ON MULTICOMMUTATORS FOR SIMPLE ALGEBRAIC GROUPS

NIKOLAI GORDEEV^{∗} AND ULF REHMANN

Abstract. There are several examples of groups for which any pair of commutators can be written such that both of them have a common entry, and one can look for a similar property for n-tuples of commutators. We here answer, for simple algebraic groups over any field, the weaker question, under which condition the set of n-tuples of commutators with one common entry is Zariski dense in the set of all n-tuples of commutators. Surprisingly, there is a uniform bound on n in terms of the so called Coxeter number of G in order to answer the question positively. An analogoue result is proved for Lie algebras of simple and simply conncected algebraic groups. 2000 Mathematics Subject Classification: primary: 20G15; secondary: 20F12, 20E45 Keywords and Phrases: Groups, Commutators, Simple Algebraic Groups

1. Introduction

Let G be an abstract group and let G^{n} = G × ⋯ × G (n factors). Further, let

ϕ_{n} : G × G^{n} ⟶ G^{n}

be the map given by the formula

ϕ_{n}((g, g_{1}, …, g_{n})) = ([g, g_{1}], …, [g, g_{n}]).

We say that the group G satisfies the property C_{n}, if Im ϕ_{n} = [G, G]^{n} where [G, G] is the commutator subgroup of G, i.e., for every σ_{1}, …, σ_{n} ∈ [G, G]^{n}, there exists a sequence g, g_{1}, …, g_{n} such that

σ_{1} = [g, g_{1}], σ_{2} = [g, g_{2}], …, σ_{n} = [g, g_{n}].

(We define [g, h] = ghg^{−1}h^{−1}.)

The case n = 1 is well known in group theory and has a long history. If G is a finite simple group, the question about the property C_{1} is the well-known Ore problem, whether any element in the commutator subgroup of G is a single commutator, which is answered positively for (simple) alternating groups (Ore [O], 1951), for sporadic groups (Neubüser, Pahlings and Cleuvers [NPC], 1984) and for finite simple Lie groups over fields with at least 9 elements (see [EG]). Is is also known that in general the answer is negative, even for groups like SL_{n}(K) (cf. [THO, Th. 1]), or for finite groups G including perfect finite groups (cf. [I]).

∗ Supported by the SFB 343 “Diskrete Strukturen in der Mathematik” of the Deutsche Forschungsgemeinschaft and by the grants INTAS-99-00817 and TMR-Network ERB FMRX CT-97-0107 of the European Commission.

The question for n > 1 was posed several years ago by R. Keith Dennis (cf. [C, prob. 14, p. 605]). In [AD], it was shown, among other things, that the Schur multiplier of the group SL_{1}(ℍ) of the Hamilton quaternions ℍ is generated by the image of the Schur multiplier of the commutative subgroup of complex numbers of norm one, a result, which allowed to determine the group K_{2}(ℍ). In the proof, the property C_{2} for the group ℍ^{∗} played an important role, and it even turned out that this group as well as the multiplicative group of an arbitrary quaternion skew field satisfies C_{3}. The main idea for this is already contained in [RS, §4, proof of 4.1].

Many years ago, R. Keith Dennis and the second named author verified, by computer and using a special purpose program (written in the programming language C), that the alternating groups A_{n} with n = 5, 6, 7, 8, 9, 10 satisfy C_{2}. It was also shown that A_{5} does not satisfy C_{3}.

The first named author verified C_{2} for the group SL_{2}(ℂ) by an explicit computation using the Bruhat decomposition of that group.

Here we look at the property C_{n} for a simple algebraic group G defined over a field K. It seems to be rather difficult to investigate the property C_{n} for the group G(K) of K-rational points even if K is algebraically closed. If K is not algebraically closed, then already the property C_{1} is a problem for G(K): In [THO], examples of elements in SL_{n}(ℝ) are given which are in the commutator subgroup but no single commutators, and in [KU] there is an example of this type in some non split group. (See also [EG].) We restrict ourselves to a weaker condition. Namely, we say that a simple algebraic group G satisfies the property \overline{C}_{n} if the map ϕ_{n} is dominant, i.e., if the Zariski closure \overline{Im ϕ_{n}} of Im ϕ_{n} in G^{n} coincides with G^{n}. Thus, the property \overline{C}_{n} is the “property C_{n} up to Zariski closure”.

The main result of this paper is the following theorem.

Theorem 1. Let G be a simple algebraic group and let h = h(G) be the Coxeter number of the corresponding root system. Then G satisfies the property \overline{C}_{n} if and only if n ≤ h+1.

Remark 1. Recall that the Coxeter number h of a root system is the order of special elements of its Weyl group (Coxeter elements) (see [B]). If the root system belongs to a simple algebraic group G, we have

h + 1 = \frac{dim G}{dim T}

where T is a maximal torus of G. Thus, G satisfies the property \overline{C}_{n} if and only if n ≤ dim G/ dim T. Remark 2 At the end of this paper we show that ϕ_{n} is a separable morphism if n ≤ h and that ϕ_{n} is not separable if n = h + 1 and if the center of the corresponding Lie algebra is not trivial.

On the basis of Theorem 1 we could propose a conjecture about the property C_{k} for groups of points G(K). Say, we may suppose that C_{1} implies C_{h+1} for such groups under the assumption that G is an adjoint group. Since the property C_{1} is satisfied by a big massive of quasisplit groups of adjoint type (see [EG]) we may suppose that the property C_{h+1} is satisfied by such groups (possibly except in the case when K is a small field). At any rate, there is a strong hope that all groups G(K) satisfy the property C_{h+1}, where G is a group of adjoint type and K is algebraically closed field.

Our example below after Remark 5 shows that C_{3} is satisfied by the groups GL_{1} and PGL_{1} over any quaternion skew field: Here, the latter group is an anisotropic adjoint simple algebraic group, and we have dim PGL_{1} = 3 and dim T = 1 for any maximal torus T of PGL_{1}.

An analogue of Theorem 1 for Lie algebras is the following result.

Theorem 2. Let L be the Lie algebra of a simple and simply connected algebraic group defined over a field K and corresponding to an irreducible root system R. Let L^{n} = L ⊕ ⋯ ⊕ L (n summands) and let

Ψ_{n} : L ⊕ L^{n} → L^{n}

be the map given by the formula

Ψ_{n}((ℓ, ℓ_{1}, …, ℓ_{n})) = ([ℓ, ℓ_{1}], …, [ℓ, ℓ_{n}]).

Let R ≠ C_{r}, r ≥ 1, or char K ≠ 2. Then the map Ψ_{n} is dominant if and only if n ≤ h. (Note that A_{1} = C_{1}, B_{2} = C_{2}.)

If R = C_{r}, r ≥ 1 and char K = 2, then the map Ψ_{n} is a dominant map onto [L, L]^{n} ⊂_{≠} L^{n} for every n.

