CAYLEY GROUPS

NICOLE LEMIRE, VLADIMIR L. POPOV, AND ZINOVY REICHSTEIN

Abstract. The classical Cayley map, X ↦ (I_{n} − X)(I_{n} + X)^{−1}, is a birational isomorphism between the special orthogonal group SO_{n} and its Lie algebra 𝔰o_{n}, which is SO_{n}-equivariant with respect to the conjugating and adjoint actions respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually “no”, with a few exceptions. In particular, we show that a Cayley map for the group SL_{n} exists if and only if n ⩽ 3, answering an old question of Luna.

Contents

1. Introduction 1.1. The classical Cayley map 1.6. Basic definitions, main problem, and examples 1.25. Notational conventions. 1.30. Main results 1.40. Application to Cremona groups

2. Preliminaries 2.1. Γ-fields and Γ-varieties. 2.2. Γ-lattices 2.8. Stable equivalence and flasque resolutions 2.15. Speiser’s Lemma 2.17. Homogeneous fiber spaces

3. Cayley maps, generic tori, and lattices 3.1 Restricting Cayley maps to Cartan subgroups 3.8. Generic tori 3.13. Proof of Theorem 1.31

4. Reduction theorems 4.1. Unipotent normal subgroups 4.3. The Levi decomposition 4.7. From reductive to semisimple 4.10. From semisimple to simple

5. Proof of Theorem 1.32: an overview

6. The groups SL_{n}/μ_{d} and their character lattices 6.1. Lattices Q_{n}(d) 6.9. Properties of Q_{n}(d).

7. Stably Cayley groups of type A_{n} 7.1. Restricting Q_{n}(d) to some subgroups 7.3. Lattices Λ_{p} and Λ_{2p} 7.10. Completion of the proof of Proposition 5.1 8 Stably Cayley groups of type D_{n} 8.1. Root systems of type D_{n} 8.2. Lattices Y_{2m} and Z_{2m}

Date: September 20, 2004. 2000 Mathematics Subject Classification. 14L35, 14L40, 14L30, 17B45, 20C10. Key words and phrases. Algebraic group, Lie algebra, reductive group, algebraic torus, Weyl group, root

system, birational isomorphism, Cayley map, rationality, cohomology, permutation lattice. N. Lemire and Z. Reichstein were supported in part by NSERC research grants. V. L. Popov was supported in part by ETH, Zürich, Switzerland and Russian grant ��������������.

9. Which stably Cayley groups are Cayley?

9.1. The group SL_{3}

9.9. The group G_{2}

10. Generalization

10.1. Generalized Cayley maps

10.10. The Cayley degree

Appendix. Alternative proof of Proposition 9.2: an outline

1. Introduction

The exponential map is a fundamental instrument of Lie theory that yields local linearization of various problems involving Lie groups and their actions, see [Bou_{1}]. Let L be a real Lie group with Lie algebra 𝔩. As the differential at 0 of the exponential exp : 𝔩 → L is bijective, exp yields a diffeomorphism of an open neighborhood of 0 in 𝔩 onto an open neighborhood U of the identity element e in L. The inverse diffeomorphism λ (logarithm) is equivariant with respect to the action of L on 𝔩 via the adjoint representation Ad_{L}: L → Aut 𝔩 and on L by conjugation, i.e., λ(gug^{−1}) = Ad_{L}g (λ(u)) if g ∈ L, u ∈ U and gug^{−1} ∈ U. This shows that the conjugating action of L on its underlying manifold is linearizable in a neighborhood of e.

In this paper we study what happens if L is replaced with a connected linear algebraic group G over an algebraically closed field k: what is a natural algebraic counterpart of λ for such G and for which G does it exist?

In the sequel we assume that char k = 0 (in fact in many places this assumption is either redundant or can be bypassed by modifying the relevant proof).

1.1. The classical Cayley map. Let 𝔤 be the Lie algebra of G. One way to look at the problem is to replace the Hausdorff topology in the Lie group setting by the étale topology, i.e., to define the algebraic counterpart of λ as a G-equivariant morphism G → 𝔤 étale at e. Then, at least for reductive groups, there is no existence problem: such morphisms always exist, see the Corollary to Lemma 10.4 below. Properties of some of them has been studied by Kostant and Michor in [KM], see Example 10.5 below. Note also that a G-equivariant dominant morphism G ⟶ 𝔤 exists for every linear algebraic group G; see Theorem 10.3 below.

In the present paper we look at the problem differently. Our point of view stems from a discovery made by Cayley in 1846, [Ca], cf. [Weyl], [Pos]. It suggests that the most direct approach, i.e., replacing the Hausdorff topology by the Zariski one, leads to something really interesting. Namely, let G be the special orthogonal group,

G = SO_{n} := {X ∈ Mat_{n×n} | X^{T}X = I_{n}},

where I_{n} is the identity n × n-matrix. Then

𝔤 = 𝔬_{n} := {Y ∈ Mat_{n×n} | Y^{T} = −Y },

and the adjoint representation Ad_{G} : G → Aut 𝔤 is given by

(1.2)  Ad_{G} g (Y) = gYg^{−1}, g ∈ G, Y ∈ 𝔤.

Cayley discovered that there exists a birational isomorphism

(1.3)  λ: G −−→^{≃} 𝔤

equivariant with respect to the conjugating and adjoint actions of G on the underlying varieties of G and 𝔤 respectively, i.e., such that

(1.4)  λ(gXg^{−1}) = Ad_{G} g (λ(X))

if g and X ∈ G and both sides of (1.4) are defined. His proof is given by the explicit formula defining such λ:

(1.5)  λ: X ↦ (I_{n} − X)(I_{n} + X)^{−1}

(one immediately deduces from (1.5) that Y ↦ (I_{n} − Y)(I_{n} + Y)^{−1} is the inverse of λ, and from (1.2) that (1.4) holds).

1.6. Basic definitions, main problem and examples. Inspired by this example, we introduce the following definition for an arbitrary connected linear algebraic group G. Definition 1.7. A Cayley map for G is a birational isomorphism (1.3) satisfying (1.4). A group G is called a Cayley group if it admits a Cayley map. If G is defined over a subfield K of k, then a Cayley map defined over K is called a Cayley K-map. If G admits a Cayley K-map, G is called a Cayley K-group.

Our starting point was a question, posed in 1975 to the second-named author by Luna, [Lun_{3}]. Using Definition 1.7, it can be reformulated as follows: Question 1.8. For what n is the special linear group SL_{n} a Cayley group?

It is easy to show, see Example 1.18 below, that SL_{2} is a Cayley group. Popov in [Pop_{2}] has proved that, contrary to what was expected, see [Lun_{1}, Remarque, p. 14], SL_{3} is a Cayley group as well.

More generally, given Definition 1.7, it is natural to pose the following problem: Problem 1.9. Which connected linear algebraic groups are Cayley groups?

Before stating our main results, we will discuss several examples. Set

μ_{d} := {a ∈ G_{m} | a^{d} = 1}.

This is a cyclic subgroup of order d of the multiplicative group G_{m}. Below we use the same notation μ_{d} for the central cyclic subgroup {aI_{n} | a ∈ μ_{d}} of GL_{n}. Example 1.10. If G_{1}, …, G_{n} are Cayley, then G := G_{1} × … × G_{n} is Cayley (the converse is false, see Subsection 4.10). Indeed, if 𝔤_{i} is the Lie algebra of G_{i} and λ_{i} : G_{i} −−→^{≃} 𝔤_{i} a Cayley map, then 𝔤 = 𝔤_{1} ⊕ … ⊕ 𝔤_{n} and λ_{1} × … × λ_{n}: G −−→^{≃} 𝔤 is a Cayley map. □ Example 1.11. Consider a finite-dimensional associative algebra A over k with identity element 1. Let L_{A} be the Lie algebra whose underlying vector space is that of A and whose Lie bracket is given by

(1.12)  [X_{1}, X_{2}] := X_{1}X_{2} − X_{2}X_{1}.

The group

G := A^{∗}

of invertible elements of A is a connected linear algebraic group whose underlying variety is an open subset of that of A. This implies that 𝔤 is naturally identified with L_{A}, and the adjoint action is given by formula (1.2). Hence the natural embedding λ : A^{∗} ↪ L_{A}, X ↦ X, is a Cayley map. Therefore G is a Cayley group.

Taking A = Mat_{n×n}, we obtain that G := GL_{n} is Cayley for every n ⩾ 1. □

Example 1.13. Maintain the notation of Example 1.11. For any element a ∈ A, denote by tr a the trace of the operator L_{a} of left multiplication of A by a. Since the algebra A is associative, a ↦ L_{a} is a homomorphism of A to the algebra of linear operators on the underlying vector space of A. From this and (1.12), we deduce that k·1 is an ideal of L_{A}, the map

τ : L_{A} → k·1, a ↦ tr a·1,

is a surjective homomorphism of Lie algebras, and

(1.14)  L_{A} = Ker τ ⊕ k·1.

The subgroup k^{∗}·1 of A^{∗} is normal; set

(1.15)  G := A^{∗}/k^{∗}·1.

As the Lie algebras of A^{∗} and k^{∗}·1 are respectively L_{A} and k·1, it follows from (1.14) that one can identify 𝔤 with Ker τ. Let A^{∗} → G, a ↦ [a], be the natural projection. Then the formula

(1.16)  [a] ↦ \frac{tr 1}{tr a}a − 1

defines a rational map λ : G −−→ 𝔤 = Ker τ. Since tr xax^{−1} = tr a for any a ∈ A, x ∈ A^{∗}, it follows from (1.16) that (1.4) holds. On the other hand, (1.16) clearly implies that

(1.17)  a ↦ [a + 1]

is the inverse of λ. Thus G is a Cayley group.

If A is defined over a subfield K of k, then the group G and birational isomorphisms (1.16), (1.17) are defined over K as well. Hence G is a Cayley K-group.

For A = Mat_{n×n} this shows that PGL_{n} is a Cayley group for every n ⩾ 1. Note that in this case, \frac{tr 1}{tr a} = \frac{n}{Tr a}, where Tr a is the trace of matrix a. Let K be a subfield of k. Since every inner K-form G of PGL_{n} is given by (1.15) for A = D ⊗_{K} k, where D is an n^{2}-dimensional central simple algebra over K and the K-structure of A is defined by D, cf. [Kn], all inner K-forms of PGL_{n} are Cayley K-groups.

Setting A = ⨁_{i=1}^{s} Mat_{n_{i}×n_{i}} we conclude that ∏_{i=1}^{s} GL_{n_{i}}/k^{∗}I_{n_{1}+...+n_{s}}, is a Cayley group. Here ∏_{i=1}^{s} GL_{n_{i}} is block-diagonally embedded in GL_{n_{1}+...+n_{s}}. □ Example 1.18. The following construction was noticed by Weil in [Weil, p. 599]. Namely, maintain the notation of Example 1.11 (Weil assumed that A is semisimple, but his construction, presented below, does not use this assumption). Let ι be an involution (i.e., an involutory k-antiautomorphism) of the algebra A. Set

(1.19)  G := {a ∈ A^{∗} | a^{ι}a = 1}^{∘}

(as usual, S^{∘} denotes the identity component of an algebraic group S). The Lie algebra of G is the subalgebra of odd elements of L_{A} for ι,

𝔤 = {a ∈ L_{A} | a^{ι} = −a}.

The formula

(1.20)  a ↦ (1 − a)(1 + a)^{−1}

defines an equivariant rational map λ: G−−→𝔤, and the formula

(1.21)  b ↦ (1 − b)(1 + b)^{−1}

defines its inverse, λ^{−1} : 𝔤−−→G. Thus λ is a Cayley map and G is a Cayley group.

If A and ι are defined over a subfield K of k, then the group G and birational isomorphisms (1.20), (1.21) are defined over K as well. Hence G is a Cayley K-group.

For A = Mat_{n×n} and the involution X ↦^{ι} X^{T}, this turns into the classical Cayley construction for G = SO_{n} yielding Cayleyness of this group for every n ⩾ 1. In particular, this shows that the following groups are Cayley: G_{m} ≃ SO_{2}, see Examples 1.11 and 1.22, PGL_{2} ≃ SL_{2}/μ_{2} ≃ SO_{3}, see Example 1.13, (SL_{2} × SL_{2})/μ_{2} ≃ SO_{4} (here SL_{2} × SL_{2} is block-diagonally embedded in SL_{4}), Sp_{4}/μ_{2} ≃ SO_{5} and SL_{4}/μ_{2} ≃ SO_{6}.

For A=Mat_{2n×2n} and the involution X ↦^{ι}J_{2n}^{−1}X^{T}J_{2n}, where J_{2n}:=[ \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ], we have

G = Sp_{2n} := {X ∈ Mat_{2n×2n} | X^{T}J_{2n}X = J_{2n}},
𝔤 = 𝔰𝔭_{2n} := {Y ∈ Mat_{2n×2n} | Y^{T}J_{2n} = −J_{2n}Y },

so the contruction shows that (1.5) is a Cayley map for Sp_{2n}, cf. [Pos, Examples 6, 7]. Thus Sp_{2n} is Cayley for every n ⩾ 1. In particular, SL_{2} ≃ Spin_{3} ≃ Sp_{2}, Spin_{4} ≃ SL_{2} × SL_{2} and Spin_{5} ≃ Sp_{4} are Cayley. Below we shall prove that Spin_{n} is not Cayley for n ⩾ 6.

Let K be a subfield of k. Since every K-form G of SO_{n} or Sp_{2n} is given by (1.19) for some algebra A and its involution ι, both defined over K, see [Weil], [Kn], all K-forms of SO_{n} and Sp_{2n} are Cayley K-groups. □ Example 1.22. Every connected commutative linear algebraic group G is Cayley. In fact, in this case, Condition (1.4) is vacuous, so the existence of (1.3) is equivalent to the property that the underlying variety of G is rational. Chevalley in [Ch_{1}] proved that over an algebraically closed field of characteristic zero this property holds for any connected linear algebraic group (not necessarily commutative). In particular, the algebraic torus G_{m}^{d}, where

G_{m}^{d} := \underbrace{G_{m} × … × G_{m}}_{d} if d ⩾ 1, G_{m}^{0} = e,

is a Cayley group for every d ⩾ 0 (as G_{m} = GL_{1}, this also follows from Examples 1.10, 1.11). Example 1.23. Every unipotent linear algebraic group G is Cayley (G is automatically connected because char k = 0). Indeed, we may assume without loss of generality that G ⊂ GL_{n}, so that elements of G are unipotent n × n-matrices, elements of 𝔤 are nilpotent n × n-matrices, and Ad_{G} is given by (1.2). So we have (I_{n} − X)^{n} = Y^{n} = 0 for any X ∈ G, Y ∈ 𝔤. Hence the exponential map is given by

exp : 𝔤 ⟶ G, Y ↦ ∑_{i=0}^{n−1} \frac{1}{i!}Y^{i}.

Therefore exp is a G-equivariant morphism of algebraic varieties. Moreover, it is an isomorphism since the formula

(1.24)  λ := ln: G ⟶ 𝔤, X ↦ − ∑_{i=1}^{n−1} \frac{1}{i}(I_{n} − X)^{i}.

defines its inverse.

More generally, by the Corollary of Proposition 4.4 below, every connected solvable linear algebraic group is Cayley. □

1.25. Notational conventions. In order to formulate our main results we need some notation and definitions.

For any algebraic torus T, we denote by T̂ its character group,

T̂ := Hom_{alg}(T, G_{m}),

written additively. It is a lattice (i.e., a free abelian group of finite rank).

Let T be a maximal torus of G and let

\{ \begin{matrix}N = N_{G,T} := \{g ∈ G | gT g^{−1} = T \}, \\ C = C_{G,T} := \{g ∈ G | gtg^{−1} = t for all t ∈ T \}, \\ W = W_{G} = W_{G,T} := N/C \end{matrix} (1.26)

be respectively its normalizer, centralizer (which is the Cartan subgroup of G) and the Weyl group. The group C is the identity component of N, and if G is reductive, then C = T, see [Bor, 12.1, 13.17]. The finite group W naturally acts by automorphisms of T̂. Since all maximal tori in G are conjugate, W and the W -lattice T̂ do not depend, up to isomorphism, on the choice of T. Definition 1.27. The W -lattice T̂ is called the character lattice of G and is denoted by X_{G}. Remark 1.28. The reader should be careful about this terminology: the elements of the character lattice of G are the characters of T, not of G. Definition 1.29. A group G is called stably Cayley if G × G_{m}^{d} is Cayley for some d ⩾ 0. If G is defined over a subfield K of k and G × G_{m}^{d} is a Cayley K-group for some d ⩾ 0, then G is called a stably Cayley K-group.

We denote by T_{G} the generic torus of G, see its definition in Section 3.8, cf. [Vos], [CK].

1.30. Main results. Now we are ready to state our main results. Theorem 1.31. Let G be a connected reductive algebraic group. Then the following implications hold:

[figure: X_{G} is sign- ; permutation; (a); G is ; Cayley; (b); T_{G} is ; rational; (c); T_{G} is stably ; rational; (d); X_{G} is quasi- ; permutation; (e); G is stably ; Cayley] .

Moreover, the implications (a) and (b) cannot be reversed. In particular, a stably Cayley group may not be Cayley.

For the definitions of sign-permutation and quasi-permutation lattices, see Section 2.2. Note that it is a long-standing open question whether or not every stably rational torus is rational, see [Vos, p. 52]. In particular, we do not know whether or not implication (c) can be reversed. We also remark that (d) is well-known, see, e.g., [Vos, Theorem 4.7.2].

A proof of Theorem 1.31 will be given in Subsection 3.13. In Section 4 we will partially reduce Problem 1.8 to the case where G is a simple group.

We will then use Theorem 1.31 to translate results about stable rationality of generic tori into statements about the existence (and more often, the non-existence) of Cayley maps for various simple algebraic groups (i.e., groups having no proper connected normal subgroups). In particular, Lemire and Lorenz in [LL] and Cortella and Kunyavskiǐ in [CK] have recently proved that the character lattice of SL_{n} is quasi-permutation if and only if n ⩽ 3. (This result had been previously conjectured and proved for prime n by Le Bruyn in [LB_{1}], [LB_{2}].) Theorem 1.31 now tells us that SL_{n} is not stably Cayley (and thus not Cayley) for any n ⩾ 4. On the other hand, Example 1.18 shows that SL_{2} is Cayley, and Popov in [Pop_{2}] has proved that SL_{3} is Cayley as well (an outline of the arguments from [Pop_{2}] is reproduced in the Appendix; see also an explicit construction in Section 9). This settles Luna’s original Question 1.8 about SL_{n}.

In a similar manner, we proceed to classify the connected simple groups G with quasipermutation character lattices X_{G}. For simply connected and adjoint groups this was done by Cortella and Kunyavskiǐ in [CK]. In Sections 6 and 8 we extend their results to all other connected simple groups. Combining this classification with Theorem 1.31, we obtain the following result.

Theorem 1.32. Let G be a connected simple algebraic group. Then the following conditions are equivalent:

(a) G is stably Cayley,

(b) G is one of the following groups:

(1.33)  SL_{n} for n ⩽ 3, SO_{n} for n ≠ 2, 4, Sp_{2n}, PGL_{n}, G_{2}.

Remark 1.34. The groups SO_{2} and SO_{4} are stably Cayley (and even Cayley, see Example 1.18) but they are excluded because they are not simple. Note also that, due to exceptional isomorphisms, some groups are listed twice in (1.33). (For example, Sp_{2} ≃ SL_{2}.)

It is now natural to ask which of the stably Cayley simple groups listed in Theorem 1.32(b) are in fact Cayley. Here is the answer:

Theorem 1.35. Let G be a connected simple algebraic group.

(a) The following conditions are equivalent: (i) G is Cayley; (ii) G is one of the following groups:

(1.36)  SL_{n} for n ⩽ 3, SO_{n} for n ≠ 2, 4, Sp_{2n}, PGL_{n}.

(b) The group G_{2} is not Cayley but the group G_{2} × G_{m}^{2} is Cayley.

The first assertion of part (b) is based on the recent work of Iskovskikh [Isk_{4}]. The groups SO_{n}, Sp_{2n} and PGL_{n} were shown to be Cayley in Examples 1.18 and 1.13. The groups SL_{3} and G_{2} will be discussed in Section 9.

Remark 1.37. Question 1.8 was inspired by Luna’s interest in the existence (for reductive G) of “algebraic linearization” of the conjugating action in a Zariski neighborhood of the identity element e ∈ G, i.e., in the existence of G-isomorphic neighborhoods of e and 0 in G and 𝔤 respectively, cf. [Lun_{1}]. In our terminology this is equivalent to the existence of a Cayley map (1.3) such that λ and λ^{−1} are defined at e and 0 respectively, and λ(e) = 0. Not all Cayley maps have this property. However, note that our proof of Theorem 1.35 (in combination with [Lun_{1}, p. 13, Proposition]) shows that each of the simple groups listed in (1.36) admits a Cayley map with this property (and so does any direct product of these groups); see Examples 1.10–1.23, Subsections 9.1, 9.9 and the Appendix.

Let K be a subfield of k. It follows from Theorems 1.32, 1.35 and Examples 1.13, 1.18 that classifying simple Cayley (respectively, stably Cayley) K-groups is reduced to classifying outer K-forms of PGL_{n} for n ⩾ 3 and K-forms of SL_{3} (respectively, outer K-forms of PGL_{n} for n ⩾ 3 and K-forms of SL_{3} and G_{2}) that are Cayley (respectively, stably Cayley) K-groups. Note that not all of these K-forms are Cayley (respectively, stably Cayley) K-groups. Indeed, Definitions 1.7, 1.29 imply the following special property of Cayley (respectively, stably Cayley) K-groups: their underlying varieties are rational (respectively, stably rational) over K. For some of the specified K-forms this property does not hold:

Example 1.38. Berhuy, Monsurrò, and Tignol in [BMT] have shown that for every n ≡ 0 mod 4, the group PGL_{n} has a K-form G of outer type whose underlying variety is not stably rational over K. Hence G is not a stably Cayley K-group. □

Remark 1.39. The underlying varieties of all outer K-forms of PGL_{n} with odd n are rational over K; see [VK]. Note also that the underlying variety of any K-form of a linear algebraic group of rank at most 2 is rational over K, e.g., see [Me, p. 189], [Vos, 4.1, 4.9].

