TAME GROUP ACTIONS ON CENTRAL SIMPLE ALGEBRAS

MARCH 14, 2007

Z. REICHSTEIN AND N. VONESSEN

Abstract. We study actions of linear algebraic groups on finite-dimen-sional central simple algebras. We describe the fixed algebra for a broad class of such actions.

1. Introduction

Let k be an algebraically closed base field of characteristic zero, K/k be a finitely generated field extension, A/K be a finite-dimensional central simple algebra and G be a linear algebraic group defined over k. Suppose G acts on A by k-algebra automorphisms. We would like to describe the fixed ring A^{G} for this action. In particular, we would like to know under what conditions A^{G} is semisimple, simple or commutative, under what conditions the center of A^{G} (or all of A^{G}) is contained in K, etc.

The goal of this paper is to describe, in terms of geometric data, the finite-dimensional algebras B/F that can occur as fixed algebras A^{G}/K^{G} for a particular type of G-action on A which we call tame. We give a precise definition of tame actions in Section 2, and we show in Section 4 that many naturally occurring actions belong to this class. To every tame action one associates a subgroup S of PGL_{n} (defined up to conjugacy), which we call the associated stabilizer. If the associated stabilizer S is reductive, we say that the action is very tame.

To state our first main result, let K be a field extension of k. As usual, we shall say that an algebra A/K is a form of A_{0}/k if the two become isomorphic after extension of scalars, i.e., A ⊗_{K} L ≃ A_{0} ⊗_{k} L for some field extension k ⊂ K ⊂ L.

1.1. Theorem. Consider a tame action of a linear algebraic group G on a central simple algebra A of degree n with center K, with associated stabilizer subgroup S ⊂ PGL_{n}. Then the fixed algebra A^{G}/K^{G} is a form of M_{n}^{S}/k.

2000 Mathematics Subject Classification. 16K20, 14L30, 20G10.

Key words and phrases. Central simple algebra, algebraic group, group action, geometric action, tame action, semisimple algebra, Galois cohomology.

Z. Reichstein was supported in part by an NSERC research grant.

Here S ⊂ PGL_{n} is acting on the matrix algebra M_{n} by conjugation, and M_{n}^{S} = (M_{n})^{S} denotes the fixed subalgebra for this action.

In the case where the G-action on A is very tame, one can describe M_{n}^{S} (and thus A^{G}) more explicitly. In this case the associated stabilizer subgroup S ⊂ PGL_{n} is, by definition, reductive, and hence so is the preimage S^{∗} of S in GL_{n}. We shall view the inclusion ϕ: S^{∗} ↪ GL_{n} as an n-dimensional linear representation of S^{∗}.

1.2. Corollary. Consider a very tame action of a linear algebraic group G on a central simple algebra A of degree n with center K, and let ϕ be as above. Then

(a) A^{G} is semisimple.

(b) A^{G} is commutative if and only if ϕ is multiplicity-free (i.e., is a sum of distinct irreducible representations).

(c) A^{G} ⊂ K if and only if ϕ is irreducible.

(d) Z(A^{G}) ⊂ K if and only if ϕ is a power of an irreducible representation.

For a more detailed description of A^{G}, in terms of the irreducible decomposition of ϕ, see Section 8.

We now return to the setting of Theorem 1.1. Let G be a linear algebraic group, A be a central simple algebra of degree n and ψ be a tame action of G on A, with associated stabilizer S ⊂ PGL_{n}. Since Theorem 1.1 asserts that A^{G}/K^{G} is a form of M_{n}^{S}/k, it is natural to ask which forms of M_{n}^{S}/k can occur in this way. Note that here n and S ⊂ PGL_{n} are fixed, but we are allowing A, G and ψ to vary.

Our answer to this question will be stated in terms of Galois cohomology. If G is an algebraic group defined over k, and K/k is a field extension, we write H^{1}(K, G) for the Galois cohomology set H^{1}(Gal(K), G(K^{sep})), where K^{sep} is the separable closure of K and Gal(K) = Gal(K^{sep}/K) acts on G(K^{sep}) in the natural way; cf. [S_{1}, Chapter II].

For any field extension F/k, the F -forms of a k-algebra R are in a natural 1-1 correspondence with H^{1}(F, Aut(R)); see [KMRT, §29.B] or [S_{2}, Proposition X.4]. Here, Aut(R) is the algebraic group of algebra automorphisms of R. If B is an F -form of R, we will write [B] for the element of H^{1}(F, Aut(R)) corresponding to B.

The normalizer N = N_{PGL_{n}}(S) naturally acts on M_{n}^{S} (by conjugation). Since S acts trivially, this action gives rise to a homomorphism α: N/S → Aut(M_{n}^{S}) of algebraic groups.

1.3. Theorem. Let S ⊂ PGL_{n} be a closed subgroup and B be an F -form of M_{n}^{S}/k. Then the following are equivalent.

(a) There exists a tame action of a linear algebraic group G on a central simple algebra A of degree n with center K, with associated stabilizer S, such that A^{G}/K^{G} is isomorphic to B/F (over k).

(b) [B] lies in the image of the natural map

α_{∗}: H^{1}(F, N/S) ⟶ H^{1}(F, Aut(M_{n}^{S})) .

The rest of this paper is structured as follows. In Section 2, we introduce tame and very tame actions. In Section 3 we prove a preparatory lemma and explain how it is used later on. Sections 4 and 5 are intended to motivate the notions of tame and very tame actions. In Sections 7, 8 and 9 we prove Theorem 1.1, Corollary 1.2 and Theorem 1.3 along with some generalizations. In the last section we illustrate Theorem 1.3 with an example. This example shows, in particular, that every finite-dimensional semisimple algebra (over a finitely generated field extension of k) occurs as A^{G}/K^{G} for some very tame action of a linear algebraic group G on some central simple algebra A with center K; see Corollary 10.4.

Notation and Terminology. We shall work over an algebraically closed base field k of characteristic zero. All fields will be assumed to be finitely generated extensions of k, and all central simple algebras to be finite-dimensional over their centers. All algebraic varieties, algebraic groups, group actions, morphisms, rational maps, etc., are assumed to be defined over k. By a point of an algebraic variety we shall always mean a k-point.

We denote by M_{n} the algebra of n × n matrices over k. If S is a subgroup of PGL_{n}, we denote by M_{n}^{S} the fixed algebra for the natural action of S on M_{n}. Throughout, G will be a linear algebraic group. We shall refer to an algebraic variety X endowed with a regular G-action as a G-variety. We will say that X (or the G-action on X) is generically free if Stab_{G}(x) = {1} for x ∈ X in general position.

2. Tame actions on central simple algebras

Tame actions are a special class of geometric actions. To review the latter concept, we need to recall the relationship between central simple algebras of degree n and irreducible, generically free PGL_{n}-varieties.

