Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields
aa r X i v : . [ m a t h . R T ] F e b SIMPLE WEIGHT MODULES WITH FINITE WEIGHTMULTIPLICITIES OVER THE LIE ALGEBRA OF POLYNOMIALVECTOR FIELDS
DIMITAR GRANTCHAROV AND VERA SERGANOVA
Abstract.
Let W n be the Lie algebra of polynomial vector fields. We classify sim-ple weight W n -modules M with finite weight multiplicities. We prove that everysuch nontrivial module M is either a tensor module or the unique simple submodulein a tensor module associated with the de Rham complex on C n .2020 MSC: 17B66, 17B10Keywords and phrases: Lie algebra, Cartan type, weight module, localization. Introduction
Lie algebras of vector fields have been studied since the fundamental works of S.Lie and E. Cartan in the late 19th century and the early 20th century. A classicalexample of such Lie algebra is the Lie algebra W n consisting of the derivations of thepolynomial algebra C [ x , ..., x n ], or, equivalently, the Lie algebra of polynomial vectorfields on C n . The first classification results concerning representations of W n andother Cartan type Lie algebras were obtained by A. Rudakov in 1974-1975, [15], [16].These results address the classification of a class of irreducible W n -representationsthat satisfy some natural topological conditions. The modules of Rudakov are aparticular class of the so-called tensor modules .General tensor modules T ( P, V ) are introduced by Shen and Larson, [17], [10],and are defined for a D n -module P and gl ( n )-module V , where D n is the algebra ofpolynomial differential operators on C n (see § T ( P, V )have nice geometric interpretations. If V is finite dimensional, then we have a naturalmap from W n to the algebra of differential operators in the section of a trivial vectorbundle on C n with fiber V . This map is a specialization of a Lie algebra homomor-phism W n → D n ⊗ U ( gl ( n )). The tensor module T ( P, V ) is nothing but the pull backof the D n ⊗ U ( gl ( n ))-module P ⊗ V .Tensor W -modules and their extensions were studied extensively in the 1970’s andin the 1980’s by B. Feigin, D. Fuks, I. Gelfand, and others, see for example, [4], [6]. Date : February 19, 2021.The first author is partially supported by Simons Collaboration Grant 358245.The second author is partially supported by NSF Grant 1701532.
Important results on general tensor modules T ( P, V ) have been recently establishedby G. Liu, R. Lu, Y. Xue, K. Zhao, and others, see [18] and the references therein.In this paper we focus on the category of weight representations of W n , namelythose that decompose as direct sums of weight spaces relative to the subalgebra h of W n spanned by the derivations x ∂ ,..., x n ∂ n . The study of weight representationsof Lie algebras of vector fields is a subject of interest by both mathematicians andtheoretical physicists in the last 30 years. Two particular cases in this study haveattracted special attention - the cases of W n and of the Witt algebra Witt n . Recallthat Witt n is the Lie algebra of the derivations of the Laurent polynomial algebra C [ x ± , ..., x ± n ], or, equivalently, the Lie algebra of polynomial vector fields on the n -dimensional complex torus. In particular, Witt is the centerless Virasoro algebra.The classification of all simple weight representations with finite weight multiplicitiesof W and Witt (and hence of the Virasoro algebra) was obtained by O. Mathieuin 1992, [12]. Following a sequence of works of S. Berman, Y. Billig, C. Conley, X.Guo, C. Martin, O. Mathieu, V. Mazorchuk, V. Kac, G. Liu, R. Lu, A. Piard, S.Eswara Rao, Y. Su, K. Zhao, recently, Y. Billig and V. Futorny managed to extendMathieu’s classification result to Witt n for arbitrary n ≥ The classification of simple bounded (i.e. with a bounded set of weight multiplici-ties) modules of W n was completed in [18]. The result in [18] states that every simplebounded module is a tensor module T ( P, V ) or a submodule of a tensor module. Inorder T ( P, V ) to be bounded, P must be a weight D n -module and V must be afinite-dimensional module.In this paper we classify all simple weight W n -modules M with finite weight mul-tiplicities. The main result is surprisingly easy to formulate - every such nontrivialmodule M is either a tensor module T ( P, V ) or the unique simple submodule of T ( P, V k C n ) for k = 1 , ..., n . The necessary and sufficient condition for P and V sothat T ( P, V ) has finite weight multiplicities is given in Theorem 3.5. This conditionis expressed in terms of the subsets of roots W n and gl ( n ) that act locally finitely orinjectively on P and V , respectively. For our classification result, we first use a the-orem of [14] stating that M is parabolically induced from a bounded simple module N over a subalgebra g = W m ⋉ ( k ⊗ O m ) of W n . This subalgebra g plays the roleof a Levi subalgebra of a parabolic subalgebra of W n . The classification of simplebounded g -modules is one of the most difficult parts of the proof. By introducing theso called ( g , O m )-modules, we prove that N is either the unique submodule of a ten-sor module, or it is a special generalized tensor module F ( T ( P, V ) , S ), see Theorem5.16. The essential tool for proving this theorem is the twisted localization functorintrduced in [13]. For the main theorem we show that the parabolic induction functormaps F ( T ( P, V ) , S ) to a tensor module. Note that the Witt algebra Witt n is denoted by W n in [1]. IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 3
The content of the paper is as follows. In Section 2 we collect some importantdefinitions and preliminary results on weight modules, twisted localization, parabolicinduction, and tensor modules. In Section 3 we prove the necessary and sufficientcondition for the tensor module T ( P, V ) to be a weight module with finite weightmultiplicities. We also show that T ( P, V ) has a unique simple submodule and explainhow the restricted duality functor acts on the tensor modules. The main theorem ofthis paper is also stated in Section 3. Section 4 is devoted to a few results concerningthe parabolic induction theorem. The study of bounded g -modules and the classifi-cation of all possible g -modules N that appear in the parabolic induction theoremare included in Section 5. In Section 6 we complete the proof of the main theoremby showing that the application of the parabolic induction functor on all possible N described in the previous section leads to modules M that are either tensor modulesor the unique simple submodules of T ( P, V k C n ) for k = 1 , ..., n .2. Preliminaries
Notation and convention.
Throughout the paper the ground field is C . Allvector spaces, algebras, and tensor products are assumed to be over C unless otherwisestated.2.2. Weight modules in general setting.
Let U be an associative unital algebraand H ⊂ U be a commutative subalgebra. We assume in addition that H is apolynomial algebra identified with the symmetric algebra of a vector space h , andthat we have a decomposition U = M µ ∈ h ∗ U µ , where U µ = { x ∈ U | [ h, x ] = µ ( h ) x, ∀ h ∈ h } . Let Q U = Z ∆ U = ∆ U ∪ ( − ∆ U ) be the Z -lattice in h ∗ generated by ∆ U = { µ ∈ h ∗ | U µ = 0 } . We obviously have U µ U ν ⊂ U µ + ν .We call a U -module M a weight module , or a ( U , H ) -module , if M = L λ ∈ h ∗ M λ ,where M λ = { m ∈ M | hm = λ ( h ) m for all h ∈ h } . We call M λ the weight space of M , dim M λ the λ -weight multiplicity of M , andsupp M = { λ ∈ h ∗ | M λ = 0 } the support of M . Note that U µ M λ ⊂ M µ + λ . for every weight module M .We will call a weight U -module bounded if its set of weight multiplicities is abounded set. For a bounded U -module M , the degree d ( M ) is the maximal weightmultiplicity of M . A weight U -module M with finite weight multiplicities is cuspidal if all nonzero elements of U µ act injectively on M . If ∆ U = − ∆ U , then every cuspidal DIMITAR GRANTCHAROV AND VERA SERGANOVA U -module is bounded. We use this notion in the case when U is the Weyl algebra orthe universal enveloping algebra of a reductive Lie algebra where the latter propertyholds.In the particular case when U = U ( g ) for a Lie algebra g and H = S ( h ) for aCartan subalgebra h of g , we have that a weight U -module is a weight g -module.2.3. Twisted localization in general setting.
