Generic singularities of nilpotent orbit closures

According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular nilpotent orbit.

1. Introduction 1.1. Generic singularities of nilpotent orbit closures. Let G be a connected, simple algebraic group of adjoint type over the complex numbers C, with Lie algebra g. A nilpotent orbit O in g is the orbit of a nilpotent element under the adjoint action of G. Its closure O is a union of finitely many nilpotent orbits. The partial order on nilpotent orbits is defined to be the closure ordering.
We are interested in the singularities of O at points of maximal orbits of its singular locus. Such singularities are known as the generic singularities of O. Kraft and Procesi determined the generic singularities in the classical types, while Brieskorn and Slodowy determined the generic singularities of the whole nilpotent cone N for g of any type. The goal of this paper is to determine the generic singularities of O when g is of exceptional type.
In fact, the singular locus of O coincides with the boundary of O in O, as was shown by Namikawa using results of Kaledin [Nam], [Kal06]. This result also follows from the main theorem in this paper in the exceptional types and from Kraft and Procesi's work in the classical types [KP81], [KP82]. Therefore to study generic singularities of O, it suffices to consider each maximal orbit O ′ in the boundary of O in O. We call such an O ′ a minimal degeneration of O.
The local geometry of O at a point e ∈ O ′ is determined by the intersection of O with a transverse slice in g to O ′ at e. Such a transverse slice in g always exist and is provided by the affine space Se = e + g f , known as the Slodowy slice. Here, e and f are the nilpotent parts of an sl2-triple and g f is the centralizer of f in g. The local geometry of O at a point e is therefore encoded in SO,e = O ∩ Se, which we call a nilpotent Slodowy slice. If O ′ is a minimal degeneration of O, then SO,e has an isolated singularity at e. The generic singularities of O can therefore be determined by studying the various SO,e, as O ′ runs over all minimal degenerations and e ∈ O ′ . The isomorphism type of the variety SO,e is independent of the choice of e.
The main theorem of this paper is a classification of SO,e up to algebraic isomorphism for each minimal degeneration O ′ of O in the exceptional types. In a few cases, however, we are only able to determine the normalization of SO,e and in a few others, we have determined SO,e only up to local analytical isomorphism.
1.2. Symplectic varieties. Recall from [Bea00] that a symplectic variety is a normal variety W with a holomorphic symplectic form ω on its smooth locus such that for any resolution π : Z → W , the pull-back π * ω extends to a regular 2-form on Z. By a result of Namikawa [Nam01], a normal variety is symplectic if and only if its singularities are rational Gorenstein and its smooth part carries a holomorphic symplectic form.
The normalization of a nilpotent orbit O is a symplectic variety: it is well-known that O admits a holomorphic non-degenerate closed 2-form (see [CM93,Ch. 1.4]) and by work of Hinich [Hin91] and Panyushev [Pan91], the normalization of O has only rational Gorenstein singularities. Hence the normalization of O is a symplectic variety.
For a minimal degeneration O ′ of O and e ∈ O ′ , the nilpotent Slodowy slice SO,e has an isolated singularity at e. Since the normalization of O has rational Gorenstein singularities, the normalization SO,e of SO,e also has rational Gorenstein singularities. Its smooth locus admits a symplectic form (see [GG02]), which is this restriction of the symplectic form on O. Thus by the aforementioned result of Namikawa, SO,e is also a symplectic variety.
The term symplectic singularity refers to a singularity of a symplectic variety. A better understanding of isolated symplectic singularities could shed light on the long-standing conjecture (e.g. [LeB95]) that a Fano contact manifold is homogeneous. The importance of finding new examples of isolated symplectic singularities was stressed in [Bea00]. It is therefore of interest to determine such singularities, as a means to find new examples of isolated symplectic singularities. Our study of the isolated symplectic singularity SO,e at e contributes to this program.
1.3. Simple surface singularities and their symmetries.
1.3.1. Definition. Let Γ be a finite subgroup of SL2(C) ∼ = Sp 2 (C). Then Γ acts on C 2 and the quotient variety C 2 /Γ is an affine symplectic variety with an isolated singularity at the image of 0. This variety is known as a simple surface singularity and also as a rational double point, a du Val singularity, or a Kleinian singularity.
The set of such Γ, up to conjugacy in SL2(C), are in bijection with the simply-laced, simple Lie algebras over C. The bijection is obtained via the exceptional fiber of a minimal resolution of C 2 /Γ. The exceptional fiber (that is, the inverse image of 0) is a union of projective lines which intersect transversely. The dual graph of the resolution is given by one vertex for each projective line in the exceptional fiber and an edge joining two vertices when the corresponding projective lines intersect. The dual graph is always a connected, simply-laced Dynkin diagram, which defines the Lie algebra attached to C 2 /Γ. Hence one refers to a simple surface singularity with one of the capital letters Ak, Dk(k ≥ 4), E6, E7, E8, according to the associated simple Lie algebra.
In dimension two, an isolated symplectic singularity is equivalent to a simple surface singularity, that is, it is locally analytically isomorphic to some C 2 /Γ (cf. [Bea00, Section 2.1]). More generally, if Γ ⊂ Sp 2n (C) is a finite subgroup whose non-trivial elements have no non-zero fixed points on C 2n , then the quotient C 2n /Γ is an isolated symplectic singularity.
1.3.2. Symmetries of simple surface singularities. Any automorphism of the simple surface singularity X = C 2 /Γ fixes 0 ∈ X and induces a permutation of the projective lines in the exceptional fiber of a minimal resolution. Hence it gives rise to a graph automorphism of the dual graph ∆ of X. Let Aut(∆) be the group of graph automorphisms of ∆. Then Aut(∆) = 1 when g is A1, E7, or E8; Aut(∆) = S3 when g is D4; and otherwise, Aut(∆) = S2.
We now address the question of when the action of Aut(∆) on the dual graph comes from an algebraic action on X (cf. [Slo80,III.6]). When X is of type A2k−1(k ≥ 2), Dk+1(k ≥ 3), or E6, then Aut(∆) comes from an algebraic action on X. In fact, the action is induced from a subgroup Γ ′ ⊂ SL2(C) containing Γ ′ as a normal subgroup. Namely, there exists such a Γ ′ with Γ ′ /Γ ∼ = Aut(∆) and the induced action of Γ ′ /Γ on the dual graph of X coincides with the action of Aut(∆) on ∆ via this isomorphism. Such a Γ ′ is unique. The result also holds for any subgroup of Aut(∆), which is relevant only for the D4 case.
Slodowy denotes the pair (X, K) consisting of X together with the induced action of K = Γ ′ /Γ on X by Bk, when X = A2k−1 and K = S2, Ck, when X = Dk+1 and K = S2, F4, when X = E6 and K = S2, G2, when X = D4 and K = S3. The reasons for this notation will become clear shortly. We also refer to corresponding pairs (∆, K), where ∆ is the the dual graph and K is a subgroup of Aut(∆), in the same way. The symmetry of the cyclic group of order 3 when X = D4 is not considered.
When X = A2k, the symmetry of X did not arise in Slodowy's work. It does, however, make an appearance in this paper. In this case Aut(∆) = S2, but the action on the dual graph does not lift to an action on X. Instead, there is a cyclic group σ of order 4 acting on X, with σ acting by non-trivial involution on ∆, but σ 2 acts non-trivially on X. This cyclic action is induced from a Γ ′ ⊂ SL2(C) corresponding to D2k+3. We define the symmetry of X to be the induced action of Γ ′ on X and denote it by A + 2k . Only the singularities A + 2 and A + 4 will appear in the sequel, and then only when g is of type E7 or E8.
1.4.1. Generic singularities of the nilpotent cone. The problem of describing the generic singularities of the nilpotent cone N of g was carried out by Brieskorn [Bri71] and Slodowy [Slo80] in confirming a conjecture of Grothendieck. In their setting O is the regular nilpotent orbit and so O equals N , and there is only one minimal degeneration, at the subregular nilpotent orbit O ′ . Slodowy's result from [Slo80,IV.8.3] is that when e ∈ O ′ , SO,e is algebraically isomorphic to a simple surface singularity of the form C 2 /Γ. Moreover, as in [Bri71], when the Dynkin diagram of g is simply-laced, the Lie algebra associated to this simple surface singularity is g. On the other hand, when g is not simply-laced, the singularity SO,e is determined from the list in §1.3.2. For example, if g is of type Bk, then SO,e is a type A2k−1 singularity. This explains the notation in the list in §1.3.2. Next we explain an intrinsic realization of the symmetry of SO,e when g is not simply-laced.
1.4.2. Intrinsic symmetry action on the slice. Let ∆ be the Dynkin diagram of g and K ⊂ Aut(∆) be a subgroup. The group Aut(∆) is trivial if and only if g is simply-laced. The action of K on ∆ can be lifted to an action on g as in [OV90,Chapter 4.3]: namely, fix a canonical system of generators of g. Then there is a subgroupK ⊂ Aut(gs), isomorphic to K, which permutes the canonical system of generators, and whose induced action on ∆ coincides with K. Any two choices of systems of generators define conjugate subgroups of Aut(g). The automorphisms inK are called diagram automorphisms of g. and symmetry result can be summarized as follows: the pair (SO,e, H), of SO,e together with the action of H, corresponds to the pair (( L ∆)s, L K) [Slo80,IV.8.4].
1.5. The other nilpotent orbits in Lie algebras of classical type. Kraft and Procesi described the generic singularities of nilpotent orbit closures for all the classical groups, up to smooth equivalence (see §2.1 for the definition of smooth equivalence) [KP81], [KP82].
1.5.2. Generic singularities in the classical types. The results of Kraft and Procesi for Lie algebras of classical type can be summarized as follows: an irreducible component of a generic singularity is either a simple surface singularity or a minimal singularity, up to smooth equivalence. Moreover, when a generic singularity is not irreducible, then it is smoothly equivalent to a union of two simple surface singularities of type A2k−1 meeting transversely in the singular point. This is denoted 2A2k−1. In more detail: Theorem 1.1. [KP81], [KP82] Assume O ′ is a minimal degeneration of O in a simple complex Lie algebra of classical type. Let e ∈ O ′ . Then (a) If the codimension of O ′ in O is two, then SO,e is smoothly equivalent to a simple surface singularity of type Ak, Dk, or 2A2k−1. The latter two singularities do not occur for sln(C), and the singularity Ak for k even does not occur in the classical Lie algebras besides sln(C). (b) If the codimension is greater than two, then SO,e is smoothly equivalent to ak, bk, ck, or dk. The latter three singularities do not occur for sln(C).
1.6. The case of type G2. The case of a Lie algebra of type G2 has been studied by Levasseur-Smith [LS88] and Kraft [Kra89]. Levasseur-Smith showed that the closure of the nilpotent orbit A1 of dimension 8 is not normal and that its non-normal locus coincides with its singular locus (and hence equals the closure of the minimal nilpotent orbit). Kraft gave another proof that this orbit is not normal and showed it has bijective normalization. Kraft also showed that the closure of the subregular orbit has singularity of type A1 in the A1 orbit.
1.7. Main results. We now summarize the main results of the paper describing the classification of generic singularities in the exceptional Lie algebras. Here, O ′ is a minimal degeneration of O and e ∈ O ′ and SO,e is the nilpotent Slodowy slice defined from an sl2-triple through e.
1.7.1. Overview. Most generic singularities are like those in the classical types: the irreducible components will be either simple surface singularities or minimal singularities. But some new features occur in the exceptional groups. There is more complicated branching and several new types of singularities occur. Among these are three singularities whose irreducible components are not normal (one of these already occurs in G2), and three additional singularities of dimension four.
A key observation ( §2.4) is that all irreducible components of SO,e are isomorphic since the action of C(s) is transitive on irreducible components. This result is not true in general when O ′ is not a minimal degeneration.
For most minimal degenerations, we determine the isomorphism type of SO,e, a stronger result than classifying the singularity up to smooth equivalence. In ten of the remaining cases, all in E8 ( §10.3), we can only determine the isomorphism type of SO,e up to normalization and in four cases we only know the result up to smooth equivalence ( §12). It is possible to use the results here to go back and establish that Kraft and Procesi's results in Theorem 1.1 hold up to algebraic isomorphism (rather than smooth equivalence), but we defer the details to a later paper.
We also caluclate the symmetry action on SO,e induced from A(e), as Slodowy did when O is regular orbit. This involves extending Slodowy's result on the splitting of C(s) and introducing the notion of symmetry on a minimal singularity. Again, it is possible to carry out this program for the classical groups, but we also defer the details to a later paper. 1.7.2. Symmetry of a minimal singularity. Let g be a simple, simply-laced Lie algebra with Dynkin diagram ∆. As in §1.4.2, letK ⊂ Aut(g) be a subgroup of diagram automorphisms lifting the subgroup K ⊂ Aut(∆). We call a pair (Omin,K), consisting of Omin with the action ofK, a symmetry of a minimal singularity. We write these pairs as a + k , d + (for the action of the full automorphism group), and e + 6 . As in the surface cases, |K| = 3 in D4 does not arise. 1.7.3. Intrinsic symmetry action on a slice: general case. In §6.1 it is shown that the splitting of C(s) that Slodowy observed for the subregular orbit holds in general, with four exceptions. Namely, The choice of splitting is in general no longer unique up to conjugacy in C(s), but if we choose H to represent diagram automorphisms of the semisimple part of c(s), then the image of H in Aut(c(s)) is unique up to conjugacy in Aut(c(s)). The four exceptions to the splitting of C(s) have |A(e)| = 2, but the best possible result is that there exists H ⊂ C(s), cyclic of order 4, with C Next, imitating §1.4.2, we describe the action of H on SO,e. The four cases where C(s) does not split give rise to the symmetries which include A + 2 and A + 4 ( §1.3.2). Three of these four cases (when O ′ has type A4 +A1 in E7 and E8 or type E6(a1)+A1 in E8) are well-known: under the Springer correspondence, their Weyl group representations lead to unexpected phenomena (see, for example, [Car93,pg. 373]). The phenomena observed here for these three orbits is directly related to the fact that A(e) = S2 acts without fixed points on the irreducible components of the Springer fiber over e. It is not clear why the fourth orbit (of type D7(a2) in E8) appears in the same company as these three orbits. 1.7.4. Unexpected singularities. In the Lie algebras of exceptional type, we have identified six varieties that arise as some SO,e, which are neither simple surface singularities nor minimal singularities. There may be additional cases depending on whether certain SO,e are normal or not.
The first three cases are all non-normal.
The variety m. Let V (i) denote the irreducible representation of highest weight i of SL2(C). Consider the linear representation of SL2(C) on V = V (2) ⊕ V (3). Let v ∈ V be a highest weight vector for a Borel subgroup of SL2(C) with non-zero projection on each summand. Then the variety m is defined to be the closure in V of the orbit through v. This is a two-dimensional variety with an isolated singularity at zero. It is not normal, but has smooth normalization, equal to the affine plane A 2 . The normalization map is given by where the right side is written with respect to an appropriate basis of weight vectors. Alternatively, m = Spec R where R is the subring of C[s, t] generated by all homogeneous polynomials of degree 2 or higher.
The first case where m appears is for the minimal degeneration (O, O ′ ) = (Ã1, A1) in G2. This singularity appears at least once in each exceptional Lie algebra, always for two non-special orbits which lie in the same special piece.
The variety m ′ . This is a four-dimensional analogue of m, with SL2(C) replaced by Sp 4 (C). Consider the linear representation of Sp 4 (C) on V = V (2ω1) ⊕ V (3ω1) where V (ω1) is the defining 4-dimensional representation of Sp 4 (C), so V (2ω1) is the adjoint representation and V (3ω2) is equal to the symmetric cube of V (ω1), a representation of dimension 20. Let v ∈ V be a highest weight vector for a Borel subgroup of Sp 4 (C) with non-zero projection on each summand. Then the variety m ′ is defined to be the closure in V of the Sp 4 (C)-orbit through v. This is a four-dimensional variety with an isolated singularity at zero. It is not normal, but has smooth normalization, equal to A 4 . Alternatively, m ′ = Spec R, where R is the subring of C[s, t, u, v] generated by all homogeneous polynomials of degree 2 or higher. We will show that m ′ occurs exactly once as SO,e, for the minimal degeneration ( The variety µ. The coordinate ring of the simple surface singularity A3 is R = C[st, s 4 , t 4 ], as a hypersurface in C 3 . We define the variety µ by µ = Spec R ′ where R ′ = C[(st) 2 , (st) 3 , s 4 , t 4 , s 5 t, st 5 ]. This variety is non-normal and its normalization is isomorphic to A3 via the inclusion of R ′ in R. Using the methods of §5, the normalization of SO,e for (O, O ′ ) = (D7(a1), E8(b6)) in E8 is shown to be isomorphic to A3 with an order two symmetry arising from A(e). In [FJLSb], we will show that SO,e is smoothly equivalent to µ. The closure of O was known to be non-normal, but our result establishes that it is non-normal in codimension 2.
There are also three additional unexpected singularities, each of dimension 4 and normal.
The degeneration (A4 + A1, A3 + A2 + A1) in E7. Let S2 be the cyclic group of order two acting on sl3(C) via an outer involution. All such involutions are conjugate in Aut(sl3(C)). The quotient a2/S2 has an isolated singularity at 0 since there are no minimal nilpotent elements in sl3(C) which are fixed by an outer involution. We will prove in [FJLSb] that SO,e is smoothly equivalent to a2/S2 for the minimal degeneration (A4 + A1, A3 + A2 + A1) in E7.
The degeneration (A4 + A3, A4 + A2 + A1) in E8. Let ∆ be a dihedral group of order 10, acting on C 4 via the sum of the reflection representation and its dual. Then it turns out that the blow-up of C 4 /∆ at its singular locus has an isolated singularity at a point lying over 0. We denote this blow-up by χ. We show in §12.3 that SO,e is smoothly equivalent to χ for the minimal degeneration (A4 + A3, A4 + A2 + A1) in E8. 1.7.5. Statement of main theorem. The main result is the determination of the generic singularities of nilpotent orbit closures in a simple Lie algebra of exceptional type, up to normalization for ten cases in E8. The graphs at the end of the paper list the precise results.
Theorem 1.2. Let O ′ be a minimal degeneration of O in a simple Lie algebra of exceptional type. Let e ∈ O ′ . Taking into consideration the intrinsic symmetry of A(e), we have (a) If the codimension of O ′ in O is two, then, with one exception, SO,e is isomorphic either to a simple surface singularity of type A − G or to one of the following A + 2 , A + 4 , 2A1, 3A1, 3C2, 3C3, 3C5, 4G2, 5G2, 10G2, or m, up to normalization for ten cases in E7 or E8. Here, kXn denotes k copies of Xn meeting pairwise transversally at the common singular point. In the one remaining case, the singularity is smoothly equivalent to µ.
(b) If the codimension is greater than two, then, with three exceptions, SO,e is isomorphic either to a minimal singularity of type a − g or to one of the following types: a + 2 , a + 3 , a + 4 , a + 5 , 2a2, d ++ 4 , e + 6 , 2g2, or m ′ , where the branched cases 2a2 and 2g2 denote two minimal singularities meeting transversally at the common singular point. The singularities for the three remaining cases are smoothly equivalent to τ, a2/S2, and χ, respectively.
In the statement of the theorem, we have recorded the induced symmetry of A(e) relative to the stabilizer in A(e) of an irreducible component of SO,e. See §6.2 for a complete statement.
1.8. Brief description of methods. The first method is an adaptation of arguments used in [KP81], where the closure of the minimal nilpotent orbit in the Lie algebra of C(s) is related to SO,e. This method is successful in dealing with most higher codimension cases and when the singularity is kA1 or m. In short, then, the Cartan-Killing type of C(s) turns out to be sufficient to classify SO,e in these cases. The second approach, on the other hand, is more apt for the surface cases. The idea is to use the fact that the normalization of a transverse slice is a simple surface singularity and then obtain a minimal resolution of the singularity by restricting the Q-factorial terminalizations of the nilpotent orbit closure to the transverse slice. Then we can apply a formula of Borho-MacPherson to compute the number of projective lines in the exceptional fiber and the action of A(e) on the projective lines.
These two methods are sufficient to describe SO,e for all minimal degenerations except when µ, χ, τ , or a2/S2 occur, perhaps up to normalization in a few of the surface singularity cases. The two cases where χ or τ occur are dealt with separately in §12. The two cases where µ or a2/S2 occur are deferred to subsequent papers.
1.9.1. Isolated symplectic singularities coming from nilpotent orbits. Examples of isolated symplectic singularities include Omin and quotient singularities C 2n /Γ, where Γ ⊂ Sp 2n (C) is a finite subgroup acting freely on C 2n \ {0}. It was established in [Bea00] that an isolated symplectic singularity with smooth projective tangent cone is locally analytically isomorphic to some Omin. It turns out that all of the isolated symplectic singularities we obtain, with one exception, are finite quotients of Omin or C 2n . It seems very likely that the singularity χ described above is not equivalent to a singularity of this form.
Another byproduct is the discovery of examples of symplectic contractions to an affine variety whose generic positive-dimensional fiber is of type A2 and with a non-trivial monodromy action. These examples correspond to minimal degenerations with singularity A + 2 . The orbits O in these cases have closures which admit a Springer resolution. Examples include the even orbits A4 + A2 in E7 and E8(b6) in E8. In [Wie03], a symplectic contraction to a projective variety of the same type is constructed. As far as we know, our examples are the first affine examples that have been constructed. This disproves Conjecture 4.2 in [AW].
1.9.2. Representation theory. The topology and geometry of the nilpotent cone N have played an important role in representation theory centered around Springer's construction of Weyl group representations and the resulting Springer correspondence (e.g., [Spr76], [Lus81], [BM81], [Jos84], [Lus90]). The second author of the present paper defined a modular version of Springer's correspondence [Jut07] to the effect that the modular representation theory of the Weyl group of g is encoded in the geometry of N . In particular, its decomposition matrix is a part of the decomposition matrix for equivariant perverse sheaves on N . The connection with decomposition numbers makes it desirable to be able to compute the stalks of intersection cohomology complexes with modular coefficients. In this setting the Lusztig-Shoji algorithm to compute Green functions is not available and one has to use other methods, such as Deligne's construction which is general, but hard to use in practice. To actually compute modular stalks it is necessary to have a good understanding of the geometry. The case of a minimal degeneration is the most tractable.
The decomposition matrices of the exceptional Weyl groups are known, so here we are not trying to use the geometry to obtain new information in modular representation theory. However, it is interesting to see how the reappearance of certain singularities in different nilpotent cones leads to equalities (or more complicated relationships) between parts of decomposition matrices. In the GLn case, the row and column removal rule for nilpotent singularities of [KP81] gives a geometric explanation for a similar rule for decomposition matrices of symmetric groups ([Jam81], [Jut07]).
It would also be interesting to investigate whether the equivalences of singularities that we obtain in exceptional nilpotent cones have some significance for studying primitive ideals in finite W -algebras (see the survey article [Los10]). 1.9.3. Special pieces. For a special nilpotent orbit O, the special piece P(O) containing O is the union of all nilpotent orbits O ′ ⊂ O which are not contained in O1 for any special nilpotent orbit O1 O. This is a locally-closed subvariety of O and it is rationally smooth (see [Lus97] and the references therein). To explain rational smoothness geometrically, Lusztig conjectured in [Lus97] that every special piece is a finite quotient of a smooth variety. This conjecture is known for classical types by [KP89], but for exceptional types it is still open.
Each special piece contains a unique minimal orbit under the closure ordering. Motivated by the aforementioned conjecture of Lusztig, we studied the transverse slice of P(O) to this minimal orbit. We shall prove in [FJLSb] the following: Theorem 1.3. Consider a special piece P(O) in any simple Lie algebra. Then a nilpotent Slodowy slice in P(O) to the minimal orbit in P(O) is isomorphic to where hn is the n-dimensional reflection representation of the symmetric group Sn+1 and k and n are uniquely determined integers.
This theorem also includes the Lie algebras of classical types where n = 1, but k can be arbitrarily large. In the exceptional types Theorem 1.2 covers the cases where P(O) consists of two orbits, in which case n = k = 1 (that is, the slice is isomorphic to the A1 simple surface singularity). This leaves only those special pieces containing more than two orbits. Some of these remaining cases can be handled quickly with the same techniques, but others require more difficult calculations.
1.9.4. Normality of nilpotent orbit closures. By work of Kraft and Procesi [KP82], together with the remaining cases covered in [Som05], in classical Lie algebras the failure of O to be normal is explained by branching for a minimal degeneration, and then only with two branches. In exceptional Lie algebras, the question of which nilpotent orbit closures are normal has not been completely solved in E7 or E8, but in [Bro98a, Section 7.8], a list of non-normal nilpotent orbit closures is given, which is expected to be the complete list.
Our analysis sheds some new light on the normality question. The occurrence of m, m ′ , and µ at a minimal degeneration of O gives a new geometric explanation for why O is not normal. Previously the only geometric explanation for the failure of normality was branching (see [BS84]) and the appearance of the non-normal singularity in the closure of theÃ1 orbit in G2, which was known to be unibranch (see [Kra89]).
We also establish: (1) for many O known to be non-normal that O is normal at points in some minimal degeneration; and (2) for many O that are expected to be normal that O is indeed normal at points in all of its minimal degenerations. So we are able to make a contribution to determining the non-normal locus of O. Examples of the above phenomena are given starting in §7.3. Along these same lines, we also note that a consequence of Theorem 1.3 is that the special pieces are normal, a question studied by Achar and Sage in [AS09].
1.9.5. Duality. An intriguing result from [KP81] for g = sln(C) is the following: a simple surface singularity of type Ak is always interchanged with a minimal singularity of type ak under the orderreversing involution on the set of nilpotent orbits in g.
This result leads to a generalization in the other Lie algebras, both classical and exceptional, by restricting to the set of special nilpotent orbits, which are reversed under the Lusztig-Spaltenstein involution. For a minimal degeneration of one special orbit to another, in most cases a simple surface singularity is interchanged with a singularity corresponding to the closure of the minimal special nilpotent orbit of dual type. There are a number of complicating factors outside of sln(C), related to Lusztig's canonical quotient and the existence of multiple branches. The duality can also be formulated as one from special orbits in g to those in L g, the more natural setting for Lusztig-Spaltenstein duality.
Numerical evidence for such a duality was discovered by Lusztig in the classical groups using the tables in [KP82]. The duality is already hinted at by Slodowy's result for the regular nilpotent orbit in §1.4.2, which requires passing from g to L g. In a subsequent article [FJLSa] we will give a more complete account of the phenomenon of duality for degenerations between special orbits.
1.10. Notation. G will be a connected, simple algebraic group of adjoint type over the complex numbers C with Lie algebra g, and O and O ′ will be nilpotent Ad G-orbits in g. We use the notation in [Car93,] to refer to nilpotent orbits. For x ∈ g, Ox refers to the orbit Ad G(x), also written G · x. For x ∈ G or g we denote by G x (resp. g x ) the centralizer of x in G (resp. g). Similar notation applies to other algebraic groups which arise, often as subgroups of G.
Generally, e is a nilpotent element in an sl2(C)-subalgebra s with standard basis e, h, f . The centralizer of s in g is c(s) and its centralizer in G is C(s). If e0 ∈ c(s) is a nilpotent element, we use s0 for an sl2(C)-subalgebra in c(s) with standard basis e0, h0, f0.
for such a pair of nilpotent orbits. Often, but not always, The field of fractions of an integral domain Y will be denoted Frac(A). The symmetric group on n letters is denoted Sn. Where we refer to explicit elements of g, we use GAP's structure constants. To study singularities it is useful to introduce the notion of smooth equivalence. Given two varieties X and Y and two points x ∈ X and y ∈ Y , the singularity of X at x is smoothly equivalent to the singularity of Y at y if there exists a variety Z, a point z ∈ Z and morphisms ϕ : Z → X and ψ : Z → Y which are smooth at z and such that ϕ(z) = x and ψ(z) = y (see [Hes76,1.7]). This defines an equivalence relation on pointed varieties (X, x) and the equivalence class of (X, x) will be denoted Sing(X, x). As in [KP81, §2.1], two singularities (X, x) and (Y, y) with dim Y = dim X + r are equivalent if and only if (X × A r , (x, 0)) is locally analytically isomorphic to (Y, y).
Let O ′ and O be nilpotent orbits in g with O ′ ⊂ O. Let e ∈ O ′ . As discussed in the Introduction, the goal of this paper is to study the local geometry of O at e in the exceptional Lie algebras. This leads us to study the equivalence class of (O, e) under smooth equivalence. The equivalence class of the singularity (O, e) will be denoted Sing(O, O ′ ) since the equivalence class is independent of the choice of element in O ′ = Oe. We will primarily be interested in the case when O ′ is a minimal degeneration of O ( §1.1), but this is not assumed unless stated otherwise. Let X be a variety on which G acts, and let x ∈ X. A transverse slice in X to G · x at x is a locally closed subvariety S of X with x ∈ S such that the morphism G × S → X, (g, s) → g · s is smooth at (1, x) and such that the dimension of S is minimal subject to these requirements. It is immediate that Sing(X, x) = Sing(S, x). If X is a vector space then it is easy to construct such a transverse slice as x + u where u is a vector space complement to Tx(G · x) = [g, x] in X. More generally, this also suffices to construct a transverse slice to a G-stable subvariety Y ⊂ X containing x by taking the intersection (x + u) ∩ Y . In such a case codimY (G · x) = dimx((x + u) ∩ Y ).
These observations are especially helpful for nilpotent orbits in the adjoint representation, where there is a natural choice of transverse slice. As before, pick e ∈ O ′ . Then there exists h, f ∈ g so that {e, h, f } ⊂ g is an sl2-triple. Then by the representation theory of sl2(C), we have [e, g] ⊕ g f = g. The affine space is a transverse slice of g at e, called the Slodowy slice. The variety SO,e := Se ∩ O is then a transverse slice of O to O ′ at the point e, which we call a nilpotent Slodowy slice. In this setting Since any two sl2-triples for e are conjugate by an element of G e , the isomorphism type of SO,e is independent of the choice of sl2-triple. Moreover, the isomorphism type of SO,e is independent of the choice of e ∈ O ′ . By focusing on SO,e we reduce the study of Sing(O, O ′ ) to the study of the singularity of SO,e at e. In fact, most of our results will be concerned with determining the isomorphism type of SO,e.

