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Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem

Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.


Introduction
In the structure and representation theory of complex semi-simple Lie algebras, root systems and their Weyl groups play a fundamental rôle. For fields of positive characteristic the situation is similar, provided one studies Lie algebras g = Lie(G) that are associated with a reductive group G. By contrast, the maximal tori of arbitrary restricted Lie algebras usually are no longer conjugate, so there are many root systems available, whose structure and utility crucially depend on the initially chosen maximal torus. Accordingly, it is not clear which maximal tori are appropriate for defining finite groups that take on the rôle of Weyl groups. In his article [21], Premet studied this problem for the Jacobson-Witt algebras W (n). He identified a maximal torus t 0 ⊆ W (n), whose automorphism group allowed to establish an analogue of the Chevalley restriction theorem on polynomial invariants.
One way to address the problem of choosing tori is the systematic study of schemes of tori. Roughly speaking, this geometric approach seeks to identify generic properties of tori and their root systems. In the context of Lie algebras of complexity ≤ 2, that naturally arise in the study of restricted enveloping algebras of tame representation type, this method has produced satisfactory results, cf. [9,10]. A detailed investigation of the scheme of tori of maximal dimension µ(g) of a restricted Lie algebra g in [8] led to the definition of a group S(g) that turned out to coincide with the Weyl group in case g is of classical type.
Aside from the analogues of the complex simple Lie algebras there are usually four additional classes of restricted simple Lie algebras, the so-called restricted Lie algebras of Cartan type. These Lie algebras, which include the abovementioned Jacobson-Witt algebras, already differ significantly from their classical precursors in that their maximal tori are no longer conjugate. The main object of the present paper is to identify the corresponding finite groups along with providing applications.
Our paper is organized as follows. After recalling basic results on the scheme T g of tori of a restricted Lie algebra g, we show in Section 1 that the toral stabilizer S(g) of g is isomorphic to the "Weyl group" associated with any generic torus of g. For future reference, we also provide criteria for a given torus or Cartan subalgebra to be generic. Section 2 is concerned with the behavior of toral stabilizers under passage to subalgebras and factor algebras. In particular, we show that Let M k and Ens be the categories of commutative k-algebras (of arbitrary dimension) and sets, respectively. For a torus t ∈ Tor(g), we consider the scheme T g : M k −→ Ens, given by T g (R) := {ϕ ∈ Hom p (t⊗ k R, g⊗ k R) ; ϕ is a split injective R-linear map} for every R ∈ M k . Here Hom p (t⊗ k R, g⊗ k R) denotes the set of homomorphisms of restricted R-Lie algebras and g ⊗ k R carries the natural structure of a restricted R-Lie algebra, with bracket and p-mapping defined via [x ⊗ r, y ⊗ s] := [x, y] ⊗ rs and (x ⊗ r) [p] := x [p] ⊗ r p ∀ x, y ∈ g, r, s ∈ R, respectively. According to [9, (1.4),(1.6)], T g is a smooth affine scheme of dimension dim k g − rk(g).
In what follows, we shall be mainly concerned with the variety T g (k) of k-rational points of T g , that is, the variety of embeddings ϕ : t −→ g of restricted Lie algebras. The automorphism groups Aut p (g) and Aut p (t) naturally act on T g (k) via g.ϕ := g • ϕ and h.ϕ := ϕ • h −1 ∀ ϕ ∈ T g (k), g ∈ Aut p (g), h ∈ Aut p (t), respectively. Both actions commute, and Aut p (t) ∼ = GL µ(g) (F p ) is a finite group. We shall also consider the canonical action of G on g and write g.x := g(x) ∀ g ∈ G, x ∈ g.
Let ι : t ֒→ g be the standard embedding, defined by the inclusion t ⊆ g. Since T g (k) is smooth, its irreducible components coincide with its connected components. Hence there exists exactly one irreducible component X t (k) ⊆ T g (k) containing the inclusion ι. We let S(g, t) := Stab Autp(t) (X t (k)) be the stabilizer of the component X t (k) in Aut p (t). For future reference, we recall the following result, cf. [8, (4
We let Nor G (t) and Cent G (t) be the normalizer and the centralizer of t in G, respectively. In view of the above result, we make the following: Definition. Let (g, [p]) be a restricted Lie algebra with automorphism group G = Aut p (g) • , t ⊆ g be a torus of dimension µ(g). Then S(g) := S(g, t) is called the toral stabilizer of g. The group W (g, t) := Nor G (t)/ Cent G (t) is referred to as the Weyl group of g relative to t.
The group G naturally acts on the variety Tor(g) of tori of maximal dimension. Since W (g, g.t) ∼ = W (g, t) for every g ∈ G, the Weyl group W (g, t) only depends on the orbit G.t ⊆ Tor(g).
By virtue of Theorem 1.1, every torus t ′ ⊆ g of dimension µ(g) is of the form t ′ = ϕ(t) for some ϕ ∈ X t (k). The foregoing Lemma then provides an embeddinḡ sending the Weyl group W (g, t ′ ) of t ′ into the toral stabilizer S(g, t). If g is of classical type, then this map is an isomorphism, see [8, (4.6)] or Theorem 1.5 below.
Our main goal of this section is to provide a criterion ensuring that W (g, t) ∼ = S(g) for some torus t ∈ Tor(g).

Definition.
A torus t ∈ Tor(g) is called generic if the orbit G.t is a dense subset of Tor(g).
Let g := Lie(G) be the Lie algebra of a connected algebraic group. In view of [11, (13.3),(13.5)] (see also [8, (4.4)]), any two tori of g of maximal dimension are conjugate via the adjoint action. Accordingly, each of these tori is generic.
) be a restricted Lie algebra with connected automorphism group G := Aut p (G) • , t ⊆ g be a torus of dimension µ(g). Then for every ϕ ∈ X t (k).
Proof. We consider the morphism ω : of irreducible varieties. Given t ∈ t, we let (kt) p denote the p-subalgebra of g generated by kt. Lemma 3.2 of [8] provides a non-empty open subset U ⊆ t with t = (kt) p for every t ∈ U . Let (ϕ, t) be an element of X t (k)×U . If (ψ, s) ∈ ω −1 (ω(ϕ, t)), then ψ(s) = ϕ(t), so that By choice of t, this implies that im ϕ ⊆ im ψ, whence im ϕ = im ψ. Hence there exists an element h ∈ Aut p (t) with ψ = ϕ • h −1 . In particular, ϕ(h −1 (s)) = ψ(s) = ϕ(t), so that s = h(t). As a result, the fiber Now let ϕ ∈ X t (k) be arbitrary. Then ω induces a surjective morphism of irreducible varieties. By the above argument, the fibers of this morphism are finite on the dense open subset (G.ϕ)×U of (G.ϕ)×t. By virtue of [18, I. §8, Corollary 1], we obtain as desired.
