Classification of simple strong Harish-Chandra W(m,n) -modules
Yuly Billig, Vyacheslav Futorny, Kenji Iohara, Iryna Kashuba
aa r X i v : . [ m a t h . R T ] J un CLASSIFICATION OF SIMPLE STRONGHARISH-CHANDRA W ( m, n ) -MODULES YULY BILLIG, VYACHESLAV FUTORNY, KENJI IOHARA,AND IRYNA KASHUBA
Abstract.
We classify all simple strong Harish-Chandra modulesfor the Lie superalgebra W ( m, n ). We show that every such moduleis either strongly cuspidal or a module of the highest weight type.We construct tensor modules for W ( m, n ), which are parametrizedby simple finite-dimensional gl ( m, n )-modules and show that everysimple strongly cuspidal W ( m, n )-module is a quotient of a tensormodule. Finally, we realize modules of the highest weight type assimple quotients of the generalized Verma modules induced fromtensor modules for W ( m − , n ). Introduction
Consider a commutative algebra of Laurent polynomials R m = C [ t ± , . . . , t ± m ] and a supercommutative Grassmann (exterior) algebraΛ = Λ( ξ , . . . , ξ n ) with odd generators ξ , . . . , ξ n . We set A to be thecommutative superalgebra R m ⊗ Λ. Our main object of study in thispaper is the Lie superalgebra V = W ( m, n ) of derivations of A . Weconsider a category of Harish-Chandra modules over W ( m, n ), whichare weight modules with finite weight multiplicities and its subcategoryof strong Harish-Chandra modules with finite multiplicities of weightsubspaces with respect to the part of the Cartan subalgebra corre-sponding to even variables. The strong Harish-Chandra modules areexactly the Harish-Chandra modules in the sense of [8], where the caseof the superconformal Lie superalgebra W (1 , n ) was studied for n ≥ W ( m )was settled in [4].Crucial information about supports of simple strong Harish-Chandramodules for W ( m, n ) may be obtained from the work of VolodymyrMazorchuk and Kaiming Zhao [9]. Even though [9] only treats Liealgebra W ( m ), the results of that paper apply in super setting as well. Primary 17B10, 17B66; Secondary17B68.
We focus on an important class of cuspidal strong Harish-Chandramodules, whose weight multiplicities are uniformly bounded. The keyobjects in this class are tensor modules which generalize tensor densitymodules, also known as the intermediate series modules, for the Liealgebra of vector fields on a torus [10], [7], [4].Tensor modules belong to the subcategory of AV -modules, they areparametrized by finite-dimensional simple gl ( m, n )-modules and theirsupports λ + Z m , λ ∈ C m . For any gl ( m, n )-module V , the tensormodule T ( V, λ ) =
A ⊗ V comes with the following action of V = W ( m, n ):( t s f d j ) t r ⊗ gv =( r j + λ j ) t r + s f g ⊗ v + m X i =1 s i t r + s f ⊗ e ij gv + n X α =1 t r + s ( f ) ∗ ∂ α ⊗ e αj gv, ( t s f ∂ β ) t r ⊗ gv = t r + s f ∂g∂ξ β ⊗ v + m X i =1 s i t r + s f ⊗ e iβ gv + n X α =1 t r + s ( f ) ∗ ∂ α ⊗ e αβ gv. Tensor modules T ( V, λ ) exhaust all simple weight modules of finiterank in the category of AV -modules and provide a classification ofsimple cuspidal strong Harish-Chandra modules over W ( m, n ).We also consider a subalgebra W ( m, n ) ⋉ A d ⊂ W ( m +1 , n ) and de-fine a tensor module for this subalgebra as a tensor module for W ( m, n )on which f d acts as multiplication by λ f , f ∈ A .We formulate our main result Theorem 1.
Let m, n be non-negative integers. Every simple strongHarish-Chandra W ( m + 1 , n ) -module is either (1) a quotient of a tensor module T ( V, λ ) , corresponding to a simplefinite-dimensional gl ( m + 1 , n ) -module V and λ ∈ C m +1 , or (2) a module of a highest weight type L ( T ) θ , twisted by an automor-phism θ ∈ GL m +1 ( Z ) , where T is a simple quotient of a tensormodule for W ( m, n ) ⋉ A d . The paper has the following structure: we discuss Harish-Chandramodules in Section 2 and their supports in Section 3. In Section 4we consider the category of weight AV -modules of a finite rank andexhibit the structure of tensor modules - simple objects in this category.We study simple strongly cuspidal W ( m, n )-modules in Section 5 andsimple modules of the highest weight type in Section 6. ARISH-CHANDRA W ( m, n )-MODULES 3 After this paper was completed, we learned about a recent preprintby Yaohui Xu and Rencai L¨u [12], where the same results were ob-tained.
Acknowledgments
Y. B. is supported in part by a Discovery grant from NSERC. Partsof this work was done during the visits of Y. B. to the University of S˜aoPaulo and to the University of Lyon. Y. B. would like to thank theseuniversities for their hospitality and FAPESP for funding the visit toUSP. V. F. is supported in part by the CNPq (304467/2017-0) and bythe FAPESP (2018/23690-6).2.
Definitions and Notations
Lie superalgebra W ( m, n ) may be written as a free A -module in thefollowing way: m M i =1 A d i ⊕ n M α =1 A ∂ α , where d i = t i ∂∂t i and ∂ α = ∂∂ξ α . We will use roman letters to indexvariables t , . . . , t m and greek letters to index ξ , . . . , ξ n . We will beusing multi-index notation, t r = t r × . . . t r m m , for r ∈ Z m . We also set | r | = | r | + . . . + | r m | .In addition to the usual derivation ∂ α of Λ, satisfying ∂ α ( ξ α ξ p ) = ξ p when p α = 0, we will also use a right derivation ∗ ∂ α , satisfying( ξ p ξ α ) ∗ ∂ α = ξ p when p α = 0. We will write the symbol ∗ ∂ α on theright, in order to satisfy the usual sign conventions of superalgebras.This right derivation has the following properties:( f g ) ∗ ∂ α = f ( g ) ∗ ∂ α + ( − g ( f ) ∗ ∂ α g, ( f ) ∗ ∂ α = ( − f +1 ∂ α ( f ) . Throughout the paper we will denote by f the parity of f .We will consider a chain of Lie superalgebras W ( m, n ) ⊂ W ( m, n ) ⋉ A d ⊂ W ( m + 1 , n ) = Der( C [ t ± ] ⊗ A ) where d = t ∂∂t . We willbe using symbol V to denote one of these three algebras. It will beconvenient to state some of our results in the context of W ( m, n ),while others in the setting of W ( m + 1 , n ). Lie superalgebra V hasa Cartan subalgebra h = h ′ ⊕ h ′′ , where h ′ = Span { d , . . . , d m } for V = W ( m, n ) and h ′ = Span { d , d , . . . , d m } for V = W ( m, n ) ⋉ A d or V = W ( m + 1 , n ), h ′′ = Span { ξ ∂ , . . . , ξ n ∂ n } . Y. BILLIG, V. FUTORNY, K. IOHARA, AND I. KASHUBA
Definition 2. A V -module M is called a Harish-Chandra module ifit has a weight decomposition with respect to h with finite-dimensionalweight spaces. Definition 3.
