VARIETIES OF NILPOTENT ELEMENTS FOR SIMPLE LIE ALGEBRAS II: BAD PRIMES

UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP^{1}

1. Introduction

1.1. Let G be a simple algebraic group over an algebraically closed field k of characteristic p > 0 and let 𝔤 be the (restricted) Lie algebra of G with pth power map [p]. The spectrum of the cohomology ring of the restricted enveloping algebra u(𝔤) can be identified with the variety N_{1}(𝔤) = {x ∈ 𝔤 : x^{[p]} = 0}. When the characteristic of the field is a good prime, this variety was first described as the closure of a certain Richardson orbits by Carlson, Lin, Nakano and Parshall [CLNP]. Their methods used the techniques developed by Nakano, Parshall and Vella [NPV] which involved the verification of a conjecture of Jantzen on the support varieties of Weyl modules. More recently, the authors in [UGA] investigated a more general question. Let ρ be a finite-dimensional representation of 𝔤 which is realized as the derivative of a representation of G. For r ≥ 0, set

N_{r,ρ}(𝔤) = {x ∈ N (𝔤) : ρ(x)^{r} = 0},

where N (𝔤) is the variety of nilpotent elements of 𝔤. The variety N_{r,ρ}(𝔤) is a G-invariant subvariety of N (𝔤). Since there are finitely many G-orbits on N (𝔤), one can ask, how can this variety can be expressed as a finite union of orbit closures. The authors determined N_{r,ρ}(𝔤) when ρ is either a minimal dimensional irreducible representation or the adjoint representation of 𝔤 and the characteristic of the field k is a good prime relative to the underlying root system of 𝔤. The methods used did not involve the machinery in [NPV] and one can easily recover all the calculations in [CLNP] by setting r = p.

The main goal of this paper is to provide descriptions of the varieties of both unipotent and nilpotent elements of order r in 𝔤 when the characteristic of the underlying field is a bad prime. Let ρ be a finite-dimensional representation of G and let U(G) be the variety of unipotent elements in G. One can define a subvariety U_{r,ρ}(G) of U(G), which is analogous to N_{r,ρ}(𝔤), by letting

U_{r,ρ}(G) = {x ∈ U(G) : (ρ(x) − 1)^{r} = 0}.

An analog of the restricted nullcone for the unipotent variety would be U_{1}(G) = {x ∈ U(G) : x^{p} = 1}. We will refer to this as the restricted unipotent variety. Since there are finitely many unipotent classes it is also reasonable to try to describe U_{r,ρ}(G) as a finite union of closures of such classes. For good primes, there exists a G-equivariant isomorphism between U(G) and N (𝔤). McNinch [McN1] has shown that, under mild hypotheses on G, the Bardsley-Richardson isomorphism restricts to give a G-equivariant isomorphism between U_{1}(G) and N_{1}(𝔤). In particular, there are bijective correspondences between nilpotent orbits in N (𝔤) (resp. N_{1}(𝔤)) and unipotent classes in U(G) (resp. U_{1}(G)). For bad primes the total number of nilpotent orbits and unipotent classes may be different (see [Car, §5.11]).

1The members of the UGA VIGRE Algebra Group are David J. Benson, Philip Bergonio, Brian D. Boe, Leonard Chastkofsky, Bobbe Cooper, G. Michael Guy, Jeremiah Hower, Markus Hunziker, Jo Jang Hyun, Jonathan Kujawa, Graham Matthews, Nadia Mazza, Daniel K. Nakano, Kenyon Platt and Caroline Wright.

We provide a concrete description of the restricted nullcone N_{1}(𝔤) and the restricted unipotent variety U_{1}(G) for bad primes. Our results in conjunction with work in [CLNP] demonstrate that these varieties are indeed irreducible, thus answering an old question posed by Friedlander and Parshall [FP1, (3.4)]. Furthermore, in this process, we also prove that there is an order preserving bijection between the nilpotent orbits in N_{1}(𝔤) and unipotent classes in U_{1}(G) for all primes. Computational methods were essential for obtaining these results.

The determination of N_{1}(𝔤) for bad primes is an important step in the study of support varieties for restricted Lie algebras. For bad primes, we will present a calculation of the support varieties of Weyl modules in future work.

The paper is organized as follows. After the notation for the paper is introduced, we look at N_{r,ρ}(𝔤) and U_{r,ρ}(G) where ρ is the standard representation for classical simple groups G in Section 2. The results in this section will use early work of Groszer [Gr], Hesselink [He] and Spaltenstein [Sp1, Sp2] who determined the nilpotent orbits and unipotent classes and their closure orderings in this setting. In Section 3, we determine U_{r,ρ}(G) where ρ is the minimal or adjoint representation and N_{r,ad}(𝔤) for the exceptional groups G. The computations for U_{r,ρ}(G) use the tables in [Law1, Law2] which list the Jordan block sizes on the images under ρ of unipotent classes. For the computations of N_{r,ad}(𝔤), we first determine explicit orbit representatives for N (𝔤). The complete set of representatives was not available in the current literature so we have made our representatives available in Section 5. Many of these representatives were checked via computer calculations to ensure that they agree with characteristic zero computations for unipotent classes given by Lawther. Once these representatives were determined, we wrote a program in MAGMA [BC, BCP] to determine Jordan block sizes. This data is given in Section 6 and complements Lawther’s previous calculations. Using the information in Sections 2 and 3, explicit descriptions of N_{1}(𝔤) and U_{1}(G) are given in Section 4 when p is bad. For exceptional groups, the orbit closure relations in these varieties are given in Section 7.

We would like to acknowledge Roger Carter, Ross Lawther, Monty McGovern, Eric Sommers and Donna Testerman for several useful discussions.

1.2. Notation. Let G be a simple algebraic group defined over k with T a maximal torus of G. The root system associated to the pair (G, T) is denoted by Φ and identified with a subset of the set of weights X(T). Let Φ^{+} be the corresponding set of positive roots and Φ^{−} be the set of negative roots. The set of simple roots determined by Φ^{+} is Δ = {α_{1}, ⋯, α_{ℓ}}. The Weyl group corresponding to Φ is W. We will use throughout this paper the ordering of simple roots given in [Hum1] following Bourbaki.

In this paper we will always assume that p is a bad prime for Φ. A list of bad primes is provided below.

• Φ of type A_{l}, no primes;

• Φ of type B_{l}, C_{l}, D_{l}, p = 2;

• Φ of type E_{6}, E_{7}, F_{4}, G_{2}, p = 2, 3;

• Φ of type E_{8}, p = 2, 3, 5. Set 𝔤 = Lie G. Let N (𝔤) be the variety of nilpotent elements of 𝔤, which is often called

the nullcone. The group G acts on N (𝔤) via the adjoint action and N (𝔤) has finitely many G-orbits. The nullcone is an irreducible variety of dimension equal to |Φ|. The unipotent variety in G is denoted by U(G). When k is an algebraically closed field and the characteristic is a good prime the classification and structures of these orbits coincide with the orbit theory for complex simple Lie algebras (see [Car, CM, Hum2]). If the characteristic of the field is bad then the number of unipotent classes in U(G) and nilpotent orbits in N (𝔤) need not coincide. The unipotent classes were determined in [He, Sh, Sp1, Sp2, Mi1, Mi2, St] and the nilpotent orbits were determined in [He, Sp2, Sp3, Sp4, HS]. The definitions of N_{r,ρ}(𝔤), N_{1}(𝔤), U_{r,ρ}(G), and U_{1}(G) were provided in Section 1.1.

We should remark that one can identify N_{1}(𝔤) = N_{p,ρ}(𝔤) (resp. U_{1}(G) = U_{p,ρ}(G)) if the representation satisfies the condition N (𝔤) ∩ ker ρ = {0} (resp. U(G) ∩ ker ρ = {1}) These conditions will hold for the representations that we are considering in the paper (i.e. the minimal representation for the classical groups and the adjoint representation for the exceptional groups).

2. Classical Groups and Lie Algebras

In this section we consider nilpotent and unipotent orbits for G = Sp(N) or O(N) where char(k) = 2, which is the only bad prime for types B, C and D. The group O(N) is a disconnected algebraic group when N is even. The connected component of the identity in O(N) is denoted SO(N). Note that this definition differs from the conventional definition of SO(N) of orthogonal matrices of determinant one.

2.1. Nilpotent orbit closures. The nilpotent orbits in 𝔤 have been classified by Hesselink [He]. They are parametrized by pairs consisting of a partition μ of N and an index function χ: I → ℤ, where I is the set of (nonzero) parts of μ. The partition gives, as usual, the sizes of the Jordan blocks for any representative of the orbit. The fact that extra data (the index function) is needed to specify an orbit means that, in contrast to the situation in good characteristic (aside from the very even case for type D), two nilpotent elements of 𝔤 having the same Jordan form need not be in the same G-orbit.

There is a one-to-one correspondence between nilpotent orbits and pairs μ, χ satisfying the following conditions. Write n(m) for the multiplicity of m ∈ I as a part in μ. (2.1.1) (1) For m > l in I, χ(m) ≥ χ(l) and m − χ(m) ≥ l − χ(l).

(2) For G = Sp(N) and m ∈ I:

(a) 0 ≤ χ(m) ≤ m/2;
(b) χ(m) = m/2 if n(m) is odd.

(3) For G = O(N) and m ∈ I:

(a) m/2 ≤ χ(m) ≤ m;
(b) χ(m) = m if n(m) is odd;
(c) { m ∈ I | n(m) is odd } = {i, i − 1} ∩ ℕ for some i ∈ ℤ .

If I = {m_{1} > m_{2} > … }, Hesselink displays the pair μ, χ as a symbol

μ_{χ} = (m_{1}_{χ(m_{1})}^{n(m_{1})}, m_{2}_{χ(m_{2})}^{n(m_{2})}, …) .

The inclusion relations between nilpotent orbit closures (the Hasse diagram) have been worked out by Groszer [Gr] for Sp(N), and by Spaltenstein [Sp2] for O(N), who presents these orderings in a unified way, as follows.

If μ is any finite sequence, set |μ| = ∑_{i} μ_{i}. If μ, λ are sequences with |μ| = |λ|, extend with zeros if necessary so they have the same length, and write μ ⊴ λ if ∑_{j≤i} μ_{j} ≤ ∑_{j≤i} λ_{j} for all i (a partial order extending the usual dominance order on partitions). If now μ is a partition of N (written as usual with parts nonincreasing), let μ^{∗} be the dual partition of N, with parts μ_{i}^{∗} = #{ j : μ_{j} ≥ i }. Let P be the set of pairs (α, β) of partitions such that |α| + |β| = n; assume α and β have the same length (by extending with zeros as necessary). For (α, β) ∈ P, let α ∫ β = (α_{1}, β_{1}, α_{2}, β_{2}, …). For (α, β), (α′, β′) ∈ P, write (α, β) ≤ (α′, β′) if α ∫ β ⊴ α′ ∫ β′.

Let H_{X} denote the set of Hesselink pairs (μ, χ) for type X_{n} (where X ∈ {B, C, D} is the underlying root system for the group G). There is an injective map τ_{X}: H_{X} → P with τ_{X}(μ, χ) = (α, β) defined as follows.

If X = B, there is a unique j for which μ_{j}^{∗} is odd. Then

α_{i} = \begin{cases}χ(μ_{2i}) − 1 & if 2i < μ_{j}^{∗} \\ χ(μ_{2i}) & if 2i > μ_{j}^{∗} \end{cases} β_{i} = \begin{cases}μ_{2i} − χ(μ_{2i}) + 1 & if 2i < μ_{j}^{∗} \\ 0 & if 2i > μ_{j}^{∗}. \end{cases}

Suppose X = C. Define J = { i | χ(μ_{i}) = μ_{i}/2 }, J_{0} = { i | i ∉ J and μ_{μ_{i}}^{∗} is even }, J_{1} = { i | i ∉ J ∪ J_{0} }. Then

α_{i} = \begin{cases}χ(μ_{2i−1}) & if 2i − 1 ∈ J ∪ J_{0} \\ μ_{2i−1} − χ(μ_{2i−1}) & if 2i − 1 ∈ J_{1} \end{cases} β_{i} = \begin{cases}χ(μ_{2i}) & if 2i ∈ J ∪ J_{1} \\ μ_{2i} − χ(μ_{2i}) & if 2i ∈ J_{0}. \end{cases}

Finally if X = D then

α_{i} = χ(μ_{2i}) β_{i} = μ_{2i} − χ(μ_{2i}).

If O′ and O are G-orbits, write O′ ≤ O if O′ ⊆ \overline{O}. Write O(μ, χ) for the nilpotent orbit associated to the pair (μ, χ) ∈ H_{X}.

Theorem (Groszer, Spaltenstein). Let G = Sp(N) or O(N) be of type X ∈ {B, C, D}. Let (μ′, χ′), (μ, χ) ∈ H_{X}. Then O(μ′, χ′) ≤ O(μ, χ) if and only if τ_{X}(μ′, χ′) ≤ τ_{X}(μ, χ).

Remark. When N is even, certain O(N) orbits split into two equidimensional orbits for SO(N), specifically, those having all parts and multiplicities even and χ(μ_{i}) = μ_{i}/2 for all i [He, Proposition 5.4]. Whenever O ≤ O′ are O(N) orbits consisting of two SO(N) orbits, there exists an O(N) orbit O′′ which does not split, satisfying O ≤ O′′ ≤ O′ [Sp2, 4.1]. This means that the SO(N) poset may be deduced from the O(N) poset.

2.2. The varieties N_{r,min}(𝔤). In this subsection we characterize, in terms of nilpotent orbit closures, the varieties N_{r,ρ}(𝔤) = { x ∈ N (𝔤) : ρ(x)^{r} = 0 }, when ρ is the minimal irreducible representation. Notice that ρ is faithful, so it certainly satisfies the condition N (𝔤) ∩ ker ρ = {0} mentioned at the end of Section 1.2.

