STABLY CAYLEY GROUPS IN CHARACTERISTIC 0

M. BOROVOI, B. KUNYAVSKIĬ, N. LEMIRE, AND Z. REICHSTEIN

Abstract. A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SO_{n} defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if G × 𝔾_{m}^{r} is Cayley for some r ≥ 0. Here 𝔾_{m}^{r} denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero.

In this paper we study reductive Cayley groups over an arbitrary field k of characteristic zero. The condition of being Cayley is considerably more delicate in this setting. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple (but not necessarily absolutely simple) groups.

Contents

1. Introduction 2 2. Preliminaries on quasi-permutation lattices 5 3. Automorphisms and semi-automorphisms of split reductive

groups 8 3.1. Notational conventions 8 3.2. Semi-automorphisms 8 3.3. Automorphisms of split reductive groups 10 4. The character lattice and the generic torus 11 4.1. The character lattice of a reductive group 11 4.2. The generic torus 12 5. Forms of reductive groups 13

2010 Mathematics Subject Classification. Primary 20G15, 20C10. Key words and phrases. Linear algebraic group, Cayley group, Cayley map, algebraic

torus, integral representation, quasi-permutation lattice. Borovoi was supported in part by the Hermann Minkowski Center for Geometry. Kunyavskiĭ was supported in part by the Minerva Foundation through the Emmy

Noether Institute for Mathematics.

Lemire was supported in part by an NSERC Discovery Grant. Reichstein was supported in part by an NSERC Discovery Grant.

5.1. Choosing a “small” cocycle 13 5.2. Forms of Cayley groups 14 6. (G, S)-fibrations and (G, S)-varieties 15 6.1. (G, S)-fibrations 15 6.2. (G, S)-varieties 17 7. Proof of Theorem 1.3 19 8. Proof of Theorem 1.4 21 9. Statement of Theorem 9.1 and first reductions 21 10. Quasi-invertible lattices 23 11. A family of non-quasi-invertible lattices 26 12. More non-quasi-invertible lattices 28 13. Standard subgroups 33 14. Coordinate and almost coordinate subspaces 35 15. Coordinate subspaces and quasi-permutation lattices 37 16. Proof of Theorem 9.1 for H of types A_{n−1} (n ≥ 5), B_{n} (n ≥ 3)

and D_{n} (n ≥ 4) 38 16.1. Case A_{n−1} (n ≥ 5) 38 16.2. Case B_{n} (n ≥ 3) and D_{n} (n ≥ 4) 39 17. Proof of Theorem 9.1 for H of type A_{1} = B_{1} = C_{1} 40 18. Proof of Theorem 9.1 for H of types A_{2}, B_{2} = C_{2}, and A_{3} = D_{3} 41 18.1. Case A_{2} 41 18.2. Case B_{2} = C_{2} 42 18.3. Case A_{3} = D_{3} 42 19. Proof of Theorem 1.5 43 References 44

1. Introduction

Let k be a field of characteristic 0 and G be a connected linear algebraic k-group. A birational isomorphism ϕ: Lie(G) −−→^{≃} G is called a Cayley map if it is equivariant with respect to the conjugation action of G on itself and the adjoint action of G on its Lie algebra Lie(G), respectively. A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SO_{n} defined by Arthur Cayley [8] in 1846. A linear algebraic group G is called Cayley if it admits a Cayley map and stably Cayley if G ×_{k} 𝔾_{m}^{r} is Cayley for some r ≥ 0. Here 𝔾_{m} denotes the split 1-dimensional torus. These notions were introduced by Lemire, Popov and Reichstein [20]; for a more detailed discussion and numerous classical examples, we refer the reader to [20, Introduction]. The main results of [20] are the classifications of simple Cayley and stably Cayley groups in the case where the base field k is algebraically closed and of characteristic 0. The goal of this paper is to extend some of these results to the case where k is an arbitrary field of characteristic 0.

Example 1.1. If k is algebraically closed and G is a reductive k-group, then by [20, Theorem 1.27] G is stably Cayley if and only if its character lattice is quasi-permutation; see Definition 2.1.

Example 1.2. Let T be a k-torus of dimension d. By definition, T is Cayley (respectively, stably Cayley) over k if and only if T is k-rational (respectively, stably k-rational). If k is algebraically closed, then T ≃ 𝔾_{m}^{d}, hence T is always rational, and thus always Cayley. More generally, Voskresenskiĭ’s criterion for stable rationality [29, Theorem 4.7.2] asserts that T is stably rational if and only if the character lattice X(T) is quasi-permutation (see Definition 2.1).

It has been conjectured that every stably rational torus is rational. To the best of our knowledge, this conjecture is still open. Moreover, we are not aware of any simple lattice-theoretic criterion for the rationality of T.

Note that the term “character lattice” is used in different ways in Examples 1.1 and 1.2. In both cases the underlying ℤ-module is X(\overline{T}) (where \overline{T} = T ×_{k} k̄, k̄ is an algebraic closure of k, and T is a maximal torus of G in Example 1.1) but the group acting on X(\overline{T}) is the Weyl group W = W (G, T) in Example 1.1 and the Galois group Gal(k) in Example 1.2. A key role in this paper will be played by the character lattice X(G) of a reductive k-group G, a notion that bridges the special cases considered in these two examples. The underlying ℤ-module in this general setting is still X(\overline{T}), but the group acting on it is the extended Weyl group W^{ext} = W ⋊ A, where W is the usual Weyl group of \overline{G} and A is the image of Gal(k̄/k) under the so-called “∗-action” (see Tits [27, § 2.3] for a construction of the ∗-action). For the definition of W^{ext}, see Section 4. Equivalently, X(G) is the character lattice of the generic torus T_{gen} of G. This torus is defined over a certain transcendental field extension K_{gen} of k; see [29, §4.2]. Informally speaking, we think of the Weyl group W as “the geometric part” of W^{ext}, and of the image A of the ∗-action as “the arithmetic part”. Examples 1.1 and 1.2 represent two opposite extremes, where the group W^{ext} is “purely geometric” and “purely arithmetic”, respectively. As we pass from a reductive group G to its generic torus T_{gen}, the geometric part migrates to the arithmetic part, while the overall group W^{ext} remains the same.

We are now ready to state our first main theorem.

Theorem 1.3. Let G be a reductive k-group. The following are equivalent:

(a) G is stably Cayley;

(b) for every field extension K/k, every maximal K-torus T ⊂ G_{K} is stably rational over K;

(c) the generic K_{gen}-torus T_{gen} of G is stably rational;

(d) the character lattice X(G) of G is quasi-permutation.

Next we turn our attention to classifying stably Cayley simple groups over an arbitrary field k of characteristic zero. The following results extend [20, Theorem 1.28], where k is assumed to be algebraically closed.

Theorem 1.4. Let k be a field of characteristic 0 and G be an absolutely simple k-group. Then the following conditions are equivalent:

(a) G is stably Cayley over k;

(b) G is a k-form of one of the following groups:

SL_{3}, PGL_{n} (n = 2 or n ≥ 3 odd), SO_{n} (n ≥ 5), Sp_{2n} (n ≥ 1), G_{2},

or an inner k-form of PGL_{n} (n ≥ 4 even).

Theorem 1.5. Let G be a simple (but not necessarily absolutely simple) k-group over a field k of characteristic 0. Then the following conditions are equivalent:

(a) G is stably Cayley over k;

(b) G is isomorphic to R_{l/k}(G_{1}), where l/k is a finite field extension and G_{1} is either a stably Cayley absolutely simple group over l (i.e., one of the groups listed in Theorem 1.4(b)) or an outer l-form of SO_{4}.

Here R_{l/k} denotes the Weil functor of restriction of scalars.

A key consequence of Theorem 1.3 is that, for a reductive k-group G, being stably Cayley is a property of its character lattice. If “stably Cayley” is replaced by “Cayley”, this is no longer the case, even for simple groups. Indeed, the simple groups: SU_{3} and split G_{2}, both defined over the field ℝ of real numbers, have the same character lattice; both are stably Cayley. By a theorem of Iskovskikh [16], G_{2} is not Cayley over ℝ (not even over ℂ); cf. [20, Proposition 9.10]. On the other hand, SU_{3} is Cayley, see Borovoi– Dolgachev [4, Thm. 1.2].

For reasons illustrated by the above example the problem of classifying simple Cayley groups, in a manner analogous to Theorems 1.4 and 1.5, appears to be out of reach at the moment. In particular, we do not know which outer forms of PGL_{n} (if any) are Cayley, for any odd integer n ≥ 5. Note that all the inner forms of PGL_{n} over a field k of characteristic 0 are Cayley, see [20, Example 1.11]. The outer forms of PGL_{n} for even n ≥ 4 are not stably Cayley (and hence, not Cayley) by Theorem 1.4.

The rest of this paper is structured as follows. Sections 2–6 are devoted to preliminary material on quasi-permutation lattices, automorphisms and semi-automorphisms of algebraic groups over non-algebraically closed fields, and (G, S)-fibrations. While some of this material is known, we have not been able to find references, where the definitions and results we need are proved in full generality. We have thus opted for a largely self-contained exposition.

Theorem 1.3 is proved in Section 7. Theorem 1.4 is an easy consequence of Theorem 1.3 and previously known results on character lattices of absolutely simple groups from [20] and Cortella and Kunyavskiĭ’s paper [13]; the details of this argument are presented in Section 8. The proof of Theorem 1.5 relies on new results of character lattices and thus requires considerably more work. After passing to the algebraic closure k̄, we are faced with the problem of classifying semisimple stably Cayley groups of the form G = H^{m}/C, where H is a simply connected simple group over k̄ and C ⊂ H^{m} is a central subgroup. Our classification theorem for such groups is stated in Section 9; see Theorem 9.1. The proof of Theorem 9.1, based on case-by-case analysis, occupies Section 10–18. In Section 19 we deduce Theorem 1.5 from Theorem 9.1 by passing back from k̄ to k.

2. Preliminaries on quasi-permutation lattices

Let Γ be a finite group. By a Γ-lattice we mean a finitely generated free abelian group M viewed together with an integral representation Γ → Aut(M). We also think of M as a ℤ[Γ]-module; by a morphism (or exact sequence) of lattices we mean a morphism (or exact sequence) of ℤ[Γ]-modules. When we write “lattice”, rather than “Γ-lattice”, we mean a Γ-lattice for some finite group Γ. We say that a lattice is faithful if the underlying integral representation is faithful. In those cases where we want to emphasize the dependence on Γ, we will sometimes write a lattice as a pair (Γ, M). (The integral representation Γ → Aut(M) is assumed to be clear from the context.) This notation will be particularly useful if we view M as a Γ_{0}-lattice with respect to the different subgroups Γ_{0} of Γ.

If φ: Γ →^{∼} Γ′ is an isomorphism of finite groups, then by a φ-isomorphism of lattices (Γ, L), (Γ′, L′) we will mean an isomorphism ψ: L →^{∼} L′ such that

ψ(γx) = φ(γ)ψ(x) for all γ ∈ Γ, x ∈ L.

By abuse of notation we will sometimes say that the lattices (Γ, M) and (Γ′, M′) are isomorphic instead of “φ-isomorphic” in two special cases: if Γ = Γ′ and φ = id, or (ii) (Γ, M) and (Γ′, M′) are φ-isomorphic for some

φ: Γ →^{∼} Γ′.

Now let k be a field, T_{spl} = 𝔾_{m}^{d} be the split d-dimensional k-torus, Γ be a finite group. By a multiplicative action of Γ on T_{spl} we mean an action by automorphisms of T_{spl} as an algebraic group over k. Recall that the following objects are in a natural bijective correspondence:

(i) Γ-lattices of rank d (up to isomorphism); (ii) integral representations ϕ: Γ → GL_{d}(ℤ) (up to conjugacy in GL_{d}(ℤ)); (iii) multiplicative actions Γ → Aut_{k-grp}(T_{spl}) (up to an automorphism of

T_{spl} as an algebraic k-group). A Γ-lattice L is called permutation if it has a ℤ-basis permuted by Γ. We

say that two Γ-lattices L and L′ are equivalent, and write L ∼ L′, if there exist short exact sequences

0 → L → E → P → 0  and  0 → L′ → E → P′ → 0

with the same Γ-lattice E, where P and P′ are permutation Γ-lattices. For a proof that this is indeed an equivalence relation, see [10, Lemma 8]. Note that if there exists a short exact sequence

0 → L → L′ → Q → 0,

where Q is a permutation Γ-lattice, then the trivial short exact sequence

0 → L′ → L′ → 0 → 0

shows that L ∼ L′. In particular, if P is a permutation Γ-lattice, then the short exact sequence

0 → 0 → P → P → 0

shows that P ∼ 0. If Γ-lattices L, L′, M, M′ satisfy L ∼ L′ and M ∼ M′ then L ⊕ M ∼ L′ ⊕ M′.

Definition 2.1. A Γ-lattice L is called quasi-permutation if it is equivalent to a permutation lattice, i.e., if there exists a short exact sequence

(2.1)  0 → L → P → P′ → 0,

where both P and P′ are permutation Γ-lattices.

Lemma 2.2. Let Γ_{1} ↠ Γ be a surjective homomorphism of finite groups, and let L be a Γ-lattice. Then L is quasi-permutation as a Γ_{1}-lattice if and only if it is quasi-permutation as a Γ-lattice.

Proof. It suffices to prove “only if”. Assume that L is quasi-permutation as a Γ_{1}-lattice and let Γ_{0} denote the kernel ker[Γ_{1} → Γ]. From the short exact sequence (2.1) of Γ_{1}-lattices, where P and P′ are some permutation Γ_{1}-lattices, we obtain the Γ_{0}-cohomology exact sequence

0 → L → P^{Γ_{0}} → (P′)^{Γ_{0}} → 0

(because L^{Γ_{0}} = L and H^{1}(Γ_{0}, L) = 0), which is a short exact sequence of Γ-lattices. It is easy to see that P^{Γ_{0}} and (P′)^{Γ_{0}} are permutation Γ-lattices, thus L is a quasi-permutation Γ-lattice. □

We say that a Γ-action on an algebraic variety X, defined over k, is linearizable (respectively, stably linearizable) if X is Γ-equivariantly birationally isomorphic (respectively, Γ-equivariantly stably birationally isomorphic) to a finite-dimensional k-vector space V with a linear Γ-action.

Remark 2.3. By the no-name lemma any two faithful linear actions of a finite group Γ on k-vector spaces V_{1} and V_{2} are stably Γ-equivariantly birationally equivalent; see, e.g., [20, Lemma 2.12(c)]. This makes stable linearizability a particularly natural notion.

Lemma 2.4. Let L be a Γ-lattice, and let T_{L} be the associated split k-torus with multiplicative Γ-action (i.e., X(T_{L}) = L).

(a) If L is a permutation lattice then the Γ-action on T_{L} is linearizable.

(b) L is quasi-permutation if and only if the Γ-action on T_{L} is stably linearizable.

Proof. (a) Suppose L ≃ ℤ[S] for some finite Γ-set S. Let V be the k-vector space with basis (e_{s})_{s∈S}. Then V carries a natural (permutation) Γ-action.

The morphism T_{L} → V given by

t → ∑_{s∈S} s(t)e_{s}

is easily seen to be a Γ-equivariant birational isomorphism.

(b) By Lemma 2.2 we may assume that Γ acts faithfully on L. Let P be a faithful permutation Γ-lattice (e.g., P = ℤ[Γ]). Let V be the linear representation of G constructed in part (a). It now suffices to show that the following conditions are equivalent:

(i) L is quasi-permutation,

(ii) L ∼ P,

(iii) T_{L} and T_{P} are Γ-equivariantly stably birationally isomorphic, (iv) T_{L} and V are Γ-equivariantly stably birationally isomorphic,

(v) T_{L} is stably linearizable. Indeed, (i) and (ii) are equivalent by Definition 2.1. (ii) and (iii) are equivalent by [19, Proposition 1.4]; note that, in the terminology of [19, §1.4] k(L) is precisely the field of rational functions of T_{L}.

In the proof of part (a) we showed that T_{P} and V are Γ-equivariantly birationally isomorphic. Consequently, (iii) is equivalent to (iv). Finally, (iv) ⟹ (v) by definition, and (v) ⟹ (iv) by the no-name lemma; see Remark 2.3. □

Lemma 2.5 (cf. [20], Proposition 4.8). Let W_{1}, …, W_{m} be finite groups. For each i = 1, …, m, let V_{i} be a finite-dimensional ℚ-representation of W_{i}. Set V := V_{1} ⊕ ⋯ ⊕ V_{m}. Suppose L ⊂ V is a free abelian subgroup, invariant under W := W_{1} × ⋯ × W_{m}.

If L is a quasi-permutation W -lattice, then L_{i} := L ∩ V_{i} is a quasipermutation W_{i}-lattice, for each i = 1, …, m.

Proof. It suffices to prove the lemma for i = 1. Set V′ := V/V_{1} = V_{2} ⊕ ⋯ ⊕ V_{m} and L′ = L/L_{1} ⊂ V′. Then W_{1} acts trivially on V′ and on L′, in particular, L′ is a permutation W_{1}-lattice. It follows from the short exact sequence of W_{1}-lattices

0 → L_{1} → L → L′ → 0

that the W_{1}-lattices L_{1} and L are equivalent.

Now assume that L is a quasi-permutation W -lattice. Then it is a quasipermutation W_{1}-lattice, and hence so is L_{1}. □

Lemma 2.6 (cf. [20], Lemma 4.7). Let W_{1}, …, W_{m} be finite groups. For each i = 1, …, m, let L_{i} be a W_{i}-lattice. Set W := W_{1} × ⋯ × W_{m} and construct a W -lattice L := L_{1} ⊕ ⋯ ⊕ L_{m}.

Then L is a quasi-permutation W -lattice if and only if L_{i} is a quasipermutation W_{i}-lattice for each i = 1, …, m.

Proof. The “if” assertion is obvious from the definition. The “only if” assertion follows from Lemma 2.5. □

Lemma 2.7. Let Γ be a finite group and L a Γ-lattice of rank 1 or 2. Then L is quasi-permutation.

Proof. This is easily deduced from [29, §4.9, Examples 6, 7]. □

3. Automorphisms and semi-automorphisms of split reductive groups

3.1. Notational conventions. Let G be a split reductive group over a field k. We will write T for a maximal k-torus of G, B for a Borel subgroup, Z = Z(G) for the center of G, G^{ad} for G/Z, and T^{ad} for T/Z. We identify G^{ad} with the algebraic group Inn(G) of inner automorphisms of G. If g ∈ G^{ad}(k) (or g ∈ T^{ad}(k)), we write inn(g) for the corresponding inner automorphism of G.