1.40. Application to Cremona groups. The Cremona group Cr_{d}, i.e., the group of birational automorphisms of the affine space A^{d}, is a classical object in algebraic geometry; see [Isk_{2}] and the references therein. Classifying the subgroups of Cr_{d} up to conjugacy is an important research direction originating in the works of Bertini, Enriques, Fano, and Wiman. Most of the currently know results on Cremona groups relate to Cr_{2} nd Cr_{3} (the case d = 1 is trivial because Cr_{1} = PGL_{2}). For d ≥ 4 the groups Cr_{d} are poorly understood, and any results that shed light on their structure are prized by the experts.

Our results provide some information about subgroups of Cr_{d} by means of the following simple construction. Consider an action of an algebraic group G on a rational variety X of dimension d. Let G_{0} be the kernel of this action. Any birational isomorphism between X and A^{d} gives rise to an embedding ι_{X}: G/G_{0} ↪ Cr_{d}. A different birational isomorphism between X and A^{d} gives rise to a conjugate embedding, so i_{X} is is uniquely determined by X (as a G-variety) up to conjugacy in Cr_{d}. If Y is another rational variety on which G acts then the embeddings ι_{X} and ι_{Y} are conjugate if and only if X and Y are birationally isomorphic as G-varieties.

Now consider the special case of this construction, where G is a connected linear algebraic group, X is the underlying variety of G (with the conjugating G-action), Y = 𝔤 (with the adjoint G-action), and the kernel G_{0} (for both actions) is the center of G; see [Bor, 3.15]. Definition 1.7 can now be rephrased as follows: a connected algebraic group G is Cayley if and only if the embeddings ι_{G} and ι_{𝔤}: G/G_{0} = Ad_{G}G ↪ Cr_{dim G} are conjugate in Cr_{dim G}. In this paper we show that many connected algebraic groups are not Cayley; each non-Cayley group G gives rise to a pair of non-conjugate embeddings of the form ι_{G}, ι_{𝔤}: Ad_{G}G ↪ Cr_{dim G}.

Definition 1.29 can be interpreted in a similar manner. For every d ≥ 1 consider the embedding Cr_{d} ↪ Cr_{d+1} given by writing A^{d+1} as A^{d}×A^{1} and sending an element g ∈ Cr_{d} to g × id_{A^{1}} ∈ Cr_{d+1}. Denote the direct limit for the tower of groups Cr_{1} ↪ Cr_{2} ↪ … obtained in this way by Cr_{∞}. Suppose G is a group acting on rational varieties X and Y (possibly of different dimensions) with the same kernel G_{0}. Then it is easy to see that the embeddings ι_{X}: G/G_{0} ↪ Cr_{dim X} and ι_{Y} : G/G_{0} ↪ Cr_{dim Y} are conjugate in Cr_{∞} (or equivalently, in Cr_{m} for some m ≥ max{dim X, dim Y }) if and only if X and Y are stably isomorphic as G-varieties.

If V_{1} and V_{2} are vector spaces with faithful linear G-actions, then ι_{V_{1}} and ι_{V_{2}} are conjugate in Cr_{∞} by the “no-name lemma”, cf. Subsection 2.15. We call an embedding G ↪ Cr_{d} stably linearizable if it is conjugate, in Cr_{∞}, to i_{V} for some faithful linear G-action on a vector space V. Definition 1.29 and the “no-name lemma” now tell us that the following conditions are equivalent: (a) G is stably Cayley, (b) the embeddings ι_{G} and ι_{𝔤}: Ad_{G}G↪ Cr_{dim G} are conjugate in Cr_{∞} and (c) ι_{G} is stably linearizable. Once again, the results of this paper (and in particular, Theorem 1.32) can be used to produce many examples of pairs of embeddings of the form Ad_{G}G↪Cr_{dim G} that are not conjugate in Cr_{∞}.

Now suppose that Γ is a finite group and L and M are faithful Γ-lattices; see Section 2.2. Then Γ acts on their dual tori, which we will denote by X and Y. It now follows from Lemma 2.7 that the embeddings ι_{X}: Γ ↪ Cr_{rank L} and ι_{Y} : Γ ↪ Cr_{rank M} are conjugate in Cr_{∞} if and only if L and M are equivalent in the sense of Definition 2.4. Taking M to be a faithful permutation lattice, we conclude that the embedding ι_{X}: Γ ↪ Cr_{rank X} is stably linearlizable if and only if L is quasi-permutation (cf. Definition 2.6 and Corollary to Lemma 2.7).

In the special case where L = X_{G} is the character lattice of algebraic group G, Γ = W_{G} is the Weyl group, and X = T is a maximal torus with Lie algebra 𝔱, we see that the following conditions are equivalent: (a) G is stably Cayley, (b) X_{G} is quasi-permutation,

(c) the embeddings ι_{𝔱} and ι_{T} : W ↪ Cr_{dim T} are conjugate in Cr_{∞}, and (d) ι_{T} is stably linearizable. (Note that (a) and (b) are equivalent by Theorem 1.31, and (c) and (d) are equivalent because the W -action on 𝔱 is linear.) Consequently, every reductive non-Cayley group G gives rise to a pair of embeddings i_{T}, i_{𝔱}: W ↪ Cr_{rank G} which are not conjugate in Cr_{∞}. Example 1.41. Let G be a simple group of type A_{n−1} which is not stably Cayley, i.e., G = SL_{n}/μ_{d}, where d | n, d < n, n ≥ 4, and (n, d) ≠ (4, 2). Then the embeddings ι_{T} and ι_{𝔱}: S_{n} ↪ Cr_{n−1} are not conjugate in Cr_{∞}.

Assume further that n ≠ 6. Then by Hölder’s theorem (see [Höl]), S_{n} has no outer automorphisms. Thus the images ι_{T} (S_{n}) and ι_{𝔱}(S_{n}) are isomorphic finite subgroups of Cr_{n−1} which are not conjugate in Cr_{∞}. □ Acknowledgements. We are grateful to G. Berhuy, V. A. Iskovskikh, and D. Luna for stimulating discussions related to the subject matter of this paper.

2. Preliminaries

In this section we collect certain preliminary facts for subsequent use. Some of them are known and some are new. Throughout this section Γ will denote a group; starting from Subsection 2.2 it is assumed to be finite.

2.1. Γ-fields and Γ-varieties.

In the sequel we will use the following terminology. A Γ-field is a field K together with an action of Γ by automorphisms of K. Let K_{1} and K_{2} be Γ-fields containing a common Γ-subfield K_{0}. We say that K_{1} and K_{2} are isomorphic as Γ-fields (or Γ-isomorphic) over K_{0} if there is a Γ-equivariant field isomorphism K_{1} → K_{2} which is the identity on K_{0}. We say that K_{1} and K_{2} are stably isomorphic as Γ-fields (or stably Γ-isomorphic) over K_{0} if, for suitable n and m, K_{1}(x_{1}, …, x_{n}) and K_{2}(y_{1}, …, y_{m}) are isomorphic as Γ-fields over K_{0}. Here, x_{1}, …, x_{n} and y_{1}, …, y_{m} are algebraically independent variables over K_{1} and K_{2}, respectively; these variables are assumed to be fixed by the Γ-action.

If Γ is an algebraic group, a Γ-variety is an algebraic variety X endowed with an algebraic (morphic) action of Γ. A Γ-equivariant morphism (respectively, rational map) of Γ-varieties is a Γ-morphism (respectively, rational Γ-map). If X_{1} and X_{2} are irreducible Γ-varieties, then k(X_{1}) and k(X_{2}) are Γ-fields with respect to the natural actions of Γ. These fields are stably F -isomorphic over k if and only if there is a birational Γ-isomorphism X_{1} × A^{r} −−→ X_{2} × A^{s} for some r and s, where Γ acts on X_{1} × A^{r} and X_{2} × A^{s} via the first factors. In this case, X_{1} and X_{2} are called stably birationally Γ-isomorphic.

2.2. Γ-lattices. From now on we assume that Γ is a finite group.

A lattice L of rank r is a free abelian group of rank r. A Γ-lattice is a lattice equipped with an action of Γ by automorphisms. It is called faithful (respectively trivial) if the homomorphism Γ → Aut_{ℤ}L defining the action is injective (respectively trivial). If H is a subgroup of Γ, then L considered as an H-lattice is denoted by L|_{H}.

Given a group H and a ring R, we denote by R[H] the group ring of H over R. If K is a field and L is a Γ-lattice, we denote by K(L) the fraction field of K[L]; both K[L] and K(L) inherit a Γ-action from L. We usually think of these objects multiplicatively, i.e., we consider the set of symbols {x^{a}}_{a∈L} as a basis of the K-vector space K[L], and the multiplication being defined by x^{a}x^{b} = x^{a+b}. So σ · x^{a} = x^{σ·a} for any σ ∈ Γ. If a_{1}, …, a_{r} is a basis of L, and x_{i} := x^{a_{i}}, then K[L] = K[x_{1}, x_{1}^{−1} …, x_{r}, x_{r}^{−1}] and K(L) = K(x_{1}, …, x_{r}). Note that any group isomorphism L → Ĝ_{m}^{r} induces the K-isomorphisms of algebras K[L] → K[G_{m}^{r}] and fields K(L) → K(G_{m}^{r}), and therefore it induces a K-defined algebraic action of Γ on the torus G_{m}^{r} by its automorphisms. Any such action is obtained in this way.

An important example is L = X_{G}, the character lattice of a connected algebraic group G, and Γ = W, the Weyl group of G. In this case, k(X_{G}) is the field of rational functions on a maximal torus of G. Definition 2.3. A Γ-lattice L is called permutation (respectively, sign-permutation) if it has a basis ε_{1}, …, ε_{r} such that the set {ε_{1}, …, ε_{r}} (respectively, {ε_{1}, −ε_{1}, …, ε_{r}, −ε_{r}}) is Γ-stable.

If X is a finite set endowed with an action of Γ, we denote by ℤ[X] the free abelian group generated by X and endowed with the natural action of Γ. Permutation lattices may be, alternatively, defined as those of the form ℤ[X]. Since X is the union of Γ-orbits, any permutation lattice is isomorphic to some ⊕_{i=1}^{s}ℤ[Γ/Γ_{i}], where each Γ_{i} is a subgroup of Γ. Definition 2.4. ([C–TS_{1}]) Two Γ-lattices M and N are called equivalent, written M ∼ N, if they become Γ-isomorphic after extending by permutation lattices, i.e., if there are exact sequences of Γ-lattices

(2.5)  0 ⟶ M ⟶ E ⟶ P ⟶ 0  and  0 ⟶ N ⟶ E ⟶ Q ⟶ 0

where P and Q are permutation.

For a direct proof that this does indeed define an equivalence relation and for further background see [C–TS_{1}, Lemma 8] or [Sw]. Definition 2.6. A Γ-lattice L is called quasi-permutation if L ∼ 0 under this equivalence relation, i.e., L becomes permutation after extending by a permutation lattice. In other words, L is quasi-permutation if and only if there is an exact sequence of Γ-lattices

0 ⟶ L ⟶ P ⟶ Q ⟶ 0,

where P and Q are permutation lattices.

It is easily seen that the properties of being permutation, sign-permutation and quasipermutation are preserved under passing to Γ-isomorphic Γ-lattices and that replacing equivalent Γ-lattices with Γ-isomorphic ones yields equivalent lattices as well. Lemma 2.7. Let M and N be faithful Γ-lattices and let K be a field. Then the following properties are equivalent:

(i) K(M) and K(N) are stably isomorphic as Γ-fields over K, (ii) M ∼ N.

Proof. See [LL, Proposition 1.4]; this assertion is also implicit in [Sw], [C–TS_{1}] and [Vos, 4.7]. □

Lemma 2.7 and Definition 2.6 immediately imply the following.

Corollary. Let L be a faithful Γ-lattice and let K be a field. Then the following properties are equivalent:

(i) K(L) is stably isomorphic to K(P) (as a Γ-field over K) for some faithful permutation F -lattice P,

(ii) L is quasi-permutation.

2.8. Stable equivalence and flasque resolutions. In addition to the equivalence relation ∼ on Γ-lattices, we will also consider a stronger equivalence relation ≈ of stable equivalence. Two Γ-lattices L_{1} and L_{2} are called stably equivalent if L_{1} ⊕ P_{1} ≃ L_{2} ⊕ P_{2} for suitable permutation Γ-lattices P_{1} and P_{2}.

A Γ-lattice L is called flasque if H^{−1}(S, L) = 0 for all subgroups S of Γ. Every Γ-lattice L has a flasque resolution

(2.9)  0 ⟶ L ⟶ P ⟶ Q ⟶ 0

with P a permutation Γ-lattice and Q a flasque Γ-lattice. Moreover, Q is determined by L up to stable equivalence: If 0 → L → P′ → Q′ → 0 is another flasque resolution of L, then Q ≈ Q′. Following [C–TS_{1}], we will denote the stable equivalence class of Q in the flasque resolution (2.9) by

ρ(L).

Note that by [C–TS_{1}, Lemme 8], for Γ-lattices M, N,

(2.10)  M ∼ N ⟺ ρ(M) = ρ(N).

Dually, every Γ-lattice L has a flasque resolution

(2.11)  0 ⟶ R ⟶ P ⟶ L ⟶ 0

with P a permutation Γ-lattice and R a coflasque Γ-lattice, that is, H^{1}(S, R) = 0 holds for all subgroups S of Γ. Similarly, R is determined by L up to stable equivalence. Note that the dual of a flasque resolution for L is a coflasque resolution for L^{∗} since H^{1}(S, L) ≃ H^{−1}(S, L^{∗}). For details, see [C–TS_{1}, Lemme 5]. Note that since H^{±1} is trivial for permutation modules, H^{±1}(Γ, L) depends only on the stable equivalence class [L] of L and therefore is denoted by H^{±1}(Γ, [L]).

Following Colliot–Thélène and Sansuc, [C–TS_{1}, C–TS_{2}], we define

� ^{i}(Γ, M) = ⋂_{a∈Γ} Ker(Res_{⟨a⟩}^{Γ}: H^{i}(Γ, M) ⟶ H^{i}(⟨a⟩, M)).

for any ℤ[Γ]-module M. Of particular interest for us will be the case where M is a Γ-lattice L and i = 1 or 2.

The following lemma is extracted from [C–TS_{2}, pp. 199–202]. For a proof, see also [LL, Lemma 4.2]. Lemma 2.12. (a) For any exact sequence of ℤ[Γ]-modules

0 ⟶ M ⟶ P ⟶ N ⟶ 0

with P a permutation projective Γ-lattice, � ^{2}(Γ, M) ≃ � ^{1}(Γ, N).

(b) H^{1}(Γ, ρ(L)) ≃ � ^{2}(Γ, L) for any Γ-lattice L.

(c) If L is equivalent to a direct summand of a quasi-permutation Γ-lattice, then � ^{2}(S, L) = 0 holds for all subgroups S of Γ.

In particular, � ^{2}(Γ, ·) is constant on ∼-classes.

The following technical proposition will help us show that certain Γ-lattices are equivalent.

Proposition 2.13. Let X and Y be Γ-lattices satisfying the exact sequence

0 ⟶ X ⟶ Y ⟶ ℤ/dℤ ⟶ 0

where Γ acts trivially on ℤ/dℤ.

(a) If (d, |Γ|) = 1, then X ⊕ ℤ ≃ Y ⊕ ℤ so that X ≈ Y and X^{∗} ≈ Y^{∗}.

(b) If the fixed point sequence

0 ⟶ X^{S} ⟶ Y^{S} ⟶ (ℤ/dℤ)^{S} ⟶ 0

is exact for all subgroups S of Γ, then X^{∗} ∼ Y^{∗} as Γ-lattices.

Proof. (a) This follows directly from Roiter’s form of Schanuel’s Lemma [CR, 31.8] applied to the sequence of the proposition and

0 ⟶ ℤ ⟶^{×d} ℤ ⟶ ℤ/dℤ ⟶ 0.

(b) We claim that any coflasque resolution

0 ⟶ C_{1} ⟶ P ⟶ X ⟶ 0

for X can be extended to a coflasque resolution

0 ⟶ C_{2} ⟶ P ⊕ Q ⟶ Y ⟶ 0

for Y so that the following diagram commutes and has exact rows and columns:

(2.14)  \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & C_{1} \ar[r] \ar[d] & P \ar[r] \ar[d] & X \ar[r] \ar[d] & 0 \\ 0 \ar[r] & C_{2} \ar[r] \ar[d] & P ⊕ Q \ar[d] \ar[r] & Y \ar[r] \ar[d] & 0 \\ 0 \ar[r] & U \ar[r] \ar[d] & Q \ar[r] \ar[d] & ℤ/dℤ \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0} .

Here C_{1}, C_{2} are Γ-coflasque and P, Q are Γ-permutation. Indeed, as is described in [C–TS_{1}, Lemme 3], given a surjective homomorphism π from a permutation Γ-lattice P_{0} to a given Γ-lattice X, to form a coflasque resolution of X, we need only adjust P_{0} to P = P_{0} ⊕ ∑_{S} ℤ[Γ/S] ⊗ X^{S} where the sum is taken over all subgroups S of Γ for which π : P^{S} → X^{S} is not a surjection, and adjust π to π̂ such that π̂|_{ℤ[Γ/S]⊗X^{S}} = ϵ_{S} ⊗ id with ϵ_{S} the augmentation map. Then π̂ maps P^{S} surjectively onto X^{S} for all subgroups S of Γ so that H^{1}(S, Ker π̂) = 0 as required. To obtain a compatible coflasque resolution for Y, extend the surjection from the permutation lattice P onto X to a surjection from the permutation lattice P ⊕ Q_{0} onto Y and then adjust this surjection P ⊕ Q_{0} → Y to one with a coflasque kernel P ⊕ Q → Y as above. Then the top two rows are exact and commutative. The bottom row is obtained via the Snake Lemma.

Let S be a subgroup of Γ. Taking S-fixed points in (2.14), we obtain

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & C_{1}^{S} \ar[r] \ar[d] & P^{S} \ar[r] \ar[d] & X^{S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & C_{2}^{S} \ar[r] \ar[d] & P^{S} ⊕ Q^{S} \ar[d] \ar[r] & Y^{S} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & U^{S} \ar[r] \ar[d] & Q^{S} \ar[r] \ar[d] & (ℤ/dℤ)^{S} \ar[r] \ar[d] & 0 \\ & 0 & 0 & 0} .

Since C_{1}, C_{2} are coflasque and P, Q are permutation, we find that the first two rows and columns are exact. By hypothesis, the third column is exact. Then a diagram chase shows that the bottom row is exact. But then this means that U is coflasque since

0 ⟶ U^{S} ⟶ Q^{S} ⟶ (ℤ/dℤ)^{S} ⟶ H^{1}(S, U) ⟶ H^{1}(S, Q) = 0

is exact. Applying [LL, Lemma 1.1] to

0 ⟶ U ⟶ Q ⟶ ℤ/dℤ ⟶ 0,

we find that U is also quasi-permutation as it satisfies

0 ⟶ U ⟶ Q ⊕ ℤ ⟶ ℤ ⟶ 0.

So as U is coflasque, this sequence splits and U is in fact stably permutation with U ⊕ ℤ ≃ Q ⊕ ℤ. The first column of the first commutative diagram then shows us that C_{1} ⊕ U ⊕ ℤ ≃ C_{2} ⊕ ℤ so that C_{1} ⊕ Q ⊕ ℤ ≃ C_{2} ⊕ ℤ. Since

0 ⟶ X^{∗} ⟶ P ⟶ C_{1}^{∗} ⟶ 0, 0 ⟶ Y^{∗} ⟶ P ⊕ Q ⟶ C_{2}^{∗} ⟶ 0

are flasque resolutions of X^{∗} and Y^{∗}, this implies ρ(X^{∗}) = ρ(Y^{∗}) (i.e., that the corresponding flasque lattices are stably equivalent). By [C–TS_{1}, Lemme 8], we conclude that X^{∗} ∼ Y^{∗}. □

2.15. Speiser’s Lemma. Let π : Y → X be an algebraic vector bundle. We call it an algebraic vector Γ-bundle if Γ acts on X and Y, the morphism π is Γ-equivariant and g: π^{−1}(x) → π^{−1}(g(x)) is a linear map for every x ∈ X and g ∈ Γ.

The first of the following related rationality results is an immediate consequence the classical Speiser’s Lemma; the others follow from the first. In a broader context, when Γ is any algebraic group, results of this type appear in the literature under the names of “no-name method” ([Do]) and “no-name lemma” (see [C–T]).

Lemma 2.16. (a) Suppose E is a Γ-field and K is a Γ-subfield of E such that Γ acts on K faithfully, E = K(x_{1}, …, x_{m}) and Kx_{1} + … + Kx_{m} is Γ-stable. Then E = K(t_{1}, …, t_{m}), where t_{1}, …, t_{m} are Γ-invariant elements of Kx_{1} + … + Kx_{m}.

(b) Let π: Y → X be an algebraic vector Γ-bundle. Suppose that X is irreducible and the action of Γ on X is faithful. Then π is birationally Γ-trivial, i.e., there exists a birational Γ-isomorphism φ : Y −−→^{≃} X × k^{m}, where Γ acts on X × k^{m} via the first factor, such that the diagram

\xymatrix{Y \ar@{-->}[rr]^{φ} \ar[dr]_{π} & & X × k^{m} \ar[dl]^{π_{1}} \\ & X} ,

is commutative (π_{1} denotes projection to the first factor).

