To the memory of Moshe Flato (1937-1998)

WEIGHT REPRESENTATIONS OF THE POLYNOMIAL CARTAN TYPE LIE ALGEBRAS W_{n} AND S̄_{n}

IVAN PENKOV AND VERA SERGANOVA

Abstract. We give an explicit description of the support of an arbitrary irreducible weight module of the infinite-dimensional Lie algebra of polynomial vector fields W_{n}, as well as of its subalgebra S̄_{n} of vector fields with constant divergence.

Key words (1991 MSC): Primary 17B66, Secondary 17B10.

Introduction

The Lie algebras of Cartan type are certain infinite-dimensional simple Lie algebras of vector fields with formal power series coefficients. They are the Lie algebras of “infinite Lie groups” which arose in the work of Sophus Lie around 1870 and were further studied by Elie Cartan in 1904 − 1908. The work of Lie and Cartan has been continued and explained in modern terms by I.M. Singer and S. Sternberg, [10], in 1964. The general theory of representations of the Cartan type infinite-dimensional Lie algebras was initiated only in 1973 when A.N. Rudakov began the study of topological irreducible representations of these Lie algebras, [8, 9]. Rudakov’s main result, roughly speaking, is that all irreducible representations which satisfy a natural continuity condition can be described explicitly as induced modules or quotients of induced modules.

In this paper we take a different approach. We restrict ourselves to the polynomial Cartan type Lie algebras W_{n} and S̄_{n} (see their definition in section 3) and, without imposing any continuity condition, we investigate arbitrary irreducible weight representations of W_{n} and S̄_{n} in the spirit of the recent papers [2, 7, 3]. Our main result is an explicit description of all possible sets of weights of such representations, i.e. a description of the supports of all irreducible weight representations of W_{n} and S̄_{n}. Although a relatively small part of these modules belongs to Rudakov’s class, the answer turns out to be surprisingly simple and resembles very much the answer for the finitedimensional Lie algebra sl_{n+1}, see [2]. A key technical feature that makes our description possible (and which is not shared by the polynomial Hamiltonian and contact Lie algebras) is that any parabolic subalgebra of sl_{n+1} has a certain canonical extension to a parabolic subalgebra of W_{n} or S̄_{n} (see Lemma 3 below). The main problem in the theory of weight W_{n}- and S̄_{n}-modules now is to gain a better understanding of each class of irreducible weight modules with fixed support. O. Mathieu has informed us that his method from [6], together with the result of the present paper, should lead to a classification of all irreducible weight W_{n}- and S̄_{n}-modules with finite-dimensional weight spaces.

Section 1 is devoted to preliminaries: Lie algebras admitting a root decomposition with respect to a finite-dimensional Cartan subalgebra, parabolically induced weight modules over such algebras, shadow decompositions for reductive Lie algebras. In section 2 we introduce our main object of study, the Lie algebras W_{n} and S̄_{n}, and discuss some properties of their root decomposition. We then state our main theorem and give examples. The proof of the theorem is presented in section 3. The reader could go directly to sections 2 and 3 and then use section 1 as reference only.

Acknowledgement. The first author acknowledges partial NSF support through a GIG Grant and thanks the Max Planck Institute in Bonn (where a portion of this work was done) for support and excellent working conditions.

0. Conventions

The ground field is ℂ. If A and B are two sets, A\B is by definition the set {a ∈ A | a ∉ B}. The superscript * always stands for dual space. We denote by ℝ_{+} (respectively by ℝ_{−}) the set of non-negative (resp. non-positive) real numbers; ℤ_{±} =^{def} ℤ ∩ ℝ_{±}. Linear span is denoted by ⟨ ⟩_{ℝ}, ⟨ ⟩_{ℝ_{+}}, ⟨ ⟩_{ℤ_{+}}etc., the subscript indicating the coefficients. We use the term cone as a synonym for an additive subset of a vector space. If 𝔤 is a Lie algebra, the terms a “representation of 𝔤” and a “𝔤-module” are synonyms. We assume that a 𝔤-module is automatically non-zero; a trivial 𝔤-module is a 1-dimensional vector space with the zero action of 𝔤.

1. Lie algebras with root decomposition and generalized weight modules

Let 𝔤 denote a (complex) Lie algebra with generalized root decomposition, i.e. with a fixed self-normalizing nilpotent Lie subalgebra, called a Cartan subalgebra, such that as an 𝔥-module 𝔤 decomposes as 𝔥 ⊕ (⊕_{α∈𝔥^{∗}}𝔤^{α}), where

𝔤^{α} =^{def} {g ∈ 𝔤 | (ad (h) − α (h))^{N} (g) = 0 for every h ∈ 𝔥 and some N > 0}.

The set of non-zero linear functions α ∈ 𝔥^{∗} with 𝔤^{α} ≠ 0 is the set of roots Δ of 𝔤. In what follows we assume 𝔥 to be finite-dimensional. Some results for the case when 𝔥 is infinite-dimensional see in [3]. A 𝔤-module M is a generalized weight 𝔤-module iff as an 𝔥-module M decomposes as ⊕_{μ∈𝔥^{∗}}M^{μ}, where

M^{μ} =^{def} {m ∈ M | (h − μ (h))^{N} · m = 0 for every h ∈ 𝔥 and some N > 0}.

The set of all linear functions μ ∈ 𝔥^{∗} such that M^{μ} ≠ 0 is the support of M and will be denoted by supp M. It is not difficult to verify (using the fact that dim 𝔥 < ∞ ) that the requirement for M to be a generalized weight module is equivalent to the requirement that 𝔥 act locally finitely on M. Furthermore, obviously, 𝔤 has a root decomposition iff the adjoint module is a generalized weight 𝔤-module, i.e. iff the adjoint action of 𝔥 on 𝔤 is locally finite.

A generalized weight 𝔤-module M is a weight 𝔤-module iff 𝔥 acts semi-simply on M, i.e. iff h · m = μ (h) m for any m ∈ M^{μ} and any h ∈ 𝔥. We would like to recall

Proposition 1. Let dim 𝔤 be countable and 𝔤 be semi-simple as 𝔥-module (in particular, let 𝔥 be abelian). Then every irreducible generalized weight 𝔤-module M is a weight module.

Proof. It is based on the following infinite-dimensional version of Schur’s lemma.

Lemma 1. Let A be an associative ℂ-algebra of countable dimension and B be an irreducible Amodule. Then End_{A} (B) = ℂ.

Proof of Lemma 1. Note that End_{A} (B) is a division algebra over ℂ, and therefore End_{A} (B) ≠ ℂ implies that End_{A} (B) contains a field isomorphic to ℂ (x) (the field of rational functions of an indeterminate x). On the other hand, dim_{ℂ} (End_{A} (B)) ≤ dim_{ℂ} B ≤ dim_{ℂ} A, and thus dim_{ℂ} (End_{A} (B)) is finite or countable. But dim_{ℂ} ℂ (x) is uncountable. Hence End_{A} (B) = ℂ.

WEIGHT REPRESENTATIONS OF W_{n} AND S̄_{n} 3

To prove Proposition 1 now, let U^{0} denote the subalgebra in U (𝔤) generated by all monomials of weight zero with respect to the adjoint action of 𝔥. Since 𝔥 acts semi-simply on 𝔤, the subalgebra S (𝔥) belongs to the center of U^{0}. Furthermore the irreducibility of M implies that M^{μ} is an irreducible U^{0}-module for any μ ∈ supp M. By Lemma 1 (applied to A = U^{0} and B = M^{μ}), S(𝔥) acts via scalars on M^{μ}, i.e. M^{μ}, and therefore also M, is a semi-simple 𝔥-module.

Recall next, see [4], that a subdivision Δ = Δ^{+} ⊔ Δ^{−} is called a triangular decomposition iff ⟨−Δ^{+} ⊔ Δ^{−}⟩_{ℝ_{+}} ∩ ⟨Δ^{+} ⊔ −Δ^{−}⟩_{ℝ_{+}} = {0}. Every triangular decomposition of Δ is determined by some (in general not unique) maximal flag of real vector subspaces in ⟨Δ⟩_{ℝ},

0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{dim⟨Δ⟩_{ℝ}} = ⟨Δ⟩_{ℝ} ,

and by a choice of labeling by + and − of the two connected components of F_{i}\F_{i−1} for each i = 1, …, n, via the formula

Δ^{±} = ⊔_{i=1}^{dim⟨Δ⟩_{ℝ}} ((F_{i}\F_{i−1})^{±} ∩ Δ) .

A generalized triangular decomposition, or a parabolic decomposition, of Δ is by definition a subdivision

Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−}

such that, if p: ⟨Δ⟩_{ℝ} → ⟨Δ⟩_{ℝ} / ⟨Δ^{0}⟩_{ℝ} is the natural projection, then p (Δ^{+}) ∩ p (Δ^{−}) = ∅, 0 ∉ p (Δ^{±}) and p (Δ\Δ^{0}) = p (Δ^{+}) ⊔ p (Δ^{−}) is a triangular decomposition of p (Δ\Δ^{0}). Clearly, a parabolic decomposition is determined by a (non-unique) flag

G_{0} ⊂ G_{1} ⊂ ⋯ ⊂ G_{k} = ⟨Δ⟩_{ℝ}, dim G_{i}/G_{i−1} = 1, k ≤ dim ⟨Δ⟩_{ℝ} ,

together with a labeling by + and − of the two connected components of G_{i}\G_{i−1}, via the formulae

Δ^{0} = Δ ∩ G_{0},

Δ^{±} = ⊔_{i=1}^{k} ((G_{i}\G_{i−1})^{±} ∩ Δ) .

