Symplectic twistor operator on R 2n and the Segal-Shale-Weil representation
aa r X i v : . [ m a t h . DG ] D ec Symplectic twistor operator on R n and theSegal-Shale-Weil representation Marie Dost´alov´a, Petr Somberg
Abstract
The aim of our article is the study of solution space of the symplectictwistor operator T s in symplectic spin geometry on standard symplecticspace ( R n , ω ), which is the symplectic analogue of the twistor operatorin (pseudo-)Riemannian spin geometry. In particular, we observe a sub-stantial difference between the case n = 1 of real dimension 2 and the caseof R n , n >
1. For n >
1, the solution space of T s is isomorphic to theSegal-Shale-Weil representation. Key words:
Symplectic twistor operator, Symplectic Dirac operator,Metaplectic Howe duality.
MSC classification:
In the case when the second Stiefel-Whitney class of a Riemannian mani-fold is trivial, there is a double cover of the frame bundle and consequentlythere is an associated vector bundle for the spinor representation of thespin structure group. There are two basic first order invariant differentialoperators acting on spinor valued fields, namely the Dirac operator andthe twistor operator. Their spectral properties are reflected in the geomet-rical properties of the underlying manifold. In Riemannian geometry, thetwistor equation appeared as an integrability condition for the canonicalalmost complex structure on the twistor space, and it plays a prominentrole in conformal differential geometry due to its larger symmetry group.In physics, its solution space defines infinitesimal isometries in Rieman-nian supergeometry. For an exposition with panorama of examples, cf.[6], [1] and references therein.The symplectic version of Dirac operator D s was introduced in [10],and its differential geometric properties were studied in [4], [8], [9]. Themetaplectic Howe duality for D s , introduced in [2], allows to character-ize the space of solutions for the symplectic Dirac operator D s on the(standard) symplectic space ( R n , ω ).The aim of the present article is to study the symplectic twistor oper-ator T s in the context of the metaplectic Howe duality, and consequentlyto determine its solution space on the standard symplectic space ( R n , ω ).The operators D s , T s were considered from a different perspective in [9],[11], [12]. From an analytic point of view, T s represents an overdetermined system of partial differential equations, acting on the space of polynomialsvalued in the vector space of the Segal-Shale-Weil representation. Fromthe point of view of representation theory, T s is mp (2 n, R )-invariant andthe initial problem is dissolved by proper understanding of the interactionof T s with the generators D s , X s of the Howe dual Lie algebra sl (2).As we shall see, as for T s there is a substantial difference between thesituation for n = 1 and n >
1. Namely, there is in Ker( T s ) an infinite num-ber of irreducible mp (2 n, R )-modules with different infinitesimal characterfor n = 1, while for n > n = 1 in a separate paper [5] using different, morecombinatorial approach, which will be useful in complete understandingof the full infinite dimensional symmetry group of our operator.The structure of our article goes as follows. In the first section, wereview the subject of symplectic spin geometry and metaplectic Howe du-ality. In the second section, we start with the definition of the symplectictwistor operator T s and compute the space of polynomial solutions of T s on ( R n , ω ). These results follow from the careful study of algebraic anddifferential consequences of T s . In the last third section we indicate thecollection of unsolved problems related to the topic of the present article.Throughout the article, we use the notation