Theorem. Let 𝔤 be of type X_{n} where X = B, C, or D, and let ρ be the minimal irreducible representation of 𝔤. Then for r ∈ ℕ, N_{r,ρ}(𝔤) = \overline{O(μ, χ)} where μ_{χ} is given as follows.

(1) If X = B write n = dr + s with d, s ∈ ℤ and 0 ≤ s < r.

μ_{χ} = \begin{cases}r_{r}^{2d} (s + 1)_{s+1} s_{s} & if r ≤ n \\ (n + 1)_{n+1} n_{n} & if r > n. \end{cases}

(2) If X = C write 2n = dr + s with d, s ∈ ℤ and 0 ≤ s < r. For r ≤ n,

μ_{χ} = \begin{cases}r_{⌊ \frac{r}{2} ⌋}^{d} s_{\frac{s}{2}} & if r or d is even \\ rd−1_{\frac{r−1}{2}} (r − 1)_{\frac{r−1}{2}} (s + 1)_{\frac{s+1}{2}} & otherwise \end{cases}

while for r > n,

μ_{χ} = \begin{cases}r_{\frac{r}{2}} (2n − r)_{\frac{2n−r}{2}} & if r is even \\ (r − 1)_{\frac{r−1}{2}} (2n − (r − 1))_{\frac{2n−(r−1)}{2}} & if r is odd and r > n + 1 \\ (r − 1)2_{\frac{r−1}{2}} & if r is odd and r = n + 1. \end{cases}

(3) If X = D write n = dr + s with d, s ∈ ℤ and 0 ≤ s < r.

μ_{χ} = \begin{cases}r_{r}^{2d} s_{s}^{2} & if r ≤ n \\ n_{n}^{2} & if r > n. \end{cases}

Proof. We will show in each case that if μ′_{χ′} is a nilpotent orbit parameter with all parts less than or equal to r, then O(μ′, χ′) ≤ O(μ, χ). Let

μ′_{χ′} = (a(1)_{b(1)}^{c(1)}, a(2)_{b(2)}^{c(2)}, …, a(k)_{b(k)}^{c(k)})

with a(1) ≤ r and corresponding Spaltenstein pair α′ = (α′_{1}, …, α′_{l}), β′ = (β′_{1}, …, β′_{l}); let α, β be the Spaltenstein pair associated to μ_{χ}.

Recall the notation α ∫ β for the (unordered) partition of n given by (α_{1}, β_{1}, α_{2}, β_{2}, …). Write α ∫_{m} β for ∑_{i=1}^{m−1}(α_{i} + β_{i}) + α_{m} and α ∫^{m} β for ∑_{i=1}^{m}(α_{i} + β_{i}).

Type B_{n}. Suppose first that r ≤ n. It is clear that μ_{χ} = r_{r}^{2d}(s + 1)_{s+1}s_{s} is a nilpotent orbit symbol, with j = s + 1 and μ_{j}^{∗} = 2d + 1 (note that s + 1 ≠ 0). Thus, its corresponding Spaltenstein pair is given by

α_{i} = \begin{cases}r − 1 & if i ≤ d \\ s & if i = d + 1 \end{cases} β_{i} = \begin{cases}1 & if i ≤ d \\ 0 & if i = d + 1. \end{cases}

Since a(1) ≤ r it is clear that α′_{1} ≤ r − 1 and thus (since α is nonincreasing) α′_{i} ≤ r − 1 for all i. Also, α′_{i} + β′_{i} ≤ μ′_{2i} ≤ r for all i. It’s then apparent that

(2.2.1)  α′_{∫m}β′ ≤ α_{∫m}β and α′^{∫ m}β′ ≤ α^{∫ m}β

for 1 ≤ m ≤ d. If s = 0 then α ∫^{d} β = n. If s ≠ 0 then α ∫_{d+1} β = n. We know α′ ∫ β′ is a partition of n, we see that (2.2.1) holds for all m. It follows that τ_{B}(μ′, χ′) ≤ τ_{B}(μ, χ), which completes the proof for the case r ≤ n.

Now assume r > n. Then (n + 1)_{n+1}n_{n} is a nilpotent orbit symbol, with μ_{j}^{∗} = 1. Consequently, α_{1} = n. Thus our result is immediate.

Type D_{n}. Again, begin with the case r ≤ n. Observe that r_{r}^{2d}s_{s}^{2} is a nilpotent orbit symbol with its corresponding Spaltenstein pair given by

α_{i} = \begin{cases}r & if i ≤ d \\ s & if i = d + 1 \end{cases} β_{i} = 0 for all i.

We have α′_{i} = χ′(μ′_{2i}) ≤ r and α′_{i} + β′_{i} = μ′_{2i} ≤ r whenever 1 ≤ i ≤ l. Thus (2.2.1) holds for 1 ≤ m ≤ d. Finally,

α_{∫d+1}β = n while α′_{∫m}β′ ≤ n and α′^{∫ m}β′ ≤ n

for all m. Consequently, τ_{D}(μ′, χ′) ≤ τ_{D}(μ, χ), as claimed.

The case r > n follows readily since the nilpotent orbit symbol n_{n}^{2} has α_{1} = n.

Type C_{n} As before, start with r ≤ n. One sees that each proposed μ_{χ} is indeed a nilpotent orbit symbol. The corresponding sequences α ∫ β are given as follows:

α∫ β = \begin{cases}(\frac{r}{2}, \frac{r}{2}, …, \frac{r}{2}, \frac{s}{2}) & if r is even \\ (\frac{r−1}{2}, \frac{r+1}{2}, \frac{r−1}{2}, …, \frac{r+1}{2}, \frac{s}{2}) & if r is odd and d is even \\ (\frac{r−1}{2}, \frac{r+1}{2}, \frac{r−1}{2}, …, \frac{r−1}{2}, \frac{s+1}{2}) & if r is odd and d is odd. \end{cases}

For each case, if some nonzero α_{q} or β_{q} is less than ⌊\frac{r}{2}⌋ then exactly one of the following occurs:

(1) α_{q} = \frac{s}{2} and d is even with q = \frac{d+2}{2}

(2) β_{q} = \frac{s}{2} and d is odd while r is even and q = \frac{d+1}{2}.

(3) β_{q} = \frac{s+1}{2} and d, r are odd with q = \frac{d+1}{2}

In each case α_{q} (resp. β_{q}) is the last nonzero entry in α ∫ β and thus α ∫_{q} β = n (resp. α ∫^{q} β = n). For the purposes of proving (2.2.1) we may therefore assume that each α_{i} and β_{i} which occurs as a summand on the right hand sides of the inequalities in (2.2.1) is ≥ ⌊\frac{r}{2}⌋.

Fix t ∈ {1, 2, …, k} and set S := ∑_{j<t} c(j). We will show that any partial sum of consecutive terms in α′ ∫ β′ involving only terms α′_{i} with μ′_{2i−1} = a(t) and β′_{i} with μ′_{2i} = a(t), and beginning with α′_{(S+2)/2} if S is even (resp. β′_{(S+1)/2} if S is odd), is less than or equal to the corresponding partial sum of terms in α ∫ β. Putting together several such partial sums will verify (2.2.1) for all m.

If c(t) is odd then all the terms α′_{i}, β′_{i} in the partial sum equal b(t) which is at most ⌊\frac{r}{2}⌋, so we get what we want. Now assume c(t) is even. First suppose S is even. Thus the terms involved are of the form α′_{i} = b(t) and β′_{i} = a(t) − b(t). Thus α′_{i} ≤ ⌊\frac{r}{2}⌋ ≤ α_{i}, while α′_{i} + β′_{i} ≤ r = α_{i} + β_{i}. Since in our alternating sum we start with an α′_{i} this case is proved. Now suppose S is odd. Thus we get β′_{i} = b(t) and α′_{i} = a(t) − b(t). Thus β′_{i} ≤ ⌊\frac{r}{2}⌋ ≤ β_{i}, while β′_{i} + α′_{i+1} ≤ r = β_{i} + α_{i+1}. Since in our alternating sum we start with a β′_{i} we have verified this case.

Finally assume r > n. Then in each case α_{1} is as large as possible for orbit symbols with parts of size at most r, while α_{1} + β_{1} = n. This completes the proof. _{□} 2.3. Orbit closures in the restricted nullcone. Our primary interest is in the restricted nullcone, N_{1}(𝔤) = { x ∈ 𝔤 | x^{2} = 0 } (since char(k) = 2): eventually (in a future paper) we plan to investigate support varieties of Weyl modules, which are G-invariant subvarieties of N_{1}(𝔤). For this reason, we will focus on the Hasse diagram of nilpotent orbits contained in N_{1}(𝔤). We give below the closure order relations between these orbits, in terms of their Hesselink parameters. Although this can be deduced from the Groszer-Spaltenstein theorem above, we give an elementary, self-contained proof, which may be of some independent interest.

We will constantly use the elementary fact that f is a continuous function if and only if f(\overline{A}) ⊂ \overline{f(A)} for any subset A of the domain of f. For example, if O is any set of matrices x satisfying x^{2} = 0, then y^{2} = 0 for all y ∈ \overline{O} in the Zariski topology. In particular, for N_{1}(𝔤), it suffices to consider partitions all of whose parts are at most 2. As another application of the fact, suppose A is an infinite subset of the field k (recall k is algebraically closed so k is infinite). If f: A → B is a continuous function, then f(t) ∈ \overline{B} for all t ∈ k, since the only closed subsets of k are either finite or all of k.

Theorem. Assume (2_{γ}^{s}, 1_{η}^{N−2s}) and (2_{δ}^{r}, 1_{η}^{N−2r}) are symbols parametrizing nilpotent orbits for G (where s, r, γ, δ, η ∈ ℕ ∪ {0}, and η must be 0 (resp. 1) if G = Sp(N) (resp. O(N))). Then O(2_{γ}^{s}, 1_{η}^{N−2s}) ≤ O(2_{δ}^{r}, 1_{η}^{N−2r}) if and only if s ≤ r and γ ≤ δ.

Before giving the proof, we need to give more details of how the index function arises. We first treat the case G = Sp(N). Following Hesselink, let V be an N-dimensional

vector space over k equipped with a non-degenerate bilinear form β satisfying β(v, v) = 0 for all v ∈ V; then N must be even. G is the group of automorphisms of V leaving β invariant. Fix x ∈ 𝔤 with x nilpotent. For each i ∈ ℕ ∪ {0} define a quadratic form α_{i}: V → k by α_{i}(v) = β(x^{i+1}v, x^{i}v). The index function χ: ℕ → ℤ is defined by

(2.3.1)  χ(m) = min{ i ≥ 0 : α_{i}|_{ker(x^{m})} ≡ 0 }.

In particular, if x^{2} = 0 then α_{1} ≡ 0, and

(2.3.2)  χ(2) = \begin{cases}0 & if α_{0} ≡ 0 \\ 1 & otherwise. \end{cases}

We take the matrix of β to be

J = (\begin{matrix}& & & & & 1 \\ & & & & 1 \\ & . & . & . \\ 1 \end{matrix})

so that x ∈ 𝔤 ⟺ x^{t}J = Jx ⟺ Jx^{t}J = x. The matrix x′ := Jx^{t}J is obtained from x by transposing across the antidiagonal, so x ∈ 𝔤 if and only if x is symmetric about the antidiagonal.

Proof of Theorem, symplectic case. Assume G = Sp(N), where N = 2n is even. Note that x ∈ 𝔤 has Jordan form corresponding to the partition (2^{r}, 1^{N−2r}) if and only if x^{2} = 0 and rk x = r. We will take x to have the block form

(2.3.3)  x = x(C) = (\begin{matrix}0 & C \\ 0 & 0 \end{matrix})

where C = (c_{ij}) is n × n and C′ = C. Then x^{2} = 0, and it is an easy calculation to show that α_{0}(v) = c_{1n}v_{2n}^{2} + c_{2,n−1}v_{2n−1}^{2} + ⋯ + c_{n1}v_{n+1}^{2}. By (2.3.2), χ(2) = 0 if and only if all antidiagonal entries of C are 0. If A_{1}, …, A_{q} are square matrices whose dimensions sum to n, write

(2.3.4)  anti(A_{1}, …, A_{q}) = (\begin{matrix}0 & & & & & A_{1} \\ & & & & A_{2} \\ & . & . & . \\ A_{q} & & & & & 0 \end{matrix}) .

For t ∈ k put I_{2}(t) = (\begin{matrix}1 & t \\ 0 & 1 \end{matrix}).

Taking C = C(t) = anti(t, 1^{r−1}, 0^{n−r}), then x(C(t)) corresponds to the symbol (2_{1}^{r}, 1_{0}^{N−2r}) if t ≠ 0, and to (2_{1}^{r−1}, 1_{0}^{N−2(r−1)}) if t = 0. Thus

O(2_{1}^{r−1}, 1_{0}^{N−2(r−1)}) ⊂ \overline{O(2_{1}^{r}, 1_{0}^{N−2r})}.

Recall that the symbol (2_{0}^{r}, 1_{0}^{N−2r}) only occurs when r is even. Suppose r = 2l ≥ 2. Take C = C(t) = anti(I_{2}(t), I_{2}^{l−1}, 0^{n−2l}). Then rk x(C(t)) = r for all t ∈ k, and χ(2) = 0 if and only if t = 0. This shows

O(2_{0}^{r}, 1_{0}^{N−2r}) ⊂ \overline{O(2_{1}^{r}, 1_{0}^{N−2r})}.

Similarly, taking C = C(t) = anti(tI_{2}, I_{2}^{l−1}, 0^{n−2l}) gives

O(2_{0}^{r−2}, 1_{0}^{N−2(r−2)}) ⊂ \overline{O(2_{0}^{r}, 1_{0}^{N−2r})}.

