THE SLICE METHOD FOR G-TORSORS

ROLAND LÖTSCHER^{(1)} AND MARK MACDONALD^{(2)}

Abstract. The notion of a (G, N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80’s for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G-torsors. Namely we show that any (G, N)- slice of a versal G-scheme gives surjective maps H^{1}(L, N) → H^{1}(L, G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G, N_{G}(H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E_{7}.

1. Introduction

In [Ka83] P.I. Katsylo introduced the notion of a slice (or section) for a regular action of a connected algebraic group G on a variety X over the field ℂ of complex numbers as follows: A (G, N)-slice of X, for a subgroup N of G, is a (locally closed) subvariety S of X satisfying the following two conditions:

(1) \overline{G · S} = X

(2) If s ∈ S and g ∈ G then gs ∈ S if and only if g ∈ N.

This definition was inspired by Seshadri’s work [Se62] from the early sixties. Katsylo observed that any (G, N)-slice induces an isomorphism ℂ(X)^{G} ≃ ℂ(S)^{N} through restriction of rational functions. Applying this observation to the case G = SL_{2} and X = V_{d} an irreducible Grepresentation, he showed that the fields ℂ(V_{d})^{SL^{2}} are projectively rational, i.e., generated by algebraically independent homomogeneous rational functions.

A closely related notion is the one of a Chevalley relative section. This is a subvariety S of X such that restriction of regular functions induces an isomorphism ℂ[X]^{G} ≃ ℂ[S]^{N}, where N = {g ∈ G | gS = S} is the normalizer of Y. Existence of Chevalley relative sections and (G, N)-slices as defined by Katsylo simplify the calculation of regular resp. rational invariants of an algebraic group G.

Notions of slices and sections are widespread in the literature, not only forming an important technical tool in invariant theory, but also studied in their own right. In disguised form these concepts were already present in the 19th century, for instance in the work of Weierstrass and Hesse on normal forms of plane cubic curves. We refer to [Po94] for a systematic treatment and survey of this topic, all over an algebraically closed base field of characteristic 0.

Our interest in slices and sections has slightly different origins. It comes from the desire to classify torsors of an algebraic group G over an arbitrary field, or equivalently, to describe the sets H_{fppf}^{1}(L, G) for an algebraic group G over F and field extension L of F. Torsors of algebraic groups often describe interesting algebraic objects like central simple algebras (for projective linear groups), quadratic forms (for orthogonal groups) or Cayley algebras (for the exceptional simple group G_{2}).

2000 Mathematics Subject Classification. 20G15, 11E72, 14L24.

Key words and phrases. G-torsor, slice, relative section, reduction of structure group, essential dimension, E_{7}, stabilizer in general position.

(1) Supported by the Deutsche Forschungsgemeinschaft, GI 706/2-1.

By getting information on the structure of torsors we also wish to improve the known upper bounds on the essential dimension of various algebraic groups. The essential dimension of an algebraic group G is a measure of how many algebraically independent parameters are needed to describe any of its torsors up to isomorphism. For example, isomorphism classes of O_{n}-torsors correspond bijectively to isometry classes of non-degenerate quadratic forms of dimension n. It is known that such quadratic forms can always be diagonalized (provided char F ≠ 2), which translates to the fact that H^{1}(L, μ_{2}^{n}) → H^{1}(L, O_{n}) is surjective for all field extensions L/F. This implies ed(O_{n}) ≤ ed(μ_{2}^{n}) = n. Indeed, diagonalization should mean that a quadratic form “is described by at most n parameters”.

Let N be a subgroup of an algebraic group over a field F. We will define a (G, N)-slice for a G-scheme X as an N-stable (locally closed) subscheme S of X such that the induced morphism (G × S)/N → V of algebraic spaces is an open embedding. Here N acts on G × S through the formula n · (g, s) = (gn^{−1}, ns). This action is free. The fppf-quotient sheaf (G × S)/N is known to be an algebraic space.

We prefer to use the term “slice” rather than “section”, following [CT88]. This is motivated by the analogy with the étale slices from Luna’s theorem [Lu73] and the analogy with the notion of a slice from differential geometry, see e.g. [Pa60, Definition 2.11]. Our definition of a (G, N)-slice can be seen as a version of Katsylo’s definition, which works well for arbitrary base fields F. See Corollary 3.5 and Proposition 5.1, which highlight the close connection between the two different definitions.

In Theorem 3.7 we show that existence of a (G, N)-slice of a versal G-scheme implies surjectivity of the maps H_{fppf}^{1}(L, N) → H_{fppf}^{1}(L, G) for infinite fields L containing F. Examples of versal G-schemes include linear representations and their associated projective spaces, as well as connected reductive groups on which G acts by group automorphisms; see Example 2.7. Every such reduction of structure group result implies the inequality ed(G) ≤ ed(N). The hope is that this will provide improved upper bounds for the essential dimension of G. Indeed in the case of E_{7}^{s.c.}, we obtain a much better upper bound on the essential dimension than was previously known.

Our principal method to construct slices of (geometrically irreducible) G-varieties comes from stabilizers in general position (SGP’s). Namely Theorem 4.1 shows that for an SGP H of a geometrically irreducible G-variety V, there exists a (G, N_{G}(H))-slice, which is open in the fixed scheme V^{H}. The concept of an SGP has its origin in invariant theory over an algebraically closed field of characteristic 0. It has only recently become a tool used for arbitrary base fields.

We will give another construction of slices for smooth algebraic groups in section 5. Proposition 5.1 shows in particular, that in characteristic 0 every (G, N)-slice in Katsylo’s sense gives a (G, N)-slice with our definition.

Theorem 3.7 could be considered a generalization of two existing results. Firstly, it generalizes to fields of characteristic other than zero an early result of Reichstein, which used (G, N)-sections (in the sense of [Re00, Definition 2.9]) of linear representations to give the inequality ed(G) ≤ ed(N); see Remark 3.9. Secondly, in [Ga09] Garibaldi, inspired by Rost’s unpublished preprint [Ro99], considered the situation when there is a zero-dimensional slice of ℙ(V) (consisting of a single point), where V is a linear representation of G, and proved the surjectivity of Theorem 3.7 when G is smooth. The surjectivities proved in Garibaldi’s paper were fruitfully applied to describe cohomological invariants of various exceptional groups and Spin groups. So our Theorem might also be considered as a generalization of this result to a much wider class of slices.

The existence of a slice of the adjoint representation was shown during Grothendieck’s proof that any smooth algebraic group G over an arbitrary field F contains a maximal torus [SGA3, XIV Theorem 1.1]. In particular, he showed that the regular elements contained in a given Cartan subalgebra form a slice of the adjoint representation, see [SGA3, XIII, Theorem 6.1(d)] (cf. Example 5.3). If T ⊂ G is a split maximal torus in G, and Aut(G) ≅ G, then the existence of a maximal torus in F̄/F -forms of G is equivalent to the surjectivity of the map H^{1}(F, N_{G}(T)) → H^{1}(F, G); in this situation, that surjectivity will now follow from Theorem 3.7.

Our results are also related to work of Chernousov, Gille and Reichstein [CGR08], who showed that every reductive algebraic group G has a finite subgroup S (contained in the normalizer of a maximal torus) for which the map H^{1}(L, S) → H^{1}(L, G) is surjective for every field extension L/F. However in practice these finite subgroups are usually not ideal for proving interesting new upper bounds on ed(G).

The rest of this paper is structured as follows: In section 2 we will fix conventions, recall some basics on algebraic spaces, torsors, twists, versal G-schemes, stabilizers in general position and essential dimension and prove some useful lemmas used later on. In section 3 we discuss (G, N)-slices of a G-scheme for an algebraic group G and subgroup N. Theorem 3.7 shows that every (G, N)-slice of a versal G-scheme (for instance a linear representation) gives rise to surjections H^{1}(L, N) → H^{1}(L, G) in fppf-cohomology for infinite fields L containing the base field. We proceed to construct slices of geometrically irreducible G-varieties out of stabilizers in general position in section 4 and give another construction for smooth algebraic groups in section 5. A simple application of the former result is given in section 6, which provides a new proof of Risman’s Theorem in the theory of central simple algebras. Finally in section 7 we take the SGP on two copies of the 56-dimensional representation of E_{7}^{s.c} to prove the upper bound ed(E_{7}^{s.c}) ≤ 11.

2. Preliminaries

2.1. Definitions and conventions. Unless otherwise specified, F will denote an arbitrary field, and F̄ an algebraic closure. An algebraic group G will be a (not necessarily smooth) affine group scheme of finite type over a field (usually F). For an algebraic group G, the symbols N_{G}(H) or Norm_{G}(Y) will denote the scheme-theoretic normalizer of G which preserves a subgroup H or closed subscheme Y respectively; the normalizer subgroup need not be smooth, even if G is smooth [Ja03]. A variety will be a reduced scheme which is separated and of finite type over a field.

We will frequently use the notion of algebraic spaces, as defined in [Stacks, 025Y]. Algebraic spaces are a generalization of schemes, that behave better under descent and appear naturally when working with quotients of free algebraic group actions on schemes. The category of schemes over F will be furnished with the big fppf (finitely presented faithfully flat) topology. Thus an algebraic space over F is a sheaf X: (Sch/F)_{fppf}^{op} → Sets such that

(1) For all schemes U, V over F and sheaf maps U → X, V → X the fiber product of sheaves U ×_{X} V is representable by a scheme.

(2) There exists a scheme U over F and a surjective étale sheaf map U → X (this means that for every scheme V over F and sheaf map V → X the morphism U ×_{X} V → V of schemes is surjective and étale).

A morphism of algebraic spaces is a natural transformation of functors. Every scheme gives rise to an algebraic space and the association X ↦ X defines a fully faithful embedding of the category (Sch/F) to the category of algebraic spaces over F. Thus we will identify every scheme over F with the algebraic space over F which it defines. If G is an algebraic group, a G-scheme will be a scheme with an action G × X → X of G.

Definition 2.1. Let G be an algebraic group over F and π: X → Y a G-invariant morphism between algebraic spaces over F (i.e. π is G-equivariant, and G acts trivially on Y). The morphism π is said to be a pseudo G-torsor if the morphism G × X → X ×_{Y} X given by (g, x) ↦ (gx, x) is an isomorphism. A pseudo G-torsor π is called an (fppf) G-torsor if the morphism π: X → Y is fppf. Equivalently (cf. [Mi80, III, Proposition 4.1]) there exists an fppf covering {Y_{i} → Y }_{i∈I} such that for every i the pull-back X ×_{Y} Y_{i} → Y_{i} is G-equivariantly isomorphic to G × Y_{i} → Y_{i}, where G acts trivially on Y_{i} and by left multiplication on G. If X → Spec F is a G-torsor, where F is a field, then we will also call X a G-torsor over F.

If an algebraic group G acts on an algebraic space X the fppf quotient sheaf X/G is defined as the sheafification of the presheaf T ↦ X(T)/G(T) on (Sch/F)_{fppf} in the sense of [Stacks, 00WG]. By the universal property of sheafification every G-invariant morphism X → Y of algebraic spaces factors through the quotient map X → X/G. Moreover, if X → Y is a Gtorsor, then the induced morphism X/G → Y is an isomorphism by [Stacks, 044M].

