TORI IN THE CREMONA GROUPS

VLADIMIR L. POPOV^{∗}

Abstract. We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected group Białynicki-Birula’s results of 1966–67. We prove “fusion theorems” for n-dimensional tori in the affine and in the special affine Cremona groups of rank n. In the final section we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups.

Contents

1. Introduction 1 2. Some subgroups of Cr_{n} 5 3. Tori in Cr_{n}, Aut A^{n}, and Aut^{∗}A^{n} 10 4. Orbits and stabilizers of the action of D_{n}(l_{1}, …, l_{n}) on A^{n} 13 5. The group N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) 16 6. Fusion theorems for tori in Aut A^{n} and Aut^{∗}A^{n} 21 7. Applications: The classifications of classes of conjugate subgroups 23 8. Jordan decomposition in Cr_{n}. Torsion primes for the Cremona

groups 26 References 29

1. Introduction

This work arose from the attempt to solve the problem posed in [Po05_{1}], [Po05_{2}]. In these papers have been introduced the notions of root α and root vector D of an affine algebraic variety X with respect to an algebraic torus T ⊆ Aut X. Namely, D is a locally-nilpotent derivation of the coordinate algebra of the variety X, and α is a character of the torus T such that t^{∗} ∘ D ∘ t^{∗}^{−1} = α(t)D for all t ∈ T. These definitions are inspired by the natural analogy with the classical definitions of the theory of algebraic groups and purport an attempt to apply to, in general, infinite dimensional group Aut X the techniques important in the theory of usual algebraic groups^{1}.

∗ Supported by grants ���� �������������, ��������������, and the program Contemporary Problems of Theoretical Mathematics of the Russian Academy of Sciences, Branch of Mathematics.

In [Po05_{1}], [Po05_{2}] the following two problems related to the classical case where X = A^{n} and T = D_{n}^{∗} is the maximal diagonal torus that leaves fixed the standard volume form (see below (9)) have been posed:

(R) Find all the roots and root vectors of the variety A^{n} with respect to

D_{n}^{∗}. (W) Describe the normalizer and centralizer of the torus D_{n}^{∗} in the group

Aut^{∗}A^{n} of automorphisms of the space A^{n} that leave fixed the standard volume form.

Problem (R) has been solved by Liendo in [Li11]. The answer is the following. Let x_{1}, …, x_{n} be the standard coordinate functions on A^{n} and let ε_{1}, …, ε_{n} be the “coordinate” characters of the standard n-dimensional diagonal torus D_{n} in Aut A^{n} (see below (8) and (10)). Then, up to multiplication by a nonzero constant, the root vectors are precisely all the derivations D of the form

x_{1}^{l_{1}} ⋯ x_{n}^{l_{n}}(∂/∂x_{i}),  (1)

where l_{1}, …, l_{n} are nonnegative integers and l_{i} = 0. The root α corresponding to root vector (1) is the restriction to D_{n}^{∗} of the character

ε_{i}^{−1} ∏_{j=1}^{n} ε_{j}^{l_{j}}.

The problem mentioned in the beginning of this introduction is Problem (W). It is clear that it is aimed at getting a description of the “Weyl group” of the root system from Problem (R). We solve it in the present paper. Namely, we prove (Theorem 14) that the normalizer (centralizer) of the torus D_{n}^{∗} in Aut^{∗}A^{n} coincides with its normalizer (centralizer) in SL_{n}, so that the Weyl group of D_{n}^{∗} in Aut^{∗}A^{n} is the same as that of D_{n}^{∗} in SL_{n} — it is the group of all permutations of the characters ε_{1}, …, ε_{n}.

In fact, this result is only one special case of the series of general results that we obtain here. Namely, D_{n}^{∗} is only one of the infinitely many nonconjugate diagonalizable algebraic subgroups G of dimension ⩾ n − 1 in the group Aut A^{n}. We prove that the normalizer of G in Aut A^{n} always is an algebraic subgroup in Aut A^{n} (Theorem 16). It is the characteristic property of the specified dimensions: in general, for the diagonalizable subgroups of dimension ⩽ n − 2 it does not hold. Moreover, in the case when nonconstant G-invariant polynomial functions on A^{n} exist, we explicitly describe the normalizer of G in Aut A^{n}, in particular, we show that in all but one cases it coincides with the normalizer of G in a group conjugated to GL_{n} (Theorem 6).

Using the found information, we obtain the new results in the Linearization Problem. In 1966–67 Białynicki-Birula proved [B-B66], [B-B67] that every algebraic action on A^{n} of an algebraic torus of dimension ⩾ n − 1 is equivalent to a linear action. We extend this statement to disconnected groups proving that every algebraic action on A^{n} of either an n-dimensional algebraic group whose connected component of identity is a torus, or an (n − 1)-dimensional diagonalizable group is equivalent to a linear action (Theorems 13 and 15).

1In [Po05_{1}], [Po05_{2}] was considered the case where X = A^{n} and T is the maximal diagonal torus that leaves fixed the standard volume form, but this restriction plays no role in the definition.

We also obtain the following classifications:

(i) the classification of diagonalizable subgroups of the group Aff_{n} of affine transformastions (see below (6)) up to conjugacy in the full Cremona group Cr_{n} = Bir A^{n} (Theorem 1);

(ii) the classification of n-dimensional diagonalizable subgroups of Aut A^{n}

up to conjugacy in Aut A^{n} (Theorem 11); (iii) the classification of (n − 1)-dimensional diagonalizable subgroups of

Aut^{∗}A^{n} up to conjugacy in Aut^{∗}A^{n} (Theorem 12); (iv) the classiciation, up to conjugacy in Aut A^{n}, of maximal n-dimensio-

nal algebraic subgroups G in Aut A^{n} such that G^{0} is a torus (Theorem 11);

(v) the classiciation, up to conjugacy in Aut^{∗}A^{n}, of maximal (n − 1)- dimensional algebraic subgroups G in Aut^{∗}A^{n} such that G^{0} is a torus (Theorem 14);

(vi) the classification of (n − 1)-dimensional diagonalizable subgroups of

Aut A^{n} up to conjugacy in Aut A^{n} (Theorem 15); (vii) the classification of diagonalizable subgroups in Aut A^{n} of dimension

⩾ n − 1 up to conjugacy in Cr_{n} (Theorems 11 and 17); (viii) the classification of one-dimensional tori of Aut A^{3} up to conjugacy

in Aut A^{3} (Theorem 18). For instance, we prove that the set of classes of (n − 1)-dimensional diag-

onalizable subgroups of Aut A^{n} that are conjugate in Aut A^{n} is bijectively parametrized by the set of nonzero nondecreasing sequences

(l_{1}, …, l_{n}) ∈ Z^{n},  (2)

such that (l_{1} …, l_{n}) ⩽ (−l_{n} …, −l_{1}) with respect to the lexicographic order. Under this parametrization, to sequence (2) corresponds the class of the subgroup ker ε_{1}^{l_{1}} ⋯ ε_{n}^{l_{n}}.

Another example: we show that diagonalizable subgroups of Aff_{n} are conjugate in Cr_{n} if and only if they are isomorphic and we specify their canonical representatives. In particular (see Corollary 5), every two isomorphic finite Abelian subgroups of Aff_{n} are conjugate in Cr_{n} (for finite cyclic subgroups this has been proved in [Bl06_{1}]).

In [Se10] Serre proved “fusion theorem” for the torus D_{n} in Cr_{n}. We prove and use “fusion theorems” for n-dimensional tori in Aut A^{n} and in Aut^{∗}A^{n+1} (Theorem 9).

In the final section, developing further the theme of analogies between the Cremona groups and algebraic groups, we introduce and discuss the notions of Jordan decomposition and torsion primes for the Cremona groups. In the course of discussion, we formulate some open questions.

Acknowledgment. I am grateful to J.-P. Serre for the comments.

Notation and conventions.

In the sequel, “variety” means “algebraic variety over the fixed algebraically closed field k of characteristic zero” in the sense of Serre’s FAC [Se55]. Apart from the standard notation and conventions of [Bo91] and [PoVi94] used without reminders we also use the following:

— Mat_{m×n}(R) is the set of all matrices with m rows, n columns, and the coefficients in R.

— N_{H}(S) and Z_{H}(S) are, respectively, the normalizer and centralizer of the subgroup S of the group H.

— μ_{d} is the subgroups of order d in G_{m}.

— X(D) is the group of rational characters of the diagonalizable algebraic group D.

— χ(X) is the Euler characteristic of the variety X relative to the ladic cohomology (for k = C, by [La81] it coincides with the Euler characteristic relative to the usual cohomology with compact supports (cf. also [KrPo85, Appendix])).

— If the group G acts on a set M and φ: G × M → M is the map defining this action, then for the subsets S ⊆ G and X ⊆ M, the subset φ(S × X) ⊆ M is denoted by S · X (each time it is clear from the context what φ is meant). In particular, by G · a is denoted the G-orbit of the point a. By G_{a} is denoted the G-stabilizer of the point a.

— x_{1}, …, x_{n} are the standard coordinate functions on A^{n}:

x_{i}(a) := a_{i}, a := (a_{1}, …, a_{n}) ∈ A^{n}.

In the sequel it is assumed that all considered algebraic groups are affine and all their homomorphisms are algebraic. Below all tori and diagonalizable groups are algebraic.

An action of a group G on a vector space V is called locally finite, if for every vector v ∈ V the linear span of the orbit G · v is finite dimensional.

The group Cr_{n} := Bir A^{n} is called the Cremona group of rank n. The map φ ↦ (φ^{∗})^{−1} identifies it with Aut_{k}k(x_{1}, …, x_{n}). Every birational isomorphism X −−→ A^{n} identifies Cr_{n} with Bir X. For every g ∈ Cr_{n} the functions

g_{i} = g^{∗}(x_{i}) ∈ k(A^{n})  (3)

determine g by the formula

g(a) = (g_{1}(a), …, g_{n}(a)) if g is defined at a ∈ A^{n};  (4)

we use the notation

(g_{1}, …, g_{n}) := g.  (5)

By means of the notion of “algebraic family” S → Cr_{n} (see [Ra64]) the group Cr_{n} is endowed with the Zariski topology (see [Se10], [Bl10]). If a homomorphism G → Cr_{n} of an algebraic group G is an algebraic family, then its image is called an algebraic subgroup of Cr_{n} (see [Po12]).

The subgroup

Aut A^{n} := {(g_{1}, …, g_{n}) ∈ Cr_{n} | g_{1}, …, g_{n} ∈ k[A^{n}] = k[x_{1}, …, x_{n}]} is called the affine Cremona group of rank n.

It contains the algebraic subgroup of affine transformations

Aff_{n} = {(g_{1}, …, g_{n}) ∈ Aut A^{n} | deg g_{1} = … = deg g_{n} = 1},  (6)

and Aff_{n}, in turn, contains the algebraic subgroup of linear transformations

GL_{n} = {g ∈ Aff_{n} | g(0) = 0}.

If g = (g_{1}, …, g_{n}) ∈ Aut A^{n} (see (5)), then we put

Jac(g) := det(∂g_{i}/∂x_{j}).

Since g ∈ Aut A^{n}, we have Jac(g) ∈ k \ {0}. Therefore, g ↦ Jac(g) is the homomorphism of Aut A^{n} in the multiplicative group of the field k. Its kernel

Aut^{∗}A^{n} := {f ∈ Aut A^{n} | Jac(g) = 1}

consists of the automorphisms of A^{n} that leaves fixed the standard volume form; it is called the special affine Cremona group of rank n − 1 (regarding the ranks in these names see Theorems 2(i) and 4(i) below). The latter group contains the algebraic subgroup

SL_{n} := GL_{n} ∩ Aut^{∗}A^{n}.

The embeddings Cr_{n} ↪ Cr_{n+1}, (g_{1}, …, g_{n}) ↦ (g_{1}, …, g_{n}, x_{n+1}) are arranged the tower Cr_{1} ↪ Cr_{2} ↪ ⋯ ↪ Cr_{n} ↪ · · ·. Its direct limit Cr_{∞} is called the Cremona group of infinite rank, see [Po12, Sect. 1].

