IS THE FUNCTION FIELD OF A REDUCTIVE LIE ALGEBRA PURELY TRANSCENDENTAL OVER

THE FIELD OF INVARIANTS FOR THE ADJOINT ACTION?

JEAN-LOUIS COLLIOT-THÉLÈNE, BORIS KUNYAVSKIĬ, VLADIMIR L. POPOV, AND ZINOVY REICHSTEIN

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Abstract. Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)^{G} and k(𝔤)/k(𝔤)^{G} are purely transcendental. We show that the answer is the same for k(G)/k(G)^{G} and k(𝔤)/k(𝔤)^{G}, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_{n} or C_{n}, and negative for groups of other types, except possibly G_{2}. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers.

As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Contents

Introduction 2 1. Preliminaries on lattices, tori, and special groups 6 1.1. Γ-lattices 6 1.2. Tori 7 1.3. Special groups 7 2. Quotients, (G, S)-fibrations, and (G, S)-varieties 7 2.1. Geometric quotients 8 2.2. (G, S)-fibrations 8 2.3. (G, S)-varieties 12 2.4. Categorical quotients 13 3. Versal actions 14 4. The conjugation action and the adjoint action 18 4.1. Quotients by the adjoint action, versal (G, S)-varieties, and Kostant’s theorem 18 4.2. The generic torus 21 4.3. Equivalent versions of the purity questions 21 5. Reduction to the case where the group G is simple and simply connected 22 6. Are homogeneous spaces of the form G/T stably rational? 23 7. Nonrationality 26 8. Weight lattices for root systems of types A_{n}, C_{n}, and G_{2} 28 9. Rationality 29 10. Appendix: G/T versus T^{0} 34 Acknowledgements 35 References 35

Date: March 4, 2010. 2000 Mathematics Subject Classification. 14E08, 17B45, 14L30, 20C10, 14F22. Key words and phrases. Algebraic group, simple Lie algebra, rationality problem, integral representation, alge-

braic torus, unramified Brauer group.

2 JEAN-LOUIS COLLIOT-THÉLÈNE, BORIS KUNYAVSKIĬ, VLADIMIR L. POPOV, AND ZINOVY REICHSTEIN

Introduction

A field extension E/F is called pure (or purely transcendental or rational ) if E is generated over F by a finite collection of algebraically independent elements. A field extension E/F is called stably pure (or stably rational ) if E is contained in a field L which is pure over both F and E. Finally, we shall say that E/F is unirational if E is contained in a field L which is pure over F. In summary,

L_{pure}E_{stably pure}F _{pure}  L E_{unirational}F . _{pure}

Let k be a field. Unless otherwise mentioned, we assume that char(k) = 0. This is in particular a standing assumption in this section.

Let G be a connected reductive algebraic group over k. Let V be a finite dimensional k-vector space and let G ↪ GL(V) be an algebraic group embedding over k. Let k(V) denote the field of k-rational functions on V and k(V)^{G} the subfield of G-invariants in k(V). It is natural to ask whether k(V)/k(V)^{G} is pure (or stably pure).

This question may be viewed as a birational counterpart of the classical problem of freeness of the module of (regular) covariants, i.e., the k[V]^{G}-module k[V]; cf. [PV, Sects. 3 and 8]. (Here k[V] is the algebra of k-regular functions on V and k[V]^{G} is the subalgebra of its G-invariant elements.) The question of rationality of k(V) over k(V)^{G} also comes up in connection with counterexamples to the Gelfand–Kirillov conjecture; see [AOVdB] and the paragraph of this introduction right after the statement of Theorem 0.4.

Recall that a connected reductive group G is called split if there exists a Borel subgroup B of G defined over k and a maximal torus in B is split.

If G is split and the G-action on V is generically free, i.e., the G-stabilizers of the points of a dense open set of V are trivial, then the following conditions are equivalent:

(i) the extension k(V)/k(V)^{G} is pure; (ii) the extension k(V)/k(V)^{G} is unirational; (iii) the group G is a “special group”.

Over an algebraically closed field, special groups were defined by Serre [Ser_{1}] and classified by him and Grothendieck [Gro] in the 1950s; cf. §1.3. The equivalence of these conditions follows from Corollary 3.6 below; see also Lemma 3.10.

The purity problem for k(V)/k(V)^{G} is thus primarily of interest in the case where the Gaction on V is faithful but not generically free. For k algebraically closed, such actions have been extensively studied and even classified, under the assumption that either the group G or the G-module V is simple; for details and further references, see [PV, Sect. 7.3].

For these G-modules, purity for k(V)/k(V)^{G} is known in some special cases. For instance, one can show that this is the case if k[V]^{G} is generated by a quadratic form. In [AOVdB, Appendix A] one can find a sketch of a proof that k(V)/k(V)^{G} is pure if G = SL_{2} and dim(V) = 4 or 5.

Theorem 0.2(b) below (with G adjoint) gives the first known examples of a connected linear algebraic group G over an algebraically closed field k with a faithful but not generically free G-module V such that k(V) is not pure (and not even stably pure) over k(V)^{G}.

Let 𝔤 be the Lie algebra of G. The homomorphism Int: G → Aut(G) sending g ∈ G to the map Int(g): G → G, x ↦ gxg^{−1}, determines the conjugation action of G on itself, G × G → G, sending (g, x) to Int(g)(x). The differential of Int(g) at the identity is the linear map Ad(g): 𝔤 → 𝔤. This defines an action of G on 𝔤, called the adjoint action. As usual, we will denote the fields of k-rational functions on G, respectively, 𝔤, by k(G), respectively, k(𝔤), and the fields of invariant k-rational functions for the conjugation action, respectively, the adjoint action, by k(G)^{G}, respectively, k(𝔤)^{G}.

The purpose of this paper is to address the following purity questions.

Questions 0.1. Let G be a connected reductive group over k and let 𝔤 be its Lie algebra.

(a) Is the field extension k(𝔤)/k(𝔤)^{G} pure? stably pure?

(b) Is the field extension k(G)/k(G)^{G} pure? stably pure?

It is worth mentioning here two other natural purity questions arising in this situation, namely, that of the purity of k(𝔤)^{G} and k(G)^{G} over k. They are, however, not directly related to the questions we are asking and both have affirmative answers: for k(𝔤)^{G} this is proved in [Kos] and, for k(G)^{G}, in [Ste] (for simply connected semisimple G) and [Pop_{5}] (for the general case).

The main case of interest for us is that of split groups, but some of our results hold for arbitrary reductive groups.

We shall give a nearly complete answer to Questions 0.1 for split groups, in particular, when k is algebraically closed. Our results can be summarized as follows.

(i) (Corollary 4.8) Let G be a connected split reductive group over k. Then the field extensions k(𝔤)/k(𝔤)^{G} and k(G)/k(G)^{G} are unirational.

This is closely related to Theorem 0.3 below.

(ii) (Theorem 4.10) For a given connected reductive group G over k, the answers to Questions 0.1(a) and (b) are the same.

(iii) (Proposition 5.1) For a connected reductive group G over k and a central k-subgroup Z of G, the answers to Questions 0.1 for G/Z are the same as for G.

Taking Z to be the radical of G, we thus reduce Questions 0.1 to the case where G is semisimple. We shall further reduce them to the case where G is simple as follows. Recall that a semisimple group G is called simple if its Lie algebra is a simple Lie algebra. Its centre is then finite but need not be trivial. In the literature such a group is sometimes referred to as an almost simple group.

(iv) (Proposition 5.3) Suppose that G is connected, semisimple, and split. Denote the simple components of the simply connected cover of G by G_{1}, …, G_{n}. Let 𝔤_{i} denote the Lie algebra of G_{i}. Then the following properties are equivalent:

(a) k(𝔤)/k(𝔤)^{G} is stably pure;

(b) k(𝔤_{i})/k(𝔤_{i})^{G_{i}} is stably pure for every i = 1, …, n.

Similarly, the following properties are equivalent:

(a) k(G)/k(G)^{G} is stably pure;

(b) k(G_{i})/k(G_{i})^{G_{i}} is stably pure for every i = 1, …, n.

If we replace “stably pure” by “pure”, we can still show that the field extension k(𝔤)/k(𝔤)^{G} (respectively, k(G)/k(G)^{G}) is pure if each k(𝔤_{i})/k(𝔤_{i})^{G_{i}} (respectively, each k(G_{i})/k(G_{i})^{G_{i}}) is pure, but we do not know whether or not the converse holds.

Finally, in the case where G is simple we prove the following theorem.

Theorem 0.2. Let G be a connected, simple algebraic group over k and let 𝔤 be its Lie algebra. Then the field extensions k(G)/k(G)^{G} and k(𝔤)/k(𝔤)^{G} are

(a) pure, if G is split of type A_{n} or C_{n};

(b) not stably pure if G is not of type A_{n}, C_{n}, or G_{2}.

To prove Theorem 0.2, we show that the two equivalent Questions 0.1(a) and (b) are equivalent to the question of (stable) K_{gen}-rationality of the homogeneous space G_{K_{gen}}/T_{gen}, where T_{gen} is the generic torus of G, defined over the field K_{gen}; see Theorem 4.10. (For the definition of T_{gen} and K_{gen} see §4.2.) We then address this rationality problem for G_{K_{gen}}/T_{gen} by using the main result of [CTK], which gives a formula for the unramified Brauer group of a homogeneous space with connected stabilizers; see §6. This allows us to prove Theorem 0.2(b) in §7 by showing that if G is not of type A_{n}, C_{n}, or G_{2}, then the unramified Brauer group of G_{K_{gen}}/T_{gen} is nontrivial over some field extension of K_{gen}. This approach also yields Theorem 0.2(a), with “pure” replaced by “stably pure” (Proposition 8.2). The proof of the purity assertion in part (a) requires additional arguments, which are carried out in §9.

A novel feature of our approach is a systematic use of the notions of (G, S)-fibration and versal (G, S)-fibration, generalizing the well known notions of G-torsor and versal G-torsor; cf., e.g., [GMS, Sect. 1.5]. Here S is a k-subgroup of G. For details we refer the reader to Sections 2 and 3.

As a byproduct of our investigations we obtain the following two results which are not directly related to Questions 0.1 but are, in our opinion, of independent interest. Recall that a connected reductive group over a field k is called quasisplit if it has a Borel subgroup defined over k.

Theorem 0.3 (Corollary 4.8(a)). Let G be a connected quasisplit reductive group over k. Then the categorical quotient map G → G//G for the conjugation action has a rational section.

In the classical case G = SL_{n} such a section is given by the companion matrices. The existence of a regular section for an arbitrary connected, split, semisimple, simply connected group G is a theorem of Steinberg [Ste, Theorems 1.4 and 1.6]. In a letter to Serre, dated January 15, 1969, Grothendieck asked whether or not a rational section exists if G is not assumed to be simply connected; see [GS, p. 240]^{1}. Theorem 0.3 answers this question in the affirmative. The example specifically mentioned by Grothendieck is PGL_{2}, or GP(1), in his notation; an explicit rational section in this case is constructed in Remark 4.9. Our proof of Theorem 0.3 does not use Steinberg’s result, but it uses Kostant’s result on the existence of sections in the Lie algebra case ([Kos], [Kot_{2}]).

If k is algebraically closed and G is not simply connected, then by [Pop_{5}] there is no regular section of the categorical quotient map G → G//G.

1The exact quote is as follows: ‘Le théorème de Steinberg […] est-il vrai uniquement pour le [groupe] simplement connexe […] ? Que se passe-t-il par exemple pour GP(1) ? Y a-t-il une section rationnelle de G sur I(G)

(“invariants”) dans ce cas ?’

Theorem 0.4 (see Propositions 7.1 and 8.1). Let G be a connected, split, simple, simply connected algebraic group defined over k and let W be the Weyl group of G. The weight lattice P (G) of G fits into an exact sequence of W -lattices

0 → P_{2} → P_{1} → P (G) → 0 ,

with P_{1} and P_{2} permutation, if and only if G is of type A_{n}, C_{n}, or G_{2}.

Recently the results and methods of this paper have played an important part in A. Premet’s (negative) solution of the Gelfand–Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than A_{n}, C_{n}, and G_{2}. The Gelfand–Kirillov conjecture is known to be true for Lie algebras of type A_{n} and is still open for the types C_{n} and G_{2}. For details, including background material on the Gelfand–Kirillov conjecture, we refer the reader to [Pr].

This paper is dedicated to Valentin Evgen’yevich Voskresenskiĭ, who turned 80 in 2007. Professor Voskresenskiĭ’s work (see [Vos_{1}, Vos_{3}]) was the starting point for many of the methods and ideas used in the present paper.

Some terminology

By definition, a k-variety is a separated k-scheme of finite type. If X is a k-variety, it is naturally equipped with its structure morphism X → Spec k. As a consequence, any Zariski open set U ⊂ X is naturally a k-variety.

The fibre product over Spec k of two k-schemes X and Y is denoted X ×_{k} Y, or simply X × Y when the context is clear.

The set of k-rational points of a k-variety X is defined by X(k) = Mor_{k}(Spec k, X).

An algebraic group over k, sometimes simply called a k-group, is a k-variety equipped with a structure of algebraic group over k. In other words, there is a multiplication morphism G×_{k} G → G and a neutral element in G(k) satisfying the usual properties. In other terms, it is a k-group scheme of finite type.

The ring k[X] is the ring of global sections of the sheaf O_{X} over the k-variety X. The group k[X]^{×} is the group of invertible elements of k[X].

If X is an integral (i.e., reduced and irreducible) k-variety, we let k(X) be the field of rational functions on X. This is the direct limit of the fields of fractions of the k-algebras k[U] for U running through the dense open subsets U of X.

When we consider two k-varieties X and Y, a k-morphism from X to Y will sometimes simply be called a morphism or even a map.

Similarly, if H and G are algebraic groups over k, if the context is clear, a k-homomorphism of k-group schemes from H to G will sometimes simply be called a homomorphism, or even a morphism.

For any field extension K/k, we may consider the K-variety X_{K} = X ×_{k} K, where the latter expression is shorthand for X ×_{Spec k} Spec K. We write K[X] = K[X_{K}]. If the K-variety X_{K} is integral, we let K(X) be the function field of X_{K}.

An integral k-variety X is called stably k-rational if its function field k(X) is stably rational over k (see the first paragraph of the Introduction) or, equivalently, if there exists a k-birational isomorphism between X×_{k}𝔸_{k}^{n} and 𝔸_{k}^{m} for some integers n, m ⩾ 0. If n = 0, X is called k-rational.

1. Preliminaries on lattices, tori, and special groups

For the details on the results of this section, see [CTS_{1}], [CTS_{2}], [Vos_{3}], or [Lor].

1.1. Γ-lattices. Let Γ be a finite group. A Γ-lattice M is a free abelian group of finite type equipped with a homomorphism Γ → Aut(M). When the context is clear, we shall say lattice instead of Γ-lattice.

In this subsection we recall some basic properties of such lattices.

The dual lattice of a lattice M is the lattice M^{0} = Hom_{ℤ}(M, ℤ) where for γ ∈ Γ, m ∈ M and φ ∈ M^{0}, we have (γ · φ)(m) = φ(γ^{−1} · m).

A permutation lattice is a lattice which has a ℤ-module basis whose elements are permuted by Γ. The dual lattice of a permutation lattice is a permutation lattice.

Any lattice M may be realized as a sublattice of a permutation lattice P with torsion-free quotient P/M ([CTS_{2}] Lemma 0.6).

Two lattices M_{1} and M_{2} are called stably equivalent if there exist permutation lattices P_{1} and P_{2} and an isomorphism M_{1} ⊕ P_{1} ≅ M_{2} ⊕ P_{2}.

A lattice M is called a stably permutation lattice if there exist permutation lattices P_{1} and P_{2} and an isomorphism M ⊕ P_{1} ≅ P_{2}.

A lattice M is called invertible if there exists a lattice N such that M ⊕ N is a permutation lattice.