Finally, if β(xv, v) = 0 for all v ∈ V and all x ∈ B ⊂ M_{N}(k), and if y ∈ \overline{B}, then β(yv, v) = 0 for all v ∈ V, by continuity. This shows that χ(2) cannot increase on closures, and completes the proof of the theorem for Sp(N). _{□}

For the orthogonal case, let V be an N-dimensional vector space over k, equipped with a quadratic form α and a bilinear form β satisfying β(v, w) = α(v + w) − α(v) − α(w) for all v, w ∈ V. Then O(N) is the algebraic group of automorphisms of V leaving α invariant. Fix x ∈ 𝔤 with x nilpotent. For each i ∈ ℕ ∪ {0} define a quadratic form α_{i}: V → k by α_{i}(v) = α(x^{i}v). The index function χ is defined as in (2.3.1). When x^{2} = 0, χ(2) is given by

(2.3.5)  χ(2) = \begin{cases}1 & if α_{1} ≡ 0 \\ 2 & otherwise. \end{cases}

We may take α(v) = v_{1}v_{N} + v_{2}v_{N−1} + …, so that the matrix of β is J when N is even, and is J_{0} = J − E_{n+1,n+1} when N = 2n + 1 is odd (E_{ij} the matrix units). A matrix x belongs to 𝔤 if and only if β(xv, v) = 0 for all v ∈ V and tr x = 0. When N is even the first condition is equivalent to x being symmetric about the antidiagonal, with antidiagonal elements all zero. If N is odd, say N = 2n + 1, then x has the block form

(2.3.6)  x = (\begin{matrix}A & 0 & C \\ w & 0 & u \\ B & 0 & A′ \end{matrix})

where A, B, C are n × n, w and u are 1 × n, and B and C are symmetric about the antidiagonal with antidiagonal elements all zero.

Proof of Theorem, orthogonal case. Assume G = O(N), first with N = 2n even. Then the symbol (2_{δ}^{r}, 1_{1}^{N−2r}) only occurs when r is even, because of condition (3c) in the definition of symbols. So assume r = 2l ≥ 2.

Taking x = x(C) as in (2.3.3), where C = C(t) = anti(tI_{2}, I_{2}^{l−1}, 0^{n−2l}), shows

O(2_{1}^{r−2}, 1_{1}^{N−2(r−2)}) ⊂ \overline{O(2_{1}^{r}, 1_{1}^{N−2r})}.

In order to get α_{0} ≢ 0, we must have some nonzero entries in x outside the upper-right n × n block. Consider x ∈ 𝔤 having the form

x = (\begin{matrix}aE_{n−1,n} & C \\ 0 & aE_{12} \end{matrix})

where C = anti(C_{1}, bI_{2}) with C_{1} an (n − 2) × (n − 2) matrix, and a, b ∈ k. Then x^{2} = 0 and α_{0}(v) = abv_{n+2}^{2}. Also rk x = 2 + rk C_{1} provided a or b is nonzero.

Assume r ≥ 4 so that l ≥ 2. Take C_{1} = anti(tI_{2}, I_{2}^{l−2}, 0^{n−2l}). With a = b = 1, and letting t be either nonzero or 0, we obtain

O(2_{2}^{r−2}, 1_{1}^{N−2(r−2)}) ⊂ \overline{O(2_{2}^{r}, 1_{1}^{N−2r})}.

Similarly, with a = 0, b = 1 we obtain

O(2_{1}^{r−2}, 1_{1}^{N−2(r−2)}) ⊂ \overline{O(2_{1}^{r}, 1_{1}^{N−2r})}.

Fixing b = t = 1 and letting a be either nonzero or 0, we obtain

O(2_{1}^{r}, 1_{1}^{N−2r}) ⊂ \overline{O(2_{2}^{r}, 1_{1}^{N−2r})}.

When r = 2, the first two orbit closure inclusions of this paragraph are trivial, and the last one is obtained in the same way using C_{1} = 0.

Next assume N = 2n + 1 is odd. Let x have the block form (2.3.6), where A = B = 0 and w = 0. Then x^{2} = 0, and α_{0} ≡ 0 if and only if u = 0. Let r = 2l ≥ 2.

Taking C = anti(I_{2}^{l}, 0^{n−2l}), u = (0^{n−1}, t) gives

O(2_{1}^{r}, 1_{1}^{N−2r}) ⊂ \overline{O(2_{2}^{r}, 1_{1}^{N−2r})}.

Taking C = anti(tI_{2}, I_{2}^{l−1}, 0^{n−2l}), u = 0 gives

O(2_{1}^{r−2}, 1_{1}^{N−2(r−2)}) ⊂ \overline{O(2_{1}^{r}, 1_{1}^{N−2r})}.

Taking C = anti(tI_{2}, I_{2}^{l−1}, 0^{n−2l}), u = (0^{n−1}, 1) gives

O(2_{2}^{r−1}, 1_{1}^{N−2(r−1)}) ⊂ \overline{O(2_{2}^{r}, 1_{1}^{N−2r})}.

Taking C = anti(I_{2}^{l}, 0^{n−2l}), u = (t, 0^{n−2}, 1) gives

O(2_{2}^{r}, 1_{1}^{N−2r}) ⊂ \overline{O(2_{2}^{r+1}, 1_{1}^{N−2(r+1)})}.

Finally, for N either even or odd, since for fixed v ∈ V, α(xv) is a polynomial in the entries of x, it follows that if α(xv) = 0 for all x ∈ B ⊂ M_{N}(k), then α(yv) = 0 for y ∈ \overline{B}. This shows that χ(2) cannot increase on closures, and completes the proof for O(N). _{□} 2.4. Unipotent orbits. In this subsection we assume G is connected; i.e., G = SO(N) in type D. Hesselink [He] also classified the unipotent orbits in the classical groups in characteristic 2. The classification is by pairs (μ, χ), analogous to the nilpotent case. The parameters for unipotent orbits in G, when G is connected, are as follows. Recall that I denotes the set of parts appearing in the partition μ. (2.4.1) (1) For m > l in I, χ(m) ≥ χ(l) and m − χ(m) ≥ l − χ(l).

(2) For G = Sp(N) and m ∈ I:

(a) (m − 2)/2 ≤ χ(m) ≤ m/2;
(b) χ(m) = m/2 if n(m) is odd.

(3) For G = SO(N) and m ∈ I:

(a) m/2 ≤ χ(m) ≤ (m + 2)/2;
(b) χ(m) = (m + 2)/2 if m ≥ 2 and n(m) is odd;
(c) ∑_{m∈I} n(m) is even if N is even.

For each pair (μ, χ) satisfying the above conditions, let C(μ, χ) be the corresponding unipotent class in U(G).

We are most interested in the restricted unipotent variety

U_{1}(G) = { u ∈ U(G) : (u − 1)^{2} = 0 } = {u ∈ U(G) : u^{2} = 1}.

One of our main results is that there is an order-preserving bijection between the G-orbits in N_{1}(𝔤) and in U_{1}(G).

Theorem. Let G be Sp(N) or SO(N) with char k = 2. There is an order-preserving bijection between the nilpotent orbits in N_{1}(𝔤) and the unipotent classes in U_{1}(G).

Proof. Since the elements x ∈ N_{1}(𝔤) (resp. u ∈ U_{1}(G)) satisfy x^{2} = 0 (resp. (u − 1)^{2} = 0), the associated partitions μ are precisely those having all parts at most 2. We will show that the same symbols arise in the nilpotent and unipotent case, provided all parts are at most 2.

First, the condition (1) is identical in (2.1.1) and (2.4.1). For G = Sp(N), condition (2a) is equivalent to χ(2) = 0 or 1, χ(1) = 0 in either case, and the conditions (2b) are identical. Assume G = SO(N). In both (2.1.1) and (2.4.1), condition (3a) becomes χ(2) = 1 or 2, χ(1) = 1. In the presence of (3a), each condition (3b) reduces to χ(2) = 2 if n(2) is odd. And in each case, condition (3c) only rules out the possibility that n(2) is odd and n(1) is even. Finally, the parameters for the nilpotent and unipotent O(N) orbits which split into two SO(N) orbits are the same.

To show that the bijection is order-preserving, it suffices to consider O(N) orbits in SO(N) (resp. 𝔰𝔬(N)), by Remark 2.1. The closure order relations between nilpotent orbits in N_{1}(𝔤) are given in Theorem 2.3. The closure order relations between unipotent classes for classical groups in characteristic 2 were worked out by Spaltenstein [Sp1]. He parametrizes the unipotent classes by pairs (λ, ε), where λ is a partition as usual, and ε: ℕ → {−1, 0, 1} ⊂ ℤ takes the place of Hesselink’s index function χ. Specifically, letting n(i) ∈ ℤ_{+} be the multiplicity of i in the partition λ,

(1) ε(i) = −1 if i is odd or if n(i) = 0,

(2) ε(i) = 1 if i is even and n(i) is odd,

(3) ε(i) = 1 if i is even, n(i) is even, and χ(i) is the larger of its two possible values (see (2a) and (3a) above), (4) ε(i) = 0 if i is even, n(i) is even, and χ(i) is the smaller of its two possible values

(see (2a) and (3a) above), Given a partition λ of N, let λ^{∗} be the dual partition of N defined by λ_{i}^{∗} = #{j: λ_{j} ≥ i}. Order the pairs (λ, ε) by (λ, ε) ≤ (μ, ϕ) if and only if, for each i,

(1) ∑_{j≤i} λ_{j}^{∗} ≥ ∑_{j≤i} μ_{j}^{∗} (i.e., λ ⊴ μ in the usual dominance ordering),

(2) ∑_{j≤i} λ_{j}^{∗} − max(ε(i), 0) ≥ ∑_{j≤i} μ_{j}^{∗} − max(ϕ(i), 0), and

(3) if ∑_{j≤i} λ_{j}^{∗} = ∑_{j≤i} μ_{j}^{∗} and λ_{i+1}^{∗} − μ_{i+1}^{∗} is odd, then ϕ(i) ≠ 0. If C_{λ,ε} is the unipotent class parametrized by (λ, ε), then C_{λ,ε} ≤ C_{μ,ϕ} if and only if (λ, ε) ≤ (μ, ϕ) [Sp1, Theorem 8.2].

Transfering the ordering on Spaltenstein pairs (λ, ε) into the corresponding ordering on Hesselink symbols, we obtain the following. Observe that if λ = (2^{s}, 1^{N−2s}), then λ^{∗} = (N − s, s). Condition (1) for (2_{γ}^{s}, 1_{η}^{N−2s}) ≤ (2_{δ}^{r}, 1_{η}^{N−2r}) becomes s ≤ r. Condition (2) (with i = 2) becomes γ ≤ δ. The remaining conditions are vacuous. These are the same conditions as in Theorem 2.3. Thus the ordering on unipotent classes in U_{1}(G) corresponds, under our bijection, to the ordering on nilpotent orbits in N_{1}(𝔤). _{□}

3. Exceptional Groups and Lie algebras

Let G be an exceptional simple algebraic group. The tables below describe U_{r,ρ}(G) when ρ is a minimal dimensional representation or the adjoint representation. The variety of nilpotent elements N_{r,ρ}(𝔤) is also provided when ρ is the adjoint representation. Both of these representations satisfy the hypotheses mentioned at the end of Section 1.2. Note that the adjoint representation for E_{8} coincides with the minimal dimensional (non-trivial) irreducible representation.

For good primes N_{r,ρ}(𝔤) was computed for minimal dimensional representations or the adjoint representation. When ρ is the adjoint representation Lawther showed that the Jordan block sizes for unipotent elements and corresponding nilpotent element coincide. Therefore, the tables in [UGA, §4.2] can be used to determine U_{r,ad}(G) for good primes. For minimal dimensional representations Lawther has also shown that the Jordan block sizes coincide except for one case when p = 5 for the regular element in E_{7}. The partition for the unipotent class is (24, 22, 10) while the partition for the nilpotent element is (23^{2}, 10). In order to obtain U_{r,min}(G) in this situation one can use the table given [UGA, §4.1] by replacing 22 (resp. 23+) by 22-23 (resp. 24+) in the p = 5 column.

The methods of computation and conventions used in the tables are the same as the ones given in [UGA, §4]. For unipotent classes, the Jordan block sizes for the images of ρ on a unipotent class are given in [Law1, Law2]. For nilpotent orbits, we have determined the Jordan block sizes by first realizing orbit representatives (given in Section 5). Then we used the MAGMA package [BC, BCP] to construct Chevalley bases for exceptional Lie algebras. From this information we deduced the Jordan block sizes of the image of the adjoint representation on the representatives. The Jordan block sizes are provided in Section 6.

In the tables the first column consists of a list of orbits via the Bala-Carter labeling. The cell values provide the rs for which U_{r,ρ}(G) (or N_{r,ρ}(𝔤)) is the union of the closures of the orbits listed in that particular row. For example, for the group E_{6}, U_{7,min}(G) = \overline{C(A_{4} + A_{1})} ∪ \overline{C(D_{4})} when p = 3, and N_{3,ad}(𝔤) = \overline{O(3A_{1})} when p = 2. (As mentioned in Section 7, the closure relations between nilpotent orbits for E_{7} and E_{8} have not been completely determined. Thus in those cases, when two or more orbits are listed in a row, it may happen that one of the orbits lies in the closure of the other.)