We retain the notation of the pre-vious subsection.Let a be an ad-nilpotent element of U . Then the set h a i = { a n | n ≥ } is an Oresubset of U which allows us to define the h a i -localization D h a i U of U . For a U -module M by D h a i M = D h a i U ⊗ U M we denote the h a i -localization of M . Note that if a isinjective on M , then M is isomorphic to a submodule of D h a i M . In the latter casewe will identify M with that submodule.We next recall the definition of the generalized conjugation of D h a i U relative to x ∈ C . This is the automorphism φ x : D h a i U → D h a i U defined by the formula φ x ( u ) = X i ≥ (cid:18) xi (cid:19) ad( a ) i ( u ) a − i . If x ∈ Z , then φ x ( u ) = a x ua − x . With the aid of φ x we define the twisted moduleΦ x ( M ) = M φ x of any D h a i U -module M . Finally, we set D x h a i M = Φ x D h a i M for any U -module M and call it the twisted localization of M relative to a and x . We willuse the notation a x · m (or simply a x m ) for the element in D x h a i M corresponding to m ∈ D h a i M . In particular, the following formula holds in D x h a i M : u ( a x m ) = a x X i ≥ (cid:18) − xi (cid:19) ad( a ) i ( u ) a − i m ! for u ∈ U , m ∈ D h a i M .If a , ..., a k are commuting ad-nilpotent elements in U and c = ( c , ..., c k ) is in C k ,then we set D h a ,...,a k i M = Q ki =1 D h a i i M and D c h a ,...,a k i M = Q ki =1 D c i h a i i M . Note thatthe products Q ki =1 D h a i i and Q ki =1 D c i h a i i are well defined because the functors involvedpairwise commute.If a ∈ U is an ad-nilpotent weight element and M is a weight module then D x h a i M is again a weight module. Lemma 2.1.
Let a ∈ U be an ad -nilpotent weight element in U , M be a simple a -injective weight U -module, and z ∈ C . If N is a simple nontrivial submodule of D z h a i M , then D h a i M ≃ D − z h a i N . In particular, if a acts bijectively on M , M ≃ D − z h a i N . Proof.
We use the fact that if M is a simple weight U -module, then D h a i M and D z h a i M are simple D h a i U -modules. Since N is submodule of D z h a i M , D h a i N is a submoduleof D z h a i M . The simplicity of N implies D h a i N ≃ D z h a i M and D − z h a i N ≃ D h a i M . If a acts bijectively, then M ≃ D h a i M . (cid:3) IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 5
We will also consider the following particular case of the twisted localization functorfor U = U ( g ), where g = W m + k ⊗ O m . Let a i ∈ k α i , i = 1 , ..., ℓ , and Γ = { α , ..., α ℓ } is a set of commuting roots of k that is linearly independent in Z ∆ k . Let also λ ∈ h ∗ be such that λ = P ℓi =1 z i α i . We set D λ Γ = D z h a i ...D z ℓ h a ℓ i . If M ≃ D λ Γ ¯ M we will saythat M is obtained by a twisted localization from ¯ M . If ¯ M is bounded, then M isbounded, [13] Lemma 4.4.2.4. The algebras O n , D n , and W n . In what follows, O n = C [ x , ..., x n ] and D n willstand for the associative algebra of differential operators in O n . In other words, D n isthe n -th Weyl algebra. We will often use the fact that D n ≃ D ⊗ ... ⊗ D ( n copies).Also, W n will stand for the Lie algebra of vector fields on C n , i.e. W n = Der( O n ).Henceforth, we fix h = Span { t ∂ , . . . , t n ∂ n } . Note that h is a Cartan subalgebraof W n and H = C [ t ∂ , . . . , t n ∂ n ] is a maximal commutative subalgebra of D n . Wewill use the setting of § § U = U ( W n ) and U = D n , and in bothcases h is the one that we fixed above. The set of roots of W n is:∆ = ( n X j =1 m j ε j , − ε i + X j = i m j ε j | m j ∈ Z ≥ , i = 1 , ..., n ) , where ε i ( t j ∂ j ) = δ ij . By ∆ ′ we denote the set of all invertible roots of W n . One cansee that ∆ ′ is a root system of type A n .A W n -module M is a ( W n , O n ) -module if M is an O n -module satisfying X ( f v ) = f X ( v ) + X ( f ) v, ∀ v ∈ M, f ∈ O n , X ∈ W n . If M is a weight W n -module with finite weight multiplicities, then the restricted dual M ∗ of M is by definition the maximal semisimple h -submodule of M ∗ . The followingproperties of the restricted dual functor are straightforward. Lemma 2.2.
Let M be a weight W n -module with finite weight multiplicities. Then (1) supp M ∗ = − supp M ; (2) dim M µ ∗ = dim M − µ ; (3) M is simple if and only if M ∗ is simple. Consider the embedding C n → C P n . The Lie algebra of vector fields on C P n isisomorphic to sl ( n + 1) and is a Lie subalgebra of W n . In other words we have acanonical embedding sl ( n + 1) ⊂ W n of Lie algebras. Lemma 2.3.
Let M be a bounded weight W n -module such that supp M ⊂ λ + Z ∆ W n for some weight λ . Then M has finite length. Proof.
The result holds for sl ( n + 1)-modules, see Lemma 3.3 in [13], and hence itholds for W n by using the natural embedding of sl ( n + 1) in W n . (cid:3) DIMITAR GRANTCHAROV AND VERA SERGANOVA
Simple weight D n -modules. According to § D n -module M is a weightmodule if M = M λ ∈ C n M λ , where M λ = { m ∈ M | x i ∂ i m = λ i m, for i = 1 , ..., n } . Below we recall the classifica-tion of the simple weight D n -modules.We will use the automorphism σ F : D n → D n defined by σ F ( t i ) = ∂ i , σ F ( ∂ i ) = − t i for all i . We call σ F the (full) Fourier transform of D n . If M is a D n -module, by M F we denote the module M twisted by σ F .The following gives the classification of all simple weight D n -modules, see for ex-ample Corollary 2.9 in [8] Proposition 2.4. (i)
Every simple weight module of D is isomorphic to one ofthe following: O = C [ x ] , O F , x λ C [ x ± ] , λ ∈ C \ Z . (ii) Every simple weight module of D n is isomorphic to P ⊗ ... ⊗ P n where P i isa simple weight D -module. We note also that every simple nontrivial weight D n -module M has degree 1, i.e.all its weight multiplicities equal 1. Moreover, for every i , x i (respectively, ∂ i ) actseither injectively, or locally nilpotently on M . Let I + ( M ) denote the subset of indicesin { , . . . , n } such that ∂ i acts locally nilpotently on M and x i acts injectively on M , I − ( M ) the subset of indices such that x i acts locally nilpotently on M and ∂ i actsinjectively on M , and I ( M ) the subset of indices such that both x i and ∂ i actinjectively on M . Note that { , . . . , n } = I − ( M ) ⊔ I ( M ) ⊔ I + ( M ). Furthermore,there exists λ ∈ supp M such thatsupp M = λ + X i ∈ I + ( M ) Z ≥ ε i + X j ∈ I ( M ) Z ε j + X k ∈ I − ( M ) Z ≤ ε k . Parabolic induction in general.
