bounded weight modules over the Lie superalgebra of Cartan W-type
aa r X i v : . [ m a t h . R T ] S e p BOUNDED WEIGHT MODULES OVER THE LIE SUPERALGEBRAOF CARTAN W-TYPE
RENCAI L ¨U, YAOHUI XUE
Abstract.
Let A m,n be the tensor product of the polynomial algebra in m evenvariables and the exterior algebra in n odd variables over the complex field C , and theWitt superalgebra W m,n be the Lie superalgebra of superderivations of A m,n . In thispaper, we classify the non-trivial simple bounded weight W m,n modules with respect tothe standard Cartan algebra of W m,n . Any such module is a simple quotient of a tensormodule F ( P, L ( V ⊗ V )) for a simple weight module P over the Weyl superalgebra K m,n , a finite-dimensional simple gl m -module V and a simple bounded gl n -module V . Introduction
We denote by Z , Z + , N , Q and C the sets of all integers, non-negative integers, positiveintegers, rational numbers and complex numbers, respectively. All vector spaces andalgebras in this paper are over C . Any module over a Lie superalgebra or an associativesuperalgebra is assumed to be Z -graded. Let 0 = ( m, n ) ∈ Z and let e , . . . , e m + n bethe standard basis of C m + n .Let A m,n ( resp. A m,n ) be the tensor superalgebra of the polynomial algebra C [ t , . . . , t m ] ( resp. Laurent polynomial algebra C [ t ± , . . . , t ± m ] ) in m even variables t , . . . , t m and the exterior algebra Λ( n ) in n odd variables ξ , . . . , ξ n . Denote by W m,n ( resp. W m,n ) the Lie superalgebra of super-derivations of A m,n ( resp. A m,n ). Cartan W -type Lie superalgebra W m,n was introduced by V. Kac in [18].Weight modules with finite-dimensional weight spaces are called Harish-Chandramodules. Many efforts have been made towards the classification of Harish-Chandramodules over various Lie (super)algebras. For finite-dimensional simple Lie algebras,O. Mathieu classified all the simple Harish-Chandra modules in [22]. M. Gorelik andD. Grantcharov completed the classification of all simple Harish-Chandra modules overall classical Lie superalgebras in [14], following the works in [8, 11, 13, 15, 17]. Suchmodules over the Virasoro algebra (which is the universal central extension of W , )were conjectured by V. Kac and classified by O. Mathieu in [21]. Y. Billig and V. Fu-torny completed the classification for W m, in [1]. Simple Harish-Chandra modules over W ,n were classified in [8]. The simple weight modules with finite-dimensional weightspaces with respect to the Cartan subalgebra of W , over the N = 2 Ramond algebra(which is a central extension of W , ) were classified in [20]. Such modules over W m,n were classified in [28], see also [3]. For more related results, we refer the readers to[2, 4, 5, 9, 10, 19, 23, 24, 25, 26] and the references therein.As we know, the classification of simple bounded weight modules is not only is animportant step in the classification of simple Harish-Chandra modules but also inter-esting on its own. Simple bounded weight modules over W , were classified in [21] andsimple bounded modules over W , were classified in [6]. Such modules over W m, wereclassified in [27]. In this paper, we classify the non-trivial simple bounded weight W m,n modules with respect to the standard Cartan algebra of W m,n . Any such module is asimple quotient of a tensor module F ( P, L ( V ⊗ V )) for a simple weight module P overthe Weyl superalgebra K m,n , a finite-dimensional simple gl m -module V and a simplebounded gl n -module V . This is achieved by widely using the results and methods in[27, 28].This paper is arranged as follows. In Section 2, we give some definitions and prelimi-naries. In Section 3, we prove that a simple weight AW -module with a finite-dimensionalweight space is a tensor module, see Lemma 3.8 and Theorem 3.9. In Section 4, weprove our main theorem, see Theorem 4.4.2. Preliminaries
A vector space V is called a superspace if V is endowed with a Z -gradation V = V ¯0 ⊕ V ¯1 . For any homogeneous element v ∈ V , let | v | ∈ Z with v ∈ V | v | . Throughoutthis paper, v is always assumed to be a homogeneous element whenever we write | v | fora vector v ∈ V .A module over a Lie superalgebra or an associative superalgebra is simple if it doesnot have nontrivial Z -graded submodules. A module M over a Lie superalgebra oran associative superalgebra g is called strictly simple if M does not have g -invariantsubspaces except 0 and M . Clearly, a strictly simple module must be simple. Denote byΠ( M ) the parity-change of M for a module M over a Lie superalgebra or an associativesuperalgebra. Lemma 2.1. [28, Lemma 2.2]
Let