N for the set of naturalnumbers including zero and N for the set of natural numbers without zero. mp (2 n, R ) , symplecticClifford algebra and a class of simple lowest weightmodules for mp (2 n, R ) In the present section we recall several algebraic and representation the-oretical results used in the next section for the analysis of the solutionspace of the symplectic twistor operator T s , see e.g., [2], [4], [7], [8], [9].Let us consider 2 n -dimensional symplectic vector space ( R n , ω = P nj =1 ǫ j ∧ ǫ n + j ), n ∈ N , and a symplectic basis { e , . . . , e n , e n +1 , . . . , e n } with respect to the non-degenerate two form ω ∈ ∧ ( R n ) ⋆ . Let E k,j bethe 2 n × n matrix with 1 on the intersection of the k -th row and the j -thcolumn and zero otherwise. The set of matrices X kj = E k,j − E n + j,n + k , Y kj = E k,n + j + E j,n + k , Z kj = E n + k,j + E n + j,k , for j, k = 1 , . . . , n is a basis of sp (2 n, R ), and can be realized by first orderdifferential operators X kj = x j ∂ x k − x n + k ∂ x n + j , Y kj = x n + j ∂ x k + x n + k ∂ x j , Z kj = x j ∂ x n + k + x k ∂ x n + j . The metaplectic Lie algebra mp (2 n, R ) is the Lie algebra of the twofoldgroup covering π : Mp(2 n, R ) → Sp(2 n, R ) of the symplectic Lie groupSp(2 n, R ). It can be realized by homogeneity two elements in the sym-plectic Clifford algebra Cl s ( R n , ω ), where the homomorphism π ⋆ : mp (2 n, R ) → sp (2 n, R ) is given by π ⋆ ( e k · e j ) = − Y kj ,π ⋆ ( e n + k · e n + j ) = Z kj ,π ⋆ ( e k · e n + j + e n + j · e k ) = 2 X kj , (1)for j, k = 1 , . . . , n . Definition 1.1
The symplectic Clifford algebra Cl s ( R n , ω ) is an asso-ciative unital algebra over C , given by the quotient of the tensor algebra T ( e , . . . , e n ) by a two-sided ideal I ⊂ T ( e , . . . , e n ) generated by v j · v k − v k · v j = − iω ( v j , v k ) for all v j , v k ∈ R n , where i ∈ C is the complex unit. The symplectic Clifford algebra Cl s ( R n , ω ) is isomorphic to the Weyl al-gebra W n of complex valued algebraic differential operators on R n , andthe symplectic Lie algebra sp (2 n, R ) can be realized as a subalgebra of W n . In particular, the Weyl algebra is an associative algebra generatedby { q , . . . , q n , ∂ q , . . . , ∂ q n } , the multiplication operator by q j and differ-entiation ∂ q j , for j = 1 , . . . , n , and the symplectic Lie algebra sp (2 n, R )has a basis {− i q j , − i ∂ ∂q j , q j ∂∂q j + } , j = 1 , . . . , n .The symplectic spinor representation is an irreducible Segal-Shale-Weil representation of Cl s ( R n , ω ) on L ( R n , e − P nj =1 q j dq R n ), the spaceof square integrable functions on ( R n , e − P nj =1 q j dq R n ) with dq R n theLebesgue measure. Its action, the symplectic Clifford multiplication c s ,preserves the subspace of C ∞ (smooth)-vectors given by the Schwartzspace S ( R n ) of rapidly decreasing complex valued functions on R n as adense subspace. The space S ( R n ) can be regarded as a smooth (Frechet)globalization of the space of ˜ K = e U( n )-finite vectors in the representa-tion, where ˜ K ⊂ Mp(2 n, R ) is the maximal compact subgroup given bythe double cover of K = U( n ) ⊂ Sp(2 n, R ). Though we shall work in thesmooth globalization S ( R n ), the representative vectors are usually chosento belong to the underlying Harish-Chandra module of ˜ K = e U( n )-finitevectors preserved by c s .The function spaces associated to the Segal-Shale-Weil representationare supported on R n ⊂ R n , a maximal isotropic subspace of ( R n , ω ). Inits restriction to mp (2 n, R ), S ( R n ) decomposes into two unitary represen-tations realized on the subspace of even resp. odd functions: ̺ : mp (2 n, R ) → End( S ( R n )) , (2)where the basis vectors act by ̺ ( e j · e k ) = iq j q k ,̺ ( e n + j · e n + k ) = − i∂ q j ∂ q k ,̺ ( e j · e n + j + e n + j · e j ) = q j ∂ q j + ∂ q j q j . (3)for all j, k = 1 , . . . , n . In this representation the Cl s ( R n , ω ) acts on L ( R n , e − P nj =1 q j dq R n ) by continuous unbounded operators with domain S ( R n ). The space of ˜ K = e U( n )-finite vectors consists of even resp. oddhomogeneity mp (2 n, R )-submodule { Pol even ( q , . . . , q n ) e − P nj =1 q j } , { Pol odd ( q , . . . , q n ) e − P nj =1 q j } . It is also an irreducible representation of mp (2 n, R ) ⋉ h ( n ), the semidirectproduct of mp (2 n, R ) and (2 n + 1)-dimensional Heisenberg Lie algebra h ( n ) spanned by { e , . . . , e n , Id } . In what follows we denote the Segal-Shale-Weil representation by S , and S ≃ S + ⊕ S − as mp (2 n, R )-module.Let us denote by Pol( R n ) the vector space of complex valued polyno-mials on R n , and by Pol l ( R n ) the subspace of homogeneity l polynomi-als. The complex vector space Pol l ( R n ) is as an irreducible mp (2 n, R )-module isomorphic to S l ( C n ), the l -th symmetric power of the complex-ification of the fundamental vector representation R n , l ∈ N . Let us review a representation-theoretical result of [3], formulated in theopposite convention of highest weight metaplectic modules. Let ω , . . . , ω n be the fundamental weights of the Lie algebra sp (2 n, R ), and let L ( λ ) de-note the simple module over the universal enveloping algebra U ( mp (2 n, R ))of mp (2 n, R ) generated by the highest weight vector of the weight λ .Algebraically, the decomposition of the space of polynomial functionson R n valued in the Segal-Shale-Weil representation corresponds to thetensor product of L ( − ω n ) resp. L ( ω n − − ω n ) with symmetric powersS k ( C n ) of the fundamental vector representation C n of sp (2 n, R ), k ∈ N . The following result is well known. Corollary 1.2 ([3]) We have for L ( − ω n )
1. In the even case k = 2 l ( l + 1 terms on the right-hand side): L ( − ω n ) ⊗ S k ( C n ) ≃ L ( − ω n ) ⊕ L ( ω + ω n − − ω n ) ⊕ L (2 ω − ω n ) ⊕ L (3 ω + ω n − − ω n ) ⊕ . . . ⊕ L ((2 l − ω + ω n − − ω n ) ⊕ L (2 lω − ω n ) ,
2. In the odd case k = 2 l + 1 ( l + 2 terms on the right-hand side): L ( − ω n ) ⊗ S k ( C n ) ≃ L ( ω n − − ω n ) ⊕ L ( ω − ω n ) ⊕ L (2 ω + ω n − − ω n ) ⊕ L (3 ω − ω n ) ⊕ . . . ⊕ L (2 lω + ω n − − ω n ) ⊕ L ((2 l + 1) ω − ω n ) , We have for L ( ω n − − ω n )
1. In the even case k = 2 l ( l + 1 terms on the right-hand side): L ( ω n − − ω n ) ⊗ S k ( C n ) ≃ L ( ω n − − ω n ) ⊕ L ( ω − ω n ) ⊕ L (2 ω + ω n − − ω n ) ⊕ · · · ⊕ L ((2 l − ω − ω n ) ⊕ L (2 lω + ω n − − ω n ) ,
2. In the odd case k = 2 l + 1 ( l + 2 terms on the right-hand side): L ( ω n − − ω n ) ⊗ S k ( C n ) ≃ L ( − ω n ) ⊕ L ( ω + ω n − − ω n ) ⊕ . . . ⊕ L (2 lω − ω n ) ⊕ L ((2 l + 1) ω + ω n − − ω n ) . A more geometrical reformulation of this statement is realized in the al-gebraic (polynomial) Weyl algebra and termed metaplectic Howe duality,[2]. The metaplectic analogue of the classical theorem on the separationof variables allows to decompose the space Pol( R n ) ⊗ S of complex poly-nomials valued in the Segal-Shale-Weil representation under the action of mp (2 n, R ) into a direct sum of simple lowest weight mp (2 n, R )-modulesPol( R n ) ⊗ S ≃ ∞ M l =0 ∞ M j =0 X js M l , (4)where we use the notation M l := M + l ⊕ M − l . This decomposition takesthe form of an infinite triangleP ⊗ S P ⊗ S P ⊗ S P ⊗ S P ⊗ S P ⊗ S M / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ M / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ M / / X s M ⊕ / / X s M ⊕ / / X s M ⊕ M / / X s M ⊕ / / X s M ⊕ M / / X s M ⊕ M (5)Let us now explain the notation used on the previous picture. First ofall, we used the shorthand notation P l = Pol l ( R n ) , l ∈ N , and