Journal of Lie Theory Volume 11 (2001) 355–379 C ◯2001 Heldermann Verlag

Subquotients in the Enveloping Algebra of a Nilpotent Lie Algebra

Bradley N. Currey III

Communicated by J. Ludwig

Abstract. For any triple (𝔤, 𝔥, f) where 𝔤 is a nilpotent Lie algebra over a field k of characteristic zero, 𝔥 is a subalgebra of 𝔤, and f is a homomorphism of 𝔲(𝔥) onto k, a subquotient 𝒟(𝔤, 𝔥, f) of 𝔲(𝔤) is studied which generalizes the algebra of invariant differential operators on a nilpotent homogeneous space. A generalized version of a conjecture of Corwin and Greenleaf is formulated using geometry of exp(ad^{∗}𝔥)-orbits in the variety L_{f} of linear functionals in 𝔤^{∗} whose restriction to 𝔥 agree with f . Certain constructions lead to a procedure by which the question of non-commutativity of 𝒟(𝔤, 𝔥, f) is reduced to a case where (𝔤, 𝔥, f) has a special structure. This reduction is then used to prove that the Corwin-Greenleaf conjecture about non-commutativity of 𝒟(𝔤, 𝔥, f) holds in certain situations, in particular when the exp(ad^{∗}𝔥)-orbits in L_{f} have dimension no greater than one.

1. Introduction

Given a real nilpotent Lie group G, an analytic subgroup H of G, and a unitary character χ of H, it is known (see [4]) that the corresponding induced representation τ of G can be of two types: uniformly bounded multiplicity a.e. (abbrev. FM), or uniformly infinite multiplicity a.e. (abbrev. IM). In the FM case, it is known that the algebra of τ -invariant differential operators is commutative, and much evidence has accumulated that indicates that the converse is true, that is, that in the IM case, there are two invariant differential operators that do not commute with each other.

In this paper we study this conjecture from a purely algebraic/geometric standpoint. We begin with a nilpotent Lie algebra 𝔤 over a field k of characteristic zero, a Lie subalgebra 𝔥 of 𝔤, and an algebra homomorphism f : 𝔲(𝔥) → k. The spectral variety L_{f} ⊂ 𝔤^{∗} is the set of all linear functionals on 𝔤 whose restriction to 𝔥 agrees with f. The triple (𝔤, 𝔥, f) is defined to be an IM-triple if the generic dimension of the coadjoint orbits in 𝔤^{∗} that meet L_{f} exceeds twice the dimension of the exp(ad^{∗}𝔥)-orbits in L_{f}; of course this reduces to the wellknown Corwin-Greenleaf condition when k = C and f is the differential of a

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag unitary character on the real analytic subgroup H. The algebra of “generalized” invariant differential operators is the subquotient 𝒟(𝔤, 𝔥, f) = c(𝔥, ℐ)/ℐ in the enveloping algebra 𝔲(𝔤) where ℐ is the left ideal in 𝔲(𝔤) generated by the kernel of f and c(𝔥, ℐ) = {W ∈ 𝔲(𝔤) : [W, 𝔥] ⊂ ℐ}. We make the following.

Conjecture 1. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent. Then 𝒟(𝔤, 𝔥, f) is not commutative.

A standard way to analyse the algebra 𝒟(𝔤, 𝔥, f) has been to take a sequence of subalgebras 𝔥 = 𝔥_{0} ⊂ 𝔥_{1} ⊂ ⋯ ⊂ 𝔥_{m} = 𝔤 where dimensions increase one at a time, and in trying to prove Conjecture 1, one quickly reduces to the situation where (𝔤, 𝔥, f) is an IM-triple while (𝔥_{m−1}, 𝔥, f) is an FMtriple. Greenleaf in [7], and Fujiwara, Lion, and Mehdi in [6] have shown in this situation (and with k = C and f the differential of a unitary character), that 𝒟(𝔤, 𝔥, f) ≠ 𝒟(𝔥_{m−1}, 𝔥, f) implies non-commutativity of 𝒟(𝔤, 𝔥, f).

Our attack on Conjecture 1 also involves a reduction, although with a bit more algebraic information. First we introduce a class of triples (𝔤, 𝔥, f) of a special form: those for which there is an ideal 𝔷 in 𝔤 and an element Y ∈ 𝔤 with the properties that [𝔤, 𝔷] ⊂ ℐ, [𝔤, Y] ⊂ 𝔷, and that the centralizer 𝔤^{Y} of Y modulo ℐ is codimension one in 𝔤. We say that triples with a pair (𝔷, Y) as just described admit a “Kirillov structure”. What is the utility of this notion? First of all, this sort of structure occurs naturally: after laying out basic algebraic framework in Section 2, Section 3 is devoted to showing that for any triple (𝔤, 𝔥, f), one can construct an associated triple (𝔤̂, 𝔥̂, f̂) that admits a Kirillov structure, and for which all the necessary information about orbit dimensions and 𝒟(𝔤, 𝔥, f) is preserved. The tradeoff is that the scaler field is extended (in fact to a subfield of the invariant rational functions on L_{f}) when passing to the associated triple; thus the necessity of considering arbitrary fields of characteristic zero. Secondly, suppose that we are given that a triple (𝔤, 𝔥, f) has a Kirillov structure, that 𝔥 ⊂ 𝔤^{Y} , and that (𝔤^{Y} , 𝔥, f) is an F M -triple. Then it is extremely easy to show that 𝒟(𝔤, 𝔥, f) ≠ 𝒟(𝔤^{Y} , 𝔥, f) (or equivalently c(𝔥, ℐ) ⊄ 𝔲(𝔤^{Y}) + ℐ) implies noncommutativity of 𝒟(𝔤, 𝔥, f). Finally, and perhaps most significantly, suppose that we are given that a triple (𝔤, 𝔥, f) has a Kirillov structure and that 𝔥 ⊄ 𝔤^{Y} . Setting 𝔥^{Y} = 𝔥 ∩ 𝔤^{Y} , a crucial result (Proposition 4.1.1) is that 𝒟(𝔤, 𝔥, f) is commutative if and only if 𝒟(𝔤^{Y} , 𝔥^{Y} , f) is commutative. Thus, in the presence of a Kirillov structure one can always pass to the triple (𝔤^{Y} , 𝔥^{Y} , f), whether 𝔥 ⊂ 𝔤^{Y} or not.

Now given an IM-triple (𝔤, 𝔥, f), the process of passing to an associated triple, and then to a codimension one centralizer, allows the construction of a sequence of triples {(𝔤(j), 𝔥(j), f(j))}_{j=0}^{N}, where

(a) (𝔤, 𝔥, f) = (𝔤(0), 𝔥(0), f(0)),

(b) (𝔤(j), 𝔥(j), f(j)) IM implies that (𝔤(j − 1), 𝔥(j − 1), f(j − 1)) is IM,

(c) 𝒟(𝔤(j), 𝔥(j), f(j)) non-commutative implies that 𝒟(𝔤(j−1), 𝔥(j−1), f(j−1)) is non-commutative, and

(d) the exp(ad^{∗}(𝔤))-orbits that meet L_{f(N)} are single points. Such a sequence is called a reducing sequence. We emphasize that the scalar fields may get larger with the index j, and in particular, 𝔤(j) is not necessarily a subalgebra of 𝔤(j−1), although its dimension (over its scalar field) is necessarily smaller. Also, the construction of the reducing sequence does not require that the subalgebras 𝔥(j)ˆ be contained in the respective codimension one centralizers; as a result it is possible that the dimension of 𝔥(j) will be smaller than that of 𝔥(j − 1).

In any case, there will necessarily be some m for which the triple (𝔤(m), 𝔥(m), f(m)) is IM but (𝔤(m+1), 𝔥(m+1), f(m+1)) is FM, and consideration of the associated triple allows one to reduce Conjecture 1 to the following. Conjecture 2. Let (𝔤, 𝔥, f) be an IM triple with 𝔤 nilpotent. Assume that (𝔤, 𝔥, f) admits a Kirillov structure (𝔷, Y) for which 𝔥 ⊂ 𝔤^{Y} and for which (𝔤^{Y} , 𝔥, f) is FM. Then c(𝔥, ℐ) ⊄ 𝔲(𝔤^{Y}) + ℐ.

Specifically, we prove the following result in Section 4.

Theorem 4.2.1. Let {(𝔤(j), 𝔥(j), f(j)) : 0 ≤ j ≤ k} be a reducing sequence for the IM-triple (𝔤, 𝔥, f) with 𝔤 nilpotent. Set

m = max{j : (𝔤(j), 𝔥(j), f(j)) is an IM-triple},

and assume that Conjecture 2 holds for ((𝔤(m)ˆ , 𝔥(m)ˆ , f(m)ˆ ). Then 𝒟(𝔤, 𝔥, f) is not commutative.

In Section 5 we apply the above theorem to prove Conjecture 1 in two situations. First, we consider the case where all of the exp(ad^{∗}𝔥)-orbit dimensions in L_{f} are no greater than one. The result is that Conjecture 1 always holds in this case.

Theorem 5.1.4. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and with the property that δ(𝔤, 𝔥, f) ≤ 1. Then 𝒟(𝔤, 𝔥, f) is not commutative.

Secondly we consider the situation where f = 0 (or “essentially zero”; see Section 5.2) and a certain generic condition is satisfied. Recall that, given increasing sequences {𝔤_{i}}_{i=1}^{n} and {𝔥_{j}}_{j=1}^{p} of exp(ad^{∗}𝔥)-modules in 𝔤 and 𝔥 respectively, with dim(𝔤_{i}) = i, 1 ≤ i ≤ n and dim(𝔥_{j}) = j, 1 ≤ j ≤ p, one has a definition of “generic” exp(ad^{∗}𝔥)-orbits.

Lemma 5.2.2. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and with f = 0. Assume that (𝔤, 𝔥, f) admits a Kirillov structure (𝔷, Y) for which 𝔥 ⊂ 𝔤^{Y} and for which (𝔤^{Y} , 𝔥, f) is FM. Assume further that there are increasing sequences of exp(ad^{∗}𝔥)-modules in 𝔤 and 𝔥 as described above and with 𝔤_{n−1} = 𝔤^{Y} , such that L_{f} meets the corresponding set of generic exp(ad^{∗}𝔥)-orbits . Then c(𝔥, ℐ) ⊈ 𝔲(𝔤^{Y}) + ℐ.

From Theorem 4.2.1 and Lemma 5.2.2, one obtains the following.

Theorem 5.2.3. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and suppose that a reducing sequence for (𝔤, 𝔥, f) is given such that the hypothesis of Lemma 5.2.2 holds for (𝔤(m)ˆ , 𝔥(m)ˆ , f(m)ˆ ). Then 𝒟(𝔤, 𝔥, f) is not commutative.

Thanks go to Chal Benson and Gail Ratcliff for many helpful conversations concerning this work. We also thank the referee for several important corrections and suggestions.

2. Basic Definitions

2.1 Let k be a field of characteristic zero, let 𝔤 be a Lie algebra over k, and let 𝔲(𝔤) be the enveloping algebra of 𝔤 over k. Let 𝔥 be a Lie subalgebra of 𝔤; we regard 𝔲(𝔥) as a subalgebra of 𝔲(𝔤) in the usual way. Let f : 𝔲(𝔥) → k be an associative algebra homomorphism, and let ℐ = ℐ(𝔤, 𝔥, f) be the left ideal in 𝔲(𝔤) generated by the kernel of f.

We use multi-indices in the same way as [2]: given an ordered set of elements X_{1}, X_{2}, …, X_{m} in 𝔤 and α ∈ Z_{+}^{m}, let X^{α} denote the element

X^{α} = X_{1}^{α_{1}}X_{2}^{α_{2}} ⋯ X_{m}^{α_{m}}

in 𝔲(𝔤).

Proposition 2.1.1. Let 𝔤 be any Lie algebra over a field k of characteristic zero, let 𝔥 be a subalgebra of 𝔤, and let f be a k-algebra homomorphism of 𝔲(𝔥) onto k. Let Y_{1}, Y_{2}, …, Y_{p}, X_{1}, X_{2}, …, X_{m} be any basis for 𝔤 over k with

𝔥 = k-span{Y_{1}, Y_{2}, …, Y_{p}}

and define U_{j} = Y_{j} − f(Y)I, 1 ≤ j ≤ p. Then one has the following.

(a) The monomials X^{α}U^{β}, α ∈ Z_{+}^{m}, β ∈ Z_{+}^{p}, form a basis for 𝔲(𝔤) over k.

(b) The left ideal ℐ generated by the kernel of f is spanned by the monomials X^{α}U^{β}, |β| > 0.

(c) Define 𝔞(𝔥, f) = k-span{Y − f(Y)1 : Y ∈ 𝔥}. Then ℐ = 𝔲(𝔤)𝔞(𝔥, f).

Proof. We imitate the proof of [2, Lemma 4.2]. Let 𝔲^{r}(𝔤) be the subspace spanned by the monomials X^{α}Y^{β}, with |α| + |β| ≤ r. For each α and β, we have

X^{α}Y^{β} = ∑_{γ≼β} \binom{β}{γ}(f(Y))^{β−γ} X^{α}V^{γ}

where γ ≼ β means that γ_{j} ≤ β_{j}, 1 ≤ j ≤ p, \binom{β}{γ} = ∏_{j=1}^{p} \binom{β_{j}}{γ_{j}}, and f(Y)^{β−γ} = ∏_{j=1}^{p} f(Y_{j})^{β_{j}−γ_{j}}. This shows that {X^{α}V^{β} : |α| + |β| ≤ r} spans 𝔲^{r}(𝔤). Part (a) then follows by counting dimensions.