2.2. Free actions, twists and versal G-schemes. Recall that an action of an algebraic group G on an algebraic space X is called free, if for every scheme T over F the action of the group G(T) on the set X(T) is free.

Lemma 2.2. Let G be an algebraic group over F acting freely on an algebraic space X over F. Let Y := X/G denote the fppf quotient sheaf. Then the sheaf Y is an algebraic space and the canonical morphism π: X → Y is an fppf G-torsor. Moreover for any algebraically closed field L/F we have Y (L) = X(L)/G(L).

Proof. By [Stacks, 06PH] Y is an algebraic space and π: X → Y is fppf. It remains to show that the map G × X → X ×_{Y} X, (g, x) ↦ (gx, x) is an isomorphism. Since the action of G on X is free, for schemes T over F, the maps

G(T) × X(T) → X(T) ×_{X(T)/G(T)} X(T), (g, x) ↦ (gx, x)

are bijective. Thus it suffices to show that the maps X(T)/G(T) → Y (T) = (X/G)(T) are injective. Since G acts freely on X the morphism G × X → X × X, (g, x) ↦ (gx, x) is a monomorphism (by [Stacks, 07S2]), hence an equivalence relation. Therefore [Sk09, Proposition 1.13] (cf. [Ja03, §5.5]) implies that the presheaf quotient (X/G)_{pre} is separated. Thus [Stacks, 00WB] implies the claim. For the last claim see [DG70, III, §1, 1.15]. □

Remark 2.3. Under some assumptions on X and G the quotient X/G in Lemma 2.2 will be a scheme. This is for example the case when G is finite and X is quasi-projective [Stacks, 07S7]. However for us it will be of no relevance, if X/G is a scheme, or only an algebraic space.

Lemma 2.4. Suppose G acts freely on an algebraic space X and f: X → Z is a G-invariant morphism. Let z: Spec(R) → Z be another morphism of algebraic spaces over F. Then G_{R} acts freely on the algebraic space X_{z} := X ×_{Z} Spec(R) over R and induces an isomorphism X_{z}/G_{R} ≃ (X/G)_{z} of algebraic spaces over R.

Proof. It is clear that G_{R} acts on X_{z} and that this action is free. Moreover we have a G_{R}- equivariant isomorphism X_{z} ≃ X ×_{X/G} (X/G)_{z}. Since the projection X ×_{X/G} (X/G)_{z} → (X/G)_{z} is a G_{R}-torsor the claim follows. □

An important example of quotients by free actions is given by the twist construction as follows:

Definition 2.5. Let X be a G-scheme and E a G-torsor. Endow E×X with the (free) diagonal G-action. Then the quotient ^{E}X := (E × X)/G is called the twist of X by E.

Note that for a G-torsor E any G-equivariant morphism of schemes X → Y gives rise to a morphism ^{E}X → ^{E}Y of algebraic spaces. Moreover if E ≃ G is the trivial torsor the action morphism G × X → X induces an isomorphism ^{E}X ≃ X.

For smooth algebraic groups acting on quasi-projective varieties, the notions of versal and p-versal were introduced in [DR15]. We generalize these definitions below. For a prime p, a field is called p-special if all of its finite extensions have degree a power of p. Every p-special field is infinite. An algebraic group G over F is said to be special, if it only has the trivial torsor over fields containing F. It is said to be p-special, if it only has the trivial torsor over p-special fields containing F.

Definition 2.6. Let G be an algebraic group and V a G-scheme over F. Let p be a prime. Then V is called versal (resp. p-versal), if for every field extension L/F with L infinite (resp. p-special), every G-torsor E over L and every non-empty G-stable open subscheme U of V, the twist ^{E}U contains an L-rational point.

Clearly any versal G-scheme is p-versal for any prime p. The following examples of versal and p-versal G-varieties are listed in [DR15]:

Example 2.7. (1) For V a linear G-representation both V and ℙ(V) are versal.

(2) If V is a connected reductive group and G acts on V by group automorphisms, then V is versal.

(3) Let V = A/B be a homogeneous space with A reductive and B a subgroup. Let G be a subgroup of A acting on V by left-translations. Suppose that V is geometrically irreducible and the image of H_{fppf}^{1}(L, G) → H_{fppf}^{1}(L, A) is contained in the image of H_{fppf}^{1}(L, B) → H_{fppf}^{1}(L, A) for every infinite field L/F. Then V is versal.

(4) Let X be a geometrically irreducible quasi-projective G-variety with a smooth F -point x having finite G-orbit G · x whose associated 0-cycle [G · x] is of degree prime to p. Then X is p-versal.

In [DR15] G is assumed to be smooth, but we will show that examples (1), (2), and (3) remain valid for non-smooth groups G as well. Later we will need that every linear representation V and its associated projective space ℙ(V) are versal for general G. For this purpose we prove the following Lemma:

Lemma 2.8. Let G → G′ and H → G′ be homomorphisms of algebraic groups over F and X be a G′-scheme over F. We view X as a G-scheme and H-scheme through the given homomorphisms to G′. Let p be a prime.

(1) If E is a G-torsor over some field extension L/F and E′ = (E × G′)/G is the induced G′-torsor over L, then ^{E}X ≃ ^{E′}X as algebraic spaces over K.

(2) Suppose that for every infinite (resp. p-special) field extension L/F the image of the map H_{fppf}^{1}(L, G) → H_{fppf}^{1}(L, G′) is contained in the image of H_{fppf}^{1}(L, H) → H_{fppf}^{1}(L, G′) and for every H-torsor T over L the twist ^{T} X has a dense set of L-rational points. Then X considered as G-scheme is versal (resp. p-versal).

(3) Suppose that G′ is special (resp. p-special) and X is a unirational variety, then X is versal (resp. p-versal) as a G-scheme.

Proof. (1) This is well known and follows e.g. from [Gi71, Proposition 1.3.5].

(2) Let E be a G-torsor over some infinite (resp. some p-special) field extension L/F. Let E′ denote the induced G′-torsor. By assumption there exists an H-torsor T which induces the same G′-torsor as E. Hence ^{E}X ≃ ^{E′}X ≃ ^{T} X, which by assumption has a dense set of L-rational points. Hence ^{E}U contains an L-rational point for every non-empty G-stable open subscheme U of X.

(3) Take for H the trivial group. Since X is unirational X_{L} has a dense set of L-rational

points for every infinite field L/F, so (2) applies.

□

Remark 2.9. It follows from [DR15, Theorem 1.1 and Lemma 8.2] that for H smooth and X an irreducible quasi-projective variety the condition of Lemma 2.8(2) on the twists ^{T} X having dense sets of rational points holds if and only if X is versal (resp. p-versal) as an H-scheme. We do not know if this is true in our more general situation as well.

Corollary 2.10. No smoothness asumption is needed in Example 2.7 (1), (2) and (3).

Proof. In (1) we take G′ = GL(V), which is special.

In (2) we take G = G′ = H and note that for every G-torsor E over a field extension L/F the twist ^{E}V is a reductive group over L, hence by Chevalley’s Theorem (see [Bo91, Theorem 18.2(ii)]) contains a dense set of rational points.

In (3) if E′ is an A-torsor induced from a B-torsor over L, then ^{E′}V = ^{E′}(A/B) has a rational point and is hence dominated by the reductive group Aut(E′) over L. Therefore ^{E′}V contains a dense set of rational points. □

Remark 2.11. We do not know if Example 2.7 (4) holds for non-smooth groups, but we will not need this kind of example in this paper.

In Example 2.7 (2) and (3), if the algebraic group V resp. A is not necessarily reductive, but only connected and smooth, then the corresponding variety V will still be versal (resp. p-versal) if char(F) = 0 (resp. char(F) ≠ p). This comes from the fact that Chevalley’s unirationality Theorem holds for smooth connected algebraic groups whenever the base field is perfect. This was also observed in [DR15].

2.3. SGP’s and generic freeness. A stabilizer in general position (SGP) for an action of an algebraic group G on a geometrically irreducible variety X (defined over F) is a subgroup H of G such that for some dense open subscheme U of X every u ∈ U(F̄) has (scheme-theoretic) stabilizer conjugate to HF̄. The G-action on X is said to be generically free, if the trivial subgroup of G is an SGP for that action. For a subgroup H of G the condition that H is an SGP for the G-action on a geometrically irreducible variety X can always be checked over an algebraic closure. Moreover we have the following Lemma

Lemma 2.12. Suppose G acts on a geometrically irreducible F -variety X, for which F -rational points are dense. Assume the GF̄-action on XF̄ has an SGP H̃. Then the G-action on X has an SGP, say H, as well, and HF̄ is conjugate to H̃.

Proof. Let Ũ ⊂ XF̄ be a dense open subscheme such that every rational point of Ũ has stabilizer conjugate to H̃. First note that Ũ has only finitely many Galois conjugates. Indeed to prove this claim, we may assume without loss of generality that X is affine, so that Ũ is the complement of the zero set of finitely many polynomials. Since every of these polynomials has only finitely many non-zero coefficients and every coefficient has only finitely many Galoisconjugates, the claim follows. Now the intersection of the Galois conjugates of Ũ is dense open in Ũ and, by [Sp98, Proposition 11.2.8], descends to a dense open subscheme U of X. Taking for H the stabilizer of an F -rational point in U we get the desired result. □

By a Theorem of Richardson [Ric72] an SGP exists for any reductive algebraic group action on an irreducible smooth affine variety over an algebraically closed field in characteristic 0. Hence by Lemma 2.12 it also exists, in characteristic 0, for reductive group actions on geometrically irreducible smooth affine varieties with a dense set of rational points. This includes the case of linear reductive group actions. Moreover for linear actions of connected semisimple algebraic groups in characteristic 0 a lot of information about their SGP’s is available in the literature, see e.g. [PV89, Section 7].

We will repeatedly make use of the following Lemma, which is due to Popov.

Lemma 2.13 ([Po89, Prop. 8]). Let G act on geometrically irreducible varieties X and Y. Let H_{2} ⊂ H_{1} ⊂ G be subgroups such that H_{1} is an SGP for the G-action on X and H_{2} is an SGP for the H_{1}-action on Y. Then H_{2} is an SGP for the G-action on X × Y.

Proof. The reference [Po89] and the proof given therein assume that the base field is algebraically closed of characteristic 0. See [Lö15, Lemma 2.3] for a version of this proof which works for arbitrary base fields. This also corrects the proof of [Mac13, Prop.1.2], which is mistaken. □

Let G act on a scheme X over F. For any commutative F -algebra R and x ∈ X(R) the functor R′ ↦ {g ∈ G(R′) | gx_{R′} = x_{R′}} from the category of commutative R-algebras to the category of groups is representable by a subgroup scheme of G_{R}, see [SGA3, I, 2.3.3] or [DG70, III, §2.2], which we denote by (G_{R})_{x} and call the stabilizer R-group scheme of x. We have (G_{R})_{gx} = g(G_{R})_{x}g^{−1} for any g ∈ G(R), x ∈ X(R). When X is separated we will denote by X^{G} the closed subscheme of X which is fixed by G.

