On polynomial automorphisms of affine spaces - Turpion

c 2001 RAS(DoM) and LMS

Izvestiya: Mathematics 65:3 569–587 Izvestiya RAN : Ser. Mat. 65:3 153–174

DOI 10.1070/IM2001v065n03ABEH000340

On polynomial automorphisms of affine spaces

V. L. Popov Abstract. In the first part of this paper we prove some general results on the linearizability of algebraic group actions on An . As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on An with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group G on the n-dimensional affine space An over an algebraically closed field k of characteristic zero is linearizable in each of the following cases: 1) n = 3; 2) n = 4 and G is not a one- or two-dimensional torus. In particular, this means that GL3 (k) is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over k.

§ 1. Introduction 1.1. Throughout the paper we denote by k an algebraically closed field of characteristic zero. Unless otherwise specified, all algebraic varieties and morphisms are defined over k, algebraic varieties are identified with the spaces of their k-points, and all actions are algebraic. 1.2. We consider algebraic actions of reductive algebraic groups G on the n-dimensional affine space An . If G is a connected group which is not a torus and n is large enough, then we know that there is a non-linearizable action of G on An . (See [18] and the literature cited there for the proofs of assertions that are stated here without references.) But if n is small, then there are some positive results, and the situation becomes more delicate. For example, by a classical result, actions of reductive algebraic groups on An are linearizable if n 2. As proved in [19], actions of connected semisimple algebraic groups on An are linearizable if n = 3. In [24] this was shown to be true for n = 4 as well. (The paper [24] is based on the ideas and methods of a preprint version of [19] that appeared before [24].) After almost fifteen years of effort, it was recently proved that actions of the one-dimensional torus on A3 are linearizable. (Separate parts of this proof are given by the results of [12], [15], [14], [9], [16] and [17]. See [10] for historical details.) This shows that dimensions n = 3, 4 (along with n = 1, 2) are exceptional as far as the linearizability of actions is concerned. This work was done with the support of CRDF (grant no. RM1–206). AMS 2000 Mathematics Subject Classification. 14L17, 14L30.

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1.3. In this paper we obtain two groups of results. The first part of the paper contains some general results on linearizable actions. We refer the reader to § 2 for their statements. As an application, we get a method of construction and concrete examples of non-linearizable actions of an infinite nonreductive insoluble algebraic group on An with a fixed point (see § 2.10). In the second part of the paper we apply the results of the first part to analyze the structure of connected algebraic subgroups of Aut A3 and Aut A4 . We get the following results. Theorem 1. Every algebraic action of a connected reductive algebraic group on A3 is linearizable. Theorem 2. Every effective algebraic action on A4 of a connected reductive algebraic group other than a one- or two-dimensional torus is linearizable. 1.4. The proof of Theorem 1 makes essential use of the fact that actions of onedimensional tori on A3 are linearizable. The results of the first part quickly lead us to this result and to the linearizability of actions of connected semisimple groups on A3 . On the other hand, the proof of Theorem 2 does not use the linearizability of actions of one-dimensional tori on A3 . 1.5. Theorems 1 and 2 may obviously be restated as follows. Theorem 1 . Every connected reductive algebraic subgroup of Aut A3 (or, equivalently, of the automorphism group of the algebra of polynomials in three variables over the field k) is conjugate to a subgroup of GL3 (k). Theorem 2 . Every connected reductive algebraic subgroup of Aut A4 (or, equivalently, of the automorphism group of the algebra of polynomials in four variables over the field k) other than a one- or two-dimensional torus, is conjugate to a subgroup of GL4 (k). 1.6. One can show (see the corollary of Theorem 4 below) that GLn (k) is a maximal reductive subgroup of the infinite-dimensional (for n 2) algebraic (see [32], [11]) group Aut An . For a finite-dimensional algebraic group H, every maximal reductive subgroup R is a Levi subgroup (that is, H is the semidirect product of R and the unipotent radical of H). Theorems 1 and 1 are obviously equivalent to the following assertion, which shows that the infinite-dimensional algebraic group Aut A3 (as well as Aut A2 ) has a fundamental property of finite-dimensional algebraic groups: its Levi subgroups are conjugate. Theorem 3. The group Aut A3 contains exactly one (up to conjugacy) maximal connected reductive algebraic subgroup, namely, GL3 (k). 1.7. The proof of linearizability of actions of one-dimensional tori on A3 is quite complicated. Therefore, if Theorems 2, 2 are valid without the assumption that G is not a one- or two-dimensional torus (and hence the analogue of Theorem 3 holds for Aut A4 ), then the proofs are expected to be difficult. We note that there is a family of non-equivalent non-linearizable algebraic actions of the orthogonal group O2 (k) (that is, the non-commutative extension of the one-dimensional torus by the group of order two) on A4 (see [22]). Hence the actions of one-dimensional tori on the space A4 lie “on the boundary” of the domain where the linearizability of