As for (b), let 𝒦 = ker(f) ⊂ 𝔲(𝔥). We claim that 𝒦 = k-span{U^{β} : |β| > 0}. Since it is obvious that 𝒦 ⊃ k-span{U^{β} : |β| > 0}, then we need only show that k-span{U^{β} : |β| > 0} has codimension one in 𝔲(𝔥). This follows from considering the fact that for r = 1, 2, 3, …, 𝔲^{r}(𝔥) = k-span{U^{β} : |β| = r} + 𝔲^{r−1}(𝔥) and then using induction on r.

The claim being verified, we see that any monomial U^{β}U^{γ}, with |γ| > 0 can be written as a linear combination of monomials U^{λ}, λ > 0, and hence any monomial X^{α}U^{β}U^{γ} is a linear combination of monomials X^{α}U^{λ}, λ > 0. But by part (a), 𝔲(𝔤) is spanned by the monomials X^{α}U^{β}, and (b) follows. Part (c) now follows immediately from part (b).

Corollary 2.1.2. Let 𝔤, 𝔥 and f be as in Proposition 2.1.1. Then ℐ(𝔤, 𝔥, f)∩ 𝔤 = 𝔥 ∩ ker(f).

Proof. From parts (a) and (b) above we see that ℐ(𝔤, 𝔥, f) ∩ 𝔤 ⊂ k-span{U_{j} : 1 ≤ j ≤ p}. On the other hand, one computes that k-span{U_{j} : 1 ≤ j ≤ p}∩𝔤 ⊂ 𝔥 ∩ ker(f): if W ∈ k-span{U_{j} : 1 ≤ j ≤ p},

W = ∑_{j=1}^{p} a_{j}U_{j} = ∑_{j=1}^{p} a_{j}Y_{j} − (∑_{j=1}^{p} a_{j}f(Y_{j})) I

can belong to 𝔤 only if ∑_{j=1}^{p} a_{j}f(Y_{j}) = 0, showing that W ∈ 𝔥 and that f(W) = 0. Hence ℐ(𝔤, 𝔥, f) ∩ 𝔤 ⊂ 𝔥 ∩ ker(f). The reverse inclusion is obvious.

Corollary 2.1.3. Let 𝔤, 𝔥 and f be as in Proposition 2.1.1 and suppose that 𝔨 is a Lie subalgebra of 𝔤. Set 𝔥_{0} = 𝔥 ∩ 𝔨, and f_{0} = f|_{𝔲(𝔥_{0})} . Then ℐ(𝔤, 𝔥, f) ∩ 𝔲(𝔨) = ℐ(𝔨, 𝔥_{0}, f_{0}).

Proof. Let T_{1}, T_{2}, ..., T_{u} be a basis of 𝔥_{0} , Z_{1}, Z_{2}, ..., Z_{t} a basis of a subspace of 𝔨 complementary to 𝔥_{0} , Y_{1}, Y_{2}, ..., Y_{s} a basis of a subspace of 𝔥 complementary to 𝔥_{0} , and X_{1}, X_{2}, ..., X_{r} a basis of a subspace of 𝔤 complementary to 𝔥+𝔨. The union of these bases is obviously a basis of 𝔤. Set V_{i} = T_{i} − f(T_{i})I, 1 ≤ i ≤ u and U_{i} = Y_{i} − f(Y_{i})I, 1 ≤ i ≤ s. From Proposition 2.1.1 (b) we have that the set {X^{α}Z^{μ}U^{β}V^{ν} : |β| + |ν| > 0} is a basis for ℐ(𝔤, 𝔥, f) and by Proposition 2.1.1 (a), that {Z^{μ}V^{ν} : μ ∈ Z_{+}^{u}, ν ∈ Z_{+}^{t}} is a basis for 𝔲(𝔨). The intersection of these two bases is {Z^{μ}V^{ν} : |ν| > 0} which by part (b) of the Proposition is a basis for ℐ(𝔨, 𝔥_{0}, f_{0}).

2.2 Let 𝔤 be a nilpotent Lie algebra over a field k of characteristic zero. For ℓ ∈ 𝔤^{∗}, define

𝔤(ℓ) = {Z ∈ 𝔤 : ℓ([X, Z]) = 0 holds for all X ∈ 𝔤}

and set d(ℓ) = dim(𝔤/𝔤(ℓ)). Let 𝔤 have the nilpotent group operation defined by the Campbell-Baker-Hausdorff formula. It is easily seen that for each ℓ ∈ 𝔤^{∗}, 𝔤(ℓ) is the stabilizer of ℓ in 𝔤 with respect to the coadjoint action: for X, Y in 𝔤,

(exp(ad^{∗}(X))ℓ)(Y) = ℓ(Y − [X, Y] + \frac{1}{2}[X, [X, Y]] − \frac{1}{6}[X, [X, [X, Y]]] + ⋯).

It is well-known that there is a finite partition 𝒫 of 𝔤^{∗} with the properties (i) each Ω in 𝒫 is exp(ad^{∗}(𝔤))-invariant, (ii) there is a total ordering < on 𝒫 such that for each Ω, ⋃{Ω′ : Ω′ ≤ Ω}

is Zariski open in 𝔤^{∗}, (iii) d is constant on each Ω, and (iv) Ω′ < Ω implies d(Ω′) ≥ d(Ω).

The construction of the partition 𝒫 depends only upon the choice of a Jordan-Hölder sequence for 𝔤.

Let 𝔥 be a Lie subalgebra of 𝔤, and set 𝔥(ℓ) = 𝔥 ∩ 𝔤(ℓ), ℓ ∈ 𝔤^{∗}. For each ℓ, 𝔥(ℓ) is the stabilizer of ℓ in 𝔥 with respect to the coadjoint action of 𝔥. Set δ(ℓ) = dim(𝔥/𝔥(ℓ)), ℓ ∈ 𝔤^{∗}. There is a similar partition 𝒫_{𝔥} of 𝔤^{∗} having properties (i) - (iv) with exp(ad^{∗}(𝔤)) replaced by exp(ad^{∗}(𝔥)) and d replaced by δ.

Given f : 𝔲(𝔥) → k an associative algebra homomorphism, set

L_{f} = {ℓ ∈ 𝔤^{∗} : ℓ|_{𝔥} = f|_{𝔥}}.

Lemma 2.2.1. There is a Zariski open subset U of L_{f} on which the dimension functions d and δ are constant.

Proof. From property (ii) of the partitions 𝒫 and 𝒫_{𝔥}, we can choose Ω ∈ 𝒫 and Ω_{𝔥} ∈ 𝒫_{𝔥} such that Ω ∩ Ω_{𝔥} ∩ L_{f} is a Zariski open subset of L_{f}; set U = Ω ∩ Ω_{𝔥} ∩ L_{f}.

We denote these “generic” values of d and δ by d(𝔤, 𝔥, f) and δ(𝔤, 𝔥, f) respectively. (Note that from property (iv) above one has that d(𝔤, 𝔥, f) = max{d(ℓ) : ℓ ∈ L_{f}} and δ(𝔤, 𝔥, f) = max{δ(ℓ) : ℓ ∈ L_{f}}, hence these values are independent of the choice of Jordan-Hölder sequence.)

Lemma 2.2.2. Let 𝔤, 𝔥, and f as above, and let d(𝔤, 𝔥, f) and δ(𝔤, 𝔥, f) be the associated generic coadjoint orbit dimensions. Then d(𝔤, 𝔥, f) ≥ 2 · δ(𝔤, 𝔥, f).

Proof. For any ℓ ∈ L_{f}, 𝔥 + 𝔤(ℓ) is a subspace of 𝔤 that is isotropic with respect to the alternate bilinear form defined by ℓ, so

d(ℓ) = dim(𝔤/𝔤(ℓ)) ≥ 2 · dim((𝔥 + 𝔤(ℓ))/𝔤(ℓ)) = 2 · dim(𝔥/𝔥(ℓ)).

A triple (𝔤, 𝔥, f), where 𝔤 is a nilpotent Lie algebra, 𝔥 is a Lie subalgebra, and f : 𝔲(𝔥) → k is an algebra homomorphism, will be called an “IM-triple” if d(𝔤, 𝔥, f) > 2 · δ(𝔤, 𝔥, f). Otherwise (𝔤, 𝔥, f) will be called an “FM-triple.”

3. Construction of an associated triple

3.1 Let 𝔤 be a Lie algebra over a field k of characteristic zero, and let 𝔲(𝔤) be its enveloping algebra. For any subsets ℳ and 𝒩 of 𝔲(𝔤), set

c(ℳ, 𝒩) = {W ∈ 𝔲(𝔤) : [W, ℳ] ⊂ 𝒩}.

Let (𝔤, 𝔥, f) be a triple with ℐ = ℐ(𝔤, 𝔥, f).

Lemma 3.1.1. Let 𝔷 be an ideal in 𝔤 such that 𝔷 ⊂ c(𝔤, ℐ). Then [𝔤, 𝔷] ⊂ 𝔷 ∩ 𝔥 ∩ ker(f). In particular, 𝔷 ∩ 𝔥 is an ideal in 𝔤.

Proof. By Corollary 2.1.2, 𝔷 ∩ ℐ = 𝔷 ∩ 𝔥 ∩ ker(f), and by definition of 𝔷, [𝔤, 𝔷] ⊂ 𝔷 ∩ ℐ.

An ideal 𝔷 in 𝔤 will be called an ℐ-central ideal if 𝔷 ⊂ c(𝔤, ℐ). Definition. A Kirillov structure for (𝔤, 𝔥, f) is a pair (𝔷, Y) where 𝔷 is an ℐcentral ideal in 𝔤 and Y is an element of 𝔤 that belongs to c(𝔤, 𝔷) and for which c(Y, ℐ) ∩ 𝔤 is codimension one in 𝔤. The element Y will be called a Kirillov element for (𝔤, 𝔥, f).

Given a Kirillov structure for (𝔤, 𝔥, f), we shall employ the notation 𝔤^{Y} = c(Y, ℐ) ∩ 𝔤. If 𝔤 is nilpotent, then any triple (𝔤, 𝔥, f) obviously has a nonzero ℐ-central ideal. However, even in cases where exp(ad^{∗}(𝔥)) acts non-trivially on L_{f}, (𝔤, 𝔥, f) may not have a Kirillov element. The goal of this section is to give a construction for any triple (𝔤, 𝔥, f) with 𝔤 nilpotent and δ(𝔤, 𝔥, f) > 0, of an associated triple (𝔤̂, 𝔥̂, f̂) that does admit a Kirillov structure, and for which important aspects of the coadjoint orbits and quotient algebras 𝒟(𝔤, 𝔥, f) are preserved.

Choose an ℐ-central ideal 𝔷, and set 𝔥_{0} = 𝔥∩𝔷. Set f_{0} = f|_{𝔲(𝔥_{0})} , and let 𝒥 be the left ideal in 𝔲(𝔤) generated by ker(f_{0}), that is, 𝒥 = ℐ(𝔤, 𝔥_{0}, f_{0}). Since 𝔥_{0} is an ideal in 𝔤, 𝒥 is a two-sided ideal in 𝔲(𝔤).

Let T_{1}, T_{2}, ..., T_{u} be a basis of 𝔥_{0} , Z_{1}, Z_{2}, ..., Z_{t} a basis of a subspace of 𝔷 complementary to 𝔥_{0} , Y_{1}, Y_{2}, ..., Y_{s} a basis of a subspace of 𝔥 complementary to 𝔥_{0} and X_{1}, X_{2}, ..., X_{r} a basis of a subspace of 𝔤 complementary to 𝔥+𝔷. The set

B = {T_{1}, T_{2}, …, T_{u}, Z_{1}, Z_{2}, …, Z_{t}, Y_{1}, Y_{2}, …, Y_{s}, X_{1}, X_{2}, …, X_{r}}

is then a basis of 𝔤, where it is understood that if any of the above vector subspaces of 𝔤 are trivial, then the corresponding basis is the empty set. Set V_{i} = T_{i} − f(T_{i})I, 1 ≤ i ≤ u and U_{i} = Y_{i} − f(Y_{i})I, 1 ≤ i ≤ s. From Proposition 2.1.1 and its corollaries we have the following.

(i) The monomials X^{α}Y^{β}Z^{μ}V^{ν}, α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, μ ∈ Z_{+}^{t}, ν ∈ Z_{+}^{u} form a basis for 𝔲(𝔤) over k,

(ii) 𝒥 is spanned by the monomials X^{α}Y^{β}Z^{μ}V^{ν}, |ν| > 0,

(iii) the monomials Z^{μ}V^{ν}, μ ∈ Z_{+}^{t}, ν ∈ Z_{+}^{u} form a basis of u(𝔷) over k,

(iv) 𝔲(𝔷) ∩ 𝒥 = ℐ(𝔷, 𝔥_{0}, f_{0}), and
(v) 𝔲(𝔷) ∩ 𝒥 is spanned by the monomials {Z^{μ}V^{ν} : |ν| > 0}.