Type E_{6}  U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤)
min  min  adj  adj  adj  adj
Orbits  p=3  p=2  p=3  p=2  p=3  p=2
1  1  1  1–2  1  1–2  1
A_{1}  2
3A_{1} 2 2–3 2–3
A_{2}  3
2A_{2} + A_{1}  3–4  3–6  3–6
D_{4}(a_{1})  5–6  4–5  7–8  4–7  7–8
D_{4}  6  4–7
A_{4} + A_{1}  7
A_{4} + A_{1}, D_{4}  7
D_{5}(a_{1})  8
D_{5}  8–12  8–15  8–15
E_{6}(a_{1})  9–14 13–15  9–18
E_{6}  15+  16+  19+  16+  9+  16+

Type E_{7}  U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤)
min  min  adj  adj  adj  adj
Orbits  p=3  p=2  p=3  p=2  p=3  p=2
1  1  1  1–2  1  1–2  1
A_{1}  2
4A_{1} 2 2–3 2–3
A_{2}, 4A_{1}  3
2A_{2} + A_{1} 3–4 3–6 3–6
D_{4}(a_{1})  5
A_{3} + A_{2} + A_{1} 6 4–5 7–8 4–7 7–8
D_{4} + A_{1}  6
D_{4} + A_{1}, (A_{3} + A_{2})^{(2)}  4–7
A_{4} + A_{2}, D_{4}  7
A_{4} + A_{2}, D_{4} + A_{1}  7
D_{5}(a_{1}) + A_{1}, A_{5} + A_{1}  8
E_{7}(a_{4})  8–12  8–15
E_{6}(a_{1})  9–14  13  9–18
D_{6}, (A_{6})^{(2)}  8–15
E_{7}(a_{3})  14–15
E_{6} 15 9–18
E_{6}, E_{7}(a_{3})  16–17
E_{7}(a_{2})  18–19  19–26  19–26
E_{7}(a_{1})  20–25 16–25  16–31
E_{7}  26+  26+  27+  32+  27+  16+

Type E_{8}  U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤)
min/adj min/adj min/adj min/adj min/adj min/adj
Orbits  p=5  p=3  p=2  p=5  p=3  p=2
1  1–2  1–2  1  1–2  1–2  1
2A_{1}  3  3
4A_{1}  4  2–3  4  2–3
2A_{2} + 2A_{1}  3–6  3–6
D_{4}(a_{1}) + A_{2}  7  7
2A_{3}  8  4–7  8
D_{4} + A_{2}, (A_{3} + A_{2})^{(2)}  4–7
A_{4} + A_{3}  5–10  5–10
E_{8}(a_{7})  11–12  11–12
A_{6} + A_{1}  13–14  13–14
D_{7}(a_{2})  15  15
A_{7}  16  8-15  16
D_{7}, (A_{6})^{(2)}, (D_{5} + A_{2})^{(2)}  8–15
E_{8}(b_{6}) 17–18 9–18 17–18
E_{6} + A_{1}, (A_{7})^{(3)}  9–18
E_{8}(a_{6})  19–22  19–22
E_{8}(b_{5})  19-22  19–22
E_{8}(a_{5})  23–24 23–26  23–24 23–26
E_{8}(a_{4})  16-31
E_{8}(a_{1})  25–50 27–54
E_{8}  51+  55+  32+  25+  27+  16+

Type F_{4} U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤)
min  min  adj  adj  adj  adj
Orbits  p=3  p=2  p=3  p=2  p=3  p=2
1  1  1  1–2  1  1–2  1
A_{1}  2
A_{1} + Ã_{1} 2 2–3 2–3
A_{2}  3
A_{1} + Ã_{2}  3–4  3–6  3–6
F_{4}(a_{3})  5–6  4–5  7–8  4–7  7–8
B_{3}  7–8  6–7  4–7
F_{4}(a_{1})  9–14  8–15  9–18  8–15  8–15
F_{4}  15+  16+  19+  16+  9+  16+

Type G_{2} U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) U_{r,ρ}(G) N_{r,ρ}(𝔤) N_{r,ρ}(𝔤)
min  min  adj  adj  adj  adj
Orbits  p=3  p=2  p=3  p=2  p=3  p=2
1  1  1  1–2  1  1–2  1
A_{1}  2
Ã_{1}  2  2–3  2–3
G_{2}(a_{1})  3–6  3–5  3–8  4–7  3–8
G_{2}  7+  6+  9+  8+  9+  4+

4. Applications

4.1. Restricted Unipotent Varieties. The following two theorems provide a description of the restricted unipotent variety for simple connected algebraic groups when the characteristic of the field is a bad prime. These results are easily deduced by setting r = p for U_{r,ρ}(G) and using the results in Sections 2 and 3.

Theorem (A). Let G be a simple classical connected algebraic group over k where char k = 2.

(i) If Φ is of type B_{l} then U_{1}(G) = \overline{C(2_{2}^{l}, 1_{1})}.

(ii) If Φ is of type C_{l} then U_{1}(G) = \overline{C(2_{1}^{l})}.

(iii) If Φ is of type D_{l} then U_{1}(G) = \overline{C(2_{2}^{l})}.

Theorem (B). Let G be an exceptional algebraic group with p a bad prime. Then (i) If Φ is of type E_{6} then U_{1}(G) = \overline{C(X)} where X = 2A_{2} + A_{1} (p = 3), 3A_{1} (p = 2). (ii) If Φ is of type E_{7} then U_{1}(G) = \overline{C(X)} where X = 2A_{2} + A_{1} (p = 3), 4A_{1} (p = 2). (iii) If Φ is of type E_{8} then U_{1}(G) = \overline{C(X)} where X = A_{4} + A_{3} (p = 5), 2A_{2} + 2A_{1}

(p = 3), 4A_{1} (p = 2). (iv) If Φ is of type F_{4} then U_{1}(G) = \overline{C(X)} where X = A_{1} + Ã_{2} (p = 3), A_{1} + Ã_{1} (p = 2).

(v) If Φ is of type G_{2} then U_{1}(G) = \overline{C(X)} where X = G_{2}(a_{1}) (p = 3), Ã_{1} (p = 2).

4.2. Restricted Nullcones. In a similar way with our computations of N_{r,ρ}(𝔤), we can determine N_{1}(𝔤) over fields of bad characteristic.

Theorem (A). Let G be a simple classical connected algebraic group over k where char k = 2.

(i) If Φ is of type B_{l} then N_{1}(𝔤) = \overline{O(2_{2}^{l}, 1_{1})}.

(ii) If Φ is of type C_{l} then N_{1}(𝔤) = \overline{O(2_{1}^{l})}.

(iii) If Φ is of type D_{l} then N_{1}(𝔤) = \overline{O(2_{2}^{l})}.

Theorem (B). Let G be an exceptional algebraic group with p a bad prime. Then (i) If Φ is of type E_{6} then N_{1}(𝔤) = \overline{O(X)} where X = 2A_{2} + A_{1} (p = 3), 3A_{1} (p = 2). (ii) If Φ is of type E_{7} then N_{1}(𝔤) = \overline{O(X)} where X = 2A_{2} + A_{1} (p = 3), 4A_{1} (p = 2). (iii) If Φ is of type E_{8} then N_{1}(𝔤) = \overline{O(X)} where X = A_{4} + A_{3} (p = 5), 2A_{2} + 2A_{1}

(p = 3), 4A_{1} (p = 2). (iv) If Φ is of type F_{4} then N_{1}(𝔤) = \overline{O(X)} where X = A_{1} + Ã_{2} (p = 3), A_{1} + Ã_{1} (p = 2).

(v) If Φ is of type G_{2} then N_{1}(𝔤) = \overline{O(X)} where X = G_{2}(a_{1}) (p = 3), Ã_{1} (p = 2).

4.3. The previous two theorems show that U_{1}(G) and N_{1}(𝔤) “coincide” for G exceptional. The G-orbits in these two varieties also coincide (see Section 7 for Hasse diagrams). Combining this with our results for classical groups, we can now state the following result.

Theorem. Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0. There exists a one-to-one correspondence between unipotent orbits in U_{1}(G) and nilpotent orbits in N_{1}(𝔤) which respects the closure ordering of orbits. This correspondence also respects dimensions of unipotent classes and nilpotent orbits.

Proof. For good primes, this is established via the computations in [UGA]. (This can also be deduced from [McN1, Theorem 35].) For bad primes, the correspondence is given by Theorems 4.1(A), 4.2(A) for the classical groups and Theorems 4.1(B), 4.2(B) for the exceptional groups. The inclusion relations are checked in Theorem 2.4 for the classical groups and case by case in Section 7 for the exceptional groups. Furthermore, one can verify directly that the orbit correspondence respects dimensions of unipotent classes and nilpotent orbits. _{□}

We should also remark that the under the correspondence in the preceding theorem the Jordan block sizes for the unipotent classes in U_{1}(G) and the nilpotent orbits in N_{1}(𝔤) coincide for the adjoint representation when G is an exceptional group. This can be verified by using [Law1] [McN2] and the tables in Section 6. In general for U and N the Jordan block sizes need not coincide. For example when Φ = G_{2}, p = 3, the partition for the regular unipotent class C(G_{2}) for the adjoint representation is (11, 3) whereas the partition for the corresponding regular nilpotent orbit O(G_{2}) is (9, 3, 2). This leads one to conjecture that there might be a G-equivariant isomorphism between U_{1}(G) and N_{1}(𝔤) for all primes. For good primes such a natural map has been constructed via an exponential process in [CLN].

In [NPV], it was shown that N_{1}(𝔤) is irreducible and the closure of a Richardson orbit when the characteristic of the field is good. Our results also show that N_{1}(𝔤) is an irreducible variety even for bad primes. However, it is not true that N_{1}(𝔤) is the closure of a Richardson orbit (e.g. when Φ is of type G_{2} and p = 2).

5. Orbit representatives

In this section, we provide orbit representatives for exceptional Lie algebras over algebraically closed fields of all characterstics. Throughout, the conventions in [Law1] will be used. If X is the Bala-Carter label for the orbit in characteristic zero then X^{(p)} will denote the new orbit that arises. For the unipotent classes such representatives can be found in [Sh, Mi1, Mi2, Law1, St]. After asking several experts in the field we were amazed to learn that such explicit representatives for nilpotent orbits were not available in the literature. Since we were interested in using such representatives for computation purposes, we wanted to have the orbits representated in the simplest form possible. For this purpose we have attempted to choose representatives which are consistent with the Bala-Carter labeling whenever possible. Several sources we used to construct representatives include [Law1, St, Sp4, Mi2, HS]. We hope that making such a list available will be useful for future computational purposes.

We should make the reader aware of several places in the literature which are pertinent to our results. First, in [CM, page 78], the algorithm for providing the orbit representative for D_{4}(a_{1}) in D_{4} is slightly misleading. In the case of types B and D one may not be able to simply take a sum of root vectors in C^{+}. If one does this for the example given the Jordan block size for this representative is (5, 1^{3}) instead of (5, 3). Rather, one needs to take a generic sum of root vectors in C^{+}. The orbit representative for D_{4}(a_{1}) that needs to be taken is obtained by deleting one of the summands. Second, in [Mi2] for the orbit E_{8}(a_{7}) (in Mizuno’s notation 2A_{4}), the root α_{47} does not exist as a root of E_{8}. This root is used as part of his description of the unipotent class representing this orbit. Finally, the representatives for the nilpotent orbits (A_{3} +A_{2})^{(2)} in E_{7} and E_{8} and (D_{4} +A_{2})^{(2)} in E_{8} are not explicitly given in [HS]. We constructed these representatives by looking at the orbits given by O(4_{3}^{2}, 2_{2}^{2}) in D_{6} (embedded into E_{7} and E_{8}) and O(5_{4}^{2}, 2_{2}^{2}) in D_{7} (embedded into E_{8}).

Type E_{6}

Orbit  Representative
E_{6}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
E_{6}(a_{1})  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{6}}
D_{5}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
E_{6}(a_{3}) x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}} + x_{α_{1}} + x_{α_{1}+α_{3}} + x_{α_{6}} + x_{α_{5}+α_{6}}
D_{5}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
A_{5}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
A_{4} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}}
D_{4}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
A_{4}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
D_{4}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
A_{3} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}}
2A_{2} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}}
A_{3}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}}
A_{2} + 2A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{6}}
2A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}}
A_{2} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}}
A_{2}  x_{α_{1}} + x_{α_{3}}
3A_{1}  x_{α_{1}} + x_{α_{4}} + x_{α_{6}}
2A_{1}  x_{α_{1}} + x_{α_{4}}
A_{1}  x_{α_{1}}
{0}  0

Type E_{7} (I)

Orbit  Representative
E_{7}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{7}(a_{1})  x_{α_{1}} + x_{α_{3}} + x_{α_{3}+α_{4}} + x_{α_{2}+α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{7}(a_{2})  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{2}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{6}+α_{7}}
E_{7}(a_{3})  x_{α_{1}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{7}}
E_{6}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
E_{6}(a_{1})  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{6}}
D_{6}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{7}(a_{4})  x_{α_{1}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{6}+α_{7}}
D_{6}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
D_{5} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}
(A_{6})^{(2)}  x_{α_{1}+α_{3}+α_{4}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{5}+α_{6}}
+x_{α_{6}+α_{7}} + x_{α_{4}+α_{5}+α_{6}+α_{7}}
A_{6}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{7}(a_{5})  x_{α_{1}+α_{3}+α_{4}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{1}+α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}}
+x_{α_{5}+α_{6}+α_{7}} + x_{α_{3}+α_{4}+α_{5}+α_{6}}
D_{5}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
E_{6}(a_{3})  x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}} + x_{α_{1}} + x_{α_{1}+α_{3}} + x_{α_{6}} + x_{α_{5}+α_{6}}
D_{6}(a_{2})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{6}+α_{7}}
D_{5}(a_{1}) + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}} + x_{α_{3}} + x_{α_{4}} + x_{α_{4}+α_{5}}
A_{5} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
(A_{5})′  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
A_{4} + A_{2}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}}
D_{5}(a_{1})  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
A_{4} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}
D_{4} + A_{1}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}

Type E_{7} (II)

Orbit  Representative
(A_{5})′′  x_{α_{2}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
A_{3} + A_{2} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
A_{4}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
(A_{3} + A_{2})^{(2)} x_{α_{7}} + x_{α_{2}+α_{3}+2α_{4}+2α_{5}+2α_{6}+α_{7}} + x_{α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}} + x_{α_{3}}
A_{3} + A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}}
D_{4}(a_{1}) + A_{1}  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{7}}
D_{4}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
A_{3} + 2A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}
D_{4}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
(A_{3} + A_{1})′  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}}
2A_{2} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}}
(A_{3} + A_{1})′′  x_{α_{2}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
A_{2} + 3A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}}
2A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{6}} + x_{α_{7}}
A_{3}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}}
A_{2} + 2A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}}
A_{2} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}}
4A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{5}} + x_{α_{7}}
A_{2}  x_{α_{1}} + x_{α_{3}}
(3A_{1})′  x_{α_{1}} + x_{α_{4}} + x_{α_{6}}
(3A_{1})′′  x_{α_{2}} + x_{α_{5}} + x_{α_{7}}
2A_{1}  x_{α_{1}} + x_{α_{4}}
A_{1}  x_{α_{1}}
{0}  0

