THE RATIONAL ADJOINT ORBITS OF A LIE ALGEBRA OF TYPE G_{2}

DAVID G. A. JACKSON

Abstract. Let G be the Chevalley group of type G_{2} over the algebraic closure of a finite field of characteristic not 2 or 3. Let F be the Frobenius endomorphism on G. In this paper we study and classify the Ad G^{F} -orbits on 𝔤^{F} and also the nilpotent Ad G-orbits on 𝔤.

1. Introduction

In 1968, Chang [5] classified the rational conjugacy classes of a Chevalley group G of type G_{2}. Here we study the analogous problem of classifying the rational adjoint orbits of G on its Lie algebra 𝔤. The approach followed here will closely parallel that of Chang. The method is based on the observation that if X ∈ 𝔤^{F} and X = X_{s} + X_{n} is its Jordan decomposition then X_{n} ∈ C_{𝔤}(X_{s}) = Lie(C_{G}(X_{s})).

First we classify the rational semisimple orbits. We deal with the regular semisimple orbits first (see Theorem 14), then the irregular semisimple orbits (see Theorem 16). If X ∈ 𝔤^{F} has a nonzero irregular semisimple part then C_{G}(X_{s}) is a nontrivial proper Levi subgroup of G, hence isomorphic to GL_{2}. Using information about the nilpotent orbits in 𝔤𝔩_{2} we may thus classify the nonsemisimple nonnilpotent orbits (see Theorem 20). Substantially more work has to be done to classify the nilpotent orbits. The results are contained in Theorems 31 and 32.

2. Notation

Let q be a power of a prime p and let 𝔽_{q} denote the finite field of q elements. Let \overline{𝔽}_{q} be the algebraic closure of 𝔽_{q}. If a ∈ 𝔽_{q}, let (\frac{a}{q}) denote the Legendre symbol which takes the value −1, 1 or 0 according as a is a nonsquare in 𝔽_{q}, a nonzero square in 𝔽_{q}, or 0. According to context, G will denote either an arbitrary connected reductive algebraic group or the connected reductive algebraic group of type G_{2} defined over 𝔽_{q}. For the general theory of reductive algebraic groups refer to Borel [1], Springer [14] and Borel & Tits [2]. We will often abbreviate the term “reductive algebraic” to merely “reductive”. We will denote the identity of G by e. Gothic lower case characters will be used to denote Lie algebras so, for example, 𝔤 (resp. 𝔲 etc.) denotes the Lie algebra of G (resp. U etc.). If H is a reductive group then H^{∘} will denote the connected component containing the identity and, for X ∈ 𝔥,

\begin{aligned}C_{H}(X) & = \{h ∈ H | Ad h(X) = X\} \\ C_{𝔥}(X) & = \{Y ∈ 𝔥 | [Y, X] = 0\}. \end{aligned}

We have Lie(C_{G}(X)) = C_{𝔤}(X) if X ∈ 𝔤 is semisimple (see Borel [1, 9.1(2)]).

We will denote the root system of G by Φ and we will let Π be a choice of simple roots.

The connected reductive group of type G_{2} over 𝔽_{q} can also be defined as the Chevalley group of type G_{2} over \overline{𝔽}_{q} and we shall use the notations of Carter [3]. Specifically, if Φ denotes the root system of type G_{2}, then in 𝔤 we have the elements e_{r} and h_{r} for r ∈ Φ and, if t ∈ \overline{𝔽}_{q}, we have in G the elements x_{r}(t) and h_{r}(t). The elements e_{r} and h_{r} satisfy the commutation relations [h_{r}, e_{s}] = A_{rs}e_{s} = \frac{2(r,s)}{(r,r)}e_{s} and [e_{r}, e_{s}] = N_{rs}e_{r+s} if r ≠ −s. If β = ∑_{α∈Π} n_{α}α is a root then we will define the height of β to be the integer ∑_{α∈Π} n_{α}. We will label the root system of type G_{2} as in the following diagram.

[figure: κ_{2}=a; 3a+b; 2a+b; 3a+2b; κ_{1}=a+b; b; −a; −3a−b; κ_{3}=−2a−b; −3a−2b; −a−b; −b]

It will be convenient to define κ_{1} = a + b, κ_{2} = a, κ_{3} = −2a − b and κ_{0} = 0. For this labelling of the root system the elements e_{r} will be chosen so that the structure constants N_{rs} are as given on page 211 of Carter [3]. The action of G on the elements e_{r}, h_{r} is given by the relations

\begin{aligned}Ad x_{r}(t)(e_{r}) & = e_{r}, \\ Ad x_{r}(t)(e_{−r}) & = e_{−r} + th_{r} − t^{2}e_{r}, \\ Ad x_{r}(t)(h_{s}) & = h_{s} − A_{sr}te_{r} \end{aligned}

and if r, s are linearly independent

Ad x_{r}(t)(e_{s}) = ∑_{0⩽i⩽m} M_{r,s,i}t^{i}e_{ir+s}

where m is the largest integer such that mr + s is a root and

M_{r,s,i} = \frac{1}{i!}N_{r,s}N_{r,r+s} … N_{r,(i−1)r+s}.

We also have the Chevalley commutator relations

x_{r}(t)x_{s}(u)x_{r}(t)^{−1} = x_{s}(u) ∏_{i,j>0} x_{ir+js}(C_{ijrs}t^{i}u^{j})

for r ≠ ±s, where C_{ijrs} ∈ {±1, ±2, ±3}. The Weyl group will be denoted by W and the reflection corresponding to the root r will be denoted by w_{r}. If we define n_{r} = x_{r}(1)x_{−r}(−1)x_{r}(1) then n_{r} is a representative in N_{G}(T) of w_{r} and

Ad n_{r}(e_{s}) = η_{r,s}e_{w_{r}(s)}

where η_{r,s} = ±1. The values of η_{r,s} are given in Ree [12, (3.4)]. For convenience, tables of A_{r,s}, N_{r,s}, η_{r,s}, Ad x_{r}(t)(e_{s}) and Ad x_{r}(t)(h_{s}) are given in full in the Appendix.

For r ∈ Φ, let H_{r} denote the subgroup of T consisting of the elements h_{r}(s) for s ∈ \overline{𝔽}_{q} and X_{r} the subgroup of G consisting of the elements x_{r}(s) for s ∈ \overline{𝔽}_{q}. If w ∈ W then U_{w} will denote the subgroup of G generated by those X_{r} for which r is a positive root but w(r) is a negative root.

There are two finite groups which feature prominently in the present work: the Weyl group W ≅ D_{12} (the dihedral group of order 12) and S_{3} (the symmetric group on three letters). A set of conjugacy class representatives for W is given by

{1, w_{a}, w_{b}, w_{6} := w_{a}w_{b}, w_{6}^{4}, w_{6}^{3}}.

Unless specified otherwise, w_{6}^{3} and w_{6}^{4} will have the values as defined above.

Given a maximal torus T, we shall denote by ẇ a representative in N = N_{G}(T) of the element w of W (G, T) = N_{G}(T)/T. We shall use the following choice of representatives. If r ∈ Φ, let ω_{r} = n_{r} represent w_{r}. We may represent w_{6} by the element ω_{6} = n_{a}n_{b} and hence w_{6}^{i} is represented by ω_{6}^{i}.

Let F : G → G be the Frobenius morphism on G (see Digne & Michel [6, §3]). The set G^{F} of fixed points of G under F is a finite group (sometimes called a finite group of Lie type or finite reductive group). For type G_{2}, this group is the Chevalley group of type G_{2} over 𝔽_{q}. An element g (resp. subgroup H) of G that is fixed by F will be said to be a rational element (resp. subgroup) of G and G^{F} is the group of rational points of G.

If X is an element of 𝔤 (resp. 𝔤^{F}) then the set {Ad g(X) | g ∈ G} (resp. {Ad g(X) | g ∈ G^{F} }) is called the Ad G- (resp. Ad G^{F} -)orbit of X. We will call an Ad G^{F} -orbit in 𝔤^{F} a rational orbit. If g, h ∈ G we will use the notation Int g(h) = ^{g}h.

Two elements a, b of a group H are said to be F -conjugate if there exists x ∈ H such that b = x^{−1}aF (x). We shall denote by H^{1}(F, H) the set of F -conjugacy classes of an algebraic group H defined over 𝔽_{q}.

If X ∈ 𝔤, we will denote the semisimple and nilpotent parts of X by X_{s} and X_{n} respectively.

3. Generalities on semisimple orbits

First we state and prove some basic general results concerning semisimple classes. For the moment G will denote an arbitrary connected reductive group. We will fix an F -stable maximal torus T and an F -stable Borel subgroup B containing T.

Lemma 1. Let X, Y ∈ 𝔱 with Y = Ad(g)(X) for some g ∈ G. If g = uẇtu′ with u ∈ U, w ∈ W, t ∈ T and u′ ∈ U_{w}, then Ad ẇ(X) = Y, Ad u′(X) = X and Ad u(Y) = Y.

Proof. We have

Ad(ẇt) Ad u′(X) = Ad u^{−1}(Y).

The map s ↦ u′su′^{−1}s^{−1} (resp. s ↦ u^{−1}sus^{−1}) takes T to U_{w} (resp. U) and hence upon differentiation we find that Ad u′(X) − X ∈ 𝔲_{w} (resp. Ad u^{−1}(Y) − Y ∈ 𝔲). Hence Ad(ẇt)(Ad u′(X) − X) ∈ 𝔲^{−}. Now

Ad(ẇt)(X) + Ad(ẇt)(Ad u′(X) − X) = Y + Ad u^{−1}(Y − Ad u(Y)).

Since 𝔤 = 𝔲^{−} ⊕ 𝔱 ⊕ 𝔲 we must thus have Ad ẇ(X) = Ad(ẇt)(X) = Y, Ad u′(X) = X and Ad u(Y) = Y.

Lemma 2. If w ∈ W, α ∈ Φ, then there exists c ≠ 0 such that, for all t, ẇx_{α}(t) = x_{w(α)}(ct) and Ad ẇ(e_{α}) = ce_{w(α)}.

Proof. Since ẇX_{α} = X_{w(α)} and Aut(G_{a}) ≡ GL_{1} (see Borel [1, §10.10]), ẇx_{α}(t_{α}) = x_{w(α)}(ct_{α}) for some c ≠ 0. Consequently, on passing to the Lie algebra, Ad ẇ(e_{α}) = ce_{w(α)}.

Proposition 3 (cf. Steinberg [16, 2.7(a), 2.8]). Let G be a connected reductive group. Let T be a fixed maximal torus. If X ∈ 𝔱 then

(i) C_{G}(X)^{∘} = ⟨T, X_{α} | α(X) = 0⟩. (ii) C_{G}(X) = ⟨T, X_{α}, ẇ | α(X) = 0, Ad ẇ(X) = X⟩.

Moreover C_{G}(X)^{∘} is reductive.

Proof. The corresponding result for centralizers of elements of T follows in a straightforward manner from the Bruhat decomposition (see, for example, Carter [Theorem 2.5.14]carter:1993). This approach must be modified in the present situation. Although well-known, there does not appear to be a proof of this result in the standard literature.

We will deal with (ii) first. Observe that T, the X_{α} with α(X) = 0 and the ẇ with Ad ẇ(X) = X all lie in C_{G}(X).

Conversely, suppose that g ∈ C_{G}(X). By the Bruhat decomposition we may write g = uẇtu′ with u ∈ U, w ∈ W, t ∈ T, u′ ∈ U_{w}. By Lemma 1 we then have Ad ẇ(X) = X, Ad u(X) = X and Ad u′(X) = X. We want to show that if Ad u(X) = X then u lies in the subgroup generated by the X_{α} for which α(X) = 0. So suppose that u does not lie in this subgroup. According to Borel [1, §14.4], U = ∏_{α∈Φ^{+}} X_{α} with uniqueness for any ordering of the roots in the product. Choose the ordering so that the roots have weakly decreasing height and write u = ∏_{α∈Φ^{+}} x_{α}(t_{α}). Then for some α, t_{α} ≠ 0 and α(X) ≠ 0. Let i be the minimal height of such a root. We may then assume that u = u_{>i}u_{i} where

u_{>i} = ∏_{\begin{matrix}α∈Φ^{+} \\ ht(α)>i \end{matrix}} x_{α}(t_{α}) and u_{i} = ∏_{\begin{matrix}α∈Φ^{+} \\ ht(α)=i \end{matrix}} x_{α}(t_{α}).

We then have

Ad u(X) − X = (Ad u_{>i}(Ad u_{i}(X) − X) − (Ad u_{i}(X) − X))

+ (Ad u_{>i}(X) − X) + (Ad u_{i}(X) − X).

By differentiating the Chevalley commutator relations we see that, if

𝔤_{>i} = ⨁_{\begin{matrix}α∈Φ^{+} \\ ht(α)>i \end{matrix}} 𝔤_{α},

with a similar definition being made for 𝔤_{⩾i}, then Ad u_{i}(X) − X ∈ 𝔤_{⩾i},

Ad u_{>i}(Ad u_{i}(X) − X) − (Ad u_{i}(X) − X) ∈ 𝔤_{>2i}

and Ad u_{>i}(X) − X ∈ 𝔤_{>i}. Since i > 0, we have 2i > i and

Ad u(X) − X ≡ Ad u_{i}(X) − X mod 𝔤_{>i}.

A similar argument then shows that

Ad u(X) − X ≡ ∑_{\begin{matrix}α∈Φ^{+} \\ ht(α)=i \end{matrix}}(Ad x_{α}(t_{α})(X) − X) mod 𝔤_{>i}

≡ ∑_{\begin{matrix}α∈Φ^{+} \\ ht(α)=i \end{matrix}} −α(X)t_{α}e_{α} mod 𝔤_{>i}.

We have assumed however that, for some α of height i, t_{α} ≠ 0 and α(X) ≠ 0 and thus Ad u(X) − X ≠ 0, a contradiction. Thus α(X) = 0 whenever t_{α} ≠ 0 and u lies in the subgroup generated by the X_{α} with α(X) = 0. A similar result holds for u′, whence g = uẇtu′ lies in the subgroup ⟨T, X_{α}, ẇ | α(X) = 0, Ad ẇ(X) = X⟩.

The group ⟨T, X_{α} | α(X) = 0⟩ is connected, being generated by connected groups T, X_{α} (see Humphreys [7, §7.5, Corollary]). Now let Ad ẇ(X) = X and suppose that α(X) = 0. Then, for all t_{α} ∈ \overline{𝔽}_{q}, Ad x_{α}(t_{α})(X) = X and hence

Ad ẇx_{α}(t_{α})(X) = Ad(ẇx_{α}(t_{α}))(X) = Ad ẇ(X) = X.

On the other hand, from Lemma 2,

Ad ẇx_{α}(t_{α})(X) = X − w(α)(X)ct_{α}e_{α}

and thus w(α)(X) = 0. Hence ẇ normalizes ⟨T, X_{α} | α(X) = 0⟩. This implies that ⟨T, X_{α} | α(X) = 0⟩ is a normal subgroup of C_{G}(X) of finite index, the index being at most |W |. This subgroup is therefore the connected component C_{G}(X)^{∘} (see Borel [1, 1.2(c)]). That C_{G}(X)^{∘} is reductive follows from Digne & Michel [6, Prop. 1.14] since {α ∈ Φ | α(X) = 0} is a closed symmetric subset of Φ.

We have the following theorem, stated in Springer & Steinberg [15, II, 3.19] without proof.

Theorem 4. Assume that G is semisimple and simply connected. If the characteristic is not a torsion prime (see Springer & Steinberg [15, I, 4.1]), then C_{G}(X) is connected and reductive for every semisimple element X of 𝔤, and conversely.

Note that for type G_{2} the only torsion prime is 2.

Proposition 5 (cf. Springer & Steinberg [15, II, 3.16]). The map O ↦ O ∩ 𝔱 yields a bijection from the F -stable semisimple orbits in 𝔤 to the F -stable W -orbits on 𝔱.

If G is as in Theorem 4 then

(i) there is a bijection between F -stable semisimple classes in 𝔤 and semisimple Ad G^{F} -orbits on 𝔤^{F} given by O ↦ O ∩ 𝔤^{F};

(ii) there is a bijection between semisimple Ad G^{F} -orbits on 𝔤^{F} and F -stable W -orbits in 𝔱 given by O ↦ Ad G(O) ∩ 𝔱.

Proof. The first statement is precisely Springer & Steinberg [15, II, 3.16]. The proof of the second statement follows from Theorem 4 and Digne & Michel [6, 3.21].

Definition. We shall say that a (not necessarily rational) element X of 𝔱 represents the Ad G^{F} -orbit O if O = Ad G(X) ∩ 𝔤^{F}.

Lemma 6. Let X ∈ 𝔱. Then

(i) the W -orbit of X is F -stable if and only if F (X) = Ad ẇ(X) for some w ∈ W; (ii) if w ∈ W satisfies F (X) = Ad ẇ(X) and if w′ ∈ W is conjugate to w then

F (X′) = Ad ẇ′(X′) for some X′ in the W -orbit of X.

Proof. Let us prove (i) first. If the orbit is F -stable, then F (X) must be an element of the orbit and hence F (X) = Ad ẇ(X) for some w ∈ W. Conversely, if F (X) = Ad ẇ(X) and w′ ∈ W then

F (Ad ẇ′(X)) = Ad F (ẇ′)(F (X)) = Ad(F (ẇ′)ẇ)(X)

and hence the W -orbit of X is F -stable.

