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Generic singularities of nilpotent orbit closures

Baohua Fu a, Daniel Juteau b, Paul Levyc, Eric Sommers d'*

a Hua Loo-Keng Key Laboratory of Mathematics and AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun East Road, Beijing, 100190, PR China b LMNO, Universite de Caen Basse-Normandie, CNRS, BP 5186, 14032 Caen Cedex, France

c Department of Mathematics and Statistics Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom

d Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA

a r t i c l e i n f o a b s t r a c t

According to a 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 poset of nilpotent orbits, 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^k-i. 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 that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL2 (C)-variety whose normalization is A2, an Sp4(C)-vari-

E-mail addresses: bhfu@math.ac.cn (B. Fu), daniel.juteau@unicaen.fr (D. Juteau), p.d.levy@lancaster.ac.uk (P. Levy), esommers@math.umass.edu (E. Sommers).

http://dx.doi.Org/10.1016/j.aim.2016.09.010

0001-8708/© 2016 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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Article history:

Received 16 July 2015

Received in revised form 1 August

Accepted 6 September 2016 Available online xxxx Communicated by Roman Bezrukavnikov

Keywords: Nilpotent orbits Symplectic singularities Slodowy slice

Corresponding author.

ety whose normalization is A4, and a two-dimensional variety whose normalization is the simple surface singularity A3. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.

© 2016 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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 [30,46]. 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 [32,33]. 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 G 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 exists and is provided by the affine space Se = e + gf, known as the Slodowy slice. Here, e and f are the nilpotent parts of an s[2-triple and gf is the centralizer of f in g. The local geometry of O at a point e is therefore encoded in So,e = OnSe, 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 G 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 analytic isomorphism.

1.2. Symplectic varieties

Recall from [4] that a symplectic variety is a normal variety W with a holomorphic symplectic form w on its smooth locus such that for any resolution n : Z ^ W, the pull-back extends to a regular 2-form on Z. If this 2-form is symplectic (i.e. if it is non-degenerate everywhere), then n is called a symplectic resolution. By a result of Namikawa [45], 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 [14, Ch. 1.4]) and by work of Hinich [26] and Panyushev [48], the normalization of O has only rational Gorenstein singularities. Hence the normalization of O is a symplectic variety.

Since the normalization of O has rational Gorenstein singularities, the normalization So,e of So,e also has rational Gorenstein singularities. The smooth locus of So,e admits a symplectic form by restriction of the symplectic form on O [22, Corollary 7.2], and this yields a symplectic form on the smooth locus of So,e since So,e is smooth in codimension one. 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. [36]) that a Fano contact manifold is homogeneous. The importance of finding new examples of isolated symplectic singularities was stressed in [4]. It is therefore of interest to determine generic singularities of nilpotent orbits, as a means to find new examples of isolated symplectic singularities. Our study of the isolated symplectic singularity So,e contributes to this program.

1.3. Motivation from 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., [7,28,39,40,55]). The second author of the present paper defined a modular version of Springer's correspondence [29] 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 cohomol-ogy 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 [32] gives a geometric explanation for a similar rule for decomposition matrices of symmetric groups [27,29].

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 ^-algebras (see the survey article [38]).

1.4. Simple surface singularities and their symmetries

1.4.1. Simple surface singularities

Let r be a finite subgroup of SL2(C) = Sp2(C). Then r acts on C2 and the quotient variety C2 /r 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.

Up to conjugacy in SL2(C), such r 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 C2/r. The exceptional fiber (that is, the inverse image of 0) is a union of projective lines which intersect transversally. 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 C2/r. Hence we denote simple surface singularities using the upper-case 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 C2/r (cf. [4, Section 2.1]). An algebraic version of this result is provided by Proposition 5.2. More generally, if r C Sp2n(C) is a finite subgroup which acts freely on C2n\{0}, then the quotient C2n/r is an isolated symplectic singularity.

1.4.2. Symmetries of simple surface singularities

Any automorphism of the simple surface singularity X = C2/r fixes 0 G 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 A of X. Let Aut(A) be the group of graph automorphisms of A. Then Aut(A) = 1 when g is A1, E7, or E8; Aut(A) = 63 when g is D4; and otherwise, Aut(A) = 62.

We now address the question of when the action of Aut(A) on the dual graph comes from an algebraic action on X (cf. [50, III.6]). When X is of type A2k-1 (k > 2), Dk+1 (k > 3), or E6, then Aut(A) comes from an algebraic action on X. In fact, the action is induced from a subgroup r' C SL2(C) containing r as a normal subgroup. More precisely, there exists such a r' with r'/r = Aut(A) and the induced action of r'/r on

the dual graph of X coincides with the action of Aut(A) on A via this isomorphism. Such a r' is unique. The result also holds for any subgroup of Aut(A), which is relevant only for the D4 case.

Slodowy denotes the pair (X, K) consisting of X together with the induced action of K = r'/r on X by

Bk, when X is of type A2k-1 and K = S2, Ck, when X is of type Dk+1 and K = S2, F4, when X is of type E6 and K = S2, G2, when X is of type D4 and K = S3.

The reasons for this notation will become clear shortly. We also refer to corresponding pairs (A, K), where A is the dual graph and K is a subgroup of Aut(A), in the same way. The symmetry of the cyclic group of order 3 when X is of type D4 is not considered.

When X is of type 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(A) = S2, but the action on the dual graph does not lift to an action on X. Instead, there is a cyclic group (ct) of order 4 acting on X, with ct acting by non-trivial involution on A, but ct2 acts non-trivially on X. This cyclic action is induced from a r' C SL2 (C) corresponding to D2k+3. We define the symmetry of X to be the induced action of r' on X and denote it by A+. Only the singularities A+ and A+ will appear in the sequel, and then only when g is of type E7 or Eg.

1.5. The regular nilpotent orbit

1.5.1. 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 [9] and Slodowy [50] 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 [50, IV.8.3] is that when e G O', the slice So,e is algebraically isomorphic to a simple surface singularity. Moreover, as in [9], 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 §1.4.2. 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 §1.4.2. Next we explain an intrinsic realization of the symmetry of So,e when g is not simply-laced.

1.5.2. Intrinsic symmetry action on the slice

Let A be the Dynkin diagram of g and K C Aut(A) be a subgroup. The group Aut(A) is trivial unless g is simply-laced. The action of K on A can be lifted to an action on

g as in [47, Chapter 4.3]: namely, fix a canonical system of generators of g. Then there is a subgroup K C Aut(g), isomorphic to K, which permutes the canonical system of generators, and whose induced action on A coincides with K. Any two choices of systems of generators define conjugate subgroups of Aut(g). The automorphisms in K are called diagram automorphisms of g.

Now given g we can associate a pair (gs, K) where gs is a simple, simply-laced Lie algebra with Dynkin diagram As and K C Aut(gs) is a lifting of some K C Aut(As) and g — (gs)K. If g is already simply-laced, then g = gs and K = 1. If g is not simply-laced, then the pair (As, K) appears in the list in §1.4.2, but according to the type of Lg, where Lg is the Langlands dual Lie algebra of g.

Recall s = sl2(C) is the subalgebra of g generated by e and f. Let C(s) be the centralizer of s in G. Then C(s) acts on So,e for any nilpotent orbit O, fixing the point e. Also the component group A(e) of the centralizer in G of e is isomorphic to the component group of C(s) (see §2.2 for more details). When e is in the subregular nilpotent orbit, Slodowy observed that C(s) is a semidirect product of its connected component C(s)° and a subgroup H = A(e). Moreover H is well-defined up to conjugacy in C(s). This is immediate except when g is of type Bk, since otherwise C(s)° is trivial. Also, A(e) = K. In particular, A(e) is trivial if g is simply-laced since G is adjoint.

Now let ((LA)s, LK) be the pair attached above to Lg. We have A(e) — H = K = LK. Then Slodowy's classification 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 ((LA)s, LK) [50, IV.8.4].

1.6. 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) [32,33].

1.6.1. Minimal singularities

Let Omin be the minimal (non-zero) nilpotent orbit in a simple Lie algebra g. Then Omin has an isolated symplectic singularity at 0 G Omin. Following Kraft and Procesi [33, 14.3], we refer to Omin by the lower case letters for the ambient simple Lie algebra: ak, bk, ck, dk(k > 4), g2, f4, e6, e7, e8. The equivalence classes of these singularities, under smooth equivalence, are called minimal singularities.

1.6.2. 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 transversally in the singular point. This is denoted 2A2k-1. In more detail:

Theorem 1.1. [32,33] Assume O' is a minimal degeneration of O in a simple complex Lie algebra of classical type. Let e G 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-i. The last 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 last three singularities do not occur for sln(C).

1.7. The case of type G2

This case was studied by Levasseur-Smith [37] and Kraft [31]. Let Og denote the a41 orbit and let O6 denote the minimal orbit. Kraft showed that the closure of the subregular orbit has A1 singularity along Og. Levasseur-Smith showed that Og has non-normal locus equal to O6 and that the natural map from the closure of the minimal orbit in so7(C) to Og is the normalization map and is bijective. From these results it follows that the singularity of Og along O6 is non-normal with smooth normalization. We describe this singularity in §4.4.3 and show that its normalization is C2.

1.8. Main results

We now summarize the main results of the paper describing the classification of generic singularities in the exceptional Lie algebras. Here and in the sequel, we may write the degeneration O' of O as (O, O'), that is, with the larger orbit appearing first. In this subsection O' is a minimal degeneration of O and e GO'.

1.8.1. Overview

Most generic singularities are like those in the classical types: the irreducible components are either simple surface singularities or minimal singularities. But some new features occur in the exceptional groups. There is more complicated branching and several singularities occur which did not occur in the classical types. Among the latter are three singularities whose irreducible components are not normal (one of these already occurs in G2 as the singularity of Al1 in the minimal orbit), and three additional singularities of dimension four.

A key observation is that all irreducible components of So,e are mutually isomorphic since the action of C(s) is transitive on irreducible components (§2.4). This result is not true in general when O' is not a minimal degeneration of O.

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 these cases, all in E8 (§10.2), we can only determine the isomorphism type of So,e up to normalization. In the remaining four cases, So,e is determined only up to smooth equivalence (§12). It is possible to use the methods here to 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 calculate the symmetry action on So,e induced from A(e), as Slodowy did when O is the regular nilpotent 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.

1.8.2. Symmetry of a minimal singularity

Let g be a simple, simply-laced Lie algebra with Dynkin diagram A. As in §1.5.2, let K C Aut(g) be a subgroup of diagram automorphisms lifting a subgroup K C Aut(A). We call a pair (Omin, K), consisting of Omin with the action of K, a symmetry of a minimal singularity. We write these pairs as a+, d+ (k > 4), d++ (for the action of the full automorphism group), and e+. As in the surface cases, |K| = 3 in D4 does not arise.

1.8.3. 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. More precisely, there exists a subgroup H C C(s) such that C(s) — C(s)° x H. So in particular H — C(s)/C(s)° = A(e). 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 C(s), cyclic of order 4, with C(s) = C(s)° • H [51, §3.4].

Next, imitating §1.5.2, 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+ and A+ (§1.4.2). 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, [12, pg. 373]). The phenomena observed here for these three orbits is directly related to the fact that A(e) = 62 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.

1.8.4. Additional singularities

In the Lie algebras of exceptional type, there are six varieties, arising as components of slice singularities, which are neither simple surface singularities nor minimal singularities.

The ten cases in type Eg where we know the singularity only up to normalization would give further examples if they turned out to be non-normal.

Non-normal cases.

The variety m. Let V(i) denote the irreducible representation of highest weight i G Z>0 of SL2(C). Consider the linear representation of SL2(C) on V = V(2) 0 V(3). Let v2 G V(2) and v3 G V(3) be highest weight vectors for a Borel subgroup of SL2(C). The variety m is defined to be the closure in V of the SL2(C)-orbit through v = v2 + v3, a two-dimensional variety with an isolated singularity at zero. It is not normal, but has smooth normalization, equal to the affine plane A2. This is an example of an S'-variety (i.e., the closure of the orbit of a sum of highest weight vectors) studied in [57], where these properties are proved (see §3.2.1). The first case where m appears is for the minimal degeneration (A1, 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 (see §1.9.2).

The variety m'. This is a four-dimensional analogue of m, with SL2(C) replaced by Sp4(C). It is an S'-variety [57] with respect to the Sp4(C)-representation on V = V(2w1)® V(3w1) where V(w1) is the defining 4-dimensional representation of Sp4(C), so that V(2w1) is the adjoint representation. Let v2 G V(2w1) and v3 G V(3w1) be highest weight vectors for a Borel subgroup of Sp4(C). The variety m' is defined to be the closure in V of the Sp4(C)-orbit through v = v2 + v3, a four-dimensional variety with an isolated singularity at zero. It is not normal, but has smooth normalization, equal to A4 (see §3.2.1). The singularity m' occurs exactly once, for the minimal degeneration (A3 + 2A1, 2A2 + 2A1) in Eg.

The variety ¡. The coordinate ring of the simple surface singularity A3 is R = C[st, s4, t4], as a hypersurface in C3. We define the variety i by i = Spec R' where R' = C[(st)2, (st)3, s4, t4, s5t, st5]. 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 (D7(a1), Eg(b6)) in Eg is shown to be isomorphic to A3 with an order two symmetry arising from A(e). In [21] we will show that So,e is smoothly equivalent to The closure of O was known to be non-normal, but our result establishes that it is non-normal in codimension two.

Normal cases. These three singularities are each of dimension four and normal.

The degeneration (2A2 + A1, A2 + 2A1) in E6. Let Z = e^P and let r be the cyclic subgroup of Sp4(C) of order three generated by the diagonal matrix with eigenvalues Z, Z, Z-1, Z-1. Then C4/r has an isolated singularity at 0, and we denote this variety by t. We show in §12.2 that So,e is smoothly equivalent to t for the minimal degeneration (2A2 + A1, A2 +2A1) in Ee.

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 [21] 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 A be a dihedral group of order 10, acting on C4 via the sum of the reflection representation and its dual. Then it turns out that the blow-up of C4/A at its singular locus has an isolated singularity at a point lying over 0. We denote this blow-up by We show in §12.3 that So,e is smoothly equivalent to x for the minimal degeneration (A4 + A3, A4 + A2 + A1) in E8.1

1.8.5. Statement of the main theorem

In our main theorem we classify the generic singularities of nilpotent orbit closures in a simple Lie algebra of exceptional type. 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 G 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+,A+,2A1,3A1,3C2,3C3,3C5,4G2,5G2,10G2, or m,

up to normalization for ten cases in 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 ¡1.

(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+ ,a+ ,a+ ,a+, 2a2 , d++ ,e+, 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 t, a2/&2, and x, 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 of the intrinsic symmetry action.

1 In private communication with the authors, Bellamy has pointed out that it can be deduced from his work [5] that the symplectic quotient of c4 by a dihedral group of order 4n + 2 has a unique q-factorial terminalization. Since o is a rigid orbit, the singularity x is also q-factorial terminal (see §5.2). Hence x can be identified with the unique q-factorial terminalization of c4/A.

1.8.6. Brief description of methods

The methods in §4 are relevant for cases when So,e has a dense C(s)-orbit and are motivated by arguments in [32]. For the higher codimension minimal degenerations (except the three which are normal of codimension four) and the codimension two minimal degenerations where the singularity is kA1 or m, we show that So,e has a dense C(s)-orbit. This allows us to show that the irreducible components of So,e are S-varieties for C(s)° (which are permuted transitively by C (s)), and we determine their isomorphism type. Proposition 3.3 contains precise information about the connection between C(s) and So,e in these cases.

The methods in §5 are applicable to 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. Proposition 3.1 summarizes the surface cases.

These two methods, as summarized in Propositions 3.1 and 3.3, are sufficient to handle all the cases in the main theorem except when x, t, or a2/S2 occur. The two cases where x or t occur are dealt with separately in §12. The two cases where i or a2/S2 occur are deferred to subsequent papers.

The determination of the symmetry action is given in §6, and the calculations supporting Propositions 3.1 and 3.3 are given in §7, §8, §9, §10.

1.9. Some consequences

1.9.1. Isolated symplectic singularities coming from nilpotent orbits

Examples of isolated symplectic singularities include Omin and quotient singularities C2n/r, where r C Sp2n(C) is a finite subgroup acting freely on C2n \ {0}. It was established in [4] 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 C2n. It seems very likely that the singularity x 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 mon-odromy action. These examples correspond to minimal degenerations with singularity A+. The orbits O in these cases have closures which admit a generalized Springer resolution. Examples include the even orbits A4 + A2 in E7 and Eg(b6) in Eg. In [58], 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 [3].

1.9.2. Special pieces

For a special nilpotent orbit O, the special piece P(O) containing O is the union of all nilpotent orbits O' CO which are not contained in O1 for any special nilpotent orbit O1 with O1 C O and O1 = O. This is a locally-closed subvariety of O and it is rationally smooth (see [41] and the references therein). To explain rational smoothness geometrically, Lusztig conjectured in [41] that every special piece is a finite quotient of a smooth variety. This conjecture is known for classical types by [34], 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 [21] 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

(hn ® hn)k/Sn+1

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.

1.9.3. Normality of nilpotent orbit closures

By work of Kraft and Procesi [33], together with the remaining cases covered in [53], 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 and in codimension two. In exceptional Lie algebras, the question of which nilpotent orbit closures are normal has not been completely solved in E7 or E8, but in [10, 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 i 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 [6]) and the appearance of the non-normal singularity in the closure of the a41 orbit in G2, which was known to be unibranch (see [31]).

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 §7.2. 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 [1].

1.9.4. Duality

An intriguing result from [32] 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 order-reversing involution on the set of nilpotent orbits in g given by transposition of partitions.

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 Lg, 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 [33]. The duality is already hinted at by Slodowy's result for the regular nilpotent orbit in §1.5.2, which requires passing from g to Lg. In a subsequent article [20] 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 [12, p. 401-407] to refer to nilpotent orbits. For x G g, Ox refers to the orbit AdG(x), also written G • x. For x G G or g we denote by Gx (resp. gx) the centralizer of x in G (resp. g). Similar notation applies to other algebraic groups which arise, including as subgroups of G. For a subalgebra 3 C g, we denote by C(3) its centralizer in G and c(z) its centralizer in g.

Generally, e is a nilpotent element in an sl2(C)-subalgebra s with standard basis e, h, f. If e0 G c(s) is a nilpotent element, we use s0 for an sl2(C)-subalgebra in c(s) with standard basis e0, h0, f0. Usually O' is a nilpotent orbit in O with O' = O and e G O'. We write (O, O') for such a pair of nilpotent orbits. Often, but not always, O' is a minimal degeneration of O. The nilpotent Slodowy slice On (e + gf) is denoted So,e.

The field of fractions of an integral domain A will be denoted Frac(A). The symmetric group on n letters is denoted Sn. Where we refer to explicit elements of g, we use the structure constants in GAP [56].

2. Transverse slices

2.1. Smooth equivalence

To study singularities it is useful to introduce the notion of smooth equivalence. Given two varieties X and Y and two points x G X and y G 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 G Z and morphisms

y : Z ^ X and ^ : Z ^ Y

which are smooth at z and such that y>(z) = x and ^(z) = y (see [25, 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 [32, §2.1], two singularities (X, x) and (Y, y) with dim Y = dim X + r are equivalent if and only if (X x Ar, (x, 0)) is locally analytically isomorphic to (Y, y).

Let O' and O be nilpotent orbits in g with O' C O. Let e G O'. The local geometry of O at e is captured by 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.

