Journal of Lie Theory Volume 10 (2000) 491–490 ◯^{C}2000 Heldermann Verlag

The closure diagrams for nilpotent orbits of real forms of F_{4} and G_{2}

Dragomir Ž. D – oković^{∗}

Communicated by K. H. Hofmann

Abstract. Let O_{1} and O_{2} be two adjoint nilpotent orbits in a real semisimple Lie algebra 𝔤_{0} . We shall write O_{1} ≥ O_{2} if O_{2} is contained in the closure of O_{1} . This defines a partial order on the set of such orbits and we refer to this order as the closure ordering. We determine the closure ordering for adjoint nilpotent orbits when 𝔤_{0} is of type G I, F I, or F II. The cases G I and F II are rather trivial and are included only for the sake of completeness.

1. Introduction

Let 𝔤 be a semisimple complex Lie algebra, G the adjoint group of 𝔤, and 𝔤_{0} a real form of 𝔤. The conjugation σ of 𝔤 with respect to 𝔤_{0} defines a real structure on 𝔤 and G. Let G_{R} be the group of real points of G and G_{0} its identity component for the Lie group topology. Thus G_{0} is the adjoint group of 𝔤_{0}.

Let N denote the nilpotent variety of 𝔤, i.e., the set of X ∈ 𝔤 such that ad(X) is nilpotent. We set N_{R} = 𝔤_{0}∩N. The variety N is irreducible and consists of finitely many G-orbits. There is an open dense G-orbit in N called the principal orbit. Let O_{1}, O_{2} be two G-orbits in N. If O_{2} is contained in the closure of O_{1}, we shall write O_{1} ≥ O_{2}. If O_{1} ≥ O_{2} and O_{1} ≠ O_{2} we write O_{1} > O_{2}. The relation “≥” is a partial order on the set of G-orbits in N to which we refer as the closure ordering. We mention that O_{1} > O_{2} implies dim(O_{1}) > dim(O_{2}). We shall use analogous definitions and terminology for G_{R}-orbits and G_{0}-orbits in N_{R}.

The closure ordering for G-orbits in N is known (see [?] and the references given there). For the convenience of the reader we reproduce in Figure 1 the closure diagrams for the two relevant cases, F_{4} and G_{2}. The numbers on the left hand side of the diagrams are the dimensions of the orbits on that level.

If 𝔤 is of classical type (A, B, C, or D) and G the corresponding classical group, for types B and D we assume that G is the full orthogonal group, the

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag closure ordering for G_{R}-orbits in N_{R} was determined in our papers [?, ?]. In most cases either G_{R} = G_{0} or the closure ordering of G_{0}-orbits in N_{R} can be easily deduced from the closure ordering of the G_{R}-orbits. There is one notable exception, when 𝔤_{0} is of type 𝔰𝔬(p, q). This particular case has been dealt with in our joint paper with J. Sekiguchi and N. Lemire [?] but not yet solved.

∗ Supported in part by the NSERC Grant A-5285.

Our main result is the description of the closure ordering of G_{0}-orbits in N_{R} when 𝔤_{0} is of type F I, i.e., the split real form of F_{4}. For the sake of completeness we also include the cases F II and G I. In all three cases G_{R} = G_{0}, i.e., G_{R} is connected.

Let 𝔤_{0} = 𝔨_{0} ⊕ 𝔭_{0} be a Cartan decomposition of 𝔤_{0} and 𝔨, 𝔭 the complexifications of 𝔨_{0}, 𝔭_{0}. We denote by θ the corresponding Cartan involution of 𝔤_{0}. We use the same letter θ to denote the complex linear extension of this involution to 𝔤, as well as the corresponding involutory automorphism of G. We set K = {a ∈ G : θ(a) = a}. The Lie algebra of K is 𝔨. In general K is not connected (e.g. when 𝔤_{0} = 𝔰𝔬(p, q)) but it is connected in the three cases that we consider in this paper: F I, F II, and G I. By K^{0} we denote the identity component of K.

Let N_{1} = N ∩ 𝔭, the nilpotent variety of 𝔭. It is well known that N_{1} is a finite union of K^{0}-orbits. Furthermore for X ∈ N_{1}, K^{0} · X is a connected component of 𝔭 ∩ G · X, and

dim(K^{0} · X) = \frac{1}{2} dim(G · X).

There is a one-to-one correspondence between the G_{0}-orbits in N_{R} and the K^{0}-orbits in N_{1} known as the Kostant-Sekiguchi correspondence. For the precise description of this correspondence we refer the reader to the book [?]. Barbasch and Sepanski [?] have shown recently that the Kostant-Sekiguchi correspondence preserves the closure ordering. This means that our original problem of describing the closure ordering of G_{0}-orbits in N_{R} is equivalent to the same problem for K^{0}-orbits in N_{1}. We shall actually solve the latter problem.

The K^{0}-orbits in N_{1} were enumerated in our two papers [?, ?] for all noncompact real forms 𝔤_{0} of simple exceptional complex Lie algebras 𝔤. In order to avoid possible confusion, we shall use the same numbering of the K^{0}-orbits in N_{1} as in those papers. (This numbering is the same as in [?].) The trivial zero orbit will be given number 0. We have only recently computed representatives of the K^{0}-orbits in N_{1} and G_{0}-orbits in N_{R} and obtained a concrete description of the Kostant-Sekiguchi correspondence [?].

I would like to thank Prof. J. Sekiguchi for his interest and several conversations concerning the problem treated in this paper.

2. Statement of the result

The real forms G I, F I, and F II are of inner type. Hence we can choose a Cartan subalgebra 𝔥 of 𝔤 such that 𝔥 ⊂ 𝔨. Let R (resp. R_{0}) be the root system of 𝔤 (resp. 𝔨) with respect to 𝔥. Let B be a base of R. For α ∈ R we denote by 𝔤^{α} the corresponding root space. We remark that R_{0} is the set of α ∈ R for which 𝔤^{α} ⊂ 𝔨. By W (resp. W_{0}) we denote the Weyl group of R (resp. R_{0}).

We define the extended base B_{e} of R to be B∪{α_{0}} where α_{0} is the negative of the highest root of R. There is a unique base B_{0} of R_{0} such that B_{0} ⊂ B_{e}.

If 𝔤_{0} is of type G I we have B = {α_{1}, α_{2}}, α_{1} being short, and B_{0} = {β_{1}, β_{2}} with β_{1} = α_{1}, β_{2} = α_{0}. We also have K = (SL_{2} × SL_{2})/Z_{2} and dim 𝔭 = 8. For the other positive roots we use the notations: α_{3} = α_{1} + α_{2}, α_{4} = 2α_{1} + α_{2}, α_{5} = 3α_{1} + α_{2}, and α_{6} = 3α_{1} + 2α_{2}.

[figure: Figure 1: Closure diagrams for G_{2} and F_{4}; G_{2}; F_{4}; 0; 0; 6; A_{1}; 8; Ã_{1}; 10; G_{2}(a_{1}); 12; G_{2}; 0; 0; 16; A_{1}; 22; Ã_{1}; 28; A_{1} + Ã_{1}; 30; A_{2}; Ã_{2}; 34; A_{2} + Ã_{1}; 36; B_{2}; Ã_{2} + A_{1}; 38; C_{3}(a_{1}); 40; F_{4}(a_{3}); 42; C_{3}; B_{3}; 44; F_{4}(a_{2}); 46; F_{4}(a_{1}); 48; F_{4}]

If 𝔤 is of type F_{4}, we write B = {α_{1}, α_{2}, α_{3}, α_{4}} using the same notation as in Bourbaki [?] and in [?]. If 𝔤_{0} is of type F I then K = (Sp_{6} × SL_{2})/Z_{2}, dim 𝔭 = 28, and B_{0} = {β_{1}, β_{2}, β_{3}, β_{4}} with β_{1} = α_{4}, β_{2} = α_{3}, β_{3} = α_{2}, and β_{4} = α_{0}. When 𝔤_{0} is of type F II then K = Spin_{9}, dim 𝔭 = 16, and B_{0} = {β_{1}, β_{2}, β_{3}, β_{4}} with β_{1} = α_{0}, β_{2} = α_{1}, β_{3} = α_{2}, and β_{4} = α_{3}.

