OUTER AUTOMORPHISMS OF ALGEBRAIC GROUPS AND DETERMINING GROUPS BY THEIR MAXIMAL TORI

SKIP GARIBALDI

Abstract. We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_{2n} over a global field, which completes some of the main results from the paper “Weakly commensurable arithmetic groups and isospectral locally symmetric spaces” (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.

One goal of this paper is the (rather technical) Theorem 16 below, which completes some of the main results in the remarkable paper [PrR 09] by Gopal Prasad and Andrei Rapinchuk. For example, combining their Theorem 7.5 with our Theorem 16 gives:

Theorem 1. Let G_{1} and G_{2} be connected absolutely simple algebraic groups over a number field K that have the same K-isomorphism classes of maximal K-tori. Then:

(1) G_{1} and G_{2} have the same Killing-Cartan type (and even the same quasisplit inner form) or one has type B_{n} and the other has type C_{n}.

(2) If G_{1} and G_{2} have the same Killing-Cartan type and that type is not A_{n} for n ≥ 2, D_{2n+1}, or E_{6}, then G_{1} and G_{2} are K-isomorphic.

This result is essentially proved by Prasad-Rapinchuk in [PrR 09], except that paper omits types D_{2n} for 2n ≥ 4 in (2). Our Theorem 16 gives a new proof of the 2n ≥ 6 case (treated by Prasad-Rapinchuk in a later paper [PrR 10, §9]) and settles the last remaining case of groups of type D_{4}. Note that in the final form stated above, Theorem 1 is complete, in that types A_{n}, D_{2n+1}, and E_{6} are genuine exceptions by [PrR 09, 7.6].

Similarly, combining our Theorem 16 with the arguments in [PrR 09] implies that their Theorems 4, 8.16, and 10.4 remain true if you delete “D_{4}” from their statements—that is, the conclusions of those theorems regarding weak commensurability, locally symmetric spaces, etc., also hold for groups of type D_{4}.

The other goal of this paper is Theorem 8, which addresses the more general setting of a semisimple algebraic group G over an arbitrary field k. That theorem gives a cohomological criterion for the existence of outer automorphisms of G, i.e., for the existence of k-points on non-identity components of Aut(G). This criterion and the examples we give of when it holds make up the bulk of the proof of Theorem 16, which concerns groups over global fields.

2010 Mathematics Subject Classification. Primary 20G30; Secondary 11E72, 20G15.

Notation. A global field is a finite extension of ℚ or 𝔽_{p}(t) for some prime p. A (non-archimedean) local field is the completion of a global field with respect to a discrete valuation, i.e., a finite extension of ℚ_{p} or 𝔽_{p}((t)) for some prime p.

We write H^{d}(k, G) for the d-th flat (fppf) cohomology set H^{d}(Spec k, G) when G is an algebraic affine group scheme over a field k. In case G is smooth, it is the same as the Galois cohomology set H^{d}(Gal(k), G(k_{sep})) where k_{sep} denotes a separable closure of k and Gal(k) denotes the group of k-automorphisms of k_{sep}.

We refer to [PlR], [Sp 98], and [KMRT] for general background on semisimple algebraic groups. Such a group G is an inner form of G′ if there is a class γ ∈ H^{1}(k, \overline{G}), for \overline{G} the adjoint group of G, such that G′ is isomorphic to G twisted by γ. We write G_{γ} for the group G twisted by the cocycle γ, following the TEX-friendly notation of [KMRT, p. 387] instead of Serre’s more logical _{γ}G. We say simply that G is inner or of inner type if it is an inner form of a split group; if G is not inner then it is outer.

For a group scheme D of multiplicative type, we put D^{∗} for its dual Hom(D, 𝔾_{m}).

1. Background: the Tits algebras determine the Tits class

Fix a semisimple algebraic group G over a field k. Its simply connected cover G̃ and adjoint group \overline{G} fit into an exact sequence

(2)  1 ⟶ Z ⟶ G̃ ⟶ \overline{G} ⟶ 1

where Z denotes the (scheme-theoretic) center of G̃. Write δ: H^{1}(k, \overline{G}) → H^{2}(k, Z) for the corresponding coboundary map.