(c) Let V_{1} and V_{2} be finite dimensional vector spaces over k endowed with faithful linear actions of Γ. Then V_{1} and V_{2} are stably Γ-isomorphic.

(d) Suppose L is a field and

0 ⟶ S ⟶^{ι} N ⟶^{τ} P ⟶ 0

is an exact sequence of Γ-lattices, where S is faithful and P is permutation. Then the Γ-field L(N) is Γ-isomorphic over L to the Γ-field L(S)(t_{1}, …, t_{r}), where the elements t_{1}, …, t_{r} are Γ-invariant and algebraically independent over L(S).

Proof. (a) follows from Speiser’s Lemma, [Spe], cf. [HK, Theorem 1] or [Sh, Appendix 3].

(b) Recall that, by definition, algebraic bundles are locally trivial in the étale topology, but algebraic vector bundles are automatically locally trivial in the Zariski topology, see [Se]. This implies that after replacing X by a Γ-stable dense open subset U and Y by π^{−1}(U) we may assume that Y = X × k^{m} (but we do not claim that Γ acts via the first factor!) and π is projection to the first factor.

Using the projections Y → X and Y → k^{m}, we shall view k(X) and k(k^{m}) as subfields of k(Y). Put E := k(Y), K := k(X) and let x_{1}, …, x_{m} be the standard coordinate functions on k^{m}. If g ∈ Γ and b ∈ X, then the definition of Γ-bundle implies that g(x_{i})|_{π^{−1}(b)} ∈ k x_{1}|_{π^{−1}(b)} + … + k x_{m}|_{π^{−1}(b)}. In turn, this implies that the assumptions of (a) hold. Part (b) now follows from part (a).

(c) Applying part (b) to the projections V_{1} ← V_{1} × V_{2} → V_{2}, we see that both V_{1} and V_{2} are stably Γ-isomorphic to V_{1} × V_{2}.

(d) Identify S with ι(S); then K := L(S) is an Γ-subfield of E := L(N). Put x_{1} = 1 ∈ E and choose x_{2} …, x_{m} ∈ N ⊂ E such that τ(x_{2}), …, τ(x_{m}) is a basis of P permuted by Γ. The elements x_{2}, …, x_{m} are algebraically independent over K. If g ∈ Γ, then for every i there is a j such that a_{ij} := g(x_{i}) − x_{j} ∈ Ker τ = S ⊂ K; so g(x_{i}) = a_{ij}x_{1} + x_{j}. This shows that the assumptions of (a) hold. The claim (with r = m − 1) now follows from part (a). □

2.17. Homogeneous fiber spaces. Let H be an algebraic group and let S be a closed subgroup of H. Consider an algebraic variety X endowed with an algebraic (morphic) action of S and the algebraic action of S on H × X defined by

(2.18)  s(h, x) = (hs^{−1}, s(x)), s ∈ S, (h, x) ∈ H × X.

Assume that there exists a geometric quotient, [MFK], [PV, 4.2],

(2.19)  H × X ⟶ (H × X)/S.

This is always the case if every finite subset of X is contained in an affine open subset of X (note that this property holds if the variety X is quasi-projective), [Se, 3.2], cf. [PV, 4.8]. The variety (H ×X)/S, called a homogeneous fiber space over H/S with fiber X, is denoted by H ×^{S}X. If H is connected and X is irreducible, then H ×^{S}X is irreducible. We denote by [h, x] the image of a point (h, x) ∈ H × X under the morphism (2.19).

The group H acts on H × X by left translations of the first factor. As this action commutes with the S-action (2.18), the universal property of geometric quotients implies that the corresponding H-action on H ×^{S}X,

h′[h, x] = [h′h, x], h′, h ∈ H, x ∈ X,

is algebraic. It also implies that since the composition of the projection H × X → H with the canonical morphism H → H/S is constant on S-orbits of the action (2.18), this composition induces a morphism

(2.20)  π = π_{H,S,X} : H ×^{S} X ⟶ H/S, [h, x] ↦ hS.

This morphism is H-equivariant and its fiber over the point o ∈ H/S corresponding to S is S-stable and S-isomorphic to X; in the sequel we identify X with this fiber. Since H acts transitively on H/S and π is H-equivariant, the H-orbit of any point of H ×^{S} X intersects X. If Z is an open (respectively closed) H-stable subset of X, and ι : Z ↪ X is the identity embedding, then H ×^{S} Z → H ×^{S} X, [h, z] ↦ [h, ι(z)], is the embedding of algebraic varieties whose image is an H-stable closed (respectively open) subset of H ×^{S}X. Every H-stable closed (respectively open) subset of H ×^{S} X is obtained in this way.

If the action of S on X is trivial, then H ×^{S} X = H/S × X and π is the projection to the first factor.

The morphism π is a locally trivial fibration in the étale topology, i.e., each point of H/S has an open neighborhood U such that the pull back of π^{−1}(U) →^{π} U over a suitable étale covering Ũ → U is isomorphic to the trivial fibration Ũ × X → Ũ, (y, x) ↦ x, see [Se, §2], [PV, 4.8]. If X is a k-vector space and the action of S on X is linear, then (2.20) is an algebraic vector H-bundle, so π is locally trivial in the Zariski topology, i.e., π^{−1}(U) →^{π} U is isomorphic to U × X → U, (u, x) ↦ x, for a suitable U, see [Se].

If ψ is a (not necessarily H-equivariant) morphism (respectively rational map) of H ×^{S}X to H ×^{S} Y such that

(2.21)  π_{H,S,X} = π_{H,S,Y} ∘ ψ,

then we say that ψ is a morphism (respectively rational map ) over H/S. Lemma 2.22. (a) If ψ : H ×^{S} X → H ×^{S} Y is an H-morphism over H/S, then ψ|_{X} is an S-morphism X → Y. The map ψ ↦ ψ|_{X} is a bijection between H-morphisms H ×^{S} X → H ×^{S} Y over H/S and S-morphisms X → Y. Moreover, ψ is dominant (respectively, an isomorphism) if and only if ψ|_{X} is dominant (respectively, an isomorphism).

(b) Let H be connected and let X and Y be irreducible. Then the statements in (a) hold with “morphism” and “isomorphism” replaced by respectively “rational map” and “birational isomorphism”.

Proof. (a) Since X = π_{H,S,X}^{−1}(o), Y = π_{H,S,Y}^{−1} (o), the first statement follows from (2.21). As every H-orbit in H ×^{S} X intersects X and ψ is H-equivariant, ψ is uniquely determined by ψ|_{X}. If φ : X → Y is an S-morphism, then H × X → H × Y, (h, x) ↦ (h, φ(x)), is a morphism commuting with the actions of S (defined for H × X by (2.18) and analogously for H × Y) and H. By the universal property of geometric quotients, the H-map ψ : H ×^{S} X → H ×^{S} Y, [h, x] ↦ [h, φ(x)], is a morphism over H/S. We have ψ|_{X} = φ. The same argument proves the last statement.

(b) Since ψ is H-equivariant, its indeterminacy locus is H-stable. As every H-orbit in H ×^{S} X intersects X, this locus cannot contain X. Consequently, ψ|_{X} : X −−→ H ×^{S} Y is a well-defined rational S-map. In view of (2.21), its image lies in Y. Now (b) follows from (a) because rational maps are the equivalence classes of morphisms of dense open subsets, and all H-stable open subsets in H ×^{S} X are of the form H ×^{S} Z where Z is an S-stable open subset of X. □

3. Cayley maps, generic tori, and lattices

3.1. Restricting Cayley maps to Cartan subgroups. Let G be a connected linear algebraic group and let T be its maximal torus. Consider the Cartan subgroup C, its normalizer N and the Weyl group W defined by (1.26). Let 𝔤, 𝔱 and 𝔠 be the Lie algebras of G, T and C respectively.

Since C is the identity component of N and the Cartan subgroups of G are all conjugate to each other, [Bor, 12.1], assigning to a point of G/N the identity component of its Gstabilizer (respectively, the Lie algebra of this G-stabilizer) yields a bijection between G/N and the set of all Cartan subgroups in G (respectively, all Cartan subalgebras in 𝔤). So G/N can be considered as the variety of all Cartan subgroups in G (respectively, the variety of all Cartan subalgebras in 𝔤).

Moreover the Cartan subgroups in G (respectively the Cartan subalgebras in 𝔤) parametrized in this way by the points of G/N naturally “merge” to form a homogeneous fiber space over G/N with fiber C (respectively, 𝔠). More precisely, consider the homogeneous fiber space G ×^{N} C over G/N defined by the conjugating action of N on C (respectively, the homogeneous fiber space G ×^{N} 𝔠 over G/N defined by the adjoint action of N on 𝔠). Then for any g ∈ G, the map π_{G,N,C}^{−1}(g(o)) → gCg^{−1}, [g, c] ↦ gcg^{−1} (respectively, the map π_{G,N,𝔠}^{−1}(g(o)) → Ad_{G} g(𝔠), [g, x] ↦ Ad_{G} g(x)), is a well defined isomorphism (we use the notation of Subsection 2.17 for H = G, S = N).

Consider the conjugating and adjoint actions of G respectively on G and 𝔤. Then the definition of homogeneous fiber space implies that

(3.2) γ_{C} : G ×^{N} C ⟶ G, [g, c] ↦ gcg^{−1},  γ_{𝔠} : G ×^{N} 𝔠 ⟶ 𝔤, [g, x] ↦ Ad_{G} g(x),

are well defined G-equivariant maps, and the universal property of geometric factor implies that they are morphisms. Lemma 3.3. (a) The morphisms γ_{C} and γ_{𝔠} in (3.2) are birational G-isomorphisms.

(b) Any rational G-maps G ×^{N} C −−→ G ×^{N} 𝔠 and G ×^{N} 𝔠 −−→ G ×^{N} C are rational maps over G/N.

Proof. (a) Since the Cartan subgroups of G are all conjugate and every element of a dense open set U in G belongs to a unique Cartan subgroup, [Bor, §12], every fiber γ_{C}^{−1}(u), where u ∈ U, is a single point. As char k = 0, this means that γ_{C} is a birational isomorphism. For γ_{𝔠} the arguments are analogous because 𝔠 is a Cartan subalgebra in 𝔤, Cartan subalgebras in 𝔤 are all Ad_{G}G-conjugate and a general element of 𝔤 is contained in a unique Cartan subalgebra, [Bou_{3}, Ch. VII].

(b) Since a general element of T (respectively 𝔱) is regular, C (respectively 𝔠) is the unique Cartan subgroup (respectively subalgebra) containing T (respectively 𝔱), [Bor, §13], see [Bou_{3}, Ch. VII]. This implies that C and 𝔠 are the fixed point sets of the actions of T on G ×^{N}C and G ×^{N}𝔠 respectively. Since the maps under consideration are G-equivariant, this immediately implies the claim. □

Remark 3.4. The group varieties of C and 𝔠 are the “standard relative sections” of respectively G and 𝔤 induced by the rational G-map π_{G,N,C} ∘ γ_{C}^{−1} : G −−→ G/N and π_{G,N,𝔠} ∘ γ_{𝔠}^{−1} : 𝔤 −−→ G/N; in particular, this yields the following isomorphisms of invariant fields:

(3.5)  k(G)^{G} ⟶^{≃} k(C)^{N}, f ↦ f|_{C},  k(𝔤)^{G} ⟶^{≃} k(𝔠)^{N}, f ↦ f|_{𝔠};

see [Pop_{3}, Definition (1.7.6) and Theorem (1.7.5)]. Lemma 3.6. (a) G is Cayley if and only if C and 𝔠 are birationally N-isomorphic.

(b) G is stably Cayley if and only if C and 𝔠 are stably birationally N-isomorphic.

Proof. (a) By Lemma 2.22, the existence of a birational N-isomorphism φ: C −−→^{≃} 𝔠 implies the existence of a birational G-isomorphism ψ : G×^{N}C −−→^{≃} G×^{N}𝔠. Then Lemma 3.3 shows that γ_{𝔠} ∘ ψ ∘ γ_{C}^{−1} : G −−→^{≃} 𝔤 is a Cayley map.

Conversely, let λ : G −−→^{≃} 𝔤 be a Cayley map. Then ψ := γ_{𝔠}^{−1}∘ λ ∘ γ_{C} : G ×^{N}C −−→^{≃} G ×^{N}𝔠 is a birational G-isomorphism. By Lemma 3.3, ψ is a rational map over G/N. Hence, by Lemma 2.22, ψ|_{C} : C −−→^{≃} 𝔠 is a birational N-isomorphism.

(b) If C and 𝔠 are stably birationally N-isomorphic, it follows from rationality of the underlying variety of any linear algebraic torus that for some natural d there exists a birational N-isomorphism

(3.7)  C × G_{m}^{d} −−→^{≃} 𝔠 ⊕ k^{d},

where k^{d} is the Lie algebra of G_{m}^{d} and N acts on C ×G_{m}^{d} and 𝔠⊕k^{d} via C and 𝔠 respectively. Clearly C ×G_{m}^{d} is the Cartan subgroup of G×G_{m}^{d} with normalizer N ×G_{m}^{d} and Lie algebra 𝔠 ⊕ k^{d}, and the birational isomorphism (3.7) is N × G_{m}^{d}-equivariant. Now (a) implies that G × G_{m}^{d} is Cayley and hence G is stably Cayley.

Conversely, assume that G × G_{m}^{d} is Cayley for some d. Then the above arguments and (a) show that there exists a birational N-isomorphism (3.7). Since the group varieties of G_{m}^{d} and k^{d} are rational, this means that C and 𝔠 are stably birationally N-isomorphic. □

For reductive groups, Lemma 3.6 translates into the statement resulting also from [Lun_{1}, p. 13, Proposition]:

Corollary. Let G be a connected reductive linear algebraic group.

(a) G is Cayley if and only if T and 𝔱 are birationally W -isomorphic.

(b) G is stably Cayley if and only if T and 𝔱 are stably birationally W -isomorphic.

Proof. Since G is reductive, C = T and 𝔠 = 𝔱. As T is commutative, this implies that the actions of N on T and 𝔱 descend to the actions of W. The claim now follows from Lemma 3.6. □

3.8. Generic tori. We now recall the definition of generic tori in a form suitable for our purposes; see [Vos, 4.1] or [CK, p. 772]. We maintain the notation of Subsections 2.17, 3.1.

Assume that G is a connected reductive linear algebraic group; then C = T and 𝔠 = 𝔱. According to the discussion in the previous subsection, G/N may be interpreted in two ways: first, as the variety of all maximal tori in G, and second, as the variety of all maximal tori in 𝔤. The maximal torus in G (respectively, in 𝔤) assigned to a point g(o) ∈ G/N is gT g^{−1} (respectively, Ad_{G} g(𝔱)); it is naturally identified with the fiber over g(o) of the morphism π_{G,N,T} : G ×^{N} T ⟶ G/N (respectively, π_{G,N,𝔱} : G ×^{N} 𝔱 ⟶ G/N).

Definition 3.9. The triples

T_{G} := (G ×^{N} T, π_{G,N,T} , G/N) and 𝔱_{𝔤} := (G ×^{N} 𝔱, π_{G,N,𝔱}, G/N)

are called respectively the generic torus of G and the generic torus of 𝔤.

We identify the field k(G/N) with its image in k(G ×^{N} T) under the embedding π_{G,N,T}^{∗}.

Definition 3.10. The generic torus T_{G} is called rational if k(G ×^{N} T) is a purely transcendental extension of k(G/N). If T_{G×G_{m}^{d}} is rational for some d, then T_{G} is called stably rational.

Equivalently, T_{G} is called rational if there exists a birational isomorphism

(3.11)  G ×^{N} T −−→^{≃} G/N × A^{r}

over G/N (then r = dim T). The arguments used in the proof of Lemma 3.6 (b) show that stable rationality of T_{G} is equivalent to the property that there exists a purely transcendental field extension E of k(G ×^{N} T) such that E is a purely transcendental extension of k(G/N). There are groups G such that the generic torus T_{G} is not stably rational (and hence not rational), [Vos], [CK].

Of course, for the generic torus 𝔱_{𝔤} in 𝔤, one could also introduce the notions analogous to that in Definition 3.10. However in the Lie algebra context the rationality problem of generic tori is quite easy: since π_{G,N,𝔱} : G ×^{N} 𝔱 ⟶ G/N is a vector bundle, it is locally trivial in the Zariski topology, and hence 𝔱_{𝔤} is always rational, i.e., there exists a birational isomorphism

(3.12)  G ×^{N} 𝔱 −−→^{≃} G/N × A^{r}

over G/N.

3.13. Proof of Theorem 1.31.

Implication (a): By the Corollary of Lemma 3.6, it is enough to construct a W -equiva-riant birational isomorphism φ: T −−→^{≃}𝔱.

Using the sign-permutation basis of T̂, we can W -equivariantly identify the maximal torus T with G_{m}^{r}, where r is the rank of G and every w ∈ W acts on G_{m}^{r} by

(3.14)  (t_{1}, …, t_{r}) ↦ (t_{σ(1)}^{ε_{1}}, …, t_{σ(r)}^{ε_{r}}),

for some σ ∈ S_{r} and some ε_{1}, …, ε_{r} ∈ {±1} (depending on w). The Lie algebra 𝔱 is the tangent space to G_{m}^{r} at e = (1, …, 1); it follows from (3.14) that we can identify it with k^{r} where w acts by

(3.15)  (x_{1}, …, x_{r}) ↦ (ε_{1}x_{σ(1)}, …, ε_{r}x_{σ(r)}).

From (3.14) and (3.15) we easily deduce that the formula

(t_{1}, …, t_{r}) ↦ ((1 − t_{1})(1 + t_{1})^{−1}, …, (1 − t_{r})(1 + t_{r})^{−1})

defines a desired birational W -isomorphism φ: T −−→^{≃} 𝔱. This completes the proof of implication (a).

To see that implication (a) cannot be reversed, consider the group G := SL_{3}. First note that this group is Cayley; see Proposition 9.2. On the other hand, W ≃ S_{3} and since the character lattice X_{G} has rank 2, it can not be sign-permutation. Indeed, if it were, then S_{3} would embed into (ℤ/2ℤ)^{2} ⋊ S_{2}, which is impossible.

Implication (b): By the Corollary of Lemma 3.6, there is a birational N-isomorphism T −−→^{≃}𝔱. By Lemma 2.22, this implies that there is a birational G-isomorphism G ×^{N} T −−→^{≃} G ×^{N} 𝔱 over G/N. Its composition with the birational isomorphism (3.12) is a birational isomorphism (3.11) over G/N. Hence T_{G} is rational.

To see that implication (b) cannot be reversed, consider the exceptional group G_{2}. The generic torus of G_{2} is rational; see [Vos, 4.9]. On the other hand, G_{2} is not a Cayley group; see Proposition 9.11.

Implication (c): This is obvious from the definition.

Equivalence (d): This is well-known, see, e.g., [Vos, Theorem 4.7.2].

Equivalence (e): Let V be any finite dimensional faithful permutation W -module over k (for instance, the one determined by the regular representation of W). Then clearly k(V) = k(P) for some permutation W -lattice P. Since the action of W on 𝔱 is faithful, [Bor], we deduce from Lemma 2.16(c) that k(𝔱) and k(P) are stably W -isomorphic over k. Therefore, since k(T) = k(T̂), applying the Corollary of Lemma 3.6 implies that G is stably Cayley if and only if k(T̂) and k(P) are stably W -isomorphic over k. On the other hand, the latter property holds if and only if the W -lattice T̂ is quasi-permutation, see the Corollary of Lemma 2.7, whence the claim. □ Example 3.16. The character lattice ℤA_{n−1} of PGL_{n} is defined by the exact sequence

0 ⟶ ℤA_{n−1} ⟶ ℤ[S_{n}/S_{n−1}] ⟶^{ϵ} ℤ ⟶ 0,

where ϵ is the augmentation map and the Weyl group W = S_{n} acts trivially on ℤ and naturally on ℤ[S_{n}/S_{n−1}], see Subsection 6.1. Thus ℤA_{n−1} is quasi-permutation. By Theorem 1.31, we conclude that PGL_{n} is stably Cayley. We know that in fact PGL_{n} is even Cayley; see Example 1.13. □

4. Reduction theorems

The purpose of this section is to show that to a certain extent classifying arbitrary Cayley groups is reduced to classifying simple ones.

As before, let G be a connected linear algebraic group. Denote by R and R_{u} respectively the radical and the unipotent radical of G. Recall that a Levi subgroup of G is a connected subgroup L, necessarily reductive, such that G = L ⋉ R_{u}; since char k = 0, Levi subgroups exist and are conjugate, [Bor, 11.22].

In this section we will address the following questions:

(a) If a Levi subgroup of G is (stably) Cayley, is G (stably) Cayley?

(b) Let G be reductive. If G/R is (stably) Cayley, is G (stably) Cayley?

(c) Let G be reductive and let H_{1}, …, H_{n} be a complete list of its connected normal simple subgroups. What is the relation between (stable) Cayleyness of G and that of H_{1}, …, H_{n}?

4.1. Unipotent normal subgroups. We will need a generalization of Example 1.23. Let U be a normal unipotent subgroup of G. Denote by 𝔲 the Lie algebra of U. The group G acts on U by conjugation and on 𝔲 by Ad_{G}|_{𝔲}. Lemma 4.2. There exists a G-isomorphism of G-varieties U → 𝔲.

Proof. We may assume without loss of generality that G ⊂ GL_{n}. Since Ad_{G} is given by (1.2), it follows from (1.24) that ln : U → 𝔲 is a G-morphism. By Example 1.23, it is an isomorphism, whence the claim. □

4.3. The Levi decomposition.

Proposition 4.4. Let L be a Levi subgroup of G.

(a) If L is Cayley, then so is G.

(b) G is stably Cayley if and only if L is stably Cayley.