In these definitions one may replace Γ by its image in the group of automorphisms of M. Because of this one may give the analogous definitions for Γ a profinite group with a continuous and discrete action.

Let M be a Γ-lattice. For any integer i ⩾ 0 one writes

� _{ω}^{i}(Γ, M) = Ker[H^{i}(Γ, M) → ∏_{γ ∈ Γ} H^{i}(⟨γ⟩, M)].

For i = 1, 2, this kernel only depends on the image of Γ in the group of automorphisms of M. So for these i it is natural to extend the above definition to the case where Γ is a profinite group and the action is continuous and discrete.

If Γ is the absolute Galois group of a field K, one refers to lattices as Galois lattices and one uses the notation � _{ω}^{i}(K, M).

If M is a permutation lattice, then for any subgroup Γ′ of Γ,

H^{1}(Γ′, M) = 0, H^{1}(Γ′, M^{0}) = 0.

Moreover,

� _{ω}^{2}(Γ′, M) = 0, � _{ω}^{2}(Γ′, M^{0}) = 0.

If there exists an exact sequence

0 → P_{1} → P_{2} → M → 0

with P_{1} and P_{2} permutation lattices, then � _{ω}^{1}(Γ′, M) = 0 for any subgroup Γ′ of Γ.

1.2. Tori. Let K be a field, let K_{s} be a separable closure of K, and let Γ denote the Galois group of K_{s}/K. A K-torus T is an algebraic K-group which over an algebraic field extension L/K is isomorphic to a product of copies of the multiplicative group 𝔾_{m,L}. The field L is then called a splitting field for T. Inside K_{s} there is a smallest splitting field for T, it is a finite Galois extension of K, called the splitting field of T.

To any K-torus T one may associate two Γ-lattices: its (geometric) character group

T^{∗} = Hom_{K_{s}-gr}(T_{K_{s}}, 𝔾_{m,K_{s}})

and its (geometric) cocharacter group

T_{∗} = Hom_{K_{s}-gr}(𝔾_{m,K_{s}}, T_{K_{s}}).

These two Γ-lattices are dual of each other.

The association T ↦ T_{∗} defines an equivalence between the category of K-tori and the category of Γ-lattices. The association T ↦ T^{∗} defines a duality (anti-equivalence) between the category of K-tori and the category of Γ-lattices.

The K-torus whose character group is T_{∗} is denoted T^{0} and is called the torus dual to T. A K-torus is called quasitrivial, respectively, stably quasitrivial, if its character group, or equiv-

alently its cocharacter group, is a permutation lattice, respectively, is a stably permutation lattice. A quasitrivial torus T is K-isomorphic to a product of tori of the shape R_{L/K}𝔾_{m}, i.e., Weil restriction of scalars of the multiplicative group 𝔾_{m,L} from L to K, where L/K is a finite separable field extension. A quasitrivial K-torus is an open set of an affine K-space, hence is K-rational.

By a theorem of Voskresenskiĭ, a K-torus of dimension at most 2 is K-rational ([Vos_{3}, §2.4.9, Examples 6 and 7]). This implies the following property. For any Γ-lattice M which is a direct sum of lattices of rank at most 2, there exist exact sequences

0 → P_{2} → P_{1} → M → 0,

where the P_{i}’s and P′_{i}’s are permutation lattices.

1.3. Special groups. Let K be a field of characteristic zero. Recall from [Ser_{1}] that an algebraic group G over K is called special if for any field extension L/K, the Galois cohomology set H^{1}(L, G) is reduced to one point. In other words, G is special if every principal homogeneous space under G over a field containing K is trivial. Such a group is automatically linear and connected [Ser_{1}]. An extension of a special group by a special group is a special group. A unipotent group is special. A quasitrivial torus is special. So is a direct factor of a quasitrivial torus (such a K-torus need not be stably K-rational). If K is algebraically closed and G is semisimple, then G is special if and only if it is isomorphic to a direct product

SL_{n_{1}} × ⋯ × SL_{n_{r}} ×Sp_{2m_{1}} × ⋯ × Sp_{2m_{s}}

for some integers r, s, n_{1}, …, n_{r}, m_{1}, …, m_{s}. That such groups are special is proved in [Ser_{1}], that only these are is proved in [Gro].

2. Quotients, (G, S)-fibrations, and (G, S)-varieties

We recall that k is a field of characteristic zero. Let \overline{k} be an algebraic closure of k, and let G be a (not necessarily connected) linear algebraic group over k.

2.1. Geometric quotients. Let us recall some standard definitions and facts. For references, see [Bor, Sect. II.6], [Hum_{1}, Sect. 12], [Spr_{1}, Sects. 5.5, 12.2], [PV, Sect. 4], and [CTS_{4}].

Let X be a k-variety endowed with an action of the k-group G. A geometric quotient of X by G is a pair (Y, π), where Y is a k-variety, called the quotient space, and π: X → Y is a k-morphism, called the quotient map, such that

(i) π is an open orbit map, i.e., constant on G-orbits and induces a bijection of X(\overline{k})/G(\overline{k}) with Y (\overline{k});

(ii) for every open subset V of Y, the natural homomorphism π^{∗}: k[V] → k[π^{−1}(V)]^{G} is an

isomorphism. If such a pair (Y, π) exists, it has the universal mapping property, i.e., for every k-morphism

α: X → Z constant on the fibres of π, there is a unique k-morphism β: Y → Z such that α = β ∘ π. In particular, (Y, π) is unique up to a unique G-equivariant isomorphism of total spaces commuting with quotient maps. Given this, we shall denote Y by X/G.

Remark 2.1. Let X and hence X/G be geometrically integral. Being constant on G-orbits, π induces an embedding of fields π^{∗}: k(X/G) ↪ k(X)^{G}. Conditions (i) and (ii) imply that, in fact, the latter is an isomorphism of fields π^{∗}: k(X/G) →^{≅} k(X)^{G}, see, e.g., [Bor, II, 6.5] (this property holds for the ground field k of arbitrary characteristic; for char(k) = 0, it follows already from condition (i), see, e.g., [PV, Lemma 2.1]).

If G acts on a reduced k-variety Z whose irreducible components are open, B is a normal kvariety, ϱ: Z → B is a k-morphism constant on G-orbits, and ϱ induces a bijection of Z(\overline{k})/G(\overline{k}) with B(\overline{k}), then (B, ϱ) is the geometric quotient of Z by G; see [Bor, Prop. 6.6].

Example 2.2. If H is a closed k-subgroup of G, the action of H on G by right translation gives rise to a geometric quotient π_{G,H}: G → G/H called the quotient of G by H. The group G acts on G/H by left translation and, up to G-isomorphism, G/H is uniquely defined among the homogeneous spaces of G by the corresponding universal property, see, e.g., [Spr_{1}, Sects. 5.5, 12.2] or [Hum_{1}, Sect. 12].

For any reduced k-variety X endowed with a G-action, a theorem of Rosenlicht [Ros_{1}], [Ros_{2}] (cf. also [PV, Sects. 2.1–2.4], [Spr_{2}, Satz 2.2], [Tho, Prop. 4.7]) ensures that there exist a Ginvariant dense open subset U of X, a k-variety Y, and a smooth k-morphism α: U → Y such that (Y, α) is the geometric quotient of U by G.

2.2. (G, S)-fibrations. Consider the category M_{G} whose objects are k-morphisms of k-schemes π: X → Y such that X is endowed with an action of G and π is constant on G-orbits, and a morphism of π_{1} : X_{1} → Y_{1} to π_{2} : X_{2} → Y_{2} is a commutative diagram

\xymatrix{X_{1} \ar[r]^{α} \ar[d]_{π_{1}} & X_{2} \ar[d]^{π_{2}} \\ Y_{1} \ar[r]^{β} & Y_{2}} ,  (2.1)

where α and β are k-morphisms and α is G-equivariant. The notion of composition of morphisms is clear. A morphism as in (2.1) is an isomorphism if and only if α and β are isomorphisms.

Let π: X → Y be an object of M_{G} and let μ: Z → Y be a k-morphism of k-schemes. Then G acts on X ×_{Y} Z via X, and the second projection X ×_{Y} Z → Z is an object of M_{G}. We say that it is obtained from π by the base change μ.

Definition 2.3. Let F be a k-scheme endowed with an action of G and let π: X → Y be an object of M_{G}. The morphism π is called

(i) trivial (over Y) with fibre F if there exists an isomorphism between π and pr_{2} : F ×_{k}Y → Y where G acts on F ×_{k} Y via F;

(ii) fibration (over Y) with fibre F if π becomes trivial with fibre F after a surjective étale base change μ: Y′ → Y. In this case, we say that π is trivialized by μ.

Example 2.4. If F = G with the G-action by left translation, then the notion of fibration over Y with fibre G coincides with that of G-torsor over Y.

The following definition extends the definition of a G-torsor (the latter corresponds to the case where S is the trivial subgroup {1}).

Definition 2.5. Let S be a closed k-subgroup of G. A fibration with fibre G/S, where G acts on G/S by left translation, is called (G, S)-fibration.

If X is a k-scheme endowed with an action of G and there is a (G, S)-fibration X → Y, then we say that X admits the structure of a (G, S)-fibration over Y.

Remark 2.6. Replacing in Definition 2.3(ii) “étale” by “smooth”, one obtains an equivalent definition. This follows from the fact that a surjective smooth morphism Y′ → Y admits sections locally for the étale topology on Y: there exists an étale surjective morphism Y′′ → Y such that Y′ ×_{Y} Y′′ → Y′′ has a section ([EGA IV, 17.16.3]).

Remark 2.7. If the k-scheme X admits the structure of a (G, S)-fibration, then it admits the structure of a (G, S′)-fibration for any k-subgroup S′ of G such that S\overline{k} and S′\overline{k} are conjugate subgroups of G\overline{k}. Such k-groups S and S′ need not be k-isomorphic.

We list some immediate properties without proof.

Proposition 2.8. Let π: X → Y be a (G, S)-fibration. Then the following properties hold: (i) π is a smooth surjective morphism; (ii) a morphism obtained from π by a base change is a (G, S)-fibration.

Assume that X is a k-variety. Then (iii) the G-stabilizers of points of X(\overline{k}) are conjugate to the subgroup S\overline{k} of G\overline{k}; (iv) (Y, π) is the geometric quotient of X by G. □

If a k-group S acts on a k-variety Z, then the functor on commutative k-algebras A ↦ Z(A)^{S(A)} defines a sheaf for the fppf topology on Spec k. This functor is representable by a closed ksubvariety Z^{S} of Z (see [Fog] and [SGA3bis]).

Let Y be a k-variety and let Z → Y be an object of M_{S}. If Y′ is a k-variety and Y′ → Y is a k-morphism, then the natural Y′-morphism Z^{S} ×_{Y} Y′ → (Z ×_{Y} Y′)^{S} is an isomorphism

Z^{S} ×_{Y} Y′ →^{≅} (Z ×_{Y} Y′)^{S}.  (2.2)

Let S be a closed k-subgroup of a k-group G and let N be the normalizer of S in G. Assume that X is a k-variety endowed with an action of G. Then the subvariety X^{S} is N-stable and, since S acts trivially on it, the action of N on X^{S} descends to an action of the group

H := N/S.  (2.3)

Let Y be a k-variety and let π: X → Y be an object of M_{G}. Put

π^{S} := π|_{X^{S}}: X^{S} → Y

and let π′: X ×_{Y} X^{S} → X^{S} be the morphism obtained from π by the base change π^{S}:

\xymatrix{X ×_{Y} X^{S} \ar[r] \ar[d]_{π′} & X \ar[d]^{π} \\ X^{S} \ar[r]^{π^{S}} & Y} .

Since π^{S} is constant on H-orbits, H acts on X ×_{Y} X^{S} via X^{S}. The actions of G and H on X ×_{Y} X^{S} commute. Therefore X ×_{Y} X^{S} is endowed with an action of G ×_{k} H. The morphism π′ is H-equivariant and constant on G-orbits.

The group H also acts on G/S by right multiplication. This action and the action of H on X^{S} determine the H-action on (G/S) ×_{k} X^{S}. It commutes with the G-action on (G/S) ×_{k} X^{S} via left translation of G/S. Therefore (G/S) ×_{k} X^{S} is endowed with an action of G ×_{k} H. The second projection (G/S) ×_{k} X^{S} → X^{S} is H-equivariant and constant on G-orbits.

The natural morphism H → G/S yields the basic isomorphism

H ⟶^{≅} (G/S)^{S}.  (2.4)

Proposition 2.9. For every (G, S)-fibration π: X → Y where X and Y are k-varieties, the following properties hold:

(i) π^{S}: X^{S} → Y is an H-torsor and every base change trivializing π trivializes π^{S} as well; (ii) for the (G, S)-fibration X ×_{Y} X^{S} → X^{S} obtained from π by the base change π^{S}, the

(G ×_{k} H)-equivariant X^{S}-map

φ: (G/S) ×_{k} X^{S} → X ×_{Y} X^{S}, (g̃, x) ↦ (g · x, x),

where g̃ = π_{G,S}(g) (see Example 2.2), is an isomorphism; (iii) the morphism

pr_{1} ∘ φ: (G/S) ×_{k} X^{S} → X, (g̃, x) ↦ g · x,  (2.5)

is an H-torsor.

Proof. (i) Let π be trivialized by a (surjective étale) base change μ: Y′ → Y, i.e., there is a G-equivariant Y′-isomorphism (G/S) ×_{k} Y′ → X ×_{Y} Y′. By (2.2) and (2.4) we then have the H-equivariant Y′-isomorphisms H ×_{k} Y′ = (G/S)^{S} ×_{k} Y′ →(X ×_{Y} Y′)^{S} →X^{S} ×_{Y} Y′. Hence, π^{S} is an H-torsor which is trivialized by μ. This proves (i).

(ii) The morphism φ is a Y -map with respect to the compositions of the second projections with π^{S}. By [EGA IV, Vol. 24, Prop. 2.7.1(viii)] it is enough to prove the claim for the morphism of varieties obtained by the base change μ considered in the above proof of (i). By virtue of (i) and (2.3) this reduces the problem to proving that the map

(G/S) ×_{k} (N/S) → (G/S) ×_{k} (N/S), (g̃, ñ) ↦ (g̃n, ñ),

is an isomorphism. But this is clear since (g̃, ñ) ↦ ({ \atop gn^{−1}}, ñ) is the inverse map. This proves (ii).

(iii) By (i) and Proposition 2.8(ii) the morphism X ×_{Y} X^{S} → X obtained from π^{S} by the base change π is an H-torsor. Since φ is an isomorphism, this proves (iii). □

Let C be an algebraic k-group. Consider a C-torsor

α: P → Y

over a k-variety Y. Let F be a k-variety endowed with an action of C. If every finite subset of F is contained in an open affine subset of F (for instance, if F is quasi-projective), then for the natural action of C on F ×_{k} P determined by the C-actions on F and P, the geometric quotient exists; it is usually denoted by

F ×^{C} P.

Moreover, the quotient map F × P → F ×^{C} P is actually a C-torsor over F ×^{C} P (see [Fl, Prop. 2.12] and use Proposition 2.8 (iv) above; see also [PV, Sect. 4.8]).

Since the composition of morphisms F ×_{k} P ⟶^{pr_{2}} P ⟶^{α} Y is constant on C-orbits, by the universal mapping property of geometric quotients this composition factors through a k-morphism

α_{F} : F ×^{C} P → Y.

Let μ: Y′ → Y be a surjective étale k-morphism such that α becomes the trivial morphism pr_{2} : C ×_{k} Y′ → Y′ after the base change μ. Then, after the same base change, α_{F} becomes the morphism

F ×^{C} (C ×_{k} Y′) = F ×_{k} Y′ ⟶^{pr_{2}} Y′.

Hence α_{F} is a fibration over Y with fibre F.