A more general statement see in the Appendix of [3].

If Δ = Δ^{+}⊔Δ^{0}⊔Δ^{−} is a parabolic decomposition, let 𝔤^{0} and 𝔤^{±} be the following Lie subalgebras:

𝔤^{0} =^{def} 𝔥 ⊕ (⊕_{α∈Δ^{0}}𝔤^{α}) , 𝔤^{±} =^{def} ⊕_{α∈Δ^{±}}𝔤^{α}.

A Lie subalgebra 𝔭 ⊆ 𝔤 is called parabolic if, as a vector space, 𝔭 equals 𝔤^{0} ⊕ 𝔤^{+} for some parabolic decomposition.

In this paper we will consider parabolic subalgebras associated with general parabolic decompositions in which Δ^{−} does not necessarily equal −Δ^{+}. The following lemma gives some rough but important information about the irreducible quotient of a corresponding generalized Verma module.

Lemma 2. Let 𝔭 be the parabolic subalgebra associated with a given parabolic decomposition Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−}.

(a) Let M^{𝔭} be an irreducible generalized weight 𝔭-module. Then the induced 𝔤-module U (𝔤)⊗_{U(𝔭)} M^{𝔭} has a unique irreducible quotient M and

supp M ⊂ μ + (⟨Δ^{0} ⊔ Δ^{−}⟩_{ℤ_{+}} ∩ ⟨Δ^{0} ⊔ Δ^{+}⟩_{ℤ_{−}})

for any μ ∈ supp M^{𝔭}.

(b) Let M be an irreducible generalized weight 𝔤-module and λ ∈ supp M be such that λ + α ∉ supp M for any α ∈ Δ^{+}. Then M is the unique quotient of the induced 𝔤-module U (𝔤) ⊗_{U(𝔭)}

M^{𝔭}, M^{𝔭} =^{def} U (𝔭) · M_{λ} being an irreducible 𝔭-module.

Proof. (a) The existence of a unique quotient follows from the fact that any proper submodule of the induced module is a generalized weight module whose support does not intersect supp M^{𝔭}. Note next that 𝔤^{+} · M^{𝔭} = 0 by the irreducibility of M^{𝔭} as a 𝔭-module, and thus M^{𝔭} is irreducible as a 𝔤^{0}-module. This implies

supp M^{𝔭} ⊆ μ + (⟨Δ^{0}⟩_{ℤ_{+}} ∩ ⟨Δ^{0}⟩_{ℤ_{−}}).

Furthermore, as M is irreducible, for any μ′ ∈ supp M one can find β_{1}, …, β_{r} ∈ Δ^{+} such that μ′ + β_{1} + ⋯ + β_{r} ∈ supp M^{𝔭}. Therefore

supp M ⊆ supp M^{𝔭} + ⟨Δ^{+}⟩_{ℤ_{−}} .

On the other hand, obviously,

supp M ⊆ supp U (𝔤) ⊗_{U(𝔭)} M^{𝔭} ⊆ supp M^{𝔭} + ⟨Δ^{−}⟩_{ℤ_{+}} .

Hence

supp M ⊆ μ + (⟨Δ^{0} ⊔ Δ^{−}⟩_{ℤ_{+}} ∩ ⟨Δ^{0} ⊔ Δ^{+}⟩_{ℤ_{−}}).

The proof of (b) is left to the reader.

We need to introduce also the notion of a shadow decomposition. Let now 𝔤 be a reductive Lie algebra with root system Δ and M be an irreducible weight 𝔤-module. Fix a point μ ∈ supp M and for any α ∈ Δ consider the set n_{α}^{μ} =^{def} {q ∈ ℝ | μ + qα ∈ supp M}. There are four possible types of sets n_{α}^{μ}: bounded in both directions, unbounded in both directions, bounded from above and unbounded from below, and unbounded from above but bounded from below. The main point is, see [2], that the type of n_{α}^{μ} depends only on α and not on μ, and therefore the module M itself determines a decomposition of Δ into four mutually disjoint subsets:

Δ = Δ_{M}^{+} ⊔ Δ_{M}^{I} ⊔ Δ_{M}^{F} ⊔ Δ_{M}^{−}, (1)

where

Δ_{M}^{+} =^{def} {α ∈ Δ | n_{α}^{μ} is bounded only from above} ,
Δ_{M}^{−} =^{def} {α ∈ Δ | n_{α}^{μ} is bounded only from below} ,
Δ_{M}^{F} =^{def} {α ∈ Δ | n_{α}^{μ} is bounded in both directions} ,
Δ_{M}^{I} =^{def} {α ∈ Δ | n_{α}^{μ} is unbounded in both directions} .

We call (1) the shadow of M and its existence means simply that the type of n_{α}^{μ} does not depend on a choice of μ ∈ supp M.

Any subdivision of the form (1) can be characterized abstractly by the property that, for Δ_{M}^{0} =^{def} Δ_{M}^{I} ⊔ Δ_{M}^{F}, Δ = Δ_{M}^{+} ⊔ Δ_{M}^{0} ⊔ Δ_{M}^{−} is a parabolic decomposition and all roots of Δ_{M}^{I} are orthogonal to all roots of Δ_{M}^{F}. A subdivision Δ = Δ^{+} ⊔ Δ^{I} ⊔ Δ^{F} ⊔ Δ^{−} satisfying this property is by definition a shadow decomposition of Δ. Moreover, the Fernando-Futorny parabolic induction theorem, see [2], implies that the shadow of M determines supp M up to the support of a finite dimensional 𝔤^{F} -module, where 𝔤^{F} =^{def} 𝔥 ⊕ (⊕_{α∈Δ_{M}^{F}} 𝔤^{α}). More precisely, supp M = supp M^{F} + ⟨Δ_{M}^{I} ⊔ Δ_{M}^{−}⟩_{ℤ_{+}}

for some finite-dimensional irreducible 𝔤^{F} -module M^{F}.

In our study of weight modules over W_{n} and S̄_{n} (the Lie algebras W_{n} and S̄_{n} are introduced in the next section) we will use certain shadow decompositions of the root system A_{n}, and furthermore our main result will imply that, if M is an irreducible weight module over W_{n} or S̄_{n}, then M itself has a well-defined shadow.

2. The Lie algebras W_{n} and S̄_{n} and the main theorem

Let P_{n} = ℂ [x_{1}, …, x_{n}] be the ring of polynomials in the indeterminates x_{1}, …, x_{n}. Then W_{n} is by definition the Lie algebra of derivations of P_{n}, i. e.

W_{n} =^{def} Der ℂ[x_{1}, …, x_{n}] = \{∑_{i=1}^{n} p_{i} \frac{∂}{∂x_{i}} | p_{i} ∈ P_{n}\} ,

and S̄_{n} ⊂ W_{n} is its Lie subalgebra of all derivations with constant divergence, i.e.

S̄_{n} =^{def} \{∑_{i=1}^{n} p_{i} \frac{∂}{∂x_{i}} | ∑_{i=1}^{n} \frac{∂^{2}p_{i}}{∂x_{i}∂x_{j}} = 0 for all j = 1, …, n\} .

It is known that W_{n} is simple for n ≥ 1, and that S_{n} =^{def} [S̄_{n}, S̄_{n}] is a simple ideal of codimension 1 in S̄_{n} for n ≥ 2.

Throughout the rest of this paper 𝔤 will denote W_{n} for n ≥ 1 or S̄_{n} for n ≥ 2. Fix the Cartan subalgebra 𝔥 = ⟨\{x_{i} \frac{∂}{∂x_{i}} | i = 1, …, n\}⟩_{ℂ} of 𝔤 and set 𝔥′ =^{def} 𝔥∩[𝔤, 𝔤]. Let ε_{1}, …, ε_{n} be the basis in 𝔥^{∗} dual to the basis x_{1} \frac{∂}{∂x_{1}} , …, x_{n} \frac{∂}{∂x_{n}} in 𝔥. We introduce a symmetric bilinear form on 𝔥^{∗} by requiring ε_{1}, …, ε_{n} to be orthonormal. The set Δ of roots of 𝔤 is the same for W_{n} and S̄_{n} and consists of the vectors ∑_{j=1}^{n} m_{j}ε_{j} and −ε_{i} + ∑_{j≠i} m_{j}ε_{j}, where all m_{j} ∈ ℤ_{+} and i = 1, …, n. In what follows we always assume that m_{j} ∈ ℤ_{+}.