If X is a PGL_{n}-variety, we shall denote the algebra of PGL_{n}-invariant rational maps X −−→ M_{n} by RMaps_{PGL_{n}}(X, M_{n}). The algebra structure in RMaps_{PGL_{n}}(X, M_{n}) is naturally induced from M_{n}, i.e., given a, b: X −−→ M_{n}, we define a + b and ab by (a + b)(x) = a(x) + b(x) and ab(x) = a(x)b(x) for x ∈ X in general position. If the PGL_{n}-action on X is generically free, then RMaps_{PGL_{n}}(X, M_{n}) is a central simple algebra of degree n, with center k(X)^{PGL^{n}}; cf. [Re, Lemmas 8.5 and 9.1]. In this case we will sometimes denote this algebra by k_{n}(X). As this notation suggests, k_{n}(X) shares many properties with the function field k(X) (see [RV_{2}]); in particular, the two coincide if n = 1. Under our hypotheses, every central simple algebra of degree n is isomorphic to k_{n}(X) for some irreducible generically free PGL_{n}- variety X; see [RV_{2}, Theorem 1.2].

Suppose that X is a G × PGL_{n}-variety, and that the PGL_{n}-action on X is generically free. Then the G-action on X naturally induces a G-action on k_{n}(X); see [RV_{3}]. Following [RV_{3}] we will say that the action of an algebraic group G on a central simple algebra A is geometric if A is G-equivariantly isomorphic to k_{n}(X) for some G×PGL_{n}-variety X as above. (In particular, that this is only possible if the center Z(A) is a finitely generated field extension of k.) The G×PGL_{n}-variety X is then called the associated variety for the G-action on A. This associated variety is unique up to birational isomorphism of G × PGL_{n}-varieties, see [RV_{3}, Corollary 3.2].

Note that the class of geometric actions on central simple algebras is rather broad. It includes, in particular, all “algebraic” actions; see [RV_{3}, Theorem 5.3]. Roughly speaking, an action of G on A is algebraic if there is a G-invariant order R in A such that G acts regularly (“rationally”) on R. For details, see [RV_{3}].

Consider a geometric action ψ of a linear algebraic group G on a central simple algebra A, with associated G × PGL_{n}-variety X. Given x ∈ X, denote by S_{x} the projection of Stab_{G×PGL_{n}}(x) ⊂ G × PGL_{n} to PGL_{n}.

2.1. Definition. (a) We shall call the action ψ tame if there is a dense open subset U ⊂ X such that S_{x} is conjugate to S_{y} in PGL_{n} for every x, y ∈ U. In this case we shall refer to S_{x} (x ∈ U) as the associated stabilizer^{1} for ψ; the associated stabilizer is defined up to conjugacy. (b) We shall call the action ψ very tame if it is tame and if its associated stabilizer is reductive.

See Lemma 4.1 for an alternate description of tame and very tame actions. There exist non-tame geometric actions, and tame actions that are not very tame (see Proposition 5.1). However, informally speaking, many interesting actions are geometric and very tame; in particular, all geometric actions on division algebras are very tame. For details, see Section 4.

3. An exact sequence of stabilizer groups

3.1. Lemma. Let Γ be a linear algebraic group, N a closed normal subgroup, and π: Γ ⟶ Γ/N the natural projection. Suppose X is a Γ-variety, Y is a birational model for X/N (as a Γ/N-variety), and q: X −−→ Y is the rational quotient map. Then for x ∈ X in general position, the sequence of stabilizer subgroups

1 ⟶ Stab_{N}(x) ⟶ Stab_{Γ}(x) ⟶^{π} Stab_{Γ/N}(q(x)) ⟶ 1

is exact.

Proof. The sequence is clearly exact at Stab_{N}(x) and Stab_{Γ}(x), so we only need to show that the map

Stab_{Γ}(x)⟶ Stab_{Γ/N}(q(x)) ,

which we denote by π, is surjective. By a theorem of Rosenlicht (see [Ro_{1}], [Ro_{2}] or [PV, Proposition 2.5]) there is a dense Γ-invariant open subset X_{0} ⊂ X such that q^{−1}(q(x)) = N · x for every x ∈ X_{0}. Thus for x ∈ X_{0}, g ∈ Γ,

1For x ∈ X in general position, S_{x} is the stabilizer of the image of x in the rational quotient X/G, see Remark 3.2(b).

π(g) ∈ G/N stabilizes q(x) ⟺ g(x) ∈ N · x
⟺ n^{−1}g ∈ Stab_{Γ}(x) for some n ∈ N
⟺ π(g) = π(n^{−1}g) ∈ π(Stab_{Γ}(x)) ,

as claimed. □

3.2. Remark. In the sequel we will mostly apply Lemma 3.1 in a situation, where Γ = G × PGL_{n} and X is a Γ-variety with a generically free PGL_{n}- action. There are two natural choices of N here, N = G and N = PGL_{n} (or, more precisely, N = G × {1} and N = {1} × PGL_{n}). Thus we will consider two rational quotient maps:

\xymatrix{ & X \ar@{-->}[dl] \ar@{-->}[dr] \\ Y = X/G &  & Z = X/PGL_{n} .}

(a) Taking N = PGL_{n} in Lemma 3.1 and remembering that the PGL_{n}- action is generically free, we see that the natural projection G×PGL_{n} ⟶ G restricts to an isomorphism

Stab_{G×PGL_{n}}(x) ⟶^{≃} Stab_{G}(z) ,

where x ∈ X is in general position and z is the image of x in X/PGL_{n}; cf. [RV_{3}, Lemma 7.1].

(b) On the other hand, taking N = G and denoting by π the natural projection G × PGL_{n} ⟶ PGL_{n} we obtain

Stab_{PGL_{n}}(y) = π(Stab_{G×PGL_{n}}(x)) ≃ Stab_{G×PGL_{n}}(x)/ Stab_{G}(x) ,

where x ∈ X is in general position and y is the image of x in X/G, cf. [RV_{4}, Lemma 2.3].

4. An alternative description of tame and very tame actions, and some examples

It follows from Remark 3.2(b) that if ψ is a geometric G-action on a central simple algebra A, with associated G × PGL_{n}-variety X, then ψ is tame if and only if the PGL_{n}-action on X/G has a stabilizer in general position S ⊂ PGL_{n}, in the sense of [PV, p. 228]. Recall that this means that Stab_{PGL_{n}}(y) is conjugate to S for a general point y ∈ X/G, and that S is only well defined up to conjugacy in PGL_{n}. Recall also that if ψ is tame, we called S the associated stabilizer for ψ; see Definition 2.1.

Note that the stabilizer in general position is known to exist under certain mild assumptions on the action; cf. [PV, §7]. Informally speaking, this means that “most” geometric actions on central simple algebras are tame; we illustrate this point by the examples below. In particular, if Stab(y) is reductive for general y ∈ Y then, by a theorem of Richardson [Ri, Theorem 9.3.1], Y has a stabilizer in general position and thus the action ψ is very tame. For later use, we record some of these remarks as a lemma.