Since the variety G/S is quasi-projective (see [Bor, Theorem 6.8]), this construction is applicable for C = H and F = G/S.

Given a k-variety Y, we now have two constructions:

· if π: X → Y is a (G, S)-fibration, then π^{S}: X^{S} → Y is an H-torsor; · if α: P → Y is an H-torsor, then α_{G/S}: (G/S) ×^{H} P → Y is a (G, S)-fibration.

Proposition 2.10. These two constructions are inverse to each other and they are functorial in Y.

Proof. Since by Proposition 2.9(iii) morphism (2.5) is an H-torsor, X is the geometric quotient for the H-action on (G/S) ×_{k} X^{S}. Hence, by the uniqueness of geometric quotient, there is a G-equivariant isomorphism (G/S) ×^{H} X^{S} → X.

Let P → Y be an H-torsor. The natural H-action on (G/S) ×_{k} P and the G-action via left translation of G/S commute. From this and (2.4) we deduce the isomorphisms

((G/S) ×^{H} P)^{S} ⟶^{≅} (G/S)^{S} ×^{H} P ⟶^{≅} H ×^{H} P ⟶^{≅} P.

Functoriality in Y is clear. □

Remark 2.11. The étale Čech cohomology set H^{1}(Y, H) classifies H-torsors over Y. On the other hand, if Aut_{G}(G/S) is the algebraic k-group of G-equivariant automorphisms of G/S, then according to the general principle outlined at the beginning of [Ser_{2}, Section 3] (see also the references there for a more rigorous treatment), the étale Čech cohomology set H^{1}(Y, Aut_{G}(G/S)) classifies (G/S)-fibrations over Y. Since the G-action on G/S by left translation commutes with the H-action by right multiplication, we have an injection H ↪ Aut_{G}(G/S). It is well known (and easy to prove) that, in fact, H = Aut_{G}(G/S). We thus get a bijection between H^{1}(Y, H) and H^{1}(Y, Aut_{G}(G/S)), i.e., between H-torsors and (G, S)-fibrations over Y. Proposition 2.10 is an explicit version of this fact.

2.3. (G, S)-varieties.

Definition 2.12. Let S be a closed k-subgroup of G and let X be a k-variety endowed with a G-action. We shall say that X is a (G, S)-variety if X contains a dense open G-stable subset U which admits the structure of a (G, S)-fibration U → Y. Generalizing a terminology introduced in [BF], it is convenient to call such an open subset U a friendly open subset of X for the action of G.

If X is geometrically integral and U is a friendly open subset of X with (G, S)-fibration π: U → Y, then π induces an isomorphism π^{∗}: k(Y) →^{≅} k(X)^{G}.

The following statement over an algebraically closed field has previously appeared in various guises in the literature (see [PV, 2.7] and [Pop_{4}, 1.7.5]).

Theorem 2.13. Let X be a geometrically integral k-variety endowed with a G-action. Let S be a closed k-subgroup of G. Then the following properties are equivalent:

(a) X is a (G, S)-variety;

(b) X contains a dense open G-stable subset such that the G\overline{k}-stabilizer of each of its \overline{k}-points is conjugate to S\overline{k}.

Proof. That (a) implies (b) is clear. Let us assume (b). One may replace X by dense G-stable open subsets to successively ensure that:

(i) The G\overline{k}-stabilizer of every \overline{k}-point of X is conjugate to S\overline{k}.

(ii) The k-variety X is smooth over k.

(iii) (Rosenlicht’s theorem, see Section 2.1.) There exist a geometrically integral k-variety Y and a k-morphism π: X → Y such that the pair (Y, π) is the geometric quotient of X for the action of G.

(iv) The morphism π is smooth. Indeed the generic fibre of π is regular, hence smooth since char(k) = 0. The statement on π can thus be achieved by replacing Y by an open set and X by the inverse image of this open set.

(v) There exists an open set U of the reduced variety (X^{S})_{red} ⊂ X^{S} such that the composition

of maps U ↪ X^{S} →^{π^{S}} Y is smooth. This follows from the surjectivity of the map π^{S}: X^{S} → Y on \overline{k}-points, itself a consequence of (i) and (iii).

(vi) If we let U ⊂ (X^{S})_{red} be the maximal open set such that the map π|_{U}: U → Y is smooth, then this map is surjective. This is achieved by replacing Y in the previous statement by a dense open set contained in the image of the previous U, and replacing X by the inverse image of this open set.

We now consider the following G-equivariant k-morphism of smooth k-varieties:

ψ: (G/S) ×_{k} U → X ×_{Y} U, (g̃, u) ↦ (gu, u)

(see the notation in Proposition 2.9(ii)). It is a U-morphism with respect to the second projections. Since every G-equivariant morphism G/S → G/S is bijective and (Y, π) is the geometric quotient, we deduce from (i) and (iii) that ψ is bijective on \overline{k}-points. As char(k) = 0, we then conclude by Zariski’s main theorem that ψ is an isomorphism. Thus it is proved that the morphism obtained from π by the base change π|_{U} is trivial over U with fibre G/S. Since π|_{U} is smooth, we now deduce from Remark 2.6 that π is a (G, S)-fibration. □

Condition (b) of Theorem 2.13 gained much attention in the literature (see [PV, §7] and references therein). If (b) holds, one says that, for the action of G on X, there exists a stabilizer S in general position or that there exists a principal orbit type for (G, X). There are actions, even of reductive groups, for which a stabilizer in general position does not exist (see [PV, 7.1, 2.7]). There are results ensuring its existence under certain conditions or, equivalently (by Theorem 2.13), the existence of a structure of (G, S)-variety. Theorem 2.15 below is such a result.

Recall the following definition introduced in [Pop_{1}]. Definition 2.14. The action of an algebraic k-group G on a k-variety X is called stable if there exists a dense open subset U of X such that the G-orbit of every point of U(\overline{k}) is closed in X\overline{k}. Theorem 2.15. Let X be an affine geometrically integral k-variety with an action of a reductive k-group G such that X(k) is Zariski dense in X. Assume that either of the following conditions hold:

(i) X is smooth; or

(ii) the G-action on X is stable. Then there is a closed k-subgroup S of G such that X is a (G, S)-variety. In case (ii) this subgroup S is reductive. Proof. If (i) holds, then by Richardson’s theorem [Rich, Prop. 5.3] (cf. also [Lun, Cor. 8], [PV, Theorem 7.2]) there is a closed \overline{k}-subgroup R of G\overline{k} such that the G\overline{k}-stabilizer of a general \overline{k}- point of X is conjugate to R. Since X(k) is Zariski dense, R can be taken as S\overline{k}, where S is the stabilizer of a k-point of X. Then property (b) from the statement of Theorem 2.13 holds, hence X is a (G, S)-variety.

If (ii) holds, then the above subgroup S still exists by [PV, Sect. 7.2, Cor.], so the same argument applies. As the general orbit is closed, it is affine, whence S is reductive by Matsushima’s criterion ([Mat], [Oni], cf. [BB], [Lun]). □

2.4. Categorical quotients. For the definition of a categorical quotient we refer the reader to [Mum, Def. 0.5], [Bor, 6.16, 8.19], and [PV, Sect. 4.3]. In this paper we shall only work with categorical quotients for reductive group actions on affine varieties, which are constructed as follows.

Let A be a finitely generated k-algebra. Assume a reductive k-group G acts on the k-variety X = Spec(A) (over k). Then (cf. [Mum, Theorem 1.1, Cor. 1.2])

(i) the ring A^{G} is a finitely generated k-algebra; (ii) the inclusion A^{G} ↪ A induces a categorical quotient map π: X → Spec(A^{G}) =: X//G; (iii) every geometric fibre of π contains a unique closed orbit.

As G-orbits are open in their closure, the latter property implies that every geometric fibre of π containing a closed G-orbit of maximal (in this fibre) dimension, coincides with this orbit. Proposition 2.16. Let X be a geometrically integral affine k-variety with an action of a reductive k-group G such that X(k) is Zariski dense in X. Let π: X → X//G be a categorical quotient. Then the following properties are equivalent:

(a) the action of G on X is stable;

(b) there exist a reductive k-subgroup S of G and a dense open subset Y of X//G such that the restriction of π to π^{−1}(Y) is a (G, S)-fibration π^{−1}(Y) → Y.

The group S in (b) may be taken as the G-stabilizer of any k-point of π^{−1}(Y).

If (a), (b) hold, then π induces an isomorphism π^{∗}: k(X//G) →^{≅} k(X)^{G}.

Proof. Assume that (a) holds. By Theorem 2.15, there exist a reductive k-subgroup S of G and a G-invariant open subset U_{1} of X such that U_{1} admits the structure of a (G, S)-fibration α: U_{1} → Z_{1}.

On the other hand, by (a) there is an open subset U_{2} of X such that the G-orbit of every point of U_{2}(\overline{k}) is closed. Since there is an open subset U_{max} of X such that the G-orbit of every point of U_{max}(\overline{k}) has maximal (in X) dimension (cf. [Mum, Chap. 0, §2] or [PV, Sect. 1.4]), we may replace U_{2} by U_{2} ∩ U_{max} and assume in addition that this maximality property holds for every point of U_{2}(\overline{k}). The openness of U_{2} in X implies that π(U_{2}) contains a smooth open subset Y_{1} of X//G. Put U_{3} := π^{−1}(Y_{1}). Then the fibre of π over every point of Y_{1}(\overline{k}) contains a closed G-orbit of maximal dimension. As we mentioned right before the statement of Proposition 2.16, this implies that this fibre is a G-orbit. In turn, as we mentioned in Section 2.1, this implies that π|_{U_{3}} : U_{3} → Y_{1} is the geometric quotient for the G-action on U_{3}.

Let U = U_{1} ∩ U_{3}. Then, since α and π|_{U_{3}} are open morphisms, Z := α(U) and Y := π(U_{3}) are open subsets of Z_{1} and Y_{1} respectively. The morphisms π|_{U}: U → Y and α|_{U}: U → Z are geometric quotient maps for the G-action on U. By uniqueness of geometric quotients, there is an isomorphism φ: Z → Y such that the diagram

\xymatrix{ & U \ar[dl]_{α|_{U}} \ar[dr]^{π|_{U}} \\ Z \ar[rr]_{φ}^{≅} &  & Y}

is commutative. Hence (b) holds.

Conversely, if (b) holds, then fibres of π over points of Y (\overline{k}) are G-orbits. Therefore these orbits are closed; whence (a).

To prove the last assertion of the proposition, we may replace X//G by Y and thus assume that π is a (G, S)-fibration. By Proposition 2.8(iv) π is a geometric quotient map; the desired conclusion now follows from Remark 2.1. □

3. Versal actions

We recall that k is a field of characteristic zero. Let \overline{k} be an algebraic closure of k, and let G be a (not necessarily connected) linear algebraic group over k.

Definition 3.1. Let S be a closed k-subgroup of G. We say that a (G, S)-fibration π: V → Y is versal if Y is geometrically integral and for every field extension L/k, every (G_{L}, S_{L})-fibration ϱ: X → Spec(L), and every dense open subset Y_{0} of Y, there exists a Cartesian diagram of the form

\xymatrix{X \ar[d]_{ϱ} \ar[rr] &  & V \ar[d]^{π} \\ Spec(L) \ar[r] & Y_{0} \ar@{^{(}->}[r] & Y} .  (3.1)

In other words, there is an L-point of Y_{0} and an L-isomorphism between X and the fibre product V ×_{Y} Spec(L).

Note that if S = {1}, i.e., π is a G-torsor, this definition coincides with the usual definition of a versal torsor; see [GMS, Sect. I.5], [BF].

Lemma 3.2. Let N be the normalizer of S in G and let H = N/S. A (G, S)-fibration π: V → Y over a smooth Y is versal if and only if the associated H-torsor π^{S} := π|_{V^{S}}: V^{S} → Y is versal.

Proof. By Proposition 2.10 there are mutually inverse functorial correspondences between (G, S)- fibrations and H-torsors over Spec(L) given by passing from a (G, S)-fibration ϱ: X → Spec(L) to the H-torsor ϱ^{S}: X^{S} → Spec(L) and from an H-torsor α: Z → Spec(L) to the (G, S)-fibration α_{G/S}: (G/S) ×^{H} Z → Spec(L). This implies that a Cartesian diagram (3.1) exists if and only if a Cartesian diagram

\xymatrix{X^{S} \ar[d]_{ϱ^{S}} \ar[rr] & & V^{S} \ar[d]^{π^{S}} \\ Spec(L) \ar[r] & Y_{0} \ar@{^{(}->}[r] & Y} .

exists. This means that π is versal if and only if π^{S} is versal. □

We say that a (G, S)-variety X is versal if there is a friendly open set U of X (see Definition 2.12) such that the associated (G, S)-fibration U → Y is versal.

It is easy to see that if X is a versal (G, S)-variety, then the (G, S)-fibration U′ → Y′ is a versal (G, S)-fibration for every friendly open set U′ of X.

The following proposition plays an important rôle in our paper. We are grateful to the referee for the present version of this statement which strengthens our earlier result used in the proof of Theorem 0.2. Recall that the notion of stabilizer in general position utilized in the formulation of this proposition has been defined in the previous section, just after the proof of Theorem 2.13.

Proposition 3.3. Suppose G and H are linear algebraic groups over k and G acts (algebraically ) on H by group automorphisms. Assume further that

(i) the group H is connected and, for this action of G on H, there exists a stabilizer S in general position;

(ii) the group H^{S} is connected. Then H is a versal (G, S)-variety.

Note that by Theorem 2.15(i) condition (i) automatically holds if G is reductive.

Our proof of Proposition 3.3 will rely on the following two lemmas.

Lemma 3.4. Let C be an algebraic group over a field K, let X be a quasi-projective K-variety endowed with a C-action, and let π: P → Spec(K) be a C-torsor. Then the following properties are equivalent:

(a) There exists a C-equivariant morphism α: P → X defined over K.

(b) X ×^{C} P has a K-point.

Recall that here X ×^{C} P is the K-variety defined after the proof of Proposition 2.9.

Proof of Lemma 3.4. (a) ⇒ (b): By the universal mapping property (see Subsection 2.1) the C-equivariant morphism α × id: P → X × P determines a K-morphism of geometric quotients Spec(K) = P/C → (X × P)/C = X ×^{C} P, i.e., a K-point of X ×^{C} P.

(b) ⇒ (a): Given a K-point μ: Spec(K) → X ×^{C} P, let ε: E → Spec(K) be the C-torsor obtained from the C-torsor X × P → X ×^{C} P by the base change μ. By construction there is a C-equivariant morphism E → X × P (see Subsection 2.2). Its composition with the projection X × P → P is a morphism of C-torsors π and ε, hence an isomorphism. This yields a Cequivariant morphism P → X × P. Its composition with the projection X × P → X is a C-equivariant morphism P → X. □

Lemma 3.5. Let C, X, and P be as in Lemma 3.4. If X is an algebraic group and C acts on X by group automorphisms, then X ×^{C} P has a natural structure of an algebraic group defined over K.

Proof of Lemma 3.5. For notational simplicity we shall write ^{P}X in place of X ×^{C} P.

Let X_{1} and X_{2} be quasi-projective K-varieties endowed with C-actions and let φ: X_{1} → X_{2} be a C-equivariant morphism. By the universal mapping property the C-equivariant morphism φ × id: X_{1} × P → X_{2} × P determines a morphism of geometric quotients ^{P}X_{1} → ^{P}X_{2} which we shall denote by ^{P}φ.

For i = 1, 2, let π_{i}: X_{1} × X_{2} → X_{i} be the projection. We claim that the K-morphism

Pπ_{1} ×^{P}π_{2} : ^{P}(X_{1} × X_{2}) → ^{P}X_{1} ×^{P}X_{2}  (3.2)

is, in fact, an isomorphism. To prove this claim we pass to a finite field extension K′ of K such that P splits over K′, next we observe that if P splits, the claim is obvious. We conclude that (3.2) is an isomorphism over K′ and hence by [EGA IV, Vol. 24, Prop. 2.7.1(viii)] over K.