A Harish-Chandra module M is called strong if it hasfinite-dimensional weight spaces with respect to h ′ . For example, the adjoint module for V is a strong Harish-Chandramodule. Note that strong Harish-Chandra modules coincide with Harish-Chandra modules in the sense of [8].One immediately gets the following result. Lemma 4.
Suppose M is a simple strong Harish-Chandra V -module.Then the action of h ′′ on M is diagonalizable and M is a Harish-Chandra module.Proof. Since M has a weight space decomposition with respect to h ′ with finite-dimensional weight spaces, the statement follows from thesimplicity of M . (cid:3) Supports of strong Harish-Chandra modules
Cuspidal modules form an important subcategory of the Harish-Chandra modules.
Definition 5. A V -module M is called cuspidal (resp. strongly cuspi-dal) if M is a Harish-Chandra (resp. strong Harish-Chandra) modulewhose dimensions of weight spaces relative to h (resp. h ′ ) are uniformlybounded. We define support supp M of a strong Harish-Chandra V -module M as a set of weights λ ∈ h ′∗ such that M λ = 0.Let us choose a basis { ǫ i } in h ′∗ , dual to the basis { d i } of h ′ , withweights ǫ i defined by ǫ i ( d j ) = δ ij . We embed in h ′∗ integral lattice Z dim h ′ spanned by { ǫ i } .If a V -module M is indecomposable then its support lies in a singlecoset λ + Z dim h ′ , λ ∈ h ′ .Let us discuss twisting of a module by an automorphism of a Liealgebra. If M is a V -module and θ is an automorphism of V then wecan construct another V -module M θ with a new action of V on thesame space x.m = θ ( x ) m , for x ∈ V , m ∈ M .The algebra of Laurent polynomials R m +1 = C [ t ± , t ± , . . . , t ± m ] isisomorphic to the group algebra C [ Z m +1 ]. The action of GL m +1 ( Z ) on ARISH-CHANDRA W ( m, n )-MODULES 5 Z m +1 extends to the action on R m +1 by θ ( t s ) = t θ ( s ) , θ ∈ GL m +1 ( Z ), s ∈ Z m +1 .This action also induces the action of GL m +1 ( Z ) on Der( R m +1 ) = W ( m +1 , n ) by θ ( η )( f ) = θ ( η ( θ − ( f ))) for η ∈ W ( m +1 , n ), f ∈ R m +1 .The action of θ on h ′ induces action on h ′∗ , which is given by thetranspose of the inverse matrix θ − T in basis { ǫ i } . If M is a W ( m +1 , n )-module and θ ∈ GL m +1 ( Z ) then supp M θ = θ − T (supp( M )).Mazorchuk-Zhao [8] described supports of simple Harish-Chandra W ( m, W ( m, n )-modules yielding the following result: Theorem 6. ( [8] , Theorem 1) Let M be a simple strong Harish-Chandra W ( m + 1 , n ) -module. Then either(1) M is a strongly cuspidal module, or(2) there exists θ ∈ GL m +1 ( Z ) and λ ∈ supp M θ such that supp( M θ ) ⊂ λ + Z + ǫ + Z ǫ + . . . + Z ǫ m and the subspace ⊕ µ ∈ λ + Z ǫ + ... + Z ǫ m ( M θ ) µ is a simple cuspidal W ( m, n ) ⋉ R m d -module. We will call a W ( m + 1 , n )-module M a module of a highest weighttype if it satisfies condition (2) of the above theorem.In the subsequent sections we will first study simple strongly cuspidalmodules and then use their structure to get a description of simplemodules of the highest weight type.4. AV -modules of a finite type We begin in a general setting. Let A be an arbitrary commutativeunital superalgebra and V be a Lie superalgebra. We assume that V isan A -module and V acts on A by derivations, and the following relationholds: [ f η, gτ ] = f η ( g ) τ − ( − ( f + η )( g + τ ) gτ ( f ) η + ( − ηg f g [ η, τ ] . A prototypical example of this setting is when V is the algebra ofderivations of A . The foundations of the theory we discuss in thissection were laid by Rinehart [11]. Definition 7. An AV -module M is a vector space with actions of aunital commutative superalgebra A and a Lie superalgebra V , which arecompatible via the Leibniz rule: η ( f m ) = η ( f ) m + ( − ηf f ( ηm ) , f ∈ A , η ∈ V , m ∈ M. Y. BILLIG, V. FUTORNY, K. IOHARA, AND I. KASHUBA
Equivalently, AV -module structure may be expressed via the smashproduct A U ( V ). View U ( V ) as a Hopf algebra with a coproduct ∆,∆( u ) = P u (1) ⊗ u (2) . The smash product A U ( V ) is the associativealgebra structure on vector space A ⊗ U ( V ) with the product( f ⊗ u )( g ⊗ v ) = X ( − u (2) g f u (1) ( g ) ⊗ u (2) v. Then AV -module structure is equivalent to the action of the asso-ciative algebra A U ( V ). Definition 8.
The algebra of differential operators D ( A , V ) is the quo-tient of A U ( V ) by the ideal generated by the elements f η − f η , f ∈ A , η ∈ V . The following Lemma follows immediately from the definition:
Lemma 9.
The subspace A V = A ⊗ V ⊂ A U ( V ) is a Lie subsu-peralgebra with the Lie bracket [ f ⊗ η, g ⊗ τ ] = f η ( g ) ⊗ τ − ( − ( f + η )( g + τ ) gτ ( f ) ⊗ η + ( − ηg f g ⊗ [ η, τ ] . Lemma 10.