In GL_{n} is distinguished the “standard” maximal torus

D_{n} := {(t_{1}x_{1}, … t_{n}x_{n}) | t_{1}, …, t_{n} ∈ k} ⊂ GL_{n}.  (7)

Its normilizer in GL_{n} is the group of all monomial transformations in GL_{n}:

N_{GL_{n}}(D_{n}) = {(t_{1}x_{σ(1)}, …, t_{n}x_{σ(n)}) | σ ∈ S_{n}, t_{1}, …, t_{n} ∈ k} ⊂ GL_{n}, (8)

where S_{n} is the symmetric group of degree n. In Aut^{∗}A^{n} is contained the torus

D_{n}^{∗} := D_{n} ∩ Aut^{∗}A^{n} = {(t_{1}x_{1}, …, t_{n}x_{n}) | t_{1}, …, t_{n} ∈ k, t_{1} ⋯ t_{n} = 1}. (9)

The “coordinate” characters ε_{1}, …, ε_{n} of the torus D_{n} defined by

ε_{i}: D_{n} → G_{m}, (t_{1}x_{1}, …, t_{n}x_{n}) ↦ t_{i},  (10)

constitute a base of the (free Abelian) group X(D_{n}).

2. Some subgroups of Cr_{n}

In the sequel we consider the elements of the group Z^{n} as rows of length n. Then the rows of any matrix A = (a_{ij}) ∈ Mat_{m×n}(Z) become the elements of this group and we use the following notation:

R_{A} := the subgroup of Z^{n} generated by the rows of matrix A, (11)
D_{n}(A) = ⋂_{i=1}^{m} ker λ_{i}, where λ_{i} := ε_{1}^{a_{i,1}} ⋯ ε_{n}^{a_{i,n}}.  (12)

If m = 1, then in place of D_{n}((l_{1} … l_{n})) we write D_{n}(l_{1}, …, l_{n}). In particular,

D_{n}(0, …, 0) = D_{n}.  (13)

Clearly, D_{n}(A) is a closed subgroup of D_{n} and (see (11))

D_{n}(A) = ⋂_{(l_{1},...,l_{n})∈R(A)} ker ε_{1}^{l_{1}} ⋯ ε_{n}^{l_{n}}.  (14)

Recall the terminology used below (see, e.g., [Vi03] and [MaMi64]).

Every finite Abelian group G decomposes as a direct sum of cyclic subgroups of orders d_{1}, …, d_{m}, where d_{i} divides d_{i+1} for i = 1, …, m − 1, and d_{1} > 1 if |G| > 1. The numbers d_{1}, …, d_{s} are uniquely determined by G and are called the invariant factors of G.

Every nonzero integer matrix A can be transformed by means of elementary transformations of its rows and columns into a matrix S = (s_{ij}) such that only the coefficients s_{ii} for i = 1, …, r are nonzero and s_{ii} divides s_{i+1,i+1} for i = 1, …, r − 1. The integers s_{11}, …, s_{rr} are uniquely determined by A (s_{ii} = f_{i}/f_{i−1}, where f_{i} is gcd of the minors of order i of the matrix A and f_{0} := 1) and are called the invariant factors of matrix A. The matrix S is called the Smith normal form of matrix A.

Lemma 1. If B is obtained from A ∈ Mat_{m×n}(Z) by means of the elementary transformations of rows and columns, then the subgroups D_{n}(A) and D_{n}(B) are conjugate in Cr_{n}.

Proof. Let τ_{1}, …, τ_{n} be some base of the group X(D_{n}). Then (see (12)) λ_{i} = τ_{1}^{c_{i,1}} ⋯ τ_{n}^{c_{i,n}} for some c_{ij} ∈ Z and

D_{n}(A) = ⋂_{i=1}^{m} ker τ_{1}^{c_{i,1}} ⋯ τ_{n}^{c_{i,n}}.  (15)

The group Aut_{gr}D_{n} of automorphisms of the algebraic group D_{n} is naturally identified with GL_{n}(Z). Its natural action on the set of bases of the group X(D_{n}) is transitive. Therefore, there is an automorphism

φ ∈ Aut_{gr}D_{n},  (16)

such that τ_{i} ∘ φ = ε_{i} for each i. From (15) it then follows that

φ^{−1}(D_{n}(A)) = D_{n}(C), C = (c_{ij}) ∈ Mat_{m×n}(Z).  (17)

Since the map of varieties D_{n} → A^{n}, (t_{1}x_{1}, …, t_{n}x_{n}) ↦ (t_{1}, …, t_{n}), is a birational isomorphism, by means of it we can identify the group Cr_{n} = Bir A^{n} with the group of birational automorphisms of the underlying variety of torus D_{n}. Then φ becomes an element of the group Cr_{n} and from (16) and (17) it is easy to deduce that in this group we have the equality

φ^{−1}D_{n}(A)φ = D_{n}(C).

Further, notice that if the base τ_{1}, …, τ_{n} is obtained from the base ε_{1}, … …, ε_{n} by an elementary transformation, then the matrix C is obtained from A by an elementary transformation of columns, and every elementary transformation of columns of A is realizable in this way. Also, notice that if the sequence φ_{1}, …, φ_{m} ∈ X(D_{n}) is obtained by an elementary transformation from the sequence λ_{1}, …, λ_{m}, then D_{n}(A) = ⋂_{i=1}^{m} ker φ_{i}, the matrix (c_{ij}) defined by the equalities φ_{i} = ε_{1}^{c_{i,1}} ⋯ ε_{n}^{c_{i,n}} is obtained from A by an elementary transformation of rows, and every elementary transformation of rows of A is realizable in this way.

Clearly, the said implies the claim of lemma. □

Corollary 1. If S is the Smith normal form of matrix A, then the subgroups D_{n}(A) and D_{n}(S) are conjugate in Cr_{n}.

Lemma 2.

(i) Let q_{1} ⩽⋯⩽q_{r} be all the invariant factors of matrix A∈Mat_{m×n}(Z). Then the group D_{n}(A) is isomorphic to

μ_{q_{1}} × ⋯ × μ_{q_{r}} × G_{m}^{n−r}.  (18)

(ii) The closed (n − m)-dimensional subgroups of D_{n} are every possible

subgroups D_{n}(A), where A ∈ Mat_{m×n}(Z), rk A = m, and only they. (iii) R_{A} = {(l_{1}, …, l_{n}) ∈ Z^{n} | D_{n}(A) ⊆ ker ε_{1}^{l_{1}} ⋯ ε_{n}^{l_{n}}} for every A ∈

Mat_{m×n}(Z). (iv) If A ∈ Mat_{s×n}(Z), B ∈ Mat_{t×n}(Z), then

(a) D_{n}(A) = D_{n}(B) if and only if R_{A} = R_{B};

(b) The following properties are equivalent: (b_{1}) D_{n}(A) and D_{n}(B) are conjugate in GL_{n}; (b_{2}) D_{n}(A) and D_{n}(B) are conjugate in N_{GL_{n}}(D_{n}); (b_{2}) by a permutation of columns, it is possible to tranform B

into a matrix C such that R_{A} = R_{C}.

Proof. (i) Let S = (s_{ij}) be the normal Smith form of matrix A. Then s_{11} = q_{1}, …, s_{rr} = q_{r} and s_{ij} = 0 in the other cases. Hence D_{n}(S) is isomorphic to group (18). But D_{n}(A) is isomorphic to D_{n}(S) by Corollary 1. This proves (i).

(ii) From (i) we deduce that

dim D_{n}(A) = n − rk A;  (19)

therefore, dim D_{n}(A) = n−m, where rk A = m. Conversely, let H be a closed subgroup of D_{n} such that dim H = n−m. Then D_{n}/H is an m-dimensional torus [Bo91, p. 114] and therefore, there is an isomorphism α: D_{n}/H → G_{m}^{m}. Let λ_{i} ∈ X(D_{n}) be the composition of homomorphisms

D_{n} ⟶^{π} D_{n}/H ⟶^{α} G_{m}^{m} ⟶^{pr_{i}} G_{m},

where π s the canonical projection and pr_{i} is the projection to the ith factor. Then H = ⋂_{i=1}^{m} ker λ_{i}. From (12) we then deduce that H = D_{n}(A) and from (19) that rk A = m. This proves (ii).

(iii) From (11), (12) it follows that the left-hand side of the equality under proof is contained in the right-hand side. Proving the inverse inclusion consider a character λ = ε_{1}^{l_{1}} ⋯ ε_{n}^{l_{n}} whose kernel contains D_{n}(A). Without changing D_{n}(A) and R_{A}, we can leave in A only the rows that form a base of the group R_{A} removing the other rows, i.e., we can reduce our considerations to the case where rk A = m. Consider then the characters λ_{1}, …, λ_{m} defined by formula (12), and the homomorphism

φ: D_{n} → G_{m}^{m}, g ↦ (λ_{1}(g), …, λ_{m}(g)).

In view of (12), we have ker φ = D_{n}(A). From this and (19) it follows that dim φ(D_{n}) = m. Therefore, φ is a surjection. Hence G_{m}^{m} is the quotient group of D_{n} by D_{n}(A) and φ is the canonical homomorphism to it. Since λ is constant on the fibers of φ, the universal property of quotient implies the existence of character μ: G_{m}^{m} → G_{m} such that λ = μ ∘ φ. Hence λ = λ_{1}^{c_{1}} ⋯ λ_{m}^{c_{m}} for some c_{1}, …, c_{m} ∈ Z, and this means that (l_{1}, …, l_{n}) ∈ R_{A}. This proves (iii).

(iv)(a) If D_{n}(A) = D_{n}(B), then R_{A} = R_{B} because of (iii). Conversely, if R_{A} = R_{B}, then D_{n}(A) = D_{n}(B) because of (14). This proves (iv)(a).

(iv)(b) By fusion theorem [Se00, 1.1.1] the subgroups D_{n}(A) and D_{n}(B) are conjugate in GL_{n} if and only if they are conjugate in N_{GL_{n}}(D_{n}). But (8) and (12) imply that D_{n}(A) and D_{n}(B) are conjugate in N_{GL_{n}}(D_{n}) if and only if by a permutation of columns one can obtain from B a matrix C such that D_{n}(A) = D_{n}(C). Because of (iii)(a), the latter equality is equivalent to the equality R_{C} = R_{A}. This proves (iv)(b). □

Corollary 2.

(i) Let (l_{1}, …, l_{n})≠(0, …, 0) and d:=gcd(l_{1}, …, l_{n}). Then D_{n}(l_{1}, …, l_{n}) is isomorphic to μ_{d} × G_{m}^{n−1}. In particular, the group D_{n}(l_{1}, …, l_{n}) is connected (i.e., is a torus) if and only if d = 1.

(ii) The closed (n − 1)-dimensional subgroups of D_{n} are every possible

subgroups D_{n}(l_{1}, …, l_{n}) with (l_{1}, …, l_{n}) ≠ (0, …, 0) and only they. (iii) D_{n}(l_{1}, …, l_{n}) = D_{n}(l′_{1}, …, l′_{n}) if and only if

(l_{1}, …, l_{n}) = ±(l′_{1}, …, l′_{n}).

(iv) The following properties are equivalent: (iv_{1}) D_{n}(l_{1}, …, l_{n}) and D_{n}(l′_{1}, …, l′_{n}) are conjugate in GL_{n}; (iv_{2}) D_{n}(l_{1}, …, l_{n}) and D_{n}(l′_{1}, …, l′_{n}) are conjugate in N_{GL_{n}}(D_{n}); (iv_{3}) there is a permutation σ ∈ S_{n} such that

(l_{1}, …, l_{n}) = ±(l′_{σ(1)}, …, l′_{σ(n)}).

The following Lemma 3 gives an effective numerical criterion for the equality R_{A} = R_{B} from Lemma 2(iii).

Let A ∈ Mat_{m×n}(Z), rk A = m. For every strictly increasing sequence of m integers i_{1}, …, i_{m} taken from the interval [1, n], put

p_{i_{1},...,i_{m}}(A) := det A_{i_{1},...,i_{m}},  (20)

where A_{i_{1},...,i_{m}} is the submatrix of matrix A obtained by intersecting rows with numbers 1, …, m and columns with numbers i_{1}, …, i_{m} (it is natural to call the p_{i_{1},...,i_{m}}(A)’s the Plücker coordinates of matrix A).