Using this isomorphism one easily checks that the multiplication map X × X → X and the inverse map X → X give rise to group operations on ^{P}X. □

Proof of Proposition 3.3. Condition (i) and Theorem 2.13 yield that, for the G-action on H, there is a friendly open subset U of H. So a geometric factor U → Y for the action of G on U exists and is a (G, S)-fibration. The action of N_{G}(S)/S on H^{S} is generically free and Proposition 2.9(i) yields that U^{S} = H^{S} ∩ U is a friendly open subset of H^{S} for this action. By Lemma 3.2 it suffices to show that U^{S} → Y is a versal N_{G}(S)/S-torsor or, equivalently, that H^{S} is a versal N_{G}(S)/S-variety. Thus, given condition (ii), after replacing G by N_{G}(S)/S and H by H^{S}, we may assume that S = {1}.

Our goal now is to show that, under this assumption, the G-action on the friendly open subset U of H satisfies the conditions of Definition 3.1, i.e., for every field extension L/k, every G-torsor P → Spec(L), and every open dense subset Y_{0} of Y, there is a G-equivariant map P → U_{0} := π^{−1}(Y_{0}). By Lemma 3.4 with C = G_{L} this map exists if and only if (U_{0})_{L} ×^{G_{L}}P has an L-point. Since U_{0} is a dense G-invariant subset of U (and hence of H), we see that (U_{0})_{L}×^{G_{L}}P is a dense open subset of H_{L} ×^{G_{L}}P. Thus, it suffices to show that L-points are dense in H_{L} ×^{G_{L}}P.

Lemma 3.5 implies that H_{L} ×^{G_{L}} P is an algebraic group over L. It is connected because H is connected. Let L′ be a finite field extension of L such that P splits over L′. Then the L′-groups (H_{L}×^{G_{L}}P)×_{L}L′ and H_{L}×_{L}L′ are isomorphic. Since H is linear, this yields that (H_{L}×^{G_{L}}P)×_{L}L′ is linear. The standard descent result [EGA IV, Vol. 24, Prop. 2.7.1(xiii)] then implies that the Lgroup H_{L} ×^{G_{L}} P is linear. But since char(L) = 0, by Chevalley’s theorem [Bor, Theorem 18.2(ii)] any connected linear algebraic group defined over L is unirational over L. Hence H_{L} ×^{G_{L}} P is unirational over L and therefore L-points are dense in it, as claimed. □

Corollary 3.6.

(a) (cf. [Rei, Prop. 7.1] and [GMS, Ex. I.5.4]) Every finite-dimensional generically free G-mo-dule V defined over k is a versal (G, {1})-variety.

(b) If G is reductive and V is a finite-dimensional G-module defined over k, then V is a versal (G, S)-variety for a suitable k-subgroup S of G. There exists a dense open subset U of V such that the G-stabilizer of each point of U(k) is a possible choice for S.

Proof. In both parts view V as the unipotent k-group G_{a}^{dim V} and apply Proposition 3.3 and, in part (b), Theorem 2.15(i). □

Lemma 3.7. Let φ: X → Y be a dominant morphism of integral k-varieties. Denote the generic point of Y by η and the generic fibre of φ by X_{η}.

(a) Suppose φ has a rational section s: Y −−→ X. Then there exists a dense open subset Y_{0} of Y defined over k such that for any morphism Z → Y_{0} of integral schemes, the natural projection φ_{Z}: X_{Z} := X ×_{Y} Z → Z has a section Z → X_{Z}.

(b) Suppose the generic fibre X_{η} is connected and is rational (respectively, stably rational ) over k(Y). Then there exists a dense open subset Y_{0} of Y defined over k such that for any extension field L/k and any point y ∈ Y_{0}(L), the fibre X_{y} of φ over y is integral and rational (respectively, stably rational ) over L.

Proof. (a) Choose Y_{0} ⊂ Y so that s is regular on Y_{0} and pull back the section s to X_{Z}.

Before we prove (b), let us discuss a more general situation. Let p : X → Y and p′ : X′ → Y be two dominant k-morphisms of geometrically integral k-varieties with geometrically integral generic fibres. Assume that the generic fibres are birationally isomorphic over k(Y). Then X and X′ are birationally isomorphic over Y. There thus exist two dense open sets U ⊂ X and U′ ⊂ X′ and a Y -isomorphism U →^{≅} U′. Let Y_{0} ⊂ p(U) be a Zariski dense open set and replace U and U′ by their restrictions over Y_{0}. Then all geometric fibres of U → Y_{0} and U′ → Y_{0} are nonempty. For any point y ∈ Y_{0} this induces a k(y)-isomorphism between the nonempty fibre U_{y} ⊂ X_{y} and the nonempty fibre U′_{y} ⊂ X′_{y}. The same therefore holds over L with k(y) ⊂ L.

To prove statement (b) in the case where X_{η} is rational, it suffices to apply this argument to X and X′ = P_{Y}^{d} where d is the dimension of X_{η} and P_{Y}^{d} is the d-dimensional projective space over Y.

To prove statement (b) in the case where X_{η} is stably rational, it suffices to apply this argument to the pair X ×_{Y} P_{Y}^{n} and P_{Y}^{n+d}, with d as above and n some positive integer. □

Definition 3.8. Given a (G, S)-variety X, we shall say that it admits a rational section if for some and hence any friendly open set U ⊂ X the quotient map U → U/G admits a section over a dense open set of U/G.

Theorem 3.9. Let G be a linear algebraic group over k, let S be a closed k-subgroup of G, and let V be a geometrically integral versal (G, S)-variety.

(a) If the (G, S)-variety V admits a rational section, then for every field extension F/k every geometrically integral (G, S)-variety X over F admits a rational section.

(b) Assume that the homogeneous space G/S is connected. If k(V)/k(V)^{G} is pure (respectively, stably pure), then for every field extension F/k and every geometrically integral (G, S)-variety X over F, the field extension F (X)/F (X)^{G} is pure (respectively, stably pure).

Proof. Note that if V is a versal (G, S)-variety over k then V_{F} is a versal (G, S)-variety over F. Since the hypothesis F (V)/F (V)^{G} pure, or stably pure, holds as soon as it does for k(V)/k(V)^{G}, it is enough to prove the theorem for F = k. After replacing V by a friendly open subset we may assume that we are given a (G, S)-fibration π: V → Y. Choose a dense open subset Y_{0} ⊂ Y as in Lemma 3.7(a).

(a) After replacing X by a friendly open subset, we may assume that X is the total space of a (G, S)-fibration α: X → Z. Let η be the generic point of Z. Since π is versal, the (G, S)-fibration α_{η}: X_{η} → η can be obtained by pull-back from π via a morphism Spec k(Z) → Y_{0}. In other words, after replacing X by a smaller friendly open set, we may assume that α: X → Z is the pull-back of π: V → Y via a morphism Z → Y_{0} ⊂ Y. The desired conclusion now follows from Lemma 3.7(a).

The proof of part (b) is exactly the same, except that we appeal to Lemma 3.7(b), rather than to Lemma 3.7(a). □

Lemma 3.10. Let G be a connected linear algebraic group over k and let X be a geometrically integral k-variety with G-action which admits a geometric quotient π: X → Y. The following properties are equivalent:

(a) π: X → Y admits a rational section;

(b) k(X) is unirational over k(X)^{G}.

Proof. We know that π induces an isomorphism π^{∗}: k(Y) →^{≅} k(X)^{G}. Let X_{η} be the k(Y)-variety which is the generic fibre of π: X → Y.

Assume (b). The hypothesis implies that there exists a dominant k(Y)-rational map φ from some projective space ℙ_{k(Y)}^{n} to X_{η}. This rational map is defined on a dense open set U ⊂ ℙ_{k(Y)}^{n}. Since rational points are Zariski dense on projective space over an infinite field, the set U(k(Y)) is nonempty. The k(Y)-morphism φ : U → X_{η} sends such a point to a k(Y)-point of X_{η}, i.e., to a rational section of X → Y. Thus (a) holds.

Assume (a) holds. By the definition of π, the generic fibre of π is a k(Y)-variety with function field k(X), and it is a homogeneous space of G_{k(Y)} which by (a) admits a k(Y)-rational point. We thus have inclusions of fields k(Y) ⊂ k(X) ⊂ k(Y)(G). By a theorem of Chevalley, over a field of characteristic zero, any connected linear algebraic group is unirational (see [Bor, Theorem 18.2] or [SGA3, Vol. II, Chap. XIV, Cor. 6.10]). Thus k(Y)(G) embeds into a purely transcendental extension of k(Y) →^{≅} k(X)^{G}. This proves (b). □

4. The conjugation action and the adjoint action

We now concentrate on the main actors. We recall that k is a field of characteristic zero. Let G be a connected reductive group over k.

4.1. Quotients by the adjoint action, versal (G, S)-varieties, and Kostant’s theorem. The radical Rad(G) of G is a central k-torus in G. The G-stabilizer of a point g ∈ G for the conjugation action of G on itself is the centralizer of g in G. There is a Zariski dense open set of G such that the centralizers of its points in G are maximal tori of G, see [Bor, 12.2, 13.1, 13.17, 12.3].

Lemma 4.1. The following properties of an element g ∈ G are equivalent:

(i) the conjugacy class of g is closed in G; (ii) g is semisimple.

Proof. For semisimple groups this is proved in [Ste, 6.13]. The general case can be reduced to that of semisimple groups in the following manner. Let (G, G) be the commutator subgroup of G. It is a closed, connected, semisimple k-subgroup of G, and G = (G, G) · Rad(G), see [Bor, 2.3, 14.2]. Let g = hz for some h ∈ (G, G), z ∈ Rad(G), and let g = g_{s}g_{u}, h = h_{s}h_{u} be the Jordan decompositions, see [Bor, 4.2]. Since Rad(G) is a central torus, g_{s} = h_{s}z, g_{u} = h_{u}. Hence g is semisimple if and only if h shares this property. Let G · g and G · h be respectively the G-conjugacy classes of g and h in G. Since z is central, G · g = (G · h)z. Hence closedness of G · g in G is equivalent to that of G · h. But G · h coincides with the (G, G)-conjugacy class of h in (G, G) and, since (G, G) is semisimple, the cited result in [Ste] shows that the latter is closed in (G, G) if and only if h is semisimple. This completes the proof. □

Corollary 4.2. The action of G on itself by conjugation is stable.

Analogous statements hold for the adjoint action of G on 𝔤, see [Kos].

Proposition 4.3. Let G be a connected reductive group over k and 𝔤 its Lie algebra. Let S ⊂ G be a maximal k-torus. Let X be either G or 𝔤 and let π: X → Y := X//G be the categorical quotient for the conjugation, respectively, the adjoint action. Then

(a) there exists a dense Zariski open subset V of Y with inverse image U = π^{−1}(V) such that π|_{U}: U → V is a (G, S)-fibration;

(b) π^{∗} induces an isomorphism k(Y) →^{≅} k(X)^{G};

(c) X is a versal (G, S)-variety.

Proof. Statements (a) and (b) follow from Corollary 4.2 and Proposition 2.16 by virtue of the identification of general G-stabilizers with maximal tori of G, which are all conjugate over \overline{k}.

For X = 𝔤, part (c) is a special case of Corollary 3.6(b).

To prove part (c) for X = G, note that

X^{S} = centralizer of S in G = S

is connected. Hence Proposition 3.3 applies to the conjugation action of G on itself, yielding the desired conclusion. □

The following well known result plays an important rôle in the sequel.

Proposition 4.4 (Kostant). Let G be a reductive linear algebraic group over k and 𝔤 be its Lie algebra. Assume that the semisimple quotient G/Rad(G) is quasisplit. Then the categorical quotient map π: 𝔤 → 𝔤//G has a (regular) section.

Proof. For algebraically closed base field k this is a theorem of Kostant [Kos, Theorem 0.6]. For an arbitrary base field k of characteristic 0, see [Kot_{2}, §4.3]. □

Proposition 4.5. Let G be a connected reductive group over k, let S be a maximal k-torus of G, and let X be a geometrically integral (G, S)-variety over k. Assume that the semisimple group G/Rad(G) is quasisplit. Then

(a) X admits a rational section;

(b) k(X) is unirational over k(X)^{G}.

Proof. Going over to a friendly open set, we may assume that we are given a (G, S)-fibration π: X → Y. Since the radical of G lies in every conjugate of S, it acts trivially on X. Thus the G-action on X descends to the semisimple group G/Rad(G), and we may assume that the k-group G is quasisplit semisimple.

By Proposition 4.3(c), the Lie algebra 𝔤 equipped with the adjoint action is a versal (G, S)- variety. By Proposition 4.4 the map π: 𝔤 → 𝔤//G admits a section. Statement (a) now follows from Theorem 3.9(a). As for (b), by Lemma 3.10 it follows from (a). □

Many instances of the following immediate corollary have appeared in the literature (cf. [Kot_{1}]).

Corollary 4.6. Let G be a connected reductive group over a field k, let K be an overfield of k. Let X be a K-variety which is a homogeneous space of G_{K}. If the semisimple quotient G/Rad(G) is quasisplit and if the geometric stabilizers of the G_{K}-action on X are maximal tori in G\overline{K}, then X has a K-rational point. □

Remark 4.7. Suppose that the base field k is algebraically closed and G is a connected simple algebraic group defined over k. Let V be a faithful simple G-module over k. Theorem 2.15 tells us that V is a (G, H)-variety for some closed subgroup H of G. The list of all pairs (G, H), with H ≠ {1}, which can occur in this setting can be found in [PV, pp. 260–262]. Then the analogue of Corollary 4.6 holds, namely every G-homogeneous space X, defined over K whose geometric stabilizers are isomorphic to H ×_{k} \overline{K} has a K-point. The proof is similar to the one above, except that instead of Kostant’s result (Proposition 4.4), one uses the existence of a regular section for the categorical quotient map V → V //G, proved in [Pop_{3}].

Part (a) of the following corollary partially generalizes a result of Steinberg [Ste], who constructed a regular section of π in the case where G is simply connected.

Corollary 4.8. Let G be a connected reductive group over k and let 𝔤 be its Lie algebra. If the semisimple quotient G/Rad(G) is quasisplit, then

(a) the categorical quotient map π: G → G//G for the conjugation action of G on itself admits a rational section;

(b) k(G) is unirational over k(G)^{G};

(c) k(𝔤) is unirational over k(𝔤)^{G}.

Proof. By Proposition 4.3 both G and 𝔤 are (G, S)-varieties. All three parts now follow from Proposition 4.5. □

Remark 4.9. Recall from [LPR] that a Cayley map for an algebraic group G over a field k is a G-equivariant birational isomorphism

φ: G −−→ 𝔤,  (4.1)

where 𝔤 is the Lie algebra of G. Here, as before, G is assumed to act on itself by conjugation and on 𝔤 by the adjoint action. We say that G is a Cayley group if it admits a Cayley map. In particular, the special orthogonal group SO_{n}, the symplectic group Sp_{2n}, and the projective linear group PGL_{n} are Cayley groups for every n ≥ 1 and every base field k of characteristic zero; see [LPR, Examples 1.11 and 1.16].

In the case where G is a Cayley group, Theorem 0.3 (or equivalently, Corollary 4.8(a)) has the following simpler proof. By Proposition 4.4 the categorical quotient map π_{𝔤}: 𝔤 → 𝔤//G has a section σ: 𝔤//G → 𝔤. Let π_{G}: G → G//G be the categorical quotient map. Then the Cayley map (4.1) induces a commutative diagram

\xymatrix{G \ar[d]_{π_{G}} \ar@{-->}[r]^{φ} & 𝔤 \ar[d]_{π_{𝔤}} \\ G//G \ar@{-->}[r]^{φ//G} & 𝔤//G, \ar[u]_{σ}}

where φ and φ//G are birational isomorphisms. The section σ pulls back to a rational section of π_{G} via this diagram.