There is a unique element ν_{G} ∈ H^{1}(k, \overline{G}) such that the twisted group \overline{G}_{ν_{G}} is quasi-split [KMRT, 31.6], and the Tits class t_{G} of G is defined to be t_{G} := −δ(ν_{G}) ∈ H^{2}(k, Z). The element t_{G} depends only on the isogeny class of G.

For γ ∈ H^{1}(k, \overline{G}), the center of the twisted group G̃_{γ} is naturally identified with (and not merely isomorphic to) Z, and a standard twisting argument shows that

(3)  t_{G_{γ}} = t_{G} + δ(γ).

Example 4. If G itself is quasi-split, then t_{G} = 0 and for every γ ∈ H^{1}(k, \overline{G}) we have t_{G_{γ}} = δ(γ).

Definition 5 (Tits algebras). A Tits algebra of G is an element

χ(t_{G}) ∈ H^{2}(k(χ), 𝔾_{m}) for χ ∈ Z^{∗},

where k(χ) denotes the subfield of k_{sep} of elements fixed by the stabilizer of χ in Gal(k), i.e., k(χ) is the smallest separable extension of k so that χ is fixed by Gal(k(χ)).

We can modify this definition to replace elements of Z^{∗} with weights. Fix a pinning for G̃ over k_{sep} involving a maximal k-torus T̃. As Z is contained in T̃, every weight λ—i.e., every λ ∈ T̃^{∗}—induces by restriction an element of Z^{∗} and we define λ(t_{G}) to be λ|_{Z}(t_{G}).

Regarding history, the Tits algebras of G were defined in [T]. The class λ(t_{G}) measures the failure of the irreducible representation of G̃ with highest weight λ—which is defined over k_{sep}—to be defined over k. Roughly speaking, a typical example of a Tits algebra is provided by the even Clifford algebra of the special orthogonal group of a quadratic form, see for example [KMRT, §27].

Obviously, the Tits class t_{G} determines the Tits algebras χ(t_{G}). The converse also holds:

Proposition 6. If G is absolutely almost simple, then the natural map

∏ λ: H^{2}(k, Z) → ∏ H^{2}(k(λ|_{Z}), 𝔾_{m})

is injective, where the products range over minuscule weights λ.

This proposition can probably be viewed as folklore. I learned it from Alexander Merkurjev and Anne Quéguiner. Below, we we will use the following restatement: Given classes γ_{1}, γ_{2} ∈ H^{1}(k, \overline{G}), if λ(δ(γ_{1})) = λ(δ(γ_{2})) for every minuscule weight λ of G, then δ(γ_{1}) = δ(γ_{2}).

Proof of Proposition 6. As G̃ is simply connected, the weight lattice P in the pinning is identified with T̃ and the root lattice Q is the kernel of the restriction T̃^{∗} → Z^{∗}. Since G is assumed absolutely almost simple, its root system is irreducible, and this surjection identifies the minuscule (dominant) weights with the nonzero elements of Z^{∗} [B, §VI.2, Exercise 5a]. Therefore, the claim is equivalent to showing that the map

(7)  ∏_{χ∈Z^{∗}} χ: H^{2}(k, Z) → ∏_{χ∈Z^{∗}} H^{2}(k(χ), Z)

is injective. This claim depends only on Z, so we may replace G with G_{ν_{G}} and so assume that G is quasi-split.

We choose the maximal k-torus T̃ and pinning in G̃ so that the usual Galois action preserves the fundamental chamber, i.e., permutes the set of simple roots Δ and the fundamental dominant weights. In the short exact sequence 1 → Z → T̃ → T̃/Z → 1, the set Δ is a basis for the lattice (T̃/Z)^{∗}, so H^{1}(k, T̃/Z) is zero and the map H^{2}(k, Z) → H^{2}(k, T̃) is injective.