For (ii), if w′ = w_{1}ww_{1}^{−1} then simply take X′ = Ad ẇ_{1}(X).

4. Semisimple orbits (type G_{2})

For the rest of this paper, G will denote the Chevalley group of type G_{2} over the field \overline{𝔽}_{q} of characteristic p ≠ 2, 3.

Lemma 7. If α ∈ Φ then h_{α} is an integral linear combination of h_{a} and h_{b} with coefficients given as follows:

h_{a+b} = h_{a} + 3h_{b}  h_{2a+b} = 2h_{a} + 3h_{b}
h_{3a+b} = h_{a} + h_{b}  h_{3a+2b} = h_{a} + 2h_{b}

the remainder being given by h_{−α} = −h_{α}.

Proof. These relations hold for the corresponding elements of 𝔤(ℂ) by virtue of the relation h_{α} = \frac{2α}{(α,α)} (see Carter [3, 3.3, 3.5]). Since the coefficients are integers they carry across to the elements h_{α} of 𝔤 = 𝔤(\overline{𝔽}_{q}).

Lemma 8 (Carter [3, Prop. 6.4.2]). If w ∈ W, α ∈ Φ then Ad ẇ(h_{α}) = h_{w(α)}.

Exploiting the symmetry among κ_{1}, κ_{2} and κ_{3}, we find that every element of 𝔱 may be represented uniquely in the form \frac{1}{3}(λ_{1}h_{κ_{1}} + λ_{2}h_{κ_{2}} + λ_{3}h_{κ_{3}}) where λ_{1} + λ_{2} + λ_{3} = 0. We will denote such an element of 𝔱 by (λ_{1}, λ_{2}, λ_{3}). Note that the restriction p ≠ 3 is important here for otherwise h_{κ_{1}}, h_{κ_{2}} and h_{κ_{3}} do not span 𝔱. According to Carter [3, 7.1.1] every group homomorphism from ℤΦ to \overline{𝔽}_{q}^{×} gives rise to an element of T. If κ_{i} gets sent to z_{i} under such a homomorphism then this element will be denoted by h(z_{1}, z_{2}, z_{3}). The relation κ_{1} + κ_{2} + κ_{3} = 0 implies that z_{1}z_{2}z_{3} = 1. If χ_{α,s} is the homomorphism from ℤΦ to \overline{𝔽}_{q}^{×} whose value on β is s^{A_{αβ}} then the corresponding element of T is h_{α}(s). Consequently, h_{a}(s) = h(s^{−1}, s^{2}, s^{−1}) and h_{b}(s) = h(s, s^{−1}, 1). Lemma 8 implies the following result.

Lemma 9. If w ∈ W and (λ_{1}, λ_{2}, λ_{3}) ∈ 𝔱 then Ad ẇ(λ_{1}, λ_{2}, λ_{3}) is given according to the following table.

\begin{matrix}w & Ad ẇ(λ_{1}, λ_{2}, λ_{3}) & w & Ad ẇ(λ_{1}, λ_{2}, λ_{3}) \\ 1 & (λ_{1}, λ_{2}, λ_{3}) & w_{3a+2b} & (λ_{3}, λ_{2}, λ_{1}) \\ w_{a} & (−λ_{3}, −λ_{2}, −λ_{1}) & w_{6} & (−λ_{2}, −λ_{3}, −λ_{1}) \\ w_{b} & (λ_{2}, λ_{1}, λ_{3}) & w_{6}^{2} & (λ_{3}, λ_{1}, λ_{2}) \\ w_{a+b} & (−λ_{1}, −λ_{3}, −λ_{2}) & w_{6}^{3} & (−λ_{1}, −λ_{2}, −λ_{3}) \\ w_{2a+b} & (−λ_{2}, −λ_{1}, −λ_{3}) & w_{6}^{4} & (λ_{2}, λ_{3}, λ_{1}) \\ w_{3a+b} & (λ_{1}, λ_{3}, λ_{2}) & w_{6}^{5} & (−λ_{3}, −λ_{1}, −λ_{2}) \end{matrix}

According to Lemma 6, in order to determine the semisimple Ad G^{F} -orbits on 𝔤^{F} it is sufficient to determine representatives X = (λ_{1}, λ_{2}, λ_{3}) ∈ 𝔱 for W -orbits in 𝔱 such that F (X) = Ad ẇ(X) where w ∈ W runs over a chosen set of conjugacy class representatives. We will choose the set {1, w_{a}, w_{b}, w_{6}^{3}, w_{6}^{4}, w_{6}} of conjugacy class representatives.

Lemma 10. If w ∈ W is a conjugacy class representative of W and X ∈ 𝔱 is such that F (X) = Ad ẇ(X) then X is of the form given in the following table.

w  X
1  (λ_{1}, λ_{2}, λ_{3}) λ_{i}^{q} = λ_{i}
w_{a} (λ, λ^{q} − λ, −λ^{q}) λ^{q^{2}} = λ
w_{b} (λ, λ^{q}, −λ − λ^{q}) λ^{q^{2}} = λ
w_{6}^{3} (λ_{1}, λ_{2}, λ_{3}) λ_{i}^{q} = −λ_{i}
w_{6}^{4} (λ, λ^{q}, −λ − λ^{q}) λ^{q^{2}} + λ^{q} + λ = 0
w_{6} (λ, −λ^{q}, λ^{q} − λ) λ^{q^{2}} − λ^{q} + λ = 0

Proof. The result follows by application of Lemma 9.

Lemma 11. The values of the roots on 𝔱 are given by

κ_{i}(λ_{1}, λ_{2}, λ_{3}) = λ_{i}
(κ_{i} − κ_{j})(λ_{1}, λ_{2}, λ_{3}) = λ_{i} − λ_{j}.

Proof. Clearly we only need to prove the first formula. If α, β ∈ Φ then β(h_{α}) = A_{αβ}. For simplicity consider the case i = 1.

κ_{1}(λ_{1}, λ_{2}, λ_{3}) = \frac{1}{3}(λ_{1}κ_{1}(h_{κ_{1}}) + λ_{2}κ_{1}(h_{κ_{2}}) + λ_{3}κ_{1}(h_{κ_{3}}))

= \frac{1}{3}(λ_{1}A_{κ_{1}κ_{1}} + λ_{2}A_{κ_{2}κ_{1}} + λ_{3}A_{κ_{3}κ_{1}})

= \frac{1}{3}(2λ_{1} − λ_{2} − λ_{3})

= λ_{1}.

The other two cases are dealt with similarly.

We are now in a position to determine the regular semisimple Ad G^{F} -orbits. Let X ∈ 𝔤^{F} be semisimple and let Y ∈ G · X ∩ 𝔱. According to Proposition 3, X is regular if and only if α(Y) ≠ 0 for all α ∈ Φ. If Y = (λ_{1}, λ_{2}, λ_{3}) then this condition becomes λ_{i} ≠ 0 and λ_{i} ≠ λ_{j} for i ≠ j.

Definition. If the regular element Y ∈ 𝔱 represents the semisimple Ad G^{F} -orbit of X and F (Y) = Ad ẇ(Y) then X and Y will be said to be regular of type w.

Lemma 12. If X_{i} ∈ 𝔱 (i = 1, 2) are regular and F (X_{i}) = Ad ẇ_{i}(X_{i}) for some w_{i} ∈ W then X_{1} and X_{2} are in the same W -orbit if and only if w_{1} and w_{2} are conjugate.

Proof. Lemma 6 gives one direction. Now suppose that X_{2} = Ad ẇ(X_{1}). Then F (Ad ẇ(X_{1})) = Ad ẇ_{2}(Ad ẇ(X_{1})) and hence we have F (X_{1}) = Ad(ẇ^{−1}ẇ_{2}ẇ)(X_{1}). But F (X_{1}) = Ad ẇ_{1}(X_{1}) and hence ẇ_{1}^{−1}ẇ^{−1}ẇ_{2}ẇ ∈ C_{G}(X_{1}). According to Proposition 3 and Theorem 4, C_{G}(X_{1}) = T and hence w_{1} = w^{−1}w_{2}w.

Some elementary calculations in the group D_{12} yield the following lemma.

Lemma 13. For type W = D_{12}, the groups C_{W} (w), where w is a conjugacy class representative of W, are given by the following table.

\begin{matrix}w & C_{W} (w) \\ 1 & W \\ w_{a} & \{1, w_{a}, w_{3a+2b}, w_{6}^{3}\} \\ w_{b} & \{1, w_{b}, w_{2a+b}, w_{6}^{3}\} \\ w_{6}^{3} & W \\ w_{6}^{4} & \{1, w_{6}, w_{6}^{2}, w_{6}^{3}, w_{6}^{4}, w_{6}^{5}\} \\ w_{6} & \{1, w_{6}, w_{6}^{2}, w_{6}^{3}, w_{6}^{4}, w_{6}^{5}\} \end{matrix}

Theorem 14. Let G be the connected reductive group of type G_{2} over the field 𝔽_{q} of characteristic p ≠ 2, 3. The regular semisimple Ad G^{F} -orbits O on 𝔤^{F} are given by the following table. If Y ∈ 𝔱 is an entry from the first column then the corresponding regular semisimple Ad G^{F} -orbit is given by O = Ad G(Y) ∩ 𝔤^{F}. In the table, X refers to any element of O.

representative type

|C_{G^{F}} (X)|
(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = 1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0  1 (q − 1)^{2}

{(λ, λ^{q} − λ, −λ^{q}), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  w_{a}  q^{2} − 1
{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  w_{b}  q^{2} − 1

(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = −1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0 w_{6}^{3} (q + 1)^{2}

{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} + λ^{q−1} + 1 = 0}  w_{6}^{4} q^{2} + q + 1
{(λ, −λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} − λ^{q−1} + 1 = 0}  w_{6} q^{2} − q + 1

In this table representatives from different rows represent different Ad G^{F} -orbits but different representatives of the same type may represent the same Ad G^{F} -orbit. For each conjugacy class representative w ∈ W there exists a function P_{w} which associates to an element X ∈ 𝔱 of type w a set P_{w}(X) such that if X_{1}, X_{2} ∈ 𝔱 are both of type w then X_{1} and X_{2} represent the same Ad G^{F} -orbit if and only if P_{w}(X_{1}) = P_{w}(X_{2}).

The parametrizing sets P_{w}(X) are given as follows.

representative type parametrizing set

no. of ; Ad G^{F} - ; orbits of ; given ; form

(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = 1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0  1 \{\{λ_{1}, λ_{2}, λ_{3}\}, \{−λ_{1}, −λ_{2}, −λ_{3}\}\} \frac{(q−1)(q−5)}{12}

{(λ, λ^{q} − λ, −λ^{q}), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  w_{a}  \{±λ, ±λ^{q}\}  \frac{(q−1)^{2}}{4}
{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  w_{b}  \{±λ, ±λ^{q}\}  \frac{(q−1)^{2}}{4}

(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = −1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0 w_{6}^{3} \{\{λ_{1}, λ_{2}, λ_{3}\}, \{−λ_{1}, −λ_{2}, −λ_{3}\}\} \frac{(q−1)(q−5)}{12}

{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} + λ^{q−1} + 1 = 0}  w_{6}^{4}  \{±λ, ±λ^{q}, ±(λ^{q} + λ)\}  \frac{q^{2}−1}{6}
{(λ, −λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} − λ^{q−1} + 1 = 0}  w_{6}  \{±λ, ±λ^{q}, ±(λ^{q} − λ)\}  \frac{q^{2}−1}{6}

Proof. Proposition 5 and Lemma 6 imply that every regular semisimple Ad G^{F} -orbit O is represented by an element of 𝔱 of type w where w is one of the chosen conjugacy class representatives in W. Proposition 5 and Lemma 12 then show that the representatives from different rows represent different Ad G^{F} -orbits. The different parametrizing sets result from Lemmas 12 and 13.

If X is a rational element of the orbit represented by an element of 𝔱 of type w then |C_{G}^{F}(X)| = |T_{w}^{F} | where T_{w} is an F -stable maximal torus twisted by w. By Carter [4, Prop. 3.3.5], |T_{w}^{F} | = f(q) where f(x) is the characteristic polynomial of w acting on Y (T) ⊗_{ℤ} ℝ, Y (T) being the cocharacter group of T. Now, the nontrivial elements of the set {1, ω_{a}, ω_{b}, ω_{6}, ω_{6}^{4}, ω_{6}^{3}} act on T as follows:

ω_{a}h(z_{1}, z_{2}, z_{3})ω_{a}^{−1} = h(z_{3}^{−1}, z_{2}^{−1}, z_{1}^{−1})
ω_{b}h(z_{1}, z_{2}, z_{3})ω_{b}^{−1} = h(z_{2}, z_{1}, z_{3})
ω_{6}h(z_{1}, z_{2}, z_{3})ω_{6}^{−1} = h(z_{3}^{−1}, z_{1}^{−1}, z_{2}^{−1})
ω_{6}^{4}h(z_{1}, z_{2}, z_{3})ω_{6}^{−4} = h(z_{3}, z_{1}, z_{2})
ω_{6}^{3}h(z_{1}, z_{2}, z_{3})ω_{6}^{−3} = h(z_{1}^{−1}, z_{2}^{−1}, z_{3}^{−1}).

The space Y (T) ⊗ ℝ is spanned over ℝ by the elements

h_{a} : λ ↦ h(λ^{−1}, λ^{2}, λ^{−1})
h_{b} : λ ↦ h(λ, λ^{−1}, 1).

As an example of the calculation of |T_{w}^{F} | let us look at the case w = w_{a}. We have ω_{a}h_{a}(λ)ω_{a}^{−1} = h_{a}(λ^{−1}) = (−h_{a})(λ), where we have used additive notation on Y (T)⊗ℝ. Similarly, ω_{a}h_{b}(λ)ω_{a}^{−1} = h_{a}(λ)h_{b}(λ) = (h_{a}+h_{b})(λ). Thus the matrix of ω_{a} with respect

to the basis (h_{a}, h_{b}) of Y (T) ⊗ ℝ is (\begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix}). The characteristic polynomial of this

matrix is f(x) = x^{2}−1 and thus |T_{w_{a}}^{F}| = q^{2}−1. Similarly, we find that |T_{1}^{F} | = (q−1)^{2}, |T_{w_{b}}^{F}| = q^{2} − 1, |T_{w_{6}}^{F}| = q^{2} − q + 1, |T_{w_{6}^{4}}^{F}| = q^{2} + q + 1, |T_{w_{6}^{3}}^{F}| = (q + 1)^{2}.

We will now determine the number of regular semisimple Ad G^{F} -orbits of type w. If (λ_{1}, λ_{2}, λ_{3}) is of type 1 or w_{6}^{3}, then λ_{3} = −λ_{1} − λ_{2} and λ_{3} ≠ λ_{1} if and only if λ_{2} ≠ −2λ_{1}. Also λ_{3} ≠ λ_{2} if and only if λ_{2} ≠ −\frac{1}{2}λ_{1} and λ_{3} ≠ 0 if and only if λ_{2} ≠ −λ_{1}. Since |P_{w}(λ_{1}, λ_{2}, λ_{3})| = 12 for w = 1, w_{6}^{3} the number of Ad G^{F} -orbits of type 1 or w_{6}^{3} is given by

\frac{1}{12}|\{(λ_{1}, λ_{2}) | λ_{i}^{q−1} = 1, λ_{2} ≠ ±λ_{1}, −\frac{1}{2}λ_{1}, −2λ_{1}\}| = \frac{1}{12}(q − 1)(q − 5)

(note that if the characteristic is not 2 or 3 then ±1, −\frac{1}{2} and −2 are all distinct).

Similarly, the number of regular semisimple Ad G^{F} -orbits of type w_{a} or w_{b} is given by

\frac{1}{4}|\{λ | λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1\}| = \frac{(q − 1)^{2}}{4} .

For types w_{6}^{4} and w_{6} the equations λ^{q^{2}−1} ± λ^{q−1} + 1 = 0 are of degree q^{2} − 1 with no repeated roots. Hence the number of Ad G^{F} -orbits of type w_{6}^{4} or w_{6} is \frac{q^{2}−1}{6}.

We are now able to determine the irregular semisimple Ad G^{F} -orbits on 𝔤^{F}. By Proposition 5 and Proposition 3, we need to find representatives X for F -stable W - orbits on 𝔱 such that some root vanishes on X.

If two linearly independent roots vanish on X = (λ_{1}, λ_{2}, λ_{3}) ∈ 𝔱 then all roots vanish on X and hence, by Lemma 11, we must have λ_{1} = λ_{2} = λ_{3} = 0 and thus X = 0.

Lemma 15. If w ∈ W, α ∈ Φ and X ∈ 𝔱 then (wα)(Ad ẇ(X)) = α(X).

Proof. On the one hand, by Lemma 2,

\begin{aligned}Ad(ẇx_{α}(t))(X) & = Ad ẇ(X − α(X)te_{α}) \\ & = Ad ẇ(X) − α(X)cte_{w(α)}. \end{aligned}

On the other hand,

Ad(ẇx_{α}(t))(X) = Ad ẇx_{α}(t)(Ad ẇ(X))
= Ad x_{w(α)}(ct)(Ad ẇ(X))
= Ad ẇ(X) − w(α)(Ad ẇ(X))cte_{w(α)}.

Since c ≠ 0, we must have w(α)(Ad ẇ(X)) = α(X).