Consider Δ′ =^{def} Δ ∩ −Δ = {±ε_{i}, ε_{i} − ε_{j} | i, j ≤ n, i ≠ j}. Clearly, Δ′ is a root system of type A_{n}. When 𝔤 = W_{n} there is a subalgebra isomorphic to sl_{n+1} in 𝔤 (arising from the infinitesimal action of the action of P SL_{n+1} on ℂP^{n}) , but when 𝔤 = S̄_{n} the root subsystem Δ′ does not correspond to a subalgebra of 𝔤. Let Δ′′ =^{def} {ε_{i} − ε_{j} | i, j ≤ n, i ≠ j}. In both cases the weight spaces 𝔤^{α} for all α ∈ Δ′′ generate a subalgebra of 𝔤 isomorphic to sl_{n}.

In the following proposition we summarize some properties of 𝔤 whose proof is straightforward.

Proposition 2. (a) The Cartan subalgebra 𝔥 acts semi-simply on 𝔤. Therefore, by Proposition 1, every irreducible generalized weight 𝔤-module is a weight module.

(b) For any nonproportional α, β ∈ Δ we have 𝔤^{α+β} = [𝔤^{α}, 𝔤^{β}].

(c) For any α ∈ Δ′′ ⊔ {−ε_{i} | 1 ≤ i ≤ n} we have dim 𝔤^{α} = 1, and the adjoint action of 𝔤^{α} on 𝔤 is locally nilpotent.

(d) For every i ≤ n we have [𝔤^{−ε_{i}}, 𝔤^{ε_{i}}] = 𝔥′, and the Lie subalgebra 𝔤 (ℤε_{i}) =^{def} 𝔤^{−ε_{i}} ⊕ 𝔥 ⊕ 𝔤^{ε_{i}} ⊕ 𝔤^{2ε_{i}} ⊕ … contains a subalgebra isomorphic to W_{1}.

The following property is not quite obvious and is very important in our description of the supports of irreducible weight 𝔤-modules.

Lemma 3. Let Δ′ = (Δ′)^{+} ⊔ (Δ′)^{0} ⊔ (Δ′)^{−} be a parabolic decomposition of Δ′. Set Δ^{0} =^{def} Δ ∩ ⟨(Δ′)^{0}⟩_{ℤ}, Δ^{+} =^{def} (Δ ∩ ⟨(Δ′)^{+} ⊔ (Δ′)^{0}⟩_{ℤ_{+}})\Δ^{0}, Δ^{−} =^{def} Δ\ (Δ^{0} ⊔ Δ^{+}). Then Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−} is a parabolic decomposition of Δ.

Proof. We start by observing that any sequence γ_{1}, …, γ_{k} of linearly independent vectors in ⟨Δ⟩_{ℝ} determines a parabolic decomposition of Δ, as well as a parabolic decomposition of Δ′. Indeed, define the following cones in ⟨Δ⟩_{ℝ}:

C^{0} (γ_{1}, …, γ_{k}) = {ξ ∈ ⟨Δ⟩_{ℝ} | (γ_{1}, ξ) = … = (γ_{k}, ξ) = 0} ,
C^{±} (γ_{1}, …, γ_{k}) = {ξ ∈ ⟨Δ⟩_{ℝ} | (γ_{1}, ξ) = … = (γ_{i−1}, ξ) = 0, (γ_{i}, ξ) >_{(<)} 0 for some i ≤ k} .

Then Δ = Δ^{+} ⊔Δ^{0} ⊔Δ^{−}, where Δ^{0} =^{def} C^{0} (γ_{1}, …, γ_{k})∩Δ, Δ^{±} =^{def} C^{±} (γ_{1}, …, γ_{k})∩Δ, is a parabolic decomposition of Δ, and respectively Δ′ = (Δ′)^{+} ⊔ (Δ′)^{0} ⊔ (Δ′)^{−} is a parabolic decomposition of Δ′, where (Δ′)^{0} =^{def} Δ′ ∩ Δ^{0}, (Δ′)^{±} =^{def} Δ′ ∩ Δ^{±}.

Fix now a parabolic decomposition of Δ′, Δ′ = (Δ′)^{+} ⊔ (Δ′)^{0} ⊔ (Δ′)^{−}. Using the fact that any maximal parabolic subalgebra of a finite-dimensional Lie algebra corresponds to a fundamental weight, we can check that our fixed parabolic decomposition is determined by a sequence of the form ω_{1}^{−}, …, ω_{k}^{−}, ω_{1}^{+}, …, ω_{l}^{+}, where

ω_{r}^{−} = − ∑_{s∈S_{r}^{−}} ε_{s} for a decreasing sequence of sets S_{1}^{−} ⊃ … ⊃ S_{k}^{−},

ω_{r}^{+} = ∑_{s∈S_{r}^{+}} ε_{s} for an increasing sequence of sets S_{1}^{+} ⊂ … ⊂ S_{l}^{+},

and S_{l}^{+} ∩ S_{1}^{−} = 0. All it remains is to verify that the parabolic decomposition of Δ determined by the sequence ω_{1}^{−}, …, ω_{k}^{−}, ω_{1}^{+}, …, ω_{l}^{+} coincides with the decomposition of Δ defined in the Lemma. For this it suffices to check that any α ∈ Δ^{+} ⊔ Δ^{0} can be written as a sum β_{1} + … + β_{p} for some β_{j} ∈ (Δ′)^{0} ⊔ (Δ′)^{+}. Note that, by definition, either (ω_{1}^{−}, α) > 0 or (ω_{1}^{−}, α) = 0. In the former case

α = −ε_{i} + ∑_{j∉S_{1}^{−}} m_{j}ε_{j}

for some i. Since −ε_{i} ∈ Δ^{+} and ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0} for all j, in this case the Lemma is proved. The latter case splits into two:

1. (ω_{1}^{−}, α) = … = (ω_{r−1}^{−}, α) = 0, (ω_{r}^{−}, α) > 0 for some r ≤ k. Then α = ε_{i} − ε_{i′} + ∑_{j∉S_{1}^{−}} m_{j}ε_{j}

for some i′ ∈ S_{r−1}^{−}\S_{r}^{−} and some i ∈ S_{r}^{−}. Since ε_{i} − ε_{i′} ∈ (Δ′)^{+} and ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0}, this case is also settled.

2. (ω_{1}^{−}, α) = … = (ω_{k}^{−}, α) = 0. Then either α = ε_{i} − ε_{i′} + ∑_{j∉S_{1}^{−}} m_{j}ε_{j} with i, i′ ∈ S_{r−1}^{−}\S_{r}^{−}

( here ε_{i} − ε_{i′} ∈ (Δ′)^{0}, ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0}), or α = ∑_{j∉S_{1}^{−}} m_{j}ε_{j} (here ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0}), or finally α = −ε_{i} + ∑_{j∉S_{1}^{−}} m_{j}ε_{j} for some i ∉ S_{1}^{−}. In the first two subcases there is nothing to check. In the last subcase (ω_{1}^{+}, α) = … = (ω_{p−1}^{+}, α) = 0, (ω_{p}^{+}, α) > 0 for some p ≤ l + 1, where, if p = l + 1, we define S_{p}^{+} as {1, …, n} \S_{1}^{−} and ω_{p}^{+} as ∑_{s∈S_{p}^{+}} ε_{s}. Assume first that i ∈ S_{p−1}^{+}. Choose r < p such that i ∈ S_{r}^{+}\S_{r−1}^{+}. Then there is q ∈ S_{r}^{+}\S_{r−1}^{+} such that α = ε_{q} − ε_{i} + ∑_{j∉S_{1}^{−}⊔S_{p−1}^{+}} m_{j}ε_{j}. Here all ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0} and ε_{q} − ε_{i} ∈ (Δ′)^{0}. Let

now i ∉ S_{p−1}^{+}. Then one can find q ∈ S_{p}^{+}\S_{p−1}^{+} with m_{q} > 0 and rewrite α as the sum (ε_{q} − ε_{i}) + ∑_{j∉S_{1}^{−}} m′_{j}ε_{j}. This settles the last subcase, as ε_{q} − ε_{i} ∈ (Δ′)^{+} ⊔ (Δ′)^{0} for all

i ∉ S_{p−1}^{+}, and ε_{j} ∈ (Δ′)^{+} ⊔ (Δ′)^{0} for all j ∉ S_{1}^{−}.

We say that a parabolic decomposition of Δ is induced from a parabolic decomposition of Δ′ if it can be obtained as in Lemma 3.

In what follows we assume that M is an irreducible weight 𝔤-module, unless stated otherwise. For any λ ∈ supp M, set

Γ_{λ} =^{def} ⟨α ∈ Δ′ | λ + ℤ_{+}α ⊆ supp M⟩_{ℤ_{+}}

and

(Δ′)_{λ}^{I} =^{def} {α ∈ Δ′ | α, −α ∈ Γ_{λ}} ,

(Δ′)_{λ}^{F} =^{def} {α ∈ Δ′ | α, −α ∉ Γ_{λ}} ,

(Δ′)_{λ}^{+} =^{def} {α ∈ Δ′ | −α ∈ Γ_{λ}, α ∉ Γ_{λ}} ,

(Δ′)_{λ}^{−} =^{def} − (Δ′)_{λ}^{+} .

