BOREL SUBALGEBRAS AND CATEGORIES OF HIGHEST

WEIGHT MODULES FOR TOROIDAL LIE ALGEBRAS

B. Cox*, V. Futorny**

Abstract. In this article we begin an investigation of the conjugacy classes of Borel subalgebras together with Verma modules induced from “standard” Borel subalgebras of a toroidal Lie algebra 𝔱 in two variables. We define, for each highest weight λ, a category O_{λ} of representations of 𝔱 that contain these Verma modules and show that when a certain central element acts nontrivially that this category is equivalent to the category O_{λ}(𝔪̂) for an extension of a suitable affine Kac-Moody algebra 𝔪̂ ⊂ 𝔱 . From this equivalence we obtain a BGG type resolution and BGG duality theorem in the setting of O_{λ}.

§0. Introduction:

As is well known the representation theory of affine Kac-Moody Lie Algebras has many interesting applications to various problems in physics and mathematics. On the other hand the representation theory of toroidal Lie algebras is still in its enfancy. The hope is that it too will have numerous applications outside the domain of its initial conception. Certainly the best understood category of modules in the affine Kac-Moody setting is the category of highest weight modules O. A natural starting place to begin a representation theory of toroidal Lie algebras would be to find an appropriate analogue of O. Of particular importance to the structure theory of O are the Verma modules induced from Borel subalgebras. What are the appropriate Borel subalgebras in the toroidal Lie algebra setting? For affine Kac-Moody algebras there are various non-conjugate Borel subalgebras to choose

Typeset by AMS-TEX

*Funded in part by NSA grant

**Funded in part by CRDF grant

1991 Mathematics Subject Classification. 17 B67.

from, each derived from a subset of the set of simple roots for a finite-dimensional simple Lie algebra. As we shall see the toroidal Lie algebra case is complicated by the fact that non-conjugate Borel subalgebras are parametrized by k-frames in n-space. Here n is the number of variables used in defining the algebra of Laurent polynomials for the said toroidal algebra.

Let us now give an overview of the contents of the paper. In the first section we breifly review some results and definitions from the theory of convex sets that will be used in the second section on the classification of the positive sets of imaginary roots. The first main result is Theorem 2.2 giving a parameterization of these sets of imaginary roots by k-frames in n-space. In section three we recall the definition of the Weyl group given by Moody and Shi in the setting of toroidal Lie algebras and its action on the set of roots. In addition we borrow the definition of the sets of positive roots given in [F], and [JK]. Namely a subset P of the set of roots is a positive set of roots if P ∩−P = {0}, P ∪−P is the set of roots, and if α, β ∈ P with α + β a root then α + β ∈ P. Since the Weyl group acts trivially on the set of imaginary roots one can immediately see that the set of representatives for non-conjugate Borel subalgebras, being induced from sets of positive roots, has a richer structure when the number of variables for the toroidal Lie algabra is greater than one. From previous work one knows that the set of representatives is paramaterized by finite subsets of the set of roots coming from the finite dimensional Lie algebra when n = 1 (see [JK], [F]). Because the set of representatives is not parameterized by a descrete set for n ≥ 2 we have chosen to limit our study for the time being to the case when n = 2 and when the positive set of imaginary roots is of normal type. In section three we show that under these conditions there are precisely four equivalence classes of positive sets of roots under the orbit of the Weyl group.

One of our main interests is to construct a representation theory for toroidal Lie algebras which we procede to do in the last three sections. In section four the central technical result is described as follows. Let 𝔱 denote the toroidal Lie algebra with central elements c_{t} and c_{t} corresponding to the two indeterminants s, t. Let 𝔤̂ denote the affine Kac-Moody subalgebra of 𝔱 induced from the Laurent polynomials in s. Then the result is that any Verma module, M_{𝔱}(λ), for 𝔱 for which the central element c_{s} acts nontrivially has the property that every submodule of N of M_{𝔱}(λ) is generated by its intersection with M𝔤̂(λ|_{𝔤}). In section five we then introduce a catogory O_{λ} of highest weight modules and show that the above property of Verma modules in which c_{s} acts nontrivially implies that the category O_{λ} is equivalent to a highest weight category for 𝔤̂. Finally in section six we show that Bernstein, Gelfand and Gelfand’s result on reciprocity and their resolution of certain irreducible modules are consequences of this equivalence of category.

We would like to thank Stephen Berman for many useful discussions.

§1. Notation and Results from the Theory of Convex Sets: A subset of ℝ^{n} is called a flat of dimension r if it is a translate of a subspace of ℝ^{n} of dimension r. A subset S of ℝ^{n} is said to have dimension r if r is the least dimension of any flat containing S. We let relint S denote the relative interior of S in the minimal flat containing S and S̄ denote the closure of S in ℝ^{n}. The relative interior of a subset is the interior of its closure.

Let f ∈ (ℝ^{n})^{∗} be a linear functional and α ∈ ℝ. A hyperplane H_{f,α} = {v ∈ ℝ^{n}| f(v) = α} is said to separate two subsets A, B ⊂ ℝ^{n} if either (1) f(a) ≤ α and f(b) ≥ α for all a ∈ A, b ∈ B or (2) f(a) ≥ α and f(b) ≤ α for all a ∈ A, b ∈ B.

We will need the following three results later:

1.1 Theorem ([L, 4.11]). Let A and B be two convex subsets of ℝ^{n} with dim(A∪B) = n. Then A and B can be separated by a hyperplane if and only if relint A ∩ relint B = ∅.

1.2 Theorem ([L], Helley’s Theorem 6.3). Let C = {C_{j}| j ∈ J} be a family of compact convex sets in ℝ^{n}. If any n + 1 members of C have nonempty intersection, then ∩_{j∈J}C_{j} is nonempty.

1.3 Theorem ([L], Theorem 1.2.30). If S is a compact set then the convex hull of S is compact.

§2. Positive Imaginary Roots Systems: See [MS] for notation. 2.1. Let Δ̇ be an irreducible reduced root system of type X_{l} = A_{l}, D_{l} or E_{l}. Let Π̇ = {α_{1}, …, α_{l}} be a fixed basis of Δ̇, Q̇ the root lattice, θ the unique highest root with respect to Π̇ and let Ȧ be the Cartan matrix with respect to Π̇.

From now on we fix a positive integer n and set

İ : = {1, … l}, J := {l + 1, …, l + n}, I = İ ∪ J,
I_{k} : = İ ∪ {k} for l + 1 ≤ k ≤ l + n.

Let Γ := Γ^{[n]} denote the free abelian group with generators α_{i}, δ_{j}, Λ_{j} with i ∈ İ, j ∈ J so that

Γ = (⨁_{i∈İ} ℤα_{i}) ⊕ (⨁_{j∈J} ℤδ_{j} ⊕ ℤΛ_{j}).

Set

Δ^{im} = ⨁_{j∈J} ℤδ_{j}, Δ_{ℝ}^{im} = ⨁_{j∈J} ℝδ_{j}, Δ_{ℝ^{+}}^{im} = ⨁_{j∈J} ℝ_{≥0}δ_{j}, and Δ_{ℝ^{−}}^{im} = ⨁_{j∈J} ℝ_{≤0}δ_{j}

where ℝ_{R0} = {r ∈ ℝ | rR0} for any relation R. A submonoid Ψ ⊂ Δ^{im} we will call a positive set of imaginary roots if

(1) Ψ ∪ −Ψ = Δ^{im},

(2) Ψ ∩ −Ψ = 0, These two conditions imply that up to changes in sign Ψ contains the basis {δ_{i}| l + 1 ≤ i ≤ l + n}. Indeed the first condition implies that for each i either δ_{i} ∈ Ψ or −δ_{i} ∈ Ψ and the second implies that both can not be in Ψ. Let Δ_{≥0}^{im} denote the semi-group generated by {δ_{i}| l + 1 ≤ i ≤ l + n}. From now one we assume that

(3) Ψ contains Δ_{≥0}^{im}. Let β denote the bilinear form on Δ_{ℝ}^{im} such that β(δ_{i}, δ_{j}) = δ_{ij} where δ_{ij} is the Kronecker delta function. For v, w ∈ Δ_{ℝ}^{im}, let v · w denote the usual scalar or dot product of v and w with respect to β. For η ∈ Δ^{im}, let H_{η} = {v ∈ ∑_{j∈J} ℝδ_{j}|v · η = 0} be the hyperplane orthogonal to η.

Our first goal is to show that all the collection of all positive sets of imaginary roots Ψ is parameterized by certain finite sets of linearly independent vectors η_{1}, …, η_{t} i.e. by t-frames in n-space. More precisely we have the following

2.2 Theorem. Let Ψ ⊂ Δ^{im} be a positive set of imaginary roots. There exist non-zero orthogonal vectors η_{1}, …, η_{t} ∈ Δ_{ℝ}^{im} with 1 ≤ t ≤ n such that

(a) Ψ = (∪_{i=2}^{t}{v ∈ Δ^{im}|v · η_{j} = 0 for 1 ≤ j ≤ i − 1, v · η_{i} > 0})∪{v ∈ Δ^{im}|v·η_{1} > 0}∪{0}

where

(b)  0 ≠ η_{1} ∈ Δ_{≥0}^{im} and (∩_{j}H_{η_{j}}) ∩ Δ^{im} = 0.

Moreover given a set η_{1}, …, η_{t} with 1 ≤ t ≤ n of orthogonal vectors satisfying (b) one has that if Ψ is defined by (a) above, then Ψ is a positive set of imaginary roots.

Proof. Assume that n > 1. The case n = 1 is left to the reader. Recall Ψ contains {δ_{1+n}, …, δ_{l+n}}. First we will show that

(4)  There exist 0 ≠ η_{1} ∈ Δ_{+}^{im} such that η_{1} · ψ ≥ 0 for all ψ ∈ Ψ

and second we will use induction on the dimension of the ambient space to prove

(5) {v ∈ Ψ| v · η_{1} = 0} = ∪_{i=2}^{t}{v ∈ Ψ|v · η_{j} = 0 for 1 ≤ j ≤ i − 1, v · η_{i} > 0} ∪ {0}

for some orthogonal η_{i} such that (∩_{j}H_{η_{j}}) ∩ Δ^{im} = 0. Consequently

(6)  {v ∈ Ψ| v · η_{1} > 0} = {v ∈ Δ_{+}^{im}| v · η_{1} > 0}

and

(7)  {v ∈ Ψ| v · η_{1} = 0} = ∪_{i=2}^{t}{v ∈ Δ^{im}|v · η_{j} = 0 for 1 ≤ j ≤ i − 1, v · η_{i} > 0}

will then follow from (1). Indeed the inclusion of the subsets on the left of (6) are clear and if 0 ≠ v is in a subset on the right then by (1) either v ∈ Ψ or v ∈ −Ψ. By (4) we can not have v · η_{1} > 0 and v ∈ −Ψ. The only other possibility is that v · η_{1} = 0. Now if v · η_{1} = 0 and v ∈ −Ψ then by (5) we must have −v · η_{j} = 0 for 1 ≤ j ≤ i − 1 and −v · η_{i} > 0 for some 2 ≤ i ≤ t. This contradicts either the assumption that v is in a set on the right hand side or (∩H_{η_{j}}) ∩ Δ^{im} = 0. A similar argument proves (7).