Remark 3. Theorem 2 can be considered as an analogue of the property \overline{C}_{n} for simple algebraic groups. It is interesting that the bound for dominance in the case of Lie algebras is equal to h, while in the case of groups it is h + 1.

For a simple (abstract) group G, the property C_{1} follows from the stronger condition that there exists a conjugacy class C ⊂ G such that C^{2} = {c_{1}c_{2} | c_{1}, c_{2} ∈ C} = G. Indeed, if g ∈ C then g is a real element, i.e., g is conjugate to g^{−1} because 1 ∈ C^{2}. Hence every element of G can be written as gxg^{−1}x^{−1} for some g ∈ C and therefore G has property C_{1}. The conjecture about the existence of such a conjugacy class C is known as Thompson’s conjecture (see [AH]). We can generalize this question in the following way. We say that the group G satisfies the property T_{n}, if there exists an element g ∈ G such that, for every sequence σ_{1}, …, σ_{n} ∈ [G, G], there exist elements x, y_{1}, …, y_{n} ∈ G with

σ_{1} = xgx^{−1}y_{1}gy_{1}^{−1}, ⋯, σ_{n} = xgx^{−1}y_{n}gy_{n}^{−1}.

We can rewrite this condition in the following way. Let g ∈ G and let

f_{n,g} : G × G^{n} ⟶ G^{n}

be the map given by the formula

f_{n,g}((x, y_{1}, …, y_{n})) = (xgx^{−1}y_{1}gy_{1}^{−1}, ⋯, xgx^{−1}y_{n}gy_{n}^{−1}).

Then the group G satisfies the property T_{n} if Im f_{n,g} = [G, G]^{n} for some g ∈ G. Obviously, T_{n} implies C_{n}.

Now let G be a simple algebraic group defined over a field K and let g ∈ G(K). Then the map f_{n,g} is a morphism of K-varieties G × G^{n} and G^{n}. We can define a property \overline{T}_{n} in the same way as the property \overline{C}_{n}. Namely, we say that G satisfies the property \overline{T}_{n} if f_{n,g} is a dominant map for some g ∈ G(K). Here we prove that a simple algebraic group G satisfies the property \overline{T}_{n} if and only if n ≤ h where h is the Coxeter number.

Actually, we prove a more general result. Let g̃ = (g, g_{1}, …, g_{n}) ∈ G^{n+1}(K) and let

f_{n,g̃} : G^{n+1} → G^{n}

be the map defined by the formula

f_{n,g̃}((x, y_{1}, …, y_{n})) = (xgx^{−1}y_{1}g_{1}y_{1}^{−1}, …, xgx^{−1}y_{n}g_{n}y_{n}^{−1}).

Theorem 3. Let G be a simple algebraic group defined over a field K and let g, g_{1}, …, g_{n} ∈ G(K) be a sequence of semisimple regular elements. Then the map f_{n,g̃} is dominant if and only if n ≤ h where h is the Coxeter number of G.

Remark 4. This theorem shows the bound for the property \overline{T}_{n}. This bound is the Coxeter number h and it is the same as the corresponding bound in Theorem 2 for Lie algebras but it is smaller than the bound h + 1 for the property \overline{C}_{n} in Theorem 1. Remark 5. The property \overline{T}_{n} does not imply the property \overline{C}_{n}, because an element g can be non-real. However, in Theorem 3 we can take a real element g or g_{1} = g_{2} = ⋯ = g_{n} = g^{−1}.

An Example. The groups GL_{1,D}(K) and PGL_{1,D}(K) associated with a quaternion skew field D over a field K satisfy C_{3}.

This fact was observed, for GL_{1,D}, by R. Alperin and R. K. Dennis as well as by the second named author many years ago, some ideas concerning the proof can be found for the case of the real Hamilton quaternions in [AD] and for general fields K in [RS].

Proof. Let D be some quaternion skew field over some field K of any characteristic. The case char K ≠ 2 is very well known, the case char K = 2 is classical as well. For a uniform discussion, we refer to [KMRT, chap. I, §2, 2.C, p. 25 ff.] or [KR, p. 52].

We fix some notations:

Let x ↦ x̄ (x ∈ D) denote the canonical involution of D. The reduced trace T : D → K is a K-linear map and obtained by T(x) = x + x̄. The reduced norm N : D → K is a quadratic form on the 4-dimensional K-vector space D and given by N(x) = xx̄. Its associated bilinear form is given by (x, y) := N(x + y) − N(x) − N(y) = T(xȳ). From the explicit formulae in [KMRT, l.c.] it is easily checked that this bilinear form is nondegenerate.

Hence, for any three elements x_{i} ∈ [D^{∗}, D^{∗}], i = 1, 2, 3, the orthogonal complements of the three subspaces K(1 − x̄_{i}) of D are of dimension 3 and therefore have a nontrivial intersection. If w is a non-zero element of this intersection, this means that 0 = (w, 1 − x̄_{i}) = T(w(1 − x_{i})), hence T(w) = T(wx_{i}). But we also have N(w) = N(wx_{i}), since N(x_{i}) = 1.

It follows that the minimal polynomials of w and of all three elements wx_{i} coincide, thus, the K-subalgebras K(w), K(wx_{i}) of D are pairwise K-isomorphic, and the theorem of Skolem-Noether [VDW, p. 105] yields the existence of x′_{i} ∈ D^{∗} such that x′_{i}wx′_{i}^{−1} = wx_{i}, and hence x_{i} = [w^{−1}, x′_{i}] for i = 1, 2, 3. This proves the statement for GL_{1,D}.

To handle the case of PGL_{1,D}, we observe that we have an exact sequence of linear algebraic K-groups

0 → 𝔾_{m} → GL_{1,D} → PGL_{1,D} → 0,

where 𝔾_{m} is the multiplicative group over K, which is mapped to the center of GL_{1,D}. This yields, by [S, ch. I, prop. 43, p. I-71], a long exact Galois-cohomology sequence

0 → 𝔾_{m}(K) → GL_{1,D}(K) → PGL_{1,D}(K) → H^{1}(K, 𝔾_{m}) → ….

By [S, ch. II, prop. 1, p. II3], we have H^{1}(K, 𝔾_{m}) = 0 and hence the map GL_{1,D}(K) → PGL_{1,D}(K) is surjective. Given three commutators [x_{i}, y_{i}], i = 1, 2, 3 from PGL_{1,D}(K), we may lift their entries x_{i}, y_{i} to preimages x̃_{i}, ỹ_{i} ∈ GL_{1,D}(K). By the preceding result we find w̃, g̃_{i} ∈ GL_{1,D}(K) such that [w̃, g̃_{i}] = [x̃_{i}, ỹ_{i}], and mapping this down to PGL_{1,D}(K) we obtain what we want. Let us remark that, by centrality, the commutators [x̃_{i}, ỹ_{i}] in GL_{1,D}(K) do not depend on the choice of the liftings, but we don’t need that here.