2.2. Transverse slices

The main tool in studying Sing(O, O') is the transverse slice. Both [50, III.5.1] and [33, §12] are references for what follows.

Let X be a variety on which G acts, and let x G X. A transverse slice in X to G • x at x is a locally closed subvariety S of X with x G S such that the morphism

G x 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 C X containing x by taking the intersection (x + u) H Y [50, III.5.1, Lemma 2]. In such a case codimy(G • x) = dimx((x + u) H 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 G O'. Then there exists h, f G g so that {e, h, f} C g is an sl2-triple. Then by the representation theory of sl2(C), we have [e, g] © gf = g. The affine space

Se = e + gf

is a transverse slice of g at e, called the Slodowy slice. The variety

So,e := Se nO

is then a transverse slice of O to O' at the point e, which we call a nilpotent Slodowy slice. In this setting

codimo (O') = dim So,e. (2.1)

Since any two sl2-triples for e are conjugate by an element of Ge, 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 G 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 Ge and C(s) acts on S©,e, fixing e.

The second group which acts is C*. Since [h, f] = —2f, ad h preserves the subspace gf and by sl2-theory all of its eigenvalues are nonpositive integers. Set

gf (i) = {x G gf : [h,x] = ix}

for i < 0. The special case gf(0) is simply c(s), the centralizer of s in g, which coincides with Lie(C (s)).

Define ( : SL2(C) ^ G such that the image of df is equal to s, with df (0 0) = e and df (J -J = h. Set Y(t) = f 0) for t G C*. On the one hand, Adj(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 gf (—i) and t G C*,

Ad Y(t)(e + X0 + ... + xm) = t-2e + X0 + txj + ... + tmxm.

Composing this action with the scalar action of t2 on g, gives an action of t G C* on e + gf by

t ■ (e + x0 + x1 + ...) = e + t2x0 + t3x1 + ..., (2.2)

which preserves Soe = On 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 gf (i). Thus C(s) x C* acts on So .e.

2.4. Branching and component group action

The C(s) x 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 n : (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 at e. 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 [6,49], as discussed in [6, 5(E)-(F)].

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) := Ge/(Ge)° of Ge via C (s) ^ Ge ^ Ge/(Ge)°. Since any two sl2-triples containing e are conjugate by an element of (Ge)°, this gives a well-defined action of A(e) on the set of irreducible components of So, e. Moreover, as noted in [6], 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 [6,49] 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 [33]. In §6.2 we will discuss the full symmetry action on So ,e induced from A(e).

3. Statement of the key propositions

In this section we state the key propositions which underlie Theorem 1.2. The propositions give more precise information about So ,e. Throughout this section O' is always a minimal degeneration of O.

3.1. Surface cases

The case of a minimal degeneration of codimension two is summarized by the following proposition.

Proposition 3.1. Let O' be a minimal degeneration of O of codimension 2. Then there exists a finite subgroup r C SL2(C) such that the normalization So , e of So , e is isomorphic to a disjoint union of k copies of X where X = C2/r.

This is proved in §5 where techniques for determining r and k are given. For most cases in Proposition 3.1 we can show that the irreducible components of So,e are normal either by knowing that O is normal, by using Lemma 4.1 to move to a smaller subalgebra where the slice is known to be normal, or by doing an explicit computation using Lemma 4.3. In the surface case, we found only two ways that an irreducible component of So,e fails to be normal:

• When r =1, we show below that So,e — m (§1.8.4). This happens for several different minimal degenerations.

• When (O, O') = (D7(a1), Eg(b6)), we have r = Z/4, but So,e is not normal. Instead, So,e is smoothly equivalent to i (§1.8.4).

A handful of cases in Eg are determined only up to normalization (see §10.2).

Remark 3.2. The isomorphism in Proposition 3.1 is compatible with the natural C*-actions on both sides. On So,e the C*-action is the one induced from §2.3; on C2/r it is the one coming from the central torus in GL2(C). This follows from Proposition 5.2.

3.2. Cases with a dense C(s)-orbit

Next we consider all the cases of codimension 4 or greater other than the three normal cases in §1.8.4. Together with the surface cases where |r| = 1 or 2, these So,e have a dense orbit for the action of C (s). More is true, their irreducible components are examples of S-varieties [57].

3.2.1. S-varieties

Let |A1, ... Ar} be a set of dominant weights for a maximal torus in a fixed Borel subgroup of C (s)°. It will also be convenient to view the Aj's as weights for the Lie algebra of this maximal torus. Let V(Aj) be the irreducible representation of C(s)° of highest weight Aj and vi G V(Aj) a non-zero highest weight vector. Then the S-variety A(A1, ... Ar) is defined to be the closure in V := V(A1) 0 • • • © V(Ar) of the C(s)°-orbit through v := (v1, ..., vr). In [57] it is shown that S-varieties are exactly the varieties which carry a dense C (s)°-orbit and every point in this dense orbit has stabilizer containing a maximal unipotent subgroup of C (s)°.

In all the situations encountered in this paper, we find Ai = biX for each i, where bi G N and A is a fixed dominant weight. In such cases Theorems 6 (and its Corollary), 8 and 10 from [57] reduce to the following, respectively: (1) the complement of C(s)° • v in A is the origin in V; (2) the determining invariant of the C(s)°-isomorphism type of A := A(b1A, ..., br A) is the monoid in N generated by b1, ..., br; and (3) the normalization of A is A(dA), where d is the greatest common divisor of b1, ..., br. More is true in (2): if b1, ..., bj generate the same monoid as b1, ..., br for j < r, then the natural projection from A to A (b1A,..., bj A) is an isomorphism. We will assume that b1 < b2 < • •• <

br. If V factors through Z C C(s)° with Lie algebra sl2(C), then we sometimes write X(b1, b2, ... ) instead of X(b1 A, b2A,...) where A is the fundamental weight for sl2(C).

3.2.2. Let no : So ,e ^ c(s) be the restriction of the C(s) x C*-equivariant linear projection of Se onto c(s) = gf (0). Let n0 j : So, e ^ gf (0) © gf (—1) be the restriction of the C(s) x C*-equivariant linear projection of Se onto gf (0) © gf ( —1). Recall v G c(s) belongs to a minimal nilpotent C(s)°-orbit if and only if v is a highest weight vector (relative to a Borel subgroup of C (s)) in a simple summand of c(s). The proof of the next proposition is given in §4.5.

Proposition 3.3. Let O' be a minimal degeneration of O of codimension at least four (other than the three normal cases in §1.8.4) or of codimension two with |r| = 1 or 2.

Then there exists J = {¿1, .. ., ir} C N so that for each i G J there exists a highest weight vector xi G gf (—i) for the action of C(s)°, and there exists xo G c(s) minimal nilpotent, such that

• e + xo + J2ieJ xi GSo,e,

• if the weight of xo is given by A, then the weight of xi equals (2 + 1)A,

• one of the irreducible components of So,e is e + X, where X is the corresponding S-variety

X := X (A, (f + 1)A, (f + 1)A,..., (f + 1)A) C gf

for C(s)°, through the vector v := xo + ij xi,

• the irreducible components of So,e are permuted transitively by C(s), whence So,e = C(s) • (e + v).

Moreover, there are two cases:

(1) All i G J are even. Then no induces an isomorphism of each irreducible component of So,e with X(A). Furthermore, X(A) is a minimal singularity, being isomorphic to the closure of the C(s)°-orbit through the minimal nilpotent element xo G V(A) C c(s). Hence each component is a normal variety.

(2) We have 1 G J. This case occurs only if c(s) contains a simple factor 3 of type s^(C) or sp4(C) and 3 = V(A). Note that A = 2w where V(w) is the defining representation of 3. Then no,1 gives an isomorphism So,e = X(A, |A) = X(2w, 3w). In the 3 = s^(C) case, So,e = m and in the 3 = sp4(C) case, So,e = m'. In both cases So,e is irreducible and non-normal.

Remark 3.4. In case (1) of Proposition 3.3, when the codimension is at least four, we find that J = 0 except for the two minimal degenerations ending in the orbit D4(a1) + A2 in E8, where J = {2} (see §10.1.2). On the other hand, when the codimension is two in

case (1), then So,e = kA1. If k > 1, then always J = 0. If k =1, then J can be either 0, {2}, or {2, 4}.

In case (2) of Proposition 3.3, the possibilities for J that arise are {1}, {1, 2}, or {1, 2, 3}; however, the singularity m' only occurs for (A3 + 2A1, 2A2 +2A1) in Eg, where

J = {1}.

Remark 3.5. From the previous remark, we see that So,e is not irreducible only when J = 0. In that case, each irreducible component of So,e corresponds to the minimal orbit closure in a unique simple summand of c(s). The direct sum of these summands is an irreducible representation for C(s) and the summands are permuted transitively by C(s)/C(s)° as expected from Proposition 2.1. The first non-irreducible example covered by the Proposition is (C3(a1), B2), see Table 1.

Remark 3.6. In each case of Proposition 3.3, the map is surjective onto the closure of a minimal nilpotent C(s)-orbit in c(s) (namely, the one through x0). It is not true, however, that every such minimal nilpotent orbit will arise in this way. For example, when e G O' = 3A1 in E6, the centralizer c(s) has type A2 + A1. If e0 belongs to the minimal orbit in the summand of type A2 and e'0 belongs to the minimal orbit in the summand of type A1, then O' C Oe+e' C Oe+eo and O' is a minimal degeneration of Oe+e' and of no other orbit. So the minimal orbit in the A2 summand is not in the image of for any minimal degeneration ending in O'.

4. Tools for establishing Proposition 3.3

In this section we give some tools for identifying So,e, which can often be applied even when the degeneration is not minimal. Therefore we do not in general assume degenerations are minimal in this section. At the end of the section in §4.5, we prove Proposition 3.3. We keep the notation that O and O' are nilpotent G-orbits in g with O' C O and e G O'. Unless specified otherwise, O' is not assumed to be a minimal degeneration of O.

4.1. Some reduction lemmas

The first two lemmas give frameworks to relate So,e to a variety attached to a proper subalgebra of g. Both lemmas are variants of [33, Cor 13.3].

4.1.1. Passing to a reductive subalgebra

Let M C G be a closed reductive subgroup and m = Lie(M). Assume that e G m nO'. Let x G m nO and suppose that M • e C M • x. Since m is reductive, we may assume the sl2-subalgebra s containing e lies in m. Let SM•x,e be the nilpotent Slodowy slice

M • x n (e + m^) in m. Of course, SM•x,e C So,e.

Lemma 4.1. Suppose that codim^^(M • e) = codim^-(O') and Sm•x,e is equidimensional. Then Smis a union of irreducible components of So,e. Moreover if O is unibranch at e or if the number of branches of O at e equals the number of branches of M • x at e, then Sm•xe = So, e.

Proof. The first conclusion follows from (2.1) and the fact that SM x , e C So ,e, together with the hypotheses of the lemma. The second conclusion follows from the fact that the irreducible components of So,e and SMXee are unibranch (§2.4). □

Example 4.2. Let g be of type E8, m a Levi subalgebra of g of type E6, and (O, O') of type (D5, Ee(a3)). Since On m is known to be normal in E6, we are able to conclude (see §8.2) that SMis geometrically a simple surface singularity of type D4. Since O is unibranch at e and the codimension hypothesis of the lemma holds (both sides equal two), it follows that So,e = SM-x,e and so So,e has the same singularity. Here, O is conjectured to be normal, but this is still an open question.

Lemma 4.1 is needed for the cases in Table 12, but we also use it to check results that can be obtained by other methods.

4.1.2. The case of a C(s)-orbit of maximal possible dimension Every x G So,e can be written as

x = e + xo + xi (4.1)

with xi G g^ (—i). Set

x+ = ^^ xi and X = C(s) • x+.

Since C(s) fixes e, we have C(s) • x = e + C(s) • x+ and thus C(s) • x = e + X. Also, e + X = X as C(s)-varieties. By construction X is equidimensional, with irreducible components permuted transitively by C(s)/C(s)°. Of course C(s) • x C So,e. Hence the same argument as in the previous lemma gives

Lemma 4.3. Let x G So,e be written as in (4.1). Suppose that dim(C(s)-x) = codimg-(O'). Then e + X is a union of irreducible components of So,e. Moreover, if the number of branches of O at e equals the number of irreducible components of X, then e + X = So,e.

The previous lemma allows us to study So,e by studying C(s) • x+, which is often easier to understand. Of course any x G So,e satisfying the dimension hypothesis in the lemma must lie in So,e n O. Furthermore we have

Lemma 4.4. Let x G So ,e be written as in (4.1). Then dim(C(s) • x) = codim^-(O') if and only if

x0 is nilpotent and dim(C(s) • x0) = codim^-(O'). (4.2)

Proof. We always have dim(C (s) • x0) < dim(C (s) • x) < dim So, e = codim—(O'), so the reverse direction is straightforward and does not use the hypothesis that x0 is nilpotent.

For the forward direction, we are given that dim(C(s) • x) = dim So ,e. Consider the C*-action (§2.3) on So, e. We have C(s) • x C (C(s) x C*) • x C So ,e and all have the same dimension, so C(s) • x is dense in (C(s) x C*) • x. Therefore for A G C*, we have A • (C(s) • x) meets C(s) • x, from which it follows that A • x G C(s) • x. So in fact C* preserves C(s) • x. But if C* • x C C(s) • x, then C* • x0 C C(s) • x0. The C*-action on c(s) is contracting, hence 0 G C(s) • x0, and x0 must be nilpotent in c(s).

Next by [22, Corollary 7.2], So ,e nO is a symplectic subvariety of O, so the sym-plectic form on Tx(O) remains non-degenerate on restriction to Tx(So, e nO). As usual, we identify Tx(O) with [x, g] and the symplectic form on Tx(O) is then expressed as ([x, u], [x, v]} := k(x, [u, v]) for u, v G g, where k is the Killing form on g. But since C(s) • x has dimension equal to So ,e n O, we can identify Tx(So ,e n O) with [x, c(s)].

Now suppose u G c(s) satisfies [x0, u] = 0. Then for any v G c(s), we have

([x,u], [x,v]} = k(x, [u,v]) = k(x0, [u,v])

since [u, v] G c(s) = gf (0) pairs nontrivially only with elements in the 0-eigenspace of adh. But then ([x, u], [x, v]} = k(x0, [u, v]) = k([x0, u], v) = 0. Hence [x, u] is in the kernel of the symplectic form and thus [x, u] = 0 by non-degeneracy of the form. This shows that c(s)x0 C c(s)x, which forces dim(C(s) • x) = dim(C(s) • x0), as desired. □

The case where xi = 0 for all i > 1 in (4.1), and (4.2) holds, is particularly common and is also easier to handle. In that case, x = e + x0 is a sum of two commuting nilpotent elements and A = C(s) • x0 is the closure of a nilpotent orbit in c(s), which is a union of irreducible components of So, e. We discuss this case in detail in §4.3. We next prove a lemma useful for when some xi is nonzero with i > 1.

4.1.3. Assume x G So ,e n O, written as in (4.1), satisfies (4.2). The next lemma uses the C*-action on So ,e to say more about the xi's which appear in (4.1). Since x0 is nilpotent, we can find an sl2-subalgebra sx0 in c(s) containing x0, with standard semisimple basis element hx0 .

Lemma 4.5. Let x GO be written as in (4.1) so that (4.2) holds. Then the following are true.

(1) [x0, xi] =0 for i > 0.

(2) [hxo, xi] = (i + 2)xi for i > 0.

(3) If xo lies in a minimal nilpotent C(s)°-orbit of c(s), then each nonzero xi is a highest weight vector for a Borel subgroup B of C(s)°. In particular, X is a union of S-varieties. If xo has weight A relative to a maximal torus of B, then xi has weight

(2 + 1)A.

Proof. (1) By Lemma 4.4 we know that dim(C(s)-xo) = dim(C(s)-x), which is equivalent to c(s)x0 = c(s)x, which is equivalent to c(s)x0 C c(s)xi for i > 0. Since xo commutes with itself, we get [xo, xi] = 0 for i > 0.

(2) From the proof of Lemma 4.4, the dimension condition in (4.2) implies that the C*-action on So,e preserves C(s) • x. Let C := C(s) x C*. The equality dim(C(s) • xo) = dim(C(s)• x) therefore implies dim(C-xo) = dim(Cx), which means we have the inclusion of identity components of centralizers

(Cx0)° C (Cxi)° for all i. (4.3)

Now write x(t) for the element (1, t) G C(s) x C*. Let 0 : C* ^ C(s) x C* be the cocharacter coming from hx0. Of course 0(C*) C C(s) commutes with x(C*). Consider now the action of the element x(t-1)0(t) on xo for t G C*. We have 0(t).xo = t2xo since [hx0, xo] = 2 and x(t).xo = to+2xo since xo G gf (0), so the one-dimensional torus {x(t-1)0(t) | t G C*} fixes xo and hence by (4.3), it also fixes each xi. The result follows from 0(t).xi = x(t).xi = ti+2xi since xi G gf(— i). Combining with part (1), we see that each nonzero xi is a highest weight vector for the Borel subalgebra of sx0 spanned by xo and hx0 .

(3) Since xo lies in a minimal nilpotent C(s)-orbit of c(s), the stabilizer of the line through xo is a parabolic subgroup P of C(s), containing 0(C*). Let B be a Borel subgroup of P containing 0(C*) and let T be a maximal torus in B with 0(C*) C T. Let U be the unipotent radical of B, which acts trivially on xo. Now xo is a root vector relative to T, so TX0 is codimension one in T. Thus T is generated by TX0 and 0(C*), so each element of B can be written as uto0(t) for u G U, to G Tx0, and t G C*. It follows that the connected subgroup {uto0(t)x(t-1)} of C(s) x C* centralizes xo and hence centralizes each xi by (4.3). Hence B (and indeed also P) preserves the line through xi since x(C*) does; in other words, xi is a highest weight vector relative to (B, T). Moreover, the weight of xi must equal rA, with r rational, since xi and xo are both acted upon trivially by TX0. On the one hand, 0(t).xi = ti+2xi from part (2), and on the other hand, 0(t).xi = (rA)(0(t))xi = (A(0(t)))rxi = (t2)rxi. We conclude that r = (2 + 1). □

Another way to phrase part (2) of Lemma 4.5 is that x must be in the 2-eigenspace for ad(h + hx0). This fact can be used to help locate an x written as in (4.1) so that (4.2) holds, see Lemma 4.10.

Part (3) of Lemma 4.5 will be used in the proof of Proposition 3.3 in §4.5, since for each minimal degeneration covered by the proposition, it turns out that there exists x G So,e satisfying (4.2) with xo in a minimal nilpotent orbit of c(s).

4.1.4. When the degeneration is codimension two and c(s) has a simple summand isomorphic to sl2(C), the next lemma gives a criterion which guarantees that So ,e has a dense C (s)-orbit, allowing us to apply the previous lemmas.

Lemma 4.6. Let O' be a degeneration of O of codimension two (necessarily minimal). Suppose C(s)° contains a simple factor Z with Lie algebra 3 — s^(C). Let C(3) be the centralizer of 3 in G, with Lie algebra c(z).

(1) If Z acts non-trivially on So ,e, then there exists x G So ,e nO, written as in (4.1), satisfying (4.2) with x0 G 3.

(2) If x G c(z), or if x G c(z) but e G C(3) • x, then Z acts non-trivially on So ,e.