Write Ψ = {δ_{1+n}, …, δ_{l+n}} ∪ {v_{ζ}| ζ ∈ Υ} as a disjoint union where Υ is some index set and let B(m) be the closed ball of radius m ≥ 2 centered at the origin of Δ_{ℝ}^{im}. Set K = {∑_{j} r_{j}δ_{j}| (∑_{j} r_{j}) ≥ 1, r_{j} ∈ ℝ_{≥0}} ∩ B(m) and for each ζ ∈ Υ set K_{ζ} = {v ∈ Δ_{ℝ}^{im}| v · v_{ζ} ≥ 0} ∩ B(m) so that K and K_{ζ} are compact and convex. We need to prove that K ∩ (∩_{ζ∈Υ}K_{ζ}) ≠ ∅. Then any η_{1} ∈ K ∩ (∩_{ζ∈Υ}K_{ζ}) will satisfy η_{1} · ψ ≥ 0 for all ψ ∈ Ψ i.e.

(4) will hold.

We use the results stated above on convex sets. Let K_{i_{1}}, …, K_{i_{n+1}} be distinct with i_{j} ∈ Υ and let C be the convex hull in Δ_{ℝ}^{im} of v_{i_{1}}, …, v_{i_{n+1}}, K and 0. By Theorem 1.3, C is convex and compact. Observe A := Δ_{ℝ^{−}}^{im}\{0} is a convex set and dim(A ∪ C) = n. Suppose relint A ∩ relint C = ∅ then Theorem 1.1 implies that A and C can be separated by a hyperplane H_{η_{1},ϵ} = {v ∈ Δ_{ℝ}^{im}| v · η_{1} = ϵ} with η_{1} ≠ 0 and ϵ ∈ ℝ. Since 0 ∈ Ā we must have 0 = 0 · η_{1} ≤ ϵ and 0 ∈ C implies ϵ ≤ 0 · η_{1} = 0. Thus H_{η_{1}} = H_{η_{1},ϵ} passes through the origin. Since δ_{l+1}, … δ_{l+n} lie in C we have δ_{i} · η_{1} ≥ 0. Now use Theorem 1.2 to obtain K ∩ (∩_{η∈Υ}K_{η}) ≠ ∅.

If relint A ∩ relint C ≠ ∅ then there exists some 0 ≠ v ∈ relint A which is a positive ℝ-linear combination of v_{i_{1}}, …, v_{i_{n+1}} and δ_{n+1}, … δ_{n+l}. Since relint A is open in Δ^{im} there exists an open ball B ⊂ A of radius ϵ with center v. Consequently there exists positive rational numbers q_{i} such that

|∑_{k=1}^{n+1} q_{i_{k}}v_{i_{k}} + ∑_{j=n+1}^{n+l} q_{j}δ_{j} − ∑_{k=1}^{n+1} r_{i_{k}}v_{i_{k}} − ∑_{j=n+1}^{n+l} r_{j}δ_{j}| < ϵ

i.e. ∑_{j=1}^{n+1} q_{i_{j}}v_{i_{j}} + ∑_{j=n+1}^{n+l} q_{j}δ_{j} ∈ B ⊂ A. Since ℝ^{+}A ⊂ A we have that there exists a nonzero linear combination ∑_{j=1}^{n+1} m_{i_{j}}v_{i_{j}} + ∑_{j=n+1}^{n+l} m_{j}δ_{j} ∈ A ∩ Ψ = A ∩ Δ^{im} ∩ Ψ = (−Δ_{≥0}^{im}) ∩ Ψ with m_{i_{j}}, m_{j} ∈ ℤ, m_{j} ≥ 0. This contradicts (2), hence relint A ∩ relint C = ∅. This completes the proof of (4).

Now we use induction on the dimension of the minimal flat containing Ψ′ := {v ∈ Ψ| v · η_{1} = 0}. If Ψ′ = 0 then Ψ ∪ −Ψ = Δ^{im} implies that H_{η_{1}} ∩ Δ^{im} = 0. Thus to prove (b) we may assume that Ψ′ ≠ 0. Now Ψ ∩ H_{η_{1}} is a nonzero submonoid of Ψ, Ψ′ ∪ −Ψ′ = Δ^{im} ∩ H_{η_{1}} and Δ^{im′} := Δ^{im} ∩ H_{η_{1}} ≅ ℤ^{r} where r ≤ n − 1. Let δ′_{l+1}, …, δ′_{l+r} be a basis of Δ^{im′}. Then Ψ′ together with δ′_{l+1}, …, δ′_{l+r} satisfies (1), (2), (3) for Δ^{im′}. By induction on n the result (5) follows.

The converse is straightforward and is left to the reader. □

§3. Weyl Group Orbits of Positive Systems of Roots: 3.1. Let (·|·) : Γ × Γ → ℤ be the symmetric bilinear form whose associated matrix is

( \begin{matrix}(α_{i}|α_{j}) & (α_{i}|δ_{k′}) & (α_{i}|Λ_{k′}) \\ (δ_{k}|α_{j}) & (δ_{k}|δ_{k′}) & (δ_{k}|Λ_{k′}) \\ (Λ_{k}|α_{j}) & (Λ_{k}|δ_{k′}) & (Λ_{k}|Λ_{k′}) \end{matrix} ) = ( \begin{matrix}Ȧ & 0 & 0 \\ 0 & 0 & 1_{n×n} \\ 0 & 1_{n×n} & 0 \end{matrix} ) .

Note that this extends the form defined by Ȧ on Q̇. The set of real roots is defined to be the set Δ^{re} = Δ̇ + Δ^{im}. For each μ ∈ Δ^{re}, let r_{μ} denote the reflection with respect to (·|·) i.e. r_{μ}λ := λ − (λ|μ)μ. Let θ denote the longest root in Δ with respect to the basis Π̇ = {α_{1}, …, α_{l}}. For each k ∈ J set α_{k} := δ_{k} − θ and for each i ∈ I set r_{i} = r_{α_{i}}. In addition

Q :=Q^{[n]} := Q̇ ⊕ Δ^{im} is the n-toroidal root lattice,
W :=W^{[n]} := ⟨r_{i}| i ∈ I⟩ is the n-toroidal Weyl group,
Π :=Π^{[n]} := {α_{1}, …, α_{l+n}} is the set of simple roots.

For j ∈ J and α ∈ Q define linear maps on Γ by

(1)  t_{α}^{(j)}λ = λ + (λ|δ_{j})α − [(λ|α) + \frac{1}{2}(α|α)(λ|δ_{j})] δ_{j}

and set H := ⟨t_{α}^{(j)}| j ∈ J, α ∈ Q⟩. Then Lemma [MS]. For l ≥ 2, one has W = Ẇ ⋉ H where the isomorphism is implemented by sending r_{j} to r_{θ}t_{θ}^{(j)} for j ∈ J. 3.2. The set of roots in Q is the subset Δ = Δ^{re} ∪ Δ^{im} ⊂ Q. The standard set of positive imaginary roots is defined to be the set

Δ_{+}^{im} := (∪_{i=1}^{n−1}{v ∈ Δ|(v|Λ_{j}) = 0 for l + 1 ≤ j ≤ l + i − 1, (v|Λ_{l+i}) > 0})∪{v ∈ Δ| (v|Λ_{l+n}) > 0}∪{0}.

A subset P ⊂ Δ we will call a positive system of roots if (i) P ∩ −P = 0, (ii) P ∪ −P = Δ, (iii) P is closed i.e. if α, β ∈ P with α + β ∈ Δ then α + β ∈ P,

and we will say P is normal if, in addition to the above, (iv) P ⊃ Δ_{+}^{im}. Observe that the Weyl group W preserves the set of positive imaginary roots Δ_{+}^{im} .

Theorem 2.2 implies that the set of positive root systems is parameterized by k-frames in n-space so if P and P′ are two sets of positive roots, then P and P′ are conjugate under the Weyl group action only if P ∩ Δ_{+}^{im} = P′ ∩ Δ_{+}^{im} .

For ξ ∈ Γ and r ∈ ℝ, define

H_{ξ,r}^{+} : = {β ∈ Δ| (β|ξ) > r},
H_{ξ,r}^{−} : = {β ∈ Δ| (β|ξ) < r}, and
H_{ξ,r} : = {β ∈ Δ| (β|ξ) = r}.

We then have the following Lemma. For w ∈ W, one has wH_{ξ,r}^{±} = H_{wξ,r}^{±}, wH_{ξ,r} = H_{wξ,r} and

t_{α}^{(j)}H_{ξ,r}^{±} ∩ (±α + Δ^{im}) = H_{ξ,r∓(α|α)(δ_{j}|ξ)}^{±} ∩ (±α + Δ^{im}).

Proof. The first two equalities are well known and follow from the fact that the form (−, −) is W -invariant. The last equality follows from 3.1.1 since

(t_{α}^{(j)}β|ξ) = (β − (β|α)δ_{j}|ξ) > r − (β|α)(δ_{j}|ξ)

if and only if β ∈ H_{ξ,r}^{+}. □

We now turn to the case of A of rank 1 and n = 2 leaving the higher rank cases to future study. Then

Δ̇ = {±α}, Δ^{im} = ℤδ_{1} + ℤδ_{2}, Q = ℤα + ℤδ_{1} + ℤδ_{2},

Δ_{+}^{im} = {nδ_{1} + mδ_{2}|n ∈ ℤ, m > 0} ∪ {nδ_{1}| n ≥ 0}, α_{i} = δ_{i} − α, and Π = {α, α_{1}, α_{2}}.

Set

Δ_{i} = Δ̇ + ℤδ_{i},
P_{i}^{n} = {α + nδ_{i}|n ∈ ℤ},
P_{i}^{s} = {α + nδ_{i}, −α + mδ_{i}| n, m ∈ ℤ, n ≥ 0, m > 0}, i = 1, 2
T_{1} = (α + Δ^{im}) ∪ Δ_{+}^{im},
T_{2} = (α + ℤδ_{1} + Δ_{+}^{im}) ∪ (−α + ℤδ_{1} + δ_{2} + Δ_{+}^{im}) ∪ Δ_{+}^{im},
T_{3} = (α + Δ_{+}^{im}) ∪ (−α + δ_{1} + Δ_{+}^{im}) ∪ Δ_{+}^{im},
T_{4} = (α + δ_{1} − δ_{2} + Δ_{+}^{im}) ∪ (−α + δ_{2} + Δ_{+}^{im}) ∪ Δ_{+}^{im}

Observe that T_{i}, i = 1, 2, 3, 4 satisfy 3.2.(iii). We will soon see that they satisfy the other axioms 3.2.(i) and (ii).