Proof. Let T be a maximal torus of L. It is a maximal torus of G as well, [Bor, 11.20]. Using the notation of (1.26) and Subsection 3.1, we have C = T ×U where U is a unipotent group, [Bor, 12.1]. Let 𝔲 be the Lie algebra of U and let d = dim U. As T and U are respectively the semisimple and unipotent parts of the nilpotent group C, they are stable under the conjugating action of N, and C, as an N-variety, is the product of the N-varieties T and U. Similarly, 𝔱 and 𝔲 are stable under the adjoint action of N, and 𝔠, as an N-variety, is the product of N-varieties 𝔱 and 𝔲. By Lemma 4.2, there exists an isomorphism of N-varieties

(4.5)  τ: U ⟶ 𝔲.

(a) Assume that L is Cayley. Then by Corollary of Lemma 3.6, there is a birational W_{L,T} -isomorphism φ: T −−→^{≃} 𝔱. Since the action of W_{L,T} on T (respectively, 𝔱) is faithful, W_{L,T} can be considered as a transformation group of T (respectively, 𝔱). By [Bor, 11.20], it coincides with the transformation group {T → T, t ↦ ntn^{−1} | n ∈ N} (respectively, {𝔱 → 𝔱, x ↦ Ad_{G}n(x) | n ∈ N}). Therefore the map φ is N-equivariant. Hence

φ × τ: C = T × U−−→𝔱 ⊕ 𝔲 = 𝔠.

is a birational N-isomorphism. Lemma 3.6 now implies that G is Cayley.

(b) Since L × G_{m}^{d} is the Levi subgroup of G × G_{m}^{d}, it follows from (a) that if L is stably Cayley, then G is stably Cayley.

To prove the converse, it suffices to show that if G is Cayley, then L is stably Cayley. In turn, Lemma 3.6 and its Corollary reduce this to proving that if there exists a birational N-isomorphism

α: C = T × U −−→^{≃} 𝔱 × 𝔲 = 𝔠,

then T and 𝔱 are stably birationally W_{L,T} –isomorphic. We shall prove this last statement.

Since T is the identity component of N_{L,T} = N ∩ L and T acts trivially on C and 𝔠, the actions of N_{L,T} on C, 𝔠, T, 𝔱, U and 𝔲 descend to actions of W_{L,T} = N_{L,T} /T. Moreover, C (respectively, 𝔠), as an W_{L,T} -variety, is the product of W_{L,T} -varieties T and U (respectively, 𝔱 and 𝔲), and α is a birational W_{L,T} -isomorphism.

Since W_{L,T} acts linearly on 𝔲, Lemma 2.16(b) implies that there are birational W_{L,T} - isomorphisms

β: T × A^{d} −−→^{≃} T × 𝔲 and γ: 𝔱 × 𝔲 −−→^{≃} 𝔱 × A^{d},

where W_{L,T} acts on T × A^{d} and 𝔱 × A^{d} via the first factors. Considering the composition of the following birational W_{L,T} -isomorphisms

\xymatrix{T × A^{d} \ar@{-->}[r]^{β} & T × 𝔲 \ar[r]^{id×τ^{−1}} & T × U \ar@{-->}[r]^{α} & 𝔱 × 𝔲 \ar@{-->}[r]^{γ} & 𝔱 × A^{d}} ,

we now see that T and 𝔱 are indeed stably birationally W_{L,T} -isomorphic. □

Remark 4.6. The converse to Proposition 4.4(a) fails for G := G_{2} × G_{a}^{2}. Indeed, the first factor is the Levi subgroup of G. By Proposition 9.11, it is not Cayley. Consider the group H := G_{2} × G_{m}^{2}. Both G and H have the same Lie algebra 𝔤. By Proposition 9.13, H is Cayley; let λ : H −−→^{≃} 𝔤 be a Cayley map. Fix a birational isomorphism of group varieties δ : G_{a}^{2} −−→^{≃} G_{m}^{2}. Since the second factors of G and H lie in the kernels of conjugating and adjoint actions, λ ∘ (id × δ): G −−→^{≃} 𝔤 is a Cayley map. Thus G is Cayley.

Corollary. Every connected solvable linear algebraic group G is Cayley.

Proof. A Levi subgroup L of G is a torus, [Bor, 10.6]. By Example 1.22, L is Cayley. Hence by Proposition 4.4(a), G is Cayley as well. □

4.7. From reductive to semisimple.

Proposition 4.8. Let G be a connected reductive group and let Z be a connected closed central subgroup of G.

(a) If G/Z is Cayley, then so is G.

(b) G is stably Cayley if and only if G/Z is stably Cayley.

Proof. Since G is reductive, R is a torus and the identity component of the center of G, see [Bor, 11.21]. Thus Z is a subtorus of R. Let T be a maximal torus of G. We have R ⊂ T, see [Bor, 11.11], T/Z is a maximal torus of G/Z and the natural epimorphism G → G/Z identifies W with W_{G/Z,T/Z} (we use the notation of (1.26) and Subsection 3.1), see [Bor, 11.20]. Since Z is central, it is pointwise fixed with respect to the action of W. Thus we have the following exact sequence of W -homomorphisms of tori

e ⟶ Z ⟶ T ⟶ T/Z ⟶ e

which in turn yields the exact sequence of W -lattices of character groups

0 ⟶ T̂/Z ⟶ T̂ ⟶ Ẑ ⟶ 0.

Note that W acts trivially on Ẑ. In particular, Ẑ is a permutation W -lattice, and the last exact sequence tells us that the character lattices T̂ and T̂/Z are equivalent, see Definition 2.4. Thus, by Lemma 2.7 and its Corollary, if one of them is quasi-permutation, then so is the other. Part (b) now follows from Theorem 1.31.

Since the W -fields k(T) and k(T/Z) are W -isomorphic to k(T̂) and k(T̂/Z) respectively, we deduce from Lemma 2.16(d) that T is birationally W -isomorphic to T/Z × A^{m}, where W acts on T/Z × A^{m} via the first factor and m = dim Z.

On the other hand, let 𝔣 and 𝔷 be the Lie algebras of T/Z and Z respectively. Then, since the Lie algebras 𝔱 and 𝔣 ⊕ 𝔷 are W -equivariantly isomorphic and W acts on 𝔷 trivially, we see that 𝔱, as a W -variety, is isomorphic to 𝔣 × A^{m}, where W acts on 𝔣 × A^{m} via the first factor.

Now to prove part (a), assume that G/Z is Cayley. Then by the Corollary of Lemma 3.6, there is a birational W -isomorphism φ : T/Z −−→^{≃} 𝔣. This gives a birational W -isomorphism

\xymatrix{T/Z × A^{m} \ar@{-->}[r]^{φ×id} & 𝔣 × A^{m}} . Applying the Corollary of Lemma 3.6 once again, we conclude that G is Cayley. This completes the proof of part (a). □

Setting Z = R, we obtain

Corollary. Let G be a connected reductive group and G_{ss} := G/R.

(a) If G_{ss} is Cayley, then so is G.

(b) G is stably Cayley if and only if G_{ss} is stably Cayley. □

Remark 4.9. The converse to statement (a) of Corollary fails for G = G_{2} × G_{m}^{2}. Indeed, G is Cayley by Proposition 9.13 and G/R ≃ G_{2} is not Cayley by Proposition 9.11.

4.10. From semisimple to simple. Let G_{1}, …, G_{n} be connected linear algebraic groups and let 𝔤_{i} be the Lie algebra of G_{i}. If each G_{i} is Cayley, then so is G_{1} × … × G_{n}, see Example 1.10. The converse fails for n = 2, G_{1} = G_{2}, G_{2} = G_{m}^{2} by Proposition 9.11, Example 1.22 and Proposition 9.13.

Lemma 4.11. G_{1} × … × G_{n} is stably Cayley if and only if each G_{i} is stably Cayley.

Proof. The “if” direction follows from Definition 1.29 and Example 1.10. To prove the converse, we use the fact that the underlying variety of each G_{i} is rational over k, see [Ch_{1}]. This implies that the underlying variety of G_{1} × … × G_{n}, as a G_{i}-variety, is birationally isomorphic to G_{i}×G_{m}^{d_{i}} with the conjugating action via the first factor and d_{i} = ∑_{j≠i} dim G_{j}. The “only if” direction now follows from Definition 1.29 and the fact that the underlying variety of the Lie algebra of G_{1} × … × G_{n}, as G_{i}-variety, is isomorphic to 𝔤_{i} ⊕ k^{d_{i}} with the adjoint action via the first summand. □

As usual, given subgroups X and Y of G, we denote by (X, Y) the subgroup generated by the commutators xyx^{−1}y^{−1} with x ∈ X, y ∈ Y.

Proposition 4.12. Assume G is a connected reductive group and let H_{1}, …, H_{m} be the connected closed normal subgroups of G such that

(i) (H_{i}, H_{j}) = e for all i ≠ j, (ii) G = H_{1} … H_{m}.

Let H̃_{i} be the subgroup of G generated by all H_{j}’s with j ≠ i. If G is stably Cayley, then each G/H̃_{i} ≃ H_{i}/(H_{i} ∩ H̃_{i}) is stably Cayley.

Proof. Since H_{1}, …, H_{m} are connected, each H̃_{i} is connected, see [Bor, 2.2]. Since G is reductive, all H_{i} and H̃_{i} are reductive.

It follows from (i) and (ii) that

H_{1} × … × H_{m} → G, (h_{1}, …, h_{m}) ↦ h_{1} … h_{m},

is an epimorphism of algebraic groups. Let T_{i} be a maximal torus of H_{i}. Then T_{1} ×…×T_{m} is a maximal torus of H_{1} × … × H_{m}. Therefore its image T := T_{1} … T_{m} under the above epimorphism is a maximal torus of G, see [Bor, 11.14]. The same argument shows that the group S_{i} of T generated by all T_{j}’s with j ≠ i is a maximal torus of H̃_{i}.

It follows from (i) that N_{i} := N_{H_{i},T_{i}} is a subgroup of N = N_{G,T} and S_{i} is pointwise fixed under the conjugating action of N_{i} on T. Since the subgroup T_{i} of N_{i} acts on T trivially, this action descends to an action of W_{i} := W_{H_{i},T_{i}} = N_{i}/T_{i}. Since H_{i} is connected reductive, any maximal torus of H_{i} coincides with its centralizer in H_{i}, see [Bor, 13.17]. As T_{i} is such a torus, this yields the equality T ∩ H_{i} = T_{i}. It shows that W_{i}, considered as a transformation group of T, is the image of N_{i} under the natural projection N → N/T = W.

Let π_{i} : H_{i} → H_{i}/(H_{i} ∩ H̃_{i}) be the natural epimorphism. Then π_{i}(T_{i}) is a maximal torus of H_{i}/(H_{i} ∩ H̃_{i}). It follows from (i) and [Bor, 11.20, 11.11] that π_{i} identifies W_{i} with W_{H_{i}/(H_{i}∩H̃_{i}),π_{i}(T_{i})}, so that the natural isomorphism T_{i}/(T_{i} ∩H̃_{i}) → π_{i}(T_{i}) is W_{i}-equivariant.

The above argument applied to H̃_{i} and S_{i} instead of H_{i} and T_{i} shows that T ∩ H̃_{i} = S_{i}. This, in turn, implies that

T_{i} ∩ H̃_{i} = T_{i} ∩ S_{i}.

Thus a maximal torus of H_{i}/(H_{i}∩H̃_{i}) is W_{i}-isomorphic to T_{i}/(T_{i}∩S_{i}). In turn, T_{i}/(T_{i}∩S_{i}) is W_{i}-isomorphic to T/S_{i} because T = T_{i}S_{i}. Therefore there is an exact sequence of W_{i}- homomorphisms of tori

e ⟶ S_{i} ⟶ T ⟶ T_{i}/(T_{i} ∩ S_{i}) ⟶ e.

Passing to the character groups, we deduce from it the following exact sequence of W_{i}- lattices

0 ⟶ { \atop T_{i}/(T_{i} ∩ S_{i})} ⟶ T̂ ⟶ Ŝ_{i} ⟶ 0.

As the action of W_{i} on S_{i} is trivial, Ŝ_{i} is a trivial and, in particular, a permutation W_{i}- lattice. Hence the above exact sequence shows that { \atop T_{i}/(T_{i} ∩ S_{i})} and T̂ are equivalent W_{i}-lattices.

Assume now that G is stably Cayley. Then Theorem 1.31 implies that T̂ is quasipermutation as a W -lattice, and hence as a W_{i}-lattice because W_{i} is a subgroup of W. Therefore the equivalent W_{i}-lattice { \atop T_{i}/(S_{i} ∩ T_{i})} is quasi-permutation as well. Since the latter is the character lattice of H_{i}/(H_{i} ∩ H̃_{i}), Theorem 1.31 implies that H_{i}/(H_{i} ∩ H̃_{i}) is stably Cayley. □

Corollary. Let G be a connected semisimple group. Let H_{1}, …, H_{m} be the minimal elements among its connected closed normal subgroups. Define H̃_{i} as in Proposition 4.12. If G is stably Cayley, then each H_{i}/(H_{i} ∩ H̃_{i}) is stably Cayley.

Proof. By [Bor, 14.10], the assumptions of Proposition 4.12 hold. □

Remark 4.13. In Proposition 4.12, if G is stably Cayley, H_{i} is not necessarily stably Cayley. For example, take G = GL_{n}, m = 2, H_{1} = G_{m} diagonally embedded in GL_{n} and H_{2} = SL_{n}. Then G is Cayley by Example 1.11, and H_{2} is not stably Cayley for n > 3 by Theorem 1.32.

5. Proof of Theorem 1.32: an overview

In this section we outline a strategy for proving Theorem 1.32; the technical parts of the proof will be carried out in Sections 6–8.

By Theorem 1.31, it will suffice to determine which connected simple groups have a stably rational generic torus (or, equivalently, a quasi-permutation character lattice). Cortella and Kunyavskiǐ in [CK, Theorem 0.1] have classified all simply connected and all adjoint connected simple groups that have quasi-permutation character lattice. These are precisely SO_{2n+1}, Sp_{2n}, PGL_{n}, SL_{3}, and G_{2}. Therefore in order to complete the proof of Theorem 1.32, we need to determine which intermediate (i.e., neither simply connected nor adjoint) connected simple groups have a quasi-permutation character lattice.

Recall that intermediate connected simple groups exist only for types A_{n} and D_{n}. Connected simple groups of type A_{n−1} are precisely the groups SL_{n}/μ_{d}, where d is a divisor of n. Among them, intermediate groups are those with 1 < d < n. In Section 7 we will prove the following. Proposition 5.1. Let d be a divisor of n, where 1 < d < n and (n, d) ≠ (4, 2). Then the character lattice of the group SL_{n}/μ_{d} is not quasi-permutation.

As we saw in Example 1.18, the group SL_{4}/μ_{2} is Cayley; in particular, by Theorem 1.31, its character lattice is quasi-permutation.

The intermediate connected simple groups of type D_{n} are SO_{2n} for any n ⩾ 3 and the half-spinor groups Spin_{2n}^{1/2} for even n ⩾ 4. The latter are defined as follows. Consider the spinor group Spin_{2n} for even n ⩾ 4. Its the center is isomorphic to μ_{2} × μ_{2}, see [Ch_{2}], [KMRT, §25], and consequently contains precisely three subgroups of order 2. One of them is the kernel of the vector representation, so the quotient of Spin_{2n} modulo it is SO_{2n}. Two others are the kernels of the half-spinor representations of Spin_{2n}. They are mapped to each other by an outer automorphism of Spin_{2n}, so the images of the half-spin representations are isomorphic to the same group that is Spin_{2n}^{1/2}.

By Example 1.18, the groups SO_{2n} are Cayley. If n = 4, the group of outer automorphisms of Spin_{2n} is isomorphic to S_{3} (for n > 4, it is isomorphic to S_{2}) and acts transitively on the set of all subgroups of order 2 of the center of Spin_{2n}. Therefore Spin_{8}^{1/2} ≃ SO_{8}, whence it is Cayley. Thus we only need to consider the half-spin groups Spin_{2n}^{1/2} for even n > 4. In Section 8 we will prove the following.

Proposition 5.2. The character lattice of the half-spinor group Spin_{2n}^{1/2} for even n > 4 is not quasi-permutation.

Thus in order to complete the proof of Theorem 1.32, we need to prove Propositions 5.1 and 5.2. This will be done in the next three sections.

6. The groups SL_{n}/μ_{d} and their character lattices

6.1. Lattices Q_{n}(d). For any divisor d of n, the Weyl group W of the group G = SL_{n}/μ_{d} is isomorphic to the permutation group S_{n} of the set of integers {1, …, n}. The character lattice X_{G} is described as follows.

Let ε_{1}, …, ε_{n} be the standard basis for the permutation S_{n}-lattice ℤ[S_{n}/S_{n−1}] on which σ ∈ S_{n} acts via

(6.2)  σ(ε_{i}) = ε_{σ(i)} for all i = 1, …, n.

We naturally embed ℤ[S_{n}/S_{n−1}] into the ℚ-vector space ℤ[S_{n}/S_{n−1}]⊗_{ℤ}ℚ endowed with the Euclidean structure such that ε_{1}, …, ε_{n} is the orthonormal basis and we naturally extend the action of S_{n} to this space.

The root system of type A_{n−1} is the subset

A_{n−1} := {ε_{i} − ε_{j} | 1 ⩽ i ≠ j ⩽ n}.

of ℤ[S_{n}/S_{n−1}] ⊗_{ℤ} ℚ. The Weyl group W (A_{n−1}) of A_{n−1} is S_{n} acting by (6.2), and the standard base of A_{n−1} is α_{1}, …, α_{n−1}, where

(6.3)  α_{i} = ε_{i} − ε_{i+1}, i = 1, …, n − 1,

see [Bou_{2}]. The kernel of augmentation map

ℤ[S_{n}/S_{n−1}] ⟶^{ϵ} ℤ, ∑_{i=1}^{n} a_{i}ε_{i} ↦ ∑_{i=1}^{n} a_{i},

is the root S_{n}-lattice ℤA_{n−1} of A_{n−1},

(6.4)  ℤA_{n−1} := ℤα_{1} ⊕ … ⊕ ℤα_{n−1} = {∑_{i=1}^{n} a_{i}ε_{i} | ∑_{i=1}^{n} a_{i} = 0}.

The character lattice of SL_{n}/μ_{d} is isomorphic to the following S_{n}-lattice

(6.5)  Q_{n}(d) := ℤA_{n−1} + ℤdϖ_{1}, where ϖ_{1} = ε_{1} − \frac{1}{n} ∑_{i=1}^{n} ε_{i}.

The vector ϖ_{1} is the first fundamental dominant weight of the root system A_{n−1} with respect to the base α_{1}, …, α_{n−1}.

Observe that the character lattice of SL_{n}/μ_{n} = PGL_{n} is the root S_{n}-lattice Q_{n}(n) = ℤA_{n−1}, the character lattice of SL_{n}/μ_{1} = SL_{n} is the weight S_{n}-lattice Λ_{n} of type A_{n−1}, and that the following sequences of homomorphisms of S_{n}-lattices are exact:

0 ⟶ ℤA_{n−1} ⟶ Q_{n} (n/d) ⟶ ℤ/dℤ ⟶ 0, (6.6) 0 ⟶ Q_{n}(d) ⟶ Λ_{n} ⟶ ℤ/dℤ ⟶ 0. (6.7)

Here ℤ/dℤ denotes the cyclic group of order d with trivial S_{d}-action. Note that

(6.8)  Q_{n}(d)^{∗} ≃ Q_{n}(n/d).

In this section we will prove a number of preliminary results about the lattices Q_{n}(d). In the next section we will use these results to prove Proposition 5.1.

6.9. Properties of Q_{n}(d). We begin by recalling a simple lemma which computes the cohomology H^{1}(Γ, ℤA_{n−1}) for all subgroups Γ of S_{n}. The first part is extracted from [LL, Lemma 4.3]. Lemma 6.10. For any subgroup Γ of S_{n}, we have

H^{1}(Γ, ℤA_{n−1}) ≃ ℤ/ ∑_{O} |O|ℤ,

where O runs over the orbits of Γ in {1, …, n}. More explicitly, the connecting homomorphism of the cohomology sequence induced by the augmentation sequence

(6.11)  0 ⟶ ℤA_{n−1} ⟶ ℤ[S_{n}/S_{n−1}] ⟶^{ϵ} ℤ ⟶ 0,

is given by

ℤ = ℤ[S_{n}/S_{n−1}]/ℤA_{n−1} ⟶^{∂} H^{1}(Γ, ℤA_{n−1}), mε_{1} + ℤA_{n−1} ↦ [σ ↦ m(ε_{σ(1)} − ε_{1})],

where the image is the class of the given 1-cocycle from Γ to ℤA_{n−1}.

Proof. From the cohomology sequence that is associated with (6.11), one obtains the exact sequence ℤ[S_{n}/S_{n−1}]^{Γ} →^{ϵ} ℤ →^{∂} H^{1}(Γ, ℤA_{n−1}) → 0 which implies the asserted description of H^{1}(Γ, ℤA_{n−1}). The calculation of the connecting homomorphism ∂ follows directly from the identification of ℤ with ℤ[S_{n}/S_{n−1}]/ℤA_{n−1} and an application of the Snake Lemma. □

Lemma 6.12. For any subgroup Γ of S_{n}, the exact sequence (6.6) induces the following connecting homomorphism in cohomology:

ℤ/dℤ = Q_{n}(n/d)/ℤA_{n−1} ⟶^{∂} H^{1}(Γ, ℤA_{n−1}), m + dℤ ↦ \frac{mn}{d} + ∑_{O} |O|ℤ,

where the sum on the right runs over the orbits O of Γ in {1, …, n}. In particular, if |H^{1}(Γ, ℤA_{n−1})| divides n/d, then ∂ is the zero map.