Proof. (1) Consider the decomposition of gf under Z. Since the action of Z is non-trivial on So ,e, there exists x = e + x+ G So, e and a non-trivial irreducible Z-submodule V of some gf (— j) so that x+ has nonzero image xV under the C(s) xC*-equivariant projection of gf onto V. The equivariance of ensures that the image (So, e) is conical for the scalar action on V. Let Y be the projectivization of (So ,e), which is in P(V). Since dim So ,e = 2 by hypothesis, we know dim Y < 1. Now Y has a closed Z-orbit, necessarily irreducible. If Y contains a closed orbit which is a point, then Z preserves a line in V, contradicting that V is non-trivial irreducible. Thus, since dimY < 1, each irreducible component of Y must be a one-dimensional (closed) Z-orbit. The stabilizer of any point in Y is therefore a proper parabolic subgroup in Z, namely a Borel subgroup B. Since then dim(Z • xV) = 2 = dim So, e, we have also dim(Z • x) = dim So, e. Then Lemma 4.4 ensures that x satisfies (4.2) with x0 G c(s), when x is expressed as in (4.1). But then for dimension reasons the other connected simple factors of C(s) must preserve Z • x0, hence act trivially, which implies that x0 G 3.

(2) If Z acts trivially on So,e, then So,e C c(3). Hence x G c(3). Since s C c(3), it follows that the C*-action on So ,e preserves nilpotent C(3)-orbits. Hence e G C(3) • x. (This part did not use the assumption on the codimension of O' in O.)

Corollary 4.7. Let Z act non-trivially on So,e. Then there exist bi G N satisfying 2 < b2 < b33 < ... so that each irreducible component of So ,e is an S-variety for Z of the form A (2, b2, b3, ...).

Consequently, the components of So e are normal if and only if all the bi are even, in which case each component is isomorphic to the nilcone in s^(C). In the non-normal case, the normalization of an irreducible component is A2.

Proof. By part (1) of the lemma, there exists x G So ,e n O, when written as in (4.1), satisfying (4.2) with x0 G 3. Hence x0 must belong to the minimal nilpotent orbit in 3. By Lemma 4.5 the xi s are highest weight vectors, of weight i + 2, for a Borel subalgebra of 3. Thus the irreducible component Z • x of So, e is an S-variety for Z of the form A(2, b2, b3,...) with 2 <b2 < b3 < .... But since the degeneration is minimal, C(s) acts

transitively on the irreducible components of So,e by Proposition 2.1, and thus each irreducible component of So,e takes this same form.

By §3.2.1, X(2, b2, b3,...) is normal if and only if the 6i's are all even, in which case it is isomorphic to X (2), the nilcone in sl2(C), which is the ^-singularity. Otherwise, its normalization is X (1) = A2. □

4.1.5. An example in C3

Let g = sp6(C). Nilpotent orbits in g can be parametrized by the Jordan partition for any element in the orbit, viewed as a 6 x 6-matrix. Pick e G g with partition [23]. Then c(s) = sl2(C). Set 3 = c(s). A nonzero nilpotent eo G 3 has type [32] in g, so c(z) = s and thus the only non-zero nilpotent G-orbit that meets c(z) is the one through e. Consequently, part (2) of Lemma 4.6 applies to any O of which O' = Oe is a degeneration of codimension two. Let O[412] and O[32] be the nilpotent orbits with given partition type. Then O' is a minimal degeneration of both orbits, in each case of codimension two. So in both cases Corollary 4.7 applies. Now C(s) acts on gf with gf (0) = c(s) = V(2) and gf (-2) = V (4) ® V(0). Also e + eo belongs to O[4,2] (see §4.2), so we can say that for both O = O[412] and O = O[32], each irreducible component of So,e is of the form X(2, 4) = X(2), which is the nilcone in sl2(C). But since O is normal for both orbits O by [33], it follows that So,e is irreducible. In particular the singularity of O at e is an ^-singularity in both cases, as was already known from [33].

4.2. Locating nilpotent elements in c(s)

In order to make use of Lemma 4.4 or part (2) of Lemma 4.6, we will need to describe nilpotent elements in c(s) relative to the embedding of c(s) in g. We will also need to be able to start with nilpotent eo G c(s) and then compute the G-orbit to which e + eo belongs.

First, if eo G c(s), then eo centralizes the semisimple element h G s. Hence eo G gh, which is a Levi subalgebra of g. Assume h lies in a chosen Cartan subalgebra h C g and is dominant for a chosen Borel subalgebra b C g containing h. The type of the Levi subalgebra gh can then be read off from the weighted Dynkin diagram for h: the Dynkin diagram for the semisimple part of gh corresponds to the zeros of the diagram. Therefore in order to locate a nilpotent element in c(s), we first choose a nilpotent element eo G gh; the Gh-orbits of such elements are known by Dynkin's and Bala and Carter's results [12]. In particular we can compute the semisimple element ho G gh n h of an sl2-subalgebra so through eo in gh.

Next, we compute h + ho and see whether it corresponds to a nilpotent orbit in g: for if e and eo (or some conjugate of eo under Gh) commute, then h + ho will be the semisimple element in an sl2-subalgebra through the nilpotent element e + eo. Together with knowledge of the Cartan-Killing type of the reductive Lie algebra c(s) C gh (see [12]), this search usually suffices to locate the nilpotent orbit through eo in g for nilpotent elements eo G c(s) and the resulting nilpotent orbit through e + eo. In particular we

carried out this approach for all the minimal nilpotent C (s)-orbits in c(s). Two special situations are worth mentioning.

4.2.1. One special situation is when e0 is minimal in g, that is, of type A1. Then the semisimple part of c(s0) is the semisimple part of a Levi subalgebra of g, the one corresponding to the nodes in the Dynkin diagram which are not adjacent to the affine node in the extended Dynkin diagram. Of course e G c(s0). Consequently it is easy to locate all e which have e0 G c(s) when e0 is of type A1 in g.

We will see in Corollary 4.9 that Lemma 4.3 always applies in this setting with x = e + e0. Moreover the type of x in g is easy to determine: if we know the type of e in c(s0), call it A, then x has generalized Bala-Carter type A + A1. Then the usual type can be looked up in [51] or in Dynkin's seminal paper [16].

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 E8(a3).

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 gh has type A1, the semisimple element h0 G h is equal to the coroot of a long root 0 for gh. Therefore, a(h0) > —2 for any root of g and equality holds if and only a = —0. Now choose h0 dominant in gh (relative to b ngh). Then a(h0) > —1 for all simple roots a of g since —0 is a negative root. Moreover a(h0) = —1 only if a is not a simple root for gh. In that case a(h) > 1 since the simple roots of gh correspond to the zeros of the weighted Dynkin diagram for h. This shows that a(h + h0) > 0 for all simple roots a 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) in E8, which has weighted Dynkin diagram

2 0 10 10 2 0.

Then gh has type 4A1 and c(s) is isomorphic to sl2(C) since c(s) has rank one (because e is distinguished in a Levi subalgebra of rank 7) and c(s) contains e0, a nonzero nilpotent element. We want to know in which summand of gh the element e0 lies and what is e + e0. The diagram for h0 relative to g, and dominant for gh, is either:

0 0 0 0 -1 2 -1 0 0 -1 2 -1 0 0 -1 2 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 2 .

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 gh.

4.2.2. 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 ai be the simple root of g which is not a simple root of l. For nonzero eo G c(s), the corresponding ho centralizes l and hence lies in the one-dimensional subalgebra of h spanned by the coweight w/ for ai. Since the values in any weighted Dynkin diagram are 0, 1, or 2, if ho is dominant, then ho must be either w/ or 2w/.

For example, let e be of type A7 in E8, which has weighted Dynkin diagram 1 o o o 1 1 o. Then c(s) has type A1 and the weighted Dynkin diagram of a nonzero ho G c(s) must either be

Both of these are actual weighted Dynkin diagrams in E8, the first is 4A\ and the second is D4(a1) + A2. Only the orbit 4Ai meets gh (which has semisimple type exactly 4Ai). Therefore a nonzero nilpotent element e0 € c(s) C gh has type 4A1 in g.

4-3. The case vihere xi = 0 for i > 1 in (4.1), and (4.2) holds

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 + eo. If it does, then certainly x := e + eo satisfies (4.1) with xo = eo and xi = 0 for i > 1, and the dimension condition in (4.2) just becomes (4.4). By Lemmas 4.3 and 4.4, the union of some of the irreducible components of So,e is thus isomorphic to C(s) • eo. Next we give a condition for (4.4) to hold for the orbit O = Oe+e0 and show that this condition is always true when eo belongs to the minimal orbit in g.

As before, let so be an sl2-subalgebra in c(s) with standard basis eo, ho, fo. Clearly, s and so commute. We will now establish an equivalent condition to the dimension condition (4.4) in terms of the decomposition of g into irreducible subrepresentations for s © so = sl2(C) © sl2(C).

Let Vm,n denote an irreducible representation of s © so with h G s acting by m and ho G so acting by n on a highest weight vector u G Vm,n annihilated by both e and eo. The eigenvalues of h + ho 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

0 0 0 0 0 0 0 i

0 0 0 0 0 0 0 2

dim C(s) • e0 = codim—(Oe)

min(rn, 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.

g = 0 V^nm (4.5)

be a decomposition into irreducible subrepresentations Vmi,ni = Vmini for the action of s ©s0. The relationship between (4.4) and this decomposition in (4.5) is the following:

Proposition 4.8. Let O be the orbit through e + e0. The dimension condition (4.4) holds if and only if

mi > ni whenever mi > 0. (4.6)

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

dimgx = (min(mi,ni) + 1).

At the same time

dim ge = ^(ni + 1)

since the kernel of ad(e) on V,m},ni is isomorphic to V(ni). Here, V(ni) 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

y^ (ni — min(mi,ni)).

It is also necessary to compute dim c(s)e°. Since s0 C c(s) and c(s) is exactly kerad en kerad h, it follows that c(s) coincides with the sum of all Vmil,ni where mi = 0. The centralizer c(s)e° is then the span of the highest weight vectors of these V0(iJ). and hence its dimension is given by the number of these subrepresentations. That is,

dimc(s)e° = #{1 < i < N | mi = 0}.

dimC(s) • e0 = dimc(s) — dimc(s)e° = ^^ (ni + 1) — ^^ 1 = ^^ ni.

mi=0 mi =0 mi=0

The equality of dim C(s) • eo 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 above proof that if J = {i | ni > mi > 0}, then

dimSo,e — dimC(s) • eo = ^^J(ni — mi). (4.7)

The element eo G g is called height 2 if all the eigenvalues of ad ho on g are at most 2, and e is called even if all the eigenvalues of ad h on g are even.

Corollary 4.9. Suppose that either (1) eo belongs to the minimal nilpotent orbit in g, or (2) eo is of height 2 in g and e is even. Then the dimension condition (4.4) holds.

Proof. If eo belongs to the minimal nilpotent orbit of g, then eo is of height two and the 2-eigenspace of ad ho is spanned by eo. This is the case since ho is conjugate to the

coroot of the highest root. But since so C c(s), it follows that so = Vo,2 is the unique

subrepresentation of g isomorphic to Vmn with n > 2. Therefore all other Vmi,ni must have ni = 0 or ni = 1 and so condition (4.6) holds.

Next assume the second hypothesis. Since e is even, all Vm},ni with mi > 0 satisfy mi > 2. Since eo is of height two, ni < 2 and thus condition (4.6) is true. □

4.4. The case vihere xi = 0 for some i > 1 in (4.1), and (4.2) holds

Let eo G c(s) be nilpotent and suppose that the dimension condition (4.4) does not hold for O = Oe+e0. It may happen instead that Lemma 4.3 applies for a different nilpotent orbit O with Oe C O C Oe+eo. More precisely, it may be possible to find x G So,e, written as in (4.1), so that xo = eo and (4.4) does hold for this O. Then Lemma 4.4 ensures that Lemma 4.3 applies to So,e. Now in such a situation, Lemma 4.5 implies that x must lie in the 2-eigenspace for ad(h + ho). We now use this information to give one way to help locate such an x when it exists.

4.4.1. A smaller slice result

Let y = e + eo, which is nilpotent with corresponding semisimple element hy = h + ho. Write gj for the j-eigenspace of adhy on g. The centralizer Go := Ghy has Lie algebra go and Go acts on each gj. Then y G g2 and the Go-orbit through y is the unique dense orbit in g2. Now e G g2 since

[hy ,e] = [h + ho,e]=2e + 0 = 2e. (4.8)

We want to find a transverse slice in g2 to the Go-orbit through e. In fact, since g2 is a direct sum of ad h-eigenspaces, the decomposition g = Imad e © kerad f restricts to a decomposition

02 = [e, go] © (02 n kerad f).

(2) (2) Therefore, setting Se = e + (g2 n kerad f), it follows that the affine space Se is a

transverse slice of g2 at e with respect to the G0-action. Consequently, every G0-orbit in 02 containing e in its closure meets Si2\

Let g(r, s) denote the subspace of g where ad h has eigenvalue r and ad h0 has eigenvalue s. Define gf (r, s) = g(r, s) n kerad f. Then

02 n kerad f = 0 gf (-r,r + 2).

Next, we relate this decomposition to the decomposition (4.5) of g under s © s0. Let E = {i | ni > mi > 0 and n — mi even}

where (mi, ni) are defined in (4.5). Then E C J. For each i G E, let wi be a nonzero vector in the one-dimensional space V^nl^i n g(—mi, mi +2). Then wi is a lowest weight vector for s, but not in general a highest weight vector for s0. The set {wi | i G E} is then a basis for

00f (— r,r + 2)

since each vector in (—r, r + 2) lies in a sum of subrepresentations of type VrsS with r + 2 < s and s — r even. The subspace (0, 2) is just the 2-eigenspace of ad h0 in c(s), which coincides with c(s)ng(0, 2). It contains e0. A consequence of the above observations is the following

Lemma 4.10. Let x G 02 ■ If e G G0 • x, then some G0-conjugate of x can be expressed as

e + w + ^2 diWi (4.9)

where w G c(s) n g(0, 2) and di G C.

Given a nilpotent orbit O, Lemma 4.10 gives a way to show the existence of some x GO that can be written as in (4.1) with x0 = w nilpotent. But it does not guarantee that w is equal to the prescribed e0 or that (4.4) holds. When w = e0 and (4.4) holds, which is the case we are interested in, then we also know by Lemma 4.5 that the wi which appear in (4.9) with di = 0 must satisfy ni — mi = 2, so only the terms in E with ni — mi = 2 will contribute in this case.

4.4.2. Applying Lemma 4.10

In order to apply Lemma 4.10 for some x G g with Oe C Ox C Oy, we need to check two things, after possibly replacing x by a conjugate:

(1) x G 02_

(2) e G Go • 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 hx G h. In all cases we are interested in, there exists nilpotent ex G c(sx) with semisimple element hx G h, such that hx + hx = hy, after possibly replacing x again by a conjugate. Then just as in (4.8), x G g2 and the first condition holds.

We may further assume that hy is dominant with respect to the Borel subalgebra b C g and hx is dominant for the corresponding Borel subalgebra by of ghy. Then since [hx, x] = 2x and [hy, x] = 2x, it follows that x belongs to

Ix := g2 n0 g(hx; i),

where g(hx; i) are the eigenspaces for adhx. This subspace is preserved by the action of by. Thus Go • Ix = Go • x. We can carry out a similar process for e and obtain a subspace Ie C g2, with Go • Ie = Go • e. Then if Ie C Ix, it necessarily follows that

Go • Ie C Go • Ix

and the second condition holds. For the cases we are interested in, this approach will suffice to check the hypothesis in Lemma 4.10.

4.4.3. Example: (Ai, Ai) in type G2

Let g be of type G2 and let e G g be minimal nilpotent. Then c(s) = sl2(C). Let eo G c(s) be minimal nilpotent, which has type a41 in g. The decomposition in (4.5) is g = V (0, 2) © V (2, 0) © V (1, 3). Therefore (mi, ni) = (1, 3) for the unique i gE and (4.4) fails for O = Oe+e0 by Proposition 4.8. Indeed, e + eo has type G2(ai) and thus if O is the orbit of type Ai, then Oe C O C Oe+e0. Since s and c(s) are mutual centralizers, and O is unibranch at e, the argument in Example 4.1.5 gives that So,e takes the form of the S-variety X(2, 3) for SL2(C), which is isomorphic to m.

We can show also that Lemma 4.10 holds by checking the two conditions in §4.4.2. Fix nonzero wi G V(1, 3) satisfying [eo, wi] = 0 and [f, wi] = 0. Choose hy so that its weighted diagram is the usual weighted Dynkin diagram 2 o of y and choose hx and hx to have weighted diagrams o 1 and 2 -1, respectively. Then hy = hx + hx and thus by the above discussion we may replace x by a conjugate and assume x G g2. Similarly, let h and ho have weighted diagrams -1 1 and 3 -1, respectively. Then Ie is one-dimensional and Ie C Ix. The two conditions in §4.4.2 are met, so by Lemma 4.10 there exists x GO

with x = e + ae0 + dwi for a, d G C. Now d = 0 since x and e + ae0 are not in the same G-orbit for any value of a, and a = 0 by Lemma 4.4, and thus we get another proof that So,e takes the form X(2, 3).

4.4.4. Finding wi for i G E

We sometimes need to do explicit computations to verify (4.9) or to show that w = e0, especially for degenerations which are not minimal (e.g., §4.4.5) or where Lemma 4.6 does not apply. In these cases there arises the need for an analogue of the result describing isomorphisms between S-varieties (§3.2.1). Here we describe a way that is often helpful in finding wi for i G E, which frequently leads to an isomorphism of C(s) • x with C(s) • e0 in Lemma 4.3, when such an isomorphism exists.

Write g(h; j) for the j-eigenspace of adh. Since c(s) C 0h = g(h; 0), the g(h; j) are c(s)-modules. Also gh © g(h; 1) is isomorphic to , as c(s)-modules. Indeed, for j > 0,

(—2j) = (ad f )j(gh) n and (—2j — 1) = (ad f )j+1(g(h;1))n ,

as c(s)-modules.

Suppose that gh is a direct sum of classical Lie algebras. Then for M G 0h, the matrix power Mr is in gh for r odd, or if all the factors of gh are type A, then for any r. Of course [M, Mr] = 0 in gh, and hence in g. Set M := e0, where as before e0 G c(s) is nilpotent. The identity [h0, Mr] = 2rMr holds in gh because [h0, M] = h0M — Mh0 = 2M, where matrix multiplication takes place in gh; hence this identity also holds in g. Thus Mr, if nonzero, is a highest weight vector for s0 relative to e0 and h0. Now assume Mr is not zero. Then for some largest j,

(ad f )jMr G gf (—2j)

is nonzero. Since s and s0 commute, (ad f)jMr is both a highest weight vector for s0 and a lowest weight vector for s (relative to f and h).

Now suppose E = 0 and consider (mi, ni) for i G E. Suppose mi is even. In the cases of interest (see Lemma 4.5 and the paragraph after Lemma 4.10), we have ni — mi = 2. In such cases we often find that wi can be taken to be

(ad f)~ (M~+1).

Moreover, if x in (4.9) is a linear combination of such wi's and w = e0, then it follows that C(s) • x = C(s) • e0 via the projection n0 (§3.2.2) since the Gh-action, and thus the C(s)-action, commutes with taking matrix powers.