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actions holds. The case of actions of two-dimensional tori on A4 seems to be more accessible. 1.8. Since there are non-linearizable actions of disconnected reductive algebraic groups on the space A4 , we cannot omit the connectedness assumption in Theorems 2 and 2 . As far as I know, the following question remains open: do such examples (in particular, non-linearizable actions of finite groups) exist for n = 3? Another open question: do the cases n 4 exhaust all exceptional dimensions for the linearizability of actions of connected reductive algebraic groups other than tori? The smallest dimension known to me for which non-linearizable actions of such groups exist is n = 7. 1.9. We use [29] as the main reference for the standard notation and basic facts of the theory of algebraic actions. The only exception is that we denote the categorical quotient for an action of a reductive algebraic group G on an affine algebraic variety X by πX,G : X −→ X//G, which is now generally accepted. AutH Y is the group of H-equivariant automorphisms of an algebraic variety Y acted on by an algebraic group H. Vd is the simple algebraic (d+1)-dimensional SL2 (k)-module (which is unique up to isomorphism), d ∈ N := {0, 1, 2, . . .}. 1.10. I am grateful to P. Russell, whose question on the validity of Theorem 1 was my starting point and led to the appearance of the preprints [27], [28], and to H. Bass for useful correspondence. I am grateful for the hospitality of the Institute of Mathematics of Innsbruck University, where this paper was completed. § 2. Some general results on actions of algebraic groups 2.1. In this section we obtain a number of general results on algebraic actions. They will be used in the second part of the paper to prove Theorems 1 and 2. 2.2. The following assertions (Theorems 4–6) are well known. Theorem 4 [19]. Let G be a reductive algebraic group acting on the space An . If the categorical quotient An //G is a point (that is, k[An ]G = k), then the action is linearizable. Corollary 1. The group GLn (k) is a maximal reductive subgroup of Aut An . Proof. Since GLn (k) has an open orbit in An , this must also hold for every reductive subgroup G of Aut An containing GLn (k). Therefore k[An ]G = k. Hence, by Theorem 4, the action of G on An is linearizable, that is, G is conjugate in Aut An to a subgroup of GLn (k). Since GLn (k) ⊆ G, it follows that G = GLn (k). 2.3. We recall [2] that an action of a reductive algebraic group on an affine algebraic variety is said to be fixed pointed if fixed points are the only closed orbits. Theorem 5 [2]. Consider an action of a reductive algebraic group G on the space An . Let this action be fixed pointed. Then it is linearizable if and only if the variety (An )G of fixed points is isomorphic to the space As for some s. If s 2, then this condition holds automatically.

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2.4. The following assertion is a standard fact in the theory of homogeneous spaces (see, for example, [8], Theorem 1.2). Its proof is an easy exercise. Theorem 6. Let H be an algebraic group and let Q be a closed subgroup of H. Consider the action of H on the homogeneous space H/Q by left translation. Then the group AutH H/Q coincides with the image of the group NH (Q)/Q acting on H/Q by right translation. This action is effective, and so the groups AutH H/Q and NH (Q)/Q are isomorphic. 2.5. The results of §§2.5, 2.6 concern the relation between the linearity of an equivariant automorphism and some properties of the decomposition into isotype components. Let G be a reductive algebraic group and let V be a finite-dimensional algebraic G-module. Since each isotype component of the G-module k[V ] is a finitely generated k[V ]G-module, some (equivalently, every) non-zero isotype component is finite-dimensional if and only if k[V ]G = k. Since the action of G on V is linear, V ∗ is a G-invariant subspace of the algebra k[V ]. Theorem 7. Let G be a reductive algebraic group, V a finite-dimensional algebraic G-module with k[V ]G = k, and σ ∈ AutG V . Then σ ∈ GL(V ) if (i) V ∗ is a sum of isotype components of the G-module k[V ]. Suppose that G is semisimple. Let L1 , . . . , Ls be all mutually non-isomorphic terms in the decomposition of the G-module V into a sum of simple G-modules, mi the multiplicity of Li of V , and H the stabilizer of a generic point of V . Then (i) is equivalent to (ii) mi = dim LH i for all i. Proof. Since σ∗ is an isomorphism of the G-module k[V ], each isotype component of k[V ] is invariant under σ∗ . Hence (i) implies that σ∗ (V ∗ ) = V ∗ , that is, σ ∈ GL(V ). We now suppose that G is semisimple. Since k[V ]G = k, the variety V contains an open dense G-orbit O, and V \ O is of codimension at least 2 in V (see [29], Theorem 3.3, Corollary of Theorem 2.3, and Theorem 3.1). Therefore the inclusion ι : O → V induces a G-isomorphism of algebras ι∗ : k[V ] −→ k[O]. Then the second assertion of Theorem 7 follows from Frobenius reciprocity, which says that the multiplicity of L∗i in k[O] equals dim LH i . Corollary 2. In the notation of Theorem 7, suppose that G is semisimple and the module V is simple. Then the following properties are equivalent: (i) every G-equivariant automorphism of the algebraic variety V is a scalar multiple of the identity, (ii) AutG V ⊂ GL(V ), (iii) dim V H = 1, (iv) dim NG (H)/H = 1. Proof. Fix a point v ∈ OH . Since the orbit O is open in V , we have dim V H = dim OH . Since Gv = H, it follows that NG (H)·v = OH , and the varieties NG (H)·v and NG (H)/H are equivariantly isomorphic. This yields (iii)⇔(iv). (i)⇔(ii) follows from Schur’s lemma. (iii)⇒(ii) follows from Theorem 7.