Type E_{8} (I)

Orbit  Representative
E_{8}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
E_{8}(a_{1})  x_{α_{1}} + x_{α_{2}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
E_{8}(a_{2})  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{2}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}} + x_{α_{6}} + x_{α_{6}+α_{7}} + x_{α_{8}}
E_{8}(a_{3})  x_{α_{1}+α_{3}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{5}} + x_{α_{6}}
+x_{α_{7}} + x_{α_{8}}
E_{8}(a_{4})  x_{α_{1}+α_{3}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{5}} + x_{α_{6}} + x_{α_{6}+α_{7}} + x_{α_{7}+α_{8}}
E_{7}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{8}(b_{4})  x_{α_{1}+α_{3}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{5}+α_{6}+α_{7}}
+x_{α_{7}} + x_{α_{8}}
E_{8}(a_{5})  x_{α_{1}+α_{3}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{4}+α_{5}+α_{6}}
+x_{α_{6}+α_{7}} + x_{α_{7}+α_{8}}
E_{7}(a_{1})  x_{α_{1}} + x_{α_{3}+α_{4}} + x_{α_{2}+α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{8}(b_{5})  x_{α_{1}+α_{3}+α_{4}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{1}+α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{3}+α_{4}+α_{5}+α_{6}}
+x_{α_{5}+α_{6}+α_{7}} + x_{α_{8}}
(D_{7})^{(2)}  x_{α_{2}+α_{3}+α_{4}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{5}+α_{6}+α_{7}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}}
+x_{α_{6}+α_{7}+α_{8}} + x_{α_{1}} + x_{α_{1}+α_{2}+α_{3}+2α_{4}+2α_{5}+α_{6}}
D_{7}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
E_{8}(a_{6})  x_{α_{1}+α_{3}+α_{4}} + x_{α_{1}+α_{2}+α_{3}+α_{4}} + x_{α_{1}+α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}}
+x_{α_{1}+α_{2}+α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{6}+α_{7}} + x_{α_{7}+α_{8}}
E_{7}(a_{2})  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{2}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{6}+α_{7}}
E_{6} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{8}}
(D_{7}(a_{1}))^{(2)}  x_{α_{1}+α_{3}+α_{4}+α_{5}} + x_{α_{1}+α_{2}+α_{3}+α_{4}} + x_{α_{2}+α_{3}+2α_{4}+α_{5}} + x_{α_{7}+α_{8}} + x_{α_{5}+α_{6}}
+x_{α_{6}+α_{7}} + x_{α_{4}+α_{5}+α_{6}+α_{7}+α_{8}} + x_{α_{1}+α_{2}+2α_{3}+3α_{4}+2α_{5}+α_{6}}
D_{7}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
E_{8}(b_{6})  x_{α_{1}+α_{2}+α_{3}+α_{4}} + x_{α_{1}+α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}+α_{5}} + x_{α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{4}+α_{5}+α_{6}}
+x_{α_{4}+α_{5}+α_{6}+α_{7}} + x_{α_{7}+α_{8}} + x_{α_{6}+α_{7}+α_{8}}
E_{7}(a_{3})  x_{α_{1}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{5}} + x_{α_{5}+α_{6}} + x_{α_{7}}
E_{6}(a_{1}) + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{6}} + x_{α_{8}}
(A_{7})^{(3)}  x_{α_{1}+α_{3}+α_{4}+α_{5}} + x_{α_{1}+α_{2}+α_{3}+α_{4}} + x_{α_{2}+α_{3}+2α_{4}+α_{5}} + x_{α_{5}+α_{6}+α_{7}} + x_{α_{6}+α_{7}+α_{8}}
+x_{α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{4}+α_{5}+α_{6}} + x_{α_{4}+α_{5}+α_{6}+α_{7}+α_{8}}
A_{7}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
D_{7}(a_{2})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}} + x_{α_{8}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{6}+α_{7}}
E_{6}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
D_{6}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
(D_{5} + A_{2})^{(2)}  x_{α_{1}+α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{3}+α_{4}+α_{5}+α_{6}+α_{7}} + x_{α_{1}+α_{2}+α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}+α_{5}+α_{6}}
+x_{α_{2}+α_{4}+α_{5}+α_{6}+α_{7}} + x_{α_{2}+α_{3}+2α_{4}+α_{5}} + x_{α_{7}+α_{8}} + x_{α_{6}+α_{7}+α_{8}}
D_{5} + A_{2}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}} + x_{α_{8}}

22 UNIVERSITY OF GEORGIA VIGRE ALGEBRA GROUP

Type E_{8} (II)

Orbit  Representative
E_{6}(a_{1})  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{2}+α_{4}} + x_{α_{3}+α_{4}} + x_{α_{6}}
E_{7}(a_{4})  x_{α_{1}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}} + x_{α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{6}+α_{7}}
A_{6} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
D_{6}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
(A_{6})^{(2)}  x_{α_{1}+α_{2}+α_{3}} + x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{5}+α_{6}}
+x_{α_{6}+α_{7}} + x_{α_{4}+α_{5}+α_{6}+α_{7}}
A_{6}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
E_{8}(a_{7})  x_{α_{1}+α_{2}+α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{3}+2α_{4}+α_{5}+α_{6}}
+x_{α_{1}+α_{2}+α_{3}+2α_{4}+α_{5}+α_{6}} + x_{α_{3}+α_{4}+α_{5}+α_{6}+α_{7}} + x_{α_{1}+α_{2}+α_{3}+2α_{4}+α_{5}+α_{6}+α_{7}}
+x_{α_{1}+α_{2}+α_{3}+2α_{4}+α_{5}+α_{6}+α_{7}+α_{8}} + x_{α_{4}+α_{5}+α_{6}+α_{7}+α_{8}}
D_{5} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}}
E_{7}(a_{5})  x_{α_{1}+α_{3}+α_{4}} + x_{α_{2}+α_{3}+α_{4}} + x_{α_{1}+α_{3}+α_{4}+α_{5}+α_{6}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}+α_{5}+α_{6}}
+x_{α_{5}+α_{6}+α_{7}} + x_{α_{3}+α_{4}+α_{5}+α_{6}}
E_{6}(a_{3}) + A_{1}  x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}} + x_{α_{1}} + x_{α_{1}+α_{3}} + x_{α_{6}} + x_{α_{5}+α_{6}} + x_{α_{8}}
D_{6}(a_{2})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{5}+α_{6}} + x_{α_{6}+α_{7}}
D_{5}(a_{1}) + A_{2}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{7}} + x_{α_{8}}
A_{5} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{8}}
A_{4} + A_{3}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
D_{5}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
E_{6}(a_{3})  x_{α_{3}+α_{4}+α_{5}} + x_{α_{2}+α_{4}} + x_{α_{1}} + x_{α_{1}+α_{3}} + x_{α_{6}} + x_{α_{5}+α_{6}}
(D_{4} + A_{2})^{(2)}  x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}} + x_{α_{8}} + x_{α_{2}+α_{3}+2α_{4}+2α_{5}+2α_{6}+2α_{7}+α_{8}} + x_{α_{2}+α_{4}+α_{5}}
D_{4} + A_{2}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}} + x_{α_{8}}
A_{4} + A_{2} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
D_{5}(a_{1}) + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{7}}
A_{5}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{6}}
A_{4} + A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
A_{4} + 2A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
D_{5}(a_{1})  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
2A_{3}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}} + x_{α_{8}}
A_{4} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}

Type E_{8} (III)

Orbit  Representative
D_{4}(a_{1}) + A_{2}  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{7}} + x_{α_{8}}
D_{4} + A_{1}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}} + x_{α_{7}}
A_{3} + A_{2} + A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{7}}
A_{4}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
(A_{3} + A_{2})^{(2)} x_{α_{7}} + x_{α_{2}+α_{3}+2α_{4}+2α_{5}+2α_{6}+α_{7}} + x_{α_{5}} + x_{α_{2}+α_{4}+α_{5}} + x_{α_{4}} + x_{α_{3}}
A_{3} + A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{7}}
D_{4}(a_{1}) + A_{1}  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}} + x_{α_{7}}
A_{3} + 2A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}} + x_{α_{8}}
2A_{2} + 2A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{8}}
D_{4}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}} + x_{α_{5}}
D_{4}(a_{1})  x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{3}+α_{4}} + x_{α_{4}+α_{5}}
A_{3} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}} + x_{α_{6}}
2A_{2} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}} + x_{α_{8}}
2A_{2}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{6}}
A_{2} + 3A_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}}
A_{3}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}}
A_{2} + 2A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{5}} + x_{α_{7}}
A_{2} + A_{1}  x_{α_{1}} + x_{α_{4}} + x_{α_{5}}
4A_{1}  x_{α_{1}} + x_{α_{4}} + x_{α_{6}} + x_{α_{8}}
A_{2}  x_{α_{1}} + x_{α_{3}}
3A_{1}  x_{α_{1}} + x_{α_{4}} + x_{α_{6}}
2A_{1}  x_{α_{1}} + x_{α_{4}}
A_{1}  x_{α_{1}}
{0}  0

Type F_{4}

Orbit  Representative
F_{4}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}} + x_{α_{4}}
F_{4}(a_{1})  x_{α_{1}} + x_{α_{2}} + x_{α_{2}+α_{3}} + x_{α_{3}+α_{4}}
F_{4}(a_{2})  x_{α_{1}+α_{2}} + x_{α_{2}+2α_{3}} + x_{α_{4}} + x_{α_{3}+α_{4}}
(C_{3})^{(2)}  x_{α_{4}} + x_{α_{1}+α_{2}+α_{3}} + x_{α_{2}+2α_{3}} + x_{α_{1}+2α_{2}+2α_{3}+2α_{4}}
C_{3}  x_{α_{2}} + x_{α_{3}} + x_{α_{4}}
B_{3}  x_{α_{1}} + x_{α_{2}} + x_{α_{3}}
F_{4}(a_{3})  x_{α_{2}} + x_{α_{1}+α_{2}} + x_{α_{2}+2α_{3}} + x_{α_{1}+2α_{2}+4α_{3}+2α_{4}}
C_{3}(a_{1})^{(2)} x_{α_{1}+α_{2}} + x_{α_{2}+2α_{3}+α_{4}} + x_{α_{1}+α_{2}+2α_{3}+2α_{4}} + x_{α_{1}+2α_{2}+4α_{3}+2α_{4}}
C_{3}(a_{1})  x_{α_{2}} + x_{α_{4}} + x_{α_{2}+2α_{3}}
(Ã_{2} + A_{1})^{(2)} x_{α_{1}+α_{2}+α_{3}} + x_{α_{2}+2α_{3}+α_{4}} + x_{α_{1}+α_{2}+2α_{3}+2α_{4}} + x_{α_{1}+2α_{2}+2α_{3}}
Ã_{2} + A_{1}  x_{α_{1}} + x_{α_{3}} + x_{α_{4}}
(B_{2})^{(2)}  x_{α_{1}+α_{2}} + x_{α_{1}+α_{2}+2α_{3}} + x_{α_{2}+2α_{3}+2α_{4}}
B_{2}  x_{α_{2}} + x_{α_{3}}
A_{2} + Ã_{1}  x_{α_{1}} + x_{α_{2}} + x_{α_{4}}
Ã_{2}  x_{α_{3}} + x_{α_{4}}
(A_{2})^{(2)}  x_{α_{1}+2α_{2}+2α_{3}} + x_{α_{1}+α_{2}+2α_{3}+2α_{4}}
A_{2}  x_{α_{1}} + x_{α_{2}}
(Ã_{1})^{(2)}  x_{α_{1}+2α_{2}+3α_{3}+2α_{4}} + x_{2α_{1}+3α_{2}+4α_{3}+2α_{4}}
A_{1} + Ã_{1}  x_{α_{1}} + x_{α_{3}}
Ã_{1}  x_{α_{3}}
A_{1}  x_{α_{1}}
{0}  0

Type G_{2}
Orbit Representative p ≥ 3 Representative p = 2
G_{2}  x_{α_{1}} + x_{α_{2}}  x_{α_{1}} + x_{α_{2}}
G_{2}(a_{1})  x_{α_{2}} + x_{2α_{1}+α_{2}}  x_{α_{1}} + x_{2α_{1}+α_{2}}
(Ã_{1})^{(3)}  x_{α_{2}} + x_{α_{1}+α_{2}}
Ã_{1}  x_{α_{1}}  x_{α_{1}}
A_{1}  x_{α_{2}}  x_{α_{2}}
{0}  0  0

6. Jordan Blocks

We provide below the Jordan block sizes of ad(x) for x a nilpotent element in an exceptional Lie algebra 𝔤 when p is a bad prime. Lawther [Law1, Law2] computed this information for unipotent elements for the minimal and adjoint representations for exceptional groups for all primes.

For good primes the nilpotent orbits and the unipotent classes are in bijective correspondence. Under this bijection, Lawther has also shown that the Jordan block sizes for the unipotent classes and the nilpotent orbits coincide for the adjoint representation. Consequently, our computations in conjunction with Lawther’s results complete the determination of the Jordan block sizes on the adjoint representation for exceptional Lie algebras for all primes. Furthermore, our work shows that the Jordan block sizes for corresponding unipotent and nilpotent classes need not coincide in bad characteristic.