2.3.
Group actions on SO,e. An important feature of the transverse slice SO,e is that it carries the action of two commuting algebraic groups, which both fix e. Let s be the subalgebra spanned by {e, h, f } and C(s) the centralizer of s in G. Then C(s) is a maximal reductive subgroup of G e and C(s) acts on SO,e, fixing e.
The second group which acts is C * . Since [h, f ] = −2f , ad h preserves the subspace g f and by sl2-theory all of its eigenvalues are nonpositive integers. Set The special case g f (0) is simply c(s), the centralizer of s in g, which coincides with Lie(C(s)).
Define φ : SL2(C) → G such that the image of dφ is equal to s, with dφ 0 1 0 0 = e and dφ 1 0 for t ∈ C * . On the one hand, Ad γ(t) preserves O and so does the scalar action of C * on g since O is conical in g. On the other hand, for xi ∈ g f (−i) and t ∈ C * , Ad γ(t)(e + x0 + . . . + xm) = t −2 e + x0 + tx1 + . . . + t m xm.
Composing this action with the scalar action of t 2 on g, gives an action of t ∈ C * on e + g f by (2.2) t · (e + x0 + x1 + . . .) = e + t 2 x0 + t 3 x1 + . . . , which preserves SO,e = O ∩ Se. The C * -action fixes e and commutes with the action of C(s), since C(s) commutes with ad h and so preserves each g f (i). Thus C(s) × C * acts on SO,e.