Proposition 3.1.2. Let 𝔷 be an ℐ-central ideal in 𝔤, let W be any element of 𝔲(𝔤), and let T belong to 𝔲(𝔷) such that W T ∈ ℐ. Then either W ∈ ℐ or T ∈ ℐ. Proof. In accordance with Proposition 2.1.1 (b) (applied to 𝔥), we write

W = ∑_{α,β,μ,ν} a_{αβμν}X^{α}Z^{μ}U^{β}V^{ν},

and similarly

T = ∑_{σ,λ,} b_{σλ}Z^{σ}V^{λ}.

Now by Lemma 3.1.1, 𝔲(𝔷) ⊂ c(𝔲(𝔤), 𝒥), so for each set α, β, μ, ν, σ, λ of multiindices, there is Q ∈ 𝒥 such that

X^{α}Z^{μ}U^{β}V^{ν}Z^{σ}V^{λ} = X^{α}Z^{μ+σ}U^{β}V^{ν+λ} + Q.

Hence we can write

(*)  W T = ∑_{α,β,ϕ,ψ} ( ∑_{\begin{matrix}μ+σ=ϕ \\ ν+λ=ψ \end{matrix}} a_{αβμν}b_{σλ})X^{α}Z^{ϕ}U^{β}V^{ψ} + Q

where Q ∈ 𝒥.

Now assume that niether W nor T belongs to ℐ. Then there are multiindices α, β, μ, ν, σ, λ for which a_{αβμν} ≠ 0 and |β| + |ν| = 0, and for which b_{σλ} ≠ 0 and |λ| = 0. Set N = {μ ∈ Z_{+}^{t} : a_{αβμν} ≠ 0 and |β| + |ν| = 0} and L = {σ ∈ Z_{+}^{t} : b_{σλ} ≠ 0 and |λ| = 0}. We order Z_{+}^{t} lexographically, and set μ_{0} = max(N), σ_{0} = max(L), and ϕ_{0} = μ_{0} + σ_{0} . If a_{αβμν}b_{σλ} ≠ 0 and |β| + |ν| + |λ| = 0 then μ + σ ≤ ϕ_{0} with equality only if μ = μ_{0} and σ = σ_{0} . Hence there is a term in ∗ whose monomial has the form X^{α}Z^{ϕ_{0}} and whose coefficient is a_{αβμ_{0}ν}b_{σ_{0}λ} ≠ 0. By Proposition 2.1.1, this means that W T ∉ ℐ.

Lemma 3.1.3. Let W be an element of u(𝔤) of the form

W = ∑_{α,β} X^{α}Y^{β}Q_{αβ}

where each Q_{αβ} belongs to u(𝔷). Then W ∈ 𝒥 if and only if Q_{αβ} ∈ u(𝔷) ∩ 𝒥 holds for all α and β.

Proof. The “if” part of the lemma is obvious, so let us suppose that W belongs to 𝒥. By (ii) above we have

W = ∑_{\begin{matrix}α,β,μ,ν \\ |ν|>0 \end{matrix}} a_{αβμν}X^{α}Y^{β}Z^{μ}V^{ν}.

Set

Q′_{αβ} = ∑_{\begin{matrix}μ,ν \\ |ν|>0 \end{matrix}} a_{αβμν}Z^{μ}V^{ν};

Now by [5, 2.2.7. Proposition], {X^{α}Y^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}} is a basis for 𝔲(𝔤) as a right module over 𝔲(𝔷). Hence for each α and β, Q_{αβ} = Q′_{αβ}, and Q_{αβ} ∈ 𝒥.

3.2 Let (𝔤, 𝔥, f) be a triple, with 𝔷 an ℐ-central ideal, and with 𝔥_{0} , f_{0} , and 𝒥 as in the previous subsection. Set 𝒜 = u(𝔤)/𝒥, and let π : u(𝔤) → 𝒜 be the canonical homomorphism. By Corollary 2.1.2, 𝔤 ∩ 𝒥 = 𝔥_{0} ∩ ker(f_{0}), and hence π(𝔤) ≃ 𝔤/(𝔥_{0} ∩ ker(f_{0})). Set ℛ = π(u(𝔷)).

Lemma 3.2.1. The subalgebra ℛ of 𝒜 is contained in the center of 𝒜, and the elements of ℛ are not divisors of zero in 𝒜.

Proof. The ideal 𝔥_{0} ∩ ker(f) generates a two-sided ideal 𝒥_{0} in 𝔲(𝔤) that is contained in 𝒥. Since 𝔷/𝔥_{0} ∩ ker(f) is central in 𝔤/𝔥_{0} ∩ ker(f), then the canonical image of 𝔲(𝔷) in 𝔲(𝔤)/𝒥_{0} = 𝔲(𝔤/𝔥_{0} ∩ ker(f)) is central. It follows that ℛ is central in 𝒜. That elements of ℛ are not divisors of zero follows immediately from Proposition 3.1.2.

Lemma 3.2.2. The monomials {π(X^{α}Y^{β}Z^{μ}) : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, μ ∈ Z_{+}^{t}} form a basis of 𝒜, and π(ℐ) is spanned by the monomials π(X^{α}U^{β}Z^{μ}), |β| > 0.

Proof. The first statement follows immediately from (ii) above. For the second, Lemma 3.2.1 shows that π(X^{α}U^{β}Z^{μ}) = π(X^{α}Z^{μ}U^{β}). On the other hand by Proposition 2.1.1 (b), ℐ = k-span{X^{α}Z^{μ}U^{β}V^{ν}, |β| + |ν| > 0}, and hence π(ℐ) is spanned by the monomials π(X^{α}Z^{μ}U^{β}), |β| > 0.

3.3 Retaining the setup and notation from the previous subsection, we now bring in the language and results from [5, Sect. 3.6] : by Lemma 3.2.1, the set ℛ^{∗} = ℛ\{0} allows an arithmetic of fractions in 𝒜, and we let ℬ = 𝒜_{ℛ^{∗}} be the resulting fraction algebra. As in loc. cit., ℬ = 𝒜 × ℛ^{∗}/ ∼ where ∼ is the equivalence relation on 𝒜 × ℛ^{∗} defined by

(a, r) ∼ (b, s) if and only if as = br.

(Since ℛ^{∗} ⊆ cent(𝒜) , the definition of ∼ in loc. cit. reduces to the above.) We have the natural ring homomorphisms

u(𝔤) ⟶^{π} 𝒜 ⟶ 𝒜 × ℛ^{∗} ⟶^{ϵ} ℬ.

The above produces a canonical injection of 𝒜 into ℬ, and we identify 𝒜 with its image in ℬ. For an element of 𝔲(𝔤) denoted by an upper case Roman letter, denote its image in 𝒜 by the lower case counterpart; in particular denote π(I) by i. For X ∈ 𝔲(𝔤) and Z ∈ 𝔲(𝔷), denote the equivalence class in ℬ of (x, z) by x/z. In keeping with the above mentioned identification, denote x/i by x, x ∈ 𝒜. For any subset ℳ of 𝒜, let σ(ℳ) denote the subset of fractions in ℬ having numerators in ℳ, that is, σ(ℳ) = ϵ(ℳ × ℛ^{∗}), and in particular, ℬ = σ(𝒜).

Set K = ϵ(ℛ × ℛ^{∗}). Since ℛ is a commutative integral domain, K is a field of characteristic zero and ℬ has in a natural way the structure of an associative K-algebra. More generally, we have the following.

Lemma 3.3.1. Let ℳ be an ℛ-submodule of 𝒜 with ℛ-basis B. Then σ(ℳ) is a K-subspace of ℬ with basis B. Moreover, if ℳ is also a subalgebra of 𝒜, then σ(ℳ) is a K-subalgebra of ℬ.

Proof. It is clear that σ(ℳ) is a K-subspace of ℬ. It is spanned by B since, for x/z ∈ σ(ℳ), x = ∑ z_{i}x_{i} where z_{i} ∈ ℛ and x_{i} ∈ B, so x/z = ∑(z_{i}/z)x_{i} ∈ K-span(B). Suppose that ∑ q_{i}x_{i} = 0 with q_{i} ∈ K and x_{i} ∈ B. There is p ∈ ℛ^{∗} such that for each i, pq_{i} ∈ ℛ. Since B is an ℛ-basis, ∑ pq_{i}x_{i} = 0 implies pq_{i} = 0 for all i, and hence q_{i} = 0 for all i. The “moreover” part follows from the fact that the canonical injection of 𝒜 into ℬ is a ring homomorphism.

Define the Lie bracket in ℬ as usual: for v and w belonging to ℬ, [v, w] = vw − wv. There is a connection between the Lie brackets in 𝔤, 𝒜, and ℬ: let X, Y ∈ 𝔤, with x = π(X), y = π(Y), and let p, q ∈ ℛ, r, s ∈ ℛ^{∗}. Then

(3.3.1) [(px/r), (qy/s)] = [(p/r)x, (q/s)y] = (pq/rs)[x, y] = (pq/rs)π([X, Y]).

This allows us to associate Lie subalgebras of 𝔤 to finite dimensional Lie algebras in ℬ.

Lemma 3.3.2. Let 𝔨 be a subalgebra of 𝔤, and let Y_{1}, Y_{2}, ..., Y_{s} be a basis of a subspace of 𝔨 complementary to 𝔨 ∩ 𝔷. Let 𝔲(𝔷)𝔨 be the 𝔲(𝔷)-module generated by 𝔨, and set

𝔨̂ = σ(π(𝔲(𝔷)𝔨) + ℛ).

Then {i, y_{1}, y_{2}, ..., y_{s}} is a K-basis for 𝔨̂. Moreover, 𝔨̂ is a Lie algebra over K, and is a Lie subalgebra of 𝔤̂ = σ(π(𝔲(𝔷)𝔤) + ℛ). Proof. We need only show that {i, y_{1}, y_{2}, …, y_{s}} is an ℛ-basis for π(𝔲(𝔷)𝔨) + ℛ. Let v + p ∈ π(𝔲(𝔷)𝔨) + ℛ, with v = π(V), V ∈ 𝔲(𝔷)𝔨, p ∈ ℛ. We have V = ∑_{j=1}^{s} P_{j}Y_{j} +P_{0} , where P_{j} ∈ 𝔲(𝔷), 0 ≤ j ≤ s, so v+p = ∑_{j=1}^{s} p_{j}y_{j} +p_{0}+p ∈ ℛ -span{i, y_{1}, y_{2}, ..., y_{s}}. Now suppose that p_{0}, p_{1}, …, p_{s} are elements of ℛ for which ∑_{j=1}^{s} p_{j}y_{j} + p_{0} = 0. Then we have P_{0}, P_{1}, …, P_{s} ∈ 𝔲(𝔷) for which ∑_{j=1}^{s} Y_{j}P_{j} + P_{0} ∈ 𝒥. By Lemma 3.1.3, we have P_{j} ∈ 𝒥, hence p_{j} = 0 holds for each j.

That 𝔨̂ is a Lie algebra over K, and is a Lie subalgebra of 𝔤̂, follows from the equations (3.3.1).

Let 𝔲(𝔤̂) be the enveloping algebra of 𝔤̂ over K; denote its identity element by Î.

Proposition 3.3.3. The K-algebra ℬ is a canonical image of 𝔲(𝔤̂) with kernel 𝔲(𝔤̂)(i − Î) , and has as a K-basis the set {x^{α}y^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}}. Proof. By Proposition 2.1.1 (applied to the Lie subalgebra Ki of 𝔤̂), the set

{x^{α}y^{β}(i − Î)^{μ} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, μ ∈ Z_{+}}

is a K-basis of 𝔲(𝔤̂), and the ideal 𝔲(𝔤̂)(i − Î) is spanned by the monomials x^{α}y^{β}(i − Î)^{μ}, μ > 0. The identity map from 𝔤̂ to itself extends to a canonical homomorphism h from 𝔲(𝔤̂) into ℬ. The image of the above basis of 𝔲(𝔤̂) is the set {x^{α}y^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}} and (since h(Î) = i) the monomials x^{α}y^{β}(i − Î)^{μ}, μ > 0 are obviously contained in the kernel of h. On the other hand, it is immediate from Lemma 3.2.2 that {x^{α}y^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}} is an ℛ-basis for 𝒜, and hence by Lemma 3.3.2, it’s image in ℬ is a K-basis of ℬ. Hence the restriction of h to K-span{x^{α}y^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}} in 𝔲(𝔤̂) is a bijection. The proposition follows.

3.4 Let (𝔤, 𝔥, f) be a triple, with 𝔷 an ℐ-central ideal, and with 𝒜, ℬ and 𝔤̂ as above. A triple (𝔤̂, 𝔥̂, f̂) is associated with this data as follows. Let 𝔥̂ be the Lie-subalgebra of 𝔤̂ associated with 𝔥 via Lemma 3.3.2. An argument similar to that of Proposition 3.3.3 shows that h(𝔲(𝔥̂)) is canonically isomorphic with σ(ℛπ(𝔲(𝔥))), and has the monomials y^{β}, β ∈ Z_{+}^{s} as a K-basis. Let B_{𝔥} = {T_{1}, T_{2}, …, T_{u}, Y_{1}, Y_{2}, …, Y_{s}} be the basis of 𝔥 chosen in Section 3.1. By Lemma 3.3.2, a basis B𝔥̂ for 𝔥̂ over K is given by {i, y_{1}, …, y_{s}}. Now f drops to a homomorphism f̄ of π(𝔲(𝔥)), and π(𝔲(𝔥)) contains B𝔥̂. Let f̂ be the Lie algebra homomorphism from 𝔥̂ onto K whose restriction to B𝔥̂ coincides with f̄, and extend f̂ to 𝔲(𝔥̂) in the canonical way. The triple (𝔤̂, 𝔥̂, f̂) will be referred to as the triple associated with (𝔤, 𝔥, f) and 𝔷. Let ℐ̂ = ℐ(𝔤̂, 𝔥̂, f̂) be the left ideal in 𝔲(𝔤̂) generated by the kernel of f̂.