Lemma 3. For every two matrices A and B ∈ Mat_{m×n}(Z) of rank m the following properties are equivalent:

(i) R_{A} = R_{B}.

(ii) Two conditions hold:

(a) either p_{i_{1},...,i_{m}}(A) = p_{i_{1},...,i_{m}}(B) for all i_{1}, …, i_{m}, or p_{i_{1},...,i_{m}}(A) = −p_{i_{1},...,i_{m}}(B) for all i_{1}, …, i_{m};

(b) for every sequence i_{1}, …, i_{m} such that p_{i_{1},...,i_{m}}(A) ≠ 0, the following inclusion holds:

B_{i_{1},...,i_{m}}(A_{i_{1},...,i_{m}})^{−1} ∈ Mat_{m×m}(Z).  (21)

Proof. Whereas rk A = rk B = m, the rows of A and B form the bases in R_{A} and R_{B} respectively. Therefore, R_{A} = R_{B} if and only if when there is a matrix Q ∈ GL_{m}(Z) such that

A = QB.  (22)

Let R_{A} = R_{B}. Then (22) implies that

A_{i_{1},...,i_{m}} = QB_{i_{1},...,i_{m}}  (23)

for every i_{1}, …, i_{m}, and therefore, p_{i_{1},...,i_{m}}(A) = det Qp_{i_{1},...,i_{m}}(B) because of (20). Since Q ∈ GL_{m}(Z), we have det Q = ±1. Hence condition (ii)(a) holds. If p_{i_{1},...,i_{m}}(A) ≠ 0, then p_{i_{1},...,i_{m}}(B) ≠ 0 as well, hence B_{i_{1},...,i_{m}} is nondegenerate and (23) implies that Q = A_{i_{1},...,i_{m}}(B_{i_{1},...,i_{m}})^{−1}. Hence condition (ii)(b) holds. This proves (i)⇒(ii).

Proving the inverse implication, consider Z^{n} as a subset of the coordinate vectors space (of rows) Q^{n}. Condition (ii)(a) shows that the Q-linear spans of subsets R_{A} and R_{B} of Q^{n} have the same Plücker coordinates. Hence these spans are one and the same linear subspace L (see, e.g., [ShRe09, Theorem 10.1]). Since the rows of A and rows of B form two bases in L, there exists a matrix P ∈ GL_{m}(Q) such that A = P B. Therefore, A_{i_{1},...,i_{m}} = P B_{i_{1},...,i_{m}} for every i_{1}, …, i_{m} and hence P = A_{i_{1},...,i_{m}}(B_{i_{1},...,i_{m}})^{−1} if p_{i_{1},...,i_{m}}(B) ≠ 0. Then (ii)(b) implies that P ∈ GL_{n}(Z). Therefore, R_{A} = R_{B}. This proves (ii)⇒(i). □

Remark 1.

1. In the proof of Lemma 3 it is established that, in fact, (ii)(a) implies that the matrix B_{i_{1},...,i_{m}}(A_{i_{1},...,i_{m}})^{−1} is independent of the choice of a sequence i_{1}, …, i_{m} for which p_{i_{1},...,i_{m}}(A) ≠ 0. Therefore, (ii)(b) follows from (ii)(a) and feasibility of (21) for any one such a sequence.

2. If m=1, then (ii)(b) follows from (ii)(a) (but for m > 1 this is not so).

Theorem 1 (Classificaton of diagonalizable subgroups of Aff_{n} up to conjugacy in Cr_{n}).

(i) Two diagonalizable subgroups of the group Aff_{n} are conjugate in Cr_{n} if and only if they are isomorphic.

(ii) Any diagonalizable subgroup G of the group Aff_{n} is conjugate in Cr_{n} to a unique closed subgroup of the torus D_{n} that has the form

ker ε_{r+1}^{d_{1}} ∩ … ∩ ker ε_{r+s}^{d_{s}} ∩ ker ε_{r+s+1} ∩ … ∩ ker ε_{n}  (24)

where 0 ⩽ r ⩽ n, 0 ⩽ s ⩽ n, r + s ⩽ n, 2 ⩽ d_{1} and d_{i} divides d_{i+1} for every i < s. The integers determining subgroup (24) have the following meaning: r = dim G and d_{1}, …, d_{s} are all the invariant factors of the finite Abelian group G/G^{0}.

Proof. Since maximal reductive subgroups of algebraic group are conjugate (see [BoSe64, 5.1]) and GL_{n} is one of them in Aff_{n}, every diagonalizable subgroup of Aff_{n} is conjugate to a subgroup of GL_{n}. In turn, every diagonalizable subgroup of GL_{n} is conjugate to a subgroup of the torus D_{n} (see [Bo91, I.4.6]). In view of Lemma 2(ii) this shows that it suffices to prove (i) for the subgroups D_{n}(A) and D_{n}(B) of the torus D_{n}. Further, appending, if necessary, zero rows we may assume that A and B have the same number of rows. Let now D_{n}(A) and D_{n}(B) be isomorphic. Then their dimensions are equal and the groups D_{n}(A)/D_{n}(A)^{0} and D_{n}(B)/D_{n}(B)^{0} have the same invariant factors. This and Lemma 2(i) imply that the matrices A and B have the same invariant factors (because the latter are obtained by appending the same number of 1’s to the invariant factors of the specified groups). Hence the normal Smith forms of A and B are equal. By Corollary 1 this implies that D_{n}(A) and D_{n}(B) are conjugate in Cr_{n}. This proves (i).

It is clear that the integers determining subgroup (24) have the meaning specified in (ii). Since every diagonalizable group is a direct product of a finite Abelian group and a torus, it is uniquely, up to isomorphism, determined by its dimension and the invariant factors of the group of connected components. This and (i) implies (ii). □

Corollary 3. D_{n}(A) and D_{n}(B) are conjugate in Cr_{n} if and only if A and B have the same invariant factors.

Corollary 4. Every torus T in Aff_{n} is conjugate in Cr_{n} to the torus D_{r}, r = dim T.

Proof. This follows from Theorem 1(ii). □

Corollary 5. Every two isomorphic finite Abelian subgroups of Aff_{n} are conjugate in Cr_{n}.

Proof. Since char k = 0, every element in Aff_{n} of finite order is semisimple. Hence every finite Abelian subgroup of Aff_{n} is reductive, and therefore, is conjugate in Aff_{n} to a subgroups of GL_{n} (see the proof of Theorem 1). But every commutative subgroup of GL_{n} consisting only of semisimple elements is diagonalizable (see [Bo91, Prop. 4.6(b)]). The claim now follows from Theorem 1(i). □

Corollary 6 ([Bl06_{1}, Thm. 1]). Every two elements in Aff_{n} of the same finite order are conjugate in Cr_{n}.

3. Tori in Cr_{n}, Aut A^{n}, and Aut^{∗}A^{n}

Theorem 2 (Tori in Cr_{n}).

(i) In Cr_{n} there are no tori of dimension > n.

(ii) In Cr_{n} every r-dimensional torus for r = n, n − 1, n − 2 is conjugate

to the torus D_{r}. (iii) If n ⩾ 5, then in Cr_{n} there are (n − 3)-dimensional tori that are not

conjugate to subtori of the torus D_{n}. (iv) Every r-dimensional torus in Cr_{n} is conjugate in Cr_{n+r} to the torus D_{r}.

(v) In Cr_{∞} every r-dimensional torus is conjugate to the torus D_{r}.

Proof. (i) This is proved, e.g., in [De70], see also [Po12, Cor. 2.2].

(ii) According to [B-B66, Cor. 2] (see also [Po12, Cor. 2.4(b)]), every rdimensional torus in Cr_{n} for r = n, n − 1, n − 2 is conjugate to a subtorus of the torus D_{n}. Therefore, the claim follows from Corollary 4.

(iii) This is proved in [Po12, Cor. 2.5].

(iv) According to [Po12, Thm. 2.6], every r-dimensional torus in Cr_{r} is conjugate in Cr_{n+r} to a subtorus of the torus D_{n+r}. Therefore, the claim follows from Corollary 4.

(v) This follows from (iv). □

Corollary 7.

(i) In Cr_{n} every n-dimensional torus is maximal.

(ii) In Cr_{n} there are no maximal (n − 1)- and (n − 2)-dimensional tori. (iii) For n ⩾ 5, in Cr_{n} there are maximal (n − 3)-dimensional tori.

Remark 2. For n ⩽ 3, Theorem 2 gives the classification of tori in Cr_{n} up to conjugacy: the classes of conjugate nontrivial tori are exhausted by that of the tori D_{1}, …, D_{n}.

Let (l_{1}, …, l_{n}) ∈ Z^{n} be a nonzero element with gcd(l_{1}, …, l_{n})=1. Clearly, the homomorphism

G_{m} → D_{n}, t ↦ (t^{l_{1}}x_{1}, …, t^{l_{n}}x_{n}),  (25)

is an embedding and every embedding G_{m} ↪ D_{n} is of this form. Denote by T (l_{1}, …, l_{n}) the image of embedding (25). It is a one-dimensional torus in D_{n} and every one-dimensional torus in D_{n} is of this form.

Lemma 4. The following properties are equivalent:

(i) T (l_{1}, …, l_{n}) = T (l′_{1}, …, l′_{n}).

(ii) (l_{1}, …, l_{n}) = ±(l′_{1}, …, l′_{n}).

Proof. (i)⇒(ii) This is clear.

(ii)⇒(i) Assume that (i) holds. Take an element t ∈ G_{m} of infinite order. From (i) and the definition of T (l_{1}, …, l_{n}) we deduce that there is an element s ∈ G_{m} such that t^{l_{i}} = s^{l′_{i}} for every i = 1, …, n. Hence t^{l_{i}l′_{j}} = s^{l′_{i}l′_{j}} = t^{l_{j}l′_{i}} for every two nonequal integers i and j taken from the interval [1, n]. Since the order of t is infinite, this implies that l_{i}l′_{j} − l_{j}l′_{i} = 0. Hence

rk (\begin{matrix}l_{1} & … & l_{n} \\ l′_{1} & … & l′_{n} \end{matrix}) = 1.

Therefore, (l_{1}, …, l_{n}) = γ(l′_{1}, …, l′_{n}) for some γ ∈ Q, or, equivalently, p(l_{1}, …, l_{n}) = q(l′_{1}, …, l′_{n}), where p, q ∈ Z, gcd(p, q) = 1. Hence p divides each of l′_{1}, …, l′_{n}, and q divides each of l_{1}, …, l_{n}. Since the integers in each of these two sets are coprime, this implies that γ = ±1, i.e., (ii) holds. □

Theorem 3 (Tori in Aut A^{n}).

(i) In Aut A^{n} every n-dimensional torus is conjugate to the torus D_{n}.

(ii) In Aut A^{n} all (n − 1)-dimensional tori are exhausted, up to conjugacy in Aut A^{n}, by all the groups of the form D_{n}(l_{1}, …, l_{n}) with (l_{1}, …, l_{n}) ≠ (0, …, 0) and gcd(l_{1}, …, l_{n}) = 1.

(iii) In Aut A^{3} all one-dimensional tori are exhaused, up to conjugacy, by all the groups of the form T (l_{1}, l_{2}, l_{3}).

Proof. According to [B-B66] and [B-B67], every r-dimensional torus in the group Aut A^{n} for r = n and n − 1, respectively, is conjugate to a subtorus of the torus D_{n}. This and Corollary 2 imply (i) and (ii).

According to [KKM-LR97], every one-dimensional torus in Aut A^{3} is conjugate to subtorus of the torus D_{3}. This implies (iii). □

Theorem 4 (Tori in Aut^{∗}A^{n}).

(i) In Aut^{∗}A^{n} there are no tori of dimension > n − 1.

(ii) In Aut^{∗}A^{n} every (n − 1)-dimensional torus is maximal and conjugate to the torus D_{n}^{∗} (see (9)).

Proof. (i) If Aut^{∗}A^{n} contains an n-dimensional torus T, then by Theorem 3(i) there is an element g ∈ Aut A^{n} such that

T = gD_{n}g^{−1}.  (26)

Replacing g by gz, where z ∈ D_{n} is an element such that Jac(z) = det z = 1/Jac(g), we may assume that g ∈ Aut^{∗}A^{n}. This and (26) imply that D_{n} ⊂ Aut^{∗}A^{n}, — a contradiction. This proves (i).