Let A = R m ⊗ Λ and V = W ( m, n ) . Let M be an AV -module with a weight decomposition relative to h ′ . Then M is finitelygenerated over A if and only if it is a strong Harish-Chandra module.Proof. Without loss of generality we may assume that M is indecom-posable. Since A is finitely generated over R m , being finitely generatedover A is equivalent to being finitely generated over R m . Considerweight decomposition of M relative to h ′ : M = ⊕ µ ∈ h ′∗ M µ . Note that t s ξ r M µ ⊂ M µ + s , t s ξ r d i M µ ⊂ M µ + s , t s ξ r ∂ α M µ ⊂ M µ + s .We view elements of Z m as linear functionals on h ′ via s ( d i ) = s i for1 ≤ i ≤ m . Since M is indecomposable, its support consists of asingle coset λ + Z m . Suppose that M is finitely generated as an R m -module. We may assume that all generators are weight vectors. Sincemonomials t s ∈ R m act bijectively and transitively on the set of weightspaces, we may further assume that all generators belong to the sameweight space U = M λ . Then M = A ⊗ Λ U ∼ = R m ⊗ U and we seethat being finitely generated over R m is equivalent to being a strongHarish-Chandra module. (cid:3) For the rest of this section we will assume that V = W ( m, n ) or V = W ( m, n ) ⋉ A d . we will be proving many results simultaneouslyfor both of these algebras. When considering the case V = W ( m, n ), allstatements containing expressions with a subscript 0, e.g., d , should ARISH-CHANDRA W ( m, n )-MODULES 7 be ignored. Throughout the section index i will be assumed to runfrom 1 to m , and index α from 1 to n .Let us assume that M is a simple strong Harish-Chandra AV -module.From the proof of the previous Lemma we see that M ∼ = A ⊗ Λ U , where U = M λ with λ ∈ h ′∗ , dim U < ∞ . Without loss of generality we mayassume that λ = 0. Note that U is a module over Λ with a weightdecomposition with respect to h ′′ .Lie superalgebra A V has a Cartan subalgebra 1 ⊗ h . The actionof 1 ⊗ h ′ induces a Z m -grading on A V , let us denote by ( A V ) thehomogeneous component of degree zero, which is a Lie subsuperalgebra.Lie superalgebra ( A V ) is spanned by the following elements: f D i ( g, r ) = t − r f ⊗ t r gd i , f ∆ α ( g, r ) = t − r f ⊗ t r g∂ α , and f D ( g, r ) = t − r f ⊗ t r gd , where f, g ∈ Λ, r ∈ Z m .In the following Proposition, we record Lie brackets for the genera-tors of ( A V ) as Λ-module. The proof is a straightforward calcula-tion, and we omit it. Proposition 11.
Elements D i ( f, r ) , ∆ α ( f, r ) , D ( f, r ) ∈ ( A V ) ,satisfy the following commutator relations: [ D i ( f, r ) , D j ( g, s )] = s i D j ( f g, r + s ) − s i f D j ( g, s ) − r j D i ( f g, r + s ) + ( − fg r j gD i ( f, r ) , (1) [ D i ( f, r ) , ∆ α ( g, s )] = s i ∆ α ( f g, r + s ) − s i f ∆ α ( g, s ) − ( − f + g D i ( ∂ α ( f ) g, r + s ) , (2)[∆ α ( f, r ) , ∆ β ( g, s )] = ∆ β ( f ∂ α ( g ) , r + s ) − ( − f +1 ∆ α ( ∂ β ( f ) g, r + s ) , (3)(4) [ D i ( f, r ) , D ( g, s )] = s i D ( f g, r + s ) − s i f D ( g, s ) , (5) [∆ α ( f, r ) , D ( g, s )] = D ( f ∂ α ( g ) , r + s ) , (6) [ D ( f, r ) , D ( g, s )] = 0 . We have actions of a commutative superalgebra Λ and a Lie super-algebra ( A V ) on a finite-dimensional space U = M λ , which satisfythe following compatibility properties: η ( f u ) = η ( f ) u + ( − ηf f ( ηu ) , (7) f ( ηu ) =( f η ) u, (8)for η ∈ ( A V ) , f ∈ Λ, u ∈ U . This means that U admits the actionof the algebra of differential operators D (Λ , ( A V ) ). Y. BILLIG, V. FUTORNY, K. IOHARA, AND I. KASHUBA
Proposition 12.
The action of A U ( V ) on M may be recovered fromthe joint actions of Λ , ( A V ) on U : t r f d i ( t s ⊗ u ) = t r d i ( t s ) ⊗ f u + t r + s ⊗ D i ( f, r ) ut r f ∂ α ( t s ⊗ u ) = t r + s ⊗ ∆ α ( f, r ) u,t r f d ( t s ⊗ u ) = t r + s ⊗ D ( f, r ) u Proof.
This follows immediately from the definition of ( A V ) andfrom AV -module structure on M . (cid:3) We get an obvious corollary:
Corollary 13. M = A ⊗ Λ U is an irreducible AV -module if and onlyif U is an irreducible D (Λ , ( A V ) ) -module. Remark 14.
It is not difficult to show that we have in fact an isomor-phism of associative algebras: (9) A U ( V ) ∼ = A ⊗ Λ D (Λ , ( A V ) ) . Here the algebra on the right is the quotient of A U (( A V ) ) by theideal generated by af x − a f x , with a ∈ A , x ∈ ( A V ) , f ∈ Λ .Isomorphism (9) allows us to interpret M = A ⊗ Λ U as an A U ( V ) -module induced from D (Λ , ( A V ) ) -module U . Since U is a weight space for V , we also have D i (1 , | U = λ i Id U , D (1 , | U = λ Id U . Theorem 15.
Let U be a finite-dimensional ( A V ) -module. Thenthe action of ( A V ) is polynomial, i.e., D i ( f, r ) = X k ∈ Z m + r k k ! d i ( f, k − ǫ i ) , (10) ∆ α ( f, r ) = X k ∈ Z m + r k k ! ∂ α ( f, k ) , (11) D ( f, r ) = X k ∈ Z m + r k k ! d ( f, k ) , (12) where d i ( f, k ) , ∂ α ( f, k ) , d ( f, k ) ∈ End U with d i ( f, k ) = 0 , ∂ α ( f, k ) = 0 , d ( f, k ) = 0 for k ≫ . We use a shift by ǫ i in the definition of d i ( f, k ) in order to exhibita natural Z m -grading on a new Lie superalgebra that will be definedusing the elements in the right hand sides of (10)-(12). ARISH-CHANDRA W ( m, n )-MODULES 9 Proof.