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(i)⇒(iv). Since ι∗ is an isomorphism, restriction to the orbit O defines an isomorphism between the groups AutG V and AutG O. Hence, by Theorem 6, the orbit NG (H) · v coincides with the one-dimensional orbit of v under the group of scalar multiples of the identity. Remark 1. Consider the general case when G is semisimple and k[V ]G = k but the G-module V is not necessarily simple. Then the previous arguments and Theorem 6 show that (a) restriction to the orbit O defines an isomorphism between the group AutG V and the group AutG O (hence also the group NG (H)/H), (b) dim NG (H)/H = dim V H . Assertions (a), (b) were proved by a similar method in [20], which appeared at the same time as the preprint [28] of the present paper. Remark 2. For connected semisimple groups G, there is an explicit classification of all finite-dimensional algebraic G-modules V such that k[V ]G = k and either the group G or the module V is simple. The G-stabilizer H of a generic point in V has been identified in these cases (see [31], [13], [6], [7]). In more detail, since the action of G on the space V is unstable, the group H is infinite [25]. The papers [6], [7] give a classification of all pairs (G, V ) such that the G-stabilizer G∗ of a generic point in V is infinite and either the group G or the module V is simple. They also describe the Lie algebra Lie(G∗ ) in these cases (and the Lie algebra Lie(NG (G∗ )/G∗) and the subspace V G∗ in some of them). In this classification, the cases with k[V ]G = k are distinguished by the condition that dim V = dim G − dim Lie(G∗ ). This together with the corollary of Theorem 7 reduces the search for examples of simple Gmodules V with non-linear G-equivariant automorphisms to a computing problem. The paper [20] presents an example which has the minimal dimension in the class of simple modules. 2.6. Let V be a finite-dimensional algebraic module of a reductive algebraic group G. We recall that V is called spherical if the space V contains an open dense orbit of a Borel subgroup of G. This is equivalent to saying that the G-module k[V ] has no multiplicities, that is, its non-zero isotype components are simple G-modules. Theorem 8. Let G be a reductive algebraic group and let V be a spherical Gmodule. Then AutG V ⊂ GL(V ). Proof. Since each non-zero isotype component of the G-module k[V ] is a simple G-module, condition (i) of Theorem 7 holds. This yields the desired assertion. Schur’s lemma yields the following corollary. Corollary 3. With the notation and hypotheses of Theorem 8, if the G-module V is simple, then the group AutG V consists of scalar multiples of the identity. 2.7. We denote by A× the group of invertible elements of a ring A. Theorem 9. Let G be an algebraic group, V a finite-dimensional algebraic Gmodule, and X an irreducible algebraic variety. Consider the action of G on the variety Y := V × X via the first factor. Let H be an algebraic subgroup of AutG Y . Suppose that (i) the closure of any G-orbit in the space V contains the zero vector 0,

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(ii) AutG V coincides with the group of scalar multiples of the identity, (iii) k[Y ]× ⊂ k[Y ]H . Then we can find an algebraic character χ : H −→ k × and an action of H on X such that the natural action of G × H on Y is given by (g, h) · (v, x) = (χ(h)g · v, h · x),

(g, h) ∈ G × H,

(v, u) ∈ Y.

Proof. It follows from (i) that V G = {0}, whence Y G = {0}×X. Therefore {0}×X is an H-invariant subvariety of Y . Let πV : Y −→ V,

πX : Y −→ X

be the natural projections. The restriction of the morphism πX to the subvariety {0} × X is an isomorphism of the varieties {0} × X and X. Hence there is a unique action of H on X such that this isomorphism is H-equivariant. −1 Consider a point x ∈ X. It follows from (i) that a point y ∈ Y lies in πX (x) if and only if the point (0, x) lies in the closure of the orbit G · y. Since the action of H on Y commutes with that of G, it follows that πX is an H-equivariant morphism. ∗ This establishes two properties. Firstly, since πX is an H-equivariant embedding of the algebra k[X] in the algebra k[Y ], condition (iii) shows that k[X]× ⊂ k[X]H .

(1)

Secondly, for each element h ∈ H and each point x ∈ X, the map V −→ V , v → πV (h · (v, x)), is a G-equivariant automorphism of V . Therefore, by (ii), there is an element µh,x ∈ k × (depending on h and x) such that the action of H on Y is given by h · (v, x) = (µh,x v, h · x),

h ∈ H,

v ∈ V,

x ∈ X.

(2)

The definition of the action implies that µh1 h2 ,x = µh1 ,h2 ·x µh2 ,x ,

µe,x = 1,

h1 , h2 ∈ H,

x ∈ X.

(3)

For any fixed element h ∈ H, the function X −→ k × , x → µh,x , is an element of the group k[X]× . Hence (1) implies that µh1 ,h2 ·x = µh1 ,x ,

h1 , h2 ∈ H,

x ∈ X.

(4)

We now obtain from (3) and (4) that µh1 h2 ,x = µh1 ,xµh2 ,x ,

h1 , h2 ∈ H,

x ∈ X.

(5)

Thus, (5) shows that the map χx : H −→ k × ,

h → µh,x ,

(6)


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is an algebraic character of H for each x ∈ X. According to (2), it remains to establish that all these characters coincide, that is, χx1 = χx2 for all x1 , x2 ∈ X. To prove this, we consider the morphism τ : H × X −→ k × ,

τ ((h, x)) = µh,x ,

h ∈ H,

x ∈ X.

(7)

Let H 0 be the connected component of the identity in H. Since the variety H 0 × X is irreducible, and the restriction of the morphism τ to H 0 × X is an element of the group k[H 0 × X]× , it follows from [30] that we can find an algebraic character ε : H 0 −→ k × and a function ϑ ∈ k[X]× such that τ ((h, x)) = ε(h)ϑ(x),

h ∈ H 0,

x ∈ X.

(8)

Substituting h = e in (8) and using (7), (3) and the formula ε(e) = 1, we obtain that ϑ ≡ 1. This means that the restrictions of all the characters χx to H 0 are equal to ε and therefore coincide with each other. Hence, if we fix a point x0 ∈ X × and let x ∈ X be any point, then the image of the character χx χ−1 x0 : H −→ k lies in the group 0 F := {a ∈ k × | aord(H/H ) = 1}. Hence the image of the map τ : H × X −→ k × ,