4.4.5. Example: the non-minimal degeneration (C3, A2) in F4

We illustrate the previous discussion in F4 in proving that So,e contains an irreducible component isomorphic to the nilpotent cone Ng2 in G2, when O is of type C3 and e

lies in the A2 orbit. For this choice of e, the centralizer C(s) is connected, simple of type G2. Let eo G c(s) be regular nilpotent. Then e + eo lies in the orbit F4(a2) and the decomposition of g in (4.5) is

V(0, 2) © V(0,10) © V(2, 0) © V(4, 6),

so E has a single element, with (mi, ni) = (4, 6). Then Oe C O C Oe+e0 and we could use §4.4.2 to show that there exists x G O satisfying (4.9) with some additional work. Instead, we report on a direct computation using GAP. Let

e = eooio + eoooi, f = 2fooio +2foooi, h =[e, f],

eo = eoiii — eoi2o + eiooo-

The space g(-4, 6) is one-dimensional, spanned by wi := ei22o. This is also a highest weight vector for the full action of C(s) on gf (-4) = V(w2), the 7-dimensional irreducible representation of G2. We computed in GAP that there is an x GO with x = e + eo — | wi, which establishes (4.1) with xo = eo and x4 = wi. Since dimSo,e = dimC(s) • eo, Lemma 4.3 applies and thus So,e contains e + X as an irreducible component, where

X := C(s) • (eo — 4wi). ' _

We now show that X is isomorphic to C(s) • eo, which is the nilcone of c(s), by relating the choice of wi to the discussion in §4.4.4. We have gh = so7(C) © C and the so7(C) component contains c(s) and decomposes under c(s) into c(s) © V(w2). Now ad f annihilates c(s), while (ad f )2 carries the V(w2) summand isomorphically onto gf (—4). Let M = eo G c(s) C so7(C). Then M3 G so7(C) and M3 = 0 since eo has type B3 in g (i.e., the embedding of c(s) of type G2 in so7(C) is the expected one). Since M3 is centralized by eo and is an eigenvector for ad ho with eigenvalue 6, we have M3 G V(w2) since only the eigenvalues 2 and 10 are possible for the c(s) summand. Moreover, (ad f )2(M3) is a nonzero vector in g(—4, 6) and so must be a multiple of wi. Although X is not an S-variety (since eo is not minimal in c(s)), it is the closure of the C(s)-orbit through (eo, wi) G c(s) © gf (—4), which can now be described as the set of elements (M, M3) G c(s) © V(w2) = so7(C) with M G c(s) nilpotent. Hence, there is a C(s)-equivariant isomorphism of X with C(s) • eo = NG2 coming from no.

Remark 4.11. There are two branches of O in a neighborhood of e. These two branches are not conjugate under the action of Ge, which shows that Proposition 2.1 does not generally hold for degenerations which are not minimal. The other branch of O at e splits into three separate branches in a neighborhood of a point in the orbit F4(a3) (see §7.3).

4.5. Proof of Proposition 3.3

The proof is case-by-case until we exhaust all minimal degenerations covered by the Proposition. First, we consider those e for which there exists e0 G c(s) that is minimal nilpotent in g, and then compute the G-orbit O to which e + e0 belongs (§4.2.1). Corollary 4.9 ensures that (4.4) holds for this O, and then applying Lemma 4.3 to x := e + e0, we conclude that e + C(s) • e0 is a union of irreducible components of So,e. Such degenerations turn out always to be minimal degenerations, and so C(s) acts transitively on the irreducible components of So,e by Proposition 2.1. Hence So,e = e + C(s) • e0. 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 belongs to a minimal nilpotent C(s)-orbit in c(s) and check whether or not (4.4) holds for O = Oe+e0. In the cases where it does hold, the degeneration (O, Oe) turns out to be a minimal degeneration, and thus So,e = e + C(s) • e0 as in the first step. The results are recorded in the first lines of Tables 2, 4, 7, and 10. These two sets of calculations cover all the minimal degenerations in Proposition 3.3 where J = 0.

For the remaining cases, we study those e0 which are minimal in c(s), but where (4.4) does not hold for the orbit through e + e0. For such e and e + e0, we look for nilpotent orbits O with Oe C O C Oe+eo such that Oe is a minimal degeneration of O and dimC(s) • e0 = codim^-(Oe). Then O' = Oe and O are candidate orbits to apply Lemma 4.3. For the cases where the degeneration is dimension two, which is the vast majority, we can show that Lemma 4.6 (and hence Corollary 4.7) applies. Sometimes, though, we have to restrict to a subalgebra as in Lemma 4.1 or carry out a computer calculation to determine for which i G N the corresponding xi is nonzero in (4.1). There are just three others cases, all of dimension four, and for these we can show that there exists x G O satisfying (4.2) by restricting to a subalgebra as in Lemma 4.1 (§10.1.1, §10.1.2). Thus for all the remaining cases, which are the ones in the Proposition where J = 0, we find that Lemma 4.3 applies and Lemma 4.5 ensures that the xi's are highest weight vectors for C(s) with weights as prescribed in the Proposition. The possibilities for J turn out to be {2}, {2, 4}, {1}, {1, 2}, and {1, 2, 3}, as noted in Remark 3.4. By §3.2.1, the first two possibilities give the isomorphism under n0 in (1) of the Proposition, and the last three possibilities give the isomorphism under n01 in (2) of the Proposition.

Comparing with the surface cases treated in §5, in order to know which surface cases have |r| = 1 or 2, we find that all the cases in Proposition 3.3 have been addressed. The results are recorded in Tables 2, 4, 7, and 10, where E = 0. The set J consists of those mi with i G E and di = 0 in (4.9), or equivalently, xmi = 0 in (4.1). Such mi are the ones in the boldface pairs (mi, ni) in these tables. They all must satisfy ni — mi = 2 by Lemma 4.5.

5. Geometric method for surface singularities

In this section we consider a minimal degeneration O' of O such that O' is of codimen-sion 2 in O. Let e GO'. We show that the normalization of each irreducible component of

Soe is isomorphic to C2/r for some finite subgroup r C SL2(C). Our method allows us to determine the group r, hence we determine So ,e up to normalization. As mentioned in §3.1, we can often use results on normality of nilpotent orbit closures or other methods (e.g. Lemma 4.1) 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 slices

Recall that a contracting C*-action on a variety X is a C*-action on X with a unique fixed point o G X such that for any x G X, we have lim^0 A • x = o. Recall from [4] that a symplectic variety is a normal variety W with a holomorphic symplectic form w on its smooth locus such that for any resolution n : Z ^ W, the pull-back n*w 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 [26] and [48], 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 [45] (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. □

The two-dimensional symplectic singularities are exactly rational double points (cf. [4, Section 2.1]). The following is immediate from [17, Lemma 2.6].

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 C2/r for some finite subgroup r C SL2(C).

Note that by Proposition 2.1, the irreducible components of So ,e are mutually iso-morphic. 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 r C SL2(C) such that each irreducible component of the normalization So,e is isomorphic to C2/r.

Hence to determine So ,e, we only need to determine the subgroup r. In the following, we shall describe a way to construct the minimal resolution of So, e. Then the configuration of exceptional P1's in the minimal resolution will determine r.

5.2. Q-factorial terminalization for nilpotent orbit closures

A general reference for the minimal model program in algebraic geometry is [44]. Here we recall some basic definitions.

Let X be a normal variety. A Weil divisor D on X is called Q-Cartier if ND 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 n : Z ^ X such that KZ = n*KX + i=i aiEi with ai > 0 for all i, where Ei, i = 1, • •• , k are the irreducible components of the exceptional divisor of n. A Q-factorial terminalization of a Q-Gorenstein variety X is a projective birational morphism n : Z ^ X such that KZ = n*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 [44]), hence a normal variety X with only terminal singularities is smooth in codimension 2, that is, codimXSing(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 G Lie(L), we denote by Of its orbit under L in Lie(L). Let n(q) be the nilradical of Lie(Q). Then the natural map p : G xQ (n(q) + Of) ^ 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 (L, Of) [42]. When t = 0, then O is called the Richardson orbit for Q and G xQ n(q) identifies with the cotangent bundle T*(G/Q); if in addition p is birational, then we call p a generalized Springer resolution. By [18], those are the only symplectic resolutions of nilpotent orbit closures. More generally, if p is birational and the normalization of Ot is Q-factorial terminal, then the normalization of p gives a Q-factorial terminalization of O, the normalization of O. In [19], 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 [19] 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: 2Ai, A2 + Ai,A2 +2Ai in Ee; A2 + Ai, A4 + Ai in E7; A4 + Ai, A4 +2AiJn Eg.

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 C2 by Proposition 5.2 and we are done.

Example 5.4. Consider again the minimal degeneration (A1, A1) in G2 from §4.4.3. As O = Oa is a rigid orbit, its normalization has Q-factorial terminal singularities by [19]. In particular, the singular locus of O has codimension at least 4. Since the orbit A1 is of codimension two in O, this implies that O is non-normal and So,e = C2 for e G Oax , which is consistent with the description So,e = m in §4.4.3.

Next, assume that the normalization O is not terminal. Then by [19], O is an induced orbit and O admits a Q-factorial terminalization n : Z ^ O given by the normalization of a generalized Springer map. We denote by U the open subset O U Oe of O and v : U ^ U the normalization map. As Z has only terminal singularities, it is smooth in codimension two. As n is G-equivariant and Oe CO is of codimension two, we get that n(Sing(Z)) n v-1 (Oe) = 0. We deduce that V := n-1 (U) is smooth. In particular, we obtain a symplectic resolution n|V : V ^ U. By restriction, we get a resolution n : n-1(So,e) ^ So,e, which is a symplectic, hence minimal, resolution.

Let y G v-1(e). If we know: (1) the number of P1's in n-1(y) and in n-1(v-1(e)); and (2) the action of A(e) on the P1's in n-1(v-1 (e)), then in most cases we can determine the configuration of P1's in n-1(v-1(e)), and hence in n-1(y), and therefore determine So,e. We next introduce some methods to compute this information.

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, 0) consisting of a nilpotent orbit O in g and an irreducible representation 0 of the component group A(x) where x G O. The corresponding irreducible W-module will be denoted by P(x,0).

Let WL 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 be the Springer fiber of t for the group L. If Of is the orbit of L through the nilpotent element t G Lie(L), we denote by p^. 1) the WL-module corresponding to the pair (Of, 1) via the Springer correspondence for L.

Lemma 5.5. Let Z = G xQ (n(q) + Of). Let p : Z ^ O be the generalized Springer map. Let O' C O be a nilpotent orbit of codimension 2d. Assume that Z is rationally smooth at all points of p-1(e) for e G O'. Then the number of irreducible components of p-1(e) of dimension d is given by the formula

deg PLt , 1)

0 deg 0 • [Res^ p(e, 0) : pLt, 1)],

dim H toP(BL)

0Glrr A(e)

where the sum is over the irreducible representations 0 of A(e) appearing in the Springer correspondence for G.

Proof. By [8, Thm. 3.3], we have Htop(p-i(e)) ® Htop(BtL) = Htop(Be)p<M), where the right hand side denotes the pf ^-isotypical component of the restriction of Htop(Be) to WL. Recall that Htop(Be) = ©0P(e,0) <8> p, which gives

htop(p-i(e)) • htop(Bf) = degpfU) £ deg ( • [Res^ P(e,0) : pft,i)],

where htop(X) denotes the dimension of Htop(X). □

Now the component group A(e) acts on the left-hand side of

H top(p-i(e)) <g> Htop(Bf) = Htop(Be)p(t.D

where it acts trivially on Htop(Bf). It also acts on the right-hand side since the A(e)-action commutes with the W-action, and hence the Wf-action. Note that the action of A(e) is compatible with the isomorphism (see Corollary 3.5 [8]). This gives the following

Corollary 5.6. The permutation action of A(e) on the irreducible components of dimension d of p-i(e) gives rise to the linear representation

0 deg PLt,i) [ResWL P(e,0) : pLt,i)]p (5.1)

0Glrr A(e)

In particular the number of orbits of A(e) on the irreducible components of p-i(e) of dimension d equals the multiplicity of the trivial representation of A(e) in (5.1). The number of A(e)-orbits is therefore equal to deg pLt i)[Res|^L P(e,i) : pLt i)].

Example 5.7. Let g be of type F4. Let O be the nilpotent orbit of type B3 and O' of type F4(a3). Then O' CO is codimension two. Since O is even, its weighted Dynkin diagram shows that O is Richardson for the parabolic subgroup Q with Levi subgroup L of semisimple type A2. This gives rise to the generalized Springer map p : G xQn(q) ^ O as in Lemma 5.5, with t = 0. The map p is birational because e is even. Since O is normal and p is birational, the restriction of p gives a minimal resolution of So,e = So,e where e GO' as in §5.3.

Now A(e) = S4. Since t = 0, the representation pf, i) is the sign representation of WL. By the Springer correspondence for F4, p(e,[2i*]) = (i,i2, P(e,[2*]) = (e',e, P(e,[3i]) = (9,6 and P(e,[4]) = pi2,4 (see [12, pg. 428]). The multiplicity of the sign representation in the restriction of p(e,[22]) to WL is 1 and in the restriction of P(e,[4]) is 2 and it is zero otherwise. By Lemma 5.5, the number of Pi's in p-i(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 (§4.1.1). 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 n

The next lemma (see [19, Lemma 4.3]) can sometimes be used to simplify computations.

Lemma 5.8. Let O be a nilpotent orbit with Pic(O) finite and such that there is a generalized Springer resolution n : G xQ n(q) ^ O (see §5.2). Then the number of irreducible exceptional divisors of n equals b2(G/Q), the second Betti number of G/Q, which is equal to the rank of G minus the semisimple rank of a Levi subgroup of Q.

From [19, Prop 4.4] it follows that Pic(Ox) is finite whenever the character group of Gx is finite. Picking an sl2-subalgebra sx containing x, the latter is equivalent to the finiteness of the character group of C (sx), or equivalently, to the finiteness of the center of C(sx). The latter can be read off from the tables in [2] or deduced from the tables in [51]. Such calculations are closely related to those in §6. In the exceptional groups, Pic(O) is finite unless O is one of the following orbits in E6: 2A1, A2 + A1, A2 + 2A1, A3, A3 + A1, A4, A4 + A1, D5(a1), D5. For these orbits in E6, the number of irreducible exceptional divisors of a generalized Springer resolution or a Q-factorial terminalization has been explicitly computed in the proof of [19, Prop 4.4].

Let O1, ... Os be the maximal orbits in the complement of O in O. We restrict to the case where all Oi's are codimension two in O. Then the irreducible exceptional divisors of n 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 n where n := n(b) is the nilradical of the Borel subalgebra b. It is known that each orbital variety has dimension 2 dim Oi. Let X be an orbital variety for Oi which is contained in n(q). Then X is of codimension one in n(q) since Oi is of codimension two in O and dimn(q) = 1 dimO. Moreover X is stable under the action of the connected group Q since X C Q • X C Oi n n and X is maximal irreducible in Oi n n.

Let nX be the restriction of n to G xQX. The image of nX is Oi since X is irreducible and Q is a parabolic. By dimension considerations, 1(Oi) = G xQ X is an irreducible exceptional divisor of n. Conversely, any irreducible exceptional divisor of n equals G xQ Y for some irreducible component Y of Oi n n( q). Now dim Y can only equal dim n( q) — 1 or dimn(q) — 2 since ImnY = Oi. In the former case, Y is an orbital variety of X contained in n(q). In the latter case, 1(ei) is finite where ei G Oi, contradicting the fact, from above, that the irreducible components of n-1(ei) are P1's. This shows that

the irreducible exceptional divisors of n are exactly the G xQ X where X is an orbital variety of some Oi lying in n(q).

Next, the map G xB X ^ G xQ X has connected fibers isomorphic to Q/B. It follows from [54] that the Pi's in n—i(ei) are permuted transitively under the induced action of A(ei) since the analogous statement holds for the irreducible components of pX1(ei) where pX : G xB X ^ N. Consequently, if Pic(O) is finite and ri equals the number of A(ei)-orbits on n-i(ei), then ^ri = b2(G/Q) by Lemma 5.8. See, for example, [58, 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 Ai + Ai in F4. The codimension of O' in O is two. The orbit O is Richardson for the parabolic subgroup Q whose Levi subgroup has type B3. Moreover the map n : Z := G xQ n(q) ^ O is birational, hence a generalized Springer 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 n-i(O'). Since A(e) = 1 for e GO', there is only one irreducible component in n-i(e). Since O is normal, the singularity of O at e is of type Ai.

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 (Ee(ai), D5) in Ee, (E7(ai), E7(a2)) in E7, and (Eg(ai), Eg(a2)) in Eg. 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 GO. Hence for any parabolic subgroup Q with Levi factor Ai the map n : G xQ n(q) ^ O is birational. Moreover in each case the smaller orbit O' is the unique maximal orbit in O\O. Since A(e) = 1 for e G O', there are rank(g) — 1 Pi's in n-i(e) by §5.5. At the same time, this uniqueness means that O' is the Richardson orbit for any parabolic Q' with Levi factor of semisimple type Ai x Ai, so if Q' is such a parabolic, then n(q') is an orbital variety for O'. Hence if we fix Q corresponding to a simple root a, then we find an orbital variety n(q') C n(q) for O' for each simple root ¡3 not connected to a in the Dynkin diagram. Since A(e) is trivial, each of these n(q') gives rise to a unique Pi in n-i(e). By looking in the Levi subalgebra corresponding to the simple roots not connected to a, it is possible to determine the intersection pattern of these Pi 's.

5.6.1. The case of (Ee(ai), D5) in Ee

There are 5 Pi's in n-i(e). The singularity could only be A5 or D5 since O is normal. If we choose a 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 Pi'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 (E?(a1), E7(a2)) in E7

There are 6 P1's in n-1(e). The singularity could only be A6, D6, or E6 since O is normal. Choosing a so that the remaining simple roots form a system of type E6, there are 5 orbital varieties of the form n(q') in n(q). Then the 5 P1's have intersection diagram of type D5. This eliminates A6 as a possibility. If we choose a 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 P1'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 P1's in n-1(e). The singularity could only be A7, D7, or E7 since O is normal. If we choose a 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 P1's have intersection diagram of type E6. Hence So,e = E7.

Remark 5.10. Ben Johnson and the fourth author have also confirmed these three results using Broer's description of the ideal defining the closure of the subregular nilpotent orbit and the Magma algebra system.

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 §1.8.3. That is, we determine when

C(s) = C(s)° x H

for some H C 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 C A(e) be a conjugacy class. There exists s G C(s) whose image s in A(e) lies in C such that the order of s equals the order of s, except when e belongs to one the following four orbits:

A4 + A1 in E7; A4 + A1, D7(a2) and E6(a1)+A1 in E8. (6.1)

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) [51, §3.4]. Hence the splitting holds for all other orbits where A(e) = S2, with H = {1, s}.

This leaves the cases where A(e) = 63, 64, or 65. If e is distinguished, meaning C (s)° = 1, there is nothing to check. This leaves a handful of cases where A(e) = 63 and e is not distinguished. The first such case is e = D4(ai) in Ee, which we now explain.

6.1.1. 63 cases

Let G be of type Ee and s G G be an involution with Gs of semisimple type A5 + Ai. Then there exist e G gs nilpotent of type 2A2. Let s C gs be an sl2-triple through e. Then c(e) has type G2. It is easy to compute gs n c(e) inside of A5 + Ai; it is a semisimple subalgebra of type Ai + Ai. Let e0 be regular nilpotent in gs n c(e). Then e0 is in the subregular nilpotent orbit in c(s). Clearly s belongs to the centralizer of e0 in C(s), which is a finite group H = 63, from the case of the subregular orbit in G2. Next, a calculation in A5 + Ai shows that e + e0 has generalized Bala-Carter type A3 + 2Ai. From this we conclude that e = e + e0 belongs to the nilpotent orbit D4 (ai) in Ee and s represents an involution in A(e) [51, §4].