Let S g denote the set of semi-simple elements of g. The following result relates dense orbits in Tor(g) and X t (k) to dense G-saturations of tori in S g relative to the canonical action of G on g.
) be a restricted Lie algebra, G = Aut p (g) • be its connected automorphism group, and t ⊆ g be a torus of maximal dimension. For a torus t 0 ∈ Tor(g), the following statements are equivalent: (1) The torus t 0 is generic. Proof.
(3) ⇒ (1). Thanks to Theorem 1.1, there exists an element ϕ 0 ∈ X t (k) with t 0 = ϕ 0 (t). According to Lemma 1.3, we have so that the orbit G.ϕ 0 lies dense in X t (k). Hence and t 0 is a generic torus. Theorem 1.5. Let (g, [p]) be a restricted Lie algebra with connected automorphism group G := Aut p (g) • . If t ∈ Tor(g) is a generic torus, then the map Proof. Thanks to Proposition 1.4, there exists ϕ 0 ∈ X t (k) with ϕ 0 (t) = t such that the orbit G.ϕ 0 lies dense in X t (k). It follows that G.ϕ 0 is an open subset of the irreducible variety X t (k). Consequently, G.ϕ 0 is the unique dense orbit of X t (k).
Given h ∈ S(g, t), we consider the automorphism is a dense orbit of X t (k). By our observation above, we have G.w h (ϕ 0 ) = G.ϕ 0 , and there exists an element g ∈ G with In particular, we obtain is bijective, so that the groups W (g, t) and S(g, t) have the same order. We may now apply Lemma 1.2 to the canonical embedding ι : t ֒→ g to see thatΘ ι is the desired isomorphism.
We provide another density result which compensates for the lack of a generic torus to some extent. Given a torus t ⊆ g, we let C g (t) be the centralizer of t in g.
Lemma 1.6. Let (g, [p]) be a restricted Lie algebra. Then T := t∈Tor(g) t is a dense subset ofS g .
Proof. We put A = k[X t ] andg = g ⊗ k A. Let j : t ֒→g be the universal embedding. The torus t acts ong via j; we denote byg 0 ⊆g the corresponding zero weight space. For any homomorphism x ∈ Spec k (A)(k), let ϕ x := (id g ⊗x) • j ∈ T g (k) denote the corresponding embedding. Then g 0 (x) = (id g ⊗x)(g 0 ) is the zero weight space relative to the torus ϕ x (t). Consider the variety Owing to [8, (3.4)], the projection π g : V −→ g is a dominant morphism with image Also, for n = dim k g, the morphism ζ : g −→S g ; y → y [p] n is dominant. Thus, ζ • π g is dominant, and we have Since the right hand set is dense inS g and contained in T, our assertion follows.
Remark. In the proof, we used the fact that the morphism g −→ g ; y → y [p] n is dominant for n = dim k g. In fact, there are lower values of n for which this property holds, see the Remark p. 7 and Corollary 6.3.4.
Remark. Let ℓ := rk(g) − µ(g), and ζ ℓ : g −→ g ; y → y [p] ℓ . Then ζ ℓ induces a dominant morphism g →S g . Indeed, since π g : V −→ g is dominant, it is enough to show that ζ ℓ • π g is dominant. We first check it under for g nilpotent. Then the set N (g) of p-unipotent elements form a p-subalgebra of g of dimension ℓ. Hence, for all x n ∈ N (g), we have x [p] ℓ n = 0. Furthermore, g affords a unique torus t of dimension µ(g). Then it is then easy to see that x [p] ℓ ∈ t, whence g [p] ℓ = t = S g as we needed. Now when g is arbitrary, we have Since each h := C g (ϕ x (t)) is a nilpotent subalgebra of dimension rk(g) with µ(h) = µ(g), we have h [p] ℓ = ϕ x (t). Comparing with Equation (1.2) yields im ζ ℓ • π g = im ζ • π g , a dense subset ofS g .

Definition.
A Cartan subalgebra h ∈ Car(g) is called generic if the orbit G.h is dense in Car(g).
As before, the Cartan subalgebras of the Lie algebras that are associated with algebraic groups are generic. Proposition 1.7. Let (g, [p]) be a restricted Lie algebra with connected automorphism group G := Aut p (g) • . Suppose that t 0 ∈ Tor(g) is a torus, and let h 0 := C g (t 0 ) ∈ Car(g) be the corresponding Cartan subalgebra. Then the following statements are equivalent: (1) The Cartan subalgebra h 0 is generic.
As Tor(g) is irreducible, we conclude G.t 0 = Tor(g), showing that t 0 is a generic torus.
(3) ⇒ (1). Suppose that g = G.h 0 . Setting ℓ := dim k g, we recall that is a G-equivariant morphism, whose image coincides with S g . Moreover, and Proposition 1.4 implies that t 0 is generic. Thus, proving that h 0 is a generic Cartan subalgebra of g.
) be a restricted Lie algebra, V ⊆ g be a subspace of g.

Basic Properties of Toral Stabilizers
Our computations of toral stabilizers necessitate information concerning their behavior with respect to inclusions and passage to factor algebras. Let (g, [p]) be a restricted Lie algebra with torus t ⊆ g of maximal dimension. Recall that S(g, t) is a subgroup of Aut p (t).
Lemma 2.1. Let h ⊆ g be a p-subalgebra of the restricted Lie algebra (g, [p]).
(2) If h ✂ g is an ideal such that g/h is p-unipotent, then µ(h) = µ(g) and S(h, t) = S(g, t) for every torus t ⊆ h of dimension µ(h).
Corollary 2.2. Let t ⊆ g be a torus of dimension µ(g), s ⊆ t be a subtorus. Then we have a natural inclusion S(C g (s), t) ⊆ {g ∈ S(g, t) ; g| s = id s }.
Proof. We consider the p-subalgebra h := C g (s). By definition, µ(h) = µ(g), and Lemma 2.1 provides an inclusion S(h, t) ⊆ S(g, t). In view of [8, (5.11)], each element of S(h, t) belongs to the centralizer of s in Aut p (t).
For a direct sum t 1 ⊕ t 2 of vector spaces, we denote by ι t i and π t i the canonical injection from t i and the projection onto t i , respectively. Recall the identification where the maps from End k (t 1 ⊕ t 2 ) to any of the blocks arise via composition with ι's and π's: For example, the projection End k (t 1 ⊕ t 2 ) ։ Hom k (t 2 , t 1 ) is given by The groups GL(t 1 ) and GL(t 2 ) canonically act on Hom k (t 2 , t 1 ) from the left and right, respectively. The semi-direct product GL(t 2 ) ⋉ Hom k (t 2 , t 1 ) can be identified with a subgroup of GL(t 1 ⊕ t 2 ), namely: (Note that the sizes of boxes are given by the dimensions of the relevant subspaces of t, so that there is no entry in the first row and column for t 1 = (0), say.) For tori t, t ′ , let Lie p (t, t ′ ) denote the additive group of restricted homomorphisms t → t ′ . It affords a natural right action by Aut p (t) and a natural left action by Aut p (t ′ ). We first generalize a basic result from [8].