2.4.
Branching and component group action. The C(s) × C * -action on SO,e has consequences for the irreducible components of SO,e. An irreducible variety X is unibranch at x if the normalization π : (X,x) → (X, x) of (X, x) is locally a homeomorphism at x. Since the C * -action on SO,e in (2.2) is attracting to e, SO,e is connected and its irreducible components are unibranch (see [CG10], proof of Proposition 3.7.15). Consequently the number of irreducible components of SO,e is equal to the number of branches of O at e. The latter can be determined from the tables of Green functions in [BS84,Sho80], as discussed in [BS84,5 The identity component C(s) • of C(s), being connected, preserves each irreducible component of SO,e, hence there is a natural action of C(s)/C(s) • on the irreducible components of SO,e. The finite group C(s)/C(s) • is isomorphic to the component group A(e) := G e /(G e ) • of G e via C(s) ֒→ G e ։ G e /(G e ) • . Since any two sl2-triples containing e are conjugate by an element of (G e ) • , this gives a well-defined action of A(e) on the set of irreducible components of SO,e. Moreover, as noted in op. cit., the permutation representation of A(e) on the branches of O at e, and hence on the irreducible components of SO,e, can be computed. For a minimal degeneration, the situation is particularly nice. We observe by looking at the tables in [BS84,Sho80] that Proposition 2.1. When O ′ is a minimal degeneration of O in an exceptional Lie algebra, the action of A(e) on the set of irreducible components of SO,e is transitive. In particular, the irreducible components of SO,e are mutually isomorphic.
The proposition also holds in the classical types, which can be deduced using the results in [KP82]. In §6.2 we will discuss the full symmetry action on SO,e induced from A(e).
2.5. Passing to a reductive subalgebra. Often the isomorphism type of SO,e can be determined by knowing the isomorphism type of a nilpotent Slodowy slice in a reductive subalgebra of g. Proof. The first statement follows from (2.1) and the fact that SM·x,e ⊂ SO,e. The second statement follows from the fact that the irreducible components of SO,e and SM·x,e are unibranch ( §2.4).
There are two main situations where we apply Lemma 2.2: (1) when m is the centralizer of a semisimple element of G; or (2) when m = s ⊕ c(s). Consider the case when m = s ⊕ c(s). Then x can be written as x = x ′ + e0 with x ′ ∈ s and e0 ∈ c(s). Since x is nilpotent, both x ′ and e0 are nilpotent. Since M · e ⊂ M · x, x ′ belongs to the nonzero M -orbit through e, so without loss of generality x ′ = e. The Slodowy slice in m to e is then e + c(s). Therefore the nilpotent Slodowy slice SM·x,e in m is SM·x,e = e + C(s) · e0.
It follows that e + C 0 (s) · e0 is an irreducible component of SM·x,e and thus C(s) acts transitively on the irreducible components of SM·x,e. Hence SM·x,e is equidimensional. A reformulation of Lemma 2.2 in this setting is the following: In this case, the natural map from C(s) · e0 to SO,e is a C(s)-equivariant isomorphism.
2.6. A generalization of Corollary 2.3. Corollary 2.3 is a tool for showing that many SO,e are isomorphic to closures of nilpotent orbits in c(s). Sometimes we need a more general result.
As before (O, O ′ ) is a pair of nilpotent orbits with O ′ ⊂ O and e ∈ O ′ . Suppose that there exists x ∈ O of the form with e0 ∈ c(s) and x ′ ∈ ⊕i≤−1g f (i). In particular x ∈ SO,e. Assume further that e0 is nilpotent. The next lemma gives a condition for the closure of the C(s) · (e0 + x ′ ) in g to be isomorphic to SO,e. Let By construction Z is equidimensional with irreducible components permuted transitively by C(s)/C(s) • .
Certainly Z ⊂ g f and so e + Z ⊂ Se = e + g f . The case when x ′ = 0 is just Corollary 2.3. In all the cases encountered in this paper, it turns out that c(s) x ′ ⊂ c(s) e 0 so that dim C(s) · e0 = dim Z and the condition in the Lemma that must be checked Lemma 2.4 is a tool for determining SO,e for some minimal degenerations not handled by the x ′ = 0 case of Corollary 2.3 (see §3.3 and §3.4). Both Corollary 2.3 and the more general Lemma 2.4 also apply in many cases when e0 is not in the minimal nilpotent orbit of c(s).

Statement of the main results
In this section we state the main propositions which underlie Theorem 1.2. The propositions give more precise information about SO,e. We defer discussion of the intrinsic symmetry action until §6.1. Proposition 3.1. Let O ′ be a minimal degeneration of O of codimension at least 4. Assume that c(s) has a simple summand different from sl2(C) and that O ′ is neither of type D4(a1) + A2 nor 2A2 + 2A1 in E8. Then there exists a minimal nilpotent element e0 ∈ c(s) such that e + e0 ∈ O and SO,e ∼ = C(s) · e0 under the natural C(s)-equivariant inclusion.
Proof. The proof is case-by-case until we exhaust all such minimal degenerations. We first locate a minimal nilpotent element e0 ∈ c(s) such that e + e0 ∈ O and then check that dim SO,e = dim C(s) · e0. This is carried out using the ideas in §4. The checking is greatly simplified by Corollary 4.2 in most cases. Since O ′ is a minimal degeneration of O, A(e) = C(s)/C 0 (s) acts transitively on the irreducible components of SO,e by Proposition 2.1. At the same time, C(s)/C 0 (s) acts transitively on the irreducible components of C(s) · e0. Hence the full statement of Corollary 2.3 applies.
In the cases cover by the previous proposition, SO,e is a minimal singularity or a union of minimal singularities meeting transversely at e.
3.2. Surface cases. The case of a minimal degeneration of codimension two is summarized by the following proposition.
Proposition 3.2. Let O ′ be a minimal degeneration of O of codimension 2. Then there exists a finite subgroup Γ ⊂ SL2(C) such that the normalization SO,e of SO,e is isomorphic to a disjoint union of k copies of X where X = C 2 /Γ. This is proved in §5 where techniques for determining Γ and k are given. For most cases in Proposition 3.2 we can show that the irreducible components of SO,e are normal either by using that O is normal, by using Lemma 2.2 to move to a smaller subalgebra where the slice is known to be normal, or by doing an explicit computation (often with the help of Lemma 2.4). In the surface case, we found only two ways that an irreducible component of SO,e fails to be normal: • When Γ = 1, we show below that SO,e ∼ = m ( §1.7.4). This happens for several different minimal degenerations.
smoothly equivalent to µ ( §1.7.4). A handful of cases are left unresolved up to normalization.
There is a more precise statement when |Γ| = 1 or 2. In these cases there always exists x ∈ O satisfying Lemma 2.4. As a result, SO,e is either isomorphic to m in the former case or to kA1 in the latter. Let π0,1 : Then c(s) contains a simple factor z ∼ = sl2(C) and there exists Proof. This is again case-by-case. The hard part is showing the existence of x = e + e0 + x1 + x + ∈ O. We give some general techniques starting in §4.4, but sometimes have to resort to explicit computer calculations. Once such an x is found, necessarily dim C(s) · e0 = 2, so that dim SO,e = dim Z, from which the full statement of Lemma 2.4 holds. The rest of the proposition follows from §4.5. (1) (A3 + 2A1, 2A2 + 2A1) in E8. Here, SO,e ∼ = m ′ ( §1.7.4).
(2) (A4 + A1, D4(a1) + A2) and (2A3, D4(a1) + A2) in E8. For these both, SO,e ∼ = a2. ( share the property that c(s) contains a factor isomorphic to sl2(C), but C(s) does not act transitively on the smooth part of SO,e, in contrast to the other four-dimensional or higher cases covered by Proposition 3.1. They do share a common feature amongst each other: they are covers of slices in smaller groups. Cases (3) and (5) are handled in §12 and (4) is deferred to [FJLSb].
Proof. The proof is along the same lines as Proposition 3.3. See Table 10.
Remark 3.5. Propositions 3.1 and 3.2 also hold in the classical groups, as does Proposition 3.3 (the singularity m does not appear). In the classical groups all irreducible components of SO,e are normal by Kraft and Procesi's work. It follows that smooth equivalence in Kraft and Procesi's Theorem 1.1 can be replaced by algebraic isomorphism.
Remark 3.6. In Propositions 3.1, 3.3, and 3.4, C(s) has two orbits on SO,e, namely, e and its smooth complement.
Remark 3.7. In the settings of Propositions 3.1, 3.3, and 3.4, the minimal nilpotent orbits in the summands of c(s) closely control SO,e. It is not the case, however, that every summand of c(s) contributes to this story. For example, when e ∈ O ′ = 3A1 in E6, the centralizer c(s) has type A2 + A1. If e0 belongs to the minimal nilpotent orbit in the simple summand of type A2, then O ′ is not a minimal degeneration of Oe+e o . We are not able to explain these missing summands.
Remark 3.8. The isomorphism in Proposition 3.2 is compatible with the natural C * -actions on both sides. On SO,e the C * -action is the one induced from §2.3; on C 2 /Γ it is the one coming from the central torus in GL2(C). This follows from Proposition 5.2. Proposition 3.3 shows that the minimal degenerations of type A1 share the feature with the larger codimension minimal degenerations that the singularity is controlled by a summand(s) of c(s). We have observed that the minimal degenerations of type Ak (or the geometrically equivalent Bk) are also related to c(s): in all these cases C(s) contains a central torus. The singularity Ak described as C 2 /Γ (where Γ is cyclic of order k + 1 in SL2) is acted upon by the two-dimensional torus of GL2(C). This suggests that when SO,e is a surface singularity of type Ak, then the isomorphism in Proposition 3.2 extends to one compatible with the action of a central torus in C(s), in addition to the C * -action from §2.3. Along these lines, the absence of a central torus in C(s) for e = E8(b6), where SO,e is of type A3, led us to suspect that SO,e = SO,e and then to the µ singularity. 4. Tools for establishing Propositions 3.1, 3.3, and 3.4 In this section we give a way to verify Corollary 2.3 and Lemma 2.4, so that the remaining details in the proofs of Propositions 3.1, 3.3, and 3.4 can be checked. The first step is locating nilpotent elements in c(s) relative to the embedding of c(s) in g.

4.1.
Locating nilpotent elements in c(s). We want to be able to find nilpotent e0 ∈ c(s) and then compute e + e0 in order to verify cases where Corollary 2.3 hold.
First, if e0 ∈ c(s), then e0 centralizes the semisimple element h ∈ s. Hence e0 ∈ g h , which is a Levi subalgebra of g. Assume h lies in a chosen Cartan subalgebra h ⊂ g and is dominant for a chosen Borel subalgebra b ⊂ g containing h. The type of the Levi subalgebra g h can then be read off from the weighted Dynkin diagram for h: the Dynkin diagram for the semisimple part of g h corresponds to the zeros of the diagram. Therefore in order to locate a nilpotent element in c(s), we first choose a nilpotent element e0 ∈ g h ; the G h -orbits of such elements are known by Dynkin's and Bala and Carter's results [Car93]. In particular we can compute the semisimple element h0 ∈ g h ∩ h of an sl2-subalgebra s0 through e0 in g h .
Next, we compute h + h0 and see whether it corresponds to a nilpotent orbit in g: for if e and e0 commute (or some conjugate of e0 under G h ), then h + h0 will be the semisimple element in an sl2subalgebra through the nilpotent element e + e0. Together with knowledge of the Cartan-Killing type of the reductive Lie algebra c(s) ⊂ g h (see [Car93]), this search usually suffices to locate the nilpotent orbit through e0 in g for nilpotent elements e0 ∈ c(s) and the resulting nilpotent orbit through e + e0. In particular we carried out this approach for all the minimal nilpotent C(s)-orbits in c(s). In a few cases we needed to employ ad hoc methods.
Two special situations are worth mentioning.
4.1.1. One special situation is when e0 is minimal in g, that is, of Bala-Carter type A1. Then c(s0) is itself a Levi subalgebra of g and can be computed directly from the extended Dynkin diagram of g.
Of course e ∈ c(s0). Consequently it is easy to locate all e which have e0 ∈ c(s) when e0 is of type A1 in g.
We will see in Corollary 4.2 that in this setting x = e + e0 always satisfies Corollary 2.3. Moreover the Bala-Carter type of x in g is easy to determine: if we know the type of e in the Levi subalgebra c(s0), call it X, then x has generalized Bala-Carter type X + A1. Then the usual Bala-Carter type can be looked up in [Som98] or in Dynkin's seminal paper [Dyn52].
For example, in E8 when e0 is of type A1, then c(s) is of Cartan-Killing type E7. Any nilpotent element e in a Levi subalgebra of type E7 will have a conjugate of e0 in c(s). If, for instance, e is a regular nilpotent element, then e + e0 has generalized Bala-Carter type E7 + A1, which is the same as There is another way to determine e + e0 when e0 is minimal in g. It has the advantage of locating the simple summand of c(s) in which e0 lies. As above, assume h is dominant relative to b. Since e0 ∈ g h has type A1, the semisimple element h0 ∈ h is equal to the coroot of a long root θ for g h . Therefore, α(h0) ≥ −2 for any root of g and equality holds if and only α = −θ. Now choose h0 dominant in g h (relative to b ∩ g h ). Then α(h0) ≥ −1 for all simple roots α of g since −θ is a negative root. Moreover α(h0) = −1 only if α is not a simple root for g h . In that case α(h) ≥ 1 since the simple roots of g h correspond to the zeros of the weighted Dynkin diagram for h. This shows that α(h + h0) ≥ 0 for all simple roots α of g and thus h + h0 yields the weighted Dynkin diagram for e + e0 without having to conjugate by an element of the Weyl group.
For example, let e belong to the orbit E7(a3), which has weighted Dynkin diagram Then g h has type 4A1 and c(s) must have type A1 since c(s) has rank one and contains e0, a nonzero nilpotent element. We want to know in which summand of g h the element e0 lies and what is e + e0.
The diagram for h0 relative to g, and dominant for g h , is either: Only the second choice leads to a weighted Dynkin diagram for h + h0, namely for D7(a1). Hence we know the type of e + e0 and the embedding of c(s) in g h . 4.1.2. The other special situation occurs when c(s) has rank 1. Let l be a minimal Levi subalgebra containing e. Then l has semisimple rank equal to the rank of g minus one. Assume that l is a standard Levi subalgebra. Let αi be the simple root of g which is not a simple root of l. For nonzero e0 ∈ c(s), the corresponding h0 centralizes l and hence lies in the one-dimensional subalgebra of h spanned by the coweight ω ∨ i for αi. Since the values in any weighted Dynkin diagram are 0, 1, or 2, if h0 is dominant, then h0 must be either ω ∨ i or 2ω ∨ i . For example, let e be of type A7 in E8, which has c(s) of type A1. The weighted Dynkin diagram of a nonzero h0 ∈ c(s) must either be Both of these are actual weighted Dynkin diagrams in E8, but only the orbit for the first diagram, which corresponds to 4A1, meets g h , which has semisimple type exactly 4A1. Therefore a nonzero nilpotent element in c(s) has type 4A1 in g.

4.2.
Establishing the dimension condition in Corollary 2.3. Once a nilpotent e0 ∈ c(s) is located, as in the previous section, with corresponding semisimple element h0 ∈ c(s), we can compute h + h0 and check by hand whether the dimension condition holds for the orbit O through e + e0. If it holds, then the first hypothesis in Corollary 2.3 is true with O ′ = Oe and x = e + e0.
As before, let s0 be an sl2-subalgebra in c(s) with standard basis {e0, h0, f0}. Clearly, s and s0 commute. There is a statement equivalent to the dimension condition (4.1) in terms of the decomposition of g into irreducible subrepresentations for s ⊕ s0 ∼ = sl2(C) ⊕ sl2(C).
Let Vm,n denote an irreducible representation of s ⊕ s0 with h ∈ s acting by m and h0 ∈ s0 acting by n on a highest weight vector u ∈ Vm,n annihilated by both e and e0. The eigenvalues of h + h0 on Vm,n are either all even if m and n have the same parity or all odd if m and n have opposite parities. In the former case the quantity min(m, n) + 1 is equal to the dimension of the 0-eigenspace of h + h0; in the latter case, it is equal to the dimension of the 1-eigenspace of h + h0. This is analogous to what occurs in the proof of the Clebsch-Gordan formula. Let The relationship between (4.1) and this decomposition in (4.2) is the following: Proof. By sl2(C)-theory, the sum of the dimensions of the 0-eigenspace and the 1-eigenspace for ad(h+h0) on g equals the dimension of the centralizer of x = e + e0 in g. It therefore follows that (min(mi, ni) + 1) .