We will sometimes refer to a pair (T, B), where T is a split maximal ktorus and T ⊂ B ⊂ G is a Borel subgroup defined over k, as a Borel pair. It is well known that the natural action of G^{ad}(k) on the set of Borel pairs is transitive and that the stabilizer in G^{ad}(k) of a Borel pair (T, B) is T^{ad}(k).

Given a split maximal torus T ⊂ G, let RD(G, T) := (X, X^{∨}, R, R^{∨}) be the root datum of (G, T). Here X = X(T) is the character group of T, X^{∨} = Hom(X, ℤ) is the cocharacter group of T, R = R(G, T) ⊂ X is the root system of G with respect to T, and R^{∨} ⊂ X^{∨} is the coroot system of G with respect to T. The bijection R → R^{∨} sending a root to the corresponding coroot is a part of the root datum structure. For details, see [25, §1.1] or [26, §7.4].

Given a Borel pair (T, B), let BRD(G, T, B) := (X, X^{∨}, R, R^{∨}, Δ, Δ^{∨}) be the based root datum of (G, T, B). Here Δ ⊂ R is the basis of R defined by B, and Δ^{∨} ⊂ R^{∨} is the corresponding basis of R^{∨}. For details, see [25, §1.9].

The automorphism group Aut(G) is known to carry the structure of a kgroup scheme; note however, that this k-group scheme may not be of finite type. The automorphism groups Aut RD(G, T) and Aut BRD(G, T, B) are closed group subschemes of Aut(G) defined over k. These k-group schemes are discrete, in the sense that their identity components are trivial.

3.2. Semi-automorphisms. Let \overline{G} be a connected reductive group over an algebraic closure k̄ of k. We denote by SAut(\overline{G}) the group of k̄/ksemi-automorphisms of \overline{G}. We view SAut(\overline{G}) as an abstract group. For a definition of a semi-automorphism, see [3, §1.1] or [14, §1.2]. (Note that in these papers semi-automorphisms are called “semialgebraic” and “semilinear” automorphisms, respectively.) If G is a k-form of \overline{G}, then any element σ ∈ Gal(k̄/k) defines a σ-semi-automorphism σ_{∗}: \overline{G} → \overline{G}, and any semiautomorphism of \overline{G} is of the form a = α ∘ σ_{∗} where σ ∈ Gal(k̄/k) and α: \overline{G} → \overline{G} is a k̄-automorphism of the k̄-group \overline{G}.

Fix (\overline{T} , \overline{B}) as above. For any a ∈ SAut(\overline{G}) there exists g ∈ \overline{G}^{ad}(k̄) such that inn(g)(a(\overline{T}), a(\overline{B})) = (\overline{T} , \overline{B}). The semi-automorphism inn(g)a of \overline{G} defines a semi-automorphism of \overline{T} depending only on a (since the coset \overline{T}^{ad}g^{−1} is uniquely determined). The automorphism of X = X(\overline{T}) induced by inn(g)a preserves R = R(\overline{G}, \overline{T}) and \overline{B} and thus permutes the elements of the basis Δ of R defined by \overline{B}. In other words, it gives rise to an automorphism BRD(\overline{G}, \overline{T} , \overline{B}) → BRD(\overline{G}, \overline{T} , \overline{B}), depending only on a, which we denote by φ_{\overline{T} ,\overline{B}}(a).

Proposition 3.1. (a) φ_{\overline{T} ,\overline{B}}: SAut(\overline{G}) → Aut BRD(\overline{G}, \overline{T} , \overline{B}) is a group ho-

momorphism.

(b) Inn(\overline{G}) ⊂ Ker(φ_{\overline{T} ,\overline{B}}).

(c) Suppose (\overline{T}′, \overline{B}′) is another Borel pair for \overline{G}. Choose u ∈ \overline{G}^{ad}(k̄) so that (\overline{T} , \overline{B}) = inn(u)(\overline{T}′, \overline{B}′). Then the following diagram commutes

\xymatrix{ & Aut BRD(\overline{G}, \overline{T} , \overline{B}) \ar[dd]^{inn(u)^{∗}} \\ SAut(\overline{G}) \ar[dr]^{φ_{\overline{T}′,\overline{B}′}} \ar[ur]^{φ_{\overline{T} ,\overline{B}}} \\  & Aut BRD(\overline{G}, \overline{T}′, \overline{B}′).}

Moreover, the automorphism inn(u)^{∗} in this diagram is independent of the choice of u.

Proof. (a) Given a_{1}, a_{2} ∈ SAut(\overline{G}), choose g_{1}, g_{2} ∈ \overline{G}^{ad} so that inn(g_{i}) a_{i}(\overline{T} , \overline{B}) = (\overline{T} , \overline{B}). Then inn(g_{1}) (a_{1} inn(g_{2}) a_{1}^{−1}) ∈ Inn(\overline{G}); denote this inner automorphism by inn(g) for some g ∈ \overline{G}^{ad}. Then inn(g)a_{1}a_{2}(\overline{T} , \overline{B}) = (\overline{T} , \overline{B}) and thus

φ_{\overline{T} ,\overline{B}}(a_{1}a_{2}) = inn(g) a_{1}a_{2} = inn(g_{1}) a_{1} inn(g_{2}) a_{2} = φ_{\overline{T} ,\overline{B}}(a_{1}) φ_{\overline{T} ,\overline{B}}(a_{2}) .

Therefore, φ_{\overline{T} ,\overline{B}} is a homomorphism.

(b) is obvious from the definition.

(c) Let a ∈ SAut(\overline{G}). By our choice of u ∈ \overline{G}^{ad}, we have (\overline{T} , \overline{B}) = inn(u)(\overline{T}′, \overline{B}′). Choose g ∈ \overline{G}^{ad}, as before, so that inn(g) a(\overline{T} , \overline{B}) = (\overline{T} , \overline{B}). Then

inn(u^{−1}) inn(g) (a inn(u) a^{−1}) ∈ Inn(\overline{G});

denote this automorphism by inn(g′) for some g′ ∈ \overline{G}^{ad}. One readily checks that inn(g′)a(\overline{T}′, \overline{B}′) = (\overline{T}′, \overline{B}′) and thus

φ_{\overline{T}′,\overline{B}′}(a) = inn(g′) a = inn(u^{−1}) inn(g) a inn(u) = inn(u^{−1}) φ_{\overline{T} ,\overline{B}}(a) inn(u),

as desired. To prove the last assertion of part (c), note that the coset uT is independent of the choice of u. Hence, so is the map inn(u)^{∗} in the diagram. □

3.3. Automorphisms of split reductive groups.

Proposition 3.2 (cf. [24, Exposé XXIV, Thm. 1.3]). Let G be a split connected reductive group defined over k, T ⊂ G be a split maximal torus, and B ⊃ T be a Borel subgroup of G defined over k. Set \overline{G} := G ×_{k} k̄.

(a) The composite homomorphism of abstract groups

ϕ_{T,B}: Aut(\overline{G}) ↪ SAut(\overline{G}) ⟶^{φ_{\overline{T} ,\overline{B}}} Aut BRD(G, T, B)

admits a Gal(k̄/k)-equivariant splitting (homomorphic section) ψ of the form

ψ: Aut BRD(G, T, B) ↪ Aut(G, T, B) ↪ Aut(\overline{G}).

Here Aut(G, T, B) denotes the subgroup of Aut(G) consisting of automorphisms that preserve the Borel pair (T, B).

(b) The homomorphism ϕ_{T,B} of part (a) fits into a split short exact sequence of abstract groups

1 ⟶ Inn(\overline{G}) ⟶ Aut(\overline{G}) ⟶^{ϕ_{T,B}} Aut BRD(G, T, B) ⟶ 1,

which comes from a split short exact sequences of group schemes over k

(3.1)  1 ⟶ G^{ad} ⟶ Aut(G) ⟶^{ϕ} Aut BRD(G, T, B) ⟶ 1.

Note that since T is split over k, the Gal(k̄/k)-action on Aut BRD(G, T, B) is trivial.

Proof. (a) Recall that a pinning of (G, T, B) is a choice of a nonzero X_{α} ⊂ 𝔤_{α} for each α ∈ Δ, where

Lie(G) = Lie(T) ⊕ ⨁_{α∈R} 𝔤_{α}

is the root decomposition, and Δ is the basis of R = R(G, T) associated with B. By the isomorphism theorem, see [24, Exposé XXIII, Thm. 4.1] or [12, Proposition 1.5.5], the canonical homomorphism

Aut(G, T, B, (X_{α})_{α∈Δ}) → Aut BRD(G, T, B)

is an isomorphism. Composing the inverse isomorphism with the natural embeddings

Aut(G, T, B, (X_{α})_{α∈Δ}) ↪ Aut(G, T, B) ↪ Aut(G) ↪ Aut(\overline{G}),

we obtain a section ψ of ϕ_{T,B} of the desired form.

(b) See [24, Exposé XXIV, Proof of Thm. 1.3]. □

Corollary 3.3. Every abstract subgroup 𝔐 ⊂ Aut(\overline{G}), containing Inn(\overline{G}) as a subgroup of finite index, is of the form 𝔐 = M(k̄) for some linear algebraic k-group M = Inn(G) ⋊ A ⊂ Aut(G). Here A ⊂ Aut(G, T, B) is a finite k-group, all of whose k̄-points are defined over k.

Proof. Set A′ := ϕ_{T,B}(𝔐) ⊂ Aut BRD(G, T, B). Then A′ is a finite algebraic k-group all of whose k̄-points are defined over k. Set M = ϕ^{−1}(A′) ⊂ Aut(G), where ϕ: Aut(G) → Aut BRD(G, T, B) is a homomorphism of group k-schemes, as in (3.1). Then M is a group k-scheme and M(k̄) = 𝔐. Set A = ψ(A′) ⊂ Aut(G, T, B), where ψ is the splitting of Proposition 3.2, then M = Inn(G) ⋊ A. Since M has finitely many connected components, and the identity component G^{ad} of M is an affine k-group, we conclude that M is affine as well. In other words, M is a linear algebraic k-group, as desired. □

4. The character lattice and the generic torus

Throughout this section G will denote a connected reductive k-group, not necessarily split, and T ⊂ G will denote a maximal k-torus. We write \overline{G} := G ×_{k} k̄, \overline{T} = T ×_{k} k̄, and choose a Borel subgroup \overline{B} ⊃ \overline{T} of \overline{G}.

4.1. The character lattice of a reductive group.

Definition 4.1. (a) We define A_{T,\overline{B}} to be the image of the composite homomorphism

Gal(k̄/k) ↪ SAut(\overline{G}) ⟶^{φ_{\overline{T} ,\overline{B}}} Aut BRD(\overline{G}, \overline{T} , \overline{B}) ↪ Aut X(\overline{T}).

(b) We define the extended Weyl group W^{ext}(G, T, \overline{B}) by

W^{ext}(G, T, \overline{B}) := W (\overline{G}, \overline{T}) · A_{T,\overline{B}} ⊂ Aut X(\overline{T}).

Note that W^{ext}(G, T, \overline{B}) is a subgroup of Aut RD(\overline{G}, \overline{T}) (and thus of Aut X(\overline{T})), because W (\overline{G}, \overline{T}) is normal in Aut RD(\overline{G}, \overline{T}). We call the pair (W^{ext}(G, T, \overline{B}), X(\overline{T})) the character lattice of G.

Remark 4.2. Let T′ ⊂ G be another maximal k-torus, and \overline{B}′ ⊃ \overline{T}′ be a Borel subgroup of \overline{G}. Then it is easy to see that for u as in Proposition 3.1(c), the isomorphism inn(u)^{∗}: X(\overline{T}) → X(\overline{T}′) induces an isomorphism of groups

A_{\overline{T} ,\overline{B}} →^{∼} A_{\overline{T}′,\overline{B}′}

and an isomorphism of lattices (W^{ext}(G, T, \overline{B}), X(\overline{T})) →^{∼} (W^{ext}(G, T′, \overline{B}′)), X(\overline{T}′)). In other words, the character lattice (W^{ext}(\overline{G}, \overline{T} , \overline{B}), X(\overline{T})) is defined uniquely up to a canonical isomorphism.

Moreover, if T = T′ then A_{T,\overline{B}′} = wA_{T,\overline{B}}w^{−1} for some w ∈ W (\overline{G}, \overline{T}). Thus different choices of \overline{B} give rise to the same (and not just isomorphic) subgroups W^{ext}(G, T, \overline{B}) or Aut X(\overline{T}). For this reason we will write W^{ext}(G, T) in place of W^{ext}(G, T, \overline{B}) from now on.

Lemma 4.3. W^{ext}(G, T) = W (\overline{G}, \overline{T})·im λ_{T}, where λ_{T} : Gal(k̄/k) → Aut X(\overline{T}) is the usual action of the Galois group on the characters of T.

Proof. By the definition of φ_{\overline{T} ,\overline{B}}, for any σ ∈ Gal(k̄/k) there exists w_{σ} ∈ W (\overline{G}, \overline{T}) such that φ_{\overline{T} ,\overline{B}}(σ) = w_{σ} λ_{T} (σ). □

Corollary 4.4. Suppose G is a connected reductive k-group, T is a maximal k-torus, and K/k is a field extension such that k is algebraically closed in K. Then W^{ext}(G, T) = W^{ext}(G_{K}, T_{K}) as subgroups of Aut X(\overline{T}) = Aut X(T\overline{K}).

Proof. By Lemma 4.3, it suffices to show that im λ_{T} = im λ_{T_{K}}. Since T splits over k̄, the action of the Galois group Gal(\overline{K}/K) on X(\overline{T}) factors through the natural homomorphism Gal(\overline{K}/K) → Gal(k̄/k). By [18, Thm. VI.1.12], this homomorphism is surjective. Thus im λ_{T} = im λ_{T_{K}}, as desired. □

Lemma 4.5. Let T be a maximal k-torus of G and \overline{B} ⊃ \overline{T} be a Borel subgroup of \overline{G}. Then W^{ext}(G, T) is a semi-direct product: W^{ext}(G, T) = W (\overline{G}, \overline{T}) ⋊ A_{T,\overline{B}}.

Proof. Since A_{T,\overline{B}} ⊂ Aut BRD(\overline{G}, \overline{T} , \overline{B}), every element of A_{T,\overline{B}} preserves the basis Δ of R(\overline{G}, \overline{T}) corresponding to \overline{B}, while in W (\overline{G}, \overline{T}) only the identity element 1 preserves Δ. Thus W (\overline{G}, \overline{T}) ∩ A_{T,\overline{B}} = \{1\}. By definition W (\overline{G}, \overline{T}) · A_{T,\overline{B}} = W^{ext}(G, T), and the lemma follows. □

4.2. The generic torus. Let T be a maximal k-torus of G and T_{gen} be the generic torus of G. Recall that T_{gen} is defined over the field K_{gen} := k(G/N_{G}(T)), where N_{G}(T) denotes the normalizer of T in G. For details of this construction, see [29, §4.2]. For notational simplicity we will write K in place of K_{gen} for the remainder of this section.

Proposition 4.6. Let G be a connected reductive k-group and T be a maximal k-torus. Then

(a) the image 𝔄 of Gal(\overline{K}/K) in Aut X(\overline{T_{gen}}) coincides with W^{ext}(G_{K}, T_{gen}).

(b) The character lattice (𝔄, X(\overline{T_{gen}})) of the generic torus is isomorphic to the character lattice of G.

If G is semisimple then the proposition is an immediate consequence of a theorem of Voskresenskiĭ’s [29, Theorem 4.2.2]; cf. Lemma 4.5.

Proof. (a) We claim that the image of the Galois group Gal(\overline{K}/k̄K) in Aut X(\overline{T_{gen}}) coincides with the Weyl group W (\overline{G_{K}}, \overline{T_{gen}}). If G is semisimple this is Theorem 4.2.1 in [29]. In the general case, we consider the derived subgroup G^{der} = [G, G] of G which is a connected semisimple group. Consider the radical R of G (the identity component of the center). Since G is reductive, R is a k-torus. The generic torus T_{gen} of G and the generic torus T′_{gen} ⊂ G_{K}^{der} of G^{der} are defined over the same field K = k(G/N_{G}(T)) = k(G^{der}/N_{G^{der}}(T ∩ G^{der})). Note that T_{gen} = T′_{gen} · R_{K} and T′_{gen} ∩ R_{K} is finite. Hence, there is a canonical isomorphism

X(T_{gen}) ⊗ ℚ = X(T′_{gen}) ⊗ ℚ ⊕ X(R\overline{K}) ⊗ ℚ ,

where X(T_{gen}) stands for X(\overline{T_{gen}}). Let

ρ: Gal(\overline{K}/k̄K) → Aut X(T_{gen}) ⊗ ℚ,
ρ′: Gal(\overline{K}/k̄K) → Aut X(T′_{gen}) ⊗ ℚ

be the corresponding actions. Since R_{K} splits over k̄K, the Galois group Gal(\overline{K}/k̄K) acts trivially on X(R\overline{K}). Hence, for every σ ∈ Gal(\overline{K}/k̄K) we have

ρ(σ) = (ρ′(σ), 1) ∈ Aut X(T′_{gen}) ⊗ ℚ × Aut X(R\overline{K}) ⊗ ℚ ⊂ Aut X(T_{gen}) ⊗ ℚ.

By Voskresenskiĭ’s theorem [29, Theorem 4.2.1], we have im ρ′ = W (Gder ; \overline{K}, T′_{gen}) and hence

im ρ = W (Gder ; \overline{K}, T′_{gen}) × \{1\} = W (G\overline{K}, T_{gen}).

This proves the claim.

Now recall that by Lemma 4.3, W^{ext}(G_{K}, T_{gen}) is generated by 𝔄 and W (\overline{G_{K}}, T_{gen}). The claim tells us that, in fact, W (\overline{G_{K}}, T_{gen}) ⊂ 𝔄. Hence, W^{ext}(G_{K}, T_{gen}) = 𝔄.