A similar argument works if s G G is an element of order 3 with Gs of semisimple type 3A2. Therefore the centralizer H = 63 of e0 in C(s) also centralizes e + e0 and the image of H in A(e) is all of A(e). This proves the splitting for e = D4(ai) in Ee. The same procedure works for the other 63 cases.

6.1.2. We have shown

Proposition 6.1. There exists H C C(s) such that

C(s) = C(s)° x H,

except when e belongs to one of the four orbits in (6.1). For those four cases, A(e) = 62 and

C(s) = C(s)° • H

vihere H C C(s) is cyclic of order 4.

While the above splitting is unique up to conjugacy in C(s) in the subregular case (§1.5.2), this is not the case in general, as the next example shows.

Example 6.2. Let e be in the A2 orbit in g of type E8. Then c(s) has type Ee and A(e) = 62. The generalized Bala-Carter notation for the non-trivial class C in A(e) is (4Ai)''. From this it follows that both conjugacy classes of involution in G can represent C. For one choice of involution si G C(s) lifting C, gSl has type D8. The partition of e in gSl is [28], so the reductive centralizer of e in gSl is sp8. For the other choice s2 G C(s) lifting C, gs2 has type E7 + Ai and e corresponds to (3Ai)'' + Ai. Hence the reductive centralizer of e in gs2 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 Ima = Int(c(s)ss) x K for some subgroup of diagram automorphisms (§1.5.2). 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) has the desired property, since F4 is the fixed subalgebra under the non-trivial diagram automorphism of E6. We note that [2] is the original source for computing the image of the map a.

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 G 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) = ©2 cases

Let So,e be an irreducible minimal singularity admitting an involution as in §1.8.2. 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 G H realizes the involution on So,e and s2 acts trivially on So,e. We will still refer to this singularity with induced symmetry by a+.

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 [2A1] +, [2a2] + , and [2^2] + .

6.2.2. Minimal singularities: A(e) = ©3 cases

If So,e is the unique irreducible minimal singularity admitting an action of ©3 as in §1.8.2, then H realizes the full symmetry d++. This only occurs once, in E8.

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) = ©2 cases

If So,e is an irreducible simple surface singularity admitting an involution as in §1.4.2 (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 [13] imply that H corresponds to the r' C SL2(C) which defines the symmetry involution.

The only reducible surface singularities with A(e) = 62 are those with So,e = 2Ai, hence covered previously.

6.2.4. Simple surface singularities: A(e) = 63 cases

If So,e is an irreducible simple surface singularity admitting an 63 action as in §1.4.2, then H realizes the symmetry action and so So,e = G2.

An unusual situation occurs for the minimal degeneration (D7(ai), E8(be)). Here, A(e) = 63, but So,e only admits a two-fold symmetry, compatible with its normalization So,e which is A3. Here, r C SL2(C) corresponding to So,e is cyclic of order 4. The normal cyclic subgroup of H = 63 is generated by an element s with gs of type Ee + A2 and hence s acts without fixed point on the orbit D7(ai) since the latter orbit does not meet the subalgebra Ee + 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 [13] imply that H acts on So,e = C2/r via the action of r' C SL2(C), the binary dihedral group of order 24 containing r 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 [3Ai]++. 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 whether an irreducible component is normal.

6.2.5. Simple surface singularities: A(e) = 64 case

This only occurs in F4. One degeneration has So,e = G2 (see §5.7). Here, the Klein 4-group in H acts trivially on So,e and the quotient action realizes the full symmetry of 63 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 §7.2). Here, H permutes the four components transitively and the stabilizer of any component is an 63, which acts by the indicated symmetry.

6.2.6. Simple surface singularities: A(e) = 65 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 63 x 62. The 62 factor acts trivially on the given component and the 63 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 64. The 64 factor acts on the given component as in the F4 case above.

44 B. Fu et al. / Advances in Mathematics 305 (2017) 1-77 Table 1

F4: cases with eo e c(s) of type Ai in g.

e e + eo eo c(s) Isomorphism type of Sa>e

Ai 2Ai = Aii Cs cs

Ali Ali + Ai As a+

Ai + Ali 2Ai + Ai = A2 Ai + Ai Ai

Al2 A2 + Ai G2 g2

B2 B2 + Ai = Cs (ai) 2Ai [2Ai] +

Cs(ai) Cs(ai)+Ai = Fi(as) Ai Ai

Cs Cs + Ai = F4(a2) Ai Ai

Remark 6.3. Even when A(e) is non-trivial, it might not induce a non-trivial symmetry on any So,e- For example, when e = 63(01) in F4, the only degeneration above Oe has So,e = A1. Here, H acts trivially on So,e, reflecting the fact that SL2(C) has no outer automorphisms. Indeed, C(s) is just the direct product C(s)° x H.

7. Results for F4

7.1. Details in the proof of Proposition 3.3

Here we record the details for establishing Proposition 3.3 for g of type F4, as outlined in §4.5. First, we enumerate the G-orbits of those e such that c(s) has non-trivial intersection with the minimal nilpotent orbit in g. To that end, let e0 G 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 G c(s0) are therefore those in the G-orbits

Ai,Ai,Ai + Ai,A2,B2,C3(ai) and C3

and hence Corollary 4.9 applies to these elements. The computation of e + e0 G O proceeds as in §4.2. The results are in Table 1. We use boldface font in Table 1 to locate the simple factors whose minimal nilpotent orbit is of type A1 in g. Where more than one such simple factor is in boldface, this indicates that the factors are conjugate under the action of C(s). The first two lines of Table 2 have the remaining cases where Lemma 4.3 applies with x = e + e0 for an element e0 in a minimal nilpotent orbit of c(s). This now exhausts all minimal degenerations covered by Proposition 3.3 with J = 0.

The remaining minimal degenerations in the proposition, of which there are four, are all codimension two and unibranch. We use §7.2 to determine that these exhaust the remaining codimension two cases with |r| = 1 or 2. We now show that all four cases are S-varieties for SL2(C) of the form X(2, i1 +2, i2 +2, ...), with J = {i1, i2,... } among those listed in Remark 3.4. All cases can be handled with Lemma 4.6 and Corollary 4.7, but in one case we need to pass to a subalgebra (as in Lemma 4.1) and in another, do an explicit computation to find the exact form of Soe. The values of (mj, nj) for j G E are

Table 2

F4: Remaining relevant cases with eo minimal in c(s).

e + eo

i) for i e£

Isomorphism type

of So,e

A 2 + Ai

A2 + aAi

Ai + Ai

A2 + Ai Fi(a2)

Cs(ai) F4(a3)

Fi(a3)

A2 + Ai F4(a2)

A2 + Ai B2

Cs(ai)

Ai + Ai

(2, 4)

(1, 3), (2, 4) (1, 3), (2, 4) (1, 3)

listed in Table 2. Boldface is used for those (mj, nj) where xmj = 0 in (4.1). Equivalently, the set J consists of the mj's in boldface.

7.1.1. The degeneration (A2, Ai + Ai)

For e of type A1 + A1 , c(s) = sl2(C) ©sl2(C). The nonzero nilpotent elements in one simple factor of c(s) are minimal in g and this case was handled earlier. The nonzero nilpotent elements in the other simple factor 3 are of type A2 in g. Let e0 € 3 be such an element. The centralizer c(z) is contained in a Levi subalgebra of g whose semisimple type is B3, and thus c(z) does not meet the G-orbit O of type A2. Hence Lemma 4.6 applies to So,e. Now e + e0 is of type C3(a1) and (m^, nj) = (2, 4) for the unique element i € E. The argument in Example 4.1.5 then gives that So,e = e + X(2, 4) = X(2) is an A1-singularity.

In fact Example 4.1.5 can be used more directly. This also illustrates the process of passing to a subalgebra to establish that So,e has the desired form as an S-variety. Let l be a Levi subalgebra of g whose semisimple type is C3. The G-orbit through e meets l in the orbit [23], so we may assume e € s C l. Then c(s) f l coincides with 3 and O f l coincides with the orbit in l of type [32]. For dimension reasons it follows that Son i,e C l equals So,e. Thus Example 4.1.5 directly gives So,e = e + X(2, 4).

7.1.2. The degeneration (C3(a1), A2 + A1)

For e of type A2 + A1, c(s) = sl2(C). Let e0 € c(s) be a nonzero element in 3 = c(s), which has type A1 +Al1 in g. The orbit O of type C3(a1) does not meet c(z), so Lemma 4.6 applies. The sum e + e0 is of type F4(a3) and (mj, n) = (1, 3) for the unique element i €E. Hence as in Example 4.4.3, we have So,e = e + X(2, 3) = m. The result can also be obtained by reducing to the subalgebra s' © c(s'), where s' is the sl2-subalgebra through an element e! of type A2. A key factor making this work is that c(s') has type G2 and we can directly use Example 4.4.3. We omit the details.

7.1.3. The degenerations (B2, A2 + A^ and (A2 + A1, A2 + A^

For e of type A2 + A1, c(s) = sl2(C). The nonzero nilpotent orbit in c(s) also has type A2 + Al1 in g. Hence for 3 := c(s), we have c(z) = s and so Lemma 4.6 applies for O both of type B2 and of type A2 + A1. Let e0 € 3 be nonzero nilpotent. The sum e + e0

is of type F4(a3) and {(1, 3), (2, 4)} are the values for the two elements in E. Indeed the decomposition of g in (4.5) is

V(0, 2) © V(1,3) © V(2,0) © V(2,4) © V(3,1) © V(4,2).

So the only remaining question to determine the isomorphism type of So,e is whether x1 is nonzero when expressing x GO as in (4.1).

For O of type B2 in g, we see that O meets the maximal simple subalgebra l = so9(C) in F4 in the orbit with partition [42, 1], while the orbit Oe meets l in the orbit with partition [33]. So we may assume s C l and consider Sont,e. The centralizer of s in l remains sl2(C), so we may also assume that s0 = c(s) C l. Calculating E for s and s0 relative to l, we find that only (2, 4) occurs. Hence we can identify Soni,e with So,e since both are dimension two. We conclude that x1 = 0 in (4.1). It follows that So,e = e + X(2, 4) ^ X(2).

On the other hand, for the orbit O of type A2 + A1, we have to carry out an explicit computation in GAP. We find that both x1 and x2 are nonzero in (4.1) and thus So,e = e + X(2, 3, 4), which is isomorphic to m by §3.2.1.

7.2. Remaining surface singularities

This section summarizes the calculations of the singularities of the minimal degenerations of codimension two, using the methods in §5.

For the cases in Proposition 3.3, we did not need to know whether a nilpotent orbit has closure which is normal 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+a41 since the singularity of that degeneration is of type A1. Similarly for the orbit A2. The remaining non-normal orbit closures, of which there are three [11], 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 for A2 + A1). In what follows we use the fact that the orbit F4(a1) has closure which is normal [11] to classify the type of its minimal degeneration. This is the only case where we need to know whether the closure is normal in order to resolve the type of a minimal degeneration in F4.

(1) (O, O') = (F4(a1), F4(a2)). The even orbit F4(a1) is Richardson for the parabolic subgroup Q with Levi factor of type and the resulting map p : G xQ n(q) ^ O is birational, hence a generalized Springer resolution. The hypotheses of Lemma 5.8 hold and b2(G/Q) = 3. Since O' is the unique orbit of codimension two in O, it follows from §5.5 that there are 3 orbits of A(e) = S2 on the irreducible components of p-1(e). On the other hand, there are a total of four irreducible components of p-1(e) by §5.4. Thus the singularity must be C3, given that O is normal.

Table 3

Ee: cases with eo e c(s) of type A1 in g.

e e + eo eo c(s) Isomorphism type of Sa>e

Ai 2Ai A5 a5

2Ai 3Ai B3 + Ti b3

3A1 4Ai = A2 A2 - Ai Ai

A2 A2 - Ai 2A2 [2a2] +

A2 + Ai A2 - f 2Ai A2 + Ti a2

2A2 2A2 - Ai G2 92

A3 A3 - Ai B2 - Ti b2

A3 + Ai A3 - f 2Ai = ^4(ai) Ai - Ti Ai

A4 A4 - Ai Ai - Ti Ai

A5 A5 - h Ai = E6(a3) Ai Ai

(2) (O, O') = (C3, F4(a3)). The orbit O is Richardson for the parabolic subgroup Q with Levi factor of type A2. The map p : G xQ n(q) ^ O is birational, hence a generalized Springer resolution, since A(x) = 1 for x G O. If e G O', then A(e) = S4. By Lemma 5.5 and Corollary 5.6, there are 16 irreducible components in p-1(e) with two orbits under A(e). The number of orbits can also be deduced from §5.5. Looking at the possibilities for the dual graph, it is clear that O is non-normal and the normalization map v : O ^O restricts to a degree 4 map over O'. This also follows from [49] (§2.4). By Corollary 5.6, there is a fixed component of n-1(y) under the A(e)-action for y = v-1(e). This implies that the singularity of O at y is G2. We show in §7.3 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.

(3) (O, O') = (B3, F4(a3)). The singularity is G2 by §5.7.

7.3. The degeneration (C3, F4(a3)) is 4G2

We now show each irreducible component of this slice is normal. By §4.4.5 the nilpotent Slodowy slice S of C3 at A2 contains an irreducible component isomorphic to the nilpotent cone Ng2 . Recall e belongs to the A2 orbit, with corresponding sl2-subalgebra s. Let e0 G c(s) be subregular nilpotent. Then a calculation shows that e' := e + e0 lies in the F4(a3) orbit in g and also that e' lies in the component of S isomorphic to NG2. Hence the nilpotent Slodowy slice of C3 at F4(a3) contains a component that is smoothly equivalent to the nilpotent Slodowy slice in c(s) of G2 at G2(a1). But then this component must be isomorphic to the simple surface singularity D4 by Lemma 5.3. Incorporating the symmetry of A(e') = S4, the nilpotent Slodowy slice of C3 at F4(a3) is isomorphic to 4G2.

8. Results for E6

8.1. Details in the proof of Proposition 3.3

In Table 3 we list the cases where Corollary 4.9 holds for e0 in the minimal orbit of E6. Here c(so) is the semisimple part of a Levi subalgebra and has type A5. The relevant

Table 4

Eg: Remaining relevant cases with eo minimal in c(s).

e eo e + eo O c(s) (mi, ni) for i GE Isomorphism type of So,e

D4 2A1 D5(ai) D5(ai) A2 0 a-2

A 2 + 2Ai A2 + 2Ai D4(ai) A3 Ai + Ti (1, 3), (1, 3), (2, 4) Ai

2A2 + Ai 3Ai D4(ai) A3 + Ai Ai (1, 3) m

Table 5

Surface singularities using §5: Eq.

Degeneration Induced from ^P1's A(e) Ö orbits of A(e) So,e

(E6(ai),D5) (Ai, 0) 5 1 A5

(D5,Eq (a3)) (2Ai, 0) 4 S2 3 C3

(D5(ai), A4 + Ai) (A2 + Ai, 0) 2 1 A2

(A5, A4 + Ai) (D4, 3221) 2 1 A2

(D4,D4(ai)) (2A2, 0) 4 S3 2 G2

(A4,D4(ai)) (A3, 0) 9 S3 2 3C2

nonzero nilpotent G-orbits are those that have non-trivial intersection with c(s0). In the first line of Table 4 is the remaining case where Lemma 4.3 applies with x = e + e0 for an element e0 in a minimal nilpotent orbit of c(s). This exhausts all minimal degenerations covered by the proposition with J = 0. There are only two cases where J = 0, both of codimension two. The degeneration (A3, A2 + 2A\) follows from working in the Levi subalgebra of semisimple type D5, similar to §7.1.1. The degeneration (A3 +Ai, 2A2 +Ai) is similar to §7.1.2. Details are given in Table 4.

8.2. Remaining surface singularities, and an exceptional degeneration

The results are listed in Table 5. In the first four entries of the table, we use the fact that the larger orbit has closure which is normal [52]. The entry for (E6(a1), D5) 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 of semisimple type D5 and using Lemma 4.1 and [33]. The entry for (D4, D4(a1)) is also clear from working in the Levi subalgebra of semisimple type D4. The degenerations (E6(a3), D5(a1)) and (2A2, A2 + A1) are both A2 using larger slices (see Table 13). Note that the 2A2 orbit is unibranch at A2 + A1, but its closure is not normal.

The exceptional degeneration (2A2 + A1: A2 +2A1) of codimension four is treated in §12.

9. Results for E7

9.1. Details in the proof of Proposition 3.3

In Table 6 we list the cases where Corollary 4.9 applies. Here c(so) is the semisimple part of a Levi subalgebra and has type D6. The relevant nonzero nilpotent G-orbits are

Table 6

E7: cases with eo E c(s) of type Ai in g.

e e + eo GO c(s) Isomorphism type of Sa>e

Ai 2Ai Dq dQ

2Ai iA iA B4 + Ai Ai

B4 + Ai b4

iA iA 4Ai F4 f4

4Ai C3 + Ai C3

A2 C3 + Ai Ai

A2 A2 + Ai A5 a+

4Ai 5Ai = A2 + Ai C3 C3

A 2 + Ai A2 + 2Ai A3 + Ti a3

A 2 + 2Ai A2 + 3Ai Ai + Ai + Ai Ai

A3 (A3 + Ai)'' (A3 + Ai)' B3 + Ai Ai

B3 + Ai b3

2A2 2A2 + Ai g2 + Ai 92

(A3 + Ai)' A3 + 2Ai Ai + Ai + Ai Ai

(A3 + 2Ai)' = D4 (ai) Ai + Ai + Ai Ai

(A3 + Ai)'' A3 + 2Ai B3 b3

Di(ai) D4 (ai) + Ai 3Ai [3Ai]++

A3 + 2Ai A3 + 3Ai = D4 (ai) + Ai Ai + Ai Ai

D4 D4 + Ai C3 C3

Di(ai) + Ai D4 (ai) + 2Ai = A3 + A2 2Ai [2Ai] +

A3 + A2 A3 + A2 + Ai Ai + Ti Ai

A4 A4 + Ai A2 + Ti a++

D4 + Ai D4 + 2Ai = D5 (ai) B2 b2

(A5)'' A5 + Ai G2 92

D5 (ai) (A5)' D5 (ai) + Ai Ai + Ti Ai

(A5 + Ai)' = E6(a3) Ai + Ai Ai

De(a2) DQ (a2) + Ai = E7 (a5) Ai Ai

D5 D5 + Ai Ai + Ai Ai

De(ai) Dq (ai) + Ai = E7 (a4) Ai Ai

Dq Dq + Ai = E7(a3) Ai Ai

those that have non-trivial intersection with c(s0). In the first several lines of Table 7 are the remaining cases where Lemma 4.3 applies with x = e + e0 for an element e0 in a minimal nilpotent orbit of c(s).

The nine remaining cases (all of codimension two), involving e from six different G-orbits, are listed in Table 7. The cases where e is type A2 + 2A1 or 2A2 + A\ follow by restricting to a subalgebra of type E6. The case where e is type A5 + Ai proceeds as in Example 4.4.3. The two cases where e is type D5(a1) + A1 are similar to Example 4.1.5. The three minimal degenerations lying above the orbit A4 + A2 and the one above the orbit A3 + A2 + A1 satisfy part (2) of Lemma 4.6. Since all the mi are even for i 6 E, Corollary 4.7 gives that these four degenerations are A1-singularities and satisfy the proposition. Still, we carry out an explicit computer calculation in GAP to show that both x2 and x4 are nonzero for these degenerations, so that in each of these cases, So,e takes the form e + X (2, 4, 6). The details are omitted.