Theorem 16. Let G be the connected reductive group of type G_{2} over the field 𝔽_{q} of characteristic p ≠ 2, 3. The irregular semisimple Ad G^{F} -orbits O on 𝔤^{F} are given by the following table. If Y ∈ 𝔱 is an entry from the first column then the corresponding semisimple Ad G^{F} -orbit is given by O = Ad G(Y) ∩ 𝔤^{F}. In the table, X refers to any element of O.

representative |C_{G^{F}} (X)|

no. of reps of given form

(λ, λ, −2λ), λ^{q−1} = 1 q(q − 1)^{2}(q + 1)  \frac{q−1}{2}
(λ, λ, −2λ), λ^{q−1} = −1 q(q − 1)(q + 1)^{2}  \frac{q−1}{2}
(λ, 0, −λ), λ^{q−1} = 1 q(q − 1)^{2}(q + 1)  \frac{q−1}{2}
(λ, 0, −λ), λ^{q−1} = −1 q(q − 1)(q + 1)^{2}  \frac{q−1}{2}
(0, 0, 0)  q^{6}(q^{2} − 1)(q^{6} − 1)  1

In this table representatives from different rows represent different Ad G^{F} -orbits and two elements from the same row corresponding to λ = λ_{1} and λ = λ_{2} represent the same Ad G^{F} -orbit if and only if λ_{1} = ±λ_{2}.

Proof. Since there are two orbits of W on Φ, namely the sets of long and short roots, Lemma 15 implies that if one root vanishes on X, we may assume that this root is either a or b.

If b is the only root which vanishes on X then λ_{i} ≠ 0 and λ_{1} = λ_{2} and hence X is of the form (λ, λ, −2λ) with λ ≠ 0. The W -orbit of X is then

{(λ, λ, −2λ), (λ, −2λ, λ), (−2λ, λ, λ), (−λ, −λ, 2λ), (−λ, 2λ, −λ), (2λ, −λ, −λ)}.

But the restrictions on the characteristic imply that λ ≠ ±2λ and hence for the W - orbit of X to be F -stable, there are only two possibilities, namely F (X) = X in which case λ^{q−1} = 1 and F (X) = −X in which case λ^{q−1} = −1.

If a is the only root which vanishes on X then λ_{2} = 0 and λ_{1}, λ_{3} ≠ 0 so that X is of the form (λ, 0, −λ) with λ ≠ 0. A similar argument to the above then shows that λ^{q−1} = ±1.

It remains to determine the centralizers. Let Y = (λ, λ, −2λ) with λ^{q−1} = 1. Then we may take X = Y and by Proposition 3,

C_{G}(X) = ⟨T, X_{b}, X_{−b}⟩ = H_{2a+b} · ⟨X_{b}, X_{−b}⟩.

The product here is not direct since H_{2a+b} ∩ ⟨X_{b}, X_{−b}⟩ = {h_{2a+b}(±1)}. By Digne & Michel [6, Prop. 3.19(ii)], since W is F -stable, we see that |C_{G^{F}} (X)| = q(q−1)^{2}(q+1). Similarly, if X = (λ, 0, −λ) with λ^{q−1} = 1 then

C_{G}(X) = ⟨T, X_{a}, X_{−a}⟩ = H_{3a+2b} · ⟨X_{a}, X_{−a}⟩

and by the same reasoning as above, |C_{G^{F}} (X)| = q(q − 1)^{2}(q + 1).

Now suppose that Y = (λ, λ, −2λ) with λ^{q−1} = −1. Then F (Y) = Ad ẇ_{2a+b}(Y). According to the Lang-Steinberg Theorem (see, for example, Digne & Michel [6, 3.10]) x^{−1}F (x) = ẇ_{2a+b}^{−1} for some x ∈ G, in which case we may take X = Ad x(λ, λ, −2λ) ∈ 𝔤^{F}. From the above, we find that C_{G}(X) = ^{x}H_{2a+b} ·^{x}⟨X_{b}, X_{−b}⟩. Now w_{2a+b}(b) = b and hence ^{x}(T X_{b}) is an F -stable Borel subgroup of C_{G}(X) containing the F -stable maximal torus ^{x}T. Also, if w = 1 or w = w_{b} then F (^{x}ẇ_{b}) = ^{F (x)}ẇ_{b} = ^{xẇ_{2a+b}^{−1}} ẇ_{b} = ^{x}ẇ_{b}^{x}t for some t ∈ T, since w_{2a+b} and w_{b} commute. Hence the Weyl group of C_{G}(X) with respect to ^{x}T is F -stable and, by Digne & Michel [6, Prop. 3.19(ii)],

|C_{G^{F}} (X)| = q(q + 1)|(^{x}T)^{F} | = q(q + 1)|T_{w_{a}}^{F}| = q(q − 1)(q + 1)^{2}.

An analogous calculation applies when Y = (λ, 0, −λ) with λ^{q−1} = −1 by replacing b by a and w_{2a+b} by w_{3a+2b}.

5. Regular nilpotent elements

Lemma 17. Let A be a finite abelian group. If F is an automorphism of A then |H^{1}(F, A)| = |A^{F} |.

Proof. Since A is abelian, if L : A → A is the map x ↦ x^{−1}F (x) then two elements of A are F -conjugate if and only if they represent the same coset in A/L(A). Also, as the kernel of L is A^{F}, L(A) ≅ A/A^{F}. Since A is finite we find that |H^{1}(F, A)| = |A/L(A)| = |A|/|A/A^{F} | = |A^{F} |.

We now prove what is essentially the Lie algebra analogue of Digne & Michel [6, Proposition 14.24].

Proposition 18. Let G be a connected reductive algebraic group. Regardless of characteristic, the following results hold:

(i) A nilpotent element X ∈ 𝔤 is regular if and only if it is conjugate under Ad G to an element of the form ∑_{α∈Φ^{+}} X_{α} where X_{α} ∈ 𝔤_{α} for α ∈ Φ^{+} and X_{α} ≠ 0 for α ∈ Π.

(ii) If X ∈ 𝔤^{F} is regular nilpotent then

|C_{G^{F}} (X)| = q^{l}|Z(G)^{F} |

where l is the semisimple rank of G.

Proof. That regular nilpotent elements exist is proved by Springer [13, 5.9(b)] for good characteristic and by Keny [11] for bad characteristic. Moreover by Springer [13, 5.8, 5.9(c)] and Keny [11] any element of the form given in (i) is regular and all regular nilpotent elements of 𝔤 are conjugate. This deals with (i).

Now according to Digne & Michel [6, 3.21], the Ad G^{F} -orbits of regular nilpotent elements in 𝔤^{F} which are Ad G-conjugate to X are parametrized by H^{1}(F, A(X)) where A(X) = C_{G}(X)/C_{G}(X)^{∘} and moreover these Ad G^{F} -orbits have the same cardinality. By Springer & Steinberg [15, 3.7(a)], we have C_{G}(X) = Z(G) · C_{U}(X) (one does not need the restriction that p be good for this). By Springer & Steinberg [15, 3.7(b)] and Keny [11, p. 197, line 8], C_{U}(X) is connected regardless of the characteristic and hence A(X) ≅ Z(G)/Z(G)^{∘}. Thus H^{1}(F, A(X)) ≅ H^{1}(F, Z(G)/Z(G)^{∘}). Since Z(G)/Z(G)^{∘} is a finite abelian group we may apply Lemma 17 and Digne & Michel [6, 3.13] to deduce that

|H^{1}(F, A(X))| = |Z(G)^{F} |/|Z(G)^{∘F} |.

The proof of Digne & Michel [6, 14.23] carries across directly to the Lie algebra and we find that the number of regular nilpotent elements in 𝔤^{F} is |G^{F} |/|Z(G)^{∘}^{F} |q^{l}. Thus |C_{G^{F}} (X)| = |Z(G)^{∘}^{F} |q^{l} · |H^{1}(F, A(X))| = q^{l}|Z(G)^{F} |.

6. Non-semisimple, non-nilpotent orbits

Lemma 19. If X′ = (λ, λ, −2λ) or X′ = (λ, 0, −λ) with λ^{q−1} = ±1 then there is exactly one non-semisimple Ad G^{F} -orbit O in 𝔤^{F} such that if X ∈ O then the semisimple part of X is Ad G-conjugate to X′.

Proof. For both values of X′ we have C_{G}(X′) ≅ GL_{2}. The result follows from the fact that in GL_{2} there is only one non-semisimple Ad GL_{2}-orbit on 𝔤𝔩_{2} with semisimple part X′.

Theorem 20. Let G be the connected reductive group of type G_{2} over the field 𝔽_{q} of characteristic p ≠ 2, 3. The non-nilpotent non-semisimple Ad G^{F} -orbits O on 𝔤^{F} are given by the following table. If Y ∈ 𝔱 is an entry from the first column then the corresponding Ad G^{F} -orbit is given by O = G · Y ∩ 𝔤^{F}. In the table, X refers to any element of O.

representative |C_{G^{F}} (X)|

no. of reps of given form

(λ, λ, −2λ) + e_{b}, λ^{q−1} = 1 q(q − 1)  \frac{q−1}{2}
(λ, λ, −2λ) + e_{b}, λ^{q−1} = −1 q(q + 1)  \frac{q−1}{2}
(λ, 0, −λ) + e_{a}, λ^{q−1} = 1 q(q − 1)  \frac{q−1}{2}
(λ, 0, −λ) + e_{a}, λ^{q−1} = −1 q(q + 1)  \frac{q−1}{2}

Representatives from different rows represent different Ad G^{F} -orbits and two elements from the same row corresponding to λ = λ_{1} and λ = λ_{2} represent the same Ad G^{F} -orbit if and only if λ_{1} = ±λ_{2}.

Proof. The entries in the first column result from Lemma 19. The relationships amongst the different entries in the first column result from Theorem 16. It remains to calculate |C_{G^{F}} (X)|. Suppose that Y_{s} = (λ, λ, −2λ) with λ^{q−1} = 1. According to Proposition 18,

|C_{G^{F}} (X)| = |C_{C_{G^{F}} (X_{s})}(X_{n})|
= q|Z(C_{G}(X_{s}))^{F} |
= q|H_{2a+b}^{F}|
= q(q − 1).

Now suppose that Y_{s} = (λ, λ, −2λ) with λ^{q−1} = −1. Then,

|C_{G^{F}} (X)| = q|Z(C_{G^{F}} (X_{s}))^{F} | = q|(^{x}H_{2a+b})^{F} |

where x^{−1}F (x) = ẇ_{2a+b}^{−1}. Now

\begin{aligned}F (^{x}h_{2a+b}(t)) & = ^{xẇ_{2a+b}^{−1}}h_{2a+b}(t^{q}) \\ & = ^{x}h_{2a+b}(t^{−q}). \end{aligned}

Thus |(^{x}H_{2a+b})^{F} | = |\{t ∈ \overline{𝔽}_{q} | t = t^{−q}\}| = q + 1 and |C_{G^{F}} (X)| = q(q + 1). Similar calculations apply when Y = (λ, 0, −λ).

7. Nilpotent orbits

7.1. A grading of 𝔤. We shall first define a useful grading of 𝔤. Define the function h : Φ → ℤ by

h(ia + jb) = 2j.

For i ∈ ℤ, let

Φ_{i} = {α ∈ Φ | h(α) = i}, Φ_{⩾i} = {α ∈ Φ | h(α) ⩾ i},

𝔤_{i} = \begin{cases}⨁_{α∈Φ_{i}} 𝔤_{α} & if i ≠ 0, \\ 𝔱 ⊕ ⨁_{α∈Φ_{0}} 𝔤_{α} & if i = 0, \end{cases}

𝔤_{⩾i} = ⨁_{j⩾i}𝔤_{j}.

Let

G_{0} = ⟨T, X_{α} | α ∈ Φ_{0}⟩

and if i ⩾ 0

G_{⩾i} = \begin{cases}⟨X_{α} | α ∈ Φ_{⩾i}⟩ & if i > 0, \\ ⟨T, X_{α} | α ∈ Φ_{⩾0}⟩ & if i = 0. \end{cases}

Then G_{0} and G_{⩾i} are closed connected subgroups of G such that Lie G_{0} = 𝔤_{0} and Lie G_{⩾i} = 𝔤_{⩾i}. Moreover G_{⩾i} for i > 0 are subgroups of U and are hence unipotent subgroups.

Note. According to Dynkin-Kostant theory in positive characteristic (see, for example Humphreys [8, Chapter 7]) the nilpotent orbits of 𝔤 are parametrized by certain weighted Dynkin diagrams each of which gives rise to a grading of 𝔤. The grading introduced here is precisely that corresponding to the Ad G-orbit of e_{1} (see Section 7.7.3 for the definition of e_{1}).

Lemma 21. The following relations hold:

(i) Ad G_{0}(𝔤_{i}) ⊆ 𝔤_{i}. (ii) Ad G_{⩾0}(𝔤_{j}) ⊆ 𝔤_{⩾j}. (iii) Int G_{0}(G_{⩾i}) = G_{⩾i} if i ⩾ 0.

Proof. All of these relations are simple consequences of the Chevalley relations.

7.2. The action of G_{0} on 𝔤_{2}.

Definition. If r ∈ Φ, let ϕ_{r} : SL_{2} → ⟨X_{r}, X_{−r}⟩ be the isomorphism that sends

(\begin{matrix}1 & t \\ 0 & 1 \end{matrix}) to x_{a}(t), (\begin{matrix}1 & 0 \\ t & 1 \end{matrix}) to x_{−a}(t) and (\begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix}) to h_{a}(t).

Note. With this definition of ϕ_{r},

ω_{r} = x_{r}(1)x_{−r}(−1)x_{r}(1) = ϕ_{r} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

THE RATIONAL ADJOINT ORBITS OF A LIE ALGEBRA OF TYPE G_{2} 15

Proposition 22 (cf. Chang [5, 3.5]). Let t_{0}e_{b} + t_{1}e_{a+b} + t_{2}e_{2a+b} + t_{3}e_{3a+b} ∈ 𝔤_{2} be

represented by the column vector (\begin{matrix}t_{0} \\ t_{1} \\ t_{2} \\ t_{3} \end{matrix}). Then, by Lemma 21, ⟨X_{a}, X_{−a}⟩ acts on 𝔤_{2}

and the action of ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ ⟨X_{a}, X_{−a}⟩ on (\begin{matrix}t_{0} \\ t_{1} \\ t_{2} \\ t_{3} \end{matrix}) is given by

ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) · (\begin{matrix}t_{0} \\ t_{1} \\ t_{2} \\ t_{3} \end{matrix}) = (\begin{matrix}δ^{3} & −3γδ^{2} & 3γ^{2}δ & γ^{3} \\ −βδ^{2} & δ(αδ + 2βγ) & −γ(2αδ + βγ) & −αγ^{2} \\ β^{2}δ & −β(2αδ + βγ) & α(αδ + 2βγ) & α^{2}γ \\ β^{3} & −3αβ^{2} & 3α^{2}β & α^{3} \end{matrix}) (\begin{matrix}t_{0} \\ t_{1} \\ t_{2} \\ t_{3} \end{matrix})

Proof. The Bruhat decomposition in SL_{2} is given explicitly by

(\begin{matrix}α & β \\ γ & δ \end{matrix}) = \begin{cases}(\begin{matrix}1 & αγ^{−1} \\ 0 & 1 \end{matrix}) (\begin{matrix}−γ^{−1} & 0 \\ 0 & −γ \end{matrix}) (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) (\begin{matrix}1 & γ^{−1}δ \\ 0 & 1 \end{matrix}) & if γ ≠ 0, \\ (\begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix}) (\begin{matrix}1 & βα^{−1} \\ 0 & 1 \end{matrix}) & if γ = 0, \end{cases}

and hence

ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) = \begin{cases}x_{a}(αγ^{−1})h_{a}(−γ^{−1})ω_{a}x_{a}(γ^{−1}δ) & if γ ≠ 0, \\ h_{a}(α)x_{a}(βα^{−1}) & if γ = 0. \end{cases} (7.1)

Now

Ad x_{a}(z)(t_{0}e_{b} + t_{1}e_{a+b} + t_{2}e_{2a+b} + t_{3}e_{3a+b}) = t_{0}e_{b} + (t_{1} − t_{0}z)e_{a+b}

+ (t_{2} − 2t_{1}z + t_{0}z^{2})e_{2a+b} + (t_{3} + 3t_{2}z − 3t_{1}z^{2} + t_{0}z^{3})e_{3a+b}

and hence x_{a}(z) is represented by the matrix

(\begin{matrix}1 & 0 & 0 & 0 \\ −z & 1 & 0 & 0 \\ z^{2} & −2z & 1 & 0 \\ z^{3} & −3z^{2} & 3z & 1 \end{matrix}) .

We have

Ad ω_{a}(t_{0}e_{b} + t_{1}e_{a+b} + t_{2}e_{2a+b} + t_{3}e_{3a+b}) = −t_{3}e_{b} − t_{2}e_{a+b} + t_{1}e_{2a+b} + t_{0}e_{3a+b} and hence ω_{a} is represented by the matrix

(\begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) .

Note. The analogous formula of Chang [5, (3.4)] is incorrect and should read (t_{0}, t_{1}, t_{2}, t_{3})ω_{a} = (t_{3}, t_{2}, −t_{1}, −t_{0}).

Finally, if ia + b ∈ Φ then Ad h_{a}(z)e_{ia+b} = z^{2i−3}e_{ia+b} and hence h_{a}(z) is represented by the matrix

(\begin{matrix}z^{−3} & 0 & 0 & 0 \\ 0 & z^{−1} & 0 & 0 \\ 0 & 0 & z & 0 \\ 0 & 0 & 0 & z^{3} \end{matrix}) .