(b) Consider two maximal tori in G_{K}, T_{gen} and T_{K} = T ×_{k} K. By Remark 4.2 the lattices

(W^{ext}(G_{K}, T_{gen}), X(\overline{T_{gen}})) and (W^{ext}(G_{K}, T_{K}), X(\overline{T_{K}}))

are isomorphic. By part (a), the character lattice (𝔄, X(\overline{T_{gen}})) of the generic torus coincides with (W^{ext}(G_{K}, T_{gen}), X(\overline{T_{gen}})). On the other hand, since the k-variety G/N_{G}(T) is absolutely irreducible, k is algebraically closed in K = k(G/N_{G}(T)). Thus by Corollary 4.4, (W^{ext}(G_{K}, T_{K}), X(\overline{T_{K}})) coincides with (W^{ext}(G, T), X(\overline{T})), which is the character lattice of G. □

5. Forms of reductive groups

Let G_{spl} be a split connected reductive k-group. Recall that any k-form G of G_{spl} is k-isomorphic to a twisted group _{z}G_{spl} for some cocycle z ∈ Z^{1}(k, Aut(\overline{G_{spl}})). Sending z to _{z}G_{spl} gives rise to a natural bijective correspondence between the non-abelian Galois cohomology set H^{1}(k, Aut(\overline{G_{spl}})) and the isomorphism classes of k-forms of G_{spl}. For details on this, see e.g. [26, §§11.3 and 12.3].

5.1. Choosing a “small” cocycle. Let G be a connected reductive kgroup, not necessarily split. Let T ⊂ G be a maximal torus, and let \overline{B} ⊃ \overline{T} be a Borel subgroup. Let G_{spl} be a split k-form of G. We choose and fix a k̄-isomorphism θ: \overline{G_{spl}} → \overline{G}. Choose a Borel pair (T_{spl}, B_{spl}) in G_{spl}. After composing θ with an inner automorphism of \overline{G}, we may (and shall) assume that θ takes (\overline{T_{spl}}, \overline{B_{spl}}) to (\overline{T} , \overline{B}). Then θ induces isomorphisms Aut(\overline{G}) → Aut(\overline{G_{spl}}), BRD(\overline{G}, \overline{T} , \overline{B}) → BRD(\overline{G_{spl}}, \overline{T_{spl}}, \overline{B_{spl}}) = BRD(G_{spl}, T_{spl}, B_{spl}), etc.

Definition 5.1. Let G, G_{spl} and θ be as above. Let A_{T,\overline{B}} denote the image of Gal(k̄/k) in Aut BRD(\overline{G}, \overline{T} , \overline{B}), as in Definition 4.1, it is a finite group. Note that θ induces an isomorphism

θ_{∗}: Aut BRD(\overline{G}, \overline{T} , \overline{B}) →^{∼} Aut BRD(G_{spl}, T_{spl}, B_{spl}).

Set ^{θ}A := θ_{∗}(A_{T,\overline{B}}) ⊂ Aut BRD(G_{spl}, T_{spl}, B_{spl}). We define M_{G} ⊂ Aut(G_{spl}) to be the preimage of the finite group ^{θ}A in Aut(G_{spl}) under

ϕ: Aut(G_{spl}) → Aut BRD(G_{spl}, T_{spl}, B_{spl}) ;

see exact sequence (3.1) of Proposition 3.2(b) (for G_{spl}). Then M_{G} is an algebraic group defined over k; see Corollary 3.3. Set

θW^{ext} := θ_{∗}(W^{ext}(G, T)) ⊂ Aut X(T_{spl}) ,

so that ^{θ}W^{ext} = W (G_{spl}, T_{spl}) ·^{θ}A. Note that the group ^{θ}W^{ext} acts multiplicatively (i.e., by group automorphisms) on the split k-torus T_{spl}.

Proposition 5.2. With the notation of Definition 5.1, G is isomorphic to zG_{spl} for some cocycle z ∈ Z^{1}(k, M_{G}).

Proof. For σ ∈ Gal(k̄/k) denote by β(σ) the semi-automorphism of \overline{G} and by β_{spl}(σ) the semi-automorphism of \overline{G_{spl}} induced by σ. Under the usual correspondence between k-forms of G_{spl} and H^{1}(k, Aut(\overline{G_{spl}})), G is k-isomorphic to _{z}G, for the cocycle z(σ) := ^{θ}β(σ) ∘ β_{spl}(σ)^{−1} : \overline{G_{spl}} → \overline{G_{spl}}, where ^{θ}β(σ) is the image of β(σ) under the isomorphism Aut(\overline{G}) ⟶^{≃} Aut(\overline{G_{spl}}) induced by θ.

It remains to show that z(σ) ∈ M_{G}(k̄), or equivalently, z_{BRD}(σ) := φ_{\overline{T_{spl}},\overline{B_{spl}}} ∘ z(σ) lies in ^{θ}A, for every σ ∈ Gal(k̄/k). Consider the diagram

\xymatrix{ & SAut(\overline{G}) \ar[dd]_{≃} \ar[r]^{φ_{\overline{T} ,\overline{B}}} & Aut BRD(\overline{G}, \overline{T} , \overline{B}) \ar[dd]_{≃}^{θ_{∗}} \\ Gal(k̄/k) \ar@{^{(}->}[ur]^{β} \ar@{^{(}->}[dr]^{β_{spl}} \\  & SAut(\overline{G_{spl}}) \ar[r]^{φ_{\overline{T_{spl}},\overline{B_{spl}}}} & Aut BRD(\overline{G_{spl}}, \overline{T_{spl}}, \overline{B_{spl}}),}

where the vertical isomorphisms are induced by θ. The commutativity of this diagram tells us that

z_{BRD}(σ) = θ_{∗}(γ(σ)) ∘ γ_{spl}(σ)^{−1},

where γ := φ_{\overline{T} ,\overline{B}} ∘β and γ_{spl} := φ_{\overline{T_{spl}},\overline{B_{spl}}} ∘β_{spl} denote the actions of Gal(k̄/k) on BRD(\overline{G}, \overline{T} , \overline{B}) and on BRD(\overline{G_{spl}}, \overline{T_{spl}}, \overline{B_{spl}}) = BRD(G_{spl}, T_{spl}, B_{spl}), respectively. Since the Galois group Gal(k̄/k) acts trivially on BRD(G_{spl}, T_{spl}, B_{spl}), we see that γ_{spl}(σ) = id and z_{BRD}(σ) = θ_{∗}(γ(σ)). By definition, γ(σ) ∈ A_{T,\overline{B}}. Thus z_{BRD}(σ) ∈ θ_{∗}(A_{T,\overline{B}}) = ^{θ}A, as desired. □

5.2. Forms of Cayley groups.

Lemma 5.3. Let G be a split reductive k-group, M be a closed k-subgroup of Aut(G) containing Inn(G), and z ∈ Z^{1}(k, M).

(a) If there exists an M-equivariant birational isomorphism f: G −−→ Lie(G), then _{z}G is a Cayley group.

(b) If there exists an M-equivariant birational isomorphism f: G ×_{k} 𝔸^{r} −−→ Lie(G) ×_{k} 𝔸^{r} for some r ≥ 0, where M acts trivially on the affine space 𝔸^{r}, then _{z}G is a stably Cayley group.

(c) If G is Cayley, then any inner form of G is also Cayley.

(d) If G is stably Cayley, then any inner form of G is also stably Cayley.

Proof. (a) Since f is M-equivariant, we can twist f by z and obtain an zM-equivariant birational isomorphism

zf: _{z}G −−→_{z} Lie(G) .

By functoriality of the twisting operation, _{z}Inn(G) = Inn(_{z}G) ⊂ _{z}M ([26, Lemma 16.4.6]) and _{z}Lie(G) = Lie(_{z}G). Thus _{z}f is an _{z}M-equivariant (and, in particular, Inn(_{z}G)-equivariant) rational map _{z}G −−→ Lie(_{z}G). Twisting f^{−1} by z in a similar manner, we see that _{z}f is, in fact, a birational isomorphism, i.e., a Cayley map for _{z}G.

(b) Replace G by G × 𝔾_{m}^{r} and apply part (a).

(c) An inner form of G is, by definition, a twisted form _{z}G, where z ∈ Z^{1}(k, Inn(G)). If G is a Cayley group, then there exists an Inn(G)-equivariant birational isomorphism f: G −−→ Lie(G), hence by (a), _{z}G is a Cayley group.

(d) If G is a stably Cayley group, then G ×_{k} 𝔾_{m}^{r} is Cayley for some r, and we may identify Inn(G) with Inn(G ×_{k} 𝔾_{m}^{r}). If z ∈ Z^{1}(k, Inn(G)) = Z^{1}(k, Inn(G×_{k}𝔾_{m}^{r})), then by (b), the twisted group_{z}(G×_{k}𝔾_{m}^{r}) = _{z}G×_{k}𝔾_{m}^{r} is Cayley, hence _{z}G is stably Cayley. □

6. (G, S)-fibrations and (G, S)-varieties

The proof of Theorem 1.3 in the next section relies on the notions of (G, S)-fibration and (G, S)-variety. This section will be devoted to preliminary material on these notions.

6.1. (G, S)-fibrations. Let G be a linear algebraic k-group and S be a ksubgroup. Recall that a (G, S)-fibration is a morphism of k-varieties π: X → Y, where G acts on X on the left, π is constant on G-orbits, and after a surjective étale base change Y′ → Y there is a G-equivariant isomorphism between G/S ×_{k} Y′ and X ×_{Y} Y′ over Y′, cf. [9, §2.2]. If S = {1}, then a (G, S)-fibration is the same thing as a left G-torsor. Note that in general, X → Y can be both a (G, S_{1})-fibration and a (G, S_{2})-fibration for nonisomorphic k-subgroups S_{1}, S_{2} ⊂ G. However over an algebraic closure of k, S_{1} and S_{2} become conjugate.

The following lemma generalizes well-known properties of torsors to the category of (G, S)-fibrations.

Lemma 6.1. Let π: X → Y, π_{1} : X_{1} → Y_{1} and π_{2} : X_{2} → Y_{2} be (G, S)- fibrations.

(a) Every G-equivariant morphism f: X_{1} → X_{2} is a morphism of (G, S)- fibrations, i.e., gives rise to a Cartesian diagram

\xymatrix{X_{1} \ar[d]_{π_{1}} \ar[r]^{f} & X_{2} \ar[d]^{π_{2}} \\ Y_{1} \ar[r]^{\overline{f}} & Y_{2} .}

In other words, X_{1} = X_{2} ×_{Y_{2}} Y_{1}, where the G-action on X_{2} ×_{Y_{2}} Y_{1} is induced by the G-action on X_{2}.

(b) Every G-invariant closed (respectively, open) subvariety X_{0} ⊂ X is of the form π^{−1}(Y_{0}) for some closed (respectively, open) subvariety Y_{0} of Y. In particular, X_{0} is itself the total space of a (G, S)-fibration π_{|X_{0}} : X_{0} → Y_{0}.

(c) The map f in part (a) is dominant if and only if \overline{f} is dominant.

Proof. (a) We first define the map \overline{f}: Y_{1} → Y_{2} locally in the étale topology on Y_{1}. Let {U_{α}} be an étale open cover of Y_{1} such that X_{1} is G-equivariantly isomorphic to G/S ×_{k} U_{α}, over each U_{α}. Then over each U_{α}, the map π_{1} has a section s_{α}: U_{α} → π_{1}^{−1}(U_{α}), and we can define \overline{f} by composing s, f and π_{2}. The resulting local map is independent of the choice of s; these maps patch up to a k-morphism \overline{f}: Y_{1} → Y_{2} by étale descent.

By the universal property of fibered products there exists a morphism ϕ: X_{1} → X_{2} ×_{Y_{2}} Y_{1} over Y_{1}. This morphism is unique and hence, Gequivariant. Thus it suffices to show that ϕ is an isomorphism. Note that ϕ is a G-equivariant morphism between (G, S)-fibrations over Y_{1}. We want to show that if Y_{1} = Y_{2} and \overline{f} = id in the above diagram then f is an isomorphism. We do this by constructing f^{−1}. Let {U_{α}} be an étale local cover of Y_{1}, trivializing both X_{1} and X_{2}. That is, over each U_{α}, X_{1} and X_{2} are both G-equivariantly isomorphic to G/S ×_{k} U_{α}. Hence, f^{−1} is (uniquely) defined and is G-equivariant over each U_{α}. Once again, using étale descent, we see that these local inverses patch together to a well-defined G-equivariant k-morphism f^{−1} : X_{2} → X_{1}.

(b) Since open subsets are complements of closed subsets, it suffices to consider the case where X_{0} is closed. We claim that π(X_{0}) is closed in Y. It is enough to check this claim locally in the étale topology, so we may assume that X = G/S ×_{k} Y and π is the projection onto the second factor. Since X_{0} is G-equivariant, X_{0} contains {1} ×_{k} π(X_{0}). Moreover, since X_{0} is closed, X_{0} contains \{1\} × \overline{π(X_{0})}. We conclude that \overline{π(X_{0})} is contained in π(X_{0}), i.e., π(X_{0}) is closed, as claimed.

After replacing Y by π(X_{0}) and X by π^{−1}(π(X_{0})), it now suffices to show that if X_{0} ⊂ X is closed and G-invariant and π(X_{0}) = Y then X_{0} = X. To do this, we construct the inverse to the inclusion map X_{0} ↪ X. We first do this étale-locally, where we may assume X = G/S ×_{k} Y and hence, X_{0} = X, then use étale descent to patch together local inverses into a morphism X → X_{0} defined over Y.

(c) By part (b), the closure of f(X_{1}) in X_{2} is of the form π_{2}^{−1}(C) for some closed subset C ⊂ Y_{2}. Thus f is dominant if and only if C = Y_{2}, that is, if and only if \overline{f} is dominant. □

Let N := N_{G}(S) be the normalizer of S in G, W := N/S, and X → Y be a (G, S)-fibration. Note that W is again a linear algebraic group over k. Denote the S-fixed point locus in X by X^{S}. The G-action on X induces an N-action on X^{S}. Since S acts trivially on X^{S}, this N-action descends to a W -action on X^{S}. By trivializing the (G, S)-fibration X → Y over an étale cover Y′ → Y, we see that X^{S} → Y is in fact a W -torsor; see [9, Proposition 2.9]. Conversely, starting with a W -torsor Z → Y, we can build a (G, S)-fibration X → Y by setting X to be the “homogeneous fiber space” G ×^{N} Z, i.e., the quotient of G ×_{k} Z by the left N-action given by n · (g, x) → (gn^{−1}, nx). This quotient can either be constructed locally, in the étale topology on Y, by descent, or globally as a geometric quotient in the sense of geometric invariant theory. For details on these constructions, we refer the reader to [9, §2.2].

Proposition 6.2. Let V ar_{k} be the category of quasi-projective varieties, and F ib_{(G,S)} be the functor from V ar_{k} to the category of sets which associates to a quasi-projective variety Y the set of isomorphism classes of (G, S)- fibrations over Y, and to a k-morphism of varieties Ỹ → Y the pull-back morphism which base-changes (G, S)-fibrations over Y to Ỹ. If S = {1}, we will write T or_{G} in place of F ib_{(G,S)}.

Then the two constructions described above give rise to an isomorphism between the functors F ib_{(G,S)} and T or_{W}.

Proof. See [9, Proposition 2.10]. □

6.2. (G, S)-varieties. A k-variety X with a left action of G is called a (G, S)-variety if it contains a dense open subset X′ ⊂ X which is the total space of a (G, S)-fibration X′ → Y.

Lemma 6.3. Let G be a reductive k-group, T ⊂ G a maximal k-torus, and M be an algebraic subgroup of the group k-scheme Aut(G) containing Inn(G). Then G and its Lie algebra Lie(G) are both (M, T^{ad})-varieties.

In the case where M = Inn(G), the lemma was proved in [9, Proposition 4.3].

Proof. Being a (G, S)-variety is a geometric notion. That is, suppose k′/k is a field extension. Then X is a (G, S)-variety over k if and only if X_{k′} is a (G_{k′}, S_{k′})-variety over k′. Thus, after replacing k by a suitable k′, we may assume that G and T are split.

We will only consider the M-action on G; the case of the M-action on Lie(G) is similar. By Corollary 3.3, M = Inn(G) ⋊ A, where A is a finite group of automorphisms of G and every element of A preserves T.

Our proof will rely on [9, Proposition 2.16]. To apply this proposition we need to check that the M-action on G is stable, i.e., the M-orbit of x ∈ G(k̄) is closed for x in general position. By [9, Corollary 4.2], the conjugation action of G on itself is stable. Since A is a finite group, the group M contains G^{ad} as a subgroup of finite index, and therefore the Maction on G is also stable.

By [9, Proposition 2.15(i)], we can now conclude that G is an (M, S)- variety for some subgroup S ⊂ M. Moreover, by [9, Proposition 2.16], in order to show that we may take S = T^{ad}, it suffices to exhibit a dense subset D ⊂ G(k) defined over k such that the stabilizer of every p ∈ D in M is conjugate to T^{ad}.

In fact, it suffices to construct a dense open subset U ⊂ T defined over k such that the stabilizer of every p ∈ U(k) is conjugate to T^{ad}; we can then take D to be the union of Inn(G)-translates of U(k).

Consider the set T^{reg} of regular points of T. By [2, §12.2], T^{reg} is a dense open subset of T defined over k. We claim that for t ∈ T^{reg} in general position, Stab_{M}(t) = T^{ad}. Indeed, suppose g ∈ M stabilizes t. Since t lies in a unique maximal torus of G (see [2, Proposition 12.2(4)]), g(T) = T. Equivalently, g lies in N_{G^{ad}}(T^{ad}) ⋊ A ⊂ M. The latter group acts on T via its finite quotient W ⋊ A, and the W ⋊ A-action on T is faithful (see the proof of Lemma 4.5). The fixed points of each element of W ⋊ A form a proper closed subvariety of T^{reg}. Removing these closed subvarieties from T^{reg}, we obtain a dense open subset U ⊂ T such that Stab_{W ⋊A}(t) = {1} or equivalently, Stab_{M}(t) = T^{ad} for every t ∈ U, as desired. □

Proposition 6.4. Suppose X_{1} and X_{2} are (G, S)-varieties such that the fixed point loci X_{1}^{S} and X_{2}^{S} are irreducible. Set N := N_{G}(S), W := N/S. Then

(a) every G-equivariant dominant rational map α: X_{1} −−→ X_{2} restricts to a W -equivariant dominant rational map β: X_{1}^{S} −−→ X_{2}^{S}.

(b) Every W -equivariant dominant rational map β: X_{1}^{S} −−→ X_{2}^{S} lifts to a unique G-equivariant dominant rational map α: X_{1} −−→ X_{2}.