A normal triple is an ordered triple (E, H, F) of elements E, F ∈ 𝔭 and H ∈ 𝔨 spanning a three dimensional simple subalgebra of 𝔤 and satisfying the relations:

[H, E] = 2E, [H, F] = −2F, [F, E] = H.

Given a K^{0}-orbit O ⊂ 𝔭, one can choose a normal triple (E, H, F) such that E ∈ O, H ∈ 𝔥, and β_{i}(H) are nonnegative integers. Furthermore the integers β_{i}(H) determine O uniquely. We can choose w ∈ W such that α_{i}(w(H)) are nonnegative integers. These integers determine uniquely the nilpotent G-orbit containing O. We shall write H′ = w(H).

The positive roots of F_{4} and its structure constants with respect to a Chevalley basis are listed in the Appendix. The bases that we use are the same as in [?], but our table of the structure constants for F_{4} is given in the Appendix in a more “user friendly way” than in the mentioned paper. The positive roots are enumerated by integers from 1 to 24. The negative root −α_{i} is also written as α_{−i}. The root vector in the Chevalley basis corresponding to the root α_{i} is denoted by X_{i}.

In Tables 1-3 below we shall list the non-zero nilpotent K -orbits O in 𝔭 for the three cases that we consider. For each of these orbits we list the integers β_{i}(H), the integers α_{i}(H′), a representative E of the orbit O, and the complex dimension of O. In some cases two or more representatives (of different types) are given. For instance the orbit 4 in Table 1 has a representative X_{−2} + X_{−4} of type A_{1} + Ã_{1} meaning that {α_{−2}, α_{−4}} is a base of a closed root subsystem of type {α_{−2}, α_{−4}} with α_{−2} a long root and α_{−4} a short root. The same orbit also has the representative X_{−2} + X_{−5} of type A_{2}. In the last column we give the label from Figure 1 for the G-orbit in N containing the given K -orbit in N_{1}.

Table 1: G I table

O β_{i}(H) α_{i}(H′) E ∈ O  dim O Type of E G · O
1 11  01 X_{−2}  3  A_{1}  A_{1}
2 13  10 X_{−3}  4  Ã_{1}  Ã_{1}
3 22  02 X_{5} + X_{−3}  5  A_{1} + Ã_{1}  G_{2}(a_{1})
4 04  02 X_{−2} + X_{−4}  5  A_{1} + Ã_{1}  G_{2}(a_{1})
X_{−2} + X_{−5}  A_{2}
5 48  22 X_{−4} + X_{5}  6  G_{2}  G_{2}

We can now state our main result.

Theorem 2.1. Let 𝔤_{0} be of type G I, F I, or F II. Then the closure ordering of the nilpotent K -orbits in 𝔭 is as given in Figure 2 on the following page.

[figure: F I; 0; 0; 8; 1; 11; 2; .; .; .; .; .; .; 3; 14; 4; .; .; .; .; .; .; 5; 15; 6; .; .; .; 8; .; .; .; 7; 9; 17; 10; 18; 11; .; .; .; .; .; .; 12; 13; 19; 15; .; .; .; .; .; .; 14; 20; 17; .; .; .; .; .; .; 18; .; .; .; .; .; .; 16; 21; 19; .; .; .; .; .; .; 20; 21; 22; 22; .; .; .; .; .; .; 23; 23; 24; .; .; .; .; .; .; 25; 24; 26; G I; 0; 0; 3; 1; 4; 2; 5; 3; .; .; .; .; .; .; 4; 6; 5; F II; 0; 0; 11; 1; 15; 2]

Figure 2: Closure diagrams for F I, F II and G I.

The horizontal dotted lines in Figure 2 indicate that the K^{0}-orbits joined by these lines are contained in the same G-orbit. Again the numbers on the left hand side of the diagrams are the dimensions of the orbits on that level.

496 D – oković

Table 2: F I table

O β_{i}(H) α_{i}(H′) E ∈ O  dim Type  G · O
1 001 1 1000 X_{−1}  8 A_{1}  A_{1}
2 100 2 0001 X_{−8}  11 Ã_{1}  Ã_{1}
X_{−1} + X_{−14}  2A_{1}
3 010 0 0001 X_{−1} + X_{22} 11 2A_{1} Ã_{1}
4 001 3 0100 X_{−5} + X_{−15}  14 A_{1} + Ã_{1} A_{1} + Ã_{1}
X_{−5} + X_{−11} + X_{−18}  3A_{1}
5 101 1 0100 X_{23} + X_{−8}  14 A_{1} + Ã_{1} A_{1} + Ã_{1}
X_{23} + X_{−5} + X_{−11}  3A_{1}
6 000 4 2000 X_{−1} + X_{−23} 15 A_{2} A_{2}
X_{−1} + X_{−14} + X_{−21}  2A_{1} + Ã_{1}
X_{−1} + X_{−14} + X_{−20} + X_{−22}  4A_{1}
7 200 0 2000 X_{−1} + X_{18} 15 A_{2} A_{2}
X_{18} + X_{23} + X_{−7}  2A_{1} + Ã_{1}
X_{18} + X_{23} + X_{−5} + X_{−11}  4A_{1}
8 002 0 2000 X_{−18} + X_{23} + X_{−8}  15 2A_{1} + Ã_{1} A_{2}
X_{23} + X_{−5} + X_{−11} + X_{−18}  4A_{1}
9 020 0 0002 X_{−8} + X_{19} 15 Ã_{2} Ã_{2}
10 110 2 0010 (X_{22} + X_{−11}) + X_{−12}  17 A_{2} + Ã_{1} A_{2} + Ã_{1}
11 102 4 2001 X_{−18} + X_{−14} + X_{23}  18 A_{3}  B_{2}
12 012 2 2001 X_{22} + X_{−15}  18 B_{2}  B_{2}
X_{−11} + X_{22} + X_{−18}  A_{3}
13 111 1 0101 (X_{21} + X_{−12}) + X_{−11} 18 Ã_{2} + A_{1} Ã_{2} + A_{1}
14 103 1 1010 (X_{20} + X_{−12}) + X_{22}  19 B_{2} + A_{1} C_{3}(a_{1})
15 111 3 1010 (X_{22} + X_{−15}) + X_{−14}  19 B_{2} + A_{1} C_{3}(a_{1})
16 004 0 0200 (X_{−5} + X_{17}) + X_{−11} + X_{−18}  20 B_{2} + 2A_{1} F_{4}(a_{3})
17 020 4 0200 (X_{22} + X_{−15}) + X_{−14} + X_{−20} 20 B_{2} + 2A_{1} F_{4}(a_{3}) Ex.3.3
18 202 2 0200 (X_{20} + X_{−12}) + X_{−14} + X_{22}  20 B_{2} + 2A_{1} F_{4}(a_{3})
19 004 8 2200 X_{−14} + X_{23} + X_{−21} 21 B_{3} B_{3}
X_{−14} + X_{23} + X_{−20} + X_{−22}  D_{4}
20 204 4 2200 X_{−14} + X_{−18} + X_{21} 21 B_{3} B_{3}
X_{20} + X_{−18} + X_{22} + X_{−14}  D_{4}
21 131 3 1012 X_{−15} + X_{19} + X_{−14} 21 C_{3} C_{3}
22 040 4 0202 (X_{−15} + X_{21} + X_{−20}) + X_{−14} 22 C_{3} + A_{1} F_{4}(a_{2})
23 222 2 0202 (X_{−15} + X_{19} + X_{−14}) + X_{20}  22 C_{3} + A_{1} F_{4}(a_{2})
24 224 4 2202 X_{20} + X_{−18} + X_{−14} + X_{19}  23 B_{4}  F_{4}(a_{1})
25 404 8 2202 X_{20} + X_{−18} + X_{22} + X_{−19}  23 B_{4}  F_{4}(a_{1})
26 444 8 2222 X_{−18} + X_{20} + X_{−17} + X_{19}  24 F_{4}  F_{4}

Table 3: F II table

O β_{i}(H) α_{i}(H′) E ∈ O  dim O Type of E G · O
1 0001 0001 X_{−4}  11 Ã_{1}  Ã_{1}
2 4000 0002 X_{−4} + X_{−19} 15 Ã_{2}  Ã_{2}

3. The proof: Part I

Let O^{i} denote the i-th orbit as listed in Table 2. In this section we shall prove that if O^{i} and O^{j} are joined by a solid line in Figure 2 (diagram F I), with i above j, then O^{i} > O^{j}. The proofs are based on the theorem below, which is valid for any simple complex Lie algebra 𝔤 and its non-compact real form 𝔤_{0}.