An Application. If C_{2} is satisfied by some (abstract) group G, then there is an easy description of its Schur multiplier.

By a result of C. Miller [MI], the Schur multiplier H_{2}(G, ℤ) of G can be described as follows. Let U(G) denote the group generated by symbols c(x, y), x, y ∈ G subject to just the “formal commutator relations” which are generated by

• c(x, x) = 1, c(x, y)c(y, x) = 1,

• c(^{x}y,^{x}z)c(x, z) = c(xy, z),

• c(^{x}y,^{x}z)c(z, x) = c(x, [y, z]).

Here ^{x}y = xyx^{−1}. Then the map induced by c(x, y) ↦ [x, y] is a central extension of [G, G], and its kernel is canonically isomorphic to H_{2}(G, ℤ), so that we have an exact sequence

0 → H_{2}(G, ℤ) → U(G) → G → H_{1}(G, ℤ) → 0.

It is known that, in case G is perfect, U(G) is the universal central extension of G = [G, G]. The fact we want to mention here is the following:

If G satisfies C_{2}, then the Schur multiplier is generated by all elements c(x, y)c(x′, y′) such that [x, y] = [y′, x′].

That is, the Schur multiplier is generated by relations induced from Abelian subgroups of G (these are the length 1 relators) and from relators of length 2. In many cases, for example for the group of invertible elements of quaternions [AD,RS], but also for SL_{n}(K) and other almost simple split linear groups, it is even true that the relators of length 1 are sufficient to generate the Schur multiplier, so the length 2 relations are not necessary. This follows directly from Matsumoto’s theorem on the presentation of K_{2}(K) by symbols [M]. Analogous results hold also for SL_{n}(D), n ≥ 2, for any skew field D over K [R1, R2], and for Kac-Moody groups [MR].

Proof of the fact mentioned above: Let x_{i}, y_{i}, i = 1, …, n be elements in G such that [x_{1}, y_{1}] · · · · · [x_{n}, y_{n}] = 1. By C_{2}, we may find elements u, v, z ∈ G such that [x_{n−1}, y_{n−1}] = [u, z] and [x_{n}, y_{n}] = [z, v].

From the relations above we obtain [u, z][z, v] = [uv^{−1}, vzv^{−1}].

That is, any element c(x_{1}, y_{1}) · · · · · c(x_{n}, y_{n}) ∈ H_{2}(G, ℤ) can be replaced, modulo products of length at most 2, in H_{2}(G, ℤ) by a product of length n − 1. An induction now gives the result.

Acknowledgments: The proof of Theorem 2 was obtained by E. B. Vinberg for fields of characteristic zero, using the fact that the Killing form is non-degenerate. It was communicated by him to the first author in the discussion of this topic. The authors are grateful to E. B. Vinberg for his kind permission to use his result here. The authors are also grateful to Roger Alperin and Keith Dennis for valuable comments.

2. Notation and Terminology

2.1. R denotes an irreducible root system generated by a simple root system Δ = {α_{1}, α_{2}, …, α_{n}},

W = W (R) is the Weyl group of R;

w_{α} ∈ W is the reflection corresponding to α ∈ R; w_{c} = w_{α_{1}}w_{α_{2}} … w_{α_{n}} is a fixed Coxeter element of W; Γ = ⟨w_{c}⟩; h = |Γ| is the Coxeter number of R.

2.2. G denotes a simple algebraic group defined over the field K corresponding to the system R;

T ≤ G is a maximal torus (also defined over K); we identify the set R with a subset of characters of T, and for our purposes we may assume that T and hence G is split over K, that is, all characters are defined over K.

N is the normalizer of T in G; thus N/T ≅ W; by ẇ we denote an element of N with the image w ∈ W;

B is a Borel subgroup of G (below we assume T ≤ B); ϕ_{n}, f_{n,g}, f_{n.g̃} are the functions defined in the Introduction. An element g ∈ G is called regular if dim C_{G}(g) = dim T.

2.3. L denotes the Lie algebra of a simple and simply connected algebraic group G defined over a field K and corresponding to the root system R.

If K is an algebraically closed field then L = H + U is a Cartan decomposition where H is a Cartan subalgebra and

U = ∑_{α∈R} U_{α}

where U_{α} is the one-dimensional subspace of L corresponding to a root α ∈ R. Since G is simply connected one can chose a Chevalley basis h_{α_{1}}, ⋯, h_{α_{r}}, {u_{α}} of L (see [St1]) where {h_{α_{i}}} is a basis of H and {u_{α}} is a basis of U. Note that, for every k = 1, ⋯, r, we have h_{α_{k}} = [u_{α_{k}}, u_{−α_{k}}].([St1, Lemma 2]). Further, for every root α ∈ R one can define

h_{α} = [u_{α}, u_{−α}].

The adjoint action of G(K) on L will be denoted by g(ℓ) where g ∈ G(K), ℓ ∈ L. We assume that L^{T} = H, where L^{T} denotes the invariant elements of L under the (adjoint) T -action.

A semisimple element l ∈ L is called regular if dim C_{L}(l) = dim C_{L}(H).

2.4. Let f : X → Y be a morphism of irreducible affine k-varieties. We say that f is dominant if the comorphism f^{∗} : K[Y] → K[X] is an injection. Obviously, this is equivalent to the condition \overline{f(X(K̄))} = Y (K̄) where K̄ is the algebraic closure of K and \overline{f(X(K̄))} is the Zariski closure of f(X(K̄)) in Y (K̄).

Thus we may assume in the proofs of theorems 1, 2, 3 that K is an algebraically closed field. Also, it is enough to prove theorems 1 and 3 for the cases where G is a simply connected group.

Below, we suppose that K is algebraically closed and G is simply connected. From the context, it will be clear, that some of the statements hold under weaker assumptions, e.g., in the case that G is split over K, or in the case that K is sufficiently large.

3. Some technical results

3.1. Let

R = S_{1} ∪ S_{2} ⋯ ∪ S_{r}

be the decomposition of R into the union of Γ-orbits. Then r = rank R and there exists a sequence of representatives

θ_{1} ∈ S_{1}, θ_{2} ∈ S_{2}, …, θ_{r} ∈ S_{r}

which is a basis of the group Q(R) ([B, IV, 21, Proposition 33, p. 170]). (Recall that Q(R) is the lattice generated by the roots.)

The roots θ_{1}, …, θ_{r} are defined in the following way

θ_{i} = w_{α_{r}}w_{α_{r−1}} ⋯ w_{α_{i+1}}(α_{i})  (1)

(see [B, l.c.]), and we have

θ_{i} > 0

for every i = 1, …, r. Moreover, in [St2, Lemma 7.2.c, p. 298] it is proved that θ_{i}+θ_{j} ∉ R for every i, j. The following lemma is proved by the same arguments (which was essentially don in [St2] as well):

Lemma 1. Let k_{1}, …, k_{r} be non-negative integers such that k_{1} + ⋯ + k_{r} > 1. Then

θ = k_{1}θ_{1} + ⋯ + k_{r}θ_{r} ∉ R.