(c) Moreover, β is a birational isomorphism if and only if so is α.

Proof. For i = 1, 2 let X′_{i} be a G-invariant dense open subset of X_{i} which is the total space of a (G, S)-fibration, X′_{i} → Y_{i}. Since each X_{i}^{S} is irreducible, the non-empty open subset (X′_{i})^{S} is dense in X_{i}^{S}. Hence, the dominant rational map X_{1}^{S} −−→ X_{2}^{S} restricts to a dominant rational map (X′_{1})^{S} −−→ (X′_{2})^{S}, and we may, without loss of generality, replace X_{i} by X′_{i} and thus assume that X_{i} is the total space of a (G, S)-fibration X_{i} → Y_{i}. Lemma 6.1(b) now tells us that after removing a proper closed subset from Y_{1} (and its preimages from X_{1} and X_{1}^{S}), we may assume that f is regular. By Proposition 6.2, X_{i}^{S} → Y is a W -torsor for i = 1, 2. By Lemma 6.1(a), α is a morphism of (G, S)-fibrations, and β = α_{| X_{1}^{S}} : X_{1}^{S} → X_{2}^{S} is a morphism of W -torsors. By Proposition 6.2, X_{i}^{S} → Y is a W -torsor for i = 1, 2. We thus obtain the following diagram

\xymatrix{X_{1} \ar[dd] \ar[rrr]^{α} &  &  & X_{2} \ar[dd] \\  & X_{1}^{S} \ar@{^{(}->}[ul] \ar[dl] \ar[r]^{β} & X_{2}^{S} \ar[dr] \ar@{^{(}->}[ur] \\ Y_{1} \ar[rrr]^{\overline{α}=\overline{β}} &  &  & Y_{2} .}

By Proposition 6.2, α restricts to β and β lifts to α in a unique way. Moreover, α and β induce the same morphism \overline{α} = \overline{β}: Y_{1} → Y_{2}.

By Lemma 6.1(c), α is dominant if and only if \overline{α} = \overline{β} is dominant if and only if β is dominant. This proves (a) and (b).

(c) If α is a birational isomorphism, then restricting α^{−1} to X_{1}^{S}, we obtain an inverse for β. Similarly, if β is a birational isomorphism, then extending β^{−1} to X_{2} −−→ X_{1}, we obtain an inverse for α. □

Corollary 6.5. Let G be a connected reductive k-group and T ⊂ G be a maximal k-torus. Then G is Cayley if and only if there exists a W (G, T)- equivariant birational isomorphism T −−→^{≃} Lie(T) defined over k.

Note that here, as before, we view the Weyl group W (G, T) as an algebraic group over k.

Proof. By Lemma 6.3, with M = Inn(G), X_{1} = G and X_{2} = Lie(G) are both (Inn(G), T^{ad})-varieties. The fixed point loci, X_{1}^{T^{ad}} = T and X_{2}^{T^{ad}} = Lie(T), are irreducible. The desired conclusion is now a direct consequence of Proposition 6.4: there exists a G-equivariant birational isomorphism

α: G = X_{1} −−→^{≃} X_{2} = Lie(G)

(i.e., a Cayley map for G) if and only if there exists a W (T)-equivariant birational isomorphism β: T = X_{1}^{T^{ad}} −−→^{≃} X_{2}^{T^{ad}} = Lie(T). □

7. Proof of Theorem 1.3

(a) ⟹ (b). First suppose G is Cayley over k. Then G_{K} is Cayley over K for every field extension K/k. Then by Corollary 6.5, every maximal K-torus T of G_{K} is K-rational.

Now suppose G is stably Cayley over k, i.e., G × 𝔾_{m}^{r} is Cayley for some r ≥ 0. Then the above argument shows that for every K-torus T of G, T × 𝔾_{m}^{r} is K-rational. Hence, T is stably K-rational, as claimed.

(b) ⟹ (c) is obvious.

(c) ⟺ (d). By Proposition 4.6, the character lattice X(G) of G is isomorphic to the character lattice of the generic torus T_{gen} of G. Since a torus T is stably rational if and only if its character lattice X(T) is quasi-permutation (see [29, Theorem 4.7.2]), (c) and (d) are equivalent.

(d) ⟹ (a). Let G_{spl} be a split k-form of G. Let (T_{spl}, B_{spl}) be a Borel pair in G_{spl} defined over k, T be a maximal k-torus of G, and \overline{B} be a Borel subgroup defined over the algebraic closure k̄ and containing \overline{T}. We choose and fix an isomorphism θ: \overline{G_{spl}} → \overline{G} taking (\overline{T_{spl}}, \overline{B_{spl}}) to (\overline{T} , \overline{B}), and we construct the subgroup M_{G} ⊂ Aut(G_{spl}) using θ, as in Subsection 5.1. By Proposition 5.2, G is isomorphic to _{z}G_{spl} for some cocycle z ∈ Z^{1}(k, M_{G}). By Lemma 5.3(b), in order to show that G is stably Cayley, it suffices to construct an M_{G}-equivariant birational isomorphism

(7.1)  G_{spl} × 𝔾_{m}^{r} −−→^{≃} Lie(G_{spl}) × 𝔸^{r}

for some r ≥ 0, where M_{G} acts trivially on the split torus 𝔾_{m}^{r} and the affine space 𝔸^{r}. By Lemma 6.3, X_{1} := G_{spl} × 𝔾_{m}^{r} and X_{2} := Lie(G_{spl}) × 𝔸^{r} are both (M_{G}, S)-varieties, where S := (T_{spl})^{ad}. By Proposition 6.4, in order to construct an M_{G}-equivariant birational isomorphism (7.1), it suffices to construct an N_{M_{G}}(S)/S-equivariant birational isomorphism X_{1}^{S} −−→ X_{2}^{S}, where X_{1}^{S} = T_{spl} × 𝔾_{m}^{r}, X_{2}^{S} = Lie(T_{spl}) × 𝔸^{r}. Note that N_{M_{G}}(S)/S is isomorphic to the group ^{θ}W^{ext} ⊂ Aut X(T_{spl}) (see Subsection 5.1).

It thus remains to show that there exists a ^{θ}W^{ext}-equivariant birational isomorphism

(7.2)  T_{spl} × 𝔾_{m}^{r} −−→^{≃} Lie(T_{spl}) × 𝔸^{r}

for some r ≥ 0. By the definition of ^{θ}W^{ext}, the lattice (^{θ}W^{ext}, X(T_{spl})) is isomorphic to the character lattice (V (G, T), X(\overline{T})) of G. By condition (d) of the theorem, the character lattice of G is quasi-permutation, hence so is the lattice (^{θ}W^{ext}, X(T_{spl})). By Lemma 2.4(b), this implies that the ^{θ}W^{ext}- action on the split torus T_{spl} is stably linearizable. In other words, T_{spl} is θW^{ext}-equivariantly stably birationally isomorphic to a faithful linear representation V of the finite group ^{θ}W^{ext}. On the other hand, by Remark 2.3, the vector space V is ^{θ}W^{ext}-equivariantly stably birationally isomorphic to Lie(T_{spl}). Composing these two ^{θ}W^{ext}-equivariant birational isomorphisms, we see that T_{spl} and Lie(T_{spl}) are ^{θ}W^{ext}-equivariantly stably birationally isomorphic. In other words, (7.2) holds for a suitable r ≥ 0, as claimed. This completes the proof of Theorem 1.3. □

Corollary 7.1. Let k be a field of characteristic 0. Then every reductive k-group G of rank ≤ 2 is stably Cayley.

Proof. By Lemma 2.7 the character lattice of G is quasi-permutation. Thus G is stably Cayley by Theorem 1.3.

Alternatively, the generic torus of G is of dimension ≤ 2 and hence, is rational; see [29, §4.9, Examples 6, 7]. Once again, we conclude that G is stably Cayley by Theorem 1.3. □

8. Proof of Theorem 1.4

To show that (a) ⟹ (b), suppose G is stably Cayley over k. Then Gk̄ is stably Cayley over k̄, where k̄ denotes an algebraic closure of k. By [20, Theorem 1.28], Gk̄ is one of the following groups:

(8.1)  SL_{3}, SO_{n} (n ≠ 2, 4), Sp_{2n} (n ≥ 1), PGL_{n} (n ≥ 2), G_{2}.

In other words, G is a k-form of one of these groups. (Note that the group SL_{2}, which appears in the statement of [20, Theorem 1.28], is isomorphic to Sp_{2}.) If G is an outer form of PGL_{n} where n ≥ 4 is even, then by [13, Theorem 0.1] the generic torus of G is not stably rational, and by Theorem 1.3, G is not stably Cayley. Thus if G is stably Cayley, then G is one of the groups listed in part (b).

It remains to prove that (b) ⟹ (a), i.e., that all groups listed in part (b) are stably Cayley.

The classical Cayley transform shows that all forms of SO_{n} and Sp_{2n} are Cayley; see [20, Example 1.16]. All forms of the groups SL_{3} and G_{2} are of rank 2, hence their generic tori are rational by [29, Example 4.9.7], and by Theorem 1.3, these groups are stably Cayley. Every inner form of PGL_{n} is Cayley by [20, Example 1.11]; cf. also Lemma 5.3(c). Finally, the generic torus of any form of PGL_{n} for n odd is rational, hence stably rational by [30, Corollary of Theorem 8]. By Theorem 1.3, we conclude that outer forms of PGL_{n} for n odd are stably Cayley. This completes the proof of Theorem 1.4. □

9. Statement of Theorem 9.1 and first reductions

In view of Theorem 1.4 it is natural to ask for a classification of stably Cayley semisimple groups, initially over an algebraically closed field of characteristic zero. This problem turns out to be significantly more complicated; a complete solution is out of reach at the moment; cf. Remark 9.3. Fortunately, for the purpose of proving Theorem 1.5, we can limit our attention to semisimple groups all of whose simple components are of the same type. Theorem 9.1 stated below gives a classification of stably Cayley groups of this form; this theorem will be a key ingredient in our proof of Theorem 1.5 in Section 19. The proof of Theorem 9.1 will occupy much of the remainder of this paper.

Theorem 9.1. Let k be an algebraically closed field of characteristic 0 and G be a semisimple k-group of the form H^{m}/C, where H is a simple and simply connected k-group and C is a central k-subgroup of H^{m}. (In other words, the universal cover of G is of the form H^{m}.) Then G is stably Cayley if and only if G is isomorphic to a direct product G_{1} ×_{k} ⋯ ×_{k} G_{s}, where each G_{i} is either a stably Cayley simple k-group (i.e., is one of the groups listed in (8.1)) or SO_{4}.

Note that SO_{4} is semisimple but not simple. The “if” direction of Theorem 9.1 is obvious, since the direct product of stably Cayley groups is stably Cayley. (As we mentioned in the previous section, SO_{4} is Cayley via the classical Cayley transform.) Thus we only need to prove the “only if” direction. The proof will proceed by case-by-case analysis, depending on the type of H. We begin with the following easy reduction.

Lemma 9.2. Let H be a simply connected simple group over an algebraically closed field k and C be a central subgroup of H^{m} for some m ≥ 1. Let H_{i} denote the i^{th} factor of H^{m}, π_{i} denote the natural projection H^{m} → H_{i}, and C_{i} := π_{i}(C) ⊂ Z(H_{i}), where Z(H_{i}) denotes the center of H_{i}. Assume H^{m}/C is stably Cayley. Then

(a) H_{i}/C_{i} is stably Cayley;

(b) H is of type A_{n} (n ≥ 1), B_{n} (n ≥ 2), C_{n} (n ≥ 3) , D_{n} (n ≥ 4), or G_{2}.

Proof. Part (a) is a direct consequence of [20, Prop. 4.8]. To prove part (b), note that by [20, Thm. 1.28], H_{1}/C_{1} is of one of the types listed in the statement of the lemma. □

We will now settle two easy cases of Theorem 9.1, where H is of type C_{n} (n ≥ 3) and G_{2}.

Proof of Theorem 9.1 for H = G_{2}. Here Z(H) = {1}, so C ⊂ Z(H)^{m} is trivial, and

H^{m}/C = H^{m} = G_{2} ×_{k} ⋯ ×_{k} G_{2} (m times)

is a product of stably Cayley simple groups. □

Proof of Theorem 9.1 for H of type C_{n} (n ≥ 3). Let H = Sp_{2n} and C be a subgroup of Z(H)^{m} = μ_{2}^{m}. We will show that if H^{m}/C is stably Cayley, then C = {1}.

Indeed, if H^{m}/C is stably Cayley, then, by Lemma 9.2, so is H_{i}/C_{i}. Here H_{i} = Sp_{2n}, and C_{i} is a central subgroup (either μ_{2} or {1}). On the other hand, by [20, Theorem 1.28], if the group Sp_{2n} /C_{i} is stably Cayley for some n ≥ 3 then C_{i} = {1}. Thus C projects trivially to every H_{i}, which is only possible if C = {1}. We conclude that

H^{m}/C = H^{m} = Sp_{2n} ×_{k} ⋯ ×_{k} Sp_{2n} (m times)

is a product of Cayley simple groups, as desired. □

Remark 9.3. We conjecture that Theorem 9.1 remains true for every semisimple k-group G over an algebraically closed field k of characteristic 0, without any additional assumption on the universal cover of G. That is, a semisimple k-group is stably Cayley, if and only if it is isomorphic to a direct product G_{1} ×_{k} ⋯ ×_{k} G_{s}, where each G_{i} is either a stably Cayley simple group or SO_{4}.

10. Quasi-invertible lattices

The proof of the “only if” direction of Theorem 9.1 in the remaining cases, where H is of type A_{n}, B_{n} or D_{n}, is more involved. In this section, in preparation for this proof, we will describe a general method for showing that certain lattices are not quasi-permutation (and more generally, cannot even be direct summands of quasi-permutation lattices). Our approach is originally due to Voskresenskiĭ. Proposition 10.6 is essentially [28, Theorem 7 and its corollary]; see also [10, Proposition 1(ii)] and [11, Proposition 9.5(ii)]. For the sake of completeness we supply short proofs for Lemmas 10.4 and 10.5 below.

Definition 10.1. A Γ-lattice L is called quasi-invertible if it is a direct summand of a quasi-permutation Γ-lattice.

Lemma 10.2 (J.-L. Colliot-Thélène). A Γ-lattice L is quasi-invertible if and only if it fits into a short exact sequence

(10.1)  0 → L → P → I → 0,

where P is a permutation Γ-lattice and I is an invertible Γ-lattice, i.e. a direct summand of a permutation Γ-lattice.

Proof. For a Γ-lattice L we have a flasque resolution

0 → L → P → F → 0,

where P is a permutation Γ-lattice and F is a flasque Γ-lattice, see [10, § 1] or [21, Ch. 2] for the theory of flasque resolutions. We write [L]^{fl} for the class of F up to addition of a permutation lattice (note that F is defined up to addition of a permutation lattice). We have [L ⊕ L′]^{fl} = [L]^{fl} ⊕ [L′]^{fl}. If L is quasi-invertible, then L ⊕ L′ is quasi-permutation for some L′, hence [L]^{fl} ⊕ [L′]^{fl} = [L ⊕ L′]^{fl} = 0. We see that [L]^{fl} is invertible, hence L fits into an exact sequence (10.1) with I invertible.

Conversely, if L fits into an exact sequence (10.1) with I invertible, say I ⊕ J = P′ is permutation, then adding I to (10.1) twice on the left, and then adding J twice on the right, we obtain an exact sequence

0 → L ⊕ I → P ⊕ I ⊕ J → I ⊕ J → 0,

which shows that L is quasi-invertible. □

Lemma 10.3 (J.-L. Colliot-Thélène). Let Γ_{1} ↠ Γ be a surjective homomorphism of finite groups, and let L be a Γ-lattice. Then L is quasi-invertible as a Γ_{1}-lattice if and only if it is quasi-invertible as a Γ-lattice.

Proof. We argue as in the proof of Lemma 2.2. It suffices to prove “only if”. Assume that L is quasi-invertible as a Γ_{1}-lattice, then by Lemma 10.2, L fits into a short exact sequence (10.1) of Γ_{1}-lattices, where P is a permutation Γ_{1}-lattice and I is an invertible Γ_{1}-lattice. Set Γ_{0} = ker[Γ_{1} → Γ]. From (10.1) we obtain the Γ_{0}-cohomology exact sequence

0 → L → P^{Γ_{0}} → I^{Γ_{0}} → 0

(because L^{Γ_{0}} = L and H^{1}(Γ_{0}, L) = 0), which is a short exact sequence of Γ-lattices. It is easy to see that P^{Γ_{0}} is a permutation Γ-lattice and I^{Γ_{0}} is an invertible Γ-lattice, hence by Lemma 10.2, L is a quasi-permutation Γ-lattice. □

Suppose (Γ, L) and (Γ′, L′) are φ-isomorphic for some isomorphism φ: Γ → Γ′; for a definition of φ-isomorphism, see the beginning of Section 2. Then clearly L is permutation (respectively, quasi-permutation, respectively, quasiinvertible) if and only if so is L′.

The Tate–Shafarevich group of a Γ-lattice L is defined as

� ^{2}(Γ, L) = ker [H^{2}(Γ, L) → ∏_{Γ_{c}⊂Γ} H^{2}(Γ_{c}, L)] ,

where Γ_{c} runs over the set of all cyclic subgroups of Γ. If L is a quasiinvertible Γ-lattice, then for any subgroup Γ′ ⊂ Γ we have � ^{2}(Γ′, L) = 0, cf. [21, Prop. 2.9.2(a)]. Note however, that there exist Γ-lattices L such that � ^{2}(Γ′, L) = 0 for every subgroup Γ′ of Γ but L is not quasi-invertible; see the end of the proof of Prop. 12.4.

The following lemmas can be used to show that a given lattice is not quasiinvertible. Let Γ be a finite group. Consider the norm homomorphism

N_{Γ} : ℤ → ℤ[Γ], N_{Γ}(a) = a ∑_{s∈Γ} s for a ∈ ℤ,

and the short exact sequence

(10.2)  0 → ℤ → ℤ[Γ] → J_{Γ} → 0,

where J_{Γ} = coker N_{Γ}.

Lemma 10.4. Let Γ be a finite group, and Γ′ ⊂ Γ any subgroup. Then � ^{2}(Γ′, J_{Γ}) ≅ H^{3}(Γ′, ℤ).