To help facilitate the arguments below the α-plane, (α + Δ^{im}) ∩ T_{i}, of T_{i} are as follows:

δ_{1} δ_{2}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}  ^{δ_{1} δ_{2}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}}
T_{1} T_{2}

δ_{1} δ_{2}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•}  ^{δ_{1} δ_{2}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•}}
T_{3} T_{4}

The points (m, n) above give the integral coordinates of the vectors α + mδ_{1} + nδ_{n} ∈ T_{i}. The reader is encouraged to write down the −α-plane to see that if β ∈ (α + Δ^{im})\T_{i} then −β ∈ T_{i} ∩(−α+Δ^{im}). This shows axiom 3.2.(ii) is satisified and the proof for (i) is similar. 3.3 Remark. The T_{i} are normal positive root systems for i = \overline{1, 4}.

3.4 Theorem.

(i) The T_{i} are W -non-equivalent.

(ii) If P is a normal positive system in Δ, then there exists w ∈ W such that wP = T_{i} for some i = \overline{1, 4}.

Proof. (i). We will now show that T_{4} is not equivalent to T_{i} for i = \overline{1, 3}. Assume that wT_{4} = T_{i} for some w ∈ W and i ∈ \overline{1, 3}. By Corollary 1.9 [MS], we have w = \overline{w} ∏ t_{β_{j}}^{j}, where \overline{w} ∈ \{1, s_{α}\} and β_{j} ∈ ℤα. Consider the elements α + δ_{1} − δ_{2} and −α + δ_{1} + δ_{2} in T_{4}. Let w′ = ∏ t_{β_{j}}^{(j)} so that w = \overline{w}w′. Then

w′(α + δ_{1} − δ_{2}) = α + δ_{1} − δ_{2} + 2k_{1}δ_{1} + 2k_{2}δ_{2}
= α + (2k_{1} + 1) δ_{1} + (2k_{2} − 1) δ_{2}

for some k_{1}, k_{2} ∈ ℤ and

w′(−α + δ_{1} + δ_{2}) = −α + δ_{1} + δ_{2} − 2k_{1}δ_{1} − 2k_{2}δ_{2}
= −α + (1 − 2k_{1}) δ_{1} − (2k_{2} − 1) δ_{2}.

Since the coefficients in front of δ_{2} are of different signs, one has that w(α + δ_{1} − δ_{2}) and w(−α + δ_{1} + δ_{2}) can not both be in any of the T_{i}, i ∈ \overline{1, 3}. In a similar manner it is straightforward to show that T_{i} is not equivalent to T_{j} if i, j ∈ {1, 2, 3}, with i ≠ j. This completes the proof of the first part of the Theorem. (ii). Let P̃ be a normal positive system in Δ and P = P̃\Δ_{+}^{im}. Let W_{i} denote the Weyl group of Δ_{i}, i = 1, 2. Consider P̃ ∩Δ_{2} which is a positive system for Δ_{2} and therefore there exists w ∈ W_{2} such that w(P ∩ Δ_{2}) = P_{2}^{(i)} for i ∈ {1, 2} (see [JK] and [F]).

Assume that i = 1. Then we immediately obtain that

wP = P_{2}^{n} + Δ_{+}^{im} and wP̃ = T_{1}.

If i = 2, then P_{2}^{s} + Δ_{+}^{im} ⊂ wP.

On the other hand −α + δ_{2} ∈ wP̃ and since P̃ is a positive set of roots, we have

∅ = −wP̃ ∩ (−α + δ_{2} + Δ_{+}^{im})

which implies ∅ = wP̃∩(α−δ_{2}−Δ_{+}^{im}) so that wP̃∩(α+Δ^{im}) ⊆ α−δ_{2}+Δ_{+}^{im} = T_{4}∩(α+Δ^{im}). Now we have that the α-plane for wP̃ lies between the α-plane for T_{3} and T_{4}. (a). If α + nδ_{1} − δ_{2} ∈ wP for some n > 0, then P_{2}^{s} + ℤδ_{1} ⊂ wP. Adding suitable elements from Δ_{+}^{im} to α + nδ_{1} − δ_{2} we see that all points on the α-plane lying on the δ_{1}-axis and all points to the right of α + nδ_{1} − δ_{2} are in wP̃. Translating wP̃ ∩ (α + Δ^{im}) to the left by suitable multiples of 2δ_{1} (see Lemma 3.2) we can conclude that wP̃ is equivalent to either T_{4} or a positive root system whose α-plane looks like

δ_{1} δ_{2}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•} ^{•} ^{•} ^{•} ^{•} ^{•}^{•} ^{•}

Translating this last α-plane up and to the left by −2δ_{1} + 2δ_{2} and then applying the reflection s_{α}, we obtain that wP̃ is equivalent to T_{4}.

(b). Consider now when α + nδ_{1} − δ_{2} ∉ wP̃ for all n > 0. Then either wP ∩ Δ_{1} = P^{n} ∩ Δ_{1} or wP ∩ Δ_{1} is equivalent under a suitable Weyl group reflection from the affine Weyl group for Δ_{1} to P^{s} ∩ Δ_{1} (see Lemma 3.2 or [JK] and [F]). The first case (resp. second) gives us that wP̃ is equivalent to T_{2} (resp. T_{3}). □

§4. Submodules of Verma modules:

4.1 For any finite dimensional simple Lie algebra 𝔤 and any m ≥ 1 we define the algebras 𝔗_{[m]}(𝔤) = R_{m} ⊗ 𝔤 where A = R_{m} = ℂ[t_{1}^{±1}, …, t_{m}^{±1}] and 𝔗̂ = 𝔗̂_{[m]}(𝔤) = 𝔗_{[m]}(𝔤) ⊕ (Ω_{A}/dA). Letting z be the central ideal of 𝔗̂ consisting of the span of the homogeneous elements of non-zero degree in Ω_{A}/dA we obtain the algebra

\overline{𝔗_{[m]}(𝔤)} = \overline{𝔗} = 𝔗 ⊕ (⊕_{i=1}^{m}ℂc_{i})

where c_{i} denotes the class of the element \overline{t_{i}^{−1} dt_{i}} ∈ Ω/dA in \overline{𝔗}. The algebra \overline{𝔗} is still graded by ℤ^{l+m} with deg c_{i} = 0, for 1 ≤ i ≤ m, and we let d_{i} be the i-th degree derivation on \overline{𝔗} so that d_{i}(t_{1}^{n_{1}} … t_{m}^{n_{m}} x) = n_{i}(t_{1}^{n_{1}} … t_{m}^{n_{m}} x) where t_{1}^{n_{1}} … t_{m}^{n_{m}} x := t_{1}^{n_{1}} … t_{m}^{n_{m}} ⊗ x and d_{i}(c_{j}) = 0 for 1 ≤ i, j ≤ m. Set deg_{i}(t_{1}^{n_{1}} … t_{m}^{n_{m}} x) = n_{i} and let D be the abelian Lie algebra spanned by d_{1}, …, d_{m}. We form the semi-direct product of \overline{𝔗} and D and denote this by 𝔱 = 𝔱_{[m]}(𝔤). Thus

𝔱 = 𝔗 ⊕ (⊕_{i=1}^{m}ℂc_{i}) ⊕ (⊕_{i=1}^{m}ℂd_{i}).

We have that [d_{i}, t_{1}^{n_{1}} … t_{m}^{n_{m}} x] = n_{i}(t_{1}^{n_{1}} … t_{m}^{n_{m}} x) and [d_{i}, c_{j}] = 0 for 1 ≤ i, j ≤ m. 𝔱 is graded by ℤ^{l+m} and the space of degree 0, 𝔱_{0}, is of dimension 2m + l where l is the rank of 𝔤. For 1 ≤ k ≤ m we let R_{[k]} = ℂ[t_{1}^{±1}, …, t_{k}^{±1}] and let R_{[0]} = ℂ. Also, define

R_{[k]}^{+} = ℂ[t_{1}^{±1}, …, t_{k−1}^{±1}, t_{k}]t_{k},

and

R_{[k]}^{−} = ℂ[t_{1}^{±1}, …, t_{k−1}^{±1}, t_{k}^{−1}]t_{k}^{−1}

so that R_{[k]}^{±} = R_{k−1}[t_{k}^{±1}]t_{k}^{±1} for 1 ≤ k ≤ m and we view all of these in R_{[m]}. Let Π = {α_{1}, …, α_{l}} be a simple system of roots of 𝔤 with respect to a Cartan subalgebra ℌ and let Δ̇ = Δ̇_{+} ∪ Δ̇_{−} be the root system so that we have the usual triangular decomposition 𝔤 = 𝔫^{+} ⊕ ℌ ⊕ 𝔫^{−} where 𝔫^{±} = ⊕_{α∈Δ̇_{±}}𝔤_{α}. We define

𝔱_{+} = 𝔫^{+} ⊕ (⊕_{k=1}^{m}(R_{[k]}^{+} ⊗ 𝔤))

and

𝔱_{−} = 𝔫^{−} ⊕ (⊕_{k=1}^{m}(R_{[k]}^{−} ⊗ 𝔤)).

Thus, we have the decomposition 𝔱 = 𝔱_{+} ⊕ 𝔱_{0} ⊕ 𝔱_{−}. For example, all roots of 𝔱 are of the form γ = (β, (n_{1}, …, n_{m})) where β ∈ Δ̇ ∪ {0} and n_{i} ∈ ℤ, 1 ≤ i ≤ m and we have that the corresponding root space is in 𝔱_{±} if and only if either there is some j satisfying 1 ≤ j ≤ m and n_{j} > 0 but n_{j+k} = 0 for 1 ≤ k ≤ m − j or n_{1} = ⋯ = n_{m} = 0 but β ∈ Δ̇_{+}.

4.2 For any Lie algebra 𝔩, let U(𝔩) denote its universal enveloping algebra. For V a 𝔱module and λ ∈ 𝔱_{0}^{∗}, let V_{λ} := {v| hv = λ(h)v for all h ∈ 𝔱_{0}} be the λ-th weight space of V. Let P (V) = {λ ∈ 𝔱_{0}^{∗}| V_{λ} ≠ 0} be the set of weights of V.