B, B ′ be two unital associative superalgebras such density that B ′ has a countable basis, R = B ⊗ B ′ . Then (1) Let M be a B -module and M ′ be a strictly simple B ′ -module. Then M ⊗ M ′ isa simple R -module if and only if M is simple. (2) Suppose that V is a simple R -module and V contains a strictly simple B ′ = C ⊗ B ′ -submodule M ′ . Then V ∼ = M ⊗ M ′ for some simple B -module M . Write A := A m,n , W := W m,n and omit ⊗ in A for convenience.For any α = ( α , . . . , α m ) ∈ Z m + and i , . . . , i k ∈ { , . . . , n } , write t α := t α · · · t α m m and ξ i ,...,i k := ξ i · · · ξ i k . Also, for any subset I = { i , . . . , i k } ⊂ { , . . . , n } , write I = ( l , . . . , l k ) if { l , . . . , l k } = { i , . . . , i k } and l < · · · < l k . Denote ξ I := ξ l ,...,l k andset ξ ∅ = 1.Let i , . . . , i k be a sequence in { , . . . , n } . Denote by τ ( i , . . . , i k ) the inverse order ofthe sequence i , . . . , i k . Let I, J ⊂ { , . . . , n } with I ∩ J = ∅ and I = ( k , . . . , k p ) , J = ( l , . . . , l q ). We write τ ( I, J ) = ( k , . . . , k p , l , . . . , l q ). Set τ ( ∅ , ∅ ) = 0. Then ξ I ∪ J =( − τ ( I,J ) ξ I ξ J for all I ∩ J = ∅ . W has a standard basis { t α ξ I ∂∂t i , t α ξ I ∂∂ξ j | α ∈ Z m + , I ⊂ { , . . . , n } , i ∈ { , . . . , m } , j ∈ { , . . . , n }} . Define the extended Witt superalgebra ˜ W = W ⋉ A by(2.1) [ a, a ′ ] = 0 , [ x, a ] = − ( − | x || a | [ a, x ] = x ( a ) , ∀ x ∈ W, a, a ′ ∈ A. Write d i := t i ∂∂t i for any i ∈ { , . . . , m } and δ j := ξ j ∂∂ξ j for any j ∈ { , . . . , n } . Then H = span { d i , δ j | i = 1 , . . . , m, j = 1 , . . . , n } is the standard Cartan subalgebra of W .Let g be any Lie super-subalgebra of ˜ W that contains H and let M be a g -module. M is called a weight module if the action of H on M is diagonalizable. Namely, M is aweight module if M = ⊕ λ ∈ C m ,µ ∈ C n M ( λ,µ ) , where M ( λ,µ ) = { v ∈ M | d i ( v ) = λ i v, δ j ( v ) = µ j v, i ∈ { , . . . , m } , j ∈ { , . . . , n }} is called the weight space with weight ( λ, µ ). Denote bySupp( M ) = { ( λ, µ ) ∈ C m + n | M ( λ,µ ) = 0 } the support set of M . A weight g -module is called if the dimensions of its weight spacesare uniformly bounded by a constant positive integer.It’s easy to see that ˜ W (resp. W ) itself is a weight module over ˜ W (resp. W ) withSupp( ˜ W ) (resp. W ) ⊂ Z m + n . So for any indecomposable weight module M over ˜ W or W we have Supp( M ) ∈ ( λ, µ ) + Z m + n with some ( λ, µ ) ∈ C m + n .Let gl ( m, n ) = gl ( C m | n ) be the general linear Lie superalgebra realized as the spacesof all ( m + n ) × ( m + n ) matrices. Denote by E i,j , i, j = 1 , , . . . , m + n be the ( i, j )-thmatrix unit. gl ( m, n ) has a Z -gradation gl ( m, n ) = gl ( m, n ) − ⊕ gl ( m, n ) ⊕ gl ( m, n ) ,where gl ( m, n ) − = span { E m + j,i | i ∈ { , . . . , m } , j ∈ { , . . . , n }} , gl ( m, n ) = span { E i,m + j | i ∈ { , . . . , m } , j ∈ { , . . . , n }} and gl ( m, n ) = gl ( m, n ) ¯0 . Obviously, this Z -gradation is consistent with the Z -gradation of gl ( m, n ).A gl ( m, n )-module M is a weight module if M = ⊕ λ ∈ C m ,µ ∈ C n M ( λ,µ ) , where M ( λ,µ ) = { v ∈ M | E i,i ( v ) = λ i v, E m + j,m + j ( v ) = µ j v, i ∈ { , . . . , m } , j ∈ { , . . . , n }} is called the weight space with weight ( λ, µ ). Denote bySupp( M ) = { ( λ, µ ) ∈ C m + n | M ( λ,µ ) = 0 } the support set of M . A weight gl ( m, n )-module is called if the dimensions of its weightspaces are uniformly bounded by a constant positive integer.For any module V over the Lie algebra gl ( m, n ) , V could be viewed as modules overthe Lie superalgebra gl ( m, n ) with V ¯0 = V . Let V be a gl ( m, n ) -module and extend RENCAI L ¨U, YAOHUI XUE V trivially to a gl ( m, n ) ⊕ gl ( m, n ) -module. The Kac module of V is the inducedmodule K ( V ) := Ind gl (m , n) gl (m , n) ⊕ gl (m , n) (V). It’s easy to see that K ( V ) is isomorphic toΛ( gl ( m, n ) − ) ⊗ V as superspaces. Lemma 2.2. [7, Theorem 4.1]
For any simple gl ( m, n ) -module V , the module K ( V ) L(V) has a unique maximal submodule. The unique simple top of K ( V ) is denoted L ( V ) .Any simple gl ( m, n ) -module is isomorphic to L ( V ) for some simple gl ( m, n ) -module V up to a parity-change. Clearly, L ( V ) is a weight gl ( m, n )-module if and only if V is a weight gl ( m, n ) -module. 3. AW -modules A ˜ W -module M is called an AW -module if the action of A on M is associative, i.e., a · a ′ · v = ( aa ′ ) · v, t · v = v, ∀ a, a ′ ∈ A, v ∈ M. In this section, we will classify all simple bounded AW -modules.For any Lie (super)algebra g , let U ( g ) be the universal enveloping algebra of g . Bythe PBW Theorem, U ( ˜ W ) = U ( A ) · U ( W ). Let J be the left ideal of U ( ˜ W ) generatedby { t − , t α ξ I · t β ξ J − t α + β ξ I ξ J | α, β ∈ Z m + , I, J ⊂ { , . . . , n }} . It is easy to see that J is an ideal of U ( ˜ W ). Let ¯ U be the quotient algebra U ( ˜ W ) / J .Identify A and W with their images in ¯ U , then ¯ U = A · U ( W ). A · W is a Lie super-subalgebra of ¯ U , of which the bracket is given by(3.1) [ a · x, b · y ] = ax ( b ) · y − ( − | a · x || b · y | by ( a ) · x + ( − | x || b | ab · [ x, y ] , ∀ a, b ∈ A, x, y ∈ W. For any α ∈ Z m + , I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } , define X α,I,∂ = X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) t β ξ J · t α − β ξ I \ J ∂. Then X , ∅ ,∂ = ∂ . Let T = span { X α,I,∂ | α ∈ Z m + , I ⊂ { , . . . , n } , | α | + | I | > , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n }} and ∆ = span { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } . compute Lemma 3.1. (1) [