4.1. Lemma. Consider a geometric action of a linear algebraic group G on a central simple algebra A with associated G × PGL_{n}-variety X.

(a) The action of G on A is tame if and only if the PGL_{n}-action on X/G has a stabilizer in general position.

(b) The action of G on A is very tame if and only if Stab_{PGL_{n}}(y) is reductive for y ∈ X/G in general position. □

4.2. Example. A geometric action ψ on a central simple algebra A is very tame if

(a) A is a division algebra, or

(b) the connected component G^{0} of G is a torus (this includes the case where G is finite), or

(c) Stab_{G}(z) is reductive for z ∈ X/PGL_{n} in general position, or

(d) G is reductive and the G-action on A is inner, or

(e) A has a G-invariant maximal étale subalgebra.

Here by an étale subalgebra of A we mean a subalgebra of A which is a (finite) direct sum of finite field extensions of the center of A. (Since we are working in characteristic zero, these field extensions are necessarily separable.)

Proof. Using Richardson’s theorem (see the remarks before Lemma 4.1), part (a) follows from [RV_{4}, Section 3], so we will focus on the other parts. Let X be the associated G × PGL_{n}-variety, let x ∈ X be a point in general position, and let y and z be the images of x in Y = X/G and Z = X/PGL_{n}, respectively, as in Remark 3.2. By Lemma 4.1, it suffices to show that Stab_{PGL_{n}}(y) is reductive for y ∈ Y in general position.

(b) For x ∈ X in general position, Stab_{G×PGL_{n}}(x) is isomorphic to a closed subgroup of G and is thus reductive; see Remark 3.2(a). Hence,

Stab_{PGL_{n}}(y) ≃ Stab_{G×PGL_{n}}(x)/ Stab_{G}(x)

is also reductive for y ∈ Y in general position; cf. Remark 3.2(b).

(c) Let x ∈ X in general position. Since Stab_{G×PGL_{n}}(x) ≃ Stab_{G}(z), it is a reductive group, and so is its image Stab_{PGL_{n}}(y). The desired conclusion follows from Lemma 4.1.

(d) Here G acts trivially on Z(A) = k(X/PGL_{n}) and hence, on X/PGL_{n}. Thus Stab_{G}(z) = G is reductive for every z ∈ X/PGL_{n}, and part (c) applies.

(e) By [RV_{3}, Theorem 1.5(b)] there exists a maximal torus T ⊂ PGL_{n} (depending on x) such that Stab_{G×PGL_{n}}(x) ⊂ G × N(T) where N(T) denotes the normalizer of T in PGL_{n}. Thus the image π(Stab_{G×PGL_{n}}(x)) of Stab_{G×PGL_{n}}(x) under the natural projection π: G × PGL_{n} ⟶ PGL_{n} consists of semisimple elements and, consequently, is reductive. The desired assertion now follows from the fact that Stab_{PGL_{n}}(y) and π(Stab_{G×PGL_{n}}(x)) are equal; see Remark 3.2(b). □

5. Examples of wild actions

The purpose of this section is to show that Definition 2.1 is not vacuous.

5.1. Proposition. (a) There exists a geometric action of an algebraic group on a central simple algebra which is tame but not very tame.

(b) There exists a geometric action of an algebraic group on a central simple algebra which is not tame.

Our proof of Proposition 5.1 will rely on the following lemma.

5.2. Lemma. Let G be an algebraic group and H a closed subgroup. Then for every H-variety Y (not necessarily generically free) there exists a G×Hvariety X such that

(i) the action of H = {1} × H on X is generically free, and (ii) X/G is birationally isomorphic to Y (as an H-variety).

Proof. Set X = G × Y, with G acting on the first factor by translations on the left and H acting by h: (g, y) ↦ (gh^{−1}, hy). The G-action and the H-action on X commute and thus define a G × H-action. Conditions (i) and (ii) are now easy to check. □

Proof of Proposition 5.1. (a) In view of Lemma 4.1, we need to construct a G × PGL_{n}-variety X, with PGL_{n} acting generically freely, and such that the PGL_{n}-action on the rational quotient variety X/G has a non-reductive stabilizer in general position. By Lemma 5.2 (with G = H = PGL_{n}) it suffices to construct a PGL_{n}-variety Y with a non-reductive stabilizer in general position. To construct such a Y, take any non-reductive subgroup S ⊂ PGL_{n} and let Y be the homogeneous space PGL_{n}/S.

(b) Arguing as in part (a), we see that it suffices to construct a PGL_{n}- variety Y which does not have a stabilizer in general position. Richardson [Ri, §12.4] showed that (for suitably chosen n) there exists an SL_{n}-variety Z with the following property: for every Zariski open subset U ⊂ Z there are infinitely many stabilizers Stab_{SL_{n}}(z), z ∈ U, with pairwise non-isomorphic Lie algebras. Clearly such an SL_{n}-variety Z cannot have a stabilizer in general position. We will now construct a PGL_{n}-variety Y with no stabilizer in general position by modifying this example. (Note that in [Ri] the variety Z is only constructed over the field ℂ of the complex numbers. However, by the Lefschetz principle this construction can be reproduced over any algebraically closed base field k of characteristic zero.)

Consider the rational quotient map π: Z −−→ Y = Z/μ_{n}, for the action of the center μ_{n} of SL_{n} on Z. Recall that (after choosing a suitable birational model for Y), the SL_{n}-action on Z descends to a PGL_{n}-action on Y. In fact, this is exactly the situation we considered in the setting of Lemma 3.1 (with Γ = SL_{n} and N = μ_{n}); this lemma tells us that, after replacing Y by a dense open subset Y_{0}, and Z by π^{−1}(Y_{0}),

Stab_{PGL_{n}}(y) ≃ Stab_{SL_{n}}(z)/ Stab_{μ_{n}}(z)

for every z ∈ Z and y = π(z). Since Stab_{μ_{n}}(z) is a finite group we see that Stab_{PGL_{n}}(y) and Stab_{SL_{n}}(z) have isomorphic Lie algebras. This shows that for every open subset V of Y there are infinitely many stabilizers Stab_{PGL_{n}}(y), y ∈ V, with pairwise non-isomorphic Lie algebras. In particular, the PGL_{n}-action on Y has no stabilizer in general position. □

6. Homogeneous fiber spaces

In this section we briefly recall the definition and basic properties of homogeneous fiber spaces. Let φ: H → G be a homomorphism of linear algebraic groups (defined over k) and Z be an H-variety. The homogeneous fiber space X = G ∗_{H} Z is defined as the rational quotient of G × Z for the action of {1} × H, where G × H acts on G × Z via (g, h) · (g̃, z) = (gg̃φ(h)^{−1}, hz). As usual, we choose a model for this rational quotient, so that the G-action on G × Z descends to a regular action on this model. Note that {1_{G}} × Z projects to an H-subvariety of X birationally isomorphic to Z/ ker(φ).