In the case of G = PGL_{2} explicitly mentioned by Grothendieck (see the footnote in the introduction), we have 𝔤 = 𝔰𝔩_{2} and

(i) G//G is the affine line 𝔸^{1} and π_{PGL_{2}}: PGL_{2} → 𝔸^{1} is given by [g] ↦ (tr g)^{2}/ det g, where GL_{2} → PGL_{2}, g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ [g] := [ \begin{matrix}a & b \\ c & d \end{matrix} ], is the natural projection;

(ii) 𝔤//G is the affine line 𝔸^{1} and π_{𝔤}: 𝔤 → 𝔸^{1} is given by g ↦ det g.

In this case a Cayley map φ: PGL_{2} −−→ 𝔰𝔩_{2} is given by

[g] ↦ \frac{2}{tr g} g − I_{2},

where I_{2} is the 2 × 2 identity matrix (see [LPR, Example 1.11]), the map φ//G: 𝔸^{1} −−→ 𝔸^{1} is given by t ↦ −1 + 4/t, and the section σ: 𝔸^{1} → 𝔰𝔩_{2} is given by t ↦ ( \begin{matrix}0 & 1 \\ −t & 0 \end{matrix} ). The above strategy then leads to the rational section

𝔸^{1} −−→ PGL_{2}, t ↦ [\begin{matrix}1 & 1 \\ 1 − 4/t & 1 \end{matrix}] .

4.2. The generic torus. The conjugation action of a connected reductive group G on itself leads to another construction, that of the generic torus. Let S be a maximal k-torus of the connected reductive k-group G and let N be the normalizer of S in G. Consider the natural map φ: S ×_{k} (G/S) → G ×_{k} (G/N) given by (s, gS) ↦ (gsg^{−1}, gN). Its image H ⊂ G ×_{k} (G/N) is closed (see [Hum_{2}, p. 10]). The point φ(s, g) defines the maximal torus gSg^{−1} and the point gsg^{−1} in that torus. The second projection π: H → G/N gives H the structure of a torus over G/N, and the family of fibres of this projection is the family of maximal tori in G. To be more precise, the morphism H ↪ G×_{k} (G/N) is a morphism of G/N-group schemes, where H is a (G/N)-torus and G ×_{k} (G/N) is the constant (G/N)-group scheme induced by base change from G → Spec k. The (G/N)-torus H is thus a maximal torus in the (fibrewise connected) reductive (G/N)-group G×_{k} G/N. The variety G/N is the “variety of maximal tori in G”. Given any field extension L/k and any maximal L-torus S in G_{L}, there exists an L-point s ∈ (G/N)(L) such that π^{−1}(s) = S. The actions of G by conjugation on itself and by left translation on G/N induce a G-action on G ×_{k} G/N with respect to which H is stable and π is G-equivariant. Since G(L) is dense in G, this implies that the set of L-points of G/N whose fibre under H → G/N is isomorphic to a given L-torus is Zariski dense in G/N. The field k(G/N) is denoted K_{gen}. The generic torus T_{gen} is by definition the generic fibre of π. For the details of this construction, see [Vos_{3}, §4.1–4.2].

Assume that G is split over k and S is a split maximal torus of G. Then the K_{gen}-torus T_{gen} is split by the extension k(G/S) of k(G/N), which is a Galois extension with Galois group the Weyl group W = N/S. If, moreover, G is simple, simply connected of type R, then the character lattice of the generic torus is the weight W -lattice P (R). For proofs of these assertions, see [Vos_{2}].

4.3. Equivalent versions of the purity questions. We consider the purity Questions 0.1(a) and (b) from the Introduction.

Theorem 4.10. Let G be a connected reductive group over k and let S be a maximal k-torus of G. Then the following three conditions are equivalent:

(a) k(G)/k(G)^{G} is pure (respectively, stably pure);

(b) k(𝔤)/k(𝔤)^{G} is pure (respectively, stably pure);

(c) for every field extension F/k and every integral (G, S)-variety X over F, the extension F (X)/F (X)^{G} is pure (respectively, stably pure).

The following two conditions are equivalent and are implied by the previous conditions:

(d) for every field extension F/k and every maximal F -torus T of G_{F}, the F -variety G_{F} /T is rational (respectively, stably rational) over F;

(e) the K_{gen}-variety G_{K_{gen}}/T_{gen} is K_{gen}-rational (respectively, stably rational).

If G is quasisplit, then all five conditions are equivalent.

Proof. By Proposition 4.3, both G and 𝔤 are versal (G, S)-varieties, where S is a maximal k-torus of G. The equivalence of (a), (b) and (c) now follows from Theorem 3.9(b).

The implication (d) ⇒ (e) is clear. Let us prove that (e) ⇒ (d). By assumption, the generic fibre of the quotient^{2} of the (G/N)-group scheme G ×_{k} (G/N) by the maximal torus H over G/N is k(G/N)-rational.

By Lemma 3.7(b) it follows that over a dense Zariski open set of G/N, all fibres are rational. The F -torus T may be represented by an F -point of this open set. Hence the result.

The implication (c) ⇒(d) is immediate, since G_{F} /T is a (G, S)-variety over F.

Let us show that (d) ⇒(c) if G is quasisplit. Let X be a (G, S)-variety. We may assume that X is a (G, S)-fibration. By Proposition 4.5(a) the quotient map π: X → Y = X/G has a rational section s: X/G −−→ X. Let K = k(Y) be the function field of Y. Let η denote the generic point of Y and let X_{η} be the fibre of π over η. The K-variety X_{η} is a G_{K}-homogeneous space with a K-point, namely s(η) ∈ X_{η}(K). Let T ⊂ G_{K} be the stabilizer of s(η). This is a maximal K-torus in G_{K}. The G_{K}-homogeneous space X_{η} is isomorphic to G_{K}/T. By our construction, K(G_{K}/T) = K(X_{η}) = k(X) and K = k(X)^{G}. By (d), K(G_{K}/T) is rational (respectively, stably rational) over K. Hence, k(X) is rational (respectively, stably rational) over k(X)^{G}. □

Remark 4.11. From the equivalence between (a) and (b) we conclude that the answers to Questions 0.1(a) and (b) stated in the Introduction depend only on the isogeny class of the group G.

Remark 4.12. If G_{F} is split and the maximal F -torus T ⊂ G_{F} is also split then it easily follows from the Bruhat decomposition of G that the homogeneous space G_{F} /T which appears in part (d) of Theorem 4.10 is F -rational; cf. the last paragraph on p. 219 in [Bor]. For general T ⊂ G_{F}, we shall see in this paper that the quotient G_{F} /T need not be F -rational.

5. Reduction to the case where the group G is simple and simply connected

In the previous section we have seen that the answers to Questions 0.1(a) and (b) stated in the Introduction are the same. Moreover, these answers remain unchanged if we replace G by an isogenous group. In this section we reduce these questions for a general connected split reductive group G to the case where G is split, simple and simply connected.

Proposition 5.1. Let G be a connected reductive algebraic group over k, Z be a central ksubgroup of G and G̃ = G/Z. Denote the Lie algebras of G and G̃ by 𝔤 and 𝔤̃ respectively. Then the following properties are equivalent:

(i) k(𝔤)/k(𝔤)^{G} is pure (respectively, stably pure); (ii) k(𝔤̃)/k(𝔤̃)G̃ is pure (respectively, stably pure).

Proof. We may assume without loss of generality that Z is the centre of G.

Since Z acts trivially on 𝔤, the adjoint action of G on 𝔤 descends to a G̃-action, making 𝔤 into a (G̃, S̃)-variety, where S̃ is a maximal torus in the semisimple group G̃. Thus both 𝔤 and 𝔤̃ are (G̃, S̃)-varieties. The action of G̃ on each of 𝔤 and 𝔤̃ is linear. By Corollary 3.6 both are versal. The desired conclusion now follows from Theorem 3.9(b). □

Taking Z to be the radical (connected centre) of G, we see that Proposition 5.1 reduces Questions 0.1 to the case where G is semisimple. Proposition 5.3 below further reduces it to the case where G is simple.

2This quotient exists as affine (G/N)-scheme. This is a consequence of the fact that H is a (fibrewise connected)

reductive (G/N)-group, the scheme G ×_{k} (G/N) is affine over G/N, and G/N is a scheme of characteristic zero.

For our purposes, it is enough to know this over a dense open subset of G/N, and that is a consequence of the existence of the quotient over the field k(G/N).

Lemma 5.2. Let G = G_{1} × ⋯ × G_{n}, where each G_{i} is a connected reductive k-group. Denote the Lie algebra of G_{i} by 𝔤_{i} and the Lie algebra of G by 𝔤 = 𝔤_{1} × ⋯ × 𝔤_{n}.

(a) If k(𝔤_{i})/k(𝔤_{i})^{G_{i}} is pure (respectively, stably pure) for every i = 1, …, n, then k(𝔤)/k(𝔤)^{G} is pure (respectively, stably pure).

(b) If each G_{i} is split and k(𝔤)/k(𝔤)^{G} is stably pure, then k(𝔤_{i})/k(𝔤_{i})^{G_{i}} is stably pure for every i = 1, …, n.

Proof. Denote the categorical quotient map for the adjoint G_{i}-action by π_{i}: 𝔤_{i} → 𝔤_{i}//G_{i}. Then the categorical quotient map for the adjoint G-action is

π = π_{1} × ⋯ × π_{n}: 𝔤 = 𝔤_{1} × ⋯ × 𝔤_{n} → (𝔤_{1}//G_{1}) × ⋯ × (𝔤_{n}//G_{n}) = 𝔤//G .

Clearly if each 𝔤_{i} is rational (respectively, stably rational) over 𝔤_{i}//G_{i} then 𝔤 is rational (respectively, stably rational) over 𝔤//G. This proves part (a).

Let us prove (b). Suppose that k(𝔤)/k(𝔤)^{G} is stably pure. By symmetry it suffices to show that k(𝔤_{1})/k(𝔤_{1})^{G_{1}} is stably pure.

Let S_{i} be a split maximal torus in G_{i}. Then S = S_{1} × ⋯ × S_{n} is a maximal torus in G. Consider the (G, S)-variety X = 𝔤_{1} × (G_{2}/S_{2}) × … (G_{n}/S_{n}), where the G-action on X is the direct product of the adjoint action of G_{1} on 𝔤_{1} and the left translation action G_{i} on G_{i}/S_{i}. Clearly X//G = 𝔤_{1}//G_{1} and the quotient map for X is the composition of the projection pr_{1} : X → 𝔤_{1} to the first component and the quotient map π_{1} : 𝔤_{1} → 𝔤_{1}//G_{1}:

X = 𝔤_{1} × (G_{2}/S_{2}) × ⋯ × (G_{n}/S_{n}) ⟶^{pr_{1}} 𝔤_{1} ⟶^{π_{1}} 𝔤_{1}//G_{1}.

Because G_{i} and S_{i} are both split, each G_{i}/S_{i} is rational over k; see Remark 4.12. Hence, X is rational over 𝔤_{1}.

By Proposition 4.3(c) 𝔤 is a versal (G, S)-variety. Since k(𝔤)/k(𝔤)^{G} is stably pure, Theorem 3.9(b) tells us that X is stably rational over X//G ≅ 𝔤_{1}//G_{1}. Consequently, 𝔤_{1} is stably rational over 𝔤_{1}/G_{1}, i.e., k(𝔤_{1}) is stably pure over k(𝔤_{1})^{G_{1}}, as desired. □

Proposition 5.3. Let G be a split semisimple group over k. Let G_{1}, …, G_{n} denote the simple components of the simply connected cover of G. Denote the Lie algebras of G, G_{1}, …, G_{n} by 𝔤, 𝔤_{1}, …, 𝔤_{n} respectively.

(a) The following properties are equivalent:

(i) k(𝔤)/k(𝔤)^{G} is stably pure; (ii) k(𝔤_{i})/k(𝔤_{i})^{G_{i}} is stably pure for every i.

(b) If k(𝔤_{i})/k(𝔤_{i})^{G_{i}} is pure for every i, then k(𝔤)/k(𝔤)^{G} is pure.

Proof. The fields k(𝔤), k(𝔤)^{G}, k(𝔤_{i}) and k(𝔤_{i})^{G_{i}} remain unchanged if we replace G by its simply connected cover. Hence we may assume G = G_{1} × ⋯ × G_{n}, and the proposition follows from Lemma 5.2. □

6. Are homogeneous spaces of the form G/T stably rational?

The rest of this paper will be devoted to proving Theorem 0.2. Using Theorem 4.10 and Proposition 5.1 we may assume without loss of generality that G is simply connected and restate Theorem 0.2 in the following equivalent form.

Theorem 6.1. Let G be a split, simple, simply connected algebraic group over k and let T_{gen} be the generic torus of G. (Recall that T_{gen} is defined over the field K_{gen} = k(G/N); see §4.2.) Then the homogeneous space G_{K_{gen}}/T_{gen} is

(a) rational over K_{gen} if G is split of type A_{n} (n ⩾ 1) or C_{n} (n ⩾ 2);

(b) not stably rational over K_{gen} if G is not of type A_{n} (n ⩾ 1) or C_{n} (n ⩾ 2) or G_{2}.

Remark 6.2. For k algebraically closed, the question whether or not the generic torus T_{gen} of G is itself K-rational has been studied in some detail. The (almost) simple groups whose generic torus is (stably) rational are classified in [LL] (in type A only), [CK] (for simply connected or adjoint G of all types) and [LPR] (for all G). A comparison of Theorem 6.1 and [CK, Theorem 0.1] shows that there is no obvious relation between the (stable) rationality of the homogeneous space G_{K}/T_{gen} and that of the generic torus T_{gen}. On the other hand, in view of the results of this section one might wonder if the stable rationality of G_{K}/T_{gen} is related to that of the dual torus T_{gen}^{0}. We shall return to this question in Appendix.

Let X be a K-variety. As usual, we let \overline{K} denote the algebraic closure of K and set \overline{X} := X\overline{K}. The ring \overline{K}[X] and the abelian group Pic \overline{X} then come equipped with the actions of the Galois group of \overline{K} over K.

Theorem 6.3 ([CTK]). Let K be a field of arbitrary characteristic, let G be a semisimple, simply connected linear algebraic K-group and let T ⊂ G be a K-torus. Denote the character lattice of T by T^{∗}.

(a) If a K-variety X_{c} is a smooth compactification of X = G/T, then there is an exact sequence of Galois lattices

0 → P → Pic \overline{X}_{c} → T^{∗} → 0

with P a permutation lattice.

(b) If G/T is stably K-rational, then there exists an exact sequence of Galois lattices

0 → P_{2} → P_{1} → T^{∗} → 0

with P_{2} and P_{1} permutation lattices.

(c) If G/T is stably K-rational, then � _{ω}^{1}(K, T^{∗}) = 0.

Proof. For K of characteristic zero, this is an immediate consequence of the more general result [CTK, Theorem 5.1]. This special case is easier to prove.

We let here \overline{K} denote a separable closure of K. Associated to the T -torsor G → G/T = X there is a well known exact sequence of Galois lattices (see [CTS_{3}, Prop. 2.1.1]):

0 → \overline{K}[X]^{×}/\overline{K}^{×} → \overline{K}[G]^{×}/\overline{K}^{×} → T^{∗} → Pic \overline{X} → Pic \overline{G}.

Since G is semisimple and simply connected, we have \overline{K}^{×} = \overline{K}[G]^{×} and Pic \overline{G} = 0 (see [Pop_{2}, Prop. 1]). We thus get \overline{K}^{×} = \overline{K}[X]^{×} and T^{∗} ⟶^{≅} Pic \overline{X} (this is a special case of [Pop_{2}, Theorem 4] where Pic \overline{G/H} for arbitrary subgroup H is described). The open immersion of smooth Kvarieties X ⊂ X_{c} gives rise to an exact sequence of Galois lattices

0 → \overline{K}[X]^{×}/\overline{K}^{×} → Div_{∞} \overline{X}_{c} → Pic \overline{X}_{c} → Pic \overline{X} → 0.