T, A ] = [ T, ∆] = 0 . (2) t α ξ I ∂ = P β αJ ⊂ I ( − τ ( J,I \ J ) (cid:0) αβ (cid:1) t β ξ J · X α − β,I \ J,∂ , ∀ α ∈ Z m + , I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } . Proof. (1)Let α ∈ Z m + , I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } . For any i ∈{ , . . . , m } , we have[ ∂∂t i , X α,I,∂ ] = X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) β i t β − e i ξ J · t α − β ξ I \ J ∂ + X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) ( α − β ) i t β ξ J · t α − β − e i ξ I \ J ∂ = X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) β i t β − e i ξ J · t α − β ξ I \ J ∂ − X e i β α + e i J ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ − e i (cid:19) ( α − β + e i ) i t β − e i ξ J · t α − β ξ I \ J ∂ = 0 . Clearly, for any j ∈ { , . . . , n } \ I , [ ∂∂ξ j , X α,I,∂ ] = 0. Now let I = ( l , . . . , l k ) and s ∈ { , . . . , k } .For any J ⊂ I with l s ∈ J , let J = ( l i , . . . , l i p ) and s = i q for some q ∈ { , . . . , p } .Then there are ( s − − ( q −
1) = s − q elements in I \ J that are less than l s . It followsthat τ ( J, I \ J ) = s − q + τ ( J \ { l s } , I \ J ). So( − | J | + τ ( J,I \ J ) ∂∂ξ l s ( ξ J ) = ( − | J | + s − q + τ ( J \{ l s } ,I \ J ) ( − q − ξ J \{ l s } = ( − s + | J \{ l s }| + τ ( J \{ l s } ,I \ J ) ξ J \{ l s } . For any J ⊂ I with l s / ∈ J , let I \ J = ( l i p +1 , . . . , l i k ) and s = i q for some q ∈{ p + 1 , . . . , k } . Then there are ( k − s ) − ( k − q ) = q − s elements in J that are greaterthan l s . It follows that τ ( J, I \ J ) = q − s + τ ( J, I \ ( J ∪ { l s } )). So( − | J | ( − | J | + τ ( J,I \ J ) ∂∂ξ l s ( ξ I \ J ) = ( − q − s + τ ( J,I \ ( J ∪{ l s } )) ( − q − −| J | ξ I \ ( J ∪{ l s } ) = ( − s +1+ | J | + τ ( J,I \ ( J ∪{ l s } )) ξ I \ ( J ∪{ l s } ) . Thus,[ ∂∂ξ l s , X α,I,∂ ] = X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) t β ∂∂ξ l s ( ξ J ) · t α − β ξ I \ J ∂ + X β αJ ⊂ I ( − | J | ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) t β ξ J · t α − β ∂∂ξ l s ( ξ I \ J ) ∂ RENCAI L ¨U, YAOHUI XUE = X β αl s ∈ J ⊂ I ( − | β | + s + | J \{ l s }| + τ ( J \{ l s } ,I \ J ) (cid:18) αβ (cid:19) t β ξ J \{ l s } · t α − β ξ I \ J ∂ + X β αl s / ∈ J ⊂ I ( − | β | + s +1+ | J | + τ ( J,I \ ( J ∪{ l s } )) (cid:18) αβ (cid:19) t β ξ J · t α − β ξ I \ ( J ∪{ l s } ) ∂ = ( − s X α,I \{ l s } ,∂ + ( − s +1 X α,I \{ l s } ,∂ = 0 . Therefore, [ X α,I,∂ , ∆] = 0.Now let | α | + | I | > a ∈ A . We have[ X α,I,∂ , a ] = X β αJ ⊂ I ( − | β | + | J | + τ ( J,I \ J ) (cid:18) αβ (cid:19) t β ξ J · t α − β ξ I \ J ∂ ( a )= X β αJ ⊂ I ( − | β | + | J | (cid:18) αβ (cid:19) t α ξ I ∂ ( a )= (1 − | α | + | I | t α ξ I ∂ ( a )= 0 . Hence, [
T, A ] = [ T, ∆] = 0.(2)For any ∅ = K ⊂ I , ξ K ξ I \ K = 0 and X J ⊂ K ( − τ ( J,I \ J )+ | K \ J | + τ ( K \ J,I \ K )+ τ ( J,K \ J ) ξ K ξ I \ K = X J ⊂ K ( − τ ( J,I \ J )+ | K \ J | + τ ( K \ J,I \ K ) ξ J ξ K \ J ξ I \ K = X J ⊂ K ( − τ ( J,I \ J )+ | K \ J | ξ J ξ I \ J = X J ⊂ K ( − | K \ J | ξ I = 0 . So P J ⊂ K ( − τ ( J,I \ J )+ | K \ J | + τ ( K \ J,I \ K )+ τ ( J,K \ J ) = 0.Then we have X β αJ ⊂ I ( − τ ( J,I \ J ) (cid:18) αβ (cid:19) t β ξ J · X α − β,I \ J,∂ = X β αJ ⊂ I X β ′ α − βJ ′ ⊂ I \ J ( − τ ( J,I \ J )+ | β ′ | + | J ′ | + τ ( J ′ ,I \ ( J ∪ J ′ )) (cid:18) αβ (cid:19)(cid:18) α − ββ ′ (cid:19) t β ξ J · t β ′ ξ J ′ · t α − β − β ′ ξ I \ ( J ∪ J ′ ) ∂ = X β αK ⊂ I X β ′ α − βJ ⊂ K ( − τ ( J,I \ J )+ | β ′ | + | K \ J | + τ ( K \ J,I \ K )+ τ ( J,K \ J ) (cid:18) αβ + β ′ (cid:19)(cid:18) β + β ′ β (cid:19) t β + β ′ ξ K · t α − β − β ′ ξ I \ K ∂ = X K ⊂ IJ ⊂ K X γ α β γ ( − τ ( J,I \ J )+ | γ − β | + | K \ J | + τ ( K \ J,I \ K )+ τ ( J,K \ J ) (cid:18) αγ (cid:19)(cid:18) γβ (cid:19) t γ ξ K · t α − γ ξ I \ K ∂ = X K ⊂ IJ ⊂ K ( − τ ( J,I \ J )+ | K \ J | + τ ( K \ J,I \ K )+ τ ( J,K \ J ) X γ α (cid:18) αγ (cid:19) (1 − | γ | ( − | γ | t γ ξ K · t α − γ ξ I \ K ∂ = t α ξ I ∂. (cid:3) Talg
Lemma 3.2. (1) B = { X α,I,∂ | α ∈ Z m + , I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n }} is an A -basis of the free left A -module A · W . (2) T = { x ∈ A · W | [ x, A ] = [ x, ∆] = 0 } . Thus T is a Lie subalgebra of A · W .Proof. (1)By Lemma 3.1 (2), B is a generating set of the free left A -module A · W . Andit is easy to see that B is A -linearly independent.(2)Let T = { x ∈ A · W | [ x, A ] = [ x, ∆] = 0 } . Then T ⊂ T by Lemma 3.1(1). Let x ∈ T , from (1) we know that x = P ki =1 a i · x i + x ′ , where x , . . . , x k arelinearly independent elements in T , a , . . . , a k ∈ A and x ′ ∈ A · ∆. For any a ∈ A ,0 = [ x, a ] = [ x ′ , a ]. So x ′ = 0. For any y ∈ ∆, 0 = [ y, x ] = P ki =1 y ( a i ) · x i . So a , . . . , a k ∈ C . Therefore T ⊂ T and consequently T = T . (cid:3) Let K m,n be the associative subalgebra of ¯ U generated by A and ∆. Ksimple
Lemma 3.3.