Type E_{6}

Orbit Prime  Partition  Orbit Prime  Partition
E_{6}  3  (9^{6}, 8^{3})  2A_{2} + A_{1}  3  (3^{24}, 2^{3})
2  (16^{4}, 4^{2}, 3^{2})  2  (4^{16}, 2^{6}, 1^{2})
E_{6}(a_{1})  3  (9^{6}, 8^{3})  A_{3}  3 (7, 5^{5}, 4^{8}, 3, 1^{11})
2  (16^{2}, 11^{2}, 8^{2}, 4^{2})  2  (4^{12}, 3^{8}, 1^{6})
D_{5}  3  (9^{5}, 8^{4}, 1)  A_{2} + 2A_{1}  3  (3^{23}, 2^{4}, 1)
2  (8^{8}, 4^{2}, 3^{2})  2  (4^{10}, 3^{2}, 2^{16})
E_{6}(a_{3})  3  (9^{3}, 8^{2}, 6^{4}, 3^{3}, 2)  2A_{2}  3  (3^{23}, 2, 1^{7})
2  (8^{8}, 4^{2}, 3^{2})  2  (4^{16}, 1^{14})
D_{5}(a_{1})  3  (9^{3}, 8^{2}, 6^{4}, 3^{2}, 2^{2}, 1)  A_{2} + A_{1}  3  (3^{22}, 2^{2}, 1^{8})
2  (8^{2}, 7^{4}, 5^{2}, 4^{2}, 3^{4}, 1^{4})  2  (4^{8}, 3^{6}, 2^{10}, 1^{8})
A_{5}  3  (9^{3}, 7^{4}, 6^{2}, 3^{2}, 2, 1^{3})  A_{2}  3  (3^{21}, 1^{15})
2  (8^{8}, 2^{6}, 1^{2})  2  (4^{2}, 3^{18}, 1^{16})
A_{4} + A_{1}  3  (9, 8^{2}, 7, 6^{2}, 5^{3}, 3^{6}, 1)  3A_{1}  3  (3^{13}, 2^{14}, 1^{11})
2  (8^{4}, 6^{2}, 5^{2}, 4^{4}, 2^{4})  2  (2^{38}, 1^{2})
D_{4}  3  (9^{2}, 7^{7}, 3, 1^{8})  2A_{1}  3  (3^{8}, 2^{16}, 1^{22})
2  (4^{18}, 3^{2})  2  (2^{32}, 1^{14})
A_{4}  3  (9, 7^{5}, 5^{3}, 3^{5}, 1^{4})  A_{1}  3  (3, 2^{20}, 1^{35})
2  (8^{2}, 7^{4}, 5^{2}, 4^{2}, 3^{4}, 1^{4})  2  (2^{22}, 1^{34})
D_{4}(a_{1})  3  (7, 6^{2}, 5^{6}, 3^{9}, 1^{2})  {0}  3,2  (1^{78})
2  (4^{12}, 3^{8}, 1^{6})
A_{3} + A_{1}  3 (7, 6^{2}, 5^{3}, 4^{6}, 3^{4}, 2^{2}, 1^{4})
2  (4^{16}, 2^{6}, 1^{2})

Type E_{7} (I)

Orbit  Prime  Partition
E_{7}  3  (27^{3}, 9^{4}, 8^{2})
2  (16^{6}, 15^{2}, 2, 1^{5})
E_{7}(a_{1}) 3 (27, 19, 17^{3}, 9^{4})
2  (16^{6}, 15^{2}, 2, 1^{5})
E_{7}(a_{2}) 3 (19, 17^{3}, 9^{7})
2  (16^{4}, 13^{4}, 4^{2}, 3^{2}, 2, 1)
E_{7}(a_{3}) 3 (19, 17, 15^{2}, 11^{2}, 9^{3}, 6^{2}, 3^{2})
2  (16^{2}, 15^{2}, 9^{4}, 8^{4}, 2, 1)
E_{6} 3 (9^{13}, 8^{2})
2  (16^{4}, 13^{4}, 4^{2}, 3^{2}, 1^{3})
E_{6}(a_{1}) 3 (9^{13}, 8^{2})
2  (16^{2}, 13^{2}, 11^{2}, 9^{2}, 8^{2}, 5^{2}, 4^{2}, 1)
D_{6}  3 (19, 16^{2}, 15, 11, 10^{2}, 9^{2}, 6^{2}, 3, 1^{3})
2  (8^{14}, 7^{2}, 2, 1^{5})
E_{7}(a_{4}) 3 (9^{11}, 8^{2}, 6^{2}, 3^{2})
2  (8^{14}, 7^{2}, 2, 1^{5})
D_{6}(a_{1}) 3 (9^{11}, 7^{4}, 3, 1^{3})
2  (8^{8}, 7^{4}, 6^{4}, 2^{6}, 1^{5})
D_{5} + A_{1} 3 (9^{11}, 7^{4}, 3, 1^{3})
2  (8^{12}, 5^{4}, 4^{2}, 3^{2}, 2, 1)
(A_{6})^{(2)}  2  (8^{14}, 7^{2}, 2, 1^{5})
A_{6}  3  (9^{11}, 7^{4}, 3, 1^{3})
2  (8^{14}, 7^{2}, 1^{7})
E_{7}(a_{5}) 3 (9^{9}, 7, 6^{2}, 5^{3}, 3^{6})
2  (8^{12}, 5^{4}, 4^{2}, 3^{2}, 2, 1)
D_{5} 3 (9^{7}, 8^{8}, 1^{6})
2  (8^{12}, 5^{4}, 4^{2}, 3^{2}, 1^{3})
E_{6}(a_{3}) 3 (9^{5}, 8^{2}, 6^{8}, 3^{8})
2  (8^{12}, 5^{4}, 4^{2}, 3^{2}, 1^{3})
D_{6}(a_{2}) 3 (9^{9}, 3^{1}5, 1^{7})
2  (8^{8}, 7^{4}, 6^{4}, 2^{7}, 1^{3})

Type E_{7} (II)

Orbit  Prime  Partition
D_{5}(a_{1}) + A_{1}  3  (9^{5}, 7^{4}, 6^{6}, 3^{7}, 1^{3})
2  (8^{4}, 7^{2}, 6^{4}, 5^{4}, 4^{6}, 3^{2}, 2^{6}, 1)
A_{5} + A_{1} 3 (9^{9}, 3^{1}6, 2^{2})
2  (8^{8}, 7^{4}, 6^{4}, 2^{7}, 1^{3})
(A_{5})′ 3 (9^{5}, 7^{4}, 6^{6}, 3^{7}, 1^{3})
2  (8^{8}, 7^{4}, 6^{4}, 2^{6}, 1^{5})
A_{4} + A_{2} 3 (9^{3}, 7, 6^{8}, 5^{3}, 3^{12})
2  (8^{6}, 7^{2}, 5^{4}, 4^{12}, 1^{3})
D_{5}(a_{1}) 3 (9^{3}, 8^{4}, 7^{2}, 6^{6}, 3^{4}, 2^{4}, 1^{4})
2  (8^{2}, 7^{6}, 5^{8}, 4^{2}, 3^{6}, 1^{9})
A_{4} + A_{1}  3 (9, 8^{2}, 7^{3}, 6^{4}, 5^{5}, 4^{2}, 3^{8}, 2^{2}, 1^{2})
2  (8^{4}, 7^{2}, 6^{4}, 5^{4}, 4^{6}, 3^{2}, 2^{6}, 1)
D_{4} + A_{1} 3 (9^{2}, 8^{4}, 7^{5}, 6^{4}, 3^{2}, 2^{4}, 1^{10})
2  (4^{30}, 3^{2}, 2, 1^{5})
(A_{5})′′ 3 (9^{9}, 3^{15}, 1^{7})
2  (8^{4}, 7^{12}, 2, 1^{15})
A_{3} + A_{2} + A_{1} 3 (7, 6^{8}, 5^{3}, 3^{21})
2  (4^{30}, 3^{2}, 2, 1^{5})
A_{4} 3 (9, 7^{7}, 5^{9}, 3^{7}, 1^{9})
2  (8^{2}, 7^{6}, 5^{8}, 4^{2}, 3^{6}, 1^{9})
(A_{3} + A_{2})^{(2)}  2  (4^{30}, 3^{2}, 2, 1^{5})
A_{3} + A_{2}  3  (7, 6^{8}, 5, 4^{4}, 3^{18}, 1^{3})
2  (4^{30}, 3^{2}, 1^{7})
D_{4}(a_{1}) + A_{1} 3 (7, 6^{6}, 5^{4}, 4^{4}, 3^{16}, 1^{6})
2  (4^{24}, 3^{4}, 2^{10}, 1^{5})
D_{4} 3 (9^{2}, 7^{13}, 3, 1^{21})
2  (4^{30}, 3^{2}, 1^{7})
A_{3} + 2A_{1} 3 (7, 6^{4}, 5^{7}, 4^{4}, 3^{11}, 2^{6}, 1^{6})
2  (4^{24}, 3^{4}, 2^{11}, 1^{3})
D_{4}(a_{1}) 3 (7, 6^{2}, 5^{12}, 3^{15}, 1^{9})
2  (4^{20}, 3^{12}, 1^{17})

Type E_{7} (III)

Orbit  Prime  Partition
(A_{3} + A_{1})′  3 (7, 6^{2}, 5^{7}, 4^{10}, 3^{6}, 2^{6}, 1^{9})
2  (4^{24}, 3^{4}, 2^{10}, 1^{5})
2A_{2} + A_{1} 3 (3^{43}, 2^{2})
2  (4^{24}, 3^{4}, 2^{10}, 1^{5})
(A_{3} + A_{1})′′ 3 (7, 5^{15}, 3^{10}, 1^{21})
2  (4^{20}, 3^{12}, 2, 1^{15})
A_{2} + 3A_{1} 3 (3^{42}, 1^{7})
2  (4^{14}, 3^{6}, 2^{29}, 1)
2A_{2} 3 (3^{42}, 1^{7})
2  (4^{20}, 3^{12}, 1^{17})
A_{3} 3 (7, 5^{7}, 4^{16}, 3, 1^{24})
2  (4^{20}, 3^{12}, 1^{17})
A_{2} + 2A_{1} 3 (3^{37}, 2^{8}, 1^{6})
2  (4^{14}, 3^{6}, 2^{28}, 1^{3})
A_{2} + A_{1} 3 (3^{34}, 2^{8}, 1^{15})
2  (4^{10}, 3^{14}, 2^{18}, 1^{15})
4A_{1} 3 (3^{28}, 2^{14}, 1^{21})
2  (2^{63}, 1^{7})
A_{2} 3 (3^{33}, 1^{34})
2  (4^{2}, 3^{30}, 1^{35})
(3A_{1})′ 3 (3^{19}, 2^{26}, 1^{24})
2  (2^{62}, 1^{9})
(3A_{1})′′ 3 (3^{27}, 1^{52})
2  (2^{53}, 1^{27})
2A_{1} 3 (3^{10}, 2^{32}, 1^{39})
2  (2^{52}, 1^{29})
A_{1} 3 (3, 2^{32}, 1^{66})
2  (2^{34}, 1^{65})
{0}  3,2  (1^{133})

VARIETIES OF NILPOTENT ELEMENTS FOR SIMPLE LIE ALGEBRAS II 29

Type E_{8} (I)

Orbit  Prime  Partition
E_{8}  5  (25^{8}, 24^{2})
3  (27^{7}, 26^{2}, 3, 2^{2})
2  (16^{8}, 15^{8})
E_{8}(a_{1}) 5 (25^{8}, 24^{2})
3  (27^{7}, 26^{2}, 3, 2^{2})
2  (16^{8}, 15^{8})
E_{8}(a_{2}) 5 (25^{6}, 24^{2}, 20^{2}, 5^{2})
3  (27^{5}, 26^{2}, 20^{2}, 9^{2}, 3)
2  (16^{8}, 15^{8})
E_{8}(a_{3}) 5 (25^{4}, 24^{2}, 22^{2}, 15^{2}, 10^{2}, 3^{2})
3  (27^{3}, 20^{2}, 18^{4}, 9^{4}, 8^{2}, 3)
2  (16^{8}, 15^{8})
E_{8}(a_{4}) 5 (25^{2}, 24^{2}, 20^{2}, 15^{4}, 10^{4}, 5^{2})
3  (27, 26^{2}, 21^{2}, 18^{2}, 15^{2}, 12^{2}, 9^{2}, 8^{2}, 3)
2  (16^{8}, 15^{8})
E_{7} 5 (25^{4}, 23^{4}, 15^{2}, 10^{2}, 3, 1^{3})
3  (27^{3}, 19^{4}, 18^{2}, 9^{4}, 8^{2}, 1^{3})
2  (16^{6}, 15^{2}, 14^{8}, 2^{2}, 1^{6})
E_{8}(b_{4}) 5 (25^{2}, 23, 21, 19, 17^{2}, 15^{2}, 13, 11^{3}, 7^{2}, 5, 3^{2})
3  (27, 20^{2}, 19, 18^{2}, 17^{3}, 9^{8}, 3)
2  (16^{6}, 15^{2}, 14^{8}, 2^{2}, 1^{6})
E_{8}(a_{5}) 5 (23^{2}, 21, 19, 17, 15^{3}, 13^{2}, 11^{3}, 10^{2}, 7, 5, 3^{3})
3  (23^{2}, 21, 19, 17, 15^{3}, 12^{4}, 11^{2}, 9, 7, 5, 3^{3})
2  (16^{4}, 15^{4}, 12^{8}, 4^{4}, 3^{4})
E_{7}(a_{1}) 5 (25^{2}, 22^{2}, 19, 17, 16^{2}, 15, 12^{2}, 11^{2}, 7, 6^{2}, 3, 1^{3})
3  (27, 19^{5}, 17^{3}, 9^{8}, 1^{3})
2  (16^{6}, 15^{2}, 14^{8}, 2^{2}, 1^{6})
E_{8}(b_{5}) 5 (23, 19^{2}, 17^{3}, 15^{3}, 11, 10^{4}, 9, 7^{2}, 5, 3^{4})
3  (19, 18^{2}, 17^{3}, 11^{3}, 9^{11}, 7, 3)
2  (16^{4}, 15^{4}, 12^{8}, 4^{4}, 3^{4})
(D_{7})^{(2)} 2 (16^{2}, 15^{6}, 8^{14}, 7^{2})
D_{7}  5 (23, 22^{2}, 19, 16^{2}, 15, 13^{3}, 12^{2}, 11, 10^{2}, 7, 4^{2}, 3, 1^{3})
3 (23, 22^{2}, 19, 16^{2}, 15, 13^{3}, 12^{2}, 11, 10^{2}, 7, 4^{2}, 3, 1^{3})
2  (8^{24}, 7^{8})
E_{8}(a_{6}) 5 (19^{2}, 15^{5}, 11, 10^{8}, 9, 5^{7})
3  (19, 18^{2}, 15^{3}, 13^{3}, 9^{9}, 7, 6^{2}, 3^{3})
2  (16^{2}, 15^{6}, 8^{14}, 7^{2})