By Theorem 3.1 in [2], for i = 1 , . . . , m , the action of { D i (1 , s ) } on U is polynomial in s . Using Proposition 17, we compute:[ D i (1 , s ) , D i ( f, − s i D i ( f, s ) + s i f D i (1 , s ) . Since the left hand side and the last summand of the right hand sideare polynomial in s , we conclude that s i D i ( f, s ) is polynomial in s .Next,[ D i (1 , s − ǫ i ) , D i ( f, ǫ i )] =2 D i ( f, s ) − D i ( f, ǫ i ) − s i D i ( f, s ) + ( s i − f D i (1 , s − ǫ i ) . In the above equality, all terms except D i ( f, s ) are known to be poly-nomial, hence D i ( f, s ) is a polynomial in s .From [ D i (1 , s − ǫ i ) , ∆ α ( f, ǫ i )] = ∆ α ( f, s ) − ∆ α ( f, ǫ i ) , we conclude that ∆ α ( f, s ) is also a polynomial in s .In case when V = W ( m, n ) ⋉ A d , we also need to establish polyno-miality of D ( f, s ). This follows from the equality[∆ α ( f, s ) , D ( ξ a , D ( f, s ) . (cid:3) Theorem 16.
Operators d i ( f, k ) , ∂ α ( f, k ) , and d ( f, k ) acting on U satisfy the following commutator relations: (13) [ d i ( f, − ǫ i ) , d j ( g, − ǫ j )] = 0 , [ d i ( f, − ǫ i ) , d j ( g, k )] =( k i + δ ij ) d j ( f g, k − ǫ i )(14) − ( k i + δ ij ) f d j ( g, k − ǫ i ) , f or k = − ǫ j , [ d i ( f, ℓ ) , d j ( g, k )] = k i d j ( f g, ℓ + k )(15) − ℓ j d i ( f g, ℓ + k ) , f or ℓ = − ǫ i , k = − ǫ j , [ d i ( f, − ǫ i ) , ∂ α ( g, k )] = k i ∂ α ( f g, k − ǫ i ) − k i f ∂ α ( g, k − ǫ i )(16) − ( − f + g d i ( ∂ α ( f ) g, k − ǫ i ) , [ d i ( f, ℓ ) , ∂ α ( g, k )] = k i ∂ α ( f g, ℓ + k )(17) − ( − f + g d i ( ∂ α ( f ) g, ℓ + k ) , f or ℓ = − ǫ i , [ ∂ α ( f, ℓ ) , ∂ β ( g, k )] = ∂ β ( f ∂ α ( g ) , ℓ + k )(18) − ( − f +1 ∂ α ( ∂ β ( f ) g, ℓ + k ) , [ d i ( f, − ǫ i ) , d ( g, k )] = k i d ( f g, k − ǫ i ) − k i f d ( g, k − ǫ i )(19) [ d i ( f, ℓ ) , d ( g, k )] = k i d ( f g, ℓ + k ) , f or ℓ = − ǫ i (20) [ ∂ α ( f, ℓ ) , d ( g, k )] = d ( f ∂ α ( g ) , ℓ + k ) , (21) [ d ( f, ℓ ) , d ( g, k )] = 0 , (22)(23) [ ∂ α ( f, , g ] = f ∂ α ( g ) , (24) [ ∂ α ( f, k ) , g ] = 0 , f or k = 0 , (25) [ d i ( f, k ) , g ] = 0 , [ d ( f, k ) , g ] = 0 . Proof.
Let us derive relations (13)-(15). We take polynomial expan-sions of (1): X p ∈ Z m + s p p ! d i ( f, p − ǫ i ) , X q ∈ Z m + r q q ! d j ( g, q − ǫ j ) = r i X l ∈ Z m + ( r + s ) l l ! d j ( f g, l − ǫ j ) − r i X l ∈ Z m + r l l ! f d j ( g, l − ǫ j ) − s j X l ∈ Z m + ( r + s ) l l ! d i ( f g, l − ǫ j ) + ( − fg s j X l ∈ Z m + s l l ! gd i ( f, l − ǫ i ) . Two polynomials in 2 m variables that have equal values on Z m + mustbe equal as polynomials, which allows us to equate their coefficients.Equating the terms with s r we immediately get (13). Extractingterms with s r q with q = 0 we get1 q ! [ d i ( f, − ǫ i ) ,d j ( g, q − ǫ j )]= 1( q − ǫ i )! d j ( f g, q − ǫ i − ǫ j ) − q − ǫ i )! f d j ( g, q − ǫ i − ǫ j ) . Multiplying both sides by q ! and setting k = q − ǫ j , we obtain (14). Inthe same way, equating the terms with s p r q where p, q = 0, we establish(15). Derivation of the remaining relations of Theorem 16 is analogous,and we omit these calculations. (cid:3) Denote by L Lie superalgebra with basis { ξ r d i ( ξ s , k − ǫ i ) , ξ r d ( ξ s , k ) , ξ r ∂ α ( ξ s , k ) } where r, s ∈ { , } n , k ∈ Z m + , 1 ≤ i ≤ m , 1 ≤ α ≤ n } and Lie bracketsdefined by (13)-(25). In fact, L may be defined as a jet Lie superalgebracorresponding to ( A V ) (see [3] for the general construction of thejet Lie algebra). ARISH-CHANDRA W ( m, n )-MODULES 11 Note that elements { d i (1 , − ǫ i ) , d (1 , , } are central in L .Lie algebra L has a Cartan subalgebra H = Span { d i (1 , , ∂ α ( ξ α , , d (1 , } which is diagonalizable in the adjoint representation:[ d i (1 , , ξ r d j (1 , − ǫ j )] =0 , [ ∂ α ( ξ α , , ξ r d j (1 , − ǫ j )] = r α ξ r d j (1 , − ǫ j ) , [ d i (1 , , ξ s d j ( ξ r , − ǫ j ) − ξ s ξ r d j (1 , − ǫ j )] = − δ ij ( ξ s d j ( ξ r , − ǫ j ) − ξ s ξ r d j (1 , − ǫ j )) , [ ∂ α ( ξ α , , ξ s d j ( ξ r , − ǫ j ) − ξ s ξ r d j (1 , − ǫ j )] =( s α + r α ) ( ξ s d j ( ξ r , − ǫ j ) − ξ s ξ r d j (1 , − ǫ j )) , [ d i (1 , , ξ r d j ( ξ s , k )] = k i ξ r d j ( ξ s , k ) f or k = − ǫ j , [ ∂ α ( ξ α , , ξ r d j ( ξ s , k )] =( r α + s α ) ξ r d j ( ξ s , k ) , f or k = − ǫ j , [ d i (1 , , ξ r ∂ β ( ξ s , k )] = k i ξ r ∂ β ( ξ s , k ) , [ ∂ α ( ξ α , , ξ r ∂ β ( ξ s , k )] =( r α + s α − δ αβ ) ξ r ∂ β ( ξ s , k ) , [ d i (1 , , ξ r d ( ξ s , k )] = k i ξ r d ( ξ s , k ) , [ ∂ α ( ξ α , , ξ r d ( ξ s , k )] =( r α + s α ) ξ r d ( ξ s , k ) , and d (1 ,
0) is central.