It is easy to see that G∗_{H}Z is birationally isomorphic to G∗_{φ(H)}(Z/ ker(φ)) as a G-variety; thus in carrying out this construction we may assume that H is a subgroup of G and φ is the inclusion map. For a more detailed discussion of homogeneous fiber spaces in this setting and further references, we refer the reader to [RV_{1}, §3].

We also recall that a G-variety X is called primitive if G transitively permutes the irreducible components of X or, equivalently, if k(X)^{G} is a field; see [Re, Lemma 2.2(b)].

Our goal in this section is to prove the following variant of [LPR, Lemma 2.17], which will be repeatedly used in the sequel.

6.1. Lemma. Let φ: H → G be a homomorphism of linear algebraic groups (defined over k), Z be an H-variety and R be a finite-dimensional k-algebra. Let X be the homogeneous fiber space G ∗_{H} Z. Assume further that G acts on R via k-algebra automorphisms (i.e., via a morphism G ⟶ Aut(R) of algebraic groups). Then

(a) (cf. [PV, p. 161]) k(X)^{G} and k(Z)^{H} are isomorphic as k-algebras. In particular, X is primitive if and only if Z is primitive.

(b) If X and Z are primitive, the algebras RMaps_{G}(X, R)/k(X)^{G} and RMaps_{H}(Z, R)/k(Z)^{H} are isomorphic.

Here, as usual, RMaps_{H}(Z, R) denotes the k(Z)^{H}-algebra of H-equivariant rational maps Z −−→ R.

Proof. Let f: X = G ∗_{H} Z −−→ R be a G-equivariant rational map. The indeterminacy locus of such a map is a G-invariant subvariety of X. Let χ: Z −−→ X be the rational map sending z ∈ Z to the image of (1_{G}, z) ∈ G×Z in X = G∗_{H} Z. Since G·χ(Z) is dense in X, χ(Z) cannot be contained in the indeterminacy locus of f. Thus the map

ψ: RMaps_{G}(X, R) → RMaps_{H}(Z, R)

given by ψ(f) = f ∘ χ is a well-defined k-algebra homomorphism. Since G · χ(Z) is dense in X, ψ is injective.

It is easy to see that ψ has an inverse. Indeed, given H-equivariant rational map λ: Z −−→ R, we define a G-equivariant map λ′: G×Z −−→ R by (g, z) ↦ gλ(z). Since λ′(gφ(h)^{−1}, hz) = λ′(g, z) for every g ∈ G and z ∈ Z, and X = (G × Z)/H, the universal property of rational quotients tells us that λ′ descends to a G-equivariant rational map f: X −−→ R. Moreover, by our construction ψ(f) = f ∘ χ = λ, so that the map RMaps_{H}(Z, R) → RMaps_{G}(X, R) taking λ to f is the inverse of ψ.

We are now ready to complete the proof of Lemma 6.1. To prove part (a), we apply the above construction in the case where R = k, with trivial G-action. Here RMaps_{G}(X, k) = k(X)^{G}, RMaps_{H}(Z, k) = k(Z)^{H}, and ψ is the desired isomorphism of k-algebras. To prove part (b), note that if we identify elements of k(X)^{G} (respectively, k(Z)^{H}) with G-equivariant rational maps X −−→ R (respectively, H-equivariant rational maps Z −−→ R) whose image is contained in k ·1_{R} then ψ is an algebra isomorphism between RMaps_{G}(X, R)/k(X)^{G} and RMaps_{H}(Z, R)/k(Z)^{H}. □

7. Proof of Theorem 1.1

We begin with a simple observation. Let X be the associated G × PGL_{n}- variety for the G-action on A. Let Y be a PGL_{n}-variety, which is a birational model for X/G. Then by the universal property of rational quotients (see, e.g, [PV, §2.3-2.4], [Re, Remark 2.4] or [RV_{3}, Remark 6.1]),

(7.1)  A^{G} = RMaps_{PGL_{n}}(X, M_{n})^{G} ≃ RMaps_{PGL_{n}}(Y, M_{n}) .

Moreover,

Z(A)^{G} = (k(X)^{PGL^{n}})^{G} = k(X)^{G×PGL_{n}} ≃ k(Y)^{PGL^{n}} .

Thus Theorem 1.1 is a special case of the following result (applied to the conjugation action of Γ = PGL_{n} on R = M_{n}).

7.2. Proposition. Let R be a finite-dimensional k-algebra, and Γ a linear algebraic group, acting on R by k-algebra automorphisms (i.e., via a morphism Γ ⟶ Aut(R) of algebraic groups). If a primitive Γ-variety Y has a stabilizer S ⊂ Γ in general position then RMaps_{Γ}(Y, R)/k(Y)^{Γ} is a form of R^{S}/k.

Here we view RMaps_{Γ}(Y, R) as a K-algebra, where K = k(Y)^{Γ}. Note that K is a field since Y is a primitive Γ-variety (the latter means that Γ transitively permutes the irreducible components of Y).

Proof. We will proceed in two steps.

Step 1. Suppose first that S = {1}, i.e., that the Γ-action on Y is generically free. Let Y_{0} be an irreducible component of Y, and let ϕ: Γ×Y_{0} ⟶ Y be the natural Γ-equivariant map, where Γ acts on itself by left multiplication and trivially on Y_{0}. Then ϕ induces an injective homomorphism RMaps_{Γ}(Y, R) ↪ RMaps_{Γ}(Γ × Y_{0}, R) via f ↦ f ∘ ϕ. Clearly Γ-equivariant rational maps from Γ×Y_{0} to R are in 1-1 correspondence with rational maps from Y_{0} to R. In other words,

RMaps_{Γ}(Y, R) ↪ RMaps_{Γ}(Γ × Y_{0}, R) ≃ RMaps(Y_{0}, R) ≃ R ⊗_{k} k(Y_{0}) .

On the other hand, by [Re, Lemma 7.4(b)], dim_{K} RMaps_{Γ}(Y, R) = dim_{k}(R). By [Re, Lemma 7.4(a)], the induced map

RMaps_{Γ}(Y, R) ⊗_{K} k(Y_{0}) → RMaps_{Γ}(Γ × Y_{0}, R)

is injective, so an isomorphism. That is, RMaps_{Γ}(Y, R) is a form of R. This completes the proof of Proposition 7.2 in the case that S = {1}.

Step 2. We will now reduce the general case to the case considered in Step 1. Let Z be the union of the components of Y^{S} of maximal dimension, and let N = N_{Γ}(S) be the normalizer of S in Γ. Note that Z is an N-variety. By [RV_{1}, Lemma 3.2], Y is birationally isomorphic to Γ∗_{N} Z, as a Γ-variety. By Lemma 6.1, Z is primitive since Y is, and

(7.3)  RMaps_{Γ}(Y, R)≃ RMaps_{N}(Z, R) and k(Y)^{Γ} ≃ k(Z)^{N} .