Lemma 2.14. Suppose H is an SGP for the G-action on a geometrically irreducible variety X and U is a dense open subscheme as in the definition of an SGP. Then

(1) The subscheme U contains a dense open subscheme U′ which is G-stable.

(2) Suppose U is G-stable. Then S := U ∩ X^{H} is a non-empty Norm_{G}(X^{H})-stable open subscheme of X^{H} such that for every commutative F -algebra R and every s ∈ S(R) ⊂ X(R) we have (G_{R})_{s} = H_{R}.

Proof. (1) This is well known. See e.g. [Lö15, §2, before Lemma 2.3].

(2) Clearly S is non-empty and Norm_{G}(X^{H})-stable. For any commutative F -algebra R and s ∈ S(R) we want to show that (G_{R})_{s} = H_{R}. Note that the inclusion (G_{R})_{s} ⊃ H_{R} is obvious. If R = F̄ by the definition of an SGP (GF̄)_{s} is conjugate to HF̄ and contains HF̄. The endomorphism (GF̄)_{s} ≃ HF̄ ↪ (GF̄)_{s} is a monomorphism of schemes, hence, by [EGAIV, Proposition 17.9.6] and since (GF̄)_{s} if of finite type over F̄, an automorphism. This implies (GF̄)_{s} = HF̄ as claimed.

To prove this equality for arbitrary commutative F -algebras R we consider the inertia scheme X^{T} = (G × X) ×_{X×X} X for the G-action on X. Here the morphism X → X × X is the diagonal and G × X → X × X is given by (g, x) ↦ (gx, x). By [DG70, III, §2.2] the fiber of the morphism X^{T} → X over any x ∈ X(R) is isomorphic to the stabilizer R-group scheme (G_{R})_{x}.

Now let S^{T} = X^{T} ×_{X} S = (G × X) ×_{X×X} S. The H-action on G × X × S through h · (g, x, s) = (gh^{−1}, hx, s) induces a free H-action on S^{T}. The morphism S^{T} → S is H-invariant. By Lemma 2.4 the fiber (S^{T} /H)_{s} of the induced morphism f: S^{T} /H → S over s ∈ S(R) is isomorphic to (S^{T})_{s}/H_{R} ≃ (G_{R})_{s}/H_{R}. In particular for every s ∈ S(F̄) the morphism (S^{T} /H)_{s} → Spec(F̄) is an isomorphism.

Note that S^{T} /H is in fact a scheme of finite type over F, since S^{T} ≃ (G×S)×_{X×S} S, hence S^{T} /H ≃ (G/H × S) ×_{X×S} S and G/H is a finite type scheme over F by [DG70, III, §3, 5.4]. Thus by [DG70, III §2.2, Lemme] we conclude that f is a monomorphism. Since f has a section, it follows that f is an isomorphism. Hence for s ∈ S(R) the morphism (G_{R})_{s}/H_{R} → Spec(R) is an isomorphism too. So (G_{R})_{s} → Spec(R) is an H_{R}-torsor. The existence of the identity section shows that (G_{R})_{s} = H_{R} as claimed.

□

2.4. Essential dimension. For the definition of essential dimension and essential p-dimension of an algebraic group G, see [Me09]; it is the essential dimension (resp. p-dimension) of the fppf cohomology functor L ↦ H_{fppf}^{1}(L, G). The set H_{fppf}^{1}(L, G) classifies fppf G-torsors over L (i.e. over Spec(L)).

When G is smooth every fppf G-torsor is locally trivial in the étale topology and these sets can be identified with the Galois cohomology set:

H_{fppf}^{1}(L, G) = H_{ét}^{1}(L, G) = H^{1}(Gal(L_{sep}/L), G(L_{sep})).

So when G is smooth, we will simply write H^{1}(L, G) for any of these sets.

Lemma 2.15. Let H → G be a homomorphism of algebraic groups (e.g. the inclusion of a subgroup) and p a prime. If the map H_{fppf}^{1}(L, H) → H_{fppf}^{1}(L, G) is surjective for every infinite (resp. p-special) field L/F, then ed(G) ≤ ed(H) (resp. ed_{p}(G) ≤ ed_{p}(H)).

Proof. For (absolute) essential dimension this is shown as in the proof of [Me09, Proposition 1.3]. Note that elements of H_{fppf}^{1}(L, G) with L finite do not contribute to the essential dimension of G. For essential p-dimension, see [LMMR13, Proposition 2.4]. □

Lemma 2.16. [Me09, Corollary 4.2] If an algebraic group G acts on an F -vector space V linearly and generically freely then ed(G) ≤ dim(V) − dim(G).

3. Reduction of structure group from slices of versal G-schemes

Given a G-scheme V and a (locally closed) subscheme S_{0} ⊂ V stable under the action of a subgroup N of G we have a free action of N on G × S_{0} via the formula n · (g, s_{0}) = (gn^{−1}, ns_{0}). The action morphism

G × S_{0} → V, (g, s_{0}) ↦ gs_{0}

is N-invariant. Hence it gives rise to a morphism (G×S_{0})/N → V of algebraic spaces, which is G-equivariant. Here we equip (G×S_{0})/N with the G-action induced from g ·(g′, s_{0}) = (gg′, s_{0}).

Definition 3.1. Let N be a subgroup of an algebraic group G. For us a (G, N)-slice of a Gscheme V will be a non-empty N-stable subscheme S_{0} ⊂ V such that the induced G-equivariant morphism (G × S_{0})/N → V of algebraic spaces is an open immersion.

Example 3.2. Suppose a G-scheme V contains an F -rational point x with an open orbit in V. Then S_{0} = {x} is a (G, N)-slice of V, where N is the stabilizer of x in G. Indeed, in this case the morphism (G × S_{0})/N → V is given by the inclusion of the open orbit in V.

Lemma 3.3. Let N be a subgroup of G. The following conditions on a non-empty N-stable subscheme S_{0} of a G-scheme V are equivalent:

(1) S_{0} is a (G, N)-slice.

(2) The morphism m: G × S_{0} → V is open and for V_{0} = m(G × S_{0}) = G · S_{0} the induced morphism G × S_{0} → V_{0} is an N-torsor.

(3) The morphism m: G × S_{0} → V is flat and locally of finite presentation, and for every commutative F -algebra R, every g ∈ G(R) with gS_{0}(R) ∩ S_{0}(R) ≠ ∅ is contained in N(R).

Proof. If (1) holds, then both G × S_{0} → (G × S_{0})/N and (G × S_{0})/N → V are flat and locally of finite presentation, hence so is m: G × S_{0} → V. Moreover for g ∈ G(R), s_{0}, s′_{0} ∈ S_{0}(R) such that gs_{0} = s′_{0} the elements (g, s_{0}) and (e, s′_{0}) of (G × S_{0})(R) have the same image in V (R), hence also in ((G × S_{0})/N)(R). Therefore there exists n ∈ N(R) with (e, s′_{0}) = (gn^{−1}, ns_{0}). Hence g = n ∈ N(R), and (3) follows.

If (3) holds, then m: G × S_{0} → V is open (by [Stacks, 01UA]). The induced morphism G × S_{0} → V_{0} is fppf, so it suffices to show that for every commutative F -algebra R and g, g′ ∈ G(R), s_{0}, s′_{0} ∈ S_{0}(R) with gs_{0} = g′s′_{0} there exists a unique n ∈ N(R) such that (g′, s′_{0}) = (gn^{−1}, ns_{0}). Uniqueness is clear and existence follows from (g′)^{−1}gS_{0}(R)∩S_{0}(R) ≠ ∅. So (2) follows.

If (2) holds then (G × S_{0})/N ≃ V_{0} and the morphism (G × S_{0})/N → V is given by the inclusion of V_{0} in V. □

Remark 3.4. Let S_{0} be a (G, N)-slice of a G-scheme V over F.

(1) If V and N are smooth then G and S_{0} are smooth as well, since the composite

G × S_{0} → (G × S_{0})/N → V → Spec F

of smooth morphisms is smooth.

(2) Let S := \overline{S_{0}} be the scheme-theoretic image of S_{0} in V. If V is of finite type over F and Norm_{G}(S) is smooth then N = Norm_{G}(S). In fact N normalizes S_{0}, hence also S (since the image of N × S_{0} under the action morphism N × V → V lies in \overline{S_{0}} and \overline{N × S_{0}} = N × \overline{S_{0}} in N × V), from which we get N ⊂ Norm_{G}(S). Conversely let g ∈ Norm_{G}(S)(F̄). Since gS(F̄) = S(F̄) and S_{0}(F̄) is dense and open in S(F̄) we have gS_{0}(F̄) ∩ S_{0}(F̄) ≠ ∅. Hence g ∈ N(F̄) by Lemma 3.3. Since N is smooth we get the desired result.

Corollary 3.5. Let N be a subgroup of an algebraic group G and V be a reduced G-scheme of finite type over F. Let S be a non-empty N-stable subscheme of V. Suppose that the morphism m: G × S → V, (g, s) → gs is dominant and that for every commutative F -algebra R, every g ∈ G(R) with gS(R) ∩ S(R) ≠ ∅ lies in N(R). Then the flat locus of m is of the form G × S_{0} for some N-stable non-empty open subscheme S_{0} of S and S_{0} is a (G, N)-slice of V.

Proof. By [Stacks, 052B and 0399] the flat locus of m: G × S → V is non-empty and open in G×S. Note that m is equivariant for the G×N-action on G×S and V through (g, n)·(g′, s) = (gg′n^{−1}, ns) and (g, n) · v = gv, respectively. Since the formation of the flat locus commutes with flat base change [Stacks, 047C], the flat locus of m is of the form G × S_{0} for some Nstable non-empty open subscheme S_{0} of S. Moreover G × S_{0} → V is flat and locally of finite presentation. Applying Lemma 3.3 we get that S_{0} is a (G, N)-slice of V. □

The following observation will be useful for non-faithful G-actions:

Lemma 3.6. Let π: G → G′ be a surjective homomorphism of algebraic groups and N′ be a subgroup of G′. Set N = π^{−1}(N′). Let V be a G′-scheme and S_{0} ⊂ V a N′-stable subscheme. Then S_{0} is a (G′, N′)-slice of V if and only if it is a (G, N)-slice of V.

Proof. The composition G × S_{0} → G′ × S_{0} → (G′ × S_{0})/N′ is N-invariant, hence it induces a morphism (G × S_{0})/N → (G′ × S_{0})/N′. We claim it is an isomorphism. Since ker(π) acts trivially on S_{0} the morphism (G × S_{0}) → (G × S_{0})/N factors through G′ × S_{0}. We get an N′-invariant morphism G′ × S_{0} → (G × S_{0})/N, which is easily seen to induce an inverse of (G × S_{0})/N → (G′ × S_{0})/N′.

Now we conclude the proof by observing that the morphism (G × S_{0})/N → V is the composition of the above isomorphism with the canonical morphism (G′ × S_{0})/N′ → V. □

The definition of a (G, N)-slice was designed in order to make the following result work. Its usefulness will be seen in the following sections.