Proof. Let i = max{m | k_{m} ≠ 0}. From (1) we have

θ′ = w_{α_{i}}w_{α_{i+1}} … w_{α_{r}}(θ) = −k_{i}α_{i} + k_{i−1}α_{i−1} + ∑_{j<i−1} k_{j}w_{α_{i−1}}w_{α_{i−2}} … w_{α_{j+1}}(α_{j})

Applying formulas (1) to the irreducible root system generated by roots {α_{1}, …, α_{i−1}} we get

w_{α_{i−1}}w_{α_{i−2}} … w_{α_{j+1}}(α_{j}) = ∑_{s≤i−1} ℓ_{s}α_{s}

where ℓ_{s} ≥ 0 for every s ≤ i − 1 (indeed, all θ_{i} in (1) are positive). Now we have

θ′ = −k_{i}α_{i} + ∑_{s≤i−1} m_{s}α_{s}

where m_{s} ≥ 0 for every s ≤ i − 1. Since {α_{1}, …, α_{i}} is a simple root system the vector θ′ belongs to R only if m_{1} = m_{2} = ⋯ = m_{i−1} = 0 and k_{i} = 1. But in this case k_{j} = 0 for every j < i and, therefore, k_{1} + ⋯ + k_{r} = 1. This contradicts to the condition of the lemma.

3.2.

Lemma 2. {h_{θ_{1}}, …, h_{θ_{r}}} is a basis of H.

Proof. If r = 1 then θ_{1} = α_{1} and {h_{α_{1}}} is a basis of H. Suppose our assertion holds for root systems of rank < r. Let ε_{1} = w_{α_{r}}(θ_{1}), …, ε_{r−1} = w_{α_{r}}(θ_{r−1}). It follows from (1) that ε_{1}, …, ε_{r−1} belong to the root system generated by {α_{1}, …, α_{r−1}}. Moreover, the elements ε_{1}, …, ε_{r−1} are defined in the same way as θ_{1}, …, θ_{r} for the root system R. Therefore the assumption of the induction implies that {h_{ε_{1}}, …, h_{ε_{r−1}}} is a basis of subspace H′ ⊂ H generated by {h_{α_{1}}, …, h_{α_{r−1}}}. Further, θ_{r} = α_{r} and ε_{r} = w_{α_{r}}(θ_{r}) = w_{α_{r}}(α_{r}) = −α_{r}. Hence h_{ε_{r}} = −h_{α_{r}}. Thus, {h_{ε_{1}}, …, h_{ε_{r−1}}, h_{ε_{r}}} is a basis of H. But h_{ε_{i}} = w_{α_{r}}(h_{θ_{i}}) and therefore {h_{θ_{1}}, …, h_{θ_{r}}} is also a basis of H.

3.3. For every reflection w_{α} ∈ W we can find a preimage ẇ_{α} ∈ N such that ẇ_{α}(u_{β}) = ±u_{w_{α}(β)} for every root β ∈ R ([St1, Lemma 19, (a)]). Now we fix such preimages ẇ_{α} and put

ẇ_{c} = ẇ_{α_{1}} ẇ_{α_{2}} ⋯ ẇ_{α_{r}}.

Let x_{α}(s) = exp(su_{α}) ∈ G(K) be the corresponding root element where s ∈ K. Put

γ = x_{θ_{1}}(1)x_{θ_{2}}(1) … x_{θ_{r}}(1), γ_{j} = ẇ_{c}^{j−1}γẇ_{c}^{−j+1} for i = 1, …, h  (2)

Further, put

θ_{ij} = w_{c}^{j−1}θ_{i}w_{c}^{−j+1} for i = 1, …, h.  (3)

Lemma 3. γ_{m}(u_{−θ_{ij}}) ≡ 0 (mod U) if m ≠ j and γ_{j}(u_{−θ_{ij}}) ≡ ±h_{θ_{ij}} (mod U).

Proof. Let g ∈ G(K). From the definition of ẇ_{c} we have

ẇ_{c}(g(u_{α})) = ẇ_{c}gẇ_{c}^{−1}(±u_{w_{c}(α)})  (4)

for every α ∈ R. Moreover, ẇ_{c}(U) = U and ẇ_{c}(0) = 0. Acting on both sides of the congruences by an appropriate power of ẇ_{c} and using (2), (3), (4) we can get the equivalent congruences

γ_{1}(u_{−θ_{ij}}) ≡ 0 (mod U) for j ≠ 1,
γ_{1}(u_{−θ_{i1}}) ≡ ±h_{θ_{i1}} (mod U). (5)

Thus, it is enough to prove (5).

Let ε, δ ∈ R. If ε ≠ −δ then

x_{ε}(1)(u_{δ}) = u_{δ} + ∑_{δ+iε∈R} ℓ_{i}u_{δ+iε}  (6)

where ℓ_{i} ∈ K and

x_{ε}(1)(u_{−ε}) = u_{−ε} ± h_{ε} ∓ u_{ε}.  (7)

Equations (6) and (7) follow from [St1, Lemma 72, p. 209].

Further, if k_{1}, …, k_{r} are non-negative integers then for every i = 1, …, r and for every β ∈ R the equality

−θ_{i} = −β + k_{1}θ_{1} + ⋯ + k_{r}θ_{r}

is possible only for k_{1} = k_{2} = ⋯ = k_{r} = 0. This follows from Lemma 1. For β ∈ R put

M_{β} = {−β + k_{1}θ_{1} + ⋯ + k_{r}θ_{r} ∈ R | k_{i} ≥ 0, k_{1} + k_{2} + ⋯ + k_{r} ≥ 1}.

Thus, for every i = 1, …, r the set M_{β} does not contain the root −θ_{i}.

Let β = θ_{ij} where j > 1. Since the group Γ = ⟨w_{c}⟩ acts free on the Γ-orbits S_{1}, …, S_{r} we have β ≠ θ_{1}, …, θ_{r}. From the definition γ_{1} = γ and (6) we get

γ_{1}(u_{−β}) = u_{−β} + ∑_{α∈M_{β}} ℓ_{α}u_{α}  (8)

where ℓ_{α} ∈ K.

Let β = θ_{i1} = θ_{i} and M_{θ} = {θ_{i}, i = 1, …, h}. From (6) and (7) we obtain

γ_{1}(u_{−β}) = u_{−β} ± h_{β} + ∑_{α∈M_{β}∪M_{θ}} ℓ_{α}u_{α}  (9)

where ℓ_{α} ∈ K. Now (8) and (9) imply (5). 3.4. We define the subspace Ũ of L^{h} = \underbrace{L ⊕ ⋯ ⊕ L}_{h−times} by

Ũ := {(γ_{1}(u) + u_{1}, γ_{2}(u) + u_{2}, …, γ_{h}(u) + u_{h}) | u, u_{i} ∈ U}.