At the same time
is an irreducible representation of s0 ∼ = sl2(C) of highest weight ni, hence of dimension ni + 1. Putting the two formulas together, the codimension of Oe in Ox is equal to It is also necessary to compute dim c(s) e 0 . Since s0 ⊂ c(s) and c(s) is exactly ker ad e ∩ ker ad h, it follows that c(s) coincides with the sum of all V (i) m i ,n i where mi = 0. The centralizer c(s) e 0 is then the span of the highest weight vectors of these V (i) 0,n i and hence its dimension is given by the number of these subrepresentations. That is, The equality of dim C(s) · e0 and the codimension of Oe in Ox is therefore equivalent to min(mi, ni) = ni for all i with mi = 0.
It follows from the proof that if J = {i | ni > mi > 0}, then The element e0 ∈ g is called height 2 if all the eigenvalues of ad h0 on g are at most 2, and e is called even if all the eigenvalues of ad h on g are even.
Corollary 4.2. Suppose that either (1) e0 belongs to the minimal nilpotent orbit in g, or (2) e0 is of height 2 in g and e is even. Then Proof. If e0 belongs to the minimal nilpotent orbit of g, then e0 is of height two and the 2-eigenspace of ad h0 is spanned by e0. This is the case since h0 is conjugate to the coroot of the highest root. But since s0 ⊂ c(s), it follows that s0 ∼ = V0,2 is the unique subrepresentation of g isomorphic to Vm,n with n ≥ 2. Therefore all other V (i) m i ,n i must have ni = 0 or ni = 1 and so condition (4.3) holds. Next assume the second hypothesis. Since e is even, all V (i) m i ,n i with mi > 0 satisfy mi ≥ 2. Since e0 is of height two, ni ≤ 2 and thus condition (4.3) is true and hence also (4.1).

4.3.
Calculations in the proof of Proposition 3.1. First, we consider the cases where e0 is a minimal nilpotent element in g and compute e + e0 ( §4.1.1). Then Corollary 4.2 ensures that (4.1) holds. The degeneration (Oe+e o , Oe) turns out always to be a minimal degeneration so the explanation in the proof of Proposition 3.1 shows that SO,e ∼ = C(s) · e0 for O = Oe+e o . The results are recorded in Tables 1, 3, 6, and 9 for each of the exceptional groups F4, E6, E7, and E8, respectively. Next, we consider all other cases where e0 is a minimal nilpotent element in c(s) and check whether or not (4.1) holds for e + e0. In the cases when it does hold, the degeneration (Oe+e o , Oe) turns out again always to be a minimal degeneration and thus SO,e ∼ = C(s) · e0. The results are recorded in the first lines of Tables 2, 4, 7, and 10. These two sets of calculations include all the minimal degenerations covered by Proposition 3.1. They also include those cases in Proposition 3.3 where |Γ| = 2 and x + = 0. Let e0 ∈ c(s) be nilpotent, but not necessarily minimal nilpotent, and suppose that the dimension condition (4.1) does not hold for e+e0. Sometimes it may happen that Lemma 2.4 holds for a nilpotent orbit O with Oe ⊂ O ⊂ Oe+e • . That is, it may be possible to locate x ′ ∈ ⊕i≤−1g f (i) with (4.5) and so that the dimension condition dim C(s) · (e0 + x ′ ) = dim SO,e holds. We now discuss how such an x ∈ SO,e might be located. 4.4.1. A smaller slice result. Let y = e + e0, which is nilpotent with corresponding semisimple element hy = h + h0. Write gj for the j-eigenspace of ad hy on g. The centralizer G := G hy has Lie algebra g0 and G acts on each gj. Then y ∈ g2 and the G-orbit through y is the unique dense orbit. Now e ∈ g2 since (4.6) We want to find a transverse slice for the G-action on g2 to the G-orbit through e. In fact, since g2 is a direct sum of ad h-eigenspaces, the decomposition g = Im ad e ⊕ ker ad f restricts to a decomposition g2 = [e, g0] ⊕ (g2 ∩ ker ad f ).
Therefore, setting S (2) e = e + (g2 ∩ ker ad f ), it follows that the affine space S (2) e is a tranverse slice of g2 at e with respect to the G-action. Consequently, every G-orbit in g2 containing e in its closure meets Let g(r, s) denote the subspace of g where ad h has eigenvalue r and ad h0 has eigenvalue s. Define g f (r, s) = g(r, s) ∩ ker ad f . Then g2 ∩ ker ad f = r≥0 g f (−r, r + 2).
Next, we relate this decomposition to the decomposition (4.2) of g under s ⊕ s0. Let E = {i | ni > mi > 0 and ni − mi even} where (mi, ni) are defined in (4.2). Then E ⊂ J and E is empty if Proposition 4.1 is true. For each i ∈ E, let vi be a nonzero vector in the one-dimensional space V (i) m i ,n i ∩ g(−mi, mi + 2). Then vi is a lowest weight vector for s, but not in general a highest weight vector for s0. The set {vi | i ∈ E} is then a basis for r≥1 g f (−r, r + 2) since each vector in g f (−r, r + 2) lies in some subrepresentation of type Vr,s with r + 2 ≤ s and s − r even. The subspace g f (0, 2) is just the 2-eigenspace of ad h0 in c(s), which coincides with c(s) ∩ g(0, 2). It contains e0. A consequence of the above observations is the following Lemma 4.3. Let x ∈ g2. If e ∈ G · x, then some G-conjugate of x can be expressed as The cases in which we are interested are those where dim(C(s) · e0) = dim SO x,e . In all these cases, it turns out that (4.7) holds with a = 0, w = e0, and [e0, vi] = 0 for all i ∈ E with bi = 0. In particular [e0, x] = 0. Hence the vi's turn out to be highest weight vectors for s0 in Proof. By assumption, after rescaling, we may write x = e + e0 + vi. Consider the natural map C(s)equivariant map π : SO,e → c(s). Then π is surjective onto C(s) · e0. The dimension assumption and the C(s)-equivariance of π implies that π is finite over each point in the dense orbit C(s) · e0. Now if g is in the unipotent subgroup U ⊂ C(s) corresponding to the line through e0, then g.x = e + e0 + g.vi. Thus U.x lies in the fiber of π over e0. Therefore U.x is finite and hence a single point since U is irreducible. It follows that U.vi = vi and hence [e0, vi] = 0 for all i with bi = 0 and so [e0, x] = 0. 4.4.2. Applying Lemma 4.3. In order to apply Lemma 4.3 for some x ∈ g with Oe ⊂ Ox ⊂ Oy, we need to check two things, after possibly replacing x by a conjugate: (1) x ∈ g2 (2) e ∈ G · x The first condition can often be shown as follows. Let sx be an sl2-subalgebra through some conjugate of x with standard semisimple element h x ∈ h. In all cases we are interested in, there exists nilpotent e x 0 ∈ c(sx) with semisimple element h x 0 ∈ h, such that h x + h x 0 = hy, after possibly replacing x again by a conjugate. Then just as in (4.6), x ∈ g2 and the first condition holds.
We may further assume that hy is dominant with respect to the Borel subalgebra b ⊂ g and h x is dominant for the corresponding Borel subalgebra by of g hy . Then since [h x , x] = 2x and [hy, x] = 2x, if follows that x belongs to where g(hx; i) are the eigenspaces for ad hx. This subspace is preserved by the action of by. Thus G · Ix = G · x. We can carry out a similar process for e and obtain a subspace Ie ⊂ g2, with G · Ie = G · e. Then if Ie ⊂ Ix, it necessarily follows that G · Ie ⊂ G · Ix and the second condition holds. For the cases we are interested in, this approach will suffice to check the condition in the Lemma. 4.4.3. An example in C3. Let g = sp 6 (C). Nilpotent orbits in g can also be parametrized by the Jordan partition for any element in the orbit, viewed as a 6 × 6-matrix. Pick e ∈ g with partition [2, 2, 2] (so e is of type A1 +Ã1). Then c(s) is isomorphic to so3(C) ∼ = sl2(C). Let e0 ∈ c(s) be a nonzero nilpotent element. Then viewed as an element in g, e0 has partition [3, 3]. We complete e and e0 to commuting sl2(C)-subalgebras s and s0, respectively, with standard semisimple elements h and h0, respectively. Let y = e + e0. Then the eigenvalues of hy := h + h0 acting on the natural six-dimensional representation of g are {3, 1, 1, −1, −1, −3} and so y has partition [4,2], which corresponds to the subregular nilpotent orbit in g. Moreover, the decomposition of g under s ⊕ s0 in (4.2) is g ∼ = V (0, 2) ⊕ V (2, 0) ⊕ V (2, 4). Therefore (m, n) := (mi, ni) = (2, 4) for the unique i ∈ E and thus, by Proposition 4.1, the dimension identity (4.1) does not hold for e + e0. If we choose a vector v = v1 ∈ V (2, 4) as in Lemma 4.3, then since m = n + 2, There are two G-orbits between Oe and Oy, one with partition [3,3] and the other with [4, 1, 1]. We would like to show that Lemma 4.3 holds for both orbits. Let O be the orbit [3,3] and let x ∈ O. We need to show that x = e + ae0 + bv, with a and b nonzero, for some conjugate of x. Now assume hy is dominant for g so its weighted diagram {αi(hy)} is the usual weighted Dynkin diagram 2 0 2 . Recall hy = h + h0 and we may assume that h and h0 have weighted diagrams h = 0 0 2 and h0 = 2 2 −4 , respectively. Then Ie is a 2-dimensional subspace of g2. Similarly we can pick h x and h x 0 to have weighted diagrams 0 2 0 and 2 −2 2 , respectively. Then hy = h x + h x 0 and thus without loss of generality we can assume x ∈ g2. Moreover, Ix is a 3-dimensional subspace of g2 containing Ie and therefore the two conditions above hold. Hence by Lemma 4.3 x = e + ae0 + bv, after perhaps replacing x by a conjugate. Now b = 0 since otherwise x would be conjugate to e + e0, which is not the case. We also want to eliminate the case where a = 0. Since [e0, v] = 0, the stabilizer of x in c(s) is the line through e0 and so dim C(s) · x = 2 Also Ox is normal, hence unibranched at e [KP82]. Thus by Lemma 2.4, SO,e = C(s) · x. Therefore if a = 0, SO,e would be isomorphic to the closure of the SL2(C)-orbit in V (4) through the highest weight. But the latter variety is normal, but not Gorenstein (cf. [FZ03, Theorem 2.19]). Since SO,e is normal and Gorenstein ( §1.2), we deduce that a = 0. Now it is easy to see that SO,e ∼ = C(s) · e0, the nilpotent cone in sl2(C) (or see Lemma 4.5). Thus the singularity (O, Oe) is an A1-singularity, as was already known from [KP82]. This is an example where Proposition 3.3 holds with x + = 0. The result also holds for the orbit [4, 1, 1] by the same kind of argument. 4.4.4. Example: (Ã1, A1) in type G2. Let g be of type G2 and let e be a minimal nilpotent element. Then c(s) ∼ = sl2(C). Let e0 be a minimal nilpotent element in c(s). Then e0 has typeÃ1 in g and y = e + e0 has type G2(a1). The decomposition of g in (4.2) is V (0, 2)⊕V (2, 0)⊕V (1, 3). Therefore (m, n) := (mi, ni) = (1, 3) for the unique i ∈ E and (4.1) does not hold for e + e0. Fix nonzero v = v1 ∈ V (1, 3) satisfying [e0, v] = 0 and [f, v] = 0. Between Oe and Oy there is a unique G-orbit O, the one of typeÃ1. We will show show that there exits x ∈ O satisfying x = e + ae0 + bv.
Choose hy so that its weighted diagram is the usual weighted Dynkin diagram 2 0 and h x and h x 0 to have weighted diagrams 0 1 and 2 −1 , respectively. Then hy = h x + h x 0 and thus we may assume x ∈ g2. Similarly, let h and h0 have weighted diagrams −1 1 and 3 −1 , respectively. Then Ie is one-dimensional and Ie ⊂ Ix. The two conditions above are met, so indeed there exists x ∈ O with x = e + ae0 + bv by Lemma 4.3. Now b = 0 since x and e + e0 are not in the same G-orbit. If a = 0, then since O is unibranch at e, we would have that SO,e = C(s) · x is isomorphic to the closure of SL2(C)-orbit of a highest weight vector in V (3). As in the previous example, the latter is normal, but not Gorenstein, a contradiction of 1.2. Since e0 ∈ V (2) and v ∈ V (3) are highest weight vectors for SL2(C) (relative to e0), and both a and b are nonzero, SO,e ∼ = m, by the definition of m.

4.5.
A lemma for Propositions 3.3 and 3.4. Recall that V (i) denotes an irreducible module for SL2 with highest weight i ≥ 0. Consider the SL2-representation where I := {i1, i2, . . . , ir}. Let vj be a highest weight vector in V (ij).
Define XI to be the closure of the orbit SL2 · (v1 + v2 + . . . + vr) in VI . If any of the natural numbers in I are repeated, then it is possible to pass to a subrepresentation V I ′ of VI where I ′ contains one copy of each distinct number in I. It is clear that X I ′ ∼ = XI. Therefore it is enough to consider the case of distinct ij's. Let Imin be the minimal generating set of the monoid in N generated by i1, . . . , ir.
Lemma 4.5. Let π be the SL2-linear projection of VI onto VI min . a) Then π restricts to an SL2-equivariant isomorphism of XI onto XI min . b) Let d be the greatest common divisor of i1, . . . , ir. Then X{d} is the normalization of XI.
Proof. Since the elements of I are distinct, VI is isomorphic to a subrepresentation of the symmetric algebra S * C 2 on V (1) ∼ = C 2 . After fixing ℓ ∈ C 2 , we may identify vj ∈ V (ij ) with ℓ i j ∈ S i j C 2 . Then SL2 · (v1 + v2 + . . . + vr) corresponds to the set of elements i∈I L i where L ∈ C 2 is nonzero. It follows that XI identifies equivariantly with i∈I L i for L ∈ C 2 . It follows that the restriction of π, which maps XI to XI min , corresponds to the map This is an isomorphism since L i for i / ∈ Imin depends polynomially on those L j for j ∈ Imin. For part (b) the map L d → i∈I (L d ) i/d from X{d} onto XI is regular and clearly surjective. It is also injective since d is the greatest common divisor of i1, . . . , ir. Thus the map restricts to an isomorphism between the open orbits, hence is birational. Since X{d} is normal (being a quotient of C 2 by a cyclic group), Zariski's main theorem implies that X{d} is the normalization of XI.
An analogous result holds for any simple G. Let V (µ) denote the irreducible highest weight module for G with weight µ. Let λ be a dominant weight for G relative to a Borel subgroup B and consider VI = V (i1λ) ⊕ V (i2λ) ⊕ . . . ⊕ V (irλ). Define vi and XI in analogy with the SL2-case and the conclusions of the lemma go through for this situation. The XI, when normal, are spherical varieties for the Borel subgroup opposite to B. Hence Lemma 4.5 and its generalization to simple G follow from known results about spherical varieties (see [Bri10,Thm 1.4

]).
In all the cases where we apply Lemma 4.5 or its generalization, it turns out that i1 = 2 and either the ij's are all even or i2 = 3. In the former case when G = SL2(C), the lemma implies that XI identifies with the minimal nilpotent orbit closure in sl2(C), hence is isomorphic to the A1 singularity. In the latter case for G = SL2(C) or Sp4(C), the lemma implies that XI is not normal, XI identifies with either m or m ′ , respectively, and XI has normalization C 2 or C 4 , respectively. 4.6. Finding vi for i ∈ E. We sometimes need to do explicit computations to verify (4.7) or to show that w = e0 and a = 0, especially for degenerations which are not minimal (e.g., starting with §7.4). In these cases there arises the need for an analog of Lemma 4.5. Here we describe a way to find vi for i ∈ E which frequently leads to an isomorphism of Z = C(s) · (e0 + x ′ ) with C(s) · e0 in Lemma 2.4, when such an isomorphism exists.
Write g(h; j) for the j-eigenspace of ad h. Since c(s) ⊂ g h = g(h; 1), the g(h; j) are c(s)-modules. The g h ⊕ g(h; 1) isomorphic as c(s)-modules to g f . In particular for j ≥ 0, as c(s)-modules.
Take, as usual, e0 ∈ c(s) and y = e + e0. If E = ∅, then the following method suffices in most cases to locate vi ∈ E when mi is even. Suppose that g h contains a simple factor of type slN . Then setting X = e0, we may consider the powers X r ∈ slN ⊂ g h . Of course [X, X r ] = 0 in slN ⊂ g h and hence in g, which means that X r , if nonzero, is a highest weight vector for s0 relative to e0. Moreover in slN the identity [h0, X r ] = 2rX r holds because [h0, X] = h0X − Xh0 = 2X; hence this also holds in g. Assume X r is not zero. Then for some largest j, is nonzero. Since s and s0 commute, Xj,r also commutes with e0, so in fact Xj,r is a nonzero element of g(−2j, 2r) ∩ g f ∩ g e 0 .
Now let (mi, ni) for i ∈ E with mi even. In the cases of interest ni = mi + 2, as suggested by Lemma 4.4, and we we often find that Moreover, if x in (4.7) is a linear combination of these vi's and e0, then it follows that C(s) · x ∼ = C(s) · e0 via the natural projection (see §3.2) since C(s)-commutes with taking powers. This phenomenon appears more generally when e0 is regular nilpotent in g h (or some simple factor of g h ); then any element in g h ∩ g e 0 is polynomial function on g evaluated at e0. 4.6.1. Example in C3 (continued). Continuing with Example 4.4.3, to find an explicit v, let X = e0.
Here g h has semisimple part sl3(C). Then if X = e0, X 2 = 0 since X is regular in sl3(C) and X 2 ∈ so3. Since [h, X 2 ] = 4X 2 , it must be that X 2 ∈ V (2, 4) and so (ad f )(X 2 ) is in the span of v. Consequently, SO,e identifies with the closure of the C(s)-orbit of (X, X 2 ) ∈ c(s) ⊕ V (4) = sl3(C) where X ∈ c(s) is nilpotent. The latter is isomorphic, as noted before, to the nilpotent cone in sl2(C).