Let g be any Lie algebra with Cartan sub-algebra h such that g = h ⊕ L α ∈ ∆ g α . Let γ ∈ h ∗ . Then the subalgebra p = h ⊕ M Re h γ,α i≥ g α is called the parabolic subalgebra of g corresponding to γ . The Levi subalgebra of p is l = h ⊕ M Re h γ,α i =0 g α , and the nilradical of p is n = M Re h γ,α i > g α , We are going to use extensively the following standard result.
IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 7
Proposition 2.5. (a) Let N be a simple l -module, considered also as simple p -modules by letting n act trivially on N . Then the g -module U ( g ) ⊗ U ( p ) N has aunique simple quotient.(b) If L is a simple g -module such that L n = 0 , then L n is a simple l -module.(c) If L and M are simple g -modules such that M n ≃ L n as l -modules, then M ≃ L as g -modules. Remark . If M is a simple weight g -module then M n = L λ ∈ S M λ where S is thesubset of supp M such that λ + α / ∈ supp M for any α ∈ ∆( n ). For an arbitraryweight module M we call ⊕ λ ∈ S M λ the p -top of M and denote it by M top .2.7. Parabolic induction for W n . In this subsection we recall one of the mainresults in [14]. Recall the definitions of ∆ and ∆ ′ from § γ = a ε + · · · + a n ε n for some a i ∈ R . Set∆ = { α ∈ ∆ | ( γ, α ) = 0 } , ∆ ± = { α ∈ ∆ | ( γ, α ) > ( < } , ∆ ′ = ∆ ∩ ∆ ′ , ∆ ′± = ∆ ± ∩ ∆ ′ . Let p = h ⊕ M α ∈ ∆ ∪ ∆ + ( W n ) α , g = h ⊕ M α ∈ ∆ ( W n ) α . Theorem 2.7.
Let M be a simple weight W n -module.(a) There exists a weight λ ∈ supp M and γ such that supp M ⊂ λ + Z ≥ (∆ ′− ∪ ∆ ′ ) . (b) One can choose γ in such a way that Z ∆ ′ = Z ∆ and λ + Z ∆ ⊂ supp M. (c) M is a unique simple quotient of the parabolically induced module U ( W n ) ⊗ U ( p ) M for some simple weight g -module M that is extended in the natural way to asimple p -module. Tensor modules over W n . Let V be a gl ( n )-module and ˜ V := O n ⊗ V . Onecan look at ˜ V as the space of sections of the gl ( n )-bundle on C n with fiber V . Thus,˜ V has the natural structure of a ( W n , O n )-module.For a D n -module P and a gl ( n )-module V , we define the tensor ( W n , O n )-moduleby T ( P, V ) := P ⊗ O n ˜ V and call it the tensor W n -module relative to P and V . If P is a weight D n -moduleand V is a weight gl ( n )-module then P ⊗ O n ˜ V is a weight module andsupp( P ⊗ O n ˜ V ) = supp P + supp V. DIMITAR GRANTCHAROV AND VERA SERGANOVA
Alternatively, we can define T ( P, V ) as follows. Consider T ( P, V ) as the vectorspace T ( P, V ) = P ⊗ C V and define W n -action and O n -action by the formulas x α ∂ j · ( f ⊗ v ) = x α ∂ j f ⊗ v + n X i =1 ∂ i ( x α ) f ⊗ E ij v,x α · ( f ⊗ v ) = x α f ⊗ v, for f ∈ P , v ∈ V .In what follows, the k -th exterior power V k C n of the natural representation of gl ( n )will be called the k -th fundamental representation. We have the following result from[11] (Theorem 3.1 and Lemma 3.7): Proposition 2.8. (i)
Let P be a simple D n -module and V be a simple gl ( n ) -module that is not isomorphic to a fundamental representation. Then T ( P, V ) is a simple W n -module. (ii) Let P and P be simple D n -modules and let V and V be simple gl ( n ) -modulessuch that neither of them is isomorphic to a fundamental representation. Then T ( P , V ) ≃ T ( P , V ) if and only if P ≃ P and V ≃ V . We next consider tensor modules T ( P, V ) for which V is a fundamental represen-tation. For any D n -module P , the differential map d : T ( P, ^ C n ) → T ( P, ^ C n ) , is defined by d ( f ⊗ v ) = P ni =1 ( ∂ i f ) ⊗ ( e i ∧ v ), where ( e , ..., e n ) is the standard basisof C n associated to the coordinates x , ..., x n of C n . The map d is a homomorphsimof W n -modules but not O n -modules. One readily sees that d = 0. As a result wehave the following generalized de Rham complex:0 d −→ T ( P, ^ C n ) d −→ T ( P, ^ C n ) d −→ · · · d −→ T ( P, ^ n C n ) d −→ . By Theorem 3.5 in [11] we have the following.
Proposition 2.9.
Let P be a simple D n -module. (i) If k = 0 , ..., n − , then the module T ( P, V k C n ) has a simple quotient isomor-phic to dT ( P, V k C n ) . (ii) The module T ( P, V C n ) is simple if and only if P is not isomorphic to O n . If P ≃ O n then T ( P, V C n ) contains a trivial W n -submodule C and dT ( P, V C n ) ≃ T ( P, V C n ) / C . (iii) The module T ( P, V n C n ) is simple if and only if P i ∂ i P = P . We finish this subsection by stating the main result from [18] concerning the clas-sification of the simple bounded W n -modules. Theorem 2.10.
Let M be a nontrivial simple bounded W n -module. Then M isisomorphic to one of the following: IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 9 (a) the module T ( P, V ) , where P is a simple weight D n module and V is a sim-ple finite-dimensional gl ( n ) -module that is not isomorphic to a fundamentalrepresentation; (b) a simple submodule of T ( P, V k C n ) , where k ∈ { , , ..., n } , and P is a simpleweight D n module. Remark . Proposition 3.7 implies that T ( P, V k C n ) has a unique simple submod-ule. 3. Tensor modules with finite weight multiplicities
Tensor product of weight gl ( n ) -modules. Let M be a simple weight gl ( n )-module with finite weight multiplicities. Recall from [3] that M has the following shadow decomposition : ∆( gl ( n )) = ∆ FM ⊔ ∆ IM ⊔ ∆ + M ⊔ ∆ − M , such that the α -root vectors X α act locally nilpotently on M for all roots α ∈ ∆ + M ⊔ ∆ FM and injectively for all roots α ∈ ∆ − M ⊔ ∆ IM . Moreover, ∆ IM ⊔ ∆ FM and ∆ + M are the roots of the Levi subalgebra g I + g F and the nilradical g + , respectively,of a parabolic subalgebra p ⊂ gl ( n ), and M is a quotient a parabolically inducedmodule Ind gl ( n ) p (cid:0) M F ⊗ M I (cid:1) , for some cuspidal simple g I -module M I and some finite-dimensional simple g F -module M F . Lemma 3.1.