Here Div_{∞} \overline{X}_{c} is the free abelian group on points of codimension 1 of \overline{X}_{c} with support in the complement of \overline{X}. This is a permutation lattice, call it P. All in all, we get an exact sequence of torsion-free Galois lattices

0 → P → Pic \overline{X}_{c} → T^{∗} → 0.

This proves (a).

If char(K) = 0, then G/T admits a smooth K-compactification. Statement (b) is then a consequence of (a) and the well known fact that if the smooth, proper K-variety X_{c} is stably K-rational, then the Galois lattice Pic(\overline{X}_{c}) is a stably permutation lattice (see [CTS_{3}, Prop. 2.A.1]). Statement (c) then follows from (b) (see §1).

There is however no need to use a smooth compactification to prove (b) and (c). If X = G/T is stably K-rational, there exist natural integers r and s and dense open sets U ⊂ Y = X ×_{K} 𝔸_{K}^{r} and V ⊂ 𝔸_{K}^{s} together with a K-isomorphism U →^{≅} V. The natural maps \overline{K}^{×} → \overline{K}[𝔸^{s}]^{×} and \overline{K}[X]^{×} → \overline{K}[X ×_{K} 𝔸^{r}]^{×} are isomorphisms. By what we have seen above, \overline{K}^{×} → \overline{K}[X]^{×} is an isomorphism. Thus \overline{K}^{×} →^{≅} \overline{K}[Y]^{×}. We have Pic \overline{𝔸}^{s} = 0, hence Pic \overline{V} = 0, hence Pic \overline{U} = 0. The pull-back map associated to the projection Y = X ×_{K} 𝔸^{r} → X induces an isomorphism of Galois modules Pic \overline{X} →^{≅} Pic \overline{Y}. From what we have proved above this induces an isomorphism T^{∗} ≃ Pic \overline{Y}.

The open immersion V ⊂ 𝔸_{K}^{s} induces an isomorphism of Galois modules between the permutation module on irreducible divisors of 𝔸s ; \overline{K} with support in the complement of \overline{V} and the Galois

module \overline{K}[V]^{×}/\overline{K}^{×}, hence the Galois module \overline{K}[U]^{×}/\overline{K}^{×} is a permutation module. The open immersion U ⊂ Y induces an exact sequence

0 → \overline{K}[U]^{×}/\overline{K}^{×} → Δ → Pic \overline{Y} → 0,

where Δ is the permutation module on irreducible divisors of \overline{Y} with support in the complement of \overline{V}. This completes the proof of (b), hence also of (c). □

Remark 6.4. In certain circles, the (unramified) Brauer group is a well known K-birational invariant of smooth, projective, geometrically integral K-varieties. For X ⊂ X_{c} as above, this is the group Br X_{c}. The connection with the above proposition is given by an isomorphism Br X_{c}/ Br K →^{≅} H^{1}(K, Pic \overline{X}_{c}), which one combines with an isomorphism H^{1}(K, Pic \overline{X}_{c}) →^{≅} � _{ω}^{1}(K, T^{∗}) deduced from Statement (a) to produce an isomorphism

Br X_{c}/ Br K →^{≅} � _{ω}^{1}(K, T^{∗}).

The interested reader is referred to [CTK] for details.

Under a strong assumption on the group G, we shall now establish a converse to statement (b) in Theorem 6.3. We first prove a lemma.

Lemma 6.5. Let K be a field of characteristic 0 and let H be a special linear algebraic K-group. If X is a geometrically integral K-variety with a generically free action of H, then X admits a dense H-stable open set U which is H-isomorphic to H ×_{K} Y for a K-variety Y whose function field K(Y) is K-isomorphic to K(X)^{H} and for the H-action induced by left translation of H.

Proof. After replacing X by a friendly open set, we may assume that X is the total space of an H-torsor X → Y = X/H. Since H is special, this torsor splits over a dense open set of Y. If we replace Y by this open set, we have X = H ×_{K} Y as varieties with an H-action. □

Proposition 6.6. Let K be a field of characteristic zero, and let G be a special, K-rational K-group. Let T ⊂ G be a K-torus. If there exists an exact sequence of Galois modules

0 → P_{2} → P_{1} → T^{∗} → 0

with P_{2} and P_{1} permutation modules, then G/T is stably K-rational.

Proof. By assumption we have an exact sequence of K-tori

1 → T → T_{1} → T_{2} → 1

with T_{1} and T_{2} quasitrivial. We now identify T with the diagonal subgroup T of G × T_{1}, and G with the subgroup G × {1} of G × T_{1}. The first projection G × T_{1} → G induces a map (G × T_{1})/T → G/T which makes (G × T_{1})/T into a right T_{1}-torsor over G/T. The second projection G × T_{1} → T_{1} induces a map (G × T_{1})/T → T_{1}/T which makes (G × T_{1})/T into a left G-torsor over T_{1}/T = T_{2}. In summary, we have the following diagram:

\xymatrix{ & (G × T_{1})/T \ar@{-->}[dr]^{left G-torsor} \ar[dl]_{right T_{1}-torsor} \\ G/T &  & T_{1}/T = T_{2}}

Since T_{1} is a quasitrivial torus hence a special K-group, (G×T_{1})/T is K-birationally isomorphic to T_{1} ×(G/T); see Lemma 6.5. Similarly, since G is special, (G×T_{1})/T is K-birationally isomorphic to G × T_{2}. Thus T_{1} × (G/T) is K-birationally isomorphic to G × T_{2}. Since T_{1}, T_{2} and G are K-rational varieties, we conclude that G/T is stably K-rational. □

We now specialize Theorem 6.3 and Proposition 6.6 to the setting of Theorem 6.1.

The weight lattice P (R) of a root system of type R is equipped with a natural action of the Weyl group W.

Corollary 6.7. Let G be a split, simple, simply connected algebraic group of type R, defined over k and T_{gen} be the generic torus of G. Recall that T_{gen} is defined over the field K_{gen} = k(G/N); cf. §4.2.

(a) If the homogeneous space G_{K_{gen}}/T_{gen} is stably rational over K_{gen}, then there exists an

exact sequence

0 → P_{2} → P_{1} → P (R) → 0  (6.1)

of W -lattices, where P_{1} and P_{2} are permutation.

(b) Suppose G is special. Then the converse to part (a) holds. That is, if there exists an exact sequence (6.1) of W -lattices with P_{1} and P_{2} permutation lattices, then G_{K_{gen}}/T_{gen} is stably rational over K_{gen}.

Proof. As recalled in §4.2, the K_{gen}-torus T_{gen} splits over a Galois extension of K_{gen} with Galois group the Weyl group W, and the character lattice of T_{gen} is isomorphic to the weight lattice P (R) with its natural W -action. Note also that since G is split, G is rational over k and hence G_{K_{gen}} is rational over K_{gen}. Applying Theorem 6.3 (b) and Proposition 6.6 to G_{K_{gen}} and T_{gen} and using the correspondence between tori and lattices (§1) we get (a) and (b). □

7. Nonrationality

In this section we shall prove Theorem 0.2(b) or equivalently, Theorem 6.1(b). To prove the latter, one may assume that k is algebraically closed. In view of Corollary 6.7(a), it suffices to establish the following proposition.

Proposition 7.1. Let R be a reduced, irreducible root system in a real vector space V that it spans. Let P (R) be the weight lattice equipped with the action of the Weyl group W = W (R). If R is not of type A_{n}, C_{n} or G_{2}, then there exists a subgroup H ≅ (ℤ/2ℤ)^{2} in W such that � _{ω}^{1}(H, P (R)) ≠ 0. In particular, there does not exist an exact sequence of W -lattices

0 → P_{2} → P_{1} → P (R) → 0

with P_{1} and P_{2} permutation lattices.

Proof. Let B be a basis of R. Let W = W (R) be the Weyl group. The abelian group Q(R) ⊂ V spanned by R is the root lattice, its rank is l := dim(V). There is an inclusion Q(R) ⊂ P (R) ⊂ V, where P (R) is the weight lattice. See [Bou, VI. 1.9]. Both Q(R) and P (R) are W -lattices.

Let B′ ⊂ B be a subset of B of cardinality l′, let V′ ⊂ V be the vector space spanned by B′ and let R′ = R ∩ V′. This R′ is a root system in V′, B′ is a basis of R′, and Q(R′) = Q(R) ∩ V′. See [Bou, VI.1.7, Cor. 4, p. 162].

This implies that Q(R′) is a direct factor of Q(R) (as an abelian group). Moreover, since W (R′) is generated by the reflections in the hyperplanes orthogonal to the roots α ∈ R′, W (R′) can naturally be viewed as a subgroup of W (R). From the formula s_{α}(β) = β − n_{β,α}α we see that W (R′) acts trivially on V/V′ and hence on Q(R)/Q(R′). We write Q(R)/Q(R′) = ℤ^{l−l′}, the trivial W (R′)-lattice. In other words, there is a short exact sequence of W (R′)-lattices

0 → Q(R′) → Q(R) → ℤ^{l−l′} → 0.  (7.1)

Our proof of Proposition 7.1 will rely on the following claim.

Claim 7.2. For a root system R dual (inverse in the terminology of [Bou, VI 1.1]) to one of the root systems occurring in the statement of the proposition, there exist a subsystem R′ ⊂ R (as above), a subgroup H ≅ (ℤ/2ℤ)^{2} in W (R′) and a direct factor J_{H} of the H-lattice Q(R′), where J_{H} is the cokernel of the map ℤ → ℤ[H] given by the norm.

Indeed, assume Claim 7.2 is established. Consider the exact sequences

0 → ℤ → ℤ[H] → J_{H} → 0,
0 → I_{H} → ℤ[H] → ℤ → 0,

where the map ℤ[H] → ℤ is augmentation. The latter sequence yields � _{ω}^{1}(H, I_{H}) ≅ ℤ/2ℤ.

The Weyl groups of a root system R and of its dual R^{∨} are identical. Exact sequence (7.1) induces an exact sequence of W (R′)-lattices (the last two are weight lattices)

0 → ℤ^{l−l′} → P (R^{∨}) → P (R^{′∨}) → 0,  (7.2)

which we view as an exact sequence of H-lattices. Here R^{∨} is a root system as occurring in the proposition to be proved, i.e., a root system not of type A_{n}, C_{n} or G_{2}. (Recall that by the assumption G is simply connected, hence the W -lattice given by the character group of a maximal torus is the weight lattice.)

Using Claim 7.2 we conclude that the H-lattice P (R^{′∨}) (dual to Q(R′)) contains the H-lattice I_{H} (dual to J_{H}) as a direct factor, hence � _{ω}^{1}(H, P (R^{′∨})) ≠ 0. From the exact sequence (7.2) we get, by a standard computation,

� _{ω}^{1}(H, P (R^{∨})) ≅ � _{ω}^{1}(H, P (R^{′∨}))

hence

� _{ω}^{1}(H, P (R^{∨})) ≠ 0.

To complete the proof of Proposition 7.1 it remains to establish Claim 7.2.

Proof of Claim 7.2: The root systems R dual to those considered in the proposition are those of types C_{n} (n ⩾ 3), D_{n} (n ⩾ 4), E_{r} (r = 6, 7, 8) and F_{4}.

Any Dynkin diagram of type C_{n} (n ⩾ 3) contains a subdiagram of type C_{3}. All the other ones in the list above, except F_{4}, contain a subdiagram of type D_{4}. The case of F_{4} can be reduced to D_{4} because the weight lattices P (F_{4}) and P (D_{4}) coincide (as abelian groups in ℝ^{4}) and W (D_{4}) ⊂ W (F_{4}) ⊂ GL_{4}(ℝ) (compare Planche IV and Planche VIII in [Bou]).

For the reader’s convenience, we reproduce some of the calculations from [CK].

Recall that W (C_{n}) is a semidirect product (ℤ/2ℤ)^{n} ⋊ S_{n}. We denote by c_{1}, …, c_{n} the natural generators of (ℤ/2ℤ)^{n}.

Let us first discuss the case where R is of type C_{3}. We choose H = ⟨c_{1}c_{3}, c_{2}(13)⟩ = ⟨a, b⟩ ⊂ W (C_{3}). In the basis α_{1} = ε_{1} − ε_{2}, α_{2} = ε_{2} − ε_{3}, α_{3} = ε_{2} + ε_{3}, the group H acts on M = Q(C_{3}) as follows:

a: \{ \begin{matrix}α_{1} & ↦ & −α_{1} − α_{2} − α_{3}, \\ α_{2} & ↦ & α_{3}, \\ α_{3} & ↦ & α_{2}, \end{matrix} b: \{ \begin{matrix}α_{1} & ↦ & α_{3}, \\ α_{2} & ↦ & −α_{1} − α_{2} − α_{3}, \\ α_{3} & ↦ & α_{1}. \end{matrix}

This coincides with the standard formulas for J_{H}.

Let us now discuss the case where R is of type D_{4}. In ℝ^{4} equipped with the standard basis ε_{1}, …, ε_{4}, we consider M = Q(D_{4}) with ℤ-basis α_{1} = ε_{1} − ε_{2}, α_{2} = ε_{2} − ε_{3}, α_{3} = ε_{3} − ε_{4}, α_{4} = ε_{3} + ε_{4}. The Weyl group W (D_{4}) can be identified with the subgroup of W (C_{4}) consisting of the elements with even numbers of c_{i}’s. We choose H = ⟨c_{3}c_{4}, c_{1}c_{2}(34)⟩. The group H acting on M respects V′ = ⟨ε_{2}, ε_{3}, ε_{4}⟩ = ⟨α_{2}, α_{3}, α_{4}⟩, and R′ = R ∩ V′ is of type D_{3}. Moreover, H respects the one-dimensional ℤ-module generated by α_{1}: indeed, c_{3}c_{4} fixes α_{1} and c_{1}c_{2}(34) sends α_{1} to −α_{1}. Therefore the H-lattice M decomposes into a direct sum of a one-dimensional lattice and a three-dimensional lattice. It remains to note that the latter three-dimensional lattice J is isomorphic to J_{H}. To see that, we observe that the action of c_{1}c_{2}(34) on J coincides with the action of c_{2}(34) ∈ W (C_{3}) on Q(C_{3}) = Q(D_{3}), and we are led (up to permutation of indices) to the former case.

This completes the proof of Claim 7.2, hence of Proposition 7.1, hence of Theorems 6.1(b) and 0.2(b). □

Remark 7.3. Our proof of Theorem 6.1(b) actually establishes the following stronger assertion.

Proposition 7.4. Let G be a simple simply connected linear algebraic group over k which is not of type A_{n}, C_{n} or G_{2}. Let T_{gen} be the generic torus of G. Recall that T_{gen} is defined over the field K_{gen} = k(G/N); cf. §4.2. Then (G_{K_{gen}}/T_{gen}) ×_{K_{gen}} Y is not rational over K_{gen} for any K_{gen}-variety Y.

Indeed, let X_{c} be a smooth K_{gen}-compactification of X = G_{K_{gen}}/T_{gen}. Combining Theorem 6.3, Remark 6.4 and Proposition 7.1 we find that there exists a finite field extension M/K_{gen} such that Br(X_{c})_{M}/ Br M ≠ 0. On the other hand, if there exists a K_{gen}-variety Y such that X ×_{K_{gen}} Y is K_{gen}-birationally isomorphic to projective space, then Br(X_{c})_{M}/ Br M = 0 for any field extension M/K_{gen}. As a matter of fact, the nonvanishing of Br(X_{c})_{M}/ Br M implies that the K_{gen}-variety X = G_{K_{gen}}/T_{gen} is not even retract rational (a concept due to D. Saltman); see [CTS_{4}, §1, Prop. 5.7 and Remark 5.8]. □

8. Weight lattices for root systems of types A_{n}, C_{n}, and G_{2}

In this section we shall prove the following converse to Proposition 7.1.