Type E_{8} (II)

Orbit  Prime  Partition
E_{7}(a_{2})  5  (23, 19, 18^{2}, 17, 16^{2}, 15^{2}, 11, 10^{4}, 8^{2}, 7, 4^{2}, 3^{2}, 1^{3})
3  (19, 18^{2}, 17^{3}, 10^{6}, 9^{7}, 8^{2}, 1^{3})
2  (16^{4}, 14^{4}, 13^{4}, 12^{4}, 4^{2}, 3^{2}, 2^{6}, 1^{2})
E_{6} + A_{1} 5 (23, 18^{2}, 17^{3}, 16^{2}, 15, 11, 10^{2}, 9^{3}, 8^{2}, 3^{2}, 2^{4}, 1^{3})
3  (9^{25}, 8^{2}, 3, 2^{2})
2  (16^{4}, 14^{4}, 13^{4}, 12^{4}, 4^{2}, 3^{2}, 2^{6}, 1^{2})
(D_{7}(a_{1}))^{(2)}  2  (16^{2}, 15^{2}, 14^{4}, 9^{4}, 8^{8}, 6^{4}, 2^{2}, 1^{2})
D_{7}(a_{1})  5  (19, 17^{2}, 15^{3}, 13, 11^{3}, 10^{6}, 7^{2}, 5^{4}, 3^{3}, 1)
3  (19, 17^{2}, 15^{3}, 12^{2}, 11^{4}, 9^{5}, 6^{4}, 3^{4}, 1)
2  (8^{14}, 7^{18}, 1^{10})
E_{8}(b_{6}) 5 (17, 15^{3}, 13^{2}, 11^{3}, 10^{6}, 7^{3}, 5^{8}, 3^{2})
3  (9^{25}, 8^{2}, 1^{7})
2  (16^{2}, 15^{2}, 12^{4}, 11^{4}, 8^{6}, 7^{2}, 4^{8})
E_{7}(a_{3}) 5 (19, 17, 16^{2}, 15^{2}, 12^{2}, 11^{2}, 10^{6}, 7, 6^{2}, 5^{3}, 3, 2^{2}, 1^{3})
3  (19, 17, 16^{2}, 15^{2}, 12^{2}, 11^{2}, 10^{4}, 9^{3}, 6^{4}, 3^{2}, 2^{2}, 1^{3})
2  (16^{2}, 15^{2}, 14^{4}, 9^{4}, 8^{8}, 6^{4}, 2^{2}, 1^{2})
E_{6}(a_{1}) + A_{1}  5 (17, 15, 14^{2}, 13^{2}, 12^{2}, 11, 10^{4}, 9^{2}, 8^{2}, 7, 5^{7}, 3^{2}, 2^{2}, 1)
3  (9^{25}, 8^{2}, 3, 2^{2})
2  (16^{2}, 14^{2}, 13^{2}, 12^{2}, 11^{2}, 10^{2}, 9^{2}, 8^{4}, 6^{2}, 5^{2}, 4^{4}, 2^{4})
(A_{7})^{(3)}  3  (9^{25}, 8^{2}, 3, 2^{2})
A_{7}  5  (16^{2}, 15, 13^{3}, 12^{2}, 11, 10^{2}, 9^{3}, 8^{4}, 7, 5^{7}, 3, 1^{3})
3  (9^{25}, 8^{2}, 1^{7})
2  (8^{24}, 7^{8})
D_{7}(a_{2}) 5 (15, 14^{2}, 11, 10^{10}, 9^{2}, 5^{15}, 1)
3  (9^{19}, 7^{8}, 3^{5}, 1^{6})
2  (8^{24}, 7^{8})
E_{6} 5 (23, 17^{7}, 15, 11, 9^{7}, 3, 1^{14})
3  (9^{25}, 8^{2}, 1^{7})
2  (16^{4}, 13^{12}, 4^{2}, 3^{2}, 1^{14})
D_{6} 5 (19, 16^{4}, 15, 11^{6}, 10^{4}, 6^{4}, 5^{2}, 1^{10})
3  (19, 16^{4}, 15, 11^{5}, 10^{4}, 9^{2}, 6^{4}, 3, 1^{10})
2  (8^{22}, 7^{2}, 6^{8}, 2^{2}, 1^{6})
(D_{5} + A_{2})^{(2)}  2  (8^{22}, 7^{2}, 6^{8}, 2^{2}, 1^{6})
D_{5} + A_{2}  5  (15, 13^{2}, 11^{3}, 10^{8}, 9, 7^{2}, 5^{11}, 3^{5}, 1)
3  (9^{19}, 8^{4}, 6^{4}, 3^{7})
2  (8^{20}, 7^{4}, 4^{12}, 3^{4})
E_{6}(a_{1}) 5 (17, 15, 13^{6}, 11, 10^{2}, 9^{6}, 7, 5^{7}, 3, 1^{8})
3  (9^{25}, 8^{2}, 1^{7})
2  (16^{2}, 13^{6}, 11^{2}, 9^{6}, 8^{2}, 5^{6}, 4^{2}, 1^{8})

VARIETIES OF NILPOTENT ELEMENTS FOR SIMPLE LIE ALGEBRAS II 31

Type E_{8} (III)

Orbit  Prime  Partition
E_{7}(a_{4})  5  (15, 13, 12^{2}, 11^{2}, 10^{8}, 8^{2}, 7, 5^{10}, 4^{2}, 3^{2}, 2^{2}, 1^{3})
3  (9^{19}, 8^{2}, 7^{4}, 6^{2}, 3^{6}, 1^{3})
2  (8^{22}, 7^{2}, 6^{8}, 2^{2}, 1^{6})
A_{6} + A_{1}  5 (13^{3}, 12^{2}, 11, 10^{2}, 9^{3}, 8^{4}, 7^{5}, 6^{4}, 5^{3}, 4^{2}, 3^{2}, 2^{2}, 1^{3})
3  (9^{19}, 8^{2}, 7^{4}, 6^{2}, 3^{6}, 1^{3})
2  (8^{22}, 7^{2}, 6^{8}, 2^{2}, 1^{6})
D_{6}(a_{1}) 5 (15, 12^{4}, 11, 10^{6}, 9^{4}, 7, 5^{10}, 3^{5}, 1^{6})
3  (9^{19}, 7^{8}, 3^{5}, 1^{6})
2  (8^{8}, 7^{12}, 6^{12}, 2^{6}, 1^{16})
(A_{6})^{(2)}  2  (8^{22}, 7^{2}, 6^{8}, 2^{2}, 1^{6})
A_{6}  5  (13^{3}, 11^{5}, 9^{3}, 7^{13}, 5^{3}, 3^{5}, 1^{6})
3  (9^{19}, 7^{8}, 3^{5}, 1^{6})
2  (8^{14}, 7^{18}, 1^{10})
E_{8}(a_{7}) 5 (11, 10^{6}, 9^{3}, 7^{5}, 5^{20}, 3^{5})
3  (9^{14}, 7^{2}, 6^{8}, 5^{6}, 3^{10})
2  (8^{20}, 7^{4}, 4^{12}, 3^{4})
D_{5} + A_{1} 5 (15, 12^{2}, 11^{5}, 10^{4}, 9^{3}, 8^{2}, 5^{10}, 3, 2^{6}, 1^{6})
3  (9^{15}, 8^{8}, 7^{4}, 3, 2^{6}, 1^{6})
2  (8^{20}, 6^{4}, 5^{4}, 4^{6}, 3^{2}, 2^{6}, 1^{2})
E_{7}(a_{5}) 5 (11, 10^{6}, 9, 8^{4}, 7^{2}, 6^{2}, 5^{18}, 4^{2}, 3^{3}, 1^{3})
3  (9^{13}, 8^{2}, 7, 6^{8}, 5^{3}, 4^{6}, 3^{6}, 1^{3})
2  (8^{20}, 6^{4}, 5^{4}, 4^{6}, 3^{2}, 2^{6}, 1^{2})
E_{6}(a_{3}) + A_{1} 5 (11, 10^{4}, 9^{3}, 8^{4}, 7^{3}, 6^{2}, 5^{17}, 3^{3}, 2^{4}, 1^{3})
3  (9^{9}, 8^{2}, 6^{16}, 3^{17}, 2^{2})
2  (8^{20}, 6^{4}, 5^{4}, 4^{6}, 3^{2}, 2^{6}, 1^{2})
D_{6}(a_{2}) 5 (11, 10^{6}, 8^{4}, 7^{5}, 5^{17}, 4^{4}, 3, 1^{6})
3  (9^{9}, 7^{4}, 6^{14}, 3^{15}, 1^{10})
2  (8^{12}, 7^{4}, 6^{16}, 2^{12}, 1^{4})
D_{5}(a_{1}) + A_{2} 5 (11, 10^{2}, 9^{3}, 8^{6}, 7^{4}, 6^{4}, 5^{10}, 4^{2}, 3^{7}, 2^{4}, 1^{3})
3  (9^{9}, 8^{2}, 6^{16}, 3^{17}, 2^{2})
2  (8^{6}, 7^{10}, 5^{4}, 4^{20}, 3^{8}, 1^{6})
A_{5} + A_{1} 5 (11, 10^{4}, 9^{3}, 8^{2}, 7^{4}, 6^{6}, 5^{13}, 4^{2}, 3, 2^{4}, 1^{6})
3  (9^{9}, 7^{4}, 6^{14}, 3^{16}, 2^{2}, 1^{3})
2  (8^{12}, 7^{4}, 6^{16}, 2^{12}, 1^{4})
A_{4} + A_{3} 5 (5^{48}, 4^{2})
3  (9^{7}, 8^{2}, 7^{6}, 6^{6}, 5^{6}, 4^{6}, 3^{10}, 2^{2}, 1^{3})
2  (8^{10}, 7^{6}, 4^{30}, 3^{2})

Type E_{8} (IV)

Orbit  Prime  Partition
D_{5}  5  (15, 11^{9}, 9^{7}, 5^{10}, 1^{21})
3  (9^{11}, 8^{16}, 1^{21})
2  (8^{20}, 5^{12}, 4^{2}, 3^{2}, 1^{14})
E_{6}(a_{3}) 5 (11, 10^{2}, 9^{7}, 7^{7}, 5^{17}, 3^{2}, 1^{14})
3  (9^{9}, 8^{2}, 6^{16}, 3^{16}, 1^{7})
2  (8^{20}, 5^{12}, 4^{2}, 3^{2}, 1^{14})
(D_{4} + A_{2})^{(2)}  2  (8^{10}, 7^{2}, 6^{4}, 5^{4}, 4^{24}, 2^{6}, 1^{2})
D_{4} + A_{2}  5  (11, 9^{6}, 7^{14}, 5^{7}, 3^{14}, 1^{8})
3  (9^{8}, 7^{7}, 6^{12}, 3^{16}, 1^{7})
2  (4^{56}, 3^{8})
A_{4} + A_{2} + A_{1} 5 (5^{46}, 4^{2}, 3^{2}, 2^{2})
3  (9^{3}, 8^{2}, 7, 6^{18}, 5^{3}, 3^{25})
2  (8^{10}, 7^{2}, 6^{4}, 5^{4}, 4^{24}, 2^{6}, 1^{2})
D_{5}(a_{1}) + A_{1}  5 (11, 9^{3}, 8^{6}, 7^{8}, 6^{6}, 5^{3}, 4^{2}, 3^{7}, 2^{10}, 1^{6})
3  (9^{5}, 8^{6}, 7^{4}, 6^{12}, 3^{11}, 2^{8}, 1^{6})
2  (8^{4}, 7^{6}, 6^{8}, 5^{8}, 4^{10}, 3^{6}, 2^{10}, 1^{8})
A_{5} 5 (11, 10^{2}, 9^{7}, 6^{14}, 5^{9}, 4^{2}, 1^{17})
3  (9^{9}, 7^{4}, 6^{14}, 3^{15}, 1^{10})
2  (8^{8}, 7^{12}, 6^{12}, 2^{6}, 1^{16})
A_{4} + A_{2} 5 (5^{46}, 3^{5}, 1^{3})
3  (9^{3}, 7^{5}, 6^{16}, 5^{3}, 3^{24}, 1^{3})
2  (8^{6}, 7^{10}, 5^{4}, 4^{20}, 3^{8}, 1^{6})
A_{4} + 2A_{1} 5 (5^{45}, 3^{4}, 2^{4}, 1^{3})
3  (9^{1}, 8^{4}, 7^{3}, 6^{12}, 5^{7}, 4^{4}, 3^{17}, 2^{4}, 1^{4})
2  (8^{6}, 7^{2}, 6^{12}, 5^{4}, 4^{14}, 3^{2}, 2^{16})
D_{5}(a_{1}) 5 (11, 9, 8^{8}, 7^{8}, 6^{8}, 5, 3^{8}, 2^{8}, 1^{15})
3  (9^{3}, 8^{8}, 7^{6}, 6^{10}, 3^{8}, 2^{8}, 1^{15})
2  (8^{2}, 7^{10}, 5^{20}, 4^{2}, 3^{10}, 1^{24})
2A_{3} 5 (5^{38}, 4^{12}, 1^{10})
3  (8^{4}, 7^{6}, 6^{4}, 5^{10}, 4^{16}, 3^{6}, 2^{4}, 1^{10})
2  (4^{56}, 3^{8})
A_{4} + A_{1} 5 (5^{45}, 3, 2^{6}, 1^{8})
3  (9, 8^{2}, 7^{7}, 6^{8}, 5^{9}, 4^{6}, 3^{1}2, 2^{6}, 1^{9})
2  (8^{4}, 7^{6}, 6^{8}, 5^{8}, 4^{10}, 3^{6}, 2^{10}, 1^{8})
D_{4}(a_{1}) + A_{2} 5 (5^{36}, 3^{20}, 1^{8})
3  (7, 6^{14}, 5^{6}, 3^{42}, 1)
2  (4^{46}, 3^{18}, 1^{10})