Thus it suffices to show that RMaps_{N}(Z, R)/k(Z)^{N} is a form of R^{S}/k.

Recall that S acts trivially on Z (because Z is, by definition, a subset of Y^{S}); hence, the image of every N-equivariant map from Z to R will actually lie in R^{S}. In other words,

(7.4)  RMaps_{N}(Z, R) = RMaps_{N}(Z, R^{S}) = RMaps_{N/S}(Z, R^{S}) .

Moreover, k(Z)^{N} = k(Z)^{N/S}. We now show that the N/S-action on Z is generically free; the desired conclusion will then follow from the result of Step 1. Equivalently, we want to show that Stab_{N}(z) = S for z ∈ Z in general position. Clearly, S ⊂ Stab_{N}(z) for any z ∈ Z ⊂ Y^{S}. To prove the opposite inclusion, it is enough to show that S = Stab_{Γ}(z) for z ∈ Z in general position. Indeed, by the definition of S, Stab_{Γ}(y) is conjugate to S for y ∈ Y in general position. Since ΓZ is dense in Y, Stab_{Γ}(z) is conjugate to S for z ∈ Z in general position. Since we know that S ⊂ Stab_{Γ}(z) we conclude that S = Stab_{Γ}(z) for z ∈ Z in general position, as desired. □

We conclude this section with a simple corollary of Theorem 1.1.

7.5. Corollary. Consider a tame action of G on a central simple algebra A with associated stabilizer S ⊂ PGL_{n}. Let N be the normalizer of S in PGL_{n}. If A^{G} is a division algebra then M_{n}^{N} = k.

Proof. Let X be the associated G×PGL_{n}-variety. Set Y = X/G, and denote by Z the union of the irreducible components of Y^{S} of maximal dimension. Recall from the proof of Theorem 1.1 that

A^{G} ≃ RMaps_{PGL_{n}}(Y, M_{n}) ≃ RMaps_{N}(Z, M_{n}^{S}) .

Now observe that if m ∈ M_{n}^{N}, then the constant map f_{m}: Z ⟶ M_{n}^{S}, given by f_{m}(z) = m for every z ∈ Z, is N-equivariant. Thus f_{m} may be viewed as an element of A^{G}, and m ↦ f_{m} defines an embedding M_{n}^{N} ↪ A^{G}. Since we are assuming that A^{G} is a division algebra, this implies that M_{n}^{N} cannot have zero divisors. On the other hand, since M_{n}^{N} is a finite-dimensional k-algebra and k is algebraically closed, this is only possible if M_{n}^{N} = k. □

8. Proof of Corollary 1.2

In this section we spell out what Theorem 1.1 says in the case where the Gaction on A is very tame. Here the associated stabilizer subgroup S ⊂ PGL_{n} is reductive, and hence so is the preimage S^{∗} of S in GL_{n}. Thus the natural n-dimensional linear representation ϕ: S^{∗} ↪ GL_{n} decomposes as a direct sum of irreducibles. Suppose ϕ has irreducible components ϕ_{1}, …, ϕ_{r} of dimensions d_{1}, …, d_{r}, occurring with multiplicities e_{1}, …, e_{r}, respectively. Note that d_{1}e_{1}+⋯+d_{r}e_{r} = n. Our main result is the following proposition. Corollary 1.2 is an immediate consequence of parts (a), (b) and (c).

8.1. Proposition. (a) A^{G} is a form of M_{e_{1}}(k) × ⋯ × M_{e_{r}}(k). In par-

ticular, A^{G} is semisimple.

(b) The maximal degree of an element of A^{G} over K is e_{1} + ⋯ + e_{r}.

(c) The maximal degree of an element of Z(A^{G}) over K is r.

(d) If A^{G} is a simple algebra, then the normalizer N(S^{∗}) of S^{∗} in GL_{n} transitively permutes the (isomorphism types of ) the irreducible representations ϕ_{1}, …, ϕ_{r} of S^{∗} occurring in ϕ. In particular, in this case, d_{1} = ⋯ = d_{r} and e_{1} = ⋯ = e_{r}.

Proof. (a) By a corollary of Schur’s Lemma (see, e.g., [FD, Proposition 1.8]),

M_{n}^{S} = (M_{n})^{S^{∗}} = End_{S^{∗}}(k^{n}) ≃ M_{e_{1}}(k) × ⋯ × M_{e_{r}}(k) .

Part (a) now follows from Theorem 1.1.

(b) The degree of x ∈ A^{G} over K is dim_{K}(K[x]). In view of part (a), the maximal degree of an element of A^{G} over K is the same as the maximal degree of an element of M_{e_{1}}(k) × ⋯ × M_{e_{r}}(k) over k. The latter is easily seen to be e_{1} + ⋯ + e_{r}.

(c) Once again, in view of part (a), we only need to show that the maximal degree of a central element of M_{e_{1}}(k) × ⋯ × M_{e_{r}}(k) over k is r. The center of M_{e_{1}}(k) × ⋯ × M_{e_{r}}(k) is isomorphic, as a k-algebra, to k × ⋯ × k (r times), and part (c) follows.

(d) Let V_{i} be the unique S^{∗}-subrepresentation of k^{n} isomorphic to ϕ_{i}^{e_{i}}. The normalizer N(S^{∗}) clearly permutes the subspaces V_{i} (and hence, the representations ϕ_{i}); we want to show that this permutation action is transitive. Assume the contrary: say, {ϕ_{1}, …, ϕ_{s}} is an N(S^{∗})-orbit for some s < r. Then W_{1} = V_{1} ⊕⋯⊕V_{s} and W_{2} = V_{s+1} ⊕⋯⊕V_{r} are complementary N(S^{∗})-invariant subspaces of k^{n} and

M_{n}^{S} = End_{S^{∗}}(k^{n}) ≃ End_{S^{∗}}(W_{1}) ⊕ End_{S^{∗}}(W_{2}) ,

where ≃ denotes an N(S)-equivariant isomorphism of k-algebras. Now let X be an associate G × PGL_{n}-variety for the action of G on A, Y = X/G, and Z be the union of components of maximal dimension of Y^{S}, as in the proof of Proposition 7.2. Then,

A^{G} =^{(7.1)} RMaps_{PGL_{n}}(Y, M_{n}) =^{(7.3),(7.4)} RMaps_{N(S)}(Z, M_{n}^{S})
= RMaps_{N(S)}(Z, End_{S^{∗}}(W_{1})) ⊕ RMaps_{N(S)}(Z, End_{S^{∗}}(W_{2})) .