Any simple K m,n -module is strictly simple.Proof. Let V be a simple K m,n -module and V ′ be a nonzero K m,n -invariant subspaceof V with v ∈ V ′ \ { } . Since ∂∂ξ , . . . , ∂∂ξ n act nilpotently on the finite-dimensionalsubspace C [ ∂∂ξ , . . . , ∂∂ξ n ] v of V ′ , there is a w ∈ C [ ∂∂ξ , . . . , ∂∂ξ n ] v \ { } such that ∂∂ξ i · w =0 , i = 1 , . . . , n . If w is homogeneous, K m,n w is a submodule of V . By the simplicity of V , K m,n w = V ′ = V .Suppose that w is not homogeneous. Then w = w + w with w ∈ V ¯0 \ { } and w ∈ V ¯1 \ { } . Clearly, ∂∂ξ i · w = 0 , i = 1 , . . . , n . Also, K m,n w = V . Conse-quently, there is an odd element x in the subalgebra of K m,n that is generated by RENCAI L ¨U, YAOHUI XUE t , . . . , t m , ξ , . . . , ξ n , ∂∂t , . . . , ∂∂t m , such that xw = w . Note that x n +1 = 0. Let k be the smallest positive integer such that x k w = 0 and x k +1 w = 0. Then x k w = x k w + x k +1 w = x k w ∈ V ′ is a nonzero homogeneous element in V ′ . By the simplicityof V , K m,n x k w = V ′ = V .Therefore V is strictly simple. (cid:3) Note that K m,n ∼ = K (1) ⊗ K (2) ⊗ · · · ⊗ K ( m + n ) , where K ( i ) is the subalgebra of K m,n generated by t i , ∂∂t i for i ∈ { , . . . , m } , and K ( m + j ) is the subalgebra of K m,n generatedby ξ j , ∂∂ξ j for j ∈ { , . . . , n } .From [12] and Lemma 3.5 in [27], we have Kweight
Lemma 3.4. (1) Let P be any simple weight K m,n -module. Then P ∼ = V ⊗ · · · ⊗ V m ⊗ C [ ξ ] ⊗· · ·⊗ C [ ξ n ] , where every V i is one of the following simple weight C [ t i , ∂∂t i ] -modules: t λ i C [ t ± i ] , C [ t i ] , C [ t ± i ] / C [ t i ] , where λ i ∈ C \ Z .(2) Any weight K m,n -module V must have a simple submodule V ′ . Moreover, Supp( V ′ ) = X × · · · × X m × S , where each X i ∈ { λ i + Z , Z + , − N } for some λ i ∈ C \ Z and S = { k e + · · · + k n e n | k , . . . , k n = 0 , } . Lemma 3.5.
There is an associative superalgebra isomorphism π : K m,n ⊗ U ( T ) → ¯ U , π ( x ⊗ y ) = x · y, ∀ x ∈ K m,n , y ∈ U ( T ) . Proof.
Since T is a Lie super-subalgebra of ¯ U , the restriction of π on U ( T ) is well-defined. By Lemma 3.1 (1), π ( K m,n ) and π ( U ( T )) are super-commutative. So π is awell-defined homomorphism of associative superalgebras.Let g = A ⊗ T + ( A ∆ + A ) ⊗ C . From Lemma 3.2 (1), it is easy to see that ι := π | g : g → A · W + A is a Lie superalgebra isomorphism. Then the restriction of ι − on ˜ W = W + A gives a Lie superalgebra homomorphism η : ˜ W → K m,n ⊗ U ( T ). ByLemma 3.1 (2), η ( t α ξ I ) = t α ξ I ⊗ , η ( t α ξ I ∂ ) = P β αJ ⊂ Iα = β or J = I ( − τ ( J,I \ J ) (cid:0) αβ (cid:1) t β ξ J ⊗ X α − β,I \ J,∂ + t α ξ I ∂ ⊗ , ∀ α ∈ Z m + , I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } .η induces an associative superalgebra homomorphism ˜ η : U ( ˜ W ) → K m,n ⊗ U ( T ).Clearly, ˜ η ( J ) = 0. Then we have the induced associative superalgebra homomorphism¯ η : ¯ U → K m,n ⊗ U ( T ). It’s easy to see that π = ¯ η − . Hence π is an isomorphism. (cid:3) Let m be the ideal of A generated by t , . . . , t m , ξ , . . . , ξ n . Then m ∆ is a super-subalgebra of W . For any k ∈ N , m k ∆ is an ideal of m ∆. Lemma 3.6.
The linear map π : m ∆ → T defined by π ( t α ξ I ∂ ) = X α,I,∂ , ∀ t α ξ I ∈ m , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } , is an isomorphism of Lie superalgebras.Proof. π is clearly an isomorphism of superspaces. Consider the following combinationof natural Lie superalgebra homomorphisms: m ∆ ⊂ m · ∆ + A · T → ( m · ∆ + A · T ) / ( m · ∆ + m · T ) → ( A · T ) / ( m · T ) → T This homomorphism, which maps t α ξ I ∂ to X α,I,∂ , is just the linear map π . (cid:3) pi3 Lemma 3.7. (1) m ∆ / m ∆ ∼ = gl ( m, n ) . (2) Suppose that V is a simple weight m ∆ -module. Then m ∆ · V = 0 . Thus V could be viewed as a simple weight gl ( m, n ) -module.Proof. (1)Define the linear map π : gl ( m, n ) → m ∆ / m ∆ by π ( E i,j ) = t i ∂ j + m ∆ , π ( E m + p,j ) = ξ p ∂ j + m ∆ , π ( E i,m + q ) = t i ∂∂ξ q + m ∆ ,π ( E m + p,m + q ) = ξ p ∂∂ξ q + m ∆ , ∀ i, j ∈ { , . . . , m } , p, q ∈ { , . . . , n } . It is easy to verify that π is an isomorphism of Lie superalgebras.(2)It is clear that the adjoint action of d = d + · · · + d m + δ + · · · + δ n on W isdiagonalizable. For any k ∈ Z , let W k = { x ∈ W | [ d, x ] = kx } . Then W k − = span { t α ξ I ∂∂t i , t α ξ I ∂∂ξ j | α ∈ Z m + , I ⊂ { , . . . , n } , | α | + | I | = k, i ∈ { , . . . , m } , j ∈ { , . . . , n }} , ∀ k ∈ Z + and W = ∞ X l = − W l , m k ∆ = ∞ X l = k − W l , ∀ k ∈ N . Let v be a nonzero homogeneous weight vector of V . Then d · v = cv for some c ∈ C .By the simplicity of V , V = U ( m ∆) v . So the action of d on V is diagonalizable and theeigenvalues of d on V are contained in c + Z + . Since m ∆ is an ideal of m ∆, m ∆ V isa submodule of V . The eigenvalues of d on m ∆ V are contained in c + N , which doesnot contain c . Thus m ∆ V = V . By the simplicity of V , m ∆ V = 0. (cid:3) We now have the associative superalgebra homomorphism π : ¯ U π − −→ K m,n ⊗ U ( T ) ⊗ π − −→K m,n ⊗ U ( m ∆) −→ K m,n ⊗ U ( m ∆ / m ∆) ⊗ π − −→ K m,n ⊗ U ( gl ( m, n )) with π ( t α ξ I ∂∂t i ) = t α ξ I ∂∂t i ⊗ P mk =1 α k t α − e k ξ I ⊗ E k,i + ( − | I |− P nk =1 ∂∂ξ k ( t α ξ I ) ⊗ E m + k,i ,π ( t α ξ I ∂∂ξ j ) = t α ξ I ∂∂ξ j ⊗ P mk =1 α k t α − e k ξ I ⊗ E k,m + j +( − | I |− P nk =1 ∂∂ξ k ( t α ξ I ) ⊗ E m + k,m + j ,π ( t α ξ I ) = t α ξ I ⊗ , ∀ α ∈ Z m + , I ⊂ { , . . . , n } , i ∈ { , . . . , m } , j ∈ { , . . . , n } . Let P be a K m,n -module and M be a gl ( m, n )-module. Define the AW -module F ( P, M ) = P ⊗ M by x · ( u ⊗ v ) = π ( x ) · ( u ⊗ v ) , x ∈ ¯ U , u ∈ P, v ∈ M. Note that π ( d i ) = d i ⊗ ⊗ E i,i , π ( δ j ) = δ j ⊗ ⊗ E m + j,m + j . Then F ( P, M ) is a weight AW -module if P is a weight K m,n -module and M is a weight gl ( m, n )-module. cusF(P,M) Lemma 3.8.