(ii) Let S be an (n − 1)-dimensional torus in Aut^{∗}A^{n}. By Theorem 3(i) and Corollary (ii) of Lemma 2, there are g ∈ Aut A^{n} and (l_{1}, …, l_{n}) ∈ Z^{n} such that

S = gD_{n}(l_{1}, …, l_{n})g^{−1}.  (27)

As in the proof of (i), we may assume that g ∈ Aut^{∗}A^{n}. From (27) it then follows that D_{n}(l_{1}, …, l_{n}) ⊂ Aut^{∗}A^{n}. This and D_{n} ∩ ker Jac = D_{n}^{∗} := D_{n}(1, …, 1) imply the inclusion D_{n}(l_{1}, …, l_{n}) ⊆ D_{n}^{∗}. By Corollary (i) of Lemma 2, both sides of this inclusion are (n − 1)-dimensional tori. Hence it is the equality. This proves (ii) □

Remark 3. For Aut A^{n} and Aut^{∗}A^{n}, unlike for Cr_{n} (see Corollary 7(iii)), at present nothing is known on the existence of tori of nonmaximal dimension. This problem is intimately related to the Cancellation Problem: Is there an affine variety X not isomorphic to A^{m}, m = dim X, such that X × A^{d} is isomorphic to A^{m+d} for some d? If the answer is positive, then in Aut A^{n} and Aut^{∗}A^{n} with n = m+d there exists a maximal torus T of a nonmaximal dimension. Indeed, multiplying, if necessary, X×A^{d} by A^{1}, one may assume that d ⩾ 2. Let λ be the character t ↦ t of the torus G_{m}. Consider the linear action of G_{m} on A^{d} with precisely two isotypic components: one is (d − 1)-dimensional of type λ, another is one-dimensional of typer λ^{1−d}. It determines the action of G_{m} on X × A^{d} via the second factor and hence, an action of G_{m} on A^{n}. Consider in Aut A^{n} the torus that is the image of G_{m} under the homomorphism determined by this action. The construction implies that this torus lies in Aut^{∗}A^{n}. Let T (respectively, T′) be a maximal torus of Aut A^{n} (respectively, of Aut^{∗}A^{n}) that contains this image. If T is n-dimensional (respectively, T′ is (n−1)-dimensional), then by Theorem 3(i) (respectively, by Theorem 4) it is conjugate to a subtorus of the torus D_{n} and, therefore, the specified action of T (respectively, T′) on A^{n} is equivalent to a linear one. Hence the action of G_{m} on A^{n} is equivalent to a linear one as well, and therefore, the set F of its fixed points is isomorphic to an affine space. But, by the construction, in A^{d} there is a single G_{m}-fixed point, consequently, F is isomorphic to X — a contradiction.

4. Orbits and stabilizers of the action of D_{n}(l_{1}, …, l_{n}) on A^{n}

Here we establish some properties, needed for what follows, of orbits and stabilizers of the natural action on A^{n} of the group

G := D_{n}(l_{1}, …, l_{n}).  (28)

Clearly, every coordinate hyperplane

H_{i} := {a ∈ A^{n} | x_{i}(a) = 0}  (29)

is G-invariant.

Lemma 5. The G-stabilizer of every point of A^{n} \ ⋃_{i=1}^{n} H_{i} is trivial.

(i) If (l_{1}, …, l_{n}) = (0, …, 0), then A^{n} \⋃_{i=1}^{n} H_{i} is a G-orbit, and dimension of the G-stabilizer of every point of ⋃_{i=1}^{n} H_{i} is positive.

(ii) If (l_{1}, …, l_{n}) ≠ (0, …, 0), then dim G · a = n − 1 for every point a ∈ A^{n} \ ⋃_{i=1}^{n} H_{i}.

Proof. This immediately follows from (28), (12), (4) and Lemma 2. □

We now consider the case

(l_{1}, …, l_{n}) ≠ (0, …, 0).

Lemma 6. If l_{i} ≠ 0, then the open subsubset O_{i} := H_{i} \ ⋃_{j≠i} H_{j} in H_{i} is a G-orbit.

Proof. By virtue of G-invariance of O_{i}, it suffices to show that O_{i} is contained in a G-orbit. Since l_{i} ≠ 0, the equation x^{l_{i}} = α has a solution for any α ∈ k, α ≠ 0. This and (28), (12) imply that for any two points

b=(b_{1}, …, b_{i−1}, 0, b_{i+1}, …, b_{n}), c=(c_{1}, …, c_{i−1}, 0, c_{i+1}, …, c_{n})∈H_{i} \ ⋃_{j≠i} H_{j}

there exists an element g = (t_{1}x_{1}, …, t_{n}x_{n}) ∈ G such that t_{j} = b_{j}^{−1}c_{j} for every j ≠ i. From (4) we then obtain that g · b = c, as required. □

Lemma 7. The following properties are equivalent:

(i) all the numbers l_{1}, …, l_{n} are nonzero and have the same sign; (ii) G-orbit of every point of A^{n} \ ⋃_{i=1}^{n} H_{i} is closed in A^{n}; (iii) G-orbit of some point of A^{n} \ ⋃_{i=1}^{n} H_{i} is closed in A^{n}.

Proof. Consider a point

a = (a_{1}, …, a_{n}) ∈ A^{n} \ ⋃_{i=1}^{n} H_{i}.

Assume that (i) holds. Suppose that the G-orbit of the point a is not closed in A^{n}. Then its boundary can be accessed by a one-parameter subgroup, i.e., there is homomorphism φ: G_{m} → G such that there is a limit

lim_{t→0}φ(t) · a = b ∈ \overline{G · a} \ G · a  (30)

(see [PoVi94, Thm. 6.9]); the latter means that the morphism G_{m} = A^{1} \ {0} → A^{n}, t ↦ φ(t) · a, extends to a morphism A^{1} → A^{n} that maps 0 to the point b. As φ is algebraic, there is a vector (d_{1}, …, d_{n}) ∈ Z^{n} such that φ(t) = (t^{d_{1}}x_{1}, …, t^{d_{n}}x_{n}) for every t ∈ G_{m}. Since φ(t)·a = (t^{d_{1}}a_{1}, …, t^{d_{n}}a_{n}) and a_{i} ≠ 0 for every i, the existence of the specified limit means that

d_{1} ⩾ 0, …, d_{n} ⩾ 0.  (31)

On the other hand, it follows from φ(t) ∈ G and (28), (12) that t^{d_{1}l_{1}+···+d_{n}l_{n}} = 1 for every t, i.e.,

d_{1}l_{1} + ⋯ + d_{n}l_{n} = 0.  (32)

But (31), (32) and condition (i) imply that d_{1} = ⋯ = d_{n} = 0. Hence b = a contrary to (30). This contradiction proves (i)⇒(ii).

Conversely, assume that (i) is not fulfilled, i.e., among l_{1}, …, l_{n} there are either two nonzero numbers with different signs, or one number equal to zero. In the first case, let, for instance, be l_{1} > 0, l_{2} < 0. Then (12) implies that the image of the homomorphism φ: G_{m} → D_{n}, t ↦ (t^{−l_{2}}x_{1}, t^{l_{1}}x_{2}, x_{3}, …, x_{n}), lies in G. Since

lim_{t→0}φ(t) · a = (0, 0, a_{3}, …, a_{n}) ∉ G · a,

this shows that the orbit G · a is not closed. In the second case, let, say, a_{1} = 0. Then G contains the image of the homomorphism φ: G_{m} → D_{n}, t ↦ (tx_{1}, x_{2}, …, x_{n}) and, since

lim_{t→0}φ(t) · a = (0, a_{2}, …, a_{n}) ∉ G · a,

the orbit G · a is not closed. This proves (iii)⇒(i). □

¿From Lemmas 5, 6, and 7 we infer

Corollary 8. If all the numbers l_{1}, …, l_{n} are nonzero and have the same sign, then in A^{n} there are precisely n nonclosed (n − 1)-dimensional Gorbits — these are the orbits O_{1}, …, O_{n} from Lemma 6.

Remark 4. Recall from [Po70] that an action of an algebraic group on an algebraic variety is called stable, if orbits of points in general position are closed. Lemma 7 shows that the following properties are equivalent:

(i) all the numbers l_{1}, …, l_{n} are nonzero and have the same sign; (ii) the action G on A^{n} is stable.

Lemma 8. Assume that none of the numbers l_{1}, …, l_{n} are equal to ±1. Then the following properties of the point a = (a_{1}, …, a_{n}) ∈ A^{n} are equivalent:

(i) a has a nontrivial G-stabilizer;

(ii) a ∈ ⋃_{i=1}^{n} H_{i}.

Proof. Lemma 5 implies (i)⇒(ii). Now assume that (ii) holds. Then there are the indices i_{1}, …, i_{s}, where s ⩾ 1, such that a_{j} = 0 for j = i_{1}, …, i_{s} and a_{j} ≠ 0 for other j’s. It follows from (4), (28), and (12) that an element (t_{1}x_{1}, …, t_{n}x_{n}) ∈ D_{n} lies in the G-stabilizer of the point a if and only if t_{j} = 1 for j ≠ i_{1}, …, i_{s} and

t_{i_{1}}^{l_{i_{1}}} ⋯ t_{i_{s}}^{l_{is}} = 1.  (33)

Since there is no ±1 among the numbers l_{1}, …, l_{n} and k is an algebraically closed field of characteristic zero, equality (33), considered as the equation in t_{i_{1}}, …, t_{i_{s}}, has at least two solutions. Hence the G-stabilizer of the point a is nontrivial. This proves (ii)⇒(i). □

Lemma 9. If among l_{1}, …, l_{n} there are two nonzero numbers with different signs, then every G-orbit contains (0, …, 0) in its closure.

Proof. For definiteness, let, for instance, be

l_{1} > 0, l_{2} ⩾ 0, …, l_{s} ⩾ 0, l_{s+1} < 0, l_{s+2} ⩽ 0, …, l_{n} ⩽ 0.

Choose a positive integer d so large that

q := l_{2} + ⋯ + l_{s} + dl_{s+1} + l_{s+2} + ⋯ + l_{n} < 0.

Since −ql_{1} + l_{1}l_{2} + ⋯ + l_{1}l_{s} + l_{1}dl_{s+1} + l_{1}l_{s+2} + … + l_{1}l_{n} = 0, from (28) and (12) we infer that the image of the homomorphism

φ: G_{m} → D_{n}, t ↦ (t^{−q}x_{1}, t^{l_{1}}x_{2}, …, t^{l_{1}}x_{s}, t^{l_{1}d}x_{s+1}, t^{l_{1}}x_{s+2} …, t^{l_{1}}x_{n}) (34)

lies in G. On the other hand, since the numbers −q, l_{1} and d are positive,

(4) implies that for every point a ∈ A^{n} the limit lim_{t→0}φ(t) · a exists and is equal to (0, …, 0). □

Now consider the case where 0 and ±1 are contained among l_{1}, …, l_{n}, there are at least two nonzero l_{i}’s, and all of them have the same sign. By (12), without loss of generality we may assume that this sign is positive. Up to replacing the group G by its conjugate by means of an element of N_{GL_{n}}(D_{n}), we may then assume that

l_{1} = … = l_{p} = 1, l_{p+1} ⩾ 2, …, l_{q} ⩾ 2, l_{q+1} = … = l_{n} = 0,
where p ⩾ 1, n > q ⩾ p and q ⩾ 2.

(35)

Lemma 10. Assume that (35) holds. Take a point a = (a_{1}, …, a_{n}) ∈ A^{n}.

(i) Let a ∉ ⋃_{i=1}^{n} H_{i}. Then the orbit G · b, where b = (a_{1}, …, a_{q}, 0, …, 0), lies in the closure of the orbit G · a, is closed, and dim G · b = q − 1.

(ii) Let a ∈ H_{i}. Then the group G_{a} is (a) trivial if 1 ⩽ i ⩽ p and a ∈ O_{i} (see Lemma 6);

(b) nontrivial and finite if p + 1 ⩽ i ⩽ q and a ∈ O_{i};

(c) has positive dimension if i > q.