We fix a set S of representatives of the Gal(k)-orbits in Δ and write α_{s} (resp., λ_{s}) for the simple root (resp., fundamental dominant weight) corresponding to s ∈ S. Because G̃ is simply connected,

T̃ ≅ ∏_{s∈S} R_{k(λ_{s}|_{Z})/k}(im h_{α_{s}})

where h_{α_{s}} denotes the homomorphism 𝔾_{m} → T̃ corresponding to the coroot α_{s}^{∨} [St, p. 44, Cor.]—note that the Weil restriction term makes sense because the stabilizers of α_{s}, λ_{s}, and λ_{s}|_{Z} all agree because of our particular choice of pinning. The inclusion of Z in T̃ amounts to the product ∏_{s∈S} λ_{s}, hence (7) is injective and the claim is proved. □

2. Outer automorphisms of semisimple groups

We maintain the notation of the previous section, so that G is semisimple over a field k and Δ is a set of simple roots, equivalently, the Dynkin diagram of G. The Galois action on Δ induces an action on Aut(Δ), and in this way we view Aut(Δ) as a finite étale (but not necessarily constant) group scheme. There is a map

α: Aut(G)(k) → Aut(Δ)(k)

described for example in [Sp 98, §16.3], and one can ask if this map is surjective. That is, does every connected component of Aut(G) × k_{sep} that is defined over k necessarily have a k-point?

One obstruction to α being surjective can come from the fundamental group, so we assume that G is simply connected. (One could equivalently assume that G is adjoint.) Another obstruction comes from the Tits class, as we now explain. There is a commutative diagram

\xymatrix{Aut(G) \ar[d] \ar[r]^{α} & Aut(Δ) \ar[dl] \\ Aut(Z)}

where the diagonal arrow comes from the natural action of Aut(Δ) on the coroot lattice. Hence Aut(Δ)(k) acts on H^{2}(k, Z) and we have:

Theorem 8. Recall that G is assumed semisimple and simply connected. Then there is an inclusion

(9)  im [α: Aut(G)(k) → Aut(Δ)(k)] ⊆ {π ∈ Aut(Δ)(k) | π(t_{G}) = t_{G}}.

Furthermore, the following are equivalent:

(a) Equality holds in (9).

(b) The sequence H^{1}(k, Z) → H^{1}(k, G) → H^{1}(k, Aut(G)) is exact.

(c) ker δ ∩ ker [H^{1}(k, \overline{G}) → H^{1}(k, Aut(\overline{G}))] = 0.

Proof. We consider the interlocking exact sequences

H^{1}(k, Z)
↓
H^{1}(k, G)
↓^{q}
Aut(G)(k) ⟶^{α} Aut(Δ)(k) ⟶^{β} H^{1}(k, \overline{G}) ⟶^{ε} H^{1}(k, Aut(G))
↓^{δ}
H^{2}(k, Z)

The crux is to prove that

(10)  π(t_{G_{β(π)}}) = t_{G} for π ∈ Aut(Δ)(k).

Since \overline{G} and Aut(G) are smooth, we may view their corresponding H^{1}’s as Galois cohomology. Put γ := β(π), so γ_{σ} = f^{−1} ^{σ}f for some f ∈ Aut(G)(k_{sep}) and every σ ∈ Gal(k). The group G_{γ} has the same k_{sep}-points as G, but a different Galois action ∘ given by σ ∘g = γ_{σ}σg for g ∈ G(k_{sep}), σ ∈ Gal(k), and where juxtaposition denotes the usual Galois action on G.

The map f gives a k-isomorphism G_{γ} ⟶^{∼} G. Sequence (2) gives a commutative diagram

H^{1}(k, \overline{G}_{γ}) ⟶^{δ_{γ}} H^{2}(k, Z)
f↓  ^{f}↓
H^{1}(k, \overline{G}) ⟶^{δ} H^{2}(k, Z).

Let η ∈ Z^{1}(k, \overline{G}_{γ}) be a 1-cocycle representing ν_{G_{γ}}. Then f(η) is a 1-cocycle in Z^{1}(F, \overline{G}) and f is a k-isomorphism f : (G_{γ})_{η} ⟶^{∼} G_{f(η)}. Since (G_{γ})_{η} is k-quasi-split, we have f(ν_{G_{γ}}) = f(η) = ν_{G}. The commutativity of the diagram gives f(t_{G_{γ}}) = t_{G}, proving (10).

It follows that π ∈ Aut(Δ)(k) satisfies π(t_{G}) = t_{G} if and only if t_{G_{β(π)}} = t_{G}, if and only if δ(β(π)) = 0. That is, in (9), the left side is ker β and the right side is ker δβ, which makes the inclusion in (9) and the equivalence of (a) and (c) obvious. Statement (b) says that ker εq = ker q, i.e., ker ε ∩ im q = 0, which is (c). □

It is easy to find non-simple groups, even over ℝ, for which the inclusion (9) is proper, because the Tits index also provides an obstruction to equality. Here is an example to show that these are not the only obstructions, even over a number field.