This shows that A^{G} is not simple. □

9. Proof of Theorem 1.3

We begin with some preliminaries on Galois cohomology. Let F/k be a finitely generated field extension and Γ/k be a linear algebraic group. Let Γ-Var(F) be the set of generically free Γ-varieties X with k(X)^{Γ} = F. (Since k(X)^{Γ} is a field, X is necessarily primitive, i.e., Γ transitively permutes the irreducible components of X; see Section 6.) Recall that H^{1}(F, Γ) is in a natural bijective correspondence with Γ-Var(F); see, e.g., [P, (1.3)]. We can thus identify H^{1}(F, Γ) with Γ-Var(F). If α: Γ → Γ′ is a homomorphism of algebraic groups then the induced map α_{∗}: H^{1}(F, Γ) → H^{1}(F, Γ′) is given by Z ↦ Γ′ ∗_{Γ} Z. (This follows from [P, Theorem 1.3.3(b)]; see also [KMRT, Proposition 28.16].)

If Γ is the automorphism group of some finite-dimensional k-algebra R then, as we remarked before the statement of Theorem 1.3, elements of H^{1}(F, Γ) are also in a natural bijective correspondence with the set of F - forms of R. These two interpretations of H^{1}(F, Γ) can be related as follows: the algebra corresponding to the Γ-variety W is B = RMaps_{Γ}(W, R); cf. [Re, Proposition 8.6]. Here, as usual, RMaps_{Γ} stands for Γ-equivariant rational maps.

We are now ready to proceed with the proof of Theorem 1.3. We fix n ≥ 1 and a subgroup S ⊂ PGL_{n}. For notational convenience, set H = Aut(M_{n}^{S}). Consider the following diagram:

[figure: H^{1}(F, N/S); α_{∗}; H^{1}(F, H); N/S-Var(F); H-Var(F); ≃; {F -forms of M_{n}^{S}}; Z; H ∗_{N/S} Z; W; RMaps_{H}(W, M_{n}^{S})]

This diagram shows that the image of the map α_{∗}: H^{1}(F, N/S) → H^{1}(F, H) consist of all classes in H^{1}(F, H) corresponding to algebras B/F of the form

B = RMaps_{H}(H ∗_{N/S} Z, M_{n}^{S}) ,

where F = k(H ∗_{N/S} Z)^{H} and Z is a primitive N/S-variety. But

RMaps_{H}(H ∗_{N/S} Z, M_{n}^{S}) ≃ RMaps_{N/S}(Z, M_{n}^{S})

and k(H ∗_{N/S} Z)^{H} ≃ k(Z)^{N/S}; see Lemma 6.1. Consequently, [B] lies in the image of α_{∗} if and only if there exists a generically free primitive N/Svariety Z such that B ≃ RMaps_{N/S}(Z, M_{n}^{S}) and F ≃ k(Z)^{N/S}, i.e., if and only if condition (c) of the following lemma is satisfied.

We will say that an F -algebra B is a fixed algebra with associated stabilizer S ⊂ PGL_{n} if there exists a tame action of an algebraic group G on a central simple algebra A/K of degree n with associated stabilizer S such that B ≃ A^{G} and F ≃ K^{G}. In other words, B/F is a fixed algebra with associated stabilizer S if it satisfies condition (a) of Theorem 1.3. Hence Theorem 1.3 follows from the following result:

9.1. Lemma. Let B be an F -algebra, S a subgroup of PGL_{n}, and N = N_{PGL_{n}}(S). The following are equivalent:

(a) B/F is a fixed algebra with associated stabilizer S.

(b) There exists an irreducible PGL_{n}-variety Y with stabilizer S in general position such that B ≃ RMaps_{PGL_{n}}(Y, M_{n}) and F ≃ k(Y)^{PGL^{n}}.

(c) There exists a generically free primitive N/S-variety Z such that B ≃ RMaps_{N/S}(Z, M_{n}^{S}) and F ≃ k(Z)^{N/S}.

The isomorphisms in the various parts of the lemma are compatible: the second is always a restriction of the first.

Proof. (a) ⟹ (b): By definition the algebras B/F satisfying condition (a) of Theorem 1.3 are those of the form B = RMaps_{PGL_{n}}(X, M_{n})^{G}, where F = k(X)^{G×PGL_{n}}. Here X is an irreducible G×PGL_{n}-variety, such that the PGL_{n}-action on X is generically free, and the projection of Stab_{G×PGL_{n}}(x) to PGL_{n} is conjugate to S for x ∈ X in general position. To write B and F as in (b), take Y = X/G; see (7.1).

(b) ⟹ (c): Take Z to be the union of the components of Y^{S} of maximal dimension, as in Step 2 of the proof of Proposition 7.2.

(c) ⟹ (b): Take Y to be the homogeneous fiber space PGL_{n} ∗_{N} Z, and use Lemma 6.1 and (7.4). Note that Y is irreducible since k(Y)^{PGL^{n}} is a field and PGL_{n} is connected. A simple computation shows that S is the stabilizer in general position for the PGL_{n}-action on Y.

(b) ⟹ (a): Use Lemma 5.2, with H = PGL_{n} to reconstruct X from Y. Note that here G can be any irreducible group containing PGL_{n}.

This completes the proof of Lemma 9.1 and thus of Theorem 1.3. □

10. An example

We will now illustrate Theorem 1.3 with an example. Fix an integer n ≥ 1 and write n = d_{1}e_{1} +⋯+d_{r}e_{r} for some positive integers d_{1}, e_{1}, …, d_{r}, e_{r}. A partition of n of this form naturally arises when an n-dimensional representation is decomposed as a direct sum of irreducibles of dimensions d_{1}, …, d_{r} occurring with multiplicities e_{1}, …, e_{r}, respectively. (In particular, the integers d_{i} and e_{i} came up in this way at the beginning of Section 8.) For this reason τ = [(d_{1}, e_{1}), …, (d_{r}, e_{r})] is often referred to as a “representation type”; cf., e.g., [LBP].

Choose vector spaces V_{1}, W_{1}, …, V_{r}, W_{r}, so that dim V_{i} = d_{i} and dim W_{i} = e_{i}. Write

k^{n} = (V_{1} ⊗ W_{1}) ⊕ ⋯ ⊕ (V_{r} ⊗ W_{r})

and let S_{τ}^{∗} ≃ GL_{d_{1}} × ⋯ × GL_{d_{r}} be the subgroup

S_{τ}^{∗} = (GL(V_{1}) ⊗ I_{e_{1}}) ⊕ ⋯ ⊕ (GL(V_{r}) ⊗ I_{e_{r}}) ⊂ GL_{n} .

Let N_{τ}^{∗} be the normalizer of S_{τ}^{∗} in GL_{n}. Denote the image of S_{τ}^{∗} under the natural projection GL_{n} → PGL_{n} by S_{τ} and the normalizer of S_{τ} in PGL_{n} by N_{τ}. The purpose of this section is to spell out what Theorem 1.3 tells us for S = S_{τ} ⊂ PGL_{n}. Note that

M_{n}^{S_{τ}} = (I_{d_{1}} ⊗ End(W_{1})) ⊕ ⋯ ⊕ (I_{d_{r}} ⊗ End(W_{r})) ≃ M_{e_{1}} × ⋯ × M_{e_{r}} .

