DIFFERENTIAL OPERATORS COMMUTING WITH INVARIANT FUNCTIONS

T. LEVASSEUR AND J. T. STAFFORD

Abstract. Let 𝔤 be a reductive, complex Lie algebra, with adjoint group G, let G act on the ring of differential operators D(𝔤) via the adjoint action and write τ : 𝔤 → D(𝔤) for the differential of this action. We prove that the commutant, in D(𝔤), of O(𝔤)^{G} is the algebra generated by O(𝔤) and τ(𝔤), thereby answering a question of Barlet.

1. Introduction

Fix a reductive, complex Lie algebra 𝔤, with adjoint group G, let G act on the ring of differential operators D(𝔤) via the adjoint action and write τ : 𝔤 → D(𝔤) for the differential of this action. We identify O(𝔤), the ring of regular functions on 𝔤, with S(𝔤^{∗}) and let O(𝔤)^{G} denote the subalgebra of G-invariant functions. The aim of this note is to prove:

Theorem 1.1. The commutant 𝒞 = 𝒞_{D(𝔤)}(O(𝔤)^{G}), in D(𝔤), of O(𝔤)^{G} is the algebra generated by O(𝔤) and τ(𝔤).

At the level of vector fields, this result follows from [5, Theorem 2.1], in the sense that Dixmier’s result implies that 𝒞 ∩ Der O(𝔤) = O(𝔤)τ(𝔤). In [2], D. Barlet raised the question of whether Theorem 1.1 is true, since this would form a natural generalization of Dixmier’s result. In the same paper, Barlet was able to prove the theorem in the case when 𝔤 = 𝔤𝔩(n, ℂ). We would like to thank M. Raïs for bringing Barlet’s question to our attention.

In the process of proving Theorem 1.1, we obtain a considerable amount of information about the structure of 𝒞. Some particular properties are given in the next result. The unexplained definitions can be found in Section 3.

Proposition 1.2. 𝒞 is an Auslander-Gorenstein, CM domain and a maximal order in its quotient division ring.

In fact, the main theorem of this paper is a result about commutative rings. To state this, let A denote the subalgebra of D(𝔤) generated by O(𝔤) and τ(𝔤) and set E = O(𝔤)τ(𝔤) ⊂ Der O(𝔤). If one filters D(𝔤) and its subalgebras by degree of differential operators, then it is easy to see that the associated graded rings gr A and gr 𝒞 are domains with the same quotient field as the symmetric algebra Sym_{O(𝔤)}(E). Then, Theorem 1.1 and Proposition 1.2 follow easily from the following result.

1991 Mathematics Subject Classification. 22E47, 14L30, 16S32, 17B20.

Key words and phrases. Invariant differential operators, commuting differential operators, semi-simple Lie algebras, symmetric algebras.

The research of the second author was supported in part by an NSF grant.

Theorem 1.3. (i) Let E = O(𝔤)τ(𝔤) ⊂ Der O(𝔤). Then Sym_{O(𝔤)}(E) is a factorial, complete intersection of Krull dimension 2 dim 𝔤 − rk 𝔤.

(ii) gr A = gr 𝒞 = Sym_{O(𝔤)}(E).

This research was conducted while the second author was visiting and partially supported by the University of Poitiers. He would like to thank that institution for its hospitality and financial support.

2. The symmetric algebra of the module generated by τ(𝔤)

In this section we prove Theorem 1.3 from the introduction. We begin with some preliminary notation and results.

As before, we fix a complex, reductive Lie algebra 𝔤 of dimension n and rank ℓ. Write G for the adjoint group of 𝔤. Define the categorical quotient of 𝔤 by O(𝔤//G) = Spec O(𝔤)^{G} and let u : 𝔤 → 𝔤//G denote the quotient morphism. We will write O for O(𝔤). Define

𝒳_{i} = {y ∈ 𝔤 : rk d_{y}u ≤ i},

where d_{y}u : T_{y}𝔤 → T_{u(y)}𝔤//G denotes the differential of u. Observe that each 𝒳_{i} is a closed G-subvariety of 𝔤. Recall that y ∈ 𝔤 is called regular if its centralizer in 𝔤 is of dimension ℓ. Then [10, Theorem 10.1], rk d_{y}u = ℓ if and only if y is regular.

We would like to thank D. Panyushev for the proof of the following proposition, which is considerably easier than our original proof.