4.7.
Calculations in the proof of Propositions 3.3 and 3.4. We pick up where §4.3 left off. That is, e0 is a minimal nilpotent element in c(s) such that the dimension condition (4.1) does not hold for e + e0. For such e and e + e0, we look for nilpotent orbits O with Oe ⊂ O ⊂ Oe+e • such that Oe is a minimal degeneration of O and dim(C(s) · e0) = codim O (Oe). Then O ′ = Oe and O are candidate orbits for Lemma 2.4 to hold. We proceed as in §4.4.1 and §4.4.2, but sometimes have to carry out ad hoc computer calculations. When we find candidates where Lemma 2.4 holds, then Lemma 4.5 and its generalization turn out always to be enough to describe SO,e. Comparing with the surface cases treated in §5, we find that all the cases in Propositions 3.3 and 3.4 are now handled. The results are recorded in the latter lines of Tables 2, 4, 7, and 10 where E = ∅. As noted in Remark 3.7, there are some e0, minimal in a c(s), that do not contribute to describing any minimal degeneration.

Geometric method for surface singularities
In this section we consider a minimal degeneration We show that the normalization of each irreducible component of SO,e is isomorphic to C 2 /Γ for some finite subgroup Γ ⊂ SL2(C). Our method allows us to determine the group Γ, hence we determine SO,e up to normalization. As mentioned in §3.2, we can often use results on normality of nilpotent orbit closures or other methods (e.g. Lemma 2.2) to decide whether the irreducible components of SO,e are normal. Sometimes we have to state our results up to normalization. 5.1. Two-dimensional Slodowy transverse slices. Recall that a contracting C * -action on a variety X is a C * -action on X with a unique fixed point o ∈ X such that for any x ∈ X, we have limλ→0 λ·x = o. Recall from [Bea00] that a symplectic variety is a normal variety W with a holomorphic symplectic form ω on its smooth locus such that for any resolution π : Z → W , the pull-back π * ω extends to a regular 2-form on Z. For a nilpotent orbit, we write O for the normalization of O.
Lemma 5.1. The normalization SO,e of SO,e is an affine normal variety with each irreducible component having at most an isolated symplectic singularity and endowed with a contracting C * -action.
Proof. As O has rational Gorenstein singularities by [Hin91] and [Pan91], SO,e has only rational Gorenstein singularities. On the other hand, there exists a symplectic form on its smooth locus, hence SO,e has only symplectic singularities by [Nam01] (Theorem 6). By construction, the contracting C * -action on SO,e in §2.3 has positive weights, hence it lifts to a contracting C * -action on SO,e.
Proposition 5.2. Let X be an affine irreducible surface with an isolated rational double point at o. If there exists a contracting C * -action on X, then X is isomorphic to C 2 /Γ for some finite subgroup Note that by Proposition 2.1,the irreducible components of SO,e are mutually isomorphic. As an immediate corollary, we get Corollary 5.3. Let SO,e be a two-dimensional nilpotent Slodowy slice. Then there exists a finite subgroup Γ ⊂ SL2(C) such that each irreducible component of the normalization SO,e is isomorphic to Hence to determine SO,e, we only need to determine the subgroup Γ. In the following, we shall describe a way to construct the minimal resolution of SO,e. Then the configuration of exceptional P 1 's in the minimal resolution will determine Γ.
5.2. Q-factorial terminalization for nilpotent orbit closures. A general reference for minimal model program in algebraic geometry is [Mat02]. Here we recall some basic definitions.
Let X be a normal variety. A Weil divisor D on X is called Q-Cartier if N D is a Cartier divisor for some non-zero integer N . We say that X is Q-Gorenstein if its canonical divisor KX is Q-Cartier. The variety X is called Q-factorial if every Weil divisor on X is Q-Cartier. A Q-Gorenstein variety X is said to have terminal singularities if there exists a resolution π : Z → X such that KZ = π * KX + k i=1 aiEi with ai > 0 for all i, where Ei, i = 1, · · · , k are the irreducible components of the exceptional divisor of π. A Q-factorial terminalization of a Q-Gorenstein variety X is a projective birational morphism π : Z → X such that KZ = π * KX and Z is Q-factorial with only terminal singularities.
It is well-known that two-dimensional terminal singularities are necessarily smooth (cf. Theorem 4-6-5 [Mat02]), hence a normal variety X with only terminal singularities is smooth in codimension 2, that is, codimX Sing(X) ≥ 3.
For the normalization of the closure of a nilpotent orbit, one way to obtain its Q-factorial terminalization is by the following method. Consider a parabolic subgroup Q in G. Let L be a Levi subgroup of Q. For a nilpotent element t ∈ Lie(L), we denote by O L t its orbit under L in Lie(L). Let n(q) be the nilradical of Lie(Q). Then the natural map p : G × Q (n(q) + O L t ) → g has image equal to O for some nilpotent orbit O and p is called a generalized Springer map for O. Then O is said to be induced from [LS79]. When t = 0, then O is called the Richardson orbit for Q and G × Q n(q) identifies with the cotangent bundle T * (G/Q). If p is birational and the normalization of O L t is Q-factorial terminal, then the normalization of p gives a Q-factorial terminalization of O, the normalization of O. In [Fu10], it was proved in confirming a conjecture of Namikawa that for a nilpotent orbit O in an exceptional Lie algebra, either O is Q-factorial terminal or every Q-factorial terminalization of O is given by a generalized Springer map. 5.3. Minimal resolutions of two-dimensional nilpotent Slodowy slices. We now use the generalized Springer maps to construct a minimal resolution of SO,e when SO,e is two-dimensional.
Recall from [Fu10] that in a simple Lie algebra of exceptional type, O has only terminal singularities if and only if O is either a rigid orbit or it belongs to the following list: 2A1, A2 + A1, A2 + 2A1 in E6; A2 + A1, A4 + A1 in E7; A4 + A1, A4 + 2A1 in E8.
First consider the case where O has only terminal singularities. Then O is smooth in codimension two by the previous subsection. This implies that the singularities of O along Oe are smoothable by its normalization. In other words, SO,e is smooth, which is then isomorphic to C 2 by Proposition 5.2 and we are done.
Example 5.4. Consider again the minimal degeneration ( A1, A1) in G2 from §4.4.4. As O = O A 1 is a rigid orbit, its normalization has Q-factorial terminal singularities by [Fu10]. In particular, the singular locus of O has codimension at least 4. Since the orbit A1 is codimension two in O, this implies that O is non-normal and SO,e ∼ = C 2 for e ∈ OA 1 , which is consistent with the description SO,e ∼ = m in §4.4.4.
Next, assume that the normalization O is not terminal. Then by [Fu10], O is an induced orbit and O admits a Q-factorial terminalization π : Z → O given by the normalization of a generalized Springer map. We denote by U the open subset O ∪ Oe of O and ν : U → U the normalization map. As Z has only terminal singularities, it is smooth in codimension two. As π is G-equivariant and Oe ⊂ O is of codimesion two, we get that π(Sing(Z)) ∩ ν −1 (Oe) = ∅. We deduce that V := π −1 ( U ) is smooth. In particular, we obtain a symplectic resolution π|V : V → U . By restriction, we get a resolution π : π −1 ( SO,e) → SO,e, which is a symplectic, hence minimal, resolution.
5.4. The method of Borho-MacPherson. Let W be the Weyl group of G. The Springer correspondence assigns to any irreducible W -module a unique pair (O, φ) consisting of a nilpotent orbit O in g and an irreducible representation φ of the component group A(x) where x ∈ O. The corresponding irreducible W -module will be denoted by ρ(x,φ).
Let W (L) denote the Weyl group of L, viewed as a subgroup of W . Let Bx denote the Springer fiber over x for the resolution of the nilpotent cone N in g and let B L t be the Springer fiber of t for the group L. If O L t is the orbit of L through the nilpotent element t ∈ Lie(L), we denote by ρ L (t,1) the W (L)-module corresponding to the pair (O L t , 1) via the Springer correspondence for L. Lemma 5.5. Let Z = G × Q (n(q) + O L t ). Let p : Z → O be the generalized Springer map. Let O ′ ⊂ O be a nilpotent orbit of codimension 2d. Assume that Z is rationally smooth at all points of p −1 (e) for e ∈ O ′ . Then the number of irreducible components of p −1 (e) of dimension d is given by the formula where the sum is over the irreducible representations φ of A(e) appearing in the Springer correspondence for G. 1) , where the right hand side denotes the ρ L (t,1) -isotypical component of the restriction of H top (Be) to W (L). Recall that H top (Be) = ⊕φρ(e,φ) ⊗ φ, which gives

Proof. By [BM83, Thm. 3.3], we have
where h top (X) denotes the dimension of H top (X).

Now the component group A(e) acts on the left-hand side of
where it acts trivially on H top (B L t ). It also acts on the right-hand side since the A(e) action commutes with the W and hence W (L) action. Note that the action of A(e) is compatible with the isomorphism (see Corollary 3.5 [BM83]). This gives the following is 2 and it is zero otherwise. By Lemma 5.5, the number of P 1 's in p −1 (e) is 1 · 2 + 2 · 1 = 4 and by Corollary 5.6, the group A(e) fixes one component and permutes the remaining three components transitively. Consequently the dual graph of SO,e = SO,e is the Dynkin diagram of type D4 and A(e) acts on the dual graph via the unique quotient of A(e) isomorphic to S3. Hence the singularity is G2.
The fact that the dual graph is D4 could also be obtained by restricting to a maximal subalgebra of type B4 ( §2.5). In this way we would only need to know that the degeneration in F4 is unibranch, instead of the stronger statement that O is normal. 5.5. Orbital varieties and the exceptional divisor of π. The next lemma can sometimes be used to simplify computations. Its proof follows from that of [Fu10, Lemma 4.3].
Lemma 5.8. Let X be an affine variety with rational singularities such that Pic(Xsm)Q = 0. Then for any resolution π : Z → X, the number of irreducible components in Exc(π) equals b2(Z). Here, Exc(π) denotes the exceptional divisor of π.
Let us consider a Q-factorial terminalization of a nilpotent orbit closure π : Z → O. Since codimZ Sing(Z) ≥ 3, we get a resolution π0 : Z \ Sing(Z) → O, which shares the same configuration of exceptional divisors as π. If Pic(O) is finite, then the previous lemma applies to π0, which gives the number of irreducible components of Exc(π0), hence of Exc(π). If O \ O consists of several irreducible components ∪iOi of codimension 2, then b2(Z) gives the sum of number of irreducible components of π −1 (Oi). It is especially convenient to apply the lemma when O is the Richardson orbit for a parabolic subgroup Q and the corresponding generalized Springer map π : G × Q n(q) → O is birational. Then we can take Z = G × Q n(q) above and so b2(G × Q n(q)) = b2(G/P ) equals the rank of G minus the semisimple rank of a Levi subgroup of Q. Moreover in this situation the irreducible components of Exc(π) have a description in terms of the orbital varieties for the Oi's.
Recall that an orbital variety for Oi is an irreducible component of Oi ∩ n where n := n(b) is the nilradical of the Borel subalgebra b. It is known that each orbital variety has dimension 1 2 dim Oi. Let X be an orbital variety for Oi which is contained in n(q). Then X is codimension one in n(q) since Oi is codimension two in O and dim n(q) = 1 2 dim O. Moreover X is stable under the action of the connected group Q since X ⊂ Q · X ⊂ Oi ∩ n and X is maximal irreducible in Oi ∩ n.
Let πX be the restriction of π to G × Q X. The image of πX is Oi since X is irreducible and Q is a parabolic. By dimension considerations, π −1 X (Oi) = G × Q X is an irreducible component of Exc(π).
Conversely, any irreducible component of Exc(π) equals G × Q Y for some irreducible component of Oi ∩ n(q). Now dim Y can only equal dim n(q) − 1 or dim n(q) − 2 since Im πY = Oi. In the former case, Y is an orbital variety of X contained in n(q). In the latter case, π −1 Y (ei) is finite for ei ∈ Oi, contradicting the fact that the irreducible components of π −1 (ei) are P 1 's. This shows that the irreducible components of Exc(π) are exactly the G × Q X where X is an orbital variety of some Oi lying in n(q).
Next, the map G × B X → G × Q X has connected fibers isomorphic to Q/B. It follows from [Spa82] that the P 1 's in π −1 X (ei) are permuted transitively under the induced action of A(ei) since the analogous statement holds for the irreducible components of p −1 X (ei) where pX : G × B X → N . Consequently, if mi equals the number of A(ei)-orbits on π −1 (ei), then mi = b2(G/P ). See, for example, [Wie03, Thm 1.3]) for a more general setting where this phenomenon occurs.
Example 5.9. Consider the minimal degeneration where O has type A2 and O ′ has type A1 + A1 in F4. The codimension of O ′ in O is two. The orbit O is even and so it is Richardson for the parabolic subgroup Q whose Levi subgroup has type B3. Moreover the map π : Z := G × Q n(q) → O is birational, hence a minimal resolution. The hypotheses of Lemma 5.8 hold. Since b2(Z) = 1 and there is no other minimal degeneration of O, there must be exactly one irreducible component in π −1 (O ′ ). Since A(e) = 1 for e ∈ O ′ , there is only one irreducible component in π −1 (e). Hence the singularity is of type A1 since O is normal. 5.6. Three remaining cases. There are three cases where the information in Lemma 5.5 and Corollary 5.6 is not sufficient to determine a minimal surface degeneration, even up to normalization. They are (E6(a1), D5) in E6, (E7(a1), E7(a2)) in E7, and (E8(a1), E8(a2)) in E8. In this section we give an ad hoc way to determine the singularity.
In each of the three cases, the larger orbit O is the subregular nilpotent orbit and so O is normal. Since g is simply-laced, A(x) is trivial for x ∈ O. Hence for any parabolic subgroup Q with Levi factor A1 the map π : G × Q n(q) → O is a minimal resolution. Moreover in each cases the smaller orbit O ′ is the unique maximal orbit in O\O. Since A(e) = 1 for e ∈ O ′ , there are rank(g) − 1 P 1 's in π −1 (e) by §5.5. At the same time, this uniqueness means that O ′ is the Richardson orbit for any parabolic Q ′ with Levi factor A1 × A1. In other words, n(q ′ ) is an orbital variety for O ′ . Hence if we fix Q corresponding to a simple root α, then we find an orbital variety n(q ′ ) ⊂ n(q) for O ′ for each simple root β not connected to α in the Dynkin diagram. Since A(e) is trivial, each of these n(q ′ ) gives rise to a unique P 1 in π −1 (e). By looking in the Levi subalgebra corresponding to the simple roots not connected to α, it is possible to determine the intersection pattern of these P 1 's. 5.6.1. The case of (E6(a1), D5) in E6. There are 5 P 1 's in π −1 (e). The singularity could only be A5 or D5 since O is normal. If we choose α so that the remaining simple roots form a root system of type A5, then there are 4 orbital varieties of the form n(q ′ ) in n(q). The 4 P 1 's have intersection diagram of type A2 + A2. This could only happen for a dual graph of type A5, so SO,e ∼ = A5. 5.6.2. The case of (E7(a1), E7(a2)) in E7. There are 6 P 1 's in π −1 (e). The singularity could only be A6, D6, or E6 since O is normal.. If we choose α so that the remaining simple roots form a system of type D6. Then there are 5 orbital varieties of the form n(q ′ ) in n(q). Then the 5 P 1 's have intersection diagram of type D5. This eliminates A6 as a possibility. If we choose α so that the remaining simple roots form a system of type A6, then there are 5 orbital varieties of the form n(q ′ ) in n(q) and the corresponding 5 P 1 's have intersection diagram of type A2 + A3. This eliminates E6, hence SO,e ∼ = D6. 5.6.3. The case of (E8(a1), E8(a2)) in E8. There are 7 P 1 's in π −1 (e). The singularity could only be A7, D7, or E7 since O is normal. If we choose α so that the remaining simple roots form a system of type E7, then there are 6 orbital varieties of the form n(q ′ ) in n(q). The corresponding 6 P 1 's have intersection diagram of type E6. Hence SO,e ∼ = E7.
6. On the splitting of C(s) and intrinsic symmetry action 6.1. The splitting of C(s). In this section we establish the splitting on C(s) discussed in §1.7.3. Namely, we determine when C(s) ∼ = C(s) • ⋊ H for some H ⊂ C(s). Necessarily H ∼ = A(e). We continue to assume that G is of adjoint type.
In the classical groups, C(s) is a product of orthogonal groups and a connected group, possibly up to a quotient by a central subgroup of order two. Since the result holds for any orthogonal group, it holds for C(s).
Let C ⊂ A(e) be a conjugacy class. There exists s ∈ C(s) whose images in A(e) lies in C such that the order of s equals the order ofs, except when e belongs to one the following four orbits: For these four orbits, which all have A(e) = S2, the best result is an s of order 4 to represent the non-trivial C in A(e) [Som98, §3.4]. Hence the splitting holds for all other orbits with A(e) = S2. This leaves the cases where A(e) = S3, S4, or S5. If e is distinguished, meaning C(s) • = 1, there is nothing to check. This leaves a handful of cases where A(e) = S3 and e is not distinguished. The first such case is e = D4(a1) in E6, which we now explain.
6.1.1. S3 cases. Let G be of type E6 and s ∈ G be an involution with G s of semisimple type A5 + A1. Then there existẽ ∈ g s nilpotent of type 2A2. Lets ⊂ g s be an sl2-triple throughẽ. Then c(s) has type G2. Now g s ∩ c(s) is easy to compute inside of A5 + A1; it is a semisimple subalgebra of type A1 + A1. Letẽ0 be regular nilpotent in g s ∩ c(s). Thenẽ0 is in the subregular nilpotent orbit in c(s). Clearly s belongs to the centralizer ofẽ0 in C(s), which is a finite group H ∼ = S3, from the case of the subregular orbit in G2. Next, a calculation in A5 + A1 shows thatẽ +ẽ0 has type A3 + 2A1. From this we conclude that e =ẽ +ẽ0 belong to the nilpotent orbit D4(a1) in E6 and s represents an involution in A(e) [Som98,§4].
A similar argument works if s ∈ G is an element of order 3 with G s of semisimple type 3A2. Therefore the centralizer H ∼ = S3 ofẽ0 in C(s) also centralizesẽ +ẽ0 and the image of H in A(e) is all of A(e). This proves the splitting for e = D4(a1) in E6. The same procedure works for the other S3 cases. While the above splitting is unique up to conjugacy in C(s) in the subregular case ( §1.4.2), this is not the case in general, as the next example shows.