Proposition 8.1. Let R be a reduced, irreducible root system and P (R) be the weight lattice equipped with the action of the Weyl group W = W (R). If R is of type A_{n}, C_{n} or G_{2}, then there exists an exact sequence of W -lattices

0 → P_{2} → P_{1} → P (R) → 0

with P_{1} and P_{2} permutation lattices.

Proof. Suppose R is of type G_{2}. Here the W -lattice M = P (G_{2}) is of rank 2. Thus, as we pointed out at the end of Section 1, M fits into an exact sequence

0 → P_{2} → P_{1} → M → 0 ,

with P_{1} and P_{2} permutation.

Now suppose R is of type A_{n}. Then W = S_{n+1}. From the Bourbaki tables [Bou] we get the exact sequence of S_{n+1}-lattices

0 → Q(A_{n}) → ⨁_{i=1}^{n+1} ℤε_{i} → ℤ → 0,

where the action of S_{n+1} on the middle term is by permutation, the action on the right hand side ℤ is trivial and the right hand side map to ℤ is augmentation, i.e., summation of coefficients.

If one dualizes this sequence one gets an exact sequence of S_{n+1}-lattices

0 → ℤ → ⨁_{i=1}^{n+1} ℤε_{i} → P (A_{n}) → 0,  (8.1)

where the action of S_{n+1} on the middle term is by permutation, the action on the left hand side ℤ is trivial and the map with source ℤ sends 1 to the sum of the ε_{i}.

Finally suppose R is of type C_{n} (n ⩾ 2). Then W is the semidirect product of S_{n} by (ℤ/2ℤ)^{n}. Let us first look at the B_{n}-table in [Bou]. There is an exact sequence of W (B_{n})-lattices

0 → Q(B_{n}) → ⨁_{i=1}^{n}(ℤa_{i} ⊕ ℤb_{i}) → ⨁_{i=1}^{n} ℤc_{i} → 0,

where a_{i} and b_{i} are mapped to c_{i}, and the action of W is as follows. On the right hand lattice ⨁_{i=1}^{n} ℤc_{i}, the action is the permutation action of the quotient S_{n}. On the middle lattice, S_{n} acts by naturally permuting the a_{i} and the b_{i}. An element (α_{1}, …, α_{n}) ∈ (ℤ/2ℤ)^{n} fixes a_{i} and b_{i} if α_{i} = 0 and it permutes them if α_{i} = 1. In Bourbaki’s notation for B_{n}, we have ε_{i} = a_{i} − b_{i}. If one dualizes the above sequence, one gets the exact sequence of W (C_{n})-lattices

0 → ⨁_{i=1}^{n} ℤγ_{i} → ⨁_{i=1}^{n}(ℤα_{i} ⊕ ℤβ_{i}) → P (C_{n}) → 0,  (8.2)

where the two left lattices are permutation lattices. □

Proposition 8.2. Let G be a split, simply connected simple group of type A_{n} or C_{n} (i.e., G = SL_{n} or Sp_{2n}). Then the field extensions k(G)/k(G)^{G} and k(𝔤)/k(𝔤)^{G} are stably pure.

Proof. Since these groups are special, Propositions 6.6 and 8.1 imply that G_{K}/T_{gen} is stably rational over K_{gen} (or equivalently, k(𝔤) is stably rational over k(𝔤)^{G}). The statement then follows from Theorem 4.10. □

Remark 8.3. This is weaker than the rationality assertion of Theorem 6.1(a) (or equivalently, of Theorem 0.2(a)), which will be proved in the next section. In the meantime, we remark that the same argument cannot be used to show that G_{K}/T_{gen} is stably rational (or equivalently, k(𝔤) is stably rational over k(𝔤)^{G}) for the split G_{2}, because this group is not special and Proposition 6.6 does not apply to it. In fact, in this case we do not know whether or not G_{K}/T_{gen} is stably rational over K_{gen}.

9. Rationality

We now turn to the proof of Theorem 6.1(a) (or equivalently, of Theorem 0.2(a)). Our argument will be based on the rationality criterion of Lemma 9.1(c) below. For groups of type A_{n}, Theorem 6.1(a) will be an easy consequence of this criterion. For groups of type C_{n} the proof proceeds along the same lines but requires a more elaborate argument.

Let K be a field of characteristic zero, let G be a linear algebraic group over K, and let H_{1}, H_{2} be closed K-subgroups. The actions of H_{1} and H_{2} on G by respectively left and right translation commute, thus giving rise to an (H_{1} × H_{2})-action on G. The action of H_{1} on G defines an H_{1}-torsor π_{1} : G → H_{1}\G, and the action of H_{2} defines an H_{2}-torsor π_{2} : G → G/H_{2}.

Using Rosenlicht’s theorem (see §2.1), one may find a (H_{1} × H_{2})-stable dense open subset U of G such that the action of H_{1} × H_{2} on U mods out to a geometric quotient U → V which factorizes through U → U_{1} → V and U → U_{2} → V, where U_{1} ⊂ H_{1}\G and U_{2} ⊂ G/H_{2} are open sets and U → U_{1}, respectively, U → U_{2} is an H_{1}-torsor, respectively, an H_{2}-torsor.

In this section we shall indulge in the following notation. We shall adopt the double coset notation H_{1}\G/H_{2} for some V as above. In particular, we have

K(H_{1}\G/H_{2}) ≅ K(G)^{H_{1}×H_{2}} .

We have a commutative diagram of rational maps, where, as usual, solid arrows denote regular maps and dotted arrows denote rational maps:

\xymatrix{ & G \ar[dl]_{π_{1}} \ar@{-->}[dd]^{π} \ar[dr]^{π_{2}} \\ H_{1}\G \ar@{-->}[dr] & & G/H_{2} \ar@{-->}[dl] \\ & H_{1}\G/H_{2}} .

Let Z(G) denote the centre of G.

Lemma 9.1. Let K be a field of characteristic zero and let G be a simple K-group. Suppose G has subgroups H and T, where T is a torus and H ∩ Z(G) = {1}. Then

(a) the H-action on G/T is generically free;

(b) dim(H \G/T) = dim(G) − dim(H) − dim(T);

(c) if H is special and both H, H \G/T are K-rational, then G/T is also K-rational.

Proof. To prove (a), we may pass to an algebraic closure \overline{K} of K and thus assume, without loss of generality, that K is algebraically closed.

Note that conditions

(a) the left H-action on G/T is generically free; (a′) the right T -action on H \G is generically free

are equivalent. Indeed, (a) says that H ∩ gT g^{−1} = {1} for g ∈ G in general position and (a′) says that T ∩ gHg^{−1} = {1} for g ∈ G in general position. Thus (a) and (a′) are equivalent, and it suffices to prove (a′).

Assume to the contrary that the T -action on H\G is not generically free. By a result of Sumihiro [Sum] there is an affine T -stable dense open set U of H \G. By the Embedding Theorem [PV, Theorem 1.5], U is a G-stable closed irreducible subvariety of a finite-dimensional T -module V not contained in a proper T -submodule of V. Hence U intersects the complement to the union of weight spaces of T. But the stabilizer of a point in this complement coincides with the kernel of the action of T on V. Thus the action of T on U, hence on H \G has a nontrivial kernel Γ ⊂ T, cf. [PV, Sect. 7.2, Prop.] Then Γ is contained in N = ⋂_{g∈G} gHg^{−1}, which is a normal subgroup of G. Since N ⊊ G and we are assuming that G is simple, we conclude that N ⊂ Z(G). Thus {1} ⊊ Γ ⊂ N = N ∩ Z(G) ⊂ H ∩ Z(G), contradicting our assumption that H ∩ Z(G) = {1}. This contradiction proves part (a).

By Theorem 2.13 (case S = {1}) and the standard formula for the dimension of a variety fibred over another variety, (b) follows from (a).

Let us now prove (c). Part (a) allows us to apply Lemma 6.5 to the left H-action on G/T. By Lemma 6.5, G/T is K-birationally isomorphic to H × (H\G/T). Since we are assuming that both H and H \G/T are K-rational, so is G/T. □

We now proceed with the proof of Theorem 6.1(a). The group G is a split simply connected simple group of type A_{n} or C_{n}, over the field k. In the sequel we will set K = K_{gen} and will work with the generic K-torus T = T_{gen} ⊂ G_{K}, as in §4.2.

Type A_{n}. Let H denote the stabilizer of a non-zero element for the natural action of G = SL_{n+1} on 𝔸_{k}^{n+1}. It is easy to see that H is isomorphic to a semidirect product U ⋊SL_{n}, for some unipotent group U and that H ∩ Z(SL_{n+1}) = 1. By Lemma 9.1(c), it suffices to show that (i) H is special, (ii) H is k-rational, and (iii) the “double coset space” H_{K}\G_{K}/T_{gen} (whose definition is explained above) is K-rational. We now proceed to prove (i), (ii) and (iii).

(i) For any field extension F/k, the natural map

H^{1}(F, H) → H^{1}(F, SL_{n})

is an isomorphism; see, e.g., [San, Lemme 1.13]. Since SL_{n} is a special group, we conclude that H^{1}(F, H) ≅ H^{1}(F, SL_{n}) = {1}, i.e., H is special.

(ii) In characteristic zero, any unipotent group is special (see [Ser_{2}, Prop. III.2.1.6]) and rational (see, e.g., [LPR, Example 1.21]). Viewing the natural projection H → SL_{n} as a U-torsor over SL_{n}, we see that H is k-birationally isomorphic to U × SL_{n}. This shows that H is k-rational.

(iii) H_{K} \G_{K}/T_{gen} is a 1-dimensional K-variety; see Lemma 9.1(b). It is clearly unirational over K (it is covered by G_{K}). By Lüroth’s theorem it is thus K-rational.

This completes the proof of Theorem 6.1 for groups of type A_{n}.

Type C_{n}. Let G := Sp_{2n}. Once again, we let H be the Sp_{2n}-stabilizer of a non-zero vector v ∈ k^{2n} for the natural action of G on 𝔸^{2n}. It is well known that H is k-isomorphic to a semidirect product of U ⋊ Sp_{2n−2}, where U is a unipotent group defined over k; see, e.g., [Wei, pp. 35–36] or [Igu, p. 384]. Once again, by Lemma 9.1(c), it suffices to show that

(i) H is special; (ii) H is k-rational; (iii) H_{K} \G_{K}/T_{gen} is K-rational.

The proofs of (i) and (ii) are exactly the same as for type A. In order to complete the proof of Theorem 6.1(a) (or equivalently, of Theorem 0.2(a)), it thus remains to establish (iii), which we now restate as a proposition.

Proposition 9.2. H_{K} \G_{K}/T_{gen} is K-rational.

To prove the proposition, we first note that H\G is, by definition, the Sp_{2n}-orbit of a non-zero element v in 𝔸^{2n}. By Witt’s extension theorem ([Art, Theorem 3.9]), this orbit is 𝔸^{2n} \{0}. Thus the group Sp_{2n} acts on H_{K} \G_{K} (on the right) via its natural, linear, action on 𝔸_{K}^{2n}. Restricting this action to T_{gen}, we reduce our problem to showing that 𝔸^{2n}/T_{gen} is K-rational.

The K-torus T_{gen} is split by a W -Galois extension and the character lattice of T_{gen} is the weight lattice P (C_{n}) with its W -action (see §4.2). Here W = (ℤ/2ℤ)^{n} ⋊ S_{n} is the Weyl group of G = Sp_{2n}. Over L we can diagonalize the T_{gen}-action on 𝔸^{2n} in some L-basis f_{1}, …, f_{2n}. Let χ_{1}, …, χ_{2n} be the associated characters of T_{gen} ×_{K} L. We have t · f_{i} = χ_{i}(t)f_{i} for every t ∈ T_{gen}(L). These characters are permuted by W; denote the associated W -permutation lattice of rank 2n by P_{1}. That is, W permutes a set of generators a_{1}, …, a_{2n} of P_{1}; sending a_{i} to χ_{i}, we obtain a morphism of W -lattices

τ: P_{1} → P (C_{n}) .

Since G = Sp_{2n} acts faithfully on 𝔸^{2n}, so does T_{gen} ⊂ G_{K}; hence τ is surjective and we obtain a sequence of W -lattices

0 → P_{2} → P_{1} → P (C_{n}) → 0

and the dual sequence

1 → T_{gen} → T_{1} → T_{2} → 1

of K-tori. The torus T_{1} has a dense open orbit in 𝔸^{2n}; identifying T_{1} with this orbit, we obtain the following birational isomorphisms of K-varieties:

𝔸_{K}^{2n}/T_{gen} ≅ T_{1}/T_{gen} ≅ T_{2}.

It thus remains to show that the n-dimensional torus T_{2} is rational over K. Since every torus of dimension ⩽ 2 is rational, we may assume without loss of generality that n ⩾ 3. We have thus reduced Proposition 9.2 to the following lemma.

Lemma 9.3. Let n ⩾ 3, let W = W (C_{n}) = (ℤ/2ℤ)^{n} ⋊ S_{n}, let P be a permutation W -lattice of rank 2n and let

0 → M → P →^{φ} P (C_{n}) → 0

be an exact sequence of W -lattices. Then there exists a W -isomorphism between this sequence and sequence (8.2). In particular, M is a permutation lattice.

Recall that P (C_{n}) has a basis h_{1}, …, h_{n} such that c = (c_{1}, …, c_{n}) ∈ (ℤ/2ℤ)^{n} acts on P (C_{n}) by h_{i} ↦ (−1)^{c_{i}}h_{i} and S_{n} permutes h_{1}, …, h_{n} in the natural way.

Proof of Lemma 9.3. Denote the normal subgroup (ℤ/2ℤ)^{n} of W by A. We shall identify the dual group A^{∗} = Hom(A, ℤ/2ℤ) with (ℤ/2ℤ)^{n} in the usual way. That is, (b_{1}, …, b_{n}) ∈ (ℤ/2ℤ)^{n} will denote the additive character χ: A → ℤ/2ℤ taking (a_{1}, …, a_{n}) ∈ A to b_{1}a_{1} + ⋯ + b_{n}a_{n} ∈ ℤ/2ℤ.

We now observe that P (C_{n})⊗ℚ decomposes as the direct sum of n one-dimensional A-invariant ℚ-subspaces, upon which A acts by the characters

χ_{1} = (1, 0, …, 0) , …, χ_{n} = (0, …, 0, 1) .

Since the exact sequence

0 → M ⊗ ℚ → P ⊗ ℚ → P (C_{n}) ⊗ ℚ → 0

of A-modules over ℚ splits, all of these characters will be present in the irreducible decomposition of P ⊗ℚ (as an A-module over ℚ). Since P is a permutation W -module, the trivial character will also occur in this decomposition; denote its multiplicity by d, where 1 ⩽ d ⩽ n. The set Λ of the remaining n−d characters is permuted by S_{n}. Since the orbit of a character (b_{1}, …, b_{n}) ∈ A^{∗} has \binom{n}{i} elements, where i is the number of times 1 occurs among b_{1}, …, b_{n} ∈ ℤ/2ℤ, and \binom{n}{i} ⩾ n for any 1 ⩽ i ⩽ n − 1, we conclude that (b_{1}, …, b_{n}) can be in Λ only if i = n, i.e., b_{1} = ⋯ = b_{n} = 1. In summary, P ⊗ ℚ, viewed as an A-module over ℚ, is the direct sum of the following characters:

(0, …, 0), with multiplicity d,

(1, …, 1), with multiplicity n − d,  (9.1)

χ_{1} = (1, 0, …, 0), …, χ_{n} = (0, …, 0, 1), each with multiplicity 1,

where d ⩾ 1 is an integer.