In order to state this theorem we need some additional notations. Let (E, H, F) be a normal triple and O = K^{0} · E ⊂ 𝔭. Then for any integer j we define

𝔤_{H}(0, j) = {X ∈ 𝔨 : [H, X] = jX},
𝔤_{H}(1, j) = {X ∈ 𝔭 : [H, X] = jX},

and

𝔭_{i}(H) = ∑_{j≥i} 𝔤_{H}(1, j).

Finally let Q_{H} be the parabolic subgroup of K^{0} having

𝔮_{H} := ∑_{j≥0} 𝔤_{H}(0, j)

as its Lie algebra.

Theorem 3.1. Let (E, H, F) be a normal triple and O = K^{0} · E ⊂ 𝔭. Denote by Z_{K}(H) the centralizer of H in K and by Z_{K}(H)^{0} its identity component. Then

(a) The pair (Z_{K}(H)^{0}, 𝔤_{H}(0, 2)) is a prehomogeneous vector space and Z_{K}(H)^{0} · E its dense open orbit.

(b) The pair (Q_{H}, 𝔭_{2}(H)) is a prehomogeneous vector space and Q_{H} ·E its dense open orbit.

(c) If O′ ⊂ N_{1} is also a K^{0}-orbit, then O ≥ O′ if and only if O′ ∩ 𝔭_{2}(H) ≠ Ø.

The assertions (a) and (b) constitute a symmetric space analog of a result of Kostant [?, Theorem 4.3] (see also [?, Lemma 4.1.4]). The proof of these two assertions given in [?, Proposition 1] for the special case 𝔤_{0} = 𝔰𝔬(p, q) can be easily extended to the general case. The assertion (c) follows from [?, Satz 2, pp. 182-184].

We shall now apply Theorem 3.1 to the case when 𝔤_{0} is of type F I. Let (E^{i}, H^{i}, F^{i}) denote the normal triple with E^{i} ∈ O^{i} and H^{i} as given in Table 2.

Each space 𝔤_{H^{i}}(1, j) with j ≠ 0 is a sum of certain root spaces, and the same is true for the space 𝔭_{2}(H^{i}). In Table 4 we list the indices k of all roots α = α_{k} such that 𝔤^{α} ⊂ 𝔭_{2}(H^{i}). We list first the indices of roots α for which 𝔤^{α} ⊂ 𝔤_{H^{i}}(1, 2) and separate them by a semi-colon from the indices of the roots β (if any) for which 𝔤^{β} ⊂ 𝔭_{3}(H^{i}).

Table 4: Root spaces in 𝔤_{H^{i}}(1, 2) and 𝔭_{3}(H^{i})
i  Indices of roots
1 -1;
2 -1, -5, -8, -11, -14;
3 -1, -5, 22, 23;
4 -5, -8, -11, -12, -15, -18; -1
5 -5, -8, -11, 23; -1
6 -1, -5, -8, -11, -12, -14, -15, -17, -18, -19, -20, -21, -22, -23;
7 -1, -5, -8, -11, -14, 18, 20, 21, 22, 23;
8 -5, -8, -11, -12, -15, -18, 23; -1
9 -8, -12, 19, 21; -1, -5, 22, 23
10 -11, -12, -14, 22, 23; -1, -5, -8
11 -14, -18, 23; -1, -5, -8, -11, -12, -15
12 -11, -15, -18, 22; -1, -5, -8, -12, 23
13 -11, -12, 21; -1, -5, -8, 22, 23
14 -12, -15, 20, 21, 22; -1, -5, -8, -11, 23
15 -14, -15, 22; -1, -5, -11, -12, 23
16 -5, -8, -11, -12, 14, -15, 17, -18, 19, 20, 21, 22; -1, 23
17 -11, -14, -15, -17, -18, -20, 22, 23; -1, -5, -8, -12
18 -12, -14, -15, 20, 21, 22; -1, -5, -8, -11, 23
19 -14, -17, -19, -20, -21, -22, 23; -1, -5, -8, -11, -12, -15, -18
20 -14, -18, 20, 21, 22; -1, -5, -8, -11, -12, -15, 23
21 -14, -15, 19; -1, -5, -8, -11, -12, 21, 22, 23
22 -11, -14, -15, -17, -18, 19, -20, 21; -1, -5, -8, -12, 22, 23
23 -14, -15, 19, 20; -1, -5, -8, -11, -12, 21, 22, 23
24 -14, -18, 19, 20; -1, -5, -8, -11, -12, -15, 21, 22, 23
25 -17, -18, -19, 20, 21, 22; -1, -5, -8, -11, -12, -14, -15, 23  Ex.3.2
26 -17, -18, 19, 20; -1, -5, -8, -11, -12, -14, -15, 21, 22, 23

Next we examine all pairs (i, j) such that i is joined to j by a solid line in Figure 2 and i is above j. For each such pair we have constructed a normal triple (E, H, F) such that E ∈ 𝔭_{2}(H^{i}) ∩ O^{j}. By Theorem 3.1, (c), it follows that O^{i} > O^{j}.

In Table 5, for each such pair (i, j), we exhibit the element E as a sum of root vectors. If the roots involved in E form a basis of a closed root subsystem of R, we indicate its type in the third column. The last column identifies the element H by listing the integers α_{1}(H), …, α_{4}(H) in that order. If these integers are not listed, then H = H^{j} and one can use Table 2 and the integers β_{1}(H), …, β_{4}(H) listed there to compute α_{1}(H), …, α_{4}(H).

We give more details in two cases. The coroot corresponding to the root α_{k} is denoted by H_{k}(∈ 𝔥). Note that H_{−k} = −H_{k}.

Example 3.2. i = 25, j = 22. Then E = (X_{−19} + X_{21} + X_{−18}) + (X_{−5}) and the type is given as C_{3} + A_{1}. The meaning of the parantheses is that the roots α_{−19}, α_{21}, α_{−18} form a base for the C_{3}-component and α_{−5} a base for the A_{1}-component. In this case

α_{1}(H) = −10, α_{2}(H) = 8, α_{3}(H) = −4, α_{4}(H) = 4.

Since

\begin{aligned}α_{19} & = & α_{1} + 2α_{2} + 3α_{3} + α_{4}, \\ α_{21} & = & α_{1} + 2α_{2} + 3α_{3} + 2α_{4}, \\ α_{18} & = & α_{1} + α_{2} + 2α_{3} + 2α_{4}, \\ α_{5} & = & α_{1} + α_{2}, \end{aligned}

we indeed have E ∈ 𝔤_{H}(1, 2). The element F is not given in the table, but it can be easily computed. In this case it is a linear combination of X_{19}, X_{−21}, X_{18}, and X_{5}.