Lemma 3.4.1. Given any triple (𝔤, 𝔥, f) with associated triple (𝔤̂, 𝔥̂, f̂), we have h(ℐ̂) = σ(π(ℐ)) and h(ℐ̂) ∩ 𝒜 = π(ℐ).

Proof. By Proposition 2.1.1, ℐ̂ has as a K-basis the set {x^{α}u^{β}(i − Î)^{μ} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, μ ∈ Z_{+}}, and this set is mapped by h onto the set {x^{α}u^{β} : α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}}. By Lemma 3.2.2, this set is an ℛ-basis for π(ℐ), and so by Lemma 3.3.1, it is a K-basis for σ(π(ℐ)). This proves the first part of the lemma.

Let v/z = w belong to h(ℐ̂) ∩ 𝒜 with v ∈ π(ℐ), z ∈ ℛ^{∗}, and w ∈ 𝒜. Taking preimages gives V = W Z ∈ ℐ with Z ∉ ℐ. By Proposition 3.1.2 we have W ∈ ℐ.

Proposition 3.4.2. Let (𝔤, 𝔥, f) be a triple with associated triple (𝔤̂, 𝔥̂, f̂). Let 𝔨 be a subalgebra of 𝔤 with 𝔨̂ the corresponding subalgebra of 𝔤̂. Then h(c(𝔨̂, ℐ̂)) = σ(π(c(𝔨, ℐ))).

Proof. From the definitions of 𝔨̂ and σ, and formula (3.3.1) above, in order to show that σ(π(c(𝔨, ℐ))) ⊂ h(c(𝔨̂, ℐ̂)), it is enough to show that [π(c(𝔨, ℐ)), π(𝔨)] ⊂ π(ℐ). But this is obvious since [c(𝔨, ℐ), 𝔨] ⊂ ℐ.

On the other hand, let w = ∑_{α,β} q_{αβ}x^{α}y^{β} be an element of ℬ that belongs to h(c(𝔨̂, ℐ̂)). Choose some p ∈ ℛ^{∗} such that x = pw ∈ 𝒜 , and let X = P W ∈ 𝔲(𝔤) be a pre-image of pw in 𝔲(𝔤). I claim that X ∈ c(𝔨, ℐ). Let Y ∈ 𝔨. Then y = π(Y) ∈ 𝔨̂, and it follows that [x, y] ∈ h(ℐ̂) ∩ 𝒜. By Lemma 3.4.1, this means [x, y] ∈ π(ℐ), whence [X, Y] ∈ ℐ and the claim is proved. Now by definition of σ we have w = x/p ∈ σ(π(c(𝔨, ℐ))).

Corollary 3.4.3. Let (𝔤, 𝔥, f) be a triple with associated triple (𝔤̂, 𝔥̂, f̂). Then c(𝔥, ℐ)/ℐ is commutative if and only if c(𝔥̂, ℐ̂)/ℐ̂ is commutative. Proof. Clearly c(𝔥, ℐ)/ℐ is commutative if and only if π(c(𝔥, ℐ))/π(ℐ) is commutative, so we may as well assume that 𝒥 = 0. Set ℳ = c(𝔥, ℐ), ℳ̄ = ℳ/ℐ, and let ℛ̄ be the image of ℛ in ℳ̄. Then ℛ̄ allows an arithmetic of fractions with respect to ℳ̄, and ℳ̄ℛ̄ is commutative if and only if ℳ̄ is. Now the map ϕ : ℳ̄ℛ̄ → σ(ℳ)/σ(ℐ) defined by ϕ(m̄/z̄) = m/z + σ(ℐ) is easily seen to be an isomorphism. Thus we have that ℳ/ℐ is commutative if and only if σ(ℳ)/σ(ℐ) is commutative.

Now set ℳ̂ = c(𝔥̂, ℐ̂). Since both ℳ̂ and ℐ̂ contain 𝔲(𝔤̂)(i−Î), by Lemma 3.4.1 and Proposition 3.4.2,

σ(ℳ)/σ(ℐ) = h(ℳ̂)/h(ℐ̂) ≃ ℳ̂/ℐ̂.

The Corollary follows.

Finally we make good on our claim that if 𝔤 is nilpotent, then (𝔤̂, 𝔥̂, f̂) admits a Kirillov structure. Actually this is not necessarily true for any associated triple; indeed if we choose 𝔷 = (0), then the associated triple is the same as its progenitor (𝔤, 𝔥, f). The point is that 𝔷 must be sufficiently large.

Proposition 3.4.4. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent, and let 𝔷 be maximal among all ℐ-central ideals in 𝔤. If 𝔷 ≠ 𝔤, then (𝔤̂, 𝔥̂, f̂) admits a Kirillov structure.

Proof. Since 𝔤 is nilpotent and 𝔷 ≠ 𝔤, c(𝔤, 𝔷)∩𝔤 ≠ 𝔷; choose Y ∈ c(𝔤, 𝔷)∩𝔤 but not in 𝔷. Then y ∈ c(𝔤̂, Ki) ∩ 𝔤̂, so the dimension of 𝔤̂/(c(y, ℐ̂) ∩ 𝔤̂) over K is no greater than one. By maximality of 𝔷, we have 𝔤 ⊈ c(Y, ℐ), so if X ∈ 𝔤\c(Y, ℐ), then x ∈ 𝔤̂\c(y, ℐ̂). Thus (𝔤̂, 𝔥̂, f̂) admits the Kirillov structure (Ki, y).

Note that if 𝔷 = 𝔤, then for any X ∈ 𝔤, one has [X, 𝔤] ⊂ 𝔥 ∩ ker(f), so X ∈ 𝔤(ℓ) holds for all ℓ ∈ L_{f} and d(𝔤, 𝔥, f) = 0. The converse is also true: if d(𝔤, 𝔥, f) = 0 then 𝔤 is ℐ-central.

3.5 Let (𝔤, 𝔥, f) be a triple with the associated triple (𝔤̂, 𝔥̂, f̂), let S(𝔤) denote the symmetric algebra of 𝔤, and let ω : S(𝔤) → 𝔲(𝔤) be the symmetrization mapping. We have a natural surjective homomorphism π̃ : S(𝔤) → S(π(𝔤)), and in view of the inclusion π(𝔤) ↪ 𝔤̂, we have a natural injection ι : S(π(𝔤)) → S(𝔤̂).

Whenever convenient we regard S(𝔤) as the algebra of polynomial functions on 𝔤^{∗} in the usual way. Set

L_{f_{0}} = {ℓ ∈ 𝔤^{∗} : ℓ|_{𝔥_{0}} = f_{0}|_{𝔥_{0}}},

and let J be the ideal in S(𝔤) of elements which vanish on L_{f_{0}} . Then J is the ideal generated by the elements in S(𝔥_{0}) for which f_{0} is a zero, and S(𝔤)/J is canonically isomorphic with the algebra k[L_{f_{0}}] of k-valued polynomial functions on L_{f_{0}} . We denote this algebra by A and the canonical map from S(𝔤) onto A by Π. For P ∈ S(𝔤), denote Π(P) by P̄. Set R = Π(S(𝔷)) and form the fraction ring B = A_{R}. We have a canonical injective homomorphism from A into B, and we thereby regard A as a subset of B. Let p_{𝔷} : 𝔤^{∗} → 𝔷^{∗} be the restriction mapping. The field of fractions K̄ = {Z̄_{1}/Z̄_{2}, Z̄_{1} ∈ R, Z̄_{2} ∈ R^{∗}} in B can be regarded as the field k[p(L_{f_{0}})] of k-valued rational functions on p(L_{f_{0}}). On the other hand, the map Z → π(ω(Z)) from S(𝔷) to ℛ is surjective with kernel S(𝔷) ∩ J. Hence R ≃ ℛ and K̄ ≃ K in a canonical way, and B is an algebra over the field K. More explicitly, let P ∈ S(𝔤) and write

P = ∑_{α,β,μ,ν} a_{αβμν}X^{α}Y^{β}Z^{μ}T^{ν}.

Then

P̄ = ∑_{α,β} (∑_{μ,ν} a_{αβμν}f(T)^{ν} Z̄^{μ}) X̄^{α}Ȳ^{β}.

This shows that the monomials X̄^{α}Ȳ^{β}, α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, span A as an Rmodule. In the same way as for the corresponding objects in 𝔲(𝔤), we see that in fact the monomials X̄^{α}Ȳ^{β}, α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s} constitute an R-basis for A, and hence a K basis for B.

In 𝔤̂^{∗}, set

L_{i^{∗}} = {ℓ ∈ 𝔤̂^{∗} : ℓ(i) = 1}.

We regard S(𝔤̂)/S(𝔤̂)(i−Î) as the algebra K[L_{i^{∗}}] of K-valued polynomial functions on L_{i^{∗}}, and we denote the image of an element p ∈ S(𝔤̂) in S(𝔤̂)/S(𝔤̂)(i−Î) by p̄. Given p ∈ S(𝔤̂), write

p = ∑_{α,β,μ} q_{αβμ}x^{α}y^{β}i^{μ};

then

p̄ = ∑_{α,β} (∑_{μ} q_{αβμ}) x̄^{α}ȳ^{β}.

In the same way as Proposition 3.3.3, we find that the monomials x̄^{α}ȳ^{β}, α ∈ Z_{+}^{r}, β ∈ Z_{+}^{s}, constitute a K-basis for S(𝔤̂)/S(𝔤̂)(i−Î). Thus we have an obvious isomorphism between B and S(𝔤̂)/S(𝔤̂)(i − Î).

Now for P ∈ S(𝔤), let P̃ be the restriction of P to L_{f}. In a similar way we find that k[L_{f}] has as a k-basis the monomials X̃^{α}Z̃^{μ}, α ∈ Z_{+}^{r}, μ ∈ Z_{+}^{t}, and that it has the monomials X̃^{α}, α ∈ Z_{+}^{r} as an R-basis. Also the monomials x̃^{α}, α ∈ Z_{+}^{r} constitute a K-basis for K[Lf̂]. Note that p_{𝔷}(L_{f}) = p_{𝔷}(L_{f_{0}}) and so for all Z ∈ S(𝔷), Z̃ = Z̄. We have a mapping

∑_{α,μ} a_{αμ}Z̃^{μ}X̃^{α} ⟶ ∑_{α} (∑_{μ} a_{αμ}z^{μ}) x̃^{α}

which is an injective homomorphism of k[L_{f}] onto the subring of R-valued elements of K[Lf̂]. We sum up the preceding in the following commutative diagram where the vertical arrows are surjective restriction mappings, and the horozontal arrows are monomorphisms.

(3.5.1)  \begin{matrix}S(𝔤) \\ π̃ & ↓ \\ S(π(𝔤)) & ⟶^{ι} & S(𝔤̂) \\ ^{η}↓ & & η̂↓ \\ k[L_{f_{0}}] & ⟶^{ι_{0}} & K[L_{i^{∗}}] \\ ^{θ}↓ & & θ̂↓ \\ k[L_{f}] & ⟶^{ϵ} & K[Lf̂] \end{matrix}

3.6 Let 𝔤 be a nilpotent Lie algebra over k and let 𝔥 be a subalgebra. It is desirable to describe in explicit terms a scheme for layering the exp(ad^{∗}(𝔥))- orbits in 𝔤^{∗}. For simplicity of notation we set exp(ad^{∗}(Y)) = α(Y), Y ∈ 𝔥. Let (0) = 𝔤_{0} ⊂ 𝔤_{1} ⊂ ⋯ ⊂ 𝔤_{n} = 𝔤 be a sequence of α(𝔥)-modules in 𝔤 with dim(𝔤_{j}) = j, 1 ≤ j ≤ n, and (0) = 𝔥_{0} ⊂ 𝔥_{1} ⊂ ⋯ ⊂ 𝔥_{p} = 𝔥 a Jordan-Hölder sequence of ideals in 𝔥. Choose X_{j} ∈ 𝔤_{j}\𝔤_{j−1} and Y_{j} ∈ 𝔥_{j}\𝔥_{j−1} .