Example.
Consider AV -module M = W ( m, n ) with the adjointaction of W ( m, n ), natural left multiplication action of A , and in casewhen V = W ( m, n ) ⋉ A d , we define the action of ad as left multipli-cation by λ a , a ∈ A , where λ is some fixed scalar, λ ∈ C . Let us fix r ∈ Z m and take as U the corresponding root space: U = m X i =1 t r Λ d i ⊕ n X α =1 t r Λ ∂ α . We see that U is a free Λ-module of rank m + n . Let us compute theaction of ( A V ) on U : D i ( f, s ) t r gd j = t − s [ t s f d i , t r gd j ] = r i t r f gd j − s j t t f gd i ,D i ( f, s ) t r g∂ β = r i t r f g∂ β − ( − f + g t r ∂ β ( f ) gd i , ∆ α ( f, s ) t r gd j = t r f ∂ α ( g ) d j − ( − g s j t r f g∂ α , ∆ α ( f, s ) t r g∂ β = t r f ∂ α ( g ) ∂ β + ( − f t r ∂ β ( f ) g∂ α ,D ( f, s ) t r gd j = λ t r f gd j ,D ( f, s ) t r g∂ β = λ t r f g∂ β . Taking formal expansions (10)-(12) of D i ( f, s ), ∆ α ( f, s ), and D ( f, s )in powers of s and comparing them with the right hand sides of the above equalities, we derive the action on U of Lie superalgebra L : d i ( f, − ǫ i ) t r gd j = r i t r f gd j ,d i ( f, − ǫ i ) t r g∂ α = r i t r f g∂ α − ( − f + g t r ∂ β ( f ) gd i ,d i ( f, ǫ a − ǫ i ) t r gd j = − δ aj t r f gd i ,d i ( f, ǫ a − ǫ i ) t r g∂ α =0 ,d i ( f, k − ǫ i ) =0 for | k | > ,∂ α ( f, t r gd j = t r f ∂ α ( g ) d j ,∂ α ( f, t r g∂ β = t r f ∂ α ( g ) ∂ β + ( − f t r ∂ ( f ) g∂ α ,∂ α ( f, ǫ a ) t r gd j = − δ aj ( − g t r f g∂ α ,∂ α ( f, ǫ a ) t r g∂ β =0 ,∂ α ( f, k ) =0 for | k | > ,d ( f, t r gd j = λ t r f gd j ,d ( f, t r g∂ α = λ t r f g∂ α ,d ( f, k ) =0 for | k | > . Denote by J r,λ the kernel in L of this representation. To makethis ideal independent of r and λ , we set J = ∩ r ∈ Z m ,λ ∈ C J r,λ . NextProposition is an immediate consequence of the above formulas: Proposition 17. (i) The quotient Lie superalgebra L /J is a free Λ -module with generators { d i (1 , − ǫ i ) , d i ( ξ β , − ǫ i ) , d i (1 , ǫ j − ǫ i ) , ∂ α (1 , , ∂ α ( ξ β , , ∂ α (1 , ǫ j ) , d (1 , } with i, j = 1 , . . . , m, α, β = 1 , . . . , n. (ii) The following relations determine L /J as a Λ -module: d i ( f, k − ǫ i ) = 0 , ∂ α ( f, k ) = 0 for | k | > ,d i ( f, ǫ j − ǫ i ) = f d i (1 , ǫ j − ǫ i ) ,∂ α ( f, ǫ j ) = f ∂ α (1 , ǫ j ) ,d i ( f, − ǫ i ) = f d i (1 , − ǫ i ) + n X β =1 ( f ) ∗ ∂ β ( d i ( ξ β , − ǫ i ) − ξ β d i (1 , − ǫ i )) ,∂ α ( f,
0) = f ∂ α (1 ,
0) + n X β =1 ( f ) ∗ ∂ β ( ∂ α ( ξ β , − ξ β ∂ α (1 , ,d ( f, k ) = 0 for | k | > ,d ( f,
0) = f d (1 , . ARISH-CHANDRA W ( m, n )-MODULES 13 Theorem 18.
Ideal J is in the kernel of any finite-dimensional simple L -module.Proof. Let U be a finite-dimensional simple L -module. Consider thefollowing elements of the Cartan subalgebra of L : I ′ = m X i =1 d i (1 , , I ′′ = m X α =1 ∂ α ( ξ α , , I = I ′ + I ′′ . Each of these elements defines a Z -grading on L , and each of thesegradings starts at − Z -grading induced by I ′ , we seethat for some N a proper ideal containing { f d i ( g, k ) , f ∂ α ( g, k ) , f d ( g, k ) | | k | > N } vanishes in U . The quotient by this ideal is a finite-dimensional Liealgebra, which acts on U , and we can now apply the following result: Lemma 19. ( [6] , Lemma 1). Let g be a finite-dimensional Lie super-algebra and let n be a solvable ideal of g . Let a be an even subalgebraof g such that n is a completely reducible ad ( a ) -module with no trivialsummand. Then n acts trivially in any irreducible finite-dimensional g -module U . From Proposition 17 we can see that ad ( I ) acts on J in a diagonalway with positive integer eigenvalues. This implies that the image of J in the finite-dimensional quotient of L is nilpotent and by previousLemma, J vanishes in every simple finite-dimensional L -module module U . (cid:3) The following Proposition may be shown by a direct computationusing Proposition 17 and Theorem 16.