Lemma 4. Ũ = L^{h}. Proof. Obviously, U^{h} = \underbrace{U ⊕ ⋯ ⊕ U}_{h−times} ⊂ Ũ. Since H^{h} +U^{h} = L^{h} we have to prove H^{h} ⊂ Ũ.

Let u = u_{−θ_{ij}}. By Lemma 3 we have γ_{m}(u) ∈ U if m ≠ j and γ_{j}(u) = ±h_{θ_{ij}} + u′ for some u′ ∈ U. Thus, for every i, j, the element

(0, 0, …, h_{θ_{ij}}, 0, …, 0)  (10)

belongs to Ũ. Since h_{θ_{1}}, …, h_{θ_{r}} is a basis of H (Lemma 2), the sequence h_{θ_{1j}}, h_{θ_{2j}} …, h_{θ_{rj}} is also a basis of H and, therefore, the set of r × h elements of the form (10) is a basis of H^{h}.

Now let us fix the sequence of elements δ_{1} ∈ T γ_{1}T, …, δ_{h} ∈ T γ_{h}T and let us define

Ũ′ = {(δ_{1} − 1)(l) + u_{1}, …, (δ_{h} − 1)(l) + u_{h}) | l ∈ L, u_{i} ∈ U}.

The same arguments as in the proof of the previous lemma give Lemma 5. Ũ′ = L^{h} 3.5. Now we formulate an analogue for lemma 4 for the action of Lie algebra on itself. Namely, let

y = u_{θ_{1}} + u_{θ_{2}} + ⋯ + u_{θ_{r}}

and

y_{1} = y, y_{2} = w_{c}(y), …, y_{h} = w_{c}^{h−1}(y).

We define, similarly to Ũ, the set

Û = {[l, y_{1}] + u_{1}, ⋯, [l, y_{h}] + u_{h} | l ∈ L, u_{i} ∈ U}.

It is easy to see that the congruences of Lemma 3 also hold if we use elements y_{m}, y_{j} instead of γ_{m}, γ_{j}. Thus, the same arguments as in the proof of Lemma 4 give

Lemma 6. Û = L^{h}.

3.6.

Lemma 7. 1.Assume char K ≠ 2 if R = C_{r}, r ≥ 1. Then there exists an element h ∈ H such that C_{L}(h) = H.

2. Let R = C_{r}, r ≥ 1 and char K = 2. Further, let R_{l} be the set of all long roots of R. Then there exists an element h ∈ H such that

C_{L}(h) = C_{L}(H) = {H + ∑_{α∈R_{l}} U_{α}}.

Proof. 1. If char K ≠ 2 or R ≠ C_{r} then for every root α ∈ H the corresponding linear function α : H ⟶ K is not trivial. Since K is an algebraically closed field it is infinite and therefore the set

H \ (∪_{α∈R}Ker α)

is not empty.

2. If R = C_{r} and char K = 2 a map α : H ⟶ K , α ∈ R is trivial if and only if α ∈ R_{l}. Thus we can get the assertion in the same way as above.

4. Proof of Theorem 2

Let char K ≠ 2 if R = C_{r}, r ≥ 1.

Obviously, if Ψ_{n} is dominant, then Ψ_{1}, Ψ_{2}, …, Ψ_{n−1} is also dominant. Thus we have to prove that Ψ_{h} is dominant but Ψ_{h+1} is not. Let (ℓ, ℓ_{1}, …, ℓ_{n}) ∈ L ⊕ L^{n}. Assume that l ∈ H is a regular element. Since for every regular element h ∈ H we have [h, L] = U (Lemma 7) then for every regular element l′ ∈ H there exist l′_{1}, ⋯, l′_{n} ∈ L such that

[l′, l′_{1}] = [l, l_{1}], ⋯, [l′, l′_{n}] = [l, l_{n}].  (11)

Further, in (11) we can replace every element l′_{i} by l′_{i} + x_{i} where x_{i} is an element of the centralizer of l′, i.e., an element of H. Thus, the dimension of the fiber Ψ_{n}^{−1}(Ψ_{n}((l, l_{1}, ⋯, l_{n}))) ≥ (n + 1) dim H and therefore

dim Ψ_{h+1}^{−1}(Ψ_{h+1}((ℓ, ℓ_{1}, …, ℓ_{h+1}))) ≥ (h + 2)rank L > dim L  (12)

(recall, dim L = (h + 1)rank L). Since the set of regular semisimple elements is dense in L the inequality (12) holds for a “generic fiber” of Ψ_{h+1}. Hence

dim Im Ψ_{h+1} < dim L^{h+1}

and therefore Ψ_{h+1} cannot be dominant.

Consider now Ψ_{h}. Let ã = (a_{1}, …, a_{h}), b = (a, a_{1}, …, a_{h}) ∈ L ⊕ L^{h} be fixed points and let c = Ψ_{h}(b) ∈ L^{h} . Consider the differential d_{b}Ψ_{h} of Ψ_{h} at the point b

d_{b}Ψ_{h} : T_{b} → T_{c}

where T_{b}, T_{c} are the corresponding tangent spaces. We identify T_{b} with L ⊕ L^{h} and T_{c} with L^{h}. Then we have

d_{b}Ψ_{h} : L ⊕ L^{h} → L^{h}

and

Im d_{b}Ψ_{h} = [a, L^{h}] + [L, ã].  (13)

(Here we used the rules for the differential of the map Ψ_{h} at the point b = (a, ã) in the following sense: d_{(a,a_{i})}[x, y](l_{1}, l_{2}) = [a, l_{2}] + [l_{1}, a_{i}] where x, y ∈ L are variables and l_{1}, l_{2} are elements in the tangent space T_{a}, T_{a_{i}} of L at the points a, a_{i}, which we identify with L.)

Let a ∈ H be a regular element and let a_{1} = y_{1}, …, a_{h} = y_{h} be the sequence defined in 3.5. We have [a, L^{h}] = U^{h} (Lemma 7) and, therefore,

[a, L^{h}] + [L, ã] = {([ℓ, y_{1}] + u_{1}, [ℓ, y_{2}] + u_{2}, …, [ℓ, y_{h}] + u_{h})| ℓ ∈ L, u_{i} ∈ U} = Û

By Lemma 6,

[a, L^{h}] + [L, ã] = L^{h}

for these particular a and ã. Hence there exists a point b ∈ L ⊕ L^{h} where the rank of differential is equal dim L^{h} and therefore it holds for points from some open subset X ⊂ L ⊕ L^{h}. We can find the point x ∈ X such that Ψ_{h}(x) is a regular point of \overline{Ψ_{h}(L ⊕ L^{h})}. Since the tangent space at the point Ψ_{h}(x) has rank = dim L^{h} we have dim \overline{Ψ_{h}(L ⊕ L^{h})} = dim L^{h}.