Fix ℓ ∈ 𝔤^{∗}. Our scheme involves the definition of a pair (ϕ(ℓ), ψ(ℓ)) of index sequences associated with ℓ and the α(𝔥)-module sequences chosen above. If the α(𝔥)-orbit of ℓ is just {ℓ}, then set ϕ(ℓ) = Ø and ψ(ℓ) = Ø. Otherwise set ϕ_{1}(ℓ) = min{j : 𝔤_{j} ⊈ 𝔥^{ℓ}}, and ψ_{1}(ℓ) = min{j : 𝔥_{j} ⊈ (𝔤_{ϕ_{1}(ℓ)})^{ℓ}}. Define the codimension one ideal 𝔥^{1}(ℓ) by 𝔥^{1}(ℓ) = 𝔥 ∩ 𝔤_{ϕ_{1}(ℓ)}^{ℓ}; then dim(α(𝔥)(ℓ)) = 1 if and only if 𝔥^{1}(ℓ) = 𝔥(ℓ) = 𝔥 ∩ 𝔤^{ℓ}, and if dim(α(𝔥)(ℓ)) = 1, then α(𝔥)(ℓ) = {α(tY_{ψ_{1}(ℓ)})(ℓ) : t ∈ k}, and we are done. If dim(α(𝔥)(ℓ)) > 1, then set

ϕ_{2}(ℓ) = min{j : 𝔤_{j} ⊈ (𝔥^{1}(ℓ))^{ℓ}}

and

ψ_{2}(ℓ) = min{j : 𝔥_{j} ∩ 𝔤_{ϕ_{1}(ℓ)}^{ℓ} ⊈ (𝔤_{ϕ_{2}(ℓ)})^{ℓ}}.

Note that ϕ_{1}(ℓ) < ϕ_{2}(ℓ). Define the subalgebra 𝔥^{2}(ℓ) by 𝔥^{2}(ℓ) = 𝔥 ∩ 𝔤_{ϕ_{2}(ℓ)}^{ℓ}; and we have dim(α(𝔥)(ℓ)) = 2 if and only if 𝔥^{2}(ℓ) = 𝔥(ℓ). If dim(α(𝔥)(ℓ)) = 2, then α(𝔥)(ℓ) = {α(t_{1}Y_{ψ_{1}(ℓ)})α(t_{2}Y_{ψ_{2}(ℓ)}(ℓ))(ℓ) : t_{1}, t_{2} ∈ k}. Here Y_{ψ_{2}(ℓ)}(ℓ) is the projection of Y_{ψ_{2}(ℓ)} into 𝔥^{1}(ℓ) parallel to Y_{ψ_{1}(ℓ)} ; explicitly,

Y_{ψ_{2}(ℓ)}(ℓ) = Y_{ψ_{2}(ℓ)} − \frac{ℓ([Y_{ψ_{2}(ℓ)}, X_{ϕ_{1}(ℓ)}])}{ℓ([Y_{ψ_{1}(ℓ)}, X_{ϕ_{1}(ℓ)}])}Y_{ψ_{1}(ℓ)}.

If dim(α(𝔥)(ℓ)) = 2, then we stop here; otherwise continue the process until the following objects are obtained.

(a) Index sequences ϕ(ℓ) and ψ(ℓ) where ♯(ϕ(ℓ)) = ♯(ψ(ℓ)) = δ(ℓ) and ϕ(ℓ) is increasing.

(b) Subalgebras 𝔥(ℓ) = 𝔥^{δ}(ℓ) ⊂ ⋯ ⊂ 𝔥^{1}(ℓ) ⊂ 𝔥, where 𝔥^{k}(ℓ) is codimension k in 𝔥. (Here δ = δ(ℓ).)

(c) Elements Y_{ψ_{1}(ℓ)} = Y_{ψ_{1}(ℓ)}(ℓ), Y_{ψ_{2}(ℓ)}(ℓ), …, Y_{ψ_{δ}(ℓ)}(ℓ) in 𝔥 with Y_{ψ_{k}(ℓ)}(ℓ) ∈ 𝔥^{k−1}(ℓ)\𝔥^{k}(ℓ), and

α(𝔥)(ℓ) = {α(t_{1}Y_{ψ_{1}(ℓ)}))α(t_{2}Y_{ψ_{2}(ℓ)}(ℓ)) ⋯ α(t_{δ}Y_{ψ_{δ}(ℓ)}(ℓ))(ℓ) : (t_{1}, t_{2}, …, t_{δ}) ∈ k^{δ}}.

Let Φ = {(ϕ(ℓ), ψ(ℓ)) : ℓ ∈ 𝔤^{∗}} and for each (ϕ, ψ) ∈ Φ, set

Ω_{ϕ,ψ} = {ℓ ∈ 𝔤^{∗} : (ϕ(ℓ), ψ(ℓ)) = (ϕ, ψ)}.

Obviously 𝒫 = {Ω_{ϕ,ψ} : (ϕ, ψ) ∈ Φ} is a partition of 𝔤^{∗} and it is easily seen that each element of 𝒫 is α(𝔥)-invariant.

We shall now define an ordering ≺ on all pairs (ϕ, ψ) of index sequences with 1 ≤ ϕ_{1} < ϕ_{2} < ⋯ < ϕ_{δ} ≤ n and 1 ≤ ψ_{j} ≤ p. For two pairs (ϕ, ψ) and (ϕ′, ψ′) of index sequences as above, we say that (ϕ, ψ) ≺ (ϕ′, ψ′) if one of the following conditions hold:

(i) ♯(ϕ) > ♯(ϕ′), or (ii) if ♯(ϕ) = ♯(ϕ′) and k = min{j : (ϕ_{j}, ψ_{j}) ≠ (ϕ′_{j}, ψ′_{j})} , then ϕ_{k} < ϕ′_{k}, or (iii) if ♯(ϕ) = ♯(ϕ′) and k = min{j : (ϕ_{j}, ψ_{j}) ≠ (ϕ′_{j}, ψ′_{j})} , then ϕ_{k} = ϕ′_{k} and ψ_{k} < ψ′_{k}.

Next we define polynomial functions Q_{ϕ,ψ} on 𝔤^{∗} that determine the layers Ω ∈ 𝒫. Fix an any index sequence pair (ϕ, ψ) with δ = ♯(ϕ), and for each k, 1 ≤ k ≤ δ, set

Q_{ϕ,ψ}^{k}(ℓ) = det [ℓ ([Y_{ψ_{r}}, X_{ϕ_{s}}])]_{1≤r,s≤k},

and define

Q_{ϕ,ψ}(ℓ) = Q_{ϕ,ψ}^{1}(ℓ)Q_{ϕ,ψ}^{2}(ℓ) ⋯ Q_{ϕ,ψ}^{δ}(ℓ).

Lemma 3.6.1. Let (ϕ, ψ) be an index sequence pair with δ = ♯(ϕ). For any ℓ ∈ 𝔤^{∗}, if Q_{ϕ,ψ}(ℓ) ≠ 0, then δ(ℓ) ≥ δ. Proof. Suppose that Q_{ϕ,ψ}(ℓ) ≠ 0. Then the scheme above can be used to define δ elements

Y_{ψ_{1}}(ℓ), Y_{ψ_{2}}(ℓ), …, Y_{ψ_{δ}}(ℓ)

in 𝔥 with the property that for each 1 ≤ k ≤ δ, ℓ([Y_{ψ_{k}}(ℓ), X_{ϕ_{j}}]) = 0 if j < k and ℓ([Y_{ψ_{k}}(ℓ), X_{ϕ_{k}}]) ≠ 0. It follows that δ(ℓ) = dim(𝔥/𝔥(ℓ)) ≥ δ.

Lemma 3.6.2. For any ℓ ∈ 𝔤^{∗}, we have

(ϕ(ℓ), ψ(ℓ)) = min{(ϕ, ψ) : Q_{ϕ,ψ}(ℓ) ≠ 0}.

Proof. Suppose that (ϕ, ψ) ≺ (ϕ(ℓ), ψ(ℓ)); we show that Q_{ϕ,ψ}(ℓ) = 0. Set δ = ♯(ϕ). If δ > δ(ℓ), then by Lemma 3.6.1, Q_{ϕ,ψ}(ℓ) = 0. Suppose that δ = δ(ℓ). Set k = min{j : (ϕ_{j}, ψ_{j}) ≠ (ϕ_{j}(ℓ), ψ_{j}(ℓ))}. By definition of ≺, we have either ϕ_{k} < ϕ_{k}(ℓ), or ϕ_{k} = ϕ_{k}(ℓ) and ψ_{k} < ψ_{k}(ℓ). Let Y_{ψ_{k}}(ℓ) be the projection of Y_{ψ_{k}} into 𝔥^{k−1}(ℓ) parallel to k-span{Y_{ψ_{1}}(ℓ), Y_{ψ_{2}}(ℓ), …, Y_{ψ_{k−1}}(ℓ)}. If ϕ_{k} < ϕ_{k}(ℓ), then X_{ϕ_{k}} ∈ (𝔥^{k−1}(ℓ))^{ℓ}, so ℓ([Y_{ψ_{k}}(ℓ), X_{ϕ_{k}}]) = 0. Otherwise ϕ_{k} = ϕ_{k}(ℓ) and ψ_{k} < ψ_{k}(ℓ), whence Y_{ψ_{k}}(ℓ) ∈ (𝔤_{ϕ_{k}})^{ℓ} so again ℓ([Y_{ψ_{k}}(ℓ), X_{ϕ_{k}}]) = 0. Now

Q_{ϕ,ψ}^{k}(ℓ) = ℓ([Y_{ψ_{1}}(ℓ), X_{ϕ_{1}}(ℓ)])ℓ([Y_{ψ_{2}}(ℓ), X_{ϕ_{2}}(ℓ)]) ⋯ ℓ([Y_{ψ_{k}}(ℓ), X_{ϕ_{k}}(ℓ)]),

which shows that Q_{ϕ,ψ}(ℓ) = 0. The Lemma follows.

The following description of the layers is immediate.

Proposition 3.6.3. With all the data retained from the above, fix an index sequence pair (ϕ, ψ) ∈ Φ. Then

Ω_{ϕ,ψ} = {ℓ ∈ 𝔤^{∗} : Q_{ϕ′,ψ′}(ℓ) = 0, for all (ϕ′, ψ′) with (ϕ′, ψ′) ≺ (ϕ, ψ) and Q_{ϕ,ψ}(ℓ) ≠ 0}.

3.7 In this subsection we show that generic orbital dimensions in L_{f} are preserved when passing to Lf̂. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent, let 𝔷 be an ℐ-central ideal in 𝔤, and let (𝔤̂, 𝔥̂, f̂) be the associated triple. Choose a Jordan-Hölder sequence (0) = 𝔤_{0} ⊂ 𝔤_{1} ⊂ ⋯ ⊂ 𝔤_{n} = 𝔤 with the following properties.

(a) For some a, 𝔤_{a} = 𝔷.

(b) For some u, 𝔤_{u} = 𝔥 ∩ 𝔷 (recall that 𝔥 ∩ 𝔷 must be an ideal in 𝔤.) There are indices 1 ≤ j_{1} < j_{2} < ⋯ < j_{p} ≤ n for which 𝔥∩𝔤_{j_{k−1}} ⊊ 𝔥∩𝔤_{j_{k}}

and for which (0) = 𝔥_{0} ⊂ 𝔥_{1} ⊂ ⋯ ⊂ 𝔥_{p} = 𝔥 is a Jordan Holder sequence for 𝔥

Choose X′_{j} ∈ 𝔤_{j}\𝔤_{j−1}, 1 ≤ j ≤ n such that X′_{j_{k}} ∈ 𝔥_{k}\𝔥_{k−1}, 1 ≤ k ≤ p. Set Y′_{k} = X′_{j_{k}}, 1 ≤ k ≤ p. Denoting this basis by B, the first u elements of B are a basis for 𝔷 ∩ 𝔥, and the first a elements of B are a basis for 𝔷. With reordering if necessary, B can be written

B = {T_{1}, T_{2}, …, T_{u}, Z_{1}, Z_{2}, …, Z_{t}, Y_{1}, Y_{2}, …, Y_{s}, X_{1}, X_{2}, …, X_{r}}

as in Section 3.1 (a = t+u). There is a corresponding Jordan-Hölder sequence

(0) = 𝔤̂_{0} ⊂ 𝔤̂_{1} ⊂ 𝔤̂_{2} ⊂ ⋯ ⊂ 𝔤̂_{m} = 𝔤̂

where 𝔤̂_{1} = 𝔷̂ = Ki, 𝔤̂_{2} = (𝔤_{a+1})ˆ , etc., with a compatible basis {x′_{l} : 1 ≤ l ≤ m} that can be written (again with reodering) as

{i, y_{1}, y_{2}, …, y_{s}, x_{1}, x_{2}, …, x_{r}}

(where x′_{1} = i). Let 𝒫_{𝔥} be the layering for the α(𝔥)-orbits as defined in Section 3.6 relative to the sequences of α(𝔥)-modules for 𝔤 and 𝔥 chosen above, and in the same way let 𝒫𝔥̂ be the layering for the α(𝔥̂)-orbits in 𝔤̂^{∗} relative to the corresponding sequences of α(𝔥̂)-modules for 𝔤̂ and 𝔥̂. Now given any index sequence pair (ϕ, ψ) with the property that ϕ_{1} > a and ψ_{j} > u, 1 ≤ j ≤ p, define (ϕ̂, ψ̂) by ϕ̂_{j} = ϕ_{j} − a + 1 and ψ̂_{j} = ψ_{j} − u + 1. This obviously defines a bijection between the set of all index sequence pairs (ϕ, ψ) relative to the chosen α(𝔥)-modules with ϕ_{1} > a and ψ_{j} > u, 1 ≤ j ≤ p, and the set of all index sequence pairs relative to the α(𝔥̂)-modules whose values are greater than one. We denote elements of S(𝔤̂) by lower case letters.

Lemma 3.7.1. Let (ϕ, ψ) be an index sequence pair relative the above α(𝔥)- modules with the property that ϕ_{1} > a and ψ_{j} > u, 1 ≤ j ≤ p. Then

η̂(q_{ϕ̂,ψ̂}) = ι_{0}(η(π̃(Q_{ϕ,ψ}))).