Proof. Since Q_{n}(n/d) has ℤ-basis \frac{n}{d}ϖ_{1}, ε_{1} − ε_{2}, …, ε_{n−2} − ε_{n−1} where ϖ_{1} is given by (6.5), we conclude that Q_{n}(n/d)/ℤA_{n−1} is generated by \frac{n}{d}ϖ_{1} + ℤA_{n−1}. Using the Snake Lemma,

one sees that the connecting homomorphism ℤ/dℤ = Q_{n}(n/d)/ℤA_{n−1} →^{∂} H^{1}(Γ, ℤA_{n−1}) sends \frac{n}{d}ϖ_{1} + ℤA_{n−1} to the class of the 1-cocycle [σ ↦ \frac{n}{d}(ε_{σ(1)} − ε_{1})] in H^{1}(Γ, ℤA_{n−1}). An application of Lemma 6.10 and the identification ℤ/dℤ = Q_{n}(n/d)/ℤA_{n−1} completes the proof of the first statement. The second statement follows directly from the first. □

Lemma 6.13. Let Γ be a subgroup of S_{n} which fixes at least one integer i ∈ {1, …, n}. Then H^{1}(Γ, Q_{n}(d)) = 0.

Proof. Note that in this case, {ε_{t} −ε_{i} | t ≠ i} is a permutation basis for ℤA_{n−1} so that both ℤA_{n−1} and Λ_{n} = (ℤA_{n−1})^{∗} are permutation Γ-lattices. This implies that H^{1}(Γ, ℤA_{n−1}) = 0 = H^{1}(Γ, Λ_{n}). Observe that ν_{i} = ε_{i} − \frac{1}{n} ∑_{t=1}^{n} ε_{t} ∈ Λ_{n}^{Γ} and that ν_{i} + Q_{n}(d) = ϖ_{1} + Q_{n}(d) since ν_{i} −ϖ_{1} = ε_{i} −ε_{1} ∈ ℤA_{n−1} ⊆ Q_{n}(d). Then applying cohomology to the exact sequence (6.7), we obtain

Λ_{n}^{Γ} ⟶ ℤ/dℤ ⟶ H^{1}(Γ, Q_{n}(d)) ⟶ H^{1}(Γ, Λ_{n}) = 0.

Since Λ_{n}/Q_{n}(d) = ℤ/dℤ is generated by ϖ_{1} + Q_{n}(d), the above argument shows that the map Λ_{n}^{Γ} → ℤ/dℤ is surjective so that H^{1}(Γ, Q_{n}(d)) = 0, as required. □

For a sequence of integers 1 ⩽ i_{1} < … < i_{r} ⩽ n, set

S_{{i_{1},...,i_{r}}} := {σ ∈ S_{n} | σ(j) = j for every j ∉ {i_{1}, …, i_{r}}}.

This is a subgroup of S_{n}; in particular, S_{{1,...,n}} = S_{n}. The map

ι_{{i_{1},...,i_{r}}} : S_{r} ⟶ S_{{i_{1},...,i_{r}}}, ι_{{i_{1},...,i_{r}}}(σ)(i_{s}) = i_{σ(s)} for all σ and s,

is an isomorphism. In the sequel, the subgroup S_{{1,...,m}} × S_{{m+1,...,2m}} of S_{2m} is denoted simply by S_{m} × S_{m}. For a sequence of integers

1 ⩽ i_{1} < … < i_{r} < j_{1} < … < j_{r} < … < l_{1} < … < l_{r} ⩽ n,

the image of the embedding

S_{r} ⟶ S_{n}, σ ↦ ι_{{i_{1},...,i_{r}}}(σ)ι_{{j_{1},...,j_{r}}}(σ) … ι_{{l_{1},...,l_{r}}}(σ),

is called the copy of S_{r} diagonally embedded in S_{{i_{1},...,i_{r},j_{1},...,j_{r},...,l_{1},...,l_{r}}}. Lemma 6.14. Let n = td. Then the following properties hold:

(a) Let X_{d} be the copy of S_{d} diagonally embedded in S_{n}. Then

ℤA_{n−1}|_{X_{d}} ≃ ℤA_{d−1} ⊕ ℤ[S_{d}/S_{d−1}]^{t−1}.

(b) Let Y_{d} := S_{{1,...,d}} ×X̃_{d} where X̃_{d} is the copy of S_{d} diagonally embedded in S_{{d+1,...,n}}. Then

ℤA_{n−1}|_{Y_{d}} ≃ ℤA_{2d−1}|_{S_{d}×S_{d}} ⊕ ℤ[(S_{d} × S_{d})/(S_{d} × S_{d−1})]^{t−2}.

Proof. For the first statement, note that

{ε_{i} − ε_{d+i} | i = 1, …, (t − 1)d} ∪ {ε_{1} − ε_{2}, …, ε_{d−1} − ε_{d}}

is a basis for ℤA_{n−1}, since {α_{i} = ε_{i} − ε_{i+1} | i = 1, …, n − 1} is a basis for ℤA_{n−1} and

ε_{i} − ε_{d+i} = ∑_{t=i}^{d+i−1} α_{k}

for i = 1, …, (t − 1)d. But then

ℤA_{n−1}|_{X_{d}} = ∑_{r=1}^{t−1}(∑_{i=(r−1)d+1}^{rd} ℤ(ε_{i} −ε_{d+i}))⊕∑_{i=1}^{d−1} ℤ(ε_{i} −ε_{i+1}) ≃ ℤ[S_{d}/S_{d−1}]^{t−1} ⊕ℤA_{d−1}.

For the second statement, similarly note that

{ε_{i} − ε_{d+i} | i = d + 1, …, (t − 1)d} ∪ {ε_{1} − ε_{2}, …, ε_{2d−1} − ε_{2d}}

is a basis for ℤA_{n−1} so that

ℤA_{n−1}|_{Y_{d}} = ∑_{r=2}^{t−1}(∑_{i=(r−1)d+1}^{rd} ℤ(ε_{i} − ε_{d+i})) ⊕ ∑_{i=1}^{2d−1} ℤ(ε_{i} − ε_{i+1})
≃ ℤ[(S_{d} × S_{d})/(S_{d} × S_{d−1})]^{t−2} ⊕ ℤA_{2d−1}|_{S_{d}×S_{d}}. □

7. Stably Cayley groups of type A_{n}

7.1. Restricting Q_{n}(d) to some subgroups. In this section we will prove Proposition 5.1. We will first show that Q_{n}(d) restricted to certain appropriate subgroups of S_{n} is equivalent in each case to a smaller more manageable sublattice. We will then show that the smaller lattices are not quasi-permutation. Proposition 7.2. Suppose d|n and let p be a prime divisor of n/d. Let X_{p} be the copy of S_{p} diagonally embedded in S_{n}, and let Y_{p} = S_{{1,...,p}} × X̃_{p}, where X̃_{p} is the copy of S_{p} diagonally embedded in S_{{p+1,...,n}}. Then the following equivalencies hold:

(a) Q_{n}(d)|_{X_{p}} ∼ Λ_{p}.

(b) Q_{n}(d)|_{Y_{p}} ∼ Λ_{2p}|_{S_{p}×S_{p}}.

Proof. Recall that we have the exact sequence (6.6). The definition of p implies that n = lp for a positive integer l. By Lemma 6.14,

ℤA_{n−1}|_{X_{p}} ≃ ℤA_{p−1} ⊕ ℤ[S_{p}/S_{p−1}]^{l−1},
ℤA_{n−1}|_{Y_{p}} ≃ ℤA_{2p−1}|_{S_{p}×S_{p}} ⊕ ℤ[(S_{p} × S_{p})/(S_{p} × S_{p−1})]^{l−2}.

Using this and Lemma 6.10, we see that H^{1}(Γ, ℤA_{n−1}) = H^{1}(Γ, ℤA_{p−1}) = 0 or ℤ/pℤ for all subgroups Γ of X_{p} and that H^{1}(Γ, ℤA_{n−1}) = H^{1}(Γ, ℤA_{2p−1}) = 0 or ℤ/pℤ for all subgroups Γ of Y_{p}. Then by Lemma 6.12 and the fact that p divides n/d shows that the connecting homomorphism (ℤ/dℤ)^{Γ} → H^{1}(Γ, ℤA_{n−1}) is zero for all subgroups Γ of X_{p} or of Y_{p}. But then the sequence above restricted to X_{p} or Y_{p} satisfies the conditions of Proposition 2.13(b). This means that

Q_{n}(d)|_{X_{p}} = Q_{n}(n/d)^{∗}|_{X_{p}} ∼ (ℤA_{n−1})^{∗}|_{X_{p}} ∼ (ℤA_{p−1})^{∗} = Λ_{p},

Q_{n}(d)|_{Y_{p}} = Q_{n}(n/d)^{∗}|_{Y_{p}} ∼ (ℤA_{n−1})^{∗}|_{Y_{p}} ∼ (ℤA_{2p−1})^{∗}|_{S_{p}×S_{p}} = Λ_{2p}|_{S_{p}×S_{p}}. □

7.3. Lattices Λ_{p} and Λ_{2p}. The following lemma is essentially a rephrasing of a result proved by Bessenrodt and Le Bruyn in [BLB]:

Lemma 7.4. Let p > 3 be prime. Then Λ_{p} is not a quasi-permutation S_{p}-lattice.

Proof. Tensoring the augmentation sequence for ℤ[S_{n}/S_{n−1}] with ℤA_{n−1}, we obtain the exact sequence

(7.5)  0 ⟶ (ℤA_{n−1})^{⊗2} ⟶ ℤA_{n−1} ⊗ ℤ[S_{n}/S_{n−1}] ⟶^{τ} ℤA_{n−1} ⟶ 0.

We have

ℤA_{n−1} ⊗ ℤ[S_{n}/S_{n−1}] ≃ ℤ[S_{n}/S_{n−2}].

One can show that {(ε_{i} − ε_{j}) ⊗ ε_{j} | i ≠ j} is the set of elements of a permutation basis for ℤA_{n−1} ⊗ ℤ[S_{n}/S_{n−1}]. The map τ then sends (ε_{i} − ε_{j}) ⊗ ε_{j} to ε_{i} − ε_{j}.

For p prime, Bessenrodt and Le Bruyn in [BLB] show that

0 ⟶ (ℤA_{p−1})^{⊗2} ⟶ ℤ[S_{p}/S_{p−2}] ⟶ ℤA_{p−1} ⟶ 0

is a coflasque resolution of ℤA_{p−1} as an S_{p}-lattice. They also show that (ℤA_{p−1})^{⊗2} is permutation projective as an S_{p}-lattice but is only S_{p}-stably permutation if p = 2, 3. By duality, the stable equivalence class of ((ℤA_{p−1})^{⊗2})^{∗} is ρ(Λ_{p}), see Subsection 2.8). The statements above then imply that Λ_{p} is not a quasi-permutation S_{p}-lattice for any p > 3. □ Proposition 7.6. Let p be a prime and let

Γ := ⟨(1, …, p), (p + 1, …, 2p)⟩ ⩽ S_{p} × S_{p} ⩽ S_{2p}.

Then

(a) � ^{2}(Γ, Λ_{2p}) = 0. In particular, a lattice in the stable equivalence class ρ(Λ_{2p}) is coflasque as an Γ-lattice.

(b) If p is odd, Λ_{2p} is not quasi-permutation as an Γ-lattice and hence is not quasipermutation as an S_{p} × S_{p}-lattice.

Proof. (a) The second statement follows from the first. Note that any proper subgroup of Γ is cyclic, so that by the claim � ^{2}(S, Λ_{2p}) = 0 for all subgroups S of Γ. Then if

0 ⟶ Λ_{2p} ⟶ Q ⟶ M ⟶ 0

is a flasque resolution of Λ_{2p} considered as an S-lattice, then H^{1}(S, M) = � ^{2}(S, Λ_{2p}) = 0 by Lemma 2.12.

To prove the first statement, we need to first compute H^{1}(Γ, Λ_{2p}) and H^{2}(Γ, Λ_{2p}).

We have H^{1}(Γ, Λ_{2p}) = H^{−1}(Γ, ℤA_{2p−1}) by duality. Then

H^{−1}(Γ, ℤA_{2p−1}) = Ker_{ℤA_{2p−1}}(N_{Γ})/I_{Γ}ℤA_{2p−1},

where N_{Γ} is the endomorphism l ↦ ∑_{a∈Γ} al, and I_{Γ} is the augmentation ideal of ℤ[Γ], [Br]. We need to compute N_{Γ} on a basis for ℤA_{2p−1}: we have N_{Γ}(ε_{i} − ε_{i+1}) = 0 for i = 1, …, p − 1, p + 1, …, 2p − 1, and N_{Γ}(ε_{p} − ε_{p+1}) = p(ε_{1} + ⋯ + ε_{p} − ε_{p+1} − ⋯ − ε_{2p}). Then

Ker N_{Γ} = Span{ε_{1} − ε_{2}, …, ε_{p−1} − ε_{p}, ε_{p+1} − ε_{p+2}, …, ε_{2p−1} − ε_{2p}}.

But I_{Γ}ℤA_{2p−1} = Ker N_{Γ} as ((1, …, p) − id)(ε_{p+1} − ε_{i}) = ε_{i} − ε_{i+1}, i = 1, …, p − 1, ((p+1, …, 2p)−id)(ε_{1} −ε_{i}) = ε_{i} −ε_{i+1}, i = p+1, …, 2p−1. This shows that H^{1}(Γ, Λ_{2p}) = H^{−1}(Γ, ℤA_{2p−1}) = 0.

To determine H^{2}(Γ, Λ_{2p}), we use the restriction of the sequence

0 ⟶ ℤ ⟶ ℤ[S_{2p}/S_{2p−1}] ⟶ Λ_{2p} ⟶ 0

to Γ. Let

(7.7) C_{1} = ⟨(1, …, p)⟩, C_{2} = ⟨(p + 1, …, 2p)⟩ and P_{1} = ℤ[Γ/C_{2}], P_{2} = ℤ[Γ/C_{1}].

Then we have the following exact sequence of Γ-lattices

0 ⟶ ℤ ⟶ P_{1} ⊕ P_{2} ⟶ Λ_{2p} ⟶ 0.

Taking cohomology of this sequence, we get

0 = H^{1}(Γ, Λ_{2p}) ⟶ H^{2}(Γ, ℤ) ⟶ H^{2}(Γ, P_{1}) ⊕ H^{2}(Γ, P_{2})
⟶ H^{2}(Γ, Λ_{2p}) ⟶ H^{3}(Γ, ℤ) ⟶ H^{3}(Γ, P_{1} ⊕ P_{2}).

But by Shapiro’s Lemma, we have H^{2}(Γ, P_{i}) = H^{2}(ℤ/pℤ, ℤ) = ℤ/pℤ and H^{3}(Γ, P_{i}) = H^{3}(ℤ/pℤ, ℤ) = 0 for i = 1, 2. Also, by the Künneth formula, [Weib, p. 166],

H^{n}(Γ, ℤ)=⨁_{i+j=n} H^{i}(ℤ/pℤ, ℤ) ⊗ H^{j}(ℤ/pℤ, ℤ)
⊕ ⨁_{i+j=n+1}Tor_{ℤ}^{1}(H^{i}(ℤ/pℤ, ℤ), H^{j}(ℤ/pℤ, ℤ)),

so that, in particular, H^{3}(Γ, ℤ) = ℤ/pℤ and H^{2}(Γ, ℤ) = (ℤ/pℤ)^{2}. This all yields an exact sequence

0 ⟶ (ℤ/pℤ)^{2} ⟶ (ℤ/pℤ)^{2} ⟶ H^{2}(Γ, Λ_{2p}) ⟶ ℤ/pℤ ⟶ 0,

and so H^{2}(Γ, Λ_{2p}) = ℤ/pℤ.

To show that � ^{2}(Γ, Λ_{2p}) = 0, it would suffice to find a cyclic subgroup C of Γ for which Res_{C}^{Γ} : H^{2}(Γ, Λ_{2p}) → H^{2}(C, Λ_{2p}) is injective.

Take C = C_{1}. Since H^{1}(Γ, Λ_{2p}) = 0, we have that the sequence

0 ⟶ H^{2}(Γ/C, Λ_{2p}^{C}) ⟶^{Inf} H^{2}(Γ, Λ_{2p}) ⟶^{Res} H^{2}(C, Λ_{2p})

is exact. So it suffices to show that H^{2}(Γ/C, Λ_{2p}^{C}) = 0.

The fundamental dominant weights for Λ_{2p} are

ϖ_{t} = ∑_{i=1}^{t} ε_{i} − \frac{t}{2p} ∑_{i=1}^{2p} ε_{i}, t = 1, …, 2p − 1.

Let ν_{i} = ε_{i} − \frac{1}{2p} ∑_{i=1}^{2p} ε_{i}, i = 1, …, 2p. Note that

ν_{1} = ϖ_{1}, ν_{t} = ϖ_{t} − ϖ_{t−1}, t = 2, …, 2p − 1, ν_{2p} = −ϖ_{2p−1}.

This shows that ν_{1}, …, ν_{p}, ϖ_{p+1}, …, ϖ_{2p−1} is another basis for Λ_{2p} and that

Λ_{2p}|_{C} = ⨁_{i=1}^{p} ℤν_{i} ⊕ ⨁_{i=p+1}^{2p−1} ℤϖ_{i} ≃ ℤ[C] ⊕ ℤ^{p−1}.

This shows that

Λ_{2p}^{C} = ℤ(∑_{i=1}^{p} ν_{i}) ⊕ ⨁_{i=p+1}^{2p−1} ℤϖ_{i} = ⨁_{i=p}^{2p−1} ℤϖ_{i} = ⨁_{i=p+1}^{2p} ℤν_{i}.

But Γ/C permutes ν_{p+1}, …, ν_{2p} cyclically so that Λ_{2p}^{C} ≃ ℤ[Γ/C]. This implies that H^{2}(Γ/C, Λ_{2p}^{C}) = 0 as required.

(b) To prove that Λ_{2p} is not Γ-quasi-permutation, we will construct a coflasque Γresolution of ℤA_{2p−1} (dually, a flasque resolution of Λ_{2p}) and show that the flasque lattice is not permutation projective as an Γ-lattice.

As α_{1}, …, α_{p−1} and α_{p+1}, … α_{2p−1} are the standard bases of the root subsystems of type A_{p−1}, we denote the Γ-sublattice of ℤA_{2p−1} generated by them simply by ℤA_{p−1} ⊕ ℤA_{p−1}. Let ι be its natural embedding into ℤA_{2p−1}. It is easily seen that α_{p} + ℤA_{p−1} ⊕ ℤA_{p−1} is Γ-stable. This implies that there is an exact sequence of Γ-lattices

0 ⟶ ℤA_{p−1} ⊕ ℤA_{p−1} ⟶^{ι} ℤA_{2p−1} ⟶ ℤ ⟶ 0.

A coflasque resolution of the Γ-lattice ℤA_{p−1} ⊕ ℤA_{p−1} is given by

0 ⟶ ℤ^{2} ⟶ P_{1} ⊕ P_{2} ⟶ ℤA_{p−1} ⊕ ℤA_{p−1} ⟶ 0

where P_{1} and P_{2} are defined by (7.7) and the generator of the Γ-lattice P_{1} (respectively P_{2}) is sent to α_{1} (respectively α_{p+1}).

We now extend ι to a coflasque resolution of the Γ-lattice ℤA_{2p−1}. Let

P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ ⟶^{ϱ} ℤA_{2p−1}

be a map of Γ-lattices where ϱ_{P_{1}⊕P_{2}} = ι, ϱ sends 1 ∈ ℤ[Γ] to α_{p} and ϱ sends the 1 ∈ ℤ to ∑_{i=1}^{p} ε_{i} − ∑_{i=p+1}^{2p} ε_{i} = 2ϖ_{p}. It is easily verified that ϱ is surjective (in fact ϱ|_{ℤΓ} is surjective).

Let L = Ker ϱ. To check that L is coflasque and hence that

0 ⟶ L ⟶ P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ ⟶^{ϱ} ℤA_{2p−1} ⟶ 0

is a coflasque resolution of ℤA_{2p−1}, we need only verify that for R := P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ, we have ϱ(R^{K}) = (ℤA_{2p−1})^{K} for all subgroups K of Γ.

For K = Γ or a cyclic subgroup generated by a disjoint product of two p-cycles, (ℤA_{2p−1})^{K} = ℤ2ϖ_{p} so that ϱ(ℤ^{K}) = ϱ(ℤ) = (ℤA_{2p−1})^{K} and so ϱ(R^{K}) = (ℤA_{2p−1})^{K}.

The only other subgroups are C_{1} and C_{2}. As the arguments are similar, we just consider C_{1}: the lattice (ℤA_{2p−1})^{H_{1}} has basis 2ϖ_{p}, α_{p+1}, …, α_{2p−1}, and we have ϱ(ℤ) = ℤ2ϖ_{2} and ϱ(P_{2}^{H_{1}}) = ϱ(P_{2}) = ⊕_{i=p+1}^{2p−1}ℤα_{i}. This shows that

0 ⟶ L ⟶ P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ ⟶^{ϱ} ℤA_{2p−1} ⟶ 0

is a coflasque resolution. Dualizing, we obtain a flasque resolution for Λ_{2p}:

0 ⟶ Λ_{2p} ⟶ P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ ⟶ L^{∗} ⟶ 0.

We have � ^{2}(Γ, Λ_{2p}) = H^{1}(Γ, L^{∗}) = 0. This shows that L is flasque and coflasque as a Γ-lattice.

