CORRIGENDUM TO: SEMI-SIMPLICITY OF INVARIANT HOLONOMIC SYSTEMS ON A REDUCTIVE LIE ALGEBRA

AMER. J. MATH. 119 (1997), 1095-1117

T. LEVASSEUR AND J. T. STAFFORD

The proof of Proposition 6.5 of [2] is inadequate (the application of the Chinese Remainder Theorem is incorrect) and so the aim of this corrigendum is to provide a correct proof of that result. All the results in [2] are correct as stated.

Recall the notation of [2]: A = D(𝔥), d = ∏_{α>0} α^{2}, λ ∈ 𝔥^{∗}, m = Ker λ ⊂ S(𝔥). Set P = A/Am. Then the A-module P identifies with O(𝔥)e^{λ} endowed with the natural action of A.

Definition 1.1. Let M be an A-module. We say that M has a C-filtration if there exists a finite chain of submodules 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = M satisfying

M_{i}/M_{i−1} ≅ A ⊗_{C_{i}} (C_{i}/I_{i}) for each i, (1.1)

where C_{i} = ℂ[y_{1}, …, y_{ℓ}] is a commutative polynomial ring such that D(C_{i}) = A, and I_{i} is an ideal of C_{i} of finite codimension.

Remarks 1.2. (1) The A-module P has a C-filtration: 0 = M_{0} ⊂ M_{1} = P.

(2) Suppose that M has a C-filtration and keep the notation of Definition 1.1. Then, M is automatically holonomic. Moreover, classical results (for example, Kashiwara’s equivalence) imply that A/AI_{i} is isomorphic to a finite direct sum of simple modules of the form A/Ay, where y is maximal ideal of C_{i}. In particular, every subquotient of M has a C-filtration.

(3) Let 0 → M′ → M → M′′ → 0 be an exact sequence of A-modules. Then, M′ and M′′ have a C-filtration if and only if M has one.

Lemma 1.3. Let D be left noetherian ring and B be a (D, A)-bimodule finitely generated as left D-module. Let M be an A-module having a C-filtration. Then Tor_{j}^{A}(B, M) = 0 for all j ⩾ 1.

Proof. Keep the notation of Definition 1.1. By [2, Lemma 6.4], B is a flat right C_{i}- module and so Tor_{j}^{A}(B, M_{i}/M_{i−1}) ≅ Tor_{j}^{C_{i}}(B, C_{i}/I_{i}) = 0, by[3, Theorem 11.53]. The lemma follows easily by induction on t from the short exact sequence

Tor_{j}^{A}(B, M_{1}) ⟶ Tor_{j}^{A}(B, M) ⟶ Tor_{j}(B, M/M_{1}).

Let M be an A-module and f ∈ O(𝔥). We denote by M_{(f)} the localization O(𝔥)[f^{−1}] ⊗_{O(𝔥)} M with respect to {f^{i}}. Recall that if M is holonomic, then M_{(f)} is also a holonomic A-module [1, Theorem 3.2.11]. In particular, M_{(f)} has finite length. This applies to P = O(𝔥)e^{λ} and P_{(f)} = ∑_{i∈ℕ} O(𝔥)f^{−i}e^{λ} (with the natural action of A by differential operators). Since {f^{−i}e^{λ}}_{i} is a system of generators of P_{(f)} this implies that P_{(f)} = Af^{−m}e^{λ} for some m ∈ ℕ. Notice that in the notation of [2], P_{(d)} = P_{𝒞}.

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

Key words and phrases. Invariant differential operators, Weyl group representations, holonomic systems, invariant eigendistributions, semisimple Lie algebras.

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

Lemma 1.4. The A-module P_{𝒞} /P has a C-filtration.

Proof. Set P_{0} = P. Let β_{1}, …, β_{k} be distinct positive roots for some k ⩾ 1. Observe that, since d = ∏_{α>0} α^{2}, we have P_{(β_{1}···β_{k})^{2}} = P_{(β_{1}···β_{k})} ⊂ P_{𝒞}. Define an A-submodule of P_{𝒞} by

P_{k} = ∑_{β_{1},...,β_{k}} P_{(β_{1}···β_{k})}

where the sum is taken over all possible sets of k distinct, positive roots. Notice that we have

P_{0} ⊂ ⋯ ⊂ P_{k} ⊂ ⋯ ⊂ P_{ν} = P_{𝒞} ,

where ν is the number of positive roots.

Set N_{γ} = (P_{(γ_{1}···γ_{k})} + P_{k−1})/P_{k−1}, where the γ_{i} are some distinct positive roots. Clearly, P_{k}/P_{k−1} = ∑ N_{γ} where the sum is over all possible γ. Thus, by Remarks 1.2, in order to prove that P_{𝒞} has a C-filtration it suffices to prove this for each P_{k}/P_{k−1} and hence for each N_{γ}. So, consider N = N_{γ}. As noticed above, P_{(γ_{1}···γ_{k})} = Aγ_{1}^{−2i} ⋯ γ_{k}^{−2i}e^{λ} for some i ∈ ℕ; hence N is generated by the class e = [γ_{1}^{−2i} ⋯ γ_{k}^{−2i}e^{λ}]. The significance behind the definition of the P_{ℓ} is that γ_{j}^{2i}e = 0 for 1 ≤ j ≤ k. Order the γ_{j}’s so that γ_{1}, …, γ_{h} are linearly independent while γ_{j} ∈ ∑_{i=1}^{h} ℂγ_{j} for j ⩾ h + 1. Pick x_{h+1}, …, x_{ℓ} ∈ 𝔥^{∗} such that {y_{1} = γ_{1}, …, y_{h} = γ_{h}, x_{h+1}, …, x_{ℓ}} is a basis of 𝔥^{∗}. Then, for j ⩾ h + 1, one has y_{j} = ∂_{x_{j}} − α_{j} ∈ m for some α_{j} ∈ ℂ. Then, [y_{j}, γ_{i}] = 0, for all j ≥ h + 1 and i ≤ k and so:

y_{1}^{2i}.e = y_{2}^{2i}.e = ⋯ = y_{h}^{2i}.e = y_{h+1}.e = ⋯ = y_{ℓ}.e = 0.

Notice that C_{i} = ℂ[y_{1}, …, y_{ℓ}] is a polynomial ring with D(C_{i}) = A. By the last displayed equation, A.e is a factor of the module A ⊗_{C_{i}} (C_{i}/I_{i}), where I_{i} = (y_{1}^{2i}, …, y_{h}^{2i}, y_{h+1}, …, y_{ℓ}). Hence the lemma.

Corollary 1.5. ([2, Proposition 6.5].) Let B be the (D(𝔤), A)-bimodule defined in [2, p.1109]. Let m be a maximal ideal of S(𝔥) and write P = A/Am. Then, Tor_{1}^{A}(B, P_{𝒞} /P) = 0.

Proof. By its construction, B is a finitely generated left D(𝔤)-module. Thus, the corollary follows from Lemma 1.4 and Lemma 1.3 with M = P_{𝒞} /P.

References

[1] J.-E. Björk, Rings of Differential Operators, North Holland, Amsterdam, 1979.

[2] T. Levasseur and J. T. Stafford, Semi-simplicity of invariant holonomic systems on a reductive Lie algebra, Amer. J. Math., 119 (1997), 1095-1117.

[3] J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.

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