If γ ≠ 0, ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) is represented by the matrix

(\begin{matrix}1 & 0 & 0 & 0 \\ −αγ^{−1} & 1 & 0 & 0 \\ α^{2}γ^{−2} & −2αγ^{−1} & 1 & 0 \\ α^{3}γ^{−3} & −3α^{2}γ^{−2} & 3αγ^{−1} & 1 \end{matrix}) (\begin{matrix}−γ^{3} & 0 & 0 & 0 \\ 0 & −γ & 0 & 0 \\ 0 & 0 & −γ^{−1} & 0 \\ 0 & 0 & 0 & −γ^{−3} \end{matrix}) ·

· (\begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) (\begin{matrix}1 & 0 & 0 & 0 \\ −γ^{−1}δ & 1 & 0 & 0 \\ −γ^{−2}δ^{2} & −2γ^{−1}δ & 1 & 0 \\ γ^{−3}δ^{3} & −3γ^{−2}δ^{2} & 3γ^{−1}δ & 1 \end{matrix})

= (\begin{matrix}δ^{3} & −3γδ^{2} & 3γ^{2}δ & γ^{3} \\ −βδ^{2} & δ(αδ + 2βγ) & −γ(2αδ + βγ) & −αγ^{2} \\ β^{2}δ & −β(2αδ + βγ) & α(αδ + 2βγ) & α^{2}γ \\ β^{3} & −3αβ^{2} & 3α^{2}β & α^{3} \end{matrix}) .

If γ = 0, ϕ_{a} (\begin{matrix}α & β \\ 0 & α^{−1} \end{matrix}) is represented by the matrix

(\begin{matrix}α^{−3} & 0 & 0 & 0 \\ 0 & α^{−1} & 0 & 0 \\ 0 & 0 & α & 0 \\ 0 & 0 & 0 & α^{3} \end{matrix}) (\begin{matrix}1 & 0 & 0 & 0 \\ −βα^{−1} & 1 & 0 & 0 \\ β^{2}α^{−2} & −2βα^{−1} & 1 & 0 \\ β^{3}α^{−3} & −3β^{2}α^{−2} & 3βα^{−1} & 1 \end{matrix})

= (\begin{matrix}α^{−3} & 0 & 0 & 0 \\ −βα^{−2} & α^{−1} & 0 & 0 \\ β^{2}α^{−1} & −2β & α & 0 \\ β^{3} & −3β^{2}α & 3βα^{2} & α^{3} \end{matrix}) .

Hence for all (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ SL_{2}, ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) is represented by the matrix

(\begin{matrix}δ^{3} & −3γδ^{2} & 3γ^{2}δ & γ^{3} \\ −βδ^{2} & δ(αδ + 2βγ) & −γ(2αδ + βγ) & −αγ^{2} \\ β^{2}δ & −β(2αδ + βγ) & α(αδ + 2βγ) & α^{2}γ \\ β^{3} & −3αβ^{2} & 3α^{2}β & α^{3} \end{matrix}) .

Lemma 23. ω_{a}h(z_{1}, z_{2}, z_{3}) = h(z_{3}^{−1}, z_{2}^{−1}, z_{1}^{−1})ω_{a}. Proof. Since ω_{a} normalizes T we know that

ω_{a}h(z_{1}, z_{2}, z_{3})ω_{a}^{−1} = h(z′_{1}, z′_{2}, z′_{3})

THE RATIONAL ADJOINT ORBITS OF A LIE ALGEBRA OF TYPE G_{2} 17

for some z′_{1}, z′_{2}, z′_{3}. Then

z′_{i} = κ_{i}(h(z′_{1}, z′_{2}, z′_{3}))
= κ_{i}(ω_{a}h(z_{1}, z_{2}, z_{3})ω_{a}^{−1})
= w_{a}^{−1}(κ_{i})(h(z_{1}, z_{2}, z_{3})).

The result then follows from the relation w_{a}(κ_{i}) = −κ_{4−i}. Lemma 24. If (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ SL_{2}, t ∈ T then

(i) ω_{a}ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) = ϕ_{a} (\begin{matrix}δ & −γ \\ −β & α \end{matrix}) ω_{a}

(ii) tϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) = ϕ_{a} (\begin{matrix}α & a(t)β \\ a(t)^{−1}γ & δ \end{matrix}) t.

Proof. (i) follows from the relations ω_{a} = ϕ_{a} (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) and

(\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) (\begin{matrix}α & β \\ γ & δ \end{matrix}) = (\begin{matrix}δ & −γ \\ −β & α \end{matrix}) (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

As for (ii), if t = h(z_{1}, z_{2}, z_{3}) then

tω_{a} = h(z_{1}, z_{2}, z_{3})ω_{a}h(z_{1}^{−1}, z_{2}^{−1}, z_{3}^{−1})

= h(z_{1}, z_{2}, z_{3})h(z_{3}, z_{2}, z_{1})ω_{a} by Lemma 23 = h(z_{2}^{−1}, z_{2}^{2}, z_{2}^{−1})ω_{a} since z_{1}z_{2}z_{3} = 1

= h_{a}(z_{2})ω_{a}
= h_{a}(a(t))ω_{a}.

Hence, if γ ≠ 0, then according to Equation (7.1),

tϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) t^{−1} = tx_{a}(αγ^{−1})h_{a}(−γ^{−1})ω_{a}x_{a}(γ^{−1}δ)t^{−1}

= ^{t}x_{a}(αγ^{−1}) ·^{t}h_{a}(−γ^{−1}) ·^{t}ω_{a} ·^{t}x_{a}(γ^{−1}δ)

= x_{a}(a(t)αγ^{−1})h_{a}(−γ^{−1})h_{a}(a(t))ω_{a}x_{a}(a(t)γ^{−1}δ)

= x_{a}(α(a(t)^{−1}γ)^{−1})h_{a}(−(a(t)^{−1}γ)^{−1})ω_{a}x_{a}((a(t)^{−1}γ)^{−1}δ)

= ϕ_{a} (\begin{matrix}α & a(t)β \\ a(t)^{−1}γ & δ \end{matrix})

again by Equation (7.1).

If γ = 0,

tϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) t^{−1} = th_{a}(α)x_{a}(βα^{−1})t^{−1}

= ^{t}h_{a}(α) ·^{t}x_{a}(βα^{−1}) = h_{a}(α)x_{a}(a(t)βα^{−1})

= ϕ_{a} (\begin{matrix}α & a(t)β \\ a(t)^{−1}γ & δ \end{matrix}) .

Lemma 25. The map

ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) h_{b}(z) ↦ (\begin{matrix}α & zβ \\ γ & zδ \end{matrix})

is an isomorphism G_{0} ≅ GL_{2}. Proof. Since G_{0} = ⟨T, X_{a}, X_{−a}⟩, we may deduce from Lemma 24(ii) and the relation h(z_{1}, z_{2}, z_{3}) = h_{a}(z_{1}z_{2})h_{b}(z_{1}^{2}z_{2}) that every element of G_{0} can be expressed in the form ϕ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) h_{b}(z). That such an expression is unique follows from the fact that ⟨X_{a}, X_{−a}⟩ ∩ H_{b} = H_{a} ∩ H_{b} = {e}.

It is clear that e gets sent to I so let g_{i} = ϕ_{a} (\begin{matrix}α_{i} & β_{i} \\ γ_{i} & δ_{i} \end{matrix}) h_{b}(z_{i}) with α_{i}δ_{i} − β_{i}γ_{i} = 1 for i = 1, 2. Then, using Lemma 24(ii),

g_{1}^{−1}g_{2} = ϕ_{a} (\begin{matrix}δ_{1}α_{2} − β_{1}γ_{2} & z_{1}(δ_{1}β_{2} − β_{1}δ_{2}) \\ z_{1}^{−1}(−γ_{1}α_{2} + α_{1}γ_{2}) & −γ_{1}β_{2} + α_{1}δ_{2} \end{matrix}) h_{b}(z_{1}^{−1}z_{2})

gets mapped to

(\begin{matrix}δ_{1}α_{2} − β_{1}γ_{2} & z_{2}(δ_{1}β_{2} − β_{1}δ_{2}) \\ z_{1}^{−1}(−γ_{1}α_{2} + α_{1}γ_{2}) & z_{1}^{−1}z_{2}(−γ_{1}β_{2} + α_{1}δ_{2}) \end{matrix}) = (\begin{matrix}α_{1} & z_{1}β_{1} \\ γ_{1} & z_{1}δ_{1} \end{matrix})^{−1} (\begin{matrix}α_{2} & z_{2}β_{2} \\ γ_{2} & z_{2}δ_{2} \end{matrix}) .

Hence the map is a group homomorphism, which, being clearly surjective, is thus an isomorphism. Corollary. The map ψ_{a} : GL_{2} → G_{0}, defined by

(\begin{matrix}α & β \\ γ & δ \end{matrix}) ↦ ϕ_{a} (\begin{matrix}α & D^{−1}β \\ γ & D^{−1}δ \end{matrix}) h_{b}(D), (7.2)

where D = αδ − βγ, is an isomorphism. Lemma 26. The action of h_{b}(z) on 𝔤_{2} is given by the matrix

(\begin{matrix}z^{2} & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z^{−1} \end{matrix}) .

Proof. This follows from the relation Ad h_{b}(z)e_{ia+b} = z^{2−i}e_{ia+b}.

Proposition 27. The action of ψ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ G_{0} on 𝔤_{2} is given by the matrix

D^{−1} (\begin{matrix}δ^{3} & −3γδ^{2} & 3γ^{2}δ & γ^{3} \\ −βδ^{2} & δ(αδ + 2βγ) & −γ(2αδ + βγ) & −αγ^{2} \\ β^{2}δ & −β(2αδ + βγ) & α(αδ + 2βγ) & α^{2}γ \\ β^{3} & −3αβ^{2} & 3α^{2}β & α^{3} \end{matrix})

where D = αδ − βγ. Proof. This follows from Equation (7.2), Proposition 22 and Lemma 26 by multiplying out the matrices.

Lemma 28 (cf. Chang [5, Equation 3.6]). Let z, t ∈ \overline{𝔽}_{q} be such that t ≠ ±1 and z = t + t^{−1}. Then

ψ_{a} (\begin{matrix}1 & −t \\ −t & 1 \end{matrix}) · (\begin{matrix}1 \\ 0 \\ 0 \\ t^{−1} \end{matrix}) = (\begin{matrix}1 \\ 0 \\ −1 \\ z \end{matrix}) .

Proof. By Proposition 27, the left hand side is given by

\frac{1}{1 − t^{2}} (\begin{matrix}1 & 3t & 3t^{2} & −t^{3} \\ t & 1 + 2t^{2} & t(2 + t^{2}) & −t^{2} \\ t^{2} & t(2 + t^{2}) & 1 + 2t^{2} & −t \\ −t^{3} & −3t^{2} & −3t & 1 \end{matrix}) (\begin{matrix}1 \\ 0 \\ 0 \\ t^{−1} \end{matrix}) = (\begin{matrix}1 \\ 0 \\ −1 \\ z \end{matrix})

Lemma 29. Let r, s, t, u ∈ \overline{𝔽}_{q} be such that u, r ≠ 0, r^{3} = t^{−1} and s^{2} = u. Then

ψ_{a}(−\frac{r}{2u} \frac{r}{2s}^{\frac{r^{2}}{2u} \frac{r^{2}}{2s}} ) · (\begin{matrix}1 \\ 0 \\ u \\ 0 \end{matrix}) = (\begin{matrix}1 \\ 0 \\ 0 \\ t^{−1} \end{matrix}) (7.3)

Proof. Writing

ψ_{a}(−\frac{r}{2u} \frac{r}{2s}^{\frac{r^{2}}{2u} \frac{r^{2}}{2s}} ) = ψ_{a} (\frac{r}{2u} (\begin{matrix}r & rs \\ −1 & s \end{matrix}))

and using Proposition 27, we see that the left hand side of Equation (7.3) is given by

2ts^{3} · \frac{r^{3}}{8u^{3}} (\begin{matrix}s^{3} & 3u & 3s & −1 \\ −rs^{3} & −ru & rs & −r \\ r^{2}s^{3} & −r^{2}u & −r^{2}s & −r^{2} \\ r^{3}s^{3} & −3r^{3}u & 3r^{3}s & r^{3} \end{matrix}) (\begin{matrix}1 \\ 0 \\ u \\ 0 \end{matrix}) = \frac{tr^{3}}{4s^{3}} (\begin{matrix}4s^{3} \\ 0 \\ 0 \\ 4r^{3}s^{3} \end{matrix}) = (\begin{matrix}1 \\ 0 \\ 0 \\ t^{−1} \end{matrix})

since r^{3} = t^{−1}.

7.3. Classification of nilpotent orbits. It is useful to define the number ϵ by

ϵ = \begin{cases}1 & if q ≡ 1 mod 3, \\ −1 & if q ≡ −1 mod 3. \end{cases}

Let λ be a fixed nonsquare in 𝔽_{q}, μ a fixed noncube in 𝔽_{q} if q ≡ 1 mod 3 and ζ ∈ 𝔽_{q} such that x^{3} − 3x − ζ is irreducible over 𝔽_{q} if q ≡ −1 mod 3. It is useful to record here the following result proved in [9].

Proposition 30. [9, Lemma 2, Corollary 3] Let q ≡ −1 mod 3. Then the polynomial x^{3} − 3x − ζ is irreducible over 𝔽_{q} if and only if ζ = t + t^{−1} where t is a noncube in 𝔽_{q^{2}} such that t^{q+1} = 1.

We define the following nilpotent elements of 𝔤^{F}.

e_{0} = e_{a} + e_{b}
e_{1} = \begin{cases}e_{b} + e_{2a+b} & if q ≡ 1 mod 3, \\ e_{b} + λe_{2a+b} & if q ≡ −1 mod 3 \end{cases}

e_{2} = \begin{cases}e_{b} + λe_{2a+b} & if q ≡ 1 mod 3, \\ e_{b} + e_{2a+b} & if q ≡ −1 mod 3 \end{cases}

e_{3} = \begin{cases}e_{b} + μe_{3a+b} & if q ≡ 1 mod 3, \\ e_{b} − e_{2a+b} + ζe_{3a+b} & if q ≡ −1 mod 3 \end{cases}
e_{4} = e_{2a+b}
e_{5} = e_{3a+2b}
e_{6} = 0.

For i = 1, 2 it will be convenient to write e_{i} = e_{b} + λ_{i}e_{2a+b} with

λ_{1} = \begin{cases}1 & if q ≡ 1 mod 3, \\ λ & if q ≡ −1 mod 3 \end{cases}

and

λ_{2} = \begin{cases}λ & if q ≡ 1 mod 3, \\ 1 & if q ≡ −1 mod 3 \end{cases}
= λ^{ϵ}λ_{1}.

Theorem 31 (cf. Stuhler [17, Satz 1′]). Let G be the connected reductive group of type G_{2} over the field 𝔽_{q} of characteristic p ≠ 2, 3. Let e_{i} be as defined above. The set {e_{0}, e_{1}, e_{4}, e_{5}, e_{6}} is a complete set of representatives for the nilpotent Ad G-orbits in 𝔤.

Proof. We will show that every nilpotent element of 𝔤 can be transformed into one of e_{0}, e_{1}, e_{4}, e_{5} or e_{6} by applying some element of G. Computation of the dimensions of their centralizers carried out later in the chapter will then show that these elements of 𝔤 lie in different Ad G-orbits.

Firstly, every nilpotent element X of 𝔤 can be transformed into an element of 𝔲 so we may suppose that X ∈ 𝔲. Write X = ∑_{α∈Φ^{+}} t_{α}e_{α}. If t_{a}, t_{b} ≠ 0 then X is regular and by Proposition 18 it is conjugate to e_{0}. Now suppose that t_{a} = 0. If t_{b} = t_{a+b} = t_{2a+b} = t_{3a+b} = 0 then after conjugating by an element of T if necessary we see that X is conjugate to e_{5} or e_{6}. So suppose that at least one of t_{ia+b} ≠ 0 for 0 ⩽ i ⩽ 3. Then, conjugating by a suitable element of X_{(3−i)a+b}, we may suppose t_{3a+2b} = 0.

Case 1. t_{b} ≠ 0. In this case we may first conjugate by an element of T to get t_{b} = 1. If t_{a+b} ≠ 0 then, conjugating by a suitable element of X_{α}, we may assume that t_{a+b} = 0. If now t_{2a+b} = t_{3a+b} = 0 then X is conjugate to e_{b}, which, by conjugating by h_{b}(−1)ω_{3a+b}, is seen to be conjugate to e_{5}. If t_{2a+b} = 0 but t_{3a+b} ≠ 0 then, conjugating by a suitable element of T, we see that X is conjugate to e_{b} + e_{3a+b}, which, by Lemma 29, is conjugate to e_{1}. If t_{2a+b} ≠ 0 and t_{3a+b} = 0 then similarly we see that X is conjugate to e_{1}. Lastly, if t_{2a+b} ≠ 0 and t_{3a+b} ≠ 0, we see that X is conjugate to e_{b} −e_{2a+b} +ze_{3a+b} with z ≠ 0, which, by Lemmas 28 and 29, is conjugate to e_{1}.

Case 2. t_{b} = t_{a+b} = t_{2a+b} = 0, t_{3a+b} ≠ 0. Conjugating by ω_{a} transforms X into −t_{3a+b}e_{b} and we are back in Case 1.