It is a simple matter to compute

β_{1}(H) = 4, β_{2}(H) = −4, β_{3}(H) = 8, β_{4}(H) = −4

and show that w(H) = H^{22} for a suitable w ∈ W_{0}. This proves that E ∈ O^{22}. Since the indices −19, 21, −18, and −5 appear in row 25 of Table 4, we also have E ∈ 𝔭_{2}(H^{25}).

Example 3.3. i = 19, j = 17. In this case E = X_{22} + X_{−18} + X_{−14} + X_{23} and (since a type is not given) the roots α_{22}, α_{−18}, α_{−14}, and α_{23} do not form a basis for a closed root subsystem. Since the integers α_{1}(H), …, α_{4}(H) are not listed, this means that H = H^{17}. From Table 2 we have

β_{1}(H) = 0, β_{2}(H) = 2, β_{3}(H) = 0, β_{4}(H) = 4.

By using −α_{0} = 2α_{1} + 3α_{2} + 4α_{3} + 2α_{4}, we find that

α_{1}(H) = −6, α_{2}(H) = 0, α_{3}(H) = 2, α_{4}(H) = 0.

Since

\begin{aligned}α_{22} & = & α_{1} + 2α_{2} + 4α_{3} + 2α_{4}, \\ α_{18} & = & α_{1} + α_{2} + 2α_{3} + 2α_{4}, \\ α_{14} & = & α_{1} + 2α_{2} + 2α_{3}, \\ α_{23} & = & α_{1} + 3α_{2} + 4α_{3} + 2α_{4}, \end{aligned}

we indeed have E ∈ 𝔤_{H}(1, 2).

The root spaces for the roots α_{20}, α_{−22}, α_{11}, α_{−23}, α_{18}, and α_{14} are contained in 𝔤_{H}(1, −2). By using the structure constants for F_{4} given in the Appendix, we obtain that

[X_{20} + X_{−22}, E] = H_{22}, [X_{11} + X_{−23}, E] = H_{23},
[X_{18}, E] = −H_{18}, [X_{14}, E] = −H_{14}.

The roots α_{22}, α_{23}, α_{18}, α_{14} are linearly independent and so 𝔥 = ad(E)𝔤_{H}(1, −2). Hence there exists F ∈ 𝔤_{H}(1, −2) such that [F, E] = H. As in the previous example, we can check that E ∈ 𝔭_{2}(H^{19}) ∩ O^{17}.

Table 5: Normal triples (E, H, F) with E ∈ 𝔭_{2}(H^{i}) ∩ O^{j}
i j Type  E  α_{1}(H), …, α_{4}(H)
26 25 B_{4}  X_{22} + X_{−18} + X_{20} + X_{−17}
26 24 B_{4}  X_{20} + X_{−18} + X_{−14} + X_{19}
25 19 B_{3} X_{−18} + X_{20} + X_{−17} −6, 4, −4, 4
25 22 C_{3} + A_{1} (X_{−19} + X_{21} + X_{−18}) + (X_{−5}) −10, 8, −4, 4  Ex.3.2
24 22 X_{19} + X_{21} + X_{−14} + X_{−18} −10, 0, 4, 0
24 23 C_{3} + A_{1} (X_{−15} + X_{19} + X_{−14}) + (X_{20}) −10, 2, 2, 2
22 20 B_{3}  X_{−14} + X_{−18} + X_{21}
22,23 21 C_{3}  X_{−15} + X_{19} + X_{−14}
19 17 X_{22} + X_{−18} + X_{−14} + X_{23} Ex.3.3
20 17 A_{3} + Ã_{1} (X_{20} + X_{−18} + X_{22}) + (X_{8})  −2, 4, 0, −2
20 18 B_{2} + 2A_{1} (X_{22} + X_{−15}) + (X_{−14}) + (X_{20})
20 16  X_{20} + X_{22} + X_{−5} + X_{−18}  −6, 4, 0, 0
21 18  X_{21} + X_{22} + X_{−14} + X_{−15}  −6, 2, 0, 2
21 16  X_{19} + X_{21} + X_{22} + X_{−15}  −6, 4, 0, 0
17 11 A_{3}  X_{−18} + X_{−14} + X_{23}
17,18 15 B_{2} + A_{1} (X_{22} + X_{−15}) + (X_{−14})
16,18 14 B_{2} + A_{1} (X_{20} + X_{−12}) + (X_{22})
14,15 12 B_{2}  X_{22} + X_{−15}
14 13 Ã_{2} + A_{1} (X_{−12} + X_{21}) + (X_{−11})
11 6 A_{2}  X_{−14} + X_{−18}
11 10 A_{2} + Ã_{1} (X_{−14} + X_{23}) + (X_{−12})
12,13 10 A_{2} + Ã_{1} (X_{−11} + X_{22}) + (X_{−12})
13 9 Ã_{2}  X_{−12} + X_{21}
10 8 2A_{1} + Ã_{1} (X_{−11}) + (X_{23}) + (X_{−12})
10 7 A_{2}  X_{−14} + X_{23}
6,8 4 A_{1} + Ã_{1} (X_{−18}) + (X_{−8})
7,8,9 5 A_{1} + Ã_{1} (X_{23}) + (X_{−8})
4,5 2 Ã_{1}  X_{−8}
5 3 2A_{1}  (X_{−5}) + (X_{23})
2,3 1 A_{1}  X_{−1}

4. The proof: Part II

In order to complete the proof of Theorem 2.1, we must show that O^{r} ≯ O^{s} whenever in Figure 2 the pair (r, s) is not connected by an edge. We need not check all possible pairs because of transitivity. For example: O^{4} ≯ O^{3} follows once we have proved O^{6} ≯ O^{3} because O^{6} > O^{4}. Therefore it suffices to show that O^{r} ≯ O^{s} when (r, s) is one of the following eight pairs:

\begin{matrix}(23, 6), & (25, 23), & (9, 4), & (19, 9), & (24, 19), \\ (16, 15), & (7, 4), & (6, 3). \end{matrix}  (†)

In this section we shall prove that O^{r} ≯ O^{s} for the pairs (r, s) listed in Table 7 below, covering the last three pairs (r, s) listed above. This will be derived from a general theorem which we prove next.

Let 𝔤 = 𝔨 ⊕ 𝔭 be a simple Lie algebra, as in the Introduction, and let V be a Z_{2}-graded finite-dimensional 𝔤-module. This means that V is equipped with a direct decomposition V = V_{0} ⊕ V_{1} such that

𝔨 · V_{0} ⊂ V_{0}, 𝔨 · V_{1} ⊂ V_{1}, 𝔭 · V_{0} ⊂ V_{1}, 𝔭 · V_{1} ⊂ V_{0}.

We refer to V_{0} (resp. V_{1}) as the even (resp. odd) subspace of V . Let ρ : 𝔤 → 𝔤𝔩(V) denote the representation of 𝔤 on V .

Now let (E, H, F) be a normal triple in 𝔤. Since E ∈ 𝔭, the linear operator ρ(E) maps V_{0} to V_{1} and V_{1} to V_{0}. We introduce the integers

d_{0}(j, E) = dim V_{0} ∩ ker ρ(E)^{j}, d_{1}(j, E) = dim V_{1} ∩ ker ρ(E)^{j},

for j ≥ 1. They depend only on the orbit O = K^{0} · E (and the module V ), and so we can set

d_{0}(j, O) = d_{0}(j, E), d_{1}(j, O) = d_{1}(j, E).

Theorem 4.1. Let O, O′ ⊂ N_{1} be K^{0}-orbits with dim O > dim O′, and let V be a finite-dimensional Z_{2}-graded 𝔤-module. If d_{0}(j, O) > d_{0}(j, O′) or d_{1}(j, O) > d_{1}(j, O′) for some j ≥ 1, then O ≯ O′.