Proof. From (10.2) we obtain a cohomology exact sequence

(10.3)  H^{2}(Γ′, ℤ[Γ]) → H^{2}(Γ′, J_{Γ}) → H^{3}(Γ′, ℤ) → H^{3}(Γ′, ℤ[Γ]).

We have H^{i}(Γ′, ℤ[Γ′]) = 0 for i ≥ 1, hence H^{i}(Γ′, ℤ[Γ]) = 0 for i ≥ 1, and we see from (10.3) that H^{2}(Γ′, J_{Γ}) ≅ H^{3}(Γ′, ℤ).

Now let Γ_{c} ⊂ Γ′ be a cyclic subgroup. We have H^{2}(Γ_{c}, J_{Γ}) ≅ H^{3}(Γ_{c}, ℤ). By periodicity for cyclic groups, cf. [1, IV.8, Thm. 5], we have

H^{3}(Γ_{c}, ℤ) ≅ H^{1}(Γ_{c}, ℤ) = Hom(Γ_{c}, ℤ) = 0.

Thus H^{2}(Γ_{c}, J_{Γ}) = 0 and consequently, � ^{2}(Γ′, J_{Γ}) = H^{2}(Γ′, J_{Γ}) ≅ H^{3}(Γ′, ℤ). □

Lemma 10.5. Let Γ = ℤ/pℤ×ℤ/pℤ, where p is a prime. Then H^{3}(Γ, ℤ) ≅ ℤ/pℤ.

Proof. For any group Γ, the group H^{3}(Γ, ℤ) is canonically isomorphic to H^{2}(Γ, ℂ^{×}). The latter group is called the Schur multiplier of Γ. For finite abelian groups, the Schur multipliers were computed by Schur in [23, §4, VIII]. In particular, by [23, §4, VIII], the Schur multiplier of ℤ/pℤ × ℤ/pℤ is a cyclic group of order p, which proves the lemma.

An alternative proof based on modern references proceeds as follows. For any finite group Γ, the group H^{3}(Γ, ℤ) is dual to H^{−3}(Γ, ℤ), cf. [7, Thm. XII.6.6] or [6, Thm. VI.7.4]. By definition H^{−3}(Γ, ℤ) = H_{2}(Γ, ℤ). For an abelian group Γ we have H_{2}(Γ, ℤ) = Λ^{2}(Γ) (the second exterior power of the ℤ-module Γ), see [22, Thm. 3] or [6, Thm. V.6.4(c)]. Clearly Λ^{2}(ℤ/pℤ × ℤ/pℤ) ≅ ℤ/pℤ, hence H_{2}(ℤ/pℤ×ℤ/pℤ, ℤ) ≅ ℤ/pℤ and H^{3}(ℤ/pℤ×ℤ/pℤ, ℤ) ≅ ℤ/pℤ. □

As an immediate consequence, we obtain the following

Proposition 10.6. Let Γ = ℤ/pℤ × ℤ/pℤ, where p is a prime. Then � ^{2}(Γ, J_{Γ}) ≅ ℤ/pℤ, and therefore the Γ-lattice J_{Γ} is not quasi-invertible.

□

The following example and subsequent proposition will be used in the proof of Theorem 1.5 in Section 19.

Example 10.7. Let H be an outer k-form of PGL_{n} for some even integer n ≥ 4. Recall (see Section 8) that by [13, Theorem 0.1] the character lattice of H is not quasi-permutation. In fact, it is shown in [13, §5.1] that the character lattice of H is not quasi-invertible. Indeed, let T be a maximal k-torus of H. Note that W^{ext}(H, T) = S_{n} ×ℤ/2ℤ and X(\overline{T}) = ℤA_{2n−1} on which W^{ext}(H, T) acts by permutations and sign changes. It is shown in [13, §5.1] that there exists a subgroup Γ of W^{ext}(H, T) isomorphic to ℤ/2ℤ × ℤ/2ℤ, and a direct summand M of the Γ-lattice X(\overline{T}) isomorphic to J_{Γ}. Then � ^{2}(Γ, M) ≠ 0 and so � ^{2}(Γ, X(\overline{T})) ≠ 0. This implies that the W^{ext}(H, T)-lattice X(\overline{T}) is not quasi-invertible.

Proposition 10.8. Let k be a field of characteristic zero and H be a reductive algebraic k-group with maximal k-torus T such that the character lattice X(H) = (W^{ext}(H, T), X(\overline{T})) is not quasi-invertible. Then G := H × H′ is not stably Cayley for any reductive algebraic k-group H′.

Proof. Let T, T′ and S = T × T′ be maximal k-tori in H, H′ and G, respectively. By Theorem 1.3, it suffices to show that the character lattice X(G) = (W^{ext}(G, S), X(\overline{S})) is not quasi-invertible (and hence, not quasipermutation). By Lemma 4.3, the extended Weyl group W^{ext}(G, S) is generated by the Weyl group W (G, S) = W (H, T) × W (H′, T′) and the image of the natural action λ_{S}: Gal(k̄/k) → Aut(X(\overline{S})). Since both T and T′ are defined over k, the image of λ_{S} preserves the direct sum decomposition

X(\overline{S}) = X(\overline{T}) ⊕ X(\overline{T}′)

and hence, so does W^{ext}(G, S). Moreover, W^{ext}(G, S) acts on X(\overline{T}) via a surjection π: W^{ext}(G, S) → W^{ext}(H, T). By Lemma 10.3, X(\overline{T}) is not quasiinvertible as a W^{ext}(G, S) lattice and, since X(\overline{T}) is a direct summand of X(\overline{S}), we conclude that that X(\overline{S}) is not quasi-invertible as a W^{ext}(G, S)- lattice. In other words, the character lattice X(G) = (W^{ext}(G, S), X(\overline{S})) is not quasi-invertible, and therefore G is not stably Cayley, as desired. □

11. A family of non-quasi-invertible lattices

We will now use the results of Section 10 to exhibit a large family of non-quasi-invertible lattices (i.e., lattices that are not direct summands of quasi-permutation lattices). These lattices will be used to complete the proof of Theorem 9.1.

Let Δ be a Dynkin diagram, Δ = ⋃_{i=1}^{m} Δ_{i}, where Δ_{i} are the connected components of Δ. We assume that each Δ_{i} is of type B_{l_{i}} (l_{i} ≥ 1) or of type D_{l_{i}} (l_{i} ≥ 3). Note that B_{1} = A_{1} and D_{3} = A_{3} are allowed. The root system R(Δ_{i}) can be realized in a standard way in the space V_{i} := ℚ^{l_{i}} with standard basis (ε_{s})_{s∈S_{i}}, where S_{i} is an index set consisting of l_{i} elements, see [5, Planches II, IV].

Let S = ˙⋃S_{i} (disjoint union). Consider the vector space V := ⨁_{i} V_{i} over ℚ with standard basis (ε_{s})_{s∈S}. Set

(11.1)  β = \frac{1}{2} ∑_{s∈S} ε_{s} .

We denote by M the additive subgroup in V generated by β and by the basis elements ε_{s} for all s ∈ S. In other words, M is generated by the vectors of the form \frac{1}{2} ∑_{s∈S} ±ε_{s}.

Denote the Weyl group W (Δ_{i}) by W_{i} and the Weyl group W (Δ) = ∏_{i=1}^{m} W_{i} by W. Consider the natural action of W on M. For s ∈ S_{i} let c_{s} denote the automorphism of V_{i} acting as −1 on ε_{s} and as 1 on all the other ε_{t} (t ∈ S_{i}, t ≠ s). The Weyl group W_{i} = W (Δ_{i}) is the semidirect product of the symmetric group S_{l_{i}}, acting by permutations of the basis vectors ε_{s}, and an abelian group Θ_{i}. If Δ_{i} ≅ B_{l_{i}}, then Θ_{i} = ⟨c_{s}⟩_{s∈S_{i}}, in particular c_{s} ∈ W_{i}. If Δ_{i} ≅ D_{l_{i}}, then Θ_{i} = ⟨c_{s}c_{s′}⟩_{s,s′∈S_{i}}. In this case c_{s} ∉ W_{i}, but c_{s}c_{s′} ∈ W_{i}.

Proposition 11.1. Let Δ, S, M, and W be as above. Assume that |Δ| ≥ 3. Then the W -lattice M is not quasi-invertible.

Remark 11.2. Note that rank(M) = dim(V) = |Δ|. If |Δ| = 1 or 2 then M is quasi-permutation by Lemma 2.7.

Proof. First we consider the case Δ ≅ D_{4}. Then M is not quasi-permutation, see [13, §7.1]. We will show that M is not quasi-invertible. Indeed, in [13, §7.1] the authors construct a subgroup U ⊂ W of order 8^{1}, such that M restricted to U is a direct sum of U-sublattices M = M_{1} ⊕ M_{3} of ranks 1 and 3, respectively. Now in [17, Theorem 1] it is stated that the U-lattice M_{3} is not quasi-permutation, but it is actually proved that [M_{3}]^{fl} is not invertible. Hence M_{3} is not a quasi-invertible U-lattice, and M is not a quasi-invertible W -lattice.

1This group of order 8 is actually denoted by W_{2} in [13]. We use the symbol U here to avoid a notational clash with the Weyl group W_{2} := W (Δ_{2}).

From now on we will assume that Δ ≅ D̸_{4}. Let Γ = ℤ/2ℤ × ℤ/2ℤ = {e, γ_{1}, γ_{2}, γ_{3}}. Then by Proposition 10.6, � ^{2}(Γ, J_{Γ}) ≅ ℤ/2ℤ. The idea of our proof is to construct an embedding

(11.2)  ι: Γ → W

in such a way that M, restricted to ι(Γ), is isomorphic to a direct sum of a submodule M_{0} ≃ J_{Γ} and |S| − 3 Γ-lattices of rank 1. This will imply that

� ^{2}(Γ, M) = � ^{2}(Γ, M_{0}) = � ^{2}(Γ, J_{Γ}) = ℤ/2ℤ ≠ 0,

and hence M is not quasi-invertible. We will now fill in the details of this argument in two steps.

Step 1. Construction of the embedding (11.2). We begin by partitioning each S_{i} for i = 1, …, m into three (non-overlapping) subsets S_{i,1}, S_{i,2} and S_{i,3}, subject to the requirement that

(11.3)  |S_{i,1}| ≡ |S_{i,2}| ≡ |S_{i,3}| ≡ l_{i} (mod 2), if Δ_{i} is of type D_{l_{i}}.

We then set U_{1} to be the union of the S_{i,1}, U_{2} to be the union of the S_{i,2}, and U_{3} to be the union of the S_{i,3}, as i ranges from 1 to m.

Lemma 11.3. If |S| ≥ 3 and Δ ≅ D̸_{4} then the subsets S_{i,1}, S_{i,2} and S_{i,3} of S_{i} can be chosen, subject to (11.3), so that U_{1}, U_{2}, U_{3} ≠ ∅.

To prove the lemma, note that if one of the Δ_{i}, say Δ_{1}, is of type D_{l}, where l ≥ 3 is odd, then we partition S_{1} into three non-empty sets of odd order. If m ≥ 2 then we partition S_{i} with i ≥ 2 as follows:

(11.4)  S_{i,1} = S_{i,2} = ∅ and S_{i,3} = S_{i}.

Clearly U_{1}, U_{2}, U_{3} ≠ ∅, as desired.

Similarly, if one of the Δ_{i}, say Δ_{1}, is D_{l}, where l ≥ 6 is even, then we partition S_{1} into three non-empty sets of even order, and partition the other S_{i} (if any) as in (11.4) for i ≥ 2. Once again, U_{1}, U_{2}, U_{3} ≠ ∅.

If one of the Δ_{i}, say Δ_{1}, is of type D_{4}, then by our assumption m ≥ 2. We can now partition S_{1} so that each of S_{1,1} and S_{1,2} has 2 elements and S_{1,3} = ∅, and partition S_{i} as in (11.4) for every i ≥ 2. Once again, U_{1}, U_{2}, U_{3} ≠ ∅.

Thus we may assume without loss of generality that every Δ_{i} is of type B_{l_{i}}. In this case condition (11.3) doesn’t come into play and the lemma is obvious. This completes the proof of Lemma 11.3. □

We now define the embedding ι of (11.2) by

ι(γ_{κ}) = ∏_{s∈S∖U_{κ}} c_{s} ∈ Aut(M) for κ = 1, 2, 3.

Recall that Γ = ℤ/2ℤ × ℤ/2ℤ = {e, γ_{1}, γ_{2}, γ_{3}}. One easily checks that the map ι: Γ → Aut(M) defined this way is a group homomorphism. By (11.3), its image is, in fact, in W. Moreover, since U_{κ} ≠ ∅ for all κ = 1, 2, 3, we have S ∖ U_{κ} ≠ ∅, hence ι(γ_{κ}) ≠ id, i.e., ι: Γ → W is injective. We identify Γ with ι(Γ) ⊂ W.

Step 2. Construction of the submodule M_{0}. Now let

β_{κ} := γ_{κ}(β) = \frac{1}{2} ( ∑_{s∈U_{κ}} ε_{s} − ∑_{s∈S∖U_{κ}} ε_{s})

for κ = 1, 2, 3, where β is as in (11.1). Since the set β, β_{1}, β_{2}, β_{3} is the orbit of β under Γ, the sublattice M_{0} := Span_{ℤ}(β, β_{1}, β_{2}, β_{3}) ⊂ M is Γ-invariant. Note that

(11.5)  β + β_{κ} = ∑_{s∈U_{κ}} ε_{s} .

Since U_{1}, U_{2} and U_{3} are non-empty and disjoint, β + β_{1}, β + β_{2}, and β + β_{3} are linearly independent. On the other hand,

β + β_{1} + β_{2} + β_{3} = 0.

Therefore, the Γ-invariant sublattice M_{0} ⊂ M is of rank 3 and is isomorphic (as a Γ-lattice) to J_{Γ} := ℤ[Γ]/ℤ.

It remains to show that M can be written as a direct sum of M_{0} and Γ-lattices of rank 1. Indeed, for each κ = 1, 2, 3 choose an element u_{κ} ∈ U_{κ} and set U′_{κ} = U_{κ} ∖ {u_{κ}}. (Note that U′_{κ} may be empty for some κ). We set S′ = U′_{1} ∪ U′_{2} ∪ U′_{3}. It follows from (11.5) that the abelian group generated by the ε_{s}, as s ranges over S′, together with β, β_{1}, β_{2}, β_{3}, contains both β and ε_{s} for every s ∈ S and hence, coincides with M. Since rank(M) = |S|, we conclude that the set {β, β_{1}, β_{2}} ∪ {ε_{s} | s ∈ S′} is a basis of M. The group Γ acts on ε_{s} by ±1. We see that the Γ-lattice M is a direct sum of M_{0} = Span_{ℤ}(β, β_{1}, β_{2}) and the Γ-lattices ℤe_{s} of rank 1, as s ranges over S′. Thus

� ^{2}(Γ, M) = � ^{2}(Γ, M_{0}) = � ^{2}(Γ, J_{Γ}) = ℤ/2ℤ,

and therefore M is not a quasi-invertible W -lattice, as desired. □

12. More non-quasi-invertible lattices

In this section we continue to create a stock of non-quasi-invertible lattices which will be used in the proof of Theorem 9.1.

Proposition 12.1. Let M = {(a_{1}, a_{2}, a_{3}) ∈ ℤ^{3} | a_{1} + a_{2} + a_{3} ≡ 0 (mod 2)} be the W := (ℤ/2ℤ)^{3}-lattice with the action of (ℤ/2ℤ)^{3} on M ⊂ ℤ^{3} coming from the non-trivial action of ℤ/2ℤ on ℤ. Then M is not quasi-invertible.

Proof. Let ε_{1}, ε_{2}, ε_{3} be the standard basis of ℤ^{3}. For i = 1, 2, 3 let c_{i} ∈ W denote the automorphism of M taking ε_{i} to −ε_{i} and taking each of the other two ε_{j} to itself. Set σ = c_{2}c_{3}, τ = c_{1}c_{2}, ρ = c_{1}c_{2}c_{3}. We consider the following basis of M:

e_{1} = ε_{2} − ε_{1}, e_{2} = ε_{2} − ε_{3}, e_{3} = −ε_{1} − ε_{3}.

A direct calculation shows that in this new basis {e_{1}, e_{2}, e_{3}}, the generators σ, τ, ρ of W are given by the following matrices:

σ = ( \begin{matrix}0 & 0 & 1 \\ −1 & −1 & −1 \\ 1 & 0 & 0 \end{matrix} ) , τ = ( \begin{matrix}−1 & −1 & −1 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , ρ = ( \begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ) .

By [17, Theorem 1, case W_{2}], our W -lattice M is not quasi-permutation. Moreover, the pair (W, M) is isomorphic to (U, M_{3}), where M_{3} is the nonquasi-invertible U-lattice we mentioned at the beginning of the proof of Proposition 11.1. Therefore, M is not quasi-invertible. □

Let ℤD_{3} denote the root lattice of D_{3}. Recall that

ℤD_{3} = {a_{1}ε_{1} + a_{2}ε_{2} + a_{3}ε_{3} | a_{i} ∈ ℤ, a_{1} + a_{2} + a_{3} ∈ 2ℤ} ⊂ ℚ^{3},

where {ε_{1}, ε_{2}, ε_{3}} is the standard basis of ℚ^{3} = ℚD_{3}. The set

{ε_{1} + ε_{2}, ε_{1} − ε_{2}, ε_{2} − ε_{3}}

is a basis of ℤD_{3}.

Let m ≥ 2. We consider (ℤD_{3})^{m} ⊂ (ℚD_{3})^{m}. Let L ⊂ (ℚD_{3})^{m} be the lattice generated by (ℤD_{3})^{m} and the vector

v_{e} := ε_{1} + ε_{4} + ε_{7} + ⋯ + ε_{3m−2}.

The group W (D_{3})^{m} acts on L.

Proposition 12.2. For m ≥ 2, the W (D_{3})^{m}-lattice L constructed above is not quasi-invertible.

Proof. We consider the subgroup Γ ⊂ W (D_{3})^{m} of order 4 generated by the following two commuting elements of order 2:

a =(12) c_{4}c_{5} c_{7}c_{8} … c_{3m−2}c_{3m−1},
b =c_{1}c_{2} (45).