9.2. Remaining surface singularities

The results using §5 are collected in Table 8. We have used the fact that E7(a1), E7(a2), E7(a3), E6, E6(a1) have closure which is normal [10, Section 7.8]. The method

Table 7

E7: Remaining relevant cases with eo minimal in c(s).

e eo e + eo O c(s) (mi, ni) for i € E Isomorphism type of So,e

A 2 + 2Ai 2Ai A2 A2 Ai + Ai + Ai 0 Ai

A 2 + 3Ai 2Ai 2A2 + Ai 2A2 + Ai G2 0 92

2A2 (3Ai)'' (A3 + Ai)'' (A3 + Ai)'' G2 + Ai 0 Ai

(a5)' (3Ai)'' De(a2) De(a2) Ai + Ai 0 Ai

D5 2Ai De(ai) De(ai) Ai + Ai 0 Ai

D5 + Ai 2Ai E7(a4) E7(a4) Ai 0 Ai

A6 A2 + 3Ai E7(a4) E7(a4) Ai 0 Ai

Ee(as) (3Ai)'' E7(a6) E7 (a^) Ai 0 Ai

Ee (3Ai)'' E7 (a2) E7 (a2) Ai 0 Ai

A 2 + 2 Ai A2 + 2Ai D4 (ai) A3 Ai + Ai + Ai (1, 3)4, (2, 4) Ai

2A2 + Ai (3Ai)' D4 (ai) (A3 + Ai)' Ai + Ai (1, 3) m

A3 + A2 + Ai A4 + A2 E7(as) D4 + Ai Ai (2, 4), (2, 8), Ai

(4, 6)

A4 + A2 A3 + A2 + Ai E7(as) A6 + Ai Ai (2, 4), (4, 6) Ai

(A5)' (2, 4), (4, 6) Ai

D6(ai) + Ai (2, 4), (4, 6) Ai

A6 + Ai (3Ai)' E7(as) De(a2) Ai (1, 3) m

Di(ai) + Ai 2A2 E7(as) E6(as) Ai (2, 4) Ai

D6(a2) (2, 4) Ai

Table 8

Surface singularities using §5: E7.

Degeneration Induced from ^P1's A(e) ft orbits of A(e) So,e

(E7(al),E7(a2)) (Ai, 0) 6 1 D6

(E7(a2),E7(a3)) (2Ai, 0) 5 S2 4 C4

(E7(a3),E6(ai)) ((3Ai)', 0) 5 S2 3 B3

(E6,E6(ai)) ((3Ai)'', 0) 6 S2 4 F4

(E6(ai),E7(a4)) (4Ai, 0) 4 S2 3 C3

(D6,E7 (a4)) (D4, 3221) 4 S2 3 C3

(Ä6,E7(a5)) (A2 +3Ai, 0) 4 S3 2 G2

(D5 + Ai,E7 (a5)) (2A2, 0) 4 S3 2 G2

(D6(ai),E7(a5)) (A3, 0) 12 S3 3 3C3

from [52] can be used to show D6 has closure which is normal. The entry for (E7(ai), E7(a2)) is from §5.6.2. For the three degenerations above E7(a5), the irreducible components of So,e are normal (see §9.3).

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 closure which is normal. 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+ by restriction to E6, see Table 12. The other four degenerations follow from Table 13.

9.3. Additional calculations: three degenerations above E7(a5)

The proofs are similar to the one in §7.3 and proceed by first showing that a larger slice is isomorphic to the whole nilcone of a smaller Lie algebra.

For (A6, E7(a5)) and (D5 + A1, E7(a5)), we first show that the degenerations (A6, A'5) and (D5 + A1, A5') are both isomorphic to NG2 . Then we use the fact that E7(a5) corresponds to the subregular orbit in G2. The result follows, as in §7.3, since these singularities are unibranch. In more detail: let e be in the orbit A5'. Then c(s) is of type G2. Let e0 be a regular nilpotent element in c(s). Then e + e0 lies in the orbit E7(a4) and (mi, ni) = (4, 6) for the unique element in E. The simple part of gh is so8(C). Let wi = (ad/)2(M3) with M = e0 6 c(s) C so7 C so8 (§4.4.4). Using GAP we showed that there is a unique scalar b = 0 such that e + e0 + bwi is in the orbit A6, and similarly for D5 + A1. The rest of the proof in §4.4.5 applies to give the result.

For the case of (D6(a1), E7(a5)), we first show that the degeneration (D6(a1), D4) has one branch which is isomorphic to NC3. (There are two branches of D6(a1) above D4.) Let e be in the orbit D4. Then c(s) = sp6. Let e0 be a regular nilpotent element in c(s). Then e + e0 lies in the orbit E7(a4) and g decomposes in (4.5) as

V(0,10) © V(0, 6) © V(0, 2) © V(2, 0) © V(6,4) © V(6,8) © V(10,0),

reflecting that c(s) decomposes under s0 as V(10) © V(6) © V(2) and gf (-6) decomposes under s0 as V(4)©V(8), which is 14-dimensional and as a representation of c(s) is V(w2). Also (mi, ni) = (6, 8) for the unique element in E.

The semisimple part of gh is isomorphic to sl6(C). If we take M = e0, then M4 6 sl6 is nonzero since M is regular in sl6. It cannot be in sp6 since only odd powers of M are. It satisfies [h0, M4] = 8M4 and so it must be a highest weight vector in V(8) for s0 with respect to e0. Hence we can choose wi = (ad/)3(M4) (§4.4.4). We checked using GAP that there is an x in the orbit D6(a1) with

x = e + e0 + bwi

with b = 0. Since the elements C(s) • (e0 + bwi) consist of pairs (M, M4) 6 sp6© V(w2) = sl6 with M 6 sp6 nilpotent, the slice for (D6(a1), D4) contains an irreducible component isomorphic to NC3 (the dimensions match). Since elements in the slice belonging to the E7(a5)-orbit correspond to the subregular elements in NC3, one branch of (D6(a1), E7(a5)) is isomorphic to C3, hence the singularity is 3C3.

10. Results for E8

10.1. Details in the proof of Proposition 3.3

In Table 9 we list the cases where Corollary 4.9 applies. 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. The first several lines of Table 10 contain the remaining cases where Lemma 4.3 applies with x = e + e0 for an element e0 in a minimal nilpotent orbit of c(s).

Table 9

Eg: cases with eo G c(s) of type Ai in g.

e e + eo GO c(s) Isomorphism type of Sa>e

Ai 2Ai E7 e7

2Ai 3Ai B6 bs

3Ai 4Ai F4 + Ai f4

A2 F4 + Ai Ai

A2 A2 + Ai E6 e6

4Ai 5Ai = A2 + Ai C4 C4

A 2 + Ai A2 + 2Ai A5 a+

A 2 + 2Ai A2 + 3Ai B3 + Ai b3

A 2 + 3Ai A2 +4Ai = 2A2 G2 + Ai Ai

A3 A3 + Ai B5 b5

2A2 2A2 + Ai 2G2 [252] +

2A2 + Ai 2A2 + 2Ai G2 + Ai 92

A3 + Ai A3 + 2Ai B3 + Ai b3

(A3 + 2Ai)" = D4 (ai ) B3 + Ai Ai

A3 + 2Ai A3 + 3Ai = D4(ai) + Ai B2 + Ai b2

Di(ai) D4(ai) + Ai D4 d++

D4 (ai) + Ai D4(ai) + 2Ai = A3 + A2 3Ai [3Ai]++

A3 + A2 A3 + A2 + Ai B2 + Ti b2

A3 + A2 + Ai A3 + A2 + 2Ai = D4(ai) + A2 Ai + Ai Ai

A4 A4 + Ai A4 a+

D4 D4 + Ai F4 f4

D4 + Ai D4 + 2Ai = D5 (ai ) C3 C3

A4 + Ai A4 + 2Ai A2 + Ti a +

D5 (ai ) D5 (ai ) + Ai A3 a +

A4 + A2 A4 + A2 + Ai Ai + Ai Ai

D5 (ai ) + Ai D5 (ai ) + 2Ai = D4 + A2 Ai + Ai Ai

A5 A5 + Ai G2 + Ai 92

A5 + Ai = E6(a3) G2 + Ai Ai

A5 + Ai A5 + 2Ai = Es (as) + Ai Ai + Ai Ai

Ee(a3) Es(a3) + Ai G2 92

De(a2) Ds(a2) + Ai = E7(a5) 2Ai [2Ai]+

D5 D5 + Ai B3 b3

E7K) E7(a5)+Ai = Es (a7) Ai Ai

D5 + Ai D5 + 2Ai = Ds(ai) Ai + Ai Ai

D6(ai) Ds(ai) + Ai = E7(a4) 2Ai [2Ai]+

As A6 + Ai Ai + Ai Ai

E7(a4) E7(a4) + Ai = D5 + A2 Ai Ai

Es(ai) Es(ai) + Ai A2 a +

D6 Ds + Ai = E7(a3) B2 b2

E6 E6 + Ai G2 92

E7(a3) E7(a3) + Ai = D7(ai) Ai Ai

E7(a2) E7(a2) +Ai = Es (65) Ai Ai

E7(ai) E7(ai) +Ai = Es (64) Ai Ai

E7 E7 + Ai = Es(a3) Ai Ai

The remaining cases of the proposition, where J = 0, are listed in Table 10 and include two non-surface cases. All cases follow by restriction to a subalgebra or by using Lemma 4.6 and Corollary 4.7, except for the two degenerations above A4 + A3. We now

Table 10

Eg: Remaining relevant cases with eo minimal in c(s).

e eo e - eo o c(s) (m for i , ni ) i £ E Isomorphism type of So,e

2 2Ai A4 + 2Ai A4 + 2Ai Bj 0 bj

d4 + A2 2Ai D5 (ai) + A2 Ds (ai) + Aj Aj 0 a +

Aa A2 + - 3Ai E7(a4) E7(a4) Ai + Ai 0 Ai

A4 + 2A1 2 iA A4 + A2 A4 + A2 Ai + Ti 0 Ai

A6 + Ai A2 + - 3Ai Ds + A2 Ds + A2 Ai 0 Ai

a7 4Ai Ee(b6) Eg(b6) Ai 0 Ai

D7 2Ai Eg(as) Eg (a^) Ai 0 Ai

A 2 + 2Ai A2 + - 2Ai D4(ai) A3 B3 - Ai (1, 3)g,(2, 4) Ai

2A2 + A1 3Ai D4(ai) A3 - Ai G2 + Ai (1, 3) m

2A2 + 2 A i 3Ai D4 (ai) + Ai A3 + 2Ai Bj (1, 3) m'

A 3 + A 2 - Ai A4 - - A2 E7(a5) D4 - Ai Ai - Ai (2, 4), (2, 8), Ai

(4, 6)

D4 (ai) + A2 A2 + - 2Ai A4 + 2Ai 2 3A Aj (2, 4) a +

A4 - Ai (2, 4) a +

A4 + A2 A3 + A2 + Ai Er(as) a5 Ai - Ai (2, 4), (4, 6) Ai

D5 (ai) + Ai (2, 4), (4, 6) Ai

Ds(ai) + Ai 2 A Er(as) E6(a3) Ai - Ai (2, 4) Ai

A4 + A2 + Ai A3 - ■ A2 + Ai Eg(a7) D4 + A2 Ai (1, 5), (2, 4), Ai

(3, 5), (4, 6)

A4 - A 2 >A + 2Ai Eg(a7) As + Ai Ai (1, 3), (2, 4), m

(3, 5)

D5 (ai) + Aj (1, 3), (2, 4), m

(3, 5)

A6 + Ai 3Ai Er(as) Da (oj) Ai - Ai (1, 3) m

D6(ai) + A2 A2 + - 2Ai Eg(a7) Ea(as) + Ai Ai (1, 3), (2, 4) m

Da (aj) (1, 3),(2, 4) Ai

Ea(a3)+ Ai 3Ai Eg(a7) £r(as) Ai (1, 3) m

Ea + Ai 3Ai Eg(bs) E7 (aj ) Ai (1, 3) m

discuss those two cases and the two non-surface cases, but omit the details for the other degenerations.

10.1.1. The degeneration (A3 +2Ai, 2A2 +2Ai)

Here e is in the orbit 2A2 +2A1 and c(s) = sp4(C). Let e0 be in the minimal nilpotent orbit of c(s). In this case E has one element corresponding to (mi;n) = (1,3). Consider the Levi subalgebra l of type E6 + A1. Then without loss of generality e G l (with nonzero component on the A1 factor) and e0 G l (contained in the E6 factor). By the results for E6, there is an x in the orbit O of type A3 +2A1 (in E8) with x = e+e0 + e1 for a choice of e1 G gf (—1) corresponding to (1, 3). Moreover, writing c(s) = V(2w1), then e1 is a highest weight vector for a c(s)-module V(3w1) C gf (—1). Hence So,e = e + X(2w1, 3w1) = m' since (4.4) holds and the singularity is unibranch.

10.1.2. The degenerations (A4 + A1, D4(a1) + A2) and (2A3, D4(a1) + A2)

Here c(s) = sl3(C). All the orbits meet the semisimple subalgebra l of g of type D5 + A3: Oe meets l in the orbit [33, 1] U [4]; A4 + A1 meets l in the orbit [5, 22, 1] U [4]; and 2A3 meets l in the orbit [42, 12] U [4]. Then just as in the case (B2, A2 + A1) in §7.1.3, there exists x G O with x = e + e0 + e2 for some e2 G gf (-2) corresponding to the pair (2, 4), for O either of type A4 + A1 or type 2A3. Identifying c(s) with V(9) where 9 is a highest root of c(s), we have e2 is a highest weight vector for a c(s)-module

V(20) C gf (-2). Hence for both orbits SOe = e + X(0, 20) = X(0), as desired (for two different choices of e2, related by a scalar).

10.1.3. The degenerations (A5 + A1, A4 + A3) and (D5(a1) + A2, A4 + A3)

Corollary 4.7 applies, but is not sufficient to pin down the singularity, so we carry out an explicit computation. In both of these cases, e lies in the orbit A4 + A3, for which c(s) = sl2(C). This case is a more complicated version of §7.1.3 in F4. Using the information in [35, p. 146] (adjusted for sign differences in GAP), let

e = — (4/ai +6fa3 +6/a4 +4/a2 +3/ag + 4/a7 + 3/as), / = eai and h =[e,f ].

A nilpositive element in c(s) is

eo = 2e 1122211 — e 1232110 + 2e 0122221 + e 1232100 + e 1222210 + e 1222111, i i i 2 i i

embedded in an s[2-triple {e0, h0, /0} for c(s). Then the three elements in E correspond to {(1, 3), (2, 4), (3, 5)}. The spaces gf (—1), gf (—2) and gf (—3) contain highest weight modules for c(s) with respective highest weights 3, 4 and 5, and highest weight vectors:

e1 = e 2443210 — e 1343211 + e 1243221 — e 1233321, e2 = e 2454321 + e 2354321, e3 = e 2465432.

2 2 2 2 2 3 3

We checked in GAP that

e + e0 +3e2 ± (2e1 +4e3) G D5(a1) + A2, and e + e0 — 6e2 ± yf8(e1 — 19e3) G A5 + A1.

Hence in both cases So,e = e + X (2, 3, 4, 5) = X (2, 3) = m.

10.2. Remaining surface singularities, and an exceptional degeneration

The results using §5 are collected in Table 11. We use the fact that E8, E8(a1), E8(a2), E8(a3), E8(a4) have closure which is normal [10, Section 7.8]. The method from [52] can be used to show E7, E8(b4), and E8(a5) have closure which is normal. The entry for (E8(a1), E8(a2)) is from §5.6.3. That each irreducible component of (D6(a1), E8(a7)) and (A6, E8(a7)) is G2 follows from the fact that the degeneration (E6, D4) contains a branch isomorphic to the nilpotent cone in F4, and then from the results in F4.

There are 19 other cases. For nine of them, the degenerations are unibranch, but either the larger orbit has non-normal closure or it is not known whether the larger orbit has closure which is normal. Nevertheless, 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

Table 11

Surface singularities using §5: Eg.

Degeneration Induced from 's A(e) ft orbits of A(e) so,e

(E8(ai),E8(a2)) (Ai, 0) 7 1 E7

(E8(a2),E8(a3)) (2Ai, 0) 7 S2 6 C6

(E8(as),E8(a4)) (3Ai, 0) 6 S2 4 F4

(E8(a4),E8(b4)) (4Ai, 0) 5 S2 4 C4

(E8(a5),E8(b5)) (A2 +3Ai, 0) 4 S3 2 G2

(E7(ai),E8(b5)) (A3, 0) 18 S3 5 3(C5)

(E8(b5),E8(ae)) (2A2 + Ai, 0) 4 S3 2 (G2)

(E7(a3),Ee(ai) +Ai) (Da, 3222 12) 4 S2 2 (A+)

(D7(a2),Ö5 + A2) (2A3, 0) 3 S2 2 (C2)

(E7,E8(b4)) (D4, 3221) 6 S2 4 F4

(D7,E8(a6)) (D4 + Aj, 3221 + 0) 4 S3 2 (G2)

(E8(b4),E8(a5)) (Aj +2Ai, 0) 4 S2 3 C3

(E7(a2),D7(ai)) (D5, 32213) 5 S2 3 (B3)

(D7(ai),E8(b6)) (A3 + Aj, 0) 3 S3 2 (C2) = ß

(E6 + Ai,E8 (b6)) (Ea, 2Aj + Ai) 4 S3 2 (G2)

(A7,D7(a2)) (D5 + Aj, 32213 - + 0) 2 S2 1 (A+)

(E6(ai) + Ai,D7(a2 )) (E7, A4 + Ai) 2 S2 1 (A+)

(D6,D5 + A2) (Da, 3241) 3 S2 2 (C2)

(D6(ai),E8(a7)) (A5, 0) 40 S5 2 IOG2

(A6,E8(a7)) (D4 + Aj, 0) 20 S5 2 5G2

Table 12

Some surface cases where Lemma 4.1 can be applied.

g e x e o Subalgebra

E7, E8 E6(a3) D5 E6 C3

E7,E8 A4 + Ai D5 (ai) E6 A+

E7,E8 A3 + A2 A4 D6 C2

(D5(a1), A4 + A1) is A+, both by restriction to E6 (see Table 12). The other degenerations follow from Table 13. For the other ten cases, the result is determined up to normalization. In four of these cases, the orbit closure is known to be non-normal: (E7(a1), Eg(b5)), (E7(a3), Ee (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 exceptional degeneration (A4 + A3, A4 + A2 + A1) of codimension four is treated in §12.

11. Slices related to entire nilcones

The main goal of the paper was to study 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,

Table 13

Slices containing a smaller nilcone.

0 Degeneration and nilcone

F4 (B3, Ä2) = N02

(C3, A2) DNO,

Eg (E6(as), Di) = Na2

(A4 , A3) d Nc2

(2A2, A2) = [2Na2] +

Ej (Ej(ai), D5) = N2A1

(D6(ai),Di) d Ncs

(E7(a5),A'5) = N2A1

(ag,a5') = N02

(D5 + Ai,A5') = N02

(A4 + A2, A4) = nA2

(D4, A2 +3Ai) = NO2

(D4, 2A2) = N02

(D4(ai) + Ai, (A3 + Ai)') = N2a1

(A5',As) DNBS

Es (E8(a5),E6) = N02

(Es(a6),D6) = Nc2

(eg,d4) d Nf4

(D5 + Ai,A5) d N02

(A6,E6(as)) D N02

(D6(ai),E6(as)) d N02

(Es(b6),E6(ai))= N+2

(A4, A3 + 2Ai) d Nc2

(D4, 2A2) = 2N02

we write down some cases where Soe, or one of its irreducible components, is isomorphic to an entire nilcone. Some of these were used to show in the surface case that Soe, or an irreducible component of Soe, is normal (e.g., starting with §7.3). These examples are relevant for the duality discussed in §1.9.4, to be explored in future work. They are also examples where C(s) acts with a dense orbit.