Proof. From (28), (12), and (35) it follows that the image of the homomorphism

φ: G_{m} → D_{n}, t ↦ (x_{1}, …, x_{q}, tx_{q+1}, …, tx_{n}),

lies in G. Hence the point

lim_{t→0}φ(t) · a = (a_{1}, …, a_{q}, 0, …, 0) = b  (36)

lies in the closure of the orbit G · a.

Now let a ∉ ⋃_{i=1}^{n} H_{i}. If the orbit G · b is not closed, then, as in the proof of Lemma 7, there exist a homomorphism

ψ: G_{m} → G, t ↦ (t^{d_{1}}x_{1}, …, t^{d_{n}}x_{n}),  (37)

such that c := lim_{t→0}ψ(t) · b ∈ \overline{G · b} \ G · b. ¿From (36) it follows that d_{1} ⩾ 0, …, d_{q} ⩾ 0, and from (35), (28), and (12) it follows that d_{1}l_{1}+⋯+d_{q}l_{q} = 0. Since l_{1}, …, l_{q} are positive, this yields d_{1} = … = d_{q} = 0. In view of (36), from this we infer that ψ(t) · b = b for every t, and therefore, c = b — a contradiction. Thus, G · b is closed. Since a_{1}, …, a_{q} are nonzero, (28), (12), (4), (35), and (36) imply that an element (t_{1}x_{1}, …, t_{n}x_{n}) ∈ D_{n} lies in G_{b} if and only if t_{1} = … = t_{q} = 1. This proves (i).

The arguments analogous to that used in the proof of Lemma 8 yield (ii). □

Finally, consider the case where one of the numbers l_{1}, …, l_{n} is equal to ±1 (in view of (12), without loss of generality we may assume that it is equal to 1), and all the others are equal to 0.

Lemma 11. Let l_{i} = 1 and let l_{j} = 0 for j ≠ i. For any s ∈ k denote by H(s) the hyperplane in A^{n} defined by the equation x_{i} + s = 0. Then:

(i) ⋃_{j≠i} H_{j} is the set of points with nontrivial G-stabilizer (that automatically has positive dimension).

(ii) The open subset H(s) \ ⋃_{j≠i} H_{j} of H(s) is an (n − 1)-dimensional G-orbit and every (n − 1)-dimensional G-orbit is of this form.

Proof. Part (i) immediately follows from (4), (28), and (12), and part (ii) follows from (i), the invariance of H(α), and the equality dim G = dim H(α) = n − 1. □

5. The group N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n}))

First, we shall prove several general statements about normalizers for the actions on arbitrary affine varieties.

Lemma 12. Let X be an irreducible affine variety and let G be an algebraic subgroup of Aut X. Then the following properties are equivalent:

(i) N_{Aut X}(G) is an algebraic subgroup of Aut X.

(ii) The natural action of N_{Aut X}(G) on k[X] is locally finite.

Proof. (i)⇒(ii) This follows from the fact that the natural action on k[X] of every algebraic subgroup of Aut X is locally finite (see [Bo91, Prop. 1.9]).

(ii)⇒(i) Assume that (ii) holds. Then in k[X] there is an N_{Aut X}(G)- invariant finite dimensional k-linear subspace V containing a system of generators of the k-algebra k[X]. Hence the homomorphism

ρ: N_{Aut X}(G) → GL(V^{∗})

determined by the action of N_{Aut X}(G) on V is an embedding. Consider the N_{Aut X}(G)-equivariant map

ι: X → V^{∗}, ι(x)(f) := f(x) for every x ∈ X, f ∈ V.

The standard argument (see [Bo91, Prop. 1.12]) shows that ρ|_{G} is a morphism of algebraic groups, and ι is a closed embedding. Identify X with ι(X) by means of ι, and N_{Aut X}(G) with ρ(N_{Aut X}(G)) by means of ρ. Then X is a closed subvariety of V^{∗}, and N_{Aut X}(G) and G are the subgroups of GL(V^{∗}); besides, G is closed, and

N_{Aut X}(G) ⊂ N_{GL(V^{∗})}(G) ∩ Tran_{GL(V^{∗})}(X, X), where  (38)
Tran_{GL(V^{∗})}(X, X) := {g ∈ GL(V^{∗}) | g · X ⊂ X}.  (39)

In fact, in the right-hand side of (39) the equality g·X = X automatically holds: indeed, X is irreducible and closed in V^{∗}, and g ∈ Aut V^{∗} implies that g · X is a closed subset of X of the same dimension as X. Hence Tran_{GL(V^{∗})}(X, X), as well as N_{GL(V^{∗})}(G), is a subgroup of GL(V^{∗}), and therefore, the right-hand side of (38) is a subgroup of GL(V^{∗}). Its elements normalize G an, being restricted to X, are the automorphisms of X; whence, they lie in N_{Aut X}(G). Therefore,

N_{Aut X}(G) = N_{GL(V^{∗})}(G) ∩ Tran_{GL(V^{∗})}(X, X).  (40)

¿From closedness of G in GL(V^{∗}) and that of X in V^{∗} we deduce, respectively, that N_{GL(V^{∗})}(G) and Tran_{GL(V^{∗})}(X, X) are closed in GL(V^{∗}) (see [Bo91, Prop. 1.7]). This and (40) imply that N_{Aut X}(G) is closed in GL(V^{∗}). Hence N_{Aut X}(G) is an algebraic subgroup of Aut X. □

Theorem 5. Let X be an irreducible affine variety and let G be a reductive algebraic subgroup in Aut X such that

k[X]^{G} = k.  (41)

In either of the following cases N_{Aut X}(G) is the algebraic subgroup of Aut X: (i) G has a fixed point in X. (ii) G^{0} is semisimple.

Proof. Take f ∈ k[X]. We shall prove that in each of cases (i) and (ii) the k-linear span of the orbit N_{Aut X}(G) · f is finite dimensional. The claim of the theorem will then follow from Lemma 12.

Let M(G) be the set of isomorphism classes of algebraic simple G-modu-les. If L is an algebraic G-module, denote by L_{μ} its isotypic component of type μ ∈ M(G).

Since G is reductive, we have (see [PoVi94, 3.13])

k[X] = ⨁_{μ∈M(G)} k[X]_{μ}  (42)

The group N_{Aut X}(G) permutes the isotypic components of the G-module k[X].

Since k[X]_{μ} is a finitely generated k[X]^{G}-module (see [PoVi94, Thm. 3.24]), (41) implies that

dim_{k} k[X]_{μ} < ∞ for every μ.  (43)

In view of (42), there are elements μ_{1}, …, μ_{s} ∈ M(G) such that

k[X]_{μ_{i}} ≠ 0 for all i, and f ∈ k[X]_{μ_{1}} ⊕ ⋯ ⊕ k[X]_{μ_{s}}.  (44)

(i) Assume that in X there is a G-fixed point a. Since closed orbits are separated by G-invariant regular functions (see [PoVi94, Thm. 4.7]), it follows from (41) that there are no other G-fixed points in X. Hence a is fixed by N_{Aut X}(G) as well. Therefore, the ideal

𝔪_{a} := {f ∈ k[X] | f(a) = 0}

is N_{Aut X}(G)-invariant. Hence every member of the decreasing filtration

𝔪_{a} ⊃ ⋯ ⊃ 𝔪_{a}^{d} ⊃ 𝔪_{a}^{d+1} ⊃ · · · .  (45)

is N_{Aut X}(G)-invariant. This filtration has the property (see [AtMa69, Cor. 10.18]) that

⋂_{d} 𝔪_{a}^{d} = 0.  (46)

In view of (45), we have a decreasing system of nested linear subspaces {k[X]_{μ} ∩ 𝔪_{a}^{d} | d = 1, 2, …}. Since they are finite dimensional (see (43)), there is d_{μ} such that k[X]_{μ} ∩ 𝔪_{a}^{d} = k[X]_{μ} ∩ 𝔪_{a}^{d+1} for every d ⩾ d_{μ}. From (46) it then follows that, in fact,

k[X]_{μ} ∩ 𝔪_{a}^{d} = 0 for every d ⩾ d_{μ}.  (47)

Let l ∈ Z, l ⩾ max{d_{μ_{1}}, …, d_{μ_{s}}}. ¿From (47) it follows that

k[X]_{μ_{i}} ∩ 𝔪_{a}^{l} = 0 for every i = 1, …, s.  (48)

Since the natural projection π: k[X] → k[X]/𝔪_{a}^{l} is an epimorphism of G-modules, we have π(k[X]_{μ}) = (k[X]/𝔪_{a}^{l})_{μ} for every μ ∈ M(G). Whereas

dim_{k} k[X]/𝔪_{a}^{l} < ∞,

(see [AtMa69, Prop. 11.4]), this implies finiteness of the set of μ ∈ M(G) such that

k[X]_{μ} ≠ 0 k[X]_{μ} ∩ 𝔪_{a}^{l} = 0. (49)

Let {λ_{1}, …, λ_{t}} be this set. Since (49) holds for μ = μ_{1}, …, μ_{s} (see (44), (48)), we may assume that

λ_{i} = μ_{i} i = 1, …, s.  (50)

Whereas the group N_{Aut X}(G) permutes the isotypic components of the Gmodule k[X] and sends 𝔪_{a}^{l} into itself, it permutes k[X]_{λ_{1}}, …, k[X]_{λ_{t}}. Hence k[X]_{λ_{1}} ⊕ ⋯ ⊕ k[X]_{λ_{t}} is an N_{Aut A^{n}}(G)-invariant subspace of k[X]. In views of (43), (44), and (50), it is finite dimensional and contains f. Hence the k-linear span of the orbit N_{Aut X}(G) · f is finite dimensional, as claimed.

(ii) Let G^{0} be semisimple. From the Weyl formula for dimension of simple G^{0}-module and finiteness of the index [G: G^{0}] it follows that, up to isomorphism, there are only finitely many algebraic simple G-modules whose dimension does not exceed any preassigned constant. This implies finiteness of the set of μ ∈ M(G) such that

k[X]_{μ} ≠ 0 dim_{k} k[X]_{μ} ⩽ max_{i} dim_{k} k[X]_{μ_{i}}.

Let {λ_{1}, …, λ_{t}} be this set. We may assume that (50) holds. Since N_{Aut X}(G), permuting isotypic components, preserves their dimensions, k[X]_{λ_{1}} ⊕ ⋯ ⊕ k[X]_{λ_{t}} is invariant with respect to N_{Aut X}(G). The proof can be now completed as in case (i). □

Corollary 9. Let X and G be the same as in Theorem 5 and let k = C. Assume that the variety X is simply connected and smooth, and χ(X) = 1. Then N_{Aut X}(G) is an algebraic subgroup of Aut X.

Proof. By corollary of theorem on étale slice (see [Lu73, III, Cor. 2] and [PoVi94, Thm. 6.7]), condition (41) and smoothness of X imply that X is a homogeneous vector bundle over the unique closed G-orbit O in X. Hence O is simply connected and χ(X) = χ(O). Being affine, O is isomorphic to G/H for some reductive subgroup H (see [PoVi94, Thm. 4.17]). The conditions that G/H is simply connected and χ(G/H) = 1 imply that G = H (see [KrPo85, 5.1]). Hence O is a fixed point. The claim now follows from Theorem 5(i). □

Corollary 10. Let G be an reductive algebraic subgroups of Aut A^{n} and k[A^{n}]^{G} = k. Then N_{Aut A^{n}}(G) is the algebraic subgroup of Aut A^{n}.

Proof. In view of char k = 0, by the Lefschetz principle [Ha77, 15.1] we may assume that k = C. Since A^{n} is simply connected and smooth, and χ(A^{n}) = 1, the claim follows from Corollary 9. □

Remark 5. The following example shows that condition (41) alone (for irreducible affine X and reductive G) does not, in general, imply that N_{Aut X}(G) is algebraic.