Proposition 20.
As a Λ -module, L /J is generated by three supercom-muting subalgebras:(1) Central ideal spanned by { d i (1 , − ǫ i ) , d (1 , } ,(2) Abelian subalgebra spanned by { ∂ α (1 , } ,(3) A subalgebra isomorphic to gl ( m, n ) and spanned by the followingelements: e ij = d j (1 , ǫ i − ǫ j ) ,e αβ = ∂ β ( ξ α , − ξ α ∂ β (1 , ,e iβ = ∂ β (1 , ǫ i ) ,e αj = d j ( ξ α , − ǫ j ) − ξ α d j (1 , − ǫ j ) . Theorem 21.
Let U be a simple finite-dimensional module for D (Λ , L ) .Assume that operators { d i (1 , − ǫ i ) } act on U as scalars λ i Id U with λ = 0 , and d (1 , acts on U as λ Id U .Then there exists a finite-dimensional irreducible gl ( m, n ) -module ( V, ρ ) such that U ∼ = Λ ⊗ V . The action of L integrates to the ac-tion of ( A V ) on Λ ⊗ V in the following way: D j ( f, s ) = λ j f ⊗ Id V + m X i =1 s i f ⊗ ρ ( e ij ) + n X α =1 ( f ) ∗ ∂ α ⊗ ρ ( e αj ) , ∆ β ( f, s ) = f ∂∂ξ β ⊗ Id V + m X i =1 s i f ⊗ ρ ( e iβ ) + n X α =1 ( f ) ∗ ∂ α ⊗ ρ ( e αβ ) ,D ( f, s ) = λ f ⊗ Id V . Proof.
Since elements f d i (1 , − ǫ i ) ∈ L act on U as multiplication by λ i f and λ i = 0 for some i , we see that Λ-action is encoded in theaction of L . Hence U is an irreducible finite-dimensional L -module.By Theorem 18 it is a simple L /J -module.We are going to show that subalgebras spanned by { f d i (1 , − ǫ i ) } and { ∂ α (1 , } define a structure of a free Λ-module on U , with f d i (1 , − ǫ i )acting as multiplications by λ i f and ∂ α (1 ,
0) acting as ∂∂ξ α .Let V be the joint kernel of { ∂ α (1 , } . It is easy to see that thissubspace is non-zero. Indeed, we can start with any non-zero vectorin U and act repeatedly by elements { ∂ α (1 , } . Since these operatorsanticommute and ∂ α (1 , = 0, we will end up with a non-zero vectorin V . Since { ∂ α (1 , } commute with the gl ( m, n ) subalgebra in L , V admits a structure of a gl ( m, n )-module.Set V p = ξ p V . By (23) we have[ ∂ α (1 , , ξ p ] = ∂ α ( ξ p ) . The standard differentiation argument shows that such subspaces forma direct sum: ⊕ p ∈{ , } n V p ∼ = Λ ⊗ V. This direct sum is invariant under L /J , hence it coincides with U .If V ′ is a gl ( m, n )-submodule in V then Λ ⊗ V ′ is an L /J -submodulein U . Hence V must be irreducible as a gl ( m, n )-module. It is also easyto see that the converse holds as well: if V is irreducible as a gl ( m, n )-module then U = Λ ⊗ V is irreducible as an L /J -module. Indeed,starting with an arbitrary non-zero vector in U we can apply a sequenceof operators { ∂ α (1 , } to get a non-zero vector in V . Then using gl ( m, n ) action we can get all of V , and finally acting with elements ARISH-CHANDRA W ( m, n )-MODULES 15 { f d i (1 , − ǫ i ) } we can generate U . This shows that any non-zero vectorin U generates U , hence U is an irreducible L /J -module.Using expansions (10), (11) and combining our results about theaction of L on U we obtain the last claim of the Theorem. (cid:3) Now we can obtain a description of simple weight AV -modules of afinite rank, recovering the action of V from the action of ( A V ) byapplying Proposition 12. Theorem 22.
Let V = W ( m, n ) , A = R m ⊗ Λ . Every simple weight AV -module of a finite rank is a tensor module. Such modules areparametrized by finite-dimensional simple gl ( m, n ) -modules V and theirsupports λ + Z m . A tensor module is a tensor product A ⊗ V with the following action of V = W ( m, n ) : ( t s f d j ) t r gv =( r j + λ j ) t r + s f gv + m X i =1 s i t r + s f ρ ( e ij ) gv + n X α =1 t r + s ( f ) ∗ ∂ α ρ ( e αj ) gv, ( t s f ∂ β ) t r gv = t r + s f ∂g∂ξ β v + m X i =1 s i t r + s f ρ ( e iβ ) gv + n X α =1 t r + s ( f ) ∗ ∂ α ρ ( e αβ ) gv. We also get an analogous statement for W ( m, n ) ⋉ A d . Theorem 23.
Let V = W ( m, n ) ⋉ A d , A = R m ⊗ Λ . Every simpleweight AV -module M of a finite rank is a tensor module for W ( m, n ) asdescribed in Theorem 22, M = A⊗ V , with A d acting by multiplicationon the left tensor factor: ( t s f d ) t r gv = λ t r + s f gv for some scalar λ ∈ C . We will refer to modules described in Theorem 23 as tensor AV -modules for V = W ( m, n ) ⋉ A d . Theorem 24.
Let V = W ( m, n ) or V = W ( m, n ) ⋉ A d and A = R m ⊗ Λ . A category of weight AV -modules of a finite rank supportedon λ + Z m and d acting as multiplication by λ is equivalent to the cat-egory of finite-dimensional D (Λ , L ) -modules with the central elements { d i (1 , − ǫ i ) } acting as multiplications by λ i and { d (1 , } acting asmultiplications by λ . Given such a D (Λ , L ) -module U , the corresponding AV -module is M = R m ⊗ U with the action of V = W ( m, n ) given as follows: ( t s f d j ) t r u = r j t r + s f u + X k ∈ Z m + s k k ! t r + s ρ ( d j ( f, k − ǫ j )) u, ( t s f ∂ β ) t r u = X k ∈ Z m + s k k ! t r + s ρ ( ∂ β ( f, k )) u, ( t s f d ) t r u = λ t r + s f u. For every finite-dimensional L -module U the sums in the right handsides of the above formulas are finite. Cuspidal W ( m, n ) -modules The goal of this section is to prove the following:
Theorem 25.