Case 3. t_{b} = t_{a+b} = 0, t_{2a+b} ≠ 0. We can conjugate by elements of T and X_{a} to transform X to e_{4} = e_{2a+b}.

Case 4. t_{b} = 0, t_{a+b} ≠ 0. Conjugating by elements of T and X_{a} we can reduce X to the form e_{a+b} + t_{3a+b}e_{3a+b}. Conjugating by ω_{a} now transforms this to −t_{3a+b}e_{b} + e_{2a+b}, bringing us back to Case 1 or 3.

Case 5. t_{b} = 0, t_{a} ≠ 0. In this case we may conjugate by elements of T, X_{b}, X_{a+b} and X_{2a+b} in turn to transform X into the form e_{a} + t_{3a+2b}e_{3a+2b}. Conjugating by ω_{6}^{−1} then transforms this to −t_{3a+2b}e_{b} − e_{2a+b} and we are back to Case 1 or 3 above.

Note. It is worth pointing out that Stuhler’s proof of this result uses a Springer morphism f from the unipotent variety to the nilpotent variety but omits some details. If we assume that the regular unipotent element x_{a}(1)x_{b}(1) gets sent to the regular nilpotent element e_{a} + e_{b} under f then f is completely determined. By letting t → 0 in the formulae

Ad h_{a}(t^{2})h_{b}(t^{3})(e_{a} + e_{b}) = te_{a} + e_{b}
Int h_{a}(t^{2})h_{b}(t^{3})(x_{a}(1)x_{b}(1)) = x_{a}(t)x_{b}(1)
Ad h_{a}(t)h_{b}(t^{2})(e_{a} + e_{b}) = e_{a} + te_{b}
Int h_{a}(t)h_{b}(t^{2})(x_{a}(1)x_{b}(1)) = x_{a}(1)x_{b}(t)
Ad h_{a}(t^{4})h_{b}(t^{6})x_{b}(t^{−1})x_{a+b}(\frac{1}{2}t^{−4})x_{3a+b}(\frac{3}{4}t^{−8})(e_{a} + e_{b}) = t^{2}e_{a} + e_{b} + te_{a+b} + e_{2a+b}

and

\begin{aligned}Int & h_{a}(t^{4})h_{b}(t^{6})x_{b}(t^{−1})x_{a+b}(\frac{1}{2}t^{−4})x_{3a+b}(\frac{3}{4}t^{−8})(x_{a}(1)x_{b}(1)) \\ & = x_{a}(t^{2})x_{b}(1)x_{a+b}(t)x_{2a+b}(1 + t^{3})x_{3a+b}(−t^{5} − \frac{3}{2}t^{2})x_{3a+2b}(\frac{3}{2}t^{2} − 2t^{4} − \frac{3}{2}t + t^{5}) \end{aligned}

we see that f(x_{b}(1)) = e_{b}, f(x_{a}(1)) = e_{a} and f(x_{b}(1)x_{2a+b}(1)) = e_{b} +e_{2a+b} which leads to the set of representatives given by Stuhler.

Theorem 32 (cf. Chang [5, 3.9]). Let G be the connected reductive group of type G_{2} over the field 𝔽_{q} of characteristic p ≠ 2, 3. Let e_{i} be as defined above. The set {e_{i} | 0 ⩽ i ⩽ 6} is a complete set of representatives for the Ad G^{F} -orbits on 𝔤^{F} and the values of dim C_{G}(e_{i}) and |C_{G^{F}} (e_{i})| are given in the following table.

representative e^{i} dim C^{G}(e^{i}) |C^{G^{F}} (e^{i})|
e_{0}  2  q^{2}
e_{1}  4  6q^{4}
e_{2}  4  2q^{4}
e_{3}  4  3q^{4}
e_{4}  6  q^{4}(q^{2} − 1)
e_{5}  8  q^{6}(q^{2} − 1)
e_{6}  14  q^{6}(q^{2} − 1)(q^{6} − 1)

Proof. The proof of this will proceed in several stages. Firstly, according to Proposition 18, every regular nilpotent element of 𝔤^{F} is Ad G^{F} -conjugate to e_{0}. Also, if X = 0 then X = e_{6}. Then, according to Theorem 31 and Proposition 18 every nonzero irregular nilpotent element of 𝔤^{F} is conjugate under Ad G to one of e_{1}, e_{4}, or e_{5}. Now, according to Digne & Michel [6, 3.21], if O_{i} is the Ad G-orbit of e_{i} then O_{i}^{F} is a union of Ad G^{F} -orbits in bijective correspondence with the F -conjugacy classes of C_{G}(e_{i})/C_{G}(e_{i})^{∘}. We will show in Proposition 38 that the centralizers of e_{4} and e_{5} are connected. Hence if X ∈ 𝔤^{F} is Ad G-conjugate to one of e_{4} or e_{5} then it is in fact Ad G^{F} -conjugate to e_{4} or e_{5}. We will also show in Proposition 39 that C_{G}(e_{1})/C_{G}(e_{1})^{∘} ≅ S_{3} with trivial F -action. Hence O_{1}^{F} is a union of three Ad G^{F} - orbits. The three elements e_{1}, e_{2} and e_{3} are Ad G-conjugate to e_{1} and if e_{i} = Ad g_{i}(e_{1}) for i = 1, 2, 3 it will be shown that the images of g_{i}^{−1}F (g_{i}) in C_{G}(e_{1})/C_{G}(e_{1})^{∘} lie in different F -conjugacy classes and thus that e_{1}, e_{2} and e_{3} are representatives of the three Ad G^{F} -orbits in O_{1}^{F}. Hence if X ∈ 𝔤^{F} is Ad G-conjugate to e_{1} then it is Ad G^{F} - conjugate to one of e_{1}, e_{2} or e_{3}. The set {e_{0}, e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}} is thus a complete set of representatives for the nilpotent Ad G^{F} -orbits on 𝔤^{F}. The dimensions and orders of the centralizers will be computed in the next section.

7.4. Centralizers of nilpotent elements. We will now prove the statements about the dimensions, orders and connectedness of the centralizers.

First of all, since e_{0} is regular we have dim C_{G}(e_{0}) = 2 and since Z(G) = 1, Proposition 18 shows that |C_{G^{F}} (e_{0})| = q^{2}. It is also clear that C_{G}(e_{6}) = G and hence dim C_{G}(e_{6}) = 14 and |C_{G^{F}} (e_{6})| = q^{6}(q^{2} − 1)(q^{6} − 1). We will thus concentrate on the elements e_{i} for 1 ⩽ i ⩽ 5.

Lemma 33. We have C_{U}(e_{5}) = U, C_{U}(e_{4}) = ⟨X_{b}, X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩ and C_{U}(e_{1}) = ⟨X_{b}, X_{2a+b}, X_{3a+2b}, V ⟩, where V = \{x_{3a+b}(3λ_{1}t)x_{a+b}(t) | t ∈ \overline{𝔽}_{q}\}. Moreover, for i = 1, 2, 3, C_{U}(e_{i}) ⊆ G_{⩾2}, dim C_{U}(e_{i}) = 4 and |C_{U^{F}} (e_{i})| = q^{4}.

Proof. Using the fact that Ad x_{α}(t)(e_{β}) = e_{β} if α + β is not a root we see that C_{U}(e_{i}) ⊇ ⟨X_{α} | α ∈ R_{i}⟩ where R_{1} = R_{2} = {b, 2a + b, 3a + 2b},

R_{3} = \begin{cases}\{a + b, 2a + b, 3a + 2b\} & if q ≡ 1 mod 3, \\ \{2a + b, 3a + 2b\} & if q ≡ −1 mod 3, \end{cases}

R_{4} = {b, 2a + b, 3a + b, 3a + 2b} and R_{5} = Φ^{+}. For each i write u ∈ U in the form

u = ∏_{β∈Φ^{+}−R_{i}} x_{β}(t_{β}) ∏_{α∈R_{i}} x_{α}(t_{α})

where the product ∏ x_{β}(t_{β}) is taken in decreasing order of β ∈ Φ^{+} − R_{i}. Then, using the formulae in the Appendix, Ad u(e_{5}) = e_{5} and

Ad u(e_{4}) = e_{4} + 3t_{a}e_{3a+b} + 3t_{a+b}e_{3a+2b}. (7.4)

For i = 1, 2,

\begin{aligned}Ad u(e_{i}) & = e_{i} − t_{a}e_{a+b} + t_{a}^{2}e_{2a+b} + (t_{a}^{3} + 3λ_{i}t_{a})e_{3a+b} \\ & + (3λ_{i}t_{a+b} + 3t_{a}^{2}t_{a+b} − t_{3a+b})e_{3a+2b}. \end{aligned} (7.5)

If q ≡ 1 mod 3 then

Ad u(e_{3}) = e_{3} − t_{a}e_{a+b} + t_{a}^{2}e_{2a+b} + t_{a}^{3}e_{3a+b} + (t_{b}μ + t_{a}^{3}t_{b} − t_{3a+b})e_{3a+2b}. (7.6)

If q ≡ −1 mod 3 then

\begin{aligned}Ad u(e_{3}) & = e_{3} − t_{a}e_{a+b} + t_{a}^{2}e_{2a+b} + (t_{a}^{3} − 3t_{a})e_{3a+b} \\ & + (3t_{a+b}(t_{a}^{2} − 1) + t_{b}(ζ − 3t_{a} + t_{a}^{3}) − t_{3a+b})e_{3a+2b}. \end{aligned} (7.7)

From the above, we see that C_{U}(e_{5}) = U, C_{U}(e_{4}) = ⟨X_{b}, X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩ and C_{U}(e_{1}) = ⟨X_{b}, X_{2a+b}, X_{3a+2b}, V ⟩ where V = \{x_{3a+b}(3λ_{1}t)x_{a+b}(t) | t ∈ \overline{𝔽}_{q}\}. We also see that if u ∈ C_{U}(e_{i}) for 1 ⩽ i ⩽ 3 then t_{a} = 0 and hence C_{U}(e_{i}) ⊆ G_{⩾2}. Inspection of the above formulae shows that, for 1 ⩽ i ⩽ 3, dim Ad U(e_{i}) = 2, |Ad U^{F} (e_{i})| = q^{2} and hence that dim C_{U}(e_{i}) = dim U − dim Ad U(e_{i}) = 4 and |C_{U^{F}} (e_{i})| = |U^{F} |/|Ad U^{F} (e_{i})| = q^{4}.

Lemma 34. For 1 ⩽ i ⩽ 5, C_{B}(e_{i}) = C_{U}(e_{i})C_{T} (e_{i}).

Proof. If u ∈ U and t ∈ T, then ut = t · t^{−1}ut and t^{−1}ut ∈ U. It is thus sufficient to show that C_{B}(e_{i}) = C_{T} (e_{i})C_{U}(e_{i}). So let tu ∈ C_{B}(e_{i}) with t ∈ T, u ∈ U. Let S_{i} be the set of roots α such that the coefficient of e_{α} in e_{i} does not vanish. Then Ad u(e_{i}) is of the form

Ad u(e_{i}) = e_{i} + ∑_{α∈S_{i}} c_{α}e_{α} + ∑_{α∉S_{i}} c_{α}e_{α}

and hence

Ad tu(e_{i}) = Ad t(e_{i}) + ∑_{α∈S_{i}} c_{α}α(t)e_{α} + ∑_{α∉S_{i}} c_{α}α(t)e_{α}.

Since α(t) ≠ 0 and tu ∈ C_{B}(e_{i}) we must have c_{α} = 0 for α ∉ S_{i}. From Equations (7.4)– (7.7) we see that in all cases this implies that c_{α} = 0 also for α ∈ S_{i} and hence u ∈ C_{U}(e_{i}) implying also that t ∈ C_{T} (e_{i}).

Lemma 35. We have C_{G_{0}}(e_{1}) = {ψ_{a}(Ω^{i}Σ^{j}) | 0 ⩽ i ⩽ 2, 0 ⩽ j ⩽ 1} ≅ S_{3} where

\begin{aligned}Ω & = (\begin{matrix}−\frac{1}{2} & −\frac{1}{2}(−3λ_{1})^{\frac{1}{2}} \\ −\frac{1}{2λ_{1}} (−3λ_{1})^{\frac{1}{2}} & −\frac{1}{2} \end{matrix} ), \\ Σ & = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) \end{aligned}

and an isomorphism is given by

e ↔ 1  ψ_{a}(Σ) ↔ (12)  ψ_{a}(ΩΣ) ↔ (23)
ψ_{a}(Ω^{2}Σ) ↔ (13)  ψ_{a}(Ω^{2}) ↔ (123)  ψ_{a}(Ω) ↔ (132).

Moreover, for 1 ⩽ i ⩽ 3, there exist g_{i} ∈ G_{0} such that e_{i} = Ad g_{i}(e_{1}) with g_{1}^{−1}F (g_{1}) = 1, g_{2}^{−1}F (g_{2}) = ψ_{a}(Σ) and g_{3}^{−1}F (g_{3}) = ψ_{a}(Ω^{k}) (k = 1 or 2).

Proof. Let g = ψ_{a} (\begin{matrix}α & β \\ γ & δ \end{matrix}). Then Proposition 27 shows that g ∈ C_{G_{0}}(e_{1}) if and only

if

D^{−1} (\begin{matrix}δ^{3} & −3γδ^{2} & 3γ^{2}δ & γ^{3} \\ −βδ^{2} & δ(αδ + 2βγ) & −γ(2αδ + βγ) & −αγ^{2} \\ β^{2}δ & −β(2αδ + βγ) & α(αδ + 2βγ) & α^{2}γ \\ β^{3} & −3αβ^{2} & 3α^{2}β & α^{3} \end{matrix}) (\begin{matrix}1 \\ 0 \\ λ_{1} \\ 0 \end{matrix}) = (\begin{matrix}1 \\ 0 \\ λ_{1} \\ 0 \end{matrix}) ,

24 DAVID G. A. JACKSON

that is, δ(δ^{2} + 3λ_{1}γ^{2}) = D (7.8)

−βδ^{2} − λ_{1}γ(2αδ + βγ) = 0 (7.9)

β^{2}δ + λ_{1}α(αδ + 2βγ) = λ_{1}D (7.10) and β(β^{2} + 3λ_{1}α^{2}) = 0 (7.11) where αδ − βγ = D. (7.12)

First suppose that β = 0. Then Equation (7.9) implies that αγδ = 0 and Equation (7.12) implies that αδ = D. Equation (7.10) then implies that α = 1, δ = D, β = 0 and γ = 0. But then Equation (7.8) implies that D^{2} = 1. Hence D = ±1 = (−1)^{j}, say, and

g = ψ_{a} (\begin{matrix}1 & 0 \\ 0 & (−1)^{j} \end{matrix}) = ψ_{a}(Σ^{j}).

Now suppose that β ≠ 0. Then Equation (7.11) implies that β^{2} + 3λ_{1}α^{2} = 0. Hence β = ±(−3λ_{1})^{\frac{1}{2}} α = −ε_{1}(−3λ_{1})^{\frac{1}{2}} α with ε_{1} = ±1 and we have fixed the choice of square root. Equation (7.12) then implies that αδ + ε_{1}(−3λ_{1})^{\frac{1}{2}} αγ = D ≠ 0. Hence α ≠ 0 and δ + ε_{1}(−3λ_{1})^{\frac{1}{2}} γ = α^{−1}D. (7.13) Equation (7.10) now implies that −3λ_{1}α^{2}δ +λ_{1}α^{2}δ +2λ_{1}αβγ = λ_{1}D, that is, 2α(βγ − αδ) = D and hence that α = −\frac{1}{2}. Hence β = \frac{1}{2}ε_{1}(−3λ_{1})^{\frac{1}{2}} and from Equation (7.13), δ = −2D − ε_{1}(−3λ_{1})^{\frac{1}{2}} γ. (7.14) Adding δ times Equation (7.10) to β times Equation (7.9) gives

δD = αδ(αδ + 2βγ) − βγ(2αδ + βγ) = α^{2}δ^{2} − β^{2}γ^{2}

and hence δ = αδ + βγ = 2βγ + D = D + ε_{1}(−3λ_{1})^{\frac{1}{2}} γ. (7.15) Subtracting Equations 7.15 and 7.14 then gives 3D + 2ε_{1}(−3λ_{1})^{\frac{1}{2}} γ = 0 and hence

γ = −\frac{3D}{2ε_{1}(−3λ_{1})^{\frac{1}{2}}} .

Substituting this into Equation (7.15) gives δ = −\frac{D}{2}. Substituting the values of α, β, γ and δ found so far into Equation (7.8) now gives

−\frac{D}{2}(\frac{D^{2}}{4} − 3λ_{1} \frac{D^{2}3λ_{1}}{4λ_{1}^{2}} ) = D

and hence D = ±1. Thus

g = ψ_{a}(\begin{matrix}−\frac{1}{2} & \frac{1}{2}ε_{1}(−3λ_{1})^{\frac{1}{2}} \\ ±\frac{1}{2λ_{1}} ε_{1}(−3λ_{1})^{\frac{1}{2}} & ∓\frac{1}{2} \end{matrix} )

= ψ_{a} ((\begin{matrix}−\frac{1}{2} & ±\frac{1}{2}ε_{1}(−3λ_{1})^{\frac{1}{2}} \\ ±\frac{1}{2λ_{1}} ε_{1}(−3λ_{1})^{\frac{1}{2}} & −\frac{1}{2} \end{matrix} ) (\begin{matrix}1 & 0 \\ 0 & ±1 \end{matrix}))

= ψ_{a}(Ω^{i}Σ^{j})

for some i = 1 or 2, j = 0 or 1. Hence

C_{G_{0}}(e_{1}) = {ψ_{a}(Ω^{i}Σ^{j}) | 0 ⩽ i ⩽ 2, 0 ⩽ j ⩽ 1}.