Our goal is to determine which forms of M_{n}^{S_{τ}} occur as fixed algebras with associated stabilizer S_{τ}.

Note that reordering the pairs (d_{i}, e_{i}) will have the effect of replacing S_{τ} by a conjugate subgroup of PGL_{n}. Since the group S in Theorems 1.1 and 1.3 is only defined up to conjugacy, we are free to order them any way we want. In particular, if there are t distinct numbers f_{1}, …, f_{t} among e_{1}, …, e_{r}, occurring with multiplicities r_{1}, …, r_{t}, respectively, we will, for notational convenience, renumber the pairs (d_{i}, e_{i}) so that τ assumes the form

τ = [(d_{11}, f_{1}), …, (d_{1r_{1}}, f_{1}), …, (d_{t1}, f_{t}), …, (d_{tr_{t}}, f_{t})] .

Suppose that for each fixed i, exactly l_{i} numbers occur among d_{i1}, …, d_{ir_{i}}, with multiplicities r_{i1}, r_{i2}, …, r_{il_{i}}. (We order the pairs (d_{i1}, f_{i}), …, (d_{ir_{i}}, f_{i}) so that the first r_{i1} of the d_{ij}s are the same, the next r_{i2} are the same, etc.) Clearly r_{i} = r_{i1} + ⋯ + r_{il_{i}},

M_{n}^{S_{τ}} ≃ M_{e_{1}} × ⋯ × M_{e_{r}} ≃ M_{f_{1}}^{r_{1}} × ⋯ × M_{f_{t}}^{r_{t}} .

and

Aut(M_{n}^{S_{τ}}) ≃ ∏_{i=1}^{t} Aut(M_{f_{i}}^{r_{i}}) .

Note that Aut(M_{f}^{r}) is the semidirect product PGL_{f}^{r} {> \atop ◁} Sym_{r}, where PGL_{f}^{r} is the group of inner automorphisms of M_{f}^{r}, and Sym_{r} acts by permuting the r factors of M_{f}.

10.1. Lemma. The natural homomorphism α: N_{τ} /S_{τ} → Aut(M_{n}^{S_{τ}}) corresponds to the inclusion map

∏_{i=1}^{t}(∏_{j=1}^{l_{i}} Aut(M_{f_{i}}^{r_{ij}})) ↪ ∏_{i=1}^{t} Aut(M_{f_{i}}^{r_{i}}) .

(Recall that r_{i} = r_{i1} + ⋯ + r_{il_{i}} for each i.)

Note that this lemma says that α embeds N_{τ} /S_{τ} in Aut(M_{n}^{S_{τ}}) as a subgroup of finite index. In particular, the image of α contains the connected component

PGL_{e_{1}} × ⋯ × PGL_{e_{r}} = PGL_{f_{1}}^{r_{1}} × ⋯ × PGL_{f_{t}}^{r_{t}}

of Aut(M_{n}^{S_{τ}}).

Proof. First of all, note that we may replace N_{τ} /S_{τ} by N_{τ}^{∗}/S_{τ}^{∗}; these two groups are isomorphic and act the same way on M_{n}^{S_{τ}}.

We claim that the homomorphism α: N_{τ}^{∗}/S_{τ}^{∗} → Aut(M_{n}^{S_{τ}}) is injective. Indeed, the kernel of the action of N_{τ}^{∗} on M_{n}^{S_{τ}} is the centralizer of

M_{n}^{S_{τ}} = I_{d_{1}} ⊗ End(W_{1}) × ⋯ × I_{d_{r}} ⊗ End(W_{r})

in GL_{n}. Using Schur’s Lemma, as in the proof of Proposition 8.1(a), we see that this centralizer is

S_{τ}^{∗} = GL(V_{1}) ⊗ I_{e_{1}} × ⋯ × GL(V_{r}) ⊗ I_{e_{r}} ,

and the claim follows. Thus α is injective, and we only need to determine which automorphisms of Aut(M_{n}^{S_{τ}}) ≃ M_{e_{1}} × ⋯ × M_{e_{r}} can be realized as conjugation by some g ∈ N_{τ}^{∗}. Taking g = 1⊗a_{1}+⋯+1⊗a_{r} for a_{i} ∈ GL(W_{i}), we see that every element of

PGL_{e_{1}} × ⋯ × PGL_{e_{r}} = PGL_{f_{1}}^{r_{1}} × ⋯ × PGL_{f_{t}}^{r_{t}}

lies in the image of α.

Let us now view k^{n} as an n-dimensional representation of the group S_{τ}^{∗} ≃ GL_{d_{1}} × ⋯ × GL_{d_{r}}, as we did at the beginning of this section. Any g ∈ N_{τ}^{∗} permutes the isomorphism types of the irreducible representations V_{1}, …, V_{r} of S_{τ}^{∗}; it also permutes the isotypical components V_{1} ⊗ W_{1}, …, V_{r} ⊗ W_{r} of this representation (as subspaces of k^{n}). If g maps the isotypical component V_{i} ⊗ W_{i} to the isotypical component V_{j} ⊗ W_{j} then their dimensions d_{i}e_{i} and d_{j}e_{j} have to be the same. The dimensions d_{i} and d_{j} of the underlying irreducible representations V_{i} and V_{j} also have to be the same. In other words, this is only possible if (d_{i}, e_{i}) = (d_{j}, e_{j}).

Conversely, if (d_{i}, e_{i}) = (d_{j}, e_{j}) then a suitably chosen permutation matrix g ∈ N_{τ}^{∗}, will interchange V_{i} ⊗ W_{i} and V_{j} ⊗ W_{j} and preserve every other isotypical component V_{m} ⊗ W_{m} for m ≠ i, j. These permutation matrices generate the finite subgroup ∏_{i=1}^{t} ∏_{j=1}^{l_{i}} Sym_{r_{ij}} of Aut(M_{n}^{S_{τ}}).

Now suppose g ∈ N_{τ}^{∗}. After composing any g with a product of permutation matrices as above, we arrive at a g_{0} ∈ N_{τ}^{∗} which preserves every isotypical component V_{i} ⊗ W_{i}. Conjugating by such a g_{0} will preserve every factor I_{d_{i}} ⊗ End(W_{i}) ≃ M_{e_{i}} of Aut(M_{n}^{S_{τ}}) and thus will lie in PGL_{f_{1}}^{r_{1}} × ⋯ × PGL_{f_{t}}^{r_{t}}. We conclude that the image of α is generated by

PGL_{f_{1}}^{r_{1}} × ⋯ × PGL_{f_{t}}^{r_{t}} and ∏_{i=1}^{t} ∏_{j=1}^{l_{i}} Sym_{r_{ij}}

and the lemma follows. □

10.2. Proposition. B/F is a fixed algebra with associated stabilizer S_{τ} if and only if B is F -isomorphic to the direct product

(B_{11} × ⋯ × B_{1l_{1}}) × ⋯ × (B_{t1} × ⋯ × B_{tl_{t}})

where B_{ij}/F is a form of M_{f_{i}}^{r_{ij}} /k.