Proposition 2.1. One has: codim 𝒳_{i} ≥ ℓ − i + 2, for 0 ≤ i ≤ ℓ − 1.

Proof. Notice that u induces a surjective morphism ϖ : 𝒳_{i} → 𝒳_{i}//G and that, for all x ∈ 𝒳_{i}, the differential d_{x}ϖ is the restriction of d_{x}u to T_{x}𝒳_{i}. Set r = max{rk d_{x}ϖ : x ∈ 𝒳_{i}}. Then, by [7, Proposition III.10.6] and the definition of 𝒳_{i}, we obtain that dim 𝒳_{i}//G ≤ r ≤ i.

Since 𝒳_{i} is stable under the ℂ^{∗}-action y ↦ λy, λ ∈ ℂ^{∗}, the point 0 belongs to each irreducible component of 𝒳_{i}. Hence, dim 𝒳_{i} ≤ dim 𝒳_{i}//G + dim ϖ^{−1}(ϖ(0)) (see [11, AI.3.3] or [7, Ex. II.3.22]). But ϖ^{−1}(ϖ(0)) = 𝒳_{i} ∩ N, where N denotes the nilpotent cone of 𝔤, and, since i ≤ ℓ − 1, 𝒳_{i} ∩ N is contained in the subvariety of non-regular nilpotent elements. Therefore dim ϖ^{−1}(ϖ(0)) ≤ n − ℓ − 2 and it follows that dim 𝒳_{i} ≤ i + n − ℓ − 2, as required. □

Remark . Proposition 2.1 generalizes the well-known fact that 𝒳_{ℓ−1} has codimension at least three (see, for example, [18, Theorem 4.12]). It is natural to conjecture that Proposition 2.1 can be improved to the statement that codim 𝒳_{i} ≥ 3(ℓ − i) for 0 ≤ i ≤ ℓ − 1. D. Panyushev informs us that he has been able to prove this by a case by case analysis.

Fix a G-invariant, non-degenerate, symmetric bilinear form κ on 𝔤 and let κ̃ : 𝔤^{∗} → 𝔤 be the induced isomorphism. Thus, κ̃ induces an isomorphism between differential one-forms on 𝔤 and vector fields on 𝔤. If f ∈ O^{G}, then we define a Ginvariant vector field grad(f) ∈ O ⊗_{ℂ} 𝔤 to be the image of df under κ̃. Equivalently, if we fix an orthonormal basis {e_{i}} of 𝔤 and write x_{i} = e_{i}^{∗} ∈ 𝔤^{∗}, then

(2.1)  grad(f) = ∑_{j=1}^{n} \frac{∂f}{∂x_{j}} ⊗ e_{j} = ∑_{j=1}^{n} \frac{∂f}{∂x_{j}} \frac{∂}{∂x_{j}} .

By Chevalley’s Theorem, O^{G} is a polynomial ring, say O^{G} = ℂ[u_{1}, …, u_{ℓ}] for homogeneous, algebraically independent polynomials {u_{i}}_{i}. Set ∇_{i} = grad(u_{i}), for 1 ≤ i ≤ ℓ. If τ : 𝔤 → Der O is the differential of the adjoint action of G on 𝔤, then write E = Oτ(𝔤). We will also write τ for the induced map:

τ : O ⊗_{ℂ} 𝔤 ⟶ E ⊆ Der O.

Notice that if θ ∈ O ⊗_{ℂ} 𝔤, the vector field τ(θ) is given by τ(θ)_{y} = [y, θ_{y}] for all y ∈ 𝔤. It follows easily that if θ is G-invariant, then τ(θ) = 0. In particular, one has τ(∇_{i}) = 0 for all i. In fact rather more is true:

Lemma 2.2. There is a short exact sequence

(2.2)  0 ⟶ ⨁_{i=1}^{ℓ}O∇_{i} ⟶ O ⊗_{ℂ} 𝔤 ⟶^{τ} E ⟶ 0.

Proof. This is [16, Theorem 2.5.4]. Using the identification of 𝔤 with 𝔤^{∗} under κ̃, it also follows from [14, Theorem 1.9]. □

Corollary 2.3. If Sym_{O}(E) denotes the symmetric algebra of the O-module E, then, Sym_{O}(E) ≅ Sym_{O}(O ⊗_{ℂ} 𝔤)/(∇_{1}, …, ∇_{ℓ}).

Proof. This follows from the universal property of symmetric algebras. □

Set Sym(E) = Sym_{O}(E). The main aim of this section is to understand the structure of Sym(E), for which we use the results from [1] and [8].

Let I_{t}(u) be the ideal generated by the t × t minors of the matrix u = [\frac{∂u_{i}}{∂x_{j}} ] and consider the following condition for s ≥ 0:

(ℱ_{s})  ht I_{t}(u) ≥ ℓ − t + 1 + s, for 1 ≤ t ≤ ℓ.

Observe that, if we regard the short exact sequence (2.2) as a sequence

0 ⟶ O^{ℓ} ⟶^{β} O^{n} ⟶ E ⟶ 0,

then (2.1) implies that I_{t}(u) is the ideal generated by the t×t minors of the map β. Thus, the ideals I_{n−t}(u) are nothing more than the Fitting ideals of E (see, for example, [17, 1.1]). In particular, they are independent of the presentation of E and our condition (ℱ_{s}) coincides with that of [8].

Proposition 2.4. (i) The condition (ℱ_{2}) is satisfied by E.

(ii) Sym(E) is a factorial domain of Krull dimension 2n − ℓ. In particular, Sym(E) is a complete intersection and is Gorenstein.

(iii) If P is a prime ideal of O with ht P ≥ 2, then ht P Sym(E) ≥ 2.

Proof. Write 𝒳̃_{i−1} for the zero set of I_{i}(u); thus

𝒳̃_{i−1} = {x ∈ 𝔤 : rk (∇_{1}(x), …, ∇_{ℓ}(x)) ≤ i − 1} .

Since the ∇_{j} are the images of the du_{j} under the isomorphism κ̃, clearly 𝒳̃_{i−1} = {x ∈ 𝔤 : rk (d_{x}u_{1}, …, d_{x}u_{ℓ}) ≤ i − 1}. Since u_{1}, …, u_{ℓ} define the quotient map u : 𝔤 → 𝔤//G, this implies that 𝒳̃_{i} = 𝒳_{i}. Hence, part (i) is a reformulation of Proposition 2.1.

By Lemma 2.2, E has projective dimension at most 1. Thus, part (ii) follows from part (i), combined with [1, Propositions 3 and 6]. By [8, Remarks, pp. 664-5], the condition of part (iii) is equivalent to the condition (ℱ_{2}). □

We end this section by giving the geometric significance of Proposition 2.4. This should be compared with [9, § 2] which proves weaker results for much more general G-varieties.

The map τ induces a homomorphism of algebras

τ̃ : O(𝔤 × 𝔤^{∗}) = Sym_{O}(O ⊗ 𝔤) ⟶ O(T^{∗}𝔤) = Sym_{O}(Der O).

Clearly, the image of τ̃ is the subring O[τ(𝔤)] of Sym_{O}(Der O) generated by O and τ(𝔤). After identification of 𝔤^{∗} with 𝔤 through κ̃, the associated morphism to τ̃ is:

ν : T^{∗}𝔤 ≅ 𝔤 × 𝔤 ⟶ 𝔤 × 𝔤, ν(x, y) = (x, [y, x])

Let T̃𝔤 denote the closure of the image of ν; thus, T̃𝔤 is an irreducible affine subvariety of 𝔤 × 𝔤 with coordinate ring O(T̃𝔤) ≅ O[τ(𝔤)].

Corollary 2.5. (i) Sym(E) = O(T̃𝔤).

(ii) The variety T̃𝔤 is a factorial complete intersection in 𝔤 × 𝔤.

Proof. By universality, τ̃ induces a surjective morphism π : Sym(E) ↠ O[τ(𝔤)]. If we prove that dim T̃𝔤 ≥ 2n−ℓ, then the corollary will follow from Proposition 2.4(ii).

Let ρ : T̃𝔤 ↠ 𝔤 denote the projection onto the first factor. By [11, AI.3.3] there exists a dense open subset U ⊆ T̃𝔤 such that dim T̃𝔤 = dim 𝔤 + dim ρ^{−1}(ρ(u)) for all u ∈ U. Since T̃𝔤 is irreducible, we can pick u = (x, y) ∈ ρ^{−1}(𝔤′) ∩ U, where 𝔤′ denotes the set of generic elements in 𝔤. Now,

ρ^{−1}(ρ(u)) ⊇ ρ^{−1}(ρ(u)) ∩ Im ν = {(x, [𝔤, x])}.

Since x is generic, dim ρ^{−1}(ρ(u)) ≥ dim[𝔤, x] = n − ℓ and the result follows. □

3. The commutant of O(𝔤)^{G}

As usual, we identify D(𝔤), as a vector space, with O ⊗_{ℂ} S(𝔤), where O = O(𝔤) and the symmetric algebra S(𝔤) is identified with the constant coefficient differential operators on 𝔤. We will always filter D(𝔤) by degree of differential operators and so, as algebras, gr D(𝔤) = Sym_{O}(Der O) ≅ O ⊗_{ℂ} S(𝔤). Write A for the subring of D(𝔤) generated by O and τ(𝔤) and let 𝒞 denote the commutant of O^{G}, as in the introduction. Obviously, A is contained in 𝒞.

Lemma 3.1. Let x ∈ 𝔤 be a regular point and set R = O_{𝔤,x} for the local ring of 𝔤 at x. Then, there exists a basis of derivations {∂_{i} : 1 ≤ i ≤ n} of Der R such that ∂_{i}(u_{j}) = δ_{ij} for all 1 ≤ i, j ≤ ℓ and Rτ(𝔤) = ⨁_{i=ℓ+1}^{n} R∂_{i}.

Proof. Let m denote the maximal ideal of R. By [10, Theorem 0.1], the {d_{x}u_{i} : 1 ≤ i ≤ ℓ} are linearly independent. The {d_{x}u_{i}} may also be regarded as elements of m/m^{2}, under the usual identification of T_{x}^{∗}𝔤 with m/m^{2}. Thus, for some scalars λ_{i}, the set {u_{1} − λ_{1}, …, u_{ℓ} − λ_{ℓ}} is part of a system of parameters, say {z_{1} = u_{1} − λ_{1}, …, z_{ℓ} = u_{ℓ} − λ_{ℓ}, z_{ℓ+1}, …, z_{n}} for m. Let ∂_{i} ∈ Der R be defined by ∂_{i}(z_{j}) = δ_{ij}.

If D ∈ Der R, then D̃ = D − ∑_{i=1}^{ℓ} D(u_{i})∂_{i} satisfies D̃(u_{j}) = 0, for 1 ≤ j ≤ ℓ. Thus, D̃(O^{G}) = 0 and so, by [5, Theorem 2.1] (or directly), D̃ ∈ Rτ(𝔤). Hence, Der R = Rτ(𝔤) ⊕ (⊕_{i=1}^{ℓ}R∂_{i}). Since

Rτ(𝔤) ⊆ {D ∈ Der R : D(u_{j}) = 0 for 1 ≤ j ≤ ℓ} = ⨁_{i=ℓ+1}^{n}R∂_{i},

the result follows. □

Theorem 3.2. Let A and 𝒞 be given the filtrations induced from that on D(𝔤). Then gr A = gr 𝒞 ≅ Sym(E).

Proof. Since gr 𝒞 ⊂ gr D(𝔤) ≅ O(T^{∗}𝔤), certainly gr A ⊆ gr 𝒞 are domains. Also, as τ(𝔤) consists of derivations, we may regard τ(𝔤) ⊆ Der O ⊆ gr D(𝔤). Hence the ring O[τ(𝔤)] is contained in gr A and, by Corollary 2.5(i), the natural map π : Sym(E) → O[τ(𝔤)] is an isomorphism.

Let x ∈ 𝔤 be a regular point and let S = {f ∈ O : f(x) ≠ 0}. Given a ring C containing O, we write C_{x} for the localization C_{S} (given that it exists). Then, we claim that

(3.1)  (gr A)_{x} = (gr 𝒞)_{x} ≅ Sym(E)_{x},

where the isomorphism is induced by π^{−1}.

By mimicking the proof of Richardson’s Lemma [11, II.3.4], one can show that this suffices to prove the theorem. In more detail, assume that (3.1) is true. Since gr 𝒞 and Sym(E) are domains, (3.1) certainly implies that

Sym(E) ↪^{π} gr A ⊆ gr 𝒞

and that gr 𝒞 and Sym(E) have the same field of fractions. Moreover, {x ∈ 𝔤 : (gr 𝒞)_{x} ≠ Sym(E)_{x}} is contained in the set of non-regular elements of 𝔤. By [10, Theorem 0.1], this is precisely the subspace 𝒳_{ℓ−1} and, by Proposition 2.1 or [18, Theorem 4.12], codim 𝒳_{ℓ−1} ≥ 3. Thus, for any b ∈ gr 𝒞, there exists an ideal I of O of height at least 3 such that bI ⊆ Sym(E). By Proposition 2.4(iii), ht_{Sym(E)} I Sym(E) ≥ 2. Hence, b ∈ Sym(E)_{p} for every height one prime p of Sym(E). Since Sym(E) is Cohen-Macaulay, it satisfies the (S_{2})-condition [12, p. 125], and therefore b ∈ Sym(E).

Thus, it remains to prove (3.1). Let R = O_{x} = O_{𝔤,x} and keep the notation of Lemma 3.1. It is immediate from that lemma that D ∈ D(𝔤)_{x} satisfies [D, u_{j}] = 0 if and only if D ∈ R⟨∂_{j} : j ≠ i⟩. Consequently, 𝒞_{x} = A_{x} = R⟨∂_{ℓ+1}, …, ∂_{n}⟩.

Let \overline{∂}_{k} denote the image of ∂_{k} in gr D(𝔤). Obviously, Lemma 3.1 also implies that Rτ(𝔤) = ⨁_{k=ℓ+1}^{n} R\overline{∂}_{k}, where Rτ(𝔤) is now regarded as a subspace of Der R ⊂ gr D(𝔤)_{x} ≅ R ⊗_{ℂ} S(𝔤). Thus,

gr 𝒞_{x} = gr A_{x} = gr R⟨∂_{ℓ+1}, …, ∂_{n}⟩ = R[\overline{∂}_{ℓ+1}, …, \overline{∂}_{n}] = R[τ(𝔤)].

Since π : Sym(E)_{x} → R[τ(𝔤)] is an isomorphism, this completes the proof of (3.1) and hence of the theorem. □

The Gelfand-Kirillov dimension of a module M will be denoted GKdim M. If a Noetherian ring A has finite injective dimension, then A is called Auslander-Gorenstein if A satisfies the following condition: For any integers 0 ≤ i < j and finitely generated (right) A-module M, one has Ext_{A}^{i}(N, A) = 0 for all (left) Asubmodules N of Ext_{A}^{j}(M, A). Set j(M) = min{j : Ext_{A}^{j}(M, A) ≠ 0}. The algebra A is CM if j(M) + GKdim M = GKdim A holds for all finitely generated, non-zero A-modules M.

Corollary 3.3. (i) The commutant 𝒞 of O^{G} in D(𝔤) is the ring generated by O and τ(𝔤). Moreover, 𝒞 is an Auslander-Gorenstein, CM, Noetherian domain and a maximal order.

(ii) As a (left or right) O-module, 𝒞 ∩ D(𝔤)_{m} is generated by the elements

{τ(ξ_{1})τ(ξ_{2}) ⋯ τ(ξ_{k}) : ξ_{i} ∈ 𝔤 and k ≤ m}.

(iii) The centre of 𝒞 is O(𝔤)^{G}.

Proof. (i) By Theorem 3.2, 𝒞 is generated by O and τ(𝔤). By that theorem and Proposition 2.4, gr 𝒞 satisfies the other conditions given in part (i). Let M = ⋃_{n∈ℕ} M_{n} be a filtered right 𝒞-module such that gr M is a finitely generated gr 𝒞module. By [3, Theorem 3.9], 𝒞 is Auslander-Gorenstein and j_{𝒞} (M) = j_{gr 𝒞} (gr M). However, [13, Corollary 1.4] implies that GKdim gr M = GKdim M and hence that 𝒞 is CM. Finally, [15] implies that 𝒞 is a maximal order.

(ii) This follows from the fact that, in gr 𝒞 = Sym(E), a homogeneous element \overline{c} of degree m can be written \overline{c} = ∑ f_{i_{1},...,i_{m}}ξ_{i_{1}} ⋯ ξ_{i_{m}}, for some f_{i_{1},...,i_{m}} ∈ O and ξ_{i_{j}} ∈ τ(𝔤).

(iii) Let Z denote the centre of 𝒞. Clearly both τ(𝔤) and O commute with O^{G} and so O^{G} ⊆ Z. Conversely, Z is contained in the commutant, in D(𝔤), of O. Hence, Z ⊆ O. Since O^{G} is the commutant, in O, of τ(𝔤), the result follows. □

Corollary 3.4. Both 𝒞 and Sym(E) are free (left or right) modules over O(𝔤)^{G}.

Proof. Set O = O(𝔤) and S = Sym(E) = ⨁_{m=0}^{∞} Sym_{m}(E). We first prove the result for 𝒞, assuming that Sym_{m}(E) is a free O^{G}-module for all m ∈ ℕ. Note that the isomorphism gr 𝒞 ≅ S of Theorem 3.2 is a graded isomorphism of O-algebras, for the natural graded structure of the two objects. In other words 𝒞_{m}/𝒞_{m−1} ≅ Sym_{m}(E), for all m, where 𝒞_{m} = 𝒞 ∩ D(𝔤)_{m}. Hence, each 𝒞_{m}/𝒞_{m−1} is a free O^{G}-module; it follows routinely that 𝒞 is also free over O^{G}.

We now prove the result for Sym_{m}(E). Note, first, that S is a quotient of the polynomial ring

T = Sym_{O}(O ⊗ 𝔤) ≅ O[y_{1}, …, y_{n}] ≅ ℂ[x_{1}, …, x_{n}, y_{1} …, y_{n}],

which we now grade by giving each generator x_{i} and y_{j} degree one. Since the u_{i} are homogeneous in O, the ∇_{i} = grad(u_{i}) are homogeneous in T and so, by Corollary 2.3, Sym_{m}(E) is a graded O^{G}-module.

Set P = ∑_{i=1}^{ℓ} u_{i}S. By [6, Proposition 2.16] and its proof (which depends upon a case by case analysis), S/P is a domain of dimension 2n − 2ℓ = dim S − ℓ. Hence, the u_{j} form a regular sequence in S, and therefore in each module Sym_{m}(E). Thus, by [4, § 8, Proposition 8 and § 9, Corollaire 2], Sym_{m}(E) is a graded free O^{G}-module. □

Corollary 3.3 and Corollary 3.4 should be compared with [9] which (as a very special case) shows that the commutant of D(𝔤)^{G} is simply ℂ⟨τ(𝔤)⟩ ( ≅ U(𝔤) when 𝔤 is semisimple). Moreover, both rings are free modules over the centre of D(𝔤)^{G} (which is also the centre of ℂ⟨τ(𝔤)⟩).

References

[1] L. L. Avramov, Complete intersections and symmetric algebras, J. Algebra, 73 (1981), 248-263.

[2] D. Barlet, Unpublished manuscript, Université de Nancy, 1978.

[3] J.-E. Björk, The Auslander condition on Noetherian rings, \underline{in} Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Math. 1404, Springer Verlag, Berlin/New York, 1989, 137-173.

[4] N. Bourbaki, Algèbre, Chapitre 10, Masson, Paris, 1980.

[5] J. Dixmier, Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples, J. Reine Angew. Math., 309 (1979), 183-190.

[6] F. Geoffriau, Sur le centre de l’algèbre enveloppante d’une algèbre de Takiff, Ann. Math. Blaise Pascal, 1 (1994), 15-31.

[7] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., Vol. 52, Springer-Verlag, Berlin/New York, 1977.

[8] J. Herzog, A. Simis and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc., 283 (1984), 661-683.

[9] F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. Math., 140 (1994), 253-288.

[10] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math., 85 (1963), 327-404.

[11] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspeckte der Mathematik, Vieweg, Braunschweig, 1994.

[12] H. Matsumura, Commutative Algebra (Second Edition), Benjamin Publ. Co., Reading, 1980.

[13] J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension and associated graded modules, J. Algebra, 125 (1989), 197-214.

[14] D. Panyushev, The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Math., 94 (1994), 181-199.

[15] M. Van den Bergh and F. Van Oystaeyen, Lifting maximal orders, Comm. in Algebra, 17 (1989), 341-349.

[16] W. V. Vasconcelos, Symmetric algebras, \underline{in} Commutative Algebra, Salvador, 1988, Lecture Notes in Math. 1430, Springer-Verlag, Berlin/New York, 1990, 115-160.

[17] W. V. Vasconcelos, Arithmetic of Blowup Algebras, Cambridge University Press, 1994.

[18] F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic p, Ann. Sci. Ecole Norm. Sup., 5 (1972), 217-240.

Département de Mathématiques, Université de Poitiers, 86022 Poitiers, France.

E-mail address: levasseu@mathlabo.univ-poitiers.fr

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.

E-mail address: jts@math.lsa.umich.edu