Next, let 𝔟^{+} = 𝔱_{+} ⊕ 𝔱_{0} and define for any λ ∈ 𝔱_{0}^{∗} a one-dimensional 𝔟^{+}-module ℂ_{λ} = ℂv_{λ} by requiring that 𝔱_{+}v_{λ} = 0 and h.v_{λ} = λ(h)v_{λ} for all h ∈ 𝔱_{0}. Then our Verma modules are the induced modules

M(λ) = U(𝔱) ⊗_{U(𝔟^{+})} ℂ_{λ}

which is as a vector space isomorphic to U(𝔱_{−}). Also it is clear that M(λ) has a weight space decomposition M(λ) = ⊕_{β∈P (M(λ))}M(λ)_{β}, where M(λ)_{β} denotes the weight space corresponding to β ∈ 𝔱_{0}^{∗}, and so does any submodule.

From now on we will study the submodule structure of the M(λ) in the case when m = 2. Here we let s = t_{1} and t = t_{2} denote the variables and c_{s}, c_{t}, d_{s} and d_{t} the corresponding central elements and degree derivations. Thus, we have

𝔱 = 𝔱_{2}(𝔤) = (R_{[2]} ⊗ 𝔤) ⊕ ℂc_{s} ⊕ ℂc_{t} ⊕ ℂd_{s} ⊕ ℂd_{t}

and

[d_{s}, s^{n_{1}}t^{n_{2}} x] = n_{1}(s^{n_{1}}t^{n_{2}} x), [d_{t}, s^{n_{1}}t^{n_{2}} x] = n_{2}(s^{n_{1}}t^{n_{2}} x)

for x ∈ 𝔤, n_{1}, n_{2} ∈ ℤ. Set also deg_{s} s^{n_{1}}t^{n_{2}} x = n_{1}, deg_{s}(𝔱_{0}\{0}) = 0 , deg_{t} s^{n_{1}}t^{n_{2}} x = n_{2} and deg_{t}(𝔱_{0}\{0}) = 0 both of which uniquely extend to degree derivations of the enveloping algebra of 𝔱. The usual affine Lie algebra, 𝔤̂, is the subalgebra of 𝔱 given by 𝔤̂ = (ℂ[s, s^{−1}] ⊗ 𝔤) ⊕ ℂc_{s} ⊕ ℂd_{s} so that 𝔤̂ = 𝔫̂^{+} ⊕ ℌ̂ ⊕ 𝔫̂^{−} where

𝔫̂^{+} = 𝔫^{+} ⊕ (ℂ[s]s ⊗ 𝔤), 𝔫̂^{−} = 𝔫^{−} ⊕ (ℂ[s^{−1}]s^{−1} ⊗ 𝔤),

and

ℌ̂ = ℌ ⊕ ℂc_{s} ⊕ ℂd_{s}.

We also let 𝔱_{±1} = 𝔫̂^{±}

𝔱_{2} = ℂ[s, s^{−1}, t]t ⊗ 𝔤, 𝔱_{−2} = ℂ[s, s^{−1}, t^{−1}]t^{−1} ⊗ 𝔤

so that 𝔱_{±} = 𝔱_{±1} ⊕ 𝔱_{±2}.

Let δ_{s}, δ_{t}, Λ_{s}, and Λ_{t} in 𝔱_{0}^{∗} be the linear functionals in 𝔱_{0}^{∗} dual to d_{s}, d_{t}, c_{s}, and c_{t} respectively. We thus have

𝔱_{0}^{∗} = ℌ^{∗} ⊕ ℂΛ_{s} ⊕ ℂΛ_{t} ⊕ ℂδ_{s} ⊕ ℂδ_{t}

and

ℌ̂^{∗} = ℌ^{∗} ⊕ ℂΛ_{s} ⊕ ℂδ_{s}.

For λ ∈ 𝔱_{0}^{∗} we let λ̂ denote the restriction of λ to ℌ̂^{∗}. Now the usual Verma module corresponding to λ̂ for 𝔤̂ is M(λ̂) = M̂(λ) = U(𝔤̂) ⊗_{U(𝔫̂^{+}⊕ℌ̂)} ℂ_{λ} and it is easy to see this may be regarded as a submodule of M(λ) when we treat M(λ) as 𝔤̂-module. Clearly,

M̂(λ) = U(𝔱_{−1})v_{λ} and M(λ) ≅ U(𝔱_{−2}) ⊗ M̂(λ).

For γ ∈ Δ let 𝔱_{γ} be a corresponding root subspace of 𝔱. Let γ ∈ Δ^{im} and suppose x_{1}, ..., x_{k} ∈ 𝔱_{γ} are linearly independent. The elements x̄_{1}, …, x̄_{k} ∈ 𝔱_{−γ} are called dual to x_{1}, …, x_{k} if [x_{i}, x̄_{j}] = 0, for i ≠ j, and [x_{i}, x̄_{i}] ≠ 0. The existence of dual elements follows from the construction of algebra 𝔱 .

If N is a submodule of M(λ) then set N̂ = N ∩ M̂(λ). It was shown in [BC] that N̂ ≠ 0 if N ≠ 0. The following result describes the submodule structure of M(λ) when λ(c_{s}) ≠ 0.

4.3 Theorem. If λ(c_{s}) ≠ 0 and N ⊂ M(λ), then N ≅ U(𝔱_{−2}) ⊗ N̂.

To make the proof more palatable we need some preliminary notation and results. First we deal with the notation:

We need a total ordering on a basis of U(𝔱_{−}) and this is done as follows. Let {x_{i}}_{i∈ℤ} be a basis of the loop algebra L(𝔤) := ℂ[s, s^{−1}] ⊗ 𝔤 where x_{i} is in the root space 𝔤̂_{β_{i}} and we demand that if β_{i} < β_{j} in the usual ordering, then i < j. Note that we can write x_{i} = s^{n_{i}} ẋ_{i} for some root vector ẋ_{i} ∈ 𝔤β̇_{i}, β̇_{i} ∈ Δ̇ ∪ {0} and n_{i} ∈ ℤ. That is to say that our indexing of the x_{i}’s is consistent with the usual ordering of roots of 𝔤̂ and 𝔤. In particular β_{i} < β_{j} if either (i) n_{i} = n_{j} and β̇_{i} < β̇_{j} or (ii) n_{i} < n_{j}. We also stipulate that {c_{s}, d_{s}, x_{i}}_{i≥0} is a basis of ℌ ⊕ 𝔫^{+} while {x_{i}}_{i≤−1} is a basis of 𝔫^{−}. The Lie algebra 𝔱_{−} has as a basis the set of elements B(𝔱_{−}) := {x_{i}}_{i≤−1} ∪ {t^{m} x_{i} | i, m ∈ ℤ, m < 0} where t^{m} x_{i} := s^{n}t^{m}ẋ_{i} if x_{i} = s^{n_{i}} ẋ_{i}. We totally order B(𝔱_{−}) by declaring t^{m}x_{i} < t^{n}x_{j} if either m < n or m = n and i < j. Let α = (α_{1}, ⋯, α_{γ}) be the usual multi-index notation and define |α| = ∑_{i=1}^{γ} α_{i}. Now by the Poincaré-Birkhoff-Witt thereom U(𝔱_{−}) has as a basis the set of monomials of the form (t^{m}x_{i})^{p}y_{j}^{q} together with 1 where

(t^{m}x_{i})^{p} = (t^{m_{1}}x_{i_{1}})^{p_{1}} ⋯ (t^{m_{r}}x_{i_{r}})^{p_{r}},

y_{j}^{q} = y_{j_{1}}^{q_{1}} ⋯ y_{j_{s}}^{q_{s}}, (1)

x_{i}, y_{j} ∈ 𝔤̂ are elements in the basis fixed above The indices m_{k}, i_{k}, p_{k}, j_{l}, q_{l} lie in ℤ while p_{k} and q_{l} are assumed to be positive and the m_{k} are negative. The factors in (1) are assumed to satisfy t^{m_{k}}x_{i_{k}} < t^{m_{k+1}}x_{i_{k+1}} and y_{j_{l}} < y_{j_{l+1}}. If r > 0, then let ϵ_{i} = (0, …, 1, …, 0) be the r-tuple with zeros everywhere except in the i-th entry. Then (t^{m}x_{i})^{p−ϵ_{k}} will denote the same basis element as t^{m}x_{i}^{p} except we have decreased the power of t^{m_{k}}x_{i_{k}} by one. We lexicographically order the basis (t^{m}x_{i})^{p}y_{j}^{q} of U(𝔱_{−}) and declare 1 the largest element in this set. (See [BC] for a more explicit description of this ordering.) In particular

(t^{m}x_{i})^{p}y_{j}^{q} < (t^{a}x_{b})^{c}y_{e}^{f}

if and only if either (i) (t^{m}x_{i})^{p} < (t^{a}x_{b})^{c} or (ii) (t^{m}x_{i})^{p} = (t^{a}x_{b})^{c} and y_{j}^{q} < y_{e}^{f}. Remark: The y_{j}^{q} together with 1 form a basis of U(𝔫_{−}). Suppose that {y_{j_{γ}}^{q_{γ}} | γ ∈ J} ∪ {1} is a subset of this basis with J finite. Then given any finite set of linearly independent root vectors {s^{−n_{ξ}}y_{ξ}|ξ ∈ Ξ} in 𝔤̂ such that the n_{ξ} are sufficiently large integers, the set {(s^{−n_{ξ}}y_{ξ})y_{j_{γ}}^{q_{γ}} | ξ ∈ Ξ, γ ∈ J} ∪ {s^{−n_{ξ}}y_{ξ}|ξ ∈ Ξ} is linearly independent. This follows from the Poincaré-Birkhoff-Witt theorem as the s^{−n_{ξ}}y_{ξ} are nonzero multiples of basis elements of 𝔤̂. This remark will play an important role in the arguments below.