In order to gain greater insight into the A-action on P and in particular to determine the exact value of d, we shall now compute the irreducible decomposition of P ⊗ ℚ (as an A-module over ℚ) in a different way. Recall that P is a permutation W -lattice. In particular, we may write

P ≅ ℤ[A/A_{1}] ⊕ ⋯ ⊕ ℤ[A/A_{r}] ,

where ≅ denotes an isomorphism of A-lattices. Now observe that, as an A-module over ℚ, ℚ[A/A_{i}] is the sum of all those characters of A which vanish on A_{i}. Denote the set of all such characters by Λ_{i} ⊂ (ℤ/2ℤ)^{n}. Thus the A-module P ⊗ ℚ, which we know is the direct sum of the 2n characters listed in (9.1), can also be written as the direct sum of the characters in Λ_{1}, …, Λ_{r}. Here some of the characters may appear with multiplicity ⩾ 2; note that |Λ_{1}| + ⋯ + |Λ_{r}| = 2n (here |Λ_{i}| denotes the order of Λ_{i}). In other words, the 2n characters listed in (9.1) can be partitioned into r subsets Λ_{1}, …, Λ_{r}.

Note that each Λ_{i} is clearly a subgroup of (ℤ/2ℤ)^{n}. On the other hand, since χ_{l} + χ_{m} is not on our list (9.1) for any l ≠ m (here we are using the assumption that n ⩾ 3), we see that no two of the characters χ_{1}, …, χ_{n} can be contained in the same Λ_{i}. This implies r ⩾ n. After possibly relabeling the subgroups A_{1}, …, A_{r} and Λ_{1}, …, Λ_{r}, we may assume that χ_{i} ∈ Λ_{i} for i = 1, …, n. Since Λ_{i} is a subgroup of (ℤ/2ℤ)^{n}, each Λ_{i} should also contain the trivial character. This shows that the irreducible decomposition of P ⊗ ℚ (as an A-module) contains at least n copies of the trivial character (0, …, 0). We conclude that n − d = 0 in (9.1), r = n, Λ_{i} = {(0, …, 0), χ_{i}}, and

A_{i} = Ker(χ_{i}) = (ℤ/2ℤ) × ⋯ × (ℤ/2ℤ) × {1} × (ℤ/2ℤ) × ⋯ × (ℤ/2ℤ),

where {1} occurs in the ith position.

We now return to the permutation W -lattice P. Let e_{1}, f_{1}, …, e_{n}, f_{n} be a ℤ-basis of P permuted by W. As we saw above, the permutation action of A on this basis is isomorphic to that on A/A_{1} ∪ ⋯ ∪ A/A_{n}. After suitably relabeling the basis elements, we may thus assume that e_{i} and f_{i} are the two elements of our basis fixed by A_{i}. Clearly A permutes {e_{i}, f_{i}}. On the other hand, since conjugation by S_{n} naturally permutes the subgroups A_{1}, …, A_{n} of A, it also naturally permutes the (unordered) pairs {e_{i}, f_{i}}.

Since A_{i} fixes e_{i} and f_{i}, the elements φ(e_{i}) and φ(f_{i}) lie in P (C_{n})^{A_{i}} = ℤh_{i}. In other words, φ(e_{i}) = m_{i}h_{i} and φ(f_{i}) = −m_{i}h_{i} for some m_{1}, …, m_{n} ∈ ℤ. Since φ is surjective, m_{i} = ±1 for each i. After interchanging e_{i} and f_{i} if necessary, we may assume that m_{1} = ⋯ = m_{n} = 1. Now S_{n} permutes both {e_{1}, …, e_{n}} and {f_{1}, …, f_{n}} in the natural way. Identifying e_{i} ∈ P with α_{i}, f_{i} with β_{i}, we see that the exact sequence

0 → M → P →^{φ} P (C_{n}) → 0

is W -equivariantly isomorphic to the sequence (8.2). □

This completes the proof of Proposition 9.2, hence of Theorem 0.2 (a).

10. Appendix: G/T versus T^{0}

Let G be a semisimple simply connected group defined over a field K and let T ⊂ G be a K-torus. As we mentioned in Remark 6.2, there is no obvious connection between the (stable) K-rationality of G/T and that of T.

However, using the results on tori recalled in Section 1.2, Theorem 6.3(d) may be rephrased in the following manner: If G/T is stably K-rational, then the dual torus T^{0} is stably K-rational. Similar, Proposition 6.6 can be rephrased as follows: Suppose G is a special, split K-group and T ⊂ G is a K-torus. If the dual torus T^{0} is stably K-rational, then G/T is stably K-rational.

One might then wonder if, when G is split (but not necessarily special), the (stable) Krationality of the dual torus T^{0} implies that of G/T. This is an open question; a positive answer would yield the stable rationality in the missing case G_{2} in our main Theorem 0.2 or (equivalently, in Theorem 6.1).

One may go even further and ask whether or not G/T and T^{0} are always stably K-birationally isomorphic (assuming G is split). The purpose of this appendix is to show that this stronger assertion is false.

Proposition 10.1. There exist a K-torus T and a split semisimple simply connected group G with T ⊂ G such that G/T is not stably K-birationally isomorphic to the dual torus T^{0}.

Proof. Let K be a field of characteristic zero and let L/K be a finite Galois extension of fields with group Γ. The augmentation map ℤ[Γ] → ℤ gives rise to the exact sequence of Γ-lattices

0 → I_{Γ} → ℤ[Γ] → ℤ → 0.

The norm map 1 → N_{Γ} = ∑_{g∈Γ} g ∈ ℤ[Γ] gives rise to the exact sequence of Γ-lattices

0 → ℤ → ℤ[Γ] → J_{Γ} → 0

which is dual to the previous sequence.

Let T/K be the torus with character group T^{∗} = I_{Γ}. The character group of T^{0} is then (T^{0})^{∗} = J_{Γ}.

The torus T is the K-torus R_{L/K}^{1}𝔾_{m} of norm 1 elements in L. For d = [L : K] this torus is a maximal torus in G = SL_{d}.

The unramified Brauer group of the K-torus T^{0} (modulo Br K) is � _{ω}^{2}(Γ, (T^{0})^{∗}); see [CTS_{2}]. We have

� _{ω}^{2}(Γ, (T^{0})^{∗}) = � ^{3}(Γ, ℤ) = H^{3}(Γ, ℤ)

(recall that the cohomology of a cyclic group has period 2 and that H^{1}(H, ℤ) = 0 for any finite group H.)

The unramified Brauer group of G/T (modulo Br K) is � _{ω}^{1}(Γ, T^{∗}); see [CTK]. We have

� _{ω}^{1}(Γ, T^{∗}) = Ker[Ĥ^{0}(Γ, ℤ) → ∏_{g∈Γ} Ĥ^{0}(g, ℤ)].

This group is

Ker[ℤ/n_{Γ} → ∏_{g∈Γ} ℤ/n_{g}]

where the projection is the natural map, n_{Γ} is the order of Γ and n_{g} the exponent of g ∈ Γ.

Let us now take Γ = (ℤ/p)^{r}, r ⩾ 2. The Künneth formula shows that H^{3}(Γ, ℤ) is killed by p; see [Spa, p. 247]. Thus the group � _{ω}^{2}(Γ, (T^{0})^{∗}) is killed by p.

The group

Ker[ℤ/n_{Γ} → ∏_{g∈Γ} ℤ/n_{g}]

is

pℤ/p^{r}ℤ ⊂ ℤ/p^{r}ℤ .

Thus for any r ⩾ 3, the group � _{ω}^{1}(Γ, T^{∗}) is not killed by p.

We conclude that the unramified Brauer group of G/T (modulo Br K) is not isomorphic to the unramified Brauer group of T^{0} (modulo Br K). Therefore G/T and T^{0} are not stably Kbirationally isomorphic, as claimed.

□

Acknowledgements

The authors are grateful to the referee for contributing Proposition 3.3 and for other helpful suggestions.

Colliot-Thélène is a researcher at C.N.R.S. (France). Kunyavskiĭ was supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics. Popov was supported in part by Russian grants ���� 08–01–00095, ��–1987.2008.1, a (granting) program of the Mathematics Branch of the Russian Academy of Sciences, by ETH Zürich, and TU München. He also thanks the Department of Mathematics of UBC for its hospitality. Reichstein was supported in part by an NSERC Discovery grant and Accelerator supplement. Colliot-Thélène, Kunyavskiĭ and Reichstein thank the organizers of the Oberwolfach meeting “Quadratic Forms and Linear Algebraic Groups” (June 2006). Colliot-Thélène and Reichstein also acknowledge the support of Emory University.

References

[AOVdB] J. Alev, A. Ooms, M. Van den Bergh, A class of counterexamples to the Gel’fand–Kirillov conjecture, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1709–1716.

[Art] E. Artin, Geometric Algebra, Interscience publishers Inc., New York-London, 1957.

[BF] G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.

[BB] A. Białynicki-Birula, On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577–582.

[Bor] A. Borel, Linear Algebraic Groups, 2nd enlarged ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, 1991.

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Chap. IV, V, VI, Hermann, Paris, 1968.

[CTK] J.-L. Colliot-Thélène, B. È. Kunyavskiĭ, Groupe de Brauer et groupe de Picard des compactifications lisses d’espaces homogènes, J. Algebraic Geom. 15 (2006), 733–752.

[CTS_{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.

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

J. Algebra 106 (1987), 148–205.

[CTS_{3}] J.-L. Colliot–Thélène, J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J.

54 (1987), 375–492.

[CTS_{4}] J.-L. Colliot–Thélène, J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), in: Proceedings of the International Colloquium on Algebraic groups and Homogeneous Spaces (Mumbai 2004), ed. V. Mehta, TIFR Mumbai, Narosa Publishing House, 2007, 113–186.

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

[SGA3] M. Demazure et A. Grothendieck, Schémas en Groupes, I, II, III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, Vols. 151, 152, 153, Springer-Verlag, Berlin, New York, 1970.

[Fl] M. Florence, On the essential dimension of cyclic p-groups, Invent. Math. 171 (2008), no. 1, 175–189.

[Fog] J. Fogarty, Fixed points schemes, Amer. J. Math. 95 (1973), 35–51.

[GMS] S. R. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series, Vol. 28, American Mathematical Society, Providence, RI, 2003.

[Gro] A. Grothendieck, Torsion homologique et sections rationnelles, in: Anneaux de Chow et Applications, Séminaire C. Chevalley, 1958, exposé 5.

[EGA IV] A. Grothendieck, avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique,

IV, Publ. Math. I.H.É.S., Vol. 20, 24, 28, 32.

[GS] Correspondance Grothendieck–Serre, éd. par P. Colmez et J-P. Serre, Documents Mathématiques, vol. 2, Soc. Math. France, 2001. Reprinted and translated into English in Grothendieck–Serre Correspondence, Bilingual Edition (P. Colmez, J-P. Serre, eds.), Amer. Math. Soc., Soc. Math. France, 2004.

[Hum_{1}] J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer-Verlag,

New York, Heidelberg, 1975.

[Hum_{2}] J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and

Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1995.

[Igu] J.-i. Igusa, Geometry of absolutely admissible representations, in: Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 373–452.

[Kos] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. [Kot_{1}] R. E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785–806. [Kot_{2}] R. E. Kottwitz, Transfer factors for Lie algebras, Representation theory 3 (1999), 127–138.

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

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

[Lor] M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 135, Invariant Theory and Algebraic Transformation Groups, VI, Springer-Verlag, Berlin, 2005.

[Lun] D. Luna, Slices étales, Mémoires de la S. M. F. 33 (1973), 81–105.

[Mat] Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J. 16 (1960), 205–218.

[Mum] D. Mumford, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Bd. 34, Springer-Verlag, Berlin, New-York, 1965. 3rd enlarged ed.: D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Bd. 34, Springer-Verlag, Berlin, New-York, 1994.

[Oni] A. L. Onishchik, Complex hulls of compact homogeneous spaces, Dokl. Akad. Nauk SSSR 130 (1960), no. 4, 726–729. Engl. transl.: Sov. Math., Dokl. 1 (1960), 88–91.

[Pop_{1}] V. L. Popov, On the stability of the action of an algebraic group on an algebraic variety, Izv. Akad.

Nauk SSSR, Ser. Mat. 36 (1972), 371–385. Engl. transl.: Math. USSR Izv. 6 (1972) (1973), 367–379.

[Pop_{2}] V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Izv. Acad. Nauk SSSR, Ser. Mat. 38 (1974), no. 2, 294–322. Engl. transl.: Math. USSR Izv. 8 (1974), no. 23, 301–327.

[Pop_{3}] V. L. Popov, Groups, Generators, Syzygies, and Orbits in Invariant Theory, Translations of Mathe-

matical Monographs, Vol. 100, American Mathematical Society, Providence, RI, 1992.

[Pop_{4}] V. L. Popov, Sections in invariant theory, in: The Sophus Lie Memorial Conference (Oslo, 1992),

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

[Pop_{5}] V. L. Popov, Cross-sections, quotients, and representation rings of semisimple algebraic groups, avail-

          able at arXiv:0908.0826 and http://www.math.uni-bielefeld.de/LAG/man/351.pdf.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

[Pr] A. Premet, Modular Lie algebras and the Gelfand–Kirillov conjecture, available at http://arxiv.org/

          abs/0907.2500.
[SGA3bis] Réédition de SGA 3, available at http://people.math.jussieu.fr/∼polo/SGA3/.

[Rei] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transf. Groups 5 (2000), no. 3, 265–304.

[Rich] R. W. Richardson, Jr., Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math. 16 (1972), 6–14.

[Ros_{1}] M. Rosenlicht, Some basic theorems on algebraic groups, American Journal of Math. 78 (1956),

401–443.

[Ros_{2}] M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci. 35 (1963), 487–489.

[San] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. für die reine und angew. Mathematik (Crelle) 327 (1981), 12–80.

[Ser_{1}] J-P. Serre, Espaces fibrés algébriques, in: Anneaux de Chow et Applications, Séminaire C. Chevalley, 1958, exposé 1. Reprinted in J-P. Serre, Exposés de séminaires 1950–1999, deuxième édition, augmentée, Documents mathématiques 1, Société mathématique de France 2008, 107–140.

[Ser_{2}] J-P. Serre, Cohomologie Galoisienne, 5ième édition, Lecture Notes in Mathematics, Vol. 5, Springer-

Verlag, Berlin, 1994.

[Spa] E. H. Spanier, Algebraic Topology, Springer-Verlag, New York, Berlin, 1981. [Spr_{1}] T. A. Springer, Linear Algebraic Groups, 2nd ed., Progress in Mathematics, Vol. 9, Birkhäuser,

Boston, MA, 1998.

[Spr_{2}] T. A. Springer, Aktionen reduktiver Gruppen auf Varietäten, in: Algebraische Transformationsgruppen und Invariantentheorie, Herausgeb. H. Kraft, P. Slodowy, T. A. Springer, DMV Seminar, Bd. 13, Birkhäuser, Basel, 1989.

[Ste] R. Steinberg, Regular elements of semi-simple groups, Publ. Math. I.H.É.S. 25 (1965), 49–80.

[Sum] H. Sumihiro, Equivariant completion, I, J. Math. Kyoto Univ. 14 (1974), 1–28.

[Tho] R. W. Thomason, Comparison of equivariant algebraic and topological K-theory, Duke Math. J. 55 (1986), no. 3, 795–825. .

[Vos_{1}] �� �� ������������� �, �������������� ����, �����, ������, 1977. [Vos_{2}] V. E. Voskresenskiĭ, Maximal tori without affect in semisimple algebraic groups, Mat. Zametki 44

(1988), no. 3, 309–318, 410. English transl.: Math. Notes 44 (1988) (1989), no. 3–4, 651–655.

[Vos_{3}] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, American Mathematical

Society, Providence, RI, 1998.

[Wei] A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87.

C.N.R.S., UMR 8628, Mathématiques, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France

   E-mail address: jlct@math.u-psud.fr

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

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

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: reichst@math.ubc.ca