Example 1. Let G be an algebraic torus of dimension n ⩾ 2. Take as X the underlying variety of the algebraic group G. Its group of automorphisms Aut_{gr}G is embedded in Aut X and is isomorphic to GL_{n}(Z). The action of G on X by left translations embeds G in Aut X. These two subgroups generate Aut X, more precisely, Aut X = Aut_{gr}G ⋉ G. Therefore, N_{Aut X}(G) = Aut X. Let g ∈ Aut_{gr}G be an element of infinite order and let f_{1}, …, f_{n} ∈ k[X] be a base of X(G). Then g^{d}·f_{i} ∈ X(G) for every d ∈ Z and i = 1, …, n, and the set C_{i} := {g^{d} · f_{i} | i ∈ Z} is finite if and only if the stabilizer of f_{i} with respect to the cyclic group generated by g is nontrivial. Assume that all C_{1}, …, C_{n} are finite. Then there is d ∈ Z, d ≠ 0, such that g^{d} ·f_{i} = f_{i} for every i = 1, …, n. Since f_{1}, …, f_{n} is a base in X(G), this means that the autormorphism g^{d} is trivial contrary to the assumption that the order of g is infinite and d ≠ 0. Hence C_{i} is infinite for some i. Since different characters are linear independent over k (see [Bo91, Lemma 8.1]), this implies that the k-linear span of the set C_{i} (and hence of the orbit N_{Aut X}(G) · f_{i}) is infinite dimensional.

We shall now use the obtained information in the proof of algebraicity of the groups N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})).

Theorem 6. N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) for every l_{1}, …, l_{n} is an algebraic subgroup of Aut A^{n}. Moreover,

(i) N_{Aut A^{n}}(D_{n}) = N_{GL_{n}}(D_{n}).

(ii) If (l_{1}, …, l_{n}) ≠ (0, …, 0), then

N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) ⊆ N_{GL_{n}}(D_{n})  (51)

in either of the following cases:

(a) all the numbers l_{1}, …, l_{n} are nonzero and have the same sign;

(b) none of the numbers l_{1}, …, l_{n} is equal to ±1;

(c) the numbers l_{1}, …, l_{n} contain 0 and ±1, at least two of them are nonzero, and all nonzero of them have the same sign.

(iii) If l_{i} = 1 and l_{j} = 0 for j ≠ i, then N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) is isomorphic to N_{GL_{n−1}}(D_{n−1}) × Aff_{1} and consists of all (g_{1}, …, g_{n}) ∈ Aff_{n} of the form

g_{j} = \begin{cases}t_{j}x_{σ(j)} & j ≠ i; \\ t_{j}x_{j} + s & j = i, \end{cases}  (52)

where t_{1}, …, t_{n}, s∈k and σ is a permutation of the set {1,…, i−1, i+ 1,…, n}.

Proof. If G is a subgroup of Aut A^{n} and g ∈ N_{Aut A^{n}}(G), a ∈ A^{n}, then

g(G · a) = G · g(a), gG_{a}g^{−1} = G_{g(a)}, g(\overline{G · a}) = \overline{g(G · a)}, and if G is algebraic, dim G · a = dim g(G · a).

(53)

Let G = D_{n}. Lemma 5(i) implies that ⋃_{i=1}^{n} H_{i} is the set of points whose G-stabilizer has positive dimension. By (53), it is g-invariant. Since the restriction of g to the variety ⋃_{i=1}^{n} H_{i} is its automorphism, g permutes its irreducible components H_{1}, …, H_{n}, i.e., there is a permutation σ ∈ S_{n} such that

g(H_{i}) = H_{σ(i)} for every i = 1, …, n.  (54)

Since the ideal in k[A^{n}] determined by H_{i} is generated by x_{i}, this shows that the polynomial g^{∗}(x_{i}) divides x_{σ(i)}, hence g^{∗}(x_{i}) = t_{i}x_{σ(i)} for some nonzero t_{i} ∈ k. Therefore, g ∈ N_{GL_{n}}(D_{n}) (see (3), (8)). This proves (i).

Further, let G = D_{n}(l_{1}, …, l_{n}), (l_{1}, …, l_{n}) ≠ (0, …, 0).

Assume that condition (a) holds. Then (53) and Corollary of Lemma 7 imply that g permutes the orbits O_{1}, … O_{n}, i.e., there exists a permutation σ ∈ S_{n} such that g(O_{i}) = O_{σ(i)} for every i. In view of Lemma 6 and (53), this implies that g has property (54), and as is shown above, the latter implies that g ∈ N_{GL_{n}}(D_{n}). Thus, (51) is proved in the case when condition (a) holds.

Assume that condition (b) holds. Then g-invariance of ⋃_{i=1}^{n} H_{i} follows from (53) and Lemma 8. Now the same argument as for G = D_{n} completes the proof of (51) in the case when condition (b) holds.

Assume that condition (c) holds. Proving (51), we may replace G by the group conjugate to G by means of an appropriate element of N_{GL_{n}}(D_{n}) and assume that (35) holds. The set {a ∈ A^{n} | dim G_{a} > 0} is closed (see [PoVi94, 1.4]). By virtue of (53), it is g-invariant, and Lemmas 5 and 10(ii) imply that its (n−1)-dimensional irreducible components are H_{q+1}, … …, H_{n}. Hence g permutes H_{q+1}, …, H_{n}. Further, it follows from Lemmas (5) and (10)(ii) that O_{p+1}, …, O_{q} are all the G-orbits O in A^{n} such that G_{a} is finite and nontrivial for a ∈ O. From (53) it then follows that g permutes the orbits O_{p+1}, …, O_{q} and, therefore, permutes their closures H_{p+1}, …, H_{q}. Finally, in view of Lemmas 5 and 10(ii), all the G-orbits O in A^{n} such that G_{a} is trivial for a ∈ O are exhausted by the orbits O_{1}, …, O_{p} and G · a, where a ∉ ⋃_{i=1}^{n} H_{i}. Since G is reductive, every G-orbit in A^{n} contains in its closure a unique closed G-orbit (see [PoVi94, Cor. p. 189]). By Lemma 6, for each of the orbits O_{1}, …, O_{p} this closed orbit is the fixed point (0, …, 0). On the other hand, by Lemma 10(i), the closed orbit lying in \overline{G · a} for a ∉ ⋃_{i=1}^{n} H_{i} has dimension q − 1 ⩾ 1 and, therefore, is not the fixed point. Hence g permutes O_{1}, …, O_{p}, and therefore, permutes their closures H_{1}, …, H_{p}. Thereby, it is proved that (54) holds for a certain permutation σ ∈ S_{n}. As above, this allows to conclude that (51) holds in case (c). Thus, (ii) is proved.

Assume now that l_{i} = 1 and l_{j} = 0 for j ≠ i. From (53) and Lemma 11(i) it follows that the closed set ⋃_{j≠i} H_{j} is invariant with respect to g. Hence g permutes its irreducible components H_{1}, …, H_{i−1}, H_{i+1}, …, H_{n}. As above, from here we conclude that for j ≠ i the equality g_{j} = t_{j}x_{σ(j)} holds for some t_{j} ∈ k and some permutation σ of the set {1, …, i − 1, i + 1, …, n}. Further, (53) and Lemmas 11(i), 11(ii) imply that g(O_{i}) is an orbit open in a certain hyperplane H(c) (see Lemma 11(ii)). Since \overline{O_{i}} = H_{i}, and x_{i} and x_{i} + c are, respectively, the generators of the ideals of hyperplanes H_{i} and H(c), we conclude that g_{i} = g^{∗}(x_{i}) differs from x_{i} + c only by a nonzero constant factor: g_{i} = t_{i}x_{i} + s for some t_{i}, s ∈ k, t_{i} ≠ 0. Thus, g is of the form (52). Conversely, clearly, every element g ∈ Aut A^{n} of the form (52) normalizes G. This proves (iii).

Finally, let us prove the first claim of this theorem. According to (i), (ii), and (iii), the group N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) is algebraic if either (l_{1}, …, l_{n}) = (0, …, 0), or (l_{1}, …, l_{n}) ≠ (0, …, 0) and any of conditions (a), (b), (c) of statement (ii) or condition of statement (iii) hold. The only case not covered by these conditions is that when among l_{1}, …, l_{n} there are numbers with different signs. However, it follows from Lemma 9 that in this case there are nonconstant D_{n}(l_{1}, …, l_{n})-invariant regular functions on A^{n}. But then by Corollary 2 of Theorem 5 we conclude that N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})) is algebraic. □

6. Fusion theorems for tori in Aut A^{n} and Aut^{∗}A^{n}

Fusion theorems describe subgroups that control fusion of subsets under conjugation. Namely, let G be an (abstract) groups and let H be its subgroup. We say that N_{G}(H) controls fusion of subsets in H under conjugation by the elements of G, if the following property holds:

(F) for any subset S ⊆ H and an element g ∈ G such that gSg^{−1} ⊆ H, there is an element w ∈ N_{G}(H), such that gsg^{−1} = wsw^{−1} for every element s ∈ S.

If property (F) holds for a pair (G, H), then we say that for H in G fusion theorem holds. Notice that the action of N_{G}(H) on H by conjugation boils down to the action of the “Weyl group” N_{G}(H)/Z_{G}(H).

Examples 7. Fusion theorem for H in G holds in the following cases:

1. G is a finite group and H is its Abelian Sylow p-subgroup. It is the classical Burnside’s result.

2. G is an affine algebraic group and H is its maximal torus. It is the classical result of the theory of algebraic groups, see, e.g., [Se00, 1.1.1].

3. (G, H) = (Cr_{n}, D_{n}). It is Serre’s result [Se10, Thm. 1.1]. Since every n-dimensional torus in Cr_{n} is maximal and conjugate to D_{n} (see Theorem 2(i),(ii)), here one can replace D_{n} by any n-dimensional torus. Recall that for n ⩾ 5 in Cr_{n} there exist (n − 3)-dimensional maximal tori (see Corollary 7(iii)).

Question 8. Does fusion theorem hold if D_{n} is replaced by such a torus?

We shall now prove that fusion theorem holds for n-dimensional tori in Aut A^{n} and (n − 1)-dimensional tori in Aut^{∗}A^{n}.

Lemma 13. For every element g = (g_{1}, …, g_{n}) ∈ Aut A^{n} there is an element g′ ∈ SL_{n} such that if s, gsg^{−1} ∈ GL_{n}, then

gsg^{−1} = g′sg′^{−1}.  (55)

Proof. Let g_{i} = g_{i}^{(0)}+g_{i}^{(1)}+…, where g_{i}^{(s)} is a form of degree s in x_{1}, …, x_{n}. Since Jac(g) ∈ k, we have Jac(g) = det(∂g_{i}/∂x_{j}|_{x_{1}=...=x_{n}=0}). But the righthand side of this equality is equal to det(∂g_{i}^{(1)}/∂x_{j}). Hence

g^{(1)} := (g_{1}^{(1)}, …, g_{n}^{(1)}) ∈ GL_{n}.  (56)

The automorphism g^{(1)} is the differential of the automorphism g at the point (0, …, 0). From (56) it follows that

(i) g = g^{(1)} if g ∈ GL_{n};
(ii) (ga)^{(1)} = g^{(1)}a and (ag)^{(1)} = ag^{(1)} if a ∈ GL_{n}. (57)

Now let s and t := gsg^{−1} be the elements of GL_{n}. In view of (56), we have g^{(1)} ∈ GL_{n}, and gs = tg and (57) imply that g^{(1)}s = tg^{(1)}. Therefore, the product of g^{(1)} by a constant α ∈ k such that α^{n} det g^{(1)} = 1 can be taken as g′. □

Theorem 9. The following pairs (G, H) have property (F):

(i) (Fusion theorem for n-dimensional tori in Aut A^{n})

(G, H) = (Aut A^{n}, an n-dimensional torus).

(ii) (Fusion theorem for (n − 1)-dimensional tori in Aut^{∗}A^{n})

(G, H) = (Aut^{∗}A^{n}, an (n − 1)-dimensional torus).