Let R = C_{r}, r ≥ 1 and char K = 2.

Denote by R_{sh} the set of all short roots in R. It is easy to check

[L, L] = H + ∑_{α∈R_{sh}} U_{α} .

Further, let

l = h + ∑_{β∈R_{l}^{+}} u_{β} ,

where h ∈ H is an element satisfying the condition C_{L}(h) = C_{L}(H) (Lemma 7), and u_{β} are elements of the Chevalley basis. We have [l, L] = [L, L].( One can check this using the definition of h, l and Lemma 7.) This implies our assertion.

Theorem 2 has been proved.

5. Proof of Theorem 3

Let C_{g}, C_{g_{i}} be the conjugacy classes of the regular semisimple elements g, g_{i}. Then dim C_{g} = dim C_{g_{i}} = dim G − rank G. We assume n > 1 and define the subset of G^{n}

M_{n,g̃} = {(gy_{1}g_{1}y_{1}^{−1}, …, gy_{n}g_{n}y_{n}^{−1}) | y_{i} ∈ G}.

Obviously dim M_{n,g̃} = n dim C_{g} = n(dim G − rank G) and GM_{n,g̃} = {(xm_{1}x^{−1}, …, xm_{n}x^{−1}) | (m_{1}, …, m_{n}) ∈ M_{n,g̃}} = Im f_{n,g̃}. Since a “generic point” m ∈ M_{n,g̃} has a stabilizer which is equal to Z(G) (because n > 1) and dim(M_{n,g̃}∩Gm) ≥ rank G we have

dim GM_{n,g} ≤ n(dim G − rank G) + (dim G − rank G) =
= n dim G + (dim G − (n + 1)rank G).  (14)

If \overline{Im f_{n,g̃}} = G^{n} then (14) implies (n + 1)rank G ≤ dim G and therefore n ≤ h.

Now we prove that inequality n ≤ h implies \overline{Im f_{n,g̃}} = G^{n}. Obviously, it is enough to prove this for n = h.

We may assume g, g_{1}, …, g_{h} ∈ T. Also we may assume that (gσ_{1}g_{1}σ_{1}^{−1}, …, gσ_{h}g_{h}σ_{h}^{−1}) is a regular point of \overline{Im f_{n,g̃}} for some σ_{1}, …, σ_{h} ∈ G(K). Moreover, the set of such sequences (σ_{1}, …, σ_{h}) contains a non-empty open set of G^{h}. Put

s_{1} = σ_{1}g_{1}σ_{1}^{−1}, …, s_{h} = σ_{h}g_{h}σ_{h}^{−1}

and consider the map

Ψ : G × G^{h} → G^{h}

given by the formula

Ψ(x, y_{1}, …, y_{h}) = (xgx^{−1}y_{1}s_{1}y_{1}^{−1}s_{1}^{−1}g^{−1}, …, xgx^{−1}y_{h}s_{h}y_{h}^{−1}s_{h}^{−1}g^{−1}).

The definitions of Ψ and s_{1}, …, s_{h} imply

dim Im Ψ = dim Im f_{h,g̃}.  (15)

Moreover, Ψ((1, …, 1)) is a regular point of Im Ψ. (Indeed, Im Ψ = (Im f_{h,g̃})s) where s = (s_{1}^{−1}g^{−1}, …, s_{h}^{−1}g^{−1}). The differential dΨ at the point (1, …, 1) gives the linear map

dΨ : L × L^{h} → L^{h}

which is

dΨ ((ℓ, ℓ_{1}, …, ℓ_{h})) = ((1 − g)ℓ + g(1 − s_{1})ℓ_{1}, …, (1 − g)ℓ + g(1 − s_{h})ℓ_{1})  (∗)

(this follows from the standard formulas for differentials).

Now we want to prove, that for some sequence s_{1}, …, s_{h} defined above

Im dΨ = L^{h}.  (16)

Then (16) with (15) give us our statement.

Since g ∈ T is a regular element, we get

(1 − g)L = U.  (17)

Further,

g(1 − s_{i})L = g(1 − s_{i})g^{−1}gL = (1 − gs_{i}g^{−1})gL = (1 − gs_{i}g^{−1})L.  (18)

Recall that s_{i} = σ_{i}g_{i}σ_{i}^{−1} and g_{i} ∈ T. Since g_{i} is a regular element of T

(1 − g_{i})L = U.  (19)

Put δ_{i} = gσ_{i}. Then (17), (18), (19) imply

g(1 − s_{i})L = (1 − δ_{i}g_{i}δ_{i}^{−1})L = δ_{i}(1 − g_{i})δ_{i}^{−1}L = δ_{i}(1 − g_{i})L = δ_{i}(U).  (20)

From (∗), (17) and (20) we obtain

Im dΨ = {(u + δ_{1}(u_{1}), …, u + δ_{h}(u_{h})) | u_{i} ∈ U}.

Put δ = (δ_{1}, …, δ_{h}). Then

δ^{−1}(Im dΨ) = {(δ_{1}^{−1}(u) + u_{1}, …, δ_{h}^{−1}(u) + u_{h} | u_{i} ∈ U}.  (21)

Let τ = (τ_{1}, …, τ_{h}) ∈ G^{h} and let
Θ_{τ} : U ⊕ U^{h} → L^{h}

be the map

Θ_{τ} ((u, u_{1}, …, u_{h})) = (τ_{1}(u) + u_{1}, …, τ_{h}(u) + u_{h}).

The set of τ ∈ G^{h} such that dim Im Θ_{τ} < dim L^{h} is closed in G^{h}. By Lemma 4, Im Θ_{τ} = L^{h} where τ_{0} = (γ_{1}, …, γ_{h}). Hence the set

X = {τ ∈ G^{h} | Im Θ_{τ} = L^{h}}

is a non-empty open subset of G^{h}. Further, the set of sequences Y = {σ = (σ_{1}, …, σ_{h}) ∈ G^{h}\}, such that (gσ_{1}g_{1}σ_{1}^{−1}, …, gσ_{h}g_{h}σ_{g}^{−1}) is a regular point of \overline{Im f_{h,g̃}}, contains a nonempty open subset of G^{h}. Since δ_{i}^{−1} = σ_{i}^{−1}g^{−1} we find an element

δ^{−1} ∈ X ∩ Y^{−1}g^{−1}.  (22)

Now, from (21) and (22) we get (16).

Theorem 3 has been proved.