We have the following commutative diagram with exact rows and columns:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & ℤ^{2} \ar[r] \ar[d] & P_{1} ⊕ P_{2} \ar[r]^{ι} \ar[d] & ℤA_{p−1} ⊕ ℤA_{p−1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & L \ar[r] \ar[d] & P_{1} ⊕ P_{2} ⊕ ℤ[Γ] ⊕ ℤ \ar[d] \ar[r]^{ϱ} & ℤA_{2p−1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & U(p) \ar[r] \ar[d] & ℤ[Γ] ⊕ ℤ \ar[r]^{θ} \ar[d] & ℤ \ar[r] \ar[d] & 0 \\ & 0 & 0 & 0} (7.8)

where U(p) is the kernel of the induced map θ. Now 2ϖ_{p} = ∑_{i=1}^{p−1} i(α_{i} + α_{2p−i}) + pα_{p}. So θ sends 1 ∈ ℤΓ to \overline{α_{p}} and sends 1 ∈ ℤ to p\overline{α_{p}}. This shows that

{(h − 1, 0) | h ∈ Γ} ∪ {(−p, 1)}

is a set of elements of a ℤ-basis for U(p). Note that U(p) also satisfies

0 ⟶ U(p) ⟶ ℤ[Γ] ⟶ ℤ/pℤ,

so that ℚU(p) ≃ ℚ[Γ].

From the above diagram, we then see that ℚL ≃ ℚ[Γ] ⊕ ℚ^{2}. By [CW, Lemmas 2, 3], to determine whether or not L is permutation projective is equivalent to checking whether 𝔽_{p}L is a permutation module for 𝔽_{p}[Γ].

Tensoring the diagram (7.8) with 𝔽_{p} leaves it exact so we have the following commutative diagram with exact rows and columns:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & 𝔽_{p}^{2} \ar[r] \ar[d] & 𝔽_{p}P_{1} ⊕ 𝔽_{p}P_{2} \ar[r]^{id⊗ι} \ar[d] & 𝔽_{p}A_{p−1} ⊕ 𝔽_{p}A_{p−1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝔽_{p}L \ar[r] \ar[d] & 𝔽_{p}P_{1} ⊕ 𝔽_{p}P_{2} ⊕ 𝔽_{p}[Γ] ⊕ 𝔽_{p} \ar[d] \ar[r]^{id⊗ϱ} & 𝔽_{p}A_{2p−1} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝔽_{p}U(p) \ar[r] \ar[d] & 𝔽_{p}[Γ] ⊕ 𝔽_{p} \ar[r]^{id⊗θ} \ar[d] & 𝔽_{p} \ar[r] \ar[d] & 0 \\ & 0 & 0 & 0} .

Suppose that 𝔽_{p}L is permutation. Then since L is coflasque, the sequence

0 ⟶ L^{Γ} ⟶^{p} L^{Γ} ⟶ (L/pL)^{Γ} ⟶ 0

is exact so that (𝔽_{p}L)^{Γ} = L^{Γ}/pL^{Γ}. Since ℚ[L] ≃ ℚ[Γ] ⊕ ℚ^{2}, rank L^{Γ} = 3. But then dim_{𝔽_{p}}(𝔽_{p}L)^{Γ} = 3. This means that 𝔽_{p}L must then have three transitive components. Since rank L = p^{2} + 2 and p > 2, this means that 𝔽_{p}L = 𝔽_{p}[Γ] ⊕ 𝔽_{p}^{2}.

Looking at the ℤ-basis for U(p) given above, it is clear that 𝔽_{p}U(p) ≃ 𝔽_{p} ⊕ 𝔽_{p}I_{Γ} where 𝔽_{p}I_{Γ} is the augmentation ideal of 𝔽_{p}[Γ]. Then the left column of the last commutative diagram implies that we have a surjective map 𝔽_{p}[Γ] ⊕ 𝔽_{p}^{2} → 𝔽_{p} ⊕ 𝔽_{p}I_{Γ}. Since (𝔽_{p}I_{Γ})^{Γ} = 0, this would imply that we have a surjective map 𝔽_{p}[Γ] → 𝔽_{p}I_{Γ} or equivalently that 𝔽_{p}I_{Γ} is a cyclic 𝔽_{p}[Γ]-module. But since 𝔽_{p}[Γ] is a local ring with unique maximal ideal 𝔽_{p}I_{Γ}, Nakayama’s Lemma implies that 𝔽_{p}I_{Γ} is a cyclic 𝔽_{p}[Γ]-module if and only if 𝔽_{p}I_{Γ}/(𝔽_{p}I_{Γ})^{2} is generated by one element over 𝔽_{p}. As dim_{𝔽_{p}} 𝔽_{p}I_{Γ}/(𝔽_{p}I_{Γ})^{2} = 2, this is impossible. By contradiction, there is no such surjective map from 𝔽_{p}[Γ] to 𝔽_{p}I_{Γ}. This implies that 𝔽_{p}L is not permutation and hence L is not permutation projective as an ℤ[Γ]-module. This implies in turn that Λ_{2p} is not quasi-permutation as a Γ-lattice or as an S_{p} × S_{p}-lattice. □

Remark 7.9. Note that this argument fails for p = 2. Indeed, we showed that rank L = p^{2}+2 and if 𝔽_{p}L were permutation, it would have three transitive components. For p > 2, we used these facts to conclude that 𝔽_{p}L = 𝔽_{p}[Γ] ⊕ 𝔽_{p}^{2}. For p = 2, this is not so; here 𝔽_{2}L may have three permutation components, each of rank 2. Indeed, if Γ = ⟨g, h⟩ ≃ ℤ/2ℤ × ℤ/2ℤ, then one can define a surjective 𝔽_{2}[Γ]-homomorphism

𝔽_{2}[Γ/⟨g⟩] ⊕ 𝔽_{2}[Γ/⟨h⟩] ⊕ 𝔽_{2}[Γ/⟨gh⟩] → 𝔽_{2}I_{Γ} ⊕ 𝔽_{2}

by sending the generator of the first component to (1 + g, 0), the generator of the second component to (1 + h, 0) and that of the third component to (0, 1).

In fact, by Proposition 7.2, we see that Q_{4}(2)|_{Γ} ∼ Λ_{4}|_{Γ}. Since Q_{4}(2) is the character lattice of the Cayley group SL_{4}/μ_{2} ≃ SO_{6}, by Theorem 1.31 it must be quasi-permutation as an S_{4}-lattice and hence as an Γ-lattice. Alternatively, one could show directly that Q_{4}(2) is a sign-permutation S_{4}-lattice and hence is quasi-permutation.

7.10. Completion of the proof of Proposition 5.1. It now suffices to prove the following proposition to complete the proof of Proposition 5.1: Proposition 7.11. Suppose n/d is divisible by a prime p.

(a) If p > 2, then the S_{n}-lattice Q_{n}(d) is not quasi-permutation.

(b) If n > p^{2}, then the S_{n}-lattice Q_{n}(d) is not quasi-permutation.

Indeed, by part (a), the S_{n}-lattice Q_{n}(d) is not quasi-permutation if the prime factorization of n/d includes a prime larger than 2. On the other hand, if n/d = 2^{k}, then, by part (b), the S_{n}-lattice Q_{n}(d) is not quasi-permutation, for any (n, d) ≠ (4, 2), and Proposition 5.1 follows.

Proof. (a) Proposition 7.2 shows that Q_{n}(d)|_{Y_{p}} is equivalent to Λ_{2p}|_{S_{p}×S_{p}} which is not quasipermutation by Proposition 7.6. Thus Q_{n}(d) is not quasi-permutation as an Y_{p}-lattice and hence as an S_{n}-lattice as well.

(b) We have n = tp with t > p. Following the proof of Proposition 4.1(i) in [LL], we define a subgroup Γ ≃ ℤ/pℤ × ℤ/pℤ of S_{n} as follows. Arrange the numbers from 1 to n into a rectangular table with p columns and t rows, so that the first row is 1, …, p, the second row is p + 1, …, 2p, etc. Let σ_{i} be the p-cycle that cyclically permutes the ith row and leaves elements of all other rows fixed. Note that σ_{1}, …, σ_{t} are commuting p-cycles; explicitly

σ_{i} = ((i − 1)p + 1, (i − 1)p + 2, …, ip).

We now set Γ := ⟨α, β⟩, where

α := ∏_{i=1}^{t−1} σ_{i} and β := ∏_{i=1}^{p−1} σ_{i}^{−i} · ∏_{i=p+1}^{t} σ_{i}.

The subgroup Γ has orbits O_{i} = {(i − 1)p + 1, (i − 1)p + 2, …, ip}, i = 1, …, t, all of length p and every cyclic subgroup C of Γ has fixed points. This means that by Lemma 6.10

H^{1}(Γ, ℤA_{n−1}) ≃ ℤ/pℤ but H^{1}(C, ℤA_{n−1}) = 0.

Also by Lemma 6.13, we find that

H^{1}(C, Q_{n} (n/d)) = 0.

Then, Lemma 6.12 and the fact that p divides n/d shows that ℤ/dℤ →^{∂} H^{1}(Γ, ℤA_{n−1}) is the zero map. The following commutative diagram

\xymatrix{ℤ/dℤ \ar[r]^{0} \ar[d]^{Res} & H^{1}(Γ, ℤA_{n−1}) \ar[r] \ar[d]^{Res} & H^{1} (Γ, Q_{n} (n/d)) \ar[d]^{Res} \\ ∏_{a∈Γ} ℤ/dℤ \ar[r]^{0} & ∏_{a∈Γ} H^{1}(⟨a⟩, ℤA_{n−1}) = 0 \ar[r] & ∏_{a∈Γ} H^{1} (⟨a⟩, Q_{n} (n/d)) = 0}

shows that

ℤ/pℤ ≃ � ^{1}(Γ, ℤA_{n−1}) ⩽ � ^{1}(Γ, Q_{n} (n/d)) .

Now if M were a flasque lattice with ρ(Q_{n}(d)) = stable equivalence class of M, then M^{∗} is a coflasque lattice satisfying

0 ⟶ M^{∗} ⟶ P ⟶ Q_{n} (n/d) ⟶ 0,

so that by Lemma 2.12(a), � ^{2}(Γ, M^{∗}) ≃ � ^{1}(Γ, Q_{n}(n/d)) ≠ 0. Lemma 2.12(c) now shows that M^{∗} cannot be a direct summand of a quasi-permutation lattice and hence not stably permutation. This implies that M cannot be stably permutation and so Q_{n}(d) cannot be quasi-permutation. □

8. Stably Cayley groups of type D_{n}

8.1. Root system of type D_{n}. Let ε_{1}, … ε_{n} be the same as in Subsection 6.1. The root system of type D_{n} is the set

D_{n} = {±ε_{i} ± ε_{j} | 1 ⩽ i < j ⩽ n}.

It has a base α_{1}, …, α_{n}, where α_{1}, …, α_{n−1} are given by (6.3) and α_{n} = ε_{n−1} + ε_{n}. The fundamental dominant weights of D_{n} with respect to this base are ϖ_{i} = ε_{1} + ⋯ + ε_{i} for i = 1, …, n − 2,

ϖ_{n−1} = \frac{1}{2} ∑_{i=1}^{n−1} ε_{i} − \frac{1}{2}ε_{n} and ϖ_{n} = \frac{1}{2} ∑_{i=1}^{n−1} ε_{i} + \frac{1}{2}ε_{n}.

The Weyl group W (D_{n}) of D_{n} is (ℤ/2ℤ)^{n−1} ⋉ S_{n}, where (ℤ/2ℤ)^{n−1} consists of all even numbers of sign changes on {ε_{1}, …, ε_{n}} and S_{n} acts via (6.2). The root and weight W (D_{n})- lattices of D_{n} are respectively ℤD_{n} and Λ(D_{n}) := ℤϖ_{1} ⊕ … ⊕ ℤϖ_{n}.

8.2. Lattices Y_{2m} and Z_{2m}. As we explained in Section 5, we are interested in the case where n is even, n = 2m, m > 2. There are precisely the following three lattices between Λ(D_{2m}) and ℤD_{2m}:

(8.3) X_{2m} := ℤD_{2m} + ℤϖ_{1}, Y_{2m} := ℤD_{2m} + ℤϖ_{2m−1} and Z_{2m} := ℤD_{2m} + ℤϖ_{2m}.

The character lattice of Spin_{4m}^{1/2} (see Section 5) is isomorphic to either of the lattices Y_{2m} and Z_{2m} while X_{2m} is isomorphic to the character lattice of SO_{4m}. Note that ε_{1}, …, ε_{n} is the sign-permutation basis for X_{2m}; this is consistent with the fact that SO_{4m} is Cayley, see Theorem 1.31(a). Also note that

\{\frac{1}{2}(ε_{1} + ε_{2} + ε_{3} − ε_{4}), \frac{1}{2}(ε_{1} + ε_{2} − ε_{3} + ε_{4}), \frac{1}{2}(ε_{1} − ε_{2} + ε_{3} + ε_{4}), \frac{1}{2}(−ε_{1} + ε_{2} + ε_{3} + ε_{4})\}

is the sign-permutation basis for Y_{4}, and

\{\frac{1}{2}(ε_{1} + ε_{2} + ε_{3} + ε_{4}), \frac{1}{2}(ε_{1} + ε_{2} − ε_{3} − ε_{4}), \frac{1}{2}(ε_{1} − ε_{2} + ε_{3} − ε_{4}), \frac{1}{2}(−ε_{1} + ε_{2} + ε_{3} − ε_{4})\}

is that for Z_{4}; this is consistent with the fact that Spin_{8}^{1/2} is Cayley, see Section 5.

Our goal is to prove Proposition 5.2. In view of the aforesaid, this is equivalent to proving the following. Proposition 8.4. The W (D_{2m})-lattices Y_{2m} and Z_{2m} are not quasi-permutation for any m > 2.

Proof. For the subgroup S_{2m} of W (D_{2m}) acting by (6.2), we consider the S_{2m}-lattices Y_{2m}|_{S_{2m}} and Z_{2m}|_{S_{2m}} and compare them to the S_{2m}-lattice Q_{2m}(m) defined by (6.5) and (6.4),

(8.5)  Q_{2m}(m) = ℤα_{1} + … + ℤα_{2m−1} + ℤβ, where β := mε_{1} − \frac{1}{2} ∑_{i=1}^{2m} ε_{i},

that is isomorphic to the character lattice of SL_{2m}/μ_{m}, see Subsection 6.1.

First we observe that

α_{1}, …, α_{2m−2}, γ, ε_{2m−2} + ε_{2m−1}, where γ := \frac{1}{2} ∑_{i=1}^{m} ε_{i} − \frac{1}{2} ∑_{i=m+1}^{2m} ε_{i},

is a basis for Y_{2m} if m is odd, and for Z_{2m} if m is even. Since α_{1}, …, α_{2m−2}, ε_{2m−2} + ε_{2m−1} is a basis for ℤD_{2m−1}, (8.3) implies that proving this claim is equivalent to proving the equality

(8.6)  ℤD_{2m−1} + ℤγ = \begin{cases}ℤD_{2m} + ℤϖ_{2m−1} & if m is odd, \\ ℤD_{2m} + ℤϖ_{2m} & if m is even. \end{cases}

Note that

ϖ_{2m−1} − γ = ∑_{i=m+1}^{2m−1} ε_{i} ∈ ℤD_{2m−1} if m is odd,
ϖ_{2m} + γ = ∑_{i=1}^{m} ε_{i} ∈ ℤD_{2m−1} if m is even.

Therefore proving (8.6) is equivalent to proving the inclusion

ℤD_{2m} ⊆ ℤD_{2m−1} + ℤγ,

which in turn is equivalent to proving the inclusions

ε_{2m−1} ± ε_{2m} ∈ ℤD_{2m−1} + ℤγ.

Finally, the last inclusions indeed hold as we have

2γ + (ε_{2m−1} + ε_{2m}) = ∑_{i=1}^{m} ε_{i} − ∑_{i=m+1}^{2m−2} ε_{i} ∈ ℤD_{2m−1},
2γ − (ε_{2m−1} − ε_{2m}) = ∑_{i=1}^{m−1}(ε_{i} − ε_{m+i}) + (ε_{m} − ε_{2m−1}) ∈ ℤD_{2m−1}.

Thus the claim is proved.

Further, the easily checked equalities

β = γ + ∑_{i=1}^{m−1}(m − i)α_{i},
α_{2m−1} = 2γ − ∑_{i=1}^{m} iα_{i} − ∑_{i=1}^{m−2}(m − i)α_{m+i}

and (8.5) imply that α_{1}, …, α_{2m−2}, γ is a ℤ-basis for Q_{2m}(m).

We thus obtain the following exact sequences of S_{2m}-lattices:

0 ⟶ Q_{2m}(m) ⟶ Y_{2m}|_{S_{2m}} ⟶ ℤ ⟶ 0

if m is odd and

0 ⟶ Q_{2m}(m) ⟶ Z_{2m}|_{S_{2m}} ⟶ ℤ ⟶ 0

if m is even. Here the S_{2m}-lattice ℤ is generated by ε_{2m−2} + ε_{2m−1}, modulo Q_{2m}(m). We claim that the S_{2m}-action on this lattice is trivial. Indeed, on the one hand, the alternating subgroup of S_{2m} has to act on this lattice trivially because it has no non-trivial one-dimensional representations. On the other hand, as m > 2, the transposition (1, 2) acts trivially on ε_{2m−2} + ε_{2m−1}. Since the alternating subgroup and the transposition (1, 2) generate S_{2m}, this proves the claim.

The above exact sequences thus tell us that Y_{2m}|_{S_{2m}} ∼ Q_{2m}(m) if m is odd, and Z_{2m}|_{S_{2m}} ∼ Q_{2m}(m) if m is even. By Proposition 5.1, the S_{2m}-lattice Q_{2m}(m) is not quasi-permutation for any m > 2. Thus for m > 2, the W (D_{2m})-lattice Y_{2m} is not quasi-permutation if m is odd, and the W (D_{2m})-lattice Z_{2m} is not quasi-permutation if m is even, as their restrictions to S_{2m} are not quasi-permutation. Since Y_{2m} ≃ Z_{2m} as W (D_{2m})-lattices, this completes the proof. □

9. Which stably Cayley groups are Cayley?

In this section we will prove Theorem 1.35. The groups G = SO_{n}, Sp_{2n} and PGL_{n} are shown to be Cayley in Examples 1.18 and 1.13. It thus remains to consider SL_{3} and G_{2}.

9.1. The group SL_{3}. Proposition 9.2. The group SL_{3} is Cayley.

The proof below is based on analysis of the explicit formulas in [Vos, 4.9] and the geometric ideas of the proof of Proposition 9.2 given in [Pop_{2}]. We present it in a form that will also help us prove that G_{2} × G_{m}^{2} is Cayley, see Proposition 9.13 below. On the other hand, the spirit of arguments in [Pop_{2}] is close to that in [Isk_{4}]. Since [Isk_{4}] is the main ingredient we will use in showing that G_{2} is not Cayley, see Lemma 9.11 and Proposition 9.12 below, we will give an outline of the proof of Proposition 9.2 from [Pop_{2}] in the Appendix.

Proof. The Weyl group W of SL_{3} is S_{3}. Consider the following subalgebra D of Mat_{3×3}:

(9.3)  D := {diag(a_{1}, a_{2}, a_{3}) ∈ Mat_{3×3} | a_{i} ∈ k}

and the action of S_{3} on D given by

(9.4)  σ(diag(a_{1}, a_{2}, a_{3})) := diag(a_{σ(1)}, a_{σ(2)}, a_{σ(3)}) where σ ∈ S_{3}.

The S_{3}-stable subvarieties

(9.5) T = {X ∈ D | det X = 1} and 𝔱 = {Y ∈ D | tr Y = 0}

are respectively the maximal torus of SL_{3} and its Lie algebra, considered as W -varieties. By the Corollary of Lemma 3.6, it suffices to show that T and 𝔱 are birationally S_{3}-isomorphic.

Let D \ {0} → P(D), X ↦ [X], be the natural projection. Denote by P_{S_{3}-natural}^{2} and P_{S_{3}-twisted}^{2} the projective plane P(D) endowed respectively with the natural and “twisted” rational actions of S_{3} given by

σ([X]) := [σ(X)] and σ([X]) := [σ(X)^{sign σ}], where σ ∈ S_{3}, X ∈ D.

Let π : SL_{3} → PGL_{3} be the natural projection. Since d_{e}π is an isomorphism between the Lie algebras of SL_{3} and PGL_{3}, and PGL_{3} is a Cayley group, see Example 1.13, the Corollary of Lemma 3.6 tells us that 𝔱 is birationally S_{3}-isomorphic to the maximal torus π(T) of PGL_{3}. In turn, we have the following birational S_{3}-isomorphisms of S_{3}-varieties:

π(T) −−→^{≃} P_{S_{3}-natural}^{2}, π(X) ↦ [X],

P_{S_{3}-twisted}^{2} −−→^{≃} T, [diag(a_{1}, a_{2}, a_{3})] ↦ diag(a_{2}/a_{3}, a_{3}/a_{1}, a_{1}/a_{2}).

Thus we only need to show that P_{S_{3}-natural}^{2} and P_{S_{3}-twisted}^{2} are birationally S_{3}-isomorphic. We shall establish this in three steps.

Step 1. Consider the action of S_{3} on 𝔱 × 𝔱 given by

(9.6)  σ(Y, Z) := \begin{cases}(σ(Y), σ(Z)) & if σ is even, \\ (σ(Z), σ(Y)) & if σ is odd, \end{cases} where σ ∈ S_{3}, Y, Z ∈ 𝔱.

It determines the action of S_{3} on the surface P(𝔱) × P(𝔱). Denote by (P(𝔱) × P(𝔱))_{S_{3}-twisted} the surface endowed with this action.