Example 11. Fix a prime p and write x_{1}, x_{2} for the two square roots of p in k := ℚ(\sqrt{p}). For i = 1, 2, let H_{i} be the group of type G_{2} associated with the 3-Pfister quadratic form ϕ_{i} := ⟨⟨−1, −1, x_{i}⟩⟩. For G = H_{1} × H_{2}, the Tits index is

[figure: <; <]

and Aut(Δ)(k) = ℤ/2ℤ, but H_{1} is not isomorphic to H_{2}, so no k-automorphism of G interchanges the two components.

Nonetheless, we now list “many” cases in which equality holds in (9).

Example 12. If G is quasi-split, then α maps Aut(G)(k) onto Aut(Δ)(k) by [SGA3, XXIV.3.10] or [KMRT, 31.4], so equality holds in (9).

Example 13. If H^{1}(k, G) = 0, then trivially Th. 8(b) holds. That is, (a)–(c) hold for every semisimple simply connected G if k is local (by Kneser-Bruhat-Tits), global with no real embeddings (Kneser-Harder-Chernousov), or the function field of a complex surface (de Jong-He-Starr-Gille), and conjecturally if the cohomological dimension of k is at most 2 (Serre).

Example 14. Suppose G is absolutely almost simple (and simply connected). Conditions (a)–(c) of the proposition hold trivially if Aut(Δ)(k) = 1, in particular if G is not of type A, D, or E_{6} or if G has type ^{6}D_{4}. Conditions (a)–(c) also hold:

(i) if G is of inner type. If G is of inner type A (n ≥ 2), then Aut(Δ) = ℤ/2ℤ and the nontrivial element π acts via z ↦ z^{−1} on Z, hence π(t_{G}) = −t_{G}. If 2t_{G} = 0, then G is SL_{1}(D) for D a central simple algebra of degree n + 1 such that there is an anti-automorphism σ of D, hence g ↦ σ(g)^{−1} is a kautomorphism of G mapping to π. (By a theorem of Albert [Sch, Th. 8.8.4] one can even arrange for σ to have order 2, hence for this automorphism of G to have order 2.)

Next let G be of type ^{1}D_{n} for n ≥ 5 and suppose that the nonidentity element π ∈ Aut(Δ)(k) fixes the Tits class t_{G}. The group G is isomorphic to Spin(A, σ, f) for some central simple k-algebra A of degree 2n and quadratic pair (σ, f) on A such that the even Clifford algebra C_{0}(A, σ, f) is isomorphic to a direct product C_{+}×C_{−} of central simple algebras. Since π fixes the Tits class, the algebras C_{+} and C_{−} are isomorphic. The equation [A] + [C_{+}] − [C_{−}] = 0 holds in the Brauer group of k by [KMRT, 9.12] (alternatively, as a consequence of the fact that the cocenter is an abelian group of order 4). Therefore, A is split. Let ϕ ∈ O(A, σ, f)(k) be a hyperplane reflection as in [KMRT, 12.13]; it does not lie in the identity component of O(A, σ, f). The automorphism of SO(A, σ, f) given by g ↦ ϕgϕ^{−1} lifts to an automorphism of Spin(A, σ, f) that is outer, i.e., that induces the automorphism π on Δ. (To recap: given a nonzero π ∈ Aut(Δ)(k) that preserves the Tits class, we deduced that A is split and therefore (A, σ, f) has an improper isometry. Conversely, Lemma 1b from [Kn, p. 42] shows: if char k ≠ 2 and (A, σ) has an improper isometry, then A is split and obviously such a π exists.)

For the remaining cases, we point out merely that an outer automorphism of order 3 in the D_{4} case exists when t_{G} = 0 by triality [SpV, 3.6.3, 3.6.4] and an outer automorphism of order 2 in the E_{6} case when t_{G} = 0 is provided by the “standard automorphism” of a J-structure [Sp 73, p. 150].

(ii) if G is the special unitary group of a hermitian form relative to a separable quadratic extension K/k, i.e., G is of type ^{2}A_{n} and res_{K/k}(t_{G}) is zero in H^{2}(K, Z). We leave the details in this case as an exercise.