Proof. It is enough to show that

η̂(q_{ϕ̂,ψ̂}^{k}) = ι_{0}(η(π̃(Q_{ϕ,ψ}^{k})))

holds for each 1 ≤ k ≤ δ = ♯(ϕ). We have

Q_{ϕ,ψ}^{k} = ∑_{σ∈S_{k}} sign(σ)[Y′_{ψ_{1}}, X′_{ϕ_{σ(1)}}][Y′_{ψ_{2}}, X′_{ϕ_{σ(2)}}] ⋯ [Y′_{ψ_{k}}, X′_{ϕ_{σ(k)}}]

and so

ιπ̃(Q_{ϕ,ψ}^{k})

= ι ( ∑_{σ∈S_{k}} sign(σ)[π(Y′_{ψ_{1}}), π(X′_{ϕ_{σ(1)}})][π(Y′_{ψ_{2}}), π(X′_{ϕ_{σ(2)}})] ⋯ [π(Y′_{ψ_{k}}), π(X′_{ϕ_{σ(k)}})])

= ι ( ∑_{σ∈S_{k}} sign(σ)[y′ ; ψ̂_{1}, x′ ; ϕ̂_{σ(1)}][y′ ; ψ̂_{2}, x′ ; ϕ̂_{σ(2)}] ⋯ [y′ ; ψ̂_{k}, x′ ; ϕ̂_{σ(k)}])

= q_{ϕ̂,ψ̂}^{k}.

Now by commutativity of the diagram 3.5.1, we have

ι_{0}(η(π̃(Q_{ϕ,ψ}^{k}))) = η̂(ι(π̃(Q_{ϕ,ψ}^{k}))) = η̂(q_{ϕ̂,ψ̂}^{k}).

A version of the above lemma also holds for the index sequence pairs relative to the α(𝔤)-action and the α(𝔤̂)-action. We omit the proof.

Lemma 3.7.2. Let (ϕ, ψ) be an index sequence pair relative the α(𝔤)-modules chosen above with the property that ϕ_{1} > a and ψ_{j} > u, 1 ≤ j ≤ p. Then η̂(q_{ϕ̂,ψ̂}) = ι_{0}(η(π̃(Q_{ϕ,ψ}))).

Proposition 3.7.3. Let (𝔤, 𝔥, f) be a triple with associated triple (𝔤̂, 𝔥̂, f̂). Then d(𝔤, 𝔥, f) = d(𝔤̂, 𝔥̂, f̂) and δ(𝔤, 𝔥, f) = δ(𝔤̂, 𝔥̂, f̂).

Proof. We have d(𝔤, 𝔥, f) = ♯(ϕ_{0}), where (ϕ_{0}, ψ_{0}) is the minimum index sequence pair (ϕ, ψ) for which Ω_{ϕ,ψ} ∩ L_{f} ≠ Ø. By Proposition 3.6.3, this is equal to the minimum index sequence pair (ϕ, ψ) for which θ(η(π̃(Q_{ϕ,ψ}))) ≠ 0. A similar statement holds for d(𝔤̂, 𝔥̂, f̂). Applying Lemma 3.7.2 and diagram 3.5.1, we see that the minimum index sequence pair that determines d(𝔤̂, 𝔥̂, f̂) is (ϕ̂_{0}, ψ̂_{0}). Hence d(𝔤, 𝔥, f) = d(𝔤̂, 𝔥̂, f̂). A parallel argument shows that δ(𝔤, 𝔥, f) = δ(𝔤̂, 𝔥̂, f̂).

Finally we examine the case where (𝔤, 𝔥, f) admits a Kirillov structure.

Proposition 3.7.4. Suppose that (𝔤, 𝔥, f) is a triple with 𝔤 nilpotent, and that admits a Kirillov structure (𝔷, Y). Set 𝔤^{Y} = c(Y, ℐ)∩𝔤 and 𝔥^{Y} = c(Y, ℐ)∩𝔥. Then d(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) = d(𝔤, 𝔥, f) − 2 and δ(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) is equal either to δ(𝔤, 𝔥, f) or δ(𝔤, 𝔥, f) − 1.

Proof. Choose a Jordan-Hölder sequence for 𝔤 that includes 𝔷, 𝔷 + kY , and 𝔤^{Y} , and let a = dim(𝔷). Apply the constructions of Section 3.6 with 𝔥 = 𝔤, and we find that for any ℓ ∈ L_{f}, ϕ_{1}(ℓ) > a, and that ϕ_{1}(ℓ) = a + 1 if and only if ℓ([X, Y]) ≠ 0, and in this case ψ_{1}(ℓ) = n, and 𝔤^{Y} = 𝔥^{1}(ℓ). It follows that if Ω_{ϕ^{0},ψ^{0}} is the generic layer in (𝔤^{Y})^{∗} for L_{f|_{𝔥^{Y}}} , then the generic layer in 𝔤^{∗} for L_{f} is Ω_{ϕ,ψ}, where ϕ_{1} = a + 1, ϕ_{j} = ϕ_{j−1}^{0} and ψ_{1} = n, ψ_{j} = ψ_{j−1}^{0}, j > 1. This shows that d(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) = d(𝔤, 𝔥, f) − 2.

Turning to the α(𝔥)-action, a similar analysis shows that if 𝔥 ≠ 𝔥^{Y} , then ϕ_{1}(ℓ) = a + 1 and ψ_{1}(ℓ) = n hold for generic ℓ ∈ L_{f}, whence δ(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) = δ(𝔤, 𝔥, f) − 1. On the other hand, if 𝔥 = 𝔥^{Y} , then L_{f|_{𝔥^{Y}}} is just the set of restrictions of elements in L_{f}. For any ℓ, if ℓ_{0} denotes its restriction to 𝔤^{Y} , then 𝔤^{ℓ} is codimension one in (𝔤^{Y})^{ℓ_{0}} , and since 𝔥(ℓ) = 𝔥 ∩ 𝔤^{ℓ} (and similarly for ℓ_{0} ), the two possibilities for δ(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) are evident.

4. Non-commutativity of 𝒟(𝔤, 𝔥, f)

4.1 Let 𝔤 be a nilpotent Lie algebra over a field k of characteristic zero. Let 0 = 𝔲_{−1}(𝔤) ⊂ 𝔲_{0}(𝔤) ⊂ 𝔲_{1}(𝔤) ⊂ ⋯ ⊂ 𝔲(𝔤) be the standard filtration of 𝔲(𝔤) and fix a strong Malcev basis {X_{1}, X_{2}, …, X_{n}} for 𝔤. Let 𝒜_{m} = {(a_{1}, a_{2}, …, a_{m}) : a_{i} ∈ {1, 2, …, n}, 1 ≤ i ≤ m, a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{m}}. For each a ∈ 𝒜_{m}, set X^{a} = X_{a_{m}}X_{a_{m−1}} … X_{a_{1}} . Order 𝒜_{m} as follows: a < a′ if a_{k} < a′_{k}, where k = min{j : a_{j} ≠ a′_{j}}. Set

𝔲_{m}^{a}(𝔤) = 𝔲_{m−1}(𝔤) ⊕ span{X^{a′} : a′ ≤ a}.

We have a total ordering on the set of pairs P = {(m, a) : m ≥ 0, a ∈ 𝒜_{m}} that has the following property. For p = (m, a) ∈ P denote 𝔲_{m}^{a}(𝔤) by 𝔲_{p}. If Y ∈ 𝔤, then [Y, 𝔲_{p}] ⊂ 𝔲_{p−1} , where p − 1 is the predecessor of p.

Proposition 4.1.1. Let (𝔤, 𝔥, f) be a triple that admits a Kirillov structure (𝔷, Y). Set 𝔤^{Y} = c(Y, ℐ) ∩ 𝔤 and 𝔥^{Y} = c(Y, ℐ) ∩ 𝔥. Assume that 𝔥 ≠ 𝔥^{Y} . Then

(i) there is a monomorphism of 𝒟(𝔤, 𝔥, f) into 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}), and (ii) 𝒟(𝔤, 𝔥, f) is commutative if and only if 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) is commu-

tative.

Proof. Choose X ∈ 𝔥\𝔤^{Y} , and set Ξ = 𝔲(𝔤)(X − f(X)I). By Proposition 2.1.1, we have 𝔲(𝔤) = 𝔲(𝔤^{Y}) ⊕ Ξ. Since Ξ ⊂ c(𝔥, ℐ), it follows that c(𝔥, ℐ) = c(𝔥, ℐ) ∩ 𝔲(𝔤^{Y}) ⊕ Ξ. Let ℐ^{Y} be the left ideal in 𝔲(𝔤^{Y}) generated by the kernel of f|_{𝔥^{Y}} , and c(𝔥^{Y} , ℐ^{Y}) the centralizer in u(𝔤^{Y}) of 𝔥^{Y} modulo ℐ^{Y} . Then Corollary 2.1.3 gives ℐ^{Y} = ℐ ∩ 𝔲(𝔤^{Y}), and again by Proposition 2.1.1, ℐ = ℐ^{Y} ⊕ Ξ. Thus

𝒟(𝔤, 𝔥, f) ≃ c(𝔥, ℐ) ∩ 𝔲(𝔤^{Y})/ℐ^{Y} .

But c(𝔥, ℐ) ∩ 𝔲(𝔤^{Y}) ⊂ c(𝔥^{Y} , ℐ^{Y}). Thus 𝒟(𝔤, 𝔥, f) injects into 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}), and (i) is proved.

It is immediate from (i) that if 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) is commutative, then 𝒟(𝔤, 𝔥, f) is also. Assume then that 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) is not commutative. Let

p_{1} = min{p ∈ P : c(𝔥^{Y} , ℐ^{Y}) ∩ 𝔲_{p}(𝔤^{Y}) ⊈ c(c(𝔥^{Y} , ℐ^{Y}), ℐ^{Y})}

and let W ∈ c(𝔥^{Y} , ℐ^{Y}) ∩ 𝔲_{p_{1}}(𝔤^{Y})\c(c(𝔥^{Y} , ℐ^{Y}), ℐ^{Y}). Now let

q_{1} = min{q ∈ P : c(𝔥^{Y} , ℐ^{Y}) ∩ 𝔲_{q}(𝔤^{Y}) ⊈ c(W, ℐ^{Y})}

and choose V ∈ c(𝔥^{Y} , ℐ^{Y}) ∩ 𝔲_{q_{1}}(𝔤^{Y})\c(W, ℐ^{Y}). Set

r = max{j : (adX)^{j}V ≠ 0} and s = max{j : (adX)^{j}W ≠ 0}.

Now by definition of Kirillov structure, [X, Y] ∈ 𝔷\ℐ. For any positive integer j we have (adX)^{j}(Y^{j}) = j![X, Y]^{j} modulo ℐ, hence (adX)^{j}(Y^{j}) ∉ ℐ and (adX)^{j+1}(Y^{j}) ∈ ℐ ∩ 𝔲(𝔷). Define

\begin{matrix}ρ(V) = V (adX)^{r}(Y^{r}) − adX(V)(adX)^{r−1}(Y^{r})+ & (adX)^{2}(V)(adX)^{r−2}(Y^{r}) \\ − ⋯ + (−1)^{r}(adX)^{r}(V)Y^{r} \end{matrix}

and

\begin{matrix}ρ(W) = W (adX)^{s}(Y^{s}) − adX(W)(adX)^{s−1}(Y^{s})+ & (adX)^{2}(W)(adX)^{s−2}(Y^{s}) \\ − ⋯ + (−1)^{s}(adX)^{s}(W)Y^{s}. \end{matrix}

Then it is easily seen that [X, ρ(V)] = [X, ρ(W)] = 0. Now I claim that ρ(V) belongs to c(𝔥^{Y} , ℐ). For this it is enough to show that for each j, (adX)^{j}V belongs to c(𝔥^{Y} , ℐ^{Y}), since (adX)^{k}(Y^{r}) ∈ 𝔲(kY + 𝔷) ⊂ c(𝔤^{Y} , ℐ^{Y}) for all k. But this fact follows from an easy induction argument and the formula

[S, (adX)^{j}(V)] = [[S, X], (adX)^{j−1}(V)] + [X, [S, (adX)^{j−1}(V)]].

Therefore ρ(V) and ρ(W) belong to c(𝔥, ℐ). It remains to show that [ρ(V), ρ(W)] does not belong to ℐ. Since (adX)^{j}(Y^{r}) belongs to 𝔲(kY +𝔷), then (adX)^{j}(Y^{r}) lies in the center of 𝔲(𝔤^{Y}) modulo 𝒥, where 𝒥 is the (two-sided) ideal in 𝔲(𝔤) generated by ℐ ∩ 𝔲(𝔷). Hence we can write

\begin{matrix}[ρ(V), ρ(W)] = [V, W]r!s![X, Y]^{r+s} + \\ ∑_{j+k≥1}(−1)^{j+k}[(adX)^{j}(V), (adX)^{k}(W)](adX)^{r−j}(Y^{r})(adX)^{s−k}(Y^{s}) modulo 𝒥. \end{matrix}

Now by definition of p_{1} , (adX)^{k}(W) lies in c(c(𝔥^{Y} , ℐ^{Y}), ℐ^{Y}) for all k > 0. Hence every term in the above sum with k > 0 lies in ℐ^{Y} . On the other hand, if j > 0, then adX^{j}(V) lies in c(W, ℐ^{Y}), so that the terms for which j > 0 and k = 0 belong to ℐ^{Y} also. Hence

[ρ(V), ρ(W)] = [V, W]r!s![X, Y]^{r+s} modulo ℐ^{Y} .