Type E_{8} (V)

Orbit  Prime  Partition
D_{4} + A_{1}  5  (11, 8^{6}, 7^{14}, 6^{6}, 3^{2}, 2^{14}, 1^{21})
3  (9^{2}, 8^{6}, 7^{13}, 6^{6}, 3^{2}, 2^{14}, 1^{21})
2  (4^{54}, 3^{2}, 2^{10}, 1^{6})
A_{3} + A_{2} + A_{1} 5 (5^{32}, 4^{8}, 3^{10}, 2^{10}, 1^{6})
3  (7, 6^{14}, 5^{3}, 4^{6}, 3^{37}, 2^{2}, 1^{3})
2  (4^{54}, 3^{2}, 2^{10}, 1^{6})
A_{4} 5 (5^{45}, 1^{23})
3  (9, 7^{11}, 5^{21}, 3^{11}, 1^{24})
2  (8^{2}, 7^{10}, 5^{20}, 4^{2}, 3^{10}, 1^{24})
(A_{3} + A_{2})^{(2)}  2  (4^{54}, 3^{2}, 2^{10}, 1^{6})
A_{3} + A_{2}  5  (5^{30}, 4^{8}, 3^{13}, 2^{8}, 1^{11})
3  (7, 6^{12}, 5^{5}, 4^{8}, 3^{34}, 1^{10})
2  (4^{46}, 3^{18}, 1^{10})
D_{4}(a_{1}) + A_{1} 5 (5^{29}, 4^{6}, 3^{14}, 2^{14}, 1^{9})
3  (7, 6^{8}, 5^{12}, 4^{6}, 3^{28}, 2^{8}, 1^{9})
2  (4^{40}, 3^{12}, 2^{18}, 1^{16})
A_{3} + 2A_{1} 5 (5^{25}, 4^{10}, 3^{14}, 2^{14}, 1^{13})
3 (7, 6^{6}, 5^{11}, 4^{14}, 3^{19}, 2^{12}, 1^{13})
2  (4^{44}, 3^{4}, 2^{28}, 1^{4})
2A_{2} + 2A_{1} 5 (5^{18}, 4^{12}, 3^{20}, 2^{20}, 1^{10})
3  (3^{80}, 2^{4})
2  (4^{44}, 3^{4}, 2^{28}, 1^{4})
D_{4} 5 (11, 7^{26}, 3, 1^{52})
3  (9^{2}, 7^{25}, 3, 1^{52})
2  (4^{54}, 3^{2}, 1^{26})
D_{4}(a_{1}) 5 (5^{29}, 3^{25}, 1^{28})
3  (7, 6^{2}, 5^{24}, 3^{27}, 1^{28})
2  (4^{36}, 3^{20}, 1^{44})
A_{3} + A_{1} 5 (5^{21}, 4^{16}, 3^{9}, 2^{14}, 1^{24})
3 (7, 6^{2}, 5^{15}, 4^{18}, 3^{10}, 2^{14}, 1^{24})
2  (4^{40}, 3^{12}, 2^{18}, 1^{16})
2A_{2} + A_{1} 5 (5^{14}, 4^{14}, 3^{23}, 2^{18}, 1^{17})
3  (3^{79}, 2^{2}, 1^{7})
2  (4^{40}, 3^{12}, 2^{18}, 1^{16})

Type E_{8} (VI)

Orbit  Prime  Partition
2A_{2}  5  (5^{14}, 3^{50}, 1^{28})
3  (3^{78}, 1^{14})
2  (4^{28}, 3^{36}, 1^{28})
A_{2} + 3A_{1}  5 (5^{7}, 4^{14}, 3^{28}, 2^{28}, 1^{17})
3  (3^{70}, 2^{14}, 1^{10})
2  (4^{26}, 3^{6}, 2^{62}, 1^{2})
A_{3} 5 (5^{13}, 4^{32}, 1^{55})
3  (7, 5^{11}, 4^{32}, 3, 1^{55})
2  (4^{36}, 3^{20}, 1^{44})
A_{2} + 2A_{1}  5 (5^{3}, 4^{16}, 3^{27}, 2^{32}, 1^{24})
3  (3^{65}, 2^{16}, 1^{21})
2  (4^{22}, 3^{14}, 2^{52}, 1^{14})
A_{2} + A_{1} 5 (5, 4^{12}, 3^{32}, 2^{32}, 1^{35})
3  (3^{58}, 2^{20}, 1^{34})
2  (4^{14}, 3^{30}, 2^{34}, 1^{34})
4A_{1} 5 (4^{8}, 3^{28}, 2^{48}, 1^{36})
3  (3^{44}, 2^{40}, 1^{36})
2  (2^{120}, 1^{8})
A_{2} 5 (5, 3^{55}, 1^{78})
3  (3^{57}, 1^{77})
2  (4^{2}, 3^{54}, 1^{78})
3A_{1} 5 (4^{2}, 3^{27}, 2^{52}, 1^{55})
3  (3^{31}, 2^{50}, 1^{55})
2  (2^{110}, 1^{28})
2A_{1} 5 (3^{14}, 2^{64}, 1^{78})
3  (3^{14}, 2^{64}, 1^{78})
2  (2^{92}, 1^{64})
A_{1} 5 (3, 2^{56}, 1^{133})
3  (3, 2^{56}, 1^{133})
2  (2^{58}, 1^{132})
{0}  5,3,2  (1^{248})

Type F_{4}

Orbit  Prime  Partition  Orbit Prime  Partition
F_{4}  3  (9^{4}, 8^{2})  (B_{2})^{(2)}  2  (4^{7}, 3^{7}, 1^{3})
2  (16^{2}, 4^{2}, 3^{4})  B_{2}  3 (7, 5^{4}, 4^{4}, 3, 1^{6})
F_{4}(a_{1})  3  (9^{4}, 8^{2})  2  (4^{6}, 3^{4}, 2^{5}, 1^{6})
2  (8^{4}, 4^{2}, 3^{4})  A_{2} + Ã_{1}  3  (3^{16}, 2^{2})
F_{4}(a_{2})  3  (9^{2}, 8^{2}, 6^{2}, 3^{2})  2  (4^{6}, 3^{4}, 2^{8})
2  (8^{4}, 4^{2}, 3^{4})  Ã_{2}  3  (3^{15}, 1^{7})
(C_{3})^{(2)}  2  (8^{4}, 4, 3^{3}, 2^{3}, 1)  2  (4^{8}, 1^{20})
C_{3}  3  (9^{2}, 7^{4}, 3, 1^{3})  (A_{2})^{(2)}  2  (4^{2}, 3^{12}, 1^{8})
2  (8^{4}, 2^{8}, 1^{4})  A_{2}  3  (3^{15}, 1^{7})
B_{3}  3  (9^{2}, 7^{4}, 3, 1^{3})  2  (4^{2}, 3^{12}, 1^{8})
2  (4^{10}, 3^{4})  A_{1} + Ã_{1}  3  (3^{10}, 2^{8}, 1^{6})
F_{4}(a_{3})  3  (7, 6^{2}, 5^{3}, 3^{6})  2  (2^{24}, 1^{4})
2  (4^{10}, 3^{4})  (Ã_{1})^{(2)}  2  (2^{21}, 1^{10})
C_{3}(a_{1})^{(2)}  2  (4^{9}, 3^{3}, 2^{3}, 1)  Ã_{1}  3  (3^{7}, 2^{8}, 1^{15})
C_{3}(a_{1})  3 (7, 6^{2}, 5, 4^{4}, 3^{3}, 1^{3})  2  (2^{16}, 1^{20})
2  (4^{8}, 2^{8}, 1^{4})  A_{1}  3  (3, 2^{14}, 1^{21})
(A_{1} + Ã_{2})^{(2)}  2  (4^{9}, 3^{3}, 2^{3}, 1)  2  (2^{16}, 1^{20})
A_{1} + Ã_{2}  3  (3^{16}, 2^{2})  {0}  3,2  (1^{52})
2  (4^{8}, 2^{8}, 1^{4})

Type G_{2}
Orbit Prime Partition
G_{2} 3 (9, 3, 2)
2  (4^{2}, 3^{2})
G_{2}(a_{1}) 3 (3^{4}, 2)
2  (4^{2}, 3^{2})
(Ã_{1})^{(3)}  3 (3^{3}, 2^{2}, 1)
Ã_{1} 3 (3^{3}, 1^{5})
2  (2^{6}, 1^{2})
A_{1}  3 (3, 2^{4}, 1^{3})
2  (2^{6}, 1^{2})
{0}  3,2  (1^{14})

7. Hasse diagrams

In this section, we provide the Hasse diagram for U_{1}(G) and N_{1}(𝔤) when Φ is of exceptional type. Our results in Section 4 show that the number of orbits (and the labeling) in these varieties coincide. The inclusion of closures of unipotent classes over bad characteristics is given in [Sp1, Mi1, Mi2]. For nilpotent orbits the inclusions of orbit closures are known for bad characteristics for G_{2} [St], F_{4} [Sp3, HS], E_{6} [Sp2]. The situation for E_{7} and E_{8} is still undetermined. However, by using our computations in Section 6, the new orbits which arise in these cases are not contained in N_{1}(𝔤). Therefore, the inclusion of orbit closures will be the same as in characteristic zero. Understanding the structure of N_{1}(𝔤) provides vital information which can help determine the support varieties for G-modules.

We should remark that the Hasse diagram for E_{8} in [Car, p. 444] is not correct. Eric Sommers has pointed out that there should be a line between between the orbits E_{6} + A_{1} and E_{8}(b_{6}). We have also found that there should not be a line between D_{4}(a_{1}) and A_{3}, but rather a line between A_{3} + A_{1} and A_{3}. These corrections agree with the Hasse diagram produced by Spaltenstein [Sp1].

∅ A_{1} 2A_{1} 3A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} A_{2} + A_{1} 2A_{2} A_{2} + 2A_{1} 2A_{2} + A_{1}

Figure 1. E_{6} for p = 2 (left) and p = 3 (right)

∅ A_{1} 2A_{1} (3A_{1})′ (3A_{1})′′ 4A_{1}  ∅ A_{1} 2A_{1} (3A_{1})′ (3A_{1})′′  A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1}2A_{2} A_{2} + 3A_{1} 2A_{2} + A_{1}

Figure 2. E_{7} for p = 2 (left) and p = 3 (right)

∅ A_{1} 2A_{1} 3A_{1} 4A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1} A_{2} + 3A_{1} 2A_{2} 2A_{2} + A_{1} 2A_{2} + 2A_{1}  ∅ A_{1} 2A_{1} 3A_{1} A_{2} 4A_{1} A_{2} + A_{1} A_{2} + 2A_{1} A_{2} + 3A_{1} 2A_{2} 2A_{2} + A_{1} A_{3}  2A_{2} + 2A_{1} A_{3} + A_{1} A_{3} + 2A_{1} D_{4}(a_{1}) D_{4}(a_{1}) + A_{1} A_{3} + A_{2} A_{3} + A_{2} + A_{1} D_{4}(a_{1}) + A_{2} A_{4} A_{4} + A_{1}  2A_{3} A_{4} + 2A_{1} A_{4} + A_{2} A_{4} + A_{2} + A_{1} A_{4} + A_{3}

Figure 3. E_{8} for p = 2 (left), p = 3, (center), and p = 5 (right)

∅ A_{1} Ã_{1}^{(2)} Ã_{1} A_{1} + Ã_{1}  ∅ A_{1} Ã_{1} A_{1} + Ã_{1} A_{2} Ã_{2}  A_{2} + Ã_{1} A_{1} + Ã_{2}

Figure 4. F_{4} with p = 2 (left) and p = 3 (right)

∅ A_{1} Ã_{1}  ∅ A_{1} Ã_{1}^{(3)} Ã_{1} G_{2}(a_{1})

Figure 5. G_{2} for p = 2 (left) and p = 3 (right)

8. VIGRE Algebra Group at the University of Georgia

This project was initiated during 2003–2004 under the Vertical Integration of Research and Education (VIGRE) Program sponsored by the National Science Foundation (NSF) at the Department of Mathematics at the University of Georgia (UGA). We would like to acknowledge the NSF grant DMS-0089927 for its financial support of this project. The VIGRE Algebra Group consists of 4 faculty members, 4 postdoctoral fellows and 7 graduate students. The group was led by David J. Benson, Brian D. Boe and Daniel K. Nakano. The email addresses of the members of the group are given below.

Faculty:

David J. Benson  bensondj@math.uga.edu
Brian D. Boe  brian@math.uga.edu
Leonard Chastkofsky lenny@math.uga.edu
Daniel K. Nakano  nakano@math.uga.edu

Postdoctoral Fellows:

Markus Hunziker  hunziker@math.uga.edu
Jo Jang Hyun  jhjo@math.uga.edu
Jonathan Kujawa  kujawa@math.uga.edu
Nadia Mazza  nmazza@math.uga.edu

Graduate Students:

Philip Bergonio  pbergonio@math.uga.edu
Bobbe Cooper  bcooper@math.uga.edu
G. Michael Guy  guy@math.uga.edu
Jeremiah Hower  jhower@math.uga.edu
Graham Matthews  matthews@math.uga.edu
Kenyon J. Platt  platt@math.uga.edu
Caroline Wright  cwright@math.uga.edu

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[UGA] University of Georgia VIGRE Algebra Group, Varieties of nilpotent elements for simple Lie algebras I: good primes, to appear in J. Algebra.

Department of Mathematics, University of Georgia, Athens, Georgia 30602