We claim that the S_{3}-varieties P_{S_{3}-twisted}^{2} and (P(𝔱) × P(𝔱))_{S_{3}-twisted} are birationally S_{3}- isomorphic. Indeed, it is immediately seen that the rational map

φ : P_{S_{3}-twisted}^{2} −−→ (P(𝔱) × P(𝔱))_{S_{3}-twisted}, [X] ↦ ([X − \frac{tr(X)}{3}I_{3}], [X^{−1} − \frac{tr(X^{−1})}{3}I_{3}]),

is S_{3}-equivariant and we shall now construct a rational map inverse to φ. Note that for Y, Z ∈ 𝔱 in general position, Y, Z, I_{3} form a basis of the vector space D. Thus there are unique α, β, γ ∈ k such that

αZ + βY + γI = −Y Z

Note that α, β, and γ are, in fact, bihomogeneous rational functions of Y and Z of bidegree (1, 0), (0, 1) and (1, 1), respectively. We now consider the map

ψ: (P(𝔱) × P(𝔱))_{S_{3}-twisted} −−→ P_{S_{3}-twisted}^{2}, ([Y], [Z]) ↦ [Y + αI_{3}].

To compute ψ ∘ φ, note that if Y = X − \frac{tr(X)}{3}I_{3} and Z = X^{−1} − \frac{tr(X^{−1})}{3}I_{3}, then expanding

I_{3} = (Y + \frac{tr(X)}{3}I_{3})(Z + \frac{tr(X^{−1})}{3}I_{3}),

we see that α = \frac{tr(X)}{3} and thus ψ([Y], [Z]) = [X]. Thus ψ ∘ φ = id, and hence φ is a birational S_{3}-isomorphism.

Step 2. We now consider the linear action of S_{3} on 𝔱 ⊗ 𝔱 determined by the action (9.6) and the corresponding action of S_{3} on P(𝔱 ⊗ 𝔱). Then the Segre embedding

(P(𝔱) × P(𝔱))_{S_{3}-twisted} ↪ P(𝔱 ⊗ 𝔱)

is S_{3}-equivariant. Its image is a quadric Q in P(𝔱 ⊗ 𝔱) described as follows. Choose a basis D_{1} := diag(1, ζ, ζ^{2}), D_{2} := diag(1, ζ^{2}, ζ) of 𝔱, where ζ is a primitive cube root of unity. Set D_{ij} = D_{i} ⊗ D_{j}. Then

(9.7)  Q = {(α_{11} : α_{12} : α_{21} : α_{22}) | α_{11}α_{22} = α_{12}α_{21}},

where (α_{11} : α_{12} : α_{21} : α_{22}) is the point of P(𝔱 ⊗ 𝔱) corresponding to α_{11}D_{11} + α_{12}D_{12} + α_{21}D_{21} + α_{22}D_{22} ∈ 𝔱 ⊗ 𝔱.

Step 3. Decomposing 𝔱 ⊗ 𝔱 as a sum of S_{3}-submodules, we obtain

(9.8)  𝔱 ⊗ 𝔱 = V_{1} ⊕ V_{2} ⊕ V_{3},

where V_{1} = kD_{11} + kD_{22} is a simple 2-dimensional submodule and V_{2} = kD_{12}, V_{3} = kD_{21} are trivial 1-dimensional submodules. Since the S_{3}-fixed point (0 : 0 : 1 : 0) ∈ P(𝔱 ⊗ 𝔱) corresponding to V_{3} lies on Q, the stereographic projection Q −−→ P(V_{1} ⊕ V_{2}) from this point is a birational S_{3}-isomorphism.

Finally, the S_{3}-module D is isomorphic to V_{1} ⊕ V_{2}. Hence P(V_{1} ⊕ V_{2}) and P_{S_{3}-natural}^{2} are S_{3}-isomorphic.

To sum up, we have established the existence of the following birational S_{3}-isomorphisms:

\xymatrix{P_{S_{3}-twisted}^{2} \ar@{-->}[r]^{≃^{Step1}} & (P(𝔱) × P(𝔱))_{S_{3}-twisted} \ar@{-->}[r]^{≃^{Step2}} & Q \ar@{-->}[r]^{≃^{Step3}} & P_{S_{3}-natural}^{2}.}

This completes the proof of Proposition 9.2. □

9.9. The group G_{2}.

The Weyl group of G_{2} is the dihedral group S_{3} × S_{2} of order 12. The maximal torus of G_{2} and its Lie algebra are S_{3} × S_{2}-isomorphic respectively to T and 𝔱 given by (9.5), where the action of the first factor of S_{3} × S_{2} is defined, as in the case of SL_{3}, by (9.4), and that of the nontrivial element θ of the second factor by

(9.10)  θ(X) := X^{−1} for X ∈ T and θ(Y) := −Y for Y ∈ 𝔱.

We begin with the following surprising recent result due to Iskovskikh, [Isk_{4}]. Lemma 9.11. The S_{3} × S_{2}-varieties T and 𝔱 are not birationally S_{3} × S_{2}-isomorphic. Proof outline. Since T and 𝔱 are rational surfaces, the theory of rational G-surfaces, due to Manin [Ma] and Iskovskikh [Isk_{1}], [Isk_{3}], can be applied; this is precisely what is done in [Isk_{4}]. Minimal rational S_{3} × S_{2}-surfaces are known, and any equivariant birational isomorphism between two such surfaces can be written as a composition of so-called “elementary links”, which are completely enumerated in [Isk_{3}]. The argument in [Isk_{4}] amounts to constructing suitable minimal models for T and 𝔱, and explicitly checking that it is impossible to get from one to the other by a sequence of elementary links. □ Proposition 9.12. G_{2} is not a Cayley group.

Proof. By the Corollary of Lemma 3.6, this follows from Lemma 9.11. □

The following result illustrates how delicate the matter is. Proposition 9.13. G_{2} × G_{m}^{2} is a Cayley group.

Proof. By Corollary of Lemma 3.6, it suffices to show that T ×A^{2} and 𝔱×A^{2} are birationally S_{3} × S_{2}-isomorphic, where in both cases S_{3} × S_{2} acts via the first factor. We shall define a birational S_{3} × S_{2}-isomorphism between them in three steps.

Step 1. Let (𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} be the variety 𝔱 × 𝔱 endowed with the action of S_{3} × S_{2} given by

(9.14) (σ, ε)(Y, Z):=\begin{cases}(σ(Y), σ(Z)) & if sign σ = sign ε, \\ (σ(Z), σ(Y)) & otherwise, \end{cases} where (σ, ε) ∈ S_{3}×S_{2}, Y, Z ∈ 𝔱.

The latter determines the action of S_{3}×S_{2} on P(𝔱)×P(𝔱). Denote by (P(𝔱)×P(𝔱))_{S_{3}×S_{2}-twisted} the surface endowed with this action. Then

(𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} ⟶ (P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted}, (Y, Z) ↦ ([Y], [Z]),

is an algebraic vector S_{3}×S_{2}-bundle of rank 2. Since S_{3}×S_{2} acts on (P(𝔱)×P(𝔱))_{S_{3}×S_{2}-twisted} faithfully, Lemma 2.16(b) shows that (𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} and (P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted} × A^{2}, where S_{3} × S_{2} acts via the first factor, are birationally S_{3} × S_{2}-isomorphic.

Step 2. Let P_{S_{3}×S_{2}-twisted}^{2} be the projective plane P(D) endowed with the action of S_{3}×S_{2} given by

(σ, ε)([X]) := [σ(X)^{sign σ sign ε}], where (σ, ε) ∈ S_{3} × S_{2}, X ∈ D.

Then the rational maps

P_{S_{3}×S_{2}-twisted}^{2} −−→ T, [diag(a_{1}, a_{2}, a_{3})] ↦ diag(a_{2}/a_{3}, a_{3}/a_{1}, a_{1}/a_{2}), and

P_{S_{3}×S_{2}-twisted}^{2} −−→ (P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted}, [X] ↦ ([X − \frac{tr(X)}{3}I_{3}], [X^{−1} − \frac{tr(X^{−1})}{3}I_{3}]),

are birational S_{3} × S_{2}-isomorphisms — the arguments are similar to those in the proof of Proposition 9.2.

Step 3. Since

(𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} ⟶ 𝔱, (t_{1}, t_{2}) ↦ t_{1} − t_{2},

is an algebraic vector S_{3} × S_{2}-bundle of rank 2 and S_{3} × S_{2} acts on 𝔱 faithfully, applying Lemma 2.16(b) once again we conclude that (𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} is birationally S_{3} × S_{2}- isomorphic to 𝔱 × 𝔸^{2}, where S_{3} × S_{2} acts via the first factor.

To sum up, we have established the existence of the following birational S_{3} × S_{2}-isomor-phisms:

\xymatrix{T × A^{2} \ar@{-->}[r]^{≃^{Step2}} & (P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted} × A^{2} \ar@{-->}[r]^{≃^{Step1}} & (𝔱 × 𝔱)_{S_{3}×S_{2}-twisted} \ar@{-->}[r]^{≃^{Step3}} & 𝔱 × A^{2}.}

This completes the proof of Proposition 9.13. □

Remark 9.15. We do not know whether or not G_{2} × G_{m} is a Cayley group.

10. Generalization

The notions of Cayley map and Cayley group naturally lead to generalizations which will be considered in this section.

10.1. Generalized Cayley maps. Let G be a connected linear algebraic group and let 𝔤 be its Lie algebra. We consider G and 𝔤 as G-varieties with respect to the conjugating and adjoint actions respectively and denote by Rat_{G}(G, 𝔤) the set of all rational G-maps G −−→ 𝔤 endowed with the natural structure of a vector space over k(G)^{G}. Set Mor_{G}(G, 𝔤) := {φ ∈ Rat_{G}(G, 𝔤) | φ is a morphism}. Definition 10.2. An element φ ∈ Rat_{G}(G, 𝔤) (respectively, φ ∈ Mor_{G}(G, 𝔤)) is called a generalized Cayley map (respectively, generalized Cayley morphism) of G if φ is a dominant map.

We are now ready to state the main result of this subsection. Theorem 10.3. Every connected linear algebraic group admits a generalized Cayley morphism.

Our proof of Theorem 10.3 will proceed in three steps. First we will construct a generalized Cayley morphism for every reductive group (Corollary to Lemma 10.4), then a generalized Cayley map for an arbitrary linear algebraic group (Proposition 10.6), then a generalized Cayley morphism for an arbitrary linear algebraic group.

Our construction in the case of reductive groups relies on the following known fact; see [Lun_{2}, Lemme III.1], cf. [PV, 6.3]. Lemma 10.4. Assume that the group G is reductive. Let X be an affine algebraic variety endowed with an algebraic action of G and let x ∈ X be a nonsingular fixed point of G. Let T_{x} be the tangent space of X at x endowed with the natural action of G. Then there is a G-morphism ε : X → T_{x} étale at x (hence dominant) and such that ε(x) = 0.

Proof. We can consider X as a G-stable subvariety of a finite dimensional algebraic Gmodule V, see [PV, Theorem 1.5]. Since x is a fixed point of H, we can replace X by its image under the parallel translation v ↦ v − x and assume that x = 0. The tangent space T_{x} is identified with a submodule of V. Since G is reductive, the G-module V is semisimple. Hence V = T_{x} ⊕ M for some submodule M. Now we can take ε = π|_{X}, where π : V → T_{x} is the projection parallel to M. □

Taking X = G with the conjugating action and x = e, we obtain the following.

Corollary. Assume that G is reductive. Then there is a generalized Cayley morphism φ of G étale at e and such that φ(e) = 0.

The following special case of this construction was considered by Kostant and Michor, [KM]. Example 10.5. Assume that G is reductive. Consider an algebraic homomorphism ν : G → GL(S), where S is a finite dimensional vector space over k. Then the k-vector space V := End(S) has a natural G-module structure defined by g(h) := ν(g)hν(g)^{−1} for every g ∈ G and h ∈ V. If ν is injective, identify G with the image of ι ∘ ν, where ι : GL(S) ↪ V is the natural embedding. Then G is a G-stable subvariety of V and the restriction to 𝔤 = T_{e} of the G-invariant inner product (x, y) ↦ tr xy on V is nondegenerate. This yields the G-module decomposition V = 𝔤⊕𝔤^{⊥}, where 𝔤^{⊥} is the orthogonal complement to 𝔤 with respect to ( , ). The restriction to G of the projection V → 𝔤 parallel to 𝔤^{⊥} is a generalized Cayley morphism φ : G → 𝔤 étale at e such that φ(e) = 0. □ Proposition 10.6. Every connected linear algebraic group G admits a generalized Cayley map.

Proof. We use the notation of Proposition 4.4 and its proof. The group W_{L,T} is finite, hence reductive, and e ∈ T is its fixed point. Therefore Lemma 10.4 implies that there is a dominant W_{L,T} -morphism ε : T → 𝔱. The arguments in the proof of part (a) of Proposition 4.4 show that ε is N-equivariant. Consider an N-isomorphism (4.5). Then

ε × τ : C = T × U ⟶ 𝔱 ⊕ 𝔲 = 𝔠,

is a dominant N-morphism. Hence by Lemma 2.22, there is a dominant G-morphism

θ : G ×^{N} C ⟶ G ×^{N} 𝔠

such that θ|_{C} = ε × τ. Now, since, by Lemma 3.3, the G-morphisms γ_{C} and γ_{𝔠} given by (3.2) are birational G-isomorphisms, γ_{𝔠} ∘ θ ∘ γ_{C}^{−1} ∈ Rat_{G}(G, 𝔤) is a generalized Cayley map. □

Our next task is to deduce Theorem 10.3 from Proposition 10.6. Our argument will rely on the following simple lemma. Lemma 10.7. Every semi-invariant for the conjugating action of G on itself is, in fact, an invariant.

Proof. Suppose t ∈ k[G] is a semi-invariant. That is, there exists an algebraic character χ: G → G_{m} such that t(ghg^{−1}) = χ(g)t(h) for every g, h ∈ G. We may assume t is not identically zero. Setting g = h in the above formula, we obtain

t(g) = χ(g)t(g) for every g ∈ G.

Since G is connected and t is not identically zero, this implies that χ(g) = 1 for every g ∈ G, i.e., t ∈ k[G]^{G}. □

Theorem 10.3 is now an immediate consequence of Proposition 10.6 and Proposition 10.8 below. Proposition 10.8. Let φ∈Rat_{G}(G, 𝔤). Then there is f ∈k[G]^{G} such that

(i) {g ∈ G | f(g) = 0} is the indeterminacy locus of φ, (ii) fφ ∈ Mor_{G}(G, 𝔤).

Moreover, if φ is a generalized Cayley map of G, then (ii) may be replaced by (ii)′ fφ is a generalized Cayley morphism G → 𝔤.

Proof. We may assume that φ is not a morphism. Then the indeterminacy locus of φ is an unmixed closed subset X of G of codimension 1. Since, by [Pop_{1}, Theorem 6], the Picard group of the underlying variety of G is finite, this implies that there is t ∈ k[G] such that {g ∈ G | t(g) = 0} = X. As φ is G-equivariant, X is G-stable. Hence, by [PV, Theorem 3.1], t is a semi-invariant of G and therefore t ∈ k[G]^{G} by Lemma 10.7. Consequently the function f = t^{m} satisfies (i) and (ii) for a sufficiently large positive integer m. The second assertion of the proposition follows from Lemma 10.9 below. □

Lemma 10.9. Let ψ : X −−→ V be a dominant rational map, where X is an irreducible algebraic variety, V a vector space over k and dim X = dim V. Then for every nonzero function t ∈ k(X), at least one of the maps α := tψ and β := t^{2}ψ is dominant.

Proof. Put h_{i} := ψ^{∗}(x_{i}) ∈ k(X), where x_{1}, …, x_{n} are the coordinate functions on V with respect to some basis. Then K := ψ^{∗}(k(V)) = k(h_{1}, …, h_{n}), K_{1} := α^{∗}(k(\overline{α(X)})) = k(th_{1}, …, th_{n}) and K_{2} := β^{∗}(k(\overline{β(X)})) = k(t^{2}h_{1}, …, t^{2}h_{n}), where bar denotes the closure in V. All three fields contain the subfield K_{0} := k(…, h_{i}/h_{j}, …). We have trdeg_{k}K = n. Therefore trdeg_{k}K_{0} = n − 1.

Assume the contrary: neither tψ nor t^{2}ψ is dominant. Then trdeg_{k}K_{1} = trdeg_{k}K_{2} = n − 1. Since K_{1} = K_{0}(th_{i}) and K_{2} = K_{0}(t^{2}h_{i}) for any i, this implies that both th_{i} and t^{2}h_{i} are algebraic over K_{0}. Hence h_{i} = (th_{i})^{2}/t^{2}h_{i} is algebraic over K_{0}. Thus K is algebraic over K_{0}. Hence trdeg_{k}K = trdeg_{k}K_{0} = n − 1, a contradiction. □

10.10. The Cayley degree. Note that any generalized Cayley map φ : G −−→ 𝔤 has finite degree, i.e., deg φ := [k(G) : φ^{∗}(k(𝔤))] < ∞. Since we are working in characteristic zero, deg φ is equal to the number of points in the fiber φ^{−1}(x) for a general point x ∈ 𝔤; see, e.g., [Sh, II.6.3, Theorem 4].

Example 10.11. Let G be reductive. Take a nonzero integer n. Using Jordan decomposition arguments and the fact that any semisimple element of G lies in a maximal torus, [Bor, Theorem 10.6], one can show that for a general element t of a maximal torus of G, the equation x^{n} = t has exactly |n|^{rk G} solutions. Since G is reductive, its general element is semisimple, see Lemma 3.3(a). This implies that the G-morphism ϱ_{n} : G → G, g ↦ g^{n}, is a dominant map of degree |n|^{rk G}. Therefore if φ is a generalized Cayley map of G, then φ ∘ ϱ_{n} is its generalized Cayley map of degree |n|^{rk G}deg φ. In particular, this shows that

{deg φ | φ is a generalized Cayley morphism of G}

is an infinite set. □ Example 10.12. Let G = G_{1} × … × G_{n}, where G_{i} be a connected linear algebraic group with Lie algebra 𝔤_{i}. Let φ_{i} : G_{i} −−→ 𝔤_{i} be a generalized Cayley map. The interpretation of degree as the number of points in a general fiber implies that φ_{1} × … × φ_{n} : G −−→ 𝔤 = 𝔤_{1} ⊕ … ⊕ g_{n} is a generalized Cayley map and

(10.13)  deg (φ_{1} × … × φ_{n}) = deg φ_{1} … deg φ_{n}. □

By Definition 1.7, Cayley maps are exactly generalized Cayley maps of degree 1. This naturally leads to the following definition of a “measure of non-Cayleyness” of G. Definition 10.14. The Cayley degree of G is the number

(10.15)  Cay(G) := min_{φ} deg φ,

where φ runs through all generalized Cayley maps of G. If G is defined over a subfield K of k, then the Cayley K-degree of G is the number Cay_{K}(G) defined by the right-hand side of (10.15), where φ runs through all generalized Cayley maps of G defined over K.

Thus G is a Cayley group (respectively, Cayley K-group) if and only if Cay(G) = 1 (respectively, Cay_{K}(G) = 1). Therefore Theorem 1.35 yields a classification of all connected simple algebraic groups of Cayley degree 1. This result may be considered as a contribution to the solution of the following general problem: Problem 10.16. Find the Cayley degrees of connected simple algebraic groups and, more generally, the Cayley K-degrees of connected simple algebraic K-groups. Example 10.17. Let G be the K-form of PGL_{n} from Example 1.38. Then Cay(G) = 1 but Cay_{K}(G) > 1. □

From (10.13) and (10.15) we deduce that

Cay(G_{1} × … × G_{n}) ⩽ Cay(G_{1}) … Cay(G_{n}),
Cay_{K}(G_{1} × … × G_{n}) ⩽ Cay_{K}(G_{1}) … Cay_{K}(G_{n})

(10.18)

(in the last inequality, it is assumed that every G_{i} is defined over a subfield K of k).

In general, the equality in (10.18) does not hold. Example 10.19. Theorem 1.35 yields Cay(G_{2} × G_{m}^{2}) = 1 but Cay(G_{2})Cay(G_{m}^{2}) > 1. □

The following number can be considered as a natural “measure of stable non-Cayleyness” of G: Definition 10.20. The stable Cayley degree of G is the number

sCay(G) := min_{n⩾0}Cay(G × G_{m}^{n}).

If G is defined over a subfield K of k, then the stable Cayley K-degree of G is the number

sCay_{K}(G) := min_{n⩾0}Cay_{K}(G × G_{m}^{n}).

Thus G is a stably Cayley group (respectively, stably Cayley K-group) if and only if sCay(G) = 1 (respectively, sCay_{K}(G) = 1). Therefore Theorem 1.32 yields a classification of all connected simple algebraic groups of stable Cayley degree 1. This result may be considered as a contribution to the solution of the following general problem: Problem 10.21. Find the stable Cayley degrees of connected simple algebraic groups and, more generally, the stable Cayley K-degrees of connected simple algebraic K-groups.

It follows from Definition 10.20 that

(10.22)  sCay(G) ⩽ Cay(G), sCay_{K}(G) ⩽ Cay_{K}(G).

In general, the equality in (10.22) does not hold. Example 10.23. Theorem 1.35 and Definition 10.20 imply that Cay(G_{2}) > 1 (in Example 10.31 below we shall show that actually Cay(G_{2}) = 2) but sCay(G_{2}) = 1. □

Using our results, we can find Cayley degrees and stable Cayley degrees of some non-Cayley groups. Example 10.24. Let π: G → H be an isogeny (so Ker π is finite and the Lie algebra of H is 𝔤). Let φ : H −−→ 𝔤 be a generalized Cayley map. Since Ker π is a central subgroup of G and deg π = | Ker π|, the composition φ ∘ π : G −−→ 𝔤 is a generalized Cayley map of degree | Ker π|deg φ. From this and Definition 10.14 we deduce that

(10.25)  Cay(G) ⩽ |Ker π|Cay(H).

In particular, (10.25) implies the following:

(10.26)  If G is not Cayley, H is Cayley and Ker π ≃ μ_{2}, then Cay(G)=2.

If moreover G is not stably Cayley, then (10.26) and (10.22) imply that sCay(G) = 2.