Here c_{i} takes ε_{i} to −ε_{i}. Thus Γ = {e, a, b, ab} ⊂ W (D_{3})^{m}. We show that � ^{2}(Γ, L) = ℤ/2ℤ. Indeed, let V = (ℚD_{3})^{m} with the basis ε_{1}, …, ε_{3m}. Let V_{0} be the sub-

space of V spanned by

ε_{1}, ε_{2}, ε_{4}, ε_{5}, …, ε_{3m−2}, ε_{3m−1}.

It is Γ-invariant. Set L_{0} = L ∩ V_{0}. Clearly L/L_{0} injects into V/V_{0}. Since Γ acts trivially on V/V_{0}, we see that L/L_{0} ≅ ℤ^{m} with trivial Γ-action. Thus we have a short exact sequence of Γ-lattices

0 → L_{0} → L → ℤ^{m} → 0.

Since ℤ^{m} is a permutation Γ-lattice, we see that

� ^{2}(Γ, L) ≅ � ^{2}(Γ, L_{0}).

We prove that � ^{2}(Γ, L_{0}) = ℤ/2ℤ.

For γ ∈ Γ we set v_{γ} = γ · v_{e}. If m > 2 we set

δ = ε_{7} + ε_{10} + ⋯ + ε_{3m−2}.

If m = 2 we set δ = 0. We obtain

v_{e} = ε_{1} + ε_{4} + δ,
v_{a} = ε_{2} − ε_{4} − δ,
v_{b} = −ε_{1} + ε_{5} + δ,
v_{ab} = −ε_{2} − ε_{5} − δ.

Clearly

v_{e} + v_{a} + v_{b} + v_{ab} = 0.

Set M_{0} = ⟨v_{e}, v_{a}, v_{b}, v_{ab}⟩, then M_{0} ≅ J_{Γ} := ℤ[Γ]/ℤ, and by Proposition 10.6 we have � ^{2}(Γ, M_{0}) = ℤ/2ℤ.

Set β_{1} = v_{e}, β_{2} = v_{a}, β_{3} = v_{b}. We set

β_{4} = ε_{4} − ε_{5},
β_{5} = ε_{7} + ε_{8},
β_{6} = ε_{7} − ε_{8},
. . . . . . . . . . . .
β_{2m−1} = ε_{3m−2} + ε_{3m−1},
β_{2m} = ε_{3m−2} − ε_{3m−1}.

By Lemma 12.3 below, the set β := {β_{1}, …, β_{2m}} is a basis of L_{0}. We have M_{0} = ⟨β_{1}, β_{2}, β_{3}⟩. Our Γ-lattice L_{0} decomposes into a direct sum of Γ-sublattices

L_{0} = M_{0} ⊕ ⟨β_{4}⟩ ⊕ ⋯ ⊕ ⟨β_{2m}⟩.

For 4 ≤ i ≤ 2m the Γ-lattice ⟨β_{i}⟩ is of rank 1, hence quasi-permutation, and therefore � ^{2}(Γ, ⟨β_{i}⟩) = 0. It follows that � ^{2}(Γ, L_{0}) = � ^{2}(Γ, M_{0}) = ℤ/2ℤ, hence � ^{2}(Γ, L) = ℤ/2ℤ. Thus L is not a quasi-invertible W (D_{3})^{m}- lattice. □

Lemma 12.3. The set β := {β_{1}, …, β_{2m}} is a basis of L_{0}.

Proof. First note that β ⊂ L_{0}. Since the set β has 2m elements and the lattice L_{0} is of rank 2m, it suffices to show that β generates L_{0}.

Recall that L_{0} = L ∩ V_{0} and that L is generated by (ℤD_{3})^{m} and v_{e}. Since v_{e} ∈ V_{0}, we see that L_{0} is generated by v_{e} and (ℤD_{3})^{m} ∩ V_{0}. Since v_{e} = β_{1} ∈ β, it suffices to prove that (ℤD_{3})^{m}∩V_{0} ⊂ ⟨β⟩. Clearly (ℤD_{3})^{m}∩V_{0} is generated by the vectors

ε_{1} + ε_{2} , ε_{1} − ε_{2}, ε_{4} + ε_{5}, ε_{4} − ε_{5}, …, ε_{3m−2} + ε_{3m−1}, ε_{3m−2} − ε_{3m−1}.

Note that all the vectors in this list starting with ε_{4}−ε_{5} are clearly contained in β. It remains to show that the vectors ε_{1}+ε_{2} , ε_{1}−ε_{2}, ε_{4}+ε_{5} are contained in ⟨β⟩.

Note that 2δ ∈ ⟨β⟩ (because 2ε_{7} ∈ ⟨β⟩, …, 2ε_{3m−2} ∈ ⟨β⟩). We have

β_{1} + β_{2} = v_{e} + v_{a} = ε_{1} + ε_{2},

hence ε_{1} + ε_{2} ∈ ⟨β⟩. We have

β_{1} + β_{3} = v_{e} + v_{b} = ε_{4} + ε_{5} + 2δ,

hence ε_{4} + ε_{5} ∈ ⟨β⟩. Since also ε_{4} − ε_{5} ∈ β ⊂ ⟨β⟩, we see that 2ε_{4} ∈ ⟨β⟩. We have

β_{1} − β_{2} = v_{e} − v_{a} = ε_{1} − ε_{2} + 2ε_{4} + 2δ,

hence ε_{1}−ε_{2} ∈ ⟨β⟩. We conclude that (ℤD_{3})^{m}∩V_{0} ⊂ ⟨β⟩, hence β generates L_{0} and is a basis of L_{0}. This completes the proofs of Lemma 12.3 and of Proposition 12.2. □

We will now consider the root system A_{n−1}, which is embedded in ℤ^{n}, see [5, Planche I]. Let ℤA_{n−1} denote the root lattice of A_{n−1}, and let α_{1}, α_{2}, …, α_{n−1} denote the standard basis of the root system A_{n−1} and of ℤA_{n−1} (loc. cit). Let Λ_{n} denote the weight lattice of A_{n−1}, and let ω_{1}, ω_{2}, …, ω_{n−1} denote the standard basis of Λ_{n} consisting of fundamental weights (loc. cit).

Consider ℤA_{2} ⊂ Λ_{3}. The nontrivial automorphism σ of the basis Δ = {α_{1}, α_{2}} = {ε_{1} −ε_{2}, ε_{2} −ε_{3}} (loc. cit) induces the automorphism (−1)∘(1, 3) of ℤA_{2} (where −1 ∈ Aut ℤ ⊂ Aut ℤ^{3}, (1, 3) ∈ S_{3} ⊂ Aut ℤ^{3}), and an automorphism σ_{∗} of S_{3} = W (A_{2}) (namely, the conjugation by the transposition (1, 3)).

Let m ≥ 2. We consider (ℤA_{2})^{m} ⊂ (Λ_{3})^{m}. Let (ℤA_{2})^{(i)} ⊂ Λ_{3}^{(i)} be the i^{th} factor. Let ω_{1}^{(i)}, ω_{2}^{(i)} be the basis of Λ_{3}^{(i)} consisting of fundamental weights.

Let a = (a_{1}, …, a_{m}) (a row vector), where each a_{i} equals 1 or 2. In particular, let 1_{m} = (1, …, 1). Let L_{a} denote the (S_{3})^{m}-lattice generated by (ℤA_{2})^{m} and the vector

x_{a} := ∑_{i=1}^{m} a_{i}ω_{1}^{(i)}.

Proposition 12.4. For m ≥ 2 and for any a as above (i.e., each a_{i} equals 1 or 2), the (S_{3})^{m}-lattice L_{a} is not quasi-invertible.

Proof. First we note that L_{a} is ϕ-isomorphic to L_{1_{m}} with respect to some automorphism φ of (S_{3})^{m} (for a definition of φ-isomorphism, see the beginning of Section 2).

Indeed, let α_{1}, α_{2} be the standard basis of the root system A_{2} (and of ℤA_{2}). Let

ω_{1} = \frac{1}{3}(2α_{1} + α_{2}), ω_{2} = \frac{1}{3}(α_{1} + 2α_{2})

be the fundamental weights, this is the standard basis of Λ_{3} (loc. cit.). Let \overline{ω}_{1}, \overline{ω}_{2} be their images in Λ_{3}/ℤA_{2} ≅ ℤ/3ℤ. Since

ω_{1} + ω_{2} = α_{1} + α_{2} ∈ ℤA_{2},

we have \overline{ω}_{1} + \overline{ω}_{2} = 0, hence \overline{ω}_{2} = 2\overline{ω}_{1}. Thus the nontrivial automorphism σ of the Dynkin diagram A_{2} takes \overline{ω}_{1} to \overline{ω}_{2} = 2\overline{ω}_{1} when acting on Λ_{3}/ℤA_{2}.

Now let a be as above. Write Δ = (A_{2})^{m}, Δ = Δ_{1} ∪ ⋯ ∪ Δ_{m}. For each i = 1, …, m we define an automorphism τ_{i} of Δ_{i} = A_{2}. If a_{i} = 1, we set τ_{i} = id, while if a_{i} = 2, we set τ_{i} = σ_{i}, where σ_{i} is the nontrivial automorphism of Δ_{i}. Then the automorphism τ = ∏_{i} τ_{i} of Δ = (A_{2})^{m} acts on (Λ_{3})^{m} and takes L_{1_{m}} to L_{a}. We see that the (S_{3})^{m}-lattices L_{1_{m}} and L_{a} are τ_{∗}-isomorphic, where τ_{∗} is the induced automorphism of (S_{3})^{m}. Thus, in order to prove that the (S_{3})^{m}-lattice L_{a} is not quasi-invertible, it suffices to show that L_{1_{m}} is not quasi-invertible.

Let α_{1}^{(i)}, α_{2}^{(i)} be the standard basis of (ℤA_{2})^{(i)}. Let ω_{1}^{(i)}, ω_{2}^{(i)} be the standard basis of Λ_{3}^{(i)}, then

ω_{1}^{(i)} = \frac{1}{3}(2α_{1}^{(i)} + α_{2}^{(i)}).

Let α_{1}, …, α_{3m−1} be the standard basis of ℤA_{3m−1}. We denote by λ_{1}, …, λ_{3m−1} (rather than ω_{1}, …, ω_{3m−1}) the standard basis of Λ_{3m} consisting of fundamental weights. Then we have (loc. cit.)

(12.1) λ_{1} = \frac{1}{3m}((3m − 1)α_{1} + (3m − 2)α_{2} + ⋯ + 2α_{3m−2} + α_{3m−1}).

We embed (ℤA_{2})^{m} into ℤA_{3m−1} as follows:

α_{1}^{(i)} ↦ α_{3(i−1)+1}, α_{2}^{(i)} ↦ α_{3(i−1)+2}

(i.e., α_{1}^{(1)} ↦ α_{1}, α_{2}^{(1)} ↦ α_{2}, α_{1}^{(2)} ↦ α_{4}, α_{2}^{(2)} ↦ α_{5}, etc.). This embedding induces an embedding

ψ: (ℚA_{2})^{m} ↪ ℚA_{3m−1}.

Set

M = Λ_{3m} ∩ ψ((ℚA_{2})^{m}).

We show that M = ψ(L_{1_{m}}). Since by (12.1) the image of λ_{1} in Λ_{3m}/ℤA_{3m−1} is of order 3m, we see that Λ_{3m} is generated by ℤA_{3m−1} and λ_{1}, hence the set {α_{1}, …, α_{3m−1}, λ_{1}} is a generating set for Λ_{3m}. From (12.1) we see that

α_{3m−1} = 3mλ_{1} − (3m − 1)α_{1} − (3m − 2)α_{2} − ⋯ − 2α_{3m−2},

hence the set Ξ := {α_{1}, …, α_{3m−2}, λ_{1}} is a basis for Λ_{3m}. The subset

Ξ′ := {α_{1}, α_{2}, α_{4}, α_{5}, …, α_{3m−5}, α_{3m−4}, α_{3m−2}}

of Ξ is contained in M. Set N := ℤ[Ξ ∖ Ξ′] ∩ M ⊂ ℚA_{3m−1}, then clearly M = ℤΞ′ ⊕ N. Since rank M = 2m = |Ξ′| + 1, we see that rank N = 1. The element

μ := mλ_{1} − (m − 1)α_{3} − (m − 2)α_{6} − ⋯ − α_{3m−3} = \frac{1}{3}((3m − 1)α_{1}
+(3m − 2)α_{2} + (3m − 4)α_{4} + (3m − 5)α_{5} + ⋯ + 2α_{3m−2} + α_{3m−1})

is contained in N and indivisible in M, hence the one-element set {μ} is a basis of N, and Ξ′ ∪ {μ} is a basis of M. Now

μ − (m − 1)(α_{1} + α_{2}) − (m − 2)(α_{4} + α_{5}) − ⋯ − 1(α_{3(m−2)+1} + α_{3(m−2)+2})

= \frac{1}{3}((2α_{1} + α_{2}) + (2α_{4} + α_{5}) + ⋯ + (2α_{3m−2} + α_{3m−1}))

= ψ(ω_{1}^{(1)} + ω_{1}^{(2)} + ⋯ + ω_{1}^{(m)}).

We see that M is generated by ψ((ℤA_{2})^{m}) and ψ(ω_{1}^{(1)} + ω_{1}^{(2)} + ⋯ + ω_{1}^{(m)}), hence M = ψ(L_{1_{m}}), thus M is isomorphic to L_{1_{m}}. Therefore, it suffices to prove that M is not quasi-invertible.

The quotient lattice Λ_{3m}/M injects into the ℚ-vector space ℚA_{3m−1}/ψ((ℚA_{2})^{m}) with basis \overline{α_{3}}, \overline{α_{6}}, …, \overline{α_{3(m−1)}} on which (S_{3})^{m} acts trivially. Thus we obtain a short exact sequence

0 → M → Λ_{3m} → ℤ^{m−1} → 0,

where ℤ^{m−1} is a trivial, hence permutation, (S_{3})^{m}-lattice. It follows that the (S_{3})^{m}-lattices M and Λ_{3m} are equivalent, and therefore it suffices to show that Λ_{3m} is not a quasi-invertible (S_{3})^{m}-lattice.

Now we embed S_{3} × S_{3} into (S_{3})^{m} as follows: (s, t) ∈ S_{3} × S_{3} maps to (s, t, …, t) ∈ (S_{3})^{m}. With the notation of [20, (6.4)] we have Λ_{3m} = Q_{3m}(1). By [20, Proposition 7.1(b)], with respect to the above embedding S_{3} × S_{3} ↪ (S_{3})^{m}, we have

Q_{3m}(1)|_{S_{3} × S_{3}} ∼ Λ_{6}|_{S_{3} × S_{3}}.

By [20, Proposition 7.4(b)], Λ_{6} is not a quasi-permutation S_{3} × S_{3}-lattice, and it is actually proved there that [Λ_{6}]^{fl} (see [21, § 2.7] for the notation) is not an invertible S_{3} × S_{3}-lattice. It follows that Λ_{6} is not a quasi-invertible S_{3} × S_{3}-lattice (although � ^{2}(Γ′, Λ_{6}) = 0 for every subgroup Γ′ of S_{3} × S_{3}). Thus Λ_{3m} is not a quasi-invertible S_{3} × S_{3}-lattice, hence it is not a quasiinvertible (S_{3})^{m}-lattice. Thus L_{1_{m}} is not a quasi-invertible (S_{3})^{m}-lattice, and therefore L_{a} is not a quasi-invertible (S_{3})^{m}-lattice for any a as above. This completes the proof of Proposition 12.4. □

13. Standard subgroups

In this and the next sections we will collect several elementary results from combinatorial linear algebra, which will be needed to complete the proof of Theorem 9.1.

Let e_{1}, …, e_{m} be the standard ℤ/nℤ-basis of (ℤ/nℤ)^{m}. We say that a subgroup S ⊂ (ℤ/nℤ)^{m} is standard if S is generated by n_{1}e_{1}, …, n_{r}e_{r} for some 1 ≤ r ≤ m and some integers n_{1}, …, n_{r}, where n_{i} divides n_{i+1} for i = 1, …, r − 1.

Let W be a finite group, P be a W -lattice, and λ: P → ℤ/nℤ be a surjective morphism of W -modules, where W acts trivially on ℤ/nℤ. Given a subgroup S of (ℤ/nℤ)^{m}, let P_{S}^{m} denote the preimage of S in P^{m} with respect to the homomorphism λ^{m}: P^{m} → (ℤ/nℤ)^{m}. We regard P_{S}^{m} as a W -submodule of P^{m}, where W acts diagonally on P^{m}.

Lemma 13.1. Let W, P, n and λ be as above. For every subgroup S ⊂ (ℤ/nℤ)^{m} there exists a standard subgroup S_{st} ⊂ (ℤ/nℤ)^{m} with the following property: there exist an isomorphism g_{P} : P_{S}^{m} →^{∼} P_{S_{st}}^{m} of W -modules and an automorphism g of (ℤ/nℤ)^{m} taking S to S_{st} such that the following diagram commutes:

\xymatrix{P_{S}^{m} \ar[d]^{g_{P}} \ar[r]^{λ^{m}} & S \ar[d] \ar@{^{(}->}[r] & (ℤ/nℤ)^{m} \ar[d]^{g} \\ P_{S_{st}}^{m} \ar[r]^{λ^{m}} & S_{st} \ar@{^{(}->}[r] & (ℤ/nℤ)^{m}.}

Proof. The homomorphism λ^{m}: P^{m} → (ℤ/nℤ)^{m} can be written as

\xymatrix{λ^{m} = λ ⊗_{ℤ} id: P ⊗_{ℤ} ℤ^{m} \ar[r] & ℤ/nℤ ⊗_{ℤ} ℤ^{m}.}

Since for any g ∈ GL_{m}(ℤ) = Aut(ℤ^{m}) the diagram

\xymatrix{P ⊗_{ℤ} ℤ^{m} \ar[d]^{id_{P} ⊗g} \ar[r]^{λ⊗id_{ℤ^{m}}} & ℤ/nℤ ⊗_{ℤ} ℤ^{m} \ar[d]^{id_{ℤ/nℤ}⊗g} \\ P ⊗_{ℤ} ℤ^{m} \ar[r]^{λ⊗id_{ℤ^{m}}} & ℤ/nℤ ⊗_{ℤ} ℤ^{m}}

commutes, it suffices to show that for every subgroup S ⊂ (ℤ/nℤ)^{m} there exists g ∈ GL_{m}(ℤ) such that g(S) is standard.

