller and smaller”

neighborhood of X around 0. Formally, we look at all the couples (V, Y ) where V ⊂ Cn is open, Y

is a closed algebraic subvariety of V and 0 ∈ Y . We say that (V, Y ) and (V ′, Y ′) are equivalent if

Y ∩W = Y ′ ∩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 corre-

sponding 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 f∗x : 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

morphism χ : (X,x) → (U, u) between germs, with (χ−1(u), x) ∼= (X0, x0). U is called the base of

the deformation. An isomorphism of deformations is an isomorphism X ∼= X ′, where X,X ′ 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

morphism f : X → C, (x, y, t) 7→ t is a deformation of the nodal curve y2 = x2(x − 1). Indeed,

the corresponding map is C[t] ⊂ C[x, y, t]/(y2 − 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

an injection g : Y → Pm ×X such that f = p2 ◦ g where p2 : Pm ×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)

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

as follows : let T 1 = 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

proof of this fact later, see 5.21). Let b1, . . . , br ∈ C{x, y, z} be the lifts of a basis of T 1 with br = 1.

Then, the semiuniversal deformation is the morphism of germs

(x, y, z, u1, . . . , ur−1) 7→ (f(x, y, z) +

r−1∑i=1

uibi), u1, . . . , ur)

The proof of this fact is non-trivial, see [2], chapter 1, section 8.

For example, for g = sl2 the semiuniversal deformation is χ : C3 → C, (x, y, z) 7→ x2 + yz. More

generally, let us describe concretely the semiuniversal deformation of the singularity x2 + y2 +

zn+1 = 0. We have T 1 ∼= C{x, y, z}/(f, 2x, 2y, (n + 1)zn) ∼= C[z]/(zn). A basis of T 1 is given by

1, z, z2, . . . , zn−1. The semiuniversal deformation is

(x, y, z, u1, . . . , un−1) 7→ (x2 + y2 + zn+1 + u1z + · · ·+ un−1zn−1, 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 dimS = 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

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

span of the lowest weight spaces. Indeed, for simplicty assume that g ∼= V (n) =⊕−n−1≤i≤n−1Wi

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

argument applies in general, where g ∼=⊕r′

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

an action of the torus C∗ ρ : C∗×S → S given by ρ(t)(x+∑

i ciei) = t2x+∑r′

i=1 t−ni+1ciei, where

36

the ei are the lowest weight vectors (so ni is the weight). We get an action j(t) = t2ρ(t−1) which

stabilize S, indeed j(t) · (x+∑

i ciei) = x+∑

i tni+1ciei.

According to the Chevalley restriction theorem and the Chevalley-Shephard-Todd theorem, the

adjoint quotient χ : g → h/W can be realized as a morphism g → Cr, (χ1, . . . , χr) where χi are

homogeneous generators of degree dj of C[g]G.

Since each χi is G-invariant, it is H = exp(h) invariant so χi(t · s) = χi(t2ρ(t)s) = χi(t

2s) =

t2diχi(s). So if C∗ acts with weight 2di on h/W ∼= Cr, the map σ : S → h/W becomes C∗-equivariant, where σ = χ|S .

Definition 5.18. Let V ∼= Cm,W ∼= Cn 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).

For example, if we take V = W = C2 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, . . . , nr′ + 1; 2d1, . . . , 2dr) (ni are the weights of g seen as a sl2-module). Now we

look at the special case when x is a subregular nilpotent element, that is dimZG(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]

for a proof of this fact). For simplicity, we compute instead wi := ni+12 . 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

Ar 2 3 . . . . . . r + 1 1 r+12

r+12

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

This give us generators for C[g]G ∼= C[h]W 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 :

Lemma 5.21. Let (X, 0) a germ of a surface in C3, given by the equation f = 0. If X has an

isolated singularity then

dimCC{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 H i(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 = 〈f, fx, fy, fz〉, our hypothesis says that C ⊂ Z(J) i.e

J ⊂ 〈g1, g2〉, so dimCC{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 Wi, so ∂χr

∂Wi= 0 for all i. It follows that for all j < r we should have at least one

ij with∂χj

∂Wij6= 0 : comparing degree, since wi = di for i < r and that the di are strictly increasing

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

have χr = ax2 + by3 + cz2 + dx2z.

We claim that c 6= 0. Indeed, if c = 0 a little computation give C{x, y, z}/J ∼= C{x, z}/(x2, xz)

which has infinite dimension as a vector space as it contains the monomials zk for k ≥ 0. Therefore

by 5.21, c 6= 0 and a small computation gives χr = (a − d2

4c )z4 + by3 + c(x2 + d

2cz)2. 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

appear : z3y, y3, x2 for E7 and x2, y3 and z5 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.

Now, let V ∼=⊕m

j=1 Vj be the decomposition of V into C∗ eigenspaces of weight λj with λ1 < λ2 <

. . . , with dimVj = #{1 ≤ i ≤ n : ai = λj}, and similarly for U ∼=⊕m

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

in v2. But v1 = l−11 (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

(w1, . . . , wr+2; d1, . . . , dr). So if π : Cr+2 = X → U = Cr is the semiuniversal deformation, we

obtain a C∗-equivariant deformation φ : h/W → U such that the following diagram commutes :

SΦ //

χ

��

X

π

��h/W

φ // 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,

namely σ−1(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.