Theorem 3.7. Let S_{0} be a (G, N)-slice of a G-scheme V and let p be a prime. Then if V is versal (resp. p-versal), then the map H_{fppf}^{1}(L, N) → H_{fppf}^{1}(L, G) is surjective for every infinite (resp. p-special) field L/F. In particular, ed(G) ≤ ed(N) (resp. ed_{p}(G) ≤ ed_{p}(N)).

Proof. Let L be an infinite (resp. p-special) field containing F and E a G-torsor over L. By [DG70, p. 373, Prop. III.4.4.6b] in order to show that the class of E in H_{fppf}^{1}(L, G) comes from H_{fppf}^{1}(L, N) it suffices to show that the twist ^{E}(G/N) has an L-rational point. By definition of a slice the fppf quotient (G × S_{0})/N is representable by a G-stable non-empty open subscheme of V, which we call U. The morphism G × S_{0} →^{π_{1}} G → G/N induces a G-equivariant morphism U → G/N. We get an induced morphism ^{E}U → ^{E}(G/N). Since V is versal (resp. p-versal), EU contains an L-rational point. Its image under ^{E}U → ^{E}(G/N) is an L-rational point of E(G/N). □

Remark 3.8. Saying that the map H_{fppf}^{1}(L, N) → H_{fppf}^{1}(L, G) is surjective for every p-special field L/F is equivalent to saying that for every field K/F and G-torsor E over K there exists a finite extension K′/K of degree prime to p such that the class of E_{K′} in H^{1}(K′, G) lies in the image of the map H_{fppf}^{1}(K′, N) → H_{fppf}^{1}(K′, G). This follows from the continuity of the functors H_{fppf}^{1}(−, N), H_{fppf}^{1}(−, G) and the existence of a p-special closure of K, see [LMMR13, §2].

If G is connected then for any finite field L/F the map H_{fppf}^{1}(L, N) → H_{fppf}^{1}(L, G) is also surjective, since in that case H^{1}(L, G) is trivial by Lang’s Theorem.

Remark 3.9. If char F = 0, and there is a linear subspace S_{0} of a linear representation V, which is a (G, N)-section, in the terminology of [Re00], then one can also show the existence of a (G, N)-slice of V, and so by Theorem 3.7 we have ed(N) ≤ ed(G). But in this case, this inequality on essential dimensions already follows from [Re00, Lemma 4.1] (notice that for characteristic zero, Reichstein’s original definition of essential dimension is equivalent to the one above [BF03, §6]). To see this, if W is a generically free G-representation, then S_{0} × W is a (G, N)-section of V × W, so the Lemma applies. Therefore, Theorem 3.7 is a generalization of this result to fields of arbitrary characteristic, non-smooth groups, and non-linear actions.

4. Producing slices from SGP’s

In this section we will describe a technique to produce a slice of a geometrically irreducible Gvariety V admitting an SGP. Moreover we apply Theorem 3.7 to versal G-varieties V to obtain reduction of structure for G-torsors. This will in particular apply for linear representations of G. The following Theorem is a variant of [Po94, 1.7.8], where relative sections are replaced by slices in our sense.

Theorem 4.1. Let V be a geometrically irreducible G-variety. Suppose that H is an SGP for the G-action on V (in particular an SGP exists). Let N := N_{G}(H) be its normalizer. Then there exists an open N-stable subscheme S_{0} of V^{H}, which is a (G, N)-slice of V.

Proof. By Lemma 2.14 there is a G-stable non-empty open subscheme U of V such that S := U ∩ V^{H} is a N-stable non-empty open subscheme in V^{H} and for every commutative F -algebra R and s ∈ S(R) the stabilizer R-group scheme (G_{R})_{s} of s is equal to H_{R}. In particular for any g ∈ G(R) and s ∈ S(R) with gS(R) ∩ S(R) ≠ ∅ comparision of stabilizer R-group schemes yields gH_{R}g^{−1} = H_{R}, hence g ∈ N(R). Therefore the claim follows from Corollary 3.5. □

Corollary 4.2. Let V be a geometrically irreducible G-variety which admits an SGP H. Let N := N_{G}(H) and p be a prime. If V is versal (resp. p-versal), then the map H_{fppf}^{1}(L, N) → H^{1}(L, G) is surjective for every infinite (resp. p-special) field L/F. In particular ed(G) ≤ ed(N) (resp. ed_{p}(G) ≤ ed_{p}(N)).

Proof. This follows immediately from Theorems 4.1 and 3.7. □

Example 4.3. Let G be a connected reductive algebraic group over F and let V = G, which we view as a G-scheme through the conjugation action. By Example 2.7 V is versal. Let T be a maximal torus of G. We claim that T is an SGP for the G-action on V. So Corollary 4.2 shows the well known surjection H^{1}(L, N_{G}(T)) → H^{1}(L, G) for any field extension L/F.

In order to prove the claim we may assume without loss of generality that F = F̄. First note that there is a dense open subscheme T_{0} of T, such that the centralizer of every t ∈ T_{0}(F) is equal to T. In fact, this can be seen by embedding G in some GL_{n}, where T is diagonalized: let χ_{1}, …, χ_{n} denote the standard characters of the diagonal torus of GL_{n} and take for T_{0} the intersection of the open subschemes D_{(χ_{i}/χ_{j})|_{T}}, where (i, j) runs over all pairs with (χ_{i})|_{T} ≠ (χ_{j})|_{T}. The morphism G × T_{0} → G, (g, t) ↦ gtg^{−1} is dominant, hence contains an open subscheme U in its image. The stabilizer of every u ∈ U(F) is conjugate to T. This shows the claim.

Remark 4.4. The result of Example 4.3 can also be obtained by considering the adjoint representation of G. We will show this in more generality in section 5.

Corollary 4.5. Let V be a linear representation of G which admits an SGP H. Write N := N_{G}(H). Then the map H_{fppf}^{1}(L, N) → H^{1}(L, G) is surjective for every infinite field L/F. Moreover ed(G) ≤ ed(N) and ed_{p}(G) ≤ ed_{p}(N) for every prime p.

Proof. Since V is versal (hence also p-versal for every prime p), this is only a special case of Corollary 4.2. □

5. Slices for smooth algebraic groups G

In section 4 we saw how to construct a slice of a geometrically irreducible G-variety, when an SGP exists. This may not always be the case, especially for actions of non-reductive groups. So we give another construction similar to the one in Corollary 3.5, but with the flat locus replaced by the smooth locus. This will allows us to check the condition gS(R) ∩ S(R) ≠ ∅ ⇒ g ∈ N(R) only for algebraically closed field extensions R of F instead of arbitrary commutative F -algebras. Our goal consists in proving the following result:

Proposition 5.1. Let N be a subgroup of G. Let V be a G-scheme over F and S be a nonempty N-stable subscheme of V. Suppose that for every algebraically closed field extension L/F and g ∈ G(L) with gS(L) ∩ S(L) ≠ ∅ we have g ∈ N(L).

(1) If m: G × S → V, (g, s) ↦ gs is smooth, then S is a (G, N)-slice of V.

(2) Suppose G is smooth, S and V are geometrically integral and of finite type over F, and mF̄ is dominant and separable (in the sense of [Sp98, §8]). Then the smooth locus of m is of the form G × S_{0} for a (non-empty) N-stable open subscheme S_{0} of S and S_{0} is a (G, N)-slice of V.

In order to prove Proposition 5.1 we will need the following Lemma:

Lemma 5.2. Let ϕ: X → Y and ψ: Y → Z be morphism of algebraic spaces over F such that (1) Z is a scheme,

(2) ψ ∘ ϕ: X → Z is smooth,

(3) ϕ: X → Y is fppf,

(4) Y (L) → Z(L) is injective for every algebraically closed field L containing F. Then ψ: Y → Z is an open embedding.

Proof. First ψ is universally injective by the proof of [Stacks, 040X]. Since every universally injective étale morphism of algebraic spaces is an open immersion (by [Stacks, 05W5]), it remains to show ψ is étale, i.e., for every étale morphism α: U → Y with U a scheme, the composition ψ ∘ α: U → Z is étale. By [Stacks, 0AHE] ψ is smooth, and in particular locally of finite type [Stacks, 06MH]. Hence ψ ∘ α is smooth as well, so by [Stacks, 0397] it suffices to show that ψ ∘ α is quasi-finite. Now both ψ (universally injective) and α (étale) are quasi-finite by [Stacks, 06RW], hence so is their composition. □

Proof of Proposition 5.1. (1) Let ϕ: G × S → (G × S)/N and ψ: (G × S)/N → V be the canonical morphisms, so that ψ ∘ ϕ = m: G × S → V. For L/F an algebraically closed field ((G×S)/N)(L) = (G(L)×S(L))/N(L) and the injectivity of ψ on L-points translates to the condition gS(L) ∩ S(L) ≠ ∅ ⇒ g ∈ N(L), which we have assumed. Therefore Lemma 5.2 implies that ψ is an open immersion, hence S is a (G, N)-slice of V.

(2) First of all by Corollary 3.5, the flat locus of m is of the form G × S′ for a non-empty (hence dense) N-stable open subscheme S′ of S. It contains the smooth locus of m. Thus after replacing S by S′ we may assume that m is flat. By [Sp98, Theorem 17.3] and [EGAIV, (4) 17.11.1] the smooth locus of mF̄ is non-empty open. Hence by [Stacks, 02VA] the smooth locus of m is non-empty open as well and (by G × N-equivariance of m as in the proof of Corollary 3.5) of the form G × S_{0} for a (non-empty) N-stable subscheme S_{0} of S. Now the claim follows from part (1).

□

Example 5.3. Let G be a smooth (not necessarily reductive) algebraic group over F, acting on 𝔤 = Lie(G) through the adjoint representation. Let 𝔠 be a Cartan subalgebra of 𝔤. Let S denote the open subscheme of 𝔠 formed by the regular elements. By [SGA3, XIII, Corollaire 5.4] the morphism G × S → 𝔤, (g, s) ↦ Ad(g)(s) is smooth. Let N := Norm_{G}(𝔠). For L/F algebraically closed, g ∈ G(L) and x, x′ ∈ S(L) with Ad(g)(x) = x′, the element x′ is contained in both 𝔠_{L} and Ad(g)𝔠_{L}. Since every regular element is only contained in one Cartan-subalgebra [SGA3, XIII, Proposition 4.6], we get g ∈ N(L). Hence by Proposition 5.1, S is a (G, N)-slice of 𝔤. This is also directly proven in [SGA3, XIII, Theorem 6.1].

Example 5.4. Let G be a connected reductive algebraic group over a field F. Let ζ be a root of unity in F and let m denote its order. Suppose θ is an automorphism of G of order m. Then G(0) := (G^{θ})^{0} is connected reductive and G(0) preserves 𝔤(1) := {x ∈ 𝔤 | dθ(x) = ζx} ⊂ 𝔤 under the adjoint action, where 𝔤 = Lie(G). This is the setting of Vinberg’s θ-groups, introduced by E.B. Vinberg in the 70’s [Vi76].