Proof. Assume that, say, d_{0}(j, O) > d_{0}(j, O′) for some j ≥ 1. Let L be the space of linear operators u : V → V such that u(V_{0}) ⊂ V_{1} and u(V_{1}) ⊂ V_{0}. Let L_{0}(k, m) be the subset of L consisting of all u ∈ L such that

dim V_{0} ∩ ker u^{k} ≥ m.

Clearly L_{0}(k, m) is an affine subvariety of L.

If m = d_{0}(j, O), then ρ(O) ⊂ L_{0}(j, m) and since L_{0}(j, m) is closed in L, we deduce that

ρ(\overline{O}) ⊂ L_{0}(j, m).

Since m = d_{0}(j, O) > d_{0}(j, O′), ρ(O′) is not contained in L_{0}(j, m). It follows that O′ ⊄ \overline{O}, i.e., O ≯ O′.

In order to apply this theorem, we have to compute the integers d_{i}(j, E). Fortunately there is a simple method for doing these computations by using the 𝔰𝔩_{2}-theory. By restriction, V is a Z_{2}-graded module for the Z_{2}-graded subalgebra 𝔰 = 𝔰_{0} ⊕ 𝔰_{1} of 𝔤 spanned by the normal triple (E, H, F).

Note that H ∈ 𝔰_{0} while E, F ∈ 𝔰_{1}. We can decompose V into direct sum of simple Z_{2}-graded 𝔰-modules X_{ij}, see [?, Lemma 1]. We recall that i = 0 or 1 and j is a nonnegative integer. X_{ij} is also simple as ungraded 𝔰-module, it has dimension j + 1, and the highest weight vector is even if i = 0 and odd if i = 1. Let m_{ij} be the multiplicity of X_{ij} in V , i.e., symbolically we have

V = ∑_{i,j} m_{ij}X_{ij}.

Table 6: The integers d_{i}(j, k) for the module V (ω_{4})

k d_{0}(j, k)  d_{1}(j, k)
1 11,14  9,12
2 9,14  7,11,12
3 9,13,14  7,12  Case (6,3)
4 8,14  6,9,12  Case (7,4)
5 8,12,14  6,11,12
6 8,14  6,6,12  Case (6,3)
7 8,10,14  6,10,12  Case (7,4)
8 8,11,14  6,9,12
9 5,8,13,13,14  3,8,11,12
10 6,10,13,14  4,9,11,12
11 6,9,11,14  4,6,9,11,12
12 6,8,11,13,14  4,7,9,12
13 5,8,12,13,14  3,8,10,12
14 5,8,11,12,14  3,7,9,12
15 5,8,11,13,14  3,7,9,11,12  Case (16,15)
16 5,8,11,11,14  3,6,9,12  Case (16,15)
17 5,8,11,13,14  3,6,9,10,12
18 5,7,11,12,14  3,7,9,11,12
19 5,8,8,11,11,14  3,3,6,6,9,9,12
20 5,6,8,9,11,12,14  3,5,6,8,9,11,12
21 3,4,7,8,11,12,13,13,14  1,4,5,8,9,11,11,12
22 3,4,7,8,11,12,13,13,14  1,4,5,8,9,10,11,12
23 3,4,7,8,11,11,13,13,14  1,4,5,8,9,11,11,12
24 3,4,6,7,9,9,11,11,13,13,14 1,3,4,6,7,9,9,11,11,12
25 3,4,6,7,9,10,11,12,13,14 1,3,4,6,7,8,9,10,11,11,12
26 2,2,4,4,6,6,8,8,10,10,11, 0,2,2,4,4,6,6,8,8,9,9,10,
11,12,12,13,13,14  10,11,11,12

The 𝔰𝔩_{2}-theory gives the formulas:

d_{i}(1, E) = ∑_{j≥0} m_{ij},

d_{i}(2, E) − d_{i}(1, E) = ∑_{j≥1} m_{1−i,j},

d_{i}(3, E) − d_{i}(2, E) = ∑_{j≥2} m_{ij},

⋮

for i = 0, 1. Hence the problem reduces to the computation of the multiplicities m_{ij}.

The linear operator ρ(H) is semisimple and leaves the subspaces V_{0} and V_{1} invariant. Define integers

N(i, j, H) = dim{X ∈ V_{i} : ρ(H)X = jX}

for i = 0, 1 and j ∈ Z. Then, by 𝔰𝔩_{2}-theory,

m_{ij} = N(i, j, H) − N(1 − i, j + 2, H) for i = 0, 1 and arbitrary j ≥ 0. The integers N(i, j, H) can be computed by using the Weyl character formula.

Let us now return to our special case where 𝔤_{0} is of type F I. We take V to be the fundamental 𝔤-module V (ω_{4}) (by ω_{1}, …, ω_{4} we denote the fundamental weights of F_{4} as in [?]). It has dimension 26 and its weights are the short roots of R (all simple weights) and 0 which has multiplicity 2. We postulate that the highest weight vector of V is even. This gives rise to a Z_{2}-grading V = V_{0} ⊕ V_{1} with dim V_{0} = 14 and dim V_{1} = 12.

In Table 6 we list the integers d_{i}(j, k) = d_{i}(j, O^{k}) for each of the 26 nonzero K -orbits in N_{1}. By using Theorem 4.1, it follows from Table 6 that O^{r} ≯ O^{s} for the pairs (r, s), with dim O^{r} > dim O^{s}, listed in Table 7. In each case we exhibit i ∈ {0, 1} and j such that d_{i}(j, r) > d_{i}(j, s).

We did compute tables similar to Table 6 for several other simple 𝔤-modules (including the adjoint module) but no new information was gained.

Table 7: Some pairs (r, s) with O^{r} ≯ O^{s}

\begin{matrix}r & s & d_{i}(j, r) & d_{i}(j, s) \\ 23 & 19 & d_{1}(2, 23) = 4 & d_{1}(2, 19) = 3 \\ 22 & 19 & d_{1}(2, 22) = 4 & d_{1}(2, 19) = 3 \\ 19 & 18 & d_{0}(2, 9) = 8 & d_{0}(2, 18) = 7 \\ 17 & 14 & d_{0}(4, 17) = 13 & d_{0}(4, 14) = 12 \\ 16 & 15 & d_{1}(4, 16) = 12 & d_{1}(4, 15) = 11 \\ 12 & 9 & d_{1}(1, 12) = 4 & d_{1}(1, 9) = 3 \\ 11 & 9 & d_{1}(1, 11) = 4 & d_{1}(1, 9) = 3 \\ 7 & 4 & d_{1}(2, 7) = 10 & d_{1}(2, 4) = 9 \\ 6 & 3 & d_{0}(2, 6) = 14 & d_{0}(2, 3) = 13 \end{matrix}

5. The proof: Part III

In this section we prove that O^{r} ≯ O^{s} for the first five of the eight pairs (r, s) listed in (†) in the previous section. This will complete the proof.

For that purpose we examine the structure of 𝔭 as a K -module in more detail. First recall that K = (Sp_{6} × SL_{2})/Z_{2} and that 𝔭 ≅ V_{0} ⊗ V_{1} where V_{0} is the third fundamental module of Sp_{6} and V_{1} is the defining 2-dimensional module of SL_{2}. We remark that the second and third fundamental modules of Sp_{6} both have dimension 14 but the former has 0 as a weight of multiplicity 2 while 0 is not a weight of the latter.

Let R_{1} = R \ R_{0}, i.e., R_{1} is the set of roots α ∈ R such that 𝔤^{α} ⊂ 𝔭. Denote by R_{1}^{+} (resp. R_{1}^{−}) the set of positive (resp. negative) roots in R_{1}. Each of the sets R_{1}^{+}, R_{1}^{−} consists of 14 roots. The subspaces

𝔭^{+} = ∑_{α∈R_{1}^{+}} 𝔤^{α}, 𝔭^{−} = ∑_{α∈R_{1}^{−}} 𝔤^{α}

are simple Sp_{6}-modules isomorphic to V_{0}, and 𝔭 = 𝔭^{+} ⊕ 𝔭^{−}.