Let l be the Levi subalgebra of some parabolic p in gl ( n ) . Assume that M ′ and N ′ are weight l -modules and that M ′ ⊗ N ′ has finite weight multiplicities.Then (Ind gl ( n ) p M ′ ) ⊗ (Ind gl ( n ) p N ′ ) has finite weight multiplicities. Proof.
Let m denote the nilradical of the opposite to p parabolic subalgebra p − , andlet U = U ( m ). Then U has a Z ≥ -grading U = L p ≥ U p such that U = C andeach U p is a finite-dimensional l -module. This grading induces Z ≥ -gradings on both M = Ind gl ( n ) p M ′ and N = Ind gl ( n ) p N ′ so that M p = M ′ ⊗ U p and N p = N ′ ⊗ U p . Then M ⊗ N is also graded and its m th graded component is( M ⊗ N ) m = M p + q = m M ′ ⊗ N ′ ⊗ U p ⊗ U q . Hence, M ⊗ N has finite weight multiplicities. (cid:3) Lemma 3.2.
Let M and N be simple weight gl ( n ) -modules. Then M ⊗ N hasfinite weight multiplicities if and only if (∆ IM ⊔ ∆ − M ) ⊂ (∆ FN ⊔ ∆ − N ) or, equivalently, (∆ IM ⊔ ∆ − M ) ∩ (∆ IN ⊔ ∆ + N ) = ∅ . Proof.
First assume that the condition is not true. There exists a root α ∈ ∆ IM ⊔ ∆ − M such that − α ∈ ∆ IN ⊔ ∆ − N . If µ ∈ supp M and ν ∈ supp N then µ + Z ≥ α ⊂ supp M and ν − Z ≥ α ∈ supp N . Hence µ + ν has infinite multiplicity in M ⊗ N . Next assume that the condition holds. Then ∆ IM ⊂ ∆ FN , ∆ IN ⊂ ∆ FM and ∆ − M ⊂ (∆ FN ⊔ ∆ − N ). Choose γ M ∈ Q ∆ such that ( γ M , α ) = 0 for all α ∈ ∆ IM ⊔ ∆ FM and( γ M , α ) < α ∈ ∆ − M . Similarly choose γ N , and let γ = γ M + γ N . Then( γ, ∆ IM ) = ( γ, ∆ IN ) = 0 and ( γ, α ) < α ∈ ∆ − M ∪ ∆ − N . Let p be theparabolic defined by γ . Then both M and N are quotients of the parabolicallyinduced modules Ind gl ( n ) p M ′ and Ind gl ( n ) p N ′ , respectively. The Levi subalgebra l of p is isomorphic to g IM ⊕ g IN ⊕ ( g FM ∩ g FN ). Furthermore, M ′ = M i ⊗ M f where M i is a simple cuspidal g IM -module and M f is some finite-dimensional g IN ⊕ ( g FM ∩ g FN )-module. Similarly, N ′ = N i ⊗ N f where N i is a simple cuspidal g IN -module and N f issome finite-dimensional g IM ⊕ ( g FM ∩ g FN )-module. Therefore M ′ ⊗ N ′ has finite weightmultiplicities and the statement follows from Lemma 3.1. (cid:3) Weight tensor modules.Lemma 3.3.
Let P be a simple weight D n -module. Then P = L κ P κ , where P κ isthe eigenspace of P ni =1 x i ∂ i with eigenvalue κ . Furthermore, every nonzero P κ is asimple gl ( n ) -module and all nonzero P κ have the same shadow. Proof.
The first assertion is obvious. Since the adjoint action of gl ( n ) on D n is locallyfinite, every root vector X α ∈ gl ( n ) either acts locally nilpotently or injectively on allnonzero vectors of P . Therefore all P κ have the same shadow. By the classification ofsimple weight D n -modules, every P κ is multiplicity free and supp P κ ⊂ λ + Z ∆( gl ( n ))for any weight λ ∈ supp P κ . Using these and the fact that U ( gl ( n )) P λκ = P κ , weobtain that P κ is simple. (cid:3) Remark . Lemma 3.3 implies that every simple D n -module P has a well-defined gl ( n )-shadow. Below we give an explicit description of this shadow in terms of thesubsets I ± ( P ), I ( P ) of { , . . . , n } defined in § IP = { ε i − ε j | i, j ∈ I ( P ) } , ∆ FP = { ε i − ε j | i, j ∈ I + ( P ) or i, j ∈ I − ( P ) } , ∆ − P = { ε i − ε j | i ∈ I + ( P ) , j / ∈ I + ( P ) or i / ∈ I − ( P ) , j ∈ I − ( P ) } , ∆ + P = − ∆ − P . Theorem 3.5.
Let P be a simple weight D n -module and V be a simple weight gl ( n ) -module. Then the W n -module T ( P, V ) has finite weight multiplicities if and only if (∆ IP ⊔ ∆ − P ) ⊂ (∆ FV ⊔ ∆ − V ) . Proof.
For every semisimple h -module X we denote by X κ the eigenspace of P x i ∂ i with eigenvalue κ . By Lemma 3.3, P = L τ ∈ τ + Z P τ for some τ ∈ C . Then T ( P, V ) = M τ ∈ τ + Z P τ ⊗ V, and the statement follows from Lemma 3.2. (cid:3) Example 3.6.
Consider a simple highest weight module gl (4)-module V such that∆ FV ⊔ ∆ + V = { ε i − ε j | i = 1 , j = 3 , } . IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 11
Let P be a simple weight D n -module P on which x , x , ∂ , ∂ act injectively and ∂ , ∂ , x , x act locally nilpotently. Then by Remark 3.4 and Theorem 3.5, T ( P, V )has infinite weight multiplicities as ε − ε ∈ ∆ − P ∩ ∆ + V . On the other hand, if P ′ is asimple D n -module on which x , ∂ , ∂ , ∂ act locally nilpotently and ∂ , x , x , x actinjectively, then T ( P ′ , V ) has finite weight multiplicities. Proposition 3.7.
For any simple weight D n -module P and any simple weight gl ( n ) -module V , the W n -module T ( P, V ) has a unique simple submodule. Proof. If V is not a fundamental representation the statement follows from Propo-sition 2.8(i). Now let V = V k C n . It is shown in [9] that if P is cuspidal, i.e., I + ( P ) = I − ( P ) = ∅ , then T ( P, V k C n ) is simple for k = 0 , n and an indecomposable sl ( n + 1)-module of length two for k = 1 , . . . , n −
1. This implies the statement for acuspidal module P . For a general module P , consider γ = s X i ∈ I − ( P ) ε i − X j ∈ I + ( P ) ε j for some irrational s >
1. Let p be the corresponding parabolic subalgebra of W n and n be the nilradical of p . The Levi subalgebra g is isomorphic to gl ( p ) ⊕ gl ( q ) ⊕ W m where p = | I − ( P ) | , q = | I + ( P ) | , and m = | I ( P ) | . Note that P ≃ O Fp ⊗ O q ⊗ P m for some cuspidal D m -module P m . Since V is finite dimensional and simple, V n ∩ gl ( n ) ≃ V p ⊗ V q ⊗ V m is a simple module over gl ( p ) ⊕ gl ( q ) ⊕ gl ( m ). It is easy to compute that T ( P, V ) n ≃ V p ⊗ V q ⊗ T ( P m , V m ) . Since P m is cuspidal, T ( P m , V m ) has a unique simple W m -submodule and hence T ( P, V ) n has a unique simple g -submodule N . If M is a simple W n submoduleof T ( P, V ) then M n = 0 and hence N ⊂ M . That implies the uniquness of M . (cid:3) Duality for tensor modules.Lemma 3.8.