One more piece of notation: Recall that U(𝔱_{−}) has a canonical filtration {U_{n}}_{n≥0} induced from the gradation of the tensor algebra T (𝔱_{−}). This induces a filtration on the Verma module M(λ) which we denote by {M_{n}}_{n≥0}. The first result we need is 4.4 Lemma (cf. [BC]). (a). Consider a basis element (t^{m}x_{i})^{p} of U(𝔱_{−}). Let z be a root vector in 𝔤̂ with deg_{s} z + max_{k}{deg_{s} x_{i_{k}}} < deg_{s} y_{j}^{q} and n = |p| + |q|. Then

(t^{m_{1}}z)(t^{m}x_{i})^{p}y_{j}^{q}v_{λ} = ∑_{1≤k≤r:m_{k}=m_{1}} p_{k}(t^{m}x_{i})^{p−ϵ_{k}}[z, x_{i_{k}}]y_{j}^{q}v_{λ} mod M_{n−1}

where those summands on the right that are nonzero form a linearly independent set by remark 4.3. Here v_{λ} is the highest weight vector of M(λ). Moreover

(t^{m}z)(t^{m}x_{i})^{p}y_{j}^{q}v_{λ} = 0 mod M_{n−1}

if m > m_{1}. (b). Consider another basis element (t^{a}x_{b})^{c}y_{g}^{h} that has the same weight as (t^{a}x_{b})^{c} in U(𝔱_{−}). Let μ be such that a_{μ} = a_{1} but a_{μ+1} > a_{1}. In addition to the hypothesis of (a) assume that deg_{s} z + max_{k}{deg_{s} x_{b_{k}}} < deg_{s} y_{g}^{h}. Then

(1)  0 ≠ (  ∑_{1≤k≤r:m_{k}=m_{1}} ℂ(t^{m}x_{i})^{p−ϵ_{k}}[z, x_{i_{k}}]y_{j}^{q}v_{λ}) ∩ ℂ(t^{a}x_{b})^{c−ϵ_{μ}}[z, x_{b_{μ}}]y_{g}^{h}v_{λ}

if and only if

y_{j}^{q} = y_{g}^{h}, 0 ≠ [z, x_{b_{μ}}] ∈ ∑_{1≤k≤r:m_{k}=m_{1}} ℂ[z, x_{i_{k}}],

and

(t^{m}x_{i})^{p} = (t^{a}x_{b})^{c−ϵ_{μ}}(t^{a_{μ}}x_{i_{k}}) mod U_{|c|−1}

for some 1 ≤ k ≤ r with β_{i_{k}} = β_{b_{μ}} (i.e. x_{i_{k}} and x_{b_{μ}} lie in the same root space). Proof. As ad t^{m}z is a derivation on U(𝔱) we have

[t^{m}z, (t^{m}x_{i})^{p}] = ∑_{k=1}^{r}(t^{m_{1}}x_{i_{1}})^{p_{1}} ⋯ [t^{m}z, (t^{m_{k}}x_{i_{k}})^{p_{k}}] ⋯ (t^{m_{r}}x_{i_{r}})^{p_{r}}.

Part (a) of the lemma follows as (t^{m−m_{k}}[z, x_{i_{k}}])y_{j}^{q}v_{λ} = 0 for m_{k} < m and

[t^{m}z, (t^{m_{k}}x_{i_{k}})^{p_{k}}] = ∑_{a=0}^{p_{k}−1}(t^{m_{k}}x_{i_{k}})^{a}(t^{m−m_{k}}[z, x_{i_{k}}])(t^{m_{k}}x_{i_{k}})^{p_{k}−1−a}
= p_{k}(t^{m_{k}}x_{i_{k}})^{p_{k}−1}(t^{m−m_{k}}[z, x_{i_{k}}]) mod U_{p_{k}−1}.

Now suppose that (1) is true. Then since (t^{m}x_{i})^{p} and (t^{a}x_{b})^{c} have the same degree in t we have

a_{μ} = deg_{t}(t^{a}x_{b})^{c} − deg_{t}(t^{a}x_{b})^{c−ϵ_{μ}} = deg_{t}(t^{m}x_{i})^{p} − deg_{t}(t^{m}x_{i})^{p−ϵ_{k}} = m_{k}

for some k. The Poincaré-Birkhoff-Witt theorem together with the hypothesis on z imply that (t^{a}x_{b})^{c−ϵ_{μ}} = (t^{m}x_{i})^{p−ϵ_{k}}, 0 ≠ ℂ[z, x_{i_{k}}] = ℂ[z, x_{b_{μ}}] and y_{j}^{q} = y_{g}^{h}. Finally this implies that x_{i_{k}} and x_{b_{μ}} lie in the same weight space.

The converse of part (b) is left to the reader. □

4.5 Lemma. Let N be a submodule of M(λ) and suppose c_{s} acts by a non-zero scalar on M(λ). Suppose that 0 ≠ uv_{λ} ∈ N where

u = ∑_{i∈I} x_{i}u_{i}.

Here x_{i} are amongst the linearly independent basis vectors of 𝔱_{−} given in section 4.3, and u_{i} ∈ U(𝔱_{−1}). If max_{j∈I}{[deg_{s} x_{j}/2]} < min_{i∈I}{deg_{s} u_{i}} where [k] denotes the greatest integer less than or equal to k, then u_{i}v_{λ} ∈ N for all i.

Proof. The proof is by induction on |I|. Let |I| = 1 so that u = (s^{−m}ẋ_{β})u′ where ẋ_{β} is a root vector of the finite dimensional 𝔤. If β = 0, then we use the fact that (s^{m}ẋ)u′v_{λ} = 0, for any ẋ ∈ 𝔤, m ≫ 0, and λ(c_{s}) ≠ 0 to obtain u′v_{λ} ∈ N. More precisely this follows from the simple calculation

(s^{m}\overline{ẋ_{β}})uv_{λ} = (s^{−m}ẋ_{β})(s^{m}\overline{ẋ_{β}})u′v_{λ} − m(ẋ_{β}|\overline{ẋ_{β}})λ(c_{s})u′v_{λ} = mλ(c_{s})u′v_{λ}

for m ≫ 0. If β ≠ 0, then (s^{[m/2]}ẋ_{−β})u′v_{λ} = 0 for a non-zero element s^{[m/2]}ẋ_{−β} ∈ 𝔱_{−β+[m/2]δ_{s}}, since m ≫ 0. Hence, we obtain x = [s^{[m/2]}ẋ_{−β}, s^{−m}ẋ_{β}] ∈ 𝔱_{([m/2]−m)δ_{s}} and xu′v_{λ} ≠ 0 for m ≫ 0. Now

0 ≠ xu′v_{λ} = (s^{[m/2]}ẋ_{−β})(s^{−m}ẋ_{β})u′v_{λ} − (s^{−m}ẋ_{β})(s^{[m/2]}ẋ_{−β})u′v_{λ}
= (s^{[m/2]}ẋ_{−β})(s^{−m}ẋ_{β})u′v_{λ} ∈ N

and then we use the above case to finish the proof for |I| = 1.

Next consider the case that |I| > 1 and assume that our statement is proved in all cases with fewer than |I| summands. Let −N_{i} = deg_{s} x_{i}, x_{i} = s^{−N_{i}}ẋ_{β_{i}}^{(i)}, where ẋ_{β_{i}}^{(i)} is a root vector in 𝔤_{β_{i}}, n = min_{i∈I} N_{i} and let I′ ⊂ I be the subset consisting of all indices j for which N_{j} = n.

Suppose that there exists a j ∈ I such that β_{j} ≠ 0. If there exists k ∈ I′ such that β_{k} = 0, then consider a non-zero element s^{[n/2]}ẋ ∈ 𝔱_{[n/2]δ_{s}} such that [s^{[n/2]}ẋ, s^{−n}ẋ_{β_{j}}^{(j)}] ≠ 0. Applying s^{[n/2]}ẋ to uv_{λ} we get

(*)  (s^{[n/2]}ẋ)uv_{λ} == ∑_{i∈I′}(s^{[n/2]−n}[ẋ, ẋ_{β_{i}}^{(i)}])u_{i}v_{λ} + ∑_{i∉I′}(s^{[n/2]−N_{i}}[ẋ, ẋ_{β_{i}}^{(i)}])u_{i}v_{λ}

for n ≫ 0. Since [s^{[n/2]}ẋ, s^{−n}ẋ_{β_{k}}^{(k)}] = 0, we have that the summand with index k is zero in (s^{[n/2]}ẋ)uv_{λ}, while summand with index j is not as β_{j} ≠ 0. By Remark 4.3 we have that for n ≫ 0 the nonzero summands in (*) are still linearly independent after applying s^{[n/2]}ẋ, so (s^{[n/2]}ẋ)uv_{λ} ≠ 0 and it has at least one less summand then uv_{λ}.

By induction we obtain u_{j}v_{λ} ∈ N and thus (u − (s^{−n}ẋ_{β_{j}}^{(j)})u_{j})v_{λ} ∈ N. Now apply induction one more time to (u − (s^{−n}ẋ_{β_{j}}^{(j)})u_{j})v_{λ} to get the desired result.

Suppose next that β_{i} ≠ 0 for all i ∈ I′. Then fix i ∈ I′ and apply a non-zero s^{[n/2]}ẋ_{−β_{i}} ∈ 𝔱_{−β_{i}+[n/2]δ_{s}} to uv_{λ}. Again, as n ≫ 0, we have that (s^{[n/2]}ẋ_{−β_{i}})uv_{λ} ≠ 0 and the summation (*) contains a term xu_{i}v_{λ} with x ∈ 𝔱_{([n/2]−n)δ_{s}} and we have not increased the number of summands. Now we can apply the previous paragraph to finish the proof.

Finally we consider the case when β_{j} = 0 for all j ∈ I. Since s^{−n}ẋ_{β_{i}}^{(i)}, i ∈ I′, are linearly independent, consider the dual elements s^{n}\overline{ẋ_{β_{i}}^{(i)}} ∈ 𝔱_{nδ_{s}}. Fix i_{0} ∈ I′ and apply s^{n}\overline{ẋ_{β_{i_{0}}}^{(i_{0})}} to uv_{λ}. Using the fact that [s^{n}\overline{ẋ_{β_{i_{0}}}^{(i_{0})}}, s^{−N_{j}}ẋ_{β_{j}}^{(j)}] = 0 for j ≠ i_{0} and λ(c_{s}) ≠ 0, we obtain u_{i_{0}}v_{λ} ∈ N and thus (u − (s^{−n}ẋ_{β_{i_{0}}}^{(i_{0})})u_{i_{0}})v_{λ} ∈ N. Now applying induction on |I| we finish the proof of the Lemma. □

4.6 Lemma. Let N be a submodule of M(λ) and suppose c_{s} acts by a non-zero scalar on M(λ). Let n ∈ ℤ, m > 0 and suppose that 0 ≠ uv_{λ} ∈ N where

u = ∑_{i∈I}(s^{n}t^{−m}ḣ_{i})u_{i},

u_{i} (resp. ḣ_{i}) are linearly independent elements of U(𝔱_{−1}) (resp. 𝔥). Then u_{i}v_{λ} ∈ N for all i.