Suppose that P is a simple weight K m,n -module and M is a simple weight gl ( m, n ) -module. Then F ( P, M ) is a bounded AW -module if and only if M is isomorphicto the gl ( m, n ) -module L ( V ⊗ V ) for some finite-dimensional simple gl m -module V andsome simple bounded gl n -module V up to a parity-change.Proof. Clearly, P is a simple weight K m,n -module with one-dimensional weight spaces.By Lemma 3.4, Supp( P ) is of the form X × · · · × X m × S , where each X i ∈ { λ (0) i + Z , Z + , − N } for some λ (0) i ∈ C \ Z and S = { k e + · · · + k n e n | k , . . . , k n = 0 , } .Let V be a finite-dimensional simple gl m -module and V be a simple bounded gl n -module. Then there is a positive integer N such thatdim( V ) µ N, ∀ µ ∈ Supp( V ) . The gl ( m, n )-module K ( V ⊗ V ) = Λ( gl ( m, n ) − ) ⊗ ( V ⊗ V ). Note that Λ( gl ( m, n ) − ) ⊗ V is a finite-dimensional weight module over the standard Cantan subalgebra of gl ( m, n ).Then K ( V ⊗ V ) has weight spaces with dimensions no more thandim Λ( gl ( m, n ) − ) · dim V · N = 2 mn N dim V . Moreover, Supp( K ( V ⊗ V )) ⊂ S × ( µ + Z n ) for some finite set S ⊂ C m and some µ ∈ Supp( V ). So L ( V ⊗ V ) has weight spaces with bounded municipality and Supp( L ( V ⊗ V )) ⊂ S × ( µ + Z n ).For any given ( λ ′ , µ ′ ) ∈ Supp( F ( P, L ( V ⊗ V ))), there are at most | S || S | pairs( λ (1) , µ (1) ) ∈ X × · · · × X m × S, ( λ (2) , µ (2) ) ∈ S × ( µ + Z n ) such that ( λ (1) , µ (1) ) +( λ (2) , µ (2) ) = ( λ ′ , µ ′ ). Since any weight space of P is one-dimensional and L ( V ⊗ V ) isbounded, F ( P, L ( V ⊗ V )) is bounded. Clearly, F ( P, Π( L ( V ⊗ V ))) is also bounded.Thus, the sufficiency is proved.Now suppose that F ( P, M ) is a bounded AW -module. By Lemma 2.2, there is asimple weight gl ( m, n ) -module V such that M is isomorphic to the gl ( m, n )-module L ( V ) or Π( L ( V )). Note that F ( P, Π( L ( V ))) ∼ = F (Π( P ) , L ( V )), where Π( P ) is also asimple weight K m,n -module. Without loss of generality, we assume that M = L ( V ). If m = 0, V is a simple bounded gl n -module and M = V ∼ = L ( V , V ), where V is theone-dimensional trivial gl -module. Suppose m > v be a nonzero weight vector of weight ( λ ′ , µ ′ ) in V ⊂ L ( V ). Then U ( gl m ) v is aweight gl m -submodule of V with E m + i,m + i w = µ ′ i w, ∀ i ∈ { , . . . , n } , w ∈ U ( gl m ) v. We claim that U ( gl m ) v is finite-dimensional. It’s easy to see that any weight spaceof U ( gl m ) v is finite-dimensional. For any positive integer k , suppose that U ( gl m ) v has k pairwise different weights λ ′ + β (1) , . . . , λ ′ + β ( k ) for some β (1) , . . . , β ( k ) ∈ Z m . Let v , . . . , v k be nonzero weight vectors in U ( gl m ) v of weight λ ′ + β (1) , . . . , λ ′ + β ( k ) respectively. Let ˜ λ ∈ C m such that ˜ λ − β ( i ) ∈ X × · · · × X m for all i ∈ { , . . . , k } and let ˜ µ ∈ S . Then (˜ λ − β , ˜ µ ) , . . . , (˜ λ − β k , ˜ µ ) are pairwise different weights of P .Let w , . . . , w k be nonzero weight vectors in P of weight (˜ λ − β , ˜ µ ) , . . . , (˜ λ − β k , ˜ µ )respectively. Obviously, w ⊗ v , . . . , w k ⊗ v k are weight vectors in F ( P, M ) of weight(˜ λ + λ ′ , ˜ µ + µ ′ ). So dim( F ( P, M )) (˜ λ + λ ′ , ˜ µ + µ ′ ) > k . Since F ( P, M ) is a bounded AW -module, U ( gl m ) v has only finite weights. Therefore, U ( gl m ) v is finite-dimensional.Let V be a simple gl m -submodule of U ( gl m ) v , which is clearly also a finite-dimensionalweight module. By Lemma 2.1(2), there is a simple gl n -module V such that V ∼ = V ⊗ V . V is a weight module for V is, and V is a bounded module for F ( P, L ( V )) is. Thus, M ∼ = L ( V ⊗ V ) and the necessity is proved. (cid:3) F(P,M)
Theorem 3.9.