) be a restricted Lie algebra, n ✂ g be a p-ideal. Then the following statements hold: (1) If t ⊆ g is a torus of maximal dimension, then t ∩ n and (t + n)/n are tori of maximal dimension of n and g/n, respectively.
(1) According to [10, (3.3)], the p-subalgebra (t + n)/n ⊆ g/n is a torus of g/n of maximal dimension. By the same token, we have Since dim k t = dim k (t ∩ n) + dim k (t + n)/n, it follows that t ∩ n is a torus of n of maximal dimension.
Owing to ( * ), the canonical restriction map induces a morphism res : X t (k) −→ T n (k) ; ϕ → ϕ| t∩n of affine varieties. As res is continuous, we have im res ⊆ X t∩n (k).
(1) The first part of the Proposition is well-known, see for example [28,Theorem 4.3] for maximal tori and [2, Lemma 1.7.2] for tori of maximal dimension.
(2) Let t ′ ⊆ t be tori. Let {t 1 , . . . , t n } be a toral basis of t and denote by V the F p -subspace of t generated by {t 1 , . . . , t n }. Since V is the set of toral elements of t, there exists a subspace (3) One can show that the restriction morphism res : X t (k) −→ X t∩n (k) is smooth. In particular, res is open and dominant.
Corollary 2.4. Let g 1 , g 2 be restricted Lie algebras and t i ⊆ g i be tori of maximal dimension for i ∈ {1, 2}. Then t 1 ⊕ t 2 ⊆ g 1 ⊕ g 2 is a torus of maximal dimension, and Proof. We put g := g 1 ⊕ g 2 and t := t 1 ⊕ t 2 , and denote by ι : t ֒→ g the canonical embedding. Let T g be the scheme of embeddings of g and X ⊆ T g be the connected component containing ι. For i ∈ {1, 2}, we denote similarly ι i : t i ֒→ g i and the schemes X i ⊆ T g i . According to Proposition 2.3, we have ( * ) for each i ∈ {1, 2}. Consequently, Proposition 2.3 shows that is an injective homomorphism of groups. Given i ∈ {1, 2}, the map is a morphism of affine varieties. As a result, given g i ∈ S(g i , t i ), the elements g 1 ⊕ id t 2 and id t 1 ⊕g 2 belong to S(g, t), and the above map is also surjective.

Automorphisms and generic tori for Lie algebras of Cartan type
Henceforth, we will assume that k is an algebraically closed field of characteristic p ≥ 3. We shall apply the results of Section 1 in the situation where g is a restricted Lie algebra of Cartan type. By definition, these restricted simple Lie algebras possess a restricted Z-grading where we set g i := (0) for i ∈ {−r, . . . , s}. Given such an algebra g, we consider the associated descending filtration (g (i) ) i≥−r , defined via By definition, the restricted Lie algebras of Cartan type are p-subalgebras of the algebra of derivations of the truncated polynomial rings whose canonical generators will be denoted x 1 , . . . , x n . The Lie algebra W (n) := Der k (A n ) is called the n-th Jacobson-Witt algebra. Its p-map is the standard p-th power of linear operators. We let ∂ i ∈ W (n) denote the partial derivative with respect to the variable x i . Then {∂ 1 , . . . , ∂ n } is a basis of the A n -module W (n), so that dim k W (n) = np n .
Here and further, we use the standard multi-index notation and write x α := x α 1 1 · · · x αn n and We briefly describe the other Cartan type Lie algebras in terms of differential forms; the reader is also referred to [25, Chap. IV] and [14] for further details.
We set Ω 0 (A n ) = A n as well as Ω 1 (A n ) := Hom An (W (n), A n ). The latter space carries canonical structures of an A n -module and a W (n)-module (cf. [14] or [25]) via Let Ω r (A n ) := r Ω 1 (A n ) be the r-fold exterior power over A n and denote the exterior algebra by The action of W (n) on Ω 1 (A n ) extends uniquely to an action by derivations on Ω(A n ). For r ≥ 2, we have Consider the k-linear map d : Associated with the Cartan differential forms Recall that, given a 2r-dimensional k-vector space with a non-degenerate symplectic form , : is referred to as the group of similitudes of the symplectic space (V, , ). 26,27]). Let g be a restricted Lie algebra of Cartan type X(n) over a field k of characteristic p ≥ 3. If g = W (1) or H(2), assume that p ≥ 5. Then the following statements hold: (1) The group G := Aut p (g) is connected.
(2) The group G is a semidirect product G = U ⋊ G 0 , where G 0 consists of those automorphisms preserving the Z-grading of g, and U : for X = W, S, G Sp(2r), for X = H and K with n = 2r and 2r+1, respectively.
(3) We have g(g (i) ) = g (i) for every g ∈ G and i ∈ Z.
Proof. According to Wilson's results [26,Thm. 2], the theorem holds for the group of automorphisms which preserve the standard filtration. So we need to show that under the assumptions of the theorem, all automorphisms of g preserve it. Since this is the standard filtration defined with respect to the maximal subalgebra g (0) g [24, Definition 3.5.1], we have to show that any automorphism preserves this subalgebra. When p ≥ 5, this is Kreknin's theorem [16]. For p = 3, the result is still true by [24, (4 is the unique maximal subalgebra h of codimension 3 such that the h-module g/h is not irreducible. It readily follows that g (0) is stable under all automorphisms of g in all relevant cases, so the proof is complete.
Owing to (3) of Theorem 3.1, the subsets Tor(g) g (i) and Tor(g) g (i) of Tor(g) are G-invariant. It also follows that the function is constant on the G-orbits of Tor(g) g (−1) .
The following result, which states that f 0 forms a complete set of invariants for the G-orbits of Tor(g) g (−1) , is based on Demuškin's work [4,5], as corrected in [24, §7]. In our statement below, we represent the contact algebra as in [25,(IV.5)].
Theorem 3.2. Let g be a Lie algebra of Cartan type X(n), where X ∈ {W, S, H, K}. Then the following statements hold: (1) Two tori t, t ′ ∈ Tor(g) g (−1) belong to the same G-orbit if and only if f 0 (t) = f 0 (t ′ ).