From now on (in this example) assume that F is perfect and either char(F) = 0 or p = char(F) > 2 and p is good for G (this is in particular the case for any p ≥ 7). Let 𝔠 be a Cartan subspace of 𝔤(1), i.e., a maximal commutative subspace of 𝔤(1) consisting of semisimple elements. We assume that 𝔠 remains maximal over F̄. Let

𝔷_{𝔤(1)}(𝔠) = {z ∈ 𝔤(1) | [z, c] = 0 ∀ c ∈ 𝔠}.

By [Le09, Lemma 1.10] 𝔷_{𝔤(1)}(𝔠) = 𝔠 ⊕ 𝔲 for a subspace 𝔲 consisting of nilpotent elements. In particular Norm_{G(0)}(𝔠) = Norm_{G(0)}(𝔷_{𝔤(1)}(𝔠)), which we will denote by N. We claim that there is a (G(0), N)-slice S_{0} of 𝔤(1), which is an open subscheme of 𝔷_{𝔤(1)}(𝔠).

Let 𝔠_{reg} denote the open subscheme of regular elements in 𝔠 and by R(𝔠) = 𝔠_{reg} ⊕ 𝔲 its preimage in 𝔷_{𝔤(1)}(𝔠). It is N-stable as well. Let m: G(0) × R(𝔠) → 𝔤(1), (g, x) ↦ Ad(g)(x). By [Le09, Corollary 2.4] the morphism mF̄ is dominant and separable. Moreover for L/F algebraically closed, g ∈ G(0)(L) and x, x′ ∈ R(𝔠)(L) with Ad(g)(x) = x′ we have

𝔷_{𝔤_{L}}(x_{s}) = 𝔷_{𝔤_{L}}(𝔠_{L}) = 𝔷_{𝔤_{L}}(x′_{s}) = 𝔷_{𝔤_{L}}(Ad(g)x_{s}) = Ad(g)𝔷_{𝔤_{L}}(x_{s}).

Intersecting both sides with 𝔤(1)_{L} yields g ∈ N(L). Therefore Proposition 5.1 applies and we get a (G(0), N)-slice S_{0} of 𝔤(1), which is open in R(𝔠), as claimed.

Corollary 5.5. Let (H, 𝔠, V) be a triple consisting either of a smooth algebraic group H over F and a Cartan subalgebra 𝔠 of V = Lie(H), or of a θ-group H = G(0) over F as in Example 5.4 (with the same assumptions on the reductive group G and the base field F) and a Cartan subspace 𝔠 of V = 𝔤(1). Let K be a normal subgroup of H contained in the kernel of the Haction on V. Then the map H_{fppf}^{1}(L, Norm_{H/K}(𝔠)) → H_{fppf}^{1}(L, H/K) is surjective for every infinite field L containing F.

Proof. In view of Theorem 3.7 and Lemma 3.6 this follows straight from the existence of (H, Norm_{H}(𝔠))-sections in V as shown in Examples 5.3 and 5.4. □

6. Common maximal étale subalgebras

In this section we give an application of Corollary 4.5 to prove that central simple algebras of the same degree, whose tensor product is of index ≤ 2, have a common maximal étale subalgebra; see Corollary 6.3.

We start with the group G = GL_{n} × GL_{n} acting on two copies of M_{n}(F) via the formula (b, c)·a = bac^{T}. The following easy result was shown in [Lö15, Proposition 2.4]. For convienience of the reader we include a proof.

Lemma 6.1. Let image H of the homomorphism 𝔾_{m}^{n} → G, t ↦ (t, t^{−1}) is an SGP for the G-action on M_{n} ⊕ M_{n} with Lie algebra 𝔥 = {(d, −d) ∈ M_{n} × M_{n} | d ∈ D}, where D are the diagonal matrices.

Proof. Since G has an open orbit on M_{n} given by the invertible matrices, the image of the homomorphism GL_{n} → G, b ↦ (b, (b^{−1})^{T}) is an SGP for the G-action on M_{n}. Moreover this group acts on M_{n} through b · a = bab^{−1}. Hence for any semisimple regular a ∈ VF̄ the stabilizer is a maximal torus of (GL_{n})F̄. Since these are all conjugate the claim follows by Lemma 2.13. □

We now consider the following twisted version of the above representation. Let A_{1} and A_{2} be central simple algebras of the same degree such that A_{1} ⊗_{F} A_{2} has index ≤ 2. Set G = GL_{1}(A_{1}) × GL_{1}(A_{2}) and let V be a left ideal of A_{1} ⊗_{F} A_{2} of reduced dimension 2. The representation

G → GL_{1}(A_{1} ⊗_{F} A_{2}) ↪ GL(V),

where the first map takes (a_{1}, a_{2}) to a_{1} ⊗ a_{2}, decomposes into two irreducible copies over any splitting field of A_{1} and A_{2} as previously. Write 𝔤 = Lie(G) = A_{1} × A_{2}.

Corollary 6.2. The G-action on V has an SGP H. Let 𝔥 = Lie(H). The compositions

f_{i}: 𝔥 ↪ 𝔤 = A_{1} × A_{2} → A_{i}

are injective and their images are étale subalgebras E_{i} ⊂ A_{i} with E_{1} ≃ E_{2} as F -algebras.

Proof. When F is infinite, the F -rational points are dense in V, hence the existence of H follows from Lemma 2.12 and 6.1. Otherwise, if F is finite, then A_{1} and A_{2} are split, and the existence follows directly from Lemma 6.1. The statement on Lie algebras obviously holds over F̄, hence it follows over F by descent. □

Corollary 6.3 (Risman’s Theorem). (1) Let A_{1} and A_{2} be central simple algebras of the same degree such that ind(A_{1} ⊗_{F} A_{2}) ≤ 2. Then A_{1} and A_{2} contain a common maximal étale subalgebra.

(2) Let D be a division algebra and Q a quaternion algebra over F such that D ⊗_{F} Q is not a division algebra. Then D contains a separable subfield which splits Q.

Proof. (1) This follows immediately from Corollary 6.2

(2) The assumption implies that there exists a central simple algebra B of degree deg(D) Brauer-equivalent to D⊗_{F} Q^{op}. Then ind(D⊗_{F} B) = ind(Q) ≤ 2. So in view of part (1) D and B share a common maximal étale subalgebra L. Since D is a division algebra, L is a field. Moreover L splits Q.

□

Remark 6.4. Corollary 6.3 is the well known common slot theorem in case A_{1} and A_{2} (resp. D) are quaternion algebras. We refer to [Kn93] for an elegant proof of that theorem and references to a variety of other proofs (all of a different flavor than ours). For arbitrary 2-primary degree, the following weaker result has been proven by D. Krashen in [Kr10, Corollary 4.4]: There exists an étale F -algebra of degree 2^{n}m with m ≥ 1 odd, which splits both A_{1} and A_{2}.

Part (2) of Corollary 6.3 is Risman’s Theorem [Ris75, Theorem 1] for finite dimensional division algebras. In fact the two parts are equivalent. The link between part (1) and part (2) was brought to our attention by Z. Reichstein. The proof of the reverse implication goes as follows: We may assume that ind(A_{2}) ≤ ind(A_{1}). Let Q be a quaternion algebra Brauer equivalent to A_{1} ⊗_{F} A_{2} and let D a division algebra Brauer equivalent to A_{1}. Then D ⊗_{F} Q is Brauer equivalent to A_{2} which has index ≤ ind(A_{1}) = deg(D). Hence D ⊗_{F} Q is not a division algebra. Now by part (2) D contains a splitting field L of Q. We let L′ be a maximal separable subfield of D containing L and let E = (L′)^{r} ⊂ A_{1} = M_{r}(D). Since E splits D and Q, it does also split A_{2}. As dim(E) = deg(A_{1}) = deg(A_{2}) it follows that A_{2} contains E as well.

We proceed to describe the normalizer N of the SGP H on V and interpret the resulting reduction of structure result (applied to the image of N in GL(V)) from an algebraic point of view. This in fact gives another proof of Risman’s Theorem (Corollary 6.3).

Proposition 6.5. Let H be the SGP for the G-action on V from Corollary 6.2. It is the image of the homomorphism

GL_{1}(E) → GL_{1}(E) × GL_{1}(E) ↪ GL_{1}(A_{1}) × GL_{1}(A_{2}) = G,

for some common maximal étale subalgebra E of A_{1}, A_{2}, where the first map is given by e ↦ (e, e^{−1}). We have

N_{G}(H) = N_{GL_{1}(A_{1})}(GL_{1}(E)) ×_{Aut(E)} N_{GL_{1}(A_{2})}(GL_{1}(E)).

Proof. The inclusion N_{G}(H) ⊃ N_{GL_{1}(A_{1})}(GL_{1}(E)) ×_{Aut(E)} N_{GL_{1}(A_{2})}(GL_{1}(E)) is obvious, so it remains to show the reverse containment. For i = 1, 2 the image of N_{G}(H) in GL_{1}(A_{i}) under the projection homomorphism normalizes the image of H in GL_{1}(A_{i}), which is GL_{1}(E). Therefore N_{G}(H) ⊂ N_{GL_{1}(A_{1})}(GL_{1}(E))×N_{GL_{1}(A_{2})}(GL_{1}(E)). Moreover let R be a commutative R-algebra and (a_{1}, a_{2}) ∈ N_{G}(H)(R). Then for all e ∈ E_{R} we have a_{1}ea_{1}^{−1} = (a_{2}e^{−1}a_{2}^{−1})^{−1} = a_{2}ea_{2}^{−1}, hence a_{1} and a_{2} have the same image in Aut(E)(R). This shows the claim. □

Corollary 6.6. Let n ∈ ℕ and let A_{1} and A_{2} be central simple algebras of degree n containing a common maximal étale subalgebra E, such that ind(A_{1} ⊗_{F} A_{2}) ≤ 2. Let

G = GL_{1}(A_{1}) × GL_{1}(A_{2}) ↠ Ḡ := N_{GL_{1}(A_{1}⊗_{F} A_{2})}(GL_{1}(A_{1})) ⊂ GL_{1}(A_{1} ⊗_{F} A_{2})

and let \overline{N} be the image of N_{GL_{1}(A_{1})}(GL_{1}(E)) ×_{Aut(E)} N_{GL_{1}(A_{2})}(GL_{1}(E)) in \overline{G}. Then for every field extension L/F the natural map H^{1}(L, \overline{N}) → H^{1}(L, \overline{G}) is surjective.

Moreover we have natural correspondences

L-isomorphism classes of quintuples (A′_{1}, A′_{2}, E′_{1}, E′_{2}, ϕ) with A′_{1}, A′_{2} central simple of degree n, E′_{1} ⊂ A′_{1}, E′_{2} ⊂ A′_{2} maximal étale and ϕ: E′_{1} →^{∼} E′_{2} such that A′_{1} ⊗_{L} A′_{2} ≃ (A_{1} ⊗_{F} A_{2})_{L}  ⟷  H^{1}(L, \overline{N})

L-isomorphism classes of pairs of central simple L-algebras (A′_{1}, A′_{2}) of degree n such that A′_{1} ⊗_{L} A′_{2} ≃ (A_{1} ⊗_{F} A_{2})_{L}  ⟷  H^{1}(L, \overline{G}).