Every non-trivial simple cuspidal strong Harish-Chandramodule for V = W ( m, n ) or V = W ( m, n ) ⋉ A d is a quotient of a ten-sor AV -module. In order to prove this result, we construct a functor from the categoryof V -modules to the category of AV -modules, preserving the propertyof being (strongly) cuspidal. We will be following the ideas developedin [4].We begin with the coinduction functor. For a V -module M , the coin-duced AV -module is the space Hom( A , M ) with the AV -action definedin Lemma 26 below. Note that Hom( A , M ) is a Z -graded space withHom ( A , M ) consisting of parity-preserving maps, and Hom ( A , M )consisting of parity-reversing maps. Lemma 26.
Let M be a V -module. Then(a) Hom( A , M ) is an AV -module with the action defined as follows: ( f ϕ )( g ) =( − fϕ ϕ ( f g ) , ( ηϕ )( g ) = ηϕ ( g ) − ( − ηϕ ϕ ( η ( g )) , where ϕ ∈ Hom( A , M ) , η ∈ V , f, g ∈ A .(b) There exists a V -module homomorphism π : Hom( A , M ) → M ,defined by π ( ϕ ) = ϕ (1) . The proof given in Proposition 4.3 in [4], generalizes to super settingin a straightforward way.
ARISH-CHANDRA W ( m, n )-MODULES 17 The coinduced module is too big for our purposes, it is not even aweight module. It does have a maximal weight submodule M β ∈ h ∗ Hom( A , M ) β , whereHom( A , M ) β = { ϕ ∈ Hom( A , M ) | ∀ α ∈ h ∗ ϕ ( A α ) ⊂ M α + β } . Still, this weight submodule is also too big – it is not cuspidal when M is. To remedy this situation we define the AV -cover c M of a V -module as a subspace in Hom( A , M ): c M = Span { ψ ( τ, m ) | τ ∈ V , m ∈ M } , where ψ ( τ, m ) is defined as ψ ( τ, m ) g = ( − ( τ + m ) g ( gτ ) m. Again, a generalization of the argument of Proposition 4.5 in [4] tosuper setting, shows that c M is a weight AV -module.The map c M → M , ψ ( τ, m ) τ m , is a homomorphism of V -moduleswith the image V M . Theorem 27. If M is a cuspidal (resp. strongly cuspidal) V -modulethen c M is a cuspidal (resp. strongly cuspidal) AV -module. The proof of this theorem is based on the following
Proposition 28.
For any cuspidal V -module M there exists N ∈ N such that for all p, q ∈ Z m , r ∈ { , } n , i, j = 1 , . . . , m , α = 1 , . . . , n ,the following elements of U ( V ) annihilate M : N X a =0 ( − a (cid:18) Na (cid:19) ( t p t ai ξ r d j )( t q − ai d i ) , (26) N X a =0 ( − a (cid:18) Na (cid:19) ( t p t ai ξ r ∂ α )( t q − ai d i ) , (27) N X a =0 ( − a (cid:18) Na (cid:19) ( t p t ai ξ r d )( t q − ai d i ) . (28) Proof.
We begin by establishing (26) with i = j . Consider a subalgebrain V spanned by elements t ki d i , k ∈ Z . This subalgebra is isomorphicto Witt algebra W (1). With respect to the action of this subalgebra, M decomposes into a direct sum of cuspidal W (1)-modules whose di-mensions of weight spaces (with respect to d i ) have the same bound over all direct summands. By Corollary 3.4 in [4], there exists ℓ ∈ N such that for all p, q ∈ Z the following elements of U ( W (1)) annihilate M : Ω ( ℓ ) p,q = ℓ X a =0 ( − a (cid:18) ℓa (cid:19) ( t p + ai d i )( t q − ai d i ) . Let s ∈ Z m and r ∈ { , } n . Then Γ( s, p, q ) = [ t s ξ r d i , Ω ( ℓ ) p,q ] also annihi-lates M . We haveΓ( s, p, q ) = ℓ X a =0 ( − a (cid:18) ℓa (cid:19) ( p + a − s i )( t s t p + ai ξ r d i )( t q − ai d i )+ ℓ X a =0 ( − a (cid:18) ℓa (cid:19) ( q − a − s i )( t p + ai d i )( t s t q − ai ξ r d i ) . Consider expressionΦ( s, p, q ) = Γ( s + 2 ǫ i , p, q − − s + ǫ i , p, q −
1) + Γ( s, p, q ) , which also annihilates M . In this expression the second parts of Γ’swill cancel, yieldingΦ( s, p, q ) = ℓ +2 X a =2 ( − a (cid:18) ℓa − (cid:19) ( p + a − s i − t s t p + ai ξ r d i )( t q − ai d i ) − ℓ +1 X a =1 ( − a − (cid:18) ℓa − (cid:19) ( p + a − s i − t s t p + ai ξ r d i )( t q − ai d i )+ ℓ X a =0 ( − a (cid:18) ℓa (cid:19) ( p + a − s i )( t s t p + ai ξ r d i )( t q − ai d i ) . Finally, consider Φ( s − ǫ i , p + 1 , q ) − Φ( s, p, q ), which gives us the fol-lowing expression annihilating M :2 ℓ +2 X a =0 ( − a (cid:18)(cid:18) ℓa − (cid:19) + 2 (cid:18) ℓa − (cid:19) + (cid:18) ℓa (cid:19)(cid:19) ( t s t p + ai ξ r d i )( t q − ai d i )=2 ℓ +2 X a =0 ( − a (cid:18) ℓ + 2 a (cid:19) ( t s t p + ai ξ r d i )( t q − ai d i ) . This establishes (26) with i = j and N = ℓ + 2. The cases of (26) with i = j , (27) and (28) are actually slightly easier and follow exactly thesame lines, where we consider commutators [ t s ξ r d j , Ω ( ℓ ) p,q ], [ t s ξ r ∂ α , Ω ( ℓ ) p,q ],and [ t s ξ r d , Ω ( ℓ ) p,q ] as the starting points. (cid:3) ARISH-CHANDRA W ( m, n )-MODULES 19 Proof.