(2) Each of the following tori represents the unique orbit The foregoing results yield the following description of the toral stabilizers of the Lie algebras of Cartan types W, S and H. Proof. According to Lemma 1.8 and Theorem 3.1, Tor(g) g (−1) is a closed, G-invariant subset of Tor(g). By the same token, In particular, the orbit G.t 0 is open in Tor(g) g (−1) . Since g = g (−1) for g = W (n), S(n), H(2r), it follows from the irreducibility of Tor(g) that t 0 actually is a generic torus of g. Our second assertion is now a direct consequence of Theorem 1.5.
S g , and Proposition 1.4 shows that the torus t 0 is not generic. In fact, we will see in (6.2.3) that contact algebras do not afford generic tori.

Embeddings of W (n)
In preparation for our computation of the toral stabilizers and Weyl groups of the generic tori of the Lie algebras of Cartan types W, S and H, we provide in this section embeddings W (µ(g)) ֒→ g for such Lie algebras.
The map div : is called the divergence of W (n). The special algebra S(n) is the derived algebra of the subalgebra ker div, cf. [25,(IV.3)]. Let r ≤ n. In the sequel, we shall identify A r with the subalgebra of A n that is generated by x 1 , . . . , x r . A derivation D ∈ Der k (A r ) will be regarded as an element of Der k (A n ) by setting D(x j ) = 0 for r +1 ≤ j ≤ n. As before, we shall employ multi-index notation. In particular, ε j denotes the vector, whose coordinates are given by the Kronecker symbols δ ij .
is an injective homomorphism of restricted Lie algebras.
In view of Jacobson's formula, it suffices to verify the identity σ n (x [p] ) = σ n (x) [p] on a basis of W (n−1). Let a ≤ τ be an element of N n−1 This expression vanishes unless a = ε j . In the remaining case, we readily obtain We denote by P(2r) the Poisson algebra, i.e., the space A 2r , endowed with the standard Poisson bracket { , }. By definition, { , } is an associative bi-derivation, defined on the generators x 1 , . . . , x 2r by the rule: The Lie algebra P(2r) is restrictable, and any p-mapping of P(2r) renders the above exact sequence an exact sequence of restricted Lie algebras. We define a p-mapping by the following formula (cf. [24, p. 403]): x a for a ∈ {0, ε i + ε i+r ; i = 1, . . . , r}.
In particular, 1 [p] = 1, and the center is a torus.
is an injective homomorphism of restricted Lie algebras.
Proof. We check that ϕ r is a Lie algebra homomorphism. Given f, g ∈ A r and i, j ∈ {1, . . . , r}, . , x 2r ), we therefore obtain for all a ∈ N r 0 . Consequently, ϕ r is an injective homomorphism of restricted Lie algebras.
As a corollary, we readily obtain:

Toral stabilizers of Cartan type Lie algebras
In this section, we turn to the computation of the toral stabilizers of restricted Lie algebras of Cartan type. Our approach involves embeddings of suitable p-subalgebras.

Poisson algebras.
We begin by determining the toral stabilizers of the Poisson algebra with toral center. For r ≥ 1, we consider the Lie algebra kt i , whose bracket and p-mapping are given by , respectively, with all unspecified products being zero. Proof. Setting t := r i=2 kt i , we obtain a direct sum decomposition l r = l 1 ⊕ t of l r , whose second summand is a torus. Consequently, Corollary 2.4 implies According to [8, p.4215], we have S(l 1 ) ∼ = Z/pZ, as desired.
A restricted Lie algebra g is referred to as trigonalizable if it can be embedded into a restricted Lie algebra of triangular matrices. This is equivalent to every restricted simple g-module being one-dimensional.
Proof. We first show that S(P(2r)) ≃ GL r (F p ) ⋉ W , where W = (0) or W = (F p ) r with the natural action by GL r (F p ).
Owing to [21, Thm. 1] and Proposition 3.3, the group S(W (r), t ′ ) is isomorphic to GL r (F p ). Since the matrix group on the right identifies with the semi-direct product of GL r (F p ) with the standard module (F p ) r , we obtain embeddings In order to show that W = (0), we consider the subspace of P(2r). We claim that l is a p-subalgebra of P(2r) which is isomorphic to l r . Using the fact that the Poisson bracket is a bi-derivation for the associative structure, we obtain Since polynomials in x i and x i+r commute with polynomials in the remaining variables, we see that all other brackets are 0. Hence, l ≃ l r as abstract Lie algebras. We have 1 [p] = 1 by definition. Also, using Jacobson's formula, we obtain One can easily check (ad t i ) p = ad t i by comparing the action of these derivations on the associative generators x 1 , . . . , x r and 1+x r+1 , . . . , 1+x 2r of P(2r). As the center C(P(2r)) coincides with k, there exists λ ∈ k such that t Observe that both, x i and x i x i+r , belong to the subspace E := x i ·k[x i+r ]. It is easy to check that E is stable under ad x i and ad x i x i+r . Hence all the s j (x i , x i x i+r ) belong to E. Consequently, we obtain λ1 + t i = t [p] i ≡ x i x i+r mod E, forcing λ = 0, as required. According to [24, (7.5.10)], the space t := k + r i+1 kt i ⊆ l is a torus of P(2r) of maximal dimension. By Lemma 5.1.1, we have S(l, t) ≃ Z/pZ. Since the factor algebra l/k is trigonalizable with torus t ′ := t/k of maximal dimension, [8, (6.1)] implies S(l/k, t ′ ) = {1}. Proposition 2.3 now provides an embedding ξ in the top line of the following diagram: The map α is the inclusion given by Lemma 2.1, and the matrices on the right describe the endomorphisms of t = k ⊕ t ′ according to the block decomposition (2.1). By construction of all these maps, the diagram is commutative. Since im ξ ≃ S(l, t) = {1}, we have W = (0), as desired.
For the last statement of the theorem, observe that a maximal torus t always contains n = k.1 as a subtorus. Corollary 2.2 provides an inclusion S(P(2r), t) ⊆ {g ∈ Aut p (t) ; g(1) = 1}. As these groups have same order, this inclusion is an equality.

Lie algebras of Cartan type. The main result of this section reads:
Theorem 5.2.1. Let g be a Lie algebra of Cartan type. Then there is an isomorphism Proof. Assume first that g is of type W . Proposition 3.3 provides a generic torus t 0 ⊆ g and an isomorphism W (g, t 0 ) ∼ = S(g).
The rest of the proof is devoted to contact algebras. Let n = 2r+1 be an odd number. Following [25, (IV.5)], we endow the truncated polynomial ring A n with the structure of a restricted Lie algebra; the corresponding Lie bracket , is called contact bracket. In the sequel, K ′′ (n) will denote the vector space A n , endowed with this bracket. Then K(n) = K ′′ (n) (1) is the contact algebra. Let Then t is a torus of maximal dimension in K(n), and we claim that We identify Aut p (t) with GL r+1 (F p ), using the ordered basis {1+x n , x 1 x r+1 , . . . , x r x 2r }. Henceforth, all matrix representations will describe automorphisms of t by means of this basis.
By [24, (7.5.15)], we know that the centralizer c = C K ′′ (n) (1+x n ) is isomorphic to the Poisson algebra P(2r) with toral center. Since t ⊆ c, Theorem 5.1.2 yields the following matrix description: with the inclusion following from Lemma 2.1.
We will now embed lower triangular matrices into S(K ′′ (n), t), by means of a method similar to the one used in Theorem 5.1.2. Consider the subspace l := kx r x 2r ⊕ kx r x 2r (1+x n ) ⊕ k(1+x n ) ⊆ K ′′ (n).
We first prove that l is a restricted Lie algebra isomorphic to the one in [8, p. 4216] (or to l 1 in Lemma 5.1.1). We check that such an isomorphism is defined by the assignment Indeed, directly from the rules given in [25, p.173], we obtain x r x 2r , 1+x n = 0 = x r x 2r , x r x 2r (1+x n ) as well as 1+x n , x r x 2r (1+x n ) = 1, x r x 2r (1+x n ) + x n , x r x 2r (1+x n ) = 2x r x 2r + 2x r x 2r x n = 2x r x 2r (1+x n ).
In view of [25, p.177], we also have . Write t l = k(1+x n ) + kx r x 2r to ease notation. Since t ′ , l = 0, there is an embedding t ′ ⊕ l ֒→ K ′′ (n) of restricted Lie algebras, which in turn yields an embedding of finite groups. According to [8, p. 4216], the group S(l, t l ) is generated by the transformation x r x 2r → x r x 2r , (1+x n ) → x r x 2r + (1+x n ). Using the previous matrix identification, this yields In view of (5.1), the group S(K ′′ (n), t) also contains 1 0 0 GLr(Fp) , so the irreducibility of the GL r (F p )-module (F p ) r yields Using (5.1) and (5.2), we see that S(K ′′ (n), t) contains the subgroup generated by all shear matrices, which, owing to [13, Lemma 6.7.1], coincides with SL r+1 (F p ). By (5.2), it also contains matrices of arbitrary determinant, whence S(K ′′ (n), t) ∼ = GL r+1 (F p ). Since the factor algebra K ′′ (n)/K(n) is p-unipotent, Lemma 2.1 now yields the result.
Taking into account Theorem 1.1 (1), we obtain the following remarkable corollary: Corollary 5.2.2. Let g be a simple Lie algebra of Cartan type. Then the variety T g (k) is irreducible.
Proof. Let t ⊆ g be a torus of maximal dimension, so that S(g, t) ⊆ Aut p (t). By Theorem 1.1, the variety T g (k) has [Aut p (t) : S(g, t)] irreducible components. Now the previous theorem shows that [Aut p (t) : S(g, t)] = 1.

The Melikian algebra.
For the sake of completeness, we mention the case of the restricted Melikian algebra. Recall that this is a simple graded Lie algebra over a field of characteristic p = 5, which is neither classical nor of Cartan type. We refer the reader to [22] for the details.

Applications
The applications to be discussed in this final section pertain to weight space decompositions, the non-existence of generic tori and invariants for certain Lie algebras of Cartan type. While the second topic originates in Demuškin's early work [4,5] along with Strade's corrections [24], our observations on invariants are motivated by Premet's paper [21]. 6.1. Irreducibility and Weights. Our first result illustrates why it is reasonable to consider toral stabilizers as a replacement for Weyl groups when dealing with arbitrary restricted Lie algebras.
By definition, a restricted g-module M is a g-module such that for every x ∈ g, the operator m → x [p] .m is the p-th power of the transformation of M effected by x. If t ⊆ g is a torus, then M is a completely reducible t-module, giving rise to the weight space decomposition Each λ ∈ Λ M is a linear form satisfying λ(t [p] ) = λ(t) p for every t ∈ t. As a result, Λ M ⊆ Lie p (t, k) is a subset of the character group of t, i.e., the additive group of all homomorphisms t −→ k of restricted Lie algebras, where the p-map on k is the associative p-th power. The group Aut p (t) acts contragrediently on Lie p (t, k) via If t ⊆ g is a torus of maximal dimension, then S(g, t) ⊆ Aut p (t) also acts on Lie p (t, k). Proof. As in the proof of Lemma 1.6, put A := k[X t ] and let j : t ֒→ g ⊗ k A be the universal embedding of t. In particular, given a commutative k-algebra R, an element ϕ x : t → g ⊗ k R corresponds to x ∈ Spec k (A)(R) if and only if ϕ x = (id g ⊗x) • j. Consider the restricted A-Lie algebrag := g ⊗ k A. ThenM := M ⊗ k A is a restrictedg-module, and the universal embedding gives rise to a weight space decompositionM = γ∈ΓMM γ of the t-moduleM , whose constituents are A-submodules ofM . Let x ∈ Spec k (A)(k) be a k-rational point. As observed in [8, (4.2)], we have is the weight space with weight γ • ϕ −1 x relative to the torus ϕ x (t) ⊆ g. Since eachM γ is a finitely generated projective module over the integral domain A, it has constant rank and [17, p.166 In particular, these dimensions are non-zero integers which depend only on γ. Let ι : t ֒→ g be the canonical inclusion. Given g ∈ S(g, t), we have g.ι = ι • g −1 ∈ X t (k), and is the weight space decomposition of M relative to t. Thus, Λ M = g.ΓM for all g ∈ S(g, t) and dim k M g.λ = rk(M λ ) = dim k M λ for every λ ∈ Λ M . Corollary 6.1.2. Let t ⊆ g be a torus of maximal dimension, M = λ∈Λ M M λ be a restricted g-module. If the variety T g (k) is irreducible, then the following statements hold: (1) Λ M = {0}, or Λ M contains Lie p (t, k) \ {0}.
(2) All weight spaces of M belonging to non-zero weights have the same dimension.
Proof. Let t ⊆ g be a torus of maximal dimension, M = λ∈Λ M M λ be the corresponding weight space decomposition of M . By Theorem 6.1.1, the set Λ M ⊆ Lie p (t, k) is S(g, t)-stable. According to Theorem 1.1, the variety T g (k) has [Aut p (t) : S(g, t)] irreducible components. In view of our assumption, this yields S(g, t) = Aut p (t). Consequently S(g, t) ∼ = GL µ(g) (F p ) acts via two orbits on Lie Remark. Let g be such that T g (k) is irreducible. For any restricted g-module M = λ∈Λ M M λ , we have Λ M = {0}, Lie p (t, k) \ {0} or Lie p (t, k). We consider the Jacobson-Witt algebra g = W (n) acting via derivations in the truncated polynomial ring A n . Then, k ⊆ A n is a submodule, and the three possibilities occur for k, A n /k, and A n , respectively.
We record the particular case given by the adjoint representation: Corollary 6.1.3. Let g = g 0 ⊕ α∈Φ g α be the root space decomposition of g relative to a torus t ⊆ g of maximal dimension. If the variety T g (k) is irreducible, then Φ ∪ {0} = Lie p (t, k), and all root spaces of g have the same dimension.
6.2. Non-generic tori. We will prove that the Weyl groups of the Poisson algebra P(2r) do not coincide with the toral stabilizer, and obtain as a consequence that contact algebras do not afford generic tori. This result strengthens Strade's observation [24], who showed that K(2r+1) possesses infinitely many conjugacy classes of maximal tori.
To do so, we first gather some observations concerning automorphisms of P(2r). Any Lie algebra automorphism of P(2r) stabilizes the center k ⊆ P(2r). The simple Lie algebra H(2r) can be realized as the derived subalgebra of the quotient algebra P(2r)/k. There results a homomorphism π : Aut(P(2r)) −→ Aut(H(2r)) of groups. Our analysis of Aut(P(2r)) involves two classes of automorphisms. A Lie algebra automorphism σ ∈ Aut(P(2r)) is a semi-Poisson automorphism if there exists a non-zero scalar α ∈ k × such that ασ preserves the associative structure of P(2r). If α = 1, then σ is referred to as a Poisson automorphism. The set of semi-Poisson automorphisms is a subgroup of Aut(P(2r)), which we denote by Aut SP (P(2r)).
Recall the notation i ′ = i+r (resp. i−r) for i ≤ r (resp. i > r), so that {x i , x j } = δ i ′ ,j . Since the element , lies in P(2r) (1) , it is fixed by σ. Hence, for all i ∈ {1, . . . , 2r}. Since bracketing by x i lowers the degree by 1, {x i , g} is a linear combination of monomials of degree < d−1. Consequently, both sides are zero, so that α = 1 and {x i , g} = 0 for all i. The latter condition implies that g is central, i.e. g ∈ k as desired.
We turn to Weyl groups of the Poisson algebra.
Proposition 6.2.2. Let t ⊆ P(2r) be a torus of maximal dimension, H ⊆ Aut p (P(2r)) be a subgroup. Then the following statements hold: (1) There exists an embedding (3) The Poisson algebra P(2r) affords no generic torus.
Proof. (1) Let m ⊆ P(2r) denote the subspace of polynomials without constant term. Being the unique maximal ideal of the local algebra A 2r , m is stable under semi-Poisson automorphisms. Owing to [24, (7.5.10)], the torus t is conjugate under a Poisson automorphism to one of the following tori: (But the conjugating automorphism is not restricted in general.) Each of these tori has the form k ⊕ T ′ for some torus T ′ ⊆ m. Since Poisson automorphisms stabilize m, we also obtain a decomposition t = k ⊕ t 0 , where t 0 ⊆ m is a torus. Let g ∈ Nor H (t). Thanks to Theorem 6.2.1, the automorphism g is a composite g = g 0 • h, where g 0 is a semi-Poisson automorphism and h ∈ H. Now, t being a torus, we have t ⊆ P(2r) (1) , so that h acts trivially on t. Thus, g and g 0 induce the same automorphism of t, whence g(k) ⊆ k and g(t 0 ) ⊆ t ∩ m = t 0 . We obtain a well-defined mapping Nor H (t) −→ Aut p (k) × Aut p (t 0 ) ; g → (g| k , g| t 0 ), which induces the desired embedding Nor H (t)/ Cent H (t) ֒→ F * p × GL r (F p ). (2) According to Lemma 1.2, the Weyl group W (P(2r), t) is contained in S(P(2r), t). By part (1), there is an embedding W (P(2r), t) ֒→ F * p × GL r (F p ), while Theorem 5.1.2 yields S(P(2r), t) ≃ GL r (F p ) ⋉ (F p ) r .
(3) This is a direct consequence of (2) and Theorem 1.5.
Proof. We write g := K ′′ (2r + 1) to ease notation, so that g (1) = K(2r + 1). Note that g (1) is a p-ideal of g such that the factor algebra is p-unipotent. As a consequence, all tori of g are contained in g (1) , and we have S g = S g (1) . Furthermore, if t ⊆ g (1) is a torus of maximal dimension, then Lemma 2.1 and Theorem 5.2.1 yield S(g, t) = S(g (1) , t) = Aut p (t).
In the sequel, we write G := Aut p (g) • and G K := Aut p (g (1) ). Assume that there exists a generic torus t ⊆ g (1) . In view of Proposition 1.4, the saturation G K .t is dense inS g (1) . The above observations yield so that G.t is a dense subset ofS g =S g (1) . Applying Proposition 1.4 again, we conclude that t is also a generic torus of g.
Note that the argument of the remark following Proposition 3.3 applies to any generic torus of g (1) , yielding t ⊆ g (1) (−1) . Since t ⊆ g (1) , this means t ⊆ g (−1) . Owing to [24, (7.5.14)], we may assume that 1 + x n ∈ t. Then t is also a torus of maximal dimension of the centralizer c := C g (1 + x n ). By [24, (7.5.15)], there is an isomorphism ϕ : c → P(2r) of restricted Lie algebras such that ϕ(1+x n ) = 1, the canonical central element in the Poisson algebra. Applying Theorem 5.1.2, we obtain S(c, t) = Cent Aut p (t) (1+ x n ). Since t ⊆ g is generic, Theorem 1.5 in conjunction with the above observation S(g, t) = Aut p (t) ensures that the map Nor G (t) −→ Aut p (t) is surjective. Thus, given g ∈ Aut p (t), there exists ϕ ∈ G such that ϕ| t = g. Since g fixes 1+x n , we have ϕ| c ∈ Aut p (c). Setting H := Aut p (c), we obtain a surjection Nor H (t) −→ Aut p (t), which contradicts Proposition 6.2.2(1). 6.3. Invariants for Lie algebras of Cartan Type. Given a restricted Lie algebra (g, [p]) with connected automorphism group G = Aut p (g) • and maximal torus t ⊆ g, we are interested in comparing the invariants k[g] G and k[t] S(g) of the respective rings of polynomial functions. In case S(g) = GL µ(g) (F p ), the generators of the latter algebra were explicitly determined by Dickson, cf. [6].
We begin with some general observations concerning invariants of polynomial functions of a finite-dimensional associative k-algebra A. Let G ⊆ Aut(A) be a connected, closed subgroup of the automorphism group of A. Given a finite-dimensional A-module M , we let a M : M −→ M be the left multiplication effected by the element a ∈ A. Let T be an indeterminate. For a ∈ A, we consider the characteristic polynomial We record the following basic observation: Proof. Given g ∈ G, we let M (g) be the A-module with underlying k-space M and twisted action defined via a.m := g −1 (a).m ∀ a ∈ A, m ∈ M. Then we have a M (g) = g −1 (a) M , so that for every a ∈ A and i ∈ {0, . . . , dim k M }.
Let S be a simple A-module. Since the connected group G acts trivially on the set of primitive central idempotents of the algebra A/ Rad(A), it follows that Consequently, Returning to our standard set-up, we let (g, [p]) be a restricted Lie algebra with automorphism group G = Aut p (g) • . Consider the two-sided ideal I ✂ U (g) of the universal enveloping algebra that is generated by the set {x p − x [p] ; x ∈ g}. The finite-dimensional algebra U 0 (g) := U (g)/I is the restricted enveloping algebra of g, see [25, (V.5.3)] for more details. By the universal property of U 0 (g), the group G can be considered a connected subgroup of the automorphism group of U 0 (g). Any finite-dimensional U 0 (g)-module M thus gives rise to invariant polynomials ψ M,i , whose restrictions to g are elements of the ring k[g] G of invariants of polynomial functions on g. We shall henceforth consider the ψ M,i as elements of this ring.
Let (t, [p]) be a torus of dimension m. Then Aut p (t) ∼ = GL m (F p ) acts contragrediently on the ring k[t]. The following subsidiary result interprets Dickson's work [6] in this context.
with algebraically independent generators.
Proof. Let X(t) be the character group of t. By definition, X(t) is the group of algebra homomophisms λ : U 0 (t) −→ k. If {t 1 , . . . , t m } is a toral basis of t, then the map λ → (λ(t 1 ), . . . , λ(t m )) provides an isomorphism X(t) ∼ = F m p of groups. Since t is a torus, the free U 0 (t)-module of rank 1 decomposes into a direct sum of one-dimensional restricted t-modules: Consequently, we obtain for t = m i=1 x i t i ∈ t the identities P U 0 (t) (T ; t) = λ∈X(t) Our result now follows directly from Dickson's Theorem [6, (8.1.1)].
Let M be a U 0 (g)-module, t ⊆ g be a torus of maximal dimension. Thanks to Theorem 6.1.1, the restrictions ψ M,i | t belong to the subring k[t] S(g) .
Proof. We set µ := µ(g) and d := d(g) to ease notation. Also, for x ∈ g we denote by x = x s +x n its Jordan-Chevalley decomposition, where x s and x n are p-semi-simple and p-nilpotent respectively, and x s , x n ∈ (kx) p (see [25,(II.3.5)]).
(1) Let ν be the smallest integer for which there exists a non-empty open subset Ω ⊆ g such that Ω [p] ν ⊆ S g . By [20,Theorem 2], there exist polynomial functions ϕ 0 , . . . , ϕ µ ∈ k[g], with ϕ µ = 1, such that µ i=0 ϕ i (x) x [p] ν+i = 0 for all x ∈ g. We need to check that ν = ℓ. Since ϕ µ (x) = 1, we have x [p] ν+µ ∈ ν+µ−1 j=0 kx [p] j , so that ν +µ ≤ d and ν ≤ d−µ = ℓ. For the reverse inequality, we consider the subset Ω 0 := {x ∈ Ω ; dim k (kx) p = d and dim k (kx s ) p = µ} of g. We first show that it is not empty. It suffices to check that the following two sets are non-empty open subsets of g: By construction Ω ′ is not empty; for Ω ′′ it follows for example from [8, 3.5(1)]. An element x ∈ g belongs to Ω ′ if and only if the system of vectors {x, x [p] , . . . , x [p] d−1 } has maximal rank over k. This condition can be expressed in terms of non-vanishing of certain minors depending regularly on x, so Ω ′ is open. Regarding Ω ′′ , consider the morphism q : s , so that taking p ν -th powers yields Consequently, d = dim k (kx) p ≤ ν +µ, whence d−µ ≤ ν, as desired.
Finally, it follows from the construction in [20] that the polynomial Q(T ; x) is minimal for the annihilation property described in (1). The G-invariance of the coefficients ϕ i follows readily.
(2) First we show that P M (T ; x) is a p-polynomial in T . Note that for any torus t ′ ⊆ g of maximal dimension, the module M | U 0 (t ′ ) is also free of rank p r . Indeed, consider the weight space decompositions M = λ∈Λ M λ = λ ′ ∈Λ ′ M λ ′ relative to t and t ′ respectively. By [8, (4.2)], there exists an isomorphism ϕ : t ′ → t such that λ → λ • ϕ is a bijection Λ → Λ ′ satisfying dim k M λ•ϕ = dim k M λ for every λ ∈ Λ. Because restricted modules over a torus are determined by weights and multiplicities of weight spaces, it readily follows that M | U 0 (t ′ ) is free if and only if M | U 0 (t) is free, of the same rank.
Let t ⊆ g be a torus of dimension µ. Since M | U 0 (t) ∼ = U 0 (t) p r , we obtain is square-free, and its set of roots is an additive subgroup of k. Then [12, §20] implies that Suppose that i is not of the form p r+j with j ∈ {0, . . . , µ}. Then the polynomial function ψ M,i (x) vanishes on T := t∈Tor(g) t. By Lemma 1.6, T is dense inS g , so that ψ M,i vanishes onS g . Thus we obtain For arbitrary x = x s + x n ∈ g, the elements x s and x n act as commuting semi-simple and nilpotent endomorphisms on M , so that P M (T ; x) = P M (T ; x s ). Since x s ∈S g , the above formula implies ψ M,i (x) = 0 unless i = p r+j for some j ∈ {0, . . . , µ}, showing that Formula (6.1) holds for all x ∈ g.
Next we check that P M (T ; x) = Q(T, x) p r−ℓ , considered as elements of k[T ]. It suffices to verify the identity for all x in some dense open subset of g. Since P M (T ; x) and Q(T ; x) p r−ℓ are monic p-polynomials in T of the same degree, it is enough to show that they have the same set of roots.