all spacesand arrows on the picture have the following meaning. Let i ∈ C be the complex unit. The three operators X s = n X j =1 ( x n + j ∂ q j + ix j q j ) ,D s = n X j =1 ( iq j ∂ x n + j − ∂ x j ∂ q j ) ,E s = n X j =1 x j ∂ x j , (6)where D s and X s acts on the previous picture horizontally but in theopposite direction, and fulfil the sl (2)-commutation relations:[ E s + n, D s ] = − D s , [ E s + n, X s ] = X s , (7)[ X s , D s ] = i ( E + n ) . For the purposes of our article, we do not need the proper normaliza-tion of the generators D s , X s , E s making the isomorphism with standardcommutation relations in sl (2) explicit.The elements of Pol( R n ) ⊗S are called polynomial symplectic spinors.Let s ≡ s ( x , . . . , x n , q , . . . , q n ) ∈ Pol( R n ) ⊗ S , h ∈ Mp(2 n, R ) and π ( h ) = g ∈ Sp(2 n, R ) for the double covering map π : Mp(2 n, R ) → Sp(2 n, R ). We define the action of Mp(2 n, R ) on Pol( R n ) ⊗ S by˜ ̺ ( h ) s ( x , . . . , x n , q , . . . , q n ) = ̺ ( h ) s ( π ( g − )( x , . . . , x n ) T , q , . . . , q n ) , (8)with ̺ acting on the Segal-Shale-Weil representation via (2). Passingto the infinitesimal action, we get the operators representing the basiselements of mp (2 n, R ). For example, we have for j = 1 , . . . , n ˜ ̺ ( X jj ) s = ddt (cid:12)(cid:12)(cid:12) t =0 ˜ ̺ (exp( tX jj )) s ( x , . . . , x n , q , . . . , q n )= ddt (cid:12)(cid:12)(cid:12) t =0 e t s ( x , . . . , x j e − t , . . . , x n + j e t . . . , x n , q , . . . , q j e t , . . . , q n )= (cid:0) − x j ∂∂x j + x n + j ∂∂x n + j + q j ∂∂q j (cid:1) s ( x , . . . , x n , q , . . . , q n ) . These operators satisfy the commutation relation of the Lie algebra mp (2 n, R ),and preserve the homogeneity in x , . . . , x n . The operators X s and D s commute with operators ˜ ̺ ( X jk ) , ˜ ̺ ( Y jk ) and ˜ ̺ ( Z jk ), j, k = 1 , . . . , n , henceare mp (2 n, R )-intertwining differential operators.The action of mp (2 n, R ) × sl (2) generates the multiplicity free decom-position of Pol( R n ) ⊗ S and the pair of Lie algebras in the product iscalled the metaplectic Howe dual pair. The operators X s , D s acting onthe previous picture horizontally isomorphically identify the two neigh-boring mp (2 n, R )-modules. The modules M l , l ∈ N , on the most leftdiagonal of our picture are termed symplectic monogenics, and are char-acterized as l -homogeneous solutions of the symplectic Dirac operator D s .Thus the decomposition is given as a vector space by tensor product of thesymplectic monogenics multiplied the by polynomial algebra of invariants C [ X s ]. T s andits solution space on R n We start with an abstract definition of the symplectic twistor operator T s . Let ( M, ω ) be a 2 n -dimensional symplectic manifold, p : P → M a principal fiber Sp(2 n, R )-bundle of symplectic frames on M . A meta-plectic structure on ( M, ω ) is a principal fiber Mp(2 n, R )-bundle ˜ P → M together with bundle morphism ˜ P → P , equivariant with respect to thedouble covering Mp(2 n, R ) → Sp(2 n, R ). The manifold ( M, ω ) with ametaplectic structure is usually called symplectic spin manifold. Thesymplectic manifold M admits a metaplectic structure if and only if thesecond Stiefel-Whitney class w ( M ) is trivial, and the equivalence classesof metaplectic structures are classified by H ( M, Z ). There is a uniquemetaplectic structure on ( R n , ω ). Definition 2.1