Clearly,

Δ′ = (Δ′)_{λ}^{+} ⊔ (Δ′)_{λ}^{I} ⊔ (Δ′)_{λ}^{F} ⊔ (Δ′)_{λ}^{−} . (2)

Furthermore, given a shadow decomposition Δ′ = (Δ′)^{+}⊔(Δ′)^{I} ⊔(Δ′)^{F} ⊔(Δ′)^{−} and a weight λ ∈ 𝔥^{∗}, define the parabolic decomposition _{λ}Δ′ = _{λ} (Δ′)^{+} ⊔ _{λ} (Δ′)^{0} ⊔ _{λ} (Δ′)^{−} by putting

λ (Δ′)^{±} =^{def} (Δ′)^{±} ⊔ {α ∈ (Δ′)^{F} | (λ, α) >_{(<)} 0} ,

λ (Δ′)^{0} =^{def} (Δ′)^{I} ⊔ {α ∈ (Δ′)^{F} | (λ, α) = 0} .

We call a root α ∈ (Δ′)^{+} λ-indecomposable if it can not be decomposed as a sum β + β′ for some β, β′ ∈ (Δ′)^{I} ⊔ _{λ} (Δ′)^{+}. A weight λ ∈ 𝔥^{∗} is compatible with the given shadow decomposition if the following two conditions hold:

1. (λ, α) ∈ ℤ for any α ∈ Δ′′ ∩ (Δ′)^{F};

2. λ (𝔥′) ≠ 0 whenever there is at least one λ-indecomposable root α_{0} ∈ (Δ′)^{+}, and moreover (λ, α) ∉ ℤ_{+} for all λ-indecomposable roots α ∈ Δ′′ ∩ (Δ′)^{+}.

Finally we set 𝔤^{F} =^{def} 𝔥 ⊕ (⊕_{α∈Δ′′∩(Δ′)^{F}} 𝔤^{α}) and note that 𝔤^{F} is a reductive subalgebra, 𝔤^{F} ⊆ gl_{n} ⊂ 𝔤. For any weight λ, compatible with the given shadow decomposition, there is a suitable set of positive roots in (Δ′′) ∩ (Δ′)^{F} such that λ is the highest weight of a finite-dimensional irreducible 𝔤^{F} -module M_{λ}^{F}.

The following theorem is our main result.

Theorem. Let M be an irreducible weight 𝔤-module.

(a) Γ_{λ} = Γ_{μ} for any λ, μ ∈ supp M, and therefore the subdivision (2) does not depend on λ ∈ supp M. We shall denote it by

Δ′ = (Δ′)^{+} ⊔ (Δ′)^{I} ⊔ (Δ′)^{F} ⊔ (Δ′)^{−} . (3)

(b) The subdivision (3) is a shadow decomposition of Δ′, and moreover (Δ′)^{F} ⊆ Δ′′.

(c) There exists λ ∈ supp M, such that it is compatible with (3) and

(4) supp M = supp M_{λ}^{F} + ⟨(Δ′)^{I} ⊔ (Δ′)^{−}⟩_{ℤ_{+}}.

(d) For any shadow decomposition Δ′ = (Δ′)^{+} ⊔ (Δ′)^{I} ⊔ (Δ′)^{F} ⊔ (Δ′)^{−} such that (Δ′)^{F} ⊆ Δ′′, and for any compatible weight λ, there is an irreducible weight 𝔤-module M whose support is given by (4).

The proof is presented in section 3. In the rest of this section we comment on the result and discuss examples. Note, first of all, that the Theorem implies that any irreducible weight 𝔤-module M has a well-defined shadow. Indeed, using the explicit description of supp M given in claim (c), the reader will verify that for any β ∈ Δ and μ ∈ supp M the type of n_{β}^{μ} =^{def} {q ∈ ℝ | μ + qβ ∈ supp M} does not depend on μ ∈ supp M, and therefore M defines a decomposition of Δ:

Δ = Δ_{M}^{+} ⊔ Δ_{M}^{I} ⊔ Δ_{M}^{F} ⊔ Δ_{M}^{−}. (5)

Furthermore, it is natural to ask if the decomposition (5) inherits the properties of the shadow decomposition (3), and in particular is it true that, if we set Δ_{M}^{0} =^{def} Δ_{M}^{I} ⊔ Δ_{M}^{F}, the decomposition Δ_{M}^{+} ⊔ Δ_{M}^{0} ⊔ Δ_{M}^{−} is a parabolic decomposition? The answer turns out to be no as for instance 𝔤^{0} =^{def} 𝔥 ⊕ (⊕_{α∈Δ_{M}^{0}} 𝔤^{α}) may not be a Lie subalgebra. Indeed, consider the adjoint representation of 𝔤 = W_{n}, n ≥ 2. Then −ε_{1} + 2ε_{2}, −ε_{2} + 2ε_{1} ∈ Δ_{M}^{F} but (−ε_{1} + 2ε_{2}) + (−ε_{2} + 2ε_{1}) = ε_{1} + ε_{2} ∈ Δ_{M}^{−}, and in this case 𝔤_{M}^{0} is not a subalgebra.

Note also that, although the subdivision (3) is a shadow decomposition for the root system Δ′ of the Lie algebra sl_{n+1}, not all expressions of the form (4) are equal to supports of irreducible weight sl_{n+1}-modules. This is already clear in the case when 𝔤 = W_{1}, since if λ ≠ 0 is dominant integral, the support of the irreducible highest weight module with highest weight is not equal to the support of the finite-dimensional irreducible sl_{2}-module with the same highest weight, see case 2 in Lemma 4 below.

Finally, it is easy to establish an explicit relationship between the above theorem and the class of modules studied by Rudakov, see [8]. Indeed, note that Rudakov’s continuity condition ([8], Lemma 2.1) applies to the Lie algebras W_{n} and S̄_{n} considered in the present paper. Furthermore, a simple computation based on the Theorem shows that the non-trivial irreducible weight modules from Rudakov’s class are precisely those, whose shadow decomposition satisfies ε_{i} ∈ (Δ′)^{+} for all i.

Example 1. We start with the simplest example when 𝔤 = W_{1}. This case is almost trivial, but we would like to discuss it first as it will be used in the proof of the general case. The following lemma is a slightly stronger version of the main theorem for 𝔤 = W_{1}. The cases 1–4 below correspond to all possible shadow decompositions of Δ′ = {ε_{1}, −ε_{1}}.

Lemma 4. Let 𝔤 = W_{1} and M be an indecomposable weight 𝔤-module generated by a weight vector. Then one of the following is true:

1. supp M = {0} and M is a trivial module;

2. 𝔤^{−ε_{1}} acts locally nilpotently on M and then supp M = λ + ℤ_{+}ε_{1} for some λ ∈ 𝔥^{∗};

3. 𝔤^{−ε_{1}} acts freely on M and supp M = λ + ℤ_{−}ε_{1} for some λ ∈ 𝔥^{∗};

4. 𝔤^{−ε_{1}} acts freely on M and supp M = λ + ℤε_{1} for some λ ∈ 𝔥^{∗}. Furthermore, all above cases are possible.

Proof. Note first that M must be a trivial module whenever supp M is bounded. Indeed, in this case the annihilator of M is a non-zero ideal in 𝔤, and therefore coincides with 𝔤 as 𝔤 is simple.

If now 𝔤^{−ε_{1}} acts locally nilpotently on a non-trivial M one can find λ ∈ supp M and v ∈ M_{λ} such that 𝔤^{−ε_{1}} ·v = 0. Then M is a highest weight module with highest weight λ and supp M = λ+ℤ_{+}ε_{1}. If 𝔤^{−ε_{1}} acts freely on M, then supp M is ℤ_{−}ε_{1}− invariant. Therefore supp M equals either λ+ℤ_{−}ε_{1} or λ + ℤε_{1}, and in the former case M is again a highest weight module.

Finally, note that all supports listed in the Lemma can be realized. Indeed cases 2 and 3 correspond simply to highest weight modules (in this case, if M is irreducible, supp M determines M uniquely up to isomorphism). To construct a module with support as in 4 consider the space M =^{def} x_{1}^{μ}ℂ [x_{1}, x_{1}^{−1}] (dx_{1})^{λ−μ} with the natural action of W_{1}. One can easily check that M is an irreducible W_{1}-module whenever μ ∉ ℤ.