Now by choice of V and W, [V, W] ∉ ℐ^{Y} , and since ℐ^{Y} = ℐ ∩ 𝔲(𝔤^{Y}), we have [V, W] ∉ ℐ. Since [X, Y]^{r+s} ∈ 𝔲(𝔷)\ℐ, Proposition 3.1.2 obtains that [ρ(V), ρ(W)] is not in ℐ.

4.2 Let (𝔤, 𝔥, f) be an triple with 𝔤 nilpotent. Thus far we have established two principles.

(1) There is an associated triple (𝔤̂, 𝔥̂, f̂) that admits a Kirillov structure, and for which d(𝔤, 𝔥, f) = d(𝔤̂, 𝔥̂, f̂), δ(𝔤, 𝔥, f) = δ(𝔤̂, 𝔥̂, f̂), and 𝒟(𝔤, 𝔥, f) is commutative if and only if 𝒟(𝔤̂, 𝔥̂, f̂) is commutative.

(2) If (𝔤, 𝔥, f) admits a Kirillov structure (𝔷, Y), and 𝔥 + 𝔤^{Y} = 𝔤, then 𝒟(𝔤, 𝔥, f) is commutative if and only if 𝒟(𝔤^{Y} , 𝔥^{Y} , f|_{𝔥^{Y}}) is commutative.

What happens in the case where (𝔤, 𝔥, f) admits a Kirillov structure, and 𝔥 ⊂ 𝔤^{Y} ? At present, all we know in this case is that 𝒟(𝔤^{Y} , 𝔥, f) ⊂ 𝒟(𝔤, 𝔥, f) (this is easily seen using Corollary 2.1.3 with 𝔨 = 𝔤^{Y} ), and so if 𝒟(𝔤^{Y} , 𝔥, f) is not commutative, then 𝒟(𝔤, 𝔥, f) is not commutative.

Based upon the work in [2] and [7], we conjecture that if 𝔥 ⊂ 𝔤^{Y} , then

(3) if δ(𝔤^{Y} , 𝔥, f) = δ(𝔤, 𝔥, f) − 1, then 𝒟(𝔤^{Y} , 𝔥, f) = 𝒟(𝔤, 𝔥, f), while

(4) if δ(𝔤^{Y} , 𝔥, f) = δ(𝔤, 𝔥, f), then 𝒟(𝔤^{Y} , 𝔥, f) ⊊ 𝒟(𝔤, 𝔥, f).

In any case the above facts (1) and (2) suggest the following reduction procedure. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent and with d(𝔤, 𝔥, f) > 0. Choose 𝔷 an ℐ-central ideal in 𝔤 for which the associated triple (𝔤̂, 𝔥̂, f̂) admits a Kirillov structure (𝔷(1), y(1)). Define the triple (𝔤(1), 𝔥(1), f(1)) by 𝔤(1) = (𝔤̂) ^{y(1)} , 𝔥(1) = (𝔥̂) ^{y(1)} , and f(1) = f̂ |_{𝔥(1)} . Note that by Propositions 3.7.3 and 3.7.4, d(𝔤(1), 𝔥(1), f(1)) = d(𝔤, 𝔥, f) − 2, and by the preceding, if 𝒟(𝔤(1), 𝔥(1), f(1)) is not commutative, then 𝒟(𝔤, 𝔥, f) is not commutative also. Now provided d(𝔤(1), 𝔥(1), f(1)) > 0 we can apply the reduction procedure to (𝔤(1), 𝔥(1), f(1)) to produce another triple (𝔤(2), 𝔥(2), f(2)), and so on. Thus if {(𝔤(j), 𝔥(j), f(j)) : 0 ≤ j ≤ k} is any sequence of triples so obtained, then 𝔤(j) = (𝔤(j−1)ˆ ) ^{y_{j}} where y_{j} is a Kirillov element in 𝔤(j − 1)ˆ , 𝔥(j) = (𝔥(j−1))ˆ ∩𝔤(j), and f(j) = (f(j − 1))ˆ |_{𝔥(j)}, 1 ≤ j ≤ k.

Definition. Let (𝔤, 𝔥, f) be a triple. A sequence of triples {(𝔤(j), 𝔥(j), f(j)) : 0 ≤ j ≤ k} will be called a reducing sequence for (𝔤, 𝔥, f) if

(i) (𝔤(0), 𝔥(0), f(0)) = (𝔤, 𝔥, f),

(ii) for each j ≥ 1, (𝔤(j), 𝔥(j), f(j)) is obtained from (𝔤(j − 1), 𝔥(j − 1), f(j − 1)) by the reduction procedure described above, and

(iii) (𝔤(k), 𝔥(k), f(k)) is an FM-triple whose associated coadjoint orbits have zero dimension.

Let d(j) = d(𝔤(j), 𝔥(j), f(j)) and δ(j) = δ(𝔤(j), 𝔥(j), f(j)), 0 ≤ j ≤ k. From Propositions 3.7.3 and 3.7.4, we have d(j) = d(j − 1) − 2, 1 ≤ j ≤ k, and that δ(j) = δ(j −1) or δ(j −1)−1. From this and Proposition 3.4.4 the following is immediate.

Proposition 4.2.1. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent. Then (𝔤, 𝔥, f) has a reducing sequence.

It seems plausible that a determination of the conjectures (3) and (4) above in complete detail, together with the results contained herein, would provide a complete description of 𝒟(𝔤, 𝔥, f).

In this paper we are concerned with the application of the above to establish non-commutativity of 𝒟(𝔤, 𝔥, f) in certain situations. Note that if (𝔤, 𝔥, f) is an IM-triple with 𝔤 nilpotent, then any reducing sequence that terminates with a FM-triple has a minimal term that is IM; that is, there is m ≥ 0 such that (𝔤(j), 𝔥(j), f(j)) is an IM-triple for 0 ≤ j ≤ m, and (𝔤(j), 𝔥(j), f(j)) is an FM-triple for j > m. The following shows that proof of non-commutativity of 𝒟(𝔤, 𝔥, f) hinges on Conjecture 2.

Theorem 4.2.2. Let {(𝔤(j), 𝔥(j), f(j)) : 0 ≤ j ≤ k} be a reducing sequence for the IM-triple (𝔤, 𝔥, f) with 𝔤 nilpotent. Set

m = max{j : (𝔤(j), 𝔥(j), f(j)) is an IM-triple},

and assume that Conjecture 2 holds for ((𝔤(m)ˆ , 𝔥(m)ˆ , f(m)ˆ ). Then 𝒟(𝔤, 𝔥, f) is not commutative.

Proof. From Corollary 3.4.3 and Propostion 4.2.1, it is enough to show that the algebra 𝒟((𝔤(m))ˆ , (𝔥(m))ˆ , (f(m))ˆ ) is not commutative, which is to say, we may assume that m = 0 and 𝔤̂ = 𝔤.

Let (𝔷, Y) be a Kirillov structure for (𝔤, 𝔥, f). Now from Conjecture 2, we have an element V ∈ c(𝔥, ℐ) \ 𝔲(𝔤^{Y}) + ℐ. Now a standard algebraic argument (see for example Proposition 2.1 of [7]) shows that we may assume that V has the form V = P X + Q where X ∈ 𝔤\𝔤^{Y} , P ∈ 𝔲(𝔤^{Y}) \ ℐ, and Q ∈ 𝔲(𝔤^{Y}). It is clear that Y ∈ c(𝔥, ℐ)\ℐ, and that [V, Y] = P [X, Y]. Since P ∉ ℐ and [X, Y] ∈ 𝔷\ℐ, then Proposition 3.1.2 shows that [V, Y] ∉ ℐ. The Theorem follows.

5. Two Applications

5.1 Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent and δ(𝔤, 𝔥, f) = 1. As before we denote exp(ad^{∗}(Y)) by α(Y). Let (0) = 𝔤_{0} ⊂ 𝔤_{1} ⊂ ⋯ ⊂ 𝔤_{n} = 𝔤 be a sequence of α(𝔥)-modules in 𝔤 with dim(𝔤_{j}) = j, 1 ≤ j ≤ n. Set a = {j : 𝔤_{j} ⊂ 𝔤_{j−1} + 𝔥}, and write a = {a_{1} < a_{2} < ⋯ < a_{p}}; for 1 ≤ k ≤ p, set 𝔥_{k} = 𝔥 ∩ 𝔤_{a_{k}}. Then {𝔥_{k}} is a Jordan-Hölder sequence for 𝔥. Choose X_{j} ∈ 𝔤_{j}\ 𝔤_{j−1}, with X_{j} ∈ 𝔥 for j ∈ a. Let 𝒫 be the partition of 𝔤^{∗} relative to the chosen α(𝔥)-modules, and let (ϕ, ψ) be the minimal index pair for which Ω_{ϕ,ψ} meets L_{f}. For ease of notation set b = ϕ_{1}, c = ψ_{1} , Ω = Ω_{ϕ,ψ}, X = X_{b}, S = X_{a_{c}} ∈ 𝔥_{c} \ 𝔥_{c−1} , and p(ℓ) = ℓ([S, X]), ℓ ∈ 𝔤^{∗}. Then

Ω ∩ L_{f} = {ℓ ∈ L_{f} : p(ℓ) ≠ 0}.

We now make some helpful observations, based on the simple fact that ∩{ker(ℓ) : ℓ ∈ L_{f}} = 𝔥 ∩ ker(f). Only (4) requires the assumption that δ(𝔤, 𝔥, f) = 1.

(1) If 𝔷 is an ideal in 𝔤 such that 𝔷 ⊂ ∩{𝔥^{ℓ} : ℓ ∈ L_{f}}, then 𝔷 ⊂ c(𝔥, ℐ),

(2) ∩{𝔥(ℓ) : ℓ ∈ L_{f}} ⊂ c(𝔤, ℐ),
(3) b = min{j : 𝔤_{j} ⊈ c(𝔥, ℐ)}, and
(4) c = min{k : 𝔥_{k} ⊈ c(𝔤, ℐ)}.

Detailed proofs are omitted here, but we sketch an argument for (4): if 𝔥_{c−1} ⊈ c(𝔤, ℐ), then by observation (2), there would be X_{0} ∈ 𝔤 and Y_{0} ∈ 𝔥_{c−1} such that [Y_{0}, X_{0}] ∉ ℐ. Since ℓ([Y_{0}, X]) = 0 for all ℓ ∈ L_{f}, then the polynomial function Q = [S, X][Y_{0}, X_{0}] − [S, X_{0}][Y_{0}, X] would not vanish on L_{f}. Lemma 3.6.1 would then imply that δ(𝔤, 𝔥, f) > 1.

The well-known description of the collective α(𝔥)-orbit structure due to Pukanszky and summarized in [3] has the following form. For each j, 1 ≤ j ≤ n, there is a rational function F_{j} : k × Ω → k such that the collection {F_{j}} has the properties

(i) for 1 ≤ j ≤ b − 1, F_{j}(z, ℓ) = ℓ_{j},

(ii) F_{b}(z, ℓ) = z,
(iii) for b < j ≤ n, F_{j}(z, ℓ) = ℓ_{j} + F_{j}(z, ℓ_{1}, ℓ_{2}, …, ℓ_{j−1}).

Let K(𝔤) be the field of fractions of S(𝔤). We explicitly compute elements R_{j} ∈ K(𝔤), , j ≠ b, 1 ≤ j ≤ n, with the property that for any ℓ ∈ Ω ∩ L_{f}, we have R_{j}(ℓ) = F_{j}(0, ℓ). It is obvious that for j < b, one takes F_{j} = X_{j}. Fix ℓ ∈ Ω ∩ L_{f}; then the H -orbit of ℓ is 𝒪_{ℓ} = {α(tS)ℓ : t ∈ k}; for each t ∈ k, (α(tS)ℓ)_{b} = ℓ_{b} − tℓ([S, X]). The functions F_{j}(z, ℓ) are obtained by making the substitution

t = \frac{ℓ_{b} − z}{p(ℓ)}

into the functions (α(tS)ℓ)_{j}, b < j ≤ n. The result is that we obtain

R_{j} = X_{j} − (\frac{X}{[S, X]}) [S, X_{j}] + \frac{1}{2} (\frac{X}{[S, X]})^{2} [S, [S, X_{j}]] − ⋯

= ∑_{r≥0}(−1)^{r} \frac{1}{r!} (\frac{X_{b}}{[S, X]})^{r} ad(S)^{r}(X_{j}).

Note that the form of the functions R_{j} shows that for some nonnegative integer N, p^{N}R_{j} belongs to S(𝔤). Set I(L_{f}) = {W ∈ S(𝔤) : W vanishes on L_{f}}.

Lemma 5.1.1. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent and with δ(𝔤, 𝔥, f) ≤ 1. Set 𝔪 = 𝔤_{b−1} + 𝔤_{a_{c}−1} . Then ω(S(𝔪) ∩ I(L_{f})) ⊂ ℐ.