τ ((h, x)) = τ ((h, x))χx0 (h)−1 ,

h ∈ H,

x ∈ X (9)

also lies in F . Since the variety X is irreducible and the group F is finite, it follows that τ ((h, x1 )) = τ ((h, x2 )) for all h ∈ H, x1 , x2 ∈ X. (10) We now see from (9), (7) and (6) that τ ((h, x0 )) = 1 for all h ∈ H. This together with (10) shows that τ ≡ 1. Hence we see from (9), (7) and (6) that χx = χx0 for any point x ∈ X, as required. Remark 3. If Z is an irreducible algebraic variety, then k[Z]×/k × is a free Abelian group of finite rank. Therefore hypothesis (iii) of Theorem 9 holds automatically if the group H is connected and has no non-trivial characters. Corollary 4. We preserve the notation and hypotheses of Theorem 9. Suppose that X = An . If the action of H on X is linearizable, then so is the natural action of G × H on Y . Corollary 5. We preserve the notation and hypotheses of Theorem 9. Suppose that X = An and the group H is reductive. Then the natural action of G × H on Y is linearizable in each of the following cases: (i) n 2, (ii) n = 3 and the group H is connected, (iii) n = 4 and the group H is connected and different from a one- or twodimensional torus. Proof. In case (i), this follows from Corollary 4 and § 1.2. In cases (ii), (iii) it follows from Theorems 1, 2 respectively that are proved below. (Their proofs use case (i) only.)

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2.8. The following theorem concerns automorphisms of simple modules. Theorem 10. Let G be a reductive algebraic group, V a simple algebraic G-module, and H an infinite reductive algebraic subgroup of the group AutG V . Then the natural action of G × H on V is linearizable and the group H is a one-dimensional torus. Proof. We can assume that dim V > 1. Then V G = {0} and, therefore, 0 ∈ V H . The slice-module of the G-variety V at the fixed point 0 is the G-module V itself, and if we regard V as an H-variety, then this slice-module is the vector space V endowed with some linear action of the group H which commutes with the action of G. Since V is a simple G-module, Schur’s lemma implies that H acts on this last slice-module by scalar multiples of the identity. Since the group H is infinite and its action on V is effective, it follows from this and Luna’s étale slice theorem [21] that the group H is a one-dimensional torus and that the categorical quotient for the action of H on V in the theorem is a point. Therefore the categorical quotient for the natural action of the group G×H on V is also a point. The desired assertion now follows from Theorem 4. 2.9. If a reductive algebraic group G acts on an affine algebraic variety X and k[X]G = k, then AutG(X) is a finite-dimensional algebraic group. Indeed, the algebra k[X] is generated by a finite sum of its isotype components, and this sum is finite-dimensional because k[X]G = k. As an application, we get the following linearizability criterion for the action of the group G × AutG (V ). Theorem 11. Let G be a reductive algebraic group and let V be a simple algebraic G-module with k[V ]G = k. Then the following properties are equivalent: (i) the action of G × AutG (V ) on V is linearizable, (ii) the group AutG (V ) is a one-dimensional torus. Suppose that G is semisimple and its action on V is effective. Let H be the G-stabilizer of a generic point in V . Then the above properties are equivalent to each of the following assertions: (iii) the action of G × AutG (V ) on V is linear, (iv) dim V H = 1, (v) dim NG (H)/H = 1, (vi) the group NG (H)/H is a one-dimensional torus. Proof. (i)⇒(ii). Since the centralizer of any reductive subgroup in the group GL(V ) is reductive (see, for example, [23], Proposition 3.6), it follows from (i) that AutG (V ) is a reductive group. This group is infinite since it contains all scalar multiples of the identity. Hence (ii) follows from Theorem 10. (ii)⇒(i) follows from Theorem 10. (ii)⇔(vi) follows from Remark 1,(a). (iii)⇔(iv)⇔(v) follows from Corollary 2. The implications (vi)⇒(v) and (iii)⇒(i) are obvious. 2.10. Theorem 11 (see also Remark 2) provides non-linearizable actions of infinite non-reductive non-empty set of fixed points. Namely, let V be semisimple algebraic group G such that k[V ]G

a way to construct examples of insoluble algebraic groups with a simple module of a connected = k and dim NG (H)/H > 1,


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where H is the stabilizer of a generic point in V . Then the action of G × NG (H)/H on V is non-linearizable. (By Theorem 10, the group G×NG (H)/H is non-reductive in this case.) An example is given by the action of G = SL3 × SL5 × SL13 ×(k ∗ k+ ) on A195 considered in [20], § 3. § 3. Proof of Theorem 1 3.1. Let G be a connected reductive algebraic group acting on the space A3 . If G is semisimple, the assertion of Theorem 1 is proved in [19]. If G is a torus, we may assume that dim G = 1, 2, 3. In these cases, the assertion is proved in [12], [4], [3] respectively. 3.2. Thus it remains to prove the following assertion. Suppose that Z is a nonidentity algebraic torus, S is a simply connected semisimple algebraic group, and the group G := Z × S acts on A3 with a finite non-trivial kernel. Then the action is linearizable. 3.3. Every action of a connected semisimple algebraic group of rank 2 on A3 has an open dense orbit (see [26], and also [19], Remark 5.2). Thus, if S = SL2 (k), then G has an open dense orbit in A3 . Then the action of G on A3 is linearizable by Theorem 4. 3.4. We can thus assume that S = SL2 (k). Since every action of S on A3 is linearizable, the S-variety A3 coincides (up to isomorphism) with either V2 or V1 ⊕ V0 (see § 1.9). If A3 is V2 , then the assertion of Theorem 1 follows from Theorem 10. If A3 is V1 ⊕ V0 , then it follows from Theorem 8 and Corollary 5 (case (i)) because V1 is a spherical SL2 (k)-module. § 4. Proof of Theorem 2: first step 4.1. Suppose that G is a connected reductive algebraic group acting on the space A4 and G is different from a one- or two-dimensional torus. If G is semisimple, the assertion of Theorem 2 is proved in [24]. If G is a torus, we may assume that dim G 4 and, therefore, dim G = 3, 4. The assertion of Theorem 2 in these cases is proved in [4] and [3] respectively. 4.2. Thus, it remains to prove the following assertion. Suppose that Z is a nonidentity algebraic torus, S is a simply connected semisimple algebraic group, and the group G := Z × S acts on A4 with a finite non-trivial kernel. Then the action is linearizable. It may (and will) be assumed that the torus Z acts effectively on the space A4 . 4.3. Since the restriction of the action to the group S is linearizable, we may fix an appropriate linear structure on the space A4 such that A4 becomes an S-module V . Since S acts on V with a finite non-trivial kernel, it follows from [26] that the pair (S, V ) is contained (up to duality) in the following list. (Here we preserve the notation of § 3 and denote by R(λ) and ωi respectively the simple S-module with highest weight λ and the ith fundamental weight in the Bourbaki notation.) (SL2 (k), V3 ),