Suppose that V is a simple weight AW -module with dim V ( λ,µ ) < ∞ forsome ( λ, µ ) ∈ Supp( V ) . Then V is isomorphic to F ( P, M ) for some simple weight K m,n -module P and some simple weight gl ( m, n ) -module M .Proof. Let D = span { X e i , ∅ , ∂∂ti , X ,ξ j , ∂∂ξj , t i · ∂∂t i , ξ j · ∂∂ξ j | i = 1 , . . . , m, j = 1 , . . . , n } ⊂ ¯ U .Then D is an abelian Lie super-subalgebra of ¯ U and D · V ( λ,µ ) ⊂ V ( λ,µ ) . So V ( λ,µ ) contains a homogeneous common eigenvector v of D . Let ρ : K m,n ⊗ U ( m ∆) ⊗ π −→K m,n ⊗ U ( T ) π −→ ¯ U be the isomorphism of associative superalgebras. Then V couldbe viewed as a simple K m,n ⊗ U ( m ∆)-module via ρ and v is a common eigenvector of ρ − ( D ) = span { d i ⊗ , δ j ⊗ , ⊗ d i , ⊗ δ j | i = 1 , . . . , m, j = 1 , . . . , n } . It followsthat K m,n v is a weight K m,n -module. By Lemma 3.4, K m,n v has a simple submodule P .Then, from Lemma 3.3 and Lemma 2.1 (2), V ∼ = P ⊗ M for some simple U ( m ∆)-module M . Since ρ − ( D ) acts diagonally on ( K m,n ⊗ U ( m ∆)) · v = V , M is a simple weight m ∆-module. By Lemma 3.7, M could be viewed as a simple weight gl ( m, n )-module.So V is isomorphic to the simple AW -module F ( P, M ). (cid:3) Classification of Bounded Modules If m ∈ N , for any α, β ∈ Z m + , I, J ⊂ { , . . . , n } , r ∈ N , j ∈ { , . . . , m } , ∂, ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } , define(4.1) ω r,j,∂,∂ ′ α,β,I,J = r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ξ I ∂ · t β + ie j ξ J ∂ ′ ∈ U ( W ) . Then(4.2) ω r,j,∂,∂ ′ α + e j ,β,I,J − ω r,j,∂,∂ ′ α,β + e j ,I,J = ω r +1 ,j,∂,∂ ′ α,β,I,J . Lemma 4.1. [27, Lemma 4.5]
Suppose that m ∈ N and M is a weight W m, -module with Wn+omega dim M λ N for all λ ∈ Supp( M ) . Then there is an r ∈ N such that ω r,j, ∂∂ti , ∂∂t ′ i α,β, ∅ , ∅ · M = 0 for all α, β ∈ Z m + , i, i ′ , j ∈ { , . . . , m } . omega Lemma 4.2.
Suppose that m ∈ N and M is a bounded W -module. Then there is an r ∈ N such that ω r,j,∂,∂ ′ α,β,I,J · M = 0 for all α, β ∈ Z m + , I, J ⊂ { , . . . , n } , j ∈ { , . . . , m } , ∂, ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } . Proof.
Since M is bounded, there is an N ∈ N such that dim M ( λ,µ ) N for all ( λ, µ ) ∈ Supp( M ). For any ( λ, µ ) ∈ Supp( M ), M (( λ + Z m ) ,µ ) := ⊕ α ∈ Z m M (( λ + α ) ,µ ) is a weight W m, -module, the dimensions of whose weight spaces are uniformly bounded by N . By Lemma4.1, there is an r ∈ N such that ω r,j, ∂∂ti , ∂∂t ′ i α,β, ∅ , ∅ · M (( λ + α ) ,µ ) = 0 for all α, β ∈ Z m + , i, i ′ , j ∈{ , . . . , m } . It follows that ω r,j, ∂∂ti , ∂∂t ′ i α,β, ∅ , ∅ · M = 0 , ∀ α, β ∈ Z m + , i, i ′ , j ∈ { , . . . , m } . For any α, β, γ ∈ Z m + , I ⊂ { , . . . , n } , j ∈ { , . . . , m } , we have[ ω r,j, ∂∂tj , ∂∂tj α,β, ∅ , ∅ , t γ ξ I ∂∂t j ]= r X i =0 ( − i (cid:18) ri (cid:19) [ t α +( r − i ) e j ∂∂t j , t γ ξ I ∂∂t j ] · t β + ie j ∂∂t j + r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ∂∂t j · [ t β + ie j ∂∂t j , t γ ξ I ∂∂t j ]= r X i =0 ( − i (cid:18) ri (cid:19) ( γ j t α + γ +( r − i − e j ξ I ∂∂t j − ( α j + r − i ) t α + γ +( r − i − e j ξ I ∂∂t j ) · t β + ie j ∂∂t j + r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ∂∂t j · ( γ j t β + γ +( i − e j ξ I ∂∂t j − ( β j + i ) t β + γ +( i − e j ξ I ∂∂t j ) . (4.3) Then f ( α, β, γ ) := [ ω r,j, ∂∂tj , ∂∂tj α + e j ,β, ∅ , ∅ , t γ ξ I ∂∂t j ] − [ ω r,j, ∂∂tj , ∂∂tj α,β, ∅ , ∅ , t γ + e j ξ I ∂∂t j ]= − r X i =0 ( − i (cid:18) ri (cid:19) t α + γ +( r − i ) e j ξ I ∂∂t j · t β + ie j ∂∂t j + r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i +1) e j ∂∂t j · ( γ j t β + γ +( i − e j ξ I ∂∂t j − ( β j + i ) t β + γ +( i − e j ξ I ∂∂t j ) − r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ∂∂t j · (( γ j + 1) t β + γ + ie j ξ I ∂∂t j − ( β j + i ) t β + γ + ie j ξ I ∂∂t j ) ∈ ann(M) . (4.4)We have f ( α + e j , β, γ ) − f ( α, β, γ + e j )= r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i +2) e j ∂∂t j · ( γ j − β j − i ) t β + γ +( i − e j ξ I ∂∂t j − r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i +1) e j ∂∂t j · ( γ j − β j − i + 1) t β + γ + ie j ξ I ∂∂t j + r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ∂∂t j · ( γ j − β j − i + 2) t β + γ +( i +1) e j ξ I ∂∂t j . (4.5)Then( f ( α + e j , β + e j , γ ) − f ( α, β + e j , γ + e j )) − ( f ( α + e j , β, γ + e j ) − f ( α, β, γ + 2 e j ))= − ω r,j, ∂∂tj , ∂∂tj α +2 e j ,β + γ, ∅ ,I + 4 ω r,j, ∂∂tj , ∂∂tj α + e j ,β + γ + e j , ∅ ,I − ω r,j, ∂∂tj , ∂∂tj α,β + γ +2 e j , ∅ ,I = − ω r +1 ,j, ∂∂tj , ∂∂tj α + e j ,β + γ, ∅ ,I + 2 ω r +1 ,j, ∂∂tj , ∂∂tj α,β + γ + e j , ∅ ,I = − ω r +2 ,j, ∂∂tj , ∂∂tj α,β + γ, ∅ ,I ∈ ann(M) . (4.6)So omega1omega1 (4.7) ω r +2 ,j, ∂∂tj , ∂∂tj α,β, ∅ ,I ∈ ann(M)for all α, β ∈ Z m + , I ⊂ { , . . . , n } , j ∈ { , . . . , m } . Let J ⊂ { , . . . , n } . We have[ ω r +2 ,j, ∂∂tj , ∂∂tj α,β, ∅ ,I , t