An isomorphism φ : 𝔭^{+} → 𝔭^{−} of Sp_{6}-modules is given by φ(X) = [X_{−24}, X]. In Table 8 we give the φ-images of the basis of 𝔭^{+} consisting of root vectors.

When viewed as a K -module, 𝔭 has only simple weights. The weight diagram of this module is exhibited in Figure 3 where 𝔭^{+} and 𝔭^{−} are clearly visible as a left and right half of the diagram. We have also indicated by arrows how the root vectors of β_{1}, β_{2}, β_{3} act on the weight spaces of 𝔭^{+}. For β_{4} only two arrows are shown. The highest weight vector is X_{−1} and the lowest X_{1}.

Table 8: The isomorphism φ

\begin{matrix}X & φ(X) & X & φ(X) \\ X_{1} & X_{−23} & X_{17} & X_{−15} \\ X_{5} & X_{−22} & X_{18} & X_{−14} \\ X_{8} & −X_{−21} & X_{19} & −X_{−12} \\ X_{11} & X_{−20} & X_{20} & −X_{−11} \\ X_{12} & X_{−19} & X_{21} & X_{−8} \\ X_{14} & −X_{−18} & X_{22} & −X_{−5} \\ X_{15} & −X_{−17} & X_{23} & −X_{−1} \end{matrix}

If (E, H, F) is a normal triple, we set

𝔭_{2}^{±}(H) = 𝔭^{±} ∩ 𝔭_{2}(H).

Let π : 𝔭 → 𝔭^{+} be the projector with kernel 𝔭^{−} and note that it commutes with the action of Sp_{6}. Any Z ∈ 𝔭 can be written uniquely as Z = X + φ(Y) with X, Y ∈ 𝔭^{+}. The orbit SL_{2} · Z consists of all vectors

(aX + bY) + φ(cX + dY)

with ad − bc = 1. Hence

π(SL_{2} · Z) ⊂ ⟨X, Y ⟩

where ⟨X, Y ⟩ denotes the subspace of 𝔭^{+} spanned by X and Y .

From the previous two sections we know the closures of the orbits O^{6}, O^{4}, O^{2}, and O^{1}:

\overline{O_{6}} = O_{6} ∪ O_{4} ∪ O_{2} ∪ O_{1} ∪ \{0\},
\overline{O_{4}} = O_{4} ∪ O_{2} ∪ O_{1} ∪ \{0\},
\overline{O_{2}} = O_{2} ∪ O_{1} ∪ \{0\},
\overline{O_{1}} = O_{1} ∪ \{0\}.

The pair (Sp_{6}×GL_{1}, 𝔭^{+}), where GL_{1} is the maximal torus of the SL_{2} factor of K leaving 𝔭^{+} and 𝔭^{−} invariant, is a regular prehomogeneous vector space (see [?, p.145]). The orbits of this space were classified by Igusa [?]. There are four non-zero orbits:

𝔭^{+} ∩ O^{6}, 𝔭^{+} ∩ O^{4}, 𝔭^{+} ∩ O^{2}, 𝔭^{+} ∩ O^{1}

with representatives X_{1} + X_{23}, X_{8} + X_{21}, X_{8}, X_{1} and dimensions 14, 13, 10, 7, respectively.

[figure: Figure 3: The weight diagram of 𝔭; 1; β_{3}; β_{4}; 5; β_{2}; 8; β_{1}; β_{2}; 12; β_{2}; 11; 15; 14; β_{3}; β_{1}; β_{1}; 18; 17; β_{3}; β_{2}; 20; 19; β_{2}; β_{1}; 21; β_{2}; 22; β_{3}; 23; β_{4}; −23; −22; −21; −19; −20; −17; −18; −14; −15; −11; −12; −8; −5; −1]

By definition [?, Definition 1, p. 35], the singular set S of this prehomogeneous vector space is the complement of the dense open orbit, i.e., S = 𝔭^{+} \ O^{6}. Since this prehomogeneous space is regular, S is the zero set of a relative invariant. By using this invariant (which is given explicitly in [?]) it is easy to verify that the 7-dimensional subspace spanned by X_{k}, 17 ≤ k ≤ 23 (the “upper half” of 𝔭^{+} in Figure 3) is contained in S.

We start with the pairs (23, 6) and (25, 23). Let E be the reprensentative of O^{23} from Table 2:

E = X_{−15} + X_{19} + X_{−14} + X_{20} = (X_{19} + X_{20}) + φ(X_{17} + X_{18}).

We have

π(SL_{2} · E) ⊂ ⟨X_{19} + X_{20}, X_{17} + X_{18}⟩ ⊂ S,

and so O^{23} ⊂ S + 𝔭^{−}. Since S + 𝔭^{−} is closed and 𝔭^{+} ∩ O^{6} = 𝔭^{+} \ S, we have O^{23} ≯ O^{6}.

Since X_{19} + X_{20} and X_{17} + X_{18} belong to 𝔭^{+} ∩ O^{4} while their sum is in 𝔭^{+} ∩ O^{6}, it is easy to deduce (use the action of the maximal torus of K) that in fact

π(SL_{2} · E) ⊂ 𝔭^{+} ∩ (O^{6} ∪ O^{4}).

On the other hand

𝔭_{2}^{+}(H^{25}) = ⟨X_{20}, X_{21}, X_{22}, X_{23}⟩ ⊂ \overline{O^{2}}.

We conclude that O^{23} ∩ 𝔭_{2}(H^{25}) = Ø. By Theorem 3.1, O^{25} ≯ O^{23}.

We now turn to the pair (9, 4). Let E be the representative of O^{9} from Table 2:

E = X_{−8} + X_{19} = X_{19} + φ(X_{21}).

Then

π(SL_{2} · E) = ⟨X_{19}, X_{21}⟩ \ {0} ⊂ 𝔭^{+} ∩ O^{2},

and by acting with Sp_{6}, we infer that

\overline{O^{9}} ⊂ 𝔭^{+} ∩ \overline{O^{2}} + 𝔭^{−}.

Hence 𝔭^{+} ∩ \overline{O^{9}} ⊂ 𝔭^{+} ∩ \overline{O^{2}}, and so O^{9} ≯ O^{4}.

Next we consider the pair (19, 9). Since

π(O^{9}) = Sp_{6} · π(SL_{2} · E) ⊂ 𝔭^{+} ∩ O^{2}

and 𝔭_{2}^{+}(H^{19}) = ⟨X_{23}⟩ and X_{23} ∈ O^{1}, we conclude that O^{9} ∩ 𝔭_{2}(H^{19}) = Ø. Hence O^{9} ⊄ \overline{O^{19}} by Theorem 3.1, and so O^{19} ≯ O^{9}.

Finally we consider the pair (24, 19). Write an arbitrary X ∈ 𝔭_{2}(H^{24}) as

X = ξ_{1}X_{20} + ξ_{2}X_{−18} + ξ_{3}X_{−14} + ξ_{4}X_{19} + Y

where Y ∈ 𝔭_{3}(H^{24}). To simplify the notations, we set H = H^{24}. In this case, the parabolic subgroup Q_{H} (see section 3) is a Borel subgroup of K.

By taking ξ_{1} = ⋯ = ξ_{4} = 1 and Y = 0, the above element X becomes the representative E of O^{24} (see Table 2). The generic orbit Q_{H} · E is open and dense in 𝔭_{2}(H). Recall that (Q_{H}, 𝔭_{2}(H)) is a prehomogeneous vector space. Its singular set is the union of the hyperplanes S_{i} defined by ξ_{i} = 0 (i = 1, 2, 3, 4). A computation shows that each of the intersections

S_{1} ∩ O^{22}, S_{2} ∩ O^{23}, S_{3} ∩ O^{16}, S_{4} ∩ O^{20}

is an open dense set in the corresponding hyperplane. Since

dim O^{19} = dim O^{20} > dim O^{16},

it is clear that O^{19} does not meet S_{3} ∩\overline{O^{16}} or S_{4} ∩\overline{O^{20}}. Since O^{19} is not contained in \overline{O^{22}} or \overline{O^{23}} (see Table 7), we infer that

O^{19} ∩ 𝔭_{2}(H) = O^{19} ∩ S = Ø.