(SL2 (k), V2 ⊕ V0 ),

(SL3 (k), R(ω1 ) ⊕ R(0)), (SL2 (k) × SL2 (k), V1 ⊗ V1 ),

(SL2 (k), V1 ⊕ V1 ), (SL4 (k), R(ω1 )),

(SL2 (k), V1 ⊕ V0 ⊕ V0 ), (Sp4 (k), R(ω1 )),

(SL2 (k) × SL2 (k), (V1 ⊗ V0 ) ⊕ (V0 ⊗ V1 )).

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We consider all these possibilities separately. 4.4. If (S, V ) is one of the pairs SL2 (k) × SL2 (k), (V1 ⊗ V0 ) ⊕ (V0 ⊗ V1 ) , SL4 (k), R(ω1 ) or Sp4 (k), R(ω1 ) , then V contains an open dense S-orbit. Hence the desired assertion follows from Theorem 4. 4.5. If (S, V ) = (SL2 (k), V1 ⊕ V0 ⊕ V0 ) or (SL3 (k), R(ω1 ) ⊕ R(0)), then the desired assertion follows from Theorem 8 and Corollary 5(i) because V1 is a spherical SL2 (k)-module and R(ω1 ) is a spherical SL3 (k)-module. 4.6. If (S, V ) = (SL2 (k), V3 ) or (SL2 (k)×SL2 (k), V1 ⊗V1 ), then the desired assertion follows from Theorem 10. 4.7. The remaining cases (S, V ) = (SL2 (k), V2 ⊕ V0 ), (SL2 (k), V1 ⊕ V1 ) are treated in the following two sections. § 5. Proof of Theorem 2: second step 5.1. We consider the first remaining case (S, V ) = (SL2 (k), V2 ⊕ V0 ). 5.2. Since V2 //S = A1 , we have V //S = (V2 //S) × V0 = A2 .

(11)

By (11), the variety V //G = (V //S)//Z is at most two-dimensional. 5.3. If it is zero-dimensional (hence a point), then the assertion of Theorem 2 follows from Theorem 4. 5.4. If it is two-dimensional, then the generic fibre of the morphism πV //S,Z is zero-dimensional. By the connectedness, the torus Z acts trivially on this fibre. Hence it acts trivially on the variety V //S. In particular, Z acts trivially on the line πV //S,Z ({0} × V0 ). This line is an isomorphic image of the line {0} × V0 under the Z-equivariant morphism πV //S,Z . Since {0}×V0 = V G , the line {0}×V0 is invariant under Z. Therefore each of its points is fixed by Z. Hence we have {0} × V0 = V G in this case. Let Tx be the slice-module of the G-variety V at the point x ∈ {0} × V0 . Then Tx = V2 ⊕ V0 as an S-module. Since the S-submodules Tx,2 = V2 × {0} and Tx,0 = {0} × V0 in Tx are simple and non-isomorphic, they are Z-invariant and, by Schur’s lemma, Z acts on each of them by scalar multiples of the identity. Since Tx,0 is the tangent space of the variety V G at the point x, the action of Z on Tx,0 is trivial. On the other hand, the action of Z on Tx,2 is non-trivial: otherwise the action of Z on Tx would be trivial, whence (by Luna’s étale slice theorem [21]) the action of Z on V would also be trivial, which is not the case (see § 4.2). These arguments show that the action of G on Tx is fixed pointed. Then Luna’s étale slice theorem [21] yields that the action of G on V is also fixed pointed. Hence the assertion of Theorem 2 follows from Theorem 5 in this case.


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5.5. Thus, it remains to consider the case when the variety V //G is one-dimensional. Since the morphism πV,G is surjective and every irreducible unirational normal affine curve with no non-constant regular functions is A1 , we have (V //S)//Z = A1 .

(12)

To complete the proof of Theorem 2 in this case, we shall find in k[V ] a fourdimensional G-invariant linear subspace which generates k[V ] as a k-algebra. Since dim V = 4, the existence of such a subspace is equivalent to the linearizability of the action of G on V . 5.6. Proposition 1. The algebra k[V ]S contains non-constant Z-semi-invariants f1 and f2 such that (i) k[V ]S = k[f1 , f2 ], (ii) k[V ]G = k[f], where f = f1a1 f2a2 for some numbers a1 , a2 ∈ N, (iii) each non-zero isotype component of the Z-module k[V ]S has the form b1 b2 f1 f2 k[V ]G for some numbers b1 , b2 ∈ N. Proof. The existence of Z-semi-invariants f1 , f2 that satisfy (i) follows from (11) and the fact that actions of algebraic tori on the plane A2 are linearizable. Let χi ∈ Hom(Z, k ×) be the weight of the Z-semi-invariant fi . Consider the following subgroup of Z2 : D0 := {(d1 , d2) ∈ Z2 | χd11 χd22 = id}. The linear hull of the set {f1d1 f2d2 | (d1 , d2) ∈ D0 ∩N2 } is the algebra (k[V ]S )Z = k[V ]G. Since the group Z is connected, D0 is a saturated subgroup of Z2 . By (12), rk D0 = 1 and D0 ∩N2 = {(0, 0)}. (13) Hence the subgroup D0 is generated by some element (a1 , a2 ) ∈ N2 . This proves (ii). Let I be a non-zero isotype component of the Z-module k[V ]S and let χ ∈ Hom(Z, k × ) be the weight of this component. By (i), I is the linear hull of the set {f1d1 f2d2 | (d1 , d2 ) ∈ Dχ ∩N2 }, where Dχ := {(d1 , d2 ) ∈ Z2 | χd11 χd22 = χ}. Since Dχ is a coset of D0 in Z2 , it follows form (13) and (ii) that I has the form indicated in (iii), where (b1 , b2 ) is an element of Dχ ∩(N2 \ {(0, 0)}) with minimal coordinate-sum. 5.7. Consider the following linear subspaces of V ∗ : L := π2∗ (V2∗ ),