γ ξ J ∂∂t j ]= r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( − | I || J | [ t α +( r +2 − i ) e j ∂∂t j , t γ ξ J ∂∂t j ] · t β + ie j ξ I ∂∂t j + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) t α +( r +2 − i ) e j ∂∂t j · [ t β + ie j ξ I ∂∂t j , t γ ξ J ∂∂t j ]= r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( − | I || J | ( γ j − α j − r − i ) t α + γ +( r +1 − i ) e j ξ J ∂∂t j · t β + ie j ξ I ∂∂t j + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( γ j − β j − i ) t α +( r +2 − i ) e j ∂∂t j · t β + γ +( i − e j ξ I ξ J ∂∂t j . (4.8)From (4.7) we have h ( α, β, γ ) := r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( γ j − α j − r − i ) t α + γ +( r +1 − i ) e j ξ J ∂∂t j · t β + ie j ξ I ∂∂t j ∈ ann(M) . Then h ( α + e j , β, γ ) − h ( α, β, γ + e j ) = − ω r +2 ,j, ∂∂tj , ∂∂tj α + γ,β,J,I ∈ ann(M) . So omega2omega2 (4.9) ω r +2 ,j, ∂∂tj , ∂∂tj α,β,J,I ∈ ann(M)for all α, β ∈ Z m + , I, J ⊂ { , . . . , n } , j ∈ { , . . . , m } .Let ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } \ { ∂∂t j } . We have[ ω r +2 ,j, ∂∂tj , ∂∂tj α,β,J,I , t γ ∂ ]= r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( − | I || ∂ | ( γ j t α + γ +( r +1 − i ) e j ξ J ∂ − ( − | J || ∂ | t γ +( r +2 − i ) e j ∂ ( t α ξ J ) ∂∂t j ) · t β + ie j ξ I ∂∂t j + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) t α +( r +2 − i ) e j ξ J ∂∂t j · ( γ j t β + γ +( i − e j ξ I ∂ − ( − | I || ∂ | t γ + ie j ∂ ( t β ξ I ) ∂∂t j ) . (4.10) By (4.9), x ( α, β, γ ) := r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( − | I || ∂ | γ j t α + γ +( r +1 − i ) e j ξ J ∂ · t β + ie j ξ I ∂∂t j + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) γ j t α +( r +2 − i ) e j ξ J ∂∂t j · t β + γ +( i − e j ξ I ∂ ∈ ann(M) . (4.11)Then x ( α + e j , β, γ ) − x ( α, β, γ + e j )= − r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( − | I || ∂ | t α + γ +( r +2 − i ) e j ξ J ∂ · t β + ie j ξ I ∂∂t j + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) γ j t α +( r +3 − i ) e j ξ J ∂∂t j · t β + γ +( i − e j ξ I ∂ − r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( γ j + 1) t α +( r +2 − i ) e j ξ J ∂∂t j · t β + γ + ie j ξ I ∂. (4.12)Thus, y ( α, β, γ ) := ( x ( α + 2 e j , β, γ ) − x ( α + e j , β, γ + e j )) − ( x ( α + e j , β, γ + e j ) − x ( α, β, γ + 2 e j ))= r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) γ j t α +( r +4 − i ) e j ξ J ∂∂t j · t β + γ +( i − e j ξ I ∂ − r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( γ j + 1) t α +( r +3 − i ) e j ξ J ∂∂t j · t β + γ + ie j ξ I ∂ + r +2 X i =0 ( − i (cid:18) r + 2 i (cid:19) ( γ j + 2) t α +( r +2 − i ) e j ξ J ∂∂t j · t β + γ +( i +1) e j ξ I ∂ ∈ ann(M) . (4.13)We have y ( α, β + e j , γ ) − y ( α, β, γ + e j )= − ω r +2 ,j, ∂∂tj ,∂α +2 e j ,β + γ,J,I + 2 ω r +2 ,j, ∂∂tj ,∂α + e j ,β + γ + e j ,J,I − ω r +2 ,j, ∂∂tj ,∂α,β + γ +2 e j ,J,I = − ω r +3 ,j, ∂∂tj ,∂α + e j ,β + γ,J,I + ω r +3 ,j, ∂∂tj ,∂α,β + γ + e j ,J,I = − ω r +4 ,j, ∂∂tj ,∂α,β + γ,J,I ∈ ann(M) . (4.14) So omega3omega3 (4.15) ω r +4 ,j, ∂∂tj ,∂α,β,J,I ∈ ann(M)for all α, β ∈ Z m + , I, J ⊂ { , . . . , n } , j ∈ { , . . . , m } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } \{ ∂∂t j } . Similarly, omega4omega4 (4.16) ω r +4 ,j,∂, ∂∂tj α,β,J,I ∈ ann(M) . Let ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } \ { ∂∂t j } . We have[ ω r +4 ,j, ∂∂tj ,∂α,β,J,I , t γ ∂ ′ ]= r +4 X i =0 ( − i (cid:18) r + 4 i (cid:19) ( − | ξ I ∂ || ∂ ′ | ( γ j t α + γ +( r +3 − i ) e j ξ J ∂ ′ − ( − | J || ∂ ′ | t γ +( r +4 − i ) e j ∂ ′ ( t α ξ J ) ∂∂t j ) · t β + ie j ξ I ∂ + r +4 X i =0 ( − i (cid:18) r + 4 i (cid:19) t α +( r +4 − i ) e j ξ J ∂∂t j · ( t β + ie j ∂ ( t γ ξ I ) ∂ ′ − ( − | ξ I ∂ || ∂ ′ | t γ + ie j ∂ ′ ( t β ξ I ) ∂ ) . (4.17)By (4.15), r +4 X i =0 ( − i (cid:18) r + 4 i (cid:19) γ j t α + γ +( r +3 − i ) e j ξ J ∂ ′ · t β + ie j ξ I ∂ ∈ ann(M) . Taking γ = e j , we have omega5omega5 (4.18) ω r +4 ,j,∂ ′ ,∂α,β,J,I ∈ ann(M)for all α, β ∈ Z m + , I, J ⊂ { , . . . , n } , j ∈ { , . . . , m } , ∂, ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } \{ ∂∂t j } .The lemma follows from (4.9), (4.15), (4.16) and (4.18) after replacing r + 4 with r . (cid:3) Let V be a weight W -module. Then W ⊗ V is a weight W -module with( W ⊗ V ) ( λ,µ ) = ⊕ α ∈ Z m ,β ∈ Z n ( W ) ( α,β ) ⊗ V (( λ − α ) , ( µ − β )) , ∀ λ ∈ C m , µ ∈ C n . Define the action of A on W ⊗ V by a · ( x ⊗ v ) = ( ax ) ⊗ v, ∀ a ∈ A, x ∈ W, v ∈ V. It is easy to verify that W ⊗ V now is an AW -module. Define a linear map θ : W ⊗ V → V by θ ( x ⊗ v ) = x · v, ∀ x ∈ W, v ∈ V. Then θ is a W -module homomorphism. Let X ( V ) = { x ∈ Ker θ | A · x ⊂ Ker θ } . Clearly, X ( V ) is an AW -submodule of W ⊗ V . The AW -module ˆ V = ( W ⊗ V ) /X ( V ) is calledthe A -cover of V and θ induces a W -module homomorphism ˆ θ : ˆ V → V . hat Theorem 4.3.