Under these identifications the map H^{1}(L, \overline{N}) → H^{1}(L, \overline{G}) takes the isomorphism class of (A′_{1}, A′_{2}, E′_{1}, E′_{2}, ϕ′) to the isomorphism class of (A′_{1}, A′_{2}).

Proof. First of all the surjectivity of the maps H^{1}(L, \overline{N}) → H^{1}(L, \overline{G}) follows from Proposition 6.5, Corollary 4.5 and Lemma 3.6.

Let \overline{\overline{G}} = Aut_{F} (A_{1}) × Aut_{F} (A_{2}) and \overline{\overline{N}} = Aut_{F} (A_{1}, E) ×_{Aut(E)} Aut_{F} (A_{2}, E) the image of \overline{N} in \overline{\overline{G}}. Note that \overline{\overline{N}} is the automorphism group scheme of the quintuple (A_{1}, A_{2}, E, E, id_{E}) and \overline{\overline{G}} is the automorphism group scheme of the pair (A_{1}, A_{2}). Therefore we have natural correspondences:

L-isomorphism classes of quintuples (A′_{1}, A′_{2}, E′_{1}, E′_{2}, ϕ) with A′_{1}, A′_{2} central simple of degree n, E′_{1} ⊂ A′_{1}, E′_{2} ⊂ A′_{2} maximal étale and ϕ: E′_{1} →^{∼} E′_{2}  ⟷  H^{1}(L, \overline{\overline{N}})

L-isomorphism classes of pairs of central simple L-algebras (A′_{1}, A′_{2}) of degree n  ⟷  H^{1}(L, \overline{\overline{G}}).

The map H^{1}(L, \overline{\overline{N}}) → H^{1}(L, \overline{\overline{G}}) is the natural forgetful map. Moreover we have a commutative diagram with exact rows:

[figure: 1; 𝔾_{m}; \overline{N}; \overline{\overline{N}}; 1; 1; 𝔾_{m}; \overline{G}; \overline{\overline{G}}; 1]

with 𝔾_{m} central in \overline{N} and \overline{G}. The induced connecting map H^{1}(L, \overline{\overline{G}}) → H^{2}(L, 𝔾_{m}) = Br(L) takes the pair (A′_{1}, A′_{2}) to the Brauer class [A′_{1}] + [A′_{2}] − [(A_{1})_{L}] − [(A_{2})_{L}]. Hence the claim follows from the exact sequences in cohomology. □

7. Essential dimension of E_{7}^{s.c.}

In this section E_{7} will denote the split simply connected algebraic group of type E_{7}. The best published bounds on the essential dimension of this group are 7 ≤ ed(E_{7}) ≤ 29, where the lower bound is valid for fields of characteristic not 2 [CS06], and the upper bound is valid for fields of characteristic not 2 or 3 [Mac13]. We will show that ed(E_{7}) ≤ 11 for all fields of characteristic not 2 or 3.

Let V_{56} denote the smallest non-trivial linear representation of E_{7} := E_{7}^{s.c.}; it is unique and 56-dimensional, and we will give a construction below. The following three linear representations of E_{7} are not generically free, and have no trivial subrepresentations. Over the complex numbers these are the only ones [El72a], [Po86], but for the rest of this section we will allow arbitrary base fields of characteristic not 2 or 3. Firstly, the adjoint representation has a 7-dimensional slice which gives a reduction of structure to the normalizer of a maximal torus. Secondly, V_{56} has SGP E_{6}, so by Theorem 4.1 there is a 1-dimensional slice which gives a reduction of structure to Norm_{E_{7}}(E_{6}); also the 0-dimensional slice of the versal ℙ(V_{56}) gives a reduction to E_{6} ⋊ μ_{4}, which was considered in [Ga09] and [Mac13]. Thirdly, we have V := V_{56} ⊕ V_{56}, which we consider below.

We will use Theorem 4.1 to obtain a 16-dimensional slice of V, whose closure has a 37dimensional normalizer N. Then we will show that N acts generically freely on a 48-dimensional subrepresentation of V_{56}, and hence ed(E_{7}) ≤ ed N ≤ 48 − 37 = 11.

7.1. Coordinates for the E_{8} root system. We choose to describe the E_{8} root system as follows, because it will make the relevant E_{7} action easier to understand, since the slice will take a particularly simple form in these coordinates. The following are elements of the lattice ℤ^{8}, which we equip with the standard bilinear form scaled by a factor of 1/2.

α_{1} = (0, −1, 1, 0 | 0, −1, −1, 0)  α_{5} = (0, 0, 0, 1 | 1, 1, −1, 0)
α_{2} = (0, 0, 0, 1 | 1, −1, 1, 0)  α_{6} = (−1, 1, 0, 0 | − 1, −1, 0, 0)
α_{3} = (0, 0, 0, 1 | − 1, 1, 1, 0)  α_{7} = (2, 0, 0, 0 | 0, 0, 0, 0)
α_{4} = (0, 0, 0, −2 | 0, 0, 0, 0)  α_{8} = (−1, −1, −1, 0 | 0, 0, 0, 1)

This defines a system of simple roots of type E_{8}, using numbering as in Bourbaki, and we will denote the associated set of roots by Λ_{E_{8}}. So the free ℤ-module generated by these simple roots, together with the set of roots Λ_{E_{8}}, defines a root datum. Given any field F we can construct an algebraic group E_{8} over F together with a split maximal torus T_{8} whose root datum is as above [Co14, 6]. Furthermore, below we describe subroot systems of types (A_{1})^{3}, D_{4}, and E_{7}, which correspond to subgroups (SL_{2})^{3}, Spin_{8}, and E_{7} of E_{8}.

The key symmetries of Λ_{E_{8}} in these coordinates are as follows: (ℤ/2)^{8} acting by sign changes of the coordinates, and the following operation of order 3:

(a_{1}, a_{2}, a_{3}, b | c_{1}, c_{2}, c_{3}, d) ↦ (a_{2}, a_{3}, a_{1}, b | c_{2}, c_{3}, c_{1}, d).

In [Wi14] this operation is referred to as triality. The 240 E_{8} roots are partitioned into orbits under these symmetries in Table 1, were # is the orbit size.

The 126 roots labelled in Table 1 as E_{7} form a root system of type E_{7} with system of simple roots given by α_{1}, ⋯, α_{7}; they are the roots whose 8th coordinate is zero. This is a modified version of the description of E_{7} roots given in [Wi14]. The 24 roots labelled as D_{4} form a root system of type D_{4}, with system of simple roots α_{2}, α_{3}, α_{4}, α_{5}.

Representative # Inclusion Representative #  Inclusion
(0002 | 0000) 2 D_{4} ⊂ E_{7} (2000 | 0000) 6 (A_{1})^{3} ⊂ E_{7}
(0000 | 2000) 6 D_{4} ⊂ E_{7} (1110 | 0001) 16  W ⊂ V
(0001 | 1110) 16 D_{4} ⊂ E_{7} (1001 | 1001) 48  V
(0111 | 1000) 48  E_{7}  (1000 | 0111) 48  V
(0110 | 0110) 48  E_{7}  (0000 | 0002) 2  ±ρ

Table 1. Orbits of of 240 E_{8} roots

The 112 roots labelled in Table 1 as V correspond to weights of the E_{7} representation V = V_{56}^{+} ⊕ V_{56}^{−}, described below, with each copy being distinguished by the sign of the 8th coordinate. The symbol ρ denotes the highest root of E_{8} with respect to the given system of simple roots.

The Lie algebra 𝔢_{8} decomposes into root spaces, with respect to the Cartan subalgebra 𝔱 := Lie(T_{8}). For each simple root we will define the α_{i}-height as the coefficient of α_{i} when expressed as a sum of simple roots. We can define a ℤ-grading on 𝔢_{8} by defining 𝔢_{8}(i) to be the sum over root spaces whose α_{8}-height is i, with the exception of 𝔢_{8}(0), which will also contain 𝔱. This grading has dimensions 1, 56, 134, 56, 1 in degrees −2, −1, 0, 1, 2 respectively. Now we can define V_{56}^{+} := 𝔢_{8}(1) and V_{56}^{−} := 𝔢_{8}(−1). The algebraic group E_{8} acts on 𝔢_{8} via the adjoint representation. The subgroup E_{7} preserves both V_{56}^{+} and V_{56}^{−}, since this holds on the Lie algebra level. As E_{7} representations, V_{56}^{+} and V_{56}^{−} are isomorphic (we will sometimes also write V_{56} := V_{56}^{+}). The span of the weight spaces for the weights (±1, ±1, ±1|0|0, 0, 0|d) is 8-dimensional when d is either +1 or −1, and we will denote these subspaces by V_{8}^{+} ⊂ V_{56}^{+} and V_{8}^{−} ⊂ V_{56}^{−}, respectively (we will also write V_{8} := V_{8}^{+}). Below we will write W := V_{8}^{+} ⊕ V_{8}^{−}.

So our E_{7} representation of interest will be V := V_{56}^{+} ⊕ V_{56}^{−}. Choose a Chevalley basis {X_{α}, h_{i}} of 𝔢_{8}, where α runs through the roots of E_{8} and h_{1}, …, h_{8} are the elements of the Cartan subalgebra of 𝔢_{8} corresponding to the simple roots α_{1}, …, α_{8}.

Proposition 7.1. The subgroup Spin_{8} of E_{7} described above is an SGP for the E_{7}-action on V. Moreover V^{Spin^{8}} is equal to W = V_{8}^{+} ⊕ V_{8}^{−} and 𝔫_{W} := {x ∈ 𝔢_{7} | [x, W] ⊂ W } = 𝔰𝔭𝔦𝔫_{8} ⊕ (𝔰𝔩_{2})^{3}.

Proof. Let Λ_{E_{7}} denote the roots of E_{7}, Λ_{D_{4}} the roots of Spin_{8}, Λ_{V} the weights of V and Λ_{W} the weights of W. For α ∈ Λ_{E_{7}} and μ ∈ Λ_{V} we have

[X_{α}, X_{μ}] = \begin{cases}±X_{α+μ}, & if α + μ ∈ Λ_{V} \\ 0, & otherwise. \end{cases}

Hence V^{Spin^{8}} = V^{𝔰𝔭𝔦𝔫^{8}} is the subspace of V generated by all X_{μ} such that α + μ ∉ Λ_{V} for all α ∈ Λ_{D_{4}}. Similarly 𝔫_{W} is the subspace of 𝔢_{7} generated by all X_{α} such that for all μ ∈ Λ_{W} either α + μ ∈ Λ_{W} or α + μ ∉ Λ_{V}. Hence the assertions V^{Spin^{8}} = W and 𝔫_{W} = 𝔰𝔭𝔦𝔫_{8} ⊕ (𝔰𝔩_{2})^{3} are easily verified by inspecting Table 1.