M := π0∗ (V0∗ ),

(14)

where π2 : V = V2 ⊕V0 → V2 and π0 : V = V2 ⊕V0 → V0 are the natural projections. The subspaces L, M are S-submodules of the S-module V ∗ , and V ∗ = L ⊕ M,

(15)

M ⊂ k[V ]S .

(16)

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V. L. Popov

Proposition 2. The subspace L is G-invariant and the torus Z acts on L by scalar multiples of the identity. Proof. Fix a Borel subgroup B in S. Let U be its unipotent radical and let ε be the highest weight of the S-module V1 with respect to B. Then εd is the highest weight of the module Vd with respect to B. For any algebraic S-module M and any number d ∈ N we denote by Md the isotype component of type Vd in M . S Let d ∈ N. Then k[V2 ]U d is a finitely generated k[V2 ] -module without torsion. 1 Since V2 //S = A , this module is free and has a finite rank md , and the morphism πV2,S is flat. The last property implies that for every scheme fibre πV−1 (x), x ∈ 2 ,S V2 //S, we have (x)]d dim k[πV−1 2 ,S md = (17) dim Vd (see [29]). Let πV−1 (x) in (17) be a generic fibre. This fibre is reduced and isomor2 ,S phic to the variety S/T , where T is a maximal torus of the group S. (In the case under study, all fibres are actually reduced: see the first paragraph of § 5.9.) Hence the right-hand side of (17) equals dim VdT by Frobenius reciprocity (see [29]). It follows that 1 if d is even, md = (18) 0 if d is odd. We consider the subalgebra of k[V ] generated by the subspace L. By (14) and (15), this subalgebra is invariant under S and S-isomorphic to the algebra k[V2 ]. S By (16), this shows that the space k[V ]U d is a free k[V ] -module whose rank md is given by (18). Thus, for every even positive integer d there is a non-zero element hd such that S k[V ]U (19) d = hd k[V ] . Up to a scalar multiple, hd is uniquely determined as an element of the algebra U k[V ]U d with no non-constant S-invariant divisors. Indeed, k[V ]d is the subspace of d all B-semi-invariants of weight ε in the algebra k[V ]. Hence an equation hd = pq with p ∈ k[V ]S implies that q ∈ k[V ]U d . By (19), it follows that q = hd r, where r ∈ k[V ]S . Hence, rp = 1 and thus p is a constant. Our assertion now follows from (19). Since the torus Z acts by S-equivariant automorphisms of the algebra k[V ], this characterization of the elements hd shows that each hd is a Z-weight vector. We now let d = 2. It follows from (15) that every non-zero element of the space L is a simple element of the factorial ring k[V ]. In particular, it has no non-constant S-invariant divisors. Since the S-modules L and V2 are isomorphic and LU = 0, the above characterization of the elements hd shows that h2 ∈ L. To complete the proof, it remains to note that the element h2 is a Z-weight vector, and the linear hull of its S-orbit is the space L. 5.8. We fix a basis t1 , t2 , t3 of L. It follows from (15) and (16) that k[V ] = k[V ]S [t1 , t2 , t3 ]. Hence, by Proposition 1, k[V ] = k[t1 , t2 , t3 , f1 , f2 ].

(20)


581

We claim that (20) remains valid if we omit either f1 or f2 from its righthand side. Since the linear hull of the set t1 , t2 , t3 , fi is four-dimensional and (by Propositions 1 and 2) G-invariant, this will complete the proof of Theorem 2 in the case under study (see § 5.5). 5.9. Let h be a homogeneous generator (of degree 2) of the algebra k[V2 ]S . Since the group S is connected and semisimple, simple divisors of any element of k[V2 ]S in the factorial ring k[V2 ] are contained in k[V2 ]S . It follows that the polynomial h + α is a simple element of the ring k[V2 ] for any constant α ∈ k. Using (14), (15), we can identify the algebra k[V2 ] with the subalgebra of k[V ] generated by the space L. Under this identification, the polynomial h + α is again a simple element of the factorial ring k[V ] for any constant α ∈ k. The definition of the elements t1 , t2 , t3 implies that h ∈ k[t1 , t2 , t3 ].

(21)

Since the element h is homogeneous, Proposition 2 shows that h is contained in some isotype component of the Z-module k[V ]S . Hence, by Proposition 1, h = f1b1 f2b2 (αn (f1a1 f2a2 )n + αn−1(f1a1 f2a2 )n−1 + · · · + α0 )

(22)

for some n ∈ N, (b1 , b2 ), (a1 , a2 ) ∈ N2 , αi ∈ k, αn = 0. Since the element h + α is simple for any constant α ∈ k, and f1 , f2 are nonconstant elements of the algebra k[V ], it follows from (22) that one of the following cases holds: (i) b1 = 1, b2 = 0, n = 0, (ii) b1 = 0, b2 = 1, n = 0, (iii) b1 = 0, b2 = 0, n = 1, a1 = 1, a2 = 0, (iv) b1 = 0, b2 = 0, n = 1, a1 = 0, a2 = 1. By (21) and (22) we get the inclusion f1 ∈ k[t1 , t2 , t3 ] in cases (i) and (iii) and the inclusion f2 ∈ k[t1 , t2 , t3 ] in cases (ii) and (iv). This completes the proof of Theorem 2 in the case (S, V ) = (SL2 (k), V2 ⊕ V0 ). § 6. Proof of Theorem 2: last step 6.1. We consider the last remaining case, (S, V ) = (SL2 (k), V1 ⊕ V1 ).