Example 2. Let 𝔤 = W_{2}. Here Δ′ = {±ε_{1}, ±ε_{2}, ± (ε_{1} − ε_{2})} and Δ′′ = {± (ε_{1} − ε_{2})}. The Theorem implies that the following are all possible (up to a permutation of indices) supports of irreducible 𝔤-modules:

1. λ + ℤε_{1} + ℤε_{2}, λ being an arbitrary weight;

2. λ + ℤε_{1} ± ℤ_{+}ε_{2}, λ being an arbitrary weight;

3. λ + ℤ (ε_{1} − ε_{2}) ± ℤ_{+}ε_{2}, λ being an arbitrary weight;

4. λ + ℤ_{+}ε_{1} + ℤ_{−}ε_{2}, λ ≠ 0;

5. λ + ℤ_{+} (ε_{1} − ε_{2}) + ℤ_{+}ε_{2}, λ = λ_{1}ε_{1} + λ_{2}ε_{2} being such that λ_{2} − λ_{1} ∉ ℤ_{+};

6. λ + ℤ_{+} (ε_{1} − ε_{2}) + ℤ_{−}ε_{1}, λ = λ_{1}ε_{1} + λ_{2}ε_{2} being such that λ_{2} − λ_{1} ∉ ℤ_{+};

7. (λ_{1}ε_{1} + λ_{2}ε_{2} + ℤ_{+} (ε_{1} − ε_{2}) + ℤ_{+}ε_{2}) ∩ (λ_{2}ε_{1} + λ_{1}ε_{2} + ℤ_{+} (ε_{2} − ε_{1}) + ℤ_{+}ε_{1}), λ = (λ_{1}, λ_{2}) ∈ ℂ^{2}\ {0} being such that λ_{2} − λ_{1} ∈ ℤ_{+};

8. (λ_{1}ε_{1} + λ_{2}ε_{2} + ℤ_{+} (ε_{1} − ε_{2}) + ℤ_{−}ε_{1}) ∩ (λ_{2}ε_{1} + λ_{1}ε_{2} + ℤ_{+} (ε_{2} − ε_{1}) + ℤ_{−}ε_{2}), λ = (λ_{1}, λ_{2}) ∈ ℂ^{2}\ {0} being such that λ_{2} − λ_{1} ∈ ℤ_{+};

9. {0}.

The next example shows that the main theorem does not extend to the polynomial contact Lie algebras.

Example 3. Let 𝔤 be the polynomial contact algebra K_{5}. We define it as the space ℂ [t, x_{1}, y_{1}, x_{2}, y_{2}] with Lie bracket

[f, g] =^{def} −f′Dg + g′Df − ∑_{i=1}^{2}(\frac{∂f}{∂x_{i}} \frac{∂g}{∂y_{i}} − \frac{∂g}{∂x_{i}} \frac{∂f}{∂y_{i}}),

where

Dh =^{def} 2h − ∑_{i=1}^{2}(x_{i} \frac{∂h}{∂x_{i}} + y_{i} \frac{∂h}{∂y_{i}}).

Here 𝔥 =^{def} ⟨t, z_{1}, z_{2}⟩_{ℂ}, where z_{1} =^{def} x_{1}y_{1}, z_{2} =^{def} x_{2}y_{2}. Furthermore,

Δ = {kδ + aε_{1} + bε_{2} | k + 2 ∈ ℤ_{+}, a, b ∈ ℤ, |a| + |b| ≤ k + 2, a + b + k ∈ 2ℤ} .

The set Δ′ =^{def} Δ ∩ −Δ = {±2δ, ±δ ± ε_{1}, ±δ ± ε_{2}, ±ε_{1} ± ε_{2}, ±2ε_{1}, ±2ε_{2}} is a root system of type C_{3}

and its C_{2}-subsystem {±ε_{1} ± ε_{2}, ±2ε_{1}, ±2ε_{2}} plays the role of Δ′′. One can check that all statements of Proposition 2 hold after suitable reformulation. However, Lemma 3 is no longer true. As a result there are irreducible weight 𝔤-modules M such that the faces of the closure of the convex hull of supp M are not generated by roots. Here is an example. Let ω = ε_{1} + tε_{2} for some irrational real number t. Consider the parabolic decomposition Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−},

Δ^{0} =^{def} {α ∈ Δ | (ω, α) = 0} = {kδ | k + 2 ∈ ℤ_{+}} ,

Δ^{±} =^{def} {α ∈ Δ | (ω, α) >_{(<)} 0} .

Then 𝔤^{0} = ℂ [t, z_{1}, z_{2}]. Fix λ ∉ ℤ and define a 𝔤^{0}-module structure on M′ =^{def} t^{λ}ℂ [t^{±1}] by letting t^{m}z_{1}^{p}z_{2}^{q} act on M′ as the operator 2t^{m} (\frac{∂}{∂t})^{1−p−q} (where (\frac{∂}{∂t})^{−1} denotes antiderivative in M′). Obviously M′ is an irreducible 𝔤^{0}-module and supp M′ = λδ + ℤδ. Make M′ a 𝔭-module by putting 𝔤^{+} · M′ =^{def} 0 and consider the irreducible quotient M of the induced module U (𝔤) ⊗_{U(𝔭)} M′. One can check that

supp M = {λδ + lδ + k_{1}ε_{1} + k_{2}ε_{2} | l, k_{1}, k_{2} ∈ ℤ, k_{1} + tk_{2} < 0} ,

and therefore the single face of the closure of the convex hull of supp M is not generated by its intersection with the root lattice.

3. Proof of the Theorem

Let Γ =^{def} ∩_{λ∈suppM}Γ_{λ}. Clearly, supp M is Γ-invariant. For any λ ∈ supp M define K_{λ},

K_{λ} =^{def} {α ∈ Δ′ | λ + α ∉ supp M} ,

and put K̄_{λ} =^{def} Δ′\K_{λ}. We call λ ∈ supp M extremal if K_{λ} is maximal (i.e. if K_{λ} is not a proper subset of K_{μ} for any μ ∈ supp M).

Lemma 5. (a) If α ∈ Δ′′ and α, −α ∉ Γ, then 𝔤^{α} and 𝔤^{−α} act locally nilpotently on M, and supp M is invariant with respect to the Weyl group reflection r_{α}. Moreover, if α ∈ K_{λ}, then (λ, α) ∈ ℤ_{+}.

(b) If λ, μ ∈ supp M and ε_{i} ∈ K_{λ}, −ε_{i} ∈ K_{μ}, then λ (𝔥′) = 0 and −ε_{i} ∈ K_{λ}.

(c) If α ∈ Γ and λ, λ − α ∈ supp M, then K_{λ} ⊆ K_{λ−α}. In particular, if λ is extremal, then λ − α is also extremal with K_{λ} = K_{λ−α}.

(d) (Convexity property) If α ∈ Δ′ and λ, λ+kα ∈ supp M for some k ∈ ℤ_{+}, then λ+jα ∈ supp M for any 0 ≤ j ≤ k.

Proof. (a) Note that if α ∈ Δ′′ then 𝔤^{α} and 𝔤^{−α} generate an sl_{2}-subalgebra. Since furthermore 𝔤^{α} and 𝔤^{−α} act locally nilpotently on 𝔤, the existence of a vector v ∈ M with 𝔤^{α} · v = 0 (which is a consequence of the fact that α ∉ Γ) implies via a standard argument ([5], Lemma 3.4) that 𝔤^{α} acts locally nilpotently on M. Similarly 𝔤^{−α} acts locally nilpotently on M. Therefore the sl_{2} -subalgebra generated by 𝔤^{α} and 𝔤^{−α} acts locally finitely on M and in particular supp M is r_{α}-invariant. Finally, if α ∈ K_{λ}, then 𝔤^{α} · M_{λ} = 0 and λ is a highest sl_{2}-weight, i.e. (λ, α) ∈ ℤ_{+}.

(b) Since 𝔤^{−ε_{i}} acts nilpotently on 𝔤, for the same reason as in (a) 𝔤^{−ε_{i}} acts locally nilpotently on M. By Lemma 2 (d) and Lemma 4, (λ + ℤε_{i}) ∩ supp M = {λ}. Therefore 𝔤^{ε_{i}} · M_{λ} = 𝔤^{−ε_{i}} · M_{λ} = 0 and [𝔤^{ε_{i}}, 𝔤^{−ε_{i}}] · M_{λ} = 𝔥′ · M_{λ} = 0. This shows that λ (𝔥′) = 0 and therefore also that −ε_{i} ∈ K_{λ}.

(c) Since α ∈ Γ and since supp M is Γ-invariant, the inclusion λ − α + δ ∈ supp M implies λ + δ ∈ supp M. Therefore δ ∈ K̄_{λ−α} gives δ ∈ K̄_{λ}, i.e. K_{λ} ⊆ K_{λ−α}.

(d) Assume to the contrary that λ + jα ∉ supp M for some j < k. Then α ∉ Δ′′ as otherwise the sl_{2}-subalgebra generated by 𝔤^{α} and 𝔤^{−α} would have to act locally finitely on M and we immediately obtain a contradiction. Therefore α = −ε_{i} for some i. Then 𝔤^{−ε_{i}} acts locally nilpotently on M, and Lemma 4, applied to a W_{1}-subalgebra of 𝔤 (ℤε_{i}) (see Proposition 2 (d)), provides a contradiction.

Lemma 6. Let λ ∈ supp M be an extremal point and let α, β, α + β ∈ Δ′.

(a) If α, β ∈ K_{λ}, then α + β ∈ K_{λ}.

(b) If α, β ∈ K̄_{λ}, then α + β ∈ K̄_{λ}.

Proof. (a) First, assume that at least one of the two roots α, β (say α) belongs to Δ′′. Assume also that α, β ∈ K_{λ} and α + β ∉ K_{λ}. Then μ = λ + α + β ∈ supp M, α ∈ K_{λ}, −α ∈ K_{μ}. By Lemma 5 (a), we have (λ, α) ∈ ℤ_{+} and (μ, α) ∈ ℤ_{−}. Since (μ − λ, α) = (α + β, α) = 1, this is impossible.