Proof. Let |I| = 1. Then u = (s^{n}t^{−m}ḣ)u′. Since the Killing form is non-degenerate on 𝔤, there exists α ∈ Δ̇ and ẏ ∈ 𝔤_{α} such that [ẏ, ḣ] ≠ 0. Consequently we can choose M ≫ 0 and s^{−M}t^{m}ẏ ∈ 𝔱_{α−Mδ_{s}+mδ_{t}} such that 0 ≠ (s^{−M}t^{m}ẏ)uv_{λ} = −α(ḣ)(s^{n−M} ẏ)u′v_{λ} with s^{n−M} ẏ ∈ 𝔱_{α−(M−n)δ_{s}}. Now using the fact that λ(c_{s}) ≠ 0 and Lemma 4.4, conclude that u′v_{λ} ∈ N.

Suppose now that |I| > 1 and j ∈ I. Again choose M ≫ 0 and a non-zero s^{−M}t^{m}ẏ ∈ 𝔱_{α−Mδ_{s}+mδ_{t}} such that [ẏ, ḣ_{j}] ≠ 0. Then 0 ≠ (s^{−M}t^{m}ẏ)uv_{λ} = ∑_{i∈I} a_{i}u_{i}v_{λ} with a_{i} := s^{n−M}[ẏ, ḣ_{i}] ∈ 𝔱_{α−(M−n)δ_{s}} (by Remark 4.3 and the fact that the u_{i} are linearly independent the summation is nonzero as M ≫ 0). Since a_{i} = b_{i}a_{j}, for some b_{i} ∈ ℂ, i ∈ I, we have a_{j} ∑_{i∈I} b_{i}u_{i} ∈ N. Again using λ(c_{s}) ≠ 0 and Lemma 4.4, conclude that u^{∗} := ∑_{i∈I} b_{i}u_{i} ∈ N with b_{j} ≠ 0. Hence, (u − (1/b_{j})(s^{n}t^{−m}ḣ_{j})u^{∗})v_{λ} = (∑_{i∈I,i≠j}((s^{n}t^{−m}ḣ_{i}) − (b_{i}/b_{j})(s^{n}t^{−m}ḣ_{j}))u_{i})v_{λ} ∈ N. Now induction on |I| completes the proof. □

4.7 Proof of Theorem 4.3. Consider a submodule N ⊂ M(λ) and 0 ≠ uv_{λ} ∈ N where u ∈ U(𝔱_{−}). By [MP, 2.1 Prop. 1] we may assume that u is a weight vector with respect to the action of 𝔱_{0}.

Let 0 ≠ uv_{λ} ∈ N with u ∈ U(𝔱_{−}). We can write

uv_{λ} = ∑_{(m,i,p,j,q)∈I} u_{i,j}^{m,p,q}(t^{m}x_{i})^{p}y_{j}^{q}v_{λ} ∈ N

with u_{i,j}^{m,p,q} ∈ ℂ. Our goal is to show that y_{j}^{q}v_{λ} ∈ N for all j and q with nonzero coefficient. We will prove this by induction on deg_{t} u and the number of elements in I.

Let (t^{a}x_{b})^{c}y_{g}^{h} be the largest monomial in the decompostion of u above with u_{b,g}^{a,c,h} ≠ 0. Let μ be the positive integer with a_{μ} = a_{1} but a_{μ+1} ≠ 0 and let x_{μ_{1}}, … x_{μ_{ξ}} be an enumeration of the basis for the root space 𝔤̂_{β_{μ}}. Set

I_{max} := {(i, m, p, g, h)| u_{i,g}^{m,p,h} ≠ 0, (t^{m}x_{i})^{p} = (t^{a}x_{b})^{c−ϵ_{μ}}x_{μ_{ζ}} mod U_{|c|−1}for some ζ}.

First assume that deg_{t} u = −1. In this case (t^{m}x_{i})^{p} = t^{−1}x_{i_{1}} for some basis vector x_{i_{1}} in 𝔤̂. The case that |I| = 1 is straightforward and follows from Lemma 4.6 (see the case deg_{t} u < −1 and |I| = 1 below).

Now assume |I| > 1. Then we still have m_{1} = −1 and

u = ∑_{l=1}^{ξ} u_{l}(t^{−1}x_{μ_{l}})y_{g}^{h} + ∑_{(i,j,m,p,q)∉I_{max}} u_{i,j}^{m,p,q}(t^{−1}x_{i})y_{j}^{q}

= (t^{−1}x)y_{g}^{h} + ∑_{(i,j,m,p,q)∉I_{max}} u_{i,j}^{m,p,q}(t^{−1}x_{i})y_{j}^{q}

where i := i_{1}, u_{l} ∈ ℂ and x = ∑_{l} u_{l}x_{μ_{l}} ∈ 𝔤̂_{β_{μ}}.

Then choosing z ∈ 𝔤̂ a root vector with [z, x] ≠ 0 and deg_{s} z ≪ 0 we have by Lemma 4.4 that

(tz)uv_{λ} = [z, x]y_{g}^{h}v_{λ} + ∑_{(i,j,m,p,q)∉I_{max}} u_{i,j}^{m,p,q}[z, x_{i}]y_{j}^{q}v_{λ}.

By Lemma 4.4.(b) the last summation can not contribute to cancel [z, x]y_{g}^{h}v_{λ}. Now we use Lemma 4.5 to obtain y_{g}^{h}v_{λ} ∈ N. Hence

(u − u_{b,g}^{a,c,h}(t^{a}x_{b})^{c}y_{g}^{h})v_{λ} ∈ N

and u − u_{b,g}^{a,c,h}(t^{a}x_{b})^{c}y_{g}^{h} contains one less summand than u. By induction on |I| we get that y_{j}^{q} ∈ N for all indices with nonzero coefficient u_{i,j}^{m,p,q}.

Now we consider the case that deg_{t} u < −1 and |I| = 1. Here we have u = α(t^{a}x_{b})^{c}y_{g}^{h} for some nonzero α ∈ ℂ. Then again choosing z ∈ 𝔤̂_{γ}, γ ∈ Δ ∪ {0} such that [z, x_{b_{μ}}] ≠ 0 and deg_{s} z ≪ 0 we get by Lemma 4.4(a)

(t^{−a_{1}}z)uv_{λ} = ∑_{1≤k≤μ} αc_{k}(t^{a}x_{b})^{c−ϵ_{k}}[z, x_{b_{k}}]y_{g}^{h}v_{λ} mod M_{n−1}

where n = |c| + |h|. Suppose (t^{a}x_{b})^{c−ϵ_{μ}} = 1. Then (t^{a}x_{b})^{c} = t^{a_{1}}x_{b_{1}} and

(1)  t^{−a+1}z(t^{a}x_{b})^{c}y_{g}^{h}v_{λ} = α[z, x_{b_{1}}]y_{g}^{h}v_{λ} ≠ 0.

On the other hand if (t^{a}x_{b})^{c−ϵ_{μ}} ≠ 1, then (t^{a}x_{b})^{c−ϵ_{k}} < (t^{a}x_{b})^{c−ϵ_{μ}} for k < μ. Then by the Poincaré-Birkhoff-Witt theorem the other summands in (1) can not contribute to cancel α(t^{a}x_{b})^{c−ϵ_{μ}}[z, x_{b_{μ}}]y_{g}^{h}v_{λ}. Since deg_{t}(t^{−a_{1}}z)u > deg_{t} u we can use induction (if necessary) to obtain [z, x_{b_{μ}}]y_{g}^{h}v_{λ} ∈ N. Lemma 4.4 gives us that y_{g}^{h}v_{λ} ∈ N.

Finally we consider the most complicated case when deg_{t} u < −1 and |I| > 1.

4.8. We close with the following remark about integrable quotients of the modules M(λ) for the algebra 𝔱 above. Remark. We let e_{i}, f_{i}, h_{i} for 1 ≤ i ≤ l be Chevalley generators the finite dimensional simple Lie algebra 𝔤̇ associated to 𝔤 and let θ be the highest root of the finite root system Δ̇ for 𝔤̇. As in [K] let ω̇ denote the Chevalley involution of 𝔤̇ and choose ḟ ∈ 𝔤̇ such that (ḟ|ω̇(ḟ)) = −2/(θ|θ) and set ė = −ω̇(ḟ). We define a set of generators of 𝔱 by adjoining to this set the elements

e_{(s)} := ė ⊗ s = se, f_{(s)} := ḟ ⊗ s^{−1} = s^{−1}f, and
e_{(t)} := ė ⊗ t = te, f_{(t)} := ḟ ⊗ t^{−1} = t^{−1}f

we see that the elements e_{i}, f_{i}, e_{(s)}, e_{(t)}, f_{(s)}, f_{(t)}, d_{s}, d_{t} with 1 ≤ i ≤ l generate 𝔱. We will say that a 𝔱-module V is integrable if V is 𝔱_{0} semi-simple and the generators above all act locally nilpotently on V.

Fix k ∈ {1, …, l} and f := f_{k}, e := e_{k} and h := [e, f]. For m_{i} ∈ ℤ_{>0} with 1 ≤ n we have in M(λ)

(1)  Π_{i=1}^{n}(s^{m_{i}}te)(t^{−1}f)^{n+1}v_{λ} = (−1)^{n}(n + 1)!n!(s^{m}t^{−1}f)v_{λ}

where m = ∑_{i=1}^{n} m_{i}. This is easily established by induction. Indeed for n = 1 this follows as s^{m}h.v_{λ} = 0 for m > 0. Now suppose that n > 1 and the result is true for n. Then

∏_{i=1}^{n+1}(s^{m_{i}}te)(t^{−1}f)^{n+2}v_{λ} = ∑_{i=1}^{n+1}(s^{m_{1}}te) ⋯ (s^{m_{i}}h) ⋯ (s^{m_{n+1}}te)(t^{−1}f)^{n+1}v_{λ}
+ (t^{−1}f)((s^{m_{n+1}}te)(−1)^{n}(n + 1)!n!(s^{m}t^{−1}f)v_{λ}

by induction. The last summand above is zero as s^{k}h.v_{λ} = 0 for k > 0. Now

∏_{i=1}^{n+1}(s^{m_{i}}te)(t^{−1}f)^{n+2}v_{λ} =

∑_{i=1}^{n+1}(s_{i}^{m}h)(s^{m_{1}}te) ⋯ { \atop (s^{m_{i}}te)} ⋯ (s^{m_{n+1}}te)(t^{−1}f)^{n+1}v_{λ}

− 2 ∑_{i=1}^{n+1} ∑_{j=1}^{i−1}(s^{m_{i}+m_{j}}te) ⋯ { \atop (s^{m_{i}}te)} ⋯ { \atop (s^{m_{j}}te)} ⋯ (s^{m_{n+1}}te)(t^{−1}f)^{n+1}v_{λ}

= ∑_{i=1}^{n+1}(−1)^{n}n!(n + 1)!s_{i}^{m}h(s^{m−m_{i}}t^{−1}f)v_{λ}

− 2 ∑_{i=1}^{n+1} ∑_{j=1}^{i−1}(−1)^{n}n!(n + 1)!(s^{m}t^{−1}f)^{n+1}v_{λ}

by induction on n (where the hat means that the expression below it should be omitted). Now if n > 0 we take m_{i} = 1 above for all i, then we have

(ste)^{n}(t^{−1}f)^{n+1}v_{λ} = (−1)^{n}(n + 1)!n! (s^{n}t^{−1}f)v_{λ}.