Let P be a simple weight D n -module. Consider P as a W n -module viathe natural homomorphism W n → D n . Then P ∗ ≃ T ( P F , Λ n C n ) . Proof.
Recall the definition of I ± ( P ) , I ( P ). As a vector space P = Y i ∈ I ( P ) x λ i i ⊗ C [ x j ] j ∈ I + ( P ) ⊗ C [ ∂ k ] k ∈ I − ( P ) ⊗ C [ x ± ℓ ] ℓ ∈ I + ( P ) , where λ i are nonintegral for all i ∈ I ( P ). We denote the monomial basis of P by e ( µ ) where µ i ∈ λ i + Z for i ∈ I ( P ), µ i ∈ Z ≥ for i ∈ I + ( P ) ⊔ I − ( P ). We have x i e ( µ ) = ( e ( µ + ε i ) if i ∈ I ( P ) ∪ I + ( P ) − µ i e ( µ + ε i ) if i ∈ I − ( P ) , ∂ i e ( µ ) = ( µ i e ( µ − ε i ) if i ∈ I ( P ) ∪ I + ( P ) e ( µ + ε i ) if i ∈ I − ( P ) . Denote the corresponding basis of P F by f ( µ ) where µ runs over the same set as e ( µ ).Using identification of P and P F as vector spaces, we have that if X ( e ( µ )) = ce ( ν )then σ F ( X ) f ( µ ) = cf ( ν ). This observation allows us to write the action of generatorsin the basis f ( µ ): ∂ i f ( µ ) = ( − f ( µ + ε i ) if i ∈ I ( P ) ∪ I + ( P ) ,µ i f ( µ − ε i ) if i ∈ I − ( P ) ,x i f ( µ ) = ( µ i f ( µ − ε i ) if i ∈ I ( P ) ∪ I + ( P ) ,f ( µ + ε i ) if i ∈ I − ( P ) . Let ϕ ( µ ) be a function satisfying ϕ ( µ + ε i ) = ( ( µ i + 1) ϕ ( µ )if i ∈ I ( P ) ∪ I + ( P ) , − ( µ i + 1) ϕ ( µ ) if i ∈ I − ( P ) . Define a pairing P × P F → C by setting h e ( µ ) , f ( ν ) i = ϕ ( µ ) δ µ,ν . Then we have h ∂ i e ( µ ) , f ( ν ) i = −h e ( µ ) , ∂ i f ( ν ) i , h x i e ( µ ) , f ( ν ) i = h e ( µ ) , x i f ( ν ) i . Hence h g ( x ) ∂ i e ( µ ) , f ( ν ) i = −h e ( µ ) , ∂ i g ( x ) f ( ν ) i . Using that ∂ i g ( x ) = g ( x ) ∂ i + ∂ i ( g ( x )) and choosing nonzero ω ∈ V n C n , we obtain g ( x ) ∂ i ( f ( ν ) ⊗ ω ) = ( ∂ i g ( x ) f ( ν )) ⊗ ω. This leads to a nondegenerate W n -invariant pairing P × T ( P F , V n C n ) → C . (cid:3) Lemma 3.9.
Let V and P be such that T ( P, V ) has finite weight multiplicities, andlet V ∗ be the restricted dual of V . Then T ( P F , V ∗ ⊗ V n C n ) and T ( P, V ) are restricteddual to each other in the category of weight W n -modules. Proof.
We define a pairing T ( P F , V ∗ ⊗ ^ n C n ) × T ( P, V ) → C by the formula h f ⊗ v, g ⊗ w i = h f, g ih v, w i , v ∈ V, w ∈ V ∗ , f ∈ P, g ∈ T ( P F , ^ n C n ) . Then we have h x α ∂ j ( f ) ⊗ v + X i ∂ i ( x α ) f ⊗ E ij v, g ⊗ w i + h f ⊗ v, x α ∂ j ( g ) ⊗ w + X i ∂ i ( x α ) ⊗ E ij w i = h x α ∂ j ( f ) , g ih v, w i + h f, x α ∂ j ( g ) ih v, w i + IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 13 X j h ∂ j ( x α ) f, g ih E ij v, w i + h f, ∂ j ( x α ) g ih v, E ij w i = 0 , because of h x α ∂ j ( f ) , g i + h f, x α ∂ j ( f ) i = 0 , h ∂ j ( x α ) f, g i = h f, ∂ j ( x α ) g i and h E ij v, w i + h v, E ij w i = 0 . (cid:3) Statement of Main Result.
In this subsection we state and prove the mainresult in the paper. Some of the results used in the proof will be established in thenext three sections.
Theorem 3.10.
Let M be a simple weight W n -module with finite weight multiplic-ities. Then M is the unique submodule of some tensor module T ( P, V ) with finiteweight multiplicities. More precisely, exactly one of the following holds: (i) M is isomorphic to T ( P, V ) for a simple weight D n -module P and a simpleweight gl ( n ) -module V with finite weight multiplicities, such that (∆ IP ⊔ ∆ − P ) ⊂ (∆ FV ⊔ ∆ − V ) and such that V is not isomorphic to a fundamental representation. (ii) M is isomorphic to dT ( P, V k C n ) for some k = 0 , , ..., n − , and a simpleweight D n -module P . (iii) M ≃ C , which is the unique simple submodule of T ( P, V C n ) . Proof.
Let M be a simple weight W n -module with finite weight multiplicities. ByTheorem 2.7 and Proposition 4.1, M is a quotient of the parabolically induced moduleInd W n p N where N is a simple bounded g -module over the Levi subalgebra g of p .Moreover, by Corollary 5.6, N satisfies the additional conditions (1) and (2) of Section5. Theorem 5.16 provides a classification of such N . Finally, Lemma 6.2 and Lemma6.3 ensure that M is one of the modules listed in the statement. (cid:3) Applications of the parabolic induction
Recall that ∆ stands for the set of roots of W n . We use the setting of § M is a simple weight W n -module that hasfinite weight multiplicities. We will use that M is the unique simple quotient of aparabolically induced module U ( W n ) ⊗ U ( p ) M , as stated in Theorem 2.7. Let p bethe parabolic subalgebra associated with γ = P ni =1 a i ε i . We assume without loss ofgenerality that a ≥ · · · ≥ a p > a p +1 = · · · = a p + m > a p + m +1 ≥ · · · ≥ a n . Henceforth we fix p and denote by g the Levi subalgebra of p . Then g ≃ W m ⋉ ( k ⊗O m )where k is a Levi subalgebra in gl ( p ) ⊕ gl ( n − m − p ). Under this assumptions wehave the following Proposition 4.1.
The simple g -module M is bounded. Proof.
First we prove three preliminary results.
Lemma 4.2.
Let α = − ε i or α ∈ ∆( k ) . Then (1) dim g α = 1 and any nonzero X α ∈ g α can be included in the sl triple; (2) Either g α acts locally nilpotently on M or g α : M → M is injective. Proof.