Let π: ℤ^{m} → (ℤ/nℤ)^{m} be the natural projection. Then π^{−1}(S) is a finite index subgroup of ℤ^{m}. There exist a basis b_{1}, …, b_{m} of ℤ^{m} and integers n_{1} | n_{2} | … | n_{m}, such that n_{1}b_{1}, …, n_{m}b_{m} form a basis of π^{−1}(S); cf. [18, Theorem III.7.8]. Now let g ∈ GL_{m}(ℤ) be the element that takes the basis b_{1}, …, b_{m} to the standard basis of ℤ^{m}. Then g(π^{−1}(S)) is the subgroup n_{1}ℤ × ⋯ × n_{m}ℤ of ℤ^{m} and thus S_{st} := g(S) = ⟨n_{1}e_{1}, … n_{m}e_{m}⟩ = ⟨n_{1}e_{1}, …, n_{r}e_{r}⟩ is standard, where r ≤ m is the largest integer such that n does not divide n_{r}. □

Set Q = ker λ ⊂ P. For a subgroup S_{1} ⊂ ℤ/nℤ we set P_{S_{1}}^{1} = λ^{−1}(S_{1}), so that Q ⊂ P_{S_{1}}^{1} ⊂ P.

Corollary 13.2. Assume that S in Lemma 13.1 is cyclic. Then

P_{S}^{m} ≅ P_{S_{1}}^{1} ⊕ Q^{m−1}

for some subgroup S_{1} ⊂ ℤ/nℤ isomorphic to S.

Proof. By Lemma 13.1, we have P_{S}^{m} ≅ P_{S_{st}}^{m}. Since S is cyclic, say of order s, the group S_{st} is generated by (n/s)e_{1}. Set S_{1} = ⟨(n/s)e_{1}⟩ ⊂ ℤ/nℤ, then clearly

P_{S_{st}}^{m} = P_{S_{1}}^{1} ⊕ Q^{m−1},

and the corollary follows. □

Corollary 13.3. Assume that S in Lemma 13.1 contains an element of order n. Then P_{S}^{m} has a direct summand isomorphic to P.

Proof. By Lemma 13.1, P_{S}^{m} is isomorphic to P_{S_{st}}^{m} for some standard subgroup S_{st} ⊂ (ℤ/nℤ)^{m}. From the definition of a standard subgroup we see that

P_{S_{st}}^{m} = P_{S_{1}}^{1} ⊕ ⋯ ⊕ P_{S_{m}}^{1} ,

where S_{i} ⊂ ℤ/nℤ is generated by n_{i}e_{i} (for i > r we take n_{i} = 0). Since S_{st} contains an element of order n, we see that n_{1} = 1, hence S_{1} is generated by e_{1}, i.e., S_{1} = ℤ/nℤ and P_{S_{1}}^{1} = P. Thus P_{S}^{m} has a direct summand isomorphic to P. □

14. Coordinate and almost coordinate subspaces

Let F be a field, F^{m} be an m-dimensional F -vector space equipped with the standard basis e_{1} = (1, 0, …, 0), …, e_{m} = (0, …, 0, 1).

Recall that the Hamming weight of a vector v = (a_{1}, …, a_{m}) ∈ F^{m} is defined as the number of non-zero elements among a_{1}, …, a_{m}. We will say v ∈ F^{m} is defective if its Hamming weight is < m or, equivalently, if at least one of its coordinates is 0. The following lemma is well known; a variant of it is used to construct the standard open cover of the Grassmannian Gr(m, d) by d(m − d)-dimensional affine spaces, see, e.g., [15, §1.5]. For the sake of completeness, we supply a short proof.

Lemma 14.1. Let V be a vector subspace of F^{m} of dimension d ≥ 2. Then V has a basis consisting of defective vectors.

Proof. Let A be a d × m matrix whose rows form a basis of V. Then

V = {wA | w ∈ F^{d}} .

Note that for any invertible d × d matrix B, the rows of BA will also form a basis of V. Since the rows of A are linearly independent, A has a nondegenerate d × d submatrix M. Let B = M^{−1}. Then the d × m matrix BA has a d × d identity submatrix. Since d ≥ 2, this implies that every row of BA is defective. The rows of BA thus give us a desired basis of defective vectors for V. □

Definition 14.2. We will say that a subspace V ⊂ F^{m} is a coordinate subspace if V has a basis of coordinate vectors e_{i_{1}}, …, e_{i_{d}}, for some I = {i_{1}, …, i_{d}} ⊂ {1, …, m}. We will denote such a subspace by F_{I}.

In subsequent sections we will occasionally use this notation in the more general setting, where F is a commutative ring but not necessarily a field.

In this setting F_{I} will denote the free F -submodule of F^{n} generated by e_{i_{1}}, …, e_{i_{d}}.

Lemma 14.3. Let V ⊂ F^{m} be an F -subspace. Suppose V ∩F_{I} is coordinate for every I ⊊ {1, …, m}, then either

• V is the 1-dimensional subspace spanned by a vector a = (a_{1}, …, a_{m}), where a_{1} ≠ 0, …, a_{m} ≠ 0, or

• V is coordinate.

Proof. Assume that V is not of the form Span_{F} (a), where a = (a_{1}, …, a_{m}) and a_{1} ≠ 0, …, a_{m} ≠ 0. Then V has a basis of defective vectors. Indeed, if dim(V) = 1 this is obvious, since every vector in V is defective. The case where dim(V) ≥ 2 is covered by Lemma 14.1.

Clearly v ∈ F^{m} is defective if and only if v ∈ F_{I} for some I ⊊ {1, …, m}. Thus V is spanned by V ∩ F_{I}, as I ranges over the proper subsets of {1, …, m}. By our assumption, each V ∩ F_{I} is coordinate and therefore is spanned by coordinate vectors. We conclude that V itself is spanned by coordinate vectors, i.e., is coordinate, as desired. □

Definition 14.4. We will say that V ⊂ F^{m} is almost coordinate if V has a basis of the form

(14.1)  e_{i_{1}}, …, e_{i_{r}}, e_{j_{1}} + e_{h_{1}}, …, e_{j_{s}} + e_{h_{s}},

where i_{1}, …, i_{r}, j_{1}, …, j_{s}, h_{1}, …, h_{s} are distinct integers between 1 and m. We will refer to a basis of this form as an almost coordinate basis of V.

Remark 14.5. An almost coordinate subspace V ⊂ F^{m} has a unique almost coordinate basis. In other words, the set of integers {i_{1}, …, i_{r}} and the set of unordered pairs {{j_{1}, h_{1}}, …, {j_{s}, h_{s}}} in (14.1) are uniquely determined by V. Indeed, {i_{1}, …, i_{r}} is the set of subscripts i ∈ {1, …, m} such that the coordinate vector e_{i} lies in V. The set {{j_{1}, h_{1}}, {j_{2}, h_{2}}, …, {j_{s}, h_{s}}} is then the set of unordered pairs {j, h} such that j, h ∉ {i_{1}, …, i_{r}} and e_{j} + e_{h} ∈ V.

Proposition 14.6. Let F = ℤ/2ℤ, and let V ⊂ F^{m} be an F -subspace for some m ≥ 4. Assume V ∩F_{I} is almost coordinate in F_{I} ≅ (ℤ/2ℤ)^{r} for every I = {i_{1}, …, i_{r}} ⊊ {1, …, m}. Then either

• V is the 1-dimensional subspace spanned by (1, …, 1), or

• V is almost coordinate.

Proof. Assume that V is not of the form Span_{F} {(1, …, 1)}. Then, once again, Lemma 14.1 tells us that V has a basis of defective vectors, i.e., V is spanned by V ∩ F_{I}, as I ranges over the proper subsets of {1, …, m}. By our assumption, each V ∩ F_{I} is almost coordinate and therefore is spanned by vectors of Hamming weight 1 or 2. We conclude that V itself is spanned by vectors of weight 1 or 2. Choose a spanning set of the form

(14.2)  e_{i_{1}}, …, e_{i_{r}}, e_{j_{1}} + e_{h_{1}}, …, e_{j_{s}} + e_{h_{s}}

of minimal total Hamming weight, i.e., with minimal value of r + 2s. Here

i_{1}, …, i_{r}, j_{1}, h_{1}, …, j_{s}, h_{s} ∈ {1, …, m}

and j_{1} ≠ h_{1}, …, j_{s} ≠ h_{s}. We claim that (14.2) is an almost coordinate basis of V, i.e., that the subscripts

(14.3)  i_{1}, …, i_{r}, j_{1}, …, j_{s}, h_{1}, …, h_{s}

are all distinct. Clearly, Proposition 14.6 follows from this claim.

It thus remains to prove the claim. The minimality of the total Hamming weight of our spanning set (14.2) implies that we cannot remove any vectors, i.e., that it is a basis of V. In particular, the subscripts i_{1}, …, i_{r} and the pairs (j_{1}, h_{1}), …, (j_{s}, h_{s}) are distinct. If there is an overlap among subscripts (14.3), then, after permuting coordinates, we have either i_{1} = j_{1} or j_{1} = j_{2}. We will now show that neither of these equalities can occur.

If i_{1} = j_{1} then we may replace e_{j_{1}} + e_{h_{1}} by

e_{h_{1}} = (e_{j_{1}} + e_{h_{1}}) − e_{i_{1}} ∈ V .

We will obtain a new spanning set consisting of vectors of weight 1 or 2 with smaller total weight, a contradiction.

Now suppose j_{1} = j_{2}. Denote this number by j. Then V ∩ F_{{j,h_{1},h_{2}}} contains the vectors

(14.4)  e_{j} + e_{h_{1}} and e_{j} + e_{h_{2}} ∈ V.

Since we are assuming that m ≥ 4, {j, h_{1}, h_{2}} ⊊ {1, …, m} and hence, V ∩ F_{{j,h_{1},h_{2}}} is almost coordinate. The subspace in F_{{j,h_{1},h_{2}}} generated by the two vectors (14.4) is cut by the linear equation

x_{j} + x_{h_{1}} + x_{h_{2}} = 0

and clearly is not almost coordinate. It follows that V ∩F_{{j,h_{1},h_{2}}} = F_{{j,h_{1},h_{2}}}, hence V contains all three of the coordinate vectors e_{j}, e_{h_{1}} and e_{h_{2}}. Replacing e_{j} + e_{h_{1}} and e_{j} + e_{h_{2}} by e_{j}, e_{h_{1}} and e_{h_{2}} in our spanning set, we reduce the total weight by one, a contradiction. This completes the proof of the claim and thus of Proposition 14.6. □

15. Coordinate subspaces and quasi-permutation lattices

Proposition 15.1. Let W be a finite group, M be a W -lattice and let λ: M → F := ℤ/pℤ be a surjective morphism of W -modules, where p is a prime and W acts trivially on F. For any m ≥ 1, and an F -subspace S ⊂ V := F^{m}, let M_{S}^{m} be the preimage of S ⊂ F^{m} under the projection λ^{m}: M^{m} → F^{m}.

Assume that

(a) M is a quasi-permutation W -lattice;

(b) the W^{m}-lattice M_{S_{1}}^{m} is not quasi-permutation for any 1-dimensional subspace S_{1} of F^{m} of the form S_{1} = Span_{F} {(a_{1}, …, a_{n})}, where a_{1} ≠ 0, …, a_{m} ≠ 0.

Then, given a subspace S ⊂ F^{m}, M_{S}^{m} is a quasi-permutation W^{m}-lattice if and only if S is coordinate.

The following notation will be helpful in the proof of Proposition 15.1 and in the subsequent sections.

Definition 15.2. Let W be a finite group, M be a W -module and m be a positive integer. Given a subset I ⊂ {1, …, m}, we define the “coordinate subgroup” W_{I} ⊂ W^{m} as

W_{I} := {(w_{1}, …, w_{m}) ∈ W^{m} | w_{i} = id if i ∉ I}.

We will also define the W_{I}-submodule M_{I} of M^{m} as

M_{I} := {(a_{1}, …, a_{m}) ∈ M^{m} | a_{i} = 0 if i ∉ I}.

We shorten W_{{i}}, M_{{i}} to W_{i}, M_{i}.

Proof of Proposition 15.1. The “if” assertion is clear. We will prove “only if” by induction on m. In the base case, m = 1, every subspace of V is coordinate, so there is nothing to prove.

For the induction step, assume that m ≥ 2 and that our assertion has been established for all m′ < m. Suppose that for some subspace S ⊂ F^{m} the lattice M_{S}^{m} is quasi-permutation. We want to show that S is coordinate.

Since M_{S}^{m} is quasi-permutation, Lemma 2.5 tells us that M_{S}^{m} ∩ M_{I} is a quasi-permutation W_{I}-lattice for every I ⊊ {1, …, m} (cf. Definition 15.2 above). But M_{S}^{m} ∩ M_{I} = M_{S∩F_{I}}^{m}, and so by the induction hypothesis S ∩ F_{I} is a coordinate subspace in F_{I} (and hence, in F^{m}).

Now Lemma 14.3 tells us that either S is a 1-dimensional subspace of F^{m} which does not lie in any coordinate hyperplane or S is a coordinate subspace in F^{m}. Our assumption (b) rules out the first possibility. Hence, S is a coordinate subspace of F^{m}, as claimed. □

16. Proof of Theorem 9.1 for H of types A_{n−1} (n ≥ 5), B_{n} (n ≥ 3) and D_{n} (n ≥ 4)

Starting from this section, we will prove Theorem 9.1 case by case.

Notation 16.1. Let R be an irreducible reduced root system. We denote by Q = Q(R) the root lattice of R and by P = P (R) the weight lattice of R, both lattices regarded as W := W (R)-lattices. Given a positive integer m and a subset I ⊂ {1, …, m}, we define W_{I} ⊂ W^{m}, and the W_{I}-modules Q_{I}, P_{I}, etc., as in Definition 15.2. The base field k is assumed to be algebraically closed of characteristic zero.

16.1. Case A_{n−1} (n ≥ 5).

Theorem 16.2. Let G = (SL_{n})^{m}/C, where n ≥ 5 and C is a subgroup of (μ_{n})^{m} = Z(SL_{n}^{m}). Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley, (c) the character lattice X(G) is quasi-permutation,

(d) X(G) = Q^{m},

(e) G is isomorphic to (PGL_{n})^{m}.

Proof. (a) ⟹ (b) is obvious.

(b) ⟹ (c) follows from [20, Thm. 1.27].

(d) ⟹ (e): clear.

(e) ⟹ (a): clear, because the group PGL_{n} is Cayley, see [20, Thm. 1.31], and a product of Cayley groups is obviously Cayley.

The implication (c) ⟹ (d) follows from the next proposition. □

Proposition 16.3. Let R = A_{n−1}, where n ≥ 5. Suppose an intermediate W^{m}-lattice L between Q^{m} and P^{m}. is quasi-permutation. Then L = Q^{m}.

Proof. We proceed by induction on m. The base case, m = 1, follows from [20, Prop. 5.1]. For the induction step, assume that m ≥ 2 and that the proposition holds for m − 1. We will show that it also holds for m.

Set I := {2, …, m} ⊂ {1, 2, …, m} and F = P/Q = ℤ/nℤ. By Lemma 2.5, L ∩ P_{I} is a quasi-permutation W_{I}-lattice. By the induction hypothesis, L ∩ P_{I} = Q_{I}. Set S = L/Q^{m} ⊂ F^{m}, then S ∩ F_{I} = 0. It follows that the canonical projection S → F_{1} is injective. As F = ℤ/nℤ, we have S ≅ ℤ/dℤ for some divisor d of n.

In the notation of the beginning of Section 13, L = P_{S}^{m} as a W -lattice (where W acts on P^{m} diagonally). By Corollary 13.2,

(16.1)  L ≅ L_{1} ⊕ Q^{m−1},

where Q_{1} ⊂ L_{1} ⊂ P_{1}. Clearly Q^{m−1} is quasi-permutation as a W -lattice because so is Q = ker[ℤ[S_{n} / S_{n−1}] → ℤ]. By assumption, L is a quasipermutation W^{m}-lattice, hence it is quasi-permutation as a W -lattice. Since L and Q^{m−1} are quasi-permutation W -lattices, we see from (16.1) that L_{1} ∼ L_{1} ⊕ Q^{m−1} ≅ L ∼ 0, so that L_{1} is a quasi-permutation W -lattice. By [20, Prop. 5.1] it follows that L_{1} = Q_{1}, hence S = 0, and P_{S}^{m} = Q^{m}. Thus L = Q^{m}, which proves (d) for m and completes the proofs of Proposition 16.3 and Theorem 16.2. □

16.2. Case B_{n} (n ≥ 3) and D_{n} (n ≥ 4). Let n ≥ 7, R be the root system of Spin_{n} (of type B_{(n−1)/2} for n odd or of type D_{n/2} for n even) and M be the character lattice of SO_{n}. If n is odd, then M = Q; if n is even, then Q ⊊ M ⊊ P. Set F := P/M ≅ ℤ/2ℤ.

Theorem 16.4. Let G = (Spin_{n})^{m}/C, where n ≥ 7, and C is a central subgroup of (Spin_{n})^{m}. Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley,

(c) the character lattice X(G) of G is quasi-permutation,

(d) X(G) = M^{m}, where M = X(SO_{n}),

(e) G is isomorphic to (SO_{n})^{m}.

Proof. Only (c) ⟹ (d) needs to be proved; the other implications are easy.

Assume (c), i.e., X(G) is a quasi-permutation W^{m}-lattice. Clearly Q^{m} ⊂ X(G) ⊂ P^{m}. We claim that X(G) ⊃ M^{m}. If n is odd this is obvious, because M^{m} = Q^{m}. If n is even then by Lemma 2.5, X(G) ∩ P_{i} is a quasipermutation W_{i}-lattice. Now by [20, Thm. 1.28], we have X(G) ∩ P_{i} = M_{i}. Thus X(G) ⊃ M_{1} ⊕ ⋯ ⊕ M_{m} = M^{m}, as claimed.

We will now show that X(G) = M^{m}. Assume the contrary. Consider the surjection λ: P → P/M ≅ ℤ/2ℤ. Set S = X(G)/M^{m} ⊂ (ℤ/2ℤ)^{m}, then S ≠ 0. In the notation of Lemma 13.1, we have X(G) = P_{S}^{m}. Since S ≠ 0, by Corollary 13.3 X(G) has a direct W -summand isomorphic to P. By Proposition 11.1, P is not quasi-invertible, hence X(G) is not quasi-invertible as a W -lattice. It follows that X(G) is not a quasi-invertible W^{m}-lattice, which contradicts (c). Thus (d) holds, as desired. □

Remark 16.5. Alternatively, we can prove Theorem 16.4 similar to the proof of Proposition 16.3. Namely, we prove by induction that X(G) = M^{m} using Corollary 13.2. Here we make use of the fact that by Proposition 11.1, P is not quasi-permutation.