6. Proof of Theorem 1

Obviously, the property \overline{C}_{n} implies the property \overline{C}_{n−1}. Thus we have to prove \overline{C}_{h+1} for the group G and we need to show that \overline{C}_{h+2} does not hold for G. The latter follows from the inequality

dim ϕ_{h+2}^{−1}(ϕ_{h+2}((g, g_{1}, …, g_{h+2}))) ≥ (h + 2) dim (C_{G}(g)) ≥
≥ (h + 2)rank G > (h + 1)rank G = dim G.

Now we will prove the property \overline{C}_{h+1} for the group G. Recall, that we assume that G is simply connected.

Lemma 8. Let γ_{1}, …, γ_{h} be the sequence defined in (2). Then for every regular element t ∈ T and for every i = 1, ⋯, h there exists an element t_{i} ∈ T such that

[t , t_{i}γ_{i}t_{i}^{−1}] = γ_{i}

Moreover, for every regular t there is only a finite number of such t_{i}.

Proof. Obviously it is enough to prove the statement only for γ_{1} = γ.

Since t is a regular element θ_{j}(t) ≠ 1 for every j = 1, …, r. Put

v_{j} = (θ_{j}(t) − 1)^{−1}  (23)

Further, there exists an element t_{1} ∈ T such that

θ_{j}(t_{1}) = v_{j}  (24)

for every j. Indeed, every element x ∈ T can be presented in the form x = h_{α_{1}}(x_{1}) … h_{α_{r}}(x_{r}) where x_{i} ∈ K^{∗} and h_{α_{i}}(x_{i}) is the corresponding semisimple element of the α_{i}- root subgroup ([St1], Lemma 28). The system of equations θ_{j}(x) = v_{j} can be written in the form

∏_{k=1}^{r} x_{k}^{n(k,j)} = v_{j}  (25)

where n(k, j) = 2(α_{k}, θ_{j})/(α_{k}, α_{k}). Since θ_{1}, …, θ_{r} is a basis of the group Q(R) the matrix {(α_{k}, θ_{j})}_{1≤k≤r,1≤j≤r} has rank = r. Hence the matrix {n(k, j)} also has rank = r. Thus we can find the solution of (25) which gives us the element t_{1} ∈ T satisfying (24).

From (24)

t_{1}γ_{1}t_{1}^{−1} = x_{θ_{1}}(v_{1}) … x_{θ_{r}}(v_{r}).  (26)

Now [t, t_{1}γt_{1}^{−1}] = γ = γ_{1} follows from (23) ,(26) and the Chevalley commutator formula. (Note that, by Lemma 1, the sum θ_{i} + θ_{j} is not a root for every i, j and therefore x_{θ_{i}}(a)x_{θ_{j}}(b) = x_{θ_{j}}(b)x_{θ_{i}}(a) for every a, b.) On the other hand, the commutator equation for t_{1} implies the equation (26) which in turn leads us to (25). Since (25) has only a finite number of solutions (because the rank of the matrix {n_{k,j}} is equal to r) we obtain that only finitely many t_{1} are possible.

Let

X = {(t, t_{1}γ_{1}t_{1}^{−1}, …, t_{h}γ_{h}t_{h}^{−1}) | t, t_{i} ∈ T, [t, t_{i}γ_{i}t_{i}^{−1}] = γ_{i} for every i = 1, …, h}.

The set X is a constructible subset of G^{h+1}. Indeed, X is the image of a closed subset of T^{h+1} (which is defined by commutator equations) under the morphism

(t, t_{1}, …, t_{h}) ⟶ (t, t_{1}γ_{1}t_{1}^{−1}, …, t_{h}γ_{h}t_{h}^{−1}).

Let X_{0} be an irreducible component of X such that dim X_{0} = dim X. Lemma 8 implies dim X = dim T = rank G. Hence

dim X_{0} = rank G  (27)

Moreover, Lemma 8 implies that the projection of X_{0} on the first coordinate contains an open subset of T.

Further, let (1, T^{h}) be the subset of T^{h+1} consisting of the elements which have the first coordinate 1. Then we consider (1, T^{h}) as a subset of G^{h+1} and put

Y = X_{0}(1, T^{h}) = {(t, t_{1}γ_{1}t_{1}^{−1}t′_{1}, …, t_{h}γ_{h}t_{h}^{−1}t_{h}′) | t, t_{i}, t′_{i} ∈ T }  (28)

(note that the elements t_{i} in (28) depend on the first coordinate t, while elements t′_{i} run independently through the set T). Since X_{0} is an irreducible locally closed subset of G^{h} the same is Y and (27),(28) imply

dim Y = (rank G)^{h+1} = dim G.  (29)

Further, the definition of the set X_{0} and (28) imply

Y ⊂ ϕ_{h}^{−1}((γ_{1}, …, γ_{h})).  (30)

Lemma 9. The Zariski closure \overline{Y} of the set Y coincides with an irreducible component of the pre-image ϕ^{−1}((γ_{1}, …, γ_{h})).

Proof. Let y = (t, d_{1}, …, d_{h}) ∈ Y where d_{i} = t_{i}γ_{i}t_{i}^{−1}t′_{i} (see (28)). Consider the map

χ_{y} : G^{h+1} ⟶ G^{h}

given by the formula

χ_{y}((x, x_{1}, …, x_{h})) = ([xt, x_{1}d_{1}][d_{1}, t], …, [xt, x_{h}d_{h}][d_{h}, t]).

The differential of χ_{y} at the point (1, …, 1) gives the linear map

d(χ_{y}) : L^{h+1} ⟶ L^{h}.

This map can be easily calculated using usual differentiation formulas. Namely, writing the first component as

(x, x_{1}) ↦ xt(x_{1}d_{1}t^{−1}(x^{−1}d_{1}^{−1}x_{1}^{−1}d_{1})td_{1}^{−1})t^{−1},

we obtain for its differential:

(l, l_{1}) ↦ (l + t(l_{1} + d_{1}t^{−1}(−l − d_{1}^{−1}(l_{1})))) = (1 − td_{1}t^{−1})(l) + t(1 − d_{1}t^{−1}d_{1}^{−1})(l_{1})

and therefore for the whole map:

d(χ_{y})((l, l_{1}, …, l_{h})) =
((1 − td_{1}t^{−1})(l) + t(1 − d_{1}t^{−1}d_{1}^{−1})(l_{1}), …, (1 − td_{h}t^{−1})(l) + t(1 − d_{h}t^{−1}d_{h}^{−1})(l_{h}))

(recall that, for g ∈ G and l ∈ L, we write g(l) for ad g(l)). Put u_{i} = (1 − t^{−1})d_{i}^{−1}(l_{i}), l′ = t^{−1}(l). Then u_{i} ∈ U. Applying the invertible linear operator (d_{1}^{−1}t^{−1}, …, d_{h}^{−1}t^{−1}) to the image of d(χ_{y}) we get the linear space