We have shown
Example 6.2. Let e = A2 in g = E8. Then c(s) has type E6 and A(e) = S2. The generalized Bala-Carter notation for the non-trivial class C in A(e) is 4A1. From this it follows that both conjugacy classes of involution in G can represent C. For one choice of involution s1 ∈ C(s) lifting C, g s 1 has type D8. The partition of e in g s 1 is [2 8 ], so the reductive centralizer of e in g s 1 is sp 8 . For the other another choice s2 ∈ C(s) lifting C, g s 2 has type E7 + A1 and e corresponds to (3A1) ′′ + A1. Hence the reductive centralizer of e in g s 2 is of type F4. Consequently, there are two choices of splitting in Proposition 6.1 that are not only non-conjugate under C(s), but also in Aut(c(s).
Although the choice of splitting in Proposition 6.1 is not unique up to conjugacy in C(s) or even Aut(c(s)), we can restrict the choice of H further so that the image of H in Aut(c(s)) will be well-defined up to conjugacy in Aut(c(s)). Let c(s) ss be the semisimple summand of c(s). Let a : C(s) → Aut(c(s) ss ) be the natural map. Then Im a = Int(c(s) ss ) ⋊ K for some subgroup of diagram automorphisms ( §1.4.2). By a case-by-case check, H in the Proposition 6.1 can be chosen so that H maps onto K via a. Then the image of H in Aut(c(s)) is well-defined up to conjugacy in Aut(c(s)). In the above example, H = s2 since the induced automorphism is a diagram automorphism.
6.2. Computing the intrinsic symmetry. Having chosen H with a(H) = K as above, we can determine the action of H on SO,e. Here, we restrict to the exceptional groups and to a minimal degeneration O ′ of O, with e ∈ O ′ . We summarize the possibilities and record the action of H on SO,e in the graphs at the end of the paper. 6.2.1. Minimal singularities: A(e) = S2 cases. Let SO,e be an irreducible minimal singularity admitting an involution as in §1.7.2. If |H| = 2, then it turns out that H realizes this involution. There is one case of this kind when |H| = 4, when e = A4 + A1 in E8 and SO,e ∼ = a2. Let H = s . Then s ∈ H realizes the involution on SO,e and s 2 acts trivially on SO,e. We will still refer to this singularity with induced symmetry by a + 2 . If SO,e is a reducible minimal singularity, then it is turns out that SO,e has exactly two irreducible components and H interchanges the two components. The only three cases which occur are the singularities with symmetry action [ If SO,e is a reducible minimal singularity, then SO,e turns out to have 3 irreducible components and H acts by permuting transitively the three components. In other words, the stabilizer of a component acts trivially on the component. All of these cases are of the form 3A1 and the singularity with symmetry action is denoted [3A1] ++ . 6.2.3. Simple surface singularities: A(e) = S2 cases. If SO,e is an irreducible simple surface singularity admitting an involution as in §1.3.2 (or in the case of A2 and A4, admitting the appropriate cyclic action of order 4), then H realizes this symmetry. To show this, we first checked that A(e) has the appropriate action on the dual graph of a minimal resolution in Corollary 5.6. Then since C(s) acts symplectically on SO,e, Corollary 1.1 and Theorem 1.2 in [Cat87] imply that H corresponds to the Γ ′ ⊂ SL2(C) which defines the symmetry involution.
The only reducible surface singularities with A(e) = S2 are those with SO,e ∼ = 2A1, hence covered previously.
6.2.4. Simple surface singularities: A(e) = S3 cases. If SO,e is an irreducible simple surface singularity admitting an S3 action as in §1.3.2, then H realizes the symmetry action and so SO,e ∼ = G2.
An unusual situation occurs for the minimal degeneration (D7(a1), E8(b6)). Here, A(e) = S3, but SO,e only admits a two-fold symmetry, compatible with its normalization SO,e which is A3. Here, Γ ⊂ SL2(C) corresponding to SO,e is cyclic of order 4. The normal cyclic subgroup of H ∼ = S3 is generated by an element s with g s of type E6 +A2 and hence s acts without fixed point on the orbit D7(a1) since the latter orbit does not meet the subalgebra E6 + A2. On the other hand, using Corollary 5.6, we see that A(e) induces the involution on the dual graph of a minimal resolution of SO,e. Since C(s) acts symplectically on SO,e and SO,e, Corollary 1.1 and Theorem 1.2 in [Cat87] imply that H acts on SO,e = C 2 /Γ via the action of Γ ′ ⊂ SL2(C), the binary dihedral of order 24 containing Γ as normal subgroup.
If SO,e is a reducible surface singularity, then SO,e is isomorphic to 3C2, 3C3, 3(C5), or the previously covered [3A1] ++ . We have omitted the superscript in 3C2, etc. The notation means that H permutes the three components transitively and the stabilizer of any component is order 2, which acts by the indicated symmetry. The notation (C5) refers to the fact that we do not know if an irreducible component is normal.
6.2.5. Simple surface singularities: A(e) = S4 case. This only occurs in F4. One degeneration has SO,e ∼ = G2 (see §5.4). Here, the Klein 4-group in H acts trivially on SO,e and the quotient action realizes the full symmetry of S3 on SO,e. This follows either from the list of possible symplectic automorphisms of SO,e or from a direct calculation that the Klein 4-group in H fixes SO,e pointwise.
The other degeneration has SO,e ∼ = 4G2 (see §7.3). Here, H permutes the four components transitively and the stabilizer of any component is an S3, which acts by the indicated symmetry.
6.2.6. Simple surface singularities: A(e) = S5 case. This only occurs in E8. One degeneration has SO,e ∼ = 10G2. Here, H permutes the ten components transitively and the stabilizer of any component is a Young subgroup S3 × S2. The S2 factor acts trivially on the given component and the S3 factor acts by the indicated symmetry.
The other degeneration has SO,e ∼ = 5G2. Here, H permutes the five components transitively and the stabilizer of any component is a S4. The S4 factor acts on the given component as in the F4 case above.
Remark 6.3. Not every non-trivial A(e) contributes a symmetry on a SO,e. For example, e = C3(a1) in F4. The only degeneration above Oe has SO,e ∼ = A1. Here, H acts trivially on the SO,e, reflecting the fact that SL2(C) has no outer automorphisms. Indeed, C(s) is a direct product of C(s) • and H.

Results for F4
7.1. Proving Proposition 3.1. Here we record the details for establishing Proposition 3.1 for g of type F4 as outlined in §4.3. First, we enumerate the G-orbits of those e such that c(s) has nontrivial intersection with the minimal nilpotent orbit in g. To that end, let e0 ∈ g be minimal nilpotent and recall that s0 is an sl2(C)-subalgebra through e0. The centralizer c(s0) is a simple subalgebra of type C3, equal to the semisimple part of a Levi subalgebra of g. The relevant nonzero nilpotent elements e ∈ c(s0) are therefore those in the G-orbits A1,Ã1, A1 +Ã1,Ã2, B2, C3(a1) and C3 and hence Corollary 4.2 applies to these elements. The computation of e + e0 ∈ O proceeds as in §4.1. The results are in Table 1. In the cases where c(s) is not simple, we use boldface font in Table 1 to indicate those simple factors whose minimal nilpotent orbit is of type A1 in g.
In the first two lines of Table 2 we record the remaining cases where Corollary 2.3 holds for a minimal nilpotent element e0 ∈ c(s). This completes the verification of Proposition 3.1 for F4.

7.2.
Verifying the remaining cases where Proposition 3.3 holds. In the previous section there were several minimal degenerations with SO,e isomorphic to an A1 singularity. These cases are the ones with x + = 0 in Propositions 3.3. Next, we find the cases in the proposition with x + = 0. There are no other cases since by §7.3 these are the only surface minimal degenerations with |Γ| = 1 or 2.
We use the method in §4.4, with some simplifications afforded by passing to a subalgebra as in Proposition 2.2. The values of (mi, ni) for i ∈ E are listed in Table 2. Boldface is used for those (mi, ni) where the corresponding coefficient bi = 0 in (4.7). Recall that it always turns out that w = e0 and the coefficient a of e0 is nonzero in (4.7).
7.2.1. A1 +Ã1. For e of type A1 +Ã1, c(s) is semisimple of type A1 + A1. The minimal nilpotent orbit in one simple factor is minimal in g and this was handled earlier. The other simple factor has minimal nilpotent orbit which is of typeÃ2 in g. Let e0 ∈ c(s) be a representative of this orbit. The sum y = e+e0 is of type C3(a1) and (m, n) = (2, 4) for the unique element in E . Let l be a Levi subagebra of g of type C3. We can assume e, e0 ∈ l. By Example 4.4.3, there exists x ∈ l of typeÃ2 satisfying (4.7) with a, b = 0 and w = e0. Since codim O x (Oe) = 2 in F4, x satisfies Lemma 2.4, not just in l, but also in F4. It follows that SO x,e ∼ = A1 in F4. 7.2.2.Ã2 + A1. For e of typeÃ2 + A1, c(s) ∼ = sl2. The minimal nilpotent orbit in c(s) has type A1 +Ã1 in g. Let e0 ∈ c(s) be a representative of this orbit. The sum y = e + e0 is of type F3(a3) and (m, n) = (1, 3) for the unique element in E. Let x be in the C3(a1) orbit. Then Oe ⊂ Ox ⊂ Oy. We check that Ie ⊂ Ix ⊂ g(hy; 2) as in §4.4.2. It follows as in Example 4.4.4 that SO x,e ∼ = m. This result can also be deduced from Lemma 2.2 by working in the subalgebra s ′ ⊕ c(s ′ ), where s ′ is the sl2-subalgebra through an element of typeÃ2 and then using Example 4.4.4 since c(s ′ ) is of type G2. 7.2.3. A2 +Ã1. For e of type A2 +Ã1, c(s) ∼ = sl2. The minimal nilpotent orbit in c(s) also has type A2 +Ã1 in g. Let e0 ∈ c(s) be a representative of this orbit. The sum y = e + e0 is of type F3(a3) and (m1, n1) = (1, 3) and (m2, n2) = (2, 4) for the two elements in E. Indeed the decomposition of g in (4.2) is For nilpotent x ∈ g of type B2 or of typeÃ2 + A1, Oe ⊂ Ox ⊂ Oy and we checked that the two conditions in §4.4.2, and thus (4.7), hold for each orbit.
Now if x ∈ O is of type B2, then O meets the reductive subalgebra l of type B4 in F4, where it has partition [4, 4, 1] in so9(C). Also, Oy meets l in the orbit [5, 3, 1]. Assume that s, s0 are in l. Calculating E for e and e0 relative to l, one finds that only (m, n) = (2, 4) occurs (reflecting the fact that e = [3, 3, 3] is even in l). Thus v1 ∈ l and hence b1 = 0. It follows that a, b2 = 0 as in Example 4.4.3. Consequently On the other hand for x inÃ2 + A1, we have to carry out an explicit computation. Choose e := e1000 + e0100 + e0001, f := 2f1000 + 2f0100 + f0001.
Using GAP and de Graaf's package, we found that the following two elements of the Slodowy slice are of typeÃ2 + A1. Since dim SO x,e = 2 = dim C(s) · e0, Lemma 2.4 holds. Then Lemma 4.5 implies that SO x,e ∼ = m. This completes the proof of Proposition 3.3 for F4.
7.3. Remaining surface singularities. This section summarize the calculations of the singularities of the minimal degenerations of dimension two, using the methods in §5. Up until this point we have not needed to know whether a nilpotent orbit has normal closure to determine the singularity type of a minimal degeneration. Knowing the branching was sufficient. Indeed, the closure of the orbit B2 is non-normal, but it was shown above that it is normal at points in the orbit A2 +Ã1 since the singularity is A1. Similarly for the orbitÃ2. The remaining non-normal orbit closures, of which there are three [Bro98b], are detected through a minimal degeneration: either the closure is branched at a minimal degeneration (as for C3) or is isomorphic to m at a minimal degeneration (as for C3(a1) and forÃ2 + A1).
In what follows we use the fact that the orbit F4(a1) has normal closure [Bro98b] to classify the type of its minimal degeneration. This is the only case where we need to know whether the closure is normal or not in order to resolve the type of a minimal degeneration in F4.
(1) ( This implies that the singularity of O at y is G2. We show in §7.4 that SO,e is isomorphic to 4G2. In other words, the irreducible components of SO,e are normal and hence each is isomorphic to G2. F4(a3)). The singularity is G2 by the example in §5.4. (C3, F4(a3)) is 4G2. Left unresolved by the previous discussion is whether an irreducible component of the nilpotent Slodowy slice is normal for the minimal degeneration (C3, F4(a3)). In this section we prove it is. We also prove an independently interesting fact: the nilpotent Slodowy slice of C3 atÃ2 contains an irreducible component isomorphic to the nilpotent cone NG 2 in G2.

Singularity
Let e be in the orbitÃ2. Then c(s) is simple of type G2. Let e0 be a regular nilpotent element in c(s). Then y = e + e0 lies in the orbit F4(a2). The decomposition of g in (4.2) is (4, 6), so E has a single element, with (m, n) = (4, 6).
Let O be the orbit C3. Then Oe ⊂ O ⊂ Oy and dim SO,e = dim C(s) · e0. We will show that there exists x ∈ O satisfying the first part of Lemma 2.4 and therefore SO,e contains an irreducible component isomorphic to C(s) · e0, which is the nilcone of c(s). We set e = e0010 + e0001, f = 2f0010 + 2f0001, h = [e, f ], and e0 = e0111 − e0120 + e1000.
The space g(4, 6) is 1-dimensional, spanned by e1220, so v = v1 = e1220. This is also a highest weight vector for the full action of C(s) on g f (−4), which is the 7-dimensional irreducible representation of G2.
We computed in GAP that there is an x ∈ O with x = e + e0 − 1 4 v, which establishes (4.7). Next we show that C(s) · (e0 − 1 4 v) ∼ = C(s) · e0 and relate the choice of v to the discussion in §4.6. Here g h ∼ = so7 ⊕ C and so7 decomposes under c(s) into c(s) ⊕ V (ω2), where the latter is the irreducible 7-dimensional representation of c(s). Now ad f annihilates c(s), while (ad f ) 2 carries the V (ω2) summand onto g f (−4) ∼ = V (ω2). Let X = e0 ∈ c(s) ⊂ so7 ⊂ sl7. Then X 3 ∈ so7. It is nonzero since X is regular in so7.
Hence, (ad f ) 2 (X 3 ) is in the span of v. This argument gives a version of Lemma 4.5 for the closure of the C(s)-orbit (e0, v) in c(s)⊕ V (ω2) ∼ = so7: namely, the closure is exactly the set of elements (X, X 3 ) where X ∈ c(s) is nilpotent. Hence, there is a natural C(s)-equivariant isomorphism of this orbit closure with the closure of C(s) · e0, the nilpotent cone NG 2 in G2.
Thus SO,e contains an irreducible component isomorphic to NG 2 . An element x ′ in the orbit F4(a3) with x ′ ∈ SO,e corresponds to an element in the subregular nilpotent orbit in NG 2 under this identification. Thus S O,x ′ contains a component isomorphic to the simple surface singularity D4. Incorporating the symmetry of A(x ′ ) = S4, we have S O,x ′ ∼ = 4G2.
Remark 7.1. There are two branches of the C3 orbit closure in a neighborhood of e. These two branches are not conjugate under the action of G e , which shows that Proposition 2.1 does not generally hold for degenerations which are not minimal. The second branch of C3 at e splits into three separate branches in a neighborhood of a point in the orbit F4(a3).

Results for E6
8.1. Proving Proposition 3.1. In Table 3 are the cases where Corollary 4.2 holds for e0 in the minimal orbit of E6. Here c(s0) is the semsimple part of a Levi subalgebra and has type A5. The relevant nonzero nilpotent G-orbits are those that have nontrivial intersection with c(s0). They appear in the table for e.
In the first line of Table 4 is the remaining case where Corollary 2.3 holds for a minimal nilpotent element e0 ∈ c(s). This completes the verification of Proposition 3.1 for E6.

8.2.
The remaining cases where Proposition 3.3 holds. There are only two cases that remain to be checked. Both cases can be handled by passing to a subalgebra of type F4, as in Lemma 2.2. The values of (mi, ni) for i ∈ E are listed in Table 4. Boldface is used for those (mi, ni) where the corresponding coefficient bi = 0 in (4.7). is from §5.6.1. The entry for (A4, D4(a1)) is 3C2 since the irreducible components are isomorphic and one of them is isomorphic to C2 from Table 13. Alternatively, it follows from working in the Levi subalgebra D5 ( §2.5). The entry for (D4, D4(a1)) is also clear from working in the Levi subalgebra D4. The degenerations (E6(a3), D5(a1)) and (2A2, A2 + A1) are both A2 from Table 13. Note that 2A2 is unibranch at A2 + A1, but its closure is not normal.  9. Results for E7 9.1. Proving Proposition 3.1. In Table 6 are the cases where Corollary 4.2 holds for e0 in the minimal orbit of E7. Here c(s0) is the semsimple part of a Levi subalgebra and has type D6. The relevant nonzero nilpotent G-orbits are those that have nontrivial intersection with c(s0). They appear in the table for e. In the first several lines of Table 7 are the remaining cases where Corollary 2.3 holds for a minimal nilpotent element e0 ∈ c(s). This completes the verification of Proposition 3.1 for E7.

9.2.
Verifying the remaining cases where Proposition 3.3 holds. The nine remaining cases, involving e from six different G-orbits, are listed in Table 7. As before, the values of (mi, ni) are listed in boldface for those (mi, ni) for i ∈ E where the corresponding coefficient bi = 0 in (4.7). The cases where e is type A2 + 2A1 and 2A2 + A1 follow by restricting to a subalgebra of type E6 and using Lemma 2.2. The case where e is type A5 + A1 proceeds as in Example 4.4.4 or by restricting to a subalgebra s ′ ⊕ c(s ′ ), where s ′ is an sl2-subalgebra through an element of type A ′′ 5 . The two cases where e is type D5(a1) + A1 are similar to Example 4.4.3. The three minimal degenerations lying above the orbit A4 + A2 and the one above the orbit A3 + A2 + A1 require an explicit computer calculation, similar to §7.2.3, whose details are omitted.
The remaining six minimal degenerations are unibranch, but either the larger orbit has non-normal closure or it is not known whether the larger orbit has normal closure. In all cases we are able to determine that the slice is normal and hence fully determine the singularity. The corresponding action of A(e) is determined using §5. The degeneration (D5, E6(a3)) is C3 and (D5(a1), A4 + A1) is A + 2 by restriction to E6, see Table 12. The other four degenerations follow from Table 13. 9.4. Additional calculations in E7.

)
and D5 + A1, A ′′ 5 are both isomorphic to the nilpotent cone of a Lie algebra of type G2, from which the result will follow. The proof parallels §7.4, except that here the singularities are unibranch.
Let e be in the orbit A ′′ 5 . Then c(s) is a simple Lie algebra of type G2. Let e0 be a regular nilpotent element in c(s). Then y = e + e0 lies in the orbit E7(a4) and (m, n) = (4, 6) for the unique element in E. Here, the simple part of g h is so8. Still, we can choose v = (ad f ) 2 (X 3 ) with X = e0 ∈ c(s) ⊂ so7 ⊂ so8 ( §4.6). Using GAP we showed that there an element in the orbit A6 equal to e + e0 + bv for a unique b = 0, and also for an element in the orbit D5 + A1. The rest of the proof in §7.4 applies to give the result. 9.4.2. Sing(D6(a1), E7(a5)) = 3C3. We will show that the degeneration (D6(a1), D4) has one branch which is equivalent to the whole nilpotent cone of sp 6 . (There are two branches of D6(a1) above D4.) Let e be in the orbit D4. Then c(s) ∼ = sp 6 . Let e0 be a regular nilpotent element in c(s). Then y = e + e0 also lies in the orbit E7(a4) and g decomposes in (4.2) as reflecting that c(s) decomposes under s0 as V (10) ⊕ V (6) ⊕ V (2) and g f (−6) decomposes under s0 as V (4) ⊕ V (8), which 14-dimensional and as a representation of c(s) is V (ω2). Also (m, n) = (6, 8) for the unique element in E.
The semisimple part of g h is isomorphic to sl6. If we take X = e0, then X 4 ∈ sl6 is nonzero since X is regular. It cannot be in sp 6 since only odd powers of X are. It satisfies [h0, X 4 ] = 8X 4 and so it must be a highest weight vector in V (8) for s0 with respect to e0. Hence we can choose v = (ad f ) 3 (X 4 ) ( §4.6). We checked using GAP that there is an x in the orbit D6(a1) with Since the elements C(s) · (e0 + bv) consist of pairs (X, X 4 ) ∈ sp 6 ⊕ V (ω2) ∼ = sl6 with X ∈ sp 6 nilpotent, the result follows. It then follows that one branch of (D6(a1), E7(a5)) is isomorphic to C3, hence the singularity is 3C3. 9.4.3. The degeneration (A4 + A1, A3 + A2 + A1). In the Appendix of [FJLSb], we shall prove that the degeneration (A4 + A1, A3 + A2 + A1) has singularity a2/S2, where the action of S2 is given by A → −A t , where A ∈ a2 is a 3 × 3 nilpotent matrix of rank 1.

Results for E8
10.1. Proving Proposition 3.1. In Table 9 are the cases where Corollary 4.2 holds for e0 in the minimal orbit of E7. The centralizer c(s0) is the semisimple part of a Levi subalgebra of type E7. The nonzero nilpotent G-orbits meeting c(s0) are those which appear in the table.
In the first several lines of Table 10 are the remaining case where Corollary 2.3 holds for a minimal nilpotent element e0 ∈ c(s). This completes the verification of Proposition 3.1 for E8.

10.2.
Verifying the remaining cases where Propositions 3.3 and 3.4 hold. The results are listed in the latter lines of Table 10. 10.2.1. The degeneration (A3 + 2A1, 2A2 + 2A1) is m ′ . Here e is in the orbit 2A2 + 2A1 and c(s) ∼ = sp 4 . Let e0 be in the minimal nilpotent orbit of c(s). In this case E has one element with (m, n) = (1, 3).
Consider the Levi subalgebra l of type E6 + A1. Then without loss of generality e ∈ l (with nonzero component on the A1 factor) and e0 ∈ l (contained in the E6 factor). The result in §8.2.2 for E6 shows that there is an x in the orbit O of type A3 +2A1 (in E8) with x = e+e0 +v for a choice of v corresponding to (1, 3). If c(s) ∼ = V (2ω1), then v is a highest weight vector for a c(s)-module V (3ω1). Hence SO,e ∼ = m ′ . 10.2.2. The degenerations (A4 + A1, D4(a1) + A2) and (2A3, D4(a1) + A2) are a + 2 . We can work in the maximal subalgebra of type D8 and imitate §4.4.3 to show that x = e + e0 + bv for a unique b = 0, for an x in either the orbit A4 + A1 or the orbit 2A3. Since (4.7) holds in D8, it holds in g. The results follows from §4.5.
10.2.4. The case where e = A4 + A2 + A1 also needs to be handled by an explicit computation in GAP; we omit the details. The other remaining cases follow by restricting to some subalgebra.
In four of these cases, the orbit closure is known to be non-normal: (E7(a1), E8(b5)), (E7(a3), E6(a1) + A1), (D7(a2), D5 + A1), (D6, D5 + A2). The latter three are unibranched. The orbit closures, and hence the slices, for the other six are expected to be normal. We use (Y ) to denote a singularity with normalization Y . The remaining nine surface degenerations are unibranch, but either the larger orbit has non-normal closure or it is not known whether the larger orbit has normal closure. In these cases we are able to show that the slice is normal and hence fully determine the singularity. The action of A(e) is determined using §5. The degeneration (D5, E6(a3)) is C3 and the degeneration (D5(a1), A4 + A1) is A + 2 , both by restriction to E6 (see Table 12). The other degenerations follow from Table 13.   11. Slices related to entire nilcones The main goal of the paper was to describe the nilpotent Slodowy slice SO,e for a minimal degeneration. Many of the same ideas can be used to show that SO,e has a familiar description when the degeneration is not minimal. In particular, there are many cases where SO,e is isomorphic to the closure of a non-minimal orbit in a nilcone for a subalgebra of g or is isomorphic to a slice between two orbits in such a nilcone. Rather than listing all these cases here, we write down some cases where SO,e, or one of its irreducible components, is isomorphic to an entire nilcone. Some of these were used to show in the surface case that SO,e, or an irreducible component of SO,e, is normal. These examples are relevant for the duality discussed in the §1.9.5, to be explored in future work. 11.1. Exceptional groups. The results are listed in Table 13. The notation NX refers to the nilcone in the Lie algebra of type X. The proofs use Lemma 2.4, usually for x ′ = 0, and often require a computer calculation.
11.2. Slices isomorphic to entire nilcones: two slices in slN . These two examples are special cases of isomorphisms discovered by Henderson [Hen] using Maffei's work on quiver varieties [Maf05].
Here we give direct proofs that fit into the framework of Lemma 2.4 and §4.4. We are grateful to Henderson for bringing these examples to our attention.
11.2.1. First Slice. It is slightly more convenient to work in g = gl nk . Assume n ≥ 2 and k ≥ 1. Consider the nilpotent orbit O ′ with partition [n k ]. Write k = p(n + 1) + q with 0 ≤ q < n + 1, which gives kn = (pn + q − 1)(n + 1) + (n + 1 − q) for maximally dividing kn by n + 1 . Let O be the nilpotent orbit with partition [(n + 1) pn+q−1 , n + 1 − q], which is a partition of kn. Then O ′ ⊂ O by the dominance order for partitions. Moreover, X ∈ O implies X n+1 = 0 and O is maximal for nilpotent orbits in gl nk with this property.
Proposition 11.1. [Hen,Corollary 9.5] Let e ∈ O ′ . The variety SO,e is isomorphic to In particular, SO,e is isomorphic to the closure of the nilpotent orbit in gl k with partition [(n + 1) p , q], which is the whole nilcone when k ≤ n + 1.
Proof. Let Ik be the k × k identity matrix. Define e = (eij), h = (hij), and f = (fij ) to be n × n-block matrices, with blocks of size k × k, as follows: where as before Se = e + g f . Let M = e + Z({Yi}) ∈ Se. Set Y0 = − 1 n Y for a fixed matrix Y for reasons that will become clear shortly. Since M n+1 = 0, we can find constraints on the entries of M n+1 . The (n, 1)-entry of M n+1 is equal to rY1 + sY 2 0 where r is a sum of products of the coefficient in e, hence nonzero. Thus rY1 + sY 2 0 = 0 and Y1 is proportional to Y 2 . Given this fact, the (n, 2)-entry of M n+1 is equal to r ′ Y2 + s ′ Y 3 0 where r ′ is nonzero. Hence Y2 is proportional to Y 3 , and so on. In this way, we conclude that Yi = ciY i+1 for all i = 0, 1, 2, . . . , n − 1, where the ci ∈ C are uniquely determined constants (which depend on n, but not k). Consequently M ∈ SO,e takes the form e + Z({ciY i+1 }) for some Y . We were not able to find a general formula for the ci's, but in all cases that we computed, the ci's were nonzero, which we expect to be true in general. Now let T n + n−1 i=1 biT n−i ∈ C[T ] be the characteristic polynomial for the n × n-matrix e + Z({ciI1}) in the k = 1 case. A direct computation with block matrices then shows that p(T ) := T n + n−1 i=1 biY i T n−i is the characteristic polynomial of M , viewing M as an n×n-matrix over the commutative ring C[Y ], where Y acts by simultaneous multiplication on each of the block entries of M . By the Cayley-Hamilton Theorem over C[Y ], it follows that p(M ) = 0. In fact, p(T ) is the minimal polynomial of M over C[Y ]. Indeed, for 1 ≤ i ≤ n − 1, the i-th block lower diagonal of M i consists of non-zero scalar matrices while everything below that diagonal is zero. Thus M cannot satisfy a polynomial of degree less than n over The next step is to show that Y n+1 must be the zero matrix. Since p(M ) = 0, Since the minimal polynomial of M over C[Y ] has degree n, it follows that (bi − b1bi−1)Y i = 0 for i = 2, . . . , n and b1bnY n+1 = 0. Note that b1 = 1 by taking the trace of M since c0 = − 1 n . Now if Y n+1 = 0, then recursively bi = b i 1 = 1, but also b1bn = bn = 0, a contradiction. Similarly, if Y ℓ = 0 and Y ℓ−1 = 0 for some ℓ ≤ n + 1, then bi = b i 1 = 1 for i = 1, 2, . . . , ℓ − 1. We conclude that all elements in SO,e take the form e + Z({ciY i }) where Y n+1 = 0. Hence SO,e is isomorphic to a subvariety of Y via restriction of the natural projection π : Se → c(s) by the argument in §4.6. Now SO,e and Y both have dimension p 2 n 2 + 2pqn + p 2 n + q 2 − q, and the latter variety is irreducible; hence π gives an isomorphism of SO,e onto Y.
An interesting consequence is the following: since the ci's, and hence the bi's, are independent of k, choosing k > n, we deduce that all bi = 1, an interesting fact in its own right.
Remark 11.2. Fix e0 ∈ c(s) in the orbit [(n + 1) p , q]. In the language of §4.4, the vectors vi = Z({0, . . . , 0, Y i+1 , 0, . . . , 0}) have weight (i, i + 2) under s ⊕ s0, hence lie in E. The proof shows that (4.7) holds for e0 = v0 together with the vi's for i ≥ 1 with the coefficients ci in the proof. As mentioned above, we do not know whether ci = 0 in general, except for small values of i. 11.2.2. Second Slice. Next, let O be the orbit in gl nk with partition [(n + k − 1, (n − 1) k−1 ]. Then again e ∈ O. The elements in O correspond to matrices which are nilpotent and which have rank(M i ) = k(n−i) for i = 1, 2, . . . , n − 1.
Proposition 11.3. [Hen, Corollary 9.3] The variety SO,e is isomorphic to the nilcone in gl k .
Proof. Up to smooth equivalence, this result is a consequence of [KP81], by cancellation of the first n − 1 columns of the partitions for O and O ′ . Here, we show that, in fact, SO,e ∼ = NA k−1 , which also follows from [Hen, Corollary 9.3].
Keep the notation from the proof of the previous proposition. Let M ∈ Se satisfying the rank conditions rank(M i ) = k(n − i) for i = 1, 2, . . . , n − 1. The last rank condition is rank(M n−1 ) = k. The bottom, left 2 × 2-submatrix of M n−1 consists of rY 0 sY 1 tI k rY 0 , with each of r, s, t positive, since the coefficients of e are positive. Multiply the last row by r t Y0 and substract it from the second-to-last row to zero out the (n − 1, 1)-entry. Then since rank(tIk) = k, it follows that for rank(M n−1 ) = k to hold, necessarily the second-to-last row must be identically zero. In particular, the (n − 1, 2)-entry is zero, that is, Y1 is a scalar multiple of Y 2 0 . Continuing in this way for the smaller powers of M , we conclude that Yi = ciY i 0 for some ci ∈ C, as in the previous proposition. Next a direct computation shows that M n+k−1 has entry (n, 1) which is a scalar multiple of Y k 0 and all other entries are scalar multiples of Y m 0 for m > k. If any of these scalar multiples are nonzero, then since M n+k−1 = 0, it follows that Y0 is nilpotent, whence Y k 0 = 0 since Y0 ∈ gl k . These multiples are independent of k. The k = 1 case implies that the entries in M n+k−1 cannot all be zero unless all ci = 0 since e is the only nilpotent element in Se. We have therefore shown that SO,e is contained in a variety isomorphic to the nilcone of gl k . By dimension reasons, this must be an equality as in the previous proof.

11.2.3.
Example. An example of the first proposition is the degeneration [2 3 ] < [3 2 ] and of the second proposition is the degeneration [2 3 ] < [4, 1 2 ], both in sl6. Both slices are isomorphic to the nilcone of sl3. In this setting, the common intermediate orbit [3, 2, 1] corresponds to the minimal nilpotent orbit in sl3. Upon restriction to sp(6), the slice becomes isomorphic to the nilcone in so(3), which is of type A1. This gives another proof of 4.4.3, one which does not require knowing that either [3 2 ] or [4, 1 2 ] have closures which are unibranch at [2 3 ].

The remaining unexpected singularities
The singularity µ and a2/S2 will be discussed in subsequent work. Here we discuss the minimal degenerations of type τ and χ, respectively: These two cases, as well as a2/S2, are of dimension 4. All three carry an action of SL2(C) from C(s) but, unlike the other dimension 4 (or greater) cases, this action cannot be transitive on the smooth part of SO,e.
12.1. Preliminaries. Let (O, O ′ ) be either (2A2 + A1, A2 + 2A1) in type E6 or (A4 + A3, A4 + A2 + A1) in type E8. We tackle these cases by choosing a suitable orbit O ′′ < O ′ : for the first case we choose O ′′ to be the orbit labelled A2, while for the second we choose O ′′ to be A4. Let h, e, f = s be an sl2subalgebra with e ∈ O ′′ . In both cases we can choose a nilpotent element e0 ∈ c(s) such that e + e0 ∈ O ′ and S O ′ ,e = e + C(s) · e0. Then we look for a representative of O of the form x = e + x0 + x1 + . . . + xm with xi ∈ g f (−i) and x0 a nilpotent element of c(s) whose C(s)-orbit lies one step above that of e0 in the closure order. Having found such a representative, the following lemma allows us to relate the singularity of (O, O ′ ) to the singularity of the pair (x0, e0) in c(s).
Lemma 12.1. Let O ′′ ≤ O ′ be nilpotent orbits in g and let h, e, f be an sl2-subalgebra in g with e ∈ O ′′ . Let g f = i≤0 g f (i) be the (ad h)-grading of g f . Let {h0, e0, f0} be an sl2-triple in c(s) such that e + e0 ∈ O ′ .
is a transverse slice to Oe+e 0 at e + e0.
Having found a representative of O of the form e + x0 + x1 + . . . + xm, our approach then consists of the following series of steps: 1. Describe the (closure of the) set of conjugates of x0 which are in e0 + z f 0 . 2. For each such conjugate y0 of x0 found in step 1, find an element z ∈ Z such that Ad z(x0) = y0. 3. With z as in step 2, determine the values of z · x1, z · x2 etc.
In the case of (2A2 + A1, A2 + 2A1) it turns out that we can provide a purely conceptual proof that there exists a representative of O of the desired form. and let Z = C(s), z = c(s). Then z is isomorphic to sl3 ⊕ sl3, with basis of simple roots {α1, α3, α5, α6}. Let l1 be the simple Lie subalgebra of z with basis of simple roots {α1, α3} and let l2 be the subalgebra with basis of simple roots {α5, α6}, so that z = l1 ⊕ l2. Similarly, Z • = L1 × L2 ∼ = SL3 × SL3, where Lie(L1) = l1 and Lie(L2) = l2; on the other hand, Z/Z • is cyclic of order 2, generated by an element y ∈ NG(T ) which satisfies Ad y(l1) = l2 and Ad y(l2) = l1. We recall [LT11,p. 81] that g f (−2) = Cf ⊕ V ⊕ W where V is isomorphic to the tensor product of the natural representation of L1 with the dual of the natural representation of L2, and W ∼ = V * . The only other non-trivial space g f (−i) is g f (−4), which is one-dimensional, spanned by fα whereα is the highest positive root. We have the following highest weight vectors: v1 = f 01210 1 is a highest weight vector in V and w1 = fα 2 +α 4 is a highest weight vector in W . In respect of the Z • -action, we can identify V (resp. W ) with the space of 3 × 3 matrices, on which (g, h) ∈ L1 × L2 acts via: (g, h) · M = gM h −1 (resp. (g, h) · M = hM g −1 ). In what follows we will systematically make this identification: then v1 and w1 are both given by the matrix with 1 in the top right entry, and zero everywhere else.
Lemma 12.2. There exists an element of Se ∩ O of the form e +ẽ1 +ẽ2 + ξv1 + ηw1 where ξη = 0. Thus SO,e is the closure of a single Z • -orbit.
Proof. We verified the lemma by computer and found ξ = η = 3. It is also possible to give a conceptual proof.
We remark that it is straightforward to check that the connected centralizer in Z • ofẽ1 +ẽ2 acts trivially on ξv1 + ηw1. This is a necessary condition for the Z-orbit through f +ẽ1 +ẽ2 + ξv1 + ηw1 to have dimension 12. After scaling, we assume that ξ = η = 1. For later use we note the following easy consequence.
To determine the singularity type of the degeneration (2A2 + A1, A2 + 2A1), we will first consider an element of z of type (A1, A1). Proof. To see that any such element is nilpotent, we simply have to check that the determinant, the trace and the sum of all 2 × 2 minors are zero. Then equality of dimensions (and unibranchness) shows us that all nilpotent elements of the Slodowy slice must have the form Xst for some s, t.
Moreover, gst is of determinant 1 and g −1 Proof. Once again, this is a straightforward matrix calculation. It is easy to check that det gst = 1, that g −1 st is as in the statement of the Lemma and that the columns c1, c2, c3 of gst satisfy Xstc1 = 0, Xstci = ci−1 for i ≥ 2. The result follows.
Proof. By Thm. 12.6 and the preceding lemma, Sing(2A2 + A1, A2 + 2A1) is equivalent to the affine variety with coordinate ring C[st, s 3 , t 3 , uv, u 3 , v 3 , sv, tu, s 2 u, su 2 , t 2 v, tv 2 ]. It is straightforward to see that this is the invariant subring of C[s, t, u, v] for the induced action of Γ.
We note some interesting consequences of the above description: -the closed subset given by setting u = v = 0 has coordinate ring C[s 3 , t 3 , st], that is, it is a simple singularity of type A2. This is clear, since it simply describes the subset of elements of the form f + Xst + f2, that is, it describes a singularity of type A2 in the first component l1 of z. Similarly, setting s = t = 0 gives us a singularity of type A2 in the second component l2.
-the closed subset given by setting s = v, t = u has coordinate ring C[s 3 , t 3 , st, st 2 , s 2 t, s 2 , t 2 ], which is exactly the coordinate ring of the singularity m. This follows from taking fixed points in g under an appropriate outer involution of g. This gives us another proof that the singularity (Ã2 + A1, A2 +Ã1) in F4 is equivalent to m. 12.3. (A4 + A3, A4 + A2 + A1) in E8. Let ∆ = σ, τ : σ 5 = τ 2 = (στ ) 2 = 1 be a dihedral group of order 10, acting on V = C 4 by: τ (u, v) = (v, u) and σ(u, v) = (ζu, ζ −1 v), where ζ = e 2πi 5 and (u, v) ∈ C 2 ⊕ C 2 = C 4 . Denote by p, q (resp. s, t) the coordinate functions on the first (resp. second) copy of C 2 . In particular, C[V ] = C[p, q, s, t]. It is easy to show that the ring of invariants C[V ] ∆ is generated by A = pt + qs, B = −2ps, C = 2qt and the functions Fi = p 5−i q i + s 5−i t i for 0 ≤ i ≤ 5. We note that A 2 + BC = (pt − qs) 2 . Since none of the elements of ∆ act as complex reflections on V , it follows that the singular points of the quotient V /∆ are the ∆-orbits of points with non-trivial centralizer, hence are the images in V /∆ of the points of the form (u, u) (or equivalently, (u, ζ i u)) for u ∈ C 2 . Thus the singular locus is properly contained in the zero set of (A 2 + BC) in V /∆. Let D = A 2 + BC and for 0 ≤ i ≤ 5 let Gi = (p 5−i q i − s 5−i t i )/(pt − qs) ∈ Frac(C[V] ∆ ) = C(V) ∆ . It is easy to see that for 0 ≤ i ≤ 5, DGi ∈ C[V ] ∆ vanishes on the singular locus of V /∆, and that Fi = AGi + BGi+1 for i ≤ 4 (resp. Fi = CGi−1 − AGi for i ≥ 1), whence the Gi satisfy: 2AGi − CGi−1 + BGi+1 = 0 for 1 ≤ i ≤ 4. (These equations are also satisfied by the Fi.) Let Y = Spec(C[A, B, C, G0, . . . , G5]). We claim that Sing(O, O ′ ) is equivalent to Y .
Remark 12.8. a) The singularity Y can be obtained by blowing up V /∆ at its singular locus, as follows. It is not hard to show that the ideal of elements of C[V ] ∆ which vanish at the singular points is generated by D and DG0, . . . , DG5. Thus the blowup of V /∆ can be described as the subset of A 9 × P 6 which is the closure of the set of elements of the form It is an easy calculation (using the identities for the Gi mentioned above) to check that this ring is generated by B, F0, 1/G0 and G1/G0, hence by dimensions is a polynomial ring of rank four. Thus the point of Y corresponding to the maximal ideal (A, B, C, Gi) is the unique singular point of the blow-up of C 4 /∆. This justifies the more succinct description of Sing(O, O ′ ) given in the introduction. d) In general, a blow-up of a symplectic singularity is not a symplectic singularity. In our case, O inherits a symplectic structure from that of g, and so (subject to our claim) Y is a symplectic singularity. More generally, it can be shown that the blow-up (at the singular locus) of the quotient of C 4 by any dihedral group (with C 4 identified with two copies of its reflection representation) is a symplectic singularity.
Remark 12.10. The above argument proves that the degeneration [A4 + A3, A4]E 8 is equivalent to the spectrum of the ring of regular functions on the universal cover of the regular nilpotent orbit in sl5, cf. [Gra92].
Moreover, S ′ e+e 0 ∩ O is equal to the closure of the set of conjugates z ·M0, z ∈ Z such that Ad z(M0) ∈ e0 + z f 0 . We can state this more precisely as follows: Lemma 12.11. The intersection S ′ O,e+e 0 is isomorphic to the closure of the set of all (M, w ′ Proof. This follows from the above discussion.
Concretely, we can determine the intersection S ′ O,e+e 0 in the following way: if M ∈ e0 +z f 0 is nilpotent then generically M is regular and therefore there exists a basis B = {w ′ 1 , w ′ 2 , w ′ 3 , w ′ 4 , w ′ 5 } of C 5 such that M w ′ 1 = 0 and M w ′ i = w ′ i−1 for i ≥ 2. After scaling, we may assume that gB = (w ′ 1 w ′ 2 w ′ 3 w ′ 4 w ′ 5 ) has determinant one. Then the tuple (M, w ′ We may assume that where a, b, c, d, g, h, k, l ∈ C. For the purposes of our calculation, we consider the matrices in e0 + z f 0 of the form: A straightforward computer verification confirms that any such matrix satisfies Tr M 2 = Tr M 3 = 0, and that the conditions Tr M 4 = 0 and Tr M 5 = 0 are expressed in terms of the coordinates a, b, c, d, h, l as: (12.1) dh + bl + 8 3 c 2 = a(9bh − 216a 3 + 72ac), dl = c(9bh − 216a 3 + 48ac) Since every irreducible component of the set of (a, b, c, d, h, l) satisfying these two equations has dimension at least four, it follows that the set of matrices given by the coordinates satisfying (12.1) is equal to the set of nilpotent elements of e0 + z f 0 (and is therefore irreducible). It is easy to verify that the rational functions a = A/6, b = −G0/3, c = −BC/16, d = BG1/4, h = G5/3, l = CG4/4 in C(p, q, s, t) satisfy (12.1). Since A, BC, G0, G5, BG1, CG4 are regular functions on Y , we have therefore constructed a morphism from Y to N (z) ∩ (e0 + z f 0 ), corresponding to the inclusion C[A, BC, G0, G5, BG1, CG4] ⊂ C[Y ]. In fact, this morphism corresponds to quotienting Y by the action of a group of order five, as follows: let ρ be the automorphism of order five of V which sends (p, q, s, t) to (ζp, ζ −1 q, ζs, ζ −1 t). Then ρ normalizes Γ and has an induced action on Y satisfying C[Y ] ρ = C[A, BC, G0, G5, BG1, CG4]. (The invariants B 2 G2 and C 2 G3 are contained in this ring, since BG2 = CG0 − 2AG1 and CG3 = 2AG4 + BG5.) It follows that the coordinates a = A/6 etc. given above define an isomorphism from Y / ρ to N (sl5) ∩ (e0 + z f 0 ).
Remark 12.12. The above discussion indicates an interesting way to view the singularity ([5], [3, 2]) in sl5, as an affine open subset of the blow up of the quotient of C 4 by a group of order 50. Indeed, the group generated by Γ and ρ is isomorphic to the complex reflection group G(5, 1, 2), acting on C 4 = U ⊕ U * where U is the defining representation for G(5, 1, 2). Blowing up the quotient at the set of orbits of points of the form (u, u), and restricting to the affine open subset given by D = 0, one obtains the variety Y / ρ . We will first give an ad hoc justification that S ′ O,e+e 0 is as claimed, and then a more rigorous proof. Fix a matrix M as above with coordinates a = A/6, etc which we think of as depending on the point (A, B, C, G0, . . . , G5) ∈ Y . The space of (column) vectors in W which are annihilated by M is generically of dimension one, spanned by Similarly, the space of (row) vectors in W * which are annihilated by M is also generically of dimension one, spanned by u ′ such that M 4 w ′ 5 = 0, and then to multiply w ′ 5 by an appropriate scalar such that det(w ′ 1 w ′ 2 w ′ 3 w ′ 4 w ′ 5 ) = 1.
For this purpose, we first choose

Graphs
Capital letters are used to denote simple singularities and lower-case letters to denote singularities of closures of minimal nilpotent orbits. The notation m, m ′ , µ, χ, a2/S2 and τ are explained in §1.7.4. The intrinsic symmetry action induced from A(e) is explained in §6 and the notation is explained in §6.2. We use (Y ) to denote a singularity with normalization Y .