Remark 16.6. Proposition 16.3 cannot be proved by an argument analogous to the proof of Theorem 16.4. Indeed, the proof of Theorem 16.4 relies on the fact that X(Spin_{n}) is not quasi-invertible for n ≥ 7 (see Proposition 11.1). On the other hand, X(SL_{n}) is quasi-invertible (though it is not quasipermutation) whenever n is a prime; see [11, Prop. 9.1 and Rem. 9.3].

17. Proof of Theorem 9.1 for H of type A_{1} = B_{1} = C_{1}

We will continue using Notation 16.1. Let R = A_{1}. Set F = Q/P = ℤ/2ℤ.

Let G = (SL_{2})^{m}/C, where C is a subgroup of Z((SL_{2})^{m}) = (μ_{2})^{m}. We have Q^{m} ⊂ X(G) ⊂ P^{m}. Set S := X(G)/Q^{m} ⊂ F^{m} = (ℤ/2ℤ)^{m}.

Theorem 17.1. Let G = (SL_{2})^{m}/C, where C is a subgroup of Z((SL_{2})^{m}) = (μ_{2})^{m}. Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley,

(c) the character lattice X(G) is a quasi-permutation W^{m}-lattice,

(d) S := X(G)/Q^{m} is an almost coordinate subspace of F^{m} = (ℤ/2ℤ)^{m},

(e) G decomposes into a direct product of normal subgroups G_{1} ×_{k} ⋯×_{k} G_{s}, where each G_{i} is isomorphic to either SL_{2}, PGL_{2} or SO_{4}.

Remark 17.2. The set of normal subgroups G_{1}, …, G_{s} in part (e) is uniquely determined by G; see Remark 14.5.

Proof of Theorem 17.1. Only the implication (c) ⟹ (d) needs to be proved; all the other implications are easy. The implication (c) ⟹ (d) follows from the next proposition. □

Proposition 17.3. Let R = A_{1} and L be an intermediate W -lattice between Q^{m} and P^{m}, i.e., Q^{m} ⊂ L ⊂ P^{m}. Write S = L/Q^{m} ⊂ F^{m} = (ℤ/2ℤ)^{m}. Then L is quasi-permutation if and only if S is almost coordinate.

Proof. The “if” assertion follows easily from Lemmas 2.6 and 2.7. To prove the “only if” assertion, we begin by considering three special cases which will be of particular interest to us.

Case 1: m ≤ 2. Here every subspace of (ℤ/2ℤ)^{m} is almost coordinate, and condition (d) holds automatically.

Case 2: S is the line ⟨1_{m}⟩ = {0, 1_{m}} ⊂ (ℤ/2ℤ)^{m}, where 1_{m} = (1, …, 1). This S = ⟨1_{m}⟩ is not almost coordinate for any m ≥ 3. Thus we need

to show that (c) does not hold, i.e., the lattice L = P_{⟨1_{m}⟩}^{m} is not quasipermutation. This lattice is isomorphic to the lattice M described at the beginning of Section 11, in the case where Δ is the disjoint union of m copies of B_{1} (or, equivalently, of A_{1}) for m ≥ 3. By Proposition 11.1, for m ≥ 3, the lattice M ≃ L = P_{⟨1_{m}⟩}^{m}, is not quasi-invertible, hence not quasi-permutation, as claimed.

Case 3: m = 3. There are two subspaces S of (ℤ/2ℤ)^{3} that are not almost coordinate: (i) the line ⟨1_{3}⟩ and (ii) the 2-dimensional subspace cut out by x_{1} + x_{2} + x_{3} = 0. Once again we need to show that in both of these cases L is not quasi-permutation.

(i) is covered by Case 2 (with m = 3). If S is as in (ii), then L is isomorphic to the lattice M defined in the statement of Proposition 12.1. By this proposition, L is not quasi-invertible, hence not quasi-permutation, as claimed.

We now proceed with the proof of the proposition by induction on m ≥ 1. The base case, where m ≤ 3, is covered by Cases 1 and 3 above. For the induction step assume that m ≥ 4 and that the proposition has been established for all m′ ≤ m − 1.

Suppose that for some subspace S = L/Q^{m} ⊂ (ℤ/2ℤ)^{m} we know that L = P_{S}^{m} is quasi-permutation. Our goal is to show that S is almost coordinate.

Since L is quasi-permutation, by Lemma 2.5, we conclude that L ∩ P_{I} is a quasi-permutation W_{I}-lattice for every I = {i_{1}, …, i_{r}} ⊊ {1, …, m}. By the induction hypothesis, (L ∩ P_{I})/Q_{I} = S ∩ F_{I} is an almost coordinate subspace in F_{I} = (ℤ/2ℤ)^{r}.

Now Proposition 14.6 tells us that S is either the line ⟨1_{m}⟩, or almost coordinate. If S is the line ⟨1_{m}⟩, then L is not quasi-permutation by Case 2, contradicting our assumption. Thus S is almost coordinate, which completes the proofs of Proposition 17.3 and Theorem 17.1. □

18. Proof of Theorem 9.1 for H of types A_{2}, B_{2} = C_{2}, and A_{3} = D_{3}

18.1. Case A_{2}. Once again, we will continue using Notation 16.1. Set F := P/Q ≃ ℤ/3ℤ.

Theorem 18.1. Let G = (SL_{3})^{m}/C, where C is a subgroup of (μ_{3})^{m} = Z((SL_{3})^{m}). Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley,

(c) the character lattice X(G) is a quasi-permutation W^{m}-lattice,

(d) S := X(G)/Q^{m} is a coordinate subspace of F^{m} ≃ (ℤ/3ℤ)^{m},

(e) G decomposes into a direct product of normal subgroups G_{1} ×_{k} ⋯×_{k} G_{s}, where each G_{i} is isomorphic to either SL_{3} or PGL_{3}.

Proof. Once again, only the implication (c) ⟹ (d) needs to be proved; the other implications are easy.

Clearly Q^{m} ⊂ X(G) ⊂ P^{m}; assume X(G) is quasi-permutation. The W - lattices P and Q are quasi-permutation, see [20, Thm. 1.28]. If S ⊂ F^{m} is the 1-dimensional subspace ⟨a⟩ spanned by a vector a = (a_{1}, …, a_{m}) such that a_{1} ≠ 0, …, a_{m} ≠ 0, then from Proposition 12.4 it follows that X(G) = P_{⟨a⟩}^{m} is not a quasi-permutation W^{m}-lattice, a contradiction. Now by Proposition 15.1, X(G) = P_{S}^{m} is quasi-permutation if and only if S is coordinate. This shows that (c) ⟹ (d). □

18.2. Case B_{2} = C_{2}. Set F := P/Q = ℤ/2ℤ.

Theorem 18.2. Let G = (Spin_{5})^{m}/C, where C is a subgroup of the finite k-group (μ_{2})^{m} = ker[(Spin_{5})^{m} → (SO_{5})^{m}]. Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley,

(c) the character lattice X(G) is quasi-permutation,

(d) S := X(G)/Q^{m} is a coordinate subspace of F^{m} = (ℤ/2ℤ)^{m},

(e) G decomposes into a direct product of normal subgroups G_{1} ×_{k} ⋯×_{k} G_{s}, where each G_{i} is isomorphic to either Spin_{5} = Sp_{4} or SO_{5}.

Proof. As in the proof of Theorem 18.1, we only need to establish the implication (c) ⟹ (d). We have Q^{m} ⊂ X(G) ⊂ P^{m}. The W -lattices P and Q are quasi-permutation, see [20, Thm. 1.28]. If S ⊂ F^{m} is the 1-dimensional subspace ⟨1_{m}⟩ spanned by the vector 1_{m} = (1, …, 1) then by Proposition 11.1, P_{⟨1_{m}⟩}^{m} is not a quasi-invertible W^{m}-lattice. Now by Proposition 15.1, the W^{m}-lattice X(G) = P_{S}^{m} is quasi-permutation if and only if S is coordinate, which completes the proof of the theorem. □

18.3. Case A_{3} = D_{3}. Here P/Q ≃ ℤ/4ℤ.

Theorem 18.3. Let G = (Spin_{6})^{m}/C, where C is a subgroup of Z(G) = (μ_{4})^{m} = ker[(Spin_{6})^{m} → (PSO_{6})^{m}]. We have Q^{m} ⊂ X(G) ⊂ P^{m}, where P, Q and X(G) are the character lattices of PSO_{6}, Spin_{6} and G, respectively. Then the following conditions are equivalent:

(a) G is Cayley,

(b) G is stably Cayley, (c) X(G) is quasi-permutation,

(d) X(G) ⊂ (2P)^{m} and X(G)/Q^{m} is a coordinate subspace of (2P/Q)^{m} = (ℤ/2ℤ)^{m},

(e) G decomposes into a direct product of normal subgroups G_{1} ×_{k} ⋯×_{k} G_{s}, where each G_{i} is isomorphic to either SO_{6} or PSO_{6} = PGL_{4}.

Proof. Both SO_{6} and PSO_{6} = PGL_{4} are Cayley; see [20, Introduction]. Consequently, (e) ⟹ (a). Thus we only need to show that (c) ⟹ (d); the other implications are immediate. Assume that X(G) is quasi-permutation.

First we claim that X(G) ⊂ (2P)^{m}. Indeed, assume the contrary. Then X(G)/Q^{m} contains an element of order 4. By Corollary 13.3, the W^{m}-lattice X(G) restricted to the diagonal subgroup W has a direct summand isomorphic to the character lattice P of Spin_{6}. By Proposition 11.1, the W -lattice P is not quasi-invertible. We conclude that X(G) is not quasi-invertible as a W -lattice and hence not a quasi-invertible W^{m}-lattice, contradicting our assumption that X(G) is quasi-permutation. This proves the claim.

As we mentioned above, SO_{6} and PSO_{6} are both Cayley. Hence, the W - lattices 2P and Q are quasi-permutation. Set F = 2P/Q ≃ ℤ/2ℤ. If S := X(G)/Q^{m} ⊂ F^{m} is the 1-dimensional subspace ⟨1_{m}⟩ spanned by the vector 1_{m} = (1, …, 1), then by Proposition 12.2, X(G) is not a quasi-invertible W^{m}-lattice, a contradiction. Now Proposition 15.1 tells us that the W^{m}- lattice X(G)/Q^{m} is coordinate in (2P/Q)^{m}, and (d) follows. □

This completes the proof of Theorem 9.1.

19. Proof of Theorem 1.5

In this section we deduce Theorem 1.5 from Theorem 9.1. Clearly (b) implies (a), so we only need to show that (a) implies (b).

Let G be a stably Cayley simple k-group (not necessarily absolutely simple) and k̄ be a fixed algebraic closure of k. Then \overline{G} := G ×_{k} k̄ is stably Cayley over k̄ and is of the form H^{m}/C, where H is a simple and simply connected k̄-group and C is a central k-subgroup of H^{m}. By Theorem 9.1, \overline{G} = G_{1,k̄} ×k̄ ⋯ ×k̄ G_{s,k̄}, each G_{i,k̄} is either a stably Cayley simple group or is isomorphic to SO_{4,k̄}. (Recall that SO_{4,k̄} is stably Cayley and semisimple, but is not simple.) Here we write G_{i,k̄} for the factors in order to emphasize that they are defined over k̄.

If H is not of type A_{1}, then the subgroups G_{i,k̄} are simple and hence, intrinsic in \overline{G}: they are the minimal closed connected normal subgroups of dimension ≥ 1. If H is of type A_{1}, this is no longer obvious, since some of the groups G_{i,k̄} may not be simple (they may be isomorphic to SO_{4,k̄}). However, in this case the subgroups G_{i,k̄} are intrinsic in \overline{G} as well by Remark 17.2. Hence, in all cases, the Galois group Gal(k̄/k) permutes G_{1,k̄}, …, G_{s,k̄}. Since G is simple over k, this permutation action is transitive.

Let l ⊂ k̄ be the subfield corresponding to the stabilizer of G_{1,k̄} in Gal(k̄/k). Then G_{1,k̄} and G_{≥2,k̄} := G_{2,k̄}×k̄⋯×k̄G_{s,k̄} are Gal(k̄/l)-invariant, and we obtain l-forms of these two k̄-groups, which we will denote by G_{1,l} and G_{≥2,l}. Then G = R_{l/k}(G_{1,l}), where G_{1,l} is either absolutely simple or an l-form of SO_{4,l}. In the latter case G_{1,l} has to be an outer l-form of the split l-form of SO_{4}; otherwise G_{1,l} will not be l-simple and consequently, G will not be k-simple.

Now consider the case when G_{1,l} is not an l-form of SO_{4,l}. Then G_{1,l} is absolutely simple. Since G is stably Cayley over k,

(19.1)  G ×_{k} l = G_{1,l} ×_{l} G_{≥2,l} is stably Cayley over l.

It remains to show that G_{1,l} is stably Cayley over l. Assume the contrary. Note that G_{1,k̄} is stably Cayley over k̄. Comparing the classification of stably Cayley simple k̄-groups over an algebraically closed field k̄ of characteristic 0 in [20, Theorem 1.28] with the classification of stably Cayley absolutely simple l-groups over a field l of characteristic 0, not necessarily algebraically closed, in Theorem 1.4, we see that G_{1,l} is an outer l-form of PGL_{n,l} for some even integer n ≥ 4. Then by Example 10.7 the character lattice of G_{1,l} is not quasi-invertible, and by Proposition 10.8 the product G_{1,l} ×_{l} G_{≥2,l} cannot be stably Cayley over l, which contradicts (19.1). This contradiction completes the proof of Theorem 1.5. □

Acknowledgements. The authors are very grateful to Jean-Louis Colliot-Thélène for proving Lemma 10.3 and for stating and proving Lemma 10.2.

References

[1] Algebraic Number Theory, Proc. instructional conf. organized by the London Math. Soc. (J.W.S. Cassels, A. Fröhlich, eds.), Academic Press, London, 1967.

[2] A. Borel, Linear Algebraic Groups, 2nd ed., Graduate Texts in Math., vol. 126, Springer-Verlag, New York, 1991.

[3] M. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), 217–239.

[4] M. Borovoi, I. Dolgachev, Real reductive Cayley groups of rank ≤ 2, arXiv:1212.

1065v2[math.AG].

[5] N. Bourbaki, Groupes et algèbres de Lie, Ch. 4–6, Hermann, Paris, 1968.

[6] K.S. Brown, Cohomology of Groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York–Berlin, 1982.

[7] H. Cartan, S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1956.

[8] A. Cayley, Sur quelques propriétés des déterminants gauches, J. reine angew. Math. 32 (1846), 119–123; reprinted in: The Coll. Math. Papers of Arthur Cayley, Vol. I, No. 52, Cambridge Univ. Press, 1889, 332–336.

[9] J.-L. Colliot-Thélene, B.È. Kunyavskiĭ, V.L. Popov, Z. Reichstein, Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?, Compos. Math. 147 (2011), 428–466.

[10] J.-L. Colliot-Thélène, J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), 175–229.

[11] J.-L. Colliot-Thélène, J.-J. Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), 148–205.

[12] B. Conrad, Reductive group schemes (SGA3 Summer School, 2011), http://math.

stanford.edu/~conrad/papers/luminysga3.pdf.

[13] A. Cortella, B. Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), 771–793.

[14] Y.Z. Flicker, C. Scheiderer, R. Sujatha, Grothendieck’s theorem on non-abelian H^{2} and local-global principles, J. Amer. Math. Soc. 11 (1998), 731–750.

[15] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 2011.

[16] V.A. Iskovskikh, Two nonconjugate embeddings of the group S_{3}×Z_{2} into the Cremona group, (Russian) Tr. Mat. Inst. Steklova 241 (2003), Teor. Chisel, Algebra i Algebr. Geom., 105–109; translation in Proc. Steklov Inst. Math. 241, no. 2 (2003), 93–97.

[17] B.È. Kunyavskiĭ, Three-dimensional algebraic tori, in: Investigations in Number Theory, Saratov. Gos. Univ., Saratov, 1987, pp. 90–111; English transl.: Selecta Math. Sov. 21(1990), 1–21.

[18] S. Lang, Algebra, revised third edition, Graduate Texts in Math., vol. 211, Springer-Verlag, New York, 2002.

[19] N. Lemire, M. Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), 385–405.

[20] N. Lemire, V.L. Popov, Z. Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), 921–967.

[21] M. Lorenz, Multiplicative Invariant Theory, Encycl. Math. Sci., vol. 135, Invariant Theory and Algebraic Transformation Groups, VI, Springer-Verlag, Berlin, 2005.

[22] C. Miller, The second homology group of a group; relations among commutators, Proc. Amer. Math. Soc. 3 (1952), 588–595.

[23] I. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 132 (1907), 85–137, see also: Gesam. Abh., Bd. I, Springer-Verlag, Berlin-New York, 1973, 198–250.

[24] Séminaire de géométrie algébrique du Bois Marie 1962–64, Schémas en groupes (SGA3) (M. Demazure, A. Grothendieck, eds.), Lecture Notes Math., vol. 151, 152, 153, Springer-Verlag, Berlin–New York, 1970.

[25] T.A. Springer, Reductive groups, in: “Automorphic forms, representations and Lfunctions (Corvallis, Oregon, 1977)”, Proc. Sympos. Pure Math., XXXIII, Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 3–27.

[26] T.A. Springer, Linear Algebraic Groups, 2nd ed., Progr. Math., vol. 9, Birkhäuser, Boston, MA, 1998.

[27] J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62.

[28] V.E. Voskresenskiĭ, Birational properties of linear algebraic groups, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 3–19; English transl.: Math. USSR Isv. 4 (1970), 1–17.

[29] V.E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, Transl. Math. Monographs, vol. 179, Amer. Math. Soc., Providence, RI, 1998.

[30] V.E. Voskresenskiĭ, A.A. Klyachko, Toroidal Fano varieties and root systems, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 237–263; English transl.: Math. USSR Izv. 24 (1984), 221–244.

Borovoi: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel

E-mail address: borovoi@post.tau.ac.il

Kunyavskiĭ: Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

E-mail address: kunyav@macs.biu.ac.il

Lemire: Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada

E-mail address: nlemire@uwo.ca

Reichstein: Department of Mathematics, Univ. of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca