INVARIANT DIFFERENTIAL OPERATORS AND

AN HOMOMORPHISM OF HARISH-CHANDRA

T. Levasseur and J. T. Stafford

1. Introduction and Applications

Let 𝔤 be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra 𝔥 and Weyl group W. Then G acts naturally on the algebra of polynomial functions O(𝔤) and hence on the ring of differential operators with polynomial coefficients, D(𝔤). Similarly, W acts on 𝔥 and hence on D(𝔥). In [HC2], Harish-Chandra defined an algebra homomorphism δ : D(𝔤)^{G} → D(𝔥)^{W}. Recently, Wallach proved that, if 𝔤 has no factors of type E_{6}, E_{7} or E_{8}, then this map δ is surjective [Wa, Theorem 3.1]. The significance of Wallach’s result is that it enables him to give an easy proof of an important theorem of Harish-Chandra about invariant distributions and to give an elegant new approach to the Springer correspondence.

The main aim of this paper is to give an elementary proof of [Wa, Theorem 3.1] that also works for all reductive Lie algebras. Set

I = {D ∈ D(𝔤)^{G} : D(p) = 0 for all p ∈ O(𝔤)^{G}}.

Then, we prove:

Theorem 1. The sequence 0 ⟶ I ⟶ D(𝔤)^{G} ⟶^{δ} D(𝔥)^{W} ⟶ 0 is exact.

This theorem follows immediately from an abstract result that describes the fixed rings, under a finite group action, of rings of differential operators (see Theorem 5 and the comments thereafter). Before discussing that result, however, we wish to explain the significance of Theorem 1, for which we need some notation.

Fix a real form 𝔤_{0} of 𝔤 and assume that Ω is a fixed, open, completely invariant subset of 𝔤_{0}. Let N denote the cone of nilpotent elements in 𝔤_{0}. Write D′_{N} (Ω)^{G_{0}} for the set of distributions on Ω that are invariant under the action of the adjoint group G_{0} of 𝔤_{0} and supported in N. One of the key ideas in [Wa] is to relate invariant distributions to Weyl group representations. While Wallach is able to make this translation without use of Theorem 1, the result becomes simpler and more natural with that result to hand. This is summarized by the following result, which is obtained by combining Theorem 1 with [Wa, Theorems 2.7, 6.1 and Proposition 5.5]:

Typeset by AMS-TEX

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

Key words and phrases. Invariant differential operators, semi-simple Lie algebras, invariant distributions, Springer correspondence.

The research of both authors was supported in part by an NSF grant

Theorem 2. Assume that 𝔤_{0} is semisimple and let T ∈ D′_{N} (Ω)^{G_{0}}. Then, D(𝔤)^{G}T is killed by I. Thus, by Theorem 1 we may regard D(𝔤)^{G}T as a module over D(𝔥)^{W}. As such, D(𝔤)^{G}T ∈ C′_{W}, the category of all finitely generated D(𝔥)^{W} -modules on which S(𝔥^{∗})_{+}^{W} acts locally nilpotently. Moreover, C′_{W} is equivalent to the category of finite dimensional representations of W. Under this equivalence, D′_{N} (𝔤_{0})^{G_{0}} becomes a finite W -module.

By taking T to be the orbital integral corresponding to a nilpotent element X of a semisimple Lie algebra 𝔤, Wallach associates, in much this way, an irreducible representation of W to the G-orbit of X (see [Wa, Theorem 6.9]). This leads to his construction of the Springer correspondence. Wallach’s proof can be further simplified since, by Theorem 1 and [Wa, Corollary 1.4], the categories C_{W} and Č_{W} of [Wa, Section 2] are equal. Thus, [Wa, Theorem 2.4] and [Wa, Theorem 2.6] are now trivial consequences of [Wa, Theorem 1.4], respectively [Wa, Theorem 1.6], while [Wa, Lemma 2.5] is vacuously true.

In [Wa], Wallach also gives a relatively easy proof of a fundamental result of Harish-Chandra. Using Theorem 1, one can further simplify his argument. Indeed, by combining Theorem 1 with the first two paragraphs of the proof of [Wa, Theorem 5.4] one obtains an easy proof of the following result:

Theorem 3. (see [HC3, Theorem 5]) I = {D ∈ D(𝔤)^{G} : DD′(Ω)^{G_{0}} = 0}.

Moreover, the argument given at the end of [Wa, Section 5] shows that this, in turn, easily implies a second result from [HC3]:

Theorem 4. Let 𝔤′ denote the set of all generic elements of 𝔤. Suppose that T ∈ D′(Ω)^{G_{0}} satisfies dim S(𝔤)^{G}T < ∞ and T_{|Ω∩𝔤′} = 0. Then T = 0.

We end the introduction by discussing the proof of Theorem 1. This is based on a structure theorem for rings of differential operators fixed by an arbitrary finite group, that is of independent interest. Let V be an ℓ-dimensional vector space over a field k of characteristic zero and W ⊂ GL(V) a finite group. Let O(V) ≅ k[x_{1}, …, x_{ℓ}] denote the ring of polynomial functions on V and write D(V) = {∑ f_{i}(∂/∂x_{1})^{i_{1}} ⋯ (∂/∂x_{ℓ})^{i_{ℓ}} : f_{i} ∈ O(V)} for the ring of k-linear differential operators on O(V). We may identify the symmetric algebra S(V^{∗}) with O(V) and the algebra S(V) with the ring of constant coefficient differential operators on V. This provides a natural identification of left S(V^{∗})-modules: D(V) = S(V^{∗}) ⊗_{k} S(V) (see, for example, [HC2, Section 3, Corollary 4]). The diagonal action of W on S(V^{∗}) ⊗_{k} S(V) now identifies with the usual action of W on D(V) given by g(θ)(f) = g · (θ(g^{−1} · f)), for g ∈ W, θ ∈ D(V) and f ∈ O(V) (see, for example, [HC2, pp.234-5]).

Now consider D(V)^{W}. Two obvious subspaces are the W -invariant polynomial functions, O(V)^{W} = S(V^{∗})^{W}, and the W -invariant, constant coefficient differential operators, S(V)^{W}. Remarkably, these suffice to generate D(V)^{W}:

Theorem 5. Let B be the k-subalgebra of D(V) generated by S(V^{∗})^{W} and S(V)^{W}. Then, B = D(V)^{W}.

If W is a Weyl group with no factors of type E_{6}, E_{7} or E_{8}, then this theorem is proved in [Wa], by means of a case by case analysis. The relevance of Theorem 5 is that, as is shown in [Wa], it has Theorem 1 as an immediate corollary. The argument is as follows:

Proof of Theorem 1 from Theorem 5. Inside D(𝔤)^{G} one has the natural subrings S(𝔤^{∗})^{G} and S(𝔤)^{G}. Let A be the subalgebra ℂ⟨S(𝔤^{∗})^{G}, S(𝔤)^{G}⟩ that they generate. Then the restriction of Harish-Chandra’s homomorphism δ to S(𝔤)^{G} is just the Chevalley isomorphism S(𝔤)^{G} ⟶^{∼} S(𝔥)^{W}. Similarly, the restriction of δ to S(𝔤^{∗})^{G} is the Chevalley isomorphism S(𝔤^{∗})^{G} ⟶^{∼} S(𝔥^{∗})^{W} (see [HC1, Theorem 1]). Thus, with V = 𝔥 and W the Weyl group, Theorem 5 implies that δ maps A surjectively onto B = ℂ⟨S(𝔥^{∗})^{W} , S(𝔥)^{W} ⟩ = D(𝔥)^{W}. Since I = Ker(δ) (see [HC2, Theorem 1]) this completes the proof of Theorem 1, modulo proving Theorem 5. Note that this argument also shows that D(𝔤)^{G} = A + I. □

2. The Proof of Theorem Five

We will continue to use the identifications described in the introduction; in particular we will always make the identification S(V^{∗}) ⊗_{k} S(V) = D(V). Let S(V) = ⨁_{i≥0} S^{i}(V) be the usual graded structure of S(V) and write {S(V)_{n} = ⨁_{i≤n} S^{i}(V)} for the corresponding filtration of S(V). This induces a filtered structure D(V) = ⋃ D(V)_{n} on D(V) by defining D(V)_{n} = S(V^{∗}) ⊗_{k} S(V)_{n}. Since S(V) has been identified with the constant coefficient differential operators, this filtration is nothing more than the filtration of D(V) by degree of differential operators and so, in particular, is a filtration of D(V) as a k-algebra. Given any subalgebra, or even subspace, R ⊆ D(V), we will always filter R by the induced filtration: {R_{n} = R ∩ D(V)_{n}}. The associated graded object gr(R) is defined to be gr(R) = ⨁ gr_{n}R, where gr_{n}R = R_{n}/R_{n−1}. Observe that

gr_{n}D(V) = (S(V^{∗}) ⊗ S(V)_{n})/(S(V^{∗}) ⊗ S(V)_{n−1}) ≅ S(V^{∗}) ⊗ S^{n}(V). (†)

Thus, gr(D(V)) is isomorphic to the commutative ring S(V^{∗}) ⊗ S(V). Moreover, for any subalgebra R ⊆ D(V), the associated graded ring gr(R) is isomorphic to

⨁ (R ∩ D(V)_{n} + D(V)_{n−1})/ D(V)_{n−1} ⊆ gr(D(V)).

Lemma 6. Let W ⊂ GL(V) be a finite group and write B = k⟨S(V^{∗})^{W} , S(V)^{W} ⟩ for the subalgebra of D(V)^{W} generated by S(V^{∗})^{W} and S(V)^{W}. Then, B is a Noetherian domain and D(V)^{W} is finitely generated as both a left and a right Bmodule.

Proof. Observe that S(V^{∗})⊗_{k} S(V) = ⨁ S(V^{∗})⊗_{k} S^{n}(V) is a decomposition of W - modules. Thus, (†) is also an isomorphism of W -modules. Indeed, the identification D(V) = S(V^{∗}) ⊗ S(V) gives an action of W × W on D(V) by (w_{1}, w_{2}) ∘ (s_{1} ⊗ s_{2}) = w_{1}(s_{1})⊗w_{2}(s_{2}). Although this is not an action of W ×W by algebra automorphisms, (†) is still an isomorphism of W × W modules. Now consider B and its natural ksubspace L = S(V^{∗})^{W} ⊗ S(V)^{W}. Then (†) ensures that gr(L) = S(V^{∗})^{W} ⊗ S(V)^{W} as a subspace of gr(D(V)) = S(V^{∗}) ⊗ S(V). Thus, gr(L) is a ring; indeed gr(L) = {S(V^{∗}) ⊗ S(V)}^{W ×W}. In particular, gr(L) is a Noetherian domain and gr(D(V)) is a finitely generated gr(L)-module.

Clearly, gr(L) ⊆ gr(B) ⊆ gr(D(V)^{W}) ⊆ gr(D(V)). Thus, gr(B) is Noetherian and so, by [MR, Lemma 7.6.11], B is a Noetherian ring. Next, as B is a filtered subalgebra of D(V)^{W}, D(V)^{W} is a filtered (left or right) B-module and its associated graded module is just gr(D(V)^{W}). But, by the last paragraph, gr(D(V)^{W}) is a finitely generated (right or left) gr(B)-module. Thus, by [MR, Lemma 7.6.11], again, D(V)^{W} is finitely generated as both a left and a right B-module. □

Let C be a commutative ring that is a localisation of a finitely generated kalgebra. The ring of k-linear differential operators D(C) is defined to be D(C) = ⋃_{n≥0} D(C)_{n}, where D(C)_{0} = C and, inductively,

D(C)_{n} = {θ ∈ End_{k}(C) : θc − cθ ∈ D(C)_{n−1} for all c ∈ C}.

The ring D(C) is filtered by the D(C)_{n} and, when C = S(V^{∗}), this does agree with the earlier definition. Now assume that F is a finitely generated field extension of k of transcendence degree ℓ and pick a transcendence basis, say {u_{1}, …, u_{ℓ}}, for F/k. Then, by [MR, Corollaries 15.1.12(iv) and 15.2.5] the derivations ∂_{i} = \frac{∂}{∂u_{i}} extend to derivations on F and D(F) ≅ F ⟨∂_{1}, …, ∂_{ℓ}⟩. Given I = (i_{1}, …, i_{ℓ}) ∈ ℕ^{ℓ}, let ∂^{I} = ∂_{1}^{i_{1}} ⋯ ∂_{ℓ}^{i_{ℓ}}. Then, as a left F -module, D(F) has basis {∂^{I} : I ∈ ℕ^{ℓ}} ([MR, Proposition 15.3.2]). If I = (i_{1}, …, i_{ℓ}) ∈ ℕ^{ℓ}, write |I| = ∑_{j} i_{j}. We totally order ℕ^{ℓ} by I = (i_{1}, …, i_{ℓ}) > J = (j_{1}, …, j_{ℓ}) if either |I| > |J| or |I| = |J| and there exists 1 ≤ m ≤ ℓ such that i_{n} = j_{n} for n < m but i_{m} > j_{m}. Given a k-algebra R and r ∈ R, let ad(r) ∈ End_{k}(R) be defined by ad(r)(s) = [r, s] = rs − sr, for s ∈ R.

Lemma 7. Keep the above notation. Then:

(i) If J = (j_{1}, …, j_{ℓ}) ∈ ℕ^{ℓ}, define ad(u)^{J} = ad(u_{1})^{j_{1}} ⋯ ad(u_{ℓ})^{j_{ℓ}}. Then, ad(u)^{J}(∂^{I}) = λ∂^{I−J}, for some λ ∈ ℤ \ {0}. By convention, if I < J, then ∂^{I−J} = 0.

(ii) Let f ∈ D(F) \ F and write f = f_{I}∂^{I} + ∑_{J<I} f_{J}∂^{J}, with f_{I} ≠ 0. Suppose that I = (i_{1}, …, i_{ℓ}), with i_{r} > 0 and set I′ = (i_{1}, …, i_{r} − 1, …, i_{ℓ}). Then, ∂_{r} + f_{r+1}∂_{r+1} + ⋯ + f_{ℓ}∂_{ℓ} + f_{0} ∈ F · ad(u)^{I′}(f), for some f_{j} ∈ F.

Proof. (i) By construction, ad(u_{j})(∂_{i}) = −δ_{ij} and

ad(u_{j})(∂^{I}) = −(i_{j})∂_{1}^{i_{1}} ⋯ ∂_{j}^{i_{j}−1} ⋯ ∂_{ℓ}^{i_{ℓ}}.

The result now follows from the obvious induction.

(ii) Let 1_{r} = (0, …, 0, 1, 0, …, 0) ∈ ℕ^{ℓ}, where the 1 occurs in the r^{th} position. By part (i), ad(u)^{I′}(f) = μf_{I}∂_{r} + ∑_{K<1_{r}} g_{K}∂^{K}, for some non-zero integer μ and some g_{K} ∈ F. This is equivalent to the assertion of the lemma. □

Suppose that K = k(u_{1}, …, u_{ℓ}) is a field of rational functions and let R be the subalgebra of D(K) generated by K and one operator ∂_{1}^{i_{1}} ⋯ ∂_{ℓ}^{i_{ℓ}}, for which i_{j} > 0, for each j. Then an easy exercise using Lemma 7 shows that R = D(K). The same basic idea leads to the following curious result.

Lemma 8. Let F be a finitely generated field extension of k of transcendence degree ℓ. Let R ⊆ D(F) be a subalgebra such that F ⊆ R and give R the induced filtration {R_{n} = R ∩ D(F)_{n}}. Assume that

lim sup_{n→∞} {log_{n}(dim_{F} gr_{n}(R))} > ℓ − 2.  (8.1)

Then, R = D(F).

Remark. By [KL, Theorem 4.5] the condition (8.1) holds, in particular, if gr(R) contains a finitely generated, graded F -subalgebra of Krull dimension ℓ.

Proof. Let d_{i} = (∂_{i} + F)/F denote the image of ∂_{i} in gr(D(F)). By [MR, Proposition 15.3.2], gr(D(F)) ≅ F [d_{1}, …, d_{ℓ}], the (commutative) polynomial ring in the d_{j}. In particular, gr_{n}(D(F)) ≅ F [d_{1}, …, d_{ℓ}]^{n}, the vector space of homogeneous polynomials of degree n. Now, F [d_{2}, …, d_{ℓ}]^{n} has dimension O(n^{ℓ−2}). Thus, by (8.1), there exists n such that, for any 1 ≤ r ≤ ℓ, one has dim gr_{n}R > dim F [d_{1}, …, d_{r−1}, d_{r+1}, …, d_{ℓ}]^{n}.

For n chosen as in the last paragraph, pick an F -basis {t_{1}, …, t_{ν}} for gr_{n}R. As F ⊆ R, we may choose this basis such that, for 1 ≤ λ ≤ ν,

t_{λ} = d^{I(λ)} + ∑ {f_{J(λ)}d^{J(λ)} : J(λ) < I(λ) and f_{J(λ)} ∈ F }

and

if λ ≠ μ then d^{I(λ)} ≠ d^{I(μ)}.

Now pick 1 ≤ r ≤ ℓ. By the choice of n, and the fact that ν = dim_{F} (gr_{n}R), there exists 1 ≤ λ ≤ ν such that d^{I(λ)} ∉ F [d_{1}, …, d_{r−1}, d_{r+1}, …, d_{ℓ}]. For this λ, pick f ∈ R_{n} such that (f + R_{n−1})/R_{n−1} = t_{λ}. Then f = ∂^{I(λ)} + ∑_{J<I(λ)} f_{J}∂^{J} and I(λ) = (i_{1}, …, i_{ℓ}) with i_{r} ≠ 0. Since each u_{j} ∈ F ⊂ R, we may apply Lemma 7(ii) to conclude that there exist f_{j} = f_{jr} ∈ F such that

Δ_{r} = ∂_{r} + f_{r+1}∂_{r+1} + ⋯ + f_{ℓ}∂_{ℓ} ∈ R.

But D(F) = F ⟨∂_{1}, …, ∂_{ℓ}⟩ = F ⟨Δ_{1}, …, Δ_{ℓ}⟩. Thus, R = D(F). □

The next result is proved in [LS, Lemma IV.1.3] but, since it is crucial to this paper, we will reprove it here. In the case of Weyl groups, it is the only ingredient of our proof of Theorem 5 that is not implicit in [Wa].

Lemma 9. Let R ⊆ S be Noetherian domains, with the same division ring of fractions. Assume that S is a simple ring and that S is finitely generated as a left and a right R-module. Then, R = S.

Proof. We may write S = ∑_{i=1}^{m} s_{i}R, for some s_{i} ∈ S. As S ⊂ Q(R), the quotient ring of R, we may write each s_{i} over a common denominator; say s_{i} = c^{−1}r_{i}, for some c, r_{i} ∈ R. Thus, c ∈ ℓ-ann_{R}(S) = {r ∈ R : rS ⊆ R} and so N = ℓ-ann_{R}(S) ≠ 0. Observe that N is a right ideal of S. Similarly, as S is a finitely generated left R-module, M = r-ann_{R}(S) ≠ 0 and M is a left ideal of S. Thus, as S is a domain, MN is a non-zero two-sided ideal of S. Since S is a simple, S = MN ⊆ R. □

Finally, we may combine these lemmas to prove Theorem 5 of introduction:

Theorem 5. Let B be the k-subalgebra of D(V) generated by S(V^{∗})^{W} and S(V)^{W}. Then, B = D(V)^{W}.

Proof. Recall that O(V) = S(V^{∗}) and that ℓ = dim_{k} V. Let C = O(V)^{W} \ {0}. By the definition of differential operators, C is an ad-nilpotent set of elements of D(V) and hence of the subalgebras B and D(V)^{W}. Thus, by [KL, Theorem 4.9], C is a (left and right) Ore set in each of these rings. Let F denote the field of fractions of O(V)^{W} and K the field of fractions of O(V). Since O(V) is a finite O(V)^{W} -module, O(V)_{C} = K and hence, by [MR, Theorem 15.5.5], D(V)_{C} = D(K). Similarly, D(O(V)^{W})_{C} = D(F). Next, pick ℓ algebraically independent elements u_{i} ∈ O(V)^{W} ⊂ F and set ∂_{i} = ∂/∂u_{i}, as before. Then D(F) = F ⟨∂_{1}, …, ∂_{ℓ}⟩ ⊂ D(K) = K⟨∂_{1}, …, ∂_{ℓ}⟩. We also have an homomorphism D(V)^{W} → D(O(V)^{W}), given by restriction of differential operators. Since D(V)^{W} is simple (see, for example, [Mo, Theorem 2.15, p.32] or [Wa, Lemma 1.2]) this map must be an injection. Thus, there is a chain of inclusions

F = O(V)_{C}^{W} ⊂ B_{C} ⊆ D(V)_{C}^{W} ⊆ D(O(V)^{W})_{C} = D(F) ⊆ D(V)_{C} = D(K).

Notice that the various filtrations on these rings are compatible. To see this, filter D(K) by degree of differential operator. Then the original filtrations on D(V) and hence on B are just the restrictions of that on D(K); in particular, B_{n} = B∩D(K)_{n}. Similarly, for example by the choice of {∂_{j}}, D(F)_{n} = D(F) ∩ D(K)_{n}. Hence, the filtrations on B_{C} induced from D(F) and B do indeed coincide. Now, consider gr(B_{C}). Since B ⊃ S(V)^{W}, certainly gr(B_{C}) ⊇ F ⊗_{k} S(V)^{W}. Since F ⊗_{k} S(V) is a finitely generated F ⊗_{k} S(V)^{W} -module, Kdim(F ⊗_{k} S(V)^{W}) = ℓ. Thus, by Lemma 8 and the remark thereafter, B_{C} = D(F). Therefore, B_{C} = D(V)_{C}^{W} and so B and D(V)^{W} have the same quotient division ring. By Lemma 8, both B and D(V)^{W} are Noetherian domains and D(V)^{W} is finitely generated as a left or right B-module. Thus, as D(V)^{W} is a simple ring, all the hypotheses of Lemma 9 are satisfied and B = D(V)^{W}. □

Wallach’s version of Theorem 5 for Weyl groups with no factors of type E_{n} is obtained as a corollary of a theorem about Poisson algebras and we end the paper by discussing the connections between his results and the techniques of this paper. Let { , } denote the usual Poisson bracket on S(V^{∗}) ⊗ S(V) and let W ⊂ GL(V) be a finite group. Following [Wa], the pair (V, W) is called good if, whenever C is a subalgebra of S(V^{∗}) ⊗_{k} S(V) that contains S(V^{∗})^{W} ⊗ S(V)^{W} and is closed under { , }, then C ⊇ (S(V^{∗}) ⊗ S(V))^{W}. In [Wa], Wallach proves that (V, W) is good in the case when W is a Weyl group with no factors of type E_{m} and he shows that, whenever (V, W) is good, the conclusion of Theorem 5 will also hold.

This raises the natural question of whether (V, W) is good for any finite group W. However, it is easy to construct counterexamples, even in the case where W is generated by pseudo-reflections. For example, let V be 1-dimensional complex vector space and W = ⟨σ⟩ ≅ ℤ_{3} acting by σ(v) = ωv, where ω is a primitive third root of unity. We may identify S(V^{∗}) ⊗ S(V) = S(V^{∗} × V) = ℂ[x, ξ], which we grade by total degree in x and ξ. It is easy to check that R = (S(V^{∗}) ⊗ S(V))^{W} = ℂ[x^{3}, xξ, ξ^{3}]. Let R′ be the graded subalgebra ℂ + ⨁_{n≥3} R_{n} of R. Since {R_{n}, R_{m}} ⊆ R_{n+m−2}, certainly R′ is closed under the Poisson bracket. But, xξ ∈ R \ R′ and so the pair (V, W) is not good.

One of the key points in the proof of Theorem 5 is the fact that D(V)^{W} is simple. The analogous result for Poisson algebras is false. To see this, let W ⊂ GL(V) be any finite group such that V^{W} = {0}. Give S(V^{∗}) ⊗ S(V) the natural grading by total degree. Then, C = (S(V^{∗}) ⊗ S(V))^{W} will contain no homogeneous element of degree one. Hence, the augmentation ideal ⨁_{n≥1} C_{n} = ⨁_{n≥2} C_{n} will be a nontrivial Poisson ideal. It is therefore unlikely that the techniques of this paper could be used to prove that (V, W) is good when W is a Weyl group.

Acknowledgement. This research was conducted while the first author was visiting the University of Michigan and supported by the NSF. He would like to thank both institutions.

References

[HC1] Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87-120.

[HC2] Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564.

[HC3] Harish-Chandra, Invariant eigendistributions on a semi-simple Lie algebra, Inst. Hautes Etudes Sci. Publ. Math. 27 (1965), 5–54.

[KL] G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Research Notes in Math., vol. 116, Pitman, Boston, 1985.

[LS] T. Levasseur and J. T. Stafford, Rings of Differential Operators on Classical Rings of Invariants, Mem. Amer. Math. Soc. 412 (1989).

[MR] J. C. McConnell and J. C. Robson, Non-commutative Noetherian Rings, Wiley-Interscience, Chichester, 1987.

[Mo] S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lect. Notes in Math., vol. 818, Springer-Verlag, Berlin/New York, 1980.

[Wa] N. R. Wallach, Invariant differential operators on a reductive Lie algebra and Weyl group representations, J. Amer. Math. Soc. 6 (1993), 779–816.

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

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

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

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