For instance, Example 1.18 and Theorem 1.35 show that the assumptions of (10.26) hold if G = Spin_{n}, H = SO_{n}, n ⩾ 6, and π the natural projection. This and Theorem 1.32 yield

(10.27)  sCay(Spin_{n}) = Cay(Spin_{n}) = \begin{cases}2 & for n ⩾ 6, \\ 1 & for n ⩽ 5. \end{cases}

If G = SL_{n}/μ_{d}, H = PGL_{n} and π is the natural projection, then Examples 1.13, 1.18, Proposition 5.1, Theorem 1.32, and (10.25), (10.26) yield

Cay(SL_{n}/μ_{d}) ⩽ n/d,

(10.28) sCay(SL_{2m}/μ_{m}) = Cay(SL_{2m}/μ_{m}) = \begin{cases}2 & for m ⩾ 3, \\ 1 & for m ⩽ 2. \end{cases}

In turn, taking G = SL_{4}, H = SL_{4}/μ_{2} and π the natural projection, we deduce from (10.28), Theorems 1.32, 1.35 and (10.26) that

sCay(SL_{4}) = Cay(SL_{4}) = 2.

Note that as SL_{4} ≃ Spin_{6}, the last equality follows also from (10.27). □

The problem of finding Cay(G) admits the following useful reduction. Let C be a Cartan subgroup of G with the Lie algebra 𝔠. We use the notation of (1.26). By Lemma 2.22, if φ : G ×^{N} C −−→ G ×^{N} 𝔠 is a dominant rational G-map, then φ|_{C} : C −−→ 𝔠 is a dominant rational N-map and φ^{−1}(x) = φ|_{C}^{−1}(x) for a general point x ∈ 𝔠. Hence

(10.29)  deg φ = |φ^{−1}(x)| = |φ|_{C}^{−1}(x)| = deg φ|_{C}.

Thus Lemmas 3.3 and 2.22 yield a degree preserving bijection between generalized Cayley maps for G and dominant rational N-maps C −−→ 𝔠. Therefore

(10.30)  Cay(G) = min_{ψ} deg ψ,

where ψ runs through all dominant rational N-maps C −−→ 𝔠. If G is reductive, then ψ in (10.30) runs through all dominant rational W -maps T −−→ 𝔱. Example 10.31. Using (10.30), we can show that

Cay(G_{2}) = 2.

To that end it suffices, by (10.30) and Theorem 1.35, to prove the existence of a rational W -map T −−→ 𝔱 of degree 2. It can be constructed as follows.

Maintain the notation of Subsection 9.9. In the course of proving Proposition 9.13, see Step 2, we showed that T is birationally S_{3} × S_{2}-isomorphic to (P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted}. Consider the action of S_{3}×S_{2} on 𝔱⊗𝔱 determined by the action (9.14) and the corresponding action of S_{3} × S_{2} on P(𝔱 ⊗ 𝔱). Then the Segre embedding

(P(𝔱) × P(𝔱))_{S_{3}×S_{2}-twisted} ↪ P(𝔱 ⊗ 𝔱)

is S_{3} × S_{2}-equivariant. Its image is the quadric Q given by (9.7).

The points of V_{1} in (9.8) are fixed by θ and θ(D_{12}) = D_{21}. Hence D_{12} + D_{21} is a fixed point and V_{0} := V_{1} +k(D_{12} −D_{21}) is a submodule of the S_{3} ×S_{2}-module 𝔱⊗𝔱. Consequently, the point p = (0 : 1 : 1 : 0) is fixed and the plane P(V_{0}) ⊂ P(𝔱 ⊗ 𝔱) is stable for the action of S_{3} × S_{2} on P(𝔱 ⊗ 𝔱). Since the line P(V_{1}) ⊂ P(V_{0}) is stable as well, we obtain an action of S_{3} × S_{2} on the affine plane P(V_{0}) \ P(V_{1}). One easily sees that it is isomorphic to the adjoint action of S_{3} × S_{2} on 𝔱. Therefore P(V_{0}) and 𝔱 are birationally S_{3} × S_{2}-isomorphic.

It now only remains to note that since p ∉ Q and Q is a quadric, the restriction to Q of the stereographic projection P(𝔱⊗𝔱) −−→ P(V_{0}) from p is a rational S_{3} ×S_{2}-map of degree 2. □ Proposition 10.32. Let φ be a generalized Cayley map of a reductive group G. Then

deg φ = [k(G)^{G} : k(𝔤)^{G}].

If φ is a generalized Cayley morphism, Proposition 10.32 can be deduced from [Lun_{2}, Lemme Fondamental]. In the case where φ is the particular generalized Cayley morphism described in Example 10.5, a proof can be found in [KM, Corollary (3.3)].

Proof. Since W is a finite group acting on T and 𝔱 faithfully, we have [k(T) : k(T)^{W}] = [k(𝔱) : k(𝔱)^{W}] = |W |. From this we deduce that deg φ|_{T} := [k(T) : k(𝔱)] = [k(T)^{W} : k(𝔱)^{W}]. But, by (3.5), we have [k(T)^{W} : k(𝔱)^{W}] = [k(G)^{G} : k(𝔤)^{G}]. The claim now follows from (10.29). □

If X is an irreducible algebraic variety endowed with an action of an algebraic group H, and V is a vector space over k of dimension dim X endowed with a linear action of H, then rational dominant H-maps X −−→ V are described as follows. Let M be a submodule of the H-module k(X) such that (i) M is isomorphic to the H-module V^{∗}, (ii) k(X) is algebraic over the subfield k(M) generated by M over k.

By (ii), k(M)/k is a purely transcendental extension of degree dim X. Since k(V) is generated over k by V^{∗}, any isomorphism of H-modules V^{∗} → M can be uniquely extended up to an H-equivariant embedding ι: k(V) ↪ k(X) whose image is k(M). This embedding determines a dominant rational H-map ψ: X −−→ V such that ψ^{∗} = ι. We have deg ψ = [k(X) : k(M)]. Any dominant rational H-map X −−→ V is obtained in this way.

This description can be used for obtaining upper bounds for Cay(G). Example 10.33. Let G = G_{2}. Use the notation of Subsection 9.9. Let x_{i} be the restriction to T of the ith coordinate function on D, i.e., x_{i}(diag(a_{1}, a_{2}, a_{3})) = a_{i}. Then x_{1}x_{2}x_{3} = 1 and k(T) = k(x_{1}, x_{2}). Put

(10.34)  y_{i} := x_{i} − x_{i}^{−1}.

From (9.4), (9.10) and (9.5) it follows that

M := {α_{1}y_{1} + α_{2}y_{2} + α_{3}y_{3} | α_{1} + α_{2} + α_{3} = 0, α_{i} ∈ k}

is a submodule of the S_{3} × S_{2}-module k(T) that is isomorphic to the S_{3} × S_{2}-module 𝔱^{∗}. Let

(10.35)  z_{1} := y_{1} − y_{2}, z_{2} := y_{1} − y_{3}

Then z_{1}, z_{2} is a basis of M, so k(M) = k(z_{1}, z_{2}). We have k(x_{1}, z_{1}, z_{2}) = k(T) (because x_{2} = (x_{1}^{2} − 1)(x_{1}^{2}z_{1} + x_{1}z_{2} − x_{1}^{3} − x_{1}^{2} + x_{1} + 1)^{−1}). It follows from (10.34), (10.35) that

(10.36)  \begin{cases}−x_{2} + x_{2}^{−1} = z_{1} − x_{1} + x_{1}^{−1}, \\ x_{1}x_{2} − x_{1}^{−1}x_{2}^{−1} = z_{2} − x_{1} + x_{1}^{−1}. \end{cases}

Eliminating x_{2} and x_{2}^{−1} from (10.36), we obtain the following equation:

x_{1}^{6} − (z_{1} + z_{2})x_{1}^{5} + (z_{1}z_{2} − 2z_{1} − 2z_{2} − 1)x_{1}^{4} + (z_{1}^{2} + z_{2}^{2} − 5)x_{1}^{3}

+ (z_{1}z_{2} + 2z_{1} + 2z_{2} + 1)x_{1}^{2} + (z_{1} + z_{2} + 1)x_{1} + 1 = 0.

Thus for the conjugating and adjoint actions of H := S_{3} × S_{2} on X := T and V := 𝔱 respectively the above conditions (i), (ii) hold for M and [k(T) : k(M)] ⩽ 6. Hence Cay(G_{2}) ⩽ 6 (of course, by Example 10.31, we know that in fact Cay(G_{2}) = 2). □

Appendix. Alternative proof of Proposition 9.2: an outline

Step 1. Consider D, see (9.3), as an open subset of P^{3} given by x_{0} ≠ 0, and extend the S_{3}-action (9.4) up to P^{3} by

σ(a_{0} : a_{1} : a_{2} : a_{3}) = (a_{0} : a_{σ(1)} : a_{σ(2)} : a_{σ(3)}), where σ ∈ S_{3}.

The closure X of T in P^{3}, see (9.5), is the rational cubic surface given by x_{1}x_{2}x_{3}−x_{0}^{3} = 0. It has exactly three fixed points

a_{i} := (1 : ε^{i} : ε^{i} : ε^{i}), i = 1, 2, 3, ε^{3} = 1, ε ≠ 1,

and three singular (double) points

s_{1} = (0 : 1 : 0 : 0), s_{2} = (0 : 0 : 1 : 0), s_{3} = (0 : 0 : 0 : 1).

The hyperplane section of X given by x_{0} = 0 is H := l_{1} + l_{2} + l_{3}, where l_{i} is the line given by x_{0} = x_{i} = 0.

Since H is S_{3}-invariant, the S_{3}-action on X lifts to the surface X̃ obtained from X by the simultaneous blowing up μ : X̃ → X of s_{1}, s_{2}, s_{3}. The surface X̃ is smooth and T is its open S_{3}-stable subset.

Step 2. We have μ^{∗}(H) = ∑_{i} t_{i} + ∑_{ij} m_{ij} where t_{i} is the proper inverse image of l_{i} and μ^{−1}(s_{i}) = m_{ij} ∪ m_{ir}, {i, j, r} = {1, 2, 3}. The curves t_{i}, m_{ij} are isomorphic to P^{1} and form a 9-gon as shown on the figure below. Their intersections are transversal and the self-intersection indices are (t_{i}, t_{i}) = −1, (m_{ij}, m_{ij}) = −2.

Computing the canonical classes gives K_{X} = −H and KX̃ = μ^{∗}(−H). Hence

(A1)  (KX̃, KX̃) = (−H, −H) = deg X = 3.

Step 3. By the Castelnuovo criterion, the curves t_{i} are exceptional, so they can be simultaneously blown down: ν: X̃ → Y. The surface Y is smooth, and the S_{3}-invariance of t_{1} + t_{2} + t_{3} implies that the action of S_{3} on X̃ descends to Y. We can consider T as an open S_{3}-stable subset of Y.

It follows from (A1) that

(A2)  (K_{Y} , K_{Y}) = 6,

and Pic T = 0 implies that (Pic Y)^{S^{3}} is generated by D := ν_{∗}(∑_{ij} m_{ij}). Hence K_{Y} = nD for some nonzero integer n. Rationality of Y implies n < 0. From this, the Nakai–Moishezon criterion and (A2) one deduces that −K_{Y} is ample. In turn, using this fact, the Riemann– Roch theorem, the Castelnuovo rationality criterion and the Kodaira vanishing theorem, one shows that

(A3)  dim H^{0}(Y, O(−K_{Y})) = 7.

Applying the Riemann–Roch theorem again, one further deduces that the linear system | − K_{Y} | has no fixed components. Using this, (A3), Bertini’s theorem, the Riemann–Roch theorem and (A2) one shows that | − K_{Y} | has no base points.

Step 4. Thus | − K_{Y} | defines a morphism Y → P^{6} equivariant with respect to a certain action of S_{3} on P^{6}. Using (A2), one shows that in fact it is an embedding. We keep the notation Y for its image.

Consider on Y the linear system |R| of all hyperplane sections in P^{6} containing the fixed point a_{1} ∈ T ⊆ Y and singular at a_{1}. Such hyperplanes are tangent to Y at a_{1}, so

(A4)  dim |R| = 4.

This system |R| is an S_{3}-stable subsystem of |−K_{Y} |. Using Bertini’s theorem, one deduces that its general element is an irreducible rational curve whose singular locus is the double point a_{1}. This system has no fixed components, and (A4) implies that a_{1} is its unique base point.

Step 5. Let γ: Ỹ → Y be the blowing up of a_{1}. The action of S_{3} lifts to Ỹ. The proper inverse image |R̃| of |R| is a 4-dimensional S_{3}-stable linear system on Ỹ. It has no base points and separates points of an open subset of Ỹ. Hence |R̃| defines an S_{3}-equivariant morphism ψ: Ỹ → P^{3} with respect to a certain S_{3}-action on P^{3}. Its image Z := ψ(Ỹ) is an S_{3}-stable quadric in P^{3}, and ψ: Ỹ → Z is a birational S_{3}-isomorphism.

Step 6. Since the point a′_{2} := ψ ∘ γ^{−1}(a_{2}) ∈ Z is fixed by S_{3}, it follows from the complete reducibility of representations of reductive groups that there is an S_{3}-stable plane L ≃ P^{2} in P^{3} not passing through a′_{2}. Consider the stereographic projection π: Z −−→ L from a′_{2}; it is S_{3}-equivariant. The map π is defined at ψ ∘ γ^{−1}(a_{3}) and a′_{3} := π ∘ ψ ∘ γ^{−1}(a_{3}) ∈ L is a fixed point of S_{3}. Using the complete reducibility argument again, we conclude that there is an S_{3}-stable line l ⊂ L such that a′_{3} ∈ L \ l. Thus we obtain a faithful linear action of S_{3} on A^{2} ≃ L \ l. But there is a unique 2-dimensional faithful linear representation of S_{3}, namely that on 𝔱 given by (9.4), (9.5). This completes the proof. □

References

[BLB] C. Bessenrodt, L. Le Bruyn, Stable rationality of certain P GL_{n}-quotients, Invent. Math. 104 (1991), 179–199.

[BMT] G. Berhuy, M. Monsurrò, J.–P. Tignol, Cohomological invariants and R-triviality of adjoint classical groups, to appear in Math. Z. Available as preprint No. 95 at the site Linear Algebraic

         Groups and Related Structures, http://www.mathematik.uni-bielefeld.de/lag

[Bor] A. Borel, Linear Algebraic Groups: Second Enlarged Edition, Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, 1991.

[Bou_{1}] N. Bourbaki, Groupes et algèbres de Lie, Chap. II, III, Hermann, Paris, 1972. [Bou_{2}] N. Bourbaki, Groupes et algèbres de Lie, Chap. IV, V, VI, Hermann, Paris, 1968. [Bou_{3}] N. Bourbaki, Groupes et algèbres de Lie, Chap. VII, VIII, Hermann, Paris, 1975.

[Br] K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87, Springer-Verlag, New York, 1982.

[Ca] A. Cayley, Sur Quelques propriétés des déterminants gauches, J. rein. angew. Math. (Crelle) 32 (1846), 119–123. Reprinted in: The Coll. Math. Papers of Arthur Cayley, Vol. I, No. 52, Cambridge University Press, 1889, 332–336.

[Ch_{1}] C. Chevalley, On algebraic group varieties, J. Math. Soc. Japan 6 (1954), 303–324. [Ch_{2}] C. Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Springer, 1991.

[CW] G. Cliff, A. Weiss, Summands of permutation lattices for finite groups, Proc. of the AMS 110 (1999), no. 1, 17–20.

[C–TS_{1}] J.–L. Colliot–Thélène, J.–J. Sansuc, La R-équivalence sur les tores, Ann. Sci. École Norm.

Sup. (4) 10 (1977), no. 2, 175–229.

[C–TS_{2}] J.–L. Colliot–Thélène, J.–J. Sansuc, Principal homogeneous spaces under flasque tori: Ap-

plications, J. Algebra 106 (1987), 148–205.

[C–T] J.–L. Colliot–Thélène (with the collaboration of J.–J. Sansuc) The rationality problemfor fields under linear algebraic groups (with special regards to the Brauer group), IX Escuela Latinoamericana de Mathemáticas, Santiago de Chile, July 1988.

[CK] A. Cortella, B. Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), no. 2, 771–793.

[CM] V. Chernousov, A. Merkurjev, R-equivalence and special unitary groups, J. Algebra 209 (1998), 175–198.

[CR] C. Curtis, I. Reiner, Methods of Representation Theory With Applications to Finite Groups and Orders, Vol. I, Reprint of the 1981 original, Wiley Classics Library, A Wiley Interscience Publication, John Wiley and Sons, Inc, New York, 1990.

[Di] J. Dieudonné, La géométrie des groupes classiques, Erg. Math. Grenzg., Bd. 5, Springer-Verlag, 1971.

[Do] I. V. Dolgachev, Rationality of fields of invariants, in: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 3–16.

[HK] M. Hajja, M. C. Kang, Some actions of symmetric groups, J. Algebra 177 (1995), no. 2, 511–535.

[Höl] O. Hölder, Die Gruppen der Ordnungen p^{3}, pq^{2}, pqr, p^{4}, Math. Ann. 43 (1893), 301–412. [Isk_{1}] V. A. Iskovskikh, Minimal models of rational surfaces over arbitrary fields, Math. USSR–Izv.

14 (1980), no. 1, 17–39.

[Isk_{2}] V. A. Iskovskikh, Cremona group, in: Math. Encyclopaedia, Vol. 3, Sov. Encycl., Moscow, 1982,

(in Russian), p. 95.

[Isk_{3}] V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of

view of Mori theory, Russian Math. Surveys 51 (1996), no. 4, 585–652.

[Isk_{4}] V. A. Iskovskikh, Two non-conjugate embeddings of S_{3} × ℤ_{2} into the Cremona group, Proc.

Steklov Inst. of Math. 241 (2003), 93–97.

[KM] B. Kostant, P. W. Michor, The generalized Cayley map for an algebraic group to its Lie algebra, in: The Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math., Vol. 213, Birkhäuser Boston, MA, 2003, pp. 259–296.

[KMRT] M.–A. Knus, A. Merkurjev, M. Rost, J.–P. Tignol, The Book of Involutions, Colloquium Publications, Vol. 44, AMS, 1998.

[Kn] M. Kneser, Lectures on Galois Cohomology of Classical Groups, Tata Inst. of Fund. Research Lectures on Math., Vol. 47, Bombay, 1969.

[LB_{1}] L. Le Bruyn, Centers of generic division algebras, the rationality problem 1965–1990, Israel J.

Math. 76 (1991), no. 1–2, 97–111.

[LB_{2}] L. Le Bruyn, Generic norm one tori, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 401–407.

[LL] N. Lemire, M. Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), no. 4, 385–405.

[Lun_{1}] D. Luna, Sur la linearite au voisinage d’un point fixe, Unpublished manuscript, May 1972. [Lun_{2}] D. Luna, Slices étales, Bull. Soc. Math. France, Mémoire 33 (1973), 81–105. [Lun_{3}] D. Luna, Letter to V. L. Popov, January 18, 1975.

[Ma] Yu. I. Manin, Rational surfaces over perfect fields II, Math. USSR, Sb. 1 (1968), 141–168.

[Me] A. Merkurjev, R-equivalence and rationality problem for semisimple adjoint classical algebraic groups, Publ. Math. IHES 46 (1996), 189–213.

[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory: Third Enlarged Edition, Erg. Math. Grenzg., 3 Folge, 34, Springer-Verlag, 1994.

[Pop_{1}] V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional

homogeneous vector bundles, Math. USSR-Izv. 8 (1974), 301–327.

[Pop_{2}] V. L. Popov, Letter to D. Luna, March 12, 1975. [Pop_{3}] V. L. Popov, Sections in invariant theory, in: The Sophus Lie Memorial Conference (Oslo,

1992), Scand. Univ. Press, Oslo, 1994, pp. 315–361.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encycl. Math. Sci., Vol. 55, Springer Verlag, 1994, pp. 123–284.

[Pos] M. Postnikov, Lie Groups and Lie Algebras. Lectures in Geometry. Semester V, Mir, Moscow, 1986.

[Sa] D. J. Saltman, Invariant fields of linear groups and division algebras, in: Perspectives in Ring Theory, Antwerp, 1987, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 279–297.

[Se] J.–P. Serre, Espaces fibrés algébriques, in: Séminaire C. Chevalley E.N.S., 1958, Anneaux de Chow et applications, Exposé no. 1, Secr. math. 11 rue Pierre Curie, Paris 5e, 1958, 1-01– 1-37. Reprinted in: Exposés de Séminaires 1950–1999, Doc. Math. 1, Soc. Math. France, 2001, 107–139.

[Sh] I. R. Shafarevich, Basic Algebraic Geometry, Vol. 1, Second Ed., Springer-Verlag, Berlin, 1994.

[Spe] A. Speiser, Zahlentheoretische Sätze aus der Gruppentheorie, Math. Zeitschrift 5 (1919), 1–6.

[Spr] T. A. Springer, Linear Algebraic Groups: Second Edition, Progress in Math., Vol. 9, Birkhäuser, 1998.

[Sw] R. G. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148–158.

[Vos] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, AMS, Providence, RI, 1998.

[VK] V. E. Voskresenskiĭ, A. A. Klyachko, Toroidal Fano varieties and root systems, Math. USSR-Izv. 24 (1985), 221–244.

[Weib] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.

[Weil] A. Weil, Algebras with involution and the classical groups, J. Indian Math. Soc., n. Ser. 24 (1961), 589–623.

[Weyl] H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton, N.J., Princeton Univ. Press, 1935.

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

   E-mail address: nlemire@uwo.ca

Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia

   E-mail address: popovvl@orc.ru

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

   E-mail address: reichste@math.ubc.ca
   URL: www.math.ubc.ca/^{∼}reichst