For each x ∈ g, denote by Rac Q (x) and Rac P (x) the set of roots of Q(T ; x) and P M (T ; x). Since Q(T ; x) annihilates x in g, it also annihilates x M ∈ End k (M ), the action of x in M . Therefore, it must be divisible by the minimal polynomial of x M . Since the latter has the same roots as P M (T ; x), we obtain Rac P (x) ⊆ Rac Q (x) for all x ∈ g. Conversely, for x = x s + x n , we have Rac P (x) = Rac P (x s ). When x ∈ Ω 0 as above, the torus (kx s ) p has dimension µ, and Rac P (x s ) is an elementary abelian p-group of rank µ. On the other hand, since Q(T ; x) has degree p d and is the p ℓ -th power of a p-polynomial of degree p µ , Rac Q (x) is an elementary abelian p-group of rank at most d−ℓ = µ. This forces Rac P (x) = Rac Q (x) when x ∈ Ω 0 , as we needed.
(3) readily follows from the proof of (2). [p] ℓ µ ) = λβ(t 1 , . . . , t µ ) p ℓ for all t 1 , . . . , t µ ∈ t. Indeed, the p ℓ -th root of the left hand side defines an alternating µ-linear form on t, and so must be proportional to β. It follows that This shows that x → ϕ i (x) 1/p ℓ is a rational function on the Zariski closure of T ∩ Ω β . For a suitable choice of β, this is a non-empty open subset of T, so by Lemma 1.6 its Zariski closure isS g .
We record the following corollary: ) be a restricted Lie algebra and ℓ = d(g)−µ(g). Then the morphism g → g ; x → x [p] ℓ induces a dominant morphism g →S g . In particular, ℓ = 0 if and only if µ(g) = rk(g).
Let f : A −→ B be an injective homomorphism of finitely generated, commutative k-algebras. If there exists r ∈ N 0 such that B p r ⊆ im f , then f is called an inseparable isogeny of exponent ≤ r.  (1) The canonical restriction map k[g] −→ k[t] induces an algebra homomorphism res : k[S g ] G → k[t] W (g,t) , whose image contains (k[t] GL µ(g) (Fp) ) p ℓ .
(2) We have dim k[S g ] G ≥ µ(g). Equality holds if and only if res is injective. Assume further that g affords a generic torus t 0 . Then: (3) The algebra k[S g ] G is finitely generated and of Krull dimension µ(g).
is an inseparable isogeny of exponent ≤ ℓ. In particular, the varietiesS g / /G and t 0 /W (g, t 0 ) are homeomorphic.
Proof. We will write GL µ instead of GL µ(g) (F p ) to ease notation.
(3) Owing to Proposition 1.4, the G-saturation G.t 0 lies dense inS g . This readily implies that res : A is a finitely generated algebra of Krull dimension µ(g).
Proof. First we note that for g = W (n), S(n) or H(n), there exists a generic torus t ⊆ g (Proposition 3.3), and the corresponding Weyl group W (g, t) ≃ GL µ(g) )(F p ) (Theorems 1.5 and 5.2.1).
(2) Let g = H(2r) and consider the W (2r)-module A 2r . Let t 0 be a generic torus of W (2r) such that W (2r) = t 0 ⊕ W (2r) (0) . Since A 2r ∼ = U 0 (W (2r))⊗ U 0 (W (2r) (0) ) k is a free U 0 (t 0 )-module of rank 1, Lemma 6.3.3 implies with each ψ i being a homogeneous polynomial function of degree p 2r −p i on W (2r). Also note that, for any subtorus t ⊆ t 0 of dimension r, A 2r is a free U 0 (t)-module of rank p r . Let G = Aut p (g) • . Then P A 2r (T ; x) is the characteristic polynomial of x ∈ g acting on A 2r . By Lemma 6.3.1, the functions ψ i | g are G-invariant. We show that they are p r -th powers in k[g] G . Consider the Poisson algebra P(2r) with p-unipotent center. For any f ∈ P(2r), the linear map D f = {f, −} is a derivation of the associative algebra A 2r , so that D f ∈ W (2r). According to [23,Lemma 6.3], there exist polynomial functions φ 0 , . . . , φ r ∈ k[P(2r)] such that φ i (f ) p r T p r+i ∀ f ∈ P(2r).
Since ϕ p r i = ψ i is G-invariant, we readily obtain that ϕ i ∈ k[g] G as well.
Let t ⊆ g be a generic torus. Theorem 6.3.5 shows that the restriction map induces an embedding res : k[S g ] G ֒→ k[t] GL r (Fp) . As noted before, A 2r is a free U 0 (t)-module of rank p r , so where the coefficients κ i ∈ k[t] are the Dickson generators of k[t] GLr(Fp) . In view of the above, we obtain ϕ i (t) = κ i (t) for all i ∈ {0, . . . , r − 1} and t ∈ t, so that κ i = res(ϕ i ). Therefore, res is surjective.
Since d(g) = ℓ + µ(g) = n, the set Ω ′ is open and non-empty. For Ω ′′ , we have seen in the proof of Lemma 6.3.3(1) that the condition dim k t x = n − 1 defines a non-empty open subset of g. Given any such element x, the subspace t x is spanned by the linearly independent family {x [p] ℓ+1 , . . . , x [p] ℓ+n−1 }. The condition that t x ∩ g (0) = (0) is equivalent to the fact that the residue classes of {x [p] ℓ , . . . , x [p] ℓ+n−1 } in g/g (0) form a system of maximal rank, which is an open condition.
Let us compute the centralizer of t 0 in W (n). It is easy to check that {θ 1 , . . . , θ n } is a basis of the free A n -module W (n). Write a derivation D ∈ W (n) as D = n i=1 f n θ i , where f i ∈ A n for all i. Note that [t 0 , θ n ] = (0), so that 0 = [t, D] = n i=1 t(f i )θ i for all t ∈ t 0 . It readily follows that If, in addition, D ∈ (kx) p ⊆ C W (n) (t 0 ) ∩ S(n), then Now f n = f n (ζ) is a polynomial expression in ζ. Denote formally f ′ n (ζ) = df n /dζ. Then θ n (f n ) = f ′ n (ζ)θ n (ζ) = −ζf ′ n (ζ). Consequently, we must have f n (ζ) = ζf ′ n (ζ), whence f n ∈ kζ. Thus, there exists λ ∈ k such that Thanks to the proof of [3, (2.2.3)], we get that the ring of constants A x n = k. Since A x n = A f i θ i (g) + λζθ n (g) = λζθ n (g) = −λζ 2 g ′ (ζ).
Since ζ is invertible and g ∈ k, this forces λ = 0. We have thus shown that The projection g ։ g/g (0) induces a linear map (kx) p → g/g (0) . Using the inclusion (6.6) we see that its image is spanned by the images of θ 1 , . . . , θ n−1 , and hence has dimension n − 1. Therefore the kernel (kx) p ∩ g (0) has dimension 1. On the other hand, (kx) p ∩ g (0) is a p-subalgebra of (kx) p = t 0 ⊕ n x which doesn't intersect t 0 : so it must be n x , and n x ⊆ g (0) , as we wanted to show.