By Theorem 3.1, O^{24} ≯ O^{19}.

This completes the proof of Theorem 2.1.

6. Appendix

In Table 9 we give our enumeration of the positive roots of F_{4} and their corresponding coroots.

Table 9: The positive roots and coroots of F_{4}

\begin{matrix}i & α_{i} & H_{i} & i & α_{i} & H_{i} & i & α_{i} & H_{i} \\ 1 & 1000 & 1000 & 9 & 0120 & 0110 & 17 & 1221 & 2421 \\ 2 & 0100 & 0100 & 10 & 0111 & 0211 & 18 & 1122 & 1111 \\ 3 & 0010 & 0010 & 11 & 1120 & 1110 & 19 & 1231 & 2431 \\ 4 & 0001 & 0001 & 12 & 1111 & 2211 & 20 & 1222 & 1211 \\ 5 & 1100 & 1100 & 13 & 0121 & 0221 & 21 & 1232 & 2432 \\ 6 & 0110 & 0210 & 14 & 1220 & 1210 & 22 & 1242 & 1221 \\ 7 & 0011 & 0011 & 15 & 1121 & 2221 & 23 & 1342 & 1321 \\ 8 & 1110 & 2210 & 16 & 0122 & 0111 & 24 & 2342 & 2321 \end{matrix}

If α_{i} and α_{j} are two roots of F_{4} such that their sum α_{i} + α_{j} is also a root, say α_{k}, then the corresponding root vectors from our Chevalley basis satisfy [X_{i}, X_{j}] = N(i, j)X_{k} where N(i, j) is an integer. If i + j ≠ 0 and α_{i} + α_{j} is not a root then [X_{i}, X_{j}] = 0. Finally if i + j = 0 then [X_{−i}, X_{i}] = H_{i}. In Table 10, for each positive root α_{i}, i.e., for 1 ≤ i ≤ 24, we list the indices j of all roots α_{j} such that α_{i} + α_{j} is a root. Each of these indices j is followed by a semi-colon and the expression N(i, j)X_{k}. For instance, if i = 4 we have

[X_{4}, X_{3}] = −X_{7}, [X_{4}, X_{17}] = 2X_{20}, [X_{4}, X_{−15}] = −2X_{−11}.

508 D – oković

Table 10: The structure constants of F_{4}

1 2 : X_{5}, 6 : X_{8}, 9 : X_{11}, 10 : X_{12}, 13 : X_{15}, 16 : X_{18}, 23 : X_{24}

−5 : −X_{−2}, −8 : −X_{−6}, −11 : −X_{−9}, −12 : −X_{−10}, −15 : −X_{−13} −18 : −X_{−16}, −24 : −X_{−23}

2 1 : −X_{5}, 3 : −X_{6}, 7 : X_{10}, 11 : X_{14}, 15 : X_{17}, 18 : X_{20}, 22 : X_{23}

−5 : X_{−1}, −6 : X_{−3}, −10 : −X_{−7}, −14 : −X_{−11}, −17 : −X_{−15} −20 : −X_{−18}, −23 : −X_{−22}

3 2 : X_{6}, 4 : X_{7}, 5 : X_{8}, 6 : 2X_{9}, 8 : 2X_{11}, 10 : X_{13}, 12 : X_{15}

17 : X_{19}, 20 : X_{21}, 21 : 2X_{22}, −6 : −2X_{−2}, −7 : −X_{−4} −8 : −2X_{−5}, −9 : −X_{−6}, −11 : −X_{−8}, −13 : −X_{−10} −15 : −X_{−12}, −19 : −X_{−17}, −21 : −2X_{−20}, −22 : −X_{−21}

4 3 : −X_{7}, 6 : X_{10}, 8 : X_{12}, 9 : X_{13}, 11 : X_{15}, 13 : 2X_{16}, 14 : X_{17}

15 : 2X_{18}, 17 : 2X_{20}, 19 : X_{21}, −7 : X_{−3}, −10 : −X_{−6} −12 : −X_{−8}, −13 : −2X_{−9}, −15 : −2X_{−11}, −16 : −X_{−13} −17 : −2X_{−14}, −18 : −X_{−15}, −20 : −X_{−17}, −21 : −X_{−19}

5 3 : −X_{8}, 7 : X_{12}, 9 : −X_{14}, 13 : −X_{17}, 16 : −X_{20}, 22 : X_{24}

−1 : X_{2}, −2 : −X_{1}, −8 : X_{−3}, −12 : −X_{−7}, −14 : X_{−9} −17 : X_{−13}, −20 : X_{−16}, −24 : −X_{−22}

6 1 : −X_{8}, 3 : −2X_{9}, 4 : −X_{10}, 7 : X_{13}, 8 : −2X_{14}, 12 : −X_{17}

15 : X_{19}, 18 : X_{21}, 21 : −2X_{23}, −2 : −X_{3}, −3 : 2X_{2} −8 : 2X_{−1}, −9 : X_{−3}, −10 : X_{−4}, −13 : −X_{−7}, −14 : X_{−8} −17 : X_{−12}, −19 : −X_{−15}, −21 : −2X_{−18}, −23 : X_{−21}

7 2 : −X_{10}, 5 : −X_{12}, 6 : −X_{13}, 8 : −X_{15}, 10 : −2X_{16}, 12 : −2X_{18}

14 : X_{19}, 17 : X_{21}, 19 : 2X_{22}, −3 : X_{4}, −4 : −X_{3}, −10 : 2X_{−2} −12 : 2X_{−5}, −13 : X_{−6}, −15 : X_{−8}, −16 : X_{−10} −18 : X_{−12}, −19 : −2X_{−14}, −21 : −X_{−17}, −22 : −X_{−19}

8 3 : −2X_{11}, 4 : −X_{12}, 6 : 2X_{14}, 7 : X_{15}, 10 : X_{17}, 13 : −X_{19}

16 : −X_{21}, 21 : −2X_{24}, −1 : X_{6}, −3 : 2X_{5}, −5 : −X_{3}, −6 : −2X_{1} −11 : X_{−3}, −12 : X_{−4}, −14 : −X_{−6}, −15 : −X_{−7} −17 : −X_{−10}, −19 : X_{−13}, −21 : 2X_{−16}, −24 : X_{−21}

9 1 : −X_{11}, 4 : −X_{13}, 5 : X_{14}, 12 : −X_{19}, 18 : X_{22}, 20 : X_{23}, −3 : X_{6}

−6 : −X_{3}, −11 : X_{−1}, −13 : X_{−4}, −14 : −X_{−5}, −19 : X_{−12} −22 : −X_{−18}, −23 : −X_{−20}

10 1 : −X_{12}, 3 : −X_{13}, 7 : 2X_{16}, 8 : −X_{17}, 11 : −X_{19}, 12 : −2X_{20}

15 : −X_{21}, 19 : 2X_{23}, −2 : X_{7}, −4 : X_{6}, −6 : −X_{4}, −7 : −2X_{2} −12 : 2X_{−1}, −13 : X_{−3}, −16 : −X_{−7}, −17 : X_{−8} −19 : 2X_{−11}, −20 : X_{−12}, −21 : X_{−15}, −23 : −X_{−19}