Thus

(2)  (s^{−n−k}te)(ste)^{n}(t^{−1}f)^{n+1}v_{λ} = (−1)^{n}(n + 1)!n!(s^{−k}h)v_{λ}

for all k > 0. Let N denote the submodule of M(λ) generated by (t^{−1}f)^{n}v_{λ} for some fixed n > 0. The equation (2) above then shows that s^{−k}e.v_{λ} and s^{−k}h.v_{λ} are in N for all k > 0. This together with equation (1) implies that

(3)  t^{−n}h.v_{λ} = (s^{−n}e)(s^{n}t^{−1}f)v_{λ} − (s^{n}t^{−1}f)(s^{−n}e)v_{λ} ∈ N

for n ≥ 1.

Hence if λ(c_{t}) ≠ 0 or λ(c_{s}) ≠ 0 then by the Stone-von Neumann theorem and the fact that {s^{−k}h.v_{λ}, t^{−k}h.v_{λ}} ⊂ N for k > 0, the vector (t^{−1}f)^{n+1}v_{λ} will generate as a submodule all of M(λ). On the other hand if λ(c_{t}) = 0 = λ(c_{s}) and λ(h) ≠ 0 then

(4)  (s^{−n}te)(st^{−1}e)^{n}(t^{−1}f)^{n+1}v_{λ} = (−1)^{n}(n + 1)!n!λ(h)v_{λ}

and again (t^{−1}f)^{n+1}v_{λ} will generate as a submodule all of M(λ). Now if I is an integrable quotient of M(λ) then the image of (t^{−1}f)^{n+1}v_{λ} must be zero in I for some n. Thus it follows now from (1) - (4) that if λ ≠ 0 then (t^{−1}f)^{n+1}v_{λ} is in the kernel of the surjection onto I and I must be zero. Consequently M(λ) has no non-trivial integrable quotients. §5. Highest Weight Series and an Equivalence of Categories:

5.1 Denote 𝔪̂ = 𝔤̂ + ℌ and 𝔭_{±} = 𝔪̂ ⊕ 𝔱_{±2}.

For any relation R on the set of integers and b an integer we set ℤ_{Rb} = {a ∈ ℤ| aRb}. Define

Δ_{+} = {α + mδ_{s} + nδ_{t}|α ∈ Δ̇ ∪ {0}, m ∈ ℤ, n ∈ ℤ_{>0}} ∪ Δ_{±}(𝔤̂) where
Δ_{±}(𝔤̂) = Δ̇_{±} ∪ {α + nδ_{s}| α ∈ Δ̇ ∪ {0}, ±n ∈ ℤ_{>0}}.

Let Q_{+} (resp. Q_{+}(𝔤̂)) denote the monoid in ℌ^{∗} generated by Δ_{+} (resp. Δ_{+}(𝔤̂)). Define λ < μ if μ − λ ∈ Q_{+} and λ <𝔤̂ μ if μ − λ ∈ Q_{+}(𝔤̂).

Let Π𝔪̂ = Π∪{α𝔤̂} denote the set of simple roots in Δ_{+}(𝔤̂) for 𝔪̂ where α𝔤̂ = −θ +δ_{s} and θ is the maximal root of Δ̇ with respect to Π. We also let β̌ denote the coroot of β ∈ Π𝔪̂.

Let λ ∈ 𝔱_{0}^{∗}. Define λ ↓ Q_{+} = {μ ∈ 𝔱_{0}^{∗}| μ ≤ λ} and λ ↓ Q_{+}(𝔤̂) = {μ ∈ 𝔱_{0}^{∗}| μ ≤𝔤̂ λ}.

For any Lie algebra 𝔩, let 𝔩 − mod denote the category of all left 𝔩-modules. For each λ ∈ 𝔱_{0} we let O_{λ}(𝔪̂) denote the full subcategory of 𝔪̂-mod whose objects M satisfy

(1)  M = ⨁_{ξ∈λ↓Q_{+}(𝔤̂)} M_{ξ}

and

(2)  dim M_{ξ} < ∞ for all ξ ∈ 𝔱_{0}^{∗}.

For λ ∈ 𝔱_{0}^{∗} with λ(c_{s}) ≠ 0, let O_{λ} denote the full subcategory of 𝔱-mod whose objects M satisfy

(1) M = ⨁_{ξ∈λ↓Q_{+}} M_{ξ},

(2) M is generated as a 𝔤̂-module by ⨁_{ξ∈λ↓Q_{+}(𝔤̂)} M_{ξ}, and

(3) ⨁_{ξ∈λ↓Q_{+}(𝔤̂)} M_{ξ} is a module in O_{λ}(𝔪̂).

We define R : O_{λ} → O_{λ}(𝔪̂) by

R(M) = ⨁_{ξ∈λ↓Q_{+}(𝔤̂)} M_{ξ} for M ∈ O_{λ}

and R(f) = f|_{R(M)} for any f ∈ Hom 𝔪̂(M, N) in the category O_{λ}. The functor R is exact since modules in O_{λ}(𝔪̂) are 𝔱_{0}-semisimple.

Let L(λ) denotes the unique irreducible quotient of M(λ). We can view ℂ_{λ} as an 𝔫̂_{+} ⊕𝔱_{0}- module by restriction and then define 𝔪̂-module M𝔪̂(λ) = U(𝔪̂) ⨂_{U(𝔫̂_{+}⊕𝔱_{0})} ℂ_{λ}. Let L𝔪̂(λ) denotes the unique irreducible quotient of M𝔪̂(λ). Finally, if N is an 𝔪̂-module then we can make it into a 𝔭_{+}-module by letting 𝔱_{2} act by zero and inducing up to 𝔱 we obtain a 𝔱-module U(N) := U(𝔱) ⨂_{U(𝔭_{+})} N. Observe that U defines a functor O_{λ}(𝔪̂) → O_{λ}. We will show that U ∘ R and R ∘ U are both equivalent to identity functors and hence define an equivalence of categories.

5.2 Let 𝔞 be a complex Lie algebra, 𝔰 a subalgebra of 𝔞 and σ : 𝔞 → 𝔞 a linear involutive anti-automorphism (i.e. σ^{2} = 1_{𝔞}, and σ([x, y]) = [σ(y), σ(x)] for all x, y ∈ 𝔞) such that

𝔰 + σ(𝔰) = 𝔞.

Let λ : 𝔰 → ℂ be a 1-dimensional representation of 𝔰. Following [JK] we say that a representation π : 𝔞 → gl(V) is a highest weight representation of highest weight λ (with respect to 𝔰) if there exists a vector v_{λ} ∈ V such that

π(U(𝔞))v_{λ} = V and
π(x)v_{λ} = λ(x)v_{λ} for x ∈ 𝔰.

Let M be an 𝔞-module. An 𝔞-highest weight series (with respect to 𝔰) for M is an increasing chain

(0) = M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ ⋯

of submodules of M such that (i) ∪_{i=0}^{∞}M_{i} = M and (ii) M_{i}/M_{i−1} is a highest weight module (with respect to 𝔰) for all i The following is a key result.

5.3 Proposition. . Let M be a module in O_{λ} and suppose that λ(c_{s}) ≠ 0. Then M has a highest weight series {M_{i}} where M_{i}/M_{i−1} is isomorphic to M(λ_{i})/U(N_{i}) for some N_{i} in O_{λ}(𝔪̂).

Proof. The proof is nearly identical to that of [C, 2.2] and based on the results about the categories of modules for Lie algebras with triangular decompositions (cf. [DGK], [GL],

[MP] and [RW]). We repeat it here for the convenience of the reader.

First of all R(M) is a module in O_{λ}(𝔪̂) so R(M) has an 𝔪̂-highest weight series {N_{i}}_{i=0}^{∞} (see [MP]). Define M_{i} = U(𝔱)N_{i}. Since P (M) ⊂ λ ↓ Q_{+}, we have that

U(𝔱_{2})𝔱_{2}N_{i} ⊂ U(𝔱_{2}))𝔱_{2}( ⨁_{β∈Q_{+}(𝔪̂)} M_{λ−β}) = 0.

By the Poincaré-Birkhoff-Witt Theorem it now follows that M_{i} = U(𝔱_{−2})𝔱_{−2}N_{i}. We thus have a surjective 𝔱-module homomorphism β_{i} : U(𝔱) ⨂_{U(𝔭_{+})} N_{i} → M_{i} induced by u⊗n ↦ un for u ∈ U(𝔱), n ∈ N_{i}. Since N_{i}/N_{i−1} is an 𝔪̂-highest weight module of highest weight λ_{i} we have a surjective 𝔪̂-module map

(1)  M𝔪̂(λ_{i}) → N_{i}/N_{i−1}.

Here M𝔪̂(λ_{i}) is the canonical Verma 𝔪̂-module. If we let 𝔱_{2} act by zero on M𝔪̂(λ_{i}) then (1) induces a surjective 𝔱̂-module map

M(λ_{i}) → U(𝔱̂) ⨂_{U(𝔭_{+})}(N_{i}/N_{i−1}) ≅ (U(𝔱̂) ⨂_{U(𝔭_{+})} N_{i})/(U(𝔪̂) ⨂_{U(𝔭_{+})} N_{i−1}).

Combining this with the canonical map induced by β_{i} we obtain a surjective 𝔱̂-module map

M(λ_{i}) → M_{i}/M_{i−1}.

Thus M_{i}/M_{i−1} is a highest weight 𝔱̂-module of highest weight λ_{i}. Since R(M) generates M as a 𝔱̂-module and R(M) = ∪_{i=0}^{∞}N_{i} we have M = ∪_{i=0}^{∞}M_{i}. Now every submodule of M(λ_{i}) is induced from a module in O_{λ_{i}}^{X}(𝔪̂) i.e. it is of the form U(N) for some N ∈ O_{λ_{i}}^{X}(𝔪̂) (Lemma 1.4), part (b) now follows. □

5.4 Theorem. For all λ ∈ 𝔱_{0}^{∗}, λ(c_{s}) ≠ 0, U ∘ R (resp. R ∘ U) is natural equivalent to the identity functor 1_{O_{λ}} (resp. 1_{O_{λ}(𝔪̂)}).

Proof. First consider N ∈ O_{λ}^{X}(𝔪̂). Viewing N ⊂ R ∘ U(N) we immediately obtain R ∘ U(N) = N.