We know from [Mac13, Prop.1.3] that Spin_{8} is an SGP of the action of E_{7} on V. This is because the split simply connected group E_{6} is an SGP of the action of E_{7} on V_{56}, and Spin_{8} is an SGP of E_{6} acting on V_{56}. This also follows from [El72a] when F = ℂ.

□

7.2. Pinning of E_{8}. For each E_{8}-root α, we fix an isomorphism p_{α}: 𝔾_{a} → U_{α} to the unipotent root subgroup; this is known as a pinning. This choice defines constants c_{λ,α} for each λ, α ∈ Λ_{E_{8}} by the formula (the adjoint representation) p_{α}(t)X_{λ} = X_{λ} + c_{λ,α}tX_{λ+α}. Since E_{8} is simplylaced, if λ and λ + α are both weights, then c_{λ,α} = ±1. But in what follows, and in particular to prove generic freeness, it is not enough to only know the structure constants up to sign.

So we will choose our Chevalley basis in such a way that for every α = ±α_{i}, plus or minus a simple root of E_{7}, and λ, λ + α ∈ Λ_{V} we have c_{λ,α} = 1; this is possible by [Va00, §2].

We will use the notation w_{i} := p_{α_{i}}(1)p_{−α_{i}}(−1)p_{α_{i}}(1) ∈ E_{7}(F) for i = 1, …, 7. These elements generate the extended Weyl group of E_{7} and normalize the given split maximal torus of E_{7} (and also of E_{8}). The element w_{i} has the effect of swapping the coordinates of weights that are connected by an edge labelled i in the associated weight diagram (see Figure 1 below; also [Va00, Fig.2]), and negating any coordinate that was moved rightwards; we follow the convention that positive roots act in a leftwards direction.

In the following Lemma, the subgroups Spin_{8} and (SL_{2})^{3} of E_{7} are those generated by the root subgroups (images of p_{α}) for the roots in Table 1 which are labelled as D_{4} and A_{1}^{3} respectively. The subgroup S_{3} is the subgroup generated by the following two elements:

w_{(12)} := w_{6}w_{5}w_{4}w_{3}w_{2}w_{4}w_{5}w_{6}, w_{(23)} := w_{1}w_{3}w_{4}w_{5}w_{2}w_{4}w_{3}w_{1}.

Together these elements generate the symmetric group on three letters, because they are both of order 2, and one can check that w_{(12)}w_{(23)} is an element of order 3. One way to see this would be to consider its action on V_{56} via its weight diagram (see Figure 1), since this is a faithful E_{7} representation. The choice of names for w_{(12)} and w_{(23)} are motivated by the fact, that these two elements act on the E_{8} roots (a_{1}, a_{2}, a_{3}, b | c_{1}, c_{2}, c_{3}, d) by simultaneous swaping of the 1, 2 (resp. 2, 3)-coordinates. This can be seen using the weight diagram of the adjoint representation for E_{8} [PSV96, Fig.24].

Any root of E_{8} can be written as ∑ c_{i}α_{i} for integers c_{i}, and we will arrange these coefficients as in the Dynkin diagram. For example 000^{1}0000 refers to the root α_{2}, and the highest root of E_{8} is ρ = 246^{3}5432.

7.3. Reduction of structure group to the normalizer of Spin_{8}.

Lemma 7.2. N := Norm_{E_{7}}(V_{8}) = N_{E_{7}}(Spin_{8}) ≅ ((SL_{2})^{3} × Spin_{8})/μ_{2}^{2} ⋊ S_{3}.

Proof. We have an inclusion N_{E_{7}}(Spin_{8}) ⊂ Norm_{E_{7}}(V_{56}^{Spin_{8}}) = N. Moreover all of the subgroups Spin_{8}, (SL_{2})^{3}, and S_{3} normalize Spin_{8}, hence are contained in N_{E_{7}}(Spin_{8}). We know (SL_{2})^{3} and Spin_{8} commute, but their centres intersect in the subgroup (μ_{2})^{2} (this can be seen by coroot computations as in [Ti90, §1.7]). The subgroup of E_{7} generated by Spin_{8} and (SL_{2})^{3} is normalized by S_{3} and intersects trivially with S_{3}, since the non-trivial elements of S_{3} induce non-trivial automorphisms of based root systems (D_{4}, {α_{2}, α_{3}, α_{4}, α_{5}}). So the subgroup of E_{7} generated by Spin_{8}, (SL_{2})^{3} and S_{3} is isomorphic to ((SL_{2})^{3} × Spin_{8})/μ_{2}^{2} ⋊ S_{3}, yielding an inclusion ((SL_{2})^{3} × Spin_{8})/μ_{2}^{2} ⋊ S_{3} ⊂ N_{E_{7}}(Spin_{8}) ⊂ N. In order to show the reverse containment we may pass to an algebraic closure. So assume F is algebraically closed.

By [KMRT98, 21.5 (10)] Lie(N) = 𝔫_{W} and by Proposition 7.1 this is equal to 𝔰𝔭𝔦𝔫_{8} ⊕ 𝔰𝔩_{2}^{3}. Therefore the connected component N^{0} is smooth, 37-dimensional, and generated by (SL_{2})^{3} and Spin_{8}.

Assume g ∈ (N\(N^{0} ⋊ S_{3}))(F). Since all maximal tori of N^{0} are conjugate over F = F̄, by multiplying g with a suitable element of N^{0}(F) we can assume g preserves the given split torus. Then we can consider the image ḡ ∈ W (E_{7}) in the Weyl group. Since conjugation by g preserves N^{0}, ḡ preserves the root system D_{4}. Since all automorphisms of the Dynkin diagram of D_{4} are realized by elements of S_{3}, all possible automorphisms of the root system of D_{4} are induced from a torus-preserving element of N^{0} ⋊ S_{3}. Therefore we can assume our ḡ fixes the D_{4} root system. In other words, ḡ fixes all but the first three coordinates of the E_{8} roots. To see how it acts on the first three coordinates, consider:

ḡ(2000|0000) = ḡ(1000|0111)−ḡ(−1000|0111) = (±1000|0111)−(±1000|0111) = (±2000|0000).

Here we have used that the only roots of the form (∗ ∗ ∗0|0111) of E_{8} (as listed in Table 1) are the roots (±1000|0111). A similar argument applies to the 2nd and 3rd coordinates. But this means ḡ acts in the same way as an element of W (A_{1}^{3}). Since the Weyl group of E_{7} acts

18 R. LÖTSCHER AND M. MACDONALD

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7 • 6 1 5 6 3 4 3 5 4 2 2 3 4 3 4 5 4 1 4 5 2 5 3 2 6 1 7 1 3 2 1 4 2 6 4 3 2 3 5 2 4 5 4 3 4 5 3 4 2 5 1 5 6 7 6 7

Figure 1. Weight diagram of the E_{7} representation V_{56}

faithfully on the set of weights of V_{56}, this contradicts the assumption that g ∉ (N^{0} ⋊ S_{3})(F). Therefore, N ≅ N^{0} ⋊ S_{3}, which concludes the proof. □

Theorem 7.3. Let N := N_{E_{7}}(Spin_{8}). For every field L/F the map H^{1}(L, N) → H^{1}(L, E_{7}) is surjective.

Proof. When L is infinite the result follows from Proposition 7.1 and Corollary 4.5. If L is finite, then Lang’s Theorem asserts H^{1}(L, E_{7}) = 1. □

7.4. Generic freeness of V_{48}.

Theorem 7.4. The action of N on the 48-dimensional irreducible subrepresentation of V_{56} = V_{48} ⊕ V_{8} is generically free.

Proof. Figure 1 is the weight diagram of V_{56}, the E_{7}-representation (see [PSV96, Fig.21]), which can be used as a visual aid. Each node corresponds to a weight (and hence a 1-dimensional weight space), and two nodes are joined by an edge labelled i if their difference is the simple root α_{i}. The diagram is presented in such a way that adding a positive simple root move weights from right to left, so that the highest weight of the representation (ρ − α_{8}) is the node in the top left corner. The node in the lower right is the weight α_{8}. Nodes connected by the simple roots of D_{4} are solid, while the others are dotted.

By Figure 1, as a Spin_{8} representation, V_{56} decomposes into an 8-dimensional trivial representation (the isolated nodes), and 6 irreducible representations, each of dimension 8. These 8 dimensional spaces pair up, and when decomposed as an ((SL_{2})^{3}×Spin_{8})/μ_{2}^{2} representation, V_{48} decomposes into 3 representations each of dimension 16, and we will write V_{48} = V_{1} ⊕ V_{2} ⊕ V_{3}. Here V_{i} has highest weight λ_{1} := 134^{2}3221, λ_{2} := 135^{3}4321, and λ_{3} := 246^{3}5321 for i = 1, 2, 3 respectively.

First we will show it is enough to check that the connected component of the identity N^{0} acts generically freely. Namely we have a homogeneous N-invariant polynomial on V_{56} which is non-zero on the V_{i}, given by the E_{7}-invariant quartic q on V_{56}; see for example [He12, Lemma 6] for a proof that q is non-zero on each 2-dimensional space V_{λ} ⊕ V_{ρ−λ}, where ρ is the highest root of E_{8}, and λ is any weight of V_{48} (each V_{i} is a direct sum of such subspaces). The morphism

V_{48} = V_{1} ⊕ V_{2} ⊕ V_{3} → 𝔸^{3}, (u_{1}, u_{2}, u_{3}) ↦ (q(u_{1}), q(u_{2}), q(u_{3}))

is surjective and N-equivariant if we endow 𝔸^{3} with the natural (faithful) permutation action of N/N^{0} = S^{3}. Thus any u = (u_{1}, u_{2}, u_{3}) with trivial stabilizer in N^{0} contained in the preimage of the open subset of 𝔸^{3}, where all coordinates are different, has trivial stabilizer in N. This shows the claim.

Let’s label the three copies of SL_{2} in N by L_{i} for i = 1, 2, 3, where L_{i} acts trivially on V_{j} for i ≠ j. In other words L_{i} is generated by the root groups of ±β_{i}, where

β_{1} := (2, 0, 0, 0|0, 0, 0, 0) = α_{7}, β_{2} := (0, 2, 0, 0|0, 0, 0, 0) and β_{3} := (0, 0, 2, 0|0, 0, 0, 0).

We will continue to label all roots and weights as linear combinations of the simple E_{8} roots. For instance β_{2} = 012^{1}2210 and β_{3} = 234^{2}3210.

To finish the proof of Theorem 7.4 we will need the following Lemmas.

Lemma 7.5. The Spin_{8} ×L_{1} orbit of any v ∈ V_{1} such that q(v) ≠ 0 contains an element which is a non-zero scalar multiple of X_{λ_{1}} + X_{ρ−λ_{1}}. The stabilizer of X_{λ_{1}} + X_{ρ−λ_{1}} in Spin_{8} ×L_{1} is isomorphic to (Spin_{6} ×𝔾_{m})/μ_{2}.