Proof. (i) Let G = Aut A^{n}, let H be a torus, and dim H = n. Then, by Theorem 3 we may assume that H is a maximal torus in GL_{n}. Maintain the notation of the formulation of property (F). By Lemma 13 there exists an element g′ ∈ GL_{n} such that for every element s ∈ S equality (55) holds. Therefore, g′Sg^{′−1} = gSg^{−1} ⊆ H. Since g′ ∈ GL_{n}, this shows that

H′ := g^{′−1}Hg′  (58)

is another maximal torus in GL_{n} containing S. The tori H and H′ lie in the (closed [Bo91, I.1.7]) subgroup Z_{GL_{n}}(S) of the group GL_{n} and therefore, are maximal tori of Z_{GL_{n}}(S). In view of conjugacy of maximal tori in any affine algebraic group [Bo91, IV.11.3], there is an element

z ∈ Z_{GL_{n}}(S),  (59)

such that

H′ = zHz^{−1}.  (60)

From (58) and (60) it follows that w := g′z ∈ N_{GL_{n}}(H), and from (59) and (55) that gsg^{−1} = wsw^{−1} for every s ∈ S. This proves (i).

(ii) The same argument applies in the case where G = Aut^{∗}A^{n}, H is a torus, and dim H = n−1: one only has to replace GL_{n} by SL_{n}, the reference to Theorem 3 by the reference to Theorem 4, and notice that by Lemma 13 the element g′ may be taken from SL_{n}. □

7. Applications: The classifications of classes of conjugate subgroups

In this section, using the obtained results, we derive the classifications specified in the Introduction and, in particular, prove the generalizations to disconnected groups of the Białynicki-Birula’s theorems on linearization of actions on A^{n} of tori of dimension ⩾ n − 1.

Theorem 10. Let G be an algebraic subgroups of Aut A^{n} such that G^{0} is a torus.

(i) If dim G = n or n − 1, then there is an element g ∈ Aut A^{n} such that gGg^{−1} ⊂ GL_{n} and gG^{0}g^{−1} ⊂ D_{n}.

(ii) If G ⊂ Aut^{∗}A^{n} and dim G = n − 1, then there is an element g ∈ Aut^{∗}A^{n} such that gGg^{−1} ⊂ SL_{n} and gG^{0}g^{−1} = D_{n}^{∗} (see (9)).

Proof. (i) By virtue of conjugacy of maximal tori in GL_{n}, it suffices to prove the existence of g ∈ Aut A^{n} such that gGg^{−1} ⊂ GL_{n}.

The group G is reductive. Therefore, by Corollary of Theorem on étale slice [Lu73, Cor. 2, p. 98] the claim holds if k[A^{n}]^{G} = k (here only reductivity of G is used, not the stronger condition that G^{0} is a torus). Therefore, in what follows we may assume that k[A^{n}]^{G} ≠ k. The latter is equivalent to the condition

k[A^{n}]^{G^{0}} ≠ k.  (61)

In view of Theorem 3 and equality (13), replacing G by a conjugate group, we may assume that

G^{0} = D_{n}(l_{1}, …, l_{n}) ⊂ GL_{n}.  (62)

The claim on the existence of g will be proved if we show that (62) and (61) imply the inclusion

G ⊂ GL_{n}.  (63)

Let F be a finite subgroup of G that intersects every connected component of this group — such a subgroup exists, see [BoSe64, Lemme 5.11]. Then

G = F G^{0}.  (64)

Since G^{0} is a normal subgroup of G, we have F ⊂ N_{Aut A^{n}}(D_{n}(l_{1}, …, l_{n})). In view of Theorem 6 this shows that

F ⊂ N_{GL_{n}}(D_{n}) ⊂ GL_{n},  (65)

if either (l_{1}, …, l_{n}) = (0, …, 0), or (l_{1}, …, l_{n}) ≠ (0, …, 0) and any of conditions (a), (b), (c) of statement (ii) of Theorem 6 holds. By virtue of (62) and (64), this proves (63) for the specified (l_{1}, …, l_{n})’s. From Lemma 9 and (61) it now follows that it remains to consider the last possibility for (l_{1}, …, l_{n}), namely, that where l_{i} = 1 and l_{j} = 0 if j ≠ i for certain i.

Turning to its consideration, take some element g ∈ F. Then, by Theorem 6(iii), equalities (52) hold. Since F is finite, the order of g is finite. This implies that in (52) we have s = 0. This and (8) mean that in the case under consideration (65) holds as well, and therefore, (63), too. This proves (i).

(ii) The proof is the same as that of (i) if Aut A^{n} is replaced by Aut^{∗}A^{n}, GL_{n} by SL_{n} = GL_{n} ⋂ Aut^{∗}A^{n}, D_{n}(l_{1}, …, l_{n}) by D_{n}^{∗}, the reference to Theorem 3 by the reference to Theorem 4, and it is taken into account that D_{n}^{∗} is the maximal torus of SL_{n}. □

Theorem 11 (Classification of n-dimensional diagonalizable subgroups of Aut A^{n} up to conjugacy in Aut A^{n}). Up to conjugacy in Aut A^{n}, the torus D_{n} is the unique n-dimensional diagonalizable subgroup of Aut A^{n}.

Proof. By virtue of Theorem 10 this follows from the fact that in GL_{n} every diagonalizable subgroup is conjugate to a subgroup of the torus D_{n} (see [Bo91, p. 112, Prop.(d)]). □

Theorem 12 (Classification of (n − 1)-dimensional diagonalizable subgroups of Aut^{∗}A^{n} up to conjugacy in Aut^{∗}A^{n}). Up to conjugacy in Aut^{∗}A^{n}, the torus D_{n}^{∗} is the unique (n − 1)-dimensional diagonalizable subgroup of Aut^{∗}A^{n}.

Proof. By virtue of Theorem 10, this follows form the fact that in SL_{n} every diagonalizable subgroup is conjugate to a subgroup of the torus D_{n}^{∗} (this easily follows from [Bo91, p. 112, Prop.(d)]). □

Theorem 13 (Classification up to conjugacy in Aut A^{n} of maximal n-dimen-sional algebraic subgroups G of Aut A^{n} such that G^{0} is a torus). Up to conjugacy in Aut A^{n}, the group N_{GL_{n}}(D_{n}) (see (8)) is the unique maximal algebraic subgroup of Aut A^{n} whose connected component of identity is an n-dimensional torus.

Proof. By virtue of Theorem 10, this follows from the fact that D_{n} = N_{GL_{n}}(D_{n})^{0}. □

Theorem 14 (Classification up to conjugacy in Aut^{∗}A^{n} of maximal (n−1)- dimensional algebraic subgroups G in Aut^{∗}A^{n} such that G^{0} is a torus). Up to conjugacy in Aut^{∗}A^{n},

N_{SL_{n}}(D_{n}^{∗}) = N_{GL_{n}}(D_{n}) ∩ SL_{n}

is the unique maximal algebraic subgroup of Aut^{∗}A^{n} whose connected component of identity is an (n − 1)-dimensional torus.

Proof. By virtue of Theorem 10, this follows from the fact that D_{n}^{∗} = N_{SL_{n}}(D_{n}^{∗})^{0}. □

Denote by L_{n} the additive monoid of all (l_{1}, …, l_{n}) ∈ Z^{n} such that

(a) l_{1} ⩽ ⋯ ⩽ l_{n};

(b) (l_{1}, …, l_{n}) ⩽ (−l_{n}, …, −l_{1}) with respect to the lexicographical or-

der on Z^{n}.

Theorem 15 (Classification of (n−1)-dimensional diagonalizable subgroups in Aut A^{n} up to conjugacy in Aut A^{n}).

(i) Every (n−1)-dimensional diagonalizable subgroup of the group AutA^{n} is conjugate in Aut A^{n} to a unique subgroup of the form

D_{n}(l_{1}, …, l_{n}), (l_{1}, …, l_{n}) ∈ L_{n} \ {(0, …, 0)}.  (66)

(ii) Every subgroup (66) is an (n − 1)-dimensional diagonalizable group.

Proof. Let G be an (n − 1)-dimensional diagonalizable subgroup of Aut A^{n}. Taking into account Theorem 10(i), conjugacy in GL_{n} of every diagonalizable subgroup of this group to a subgroup of the torus D_{n}, and Corollary 2(ii), we conclude that G is conjugate in Aut A^{n} to a certain subgroup D_{n}(l_{1}, …, l_{n}) with (l_{1}, …, l_{n}) ≠ (0, …, 0). It follows from the definition of L_{n} and Corollary 2(iv) that we may presume that (l_{1}, …, l_{n}) ∈ L_{n}. Assume that G is also conjugate in Aut A^{n} to a subgroup D_{n}(l′_{1}, …, l′_{n}) with (l′_{1}, …, l′_{n}) ∈ L_{n}. Then D_{n}(l_{1}, …, l_{n}) and D_{n}(l′_{1}, …, l′_{n}) are conjugate in Aut A^{n}, and therefore, Theorem 9(i) implies that they are conjugate in N_{GL_{n}}(D_{n}). In view of Corollary 2(iv) and Definition L_{n}, this implies that (l_{1}, …, l_{n}) = (l′_{1}, …, l′_{n}). This proves (i).

Statement (ii) follows from Corollary 2(ii). □

¿From Theorems 6, 11, and 15 we deduce

Theorem 16. If G is a diagonalizable subgroup of Aut A^{n} of dimension ⩾ n − 1, then N_{Aut A^{n}}(G) is an algebraic subgroup of Aut A^{n}.

Remark 6. It is easy to see that n − 1 in Theorem 16 cannot be replaced by a smaller integer.

Theorem 17 (Classification of (n−1)-dimensional diagonalizable subgroups of Aut A^{n} up to conjugacy in Cr_{n}).

(i) Two diagonalizable (n−1)-dimensional subgroups of the group Aut A^{n} are conjugate in Cr_{n} if and only if they are isomorphic.

(ii) Every (n−1)-dimensional diagonalizable subgroup of the group Aut A^{n} is conjugate in Cr_{n} to a unique closed subgroup of D_{n} of the form

ker ε_{n}^{d}, d ∈ Z.

Proof. This follows from Theorems 15 and 1 and Corollary 2(i). □

For n ⩽ 3, Theorems 2(i), 11, and 15 yield the classification of all tori in Aut A^{n} up to conjugacy in Aut A^{n} except for one-dimensional tori in Aut A^{3}. The classification of the latter is given below in Theorem 18.

Theorem 18 (Classification of one-dimensional tori in Aut A^{3} up to conjugacy in Aut A^{3}). Every one-dimensional torus in Aut A^{3} is conjugate in Aut A^{3} to a unique torus of the form T (l_{1}, l_{2}, l_{3}), where (l_{1}, l_{2}, l_{3}) ∈ L_{3}.

Proof. Let G be a one-dimensional torus in Aut A^{3}. By Theorem 3 it is conjugate in Aut A^{3} to some torus T (l_{1}, l_{2}, l_{3}). In view of (8) and the equality T (l_{1}, l_{2}, l_{3}) = T (−l_{1}, −l_{2}, −l_{3}), we may presume that (l_{1}, l_{2}, l_{3}) ∈ L_{3}. Assume that G is conjugate in Aut A^{3} to another torus T (l′_{1}, l′_{2}, l′_{3}) with (l′_{1}, l′_{2}, l′_{3}) ∈ L_{3}. Then T (l_{1}, l_{2}, l_{3}) and T (l′_{1}, l′_{2}, l′_{3}) are conjugate in Aut A^{3}, and therefore, by Theorem 9(i), also in N_{GL_{3}}(D_{3}). From this, Corollary 2(iv), the definition of L_{n}, and Lemma 4, it is then not difficult to deduce that (l_{1}, l_{2}, l_{3}) = (l′_{1}, l′_{2}, l′_{n}). □

8. Jordan decomposition in Cr_{n}. Torsion primes for the Cremona groups

Although the Cremona groups are infinite dimensional (this has a precise meaning, see [Ra64]), the analogies between them and algebraic groups strike the eye: they have the Zariski topology, algebraic subgroups, tori, roots, the Weyl groups, … In [Se10, 1.2] Serre writes about the analogy

“groupe de Cremona de rang n ⟷ groupe semi-simple de rang n”.

Below we briefly touch upon two topics demonstrating that these analogies extend further. The second of them is intimately related to tori in the Cremona groups.

Jordan decomposition in Cr_{n}.

Let X be an algebraic variety. “Algebraic families” endow Bir X with the Zariski topology [Ra64], [Bl10, 2], [Se10, 1.6]: a subset of Bir X is closed if and only if its inverse image for every algebraic family S → Bir X is closed. For every algebraic subgroup G in Bir X and its subset Z, the closures of Z in this topology and in the Zariski topology of the group G coincide. In particular, G is closed in Bir X.