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 4.3, usually for x = e + e0, and often require a computer calculation.

11.2. Slices isomorphic to entire nilcones: two slices in st^

These two examples are special cases of isomorphisms discovered by Henderson [24] using Maffei's work on quiver varieties [43]. Here we give direct proofs that fit into the framework of Lemma 4.3 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 = gtnk. Assume n > 2 and k > 1. Consider the nilpotent orbit O' with partition [nk]. 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' C O by the dominance order for partitions. Moreover, X G O implies Xn+1 = 0 and O is maximal for nilpotent orbits in glnk with this property.

Proposition 11.1. [24, Corollary 9.5] Let e G O'. The variety So,e is isomorphic to

Y := (Y G fltk | Yn+1 = 0}.

In particular, So,e is isomorphic to the closure of the nilpotent orbit in flk with partition [(n + 1)p, q], which is the whole nilcone when k < n +1.

Proof. Let Ik be the k x k identity matrix. Define e = (eij), h = (hj), and f = (fij) to be n x n-block matrices, with blocks of size k x k, as follows:

lj(n — j)h i = j + 1 I 0 else

h■ ■ 7 <Hj

(2i — n — 1)Ik i = j \ Ik j = i

7 fij S

0 else 0 else

The Jordan type of e and f is [nk], and so e, f G O'. The elements (e, h, f} are a standard basis of an sl2-subalgebra s, as in the k =1 case. Also, as in the k =1 case, the centralizer gf consists of n x n-block matrices Z = (zij ) of the form

Yj-i 0

j > i otherwise

for any choice of y0, Y\,..., Yn-i G gtk. We abbreviate this matrix by Z({Yj}). In particular, c(s) = gtk consists of the matrices of the form Z({Y, 0, ..., 0}). We are interested in

So,e := Se n O = Se n{X G g | Xn+1 = 0},

where as before Se = e + gf. Let M = e + Z({Yj}) G Se. Set Y0 = — nY for a fixed matrix Y for reasons that will become clear shortly. Since Mn+1 = 0, we can find constraints on the entries of Mn+1. The ( n, 1)-entry of Mn+1 is equal to rY1 + sY02 where r is a sum of products of the coefficient in e, hence nonzero. Thus rY1 + sY02 = 0 and Y1 is proportional to Y2. Given this fact, the (n, 2)-entry of Mn+1 is equal to r'Y2 +s'Y03 where r' is nonzero. Hence Y2 is proportional to Y3, and so on. In this way, we conclude that Yi = CjYj+1 for all i = 0, 1, 2, ..., n — 1, where the ci G C are uniquely determined constants (which depend on n, but not k). Consequently M G So,e takes the form e + Z({ciYi+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 Tn+Y^n=1 aiTn-i G C[T] be the characteristic polynomial for the nxn-matrix e + Z({ciI1}) in the k = 1 case. A direct computation with block matrices then shows

that p(T) := Tn + ^Zn- aiYiTn-i is the characteristic polynomial of M, viewing M as an n x 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% 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 C[Y].

The next step is to show that Yn+1 must be the zero matrix. Since p(M) = 0,

0 = Mp(M) — b1Yp(M) = ^(a, — a1ai-1)YiMn~i+1 — a1anYn+1.

Since the minimal polynomial of M over C[Y] has degree n, it follows that (a, — a1ai-1)Y1 = 0 for i = 2, ..., n and a1anYn+1 = 0. Note that a1 = 1 by taking the trace of M since c0 = — n. Now if Yn+1 = 0, then recursively ai = a\ = 1, but also a1 an = an = 0, a contradiction. Similarly, if Y^ = 0 and Y^ 1 = 0 for some I < n +1, then ai = a1 = 1 for i = 1, 2, ..., I — 1. We conclude that all elements in So,e take the form e + Z ({c, Y1}) where Yn+1 = 0. Hence So, e is isomorphic to a subvariety of Y via restriction n0 of the natural projection Se ^ c(s) by the argument in §4.4.4. Now So,e and Y both have dimension p2n2 + 2pqn + p2n + q2 — q, and the latter variety is irreducible; hence n0 gives an isomorphism of So,e onto Y.

One consequence is the following: since the c, 's, and hence the ai s, are independent of k, choosing k > n, we deduce that all a, = 1, an interesting fact in its own right. □

Remark 11.2. Fix Y0 = e0 € c(s) in the orbit [(n + 1)p, q] and Y = —nY0. In the notation of §4.4, the vector Z({0, ..., 0, Yi+1, 0, ..., 0}) corresponds to the pair (i, i + 2), which lies in E when 1 < i < min(n, k — 1). The proof shows that there is an x €O that can be written as in (4.1) with x, := c,Z({0, ..., 0, Yi+1, 0, ..., 0}) where 0 < i < min(n, k — 1) and such that (4.2) holds.

11.2.2. Second slice

Next, let O be the orbit in g[nfe with partition [(n + k — 1, (n — 1)fe-1 ]. Then again e € O. The elements in O correspond to matrices which are nilpotent and which have rank(Mi) = k(n — i) for i = 1, 2, ..., n — 1.

Proposition 11.3. [24, Corollary 9.3] The variety So,e is isomorphic to the nilcone in gk.

Proof. Up to smooth equivalence, this result is a consequence of [32], by cancellation of the first n — 1 columns of the partitions for O and O'. Here, we show that, in fact, So,e = , which also follows from [24, Corollary 9.3].

Keep the notation from the proof of the previous proposition. Let M €Se satisfying the rank conditions rank(Mi) = k(n — i) for i = 1, 2, ..., n — 1. The last rank condition is

rank(Mn-1) = k. The bottom, left 2 x 2-submatrix of Mn-1 consists of rY0), with each of r, s, t positive, since the coefficients of e are positive. Multiply the last row by rY0 and substract it from the second-to-last row to zero out the (n — 1, 1)-entry. Then since rank(tlk) = k, it follows that for rank(Mn-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. Continuing in this way for the smaller powers of M, we conclude that Yi = diY0j for some di G C, as in the previous proposition.

Next a direct computation shows that Mn+k-1 has entry (n, 1) which is a scalar multiple of Y0k and all other entries are scalar multiples of Y0m for m > k. If any of these scalar multiples are nonzero, then since Mn+k-1 = 0, it follows that Y0 is nilpotent, whence Y0k = 0 since Y0 G glk. These multiples are independent of k. The k = 1 case implies that the entries in Mn+k-1 cannot all be zero unless all di = 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 glk. 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 [23] < [32] and of the second proposition is the degeneration [23] < [4, 12], 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 sp6, the slice becomes isomorphic to the nilcone in so3, which is of type A1. This gives another proof of §4.1.5, one which does not require knowing that either [32] or [4, 12] have closures which are unibranch at [23].

12. The remaining additional singularities

The singularities i and a2/S2 will be discussed in subsequent work. Here we discuss the minimal degenerations (2A2 + A1, A2 + 2A1) in E6 and (A4 + A3, A4 + A2 + A1) in E8 and show that they are singularities of type t and x, respectively. Both cases are related to showing that a larger slice is the cover of the nilcone in a smaller Lie algebra (compare this with the cases in §11). For the case in E6, we show for the degeneration (2A2 + A1, A2) that the slice is isomorphic to the affinization of a 3-fold cover of the regular nilpotent orbit in s[3(C)®st3(C). For the case in E8, we show for the degeneration (A4 + A3, A4) that the slice is isomorphic to the affinization of the universal cover of the regular nilpotent orbit in sl5(C).

12.1. Preliminaries

We start with a lemma that extends the results in §4.3. The lemma introduces an alternative transverse slice to some orbits, slightly different from the Slodowy slice. This

alternative slice will facilitate the determination of the singularities of the two degenerations in this section. It will also be used in subsequent work for other, non-minimal degenerations. Since this slice is different from the nilpotent Slodowy slice, we are not able to determine the isomorphism type of the nilpotent Slodowy slice, and thus the results in Theorem 1.2 are stated only up to smooth equivalence.

Lemma 12.1. Let e be a nilpotent element in g, and let s := (e, h, f} be an sl2-subalgebra containing it. Next, let eo be a nilpotent element in c(s), and let so := (eo, ho, fo} be an sl2-subalgebra of c(s). Suppose condition (4.4) is satisfied:

dimC(s) • eo = codim^ + Oe.

Si+e0 := e + eo + c(s)fo ©^ gf (i)

is a transverse slice in g to Oe+e0 at e + eo, where gf (i) denotes the ad h-eigenspace for the eigenvalue i in gf.

Proof. Decompose g under s © so as in (4.5). Then by Proposition 4.8, the dimension

hypothesis ensures that the summands Vmi,ni satisfy m, > n, whenever m, > 0.

Let V(m, n) be such a summand with m > n and m > 0, and consider the action of s © so on V(m, n). Then dimker f = n + 1 and dimker fo = m +1. As discussed in §4.3, V(m, n) decomposes into n + 1 irreducible representations under the action of the sl2-subalgebra (e + eo, h + ho, f + fo}. Therefore dimker(f + fo) = n + 1 and so dimker f = dimker(f + fo). Now ker f f Im(e + eo) = {0} on V(m, n). Indeed, if [e + eo, y] € ker f, then write y = i,j V-i,j in the common eigenbasis for h and ho, where i, j € Z. If y-m-n = 0, then [e + eo, y] has nonzero component on the (—m + 2, —n)-eigenspace since m > 0. This contradicts [e + eo, y] € ker f, since ker f coincides with the (—m)-eigenspace of h; hence y-m-n = 0. Repeating this argument for y-m+2,-n and then y-m—n+2 shows that they are both zero. Continuing inductively along the diagonals, we get y-mii = 0 for all i. Thus [e+eo, y] € ker f only if [e+eo, y] = 0, as desired. It follows that Im(e + eo) © ker f is a direct sum decomposition of V(m, n) since dimker f = dim kerf + fo).

On c(s), which is the direct sum of those Vm},ni with m, = 0, we clearly have c(s) = Im(e + eo)©c(s)fo since s acts trivially. Therefore, c(s)fo gf (i) is a complementary

subspace to [e + eo, g] in g, and we are done. □

Let (O, O') be either (2A2 + A1, A2 + 2A1) in type E6 or (A4 + A3, A4 + A2 + A1) in type E8. Let O'' be the A2 orbit in the E6 case and the A4 orbit in the E8 case. Let e € O''.

Our strategy to study the singularity of O along O' is to first describe So,e. In both cases, there exists x G O of the form in (4.1) such that (4.2) holds with x0 G c(s) regular nilpotent. Hence, So,e has a dense C(s)-orbit, and this allows us to describe So,e in a concrete way. Of independent interest, So,e is the affinization of a cover of the C(s)-orbit through x0, so unlike in §11, the projection to c(s) of a branch of So,e is not an isomorphism. The next step is to show for e0 in the unique C(s)-orbit of codimension four in the nilcone of c(s) that Lemma 12.1 applies. This allows for the singularity in question to be studied by studying O n S'e + , which is manageable since S'e + C Se = e + gf, and therefore O n S'e + = Soe n Se+e , so it is enough to work completely inside the concrete So,e.

Set Z = C (s) and z = c(s). Having found x GO of the form e + x0 + x1 + ... + xm as above, our approach then consists of the following series of steps:

1. Describe the (closure of the) set of elements in Z • x0 which are in e0 + zfo.

2. For each y0 G Z • x0 found in step 1, find an element z G Z such that z • x0 = y0.

3. With z as in step 2, determine the values of z • x1, z • x2 etc.

Then since Z • x is dense in So,e, we arrive at a parametrization of On S'e+e .

12.2. (2A2 + Ai, A2 + 2Ai) in E6

Recall O'' is of type A2. We choose e G O'' and the rest of s as follows:

e = ea2 + e 12321, f = 2fa2 +2f 12321, h =[e,f}.

Then z = sl3 © sl3, with basis of simple roots {a1, a3, a5, a6}. Let l1 be the subalgebra of z with simple roots {a1, a3} and let l2, with simple roots {a5, a6}, so that z = l1 © l2. Similarly, Z° = L1 x L2 = SL3 x SL3, where Lie(L1) = l1 and Lie(L2) = l2, and Z/Z° is cyclic of order 2, generated by an element which interchanges L1 and L2.

The Z-orbit structure of N(z) is therefore as follows: there is a unique open orbit, which is also connected. We call this the regular orbit. Its complement in N(z) has two irreducible components permuted transitively by Z/Z°, and a unique open Z-orbit, which we call the subregular orbit, consisting of pairs (x, y) where one of x, y is regular nilpotent, and the other is subregular, in sl3. The closure of this orbit contains the Z-orbit of all pairs (x, y) where both x and y are subregular nilpotent elements of sl3. There are three further Z-orbits with representatives (x, 0), as x ranges over all Jordan types in sl3.

We recall [35, p. 81} that gf (—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 gf (—i) is gf (—4), which is one-dimensional. Moreover, v1 := 3fa where ¡3 = 01110 is a highest weight vector in V and w1 := 3fa2+a4 is a highest weight vector in W, relative to the choice of simple roots

above. With respect to the Z°-action, we identify V (respectively, W) with the space of 3 x 3 matrices, on which (g, h) € L1 x L2 acts via

(g, h) • M = gMh-1 (respectively, (g, h) • M = hMg-1),

and we identify v1 and w1 with the matrix with 1 in the top right entry, and zero everywhere else.

Let e1 = eai+a3, e2 = eas+ag, <=1 = eai + ea3, ¿2 = ea5 + eag. Let xo := ¿1 +<=2, which is a regular nilpotent element in 3 and let eo := e1 + e2. Then eo satisfies the dimension hypothesis (4.4) and so we can apply Lemma 12.1 to it. On the other hand, for xo the situation in §4.4 applies:

Lemma 12.2. The element

x := e + xo + v1 + w1 lies in So,e f O. Thus Soe = Z° • x.

Proof. We verified by computer that x € O. The last part follows, as in §4.1.2, since both So,e and Z° • xo have dimension 12, and So,e is irreducible since O is unibranch

We note that xo is in the regular nilpotent Z-orbit in 3 and e1 + e2 and <=1 + e2 both lie in the subregular nilpotent Z-orbit so that Z • xo D Z • (e1 + e2) D Z • eo. Moreover, we observe that e + e1 + e2 and e + <=<1 + e2 both belong to OnSo,e. This fact can be used to give a conceptual proof of the previous lemma. It is also useful for the next proposition.

The centralizer (Z°)Xo of xo in Z° is generated by its identity component, a unipotent group of dimension four, and the nine scalar matrices in the center of Z° = SL3 x SL3. Let U be the index 3 subgroup of this centralizer containing the central cyclic group {(wiI, wiI) | i € {0, 1, 2}}, where w = e2ni/3. Let p : So,e ^ N(3) be the restriction of the Z-equivariant projection of e + gf onto 3. By the previous lemma, p is surjective onto the nilpotent cone N(3) in 3. The next proposition is not needed in the proof of the main result, but is of independent interest.

Proposition 12.3. The slice So,e is isomorphic to the affinization of the 3-fold cover Z°/U of the regular nilpotent orbit Z° • xo in sl3(C) © sl3(C), and hence is a normal variety. Moreover, p is finite and is an isomorphism when restricted to the complement of Z° • x. Finally, So,e (and hence the affinization) is smooth at points over the subregular Z-orbit in N(3).

Proof. For dimension reasons the identity component of U acts trivially on v1 and w1. A pair of scalar matrices (wiI, wjI) acts on V and W by the scalars wi-j and wj-i, respectively. Hence the subgroup of (Z°)Xo that acts trivially on x is exactly U. This

shows that Y := Z° • x identifies with the 3-fold cover Z°/U of the regular orbit Y := Z° • x0 in 3.

Now the regular functions C[S©,e] on So,e = Z° • x embed in C[Y], since Y is dense in Soe. Since p is surjective onto Y = N(3), we then have the inclusions C[Y] C C[SO,e] C C[Y]. Also C[Y] = C[N(3)] since N(3) is normal. Now from [23], the ring C[Y] is generated as a module over C[Y] by the unique copies of V and W in C[Y]. But C[So,e] contains a copy of both V and W, via the Z°-equivariant projection of Soe onto the V and W factors in gf, respectively. Hence C[S©,e] = C[Y]. This shows in particular that So e is normal and p is finite.

For any non-regular element in N(3), its centralizer in Z° will contain a torus that acts non-trivially on any line in V and W. Thus, since p is finite, So,e must be zero on the V and W components over such elements. It follows that p is an isomorphism over such elements, that is, when restricted to the complement of Z° • x in So,e.

Moreover, O<lSo,e consists of exactly two Z-orbits, corresponding to points over the regular and subregular Z-orbits in 3. Since O f So,e is smooth, it follows that So,e is smooth at points over the subregular orbit. Alternatively, this follows from the fact that transverse slice of N(3) at a subregular element is C2/r', where r C SL2 is cyclic of order three. The preimage under p of this transverse slice must then be C2. □

Before continuing, we make some observations about transverse slices in sl3. Following

up on our identification of V and W with 3 x 3 matrices, we identify [1 and [2 with sl3

/0 0 i\

so that e1 and e2 correspond to 000 .

\o 0 0/

Lemma 12.4. With the above identification of [1 with s[3, we have that

is an sl-2-triple through e\. The intersection of e\ + s[3f'1 with the nilpotent cone in s[3 is the set of elements of the form

for s, t € C.

Proof. The ideal of the nilpotent cone in sl3 is generated by the determinant and the sum of the three diagonal 2 x 2 minors. The zero set in e\ + of these two functions is exactly the elements Xst for s, t G C. □

Continuing the identification of ^ and l2 with sl3, we have ei and e2 correspond to

'010\ 0 0 1 . ,000/

Lemma 12.5. If s = 0 then gsteigt = Xst 'where

Moreover, gst G Li and gst

I —t —1/s 0

I —s2 0 0

\st2/2 —t/2 — 1/s

0— 1/s2 0

—s t/s 0 I-

2t/2 —t2 —s /

Proof. It is easy to check that det gst = 1 (hence lies in L1) and that g—1 is as described. The columns c1, c2, c3 of gst satisfy Xstc1 = 0, Xstc2 = c1, and Xstc3 = c2, from which it follows that gst e1gst1 = Xst. □

As noted above, Lemma 12.1 applies to eo = e1 + e2. Furthermore, e + eo € O'. Thus the affine linear space Si+e = e + eo + if1 + l^2 + gf (—2) + gf(—4) is transverse to O', and hence Sing(O, O') can be determined by describing the intersection OfSi+e .

Theorem 12.6. The intersection O f S'e + consists of all elements of the form:

e + Xst, X u

— 2 tu2v

— 2 s2 u2v

4 st2u2v — i st2 v2 — 2 st2u,

— 2 s2tv

— 2 s2 tu2

4s2tuv2 — 2t2uv2 — 2suv2 ,

G e + li © [2 © y © W where s, t, u, v G C.