Let ( M, ∇ , ω ) be a symplectic spin manifold of dimen-sion n , ∇ s the associated symplectic spin covariant derivative and ω ∈ C ∞ ( M, ∧ T ⋆ M ) a non-degenerate -form such that ∇ ω = 0 . We denoteby { e , . . . , e n } a local symplectic frame. The symplectic twistor opera-tor T s on M is the first order differential operator T s acting on smoothsymplectic spinors S : ∇ s : C ∞ ( M, S ) −→ T ⋆ M ⊗ C ∞ ( M, S ) ,T s := P Ker( c ) ◦ ω − ◦ ∇ s : C ∞ ( M, S ) −→ C ∞ ( M, T ) , (9) where T is the space of symplectic twistors, T ⋆ M ⊗ S ≃ S ⊕ T , given byalgebraic projection P Ker( c s ) : T ⋆ M ⊗ C ∞ ( M, S ) −→ C ∞ ( M, T ) on the kernel of the symplectic Clifford multiplication c s . In the local sym-plectic coframe { ǫ } nj =1 dual to the symplectic frame { e j } nj =1 with respectto ω , we have the local formula for T s : T s = n X k =1 ǫ k ⊗ ∇ se k + in n X j,k,l =1 ǫ l ⊗ ω kj e l · e j · ∇ se k , (10) where · is the shorthand notation for the symplectic Clifford multiplicationand i ∈ C is the imaginary unit. We use the convention ω kj = 1 for j = k + n and k = 1 , . . . , n , ω kj = − for k = n + 1 , . . . , n and j = k − n ,and ω kj = 0 otherwise. The symplectic Dirac operator D s is defined as the image of the symplecticClifford multiplication c s , and a symplectic spinor in the kernel of D s iscalled symplectic monogenic. Lemma 2.2
The symplectic twistor operator T s is mp (2 n, R ) -invariant. Proof:
The property of invariance is a direct consequence of the equiv-ariance of symplectic covariant derivative and invariance of algebraic pro-jection P Ker( c s ) , and amounts to verify T s (˜ ̺ ( g ) s ) = ˜ ̺ ( g )( T s s ) (11) for any g ∈ mp (2 n, R ) and s ∈ C ∞ ( M, S ). Using the local formula (10)for T s in a local chart ( x , . . . , x n ), both sides of (11) are equal to n X k =1 ǫ k ⊗ ̺ ( g ) ∂∂x k (cid:2) s (cid:0) π ( g ) − x (cid:1)(cid:3) + in n X j,k,l =1 ǫ l ⊗ ω kj e l · e j · h ̺ ( g ) ∂∂x k (cid:2) s (cid:0) π ( g ) − x (cid:1)(cid:3)i and the proof follows. (cid:3) In the case M = ( R n , ω ), the symplectic Dirac and the symplectictwistor operators are given by D s = n X j,k =1 ω kj e k · ∂∂x j , (12) T s = n X l =1 ǫ l ⊗ ∂∂x l + in n X j,k,l =1 ǫ l ⊗ ω kj e l · e j · ∂∂x k = n X l =1 ǫ l ⊗ (cid:16) ∂∂x l − in e l · D s (cid:17) , (13)and we restrict their action to the space of polynomial symplectic spinors. Lemma 2.3
Let s ∈ Pol( R n , S ) be a symplectic spinor in the solutionspace of the symplectic twistor operator T s . Then s is in the kernel of thesquare of the symplectic Dirac operator D s . Proof:
Let s be a polynomial symplectic spinor in Ker( T s ), T s s = n X l =1 ǫ l ⊗ (cid:16) ∂∂x l − in e l · D s (cid:17) s = 0 , (14)i.e., (cid:16) ∂∂x l − in e l · D s (cid:17) s = 0 , l = 1 , . . . , n. (15)We apply to the last equation partial differentiation operator ∂∂x m , mul-tiply it by the skew symmetric form ω ml and sum over m = 1 , . . . , n : n X l,m =1 (cid:16) ω ml ∂∂x m ∂∂x l − in ω ml e l · ∂∂x m D s (cid:17) s = 0 . (16)The first part is zero because of the skew-symmetry of ω and the symmetryin m, l , and the second part is (a non-zero multiple of) the square of thesymplectic Dirac operator D s . Hence n X l,m =1 in ω ml e l · ∂∂x m D s s = − in D s s = 0 (17)and the proof is complete. (cid:3) Let us consider mp (2 n, R )-submodules in the split composition series { } ⊂ Ker( D s ) ⊂ Ker( D s ) (18)with Ker( D s ) ≃ Ker( D s ) ⊕ (Ker( D s ) / Ker( D s )) , (19)and discuss which of them are in the solution space of the symplectictwistor operator T s . We haveKer( T s ) ≃ (Ker( T s ) ∩ Ker( D s )) ⊕ (Ker( T s ) ∩ (Ker( D s ) / Ker( D s ))) . (20) Lemma 2.4
Let n ∈ N and s ∈ Pol( R n , S ) be a symplectic spinor ful-filling s ∈ Ker( T s ) ∩ Ker( D s ) . (21) Then s is a constant (i.e., independent of x , . . . , x n ) symplectic mono-genic spinor. This is described by the following picture: • Pol( R n , S − ) : M − / / X s M − ⊕ / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / . . .M − / / . . . (22) • Pol( R n , S + ) : M +0 / / X s M +0 ⊕ / / X s M +0 ⊕ / / X s M +0 ⊕ / / . . .M +1 / / X s M +1 ⊕ / / X s M +1 ⊕ / / . . .M +2 / / X s M +2 ⊕ / / . . .M +3 / / . . . (23) Proof:
Let s ∈ Pol( R n , S ) be a solution of the symplectic twistor oper-ator, see (15), (cid:16) ∂∂x l − in e l · D s (cid:17) s = 0 , l = 1 , . . . , n, and at the same time s ∈ Ker( D s ). This implies ∂∂x l s = 0 , l = 1 , . . . , n, so s is a constant symplectic spinor. The proof is complete. (cid:3) Lemma 2.5
Let s ∈ Pol( R n , S ) be a symplectic monogenic spinor ofhomogeneity h ∈ N , i.e. D s ( s ) = 0 . Then the symplectic spinor X s ( s ) has the following property:1. If n = 1 , then X s ( s ) is in the kernel of T s for any homogeneity h ∈ N . This is described by the following picture: • Pol( R , S − ) : M − / / X s M − ⊕ / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / . . .M − / / . . . (24) • Pol( R , S + ) : M +0 / / X s M +0 ⊕ / / X s M +0 ⊕ / / X s M +0 ⊕ / / . . .M +1 / / X s M +1 ⊕ / / X s M +1 ⊕ / / . . .M +2 / / X s M +2 ⊕ / / . . .M +3 / / . . . (25)
2. If n > , then X s ( s ) is in the kernel of T s if and only if the ho-mogeneity of s is equal to h = 0 . This is described by the followingpicture: • Pol( R n , S − ) : M − / / X s M − ⊕ / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / X s M − ⊕ / / . . .M − / / X s M − ⊕ / / . . .M − / / . . . (26) • Pol( R n , S + ) : M +0 / / X s M +0 ⊕ / / X s M +0 ⊕ / / X s M +0 ⊕ / / . . .M +1 / / X s M +1 ⊕ / / X s M +1 ⊕ / / . . .M +2 / / X s M +2 ⊕ / / . . .M +3 / / . . . (27) Proof:
Let s be a non-zero symplectic spinor in the kernel of D s . Thequestion is when the system of partial differential equations acting on s ,( ∂ x k − in e k · D s ) X s s = 0 , (28)holds for all k = 1 , ..., n . In other words, we ask when X s ( s ) is inthe kernel of the symplectic twistor operator. Let us multiply the k -thequation of the system by x k and sum over all k ,( E s − in X s D s ) X s s = 0 . (29)We use the sl (2)-commutation relations for X s , D s and for E s , X s , see(7), and the fact that s is in the kernel of D s . This gives( E s X s − n X s E s − X s ) s = 0 . (30)Assuming that s is of homogeneity h , E s s = hs , the last equation reducesto ( h + 1 − hn − X s s = h (1 − n ) X s s = 0 . (31)Observe that (1 − n ) = 0 for n >
1, and X s is an mp (2 n, R )-intertwiningmap acting injectively on Pol( R n , S ) as a result of the metaplectic Howeduality (i.e., s being non-zero implies X s ( s ) is non-zero.) Because s isassumed to be non-zero, the last display implies that either1. h = 0 and n ∈ N is arbitrary, or2. n = 1 and h is arbitrary.A straightforward check for n > h = 0 gives( ∂ x k − ie k D s ) X s s = (cid:0) e k + X s ∂ x k − in e k E s − e k (cid:1) s = 0 , (32)and in the case n = 1 and arbitrary homogeneity we have( ∂ x − ie D s ) X s s = (cid:0) e + e x ∂ x + e x ∂ x − e x ∂ x − e x ∂ x − e (cid:1) s = (cid:0) x ( e ∂ x − e ∂ x ) (cid:1) s = − x D s s = 0 . (33)As for the second component ( ∂ x − ie D s ) of the symplectic twistoroperator, the computation is analogous to the first one in (33). Thiscompletes the proof. (cid:3) Let us summarize our results in the final theorem. Theorem 2.6
The solution space of the symplectic twistor operator T s on standard symplectic space ( R n , ω ) is given by mp (2 n, R ) -modules inthe boxes on the following pictures: • In the case n = 1 , we have for Pol( R , S ± ) : M ± / / X s M ± ⊕ / / X s M ± ⊕ / / X s M ± ⊕ / / . . .M ± / / X s M ± ⊕ / / X s M ± ⊕ / / . . .M ± / / X s M ± ⊕ / / . . .M ± / / . . . (34) • In the case n > , we have for Pol( R n , S ± ) : M ± / / X s M ± ⊕ / / X s M ± ⊕ / / X s M ± ⊕ / / . . .M ± / / X s M ± ⊕ / / X s M ± ⊕ / / . . .M ± / / X s M ± ⊕ / / . . .M ± / / . . . (35)An interested reader can easily verify the previous result for n > s of D s of homogeneity at least one (it is sufficientto generate such a simple solution from dimension n = 1 case) and checkthat X s ( s ) / ∈ Ker( T s ). Example 2.7