Example 5.26 (Type E6). The equation of type E6 is given by x2 + y3 + z4 = 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

z2, u4 = yz, u5 = yz2 (we order by weights). The weights on (x, y, z) are (6, 4, 3), and the total

degree is 12. The morphism is

χ : (x, y, z, u1, u2, u3, u4, u5) 7→ (x2 + y3 + z4 + u1z + u2y + u3z2 + u4yz + u5yz

2, 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

Yψ //

θ

X

χ

��U

such that ψ is proper and surjective, and induces for all u ∈ U a resolution of singularities

ψu : θ−1(u) → χ−1(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 dimZG(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.

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

singularities.

Proof. First, G×Bn is a vector bundle over the flag variety G/B =: B, with fiber n : in particular Nis smooth. We need to check that ψ0 is proper and birational. For this, we will use the isomorphism

N ∼= {(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 :

bψ //

θ

��

g

χ

��h

φ // 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+ n′) since Ad(B)h = h+ n′ for all h ∈ h and some nilpotent element n′, 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

need to show that ψh : θ−1(h)→ χ−1(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 G′ = Z(h) :

ψ0(h) : Z(h)×B(h) n(h)→ N (z(h))

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

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

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

S ∩ θ−1(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 )/TY (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

��Z

i // 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

D ⊂ F = f−1(y) be a subvariety in a fiber, then we have ND/F ⊕ Cdim(Y ) = ND/X , where Cm 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

of the proof is to reduce this computation to the computation of c1(E, T ∗P1) 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×Bn → NX/G×Bn → 0

44

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

X //

��

N

f��

Xi // 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

normal bundle is trivial, and finally c1(C, X) = c1(C,G×B 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

c1(C,G×B n) = c1(C,P ×B n)

The map C → G×B n → P/B is an isomorphism, in particular we can consider C as a section of

the bundle P ×B n, so c1(P ×B n) = c1(C,P ×B n)

Now we use the exact sequence 0→ nP → n→ n/nP → 0 which gives an exact sequence of bundles

(over P/B)

0→ P ×B nP → P ×B n→ P ×B 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

we just need to compute c1(P ×B n/nP ). This reduces immediately to c1(SL2(C)×B 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

size 1. We will take a basis v1, . . . , vn of Cn such that

x =

0 1 0 . . . . . . . . . . . . 0

0 0 1 0 . . . . . . . . . 0

0 0 0 1 0 . . . . . . 0

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

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 = 〈w1, . . . , wi〉. Of course, if ui ∈ Fi,then w′i = wi +

∑j≤i−1 λjuj determines the same flag, so we will not distinguish wk, w

′k if they are

equal in Cn/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 = 〈v1〉, which forces V2 = 〈v1, v2〉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.

This gives a copy of P1 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

[a : b] 7→ 〈v1〉 ⊂ 〈v1, v2〉 ⊂ · · · ⊂ 〈v1, v2, . . . avk + bvn〉 ⊂ 〈v1, . . . , avk + bvn, vk〉 ⊂ · · · ⊂ Cn

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 ∆′ is a non-simply laced

Dynkin diagram, then we can associate to ∆′ a simply laced Dynkin diagram ∆ and a group

46

G ⊂ Aut(∆) such that ∆′ = ∆/G. For do this, we need a key condition : two connected ver-

tices 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 ∆ ⊂ ∆. 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 ) :

χ : C4 → C2, (x, y, z, t) 7→ (z2 − x2 + 3xy2 + t(x2 + y2), 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

of Γ and the geometry of the space C2/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 intro-

duced 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 dimO− 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 alge-

bras, 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

deformation of C[V ]Γ to be a Z-filtered associative algebra A whose corresponding graded algebra

is isomorphic to C[V ]Γ.

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

〈xy − yx − λ∠ where λ ∈ Z[CΓ]. Call this algebra Hλ. Then eHλe is a noncommutative defor-

mation of C[x, y]Γ where e = 1|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 dif-

ferent 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

show that the total space of the resolution of the quadratic cone is the cotangent bundle of P1

where the resolution : f : T ∗P1 → X is just the contraction of the zero section. It implies

that E2 = −2 where E is the zero section of T ∗P1, 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 =

−e1 + e2, w = e1 + e2. We have σ∨ = σ (identitying R2 and (R2)∨ with scalar product) and the

generator of the dual lattice Z2 ∩ σ∨ are v, w and u = e2, with relations v + w = 2e2. This gives

the algebra C[x, y, z]/(xz = y2), 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

Y the corresponding toric variety : Y is obtained by gluing two copies of C2 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

the gluing map for T ∗P1 ∼= 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.

Now let us do the general case, and let σ ⊂ R2 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.

Proposition 8.1. The toric variety Xσ is the quotient of C2 by the following action of Z/mZ ∼=µm : ζ · (x, y) = (ζx, ζky).

In particular, taking k = m− 1 gives the usual action of Z/mZ, and the invariant polynomials

are a = xm, b = ym, c = xy with the relation ab = cm 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 σ′ ∈ Σ are smooth (i.e the two primitives

vectors of σ′ generates Z2). Every ray with primitive vector v correspond to an exceptional irre-

ducible 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 ∈ Zwith w1 + w2 = kv, and we have E2

v = −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.

LetG be an algebraic group. A triple (X,B, π) whereX,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

preserves the fiber, and for all b ∈ B there is a G-equivariant isomorphism φb : G → π−1(b).

Such isomorphism is by definition exactly a choice of an element φb(eG) ∈ π−1(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

works : if (g1H, g1f1) = (g2H, g2f2) for some (gi, fi) ∈ G ×H F , then there is some h ∈ H with

g1h = g2. Now (g1, f1) = (g1h, h−1f1) = (g2, g

−12 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 (E′, f) with f ∈ E′.

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 (E′, f) where E′ is conjugate to E and f ∈ E′.

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 descrip-

tion 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

lift of h, we can identify the closed orbit of χ−1(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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