Proof. One can see from the weight diagram that V_{1} is the tensor product of an 8-dimensional irreducible Spin_{8}-representation with the natural 2-dimensional vector L_{1}-representation. So by the argument in [SK77, p.110], the action has an SGP isomorphic to (Spin_{6} ×𝔾_{m})/μ_{2}. In fact, they consider the action of SO_{8} × GL_{2} instead, and in that case there is an open orbit. Since the open orbit is affine, its complement is a hypersurface. Since the quartic q is non-zero on V_{1}, the argument in [SK77, p.110] shows that hypersurface is defined by q = 0 restricted to V_{1}; see also [BGL14, Example 6.5]. The result now follows because q(X_{λ_{1}} + X_{ρ−λ_{1}}) ≠ 0. □

Lemma 7.6. Let H ≃ SL_{4} be the subgroup of E_{7} generated by the root groups for ±α_{2}, ±α_{4}, ±α_{5}, and let T_{1} ≃ 𝔾_{m} denote the image of the coroot α_{2}^{∨} + α_{5}^{∨} + 2(α_{4}^{∨} + α_{3}^{∨} − α_{7}^{∨}): 𝔾_{m} → E_{7}. Then the subgroups H and T_{1} commute and intersect in the image of μ_{2} ↪ E_{7}, t ↦ α_{2}^{∨}(t)α_{5}^{∨}(t). Moreover the subgroup generated by H and T_{1} is the stabilizer of X_{λ_{1}} + X_{ρ−λ_{1}} ∈ V_{1} in Spin_{8} ×L_{1} ⊂ E_{7}.

Proof. Looking at the bonds in the weight diagram connected to λ_{1} and ρ − λ_{1} it is clear that the stabilizer contains H. Write β^{∨} := α_{2}^{∨}+α_{5}^{∨}+2(α_{4}^{∨}+α_{3}^{∨}−α_{7}^{∨}). Using the bilinear pairing one checks that λ_{1} ∘ β^{∨} = (ρ − λ_{1}) ∘ β^{∨} = 0 and α_{i} ∘ β^{∨} = 0 for i = 2, 3, 5. Therefore T_{1} (the image of β^{∨}) lies in the stabilizer and commutes with H. Clearly the intersection H ∩ T_{1} contains the image of μ_{2} ↪ E_{7}, t ↦ α_{2}^{∨}(t)α_{5}^{∨}(t). Since the image of the coroot α_{7}^{∨} intersects Spin_{8} trivially this containment cannot be proper. So we have a 16-dimensional subgroup SL_{4} ·𝔾_{m} contained in the stabilizer. By Lemma 7.5, this must equal the whole stabilizer. □

Remark 7.7. See also [Ga09, 12.2] and [El72b, Theorem 5], which consider a similar situation.

So the subgroups H ≃ SL_{4}, T_{1} ≃ 𝔾_{m}, L_{2} ≃ SL_{2}, and L_{3} ≃ SL_{2} described above generate an SGP for the action of N^{0} on V_{1}. To finish the proof of generic freeness, we will prove the following Lemma:

Lemma 7.8. The group generated by H, T_{1}, L_{2}, and L_{3} in E_{7} acts generically freely on V_{2} ⊕ V_{3}.

Proof. As SL_{4}-representation, V_{2}⊕V_{3} decomposes into 4 standard representations, and 4 copies of the dual. By arranging the weight spaces into a pair of 4 by 4 matrices, it is easier to understand the action. For simplicity we will consider the group L_{2} × L_{3} × H × T_{1}, which is the product of four commuting subgroups of E_{7}, rather than the image of its projection to E_{7}, which has kernel μ_{2}×μ_{4}. We will identify T_{1} with 𝔾_{m} through the isomorphism 𝔾_{m} → T_{1}, ξ ↦ β^{∨}(ξ). We arrange the 32 weights as follows, written in the coordinates of the E_{8} fundamental roots.

[\begin{matrix}135^{3}4321 & 123^{2}2111 & 235^{3}4321 & 001^{1}1111 \\ 135^{2}4321 & 123^{1}2111 & 235^{2}4321 & 001^{0}1111 \\ 134^{2}4321 & 122^{1}2111 & 234^{2}4321 & 000^{0}1111 \\ 134^{2}3321 & 122^{1}1111 & 234^{2}3321 & 000^{0}0111 \end{matrix}] [\begin{matrix}123^{1}3321 & 111^{0}1111 & 245^{2}4321 & 011^{0}1111 \\ 123^{2}3321 & 111^{1}1111 & 245^{3}4321 & 011^{1}1111 \\ 124^{2}3321 & 112^{1}1111 & 246^{3}4321 & 012^{1}1111 \\ 124^{2}4321 & 112^{1}2111 & 246^{3}5321 & 012^{1}2111 \end{matrix}]

Choose a basis of the representation consisting of weight vectors X_{λ}. Then we can describe an element of V_{2} ⊕ V_{3} by its coefficients of this basis, which we will write (x, y) ∈ M_{4} ⊕ M_{4}, as a pair of matrices. Let us identify the subgroup H of E_{7} generated by the root groups of ±α_{2}, ±α_{4}, ±α_{5} with SL_{4} using the identifications

p_{α_{2}}(t) = [\begin{matrix}1 & t & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] ,  p_{α_{4}}(t) = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & t & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] ,  p_{α_{5}}(t) = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & t \\ 0 & 0 & 0 & 1 \end{matrix}] ,

p_{−α_{2}}(t) = [\begin{matrix}1 & 0 & 0 & 0 \\ t & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] , p_{−α_{4}}(t) = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & t & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] , p_{−α_{5}}(t) = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & t & 1 \end{matrix}] .

Similarly L_{i} (for i = 2, 3) is identified with SL_{2} using p_{β_{i}}(t) = [ \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ], p_{−β_{i}}(t) = [ \begin{matrix}1 & 0 \\ t & 1 \end{matrix} ].

Since we have assumed our structure constants are as in [Va00, Theorem 1], g ∈ SL_{4} acts as (x, y) ↦ (gx, D(g^{−1})^{T} Dy), where D = diag(−1, 1, −1, 1). The reason for conjugating by D is to ensure the structure constants are c_{λ,α} = 1 for simple roots α. To determine how L_{2} and L_{3} act, we use the procedure described in [Va00, Theorem 2] applied to the roots β_{2} and β_{3}. This, together with the action of T_{1}, which is determined by the bilinear pairing between characters and cocharacters, shows that L_{2} × L_{3} × H × T_{1} acts through the formula

(a, b, g, ξ) · (x, y) = (gx [\begin{matrix}ξa^{T} & 0 \\ 0 & ξ^{−1}b^{T} \end{matrix} ] , D(g^{−1})^{T} Dy [\begin{matrix}ξ^{−1}a^{T} & 0 \\ 0 & ξb^{T} \end{matrix} ]) .

A generic x is invertible, and so its H-orbit contains a scalar matrix. The stabilizer of a nonzero scalar x is the subgroup defined by the equation g = [ \begin{matrix}ξ^{−1}(a^{−1})^{T} & 0 \\ 0 & ξ(b^{−1})^{T} \end{matrix} ]. Therefore the L_{2} × L_{3} × H × T_{1}-action on the first copy of M_{4} has an SGP, which we may identify with L_{2} × L_{3} × T_{1}. It acts on the second copy of M_{4} through the formula

(a, b, ξ) · y = D [\begin{matrix}ξa & 0 \\ 0 & ξ^{−1}b \end{matrix}] Dy [\begin{matrix}ξ^{−1}a^{T} & 0 \\ 0 & ξb^{T} \end{matrix} ] .

Note that this representation decomposes into the 4 sub-representations given by the four 2 by 2 blocks in M_{4}. The representation of L_{2} × L_{3} × T_{1} on the sum M_{2} ⊕ M_{2} of the diagonal M_{2}’s is given by

(a, b, ξ) · (y_{1}, y_{2}) = (āy_{1}a^{T} , b̄y_{2}b^{T}),

where ā = dad, b̄ = dbd with d = [ \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ] = d^{−1}. Considering non-symmetric invertible matrices y_{1} and y_{2} with zeroes on the diagonal (and their L_{2} × L_{3}-conjugates) we see that the stabilizer of a generic pair of matrices (y_{1}, y_{2}) is a maximal torus of L_{2} × L_{3} × T_{1}. Since all maximal tori are conjugate we have an SGP for this action given by 𝔾_{m}^{3} embedded in L_{2} × L_{3} × T_{1} on the diagonals. The induced action of 𝔾_{m}^{3} on the upper off-diagonal copy of M_{2} has weights (1, 1, 2), (1, −1, 2), (−1, 1, 2) and (−1, −1, 2), hence it has an SGP ≃ μ_{2} × μ_{4}. Therefore we conclude that the L_{2} × L_{3} × H × T_{1} action on M_{4} ⊕ M_{4} has SGP isomorphic to μ_{2} × μ_{4}, which is the kernel of the projection L_{2} × L_{3} × H × T_{1} → E_{7}. Therefore we have shown that the subgroup of E_{7} generated by L_{2}, L_{3}, H and T_{1} acts generically freely on V_{2} ⊕ V_{3}. □

End of proof of Theorem 7.4: Putting the above Lemmas together, we see that the action of N on V_{48} = V_{1} ⊕ V_{2} ⊕ V_{3} is generically free, as required. □

Theorem 7.9. ed(E_{7}) ≤ 11.

Proof. Theorem 7.3 implies ed(E_{7}) ≤ ed(N) and by Lemma 2.16 Theorem 7.4 yields ed(N) ≤ 48 − dim N. By Lemma 7.2 dim(N) = 3 · 3 + \frac{8·7}{2} = 37, so the claim follows. □

Remark 7.10. Consider a (not necessarily split) simply connected semisimple algebraic group G of type E_{7} over F. Under what assumptions do our results on reduction of structure and the upper bound on ed(G) still hold?

(1) The upper bound ed(G) ≤ 11 still holds if G has trivial Tits algebras. Indeed such a group is of the form Aut(X) for some E_{7}-torsor X, so it has the same torsors as the split E_{7}.

(2) If G only has Tits algebras of index ≤ 2 then we still have a 112-dimensional representation V which decomposes as V_{56} ⊕ V_{56} over an algebraic closure. Assume that F is infinite. Then by Lemma 2.12 there exists an SGP H for the G-action on V, which becomes conjugate to Spin_{8} over F̄. Then applying Theorem 4.5 we still get surjectivity of the maps H^{1}(L, N) → H^{1}(L, G) for field extensions L/F and the inequality ed(G) ≤ ed(N), where N = N_{G}(H). However in that case we have no good upper bound on ed(N) at our disposal.

Acknowledgments

We would like to thank Z. Reichstein for making us aware of reference [Ris75], which allowed us to identify Corollary 6.3 with Risman’s Theorem, as outlined in Remark 6.4. We are also grateful to Skip Garibaldi who commented on an early version of this paper, and to Olivier Haution for useful discussions.

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Roland Lötscher, Mathematisches Institut, Universität München, 80333 München, Germany

   E-mail address: roland.loetscher@mathematik.uni-muenchen.de

Mark L. MacDonald, Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom

   E-mail address: m.macdonald@lancaster.ac.uk