The first assertion is obvious. The second follows from the fact that ad g α islocally nilpotent in g . (cid:3) Lemma 4.3.
Let α = − ε i for p < i ≤ p + m or α ∈ ∆( k ) . Then g α acts injectivelyon M . Proof.
Suppose that g α is locally nilpotent on M . Let h be the Cartan element inthe sl -triple containing X α ∈ g α \ { } . In particular, α ( h ) = 2. Let µ ∈ supp M .Then µ + Z α ⊂ supp M . Furthermore, for any n > k ≥ n and v ∈ M µ + kα such that g α v = 0. Let M k denote the sl (2)-submodule of M generated by v . For all sufficiently large k we have µ ∈ supp M k . Therefore dim M µ = ∞ . Acontradicton. (cid:3) Corollary 4.4.
Let α = − ε i or α ∈ ∆( k ) . For any λ ∈ supp M and X ∈ g α \ themap X : M λ → M λ + α is an isomorphism. Proof.
From the previous lemma we know that X : M λ → M λ + α is injective. Ap-plying the same lemma to M ∗ we obtain X : M − λ − α ∗ → M − λ ∗ is injective. Hence X : M λ → M λ + α is surjective. (cid:3) We are now ready to complete the proof of Proposition 4.1. Corollary 4.4 impliesdim M µ = dim M µ + γ for any γ ∈ Z ∆( W m ) + Z ∆( k ) and µ ∈ supp M . The statementfollows. (cid:3) Bounded simple g -modules Generalization of tensor modules for the Levi subalgebra g of p . Weretain the notation of the previous section. In this section we assume that m > g = W m ⋉ ( k ⊗ O m ). Without loss of generality we may assume O m = C [ x , . . . , x m ]. In this section we will classify simple bounded g -modules N satisfyingthe additional properies:(1) supp N = λ + Z ∆( g ) for any λ ∈ supp N .(2) All weight spaces of N have the same dimension d .It follows from the proof of Proposition 4.1 that g α acts injectively on N if α = − ε i or α ∈ ∆( k ).First, we generalize the notion of a ( W m , O m )-module to that of a ( g , O m )-module. IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 15
Definition 5.1. A g -module N is a ( g , O m )-module if N is a O m -module satisfying(5.1) X ( f v ) = f X ( v ) + X ( f ) v ∀ v ∈ N, f ∈ O m , X ∈ W m , (5.2) ( h ⊗ Y )( f v ) = ( hf ) Y v ∀ v ∈ N, f, h ∈ O m , Y ∈ k . From now on we assume that all ( W m , O m )-modules and all ( g , O m )-modules areweight modules. Remark . Consider the algebra A ( m ) generated by W m ⊗ ⊗ O m withrelations ( x ⊗ y ⊗ − ( y ⊗ x ⊗
1) = [ x, y ] ⊗ , (1 ⊗ f )(1 ⊗ g ) = 1 ⊗ f g, ( x ⊗ ⊗ f ) − (1 ⊗ f )( x ⊗
1) = 1 ⊗ x ( f )for x, y ∈ W m and f, g ∈ O m . Any ( W m , O m ) is an A ( m )-module, and conversly, any A ( m )-module is a ( W m , O m )-module. Furthermore, A ( m ) is isomorphic to U ( W m ) ⊗O m as a vector space by the correspondence ( X ⊗ ⊗ f ) X ⊗ f for all X ∈ U ( W m )and f ∈ O m . Let B := A ( m ) ⊗ U ( k ). Then any ( g , O m )-module is a B -module. Example 5.3.
Let S be a k -module. We define a ( g , O m )-module structure on thevector space O m ⊗ S by setting f ( h ⊗ s ) = f h ⊗ s, ( f ⊗ Y )( h ⊗ s ) = f h ⊗ Y s, X ( h ⊗ s ) = X ( h ) ⊗ s for all f, h ∈ O m , Y ∈ k , X ∈ W m and s ∈ S . One can easily verify that ˜ S := O m ⊗ S is a ( g , O m )-module. Moreover, if R is a ( W m , O m )-module then F ( R, S ) := R ⊗ O m ˜ S is a ( g , O m )-module. Remark . A simple weight ( W m , O m )-module R with finite weight multiplicitiesis a tensor module T ( P, V ) for some simple weight D m -module P and some simpleweight gl ( m )-module V , see Theorem 3.7 in [18]. Lemma 5.5. If R is a simple ( W m , O m ) -module and S is a simple weight k -modulethen F ( R, S ) is a simple ( g , O m ) -module, in the sense that it does not contain propernontrivial ( g , O m ) -submodules. Proof.
Observe that F ( R, S ) is isomorphic to R ⊗ S as a B -module. Hence it is asimple B -module. This implies the statement. (cid:3) Lemma 5.6. If N = F ( R, S ) satisfies conditions (1) and (2), then S is a simplecuspidal k -module, and R = T ( P, V ) for some simple cuspidal D m -module P a simplefinite-dimensional gl ( m ) -module V . For any λ ∈ supp N we have that dim N λ =(dim V ) d ( S ) , where d ( S ) is the degree of the cuspidal module S . Proof.
The lemma follows from the isomorphism of h -modules F ( R, S ) ≃ R ⊗ S . (cid:3) Lemma 5.7.
Let S be a simple nontrivial weight k -module, P be a simple weight D m -module, and V be a simple finite-dimensional gl ( m ) -module. Then F ( T ( P, V ) , S ) is a simple g -module. Proof.
Choose a regular u ∈ ( k ∩ h ) ∗ that acts nontrivially on S , and denote by F ( T ( P, V ) , S ) a the eigenspace of u with eigenvalue a . Let M be a proper nonzerosubmodule of F ( T ( P, V ) , S ). Then M a = M ∩ F ( T ( P, V ) , S ) a is O m -invariant forany a = 0. Using the action of the root elements of k , we obtain that M a is O m -invariant for a = 0 as well. Hence M is a ( g , O m )-submodule of F ( T ( P, V ) , S ) andwe reach a contradiction. (cid:3) Lemma 5.8.
Let N be a simple ( g , O m ) -module satisfying (1) and (2). Then N is isomorphic to F ( R, S ) for some ( W m , O m ) -module R and some simple cuspidal k -module S . Proof.
Recall the definition of B from Remark 5.2. Consider N as a B -module. Bydefinition, for any vector v ∈ N we have B v = U ( g ) v (this follows from the relation( f ⊗ Y ) v = f ( Y v )). Hence, N is a simple B -module. For a simple k -module S ′ ,the subspace Hom k ( S ′ , N ) ⊗ S ′ of N is B -stable. Hence, there is a unique up toisomorphism S ′ , such that Hom k ( S ′ , N ) = 0. The existence of such S ′ follows fromcondition on the support of N . Let S be such module. We have that R = Hom k ( S, N )is a simple A ( m )-module. Therefore, N ≃ R ⊗ S as a B -module. The condition (5.2)ensures that N ≃ F ( R, S ). (cid:3) Recall that ˜ V stands for ( W m , O m )-module O m ⊗ V . Lemma 5.9.
Let N be a simple weight g -module, such that ∂ i acts locally nilpotentlyfor all i = 1 , . . . , m . Then N is isomorphic to a simple submodule of F ( ˜ V , S ) forsome simple k -module S and a simple gl ( m ) -module V . Proof.