Proof. Recall that H^{1}(F, Aut(R)) = {F -forms of R}. Now suppose R_{1}, …, R_{m} are k-algebras, R = R_{1} × ⋯ × R_{m}, G_{i} = Aut(R_{i}) and G = Aut(R_{1}) × ⋯ × Aut(R_{m}). Then the natural inclusion G_{1} × ⋯ × G_{r} ↪ Aut(R) induces a morphism from

H^{1}(F, G_{m} × ⋯ × G_{m}) ≃ H^{1}(F, G_{1}) × ⋯ × H^{1}(F, G_{m})

to H^{1}(F, Aut(R)) which maps (A_{1}, …, A_{m}) to A = A_{1} × ⋯ × A_{m}. Here A_{i} is an F -form of R_{i} (and hence, R is an F -form of A).

In the language of G-varieties, the first isomorphism says that X is a generically free primitive G-variety if and only if X is birationally isomorphic (as a G-variety) to the fiber product X_{1}×_{Y} ⋯×_{Y} X_{m}, where X_{i} is a primitive generically free G_{i}-variety such that X_{i}/G_{i} ≃ Y is a model for F (Y is the same for each i). Note that X_{1} can be recovered from X as the rational quotient X/(G_{2} × ⋯ × G_{m}); similarly for X_{2}, …, X_{m}. The second map is given by the natural isomorphism

RMaps_{G}(X, R) ≃ RMaps_{G_{1}}(X_{1}, R_{1}) × ⋯ × RMaps_{G_{m}}(X_{m}, R_{m})

of F -algebras.

We now apply this to the homomorphism α: N_{τ} /S_{τ} → Aut(M_{n}^{S_{τ}}) described in Lemma 10.1, and the proposition follows from Theorem 1.3. □

We conclude with two corollaries of Proposition 10.2.

10.3. Corollary. Let τ = [(d_{1}, e_{1}), …, (d_{r}, e_{r})]. Then the following are equivalent.

(a) d_{i} = d_{j} whenever e_{i} = e_{j}.

(b) Every form B/F of M_{n}^{S_{τ}} appears as a fixed algebra with associated stabilizer S_{τ}.

Proof. (a) is equivalent to l_{i} = 1 for every i = 1, …, t. The implication (a) ⇒ (b) is now an immediate consequence of the proposition. To show that (b) ⟹ (a), consider an F -algebra B = D_{1} ×⋯×D_{t}, where D_{i} is a division algebra of degree f_{i} such that [Z(D_{i}) : F] = r_{i}. Here Z(D_{i}) denotes the center of D_{i}. (For example, we can take F = k(a_{1}, b_{1}, …, a_{t}, b_{t}, c), where a_{1}, b_{2}, …, a_{t}, b_{t}, c are independent variables, and D_{i} is the symbol algebra (a_{i}, b_{i})_{f_{i}} over Z(D_{i}) = F ( \sqrt[r_{i}]{c}).) The algebra B = D_{1} × ⋯ × D_{t} is then a form of

M_{n}^{S_{τ}} = M_{f_{1}}^{r_{1}} × ⋯ × M_{f_{t}}^{r_{t}} .

On the other hand, if (b) holds, Proposition 10.2 tells us that B can be written as a direct product of l_{1} +⋯+l_{t} F -algebras. But B = D_{1} ×⋯×D_{t} clearly cannot be written as a non-trivial direct product of more than t F -algebras. Thus l_{1} + ⋯ + l_{t} ≤ t, i.e., l_{1} = ⋯ = l_{t} = 1, and (a) follows. □

10.4. Corollary. Every finite-dimensional semisimple algebra B/F (where F is a finitely generated field extension of k) appears as A^{G}/K^{G} for some very tame geometric action of a linear algebraic group G on a central simple algebra A with center K.

Note that here we make no assumption on the degree of A or on the associated stabilizer. However, the proof will show that the associated stabilizer can always be taken to be the reductive group S_{τ}, where τ = [(1, e_{1}), …, (1, e_{r})] for suitable positive integers e_{1}, …, e_{r}.

Proof. Every semisimple algebra is a form of A = M_{e_{1}} × ⋯ × M_{e_{r}} for some positive integers e_{1}, …, e_{r}. Now set n = e_{1} + ⋯ + e_{r}, d_{1} = ⋯ = d_{r} = 1 and τ = [(1, e_{1}), …, (1, e_{r})]. Then M_{n}^{S_{τ}} ≃ A and the conditions of part (a) of Corollary 10.3 are satisfied for this choice of τ. Hence, every form of A appears as a fixed algebra (with associated stabilizer S_{τ}). □

References

[FD] B. Farb, R. K. Dennis, Noncommutative Algebra, Springer-Verlag, New Your, 1993.

[KMRT] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998.

[LBP] L. Le Bruyn, C. Procesi, Étale local structure of matrix invariants and concomitants, in Algebraic groups Utrecht 1986, Lecture Notes in Math., 1271 (1987), 143–175.

[LPR] N. Lemire, V. L. Popov, Z. Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), no. 4, 921–967.

[P] V. L. Popov, Sections in invariant theory, The Sophus Lie Memorial Conference (Oslo, 1992), Scand. Univ. Press, Oslo, 1994, pp. 315–361.

[PV] V. L. Popov and E. B. Vinberg, Invariant Theory, Algebraic Geometry IV, Encyclopedia of Mathematical Sciences 55, Springer, 1994, 123–284.

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

[RV_{1}] Z. Reichstein and N. Vonessen, Stable affine models for algebraic group actions, J. Lie Theory 14 (2004), no. 2, 563-568.

[RV_{2}] Z. Reichstein and N. Vonessen, Polynomial identity rings as rings of functions,

J. Algebra 310 (2007), 624-647.

[RV_{3}] Z. Reichstein and N. Vonessen, Group actions on central simple algebras: a

geometric approach, J. Algebra 304 (2006), 1160–1192.

[RV_{4}] Z. Reichstein and N. Vonessen, Group actions and invariants in algebras of

generic matrices, Adv. Appl. Math. 37 (2006), 481–500.

[Ri] R. W. Richardson Jr., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35–73.

[Ro_{1}] M. Rosenlicht, Some basic theorems on algebraic groups, American Journal of

Math., 78 (1956), 401–443.

[Ro_{2}] M. Rosenlicht, A remark on quotient spaces, Anais da Academia Brasileira de

Ciências 35 (1963), 487–489.

[S_{1}] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997. [S_{2}] J.-P. Serre, Local fields, Springer-Verlag, New York, 1979.

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

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

Department of Mathematical sciences, University of Montana, Missoula, MT 59812-0864, USA

   E-mail address: Nikolaus.Vonessen@umontana.edu
   URL: www.math.umt.edu/vonessen