(iii) if k is real closed. By the above cases, we may assume that G has type ^{2}D_{n} (for n ≥ 4) or ^{2}E_{6}. In the first case, Aut(Δ)(k) = ℤ/2ℤ (also for n = 4) and G is the spin group of a quadratic form by [KMRT, 9.14], so a hyperplane reflection gives the desired k-automorphism.

In case G has type ^{2}E_{6}, combining pages 37, 38, 119, and 120 in [J] shows that the (outer) automorphism of the Lie algebra Jacobson denotes by t is defined over k.

I don’t know any examples of absolutely almost simple G where conditions (a)–(c) fail. Furthermore, in all of the examples above, every π from the right side of (9) is not only of the form α(f) for some f ∈ Aut(G)(k), but one can even pick f to have the same order as π.

3. Groups of type D_{even} over local fields

The main point of this section is to prove the following lemma.

Lemma 15. Let G be an adjoint semisimple group over a field k, and fix a maximal k-torus T in G. If z_{1}, z_{2} are in the image of the map H^{1}(k, T) → H^{1}(k, G) such that

(1) G_{z_{1}} and G_{z_{2}} are both quasi-split; or

(2) T contains a maximal k-split torus in both G_{z_{1}} and G_{z_{2}} and (a) k is real closed, or

(b) k is a (non-archimedean) local field and G has type D_{2n} for some

n ≥ 2,

then z_{1} = z_{2}.

Proof. For short, we write G_{i} for G_{z_{i}}. In case (1), the uniqueness of the class ν_{G} ∈ H^{1}(k, G) such that G_{ν_{G}} is quasi-split (already used in §1) gives that z_{1} = ν_{G} = z_{2}. So suppose (2) holds. As T is contained in both these groups, their Tits indexes are naturally identified over k. In particular, if one is quasi-split then so is the other, and we are done as in (1). So we assume that neither group is quasi-split.

In case (2a), where k is real closed, one immediately reduces to the case where G is absolutely simple. That case is trivial because the isomorphism class of an adjoint simple group is determined by its Tits index, so G_{1} is isomorphic to G_{2}. The Tits index also determines the Tits algebras—see pages 211 and 212 of [T] for a recipe—so by Prop. 6, δ(z_{1}) = δ(z_{2}). The claim now follows from Example 14(iii) and Theorem 8(c).

So assume for the remainder of the proof that (2b) holds. In particular, δ is injective. Number the simple roots of G_{1} with respect to T as in [B]. If G_{1} has type ^{2}D_{4}, we take α_{1} to be the root at the end of the Galois-fixed arm of the Tits index. Otherwise, we assign the numbering arbitrarily in case there is ambiguity (e.g., α_{2n−1} and α_{2n}). Note that G_{1} cannot have type ^{3}D_{4} or ^{6}D_{4}, because it is not quasi-split.

As 2ω_{i} is in the root lattice for every i, the Tits algebras ω_{i}(t_{G_{1}}) for i = 2n−1, 2n define up to k-isomorphism a quaternion (Azumaya) algebra D over a quadratic étale k-algebra ℓ. By the exceptional isomorphism D_{2} = A_{1} ×A_{1} and a Tits algebra computation, PGL_{1}(D) is isomorphic to PSO(M_{2}(H), σ, f) for H the quaternion algebra underlying ω_{1}(t_{G_{1}}) and some quadratic pair (σ, f) such that the even Clifford algebra C_{0}(σ, f) is isomorphic to D, cf. [KMRT, 15.9]. Appending 2n − 2 hyperbolic planes to (σ, f), we obtain a quadratic pair (σ_{0}, f_{0}) such that C_{0}(σ_{0}, f_{0}) is Brauer-equivalent to D. As PSO(M_{2n}(H), σ_{0}, f_{0}) has the same Tits algebras as G_{1} (up to renumbering the simple roots of G_{1}), Prop. 6 and injectivity of δ implies that the two groups are isomorphic. (We have just given a characteristic-free proof of Tsukamoto’s theorem [Sch, 10.3.6], relying on the Bruhat-Tits result that δ is injective.)