Let N be the space of invariants of ∂ , . . . , ∂ m . If q is the parabolic subalgebraassociated to γ = − ( ε + · · · + ε m ), then N is the unique simple quotient of U ( g ) ⊗ U ( q ) N . Thus, N is a simple gl ( m ) ⊕ k -module, so N = V ⊗ S for some simple modules V and S . Then we have a natural homomorphism ϕ : N → F ( ˜ V , S ) of gl ( m ) ⊕ k -modules, hence, also of q -modules. The homomorphism ϕ induces a homomorphismΦ : U ( g ) ⊗ U ( q ) N → F ( ˜ V , S ) of g -modules. The image of Φ is isomorphic to N . (cid:3) Lemma 5.10.
Let N be a simple g -module satisfying (1) and (2). Assume that onecan define an O m -module structure on N in such a way that it satisfies (5.1) and f ( g ⊗ Y ) v = ( g ⊗ Y ) f v, ∀ v ∈ N, f, g ∈ O m , Y ∈ k . Then N is a ( g , O m ) -module. Proof.
We need to verify (5.2). First, we claim that verifying (5.2) is equivalent tochecking(5.3) x (1 ⊗ Y ) = ( x ⊗ Y ) , ∀ Y ∈ k . Indeed, let f ∈ O m and X = f ∂ . Then[ X, x ](1 ⊗ Y ) = f (1 ⊗ Y ) = [ X, x ⊗ Y ] = f ⊗ Y. IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 17
For f, g ∈ O m we have f ( g ⊗ Y ) = f g (1 ⊗ Y ) . Next, observe that (1) and (2) ensure that ∂ i and x i act injectively on N . Thereforewe can localize N with respect x i and define a ( W m , ˜ O m )-module structure on N ,where ˜ O m = C [ x ± , x ± , . . . , x ± m ]. Consider the twisted localization D c h x ,...,x n i N of N with U = B , and some c ∈ C m . We can choose c so that D c h x ,...,x n i N has a nonzeroweight vector annihilated by all ∂ i . Among all such vectors choose one of weight µ with maximal possible | µ | r := Re P mi =1 µ i . Let N ′ be a g -submodule of D c h x ,...,x n i N generated by u . Note that for any ν ∈ supp N ′ we have | µ | r ≥ | ν | r . Let v have weight ν with | ν | r = | µ | r . Then ∂ i v = 0 and ∂ i ( x (1 ⊗ Y ) − x ⊗ Y ) v = 0 , hence u = ( x (1 ⊗ Y ) − x ⊗ Y ) v is annihilated by all ∂ i . On the other hand, theweight η of u satisfies | η | r = | µ | r + 1 hence u = 0. Let w ∈ N ′ be a weight vector ofweight λ with minimal | λ | such that for some Y ∈ k ( x (1 ⊗ Y ) − x ⊗ Y ) w = 0 . We have ∂ i ( x (1 ⊗ Y ) − x ⊗ Y ) w = ( x (1 ⊗ Y ) − x ⊗ Y ) ∂ i w = 0 , which leads to a contradiction. Next we note that D c h x ,...,x n i N = ˜ O m · N ′ . Since x (1 ⊗ Y ) − x ⊗ Y commutes with ˜ O m we have ( x (1 ⊗ Y ) − x ⊗ Y ) N ( c ) = 0. Then( x (1 ⊗ Y ) − x ⊗ Y ) N = 0. This completes the proof. (cid:3) Lemma 5.11.
Assume that N is a simple g -module satisfying (1) and (2). Let z bea central element of k which does not act trivially on N . Then N is a ( g , O m ) -moduleand hence is isomorphic to a module F ( R, S ) . Proof.
Without loss of generality we may assume that z acts as identity on N . Definean O m -module structure on N by setting x i v := ( x i ⊗ z ) v . Then N satisfies theassumptions of Lemma 5.10. The statement follows. (cid:3) Lemma 5.12.
Let k be abelian and N be a simple g -module satisfying (1) and (2).If k N = 0 , then ( O m ⊗ k ) N = 0 . Proof.
We will show that ( f ⊗ h ) N = 0 for any f ∈ O m , h ∈ k . Note that[ ∂ , x ⊗ h ] N = hN = 0 . Therefore we have the following identites on N :[ ∂ ( x ⊗ h ) , ∂ ( x ⊗ h )] = 2 ∂ ( x ⊗ h ) = 2( ∂ ( x ⊗ h )) , [( ∂ ( x ⊗ h )) k , ∂ ( x ⊗ h )] = 2 k ( ∂ ( x ⊗ h )) k +1 . This implis that, on each weight space N λ of N , tr Nλ ( ∂ ( x ⊗ h )) k = 0 for all k > N is bounded this implies nilpotency of ∂ ( x ⊗ h ) on N . Since ∂ is invertibleon N we obtain that x ⊗ h is nilpotent on N . Let p be the nilpotency degree of x ⊗ h . There exists v ∈ N such that w := ( x ⊗ h ) p − v = 0. Then for f ∈ O m wehave 0 = f ∂ ( x ⊗ h ) p v = p ( f ⊗ h )( x ⊗ h ) p − v. In other words, w is annihilated by O m ⊗ h . The subspace N ′ of all vectors annihilatedby O m ⊗ h is g -invariant, but we just proved that N ′ = 0. By the irreducibility of N , we have N = N ′ . Thus ( O m ⊗ h ) N = 0. (cid:3) Proposition 5.13.
Let N be a g -module satisfying (1) and (2). Then for any Borelsubalgebra b ⊂ k there exists a simple bounded g -module ¯ N satisfying the followingtwo conditions: (i) There exists a weight λ ∈ h ∗ such that supp ¯ N ⊂ λ + P mi =1 Z ε i − Z ≥ ∆( b ) and λ ( h ∩ k ) = 0 . (ii) The module N is obtained from ¯ N by a twisted localization with respect tosome set of commuting roots Γ ⊂ − ∆( b ) . Proof.
Let ˆ k = k + h , n = [ b , b ]. Then N is a bounded weight ˆ k -module, hence, anycyclic ˆ k -submodule of N has finite length (see Lemma 3.3 in [13]). Let N be a simpleˆ k -submodule of N . Note that N is a cuspidal ˆ k -module. By Proposition 4.8 in [13],there exist µ ∈ ( h ∩ k ) ∗ and Γ ⊂ − ∆( b ) such that N ≃ D µ Γ M for some simplebounded b -highest weight ˆ k -module M . Since D µ Γ is well defined for g -modules andcommutes with the restriction functor Res g ˆ k , M := D − µ Γ N contains an n -primitiveweight vector v ∈ M , while ∂ i act injectively on M for all i = 1 , . . . , m . Since U (ˆ k ) v is bounded it has finite ˆ k -length and hence there is λ ′ ∈ supp U (ˆ k ) v such that λ ′ + α / ∈ supp U (ˆ k ) v for all α ∈ ∆( b ). The injectivity of the action of ∂ i implies that( λ ′ + α + m X i =1 Z ≥ ε i ) ∩ supp U ( g ) v = ∅ . If w is a nonzero vector of weight λ ′ thensupp U ( g ) w ⊂ λ ′ + m X i =1 Z ε i − Z ≥ ∆( b ) . This implies dim( U ( O m ⊗ n )) u < ∞ for any u ∈ U ( g ) w . By Lemma 2.3, the bound-edness of U ( g ) w implies that U ( g ) w has finite length. Let ¯ N be a simple submoduleof U ( g ) w . Then there is a nonzero weight vector u ∈ ¯ N annihilated by O m ⊗ n . Then¯ N satisfies (i) with λ being the weight of u , while (ii) follows from the simplicity of N .It remains to show that λ ( h ∩ k ) = 0. For the sake of contradiction, assume thatthe opposite holds. Take a simple root α ∈ ∆( b ). Then a simple computation showsthat g − α u is annihilated by O m ⊗ n . The simplicity of ¯ N hence implies that g − α u = 0for all simple roots α and thus M contains a trivial k -submodule. But the roots of Γact injectively on N and hence on M . This leads to a contradiction. (cid:3) IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 19
For a weight µ ∈ ( h ∩ k ) ∗ and a Borel subalgebra b of k , by L b ( µ ) (or simply by L ( µ ) ) we denote the simple b -highest weight k -module of highest weight µ . Lemma 5.14.