{(d_{1}^{−1} − 1)(l′) + u_{1}, …, (d_{h}^{−1} − 1)(l′) + u_{h}) | l′ ∈ L, u_{i} ∈ U}

which, according to Lemma 5, coincides with L^{h}. Thus the differential of the map χ_{y} at the point (1, …, 1) has rank = dim L^{h}. Note that the map χ_{y} is the composition of two translations and the map ϕ_{h}. This implies that the differential of ϕ_{h} at the point y also is of rank dim L^{h}. According to Theorem 3 the map ϕ_{h} is dominant. Hence ϕ_{h}(y) is a regular point of \overline{Imϕ_{h}} = G^{h}. Since the rank of the differential of ϕ_{h} at the point y is equal to dim L^{h} = dim G^{h}, there exists an irreducible component Y′ of the pre-image ϕ_{h}^{−1}(ϕ_{h}(y)) such that y ∈ Y′ and dim Y′ = dim G. We may assume that a point y belongs only to those irreducible components of ϕ_{h}^{−1}(ϕ_{h}(y)) = ϕ_{h}^{−1}(γ_{1}, …, γ_{h}) which contain the whole set Y. Thus we obtain Y ⊂ Y′, and our statement will follow from (29)

Lemma 10. Let Y′ ⊂ G^{h+1} be an irreducible component of ϕ_{h}^{−1}((γ_{1}, …, γ_{h})) which is the closure of the set Y. Then the projection of Y′ to the first component of G^{h+1} is contained in T.

Proof. Let p_{1} : G^{h+1} ⟶ G be the corresponding projection. Since Y ⊂ Y′, the set p_{1}(Y′) contains an open subset of T. This follows from the definition of Y. Further, the set \overline{p_{1}(Y′)} is an irreducible closed subset of the dimension rank G. (The latter follows from the definitions Y and Y′.) Hence this set coincides with T.

Now let t ∈ T be a regular element and s = [t, w_{c}] (recall that w_{c} is a fixed Coxeter element) and let M_{s} = ϕ_{h+1}^{−1}((γ_{1}, …, γ_{h}, s)). We want to chose the element t satisfying the following conditions:

I. Let Y′ be as in Lemma 10. There exists an element (t, d_{1}, …, d_{h}) ∈ Y′ which does not belong to any other irreducible component of ϕ_{h}^{−1}((γ_{1}, …, γ_{h})).

II. The set of all elements g ∈ G such that [t, g] = s consists only of elements of the form w_{c}t′ where t′ runs through T.

Suppose we find an element t satisfying conditions I. and II. Put

z = (t, d_{1}, …, d_{h}, w_{c}).

From the definition of t, s we get the inclusion z ∈ M_{s}. Let M_{sz} be an irreducible component of M_{s} containing the element z. Further, let P = P_{h+1}^{h+2} : G^{h+2} ⟶ G^{h+1} be the projection to the first (h + 1) components. Then P (M_{sz}) ⊂ ϕ_{h}^{−1}((γ_{1}, …, γ_{h})). Since the set M_{sz} is irreducible the set \overline{P (M_{sz})} is also irreducible and I. implies that P (M_{sz}) is contained only in the irreducible component Y′ of ϕ_{h}^{−1}((γ_{1}, …, γ_{h})). Since w_{c} is a Coxeter element there is only a finite number of t′ ∈ T satisfying the condition [t′, w_{c}] = s. Together with Lemma 10 this implies

\overline{P (M_{sz})} = \overline{\{(t, a_{1}, …, a_{h}) ∈ Y \}}  (31)

(here t is a fixed element from the torus T but the elements a_{i} run through the sets of elements of the form t_{i}γ_{i}t_{i}^{−1}t′; see the definition of Y).

From (31) and the definition of Y we get

dim P (M_{sz}) = (rank G)^{h}.

Further, II. implies that the dimension of every fiber of the projection M_{sz} ⟶ P (M_{sz}) has dimension rank G. Hence dim M_{sz} = (rank G)^{h+1} = dim G. Thus we find an irreducible component of a pre-image of a point in G^{h+1} with respect to the map ϕ_{h+1} which has dimension dim G. Therefore the dimension of the image of ϕ_{h+1} has the dimension dim G^{h+1}. This gives our assertion.

Now we have to prove the existence of a regular element t ∈ T satisfying conditions I.-II.

We can choose a point of Y which does not belong to other irreducible components of ϕ_{h}^{−1}((γ_{1}, …, γ_{h})) and which has a regular element t ∈ T as its first coordinate. This follows from the definition of Y and Lemma 10. Thus we have I.

Now we show II. for a chosen t. Let [t, g] = s for some g ∈ G. Then g ∈ BwB for some w ∈ W. Hence g = vwt′u where v, u ∈ U (here U is the product of all positive root subgroups (see [St1])) t′ ∈ T. We may assume that in u only those factors u_{α} from root subgroups are non-trivial which have the property wu_{α}w^{−1} ∈ U^{−} (here U^{−} is the product of all negative root subgroups) ([St1], Theorem 4’).

Consider the equality

[t, g] = tvwt′ut^{−1}u^{−1}(t′)^{−1}w^{−1}v^{−1} = tv(wt′ut^{−1}u^{−1}(t′)^{−1}w^{−1})v^{−1} = s

The expression in brackets lies in the group B^{−} = T U^{−}. This follows from the choice of u. The elements on both sides of the brackets lie in the Borel subgroup B. Since s ∈ T, the expression in the bracket is in T. This implies

[t, u] = 1.  (32)

On Multicommutators for Simple Algebraic Groups 21

Since t is a regular element (32) implies u = 1. The same arguments show v = 1. Thus, g = wt′. But the equality [t, w_{c}] = s = [t, wt′] implies w = w_{c} (because we assume that G is simply connected and t is a regular element of T).

Theorem 1 has been proved.

The proof of Lemma 9 shows that dim Im d_{y}(ϕ_{n}) = (dim G)^{n} if n ≤ h in the generic point y. Thus, if n ≤ h the map ϕ_{n} is always a separated morphism. Now let n = h + 1. We can consider the map χ_{y} : G^{h+2} ⟶ G^{h} which is constructed in the same way as the corresponding map in the proof of Lemma 9 changing h to h + 1. From the definition we have an equality of ranks of the differentials of χ_{y} at the point (1, ⋯, 1) and ϕ_{h+1} at the generic point y. The formula (*) in that lemma shows that this rank cannot be (dim G)^{h+1} if the center of the Lie algebra is not trivial.

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22 Nikolai Gordeev and Ulf Rehmann

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Nikolai Gordeev, Department of Mathematics of Russian State Pedagogical University, Moijka 48, Sankt Petersburg 191-186, Russia

    E-mail address: gordeev@pdmi.ras.ru

Ulf Rehmann, Fakultät für Mathematik Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

    E-mail address: rehmann@mathematik.uni-bielefeld.de