Now we can use Proposition 28 to give a proof of Theorem 27.The proofs for the cuspidal and strongly cuspidal cases are virtuallyidentical, so we will only consider the strongly cuspidal case. Withoutloss of generality we may assume that M is an indecomposable module.Then its support belongs to a single coset λ + Z m ⊂ h ′∗ . We choosethe coset representative λ in such a way that for each i = 1 , . . . , m either λ ( d i ) = 0 or λ ( d i ) Z . Clearly, the support of AV -cover c M isthe coset λ + Z m and for k ∈ Z m the weight space c M λ + k is spannedby ψ ( t k − s ξ r d j , M λ + s ), ψ ( t k − s ξ r ∂ α , M λ + s ) and ψ ( t k − s ξ r d , M λ + s ) with s ∈ Z m , r ∈ { , } n , j = 1 , . . . , m , α = 1 , . . . n . We need to show thatweight spaces of c M are in fact finite-dimensional and their dimensionsare uniformly bounded.Consider a norm on Z m , k s k = max i | s i | . We claim that c M λ + k isspanned by(29) (cid:26) ψ ( t k − s ξ r d j , M λ + s ) , ψ ( t k − s ξ r ∂ α , M λ + s ) , ψ ( t k − s ξ r d , M λ + s ) | k s k ≤ N (cid:27) , where N is the constant from Proposition 28.Let us show by induction on | s | + . . . + | s m | that elements ψ ( t k − s ξ r d j , M λ + s ), ψ ( t k − s ξ r ∂ α , M λ + s ), and ψ ( t k − s ξ r d , M λ + s ) belong tothe span of (29).If k s k ≤ N/ i such that | s i | > N/
2. Let us assume s i > N/
2, the case s i < − N/ u be an arbitrary vector from M λ + s . It follows fromour assumptions that λ ( d i ) + s i = 0. Hence there exists v ∈ M λ + s suchthat d i v = u .Let us show that ψ ( t k − s ξ r d j , u ) = − N X a =1 ( − a (cid:18) Na (cid:19) ψ ( t k − s t ai ξ r d j , ( t − ai d i ) v ) ,ψ ( t k − s ξ r ∂ α , u ) = − N X a =1 ( − a (cid:18) Na (cid:19) ψ ( t k − s t ai ξ r ∂ α , ( t − ai d i ) v ) ,ψ ( t k − s ξ r d , u ) = − N X a =1 ( − a (cid:18) Na (cid:19) ψ ( t k − s t ai ξ r d , ( t − ai d i ) v ) . Indeed, this follows immediately from the fact that operators fromProposition 28 annihilate vector v . Since vectors ( t − ai d i ) v have weights λ + s − aǫ i with a = 1 , . . . , N , by induction assumption the right handsides in the above equalities belong to the span of (29). This proves our claim, which now easily provides a uniform bound on the dimensionsof weight spaces of c M . (cid:3) To prove Theorem 25 we consider the projection π : c M → M givenby Lemma 26(b), π ( ψ ( τ, u )) = τ ( u ). Since M is a simple module witha non-trivial action of V , this map is surjective. By Theorem 27, the AV -cover c M is strongly cuspidal, hence it has a Jordan-H¨older series(0) = c M ⊂ c M ⊂ c M ⊂ . . . ⊂ c M s = c M with the quotients that are strongly cuspidal simple AV -modules. Choos-ing the largest k with c M k − ⊂ Ker π , c M k Ker π , we obtain a sur-jective V -module homomorphism from a simple strongly cuspidal AV -module c M k / c M k − onto M . By Theorem 22, a simple strongly cuspidal AV -module is a tensor module, which completes the proof of Theorem25. Remark 29.
We point out that a simple tensor AV -module for V = W ( m, n ) ⋉ A d with λ = 0 remains simple as a V -module, since inthis case the action of A is encoded by A d , hence any V -submodule isinvariant under the action of A . W ( m + 1 , n ) -modules of the highest weight type Consider a Z -grading on V = W ( m + 1 , n ) by the eigenvalues of ad ( d ). The zero component of this grading is V = W ( m, n ) ⋉ A d .This grading induces a triangular decomposition V = V − ⊕ V ⊕ V + . Let T be a strongly cuspidal V -module. We let V + act on T triviallyand define a generalized Verma module M ( T ) as the induced module M ( T ) = Ind VV ⊕V + T ∼ = U ( V − ) ⊗ T. The adjoint action of { d , d , . . . , d m } induces a Z m +1 -grading on M ( T )and every V -submodule in M ( T ) is homogeneous with respect to thisgrading. It is clear that for m > M ( T )is not Harish-Chandra.We define the radical M rad of M ( T ) as a (unique) maximal V -submodule intersecting with T trivially, and we set L ( T ) to be thequotient L ( T ) = M ( T ) /M rad . Theorem 30.
Let V = W ( m + 1 , n ) and let T be a strongly cuspidalmodule for V = W ( m, n ) ⋉ A d such that V T = T . Then L ( T ) is astrongly cuspidal V -module. ARISH-CHANDRA W ( m, n )-MODULES 21 Proof.
Consider an A ′ V ′ -cover b T of T with A ′ = R m , V ′ = W ( m, n ) ⋉ A d . By Theorem 27, b T is a strongly cuspidal module for W ( m, n ) ⋉ A d . It follows from Theorem 24 that b T is a polynomial Z m -gradedmodule (see [1] or [5] for the definition of a polynomial module). ByTheorem 1.12 in [1], L ( b T ) is a strong Harish-Chandra module (althoughthe results of [1] are proved in non-super setting, all statements andtheir proofs extend to the super case). Since T is a quotient of b T , itfollows that L ( T ) is a quotient of L ( b T ), and hence L ( T ) is a strongHarish-Chandra module. (cid:3) Now we combine all of the results of this paper to establish our mainTheorem 1. Let M be a simple strong Harish-Chandra W ( m + 1 , n )-module. By Theorem 6, M is either strongly cuspidal or of the highestweight type. If M is strongly cuspidal, by Theorem 25 it is a quotientof a tensor W ( m + 1 , n )-module, whose structure is given by Theorem22.If M is a simple module of the highest weight type, it is isomorphicto module L ( T ), twisted by an automorphism θ ∈ GL m +1 ( Z ). Here T is a simple strongly cuspidal module for W ( m, n ) ⋉ A d described inTheorem 23. References [1] S. Berman and Y. Billig,
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School of Mathematics and Statistics, Carleton University, Ot-tawa, Canada
E-mail address : [email protected] Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo,S˜ao Paulo, Brasil
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