(23)

6.2. (23) implies that V //S = A1 . Hence there are only two possibilities: either the variety V //S contains an open dense Z-orbit, or the torus Z acts trivially on V //S. If the first possibility holds, then the quotient V //G = (V //S)//Z is a point, and the assertion of Theorem 2 follows from Theorem 4. It remains to consider the second possibility. 6.3. Thus, we assume that Z acts trivially on the quotient V //S. Then each fibre of the morphism πV,S is invariant under Z and V //G = (V //S)//Z = (V //Z)//S = A1 .

(24)

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6.4. The S-stabilizer of a generic point in V is trivial and the torus S acts transitively on the generic fibre F of the morphism πV,S . Hence there is an isomorphism of S-varieties F SL2 (k). (25) (The group S = SL2 (k) acts on the variety on the right-hand side of (25) by left translation.) Since the group S acts transitively on F and this action commutes with the action of Z, all Z-orbits in F have the same dimension, say d. In particular, they are all closed in V . By § 4.2 we have d 1. Thus a generic Z-orbit in V is closed and has dimension d. Therefore, dim V //Z = 4 − d.

(26)

6.5. Let D be the generic fibre of the morphism πV //Z,S . It follows from (24) and (26) that dim D = 3 − d. (27) By general properties of categorical functors, the variety D contains a unique closed S-orbit. This orbit is affine and, by (27), its dimension does not exceed 2. Since the group SL2 has no one-dimensional affine homogeneous spaces, it follows that D is either a point or a surface. 6.6. If the first case holds, then d = 3. Since dim F = 3, this means that Z acts transitively on the fibre F . Since Z is a torus, it follows that the algebraic variety F is isomorphic to A3∗ := A∗ ×A∗ ×A∗ , where A∗ = A1 \{0}. But this contradicts (25) since k[A3∗]× = k × and k[SL2 (k)]× = k × (see [30]). Hence this case is impossible. 6.7. Thus, d = 1.

(28)

Since the stabilizers of generic points are conjugate for any action of a reductive algebraic group on a smooth irreducible affine algebraic variety, and since the action of the torus Z on the space V is effective (see § 4.2), we see that the Z-stabilizer of a generic point in V is trivial. This together with (28) implies that dim Z = 1.

(29)

6.8. The S-variety V may (and will) be identified with the space Mat2×2 (k) of all 2 × 2 matrices with entries in the field k, with S = SL2 (k) acting on this space by left multiplication. Let xpq , 1 p, q 2, be the standard pqth coordinate function on the space V , that is, a a12 xpq (v) = apq , v = [aij ] := 11 . (30) a21 a22 Then k[V ]S = k[det],

det = x11 x22 − x12 x21 .

(31)

The fibres of πV,S are the hypersurfaces Fc := {v ∈ V | det(v) = c}, c ∈ k. They are G-invariant. If c = 0, then the action of S on the fibre Fc is transitive and the S-stabilizer of a point in this fibre is trivial.


583

We also consider the action of S = SL2 (k) on the space V = Mat2×2 (k) by right multiplication. This action commutes with the initial one. Since det(ab) = det(a) det(b) for all matrices a, b ∈ Mat2×2 (k), each fibre Fc is also invariant under this action of S. 6.9. It follows from this and Theorem 6 that for each element c ∈ k, c = 0, there is a homomorphism λc : Z −→ SL2 (k) such that the action of Z on Fc factors through λc , that is, z · v = vλc (z),

z ∈ Z,

v ∈ Fc ,

(32)

(with the usual matrix multiplication on the the right-hand side). Proposition 3. The homomorphisms λc and their dependence on the parameter c are algebraic, that is, the map λ : Z × A∗ −→ SL2 (k),

(z, c) → λc (z),

(33)

is a morphism. Proof. The open set Ω := V \ F0 is invariant under G. By (32), the map α : Z × Ω −→ Ω,

(z, v) → vλdet(v) (z),

(34)

defining the action of Z on the set Ω is a morphism. Since det(v) = 0 in (34), we have λdet(v) (z) = v−1 α((z, v)), whence the map β : Z × Ω −→ SL2 (k),

(z, v) → λdet(v) (z),

(35)

is a morphism. We now consider the action of S on the variety Z × Ω via the second factor. We have πZ×Ω,S : Z × Ω −→ Z × A∗ , (z, v) → (z, det(v)). (36) It follows from (35) and (36) that the morphism β is constant on each fibre of πZ×Ω,S . Hence, by the defining property of the categorical quotient, there is a morphism γ : Z × A∗ −→ SL2 (k) such that β = γ ◦ πZ×Ω,S . On the other hand, it is clear that β = λ ◦ πZ×Ω,S . This shows that λ = γ. Hence λ is a morphism. 6.10. Let s be the standard coordinate function on the line A1 , that is, s(c) = c, c ∈ A1 . We fix a generator t of the character group of the torus Z. Then k[A1 ] = k[s],

k[A∗ ] = k[s, s−1 ],

k[Z] = k[t, t−1].