Now both G_{1} and G_{2} have the same Tits index and semisimple anisotropic kernels of Killing-Cartan type a product of A_{1}’s. As there is a unique quaternion division algebra over each finite extension of k, it follows that G_{1} and G_{2} have the same Tits class, i.e., δ(z_{1}) = δ(z_{2}). □

In the statement of (2b), we cannot replace “D_{2n} for some n ≥ 2” with “D_{ℓ} for some ℓ” because the claim fails for groups of type D_{odd}. This can been seen already for type D_{3} = A_{3}: one can find z_{1}, z_{2} ∈ H^{1}(k, PGL_{4}) so that G_{1} and G_{2} are both isomorphic to Aut(B)^{∘} for a division algebra B of degree 4, but δ(z_{1}) = −δ(z_{2}) in H^{2}(k, μ_{4}) = ℤ/4ℤ. Adding hyperbolic planes as in the proof of the lemma gives a counterexample for all odd ℓ. This counterexample is visible in the proof: for groups G_{1}, G_{2} of type D_{ℓ} with ℓ odd and ≥ 3, the semisimple anisotropic kernels have Killing-Cartan type a product of A_{1}’s and an A_{3} and the very last sentence of the proof fails.

4. Groups of type D_{even} over global fields

The following technical theorem concerning groups over a global field connects our Theorem 8 (about groups over an arbitrary field) with the results in [PrR 09]. It implies Theorem 9.1 of [PrR 10].

Theorem 16. Let G_{1} and G_{2} be adjoint groups of type D_{2n} for some n ≥ 2 over a global field K, such that G_{1} and G_{2} have the same quasi-split inner form—i.e., the smallest Galois extension of K over which G_{1} is of inner type is the same as for G_{2}. If there exists a maximal torus T_{i} in G_{i} for i = 1 and 2 such that

(1) there is a K_{sep}-isomorphism ϕ : G_{1} → G_{2} whose restriction to T_{1} is a K-isomorphism T_{1} → T_{2}; and

(2) there is a finite set V of places of K such that:

(a) For all v ∉ V, G_{1} and G_{2} are quasi-split over K_{v}.

(b) For all v ∈ V, (T_{i})_{K_{v}} contains a maximal K_{v}-split torus of (G_{i})_{K_{v}}; then G_{1} and G_{2} are isomorphic over K.

The hypotheses are what one obtains by assuming the existence of weakly commensurable arithmetic subgroups, see for example Theorems 1 and 6 and Remark 4.4 in [PrR 09]. Note that the groups appearing in the theorem can be trialitarian, i.e., of type ^{3}D_{4} or ^{6}D_{4}. We remark that Bruce Allison gave an isomorphism criterion with very different hypotheses in [A, Th. 7.7].

Proof. Write G for the unique adjoint quasi-split group that is an inner form of G_{1} and G_{2}. By Steinberg [PlR, pp. 338, 339], there is a K_{sep}-isomorphism ψ_{2} : G_{2} → G whose restriction to T_{2} is defined over K. We put ψ_{1} := ψ_{2}ϕ and T := ψ_{2}(T_{2}) = ψ_{1}(T_{1}). Then G_{i} is isomorphic to G twisted by the 1-cocycle σ ↦ ψ_{i}(^{σ}ψ_{i})^{−1}. But this 1-cocycle consists of elements of Aut(G) that fix T elementwise, hence belong to T itself. That is, for i = 1, 2, there is a cocycle z_{i} in the image of H^{1}(K, T) → H^{1}(K, G) such that G_{i} is isomorphic to G twisted by z_{i}.^{1}

Now Lemma 15 gives that res_{K_{v}/K}(z_{1}) = res_{K_{v}/K}(z_{2}) for every v, hence z_{1} = z_{2} by the Kneser-Harder Hasse Principle [PlR, p. 336, Th. 6.22] and G_{1} is isomorphic to G_{2} over K. □

Acknowledgments My research was partially supported by NSF grant DMS-0653502. I thank the University of Virginia for its hospitality during a visit there where this work began, and Andrei Rapinchuk and Gopal Prasad for their comments and suggestions.

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1This argument does not use the fact that K is a number field nor that G_{1} and G_{2} have type D_{2n}, so roughly speaking it applies generally to the situation where G_{1} and G_{2} share a maximal torus over the base field—more precisely, to the situation arising in Remark 4.4 of [PrR 09].

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Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

   E-mail address: skip@mathcs.emory.edu