Suppose that V is a simple bounded W -module. Then ˆ V is a bounded AW -module.Proof. This is obvious if V is trivial. Suppose that V is not trivial, then W · V = V bythe simplicity of V .If m = 0, W is finite-dimensional. Then the AW -module W ⊗ V is bounded for V is. So ˆ V is bounded.Suppose m ∈ N . Let ( λ, µ ) ∈ Supp( V ), then Supp( V ) ⊂ ( λ, µ ) + Z m + n . Since V isbounded, there is a k ∈ N such that V ( λ ′ ,µ ′ ) k, ∀ ( λ ′ , µ ′ ) ∈ Supp( V ) , and there is an r ∈ N such that ω r,j,∂,∂ ′ α,β,I,J · V = 0 for all α, β ∈ Z m + , I, J ⊂ { , . . . , n } , j ∈{ , . . . , m } , ∂, ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } by Lemma 4.2. Then we have (4.19) r X i =0 ( − i (cid:18) ri (cid:19) t α +( r − i ) e j ξ I ∂ ⊗ t β + ie j ξ J ∂ ′ v ∈ X ( V ) , ∀ v ∈ V. Let B = span { t α ξ I ∂ | α ∈ Z m + , α i r, i = 1 , . . . , m,I ⊂ { , . . . , n } , ∂ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n }} be a subspace of W . Then B is a finite-dimensional H -submodule of W . So B ⊗ V isa H -submodule of W ⊗ V withdim( B ⊗ V ) ( λ ′ ,µ ′ ) dim B · k, ∀ λ ′ ∈ λ + Z m , µ ′ ∈ µ + Z n . To prove that ˆ V is a bounded AW -module, we need only to prove that W ⊗ V = B ⊗ V + X ( V ). Since W · V = V , it suffices to prove that t α ξ I ∂ ⊗ t β ξ J ∂ ′ v ∈ B ⊗ V + X ( V ) , ∀ α, β ∈ Z m + , I, J ⊂ { , . . . , n } ,∂, ∂ ′ ∈ { ∂∂t , . . . , ∂∂t m , ∂∂ξ , . . . , ∂∂ξ n } , v ∈ V. We will prove this by induction on | α | = α + · · · + α m . This is clear if | α | r or α j r for all j ∈ { , . . . , m } . Suppose that α j > r for some j . By (4.19), P ri =0 ( − i (cid:0) ri (cid:1) t ( α − re j )+( r − i ) e j ξ I ∂ ⊗ t β + ie j ξ J ∂ ′ v ∈ X ( V ). So t α ξ I ∂ ⊗ t β ξ J ∂ ′ v ∈ − r X i =1 ( − i (cid:18) ri (cid:19) t ( α − re j )+( r − i ) e j ξ I ∂ ⊗ t β + ie j ξ J ∂ ′ v + X ( V ) . Note that | ( α − re j )+( r − i ) e j | < | α | for all i >
1. By the induction hypothesis, the righthand side is contained in B ⊗ V + X ( V ). Thus t α ξ I ∂ ⊗ t β ξ J ∂ ′ v ∈ B ⊗ V + X ( V ). (cid:3) main Theorem 4.4.
Let V be a simple non-trivial bounded W -module. Then V is a simple W -quotient of the AW -module F ( P, L ( V ⊗ V )) , where P is a simple weight K m,n -module, V is a finite-dimensional simple gl m -module and V is a simple bounded gl n -module.Proof. As defined above, ˆ θ : ˆ V → V is a W -module homomorphism, so ˆ θ maps any AW -submodule of ˆ V to 0 or V by the simplicity of V . In particular, since V is non-trivial,ˆ θ ( ˆ V ) = W · V = V . Let ( λ, µ ) ∈ Supp( V ), then ( λ, µ ) ∈ Supp( ˆ V ) and ˆ θ ( ˆ V ( λ,µ ) ) = V ( λ,µ ) .By Theorem 4.3, ˆ V is bounded. So ˆ V ( λ,µ ) is finite-dimensional.Let ¯ U = { x ∈ ¯ U | [ x, H ] = 0 } . Then ¯ U is a subalgebra of ¯ U and ˆ V ( λ,µ ) is a ¯ U -module. Let M be a minimal ¯ U -submodule of ˆ V ( λ,µ ) such that ˆ θ ( M ) = 0 and M ′ bea maximal ¯ U -submodule of M . Since ( ¯ U M ) ( λ,µ ) = M and ( ¯ U M ′ ) ( λ,µ ) = M ′ , we haveˆ θ ( ¯ U M ) = V and ˆ θ ( ¯ U M ′ ) = 0. It follows that V is a W -quotient of the simple bounded AW -module ¯ U M/ ¯ U M ′ .From Theorem 3.9 and Lemma 3.8, ¯ U M/ ¯ U M ′ is isomorphic to F ( P, L ( V ⊗ V )), where P is a simple weight K m,n -module, V is a finite-dimensional simple gl m -module and V is a simple bounded gl n -module. So V is a simple quotient of F ( P, L ( V ⊗ V )). (cid:3) We remark that simple weight K m,n -modules and simple bounded weight gl n modulesare known, see Lemma 3.4 in this paper and Theorem 13.3 in [22]. Ackowledgement.
This work is partially supported by NSF of China (Grant11471233, 11771122, 11971440).
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