The module ¯ N constructed in Proposition 5.13 is isomorphic to F ( T ( P, V ) , L (¯ λ )) for a cuspidal simple D m -module P , a simple finite-dimensional gl ( m ) -module V , and a simple highest weight k -module L (¯ λ ) , where ¯ λ is the restric-tion of λ to h ∩ k . Proof.
Consider γ ∈ ( k ∩ h ) ∗ which determines the Borel subalgebra b . Then γ determines also a parabolic subalgebra q in g . The q -top of the module ¯ N is asimple ( W m ⊕ h )-module. Since ¯ λ = 0, this module is isomorphic to T ( P, V ) ⊗ C ¯ λ by Lemma 5.11. A simple computation shows that it is also isomorphic to the top of F ( T ( P, V ) , L (¯ λ )). Hence the statement follows from Proposition 2.5(c). (cid:3) Corollary 5.15. If k is not abelian then N is isomorphic to F ( T ( P, V ) , S ) for somecuspidal simple D m -module P , a simple finite-dimensional gl ( m ) -module V , and asimple cuspidal k -module S . Proof.
The result follows immediately from Proposition 5.13, Lemma 5.14, and theisomorphism of g -modules D − µ Γ F ( T ( P, V ) , L (¯ λ )) ≃ F ( T ( P, V ) , D − µ Γ L (¯ λ )) . (cid:3) Theorem 5.16.
Let N be a simple bounded g -module satisfying (1) and (2). Thenwe have one of the following two mutually exclusive statements.(a) O m ⊗ k acts trivially on N and N is a unique simple submodule of T ( P, V ) forsome simple cuspidal D m -module P and a simple finite-dimensional gl ( m ) -module V .In this case k must be abelian.(b) N is isomorphic to F ( T ( P, V ) , S ) for some cuspidal simple D m -module P , asimple finite-dimensional gl ( m ) -module V and a simple nontrivial cuspidal k -module S . Proof. If k is not abelian the statement follows from Corollary 5.15. If k is abelian and k acts nontrivially on N , the statement follows from Lemma 5.11. If k acts trivially on N , then by Lemma 5.12, ( O m ⊗ k ) N = 0. Then N is a simple bounded W m -moduleand the statement is a consequence of Theorem 1.1 in [18]. (cid:3) Back to tensor modules via parabolic induction
The case of infinite-dimensional g . We retain the notation of Section 4 andassume again that M is a simple weight W n -module that is also the unique simplequotient of the parabolically induced module U ( W n ) ⊗ U ( p ) N , where N is a simplebounded g -module satisfying (1) and (2). We we will use the properties of N listedin Theorem 5.16. Recall that p and m are fixed and defined in §
4. Let p ′ = p ∩ gl ( n ). The Levisubalgebra of p ′ is isomorphic to k ⊕ gl ( m ). Consider a g -module F ( T ( P, V ) , S )where V is a finite-dimensional gl ( m )-module, P is a simple cuspidal D m -moduleand S be a simple cuspidal k -module. Note that S might be a trivial k -module in thecase when k is abelian. Let U be the one-dimensional k -module of weight P pi =1 ε i and S U = S ⊗ U . Finally, let ˆ S be the unique simple quotient of U ( gl ( n )) ⊗ U ( p ′ ) ( S U ⊗ V ).Using the isomorphism D n ≃ D p ⊗ D m ⊗ D n − p − m , define a D n -module ˜ P by˜ P = C [ x , . . . , x p ] F ⊗ P ⊗ C [ x p + m +1 , . . . , x n ] , (recall that X F is the full Fourier transform of X ). Lemma 6.1.
The p -top of T ( ˜ P , ˆ S ) is isomorphic to F ( T ( P, V ) , S ) . Proof.
The statement follows by comparing the supports of the two modules. Let p = g ⊕ n . supp ˜ P = p X i =1 Z < ε i + supp P + n X i = p + m +1 Z ≥ ε i , supp ˆ S ⊂ supp S + p X i =1 ε i + supp V − supp U ( n ′ ) , where n ′ = n ∩ gl ( n ). Then we have thatsupp F ( T ( P, V ) , S ) ⊂ supp T ( ˜ P , ˆ S ) ⊂ supp F ( T ( P, V ) , S ) + supp U ( n − ) , where n − is the nilradical of the opposite parabolic. Moreover, the multiplicity ofany µ ∈ supp F ( T ( P, V ) , S ) is the same as its multiplicity in T ( ˜ P , ˆ S ). (cid:3) Lemma 6.2.
Let N = F ( T ( P, V ) , S ) be a simple g -module. Then the unique simplequotient M of U ( W n ) ⊗ U ( p ) N is isomorphic to unique simple submodule of T ( ˜ P , ˆ S ) . Proof.
The isomorphism of p -modules N → T ( ˜ P , ˆ V ) top induces a nonzero homomor-phism of W m -modules U ( W n ) ⊗ U ( p ) N → T ( ˜ P , ˆ V ). The image of this homomorphismis simple since T ( ˜ P , ˆ V ) has a unique simple submodule. Thus, this submodule isisomorphic to M . (cid:3) Now assume that F ( T ( P, V ) , S ) is not simple. This is only possible if k is abelian, S is trivial, and V = V k C m . Lemma 6.3.
Assume that N is the simple submodule T ( P, V k C m ) for some k =0 , . . . , m − . Then the unique simple quotient M of U ( W n ) ⊗ U ( p ) N is isomorphic tothe unique simple submodule of T ( ˜ P , V p + k C n ) . IMPLE WEIGHT MODULES OVER POLYNOMIAL VECTOR FIELDS 21
Proof.
We consider the monomorphism of p -modules N → T ( ˜ P , V p + k C n ) top , and theinduced map U ( W n ) ⊗ U ( p ) N → T ( ˜ P , ^ p + k C n ) . To complete the proof, we use the same reasoning as the one in the proof of theprevious lemma. (cid:3)
The case of finite-dimensional g . In this case we have m = 0 and g is a Liesubalgebra of gl ( n ). Using arguments similar to the ones used in the previous subsec-tion, one can show that the unique simple quotient of U ( W n ) ⊗ U ( p ) S is isomorphicto the unique simple submodule of T ( ˜ P , ˆ S ).6.3. The case g = W m . This case follows from Theorem 2.10.
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