Calculations in GL_{2} show that Ω^{3} = I, Σ^{2} = I and ΣΩ = Ω^{2}Σ and hence

(7.16) ψ_{a}(Ω^{i}Σ^{j})ψ_{a}(Ω^{k}Σ^{l}) = \begin{cases}ψ_{a}(Ω^{i+k}Σ^{j+l}) & if j is even, \\ ψ_{a}(Ω^{i+2k}Σ^{j+l}) & if j is odd \end{cases}

Now S_{3} ≅ ⟨s_{1}, s_{2} | s_{1}^{2} = s_{2}^{2} = (s_{1}s_{2})^{3} = 1⟩ and we can make the identification s_{1} = (12), s_{2} = (23). Equation (7.16) implies that ψ_{a}(Σ)^{2} = I, ψ_{a}(ΩΣ)^{2} = I and (ψ_{a}(Σ)ψ_{a}(ΩΣ))^{3} = ψ_{a}(Ω^{2})^{3} = 1. But since ψ_{a}(Ω^{i}Σ^{j}) = (ψ_{a}(ΩΣ)ψ_{a}(Σ))^{i}ψ_{a}(Σ)^{j}, we see that C_{G_{0}}(e_{1}) is generated by ψ_{a}(Σ) and ψ_{a}(ΩΣ). Hence there is a homomorphism from C_{G_{0}}(e_{1}) to S_{3} sending ψ_{a}(Σ) to (12) and ψ_{a}(ΩΣ) to (23). Since |C_{G_{0}}(e_{1})| = |S_{3}| = 6 we must then have C_{G_{0}}(e_{1}) ≅ S_{3}. Writing the elements of S_{3} in terms of the generators (12) and (23), replacing (12) and (23) by ψ_{a}(Σ) and ψ_{a}(ΩΣ) and using Equation (7.16) then gives the isomorphism explicitly.

We will now find g_{i} ∈ G_{0} such that e_{i} = Ad g_{i}(e_{1}) for 1 ⩽ i ⩽ 3. Since the e_{i} are F -stable, this also implies that g_{i}^{−1}F (g_{i}) ∈ C_{G_{0}}(e_{1}). We may clearly take g_{1} = 1. Now

e_{2} = e_{b} + λ_{2}e_{2a+b}
= e_{b} + λ^{ϵ}λ_{1}e_{2a+b}
= Ad(h_{a}(λ^{ϵ})h_{b}(λ^{\frac{3ϵ}{2}} ))(e_{b} + λ_{1}e_{2a+b}).

Hence let g_{2} = h_{a}(λ^{ϵ})h_{b}(λ^{\frac{3ϵ}{2}} ). Then F (g_{2}) = h_{a}(λ^{ϵ})h_{b}(−λ^{\frac{3ϵ}{2}} ) and hence

g_{2}^{−1}F (g_{2}) = h_{b}(−1) = ψ_{a}(Σ).

If q ≡ 1 mod 3, let g_{3} = ψ_{a}(\begin{matrix}\frac{1}{2}μ^{\frac{2}{3}} & \frac{1}{2}μ^{\frac{2}{3}} \\ −\frac{1}{2}μ^{\frac{1}{3}} & \frac{1}{2}μ^{\frac{1}{3}} \end{matrix} ) where we make a fixed choice of cube

root. Since λ_{1} = 1 in this case, by substituting t = μ^{−1}, s = u = 1 and r = μ^{\frac{1}{3}} into Lemma 29 we see that e_{3} = Ad g_{3}(e_{1}). Now F (μ^{\frac{1}{3}} ) = ωμ^{\frac{1}{3}} where ω = \frac{1}{2}(−1 ± (−3)^{\frac{1}{2}} ) is a primitive cube root of unity. Then

g_{3}^{−1}F (g_{3}) = ψ_{a} ((\begin{matrix}μ^{− \frac{2}{3}} & −μ^{− \frac{1}{3}} \\ μ^{− \frac{2}{3}} & μ^{− \frac{1}{3}} \end{matrix} ) (\begin{matrix}\frac{1}{2}ω^{2}μ^{\frac{2}{3}} & \frac{1}{2}ω^{2}μ^{\frac{2}{3}} \\ −\frac{1}{2}ωμ^{\frac{1}{3}} & \frac{1}{2}ωμ^{\frac{1}{3}} \end{matrix} ))

= ψ_{a}(\begin{matrix}\frac{1}{2}(ω + ω^{2}) & \frac{1}{2}(ω^{2} − ω) \\ \frac{1}{2}(ω^{2} − ω) & \frac{1}{2}(ω + ω^{2}) \end{matrix})

= ψ_{a}(\begin{matrix}−\frac{1}{2} & ∓\frac{1}{2}(−3)^{\frac{1}{2}} \\ ∓\frac{1}{2}(−3)^{\frac{1}{2}} & −\frac{1}{2} \end{matrix} )

= ψ_{a}(Ω^{k})

where k = 1 or 2.

If q ≡ −1 mod 3 then, according to Proposition 30, ζ = t+t^{−1} where t is a noncube

in 𝔽_{q^{2}} such that t^{q+1} = 1. Let θ ∈ 𝔽_{q^{6}} be a cube root of t^{−1}, g′_{3} = ψ_{a} (\begin{matrix}1 & −t \\ −t & 1 \end{matrix}) and

g′′_{3} = ψ_{a}(−\frac{θ}{2λ} \frac{θ}{2λ^{\frac{1}{2}}}^{\frac{θ^{2}}{2λ} \frac{θ^{2}}{2λ^{\frac{1}{2}}}} ) where we make a fixed choice of square root of λ. Lemma 28

shows that e_{3} = Ad g′_{3}(e_{b} + t^{−1}e_{3a+b}) and Lemma 29 shows that e_{b} + t^{−1}e_{3a+b} = Ad g′′_{3}(e_{1}). Hence, if we let g_{3} = g′_{3}g′′_{3} = ψ_{a}(M) with

M = (−\frac{θ+θ^{−1}}{2λ} \frac{θ−θ^{−1}}{2λ^{\frac{1}{2}}}^{\frac{θ^{2}+θ^{−2}}{2λ} \frac{θ^{2}−θ^{−2}}{2λ^{\frac{1}{2}}}} ),

then e_{3} = Ad g_{3}(e_{1}). Now F (t) = t^{−1}, F (λ^{\frac{1}{2}} ) = −λ^{\frac{1}{2}} and F (θ) = ωθ^{−1} and hence

F (g_{3}) = ψ_{a}(−\frac{ω^{2}θ+ωθ^{−1}}{2λ} \frac{ω^{2}θ−ωθ^{−1}}{2λ^{\frac{1}{2}}}^{\frac{ωθ^{2}+ω^{2}θ^{−2}}{2λ} \frac{ωθ^{2}−ω^{2}θ^{−2}}{2λ^{\frac{1}{2}}}} ).

Since det M = \frac{θ^{3}−θ^{−3}}{2λ^{\frac{3}{2}}} = \frac{t^{−1}−t}{2λ^{\frac{3}{2}}}, we have

g_{3}^{−1}F (g_{3}) = ψ_{a} (\frac{2λ^{\frac{3}{2}}}{t^{−1} − t}(\frac{θ−θ^{−1}}{2λ^{\frac{1}{2}}} −\frac{θ^{2}−θ^{−2}}{2λ^{\frac{1}{2}}}_{\frac{θ+θ^{−1}}{2λ}  \frac{θ^{2}+θ^{−2}}{2λ}} )(−\frac{ω^{2}θ+ωθ^{−1}}{2λ} \frac{ω^{2}θ−ωθ^{−1}}{2λ^{\frac{1}{2}}}^{\frac{ωθ^{2}+ω^{2}θ^{−2}}{2λ} \frac{ωθ^{2}−ω^{2}θ^{−2}}{2λ^{\frac{1}{2}}}} ))

= ψ_{a} (\frac{2λ^{\frac{3}{2}}}{t^{−1} − t}(\begin{matrix}\frac{(ω+ω^{2})(θ^{3}−θ^{−3})}{4λ^{\frac{3}{2}}} & \frac{(ω−ω^{2})(θ^{3}−θ^{−3})}{4λ} \\ \frac{(ω−ω^{2})(θ^{3}−θ^{−3})}{4λ^{2}} & \frac{(ω+ω^{2})(θ^{3}−θ^{−3})}{4λ^{\frac{3}{2}}} \end{matrix} ))

= ψ_{a}(\begin{matrix}−\frac{1}{2} & ±\frac{1}{2}(−3λ)^{\frac{1}{2}} \\ ±\frac{1}{2λ}(−3λ)^{\frac{1}{2}} & −\frac{1}{2} \end{matrix} )

= ψ_{a}(Ω^{k})

for k = 1 or 2.

Lemma 36. Let X = t_{α}e_{α} + ∑_{β>α} t_{β}e_{β} and X′ = s_{α′}e_{α′} + ∑_{β′>α′} s_{β′}e_{β′} be two elements of 𝔲 with t_{α}, s_{α′} ≠ 0. If Ad ẇ(X) = X′ for some w ∈ W then w(α) ⩾ α′ and w^{−1}(α′) ⩾ α.

Proof. We have proved earlier that Ad ẇ(e_{α}) = c_{α}e_{w(α)} for some c_{α} ≠ 0. Hence

Ad ẇ(X) = t_{α}c_{α}e_{w(α)} + ∑_{β>α} t_{β}c_{β}e_{w(β)}.

If Ad ẇ(X) = X′ then t_{α}c_{α}e_{w(α)} must be one of the terms in the expansion of X′ and hence w(α) ⩾ α′. Similarly, since Ad ẇ^{−1}(X′) = X, we have w^{−1}(α′) ⩾ α.

Proposition 37. We have C_{G}(e_{i}) ⊆ G_{⩾0} = B∪Bω_{a}X_{a} for i = 1, 2, 3, 5 and C_{G}(e_{4}) ⊆ B ∪ Bω_{b}X_{b}.

Proof. Let g = bẇv ∈ C_{G}(e_{5}) with b ∈ B, w ∈ W and v ∈ X_{w}. It then follows that Ad(ẇ^{−1}b^{−1})(e_{3a+2b}) = Ad v(e_{3a+2b}) and hence that w(3a + 2b) = 3a + 2b. Thus w = 1 or w_{a} and hence C_{G}(e_{5}) ⊆ B ∪ Bω_{a}X_{a} = G_{⩾0}.

Now let g = bẇv ∈ C_{G}(e_{4}) with b ∈ B, w ∈ W and v ∈ U_{w}. Then we have Ad(ẇ^{−1}b^{−1})(e_{2a+b}) = Ad v(e_{2a+b}). Now Ad b^{−1}(e_{2a+b}) and Ad v(e_{2a+b}) are both of the form t_{2a+b}e_{2a+b} + ∑_{β>2a+b} t_{β}e_{β} with t_{2a+b} ≠ 0 and hence Lemma 36 implies that w^{−1}(2a + b) ⩾ 2a + b and w(2a + b) ⩾ 2a + b which implies that w = 1 or w_{b}. Thus C_{G}(e_{4}) ⊆ B ∪ Bω_{b}X_{b}.

For C_{G}(e_{1}) it is convenient to consider C_{G}(X) where X is the Ad G^{F} -conjugate Ad ω_{a}(e_{1}) = e_{3a+b} − λ_{1}e_{a+b} of e_{1}. If we are able to show that C_{G}(X) ⊆ G_{⩾0} then C_{G}(e_{1}) ⊆ ^{ω_{a}^{−1}}G_{⩾0} = G_{⩾0}. Since e_{i} = Ad g_{i}(e_{1}) (i = 1, 2, 3) with g_{i} ∈ G_{0}, we then have C_{G}(e_{i}) = ^{g_{i}}C_{G}(e_{1}) ⊆ ^{g_{i}}G_{⩾0} = G_{⩾0}. Let g = utẇv ∈ C_{G}(X) with u ∈ U, t ∈ T, w ∈ W and v ∈ U_{w}. Then Ad(ẇ^{−1}(ut)^{−1})(e_{3a+b} − λ_{1}e_{a+b}) = Ad v(e_{3a+b} − λ_{1}e_{a+b}) and hence by Lemma 36, w^{−1}(a + b) ⩾ a + b and w(a + b) ⩾ a + b. Hence w = 1, w_{a} or w_{3a+b}. Suppose that w = w_{3a+b}. If

u = x_{a+b}(t_{a+b})x_{3a+b}(t_{3a+b})x_{3a+2b}(t_{3a+2b})x_{a}(t_{a})x_{b}(t_{b})x_{2a+b}(t_{2a+b}),

then, since α + a + b and α + 3a + b are not roots for α ∈ {a + b, 3a + b, 3a + 2b},

Ad(ẇ^{−1}(ut)^{−1})(X) = Ad(ω_{3a+b}^{−1}t^{−1})(−λ_{1}e_{a+b} − 2λ_{1}t_{a}e_{2a+b} + (1 + 3λ_{1}t_{a}^{2})e_{3a+b}

− (3λ_{1}t_{2a+b} + t_{b} + 3λ_{1}t_{a}^{2}t_{b})e_{3a+2b})

= Ad v(X) ∈ 𝔲.

But w_{3a+b}^{−1}(2a + b) < 0 and w_{3a+b}^{−1}(3a + b) < 0 and hence −2λ_{1}t_{a} = 1 + 3λ_{1}t_{a}^{2} = 0, a contradiction. Hence C_{G}(X) ⊆ B ∪ Bω_{a}X_{a} = G_{⩾0}.

Proposition 38. We have C_{G}(e_{5}) = G_{⩾2} · ⟨X_{a}, X_{−a}⟩ and

C_{G}(e_{4}) = ⟨X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩ · ⟨X_{b}, X_{−b}⟩

as direct products. Hence dim C_{G}(e_{5}) = 8, dim C_{G}(e_{4}) = 6, |C_{G^{F}} (e_{5})| = q^{6}(q^{2} − 1), |C_{G^{F}} (e_{4})| = q^{4}(q^{2} − 1) and C_{G}(e_{5}) and C_{G}(e_{4}) are connected.

Proof. First let t = h(z_{1}, z_{2}, z_{3}) ∈ C_{T} (e_{5}). Then

Ad t(e_{3a+2b}) = (3a + 2b)(t)e_{3a+2b} = e_{3a+2b}

if and only if (3a+2b)(t) = 1. But 3a+2b = κ_{1}−κ_{3} and hence this occurs if and only if z_{1} = z_{3}, that is, t ∈ H_{a}. Hence C_{T} (e_{5}) = H_{a}. Similarly t = h(z_{1}, z_{2}, z_{3}) ∈ C_{T} (e_{4}) if and only if (2a+b)(t) = 1, that is, if and only if z_{3} = 1 and hence C_{T} (e_{4}) = H_{b}. Moreover, ω_{a} ∈ C_{G}(e_{5}), ω_{b} ∈ C_{G}(e_{4}), X_{a} ⊆ C_{G}(e_{5}) and X_{b} ⊆ C_{G}(e_{4}). Hence bω_{a}u ∈ C_{G}(e_{5}) with b ∈ B, u ∈ X_{a} if and only if b ∈ C_{G}(e_{5}) and hence

C_{G}(e_{5}) = C_{B}(e_{5}) ∪ C_{B}(e_{5})ω_{a}X_{a}
= C_{U}(e_{5})C_{T} (e_{5}) ∪ C_{U}(e_{5})C_{T} (e_{5})ω_{a}X_{a} by Lemma 34
= UH_{a} ∪ UH_{a}ω_{a}X_{a} by Lemma 33
= G_{⩾2} · (X_{a}H_{a} ∪ X_{a}H_{a}ω_{a}X_{a}) as a direct product
= G_{⩾2} · ⟨X_{a}, X_{−a}⟩.

Hence dim C_{G}(e_{5}) = 8 and |C_{G^{F}} (e_{5})| = |G_{⩾2}^{F}||⟨X_{a}, X_{−a}⟩^{F} | = q^{6}(q^{2} − 1).

Similarly

C_{G}(e_{4}) = C_{B}(e_{4}) ∪ C_{B}(e_{4})ω_{b}X_{b}
= C_{U}(e_{4})C_{T} (e_{4}) ∪ C_{U}(e_{4})C_{T} (e_{4})ω_{b}X_{b}
= ⟨X_{b}, X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩(H_{b} ∪ H_{b}ω_{b}X_{b})
= ⟨X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩(X_{b}H_{b} ∪ X_{b}H_{b}ω_{b}X_{b})
= ⟨X_{2a+b}, X_{3a+b}, X_{3a+2b}⟩ · ⟨X_{b}, X_{−b}⟩

as a direct product. Thus dim C_{G}(e_{4}) = 6 and |C_{G^{F}} (e_{4})| = q^{4}(q^{2} − 1).

Proposition 39. For 1 ⩽ i ⩽ 3, dim C_{G}(e_{i}) = 4 and

C_{G}(e_{i})/C_{G}(e_{i})^{∘} ≅ S_{3}.

Moreover F acts trivially on C_{G}(e_{1})/C_{G}(e_{1})^{∘}. We have

|C_{G^{F}} (e_{1})| = 6q^{4}, |C_{G^{F}} (e_{2})| = 2q^{4} and |C_{G^{F}} (e_{3})| = 3q^{4}.