Proof. Set 𝔪_{0} = 𝔪 ∩ I(L_{f}) = 𝔪 ∩ 𝔥 ∩ ker(f). I claim that 𝔪_{0} is an ideal in 𝔪. Let X ∈ 𝔪 and Y ∈ 𝔪_{0} . If a_{c} > b, then 𝔪_{0} is a subspace of 𝔥_{c−1} and so 𝔪_{0} is an ideal in 𝔤. On the other hand if a_{c} < b, then we have 𝔪 ⊂ c(𝔥, ℐ); in either case the claim follows.

Let η : 𝔪 → 𝔪/𝔪_{0} be the canonical map. We have maps η_{S} : S(𝔪) → S(𝔪/𝔪_{0}) and η_{u} : 𝔲(𝔪) → 𝔲(𝔪/𝔪_{0}) so that the following diagram

S(𝔪) ⟶^{η_{S}} S(𝔪/𝔪_{0})
ω↓  _{ω′}↓
𝔲(𝔪) ⟶^{η_{u}} 𝔲(𝔪/𝔪_{0})

commutes. Now for any q ∈ S(𝔪), Y ∈ 𝔪 ∩ 𝔥, η_{S}(Y − f(Y)I) is central in S(𝔪/𝔪_{0}), so

η_{u}(ω(q(Y − f(Y)I)) = ω′(η_{S}(q))ω′(η_{S}(Y − f(Y)I)) = η_{u}(ω(q))η_{u}(Y − f(Y)I).

Hence

ω(q(Y − f(Y)I)) = ω(q)(Y − f(Y)I) + W

where W ∈ u(𝔪)𝔪_{0} ⊂ ℐ. Therefore ω(q(Y − f(Y)I)) belongs to ℐ. Since every element in S(𝔪) ∩ I(L_{f}) is a sum of elements of the form q(Y − f(Y)I), the lemma follows.

Proposition 5.1.2. Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent and δ(𝔤, 𝔥, f) = 1. Then for each j, 1 ≤ j ≤ n, j ≠ b, ω(p^{N}R_{j}) belongs to c(𝔥, ℐ).

Proof. Fix j, 1 ≤ j ≤ n, j ≠ b, and fix Y ∈ 𝔥. We compute that

[Y, p^{N}R_{j}] = P_{0} + ∑_{m≥0} \frac{(−1)^{m}}{m!} · p^{N−m−1}q_{m}X^{m}

where q_{m} = (adY (adS)^{m}X_{j}) · p − ((adS)^{m+1}X_{j}) · [Y, X], m = 0, 1, 2, …, and where P_{0} has [Y, p] as a factor. Now by observation (3), p = [S, X] belongs to c(𝔥, ℐ), and hence by observation (4), [Y, p] belongs to c(𝔤, ℐ) ∩ ℐ (we regard 𝔤 as a subset both of S(𝔤) and 𝔲(𝔤), of course.) It follows that ω(P_{0}) ∈ ℐ.

Turning our attention now to the second term of the above, we begin by showing that ω(p^{N−1}q_{0}) ∈ ℐ. Since δ(𝔤, 𝔥, f) = 1, then linear algebra shows that q_{0} ∈ I(L_{f}). Now each of p, [S, X_{j}], and [Y, X] belong to 𝔪. Therefore we must have

[Y, X_{j}] = Y_{0} + Z

where Y_{0} ∈ 𝔥 and Z ∈ 𝔪. Hence

p^{N−1}q_{0} = p^{N}U_{0} + W

where U_{0} = Y_{0} − f(Y_{0})I and W ∈ S(𝔪). Since p ∈ 𝔤_{b−1} , one gets ω(p^{N}U_{0}) ∈ ℐ. Since p^{N−1}q_{0} and p^{N}U_{0} both belong to I(L_{f}), then W ∈ S(𝔪)∩I(L_{f}). Lemma 5.1.1 now gives ω(W) ∈ ℐ, and linearity of ω now gives ω(p^{N−1}q_{0}) ∈ ℐ.

Next we turn to T = ∑_{m>0} \frac{(−1)^{m}}{m!} · p^{N−m−1}q_{m}X^{m}. To show that ω(T) ∈ ℐ, observe that if a_{c} > b,then T ∈ S(𝔪)∩I(L_{f}), so again by Lemma 5.1.1 we get ω(T) ∈ ℐ. Suppose then that a_{c} < b. Set T_{m} = p^{N−m−1}q_{m}X^{m}, m ≥ 1. Since [S, X_{j}] ∈ c(𝔥, ℐ) (observation (3)), then for m ≥ 1, we have adY (adS)^{m}X_{j} and (adS)^{m+1}X_{j} both belong to 𝔥 ∩ ker(f). In fact by choice of S, they lie in 𝔥_{c−1} . Hence

T_{m} = p^{N−m}X^{m}Y_{0} + p^{N−m−1}[Y, X]X^{m}Y_{1}

where Y_{0} and Y_{1} belong to 𝔥_{c−1}∩ker(f). Now by definition of c and observation (4) above, Y_{0} and Y_{1} belong to c(𝔤, ℐ) ∩ ℐ. It is immediate from this that ω(T_{m}) ∈ ℐ holds for each m, so ω(T) ∈ ℐ.

Corollary 5.1.3. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and δ(𝔤, 𝔥, f) ≤ 1. Then Conjecture 2 holds for (𝔤, 𝔥, f).

Proof. To prove that Conjecture 2 holds for (𝔤, 𝔥, f), we assume that the IM-triple (𝔤, 𝔥, f) admits a Kirillov structure (𝔷, Y) with 𝔥 ⊂ 𝔤^{Y} , and with (𝔤^{Y} , 𝔥, f) an FM-triple. Choose X ∈ 𝔤 \ 𝔤^{Y} . If δ(𝔤, 𝔥, f) = 0, then 𝔤 ⊂ c(𝔥, ℐ), and we have that X and Y belong to c(𝔥, ℐ) and [X, Y] ∉ ℐ. Hence Conjecture 2 holds in this case.

Suppose then that δ(𝔤, 𝔥, f) = 1. Choose a sequence {𝔤_{j}} of Ad(𝔥)- modules as above with 𝔤_{n−1} = 𝔤^{Y} . Since (𝔤^{Y} , 𝔥, f) an FM-triple, we have δ(𝔤^{Y} , 𝔥, f) = 1 and so n ≠ b. Set V = ω(p^{N}R_{n}); then it is easily seen that V = P X + Q, where P and Q belong to 𝔲(𝔤^{Y}), and where P is of the form p^{N}. Now it follows from Proposition 2.1.1, Corollary 2.1.2, and the fact that p ∈ 𝔤 \ (𝔥 ∩ ker(f)), that every power of p does not lie in ℐ. Thus Conjecture 2 is proved.

Theorem 5.1.4. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and with the property that δ(𝔤, 𝔥, f) ≤ 1. Then 𝒟(𝔤, 𝔥, f) is not commutative.

Proof. Let {(𝔤(j), 𝔥(j), f(j)) : 0 ≤ j ≤ k} be a reducing sequence for (𝔤, 𝔥, f) and define the index m as in Theorem 4.2.1. Then by Corollary 5.1.3, Conjecture 2 holds for ((𝔤(m)ˆ , 𝔥(m)ˆ , f(m)ˆ ). Hence by Theorem 4.2.2, Conjecture 1 holds for (𝔤, 𝔥, f).

5.2 Let (𝔤, 𝔥, f) be a triple with 𝔤 nilpotent. Let us say that f is ”essentially zero” if there are ideals 𝔥_{1} and 𝔤_{0} in 𝔤 such that 𝔤 is the direct sum of 𝔤_{0} and 𝔥_{1} , 𝔥_{1} ⊂ c(𝔤, ℐ), and such that f vanishes on 𝔥 ∩ 𝔤_{0} .

Lemma 5.2.1. Let (𝔤, 𝔥, f) be a triple for which f is essentially zero. Let P ∈ S(𝔤_{0}) be a homogeneous element for which ω(P) ∈ ℐ. Then P vanishes on L_{f}.

Proof. Choose a basis of 𝔤_{0} that passes through 𝔥_{0} = 𝔥 ∩ 𝔤_{0} , say:

{Y_{1}, Y_{2}, …, Y_{p}, X_{1}, X_{2}, …, X_{m}}

with {Y_{1}, Y_{2}, …, Y_{p}} a basis of 𝔥_{0} . Using multi-indices in the usual way, let P_{αβ} = X^{α}Y^{β} ∈ S(𝔤_{0}) and write P as a finite sum

P = ∑_{α+β=d} a_{αβ}P_{αβ}

where d is the degree of P. Now let P̃_{αβ} be the element in 𝔲(𝔤_{0}) defined by the same expression: P̃_{αβ} = X^{α}Y^{β}. Then

ω(P) = ∑_{α+β=d} a_{αβ}P̃_{αβ} + Z

where Z ∈ 𝔲_{d−1}(𝔤_{0}) = k-span{X^{μ}Y^{ν} : |μ|+|ν| ≤ d−1}. Now a basis of ℐ∩𝔲(𝔤_{0}) is given by the monomials X^{μ}Y^{ν} with |ν| > 0. Since ω(P) ∈ ℐ∩𝔲(𝔤_{0}), it follows that Z ∈ ℐ ∩ 𝔲_{d−1}(𝔤_{0}) and that a_{αβ} ≠ 0 only if |β| > 0. Thus P vanishes on L_{f}.

We recall the constructions of Section 3.6. Given an increasing sequence {𝔤_{j} : 1 ≤ j ≤ n} of α(𝔥)-modules in 𝔤 with dim(𝔤_{j}) = j, and a Jordan-Hölder sequence for 𝔥, one has a partition of 𝔤^{∗} into the α(𝔥)-invariant algebraic subsets as defined in Section 3.6. Let Ω be the minimal element of the partition, that is, Ω = Ω_{ϕ,ψ} where (ϕ, ψ) is the smallest index pair for which Ω_{ϕ,ψ} ≠ Ø. Then Ω is Zariski open in 𝔤^{∗}.

Lemma 5.2.2. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and with f essentially zero. Assume that (𝔤, 𝔥, f) admits a Kirillov structure (𝔷, Y) for which 𝔥 ⊂ 𝔤^{Y} and for which (𝔤^{Y} , 𝔥, f) is FM. Assume further that there is an increasing sequence {𝔤_{j}} of α(𝔥)-modules in 𝔤 with 𝔤_{n−1} = 𝔤^{Y} and a Jordan-Hölder sequence for 𝔥, such that L_{f} ∩ Ω ≠ Ø. Then c(𝔥, ℐ) ⊈ 𝔲(𝔤^{Y}) + ℐ.

Proof. Let X ∈ 𝔤 \ 𝔤^{Y} . It is well-known that there is an 𝔥-invariant element P ∈ S(𝔤) of the form P = QX +R where Q and R belong to S(𝔤^{Y}), and where Q is 𝔥-invariant and never zero on Ω. From the definition of essentially zero and the construction of P we see that Q ∈ S(𝔤_{0}). Now ω(P) ∈ c(𝔥, ℐ) since P is 𝔥-invariant, and it is easily seen that ω(P) has the form

ω(P) = ω(Q)X + Z

where Z ∈ 𝔲(𝔤^{Y}). Hence [ω(P), Y] = ω(Q)[X, Y] modulo ℐ. Since Q is never zero on Ω, then Q is non-vanishing on L_{f}, and Lemma 5.2.1 allows us to conclude that ω(Q) ∉ ℐ. Now from Proposition 3.1.2 we have [ω(P), Y] ∉ ℐ. Since Y ∈ c(𝔥, ℐ) it follows that ω(P) ∈ c(𝔥, ℐ) \ 𝔲(𝔤^{Y}) + ℐ.

The following is now immediate from Theorem 4.2.2.

Theorem 5.2.3. Let (𝔤, 𝔥, f) be an IM-triple with 𝔤 nilpotent and suppose that a reducing sequence for (𝔤, 𝔥, f) is given with (𝔤(m), 𝔥(m), f(m)) defined as in Theorem 4.2.2. Assume that the hypothesis of Lemma 5.2.2 holds for (𝔤(m)ˆ , 𝔥(m)ˆ , f(m)ˆ ). Then 𝒟(𝔤, 𝔥, f) is not commutative.

References

[1] Baklouti, A., and J. Ludwig, Invariant differential operators on certain nilpotent homogeneous spaces, to appear in Monatshefte fur Mathematik.

[2] Corwin, L., and F. P. Greenleaf, Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity, Comm. on Pure and Appl. Math. XLV (1992), 681–748.

[3] Corwin, L., and F. P. Greenleaf, “Representations of nilpotent Lie groups and their applications, Part I,” Cambridge Studies in Adv. Math., vol. 18, Cambridge University Press, 1989.

[4] Corwin, L., F. P. Greenleaf, and G. Grelaud, Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups, Trans. Amer. Math. Soc. 304 (1987), 549–583.

[5] Dixmier, J. “Enveloping Algebras,” Graduate Studies in Mathematics, vol. II, American Mathematical Society, Providence, 1996.

[6] Fujiwara, H., G. Lion, and S. Mehdi, On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces, Preprint.

[7] Greenleaf, F. P., Geometry of coadjoint orbits and noncommutativity of invariant differential operators on nilpotent homogeneous spaces, Preprint.

Bradley N. Currey III Department of Mathematics and Mathematical Computer Science Saint Louis University St. Louis, MO 63103 U.S.A. curreybn@slu.edu

Received March 23, 2000 and in final form December 15, 2000