Let now α, β ∉ Δ′′. Then without loss of generality we can assume that α = −ε_{i} and β = ε_{j}. Suppose again that α, β ∈ K_{λ} and α+β ∉ K_{λ}. Then μ = λ+α+β ∈ supp M, −ε_{i}, ε_{j} ∈ K_{λ}, ε_{i}, −ε_{j} ∈ K_{μ}. By Lemma 5 (b), λ (𝔥′) = μ (𝔥′) = 0, which is impossible since (λ − μ) (𝔥′) = (ε_{j} − ε_{i}) (𝔥′) ≠ 0.

(b) As in the proof of (a) we consider two cases. First, let α ∈ Δ′′. Assume that α, β ∈ K̄_{λ} but α + β ∈ K_{λ}, i.e. λ + α, λ + β ∈ supp M, λ + α + β ∉ supp M. Note that −α ∉ Γ by Lemma 5 (c). On the other hand, α ∉ Γ because α ∈ K_{λ+β}. Therefore, 𝔤^{α} and 𝔤^{−α} act locally nilpotently on M, and (λ + β, α) ∈ ℤ_{+}. Since (β, α) = −1, we have (λ, α) ≥ 1. Furthermore, β ∈ K_{λ+α}\K_{λ}. Since λ is extremal, one can find δ ∈ K_{λ}\K_{λ+α}. Then λ + α + δ ∈ supp M and λ + δ ∉ supp M, which implies (λ + α + δ, α) ∈ ℤ_{−}. But (δ, α) ≥ −1 and therefore (α + δ, α) > 0, while we showed already that (λ, α) ≥ 1. Contradiction.

Now let α, β ∉ Δ′′, for example let α = −ε_{i}, β = ε_{j}. Assume that α, β ∈ K̄_{λ} but α + β ∈ K_{λ}, i.e. λ − ε_{i}, λ + ε_{j} ∈ supp M, λ + ε_{j} − ε_{i} ∉ supp M. Notice that −ε_{j}, ε_{i} ∉ Γ. Since ε_{j} ∈ K_{λ−ε_{i}}, by Lemma 5 (b), we have (λ − ε_{i}) (𝔥′) = 0. On the other hand, ε_{j} ∈ K_{λ−ε_{i}}\K_{λ}. Since λ is extremal, there exists δ ∈ K_{λ}\K_{λ−ε_{i}}, i.e. λ − ε_{i} + δ ∈ supp M and λ + δ ∉ supp M. Then again, by Lemma 5 (b), (λ − ε_{i} + δ) (𝔥′) = 0. Thus δ (𝔥′) = 0 for some δ ∈ Δ′, which is impossible.

Corollary 1. Let λ ∈ supp M be extremal. Put

λ (Δ′)^{+} =^{def} K_{λ}\ (K_{λ} ∩ −K_{λ}) ,

λ (Δ′)^{−} =^{def} K̄_{λ}\ (K̄_{λ} ∩ −K̄_{λ}) ,

λ (Δ′)^{0} =^{def} (K_{λ} ∩ −K_{λ}) ⊔ (K̄_{λ} ∩ −K̄_{λ}) .

Then

λ (Δ′)^{+} ⊔_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−}

is a parabolic decomposition of Δ′.

Lemma 7. If supp M contains more than one point, then {ε_{i}, −ε_{i}} ∩ Γ ≠ ∅ for all i.

Proof. Assume that ε_{i}, −ε_{i} ∉ Γ for some i. Then, by Lemma 5 (b), one can choose λ ∈ supp M such that ε_{i}, −ε_{i} ∈ K_{λ} and λ (𝔥′) = 0. Furthermore, λ can be chosen to be extremal. Consider the parabolic decomposition _{λ} (Δ′)^{+} ⊔_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−} from Corollary 1. We claim that _{λ} (Δ′)^{0} = K_{λ} ∩ −K_{λ}. This is equivalent to saying that for any α ∈ Δ′ at least one of the roots α and −α belongs to K_{λ}. The latter however is obvious because otherwise λ + α, λ − α ∈ supp M implies λ + α + ε_{i}, λ − α + ε_{i} ∈ supp M, and then, by Lemma 5 (d), one has λ + ε_{i} ∈ supp M. Consider now the parabolic decomposition Δ = _{λ}Δ^{+} ⊔ _{λ}Δ^{0} ⊔ _{λ}Δ^{−} induced by the parabolic decomposition λ (Δ′)^{+} ⊔ _{λ} (Δ′)^{0} ⊔ _{λ} (Δ′)^{−}, and let _{λ}𝔭 be the corresponding parabolic subalgebra. Then _{λ}𝔭 is generated by 𝔥 and 𝔤^{α} for all α ∈_{λ} (Δ′)^{+} ⊔_{λ} (Δ′)^{0}. Therefore 𝔭 · M^{λ} = M^{λ} and M is the unique irreducible quotient of the induced module U (𝔤) ⊗_{U(𝔭)} M^{λ}. Since λ (𝔥′) = 0, we have dim M = 1 and supp M = {λ}.

Lemma 8. Let λ ∈ supp M.

(a) (Δ′)_{λ}^{−} ⊆ Γ.

(b) The set (Δ′)_{λ}^{F} is a root subsystem of Δ′′.

(c) If α ∈ (Δ′)_{λ}^{F} and β ∈ (Δ′)_{λ}^{I}, then (α, β) = 0.

Proof. (a) Let α ∈ (Δ′)_{λ}^{−}. If α ∉ Δ′′, Lemma 7 implies that α ∈ Γ. Let now α ∈ Δ′′. Assume that α ∉ Γ. Then 𝔤^{α} and 𝔤^{−α} act locally nilpotently on M and each sl_{2}-string λ + ℤα ∩ supp M is either bounded or unbounded in both directions. Therefore α, −α ∈ (Δ′)_{λ}^{F} or α, −α ∈ (Δ′)_{λ}^{I}. Contradiction. Hence α ∈ Γ.

(b) By Lemma 7, (Δ′)_{λ}^{F} ⊆ Δ′′ and, by definition, − (Δ′)_{λ}^{F} = (Δ′)_{λ}^{F}. All we need to check is that, for any α, β ∈ (Δ′)_{λ}^{F} , α + β ∈ Δ′′ implies α + β ∈ (Δ′)_{λ}^{F}.

Assume that α + β ∉ (Δ′)_{λ}^{F}. Then α + β ∈ (Δ′)_{λ}^{I} and λ + k (α + β) ∈ supp M for all k ∈ ℤ_{+}. By possibly changing α to −α and and β to −β, we can assume that (λ, α) ≥ 0. Therefore r_{α} (λ + k (α + β)) = λ−mα+kβ for some nonnegative m. Then, by Lemma 5 (d), λ+kβ ∈ supp M for any k ∈ ℤ_{+}, which contradicts to the fact that β ∈ (Δ′)_{λ}^{F}.

(c) Assume that the statement is not true, i.e. that one can find α ∈ (Δ′)_{λ}^{F} and β ∈ (Δ′)_{λ}^{I} such that (α, β) > 0. Then the roots ±α, ±β and ±r_{α} (β) form a root system R of type A_{2}. The intersection of Γ_{λ} with R either coincides with {β, −β} or contains all roots from the half-plane of ⟨R⟩_{ℝ} bounded by the line ℝβ. In the former case (Δ′)_{λ}^{F} ∩ R = R\ {β, −β}, which contradicts to (b). In the latter case at least one of α and −α belongs to Γ_{λ}, which is also impossible.

Lemma 9. Let λ ∈ supp M be extremal. Then

K_{λ} = (Δ′)_{λ}^{+} ⊔ {α ∈ (Δ′)_{λ}^{F} | (λ, α) ≥ 0}.

Proof. Obviously,

K_{λ} ⊆ (Δ′)_{λ}^{+} ⊔ {α ∈ (Δ′)_{λ}^{F} | (λ, α) ≥ 0}.

We will prove that also

(Δ′)_{λ}^{+} ⊔ {α ∈ (Δ′)_{λ}^{F} | (λ, α) ≥ 0} ⊆ K_{λ}.

Let us show first that (Δ′)_{λ}^{+} ⊆ K_{λ}. Let α ∈ (Δ′)_{λ}^{+}. Choose k ∈ ℤ_{+} such that λ + kα ∈ supp M but λ + (k + 1) α ∉ supp M. By Lemma 8 (a), −α ∈ Γ. By Lemma 5 (c), K_{λ} ⊆ K_{λ+kα}. Since λ is extremal, k = 0 and α ∈ K_{λ}.

It remains to show that {α ∈ (Δ′)_{λ}^{F} | (λ, α) ≥ 0} ⊆ K_{λ}. Let α ∈ (Δ′)_{λ}^{F} and (λ, α) ≥ 0. Observe that K_{λ} ⊆ K_{λ+α}. Indeed, if δ ∈ K̄_{λ+α} then λ + α + δ ∈ supp M and (λ + α + δ, α) > 0. Therefore λ + α + δ − α = λ + δ ∈ supp M and δ ∈ K̄_{λ}. We complete the proof now by using the same argument as in the case when α ∈ (Δ′)_{λ}^{+}.