Proof. First we show that

C_{G}(e_{i}) = C_{G_{0}}(e_{i}) · C_{G_{⩾2}}(e_{i}). (7.17)

By Proposition 37, C_{G}(e_{i}) ⊆ G_{⩾0} = G_{0} · G_{⩾2}. So let gh ∈ C_{G}(e_{i}) with g ∈ G_{0}, h ∈ G_{⩾2}. If α ⩾ b and β ⩾ b are roots such that α + β is a root then α + β = 3a + 2b and hence Ad h(e_{i}) ≡ e_{i} mod 𝔤_{3a+2b}. Now Ad g(𝔤_{3a+2b}) = 𝔤_{3a+2b} and hence

e_{i} = Ad(gh)(e_{i}) ≡ Ad g(e_{i}) mod 𝔤_{3a+2b}.

But Ad g(𝔤_{2}) ⊆ 𝔤_{2} and hence g ∈ C_{G_{0}}(e_{i}), h ∈ C_{G_{⩾2}}(e_{i}) and Equation (7.17) holds.

By Lemma 21(iii), C_{G_{⩾2}}(e_{i}) is a normal subgroup of C_{G}(e_{i}). Moreover, C_{G_{0}}(e_{i}) ∩ C_{G_{⩾2}}(e_{i}) ⊆ G_{0} ∩ G_{⩾2} = {1} and the product in Equation (7.17) is semidirect. The index of C_{G_{⩾2}}(e_{i}) in C_{G}(e_{i}) is hence |C_{G_{0}}(e_{i})| = 6 by Lemma 35. According to Borel [1, Prop. 1.7(b)], C_{G_{⩾2}}(e_{i}) is closed in G_{⩾2}. But G_{⩾2} is itself closed in G and hence C_{G_{⩾2}}(e_{i}) is a closed subgroup of C_{G}(e_{i}). By Lemma 33, C_{G_{⩾2}}(e_{1}) = ⟨X_{b}, X_{2a+b}, X_{3a+2b}, V ⟩ with V = \{x_{3a+b}(3λ_{1}t)x_{a+b}(t) | t ∈ \overline{𝔽}_{q}\} and, being generated by closed connected subgroups, C_{G_{⩾2}}(e_{1}) is itself connected (see Humphreys [7, Cor. 7.5]). Thus C_{G_{⩾2}}(e_{i}) = ^{g_{i}}C_{G_{⩾2}}(e_{1}) is also connected. We deduce that C_{G_{⩾2}}(e_{i}) is a closed connected normal subgroup of finite index in C_{G}(e_{i}) and hence that C_{G_{⩾2}}(e_{i}) = C_{G}(e_{i})^{∘}. Thus dim C_{G}(e_{i}) = dim C_{G}(e_{i})^{∘} = dim C_{G_{⩾2}}(e_{i}) = 4 according to Lemma 33. Equation (7.17) implies that the elements of C_{G_{0}}(e_{i}) form a set of coset representatives of C_{G_{⩾2}}(e_{i}) in C_{G}(e_{i}) and hence that

C_{G}(e_{i})/C_{G}(e_{i})^{∘} ≅ C_{G_{0}}(e_{i}) ≅ S_{3}.

Now (\frac{λ_{1}}{q}) = ϵ and it is well known that (\frac{−3}{q}) = ϵ. Hence −3λ_{1} is a square in 𝔽_{q} and, by Lemma 35, F acts trivially on C_{G_{0}}(e_{1}). Thus F also acts trivially on C_{G}(e_{1})/C_{G}(e_{1})^{∘}.

Finally, according to Lemma 33,

|C_{G^{F}} (e_{i})| = |C_{G_{0}^{F}} (e_{i})||C_{G_{⩾2}^{F}}(e_{i})| = q^{4}|C_{G_{0}^{F}} (e_{i})|.

If e_{i} = Ad g_{i}(e_{1}) according to Lemma 35 then C_{G_{0}}(e_{i}) = ^{g_{i}}C_{G_{0}}(e_{1}). So let ^{g_{i}}x be an element of C_{G_{0}^{F}} (e_{i}) with x ∈ C_{G_{0}}(e_{1}). Then

g_{i}x = F (^{g_{i}}x) = ^{F (g_{i})}F (x) = ^{F (g_{i})}x,

since F acts trivially on C_{G_{0}}(e_{1}), and hence ^{g_{i}^{−1}F (g_{i})}x = x. Thus |C_{G_{0}^{F}} (e_{i})| is the number of fixed points of g_{i}^{−1}F (g_{i}) acting by conjugation on C_{G_{0}}(e_{1}) ≅ S_{3}. According to Lemma 35 we then have |C_{G_{0}^{F}} (e_{1})| = 6, |C_{G_{0}^{F}} (e_{2})| = 2 and |C_{G_{0}^{F}} (e_{3})| = 3 giving the desired values of |C_{G^{F}} (e_{i})|.

8. Summary of adjoint orbits (type G_{2})

Let λ be a fixed nonsquare in 𝔽_{q}, μ a fixed noncube in 𝔽_{q} if q ≡ 1 mod 3 and ζ ∈ 𝔽_{q} such that x^{3} − 3x − ζ is irreducible over 𝔽_{q} if q ≡ −1 mod 3. Define

e_{0} = e_{a} + e_{b}
e_{1} = \begin{cases}e_{b} + e_{2a+b} & if q ≡ 1 mod 3, \\ e_{b} + λe_{2a+b} & if q ≡ −1 mod 3 \end{cases}

e_{2} = \begin{cases}e_{b} + λe_{2a+b} & if q ≡ 1 mod 3, \\ e_{b} + e_{2a+b} & if q ≡ −1 mod 3 \end{cases}

e_{3} = \begin{cases}e_{b} + μe_{3a+b} & if q ≡ 1 mod 3, \\ e_{b} − e_{2a+b} + ζe_{3a+b} & if q ≡ −1 mod 3 \end{cases}
e_{4} = e_{2a+b}
e_{5} = e_{3a+2b}
e_{6} = 0.

Then {e_{0}, e_{1}, e_{4}, e_{5}, e_{6}} is a set of representatives for the nilpotent Ad G-orbits on 𝔤. We have C_{G}(e_{1})/C_{G}(e_{1})^{∘} ≅ S_{3} and for i = 0, 4, 5, 6 the group C_{G}(e_{i})/C_{G}(e_{i})^{∘} is trivial. The dimensions of their centralizers are given by the following table (for further information on the structure of the centralizers refer to Lemmas 33 and 35 and Propositions 37 and 38).

representative e^{i} dim C^{G}(e^{i})
e_{0}  2
e_{1}  4
e_{4}  6
e_{5}  8
e_{6}  14

If λ_{1}, λ_{2}, λ_{3} ∈ \overline{𝔽}_{q} with λ_{1} + λ_{2} + λ_{3} = 0 then let (λ_{1}, λ_{2}, λ_{3}) denote the element \frac{1}{3}(λ_{1}h_{a+b} + λ_{2}h_{a} + λ_{3}h_{−2a−b}) of Lie T. If X is a rational element of 𝔤 then the Ad G^{F} - orbit of X in 𝔤^{F} is uniquely determined by giving a representative for the Ad G-orbit of X in 𝔤. If X ∈ 𝔤^{F} then it can be shown (see, for example, [10, Proposition 3.3]) that C_{G}(X_{s})^{∘} is of the form ^{g}L_{J} for some J ⊆ Π and some g ∈ G where g^{−1}F (g) = ẇ and wF (J) = J. The genus of X is defined to be the W -class of the coset W_{J}w (see [10] for further discussion).

A complete set of representatives for these orbits and the orders of their centralizers in G^{F} are given in the following table. The table also gives the genus of X (in the form of a pair (J, w) for which W_{J}w is a representative of the genus of X). For further information, refer to Theorems 14, 16, 20, 31 and 32.

THE RATIONAL ADJOINT ORBITS OF A LIE ALGEBRA OF TYPE G_{2} 31

representative of Ad G-orbit of X

(J, w) |C_{G^{F}} (X)|

no. of ; Ad G^{F} -orbits ; of given form
(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = 1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0  (∅, 1)  (q − 1)^{2}  \frac{(q−1)(q−5)}{12}
{(λ, λ^{q} − λ, −λ^{q}), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  (∅, w_{a})  q^{2} − 1  \frac{(q−1)^{2}}{4}
{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} = 1, λ^{q−1} ≠ ±1}  (∅, w_{b})  q^{2} − 1  \frac{(q−1)^{2}}{4}
(λ_{1}, λ_{2}, λ_{3}), λ_{i}^{q−1} = −1, ; λ_{i} ≠ λ_{j} (i ≠ j), ; λ_{1} + λ_{2} + λ_{3} = 0  (∅, w_{6}^{3})  (q + 1)^{2}  \frac{(q−1)(q−5)}{12}
{(λ, λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} + λ^{q−1} + 1 = 0}  (∅, w_{6}^{4})  q^{2} + q + 1  \frac{q^{2}−1}{6}
{(λ, −λ^{q}, −λ^{q} − λ), \atop λ^{q^{2}−1} − λ^{q−1} + 1 = 0}  (∅, w_{6})  q^{2} − q + 1  \frac{q^{2}−1}{6}
(λ, λ, −2λ), λ^{q−1} = 1  (\{b\}, 1)  q(q − 1)^{2}(q + 1)  \frac{q−1}{2}
(λ, λ, −2λ) + e_{b}, λ^{q−1} = 1  (\{b\}, 1)  q(q − 1)  \frac{q−1}{2}
(λ, λ, −2λ), λ^{q−1} = −1  (\{b\}, w_{2a+b}) q(q − 1)(q + 1)^{2}  \frac{q−1}{2}
(λ, λ, −2λ) + e_{b}, λ^{q−1} = −1 (\{b\}, w_{2a+b})  q(q + 1)  \frac{q−1}{2}
(λ, 0, −λ), λ^{q−1} = 1  (\{a\}, 1)  q(q − 1)^{2}(q + 1)  \frac{q−1}{2}
(λ, 0, −λ) + e_{a}, λ^{q−1} = 1  (\{a\}, 1)  q(q − 1)  \frac{q−1}{2}
(λ, 0, −λ), λ^{q−1} = −1  (\{a\}, w_{3a+2b}) q(q − 1)(q + 1)^{2}  \frac{q−1}{2}
(λ, 0, −λ) + e_{a}, λ^{q−1} = −1 (\{a\}, w_{3a+2b})  q(q + 1)  \frac{q−1}{2}
e_{0}  ({a, b}, 1)  q^{2}  1
e_{1}  ({a, b}, 1)  6q^{4}  1
e_{2}  ({a, b}, 1)  2q^{4}  1
e_{3}  ({a, b}, 1)  3q^{4}  1
e_{4}  ({a, b}, 1)  q^{4}(q^{2} − 1)  1
e_{5}  ({a, b}, 1)  q^{6}(q^{2} − 1)  1
e_{6}  ({a, b}, 1) q^{6}(q^{2} − 1)(q^{6} − 1)  1

Appendix. Tables Relating to Chevalley Groups of Type G_{2}

In this appendix we give explicit tables for several different quantities associated to a Chevalley group of type G_{2}.

We have the following relations:

[e_{α}, e_{β}] = N_{α,β}e_{α+β} (α ≠ −β, α + β ∈ Φ)
[e_{α}, e_{−α}] = h_{α}
[h_{α}, e_{β}] = β(h_{α})e_{β} = A_{αβ}e_{β}
[h_{α}, h_{β}] = 0
Ad x_{α}(t)e_{β} = ∑_{i} M_{α,β,i}t^{i}e_{iα+β} (α ≠ −β)
Ad x_{α}(t)e_{−α} = e_{−α} + th_{α} − t^{2}e_{α}
Ad x_{α}(t)h_{β} = h_{β} − α(h_{β})te_{α} = h_{β} − A_{βα}te_{α}
Ad h_{α}(t)e_{β} = t^{β(h_{α})}e_{β} = t^{A_{αβ}}e_{β}
Ad h_{α}(t)h_{β} = h_{β}
Ad n_{α}e_{β} = η_{α,β}e_{w_{α}(β)}
Ad n_{α}h_{β} = h_{w_{α}(β)}.

We will use the choice of structure constants N_{αβ} given in [3, p. 211]. The values are given in the following table in which the rows are indexed by α, the columns by β.

a b a+b 2a+b 3a+b 3a+2b −a −b −a−b −2a−b −3a−b −3a−2b
a  −1 −2 3  3  2  −1
b  1  1  −1  −1
a+b  2  3  3 −1  −2  −1
2a+b −3  −3  2  −2  1  1
3a+b  −1  −1  1  1
3a+2b  −1 −1  1  1
−a  −3 −2  1  1  2  −3
−b  1  1 −1  −1
−a−b −3 1  2  1 −2  −3
−2a−b −2  2  −1  −1 3  3
−3a−b 1  −1  −1  1
−3a−2b  1 1 −1 −1

The values of A_{αβ} for α, β ∈ Φ^{+} are given in the following table in which the rows are indexed by α, the columns by β and the remaining values are given by A_{−α,β} = A_{α,−β} = −A_{αβ}.

a b a + b 2a + b 3a + b 3a + 2b
a  2 −3 −1  1  3  0
b  −1 2  1  0  −1  1
a + b −1 3  2  1  0  3
2a + b 1 0  1  2  3  3
3a + b 1 −1 0  1  2  1
3a + 2b 0 1  1  1  1  2

The values of η_{α,β} are given in [12]. The values of η_{α,β} for α ∈ Φ^{+} are given by the following table, the rows and columns being labelled as before. The remaining values are given by η_{α,−β} = η_{α,β}.

a b a + b 2a + b 3a + b 3a + 2b
a  −1 1  1  −1  −1  1
b  1 −1 −1  1  1  −1
a + b −1 −1 −1  1  1  1
2a + b −1 1  1  −1  −1  1
3a + b −1 −1 1  1  −1  1
3a + 2b 1 −1 −1  1  1  −1
−a  −1 −1 −1  1  1  1
−b  −1 −1 1  1  −1  1
−a − b 1 1 −1 −1  1  −1
−2a − b 1 1 −1 −1  1  −1
−3a − b 1 1  1  −1  −1  −1
−3a − 2b 1 1  1  −1  −1  −1

It will also be convenient to have a table of w(β) for w ∈ W, β ∈ Φ. The values of w(β) for β ∈ Φ^{+} are given by the following table in which the rows are labelled by w and the columns by β.

a  b  a + b 2a + b 3a + b 3a + 2b
w_{a}  −a  3a + b 2a + b a + b  b  3a + 2b
w_{b}  a + b  −b  a  2a + b 3a + 2b 3a + b
w_{a+b}  2a + b −3a − 2b −a − b  a  3a + b  −b
w_{2a+b} −a − b  b  −a −2a − b −3a − 2b −3a − b
w_{3a+b} −2a − b 3a + 2b a + b  −a  −3a − b  b
w_{3a+2b}  a  −3a − b −2a − b −a − b  −b  −3a − 2b
w_{6}  2a + b −3a − b  −a  a + b 3a + 2b  b
w_{6}^{2}  a + b −3a − 2b −2a − b −a  b  −3a − b
w_{2} = w_{6}^{3} −a  −b  −a − b −2a − b −3a − b −3a − 2b
w_{3} = w_{6}^{4} −2a − b 3a + b  a  −a − b −3a − 2b  −b
w_{6}^{5}  −a − b 3a + 2b 2a + b  a  −b  3a + b

34 DAVID G. A. JACKSON

The Chevalley relations are explicitly given as follows:

Ad x_{a}(t)e_{a}  = e_{a}
Ad x_{a}(t)e_{b}  = e_{b} − te_{a+b} + t^{2}e_{2a+b} + t^{3}e_{3a+b}
Ad x_{a}(t)e_{a+b}  = e_{a+b} − 2te_{2a+b} − 3t^{2}e_{3a+b}
Ad x_{a}(t)e_{2a+b} = e_{2a+b} + 3te_{3a+b}
Ad x_{a}(t)e_{3a+b} = e_{3a+b}
Ad x_{a}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{a}(t)e_{−a}  = e_{−a} + th_{a} − t^{2}e_{a}
Ad x_{a}(t)e_{−b}  = e_{−b}
Ad x_{a}(t)e_{−a−b} = e_{−a−b} + 3te_{−b}
Ad x_{a}(t)e_{−2a−b} = e_{−2a−b} + 2te_{−a−b} + 3t^{2}e_{−b}
Ad x_{a}(t)e_{−3a−b} = e_{−3a−b} − te_{−2a−b} − t^{2}e_{−a−b} − t^{3}e_{−b}
Ad x_{a}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{a}(t)h_{a}  = h_{a} − 2te_{a}
Ad x_{a}(t)h_{b}  = h_{b} + te_{a}
Ad x_{b}(t)e_{a}  = e_{a} + te_{a+b}
Ad x_{b}(t)e_{b}  = e_{b}
Ad x_{b}(t)e_{a+b}  = e_{a+b}
Ad x_{b}(t)e_{2a+b} = e_{2a+b}
Ad x_{b}(t)e_{3a+b} = e_{3a+b} + te_{3a+2b}
Ad x_{b}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{b}(t)e_{−a}  = e_{−a}
Ad x_{b}(t)e_{−b}  = e_{−b} + th_{b} − t^{2}e_{b}
Ad x_{b}(t)e_{−a−b} = e_{−a−b} − te_{−a}
Ad x_{b}(t)e_{−2a−b} = e_{−2a−b}
Ad x_{b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{b}(t)e_{−3a−2b} = e_{−3a−2b} − te_{−3a−b}
Ad x_{b}(t)h_{a}  = h_{a} + 3te_{b}
Ad x_{b}(t)h_{b}  = h_{b} − 2te_{b}