If M ∈ O_{λ}^{X}, then M has a 𝔱̂-highest weight series {M_{i}} with

M_{i}/M_{i−1} ≅ U(M_{𝔪}(λ_{i})/N_{i−1})

for some 𝔪̂-module N_{i−1} by Proposition 2.2. Thus for all i ≥ 1 we have R(M_{i}/M_{i−1}) ≅ M𝔪̂(λ_{i})/N_{i−1} as 𝔪̂-modules so that M_{i}/M_{i−1} ≅ U ∘ R(M_{i}/M_{i−1}). A diagram chase shows that we have a commutative diagram

0 ⟶ U ∘ R(M_{i}) ⟶ U ∘ R(M_{i+1}) ⟶ U ∘ R(M_{i+1}/M_{i}) ⟶ 0
↓  ↓  ↓
0 ⟶  M_{i}  ⟶  M_{i+1}  ⟶  M_{i}/M_{i+1}  ⟶ 0

where the vertical maps are the canonical maps induced by the obvious inclusions and projections. Lemma 1.4 implies that the left most vertical map is an isomorphism for i = 1 and the right map is an isomorphism for all i. Thus the Five Lemma and induction on i implies that

U ∘ R(M_{i}) ≅ M_{i}

for all i. Using the Five Lemma and induction one can argue as above to make the identification R(M_{i}) = N_{i} for all i. Thus

U ∘ R(M) = U( ⨁_{β∈Q_{+}(𝔪̂)} M_{λ−β}) = U(∪_{i=1}^{∞}N_{i})
= U(∪_{i=1}^{∞}R(M_{i})) ≅ ∪_{i=1}^{∞}U ∘ R(M_{i})
≅ ∪_{i=1}^{∞}M_{i} = M

since tensoring commutes with direct limits (see [R], Corollary 2.10). Let ψ_{M} denote the composition of the maps above. Now we need to check that the isomorphisms {ψ_{M}| M ∈ O_{λ}^{X}} define a natural equivalence.

Suppose that M, N are two modules in O_{λ}^{X} and f ∈ Hom 𝔱̂(M, N). Then f(M_{λ−β}) ⊂ N_{λ−β} for all β ∈ Q_{+}(𝔪̂), so that R(f) ∈ Hom 𝔪̂(R(M), R(N)). Since M is generated as a 𝔱̂-module by R(M), we have that f is completely determined by R(f). Consequently the diagram below is commutative:

U ∘ R(M) ⟶^{ψ_{M}} M
1⊗R(f)↓  ^{f}↓
U ∘ R(N) ⟶^{ψ_{N}} N.

Hence U ∘ R ≅ 1_{O_{λ}^{X}}.

Similarly if M′, N′ are two modules in O_{λ}(𝔪̂) and g ∈ Hom 𝔪̂(M′, N′), then

M′ ⟶^{∼} R ∘ U(M′)
g↓  ^{R(1⊗g)}↓
N′ ⟶^{∼} R ∘ U(N′)

is a commutative diagram where the horizontal maps are induced by inclusions and thus R ∘ U is naturally equivalent to 1_{O_{λ}(𝔪̂)}. □

§6. Consequences of the Equivalence of Categories:

6.1 Let λ ∈ 𝔱_{0}^{∗} and μ ∈ λ ↓ Q_{+}(𝔤̂). Set P𝔱̂ = {α ∈ Q_{+}(𝔤̂)| μ + α ∉ λ ↓ Q_{+}(𝔤̂)} and

V = U(𝔫̂_{+})/ ⨁_{α∈P𝔱̂} U(𝔫̂_{+})_{α}, P𝔪̂(μ) = U(𝔪̂) ⨂_{U(𝔫̂_{+}⊕𝔱_{0})}(V ⨂ ℂ_{μ})

where V ⨂ ℂ_{μ} has the tensor product structure as an 𝔫̂_{+} ⊕ 𝔱_{0}-module. Then P𝔪̂(μ) is a projective module in O_{λ}(𝔪̂) and it has a unique finitely generated indecomposable summand I𝔪̂(μ) such that L𝔪̂(μ) is an irreducible quotient of I𝔪̂(μ) (see [MP] Chapter 2, or [RW] §5 and §6). We let 𝔱_{2} act trivially on V, ℂ_{μ}, and I𝔪̂(μ), so that V ⊗ ℂ_{μ} and I𝔪̂(μ) become 𝔭_{+}-modules. Set

P (μ) = U(𝔱) ⨂_{U(𝔭_{+})}(V ⨂ ℂ_{μ}), I(μ) = U(𝔱) ⨂_{U(𝔭_{+})} I𝔪̂(μ).

A module M in O_{λ} is said to have aVerma composition series if there is a chain of submodules

(6.2)  0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{n} = M

such that M_{i+1}/M_{i} ≅ M(λ_{i}) for some λ_{i} ∈ λ ↓ Q_{+}. If M has a Verma composition series then we let [M : M(λ)] denote the number of occurences of M(λ) as a subquotient in the composition series above. If η ∈ 𝔱_{0}^{∗} and M has a local composition series (defined as in [K] or [MP]) at η then we let [M : L(μ)] denote the number of occurences of L(μ) in this series for μ ≥ η. We will see below that these numbers are independent of the series defining them. 6.2 Since 𝔤̂ is an affine Kac-Moody algebra, we let W𝔤̂ denote the Weyl group for 𝔤̂. Let l denotes the length function on W𝔤̂ and s_{β} be the reflection with respect to a real root β ∈ Δ_{+}(𝔤̂). Let W (j) ; 𝔤̂ denote the elements in W𝔤̂ of length j. We write w ← w′ if w = s_{β}w′ and l(w) = l(w′) + 1. The usual Bruhat order on W𝔤̂ is given by w ≤ w′ if w = w′ or if there exists w_{1}, …, w_{r} ∈ W𝔤̂ such that

w = w_{1} ←^{γ_{1}} w_{2} ←^{γ_{2}} ⋯ ←^{γ_{r−1}} w_{r} = w′

for some real roots γ_{i} ∈ Δ_{+}(𝔤̂). We also define the dot action of W𝔤̂ on 𝔱_{0}^{∗} by

w · μ = w(μ + ρ𝔤̂) − ρ𝔤̂

where ρ𝔤̂ ∈ 𝔱_{0}^{∗} is any fixed element such that ρ𝔤̂(β̌) = 1 for all β ∈ Π𝔪̂. In addition, let P^{+}(𝔪̂) = {λ ∈ 𝔱_{0}^{∗}| λ(α̌) ≥ 0 for all α ∈ Δ_{+}(𝔤̂_{+})} be the positive root lattice for 𝔪̂. 6.3 Proposition. Suppose λ ∈ 𝔱_{0}^{∗} with λ(c_{s}) ≠ 0.

(i) For μ ∈ λ ↓ Q_{+}(𝔤̂) the modules P (μ) and I(μ) are projective modules in O_{λ}. Moreover I(μ) is indecomposable and L(μ) is the unique subquotient of I(μ).

(ii) P (μ) has a Verma module composition series and for all ν ∈ λ ↓ Q_{+}

[P (μ) : M(ν)] = \{ \begin{matrix}dim Hom_{𝔱}(P (μ), M(ν)) & if ν ≤𝔤̂ μ ≤𝔤̂ λ \\ 0 & otherwise. \end{matrix}

(iii) (BGG duality). Let M be an object in O_{λ} and μ ∈ λ ↓ Q_{+}(𝔤̂) then

[M : L(μ)] = dim Hom_{𝔱}(I(μ), M), and P (μ) = ⨁ m_{μ}(ν)I(ν)

where m_{μ}(ν) = dim Hom_{𝔱}(P (μ), L(ν)). Moreover

[M(μ) : L(ν)] = [I(ν) : M(μ)]

(iv) Let μ ∈ P^{+}(𝔤̂), w, w′ ∈ W𝔤̂. Then

dim Hom_{𝔱}(M(w · μ), M(w′ · μ)) = 1 ⇔  w′ ≤ w
⇔ (M(w · μ) : L(w′ · μ)) ≠ 0.

(v) (Strong BGG resolution). If w ≤ w′ and μ ∈ P^{+}(𝔪̂) ∩ λ ↓ Q_{+} then by (iv) we can

fix inclusions

i_{w,w′} : M(w · μ) → M(w′ · μ).

Set C_{j} = ⨁_{w∈W (j) ; 𝔤̂} M(w · μ). Note that C_{0} = M(μ) so that there exists a canonical projection d_{0} : M(μ) → L(μ). For (w_{1}, w_{2}) ∈ W^{(j)}×W^{(j−1)} there exists c(w_{1}, w_{2}) ∈ {−1, 1} such if we define

b_{w_{1},w_{2}}^{j} = \{ \begin{matrix}c(w_{1}, w_{2}) & if w_{1} ← w_{2} \\ 0 & otherwise \end{matrix}

and if d_{j} : C_{j} → C_{j−1} is given by d_{j} = ⊕b_{w_{1},w_{2}}^{j}i_{w_{1},w_{2}} then the sequence

⋯ → C_{j} ⟶^{d_{j}} C_{j−1} ⟶ ⋯ ⟶ C_{1} ⟶^{d_{1}} C_{0} ⟶^{d_{0}} L(μ) ⟶ 0

is exact (see [RW], 9.6 for a detailed description of c(w_{1}, w_{2})).

Proof. The proof is a consequence of the equivalence of categories 2.3, Remark 1.4 together with the following results from [MP] and [RW]: (i) from §3 [RW] or [MP] Chapter 2, (ii) and (iii) from §5 and §6 of [RW] or [MP] Chapter 2, (iv) from [RW] Theorem 8.15, and (v) from [RW] Theorem 9.7 □

References

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[K] V. Kac, Infinite dimensional Lie algebras, Cambridge Univ. Press., 1985.

[L] S. Lay, Convex sets and their applications, Krieger Publishing, 1992.

[MP] R. Moody and A. Pianzola, Lie algebras with triangular decomposition, J. Wiley, 1993.

[MS] R. Moody, Z.Shi, Toroidal Weyl Groups, Nova J. of Algebra and Geometry 1 (1992), 317-337.

[RW] A. Rocha-Caridi, N. Wallach, Characters of Irreducible Representations of the Virasoro Algebra, Math. Z. 185 (1984), 1–21.

Department of Mathematical Sciences, University of Luleå, Luleå, Sweden MT. 59812-1032;

   E-mail address: ben@my1.sm.luth.se;

Department of Mathematics, Kiev University, 64 Vladimirskaya St., Kiev, 252033 Ukraine

   E-mail address: futorny@algebra.univ.kiev.ua