Corollary 1, Lemma 8 and Lemma 9 imply

Corollary 2. If λ ∈ supp M is extremal, the decomposition

Δ′ = (Δ′)_{λ}^{+} ⊔ (Δ′)_{λ}^{F} ⊔ (Δ′)_{λ}^{I} ⊔ (Δ′)_{λ}^{−}

is a shadow decomposition of Δ.

Set S^{±} =^{def} {i | ±ε_{i} ∉ Γ} and S =^{def} S^{+} ⊔ S^{−}. We say that λ ∈ supp M is minimal if λ is extremal and ±ε_{i} ∈ K_{λ} for all i ∈ S^{±}.

Lemma 10. There exists a minimal λ ∈ supp M.

Proof. It consists of three steps.

Step 1. We will show that there is an extremal point λ ∈ supp M such that ε_{l} (respectively, −ε_{l}) belongs to K_{λ} for every l ∈ S^{+} (resp. for every l ∈ S^{−}). Consider the case of S^{+} (for S^{−} the proof is the same). Assume that the statement is false. This implies that one can find i, j ∈ S^{+} and extremal weights λ, μ ∈ supp M such that ε_{i} ∈ K_{λ}, ε_{i} ∈ Γ_{μ}, ε_{j} ∈ K_{μ}, ε_{j} ∈ Γ_{λ}. Then λ + kε_{j}, μ + kε_{i} ∈ supp M and λ + ε_{i} + kε_{j}, μ + ε_{j} + kε_{i} ∉ supp M for any k ∈ ℤ_{+}. Furthermore, if α =^{def} ε_{i} − ε_{j}, then 𝔤^{α} and 𝔤^{−α} act locally nilpotently on M. Therefore, for sufficiently large k we have μ + kε_{i} − α ∈ supp M as (μ + kε_{i}, α) > 0. Contradiction.

Step 2. Choose an extremal point μ ∈ supp M such that −ε_{j} ∈ K_{μ} for all j ∈ S^{−}. Let ω^{−} = − ∑_{j∈S^{−}} ε_{j}. Consider the parabolic decomposition Δ = Δ^{−} ⊔ Δ^{0} ⊔ Δ^{+}, where Δ^{±} =^{def} {α ∈ Δ | (ω^{−}, α) >_{(<)} 0}. Any α ∈ Δ^{+} can be written as −ε_{i} + ∑_{j∉S^{−}} m_{j}ε_{j}, and, since −ε_{j} ∈ Γ for all j ∉ S^{−}, we obtain that μ + α ∉ supp M. Then, by Lemma 2 (b), M is the unique irreducible quotient of U (𝔤) ⊗_{U(𝔭)} M^{𝔭}.

Step 3. Since supp M ⊆ supp U (𝔤) ⊗_{U(𝔭)} M^{𝔭} for any ν ∈ supp M, we have (ω^{−}, ν) ≤ (ω^{−}, μ). Therefore −ε_{j} ∈ (Δ′)_{ν}^{+} for all j ∈ S^{−} and all ν ∈ supp M. In particular, if ν is extremal, then by Lemma 9 −ε_{j} ∈ K_{ν} for all j ∈ S^{−}. Thus, an extremal point λ, such that ε_{l} ∈ K_{λ} for all l ∈ S^{+}, is necessarily minimal.

Let λ ∈ supp M be a minimal point. Consider the parabolic decomposition Δ = _{λ}Δ^{+} ⊔_{λ} Δ^{0} ⊔_{λ} Δ^{+} induced by the parabolic decomposition _{λ} (Δ′)^{+} ⊔_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−} of Corollary 1, and

set _{λ}𝔭 =^{def} 𝔥 ⊕ (⊕_{α∈_{λ}(Δ′)^{0}⊔_{λ}Δ^{+}}𝔤^{α}).

Lemma 11. One can find a minimal λ ∈ supp M such that λ + α ∉ supp M for any α ∈ _{λ}Δ^{+}. Therefore, by Lemma 2 (b), M is the unique irreducible quotient of the induced module U (𝔤) ⊗_{U(_{λ}𝔭)} M′ for some irreducible _{λ}𝔭-module M′ with λ ∈ supp M′. Moreover,

(6) supp M = supp M_{λ}^{F} + ⟨ (Δ′)_{λ}^{I} ⊔ (Δ′)_{λ}^{−} ⟩_{ℤ_{+}},

and λ is compatible with the shadow decomposition

Δ′ = (Δ′)_{λ}^{+} ⊔ (Δ′)_{λ}^{I} ⊔ (Δ′)_{λ}^{F} ⊔ (Δ′)_{λ}^{−} . (7)

Proof. Set

K_{λ}^{1} =^{def} {β = ε_{i} + ε_{j} − ε_{k} | k ≠ i, j, λ + β ∉ supp M}

and

K̄_{λ}^{1} =^{def} {β = ε_{i} + ε_{j} − ε_{k} | k ≠ i, j, λ + β ∈ supp M} .

We will prove that any minimal λ with maximal K_{λ}^{1} satisfies the requirement of the Lemma. By Lemma 2 (a) it is sufficient to show that λ + α ∉ supp M for any α ∈_{λ} Δ^{+}.

Recall the proof of Lemma 3. It implies that any α ∈_{λ} Δ^{+} can be written as α = α_{0}+∑_{j∉S^{−}} m_{j}ε_{j},

(here S^{−} = S_{1}^{−}, see Lemma 3), where α_{0} ∈ Δ′′ ∩ (_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{+}) or α_{0} = 0. Since −ε_{j} ∈ Γ

for all j ∉ S^{−}, it is obvious that λ + α ∉ supp M whenever α_{0} ∈_{λ} (Δ′)^{+} or α_{0} = 0. Consider the case when α_{0} ∈_{λ} (Δ′)^{0}. In this case m_{i} > 0 for at least one i ∈ S^{+}. Assume, to the contrary, that λ + α ∈ supp M. Then λ + α_{0} + ε_{i}, λ + α_{0} ∈ supp M. Therefore α_{0} ∉ K_{λ}, which implies by Lemma 9 that α_{0} ∈ (Δ′)_{λ}^{I}. Thus μ = λ − α_{0} ∈ supp M. We claim that μ is minimal and that K_{λ}^{1} is strictly contained in K_{μ}^{1}. To prove this claim consider two cases: α_{0} ∈ Γ and α_{0} ∉ Γ. In the former case the claim is trivial. In the latter case 𝔤^{α_{0}} acts locally nilpotently on M and, since −α_{0} ∈ K_{λ+α_{0}+ε_{i}}, 𝔤^{−α^{0}} also acts locally nilpotently on M. Furthermore, (λ + α_{0} + ε_{i}, α_{0}) ∈ ℤ_{−}, i.e. (λ, α_{0}) ≤ −2. Fix an arbitrary element δ ∈ K̄_{μ} ∪ K̄_{μ}^{1}. Then λ − α_{0} + δ ∈ supp M, and (λ − α_{0} + δ, α_{0}) = (λ, α_{0}) − 2 + (δ, α_{0}) < 0 since (δ, α_{0}) ≤ 2. Therefore λ + δ ∈ supp M and δ ∈ K̄_{λ} ∪ K̄_{λ}^{1}. Hence K_{λ} ⊆ K_{μ}. Moreover, K_{λ}^{1} is strictly contained in K_{μ}^{1} as α_{0} + ε_{i} ∈ K_{μ}^{1}\K_{λ}^{1}. This contradicts the maximality of K_{λ}^{1}, and hence λ + α ∉ supp M.

To prove (6) we use Lemma 2 (a). Indeed, Lemma 3 (applied to the parabolic decomposition λ (Δ′)^{+} ⊔_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−} ) yields

⟨_{λ}Δ^{0} ⊔_{λ} Δ^{−}⟩_{ℤ_{+}} ∩ ⟨_{λ}Δ^{0} ⊔_{λ} Δ^{+}⟩_{ℤ_{−}} = ⟨_{λ}(Δ′)^{0} ⊔_{λ} (Δ′)^{−} ⟩_{ℤ_{+}}.

Therefore supp M ⊆ λ + ⟨_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−}⟩_{ℤ_{+}}. Furthermore, supp M is W_{λ}-invariant, where W_{λ}

denotes the Weyl group of the root system (Δ′)_{λ}^{F}. This gives

supp M ⊆ ∩_{w∈W_{λ}}w(λ + ⟨_{λ}(Δ′)^{0} ⊔_{λ} (Δ′)^{−}⟩_{ℤ_{+}}). (8)

On the other hand, Γ_{λ} = (_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−}) \ (Δ′)_{λ}^{F}, i.e. λ + ℤ_{+}α ⊆ supp M for any α ∈

(_{λ} (Δ′)^{0} ⊔_{λ} (Δ′)^{−}) \ (Δ′)_{λ}^{F}. As supp M satisfies the convexity property (Lemma 5 (d)), we obtain that (8) is an equality, i.e.

supp M = ∩_{w∈W_{λ}}w(λ + ⟨_{λ}(Δ′)^{0} ⊔_{λ} (Δ′)^{−}⟩_{ℤ_{+}}). (9)

Furthermore, since

supp M_{λ}^{F} = ∩_{w∈W_{λ}}w(λ + ⟨(Δ′)_{λ}^{F} ∩_{λ} (Δ′)^{−}⟩_{ℤ_{+}}),

(9), together with the W_{λ}-invariance of (Δ′)_{λ}^{I} and (Δ′)_{λ}^{−}, yields (6).