Ad x_{a+b}(t)e_{a}  = e_{a} + 2te_{2a+b} + 3t^{2}e_{3a+2b}
Ad x_{a+b}(t)e_{b}  = e_{b}
Ad x_{a+b}(t)e_{a+b}  = e_{a+b}
Ad x_{a+b}(t)e_{2a+b}  = e_{2a+b} + 3te_{3a+2b}
Ad x_{a+b}(t)e_{3a+b}  = e_{3a+b}
Ad x_{a+b}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{a+b}(t)e_{−a}  = e_{−a} + 3te_{b}
Ad x_{a+b}(t)e_{−b}  = e_{−b} − te_{a} − t^{2}e_{2a+b} − t^{3}e_{3a+2b}
Ad x_{a+b}(t)e_{−a−b} = e_{−a−b} + th_{a} + 3th_{b} − t^{2}e_{a+b}
Ad x_{a+b}(t)e_{−2a−b} = e_{−2a−b} − 2te_{−a} − 3t^{2}e_{b}
Ad x_{a+b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{a+b}(t)e_{−3a−2b} = e_{−3a−2b} − te_{−2a−b} + t^{2}e_{−a} + t^{3}e_{b}
Ad x_{a+b}(t)h_{a}  = h_{a} + te_{a+b}
Ad x_{a+b}(t)h_{b}  = h_{b} − te_{a+b}
Ad x_{2a+b}(t)e_{a}  = e_{a} − 3te_{3a+b}
Ad x_{2a+b}(t)e_{b}  = e_{b}
Ad x_{2a+b}(t)e_{a+b}  = e_{a+b} − 3te_{3a+2b}
Ad x_{2a+b}(t)e_{2a+b} = e_{2a+b}
Ad x_{2a+b}(t)e_{3a+b} = e_{3a+b}
Ad x_{2a+b}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{2a+b}(t)e_{−a}  = e_{−a} + 2te_{a+b} − 3t^{2}e_{3a+2b}
Ad x_{2a+b}(t)e_{−b}  = e_{−b}
Ad x_{2a+b}(t)e_{−a−b} = e_{−a−b} − 2te_{a} + 3t^{2}e_{3a+b}
Ad x_{2a+b}(t)e_{−2a−b} = e_{−2a−b} + 2th_{a} + 3th_{b} − t^{2}e_{2a+b}
Ad x_{2a+b}(t)e_{−3a−b} = e_{−3a−b} + te_{−a} + t^{2}e_{a+b} − t^{3}e_{3a+2b}
Ad x_{2a+b}(t)e_{−3a−2b} = e_{−3a−2b} + te_{−a−b} − t^{2}e_{a} + t^{3}e_{3a+b}
Ad x_{2a+b}(t)h_{a}  = h_{a} − te_{2a+b}
Ad x_{2a+b}(t)h_{b}  = h_{b}

Ad x_{3a+b}(t)e_{a}  = e_{a}
Ad x_{3a+b}(t)e_{b}  = e_{b} − te_{3a+2b}
Ad x_{3a+b}(t)e_{a+b}  = e_{a+b}
Ad x_{3a+b}(t)e_{2a+b}  = e_{2a+b}
Ad x_{3a+b}(t)e_{3a+b}  = e_{3a+b}
Ad x_{3a+b}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{3a+b}(t)e_{−a}  = e_{−a} − te_{2a+b}
Ad x_{3a+b}(t)e_{−b}  = e_{−b}
Ad x_{3a+b}(t)e_{−a−b} = e_{−a−b}
Ad x_{3a+b}(t)e_{−2a−b} = e_{−2a−b} + te_{a}
Ad x_{3a+b}(t)e_{−3a−b} = e_{−3a−b} + th_{a} + th_{b} − t^{2}e_{3a+b}
Ad x_{3a+b}(t)e_{−3a−2b} = e_{−3a−2b} + te_{−b}
Ad x_{3a+b}(t)h_{a}  = h_{a} − 3te_{3a+b}
Ad x_{3a+b}(t)h_{b}  = h_{b} + te_{3a+b}
Ad x_{3a+2b}(t)e_{a}  = e_{a}
Ad x_{3a+2b}(t)e_{b}  = e_{b}
Ad x_{3a+2b}(t)e_{a+b}  = e_{a+b}
Ad x_{3a+2b}(t)e_{2a+b} = e_{2a+b}
Ad x_{3a+2b}(t)e_{3a+b} = e_{3a+b}
Ad x_{3a+2b}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{3a+2b}(t)e_{−a}  = e_{−a}
Ad x_{3a+2b}(t)e_{−b}  = e_{−b} − te_{3a+b}
Ad x_{3a+2b}(t)e_{−a−b} = e_{−a−b} − te_{2a+b}
Ad x_{3a+2b}(t)e_{−2a−b} = e_{−2a−b} + te_{a+b}
Ad x_{3a+2b}(t)e_{−3a−b} = e_{−3a−b} + te_{b}
Ad x_{3a+2b}(t)e_{−3a−2b} = e_{−3a−2b} + th_{a} + 2th_{b} − t^{2}e_{3a+2b}
Ad x_{3a+2b}(t)h_{a}  = h_{a}
Ad x_{3a+2b}(t)h_{b}  = h_{b} − te_{3a+2b}

Ad x_{−a}(t)e_{a}  = e_{a} − th_{a} − t^{2}e_{−a}
Ad x_{−a}(t)e_{b}  = e_{b}
Ad x_{−a}(t)e_{a+b}  = e_{a+b} − 3te_{b}
Ad x_{−a}(t)e_{2a+b} = e_{2a+b} − 2te_{a+b} + 3t^{2}e_{b}
Ad x_{−a}(t)e_{3a+b} = e_{3a+b} + te_{2a+b} − t^{2}e_{a+b} + t^{3}e_{b}
Ad x_{−a}(t)e_{3a+2b} = e_{3a+2b}
Ad x_{−a}(t)e_{−a}  = e_{−a}
Ad x_{−a}(t)e_{−b}  = e_{−b} + te_{−a−b} + t^{2}e_{−2a−b} − t^{3}e_{−3a−b}
Ad x_{−a}(t)e_{−a−b} = e_{−a−b} + 2te_{−2a−b} − 3t^{2}e_{−3a−b}
Ad x_{−a}(t)e_{−2a−b} = e_{−2a−b} − 3te_{−3a−b}
Ad x_{−a}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{−a}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−a}(t)h_{a}  = h_{a} + 2te_{−a}
Ad x_{−a}(t)h_{b}  = h_{b} − te_{−a}
Ad x_{−b}(t)e_{a}  = e_{a}
Ad x_{−b}(t)e_{b}  = e_{b} − th_{b} − t^{2}e_{−b}
Ad x_{−b}(t)e_{a+b}  = e_{a+b} + te_{a}
Ad x_{−b}(t)e_{2a+b} = e_{2a+b}
Ad x_{−b}(t)e_{3a+b} = e_{3a+b}
Ad x_{−b}(t)e_{3a+2b} = e_{3a+2b} + te_{3a+b}
Ad x_{−b}(t)e_{−a}  = e_{−a} − te_{−a−b}
Ad x_{−b}(t)e_{−b}  = e_{−b}
Ad x_{−b}(t)e_{−a−b} = e_{−a−b}
Ad x_{−b}(t)e_{−2a−b} = e_{−2a−b}
Ad x_{−b}(t)e_{−3a−b} = e_{−3a−b} − te_{−3a−2b}
Ad x_{−b}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−b}(t)h_{a}  = h_{a} − 3te_{−b}
Ad x_{−b}(t)h_{b}  = h_{b} + 2te_{−b}

Ad x_{−a−b}(t)e_{a}  = e_{a} − 3te_{−b}
Ad x_{−a−b}(t)e_{b}  = e_{b} + te_{−a} − t^{2}e_{−2a−b} + t^{3}e_{−3a−2b}
Ad x_{−a−b}(t)e_{a+b}  = e_{a+b} − th_{a} − 3th_{b} − t^{2}e_{−a−b}
Ad x_{−a−b}(t)e_{2a+b}  = e_{2a+b} + 2te_{a} − 3t^{2}e_{−b}
Ad x_{−a−b}(t)e_{3a+b}  = e_{3a+b}
Ad x_{−a−b}(t)e_{3a+2b} = e_{3a+2b} + te_{2a+b} + t^{2}e_{a} − t^{3}e_{−b}
Ad x_{−a−b}(t)e_{−a}  = e_{−a} − 2te_{−2a−b} + 3t^{2}e_{−3a−2b}
Ad x_{−a−b}(t)e_{−b}  = e_{−b}
Ad x_{−a−b}(t)e_{−a−b} = e_{−a−b}
Ad x_{−a−b}(t)e_{−2a−b} = e_{−2a−b} − 3te_{−3a−2b}
Ad x_{−a−b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{−a−b}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−a−b}(t)h_{a}  = h_{a} − te_{−a−b}
Ad x_{−a−b}(t)h_{b}  = h_{b} + te_{−a−b}
Ad x_{−2a−b}(t)e_{a}  = e_{a} − 2te_{−a−b} − 3t^{2}e_{−3a−2b}
Ad x_{−2a−b}(t)e_{b}  = e_{b}
Ad x_{−2a−b}(t)e_{a+b}  = e_{a+b} + 2te_{−a} + 3t^{2}e_{−3a−b}
Ad x_{−2a−b}(t)e_{2a+b} = e_{2a+b} − 2th_{a} − 3th_{b} − t^{2}e_{−2a−b}
Ad x_{−2a−b}(t)e_{3a+b} = e_{3a+b} − te_{a} + t^{2}e_{−a−b} + t^{3}e_{−3a−2b}
Ad x_{−2a−b}(t)e_{3a+2b} = e_{3a+2b} − te_{a+b} − t^{2}e_{−a} − t^{3}e_{−3a−b}
Ad x_{−2a−b}(t)e_{−a}  = e_{−a} + 3te_{−3a−b}
Ad x_{−2a−b}(t)e_{−b}  = e_{−b}
Ad x_{−2a−b}(t)e_{−a−b} = e_{−a−b} + 3te_{−3a−2b}
Ad x_{−2a−b}(t)e_{−2a−b} = e_{−2a−b}
Ad x_{−2a−b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{−2a−b}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−2a−b}(t)h_{a}  = h_{a} + te_{−2a−b}
Ad x_{−2a−b}(t)h_{b}  = h_{b}

Ad x_{−3a−b}(t)e_{a}  = e_{a} + te_{−2a−b}
Ad x_{−3a−b}(t)e_{b}  = e_{b}
Ad x_{−3a−b}(t)e_{a+b}  = e_{a+b}
Ad x_{−3a−b}(t)e_{2a+b}  = e_{2a+b} − te_{−a}
Ad x_{−3a−b}(t)e_{3a+b}  = e_{3a+b} − th_{a} − th_{b} − t^{2}e_{−3a−b}
Ad x_{−3a−b}(t)e_{3a+2b} = e_{3a+2b} − te_{b}
Ad x_{−3a−b}(t)e_{−a}  = e_{−a}
Ad x_{−3a−b}(t)e_{−b}  = e_{−b} + te_{−3a−2b}
Ad x_{−3a−b}(t)e_{−a−b} = e_{−a−b}
Ad x_{−3a−b}(t)e_{−2a−b} = e_{−2a−b}
Ad x_{−3a−b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{−3a−b}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−3a−b}(t)h_{a}  = h_{a} + 3te_{−3a−b}
Ad x_{−3a−2b}(t)e_{a}  = e_{a}
Ad x_{−3a−2b}(t)e_{b}  = e_{b} + te_{−3a−b}
Ad x_{−3a−2b}(t)e_{a+b}  = e_{a+b} + te_{−2a−b}
Ad x_{−3a−2b}(t)e_{2a+b} = e_{2a+b} − te_{−a−b}
Ad x_{−3a−2b}(t)e_{3a+b} = e_{3a+b} − te_{−b}
Ad x_{−3a−2b}(t)e_{3a+2b} = e_{3a+2b} − th_{a} − 2th_{b} − t^{2}e_{−3a−2b}
Ad x_{−3a−2b}(t)e_{−a}  = e_{−a}
Ad x_{−3a−2b}(t)e_{−b}  = e_{−b}
Ad x_{−3a−2b}(t)e_{−a−b} = e_{−a−b}
Ad x_{−3a−2b}(t)e_{−2a−b} = e_{−2a−b}
Ad x_{−3a−2b}(t)e_{−3a−b} = e_{−3a−b}
Ad x_{−3a−2b}(t)e_{−3a−2b} = e_{−3a−2b}
Ad x_{−3a−2b}(t)h_{a}  = h_{a}
Ad x_{−3a−2b}(t)h_{b}  = h_{b} + te_{−3a−2b}

Ad h_{a}(t)e_{a}  = t^{2}e_{a}
Ad h_{a}(t)e_{b}  = t^{−3}e_{b}
Ad h_{a}(t)e_{a+b}  = t^{−1}e_{a+b}
Ad h_{a}(t)e_{2a+b} = te_{2a+b}
Ad h_{a}(t)e_{3a+b} = t^{3}e_{3a+b}
Ad h_{a}(t)e_{3a+2b} = e_{3a+2b}
Ad h_{a}(t)e_{−a}  = t^{−2}e_{−a}
Ad h_{a}(t)e_{−b}  = t^{3}e_{−b}
Ad h_{a}(t)e_{−a−b} = te_{−a−b}
Ad h_{a}(t)e_{−2a−b} = t^{−1}e_{−2a−b}
Ad h_{a}(t)e_{−3a−b} = t^{−3}e_{−3a−b}
Ad h_{a}(t)e_{−3a−2b} = e_{−3a−2b}
Ad h_{a}(t)h_{a}  = h_{a}
Ad h_{a}(t)h_{b}  = h_{b}

Ad h_{b}(t)e_{a}  = t^{−1}e_{a}
Ad h_{b}(t)e_{b}  = t^{2}e_{b}
Ad h_{b}(t)e_{a+b}  = te_{a+b}
Ad h_{b}(t)e_{2a+b} = e_{2a+b}
Ad h_{b}(t)e_{3a+b} = t^{−1}e_{3a+b}
Ad h_{b}(t)e_{3a+2b} = te_{3a+2b}
Ad h_{b}(t)e_{−a}  = te_{−a}
Ad h_{b}(t)e_{−b}  = t^{−2}e_{−b}
Ad h_{b}(t)e_{−a−b} = t^{−1}e_{−a−b}
Ad h_{b}(t)e_{−2a−b} = e_{−2a−b}
Ad h_{b}(t)e_{−3a−b} = te_{−3a−b}
Ad h_{b}(t)e_{−3a−2b} = t^{−1}e_{−3a−2b}
Ad h_{b}(t)h_{a}  = h_{a}
Ad h_{b}(t)h_{b}  = h_{b}

References

[1] Armand Borel. Linear Algebraic Groups. Number 126 in Graduate Texts in Mathematics. Springer-Verlag, New-York, second enlarged edition, 1991.

[2] Armand Borel and Jacques Tits. Groupes réductifs. Inst. Hautes Études Sci. Publ. Math., 27:55– 152, 1965.

[3] Roger W. Carter. Simple Groups of Lie Type. John Wiley and Sons, New York, second edition, 1989.

[4] Roger W. Carter. Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. John Wiley and Sons, Chichester, second edition, 1993.

[5] Bomshik Chang. The conjugate classes of Chevalley groups of type (G_{2}). J. Algebra, 9:190–211, [6] François Digne and Jean Michel. Representations of Finite Groups of Lie Type. Number 21 in

London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.

[7] James E. Humphreys. Linear Algebraic Groups. Number 21 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1981.

[8] James E. Humphreys. Conjugacy Classes in Semisimple Algebraic Groups. Number 43 in Mathematical Surveys and Monographs. American Mathematical Society, Providence, Rhode Island, 1995.

[9] David G. A. Jackson. The irreducibility of a cubic over 𝔽_{q}. To appear in Nieuw Archief voor Wiskunde.

[10] David G. A. Jackson. Values of the Fourier transforms of Green functions at regular elements. To appear in J. Algebra.

[11] Sharad V. Keny. Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics. J. Algebra, 108:195–201, 1987.

[12] Rimshak Ree. A family of simple groups associated with the simple Lie algebra of type (G_{2}). Amer. J. Math., 83:432–462, 1961.

[13] T. A. Springer. Some arithmetical results of semi-simple Lie algebras. Inst. Hautes Études Sci. Publ. Math., 30:115–141, 1966.

[14] T. A. Springer. Linear Algebraic Groups. Birkhäuser, Boston, 1981.

[15] T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups, number 131 in Lecture Notes in Mathematics, part E, pages 167–266. Springer-Verlag, Berlin-Heidelberg, 1986. Second Corrected Printing.

[16] Robert Steinberg. Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math., 25:49–80, 1965.

[17] U. Stuhler. Unipotente und nilpotente Klassen in einfachen Gruppen und Liealgebren vom typ G_{2}. Nederl. Akad. Wetensch. Proc. Ser. A, 74:365–378, 1971.

School of Mathematics and Statistics

University of Sydney Australia 2006