11 2 : −X_{14}, 4 : −X_{15}, 10 : X_{19}, 16 : −X_{22}, 20 : X_{24}, −1 : X_{9}

−3 : X_{8}, −8 : −X_{3}, −9 : −X_{1}, −14 : X_{−2}, −15 : X_{−4} −19 : −X_{−10}, −22 : X_{−16}, −24 : −X_{−20}

12 3 : −X_{15}, 6 : X_{17}, 7 : 2X_{18}, 9 : X_{19}, 10 : 2X_{20}, 13 : X_{21}

19 : 2X_{24}, −1 : X_{10}, −4 : X_{8}, −5 : X_{7}, −7 : −2X_{5} −8 : −X_{4}, −10 : −2X_{1}, −15 : X_{−3}, −17 : −X_{−6}, −18 : −X_{−7} −19 : −2X_{−9}, −20 : −X_{−10}, −21 : −X_{−13}, −24 : −X_{−19}

D – oković 509 Table 10 (continued)

13 1 : −X_{15}, 4 : −2X_{16}, 5 : X_{17}, 8 : X_{19}, 12 : −X_{21}, 15 : −2X_{22}

17 : −2X_{23}, −3 : X_{10}, −4 : 2X_{9}, −6 : X_{7}, −7 : −X_{6} −9 : −X_{4}, −10 : −X_{3}, −15 : 2X_{−1}, −16 : X_{−4}, −17 : −2X_{−5} −19 : −X_{−8}, −21 : X_{−12}, −22 : X_{−15}, −23 : X_{−17}

14 4 : −X_{17}, 7 : −X_{19}, 16 : −X_{23}, 18 : −X_{24}, −2 : X_{11}, −5 : −X_{9}

−6 : −X_{8}, −8 : X_{6}, −9 : X_{5}, −11 : −X_{2}, −17 : X_{−4} −19 : X_{−7}, −23 : X_{−16}, −24 : X_{−18}

15 2 : −X_{17}, 4 : −2X_{18}, 6 : −X_{19}, 10 : X_{21}, 13 : 2X_{22}, 17 : −2X_{24}

−1 : X_{13}, −3 : X_{12}, −4 : 2X_{11}, −7 : −X_{8}, −8 : X_{7} −11 : −X_{4}, −12 : −X_{3}, −13 : −2X_{1}, −17 : 2X_{−2}, −18 : X_{−4} −19 : X_{−6}, −21 : −X_{−10}, −22 : −X_{−13}, −24 : X_{−17}

16 1 : −X_{18}, 5 : X_{20}, 8 : X_{21}, 11 : X_{22}, 14 : X_{23}, −4 : X_{13},

−7 : −X_{10}, −10 : X_{7}, −13 : −X_{4}, −18 : X_{−1} −20 : −X_{−5}, −21 : −X_{−8}, −22 : −X_{−11}, −23 : −X_{−14}

17 3 : −X_{19}, 4 : −2X_{20}, 7 : −X_{21}, 13 : 2X_{23}, 15 : 2X_{24}, −2 : X_{15}

−4 : 2X_{14}, −5 : −X_{13}, −6 : −X_{12}, −8 : X_{10}, −10 : −X_{8} −12 : X_{6}, −13 : 2X_{5}, −14 : −X_{4}, −15 : −2X_{2}, −19 : X_{−3} −20 : X_{−4}, −21 : X_{−7}, −23 : −X_{−13}, −24 : −X_{−15}

18 2 : −X_{20}, 6 : −X_{21}, 9 : −X_{22}, 14 : X_{24}, −1 : X_{16}, −4 : X_{15}

−7 : −X_{12}, −12 : X_{7}, −15 : −X_{4}, −16 : −X_{1}, −20 : X_{−2} −21 : X_{−6}, −22 : X_{−9}, −24 : −X_{−14}

19 4 : −X_{21}, 7 : −2X_{22}, 10 : −2X_{23}, 12 : −2X_{24}, −3 : X_{17}

−6 : X_{15}, −7 : 2X_{14}, −8 : −X_{13}, −9 : −X_{12}, −10 : −2X_{11} −11 : X_{10}, −12 : 2X_{9}, −13 : X_{8}, −14 : −X_{7}, −15 : −X_{6} −17 : −X_{3}, −21 : X_{−4}, −22 : X_{−7}, −23 : X_{−10}, −24 : X_{−12}

20 3 : −X_{21}, 9 : −X_{23}, 11 : −X_{24}, −2 : X_{18}, −4 : X_{17}, −5 : −X_{16}

−10 : −X_{12}, −12 : X_{10}, −16 : X_{5}, −17 : −X_{4} −18 : −X_{2}, −21 : X_{−3}, −23 : X_{−9}, −24 : X_{−11}

21 3 : −2X_{22}, 6 : 2X_{23}, 8 : 2X_{24}, −3 : 2X_{20}, −4 : X_{19}, −6 : 2X_{18}

−7 : X_{17}, −8 : −2X_{16}, −10 : −X_{15}, −12 : X_{13}, −13 : −X_{12} −15 : X_{10}, −16 : X_{8}, −17 : −X_{7}, −18 : −X_{6} −19 : −X_{4}, −20 : −X_{3}, −22 : X_{−3}, −23 : −X_{−6}, −24 : −X_{−8}

22 2 : −X_{23}, 5 : −X_{24}, −3 : X_{21}, −7 : X_{19}, −9 : X_{18}, −11 : −X_{16}

−13 : −X_{15}, −15 : X_{13}, −16 : X_{11}, −18 : −X_{9} −19 : −X_{7}, −21 : −X_{3}, −23 : X_{−2}, −24 : X_{−5}

23 1 : −X_{24}, −2 : X_{22}, −6 : −X_{21}, −9 : X_{20}, −10 : X_{19}, −13 : −X_{17}

−14 : −X_{16}, −16 : X_{14}, −17 : X_{13}, −19 : −X_{10} −20 : −X_{9}, −21 : X_{6}, −22 : −X_{2}, −24 : X_{−1}

24 −1 : X_{23}, −5 : X_{22}, −8 : −X_{21}, −11 : X_{20}, −12 : X_{19} − 14 : −X_{18}

−15 : −X_{17}, −17 : X_{15}, −18 : X_{14}, −19 : −X_{12}, −20 : −X_{11} −21 : X_{8}, −22 : −X_{5}, −23 : −X_{1}

References

[1] Barbasch, D., and M. R. Sepanski, Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc. 126 (1998), 311–317.

[2] Bourbaki, N., “Groupes et Algèbres de Lie,” Chapitres IV, V, et VI, Hermann, Paris, 1968, 288 pp.

[3] Carter, R. W., “Finite Groups of Lie Type: Conjugacy Classes and Complex Characters,” J. Wiley, New York, 1985, xii+544 pp.

[4] Collingwood, D. H., and W. M. McGovern, “Nilpotent Orbits in Semisimple Lie Algebras,” Van Nostrand Reinhold, New York, 1993, xiii+186 pp.

[5] D – oković, D.Ž., Closures of conjugacy classes in classical real linear Lie groups, in “Algebra,” Carbondale, 1980, Lecture Notes in Math. 848, Springer-Verlag, New York, 1980, 63–83.

[6] —, Closures of conjugacy classes in classical real linear Lie groups, II, Trans. Amer. Math. Soc. 270 (1982), 217–252.

[7] —, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), 503–524.

[8] —, Classification of nilpotent elements in simple real Lie algebras E_{6(6)} and E_{6(−26)} and description of their centralizers, J. Algebra 116 (1988), 196–207.

[9] —, Explicit Cayley triples in real forms of G_{2}, F_{4}, and E_{6}, Pacific J. Math. 184 (1998), 231–255.

[10] D – oković, D. Ž., J. Sekiguchi, and N. Lemire, The closure ordering of adjoint nilpotent orbits in 𝔰𝔬(p, q), Preprint 1999, 48 pp.

[11] Igusa, J., A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028.

[12] Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.

[13] Kraft, H., “Geometrische Methoden in der Invariantentheorie, ” Vieweg, Braunschweig–Wiesbaden, 1984, x+308 pp.

[14] Sato, M., and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.

Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1 dragomir@herod.uwaterloo.ca

Received April 3, 2000 and in final form April 15, 2000