Let us call an element g ∈ Bir X algebraic if in Cr_{n} there is an algebraic subgroup G containing g. This is equivalent to the property that the closure of the cyclic group generated by g is an algebraic group. If G is affine, then for g the Jordan decomposition in G is defined, see [Bo91, Chap. I, §4]:

g = g_{s}g_{n}  (67)

In fact, g_{s} and g_{n} depend only of g, not of the choice of G. Indeed, let G′ be another affine algebraic subgroup of Bir X containing g, and let g = g′_{s}g′_{n} be the Jordan decomposition in G′. Since G ∩ G′ is a closed subgroup of G and of G′, there exists the Jordan decomposition g = g′′_{s}g′′_{n} in G ∩ G′. Applying theorem on behaviour of the Jordan decompositions under homomorphisms (see [Bo91, Thm. 4.4(4)]) to the embeddings G ↩ G ∩ G′ ↪ G′, we obtain g_{s} = g′′_{s}, g_{n} = g′′_{n}, g′_{s} = g′′_{s}, g′_{n} = g′′_{n}. Hence, if we call (67) the Jordan decomposition in Bir X, we get the well-defined notion.

According to [Ma63], every algebraic subgroup of Cr_{n} is affine. Therefore, in Cr_{n} every algebraic element admits the Jordan decomposition.

Jordan decompositions in algebraic groups have several known properties [St65], [Bo91], for instance:

(a) Every semisimple element of a connected group lies in its torus.

(b) The set of all unipotent elements is closed.

(c) The conjugacy class of every semisimple element of a connected reductive group is closed.

(d) The closure of the conjugacy class of every element g of a connected reductive group contains g_{s}.

Recall that Cr_{n} is connected and, for n = 1, 2, simple [Bl10], and that Aut A^{n}, Aut^{∗}A^{n} are connected and Aut^{∗}A^{n} is simple [Sh66], [Sh82]. It is natural to ask:

Question 19. Are there analogues or modifications of the mentioned properties for the groups Cr_{n}, Aut A^{n}, and Aut^{∗}A^{n}?

For instance, property (a) for Cr_{n} holds if n = 1, but does not hold if n > 1. Indeed, if n > 2, then by [Po12, Thm. 4.3] in Cr_{n} there is a semisimple element of order two that is not contained in any connected algebraic subgroup of the group Cr_{n}. If n = 2, then by virtue of Theorem 2(ii) and Corollary 5, all the elements of order d < ∞ in Cr_{n} that can be included in tori constitute a single conjugacy class, while, for instance, for even d, the set of all elements of order d is the union of infinitely many conjugacy classes [Bl07]. On the other hand, in the groups Aut A^{2} and Aut^{∗}A^{2} property (a) holds, because action of every finite group on A^{2} is linearizable [Ig77].

Torsion primes for the Cremona groups.

Let G be a connected reductive algebraic group and let p be a prime integer. Recall (see [Se00, 1.3] and references therein) that p is called torsion prime for the group G if in G there is a finite Abelian p-subgroup not contained in any torus of the group G. The set Tors(G) of all torsion primes for the group G admits various interpretations. For instance, p ∈ Tors(G) for k = C if and only if ⨁_{i} H_{i}(G, Z) contains an element of order p (this explains the name). Finding Tors(G) is reduced to the case where G is simply connected. For every simple G the set Tors(G) is explicitly described.

The possibility to speak about tori in groups of the form Bir X where X is an algebraic variety, makes it possible to replace G in the above definition by Bir X or its subgroup thereby obtaining the well-defined notion. In particular, we consider what is obtained when G is replaced by Cr_{n}, Aut A^{n}, or Aut^{∗}A^{n} as the definitions of torsion primes for these groups. Denote the sets of these primes respectively by Tors(Cr_{n}), Tors(Aut A^{n}), and Tors(Aut^{∗}A^{n}). It is natural to ask

Question 20. What are, explicitly, the sets Tors(Cr_{n}), Tors(Aut A^{n}), and Tors(Aut^{∗}A^{n})?

(About Tors(Cr_{n}) this question has been formulated and discussed in talk [Po12].)

Since Cr_{1} = PGL_{2}, we have

Tors(Cr_{1}) = {2}.  (68)

According to [Bl07], for every d = 2, 3, and 5, in Cr_{2} there are infinitely many conjugacy classes of elements of order d. On the other hand, as is explained at the end of section on Jordan decomposition, in Cr_{2} there is the unique conjugacy class of elements of order d contained in tori. Hence 2, 3, and 5 are torsion primes for the group Cr_{2}. Consider a prime integer p > 5. According to [Fe04, Thm. E], every element g ∈ Cr_{2} of order p lies in a subgroup isomorphic to Aut P^{2} = PGL_{3}. Hence g lies in a torus. Finally, according to [Bl06_{2}, Thm. B, p. 146], in Cr_{2} there is a unique up to conjugacy noncyclic finite Abelian p-group (it is denoted by 0.mn), and this group is contained in a maximal torus of a subgroup isomorphic to (Aut (P^{1}×P^{1}))^{0} = PGL_{2}×PGL_{2}. Thus we conclude that the torsion primes for Cr_{2} are the same as that for the exceptional simple algebraic group E_{8}:

Tors(Cr_{2}) = {2, 3, 5}.  (69)

¿From Aut A^{1} = Aff_{1} and [Ig77] it follows that every finite subgroup of Aut A^{n} (respectively, Aut^{∗}A^{n}) for n ⩽ 2 is contained in a subgroup isomorphic to GL_{n} (respectively, SL_{n}). Therefore,

Tors(Aut A^{n}) = Tors(Aut^{∗}A^{n}) = {∅} for n ⩽ 2.

For n ⩾ 3, there is no comprehensive information about the sets Tors(Cr_{n}), Tors(Aut A^{n}), and Tors(Aut^{∗}A^{n}). From [Po12, Thm. 4.3], (68), and (69) it follows that

Tors(Cr_{n}) = {2, …} for every n.  (70)

According to [Be07], in Cr_{2} there is a 3-elementary Abelian subgroups of rank 3 (i.e., isomorphic to (μ_{d})^{3}). Since Cr_{1} contains a cyclic subgroup of order 3, and the direct product of Cr_{2} and n − 2 copies of Cr_{1} can be embedded in Cr_{n} for n ⩾ 3, this implies that in Cr_{n} there is a 3-elementary Abelian subgroup G of rank n + 1. But from Lemma (2)(i) it follows that for every prime integer p, the rank of any elementary Abelian p-subgroup of an r-dimensional torus is at most r. From this and Theorem (2)(i) it follows that G is not contained in a torus of Cr_{n}. In view of (69) and (70), this yields

Tors(Cr_{n}) = {2, 3, …} for any n ⩾ 2.

Question 21. What is the minimal n such that 7 lies in one of the sets Tors(Cr_{n}), Tors(Aut A^{n}), and Tors(Aut^{∗}A^{n})?

Question 22. Are these sets finite?

References

[AtMa69] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Ma, 1969.

[Be07] A. Beauville, p-elementary subgroups of the Cremona group, J. Algebra 314 (2007), no. 2, 553–564.

[B-B66] A. Białynicki-Birula, Remarks on the action of an algebraic torus on k^{n},

Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 177–181.

[B-B67] A. Białynicki-Birula, Remarks on the actionof an algebraic torus on k^{n}, II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 123– 125.

[Bl06_{1}] J. Blanc, Conjugacy classes of affine automorphisms of K^{n} and linear automorphisms of P^{n} in the Cremona groups, Manuscripta Math. 119 (2006), no. 2, 225–241.

[Bl06_{2}] J. Blanc, Finite abelian subgroups of the Cremona group of the plane,

Thèse No. 3777, Université de Genève, Faculté des Sciences, 2006.

[Bl07] J. Blanc, The number of conjugacy classes of elements of the Cremona group of some given finite order, Bull. Soc. Math. France 135 (2007), no. 3, 419–434.

[Bl10] J. Blanc, Groupes de Cremona, connexité et simplicité, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 2, 357–364.

[Bo91] A. Borel, Linear Algebraic Groups, 2nd ed., Springer-Verlag, New York, 1991.

[BoSe64] A. Borel, J.-P. Serre, Théorèmes de finitude en cohomoplogie galoisienne, Comm. Math. Helv. 39 (1964), 111–164.

[De70] M. Demazure, Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. de l’E.N.S., 4e sér. 3 (1970), fasc. 4, 507–588.

[Fe04] T. de Fernex, On planar Cremona maps of prime order, Nagoya Math. J. 174 (2004), 1–28.

[Ha77] J. Harris, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 133, Springer, New Yorks, 1992.

[Ig77] T. Igarashi, Finite subgroups of the automorphism group of the affine plane, M.A. thesis, Osaka University, 1977.

[KKM-LR97] S. Kaliman, M. Koras, L. Makar-Limanov, P. Russell, ℂ^{∗}-actions on ℂ^{3} are linearizable, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 63–71 (electronic).

[KrPo85] H. Kraft, V. L. Popov, Semisimple group actions on the three dimensional affine space are linear, Comment. Math. Helvetici 60 (1985), 466–479.

[La81] G. Laumon, Comparaison de caracteristiques d’Euler–Poincaré en cohomologie l-adique, C. R. Acad. Sci. Paris 292 (1981), 209–212.

[Li11] A. Liendo, Roots of the affine Cremona group, Transform. Groups 16 (2011), no. 4, 1137–1142.

[Lu73] D. Luna, Slices étales, Bull. Soc. Math. France 33 (1973), 81–105.

[MaMi64] M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964.

[Ma63] H. Matsumura, On algebraic groups of birational transformations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8 (1963), 34, 151–155.

[Po70] V. L. Popov, Stability criteria for the action of a semisimple group on a factorial manifold, Math. USSR Izv. 4 (1970), 527–535.

[Po05_{1}] V. L. Popov, Problems for problem session, arXiv:0504517v125Apr2005. [Po05_{2}] V. L. Popov, Roots of the affine Cremona group, in: Affine Algebraic Ge-

ometry, Seville, Spain, June 18–21, 2003, Contemporary Mathematics, Vol. 369, American Mathematical Society, Providence, RI, 2005, pp. 12–13.

[Po11] V. L. Popov, Some subgroups of the Cremona groups, to appear in: Proceedings of Conference on Affine Algebraic Geometry, Osaka, March 3–6,

2011, World Scientific, arXiv:1110. 2410v111Oct2011.

[Po12] V. L. Popov, Tori in the Cremona groups, Talk delivered at the International workshop Essential Dimension and Cremona Groups, Chern Institute of Mathematics, Nankai University, Tianjin, China, June 11–15, 2012,

http://www.nim.nankai.edu.cn/activites/conferences/hy20120611/in-
dex.htm.

[PoVi94] V. L. Popov, E. B. Vinberg, Invariant theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

[Ra64] C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33.

[Ro56] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.

[Se55] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. 61 (1955), 197–278.

[Se00] J.-P. Serre, Sous-groupes finis des groupes de Lie, in: Séminaire N. Bourbaki 1998/99, Exp. no. 864, Astérisque, Vol. 266, Société Mathématique de France, 2000, pp. 415–430.

[Se10] J.-P. Serre, Le groupe de Cremona et ses sous-groupes finis, in: Séminaire N. Bourbaki 2008/09, Exp. no. 1000, Astérisque, Vol. 332, Société Mathématique de France, 2010, pp. 75–100.

[St65] R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49–80.

[Sh66] I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5) 25 (1966), nos. 1–2, 208–212.

[Sh82] I. R. Shafarevich, On some infinite-dimensional groups. II, Mathematics of the USSR-Izvestiya 18 (1982), no. 1, 185–194.

[ShRe09] �� �� ���������� �� �� �������, �������� ������� � ���������, ���������� ��� 2009. [I. R. Shafarevich, A. O. Remizov, Linear Algebra and Geometry, Fizmatlit, Moscow, 2009 (in Russian).]

[Vi03] E. B. Vinberg, A Course in Algebra, Graduate Studies in Mathematics, Vol. 56, American Mathematical Society, Providence, RI, 2003.

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

E-mail address: popovvl@mi.ras.ru