(37)

Let y1 , y2 be variables. It follows from (37) and Proposition 3 that there are rational functions fij = fij (y1 , y2 ) ∈ k[y1 , y1−1 , y2 , y2−1 ],

1 i, j 2,

(38)

such that the morphism λ is determined in the notation (30) by λ((z, c)) = [fij (t(z), c)],

(z, c) ∈ Z × A∗ .

(39)

It now follows from (33), (34) and (39) that the morphism α is determined by α((z, v)) = [xij (v)][fij (t(z), det(v))],

(z, v) ∈ Z × Ω.

(40)

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V. L. Popov

6.11. The morphism α is the restriction to Z × Ω of the morphism Z × V −→ V defining the action of Z on V , and Z × Ω is an open subset of the variety Z × V . Therefore (40) implies that all entries of the matrix [xij ][fij (t, det)] ∈ SL2 (k[Z ×Ω]) are regular (not just rational) functions on the variety Z × V , that is, xi1 f1j (t, det) + xi2 f2j (t, det) ∈ k[t, t−1, x11 , x12, x21, x22 ] = k[Z × V ], 1 i, j 2.

(41)

Proposition 4. fij ∈ k[y1 , y1−1 , y2 ], 1 i, j 2. Proof. Assume the opposite: say f1j ∈ k[y1 , y1−1 , y2 ]. Then, by (38), we have f1j (t, det) = detn (p0 + p1 det +p2 det2 + · · · ), pi ∈ k[t, t−1 ], p0 = 0, f2j (t, det) = det (q0 + q1 det +q2 det + · · · ), l

2

−1

qi ∈ k[t, t

],

q0 = 0,

(42) (43)

for some integers n, l. Put m = min{n, l}. It follows from (42) and (43) that there is an element h ∈ k[Z × V ] such that the left-hand side of the inclusion (41) equals f := detm (r + h det),

r = xi1 p0 detn−m + xi2 q0 detl−m .

(44)

The numbers n − m and l − m are non-negative, whence r ∈ k[Z × V ]. Since at least one of these numbers is zero, it follows from (31) and (44) that the restriction of the function r to the set Z × F0 is different from zero. This together with (44) implies that the function f has a pole on the set Z × F0 . This contradicts (41). 6.12. Since the set Z × Ω is open in Z × V , we have thus established that the action of Z on the space V is given by z · v = v[fij (t(z), det(v))],

z ∈ Z,

v ∈ V.

(45)

(By Proposition 4, the right-hand side of (45) is defined at every point (z, v) ∈ Z × V .) 6.13. The map (33) may be regarded as a homomorphism of group schemes Gm −→ SL2 defined over the ring k[A∗ ] = k[s, s−1 ] (see (37)). Since the torus Z acts on the space V with trivial kernel, this homomorphism is injective. By (39) and Proposition 4, it is actually defined over the ring k[A1 ] = k[s]. We now use the following assertion. Lemma 1. Let A be a factorial ring and let ϕ : Gm −→ SL2 be an injective homomorphism of group schemes defined over the ring A. Then the image T of ϕ is conjugate by an element of SL2 (A) to the diagonal torus. Proof. Let x be a generator of the character group of the torus Gm . Then the homomorphism ϕ is given by the matrix r = r(x) ∈ SL2 (A[x, x−1]), that is, ϕ(a) = r(x(a)) for a ∈ Gm . Let K be the field of fractions of the ring A. Then T is a maximal K-splittable torus in SL2 . Therefore [5], T is conjugate by an element of SL2 (K) to the diagonal torus, that is, r(t) = u diag(t, t−1 )u−1 ,

u = [uij ] ∈ SL2 (K).

(46)


585

The equation (46) remains valid if we replace u by ub, where b is any diagonal matrix in the group SL2 (K). Therefore we can assume that and u21

u11

are coprime elements of the ring A.

(47)

Since the right-hand side of the equation (46) equals

u11 u22 t − u12 u21 t−1 u21 u22 t − u21 u22 t−1

u11 u12 t−1 − u11 u12 t , u11 u22 t−1 − u12 u21 t

the condition r ∈ SL2 (A[t, t−1 ]) implies that u11 u22 ,

u12 u21 ,

u11 u12 ,

u21 u22 ∈ A.

(48)

We now easily see from (47) and (48) that u ∈ SL2 (A). This proves the lemma. Remark 4. The group scheme SL2 in the statement of Lemma 1 cannot be replaced by the group scheme SLn , n 3 [1]. 6.14. Lemma 1 implies that there are polynomials uij = uij (y) ∈ k[y], 1 i, j 2, (where y is a variable) such that [uij (s)] ∈ SL2 (k[s]) and [uij (s)][fij (t, s)][uij (s)]−1 = diag(t, t−1 ).

(49)

We define an automorphism σ of the variety V by σ : V −→ V,

v → v[uij (det(v))].

(50)

We have σ−1 (v) = v[uij (det(v))]−1 . Hence it follows from (45) and (50) that for any element z ∈ Z (regarded as an automorphism of V ) we have σ−1 ◦ z ◦ σ : V −→ V, v → v[uij (det(v))][fij (t(z), det(v ))][uij (det(v ))]−1 ,

(51)

where v = v[uij (det(v))], v = v [uij (det(v ))]. Since the matrix [uij (s)] is unimodular, we get det(v ) = det(v ) = det(v). Therefore, according to (51) and (49), the following formula holds for any element z ∈ Z: σ−1 ◦ z ◦ σ : V −→ V, v → v diag(t(z), t−1 (z)). (52) Formula (52) shows that σ−1 ◦ z ◦ σ is a linear automorphism of the space V . Since the automorphism σ is S-equivariant, we conclude that σ−1 ◦ g ◦ σ is a linear automorphism of V for any element g ∈ G. This completes the proof of linearizability in this case. Thus Theorem 2 is proved.

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