Finally, we will prove that λ is compatible with the shadow decomposition (7). First, (6) and Lemma 5 (a) imply that (λ, α) ∈ ℤ for any α ∈ (Δ′)_{λ}^{F}. To check the second condition of compatibility, choose a λ-indecomposable root α ∈ (Δ′)_{λ}^{+}. Note that β − pα ∉ ⟨(Δ′)_{λ}^{I} ⊔ _{λ}(Δ′)^{−}⟩_{ℤ_{+}} for any β ∈ _{λ} (Δ′)^{+}, β ≠ α, and any p ∈ ℤ_{+}, and hence 𝔤^{β} · M^{λ−pα} = 0. Consider first the case when α ∈ Δ′′. If k = (λ, α) ∈ ℤ_{+}, then 𝔤^{α} · M^{λ−(k+1)α} = 0, and therefore M^{λ−(k+1)α} generates a proper submodule in M. Hence M^{λ−(k+1)α} = 0, which contradicts to the fact that −α ∈ Γ_{λ}. Therefore (λ, α) ∉ ℤ_{+} for α ∈ Δ′′. Let now α ∉ Δ′′ and assume that λ (𝔥′) = 0. In the same way we can show that M^{λ−α} generates a proper submodule in M, i.e. M^{λ−α} = 0, which contradicts to the fact that −α ∈ Γ_{λ}. Thus, we have proved that λ (𝔥′) ≠ 0.

Lemma 11 (together with Lemma 8 (b)) implies claims (a), (b) and (c) of the Theorem. Indeed, the explicit expression for supp M obtained in (6) makes it obvious that the subdivision (2) does not depend on λ, and that moreover (Δ′)^{+} ⊔(Δ′)^{I} ⊔(Δ′)^{F} ⊔(Δ′)^{−} is a shadow decomposition of Δ′. The fact that (Δ′)^{F} ⊂ Δ′′ follows from Lemma 8 (b). Finally, as (3) is well-defined, (6) is equivalent to (4). Therefore all that remains is to prove claim (d) of the Theorem.

The following lemma is a straightforward corollary of the definition of the Lie algebras W_{n} and S̄_{n} and of Lemma 3.

Lemma 12. Let Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−} be a parabolic decomposition of Δ induced by some parabolic decomposition of Δ′. If 𝔤 = W_{n}, then the Lie subalgebra 𝔤^{0} is isomorphic to a semidirect sum of W_{m} = Der ℂ [x_{i_{1}}, …, x_{i_{m}}] and the ideal 𝔨 ⊗ ℂ [x_{i_{1}}, …, x_{i_{m}}] for some reductive 𝔥-invariant Lie subalgebra 𝔨 ⊆ gl_{n}. If 𝔤 = S̄_{n}, then 𝔤^{0} is isomorphic to the intersection of S̄_{n} with a semidirect sum as above.

Lemma 13. Let 𝔤^{0} be as in Lemma 12 and λ be an arbitrary weight. There exists an irreducible weight 𝔤^{0}-module M^{0} with supp M^{0} = λ + ⟨Δ^{0}⟩_{ℤ_{+}}.

Proof. Let λ = ∑_{i=1}^{n} λ_{i}ε_{i}, T = {i_{1}, …, i_{m}}. There exists an irreducible weight 𝔨-module M_{𝔨} with supp M_{𝔨} = ∑_{i∉T} λ_{i}ε_{i} + ⟨Δ_{𝔨}⟩_{ℤ}. (This follows for example from [1].) Consider the space

M^{0} =^{def} M_{𝔨} ⊗ (x_{i_{1}}^{λ_{i_{1}} −θ} · . . . · x_{i_{m}}^{λ_{im} −θ}ℂ [x_{i_{1}}^{±1}, …, x_{i_{m}}^{±1}] (dx_{i_{1}} · . . . · dx_{i_{m}})^{θ})

with the obvious action of 𝔤^{0}. One can easily check that, if λ_{i_{1}} − θ, …, λ_{i_{m}} − θ ∉ ℤ, M^{0} is an irreducible 𝔤^{0}-module which satisfies the requirement of the Lemma.

Now we are able to construct an irreducible 𝔤-module M whose support is given by (4).

Let Δ′ = (Δ′)^{+} ⊔ (Δ′)^{F} ⊔ (Δ′)^{I} ⊔ (Δ′)^{−} be a shadow decomposition of Δ′ and λ ∈ 𝔥^{∗} be a compatible weight. Consider the parabolic decomposition Δ′ = (Δ′)^{+}⊔(Δ′)^{0}⊔(Δ′)^{−}, where (Δ′)^{0} = (Δ′)^{F} ⊔ (Δ′)^{I}, and the corresponding induced parabolic decomposition Δ = Δ^{+} ⊔ Δ^{0} ⊔ Δ^{−}. Set 𝔥̄ =^{def} {(a, b) | a, b ∈ 𝔥, [a, 𝔤^{F} ] = 0, [b, 𝔤^{I}] = 0}. Note that 𝔤^{0} is isomorphic to (𝔤^{F} ⊕ 𝔤^{I}) /𝔥̄. Choose μ ∈ 𝔥^{∗} with (λ ⊕ μ) \binom{}{𝔥̄} = 0. Let M^{I} be an irreducible 𝔤^{I}-module with supp M^{I} = μ + ⟨ (Δ′)^{I} ⟩_{ℤ_{+}} (which exists by Lemma 13) and M_{λ}^{F} be the finite-dimensional irreducible 𝔤^{F} -module with highest weight λ. Then M^{0} =^{def} M^{I} ⊗ M_{λ}^{F} has a natural structure of a 𝔤^{0}-module. Furthermore, it is clear that

supp M^{0} = supp M_{λ}^{F} + ⟨ (Δ′)^{I} ⟩_{ℤ_{+}}.

Finally, put 𝔤^{+} · M^{0} =^{def} 0 and let M be the unique irreducible quotient of U (𝔤) ⊗_{U(𝔭)} M^{0}.

Lemma 14.

supp M = supp M_{λ}^{F} + ⟨ (Δ′)^{I} ⊔ (Δ′)^{−} ⟩_{ℤ_{+}}.

Proof. It follows from Lemma 2 (a) that

(10) supp M ⊆ supp M_{λ}^{F} + ⟨ (Δ′)^{I} ⊔ (Δ′)^{−} ⟩_{ℤ_{+}}.

To establish equality, note that both sides of (10) are invariant under the Weyl group W of the root system (Δ′)^{F}. Note also that

(Δ′)^{−} = ∩_{w∈W} w(_{λ} (Δ′)^{−}).

Therefore, by the convexity property of supp M, it is sufficient to show that λ + ℤ_{−}α ⊆ supp M for every simple root α ∈ (Δ′)^{+}. The latter follows easily from the compatibility of λ. Indeed, if α ∉ Δ′′ and λ (𝔥′) ≠ 0, then −α ∈ Γ by Lemma 7. If α ∈ Δ′′, then (λ, α) ∉ ℤ_{+} implies that −α ∈ Γ_{λ}.

References

1. D. J. Britten, F. W. Lemire, A classification of simple Lie modules having a 1-dimensional weight space, Transactions of AMS, 299 (1987), 683–697.

2. I. Dimitrov, O. Mathieu, and I. Penkov, On the structure of weight modules, Transactions of AMS, to appear.

3. I. Dimitrov and I. Penkov, Weight modules over direct limit Lie algebras, IMRN, to appear.

4. , Partially integrable highest weight modules, Transfomation groups, 3 (1998), 241–253.

5. V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge University Press, Cambridge, 1990.

6. O. Mathieu, Classification of irreducible weight modules, preprint (1997).

7. I. Penkov and V. Serganova, The support of an irreducible Lie algebra representation, Journ. of Algebra 209 (1998), 129–142.

8. A. N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of Cartan type, Izvestiya, Ser. Mat. (in Russian) 38 (1974), 836–866; Engl. transl.: Math USSR-Izv. 8 (1974), 836-866.

9. A. N. Rudakov, Irreducible representations of infinite-dimensional Lie algebras of types S and H, Izvestiya, Ser. Mat. (in Russian) 39 (1975), 496–511; Engl. transl.: Math USSR-Izv. 9 (1975), 465-480.

10. I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan, part I (the transitive groups), J. d’Analyse Math. 15 (1965), 1–114.

I.P.: Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA.

   E-mail address: penkov@math.ucr.edu

V.S.: Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA.

   E-mail address: serganov@math.berkeley.edu