In the case n = 2 and the homogeneity , the symplecticspinor s = e − q q ( − ix x + x x + x x + ix x ) (36) is a solution of D s . However, X s s is not a solution of the symplectictwistor operator T s because, for example, the first and the second compo-nents of T s X s ( s ) are nonzero: ( T s s ) = ǫ ⊗ e − q q q ( x + ix ) = 0 , ( T s s ) = ǫ ⊗ e − q q q ( x + ix ) = 0 . It is much harder to verify the result X s s ∈ Ker( T s ) for all polynomialsymplectic spinors s , s ∈ Ker( D s ), in the case n = 1, and we refer to [5]for a non-trivial combinatorial proof of this assertion.We would like to emphasize that the kernel of our solution spacerealizes (for n >
1) the Segal-Shale-Weil representation, a prominentSp(2 n, R )-module with far-reaching impact on harmonic analysis. In the present section we comment on the results achieved in our article.First of all, notice that in the case of (both even and odd) orthogonalalgebras and the spinor representation as an orthogonal analogue of theSegal-Shale-Weil representation, the solution space of the twistor opera-tor for orthogonal Lie algebras on R n is given by two copies of the spinorrepresentation, in complete analogy with the symplectic case, see [1] for n ≥
3. As for n = 2, we were not able to find the required result in theavailable literature, although we believe it is known to specialists. Hereone half of the Dirac operator is the Dolbeault operator and the twistoroperator is its complex conjugate, while the opposite half of the Dirac andtwistor operators are their complex conjugates, respectively. The solutionspaces for both halves of the twistor operator on R are the complex lin-ear spans of polynomials { z j } j ∈ N and { ¯ z j } j ∈ N , respectively, intersectingnon-trivially in the constant polynomials. This is an orthogonal analogueof our results in symplectic category, and indicates an infinite-dimensionalsymmetry group acting on the solutions spaces of both symplectic Diracand symplectic twistor operators in the real dimension 2.Another observation is related to the proof of Lemma 2.3 and its struc-ture on curved symplectic manifolds. Let us consider a 2 n -dimensionalmetaplectic manifold ( M, ∇ , ω ), with ∇ s the metaplectic covariant deriva-tive. Then a differential consequence of the symplectic twistor equationon M is n X l,m =1 (cid:0) ω ml ( ∇ sm , ∇ sl ) − in D s (cid:1) s = 0 , (37)where the first term (skewing of the composition of metaplectic covariantderivatives) gives the action of the symplectic curvature of the symplecticconnection ∇ s on the space of sections of a metaplectic bundle on M . Thisequation should be thought of as a symplectic analogue of the equation D s = 14 nn − Rs, n ≥ s a twistor spinor, D the Dirac opera-tor and R the scalar curvature of the Riemannian structure, cf. [1]. Theprolongation of the symplectic twistor equation then constructs a linearconnection and covariant derivative on the Segal-Shale-Weil representa-tion, in such a way that the covariantly constant sections correspond tosymplectic twistor spinors.Another ramification of our results is related to the higher order twistorequations acting on symplectic spinors, with leading part s Π hw ∇ s ( i . . . ∇ si j ) s, (39)where Π hw is projection on the highest weight (or, Cartan) componentand round brackets denote the symmetric part in the composition of sym-plectic covariant derivatives. For example, their solution spaces can bestudied by its interaction with the metaplectic Howe duality in an analo-gous way as we did in the first order case ( j = 1). Acknowledgement:
The authors gratefully acknowledge the supportof the grant GA CR P201/12/G028 and SVV-2013-267317.
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