Proof. Suppose s, u = 0. Consider the action of the element (gst, guv ) G Z° on x. From Lemmas 12.4 and 12.5 (also for the l2 version), we have

(gst,guv) • x = e + (gst,guv).(ei,e2,vi,wi) = e + [gstêigTt^guv ^g^Vhgst

= e + I Xst,Xu

— 2tu2v

— 2s2u2v 4 st2u2v

0 0 i 000 000

tv2 s2 v2

— 2 st2 v2

guv , gu

0 0 i 0 0 0 0 0 0

su — 2 st2u,

— 2 s2 tv

— 2s2tu2

4 s2tuv2 — i t2uv2 — 2 suv2,

(12.1)

By Lemma 12.2, we have x GO, so the elements in (12.1) lie in O. They also clearly are in S'e + , and so this set of elements, of dimension four, lies in OnS^+eo. But the latter is of dimension four since this is the codimension of O' in O. Moreover, O is unibranch at points in O'. Hence O n S'e+e is irreducible of dimension four, and must be the closure of the set of elements in (12.1) with s, u = 0. The closure of this latter set is evidently those in (12.1) where s, t, u, v are unrestricted. □

Let r be the subgroup of Sp4(C) generated by diag(w, o-1 , w, w-1).

Corollary 12.7. The singularity Sing(C4/r, 0) is equal to Sing(2A2 + A1, A2 + 2A1).

Proof. By the theorem, the variety OnS^+e is isomorphic to the variety with coordinate ring C[st, s3, t3, uv, u3, v3, sv, tu, s2u, su2, t2v, tv2]. It is straightforward to see that this is the invariant subring of C[s, t, u, v] for the induced action of r. Also e + e0 corresponds to the point s = t = u = v = 0. Since Sing(OnS^+^, e + e0) = Sing(2A2 + A1, A2 + 2A1), the result follows. □

We note an interesting consequence of the above description. The closed subset given by setting s = v, t = u has coordinate ring C[s3, t3, st, st2, s21, s2, t2], which is exactly the coordinate ring of the singularity m. This amounts to taking fixed points in OnSe+eo under an appropriate outer involution of g, giving us another proof that the singularity (A2 + A1, A2 + Ai) in F4 is smoothly equivalent to m.

12.3. (A4 + A3, A4 + A2 + Ai) in Eg

12.3.1. We begin by describing a concrete model for the singularity.

Let A = (a, r : a5 = r2 = (ar)2 = 1} be a dihedral group of order 10, acting on V = C4 by: r(u, v) = (v, u) and a(u, v) = ((u, (-1v), where ( = e^ and (u, v) G

C2 © C2 = C4.

Denote by p, q (resp. s, t) the coordinate functions on the first (resp. second) copy of C2. In particular, C[V] = C[p, q, s, t]. It is easy to show that the ring of invariants C[V ]A is generated by A = pt + qs, B = —2ps, C = 2qt and the functions = p5-lql + s5-ltl for 0 < i < 5. We note that A2 + BC = (pt — qs)2. Since none of the elements of A act as complex reflections on V, it follows that the singular points of the quotient V/A are the A-orbits of points with non-trivial centralizer, hence are the images in V/A of the points of the form (u, u) (or equivalently, (u, (iu)) for u G C2. Thus the singular locus is properly contained in the zero set of (A2 + BC) in V/A. Let D = A2 + BC and for 0 < i < 5 let Gi = (p5-iqi — s5-iti)/(pt — qs) G Frac(C[V]A) = C(V)A. It is easy to see that for 0 < i < 5, DGi G C[V]A vanishes on the singular locus of V/A, and that Fi = AGi + BGi+1 for i < 4 (resp. Fi = CGi-1 — AGi for i > 1), whence the Gi satisfy: 2AGi — CGi-i + BGi+1 = 0 for 1 < i < 4. (These equations are also satisfied by the Fi's.)

Let Y = Spec(C[A, B, C, G0,..., G5]).

Remark 12.8. a) The singularity Y can be obtained by blowing up V/A at its singular locus, as follows. It is not hard to show that the ideal of elements of C[V]A which vanish at the singular points is generated by D and DGo,..., DG5. Thus the blowup of V/A can be described as the subset of A9 x P6 which is the closure of the set of elements of the form (A, B, C, Fo,..., F5, [D : DGo : ... : DG5]) with at least one of D, DGo,..., DG5 = 0. Clearly, the affine open subset given by D = 0 has affine coordinates A, B, C, F,, G,, and hence is isomorphic to Y. An immediate consequence of this description is that Y is birational to V/A.

b) It can be shown that the ideal of relations satisfied by A, B, C, Go,..., G5 is generated by the expressions 2AG, + BGi+1 — CGi-1 = 0 together with ten identities of the form G,Gj — Gi+1Gj_1 — p(A, B, C) = 0, where p is a cubic polynomial. For example, GiGi+2 — G?+1 = ^B3-iCi for i < 3 and G,G,+3 — G,+1G,+2 = i-1^AB2-iCi for i < 2.

c) It can be shown that all of the remaining affine open subsets of the blowup given by DG, = 0 are smooth, in fact are isomorphic to A4. For example, the open subset given by DGo = 0 is the affine variety with coordinate ring R = C[A, B, C, Fo,..., F5, 1/Go, G1/Go,..., G5/Go]. It is an easy calculation (using the identities for the G, mentioned above) to check that this ring is generated by B, Fo, 1/Go and G1/Go, 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 C4/A. 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 C4 by any dihedral group (with C4 identified with two copies of its reflection representation) is a symplectic singularity.

We will next show that Sing(O, O') is equivalent to Y.

12.3.2. Let e = eai + ea3 + ea4 + e„2, / = 4/ai +6fa3 +6/a4 +4^, h = [e, f]. Then e € O'', the orbit of type A4, and 3 = s[5(C) with basis of simple roots {^1, fa, @3, } := {a8, a7, a6, 2435321}, and Z is isomorphic to the semidirect product of SL5(C) by an outer involution. Let xo belong to the regular nilpotent orbit in 3.

For the purposes of calculation we identify Z° with SL5(C) by identifying the basis of simple roots {^1, fa, @3, } with the basis of simple roots of SL5(C) coming from the choice of diagonal maximal torus and upper triangular Borel subgroup, with the usual ordering of simple roots. Let W be the natural module for Z°, corresponding to the defining representation of SL5. The Z°-module structure of gf includes the following spaces:

gf (—2) = W © W* © C, gf (—4) = A2(W) © A2(W*) © C

The following vectors are highest weight vectors, relative to the simple roots {ft}:

W1 = 3e oo11111 — 2e o111111 € W, U1 = 2e 1354321 — 3e 2344321 € W*, 1 o 22

y1 = e 1233321 € A2(W), Z1 = eo122221 € A2(W*). 11

Let xo := e^1 + ep2 + e^3 + ep4, a regular nilpotent element in 3. Then we verified that x = e + xo — W1 + U1 + 10y1 — 10^1 € O,

where recall O is of type A4 + A3, and so it follows that So,e = Z° • x since both sides are dimension 20 and O is unibranch at e. This leads to a description of So,e, whose details, which we omit, are similar to those in Proposition 12.3. Recall that p : So,e ^ N(3) is the Z-equivariant projection.

Proposition 12.9. The slice So,e is isomorphic to the affinization of the universal cover of the regular nilpotent orbit in 515(C), and hence is a normal variety. Moreover, p is finite and is an isomorphism when restricted to the complement of Z° • x. Finally, So,e (and hence the affinization) is smooth at points over the subregular Z°-orbit in N(3).

We also note that OfSo,e is the union of two Z°-orbits, one of which projects under p to the regular orbit and the other, to the subregular orbit in N(3).

Let eo € 3 be an element in the orbit with partition type [3, 2], which is codimension four in N(3). Then e + eo €O'. Moreover, eo satisfies the condition in Lemma 12.1 and so we can study the singularity (O, O') by studying O f S'e+e .

Lemma 12.10. The intersection O f S'e+e is isomorphic to the closure of the set of all (M, w'1, w1 Aw2, w'1 Aw2Aw3, w'1 Aw'2Aw'3Aw'4) € (eo +3fo )x(W©A2(W)©A3(W)©A4(W)) such that there is a basis {w', w'2,..., w'5 } for W with w' A ... A w'5 = 1 and Mwi = w[_ 1 (i > 2), Mw[ = 0.

Proof. We can describe So,e in the following way: let {w1, w2, w3, w4, w5} be the standard basis for C5 and let

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

so that Mow1 = 0 and Mow, = wi_1 for 2 < i < 5. Then So,e is isomorphic to the closure in sl5 © W © A2(W) © A3(W) © A4(W) of the SL5-orbit of Mo := (Mo, w1, w1 A w2, w1 A w2 A w3, w1 A w2 A w3 A w4).

To describe the subvariety O f S'e + = So,e f S'e + , we note that if M € eo + 3fo is nilpotent then generically M is regular and therefore there exists a basis B = {w', w2, w3, w4, w5 } of C5 such that Mw'1 = 0 and Mw' — w'_1 for i > 2. After scaling,

we may assume that gs :— (w1 w'2 w'3 w'4 w'5) has determinant one. Then the tuple (M , w'i, w'i A w2, w1 A w2 A w^, w1 A w2 A w3 A w4) — gs • Mo. □

Next, we concretely describe the variety N(3) i~l (e0 + 3f°). Let

0 0 0 0 0 0 0 -1 0 0

0 0 0 0 0 0 0 0 -1 0

2 0 0 0 0 , /0 = 0 0 0 0 -1

0 1 0 0 0 0 0 0 0 0

0 0 2 0 0 0 0 0 0 0

and then e0 + 3f° consists of all matrices of the form

2a b c d 9\

0 -3a h к l

2 0 2a b c

0 1 0 -3a h

0 0 2 0 2a )

where a, b, c, d, g, h, k, l G C. For the purposes of our calculation, we consider the matrices in e0 + 3f° of the form:

2a b c - 6a2 d - 2ab 40a3 - 10ac - 4 bh\

0 -3a h 9a2 - 4c l - 2ah

2 0 2a b c - 6a2

0 1 0 -3a h

0 0 2 0 2a /

A calculation confirms that any such matrix satisfies Tr M2 = Tr M3 = 0, and that the conditions Tr M4 = 0 and Tr M5 = 0 are expressed in terms of the coordinates a, b, c, d, h, l as:

dh + bl + -c2 = a(9bh - 216a3 + 72ac), dl = c(9bh - 216a3 +48ac). (12.2)

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 (12.2) is equal to the set of nilpotent elements of e0 + 3fo (and is therefore irreducible).

It is easy to verify that the rational functions a = A/6, b = -Go/3, c = -BC/16, d = BGi/4, h = G5/3, l = CG4/4 in C(p, q, s, t) satisfy (12.2). Since A, BC, Go, G5, BG1, CG4 are regular functions on Y, we have therefore constructed a morphism from Y to N(3) П (e0 + 3fo), corresponding to the inclusion C[A, BC, G0, G5, BG1, CG4] С C[Y]. In fact, this morphism corresponds to quotienting Y by the action of a group of order five, as follows: let p be the automorphism of order five of V which sends (p, q, s, t) to ((p, Z-1q, Zs, Z-1t). Then p normalizes Г and has an induced action on Y satisfying C[Y]P = C[A, BC, G0, G5, BG1, CG4]. (The invariants B2G2 and C2G3

are contained in this ring, since BG2 = CGo — 2AG1 and CG3 = 2AG4 + BG5.) It follows that the coordinates a = A/6 etc. given above define an isomorphism from Y/(p) to N(sfe) f (eo + 3fo).

Remark 12.11. 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 C4 by a group of order 50. Indeed, the group generated by r and p is isomorphic to the complex reflection group G(5, 1, 2), acting on C4 = 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/ (p).

We will first give an ad hoc justification that So e+e := OfS^+eo is isomorphic to Y, 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, Go,..., G5) € Y. The space of (column) vectors in W which are annihilated by M is generically of dimension one, spanned by

/ — 6 GoG4 + 1A2B + 32 B2C\

11 — 4 CG3 — 12 BG5

— 6 AB

Similarly, the space of (row) vectors in W* which are annihilated by M is also generically of dimension one, spanned by u1 = (C, —G1, — gAC, 11 CGo — 2AG1, 99A2C + 312BC2 + 6G1G5). It follows that if z € Z° = SL5 is such that Adz(Mo) = M, then zw1 is a scalar multiple of w'1, and u1z is a scalar multiple of u1. Our more rigorous argument below will (essentially) consist of showing that these scalars, up to multiplication by a fifth root of unity, are independent of p, q, s, t. Thus the ring of regular functions on So e+eo also contains elements which naturally correspond to B, C, G1 and G4. To continue along this line, we would have to find a vector w'2 € W such that Mw2 = w'1, and similarly for u1. Then it turns out that the coordinates of w'1 A w'2 and u1 A u'2 are contained in C[Y], and include scalar multiples of G2 and G3. Thus one obtains a morphism <p : Y ^ So e+eo, which (since all of the generators A, B, C, G, appear somewhere in the coordinates describing y>) is evidently a closed immersion, hence an isomorphism by equality of dimensions and reducedness.

For a more careful analysis, we note that finding a basis {w', ..., w5} for C5 such that Mw' — w'_ 1 for i > 2 and Mw1 — 0 is essentially equivalent to finding an element w5 € C5 such that M4w5 = 0. Moreover, any transformation of the form w5 ^ w5 +aw4 + ¡3w'3 +7w2 +5w1 preserves the elements w'1, w'1 Aw'2, w'1 Aw'2 Aw'3, w'1 Aw'2 Aw'3 Aw'4. Thus, to find z• Mo where Adz(Mo) = M, it suffices to choose an element w5 such that M4w5 = 0, and then to multiply w5 by an appropriate scalar such that det(w' w2 w3 w4 w5) = 1. For this purpose, we first choose

1 (3 A\

0 ; then w4 — 2 , w3 —

a2-1 BC\

7 ABC + 3 G0G5\

CG4 — 3 AG5

— 2 A2 _ 3 A 2

2G5 4A

and finally

( A4 + A2BC + 322 B2C2 + AG0G5 + 1BG1G5 — -3 CG0^4\

2 ACG4 + 11BCG5

( — g G0G4 + - A2B + 32 B2C\

— 4 CG3 — -J BG5

— 6 AB

Then one can show that the determinant of the matrix (w- w2 w° w4 w°) is —C5. Thus we replace each of w', 1 < i < 5 by w'' — —w'/C, which is well-defined whenever C — 0. In other words, whenever C — 0 we can construct a matrix gs — (w- w'J, w'3 w'{ w'5) of determinant 1 such that gs M0g-1 — M.

It is now a routine computer calculation to verify that, relative to the obvious basis for A2(W), we have

I -8G0G4 — C3G0 + 28WAC2Gi — —¡4A2CG2 + 118 A3G3\ 36 A2B2 + 64 B3C — i1^ AG0G3 — BG0G4

— IA2G3 — 24 ABG4 — Yg B2G5

1 ab2 + 3 G0G3

—112 C (AG2 +2BG3)

2 G3G5 — g4

— G4

11AG3 + n BG4

—2 B2 2G3

and similarly

w'' a wo' —

( —18 A3G2 + 36 A?GGi + ^^^ ABCG2 — 96 bc2gi — 18 gOg3g4\

w'1 A w2' A w''

— 9 AG3G4 + 24 BG3G5 — 1BG2

3 A2G2 + To ABG3 + 44BCG2 — To C2GQ

1 12 ——ABG3 + T^ B2G4 — 9 A2G;

—1B3 + 3 G0G2 21

22 AG2 + 2 BG3 — 3 G3G4 — t12 AC2 AG2 + BG3

—2G2

Finally, it is straightforward to show using the identification of A4(W) with W* that w'T A w'J, A w?3 A w'l = (— C Gt 6AC 3CGq — 2AGT 9A2C + 3^BC2 + 6GTG5) . What these computations amount to is the following:

Theorem 12.12. There is a morphism from the open subset of Y given by C = 0 to OnSg+eo, given by letting the matrix qb act on Mq. Moreover, this morphism extends to an isomorphism from Y to S'q e+ .

Proof. The first part follows from the above discussion, since qb has coordinates in the localized ring C[Y]C. But we can see by our calculations that in fact, the coordinates of M and wT', ..., wT' A w2' A w" A w4' all lie in C[Y]. Thus we can extend the morphism to a morphism p from Y to S^ e+e . Since each of the generators A, B, C, Gq, ..., G5 appears (up to multiplication by a scalar) as a coordinate of the map p, it follows that p is a closed immersion, and hence by dimensions, irreducibility and reducedness, is an isomorphism onto S'0 e+ . □

13. 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', x, a2/S2 and t are explained in §1.8.4. 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.

G2 IG2 G2 (ai) 1A1

I Q2 0

A2 +Ai .

F4 I F4 Fi(oi) I Cs

F4 (a2 ) A

F4 ( a 3 )

C3(ai)

B2 I Ai

" A2 +a4i

|a+ A2

Ai + a4i

A4 +Ai Ai

Ai A2 + 2Ai

E6 | E6

E6(ai) | A5

D5 | C3 E6(a3)

Ds(ai)

D4(ai)

A3 + Ai

2A2 + Ai

92 2A2

A2 + Ai

I [2a2] + A2 |Ai

3Ai I b3 2Ai | as

Ai | e6 0

E7(a1) ■D6

Al E7 (a2 ) C4

B3 E7(a3)

V "" ' A1

F4 E6(a1) CO D6

G2 1 E7(a5) A11 D6(a2)

A5 +A1

D4(a1)

2A2 + A1 32^ \Q2

A2 + 3A1 2A2

, - 1 A1 A1 A2 + 2A1 1 a+ .A2 + A1

D6(a1) 'A1 D5 1 C3 E6(ao) 'A1 D5(a1)+A1

A4 +A2 1 A+ A4 +A1 1 a2/&2 A3 +A2 +A1 1A1 A3 +A2 ' [2A1] +

D4(a1)+A1 'A1 A3 + 2A1

(A3 + A1)''

2A1 ' 64

I e7 0

E8 I Eg Eg (ai) I E7 Eg(a2)

Al Eg(a3) ^^ I F4

37__F4 Eg (a4)

Ai Eg(b4)

E7(ai) ^3(C5) G2

Eg(bs) '

m E7(a2) (B3) Eg(ae) y 2 E6 + Ai ___Jg2) Ai D7(ai)

Eg (as) ' lAi

: Eg(b6)

E7(a3) (A+)

2D6_ E6(ai) + Ai + '

I (C2)

E6(ai) C3

Ai Ds + A2

E7 (a4)

A6 + Ai

De(ai)

E7(as) I m

E6(a3) + Ai

E6 (a3) As + Ai

I 92 As

a + Ds(ai) + Ai c3 Ds (a i )

m D6(a2)

Ds(ai) + A2 i

Ai~~ A4 + A2

a+ A4 +2Ai

D4 (ai) + A2

I Ai A3 +A2 + Ai

D4 (ai) A3 +2A

i_ A3 + Ai 2A2 + 2Ai

I m ^^

.2 + Ai 92 I [292] + Ai 2A2

A2 + 2 Ai

la+ A2 + Ai ' I e+

4Ai A2

^ 3Ai i I b6 2Ai

Ai I eg 0

D4 +A2 I Ai

A4 +A2 +Ai

b3 A2 + 3Ai

Acknowledgments

The authors thank Anthony Henderson, Thierry Levasseur, Vladimir Popov, Miles Reid, Dmitriy Rumynin, Gwyn Bellamy and Michel Brion for helpful comments and/or conversations. B. Fu was supported by NSFC 11225106 and 11321101 and the KIAS Scholar Program. D. Juteau was supported by the ANR grant VARGEN (ANR-13-BS01-0001-01). P. Levy was supported in part by Engineering and Physical Sciences Research Council grant EP/K022997/1. E. Sommers was supported by NSA grant H98230-11-1-0173 and through a National Science Foundation Independent Research and Development plan. Computer calculations were carried out in GAP (including the SLA package [15]) and Python. Research visits were also supported by the CNRS and the AMSS of Chinese Academy of Sciences.

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